ࡱ>     !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entry FXI[ WordDocument y CompObj^r a mass ml in an elliptical orbit about a second mass m2. Your final answers should be functions of P, e, a, and 6 only. (b) Using the expressions for vT and ve that you derived in part (a), verify Eq. (2.34) directly from v2 = vr + ve. 2.4 Derive Eq. (2.24) from the sum of the kinetic and potential energy terms for the masses, ml and M2- 2.5 Derive Eq. (2.25) from the total angular momentum of the masses, ml and M2- 2.6 By expanding Eq. (2.38) and rearranging, obtain Eq. (2.39). 2.7 (a) Assuming that the Sun interacts only with Jupiter, calculate the total orbital angular momentum of the Sun-Jupiter system. The semimajor axis of Jupiter's orbit is a = 5.2 AU, its orbital eccen-tricity is e = 0.048, and its orbital period is P = 11.86 yr. (b) Estimate the contribution the Sun makes to the total orbital an-gular momentum of the Sun-Jupiter system. For simplicity, as-sume that the Sun's orbital eccentricity is e = 0, rather than e = 0.048. Hint: First find the distance of the center of the Sun from the center of mass. (c) Making the approximation that the orbit of Jupiter is a perfect circle, estimate the contribution it makes to the total orbital angu-lar momentum of the Sun-Jupiter system. Compare your answer with the difference between the two values found in parts (a) and (b). (d) Recall that the moment of inertia of a solid sphere of mass m and radius r is given by I = 5mr2, and that when the sphere spins on an axis passing through its center, its rotational angular momentum may be written as L=Iw, where w is the angular frequency measured in rad s-1. Assuming (incorrectly) that both the Sun and Jupiter rotate as solid spheres, calculate approximate values for the rotational angular momenta of the Sun and Jupiter. Take the rotation periods of the Sun and Jupiter to be 26 days and 10 hours, respectively. The radius of the Sun is 6.96 x 101 cm, and the radius of Jupiter is 6.9 x 109 cm.   (e) What part of the Sun-Jupiter system makes the largest contribu-tion to the total angular momentum? 2.8 (a) Using data contained in Problem 2.7 and in the chapter, calculate the escape velocity at the surface of Jupiter. (b) Calculate the escape velocity from the solar system, starting from Earth's orbit. Assume that the Sun constitutes all of the mass of the solar system. 2.9 (a) The Hubble Space Telescope is in a nearly circular orbit, approx-imately 380 miles above the surface of Earth. Estimate its orbital period. (b) Communications and weather satellites are often placed in geosyn-chronous "parking" orbits above Earth. These are orbits where satellites can remain fixed above a specific point on the surface of Earth. At what altitude must these satellites be located? (c) Is it possible for a satellite in a geosynchronous orbit to remain "parked" over any location on the surface of Earth? Why or why not? 2.10 In general, an integral average of some continuous function f (t) over an interval r is given by   (f (t)) = T ,~T f (t) dt. Beginning with an expression for the integral average, prove that  a binary system's gravitational potential energy, averaged over one period, equals the value of the instantaneous potential energy of the system when the two masses are separated by the distance a, the semi-major axis of the orbit of the reduced mass about the center of mass. Hint: You may find the following definite integral useful; ~7r de __ J0 1 + e cos 0 1 _ e2  60 Chapter 2 Celestial Mechanics 2.11 Cometary orbits usually have very large eccentricities, often approach-ing (or even exceeding) unity. Halley's comet has an orbital period of 76 yr and an orbital eccentricity of e = 0.9673. (a) What is the semimajor axis of Comet Halley's orbit? (b) Use the orbital data of Comet Halley to estimate the mass of the Sun. (c) Calculate the distance of Comet Halley from the Sun at perihelion and aphelion. (d) Determine the orbital speed of the comet when at perihelion, at aphelion, and on the semiminor axis of its orbit. (e) How many times larger is the kinetic energy of Halley's comet at perihelion when compared to aphelion? 2.12 Cܥe# Rjy g,x l,x lx x x (x x x Tx $nTimes New Roman Symbol ArialTimes New Roman TahomaCourier NewArial Narrow VerdanaLucida Console Arial An Introduction to Modern Stellar Astrophysics Dale A. Ostlie Bradley W. Carroll Weber State University Chapter 1 Problems l.l Derive the relationship between a planet's synodic period and its side-real period (Eq. 1.1). Consider both inferior and superior planets. 1.2 Devise methods to determine the relative distances of each of the plan-ets from the Sun given the information available to Copernicus (orbital configurations and synodic periods). 24 Chapter 1 The Celestial Sphere 1.3 (a) The observed orbital synodic periods of Venus and Mars are 583.9 days and 779.9 days, respectively. Calculate their sidereal peri-ods. (b) Which planet in the solar system has the shortest synodic period? Why? 1.4 List the right ascension and declination of the Sun when it is located at the vernal equinox, the summer solstice, the autumnal equinox, and the winter solstice. 1.5 (a) Calculate the altitude of the Sun along the meridian on the first day of summer for an observer at a latitude of 42 north. (b) What is the maximum altitude of the Sun on the first day of winter at the same latitude? 1.6 (a) Circumpolar stars are defined as stars that never set below the horizon of the local observer or stars that are never visible above the horizon. Calculate the range of declinations for these two groups of stars for an observer at the latitude L. (b) At what latitude(s) on Ea.rth will the Sun never set when it is at the summer solstice? (c) Is there any latitude on Earth where the Sun will never set when it is at the vernal equinox? Where? 1.7 Proxima Centauri (a Centauri C) is the closest star to the Sun and is a part of a triple star system. It has the epoch 1950.0 coordinates (a, S) _ (14h26.3m, -6228') while the center of the system is located at (a, S) _ (14h36.2m, -6038'). -11 (a) What is the angular separation of Proxima Centauri from the center of the triple star system? (b) If the distance to Proxima Centauri is 4.0 x 101$ cm, how far is the star from the center of the triple system? 1.8 (a) Using the information in Problem 1.7, precess the coordinates of Proxima Centauri to epoch 1990.0. (b) The proper motion of Proxima Centauri is 3.84" yr-1 with the position angle 282. Calculate the change in a and S due to proper motion between 1950.0 and 1990.0. (c) Which effect makes the largest contribution to changes in the coordinates of Proxima Centauri: precession or proper motion?   Suggested Readings GENERAL Kuhn, Thomas S., The Structure of Scientific Revolutions, Second Edition, Enlarged, University of Chicago Press, Chicago, 1970.  Westfall, Richard S., Never at Rest: A Biography of Isaac Newton, Cambridge University Press, Cambridge, 1980. TECHNICAL Arya, Atam P., Introduction to Classical Mechanics, Prentice Hall, Englewood Cliffs, NJ, 1990.  Clayton, Donald D., Principles of Stellar Evolution and Nucleosynthesis, McGraw-Hill, New York, 1968. Fowles, Grant R., and Cassiday, George L., Analytical Mechanics, Fifth Edi-tion, Harcourt Brace and Company, Fort Worth, 1993. , Marion, Jerry B., and Thornton, Stephen T., Classical Dynamics of Particles and Systems, Fourth Edition, Saunders College Publishing, Fort Worth, 1995. Problems 2.1 Assume that a rectangular coordinate system has its origin at the center of an elliptical planetary orbit and that the coordinate system's x axis lies along the major axis of the ellipse. Show that the equatiau for the ellipse is given by x2 2 a2+b2=1~  where a and b are the lengths of the semimajor axis and the semiminor axis, respectively. 2.2 Using the result of Problem 2.1, prove that the area of an ellipse is given by A = ~rab. $8 Chapter 2 Celestial Mechanics 2.3 (a) Beginning with Eq. (2.3) and Kepler's second law, derive general expressions for vT and ve foomputer Problem Using ORBIT, the FORTRAN computer code found in Appendix G, together with the data given in Problem 2.11, estimate the amount of time required for Halley's comet to move from perihelion to a distance of 1 AU away from the principal focus. 2.13 Computer Problem The computer code ORBIT (Appendix G) can be used to generate orbital positions, given the mass of the central star, the semimajor axis of the orbit, and the orbital eccentricity. Using ORBIT to generate the data, plot on a single sheet of graph paper the orbits for three hypothetical objects orbiting our Sun. Assume that the semimajor axis of each orbit is 1 AU and that the orbital eccentricities are: (a) 0.0. (b) 0.4. (c) 0.9. Note: Indicate the principal focus, located at x = 0.0, y = 0.0. 2.14 Computer Problem (a) From the data given in Example 2.1, use ORBIT (Appendix G) to generate an orbit for Mars. Plot at least 25 points, evenly "`- spaced in time, on a sheet of graph paper and clearly indicate the principal focus. Problexns 61 (b) Using a compass, draw a perfect circle on top of the elliptical orbit for Mars, choosing the radius of the circle and its center carefully in order to make the best possible approximation of the orbit. Be sure to mark the center of the circle you chose. (c) What can you conclude about the merit of Kepler's first attempts to use offset circles and equants to model the orbit of Mars? 2.15 Given that a geocentric universe is (mathematically) only a matter of the choice of a reference frame, explain why the Ptolemaic model of the universe was able to survive scrutiny for such a long period of time.  Problems  3.1 In 1672, an international effort was made to measure the parallax angle of Mars at the time of opposition, when it was closest to Earth; see Fig. 1.6. (a) Consider two observers who are separated by a baseline equal to Earth's diameter. If the difference in their measurements of Mars' angular position is 33.6", what is the distance between Earth and Mars at the time of opposition? Express your answer both in units of cm and AU. (b) If the distance to Mars is to be measured to within 10%, how closely must the clocks used by the two observers be synchro-nized? Hint: Ignore the rotation of Earth. The average orbital velocities of Earth and Mars are 29.79 km s-l and 24.13 km s-1, respectively. 3.2 At what distance from a 100-watt light bulb is the radiant flux equal to the solar constant? 3.3 The parallax angle for Sirius is 0.377". (a) Find the distance to Sirius in units of (i) parsecs; (ii) light-years; (iii) AU; (iv) cm. (b) Determine the distance modulus for Sirius. 3.4 Using the information in Example 3.6 and Problem 3.3, determine the absolute bolometric magnitude of Sirius and compare it with that of the Sun. What is the ratio of Sirius' luminosity to that of the Sun? 3.5 Derive the relation m = MSun - 2.51og10( F l . \ F'lo,o /  3.6 A 1.2 x 104 kg spacecraft is launched from Earth and is to be ac-celerated radially away from the Sun using a circular solar sail. The initial acceleration of the spacecraft is to be lg. Assuming a flat sail, determine the radius of the sail if it is (a) black, so it absorbs the Sun's light. 90 Chapter 3 The Continuous Spectrum of Light (b) shiny, so it reflects the Sun's light. Hint: The spacecraft, like Earth, is orbiting the Sun. Should you include the Sun's gravity in your calculation? 3.7 The average person has 1.4 m2 of skin at a skin temperature of roughly 92F (306 K). Consider the average person to be an ideal radiator standing in a room at a temperature of 68F (293 K). (a) Calculate the energy per second radiated by the average person in the form of blackbody radiation. Express your answer both in units of erg s-1 and in watts. (b) Determine the peak wavelength Amax of the blackbody radiation emitted by the average person. In what region of the electromag-netic spectrum is this wavelength found? (c) A blackbody also absorbs energy from its environment, in this case from the 293-K room. The equation describing the absorp-tion is the same as the equation describing the emission of black body radiation, Eq. (3.16). Calculate the energy per second ab-sorbed by the average person, expressed both in units of erg s-1 and in watts. (d) Calculate the net energy per second lost by the average person due to blackbody radiation. 3.8 Consider a model of a star consisting of a spherical blackbody with a surface temperature of 28,000 K and a radius of 5.16 x 1011 cm. Let this model star be located at a distance of 180 pc from Earth. Determine the following for the star: (a) Luminosity. (b) Absolute bolometric magnitude. (c) Apparent bolometric magnitude. (d) Distance modulus. (e) Radiant flux at the star's surface. (f) Radiant flux at Earth's surface (compare this with the solar con- st ant) . (g) Peak wavelength Amax. This is a model of the star Dschubba, the center star in the head of the constellation Scorpius.  3.9 Before Planck discovered the correct description of the spectrum of blackbody radiation, a formulation that was valid only for long wave-lengths was found by two English physicists, Lord Rayleigh and James Jeans. (a) Derive the Rayleigh-Jeans law by considering the Planck func-tion BA in the limit of A hc/kT. (The first-order expansion ex ;:~~ l+ x for x 1 will be useful.) Notice that Planck's constant is not present in your answer. The Rayleigh-Jeans law is a classi-cal result, so the "ultraviolet catastrophe" at short wavelengths, produced by the A 4 in the denominator, cannot be avoided. (b) On the same graph, plot the Planck function BA -and the Rayleigh-Jeans law for the Sun (To = 5770 K). At roughly what wavelength is the Rayleigh-Jeans value twice as large as the Planck function? 3.10 Derive Wien's displacement law, Eq. (3.15), by setting dBa/dA = 0. Hint: You will encounter an equation that must be solved numerically, not algebraically. 3.11 (a) Use Eq. (3.21) to find an expression for the frequency vl,,ax at which the Planck function B attains its maximum value. ( Warn-ing: 1/r,,aac =,4 C/Amax.) (b) What is the value of vmaX for the Sun? (c) Find the wavelength of a light wave having frequency vn,ax. In what region of the electromagnetic spectrum is this wavelength found? 3.12 (a) Integrate Eq. (3.24) over all wavelengths to obtain an expression for the total luminosity of a blackbody model star. Hint: u3 du 7r4 O eu -1 15 (b) Compare your result with the Stefan-Boltzmann equation (3.17), and show that the Stefan-Boltzmann constant Q is given by _ 27r5k4 15c20(c) Calculate the value of Q from this expression, and compare with the value listed in Appendix A. 92 Chapter 3 The Continuous Spectrum of Light 3.13 Use the data in Appendix E to answer the following questions. (a) Calculate the absolute and apparent visual magnitudes, Mv and V, for the Sun. (b) Determine the magnitudes MB, B, MU, and U for the Sun. (c) Locate the Sun and Sirius on the color-color diagram in Fig. 3.10. Refer to Example 3.6 for the data on Sirius. 3.14 Use the filter bandwidths for the UBV system on pages 82-83 and the effective temperature of 9500 K for Vega to determine through which filter Vega would appear brightest to a photometer [i.e., ignore the constant C in Eq. (3.27)]. Assume that S(~) = 1 inside the filter bandwidth and that S(~) = 0 outside the filter bandwidth. 3.15 Evaluate the constant Cbol in Eq. (3.28) by using mgn = -26.81. 3.16 Use the values of the constants CU_B and CB_V found in Example 3.7 to estimate the color indices U-B and B-V for the Sun. 3.17 Shaula (~ Scorpii) is a bright (V = 1.62) blue-white subgiant star located at the tip of the scorpion's tail. Its surface temperature is about 22,000 K. Use the values of the constants CU_B and CB_v found in Example 3.7 to estimate the color indices U-B and B-V for Shaula. Compare your answers with the measured values of U - B = -0.90 and B - V = -0.22. (Shaula is a pulsating star, belonging to the class of Beta Cephei variables; see Section 14.2. As its magnitude varies between V = 1.59 and V = 1.65 with a period of 5 hours 8 minutes, its color indices also change slightly.) 120 Chapter 4 The Theory of Special Relativity Problems 4.1 Use Eqs. (4.14) and (4.15) to derive the Lorentz transformation equa-tions from Eqs. (4.10)-(4.13). 4.2 Because there is no such thing as absolute simultaneity, two observers in relative motion may disagree on which of two events A and B oc-curred first. Suppose, however, that an observer in reference frame S measures that event A occurred first and caused event B. For ex-ample, event A might be pushing a light switch, and event B might be a light bulb turning on. Prove that an observer in another frame S' cannot measure event B (the effect) occurring before event A (the cause). The temporal order of cause and effect are preserved by the Lorentz transformation equations. Hint: For event A to cause event B, information must have traveled from A to B, and the fastest that anything can travel is the speed of light. 4.3 Consider the special light clock shown in Fig. 4.12. The light clock is at rest in the frame S' and consists of two perfectly reflecting mirrors separated by a vertical distance d. As measured by an observer in frame S', a light pulse bounces vertically back and forth between the two mirrors; the time interval between the pulse leaving and subsequently returning to the bottom mirror is ~t'. However, an observer in the frame S sees a moving clock and determines that the time interval between the light pulse leaving and returning to the bottom mirror is Ot. Use the fact that both observers must measure that the light pulse moves with speed c, plus some simple geometry, to derive the time-dilation equation (4.27). 4.4 A rod moving relative to an observer is measured to have its length L,o~;g contracted to one-half of its length when measured at rest. Find the value of u/c for the rod's rest frame relative to the observer's frame of reference. 4.5 An observer P stands on a train station platform as a high-speed train passes by at u/c = 0.8. The observer P, who measures the platform to be 60 m long, notices that the front and back ends of the train line up exactly with the ends of the platform at the same time. (a) How long does it take the train to pass P as he stands on the platform, as measured by his watch? Problems 121 S Y S' T Y -- Mirror ~ u w u      Figure 4.12 (a) A light clock that is moving in frame S, and (b) at rest in frame S'. (b) According to a rider T on the train, how long is the train? (c) According to a rider T on the train, what is the length of the train station platform? (d) According to a rider T on the train, how much time does it take for the train to pass observer P standing on the train station platform? (e) According to a rider T on the train, the ends of the train will not simultaneously line up with the ends of the platform. What time interval does T measure between when the front end of the train lines up with the front end of the platform, and when the back end of the train lines up with the back end of the platform? 4.6 An astronaut in a starship travels to a Centauri, a distance of approx- . imately 4 ly as measured from Earth, at a speed of u/c = 0.8. (a) How long does the trip to a Centauri take, as measured by a clock on Earth? (b) How long does the trip to a Centauri take, measured by the star-ship pilot? (c) What is the distance between Earth and c~ Centauri, measured by the starship pilot? (d) A radio signal is sent from Earth to the starship every 6 months, as measured by a clock on Earth. What is the time interval between the reception of the signals aboard the starship? 122 Chapter 4 The Theory of Special Relativity (e) A radio signal is sent from the starship to Earth every 6 months, as measured by a clock aboard the starship. What is the time interval between the reception of the signals on Earth? (f) If the wavelength of the radio signal sent from Earth is A = 15 cm, to what wavelength must the starship's receiver be tuned? 4.7 Upon reaching a Centauri, the starship in Problem 4.6 immediately reverses direction and travels back to Earth at a speed of u/c = 0.8. (Assume that the turnaround itself takes zero time.) Both Earth and the starship continue to emit radio signals at 6-month intervals, as measured by their respective clocks. Make a table for the entire trip showing at what times Earth receives the signals from the starship. Do the same for the times when the starship receives the signals from Earth. Thus an Earth observer and the starship pilot will agree that the pilot has aged 4 years less than the Earth observer during the round-trip voyage to a Centauri. 4.8 In its rest frame, quasar Q2203+29 produces a hydrogen emission line of wavelength 1216 A. Astronomers on Earth measure a wavelength of 6568 A for this line. Determine the redshift parameter and the speed of recession for this quasar. (For more information about this quasar, see McCarthy et al. 1988.) 4.9 Quasar 3C 446 is violently variable; its luminosity at optical wave-lengths has been observed to change by a factor of 40 in as little as 10 days. Using the redshift parameter z = 1.404 measured for 3C 446, determine the time for the luminosity variation as measured in the quasar's rest frame. (For more details, see Bregman et al. 1988.) 4.10 Use the Lorentz transformation equations (4.16)-(4.19) to derive the velocity transformation equations (4.40)-(4.42). 4.11 The spacetime interval, As, between two events with coordinates (xi, yi, zi, ti) and (x2, Ya, z2, t2) is defined by (Os)2 = (cOt)2 - (Ox)2 - (AY)2 - (Oz)2. (a) Use the Lorentz transformation equations (4.16)-(4.19) to show that As has the same value in all reference frames. The spacetime interval is said to be invariant under a Lorentz transformation. Problems 123 (b) If (Os)2 > 0, then the interval is timelike. Show that in this case, OT - As c is the proper time between the two events. Assuming that tl < t2, could the first event have possibly caused the second event? (c) If (AS)2 = 0, then the interval is lightlike or null. Show that only light could have traveled between the two events. Could the first event have possibly caused the second event? (d) If (Os)2 < 0, then the interval is spacelike. What is the physical significance of - (Os)2? Could the first event have possibly caused the second event? The concept of a spacetime interval will play a key role in the discussion of general relativity in Chapter 16. 4.12 General expressions for the components of a light ray's velocity as measured in reference frame S are  v., = c sin B cos o vy = c sin B sin o 11z = C COS B, where B and 0 are the angular coordinates in a spherical coordinate system. (a) Show that v= v~+vy+vz=c. (b) Use the velocity transformation equations to show that, as mea-sured in reference frame S', v' vX'2+v y '2+vz'2=c, and so confirm that the speed of light has the constant value c in all frames of reference. 4.13 Starship A moves away from Earth with a speed of v,q/c = 0.8. Star-ship B moves away from Earth in the opposite direction with a speed - of vB/c = 0.6. What is the speed of starship A as measured by starship B? What is the speed of starship B as measured by starship A? 124 Chapter 4 The Theory of Special Relativity 4.14 Use Newton's second law F = dp/dt and the formula for relativistic momentum, Eq. (4.44), to show that the acceleration vector a = dv/dt produced by a force F acting on a particle of mass m is F v a=-- 2 (Fv), ym ymc where F v is the vector dot product between the force F and the particle velocity v. Thus the acceleration depends on the particle's velocity and is not in general in the same direction as the force. 4.15 Suppose a constant force of magnitude F acts on a particle of mass m initially at rest. (a) Integrate the formula for the acceleration found in Problem 4.14 to show that the speed of the particle after time t is given by v (F/m)t c (F/m)Zt2 + c2 . (b) Rearrange this equation to express the time t as a function of v/c. If the particle's initial acceleration at time t = 0 is a = g = 980 cm s-2, how much time is required for the particle to reach a speed of v/c = 0.9? v/c = 0.99? v/c = 0.999? v/c = 0.9999? v/c = 1? 4.16 Find the value of v/c when a particle's kinetic energy equals its rest energy. 4.17 Prove that in the low-speed Newtonian limit of v/c l, Eq. (4.45) does reduce to the familiar form K = 2 mv2. 4.18 Show that the relativistic kinetic energy of a particle can be written as p2 (1 +'y)m' where p is the magnitude of the particle's relativistic momentum. This demonstrates that in the low-speed Newtonian limit of v/c 1, K = p2/2m (as expected). 4.19 Derive Eq. (4.48). 154 Suggested Readings GENERAL Feynman, Richard, The Character of Physical Law, The M.LT. Press, Cam-bridge, MA, 1965. French, A. P., and Kennedy, P. J. (eds.), Niel,s Bohr: A Centenary Volume, Harvard University Press, Cambridge, MA, 1985. Hey, Tony, and Walters, Patrick, The Quantum Universe, Cambridge Univer-sity Press, Cambridge, 1987. Pagels, Heinz R., The Cosmic Code, Simon and Schuster, New York, 1982. Segre, Emilio, From X-Rays to Quarks, W. H. Freeman and Company, San F~ancisco, 1980. TECHNICAL Harwit, Martin, Astrophysical Concepts, Second Edition, John Wiley and Sons, New York, 1990.  Resnick, Robert, and Halliday, David, Basic Concepts in Relativity and Early Quantum Theory, Second Edition, John Wiley and Sons, New York, 1985. Problems Chapter 5 The Interaction of Light and Matter 5.1 Barnard's star, named after the American astronomer Edward E. Bar-nard, is an orange star in the constellation Ophiuchus. It has the largest known proper motion (tC = 10.31" yr-1) and the second-largest parallax angle (p = 0.552"). In the spectrum of Barnard's star, the Ha absorption line is observed to have a wavelength of 6560.44 ~.  (a) Determine the radial velocity of Barnard's star. (b) Determine the transverse velocity of Barnard's star. (c) Calculate the speed of Barnard's star through space. 5.2 When salt is sprinkled on a flame, yellow light consisting of two closely ~E spaced wavelengths, 5889.97 ~ and 5895.94 ~, is produced. They are called the sodium D lines and were observed by Fraunhofer in the Sun's spectrum.  Problems 155 (a) If this light falls on a diffraction grating with 300 lines per mil-limeter, what is the angle between the second-order spectra of these two wavelengths? (b) How many lines of this grating must be illuminated for the sodium D lines to just be resolved? 5.3 Prove that hc = 12400 eV 1~. 5.4 The photoelectric effect can be an important heating mechanism for the grains of dust found in interstellar clouds (see Section 12.1). The ejection of an electron leaves the grain with a positive charge, which affects the rates at which other electrons and ions collide with and _ stick to the grain to produce the heating. This process is particularly effective for ultraviolet photons (~ ti 1000 1~) striking the smaller dust grains. If the average energy of the ejected electron is about 5 eV, estimate the work function of a typical dust grain. 5.5 Use Eq. (5.5) for the momentum of a photon, plus the conservation of relativistic momentum and energy [Eqs. (4.44) and (4.48), respec-tively], to derive Eq. (5.6) for the change in wavelength of the scattered photon in the Compton effect. 5.6 Consider the case of a "collision" between a photon and a free proton, initially at rest. What is the characteristic change in the wavelength of the scattered photon in units of angstroms? How does this compare with the Compton wavelength, ~c? 5.7 Verify that the units of Planck's constant are the units of angular momentum. 5.8 A one-electron atom is an atom with Z protons in the nucleus, and with all but one of its electrons lost to ionization. (a) Starting with Coulomb's law, determine expressions for the or-bital radii and energies for a Bohr model of the one-electron atom with Z protons. (b) Find the radius of the ground-state orbit, the ground-state energy, and the ionization energy of singly ionized helium (He II). (c) Repeat part (b) for doubly ionized lithium (Li III). 5.9 To demonstrate the relative strength of the electrical and gravitational forces of attraction between the electron and the proton in the Bohr 156 Figure 5.14 Three de Broglie wavelengths spanning an electron's orbit in the Bohr atom. 5.10 Calculate the energies and wavelengths of all possible photons that are emitted when the electron cascades from the n = 3 to the n = 1 orbit of the hydrogen atom. 5.11 Find the shortest wavelength photon emitted by a downward electron transition in the Lyman, Balmer, and Paschen series. These wave-lengths are known as the series limits. In which regions of the electro-magnetic spectrum are these wavelengths found? 5.12 An electron in a television set reaches a speed of about 5 x 109 cm s-1 before it hits the screen. What is the wavelength of this electron? 5.13 Consider the de Broglie wave of the electron in the Bohr atom. The circumference of the electron's orbit must be an integral number of wavelengths, nA; see Fig. 5.14. Otherwise, the electron wave will find itself out of phase and suffer destructive interference. Show that this requirement leads to Bohr's condition for the quantization of angular ~ ;. momentum, Eq. (5.12). 5.14 A white dwarf is a very dense star, with its ions and electrons packed extremely close together. Each electron may be considered to be lo- Chapter 5 The Interaction of Light and Matter  atom, suppose the hydrogen atom were held together solely by the force of gravity. Determine the radius of the ground-state orbit (in units of A and AU) and the energy of the ground state (in eV). Problems 157 cated within a region of size Ox .^~ 1.5 x 10-1~ cm. Use Heisenberg's uncertainty principle, Eq. (5.18), to estimate the minimum speed of the electron. Do you think that the effects of relativity will be important for these stars? 5.15 An electron spends roughly 10-8 s in the first excited state of the hydrogen atom before making a spontaneous downward transition to the ground state. (a) Use Heisenberg's uncertainty principle (Eq. 5.19) to determine the uncertainty DE in the energy of the first excited state. (b) Calculate the uncertainty 0~ in the wavelength of the photon involved in a transition (either upward or downward) between the ground and first excited states of the hydrogen atom. Why can you assume that DE = 0 for the ground state? This increase in the width of a spectral line is called natural broadening. 5.16 Each quantum state of the hydrogen atom is labeled by a set of four quantum numbers: {n, ~, m~, ms}. (a) List the sets of quantum numbers for the hydrogen atom having n=1,n=2,andn=3. (b) Show that the degeneracy of energy level n is 2n2. 5.17 The members of a class of stars known as Ap stars are distinguished by their strong magnetic fields (usually a few thousand gauss).19 The star HD215441 has an unusually strong magnetic field of 34,000 G. Find the frequencies and wavelengths of the three components of the Ha spectral line produced by the normal Zeeman effect for this magnetic field. 5.18 Computer Problem One of the most important ideas of the physics of waves is that any complex waveform can be expressed as the sum of the harmonics of simple cosine and sine waves. That is, any wave function f (x) can be written as f(x)=ao+alcosx-~a2cos2x+a3cos3x+a4cos4x+~~~ +bl sin x + b2 sin 2x + b3 sin 3x + b4 sin 4x + ~ ~ ~ . 19The letter A is the star's spectral type (to be discussed in Section 8.1), and the letter p stands for "peculiar." 158 Chapter 5 The Interaction of Light and Matter The coefficients an and b,, tell how much of each harmonic goes into the recipe for f (x). This series of cosine and sine terms is called the Fourier series for f (x). In general, both cosine and sine terms are needed, but in this problem you will use only the sine terms; all of the an - 0. On page 145, the process of constructing a wave pulse by adding a series of sine waves was described. The Fourier sine series that you will use to construct your wave employs only the odd harmonics, and is given by N + 1 (sin x - sin 3x -+ - sin 5x - sin 7x + ~ sin Nx) 2 - E (_1)(n-1)/2 sin nx, N + 1 n odd where N is an odd integer. The leading factor of 21(N + 1) does not change the shape of T, but scales the wave for convenience so its maximum value is equal to one for any choice of N. (a) Graph IF for N = 5, using values of x (in radians) between 0 and -7r. What is the width, Ox, of the wave pulse? (b) Repeat part (a) for N = 11. (c) Repeat part (a) for N = 21. (d) Repeat part (a) for N = 41. (e) If IF represents the probability wave of a particle, for which value of N is the position of the particle known with the least uncer-tainty? For which value of N is the momentum of the particle known with the least uncertainty? 194 Chapter 6 Telescopes Problems 6.1 For some point P in space, show that, for any arbitrary closed surface surrounding P, the integral over a solid angle about P gives,  ~cot = ~ d~ = 4~r. 6.2 The light rays coming from an object do not, in general, travel parallel to the optical axis of a lens or mirror system. Consider an arrow to be the object, located a distance p from the center of a simple converging lens of focal length f , such that p > f . Assume that the arrow is perpendicular to the optical axis of the system with the tail of the arrow located on the axis. To locate the image, draw two light rays coming from the tip of the arrow: (i) One ray should follow a path parallel to the optical axis until it strikes the lens. It then bends toward the focal point of the side of the lens opposite the object. (ii) A second ray should pass directly through the center of the lens undeflected. This assumes that the lens is sufficiently thin. The intersection of the two rays is the location of the tip of the image arrow. All other rays coming from the tip of the object that pass through the lens will also pass through the image tip. The tail of the image is located on the optical axis, a distance q from the center of the lens. The image should also be oriented perpendicular to the optical axis. (a) Using similar triangles, prove the relation 1 1 1 p~--q- f. (b) Show that if the distance of the object is much larger than the focal length of the lens (p f ), then the image is effectively located on the focal plane. This is essentially always the situation for astronomical observations. The analysis of a diverging lens or a mirror (either converging or diverg-ing) is similar and leads to the same rela.tion between object distance, image distance, and focal length. Problems 195 6.3 Show that if two lenses of focal lengths fl and f2 can be considered to have zero physical separation, then the effective focal length of the combination of lenses is 1 _ 1 1 feff fl + fz . Note: assuming that the actual physical separation of the lenses is x, this approximation is strictly valid only when fl x and f2 x. 6.4 (a) Using the result of Problem 6.3, show that a compound lens sys-tem can be constructed from two lenses of different indices of refraction, nla and n2A, having the property that the resultant focal lengths of the compound lens at two specific wavelengths A1 and A2, respectively, can be made equal, or feff,al = feff,a2. (b) Argue qualitatively that this condition does not guarantee that the focal length will be constant for all wavelengths. 6.5 Prove that the angular magnification of a telescope having an objective focal length of fobj and an eyepiece focal length of feye is given by Eq. (6.9) when the objective and the eyepiece are separated by the sum of their focal lengths, fobi + fey,- 6.6 The diffraction pattern for a single slit (Figs. 6.7 and 6.8) is given by (e) - Io s. m(,3/2) L Q/2 l where 0 - 27rD sin 9/A. (a) Using 1'Hopital's rule, prove that the intensity at B = 0 is given by I(0) - Io.  (b) If the slit has an aperture of 1.0 pm, what angle B corresponds to the first minimum if the wavelength of the light is 5000 A? Express your answer in degrees. 6.7 Computer Problem Suppose that two identical slits are situated next to each other in such a way that the axes of the slits are parallel and oriented vertically. Assume also that the two slits are the same distance from a flat screen. Different light sources of identical intensity -- are placed behind each slit so that the two sources are incoherent, meaning that double-slit interference effects can be neglected. 196 Chapter 6 Telescopes (a) If the two slits are separated by a distance such that the central maximum of the diffraction pattern corresponding to the first slit is located at the second minimum of the second slit's diffraction pattern, plot the resulting superposition of intensities (i.e., the total intensity at each location). Include at least two minima to the left of the central maximum of the leftmost slit and at least two minima to the right of the central maximum of the rightmost slit. Hint: Refer to the equation given in Problem 6.6 and plot your results as a function of R. (b) Repeat your calculations for the case when the two slits are sepa-rated by a distance such that the central maximum of one slit falls at the location of the first minimum of the second (the Rayleigh criterion for single slits). (c) What can you conclude about the ability to resolve two individual sources (the slits) as the sources are brought progressively closer together? 6.8 (a) Using the Rayleigh criterion, estimate the angular resolution limit of the human eye at 5500A. Assume that the diameter of the pupil is 5 mm. (b) Compare your answer in part (a) to the angular diameters of the Moon and Jupiter. You may find the data in Appendix B helpful. (c) What can you conclude about the ability to resolve the Moon's disk and Jupiter's disk with the unaided eye? 6.9 (a) Using the Rayleigh criterion, estimate the theoretical diffraction limit for the angular resolution of a typical 8-inch amateur tele-scope at 5500 A. Express your answer in arc seconds. (b) Using the information in Appendix B, estimate the minimum size of a crater on the Moon that can be resolved by an 8-inch tele-scope. (c) Is this resolution limit likely to be achieved? Why or why not? 6.10 (a) Using the information provided in the text, calculate the focal length of the primary mirror of the New Technology Telescope. (b) What is the value of the plate scale of the NTT? (c) E Bootes is a double star system whose components are separated by 2.9". Calculate the linear separation of the images on the primary mirror focal plane of the NTT. Problems 195 6.3 Show that if two lenses of focal lengths fl and f2 can be considered to have zero physical separation, then the effective focal length of the combination of lenses is 1 _ 1 1 feff fl + fa Note: assuming that the actual physical separation of the lenses is x, this approximation is strictly valid only when fl x and f2 x. 6.4 (a) Using the result of Problem 6.3, show that a compound lens sys-tem can be constructed from two lenses of different indices of refraction, nla and n2a, having the property that the resultant focal lengths of the compound lens at two specific wavelengths A1 and A2, respectively, can be made equal, or .feff,al = ,feff,a2 - (b) Argue qualitatively that this condition does not guarantee that the focal length will be constant for all wavelengths. 6.5 Prove that the angular magnification of a telescope having an objective focal length of fobj and an eyepiece focal length of fey, is given by Eq. (6.9) when the objective and the eyepiece are separated by the sum of their focal lengths, fobl + fey,- 6.6 The diffraction pattern for a single slit (Figs. 6.7 and 6.8) is given by I(e) _ _L0 rsi ~~22)12 1 where 0 - 27rD sin 9/A. (a) Using 1'Hopital's rule, prove that the intensity at B = 0 is given by I(0) - lo. (b) If the slit has an aperture of 1.0 pm, what angle B corresponds to the first minimum if the wavelength of the light is 5000 A? Express your answer in degrees.  6.7 Computer Problem Suppose that two identical slits are situated next to each other in such a way that the axes of the slits are parallel and oriented vertically. Assume also that the two slits are the same distance from a flat screen. Different light sources of identical intensity are placed behind each slit so that the two sources are incoherent, meaning that double-slit interference effects can be neglected. Problems 197 6.11 Based on the specifications for HST's WF/PC 2, estimate the angular size of the field of view of one CCD in the planetary mode. 6.12 Suppose that a radio telescope receiver has a bandwidth of 50 MHz centered at 1.430 GHz (1 GHz = 1000 MHz). Rather than being a perfect detector over the entire bandwidth, assume that receiver's frequency dependence is triangular, meaning that the sensitivity of the detector is 0% at the edges of the band and 100% at its center. This filter function can be expressed as ifve= 0.42 between 0 rad and 2 rad (0 and 90, respectively). Neglect the Doppler shift selection effect. Hint: Refer to the discussion of integral averages found in Problem 2.10. 7.3 Assume that two stars are in circular orbits about a 'mutual center of mass and are separated by a distance a. Assume also that the angle of inclination is i and their stellar radii are rl and r2. (a) Find an expression for the smallest angle of inclination that will just barely produce an eclipse. Hint: Refer to Fig. 7.8. (b) If a = 2 AU, rl = 10 Ro, and r2 = 1 Rp, what minimum value of i will result in an eclipse? 7.4 Sirius is a visual binary with a period of 49.94 yr. Its measured trigono-metric parallax is 0.377" and, assuming that the plane of the orbit is in the plane of the sky, the true angular extent of the semimajor axis of the reduced mass is 7.62". The ratio of the distances of Sirius A and Sirius B from the center of mass is aA/aB = 0.466. (a) Find the mass of each member of the system. (b) The absolute bolometric magnitude of Sirius A is 1.33, and Sir-ius B has an absolute bolometric magnitude of 8.57. Determine their luminosities. Express your answers in terms of the luminos-ity of the Sun. (c) The effective temperature of Sirius B is estimated to be approx-imately 27,000 K. Estimate its radius, and compare your answer to the radii of the Sun and Earth.  7.5 ( Phe is a 1.67-day spectroscopic binary with nearly circular orbits. The maximum measured Doppler shifts of the brighter and fainter components of the system are 121.4 km s-1 and 247 km s-1, respec-tively. page 220 Chapter 7 Binary Stars and Stellar Parameters (a) Determine the quantity m sin 3 i for each star. (b) Using a statistically chosen value for sin3 i that takes into consid-eration the Doppler-shift selection effect, estimate the individual masses of the components of ~ Phe. 7.6 Rom the light and velocity curves of an eclipsing, spectroscopic bi-nary star system, it is determined that the orbital period is 6.31 yr, and the maximum radial velocities of stars A and B are 5.4 km s-1 and 22.4 km s-1, respectively. Furthermore, the time period between first contact and minimum light (tb - ta) is 0.58 d, the length of the primary minimum (t, -tb) is 0.64 d, and the apparent bolometric mag-nitudes of maximum, primary minimum, and secondary minimum are 5.40 magnitudes, 9.20 magnitudes, and 5.44 magnitudes, respectively. From this information, and assuming circular orbits, find the (a) Ratio of stellar masses. (b) Sum of the masses (assume i - 90). : (c) Individual masses. (d) Individual radii (assume that the orbits are circular). (e) Ratio of the effective temperatures of the two stars. 7.7 The V-band light curve of YY Sgr is shown in Fig. 7.2. Neglecting bolometric corrections, estimate the ratio of the temperatures of the two stars in the system. 7.8 Referring to the synthetic light curve and model of RR Centauri shown in Fig. 7.11, (a) Indicate the approximate points on the light curve (as a function of phase) that correspond to the orientations depicted. (b) Explain qualitatively the shape of the light curve. 7.9 Computer Problem (a) Using the information in Chapter 2 [including Eqs. (2.3) and (2.34)], write a short computer program to generate orbital radial velocity data similar to Fig. 7.6 for any choice of eccentricity. As sume that Ml = 0.5 Me, M2 = 2.0 Me, a = 2.0 AU, and i = 30. Plot your results for e = 0, 0.2, 0.4, and 0.5. (You may assume that the center-of-mass velocity is zero and the orientation of the major axis is perpendicular to the line of sight.) Problems 221 (b) Verify your results for e = 0 by using the equations developed in Section 7.3. (c) Explain how you might determine the eccentricity of an orbital system. Problems 251 Problems 8.1 Show that, at room temperature, the thermal energy kT ti 1/40 eV. At what temperature is kT equal to 1 eV? to 13.6 eV? 8.2 Verify that Boltzmann's constant can be expressed as k = 8.6174 x 10-5 eV K-1. 8.3 Use Fig. 8.4, the graph of the Maxwell-Boltzmann distribution for hydrogen gas at 10,000 K, to estimate the fraction of hydrogen atoms with a speed within 105 cm s-1 of the most probable speed, vII,p. 8.4 Show that the most probable speed of the Maxwell-Boltzmann distri-bution of molecular speeds (Eq. 8.1) is given by Eq. (8.2). 8.5 For a gas of neutral hydrogen atoms, at what temperature is the num-ber of atoms in the first excited state only 1% of the number of atoms in the ground state? At what temperature is the number of atoms in the first excited state only 10% of the number of atoms in the ground state? 8.6 Consider a gas of neutral hydrogen atoms, as in Example 8.2. (a) At what temperature will equal numbers of atoms have electrons in the ground state and in the second excited state (n = 3)? (b) At a temperature of 85,400 K, when an equal number (N) of atoms are in the ground state and in the first excited state, how many atoms are in the second excited state (n = 3)? Express your answer in terms of N. (c) As the temperature T ~ oo, how will the electrons in the hydro-gen atoms be distributed, according to the Boltzmann equation? That is, what will be the relative numbers of electrons in the n = l, 2, 3, . . . orbitals? Will this in fact be the distribution that actually occurs? Why or why not? 8.7 In Example 8.3, the statement was made that "nearly all of the H I atoms are in the ground state, so Eq. (8.5) for the partition function simplifies to Zj - gl = 2(1)2 = 2." Verify that this statement is correct for a temperature of 10,000 K by evaluating the first three terms in Eq. (8.5) for the partition function. 252 Chapter 8 The Classification of Stellar Spectra 8.8 Equation (8.5) for the partition function actually diverges as n --> oo. Why can we ignore these large-n terms? 8.9 Consider a box of electrically neutral hydrogen gas that is maintained at a constant volume V. In this simple situation, the number of elec-trons must equal the number of H II ions: neV = NII. Also, the total number of hydrogen atoms (both neutral and ionized), Nt, is related to the density of the gas by Nt = pV/(mP + me) - pV/mp where mP is the mass of the proton. (The tiny mass of the electron may be safely ignored in this expression for Nt.) Let the density of the gas be 10-9 g cm-3, typical of the photosphere of an AO star. (a) Make these substitutions into Eq. (8.6) to derive a quadratic equa-tion for the fraction of ionized atoms, CNil 12 + CNII1 Cap) C27rmekTl3/2 e_Xt/kT Nt J Nt lI p J h J2 - (ap) (27rm,kT \ 3/2 e_XrlkT = 0 p h JZ (b) Solve the quadratic equation in part (a) for the fraction of ionized hydrogen, NII/Nt, for a range of temperatures between 5000 K and 25,000 K. Make a graph of your results, and compare it with Fig. 8.6. 8.10 In this problem, you will follow a procedure similar to that of Exam-ple 8.3 for the case of a stellar atmosphere composed of pure helium to find the temperature at the middle of the He I partial ionization zone, where half of the He I atoms have been ionized. (Such an atmosphere would be found on a white dwarf of spectral type DB; see Section 15.1.) The ionization energies of neutral helium and singly ionized helium are XI = 24.6 eV and xll = 54.4 eV, respectively. The partition func-tions are ZI = 1, ZII = 2, and ZIII = 1 (as expected for a completely ionized atom). Use Pe = 200 dyne cm-2 for the electron pressure. (a) Use Eq. (8.7) to find NII/NI and NIII/NII for temperatures of 5000 K, 15,000 K, and 25,000 K. How do they compare? (b) Show that NII/Ntotal = NII/(Nr + NII + NIII) can be expressed in terms of the ratios NII/NI and NIII/NII. (c) Make a graph of NII/Ntotal similar to Fig. 8.6 for a range of tem-peratures from 5000 K to 25,000 K. What is the temperature Problems 253 at the middle of the He I partial ionization zone? Because the temperatures of the middle of the hydrogen and He I partial ion-ization zones are so similar, they are sometimes considered to be a single partial ionization zone with a characteristic temperature of 1-1.5 x 104 K. 8.11 Follow the procedure of Problem 8.10 to find the temperature at the middle of the He II partial ionization zone, where half of the He II atoms have been ionized. This ionization zone is found at a greater depth in the star, and so the electron pressure is larger-use a value of Pe = 104 dyne cm-2. Let your temperatures range from 10,000 K to 60,000 K. This particular ionization zone plays a crucial role in pulsating stars, as will be discussed in Section 14.2. 8.12 Use the Saha equation to determine the fraction of hydrogen atoms that are ionized, NII/Ntotal, at the center of the Sun. Here the tem-perature is 15.8 million K and the number density of electrons is about ne = 6.4 x 1025 cm-3. (Use ZI = 2.) Does your result agree with the fact that practically all of the Sun's hydrogen is ionized at the Sun's center? What is the reason for any discrepancy? 8.13 Use the information in Example 8.4 to calculate the ratio of doubly to singly ionized calcium atoms (Ca III/Ca II) in the Sun's photosphere. The ionization energy of Ca II is XII = 11.9 eV. Use ZIII = 1 for the partition function of Ca III. Is your result consistent with the statement in Example 8.4 that, in the solar photosphere, "nearly all of the calcium atoms are available for forming the H and K lines of calcium? 8.14 Consider a giant star and a main-sequence star of the same spectral type. Appendix E shows that the giant star, which has a lower at-mospheric density, has a slightly lower temperature than the main sequence star. Use the Saha equation to explain why this is so. Note that this means that there is not a perfect correspondence between temperature and spectral type! 8.15 Figure 8.13 shows that a white dwarf star typically has a radius that is only 1% of the Sun's. Determine the average density of a 1-Mo white dwarf. 254 Chapter 8 The Classification of Stellar Spectra 8.16 The blue-white star Fomalhaut ("the fish's mouth" in Arabic) is in the southern constellation of Pisces Austrinus. Fomalhaut has an apparent visual magnitude of V = 1.19. Use the H-R diagram in Fig. 8.15 to determine the distance to this star. 9.4 The Structure of Spectral Lines 305 Atomic Log Relative Column Density Element Number Abundance (g cm-2) Hydrogen 1 12.00 1.1 Helium 2 10.99 4.3 x 10-1 Oxygen 8 8.93 1.5 x 10-2 Carbon 6 8.60 5.3 x 10-3 Neon 10 8.09 2.7 x 10-3 ' Nitrogen 7 8.00 1.5 x 10-3 Iron 26 7.67 2.9 x 10-3 Magnesium 12 7.58 1.0 x 10-3 Silicon 14 7.55 1.1 x 10-3 Sulfur 16 7.21 5.7 x 10-4 Table 9.2 The Most Abundant Elements in the Solar Photosphere. The relative abundance of an element is given by loglo(NeiINx) + 12, and the column density is based on a value of 1.1 g cm-2 for hydrogen. (Data from Grevesse and Anders, Solar Atmosphere and Interior, A. N. Cox, W. C. Livingston, and M. S. Matthews (eds.), The University of Arizona Press, Tucson, AZ, 1991.) temperature to be calculated. Similarly, it is possible to use the Saha equation to find either the electron pressure or the ionization temperature (if the other is known) in the atmosphere from the relative numbers of atoms at various stages of ionization. The ultimate refinement in the analysis of stellar atmospheres is the con-struction of a model atmosphere on a computer. Each atmospheric layer is involved in the formation of line profiles and contributes to the spectrum ob served for the star. All of the ingredients of the preceding discussion, plus the equations of hydrostatic equilibrium, thermodynamics, statistical and quan-tum mechanics, and the transport of energy by radiation and convection, are combined with extensive libraries of opacities to calculate how the tempera-ture, pressure, and density vary with depth below the surface.32 Only when the variables of the model have been "fine-tuned" to obtain good agreement with the observations can astronomers finally claim to have decoded the information carried in the light from a star. This basic procedure has led astronomers to an understanding of the abun-dances of the elements in the Sun (see Table 9.2) and other stars. Hydrogen and helium are by far the most common elements, followed by oxygen, carbon, 32 Details of the construction of a model star will be deferred to Chapter 10. 306 Chapter 9 Stellar Atmospheres and nitrogen; for every 1012 atoms of hydrogen, there are 1011 atoms of helium and about 109 atoms of oxygen. These figures are in very good agreement with abundances obtained from meteorites, giving astronomers confidence in their results.33 This knowledge of the basic ingredients of the universe provides in-valuable observational tests and constraints for some of the most fundamental theories in astronomy: the nucleosynthesis of light elements as a result of stel-lar evolution, the production of heavier elements by supernovae, and the Big Bang that produced the primordial hydrogen and helium that started it all. Suggested Readings GENERAL Aller, Lawrence H., Atoms, Stars, and Nebulae, Revised Edition, Harvard University Press, Cambridge, MA, 1971. Hearnshaw, J. B., The Analysis of Starlight, Cambridge University Press, Cam-bridge, 1986. Kaler, James B., Stars and Their Spectra, Cambridge University Press, Cam-bridge, 1989. TECHNICAL Aller, Lawrence H., The Atmospheres of the Sun and Stars, Ronald Press, New York, 1963. Bohm-Vitense, Erika, "The Effective Temperature Scale," Annual Review of Astronomy and Astrophysics, 19, 295, 1981. Bohm-Vitense, Erika, Stellar Astrophysics, Volume 2: Stellar Atmospheres, Cambridge University Press, Cambridge, 1989. Novotny, Eva, Introduction to Stellar Atmospheres and Interiors, Oxford Uni-versity Press, New York, 1973. Rybicki, George B., and Lightman, Alan P., Radiative Processes in Astro-physics, John Wiley and Sons, New York, 1979. 33A notable exception is lithium, whose solar relative abundance of 101'ls is significantly less than the value of 103~31 obtained from meteorites. The efficient depletion of the Sun's lithium, sparing only one of every 140 lithium atoms, is not yet understood. Problems 307 Problems 9.1 Evaluate the energy of the blackbody photons inside your eye. Com-pare this with the visible energy inside your eye while looking at a 100-W (109 erg s-1) light bulb that is 100 cm away. (You can assume that the light bulb is 100% efficient, although in reality it converts only a few percent of its 100 watts into visible photons. Take your eye to be a hollow sphere of radius 1.5 cm at a temperature of 37C. The area of the eye's pupil is about 0.1 cm 2.) Why is it dark when you close your eyes? 9.2 (a) Find an expression for na dA, the number density of blackbody photons (the number of blackbody photons per cm3) with a wave-length between A and A + dA. (b) Find the total number of photons inside a kitchen oven set at 400F, assuming a volume of 1 m3. 9.3 (a) Use the results of Problem 9.2 to find the total number density, n, of blackbody photons of all wavelengths. Also show that the average energy per photon, u/n, is u _ T 4kT _ n 15(2.404) 2'70kT. (9.60) (b) Find the average energy per blackbody photon at the center of the Sun, where T = 1.58 x 107 K, and in the solar photosphere, where T = 5770 K. Express your answers in units of electron volts (eV). 9.4 Derive Eq. (9.9) for the blackbody radiation pressure, 00 4 3c ~ BA (T) dA = 43c = 3aT4 = 3u. 9.5 Consider a spherical blackbody of radius R and temperature T. In-tegrate Eq. (9.6) for the radiative flux with Ia = BA over all out-ward directions to derive the Stefan-Boltzmann equation in the form of Eq. (3.17). (You will also have to .i,ntegrate over all wavelengths and ` over the surface area of the sphere.) 308 Chapter 9 Stellar Atmospheres 9.6 Using the root-mean-square speed, vrTS, estimate the mean free path of the nitrogen molecules in your classroom at room temperature (300 K). What is the average time between collisions? Take the radius of a nitro gen molecule to be 1 ~, and the density of air to be 1.2 x 10-3 g cm-3. A nitrogen molecule contains 28 nucleons (protons and neutrons). 9.7 Calculate how far you could see through Earth's atmosphere if it had the opacity of the solar photosphere. Use the value for the Sun's opacity from Example 9.2 and 1.2 x 10-3 g cm-3 for the density of Earth's atmosphere. 9.8 In Example 9.3, suppose that only two measurements of the specific intensity, h and I2, are available, made at angles Bl and B2. Determine expressions for the intensity Ia,o of the light above Earth's atmosphere and for the vertical optical depth of the atmosphere, Ta,o, in terms of these two measurements. 9.9 Use the laws of conservation of relativistic energy and momentum to prove that an isolated electron cannot absorb a photon. 9.10 By measuring the slope of the curves in Fig. 9.10, verify that the decline of the curves after the peak in the opacity follows a Kramers law, ~ a T-n, where n ~ 3.5. 9.11 According to a "standard model" of the Sun, the central density is 162 g cm-3 and the Rosseland mean opacity at the center is 1.16 cm2 g-1. (a) Calculate the mean free path of a photon at the center of the Sun. (b) If this mean free path remained constant for the photon's journey to the surface, calculate the average time it would take for the photon to escape from the Sun. 9.12 If the temperature of a star's atmosphere is increasing outward, what type of spectral lines would you expect to find in the star's spectrum at those wavelengths where the opacity is greatest? 9.13 Consider a large hollow spherical shell of hot gas surrounding a star. Under what circumstances would you see the shell as a glowing ring around the star? What can you say about the optical thickness of the shell?  Problems , 309 Problems 9.14 through 9.24 involve the optional material at the end of Section 9.3. 9.14 Verify that the emission coefficient, ja, has units of cm s-3 sr-1. 9.15 Derive Eq. (9.29) in Example 9.5, which shows how the intensity of a light ray is converted from its initial intensity Ia to the value SA of the source function. 9.16 The transfer equation, Eq. (9.28), is written in terms of the distance, s, measured along the path of a light ray. In different coordinate systems, the transfer equation will look slightly different, and care must be taken to include all of the necessary terms. (a) Show that in a spherical coordinate system, with the center of the star at the origin, the transfer equation has the form cos B' dIa - =Ia-SA, KAp dr where B' is the angle between the ray and the outward radial direction. Note that you cannot simply replace s with r! (b) Use this form of the transfer equation to derive Eq. (9.25). 9.17 Using the Eddington approximation for a plane-parallel atmosphere, show that in the Eddington approximation, the mean intensity, radia-tive flux, and radiation pressure are given by Eqs. (9.40)-(9.43). 9.18 Using the Eddington approximation for a plane-parallel atmosphere, determine the values of I;n and Iot as functions of the vertical optical depth. At what depth is the radiation isotropic to within 1%? 9.19 Using the results for the plane-parallel gray atmosphere in LTE, deter-mine the ratio of the surface temperature of a star to its temperature at the top of the atmosphere. If Te = 5770 K, what is the temperature at the top of the atmosphere? 9.20 Show that, for a plane-parallel gray atmosphere in LTE, the (constant) value of the radiative flux is equal to 7r times the source function eval-uated at an optical depth of 2/3: Frad = r'S(Tv = 2/3). This function, called the Eddington-Barbier relation, says that the radiative flux received from the surface of the star is determined by the value of the source function at Tv = 2/3. 310 Chapter 9 Stellar Atmospheres 9.21 Consider a horizontal plane-parallel slab of gas of thickness L that is maintained at a constant temperature T. Assume that the gas has optical depth r,\,., with T,\ = 0 at the top surface of the slab. Assume further that no radiation enters the gas from outside. Use the general solution of the transfer equation (Eq. 9.49) to show that, when looking at the slab from above, you see blackbody radiation if Ta,o 1 and emission lines (where ja is large) for Ta,o 1. You may assume that the source function, Sa, does not vary with position inside the gas. You may also assume thermodynamic equilibrium when 1. 9.22 Consider a horizontal plane-parallel slab of gas of thickness L that is - maintained at a constant temperature T. Assume that the gas has optical depth with Ta = 0 at the top surface of the slab. Assume further that incident radiation of intensity Ia,o enters the bottom of the slab from outside. Use the general solution of the transfer equation (Eq. 9.49) to show that, when looking at the slab from above, you see blackbody radiation if Ta,o 1. If T,\,o l, show that you see ab-sorption lines superimposed on the spectrum of the incident radiation if IA,. > Sa and emission lines superimposed on the spectrum of the incident radiation if h,o < Sa. (These latter two cases correspond to the spectral lines formed in the Sun's photosphere and chromosphere, respectively; see Section 11.2.) You may assume that the source func-tion, Sa, does not vary with position inside the gas. You may also assume thermodynamic equilibrium when -TA,,, 1. 9.23 Verify that if the source function is Sa = aa+bA Ta,v, then the emergent intensity is given by Eq. (9.52), 1,\ (0) = aa + ba cos 9. 9.24 Computer Problem In this problem, you will use the values of the density and opacity at various points near the surface of the star to calculate the optical depth of these points. The data in Table 9.3 were obtained from the stellar model building program STATSTAR, described in Section 10.5 and found in Appendix I. The first point listed is at the surface of the stellar model. (a) Find the optical depth at each point by numerically integrating Eq. (9.13). Use a simple trapezoidal rule such that dT = -Kp ds becomes Kipi + KZ+LPi+l Tz+i - Tz = - ( 2 ) (rz+l - rz) Problems 311 i r (cm) T (K) p (g cm -3) r , (cm2 g-1) 1 7.10604E+10 O.OOO00E+00 O.OOO00E+00 O.OOO00E+00 2 7.09894E+10 3.28531E+03 1.74510E-11 2.30027E+02 3 7.09183E+10 6.57721E+03 2.51661E-10 1.71702E+02 4 7.08473E+10 9.87571E+03 1.23854E-09 1.48384E+02 5 7.07762E+10 1.31808E+04 3.72025E-09 1.30444E+02 6 7.07051E+10 1.64926E+04 8.61374E-09 1.16701E+02 7 7.06341E+10 1.98110E+04 1.70083E-08 1.06028E+02 8 7.05630E+10 2.31361E+04 3.01529E-08 9.75243E+01 9 7.04920E+10 2.64680E+04 4.94472E-08 9.05811E+01 10 7.04209E+10 2.98065E+04 7.64352E-08 8.47909E+01 11 7.03498E+10 3.31518E+04 1.12801E-07 7.98764E+01 12 7.02788E+10 3.62287E+04 1.55337E-07 7.57112E+01 13 7.02077E+10 3.93132E+04 2.08540E-07 7.20713E+01 14 7.01367E+10 4.24050E+04 2.73965E-07 6.88580E+01 15 7.00656E+10 4.55039E+04 3.53262E-07 6.59958E+01 16 6.99945E+10 4.86099E+04 4.48170E-07 6.34264E+01 17 6.99235E+10 5.17229E+04 5.60522E-07 6.11040E+01 18 6.98524E+10 5.48430E+04 6.92238E-07 5.89922E+01 19 6.97814E+10 5.79700E+04 8.45332E-07 5.70616E+01 20 6.97103E+10 6.11041E+04 1.02191E-06 5.52881E+01 21 6.96392E+10 6.42451E+04 1.22416E-06 5.36521E+01 22 6.95682E+10 6.73931E+04 1.45436E-06 5.21369E+01 23 6.94971E+10 7.05481E+04 1.71491E-06 5.07286E+01 24 6.94261E+10 7.37100E+04 2.00825E-06 4.94156E+01 25 6.93550E+10 7.68790E+04 2.33696E-06 4.81877E+01 26 6.92839E+10 8.00549E+04 2.70369E-06 4.70363E+01 27 6.92129E+10 8.32377E+04 3.11117E-06 4.59540E+01 28 6.91418E+10 8.64276E+04 3.56224E-06 4.49343E+01 29 6.90708E+10 8.96245E+04 4.05984E-06 4.39714E+01 30 6.89997E+10 9.28283E+04 4.60699E-06 4.30603E+01 31 6.89286E+10 9.60392E+04 5.20682E-06 4.21968E+01 32 6.88576E+10 9.92571E+04 5.86253E-06 4.13768E+01 33 6.87865E+10 1.02482E+05 6.57744E-06 4.05969E+01 Table 9.3 A 1 MD STATSTAR Model for Problem 9.24. Te = 5500 K.  312 Chapter 9 Stellar Atmospheres ) W (~) .f 3302.98 0.067 0.0049 5895.94 0.560 0.325 Table 9.4 Data for Solar Sodium Lines for Problem 9.27. (Data from Aller, Atoms, Stars, and Nebulae, Revised Edition, Harvard University Press, Cambridge, MA, 1971.) Note that because s is measured along the path traveled by the photons, ds = dr. (b) Make a graph of the temperature (vertical axis) vs. the optical depth (horizontal axis). (c) For each value of the optical depth, use Eq. (9.48) to calculate the temperature for a plane-parallel gray atmosphere in LTE. Plot these values of T on the same graph. (d) The STATSTAR program utilizes a simplifying assumption that the surface temperature is zero (see Appendix H). Comment on the validity of the surface value of T that you found. 9.25 Suppose that the shape of a spectral line is fit with one-half of an ellipse, such that the semimajor axis a is equal to the maximum depth of the line (let Fa = 0), and the minor axis 2b is equal to the maximum width of the line (where it joins the continuum). What is the equivalent width of this line? Hint: You may find Eq. (2.4) useful. 9.26 Derive Eq. (9.55) for the uncertainty in the wavelength of a spectral line due to Heisenberg's uncertainty principle. 9.27 The two solar absorption lines given in Table 9.4 are produced when an electron makes an upward transition from the ground state orbital of the neutral Na I atom. (a) Using the general curve of growth for the Sun, Fig. 9.22, repeat the procedure of Example 9.11 to find Na, the number of absorb-ing sodium atoms per unit area of the photosphere. (b) Combine your results with those of Example 9.11 to find an av-erage value of Na. Use this value to plot the positions of the four sodium absorption lines on Fig. 9.22,.-and confirm that they do all lie on the curve of growth. Problems 313 .f 10938 (Py) 2.2 0.0554 10049 (Pb) 1.6 0.0269 Table 9.5 Data for Solar Hydrogen Lines for Problem 9.28. (Data from Aller, Atoms, Stars, and Nebulae, Revised Edition, Harvard University Press, Cambridge, MA, 1971.) 9.28 Pressure broadening (due to the presence of the electric fields of nearby ions) is unusually effective for the spectral lines of hydrogen. Using the general curve of growth for the Sun with these broad hydrogen ab sorption lines will result in an overestimate of the amount of hydrogen present. The following calculation nevertheless demonstrates just how abundant hydrogen is in the Sun. The two solar absorption given in Table 9.5 belong to the Paschen series, produced when an electron makes an upward transition from the (n = 3) orbit of the hydrogen atom. (a) Using the general curve of growth for the Sun, Fig. 9.22, repeat the procedure of Example 9.11 to find Na, the number of absorb-ing hydrogen atoms per unit area of the photosphere (those with electrons initially in the n = 3 orbit). (b) Use the Boltzmann and Saha equations to calculate the total number of hydrogen atoms above each square centimeter of the Sun's photosphere. (c) Calculate the column density of hydrogen atoms, and compare your result with the value found in Table 9.2. Problems 375 Liebert, James, and Probst, Ronald G., "Very Low Mass Stars," Annual Re-view of Astronomy and Astrophysics, 25, 473, 1987. Novotny, Eva, Introduction to Stellar Atmospheres and Interiors, Oxford Uni-versity Press, New York, 1973. Problems 10.1 Show that the equation for hydrostatic equilibrium, Eq. (10.7), can also be written in terms of the optical depth T, as dP_g dT This form of the equation is often useful in building model stellar atmospheres. 10.2 Prove that the gravitational force on a point mass located anywhere inside a hollow, spherically symmetric shell is zero. Assume that the mass of the shell is M and has a constant density p. Assume also that the radius of the inside surface of the shell is rl and that the radius of the outside surface is r2. The mass of the point is m. 10.3 Assuming that 10 eV could be released by every atom in the Sun through chemical reactions, estimate how long the Sun could shine at its current rate through chemical processes alone. For simplicity, assume that the Sun is composed entirely of hydrogen. Is it possible that the Sun's energy is entirely chemical? Why or why not? 10.4 (a) What temperature would be required for two protons to collide if quantum mechanical tunneling is neglected? Assume that nuclei having velocities ten times the root-mean-square (rms) value for the Maxwell-Boltzmann distribution can overcome the Coulomb barrier. Compare your answer with the estimated central tem-perature of the Sun. (b) Using Eq. (8.1), calculate the ratio of the number of protons hav-ing velocities ten times the rms value to those moving at the rms velocity. (c) Assuming (incorrectly) that the Sun is pure hydrogen, estimate the number of hydrogen nuclei in the Sun. Could there be enough protons moving with a speed-ten times the rms value to account - for the Sun's luminosity? , 376 Chapter 10 The Interiors of Stars 10.5 Derive the ideal gas law, Eq. (10.11). Begin with the pressure integral (Eq. 10.10) and the Maxwell-Boltzmann velocity distribution function (Eq. 8.1). 10.6 Derive Eq. (10.34) from Eq. (8.1). 10.7 Show that the form of the Coulomb potential barrier penetration prob-ability given by Eq. (10.38) follows directly from Eq. (10.37). 10.8 Prove that the energy corresponding to the Gamow peak is given by Eq. (10.41). 10.9 Calculate the ratio of the energy generation rate for the pp chain to the energy generation rate for the CNO cycle given conditions characteristic of the center of the present-day (evolved) Sun, namely T = 1.58 x 107 K, p = 162 g cm-3, X = 0.34, and XcNO = 0.013.2~ Assume that the pp chain screening factor is unity ( f~ = 1) and that the pp chain branching factor is unity (~PP = 1). 10.10 Beginning with Eq. (10.56) and writing the energy generation rate in the form E(7,) = E~~7,8 ' show that the temperature dependence for the triple alpha process, given by Eq. (10.57), is correct. E~~ is a function that is independent of temperature. Hint: First take the natural logarithm of both sides of Eq. (10.56) and then differentiate with respect to 1n Tg. Follow the same procedure with your power law form of the equation and compare the results. You may want to make use of the relation dlne _ d1nE _ d1nE d ln T8 T$ dT8 T g dT$ ~ 10.11 The Q value of a reaction is the amount of energy released (or ab-sorbed) during the reaction. Calculate the Q value for each step of the PP I reaction chain (Eq. 10.46). Express your answers in MeV. The masses of iH and 2He are 2.0141 u and 3.0160 u, respectively. Z7The interior values assumed here are taken from the standard solar model of Guzik (private communication), 1994; see Section 11.1. Problems 377 10.12 Calculate the amount of energy released or absorbed in the following reactions (express your answers in MeV): (a) isC + 1sC ~ i2 Mg +'Y (b) 16C+16C-->180+22He (c) 19F + iH -+ i80 + 2 He The mass of 16 C is 12.0000 u, by definition, and the masses of 1g O, 19F, and 12Mg are 15.99491 u, 18.99840 u, and 23.98504 u, respectively. Are these reactions exothermic or endothermic? 10.13 Complete the following reaction sequences. Be sure to include any necessary leptons. (a) i4 S1 -' isAl + e+ + ? (b) is Al + iH -~ izMg + 4? (c) i7Cl + iH ~ isAr + ? 10.14 Prove that Eq. (10.75) follows from Eq. (10.74). 10.15 Estimate the hydrogen burning lifetimes of stars on the lower and up-per ends of the main sequence. The lower end of the main sequence 28 occurs near 0.085 Me, with loglo Te = 3.438 and loglo(L/Lo) _ -3.297, while the upper end of the main sequence 29 occurs at ap-proximately 90 Me with loglo Te = 4.722 and loglo(L/Lo) = 6.045. Assume that the 0.085 Me star is entirely convective so that, through convective mixing, all of its hydrogen becomes available for burning rather than just the inner 10%. 10.16 Using the information given in Problem 10.15, calculate the radii of a 0.085 Me star and a 90 Me star. What is the ratio of their radii? 10.17 Verify that the basic stellar structure equations [Eqs. (10.7), (10.8), (10.45), (10.61)] are satisfied by the 1 Me STATSTAR model found in Appendix I. This may be done by selecting two adjacent zones and 28 Data from Grossman, Hays, and Graboske, Astron. Astrophys., 30, 95, 1974. 29Data from Cahn, Cox, and Ostlie, in Lecture Notes in Physics: Stellar Pulsation, Arthur N. Cox, Warren M. Sparks, and Sumner G. Starrfield (eds.), Springer-Verlag, Berlin, 51, 1987. 378 Chapter 10 The Interiors of Stars numerically computing the derivatives on the left-hand sides of the equations, for example dP PZ+1 - PZ  and comparing your results with results obtained from the right-hand sides using average values of quantities for the two zones [e.g., Mr = (Mi + Mi+l)/2]. Carry out your calculations for the two shells at r = 1.27 x 101 cm and r = 1.34 x 101 cm and then compare your results for the right- and left-hand sides of each equation by determining relative errors. Note that the model in Appendix I assumes complete ionization everywhere and has the uniform composition X = 0.7, Y = 0.292, Z = 0.008. Your results on the right- and left-hand sides will not agree exactly because STATSTAR uses a Runge-Kutta numerical algorithm that carries out intermediate steps not shown in Appendix I. 10.18 Computer Problem Appendix I contains an example of a theoret-ical 1.0 Mo main-sequence star produced by the stellar structure code STATSTAR, found in Appendix H. Using STATSTAR, build a second main-sequence star with a mass of 0.75 Mo that has a homogeneous composition of X = 0.7, Y = 0.292, and Z = 0.008. For these val-ues, the model's luminosity and effective temperature are 0.1877 Lo and 3839.1 K, respectively. Compare the central temperatures, pres-sures, densities, and energy generation rates between the 1.0 Mo and 0.75 Mo models. Explain the differences in the central conditions of the two models. 10.19 Computer Problem Use the stellar structure code STATSTAR found in Appendix H, together with the theoretical STATSTAR H-R diagram and mass-effective temperature data provided in Appendix I, to calculate a homogeneous, main-sequence model having the compo-sition X = 0.7, Y = 0.292, and Z = 0.008. (Note: It may be more illustrative to assign each student in the class a different mass for this problem so that the results can be compared.) (a) After obtaining a satisfactory model, plot P versus r, M,. versus r, LT versus r, and T versus r. (b) At what temperature has Lr reached approximately 99% of its surface value? 50% of its surface value? Is the temperature asso ciated with 50% of the total luminosity consistent with the rough estimate found in Eq. (10.33)? Why or why not? Problems 379 (c) What are the values of MT/,, for the two temperatures found in part (b)? M* is the total mass of the stellar model. (d) If each student in the class calculated a different mass, compare the changes in the following quantities with mass: (i) The central temperature. (ii) The central density. (iii) The central energy generation rate. (iv) The extent of the central convection zone with mass fraction and radius. (v) The effective temperature. (vi) The radius of the star. (e) If each student in the class calculated a different mass, Plot each model on a graph of luminosity versus mass (i.e., plot L*/L(D versus M,./MD). (ii) Plot loglo(L*/Lo) versus loglo(M*/Mo) for each stellar model. (iii) Using an approximate power law relation of the form L*lLo = (M*lMo)', find an appropriate value for a. a may differ for different compositions or vary somewhat with mass. This is known as the mass-luminosity relation (see Fig. 7.7). 10.20 Computer Problem Repeat Problem 10.19 using the same mass but a different composition; assume X = 0.7, Y = 0.290, Z = 0.010. (a) For a given mass, which model (Z = 0.008 or Z = 0.010) has the largest central temperature? the largest central density? (b) Referring to the appropriate stellar structure equations and con-stitutive relations, explain your results in part (a). (c) Which model has the largest energy generation rate at the center? Why? (d) How do you account for the differences in effective temperature and luminosity between your two models? 432 Chapter 11 The Sun Griffiths, David J., Introduction to Electrodynamics, Second Edition, Pren-tice-Hall, Englewood Cliffs, NJ, 1989. Hathaway, David H., and Wilson, Robert M., "Solar Rotation and the Sunspot Cycle," The Astrophysical Journal, 357, 271, 1990. Mariska, John T., "The Quiet Solar Transition Region," Annual Review of Astronomy and Astrophysics, 24, 23, 1986. Moore, Ronald and Rabin, Douglas, "Sunspots," Annual Review of Astronomy and Astrophysics, 23, 239, 1985. Parker, E. N., "Dynamics of Interplanetary Gas and Magnetic Fields," The Astrophysical Journal, 128, 664, 1958. Zirin, Harold, Astrophysics of the Sun, Cambridge University Press, C bridge, 1988. Problems 11.1 Using Fig. 11.2, verify that the change in the Sun's effective tempera-ture over the past 4.5 billion years is consistent with the variations in its radius and luminosity. 11.2 (a) At what rate is the Sun's mass decreasing due to nuclear reac-tions? Express your answer in solar masses per year. (b) Compare your answer to part (a) with the mass loss rate due to the solar wind. (c) Assuming that the solar wind mass loss rate remains constant, would either mass loss process significantly affect the total mass of the Sun over its entire main-sequence lifetime? 11.3 Using the Saha equation, calculate the ratio of the number of H-ions to neutral hydrogen atoms in the Sun's photosphere. Take the temperature of the gas to be the effective temperature, and assume that the electron pressure is 15 dyne cm-2. Note that the Pauli exclusion principle requires that only one state can exist for the ion because its two electrons must have opposite spins. 11.4 The Paschen series of hydrogen (n = 3) can contribute to the visible %v continuum for the Sun since the series limit occurs at 8208 A. However, it is the contribution from the H- ion that dominates the formation Problems 433 of the continuum. Using the results of Problem 11.3, along with the Boltzmann equation, estimate the ratio of the number of H- ions to hydrogen atoms in the n = 3 state. 11.5 (a) Using Eq. (9.58) and neglecting turbulence, estimate the full width at half-maximum of the hydrogen Ha absorption line due to random thermal motions in the Sun's photosphere. Assume that the temperature is the Sun's effective temperature. (b) Using Ha redshift data for solar granulation, estimate the full width at half-maximum when convective turbulent motions are included with thermal motions. (c) What is the ratio of vt rb to 2kT/m? (d) Determine the relative change in the full width at half-maximum due to Doppler broadening when turbulence is included. Does turbulence make a significant contribution to (0~)1~2 in the solar photosphere? 11.6 Estimate the thermally Doppler-broadened line widths for the hydro-gen Lyman a, C III, 0 VI, and Mg X lines given on page 401; use the temperatures provided. Take the masses of H, C, O, and Mg to be 1 u, 12 u, 16 u, and 24 u, respectively. 11.7 (a) Using Eq. (3.20), show that in the Sun's photosphere ln (Ba/Bb) ~ 11.5 + ~7, C ~6 ~a / where Ba/B6 is the ratio of the amount of blackbody radiation emitted at ~a = 100 1~ to the amount emitted at ~6 = 1000 1~, centered in a wavelength band 1 1~ wide. (b) What is the value of this expression for the case where the tem-perature is taken to be the effective temperature of the Sun? (c) Writing the ratio in the form Ba/Bb = 10~, determine the value of x. 11.8 Suppose that you are attempting to make observations through an op-tically thick gas that has a constant density and temperature. Assume that the density and temperature of the gas are 2.5 x 10-7 g cm-3 and 5770 K, respectively, typical of the values found at the base of the Sun's photosphere. If the opacity of the gas at one wavelength (~1) ,is r~~,l = 0.26 cm2 g-1 and the opacity at another wavelength (~2) is 434 Chapter 11 The Sun r,a2 = 0.30 cm 2 g-1, calculate the distance into the gas where the op-tical depth equals 2/3 for each wavelength. At which wavelength can you see farther into the gas? How much farther? This effect allows astronomers to probe the Sun's atmosphere at different depths (see Fig. 11.17). 11.9 (a) Using the data given in Example 11.2, estimate the pressure scale height at the base of the photosphere. (b) Assuming that the mixing length to pressure scale height ratio is 2.2, use the measured Doppler velocity of solar granulation to estimate the amount of time required for a convective bubble to travel one mixing length. Compare this value to the characteristic lifetime of a granule. : 11.10 Show that Eq. (11.7) follows directly from Eq. (11.6). 11.11 Calculate the magnetic pressure in the center of the umbra of a large sunspot. Assume that the magnetic field strength is 2000 G. Compare your answer with a typical value for the gas pressure at the base of the photosphere. 11.12 Assume that a large solar flare erupts in a region where the magnetic field strength is 300 G and it releases 1032 ergs in one hour. (a) What was the magnetic energy density in that region before the eruption began? (b) What minimum volume would be required to supply the magnetic energy necessary to fuel the flare? (c) Assuming for simplicity that the volume involved in supplying the energy for the flare eruption was a cube, compare the length of one side of the cube with the typical size of a large flare. (d) How long would it take an Alfven wave to travel the length of the flare? (e) What can you conclude about the assumption that magnetic en-ergy is the source of solar flares, given the physical dimensions and time scales involved? 11.13 (a) Calculate the frequency shift produced by the normal Zeeman effect in the center of a sunspot that,,Uas a magnetic field strength ` of 3000 G. Problems 435 (b) By what fraction would the wavelength of one component of the 6302.5 A Fe I spectral line change due to a magnetic field of 3000 G? 11.14 Argue from Eq. (11.12) and the work integral that magnetic pressure is given by Eq. (11.13). page 478 Chapter 12 The Process of Star Formation O'Dell, C. R., and Wen, Zheng, "Postrefurbishment Mission Hubble Space Telescope Images of the Core of the Orion Nebula: Proplyds, Herbig-Haro Ob-jects, and Measurements of a Circumstellar Disk," The Astrophysical Journal, 436, 194, 1994. Osterbrock, Donald E., Astrophysics of Gaseous Nebulae and Active Galactic Nuclei, University Science Books, Mill Valley, CA, 1989. Puget, J. L., and Leger, A., "A New Component of the Interstellar Matter: Small Grains and Large Aromatic Molecules," Annual Review of Astronomy and Astrophysics, 27, 161, 1989. Shu, Frank H., Adams, Fred C., and Lizano, Susana, "Star Formation in Molec-ular Clouds: Observation and Theory," Annual Review of Astronomy and As-trophysics, 25, 23, 1987. Stahler, Steven W., "Understanding Young Stars: A History," Publications of the Astronomical Society of the Pacific, 100, 1474, 1988. Zhou, Shudong, Evans, Neal J. II, Kornpe, Carsten, and Walmsley, C. M., "Evidence for Protostellar Collapse in B335," The Astrophysical Journal, 404, 232, 1993. Problems 12.1 In a certain part of the North American Nebula, the amount of in-terstellar extinction in the visual wavelength band is 1.1 magnitudes. The thickness of the nebula is estimated to be 20 pc and it is located 700 pc from Earth. Suppose that a B spectral class main-sequence star is observed in the direction of the nebula, and that the absolute visual magnitude of the star is known to be MV = -1.1 from spectroscopic data. Neglect any other sources of extinction between the observer and the nebula. (a) Find the apparent visual magnitude of the star if it is lying just in front of the nebula. (b) Find the apparent visual magnitude of the star if it is lying just behind the nebula. (c) Without taking the existence of the nebula into consideration, based on its apparent magnitude, how far away does the star in _ part (b) appear to be? What would the percentage error be in determining the distance if interstellar extinction were neglected? Problems 479 12.2 Estimate the temperature of a dust grain that is located 100 AU from a newly formed FO main-sequence star. Hint: Assume that the dust grain is in thermal equilibrium-meaning that the amount of energy absorbed by the grain in a given time interval must equal the amount of energy radiated away during the same interval of time. Assume also that the dust grain is spherically symmetric and emits and absorbs radiation as a perfect blackbody. You may want to refer to Appendix E for the effective temperature and radius of an FO main-sequence star.  12.3 The Boltzmann factor, e-~E2-E1~~kT, helps determine the relative pop-ulations of energy levels (see Section 8.1). Using the Boltzmann factor, estimate the temperature required for a hydrogen atom's electron and proton to go from being antialigned to aligned. Are the temperatures in H I clouds sufficient to produce this low-energy excited state? 12.4 An H I cloud produces a 21-cm line with an optical depth at its center of TH = 0.5 (the line is optically thin). The temperature of the gas is 100 K, the line's full width at half-maximum is 10 km s-l, and the average atomic number density of the cloud is estimated to be 10 cm-3. From this information and Eq. (12.4), find the thickness of the cloud. Express your answer in pc. 12.5 Using an approach analogous to the development of Eq. (10.36) for nuclear reaction rates, make a crude estimate of the number of ran-dom collisions per cubic centimeter per second between CO and HZ molecules in a giant molecular cloud that has a temperature of 15 K and a number density of nHz = 102 cm-3. Assume (incorrectly) that the molecules are spherical in shape with radii of approximately 1 1~, the characteristic size of an atom.  12.6 The rotational kinetic energy of a molecule is given by 1 2 L2 ~'' rot = 2I w = 2I ' where L is the molecule's angular momentum and I is its moment of inertia. The angular momentum is restricted by quantum mechanics to the discrete values   where ~ = 0,1, 2, . . .. 480 Chapter 12 The Process of Star Formation (a) For a diatomic molecule, I = mirl -I- m2r2, where ml and mz are the masses of the individual atoms and rl and r2 are their separations from the center of mass of the molecule. Using the ideas developed in Section 2.3, show that I may be written as where p is the reduced mass and r is the separation between the atoms in the molecule. (b) The separation between the C and O atoms in CO is approxi-mately 1.2 A, and the atomic masses of 1zC, 13C, and 160 are 12.000 u, 13.003 u, and 15.995 u, respectively. Calculate the mo-ments of inertia for 12 CO and 13C0. (c) What is the wavelength of the photon that is emitted by 12C0 during a transition between the rotational angular momentum states P = 3 and f = 2? To which part of the electromagnetic spectrum does this correspond? (d) Repeat part (c) for 13C0. How do astronomers distinguish be-tween different isotopes in the interstellar medium? 12.7 Calculate the Jeans length for the giant molecular cloud in Exam-ple 12.2. 12.8 (a) By using the ideal gas law, calculate IdP/drj ~~ JOP/Orl ti P~/Rj at the beginning of the collapse of a giant molecular cloud, where P, is an approximate value for the central pressure of the cloud. Assume that P = 0 at the edge of the molecular cloud and take its mass and radius to be the Jeans values found in Example 12.2 and in Problem 12.7. You should also assume the cloud temperature and density given in Example 12.2. (b) Show that, given the accuracy of our crude estimates, IdP/drl found in part (a) is comparable to (i.e., within an order of magni-tude of) GMrp/rz, as required for quasi-hydrostatic equilibrium. (c) As long as the collapse remains isothermal, show that the con-tribution of dP/dr in Eq. (10.6) continues to decrease relative to GMTp/rz, supporting the assumption made in Eq. (12.9) that dP/dr can be neglected once free-fall collapse begins. Problems : 481 12.9 Assuming that the free-fall acceleration of the surface of a collapsing cloud remains constant during the entire collapse, derive an expres-sion for the free-fall time. Show that your answer only differs from Eq. (12.16) by a term of order unity. 12.10 Using Eq. (10.76), estimate the sound speed of the giant molecular cloud discussed in Examples 12.2 and 12.3. Use this speed to find the amount of time required for a sound wave to cross the cloud, tsound = 2Rj/vsond, and compare your answer to the estimate of the free-fall time found in Example 12.3. Why would you expect the two values to be approximately the same? 12.11 Using the information contained in the text, derive Eq. (12.18). 12.12 Estimate the gravitational energy per unit volume in the giant molecu-lar cloud in Example 12.2 and compare that with the magnetic energy density that would be contained in the cloud if it has a magnetic field of uniform strength, B = 10 EcG. [Hint: Refer to Eq. (11.12).] Could magnetic fields play a significant role in the collapse of a cloud? 12.13 (a) Beginning with Eq. (12.9), adding a centripetal acceleration term, and using conservation of angular momentum, show that the col-lapse of a cloud will stop in the plane perpendicular to its axis of rotation when the radius reaches wo ro rf 2GMT where MT is the interior mass, and wo and ro are the original an-gular velocity and radius of the surface of the cloud, respectively. Assume that the initial radial velocity of the cloud is zero and that r f rp. You may also assume (incorrectly) that the cloud rotates as a rigid body during the entire collapse. Hint: Recall from the discussion leading to Eq. (11.9) that d2r/dt2 = vT dvT/dr. (Since no centripetal acceleration term exists for collapse along the rota-tion axis, disk formation is a consequence of the original angular momentum of the cloud.) (b) Assume that the original cloud had a mass of 1 Mo and an initial radius of 0.5 pc. If collapse is halted at approximately 100 AU, find the initial angular velocity of the cloud. (c) What was the original rotatiWl velocity (in cm s-1) of the edge of the cloud? 482 Chapter 12 The Process of Star Formation (d) Assuming that the moment of inertia is approximately that of a uniform solid sphere, IsPhere = 5 Mr2, when the collapse begins and a uniform disk, Idisk = 2Mr2, when it stops, determine the rotational velocity at 100 AU. (e) After the collapse has stopped, calculate the time required for a piece of mass to make one complete revolution around the cen-tral protostar. Compare your answer with the orbital period at 100 AU expected from Kepler's third law. Why would you not expect the two periods to be identical? 12.14 Estimate the Eddington luminosity of a 0.085 Mo star and compare your answer to the main-sequence luminosity given in Problem 10.15. Assume 7~ = 0.01 cm 2 g-1. Is radiation pressure likely to be significant in the stability of a low-mass main-sequence star? 12.15 Assuming a mass loss rate of 10-7 Mo yr-1 and a stellar wind velocity of 80 km s-1 from a T Tauri star, estimate the mass density of the wind at a distance of 100 AU from the star. (Hint: Refer to Example 11.1.) Compare your answer with the density of the giant molecular cloud in Example 12.2. 536 Chapter 13 Post-Main-Sequence Stellar Evolution Dupree, A. K., "Mass Loss From Cool Stars," Annual Review of Astronomy and Astrophysics, 24, 377, 1986. Hanes, Dave, and Madore, Barry (eds.), Globular Clusters, Cambridge Univer-sity Press, Cambridge, 1980. Hansen, C. J., and Kawaler, S. D., Stellar Interiors: Physical Principles, Struc-ture, and Evolution, Springer-Verlag, New York, 1994. e Harpaz, Amos, Stellar Evolution, A K Peters, Wellesley, MA, 1994. Iben, Icko Jr., "Stellar Evolution Within and Off the Main Sequence," Annual Review of Astronomy and Astrophysics, 5, 571, 1967. Iben, Icko Jr., "Single and Binary Star Evolution," The Astrophysical Journal Supplement Series, 76, 55, 1991. Iben, Icko Jr., and Renzini, Alvio, "Asymptotic Giant Branch Evolution and Beyond," Annual Review of Astronomy and Astrophysics, 21, 271, 1983. Kippenhahn, Rudolf, and Weigert, Alfred, Stellar Structure and Evolution, Springer-Verlag, Berlin, 1990. Schaerer, D., et al., "Grids of Stellar Models. IV. From 0.8 to 120 Mp at Z=0.040," Astron. Astrophys. Suppl., 102, 339, 1993.  Shklovskii, Iosif S., Stars: Their Birth, Life, and Death, W. H. Freeman and Company, San Francisco, 1978. Suntzeff, Nicholas B., et al., "The Energy Sources Powering the Late-Type Bolometric Evolution of SN 1987A," The Astrophysical Journal Letters, 384, , L33, 1992. 14 Woosley, S. E., and Weaver, Thomas A., "The Physics of Supernova Explo-sions," Annual Review of Astronomy and Astrophysics, 24, 205, 1986. Problems 13.1 (a) Beginning with Eq. (13.7), show that the radius of the isother-mal core for which the gas pressure is a maximum is given by Eq. (13.8). Recall that this solution assumes that the gas in the core is ideal and monatomic. (b) From your results in part (a), show that the maximum pressure at the surface of the isothermal core is given by Eq. (13.9). Problems 537 13.2 During the first dredge-up phase of a 5 Mo star, would you expect the composition ratio Xi3/X12 to increase or decrease? Explain your reasoning. Hint: You may find Fig. 13.6 helpful. 13.3 Use Eq. (10.33) to show that the ignition of the triple alpha process at the tip of the red giant branch ought to occur at more than 108 K. 13.4 In an attempt to identify the important components of AGB mass loss, various researchers have proposed parameterizations of the mass loss rate that are based on fitting observed rates for a specified set of stars with some general equation that includes measurable quantities associated with the stars in the sample. One of the most popular, developed by D. Reimers, is given by NI = -4 x 10-13~ R Mo yr-1, (13.23) 9 where, L, g, and R are the luminosity, surface gravity, and radius of the star, respectively (all in solar units; go = 2.74 x 104 cm s-2). r~ is a free parameter whose value is expected to be near unity. Note that the minus sign has been explicitly included here, indicating that the mass of the star is decreasing. (a) Explain qualitatively why L, g, and R enter Eq. (13.23) in the way they do. (b) Estimate the mass loss rate of a 1 Mo AGB star that has a lumi-nosity of 7000 Lp and a temperature of 3000 K. 13.5 (a) Show that the Reimers mass loss rate, given by Eq. (13.23) in Problem 13.4, can also be written in the form M = -4 x 10-is~ M Mo Yr-1~ where L, R, and M are all in solar units. (b) Assuming (incorrectly) that L, R, and r~ do not change with time, derive an expression for the mass of the star as a function of time. Let M = Mo when the mass loss phase begins. (c) Using L = 7000 Lo, R = 310 Ro, Mo = 1 Mo, and r~ = 1, make a graph of the star's mass as a function of time. (d) How long would it take for a star with an initial mass of 1 Mo to be reduced to the mass of the degenerate carbon-oxygen core (0.6 Mp)? 538 Chapter 13 Post-Main-Sequence Stellar Evolution 13.6 The Helix Nebula is a planetary nebula with an angular diameter of 15' that is located approximately 120 pc from Earth. i (a) Calculate the diameter of the nebula. (b) Assuming that the nebula is expanding away from the centra,l star at a constant velocity of 20 km s-1, estimate its age. 13.7 Using Eq. (12.16), make a crude estimate of the amount of time re-quired for the homologous collapse of the inner portion of the iron core of a massive star, marking the beginning of a Type II supernova. 13.8 (a) Show that the amount of radioactive material remaining in an initially pure sample is given by Eq. (13.20). (b) Prove that  T1/2 13.9 (a) The angular size of the Crab SNR is 4' x 2' and its distance from Earth is approximately 2000 pc (see Fig. 13.18a). Estimate the linear dimensions of the nebula. (b) Using the measured expansion rate of the Crab and ignoring any accelerations since the time of the supernova explosion, estimate the age of the nebula. 13.10 Taking the distance to the Crab to be 2000 pc, and assuming that the absolute bolometric magnitude at maximum brightness was char-acteristic of a Type II supernova, estimate its peak apparent magni tude. Compare this to the maximum brightness of the planet Venus (m - -4), which is sometimes visible in the daytime. 13.11 (a) Assuming that the light curve of a supernova is dominated by the energy released in the radioactive decay of an isotope that has a decay constant of ~, show that the slope of the light curve is given by Eq. (13.21) . (b) Prove that Eq. (13.22) follows from Eq. (13.21). 13.12 The energy released during the decay of one 26Co atom is 3.72 MeV. If 0.075 Mo of cobalt was produced by the decay of 28Ni following the explosion of SN 1987A, estimate the amount of energy released per second through the radioactive decay of cobalt (a) just after the formation of the cobalt. Problems 539 (b) one year after the explosion. (c) Compare your answers with the light curve of SN 1987A given in Fig. 13.20. 13.13 The neutrino flux from SN 1987A was estimated to be 1.3 x 1010 cm-2 at the location of Earth. If the average energy per neutrino was approx-imately 4.2 MeV, estimate the amount of energy released via neutrinos during the supernova explosion. 13.14 Using Eq. (10.28), estimate the gravitational binding energy of a neu-tron star with a mass 1.4 Mp and a radius of 10 km. Compare your answer with the amount of energy released in neutrinos during the collapse of the iron core of Sk -69 202 (the progenitor of SN 1987A). 13.15 Estimate the Eddington limit for q Car and compare your answer with the luminosity of that star. Is your answer consistent with its behavior? Why or why not? 13.16 An old version of stellar evolution, popular at the beginning of the twentieth century, maintained that stars begin their lives as large, cool spheres of gas, like the giant stars on the H-R diagram. They then contract and heat up under the pull of their own gravity to become hot, bright blue O stars. For the remainder of their lives they lose energy, becoming dimmer and redder with age. As they slowly move down the main sequence, they eventually end up as cool, dim red M stars. Explain how observations of stellar clusters, plotted on an H-R diagram, contradict this idea. 13.17 (a) Show that loglo (Lv/LB) + constant is, to within a multiplicative constant, equivalent to the color index, B - V. (b) Estimating best-fit curves through the data given in Fig. 13.30 on the next page, trace the two color-magnitude diagrams, placing them on a single diagram. Note that the abscissas have been normalized so that the lowest-luminosity stars of both clusters are located at the same positions on their respective diagrams. (c) Given that 47 Tuc is relatively metal-rich for a globular cluster (Z/Zp = 0.17, where Z(D is the solar value) and M15 is metal-poor (Z/Z(D = 0.0060), explain the difference in colors between the two clusters. Hint: You may wish to refer back to the discussion in Example 9.10 (Section 9.4). page 540 Chapter 13 Post-Main-Sequence Stellar Evolution a w m 0  9 ~_ 0 F 3 H  -2 F-  i -0.2 0 f 0.2 , ' 0.4 ' I 0.8 1 1 0.8 -0.2 0 ' 0.2 ' , 0.4 ' 1 0.8 ' ' 0.8 log(L,./18) + const log(L,./LB) + conat  Figure 13.30 (a) A color-magnitude diagram for 47 Tuc, a relatively metal-rich globular cluster with Z/Zo = 0.17. (Data from Hesser et al., Publ. Astron. Soc. Pac., 99, 739, 1987; figure courtesy of William E. Har-ris.) (b) A color-magnitude diagram for M15, a metal-poor globular cluster with Z/Zo = 0.0060. (Data from Durrell and Harris, Astron. J., 105, 1420, 1993; figure courtesy of William E. Harris.) 13.18 Using the technique of main-sequence fitting, estimate the distance to M3; refer to Figs. 13.27 and 13.29. 626 Chapter 15 The Degenerate Remnants of Stars Michel, F. Curtis, Theory of Neutron Star Magnetospheres, The University of Chicago Press, Chicago, 1991. Pacini, F., "Energy Emission from a Neutron Star," Nature, 216, 567, 1967. Shapiro, Stuart L., and Teukolsky, Saul A., Black Holes, White Dwarfs, and Neutron Stars, John Wiley and Sons, New York, 1983. Winget, D. E., et al., "An Independent Method for Determining the Age of the Universe," The Astrophysical Journal Letters, 315, L77, 1987. Winget, D. E., et al., "Hydrogen-Driving and the Blue Edge of Compositionally Stratified ZZ Ceti Star Models," The Astrophysical Journal Letters, 252, L65, 1982a. Winget, Donald E., et al., "Photometric Observations of CD 358: DB White Dwarfs Do Pulsate," The Astrophysical Journal Letters, 262, L11, 1982b. Problems 15.1 The most easily observed white dwarf in the sky is in the constellation of Eridanus (the River Eridanus). Three stars comprise the 40 Eridani system: 40 Eri A is a 4th-magnitude star similar to the Sun; 40 Eri B is a 10th-magnitude white dwarf; and 40 Eri C is an 11th-magnitude red M5 star. This problem deals only with the latter two stars, which are widely separated from 40 Eri A by 400 AU. (a) The period of the 40 Eri B and C system is 247.9 years. The system's measured trigonometric parallax is 0.201" and the true angular extent of the semimajor axis of the reduced mass is 6.89". The ratio of the distances of 40 Eri B and C from the center of mass is aB/ac = 0.37. Find the mass of 40 Ez'i B and C in terms of the mass of the Sun. (b) The absolute bolometric magnitude of 40 Eri B is 9.6. Determine its luminosity in terms of the luminosity of the Sun. (c) The effective temperature of 40 Eri B is 16,900 K. Calculate its radius, and compare your answer to the radii of the Sun, Earth, and Sirius B. (d) Calculate the average density of 40 Eri B, and compare your result with the average density of S~rius B. Which is more dense, and '~- why? :  Problems 627 (e) Calculate the product of the mass and volume of both 40 Eri B and Sirius B. Is there a departure from the mass-volume relation? What might be the cause? 15.2 The helium absorption lines seen in the spectra of DB white dwarfs are formed by excited He I atoms with one electron in the lowest (n = 1) orbital and the other in an n = 2 orbital. White dwarfs of spectral type DB are not observed with temperatures below about 11,000 K. Using what you know about spectral line formation, give a qualita-tive explanation of why the helium lines would not be seen at lower temperatures. As a DB white dwarf cools below 12,000 K, into what spectral type would it change? 15.3 Deduce a rough upper limit for X, the mass fraction of hydrogen, in the interior of a white dwarf. Hint: Use the mass and average density for Sirius B in the equations for the nuclear energy generation rate, and take T = 107 K for the central temperature. Set ~pp and fPP = 1 in Eq. (10.50) for the pp chain, and XcNO = 1 in Eq. (10.54) for the CNO cycle.  15.4 Estimate the ideal gas pressure and the radiation pressure at the center of Sirius B, using 3 x 107 K for the central temperature. Compare these values with the estimated central pressure, Eq. (15.1). 15.5 By equating the pressure of an ideal gas of electrons to the pressure of a degenerate electron gas, determine a condition for the electrons to be degenerate, and compare it with the condition of Eq. (15.5). Use the exact expression (Eq. 15.11) for the electron degeneracy pressure. 15.6 In the extreme relativistic limit, the electron speed v = c must be used instead of Eq. (15.9) to find the electron degeneracy pressure. Use this to repeat the derivation of Eq. (15.10) and find ~ 3c LCAl xJ4/3 15.7 (a) At what speed do relativistic effects become important at a level of 10%? In other words, for what value of v does the Lorentz factor, y, become equal to 1.1? (b) Estimate the density of the white dwarf for which the speed of a degenerate electron is equal to the value found in part (a). 628 Chapter 15 The Degenerate Remnants of Stars (c) Use the mass-volume relation to find the approximate mass of a white dwarf with this average density. This is roughly the mass where white dwarfs depart from the mass-volume relation. 15.8 Crystallization will occur in a cooling white dwarf when the electro-static potential energy between neighboring nuclei, ZZeZ/r in cgs units, dominates the characteristic thermal energy kT. The ratio of the two is defined to be I', r - Z2e2 rkT ' In this expression, the distance r between neighboring nuclei is custom-arily (and somewhat awkwardly) defined to be the radius of a sphere whose volume is equal to the volume per nucleus. Specifically, since the average volume per nucleus is AmH/p, r is found from 3 ~r3 - AmH . P (a) Calculate the value of the average separation r for a 0,6 MD pure carbon white dwarf of radius 0.012 Ro. (b) Much effort has been spent on precise numerical calculations of P to obtain increasingly realistic cooling curves. The results in-dicate a value of about I' = 160 for the onset of crystallization. Estimate the interior temperature, T, at which this occurs. (c) Estimate the luminosity of a pure carbon white dwarf with this interior temperature. Assume a composition like that of Exam-ple 15.2 for the nondegenerate envelope. (d) For roughly how many years could the white dwarf sustain the luminosity found in part (c), using just the latent heat of kT per nucleus released upon crystallization? Compare this amount of time (when the white dwarf cools more slowly) with Fig. 15.9. 15.9 In the liquid-drop model of an atomic nucleus, a nucleus with mass number A has a radius of roA1/3, where ro = 1.2 x 10-13 cm. Find the density of this nuclear model. 15.10 If our Moon were as dense as a neutron star, what would its diameter be? 15.11 (a) Consider two point masses, each having mass m, that are sepa-rated vertically by a distance of 1 cm just above the surface of a neutron star of radius R and mass M. Using Newton's law of grav-ity (Eq. 2.11), find an expression for the ratio of the gravitational force on the lower mass to that on the upper mass, and evaluate this expression for R = 10 km, M = 1.4 Mp, and m = 1 g. (b) An iron cube 1 cm on each side is held just above the surface of the neutron star described in (a) above. The density of iron is 7.86 g cm-3. If iron experiences a stress (force per cross-sectional area) of 4.2 x 10$ dyne cm-2, it will be permanently stretched; if the stress reaches 1.5 x 109 dyne cm-2, the iron will rupture. What will happen to the iron cube? (Hint: Imagine concentrating half of the cube's mass on each of its top and bottom surfaces.) What would happen to an iron meteoroid falling toward the surface of a neutron star? 15.12 Estimate the neutron degeneracy pressure at the center of a 1.4 MO neutron star (take the central density to be 1.5 x 1015 g cm-3), and compare this with the estimated pressure at the center of Sirius B.  15.13 (a) At a density just below neutron drip, assume that all of the neu-trons are in heavy neutron-rich nuclei such as 136 Kr. Estimate the pressure due to relativistic degenerate electrons. (b) At a density just above neutron drip, assume (wrongly.o that all of the neutrons are free (and not in nuclei). Estimate the speed of the degenerate neutrons and the pressure they would produce. 15.14 Suppose that the Sun were to collapse down to the size of a neutron star (10 km radius). (a) Assuming that no mass is lost in the collapse, find the rotation period of the neutron star. (b) Find the magnetic field strength of the neutron star. Even though our Sun will not end its life as a neutron star, this shows that the conservation of angular momentum and magnetic flux can easily produce pulsarlike rotation speeds and magnetic fields.  15.15 (a) Use Eq. (14.14) with -y = 5/3 to calculate the fundamental radial pulsation period for a one-zone model of a pulsating white dwarf (use the values for Sirius B) and a 1.4 Mp neutron star. Compare these to the observed range of pulsar periods. 630 Chapter 15 The Degenerate Remnants of Stars (b) Use Eq. (15.26) to calculate the minimum rotation period for the same stars, and compare them to the range of pulsar periods. (c) Give an explanation for the similarity of your results. 15.16 (a) Determine the minimum rotation period for a 1.4 MD neutron star (the fastest it can spin without flying apart). For convenience, assume that the star remains spherical with a radius of 10 km. (b) Newton studied the equatorial bulge of a homogeneous fluid body of mass M that is slowly rotating with angular velocity 52. He proved that the difference between its equatorial radius (E) and its polar radius (P) is related to its average radius (R) by E - P 5S22R3 R 4GM Use this to estimate the equatorial and polar radii for a 1.4 MO neutron star rotating with twice the minimum rotation period you found in part (a). 15.17 If you measured the period of PRS 1937+214 and obtained the value on page 609, about how long would you have to wait before the last digit changed from a "T' to a "6"? 15.18 Consider a pulsar that has a period Po and period derivative Po at t = 0. Assume that the product PP remains constant for the pulsar (c.f. Eq. 15.29). (a) Integrate to obtain an expression for the pulsar's period P at time t. (b) Imagine that you have constructed a clock that would keep time by counting the radio pulses received from this pulsar. Suppose you also have a perfect clock (P = 0) that is initially synchronized with the pulsar clock when they both read zero. Show that when the perfect clock displays the characteristic lifetime Po/Po, the time displayed by the pulsar clock is (v"3- - 1)P./1'o. 15.19 During a glitch, the period of the Crab pulsar decreased by JOPI ~--10-$P. If the increased rotation was due to an overall contraction of the neutron star, find the change in the star's radius. Assume that the pulsar is a rotating sphere of uniform density with an initial radius of 10 km. Problems 631 15.20 The Geminga pulsar has a period of P = 0.237 s and a period derivative of P = 1.1 x 10-14. Assuming that 8 = 90, estimate the magnetic field strength at the pulsar's poles. 15.21 (a) Find the radii of the light cylinders for the Crab pulsar and for the slowest pulsar PSR 1845-19. Compare these values to the radius of a 1.4 Me neutron star. (b) The strength of a magnetic dipole is proportional to 1/r3. Deter-mine the ratio of the magnetic field strengths at the light cylinder for the Crab pulsar and for PSR 1845-19. 15.22 (a) Integrate Eq. (15.29) to obtain an expression for a pulsar's period P at time t if its initial period was P at time t = 0. (b) Assuming that the pulsar has had time to slow down enough that P P, show that the age t of the pulsar is given approximately by P t _ 2P' where P is the period derivative at time t. (c) Evaluate this age for the case of the Crab pulsar, using the values found in Example 15.5. Compare your answer with the known age. 15.23 One way of qualitatively understanding the flow of charged particles into a pulsar's magnetosphere is to imagine a charged particle of mass m and charge e (the fundamental unit of charge) at the equator of the neutron star. Assume for convenience that the star's rotation carries it perpendicular to the pulsar's magnetic field. The moving charge experiences a magnetic Lorentz force of F,-, = evB/c (in cgs units), and a gravitational force, F9. Show that the ratio of these forces is F2 _ 27reBR Fy Pcmg ' where R is the star's radius and g is the acceleration due to gravity at the surface. Evaluate this ratio for the case of a proton at the surface of the Crab pulsar, using a magnetic field strength of 1012 G. 15.24 Find the minimum photon energy required for the creation of an elec-tron-positron pair via the pair-production process y --~ e-+e+. What , , is the wavelength of this photon? In what region of the electromagnetic - spectrum is this wavelength found? 632 Chapter 15 The Degenerate Remnants of Stars 15.25 A subpulse involves a very narrow radio beam with a width between 1 and 3. Use Eq. (4.43) for the headlight effect to calculate the minimum speed of the electrons responsible for a 1 subpulse. 676 Chapter 16 Black Holes Haswell, Carole A., and Shafter, Allen W., "A Detection of Orbital Radial Ve-locity Variations of the Primary Component the Black Hole Binary A0620-00 (= V616 Monocerotis)," The Astrophysical Journal Letters, 359, L47, 1990. Misner, Charles W., Thorne, Kip S., and Wheeler, John A., Gravitation, W. H. Freeman and Co., San Francisco, 1973. Ruffini, Remo, and Wheeler, John A., "Introducing the Black Hole," Physics Today, January 1991. Taylor, Edwin F., and Wheeler, John Archibald, Scouting Black Holes, pre-print, 1995. (For current version, contact Edwin F. Taylor, 22 Hopkins Road, Arlington, MA 02174 USA, or email eftaylor~mit.edu.) Problems 16.1 In the rubber sheet analogy of Section 16.1, a keen eye would notice that the tennis ball also depresses the sheet slightly, and so the soccer ball constantly tilts slightly toward the tennis ball as they orbit each other. Qualitatively compare this with the motion of two stars in a binary orbit. 16.2 Show that Eq. (16.3) for the gravitational redshift remains valid even if the light travels upward at an angle B measured from the vertical as long as h is taken to be the vertical distance traveled by the light pulse. 16.3 A photon near the surface of Earth travels a horizontal distance of 1 km. How far does the photon "fall" in this time? 16.4 Leadville, Colorado, is at an altitude of 3.1 km above sea level. If a person there lives for 75 years (as measured by an observer at a great distance from Earth), how much longer would gravitational time dilation have allowed that person to live if he or she had moved at birth from Leadville to a city at sea level? 16.5 (a) Estimate the radius of curvature of a horizontally traveling photon at the surface of a 1.4 Mo neutron star, and compare the result `~ . with the 10 km radius of the star. Can general relativity be neglected when studying neutron stars? Problems 677   0 ____________________[ ____; __________________________V I I a  Figure 16.26 Local inertial frames for measuring the deflection of light near the Sun (Problem 16.6). (b) If one hour passes at the surface of the neutron star, how much time passes at a great distance? Compare the times obtained from the exact and approximate expressions, Eqs. (16.10) and (16.11), respectively. 16.6 Imagine a series of rectangular local inertial reference frames suspended by cables in a line near the Sun's surface, as shown in Fig. 16.26. The frames are carefully lined up so the tops and sides of neighboring frames are parallel, and the tops of the frames lie along the z-axis. A photon travels unhindered through the frames. As the photon enters each frame, the frame is released from rest and falls freely toward the center of the Sun. (a) Show that in passing through the frame located at angle a (shown in the figure), the angular deflection of the photon's path is z where dz is the width of the reference frame and go is the New-tonian gravitational acceleration at the point of closest approach, O. The angular deflection is small, so assume that the photon is initially traveling in the z-direction as it enters the frame. (Hint: The width of the frame in the z-direction is dz, so the time for the photon to cross the frame can be taken to be dz/c.) dy= go cos3 a dz C2 678 Chapter 16 Black Holes (b) Integrate the result you found in part (a) from a = -7r/2 to +ir/2 and so find the total angular deflection of the photon as it passes through the curved spacetime near the Sun. (c) Your answer (which is also the answer obtained by Einstein in 1911 before he arrived at his field equations) is only half the cor-rect value of 1.75". Can you qualitatively account for the missing factor of two? 16.7 Assume that you are at the origin of a laboratory reference system at time t = 0 when you start your clock (event A). Determine whether the following events are within the future lightcone or past lightcone of event A, or elsewhere. (a) A flashbulb goes off 7 m away at time t = 0. (b) A flashbulb goes off 7 m away at time t = 2 s. (c) A flashbulb goes off 70 km away at time t = 2 s. (d) A flashbulb goes off 700,000 km away at time t = 2 s. (e) A supernova explodes 180,000 ly away at time t = -5.7 x 1012 s. (f) A supernova explodes 180,000 ly away at time t = 5.7 x 1012 s. (g) A supernova explodes 180,000 ly away at time t = -5.6 x 1012 s. (h) A supernova explodes 180,000 ly away at time t = 5.6 x 1012 s. For items (e) and (g), could an observer in another reference frame moving relative to yours measure that the supernova exploded after event A? For items (f) and (h), could an observer in another frame measure that the supernova exploded before event A? 16.8 T Ceti is the closest single star that is similar to the Sun. At time t = 0, Alice leaves Earth in her starship and travels at a speed of 0.95c to T Ceti, 11.7 ly away as measured by astronomers on Earth. Her twin brother, Bob, remains at home, at x = 0. (a) According to Bob, what is the interval between Alice's leaving Earth and arriving at T Ceti? (b) According to Alice, what is the interval between her leaving Earth and arriving at T Ceti? (c) Upon arriving at T Ceti, Alice immediately turns around and returns to Earth at a speed of 0.95c. (Assume that the actual turnaround takes negligible time.) What was the proper time for Alice during her round trip to T Ceti? Problems 679 (d) When she and Bob meet on her return to Earth, how much young-er will Alice be than her brother? 16.9 Consider a spherical blackbody of constant temperature and mass M whose surface lies at radial coordinate r = R. An observer located at the surface of the sphere and a distant observer both measure the blackbody radiation given off by the sphere. (a) If the observer at the surface of the sphere measures the luminos-ity of the blackbody to be L, use the gravitational time dilation formula, Eq. (16.10), to show that the observer at infinity mea-sures L~=LC1- RMI. (16.26) (b) Both observers use Wien's law, Eq. (3.15), to determine the black-body's temperature. Show that T~ = T1 1 - RM. (16.27)  (c) Both observers use the Stefan-Boltzmann law, Eq. (3.17), to de-termine the radius of the spherical blackbody. Show that R~ = R . (16.28) 1 - 2GM/Rc2 Thus using the Stefan-Boltzmann law without including the ef-fects of general relativity will lead to an overestimate of the size of a compact blackbody.  16.10 In 1792 the F~ench mathematician Simon-Pierre de Laplace (1749-1827) wrote that a hypothetical star, "of the same density as Earth, and whose diameter would be two hundred and fifty times larger than the Sun, would not, in consequence of its attraction, allow any of its rays to arrive at us." Use Newtonian mechanics to calculate the escape velocity of Laplace's star. 16.11 Qualitatively describe the effects on the orbits of the planets if the Sun were suddenly to become a black hole. 680 Chapter 16 Black Holes 16.12 Consider four black holes with masses of 1015 g, 10 Mo, 105 Mp, and 109 Mo. (a) Calculate the Schwarzschild radius for each. (b) Calculate the average density, defined by p = ll7/ ( 3 ~RS) , for each. 16.13 (a) Show that the proper distance from the event horizon to a radial coordinate r is given by OG=r 1-RS+RS ln 1+ 1_R r ' This illustrates the danger of interpreting r as a distance instead of a coordinate. Hint: Integrate Eq. (16.19). (b) Make a graph of ~G as a function of r for values of r between r = RS and r = lORs. (c) Show for large values of r that OG-r. Thus, far from the black hole, the radial coordinate r can be treated as a distance. 16.14 Verify that the area of the event horizon of a black hole is 4~rRs. (Hint: Remember that the radial coordinate r is not the distance to the center. Use the Schwarzschild metric as your starting point.) 16.15 Equation (16.22) describes the coordinate speed of a massive particle orbiting a nonrotating black hole. However, it can be shown that the orbit is not stable unless r >_ 3Rs; any disturbance will cause a particle in a smaller orbit to spiral down to the event horizon. (a) Find the coordinate speed of a particle in the smallest stable orbit around a 10 Mo black hole. (b) Find the orbital period (in coordinate time t) for this smallest stable orbit around a 10 M~ black hole. 16`:16 (a) Find an expression for the coor,~nate speed of light in the ~-direction. Problems 681 (b) Consider Eq. (16.22) in the limit that the particle's mass goes to zero and its speed approaches that of light. Use your result for part (a) to show that r = 1.5Rs for the circular orbit of a photon around a black hole. (c) Find the orbital period (in coordinate time t) for this orbit around a 10 MD black hole. (d) If a flashlight were beamed in the O-direction at r = 1.5RS, what would happen? (The surface at r = 1.5Rs is called the photon sphere.)  16.17 Use Eq. (16.25) to compare the maximum angular momentum of a 1.4 Mo black hole with the angular momentum of the fastest known pulsar, which rotates with a period of 0.00156 s. Assume that the pulsar is a 1.4 MD uniform sphere of radius 10 km. 16.18 An electron is a pointlike particle of zero radius, so it is natural to wonder whether an electron could be a black hole. However, a black hole of mass M cannot have an arbitrary amount of angular momentum L and charge Q. These values must satisfy an inequality, F 'M)' > G (1)' + (ML )2. If this inequality were violated, the singularity would be found outside the event horizon, in violation of the Law of Cosmic Censorship. Use h/2 for the electron's angular momentum to determine whether or not an electron is a black hole. 16.19 (a) The angular rotation rate, S2, at which spacetime is dragged around a rotating mass must be proportional to its angular mo-mentum L. The expression for SZ may also contain the constants G and c, together with the radial coordinate r. Show on purely dimensional grounds that SZ = constant x GL c , where the constant (which you need not determine) is of order unity. (b) Evaluate this for Earth, assuming that it is a uniformly rotating sphere. Set the leading constant equal to one, and express your answer in arcseconds per year. How much time would it take for a 682 Chapter 16 Black Holes pendulum at the north pole to rotate once relative to the distant stars because of frame dragging? (c) Repeat part (b) for the fastest known pulsar, expressing SZ in revolutions per second. 16.20 (a) Use dimensional arguments to combine the fundamental constants h, c, and G into an expression that has units of mass. Evaluate your result, which is an estimate of the least massive primordial black hole formed in the first instant after the Big Bang. What is the mass in grams? (b) What is the Schwarzschild radius for such a black hole? (c) How long would it take light to travel this distance? (d) What is the lifetime of this black hole before its evaporation? 16.21 In the x-ray binary system A0620-00, the radial orbital velocities for the normal star and the compact object are vST = 457 km s-1 and v, = 43 km s-1, respectively. The orbital period is 0.3226 days. (a) Calculate the mass function [the right-hand side of Eq. (7.8)], 3 mC 2 sin 3 i, (ms + m) where ms is the mass of the normal star, m, is the mass of its compact companion, and i is the angle of inclination of the orbit. What does this result say about the mass of the compact object? (Note that the value of vcr was not needed to obtain this result.) (b) Now use the value of the orbital radial velocity of the compact object to determine its mass, assuming i = 90. What does this result say about the mass of the compact object? (c) The x-rays are not eclipsed in this system, so the angle of in-clination must be less than approximately 85. Suppose that the angle of inclination were 45; what would the mass of the compact object be then? This simple calculation, based only on the dynamics of the binary system, provides the best evidence to date for the existence of a black hole. Problems 745 Weisberg, J. M., and Taylor, J. H., "Observations of Post-Newtonian Timing Effects in the Binary Pulsar PSR 1913+16," Physical Review Letters, 52, 1348, 1984. Problems 17.1 Use the ideal gas law to argue that, in a close binary system, the temperature of a star's photosphere is approximately constant along an equipotential surface. What effect could the proximity of the other star have on your argument? 17.2 Each of the Lagrange points L4 and L5 forms an equilateral triangle with masses Ml and M2 in Fig. 17.3. Use this to confirm the value of the effective gravitational potential at L4 and L5 given in the figure caption. 17.3 (a) Consider a gas of density p moving with velocity v across an area A perpendicular to the flow of the gas. Show that the rate at which mass crosses the area is given by Eq. (17.10). (b) Derive Eq. (17.11) for the radius of the intersection of two iden-tical overlapping spheres, when d R. 17.4 Computer Problem (a) Use the STATSTAR model data on page 311 and Eq. (17.12) to make a graph of loglo M (vertical axis) vs. loglo d (horizontal axis). Use the slope of your graph to find how the mass transfer rate, M, depends on d. (b) Use Eqs. (H.1) and (H.2) to show that M a d4.75 near the surface, and so verify that the mass transfer rate increases rapidly with the overlap distance d of two stars. Note that your answer to part (a) will be slightly different from this because of the densi-ty-dependence of TOG BF (the ratio of the guillotine factor to the gaunt factor) cal      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~culated in the STATSTAR's equation-of-state subroutine EOS. 17.5 Use Eq. (17.16) to show that the maximum disk temperature is found at r = (49/36)R, and is equal to TIax = 0.488Td;sk. 746 Chapter 17 Close Binary Star Systems 17.6 Integrate Eq. (17.13) for the ring luminosity from r = R to r = o0 [with Eq. (17.16) for the disk temperature]. Does your answer agree with Eq. (17.20) for the disk luminosity? 17.7 Consider an "average" dwarf nova that has a mass transfer rate of M = lOls.S g s-i = 5 x 10-io Mo Yr-1 during an outburst that lasts for 10 days. Estimate the total energy released and the absolute magnitude of the dwarf nova during the outburst. Use values for Z Cha's white dwarf from Example 17.4. Neglect the small amount of light contributed by the primary and secondary stars. 17.8 Assume that the absolute bolometric magnitude of a dwarf nova during quiescence is 7.5 and that it brightens by three magnitudes during outburst. Using values for Z Cha, estimate the rate of mass transfer through the accretion disk. 17.9 When the accretion disk in a cataclysmic variable is eclipsed by the secondary star, the blueshifted emission line is the first to disappear at the beginning of the eclipse, and the redshifted emission line is the last to reappear when the eclipse ends. What does this have to say about the directions of rotation of the binary system and the accretion disk? 17.10 (a) In a close binary system where angular momentum is conserved, show that the change in orbital period produced by mass transfer is given by 1 dP Ml - M2 P dt = 3M1 Ml M2 . (b) U Cephei (an Algol system) has an orbital period of 2.49 days that has increased by about 20 s in the past 100 years. The masses of the two stars are Ml = 2.9 Mo and MZ = 1.4 Mo. Assuming that this change is due to the transfer of mass between the two stars in this Algol system, estimate the mass transfer rate. Which of these stars is gaining mass?  17.11 Algol (the "demon" star, in Arabic) is a semidetached binary. Every 2.87 days, its brilliance is reduced by more than half as it undergoes ` a deep eclipse, its apparent magnitude dimming from 2.1 to 3.4. The system consists of a B8 main-sequence star and a late-type (G or K)       Problems 747 subgiant; the deep eclipses occur when the larger, cooler star (the sub-giant) moves in front of its smaller, brighter companion. The "Algol paradox," which troubled astronomers in the first half of the twentieth century, is that according to the ideas of stellar evolution discussed in Section 10.6, the more massive B8 star should have been the first to evolve off the main sequence. What is your solution to this paradox? (The Algol system actually contains a third star that orbits the other two every 1.86 years, but this has nothing to do with the solution to the Algol paradox.) Algol may be easily found in the constellation Perseus (the Hero, who rescued Andromeda in Greek mythology). Sky and Telescope pro-vides a monthly listing of the minima of Algol, which last about 2 hours. 17.12 Consider a 10-4 Me layer of hydrogen on the surface of a white dwarf. If this layer were completely fused into helium, how long would the resulting nova last (assuming a luminosity equal to the Eddington lu minosity)? What does this say about the amount of hydrogen that actually undergoes fusion during a nova outburst? 17.13 Consider a layer of 10-4 Me of hydrogen on the surface of a white dwarf. Compare the gravitational binding energy before the nova out-burst to the kinetic energy of the ejected layer when it has traveled far from the white dwarf and has a speed of 1000 km s-1. 17.14 In this problem, you will examine the fireball expansion phase of a nova shell. Suppose that mass is ejected by a nova at a constant rate of Meject and at a constant speed v. (a) Show that the density of the expanding shell at a distance r is P = Meject/47rT2v. (b) Let the mean opacity, R, of the expanding gases be a constant. Suppose that at some time t = 0, the outer radius of the shell was R, and the radius of the photosphere, where T = 2/3, was Ro. Show that 1 _ 1 1 R-Ro R~' where Roo - UMeject . S7rv (The reason for the "oo" subscript will become clear below.) 748 Chapter 17 Close Binary Star Systems (c) At some later time t, the radius of the shell will be R + vt and the radius of the photosphere will be R(t). Show that 1 _ 1 1 R + vt R(t) R~ (d) Combine the results from parts (b) and (c) to write vt(1 - Ro/R~)2 R(t) = Ro + 1 + (vt/R~)(1 - Ro/R~)' (e) Argue that terms containing Ro/R~ are very small and can be ignored, and so obtain vt R(t) = 1 + vt/R~ ~ (f) Show that initially the fireball's photosphere expands linearly with time, and then approaches the limiting value of R~, in agree-ment with Eq. (17.27). (g) Using the data given in the text following Eq. (17.28), make a graph of the R(t) vs. t for the five days after nova explodes. The "knee" in the graphs marks the end of the linear expansion pe riod; estimate when this occurs. How does this compare with the duration of the optically thick fireball phase of the nova? 17.15 Use Eq. (17.28) to estimate the photospheric temperature of a nova fireball, adopting the Eddington luminosity for the luminosity of the fireball. 17.16 Assuming that the hydrostatic burning phase of a nova lasts for 100 days, find the (constant) rate at which mass is ejected, Me~ect, for a surface layer of 10-4 Mo. 17.17 If the linear decline of a supernova light curve is powered by the ra-dioactive decay of the ejecta, find the rate of decline (in mag d-1) produced by the decay of Z6Co ~ Z6Fe, with a half-life of 77.7 days. 17.18 For each gram of a carbon-oxygen composition (30% 16C) that is burned to produce iron, 7.3 x 1017 ergs of energy is released. As-suming an initial 1.38 Mo white dwarf with a radius of 1600 km, how much iron would have to be produced to cause the star to be grav-itationally unbound? How much additional iron would have to be Problems 749 manufactured to produce a Type la supernova with an average ejecta speed of 5000 km s-1? Take the gravitational potential energy to be -5.1 x 1050 ergs (for a realistic white dwarf model), and express your answers in units of Mo. 17.19 Use Eqs. (7.4), (17.29), and (17.30) to derive Eq. (17.31), the condition for a supernova to disrupt a binary system. 17.20 (a) Show that the Alfven radius is given by Eq. (17.35). (b) Show that PIP for the spin-up of an x-ray pulsar is given by Eq. (17.38). 17.21 Find the value of the magnetic field for which the Alfven radius is equal to the radius of the white dwarf found in Example 17.3. Do the same thing for the neutron star used in that example. 17.22 Estimate the lifetime of a binary x-ray system using the information in Example 17.3. Take the lifetime to be the time required to transfer a mass of 1 Mp.  k 17.23 The x-ray pulsar 4U0115+63 has a period of 3.61 s and an x-ray lumi-nosity of about L,, = 3.8 x 1036 ergs s-1. Assuming that it is a 1.4 Me neutron star with a radius of 10 km and a surface magnetic field of 1012 G, find its mass transfer rate, M, and the value of PIP. Repeat these calculations assuming that this object is a 0.85 Me white dwarf with a radius of 6.6 x 108 cm and a surface magnetic field of 107 G. For which of these models do you obtain the better agreement with the measured value of PIP = -3.2 x 10-5 yr-1? 17.24 (a) Use Eq. (17.21) to show that the spin-up rate can be written as 2 12 1/7 loglo - p = loglo (PL~~ ) + loglo 2 I BS R l . V ~G G3M3 /  The term on the left and the first term on the right consist of quantities that can be measured observationally. The second term on the right depends on the specific model (neutron star or white dwarf) of the x-ray pulsar. (b) Make a graph of loglo(-P/P) (vertical axis) vs. loglo (PLa~7 acc) (horizontal axis). Use the values from Example 17.5 to plot two lines, one for a neutron star and one for a white dwarf. Let loglo (PL~~ ) run from 31 to 35. 750 Chapter 17 Close Binary Star Systems P La, -P/P System (s) (1037 ergs s-1) (yr -1) SMC X-1 0.714 50 7.1 x 10-4 Her X-1 1.24 1 2.9 x 10-6 Cen X-3 4.84 5 2.8 x 10-4 A0535+26 104 6 3.5 x 10-2 GX301-2 696 0.3 7.0 x 10-3 4U0352+30 835 0.0004 1.8 x 10-4 Table 17.1 X-ray Pulsar Data, for Problem 17.24. (Data from Rappaport and Joss, Nature, 266, 683, 1977, and Joss and Rappaport, Annu. Rev. Astron. Astrophys., 22, 537, 1984.) (c) Use the data in Table 17.1 to plot the positions of six binary x-ray pulsars on your graph. (You will have to convert -PIP into units of s-1.) (d) Which model of a binary x-ray pulsar is in better agreement with the data? Comment on the position of Her X-1 on your graph. 17.25 (a) Consider an x-ray burster that releases 1039 ergs in 5 seconds. If the shape of its peak spectrum is that of a 2 x 107 K blackbody, estimate the radius of the underlying neutron star. (b) In Problem 16.9 you showed that using the Stefan-Boltzmann formula to find the radius of a compact blackbody can lead to an overestimate of its radius. Use Eq. (16.28) to find a more accurate value for the radius of the neutron star. 17.26 Make a scale drawing of the SMC X-1 binary pulsar system, includ-ing the size of the secondary star. Assuming that the primary is a 1.4 MD neutron star, locate the system's center of mass and its inner Lagrangian point, L1. (You can omit the accretion disk.) 17.27 The relativistic (v/c = 0.26) jets coming from the accretion disk in SS 433 sweep out cones in space as the disk precesses. The central axis of these cones makes an angle of 79 with the line of sight, and the half-angle of each cone is 20. This means that at some point in the precession cycle, the jets are moving perpendicular to the line of sight. Yet, from Fig. 17.26, the radial velocities obtained from the Doppler-shifted spectral lines do not cross at zero radial velocity, but CONTENTS I The Tools of Astronomy 1 1 The Celestial Sphere 3 l.l The Greek Tradition . . . . . . . . . . . . . . . . . . . : . . . 3 1.2 The Copernican Revolution . . . . . . . . . . . . . . . . . . . 6 1.3 Positions on the Celestial Sphere . . . . . . . . . . . . . . . . 10 1.4 Physics and Astronomy . . . . . . . . . . . . . . . . . . . . . 21 2 Celestial Mechanics 25 2.1 Elliptical Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Kepler's Laws Derived . . . . . . . . . . . . . . . . . . . . . . 43 2.4 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . 53 b i 3 The Continuous Spectrum of Light 63 3.1 Stellar Parallax . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ? 3.2 The Magnitude Scale . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 The Wave Nature of Light . . . . . . . . . . . . . . . . . . . . 69 t 3.4 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . 75 3.5 The Quantization of Energy . . . . . . . . . . . . . . . . . . . 79 3.6 The Color Index . . . . . . . . . . . . . . . . . . . . . . . . . 82  4 The Theory of Special Relativity 93 4.1 The Failure of the Galilean Transformations . . . . . . . . . . 93 4.2 The Lorentz Transformations . . . . . . . . . . . . . . . . . . 96 4.3 Time and Space in Special Relativity . . . . . . . . . . . . . . 102 4.4 Relativistic Momentum and Energy . . . . . . . . . . . . . . 113 5 The Interaction of Light and Matter 125 5.1 Spectral Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 xiv Contents 5.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.3 The Bohr Model of the Atom . . . . . . . . . . . . . . . . . . 134 5.4 Quantum Mechanics and Wave-Particle Duality . . . . . . . 143 6 Telescopes 159 6.1 Basic Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 Optical Telescopes . . . . . . . . . . . . . . . . . . . . . . . . 173 6.3 Radio Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.4 Infrared, Ultraviolet, and X-Ray Astronomy . . . . . . . . . . 187 II The Nature of Stars 199 7 Binary Stars and Stellar Parameters 201 7.1 The Classification of Binary Stars . . . . . . . . . . . . . . . . 201 7.2 Mass Determination Using Visual Binaries . . . . . . . . . . . 205 7.3 Eclipsing, Spectroscopic Binaries . . . . . . . . . . . . . . . 208 8 The Classification of Stellar Spectra 223 8.1 The Formation of Spectral Lines . . . . . . . . . . . . . . . . 223 8.2 The Hertzsprung-Russell Diagram . . . . . . . . . . . . . . . 241 9 Stellar Atmospheres 255 9.1 The Description of the Radiation Field . . . . . . . . . . . . . 255 9.2 Stellar Opacity . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.3 Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . 276 9.4 The Structure of Spectral Lines . . . . . . . . . . . . . . . . . 293 10 The Interiors of Stars 315 10.1 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . 315 10.2 Pressure Equation of State . . . . . . . . . . . . . . . . . . . 320 10.3 Stellar Energy Sources . . . . . . . . . . . . . . . . . . . . . . 329 10.4 Energy Transport and Thermodynamics . . . . . . . . . . . . 350 10.5 Stellar Model Building . . . . . . . . . . . . . . . . . . . . . . 365 10.6 The Main Sequence . . . . . . . . . . . . . . . . . . . . . . . . 371 11 The Sun 381 11.1 The Solar Interior . . . . . . . . . . . . . . . . . . . . . . . . 381 11.2 The Solar Atmosphere . . ... . . . . . . . . . . . . . . . . . . 394 11.3 The Solar Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 416 12 The Process of Star Formation 437 12.1 Interstellar Dust and Gas . . . . . . . . . . . . . . . . . . . . 437 12.2 The Formation of Protostars . . . . . . . . . . . . . . . . . . 447 12.3 Pre-Main-Sequence Evolution . . . . . . . . . . . . . . . . . . 458 13 Post-Main-Sequence Stellar Evolution 483 13.1 Evolution on the Main Sequence . . . . . . . . . . . . . . . . 483 13.2 Late Stages of Stellar Evolution . . . . . . . . . . . . . . . . . 494 13.3 The Fate of Massive Stars . . . . . . . . . . . . . . . . . . . . 510 13.4 Stellar Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 529 14 Stellar Pulsation 541 14.1 Observations of Pulsating Stars . . . . . . . . . . . . . . . . . 541 14.2 The Physics of Stellar Pulsation . . . . . . . . . . . . . . . . . 548 14.3 Modeling Stellar Pulsation . . . . . . . . . . . . . . . . . . . . 557 14.4 Nonradial Stellar Pulsation . . . . . . . . . . . . . . . . . . . 561 14.5 Helioseismology . . . . . . . . . . . . . . . . . . . . . . . . 567 15 The Degenerate Remnants of Stars 577 15.1 The Discovery of Sirius B . . . . . . . . . . . . . . . . . . . . 577 15.2 White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 15.3 The Physics of Degenerate Matter . . . . . . . . . . . . . . . 583 15.4 The Chandrasekhar Limit . . . . . . . . . . . . . . . . . . . . 588 15.5 The Cooling of White Dwarfs . . . . . . . . . . . . . . . . . . 592 15.6 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 15.7 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 16 Black Holes 633 16.1 The General Theory of Relativity . . . . . . . . . . . . . . . . 633 16.2 Intervals and Geodesics . . . . . . . . . . . . . . . . . . . . . 648 16.3 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 17 Close Binary Star Systems 683 17.1 Gravity in a Close Binary Star System . . . . . . . . . . . . . 683 17.2 Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . . . . 692 17.3 A Survey of Close Binary Systems . . . . . . . . . . . . . . . 704 17.4 White Dwarfs in Semidetached Binaries . . . . . . . . . . . . 710 17.5 Neutron Stars and Black Holes in Binaries . . . . , . . . . . . 723 xvi Contents A Astronomical and Physical Constants A-1 B Solar System Data A-3 ' C The Constellations A-5 D The Brightest Stars A-11 E Stellar Data A-13 F The Messier Catalog A-19 G A Planetary Orbit Code A-23 H STATSTAR, A Stellar Structure Code A-27 I STATSTAR Stellar Models A-51 Chapter 1 THE CELESTIAL SPHERE 1.1 The Greek Tradition < Human beings have long looked up at the sky and pondered its mysteries. Ev-idence of the long struggle to understand its secrets may be seen in remnants of cultures around the world: the great Stonehenge monument in England, the structures and the writings of the Maya and Aztecs, and the medicine wheels of the Native Americans. However, our modern scientific view of the universe traces its beginnings to the ancient Greek tradition of natural phi-losophy. Pythagoras (ca. 550 B.C.) first demonstrated the fundamental rela-tionship between numbers and nature through his study of musical intervals and through his investigation of the geometry of the right angle. The Greeks continued their study of the universe for hundreds of years using the natural language employed by Pythagoras, namely mathematics. The modern dis-cipline of astronomy depends heavily on a mathematical formulation of its physical theories, following the process begun by the ancient Greeks. In an initial investigation of the night sky, perhaps its most obvious feature to a careful observer is the fact that it is constantly changing. Not only do the stars move steadily from east to west during the course of a night, but different stars are visible in the evening sky, depending upon the season. Of course, the Moon also changes, both in its position in the sky and its phase. More subtle, yet more complex, are the movements of the planets, or "wandering stars." Plato (ca. 350 B.C.) suggested that, to understand the motions of the heavens, one must first begin with a set of workable assumptions, or hypotheses. It seemed without question that the stars of the night sky revolve about a fixed Earth and that the heavens ought to obey the purest possible form of motion. Plato therefore proposed that celestial bodies should move about Earth with Chapter 1 The Celestial Sphere  Figure 1.1 The celestial sphere. a uniform (or constant) speed and follow a circular motion with Earth at the center of that motion. This concept of a geocentric universe was a natural consequence of the apparently unchanging relationship of the stars to one another in fixed constellations. If the stars were simply attached to a celestial sphere that rotated about an axis passing through the North and South poles of Earth and intersecting the celestial sphere at the north and south celestial poles respectively (Fig. 1.1), all of the stars' known motions could be described. The wandering stars posed a somewhat more difficult problem. A planet such as Mars moves slowly from west to east against the fixed background stars and then mysteriously reverses direction for a period of time before resuming its previous path (Fig. 1.2). Attempting to understand this backward, or ret-rograde, motion became the principal problem in astronomy for nearly 2000 years! Eudoxus of Cnidus, a student of Plato's and an exceptional mathemati-cian, suggested that each of the wandering stars occupied its own sphere and that all the spheres were connected through axes oriented at different angles and rotating at various speeds. Although this theory of a complex system of spheres initially was marginally successful at explaining retrograde motion, predictions began to deviate significantly from the observations as more data were obtained. Hipparchus (ca. 150 s.c.), perhaps the most notable of the Greek as- 1.1 The Greek Tradition 30 20 Ary~turus 0 0  A Leo  ' 15 Nov 1996, ; "~" 17 Mar 1997 ~ ~~ 1 May 1997 , /"/X 1 Jun 1997 , -10 F Hydr~ -20 14 13 12 11 10 9 8 Right ascension (hr) Figure 1.2 The retrograde motion of Mars. The coordinates of right ascension and declination are discussed on page 14 and in Fig. 1.13. tronomers, proposed a system of circles to explain retrograde motion. By placing a planet on a small, rotating epicycle that in turn moved on a larger deferent, he was able to reproduce the behavior of the wandering stars. Fur-thermore, this system was able to explain the increased brightness of the plan-ets during their retrograde phases as resulting from changes in their distances from Earth. Hipparchus also created the first catalog of the stars, developed a magnitude system for describing the brightness of stars that is still in use today, and contributed to the development of trigonometry. During the next two hundred years, the model of planetary motion put forth by Hipparchus also proved increasingly unsatisfactory in explaining many of the details of the observations. Claudius Ptolemy (ca. A.D. 100) introduced refinements to the epicycle/deferent system by adding equants (Fig. 1.3), resulting in a constant angular speed of the epicycle about the deferent (d9/dt was assumed to be constant). He also moved Earth away from the deferent center and even allowed for a wobble of the deferent itself. Predictions of the Ptolemaic model did agree more closely with observations than any previously devised scheme, but the original philosophical tenets of Plato (uniform and circular motion) were significantly compromised. Despite its shortcomings, the Ptolemaic model became almost universally accepted as the correct explanation of th~ ~,ion of the wandering stars. Wheu 6 Chapter 1 The Celestial Sphere  Figure 1.3 The Ptolemaic model of planetary motion. a disagreement between the model and observations would develop, the model was modified slightly by the addition of another circle. This process of "fixing" the existing theory led to an increasingly complex theoretical description of observable phenomena. 1.2 The Copernican Revolution By the sixteenth century the inherent simplicity of the Ptolemaic model was gone. Polish-born astronomer Nicolaus Copernicus (1473-1543), hoping to re-turn the science to a less cumbersome, more elegant view of the universe, sug-gested a heliocentric (Sun-centered) model of planetary motion (Fig. 1.4).1 His bold proposal led immediately to a much less complicated description of the relationships between the planets and the stars. Fearing severe criticism from the Catholic Church, whose doctrine then declared that Earth was the center of the universe, Copernicus postponed publication of his ideas until late in life. De Revolutionibus Orbium Coelestium (On the Revolution of the Celes-tial Sphere) first appeared in the year of his death. Faced with a radical new view of the universe, along with Earth's location in it, even some supporters of Copernicus argued that the heliocentric model merely represented a math-ematical improvement in calculating planetary positions but did not actually 'Actually, Aristarchus proposed a heliocentric model of the universe in 280 B.c. At the time his theory was presented, however, there was no compelling evidence to suggest that Earth itself was in motion. 1.3 Positions on the Celestial Sphere 13 30.0  Right ascension (hr) 18 12 6 0 18 -30.0 ' ' Dec 21 Sept 23 Jun 21 Mar 20 Dec 21 Figure 1.11 The ecliptic is the annual path of the Sun across the celestial sphere and is sinusoidal about the celestial equator. See Fig. 1.13, page 15, for definitions of right ascension and declination.  that plane out to the celestial sphere. The sinusoidal shape of the ecliptic occurs because the Northern Hemisphere alternately points toward and then away from the Sun during Earth's annual orbit. Twice during the year the Sun crosses the celestial equator, once moving northward along the ecliptic and later moving to the south. In the first case, the point of intersection is called the vernal equinox and the southern crossing occurs at the autumnal equinox. Spring officially begins when the center of the Sun is precisely on the vernal equinox; similarly, fall begins when the center of the Sun crosses the autumnal equinox. The most northern excursion of the Sun along the ecliptic occurs at the summer solstice, representing the official start of summer, and the southernmost position of the Sun is defined as the winter solstice. The seasonal variations in weather are due to the position of the Sun rela-tive to the celestial equator. During the summer months in the Northern Hemi-sphere, the Sun's northern declination causes it to appear higher in the sky, producing longer days and more intense sunlight. During the winter months the declination of the Sun is below the celestial equator, its path above the horizon is shorter, and its rays are less intense (see Fig. 1.12). The more direct the Sun's rays, the more energy per unit area strikes Earth's surface and the higher the resulting surface temperature. A coordinate system that results in nearly constant values for the positions of celestial objects, despite the complexities of diurnal and annual motions, is 14 Chapter 1 The Celestial Sphere  hec~l f .'o .. 5?, Celestial '..- equator 1.04 m 2.24 m   Figure 1.12 (a) The diurnal path of the Sun across the celestial sphere when the Sun is located at the vernal equinox (March), the summer sol-stice (June), the autumnal equinox (September), and the winter solstice (December) for an observer at latitude L. NCP and SCP designate the north and south celestial poles, respectively. (b) The direction of the Sun's rays at noon at the summer solstice (solid lines) and at the winter solstice (dashed lines) for an observer at 40 N latitude. necessarily less straightforward than the altitude-azimuth system. The equa-torial coordinate system (see Fig. 1.13) is based on the latitude-longitude system of Earth but does not participate in the planet's rotation. Declina-tion b is the equivalent of latitude and is measured in degrees north or south of the celestial equator. Right ascension a is analogous to longitude and is measured eastward along the celestial equator from the vernal equinox (T) to its intersection with the object's hour circle (the great circle passing through the object being considered and through the north celestial pole). Right as-cension is traditionally measured in hours, minutes, and seconds; 24 hours of right ascension is equivalent to 360, or 1 hour = 15. The rationale for this unit of measure is based on the 24 hours (sidereal time) necessary for an object to make two successive crossings of the observer's local meridian. The coor-dinates of right ascension and declination are also indicated in Figs. 1.2 and 1.11. Since the equatorial coordinate system is based on the celestial equator and the vernal equinox, changes in the latitude and longitude of the observer do not affect the values of right ascension and declination. Values of a and b are similarly unaffected by the annual motion of Earth around the Sun. The local sidereal time of the observer is defined as the amount of time elapsing since the vernal equinox last traversed the meridian. Local sidereal time is also equivalent to the hour angle H of the vernal equinox, where hour angle is defined as the angle between a celestial object and the observer's    Chapter 2 CELESTIAL MECHANICS 2.1 EllipticalOrbits Although the inherent simplicity of the Copernican model was aesthetically pleasing, the idea of a heliocentric universe was not immediately accepted; it lacked the support of observations capable of unambiguously demonstrating that a geocentric model was wrong. After the death of Copernicus, Tycho Brahe (1546-1601), the foremost naked-eye observer, carefully followed the motions of the "wandering stars" and other celestial objects. He carried out his work at the observatory, Uraniborg, on the island of Hveen (a facility provided for him by King Frederick II of Denmark). To improve the accuracy of his observations, Tycho used large measuring instruments, such as the quadrant depicted in the mural in Fig. 2.1(a). Tycho's observations were so meticulous that he was able to measure the position of an object in the heavens to an accuracy of better than 4', approximately one-eighth the angular diameter of a full moon. Through the accuracy of his observations he demonstrated for the first time that comets must be very distant, well beyond the Moon, rather than being some form of atmospheric phenomenon. Tycho is also credited with observing the supernova of 1572, which clearly demonstrated that the heavens were not unchanging as Church doctrine held. (This observation prompted King Frederick to build Uraniborg.) Despite the great care with which he carried out his work, Tycho was not able to find any clear evidence of the motion of Earth through the heavens, and he therefore concluded that the Copernican model must be false (see Section 3.1). At Tycho's invitation, Johannes Kepler (1571-1630), a German mathe-matician, joined him at Uraniborg (Fig. 2.1b). Unlike Tycho, Kepler was a heliocentrist, and it was his desire to find a geometrical model of the universe 25 ,y o. Yerkes Obser ~ - : - .. _ ~mat would be consistent with the best observations then available, namely Tycho's. After Tycho's death, Kepler inherited the mass of observations ac-cumulated over the years and began a painstaking analysis of the data. His initial, almost mystic, idea was that the universe is arranged with five perfect solids, nested to support the six known naked-eye planets (including Earth) on crystalline spheres, with the entire system centered on the Sun. After this model proved unsuccessful, he attempted to devise an accurate set of circular planetary orbits about the Sun, focusing specifically on Mars. Through his very clever use of offset circles and equants,l Kepler was able to obtain excel-lent agreement with Tycho's data for all but two of the points available. In particular, the discrepant points were each off by approximately 8', or twice the accuracy of Tycho's data. Believing that Tycho would not have made ob-servational errors of this magnitude, Kepler felt forced to dismiss the idea of purely circular motion. Rejecting the last fundamental assumption of the Ptolemaic model, Kepler began to consider the possibility that planetary orbits were elliptical in shape rather than circular. Through this relatively minor mathematical (though monumental philosophical) change, he was finally able to bring all of Tycho's observations into agreement with a model for planetary motion. This paradigm shift also allowed Kepler to discover that the orbital speed of a planet is not 'Recall the geocentric use of circles and equants by Ptolemy; see Fig. 1.3. 2.1 EllipticalOrbits  Figure 2.2 Kepler's second law states that the area swept out by a line between a planet and the focus of an ellipse is always the same for a given time interval, regardless of the planet's position in its orbit. The dots are evenly spaced in time. constant but varies in a precise way depending on its location in its orbit. In 1609 Kepler published the first two of his three laws of planetary motion in the book, Astronomica Nova, or The New Astronomy: Kepler's First Law A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse. Kepler's Second Law A line connecting a planet to the Sun sweeps out equal areas in equal time intervals. Kepler's first and second laws are illustrated in Fig. 2.2, where each dot on the ellipse represents the position of the planet during evenly spaced time intervals. Kepler's third law (also known as the harmonic law) was published ten years later in the book Harmonica Mundi (The Harmony of the World). His final law relates the average orbital distance of a planet from the Sun to its sidereal period: Kepler's Third Law P2 = a3,   where P is the orbital period of the planet, measured in years, and a is the average distance of the planet from the Sun, in astronomical units, or AU. An astronomical unit is, by definition, the average distance between Earth and the Sun, 1.496 x 1013 cm. A graph of Kepler's third law is shown in Fig. 2.3 using data for each planet in our solar system as given in Appendix B.  28 Chapter 2 Celestial Mechanics w . 0 .  Figure 2.3 Kepler's third law for planets orbiting the Sun. "e 'Mars ~Earth Venus Zmercury 0.1 1 '1'0 , ' 100 Period (yr) Neptune Uranus,f In retrospect it is easy to understand why the assumption of uniform and circular motion first proposed nearly 2000 years earlier was not found to be wrong much sooner; in most cases, planetary motion differs little from purely circular motion. In fact, it was actually fortuitous that Kepler chose to focus on Mars, since the data for that planet were particularly good and Mars deviates from circular motion more than most of the others. To appreciate the significance of Kepler's laws, we must first understand the nature of the ellipse. An ellipse (see Fig. 2.4) is defined by that set of points that satisfies the equation r+r'=2a, (2.1) where a is a constant, known as the semimajor axis (half the length of the long axis of the ellipse) and r and r' represent the distances to the ellipse from the two focal points, F and F', respectively. Notice that if F and F' were located at the same point, then r' = r and the previous equation would reduce to r = r' = a, the equation for a circle. Thus a circle is simply a special case of an ellipse. The distance b is known as the semiminor axis. The distance of either focal point from the center of the ellipse may be expressed as ae, where e is defined to be the eccentricity of the ellipse (0 < e < 1). For a circle, e = 0. A convenient relationship among a, b, and e may be found. Consider one of the two points at either end of the semiminor axis of an ellipse, where r = r'. In this case, r = a and, by the Pythagorean theorem, r2 = b2+-a2e2. Substitution 2.1 Elliptical Orbits 29  Figure 2.4 The geometry of an elliptical orbit. leads immediately to the expression bz = az(1 _ ez). (2.2) According to Kepler's first law, a planet orbits the Sun in an ellipse, with the Sun located at one focus of the ellipse, the principal focus (the other focus is empty space). The second law states that the orbital speed of a planet depends on its location in that orbit. To describe in detail the orbital behavior of a planet, it is necessary to specify where that planet is (its position vector) as well as how fast, and in what direction, the planet is moving (its velocity vector). It is often most convenient to express a planet's orbit in polar coordinates, indicating its distance r from the principal focus in terms of an angle B mea-sured counterclockwise from the major axis of the ellipse (see Fig. 2.4). Using the Pythagorean theorem, r 'z = rz sin 20+ (2ae + r cos B)z , which reduces to r /z =r2 -E- 4ae(ae + r cos 9) . Using the definition of an ellipse, r + r' = 2a, we find that r = a(1-e2) (0 < e < 1). (2.3) 1 + ecosB It is left as an exercise to show that the total area of an ellipse is given by A = 7rab. (2.4) 30 Chapter 2 Celestial Mechanics Example 2.1 Using Eq. (2.3), it is possible to determine the variation in dis-tance of a planet from the principal focus throughout its orbit. The semimajor axis of Mars' orbit is 1.5237 AU (or 2.2794 x 1013 cm) and the planet's orbital eccentricity is 0.0934. When B = 0, the planet is closest to the Sun, a point known as perihelion, and is at a distance given by a(1-e2) rp = l+e =a(1-e) (2.5) = 1.3814 AU. Similarly, aphelion (8 = 180), the point where Mars is farthest from the Sun, is at a distance given by a(1-e2) ra = 1-e = a (1 + e) (2.6) = 1.6660 AU. The variation in Mars' orbital distance from the Sun amounts to approximately 19% between perihelion and aphelion. An ellipse is actually one of a class of curves known as conic sections, found by passing a plane through a cone (see Fig. 2.5). Each type of conic section has its own characteristic range of eccentricities. As already mentioned, a circle is a conic section having e = 0, and an ellipse has 0 < e < 1. A curve having e = 1 is known as a parabola and is described by the equation _ 2p r 1 + cos B (2.7) where p is the distance of closest approach to the parabola's one focus, at B = 0. Curves having eccentricities greater than unity, e > 1, are hyperbolas and have the form ( 2- ) r 1 + e cos 8 (e ? =,~;; (2.8) Each type of conic section is related to a specific form of celestial motion. Chapter 3 THE CONTINUOUS SPECTRUM OF LIGHT 3.1 Stellar Parallax Measuring the intrinsic brightness of stars is inextricably linked with deter-mining their distances. This chapter on the light emitted by stars therefore begins with the problem of finding the distance to astronomical objects, one of the most important and most difficult tasks faced by astronomers. Kepler's laws in their original form describe the relative sizes of the planets' orbits in terms of astronomical units; their actual dimensions were unknown to Kepler and his contemporaries. The true scale of the solar system was first revealed in 1761 when the distance to Venus was measured as it crossed the disk of the Sun in a rare transit during inferior conjunction. The method used was trigonometric parallax, the familiar surveyor's technique of triangulation. On Earth, the distance to the peak of a remote mountain can be determined by measuring that peak's angular position from two observation points sep-arated by a known baseline distance. Simple trigonometry then supplies the distance to the peak; see Fig. 3.1. Similarly, the distances to the planets can be measured from two widely separated observation sites on Earth. Finding the distance even to the nearest stars requires a longer baseline than Earth's diameter. As Earth orbits the Sun, two observations of the same star made 6 months apart employ a baseline equal to the diameter of Earth's orbit. These measurements reveal that a nearby star exhibits an annual back-and-forth change in its position against the stationary background of much bi more distant stars. (As mentioned in Section 1.3, a star may also change its position due to its own motion through space. However, this proper motion, 63 Chapter 3 The Continuous Spectrum of Light  Figure 3.1 'I~igonometric parallax: d = B/ tan p. seen from Earth, is not periodic, and so can be distinguished from the star's periodic displacement caused by Earth's orbital motion.) As shown in Fig. 3.2, a measurement of the parallax angle p (one-half of the maximum change in angular position) allows the calculation of the distance d to the star. d ta p - p AU, where the small angle approximation tan p - p has been employed for the parallax angle p measured in radians. Using 1 radian = 57.3 = 2.063 x 105~~ to convert p to p" in units of arcseconds produces d - 2.063 x 105 AU. p~~ Defining a new unit of distance, the parsec (parallax-second, abbreviated pc), as 1 pc = 2.063 x 105 AU = 3.086 x 1018 cm leads to d = p~~ pc. (3.1) By definition, when the parallax angle p = 1", the distance to the star is 1 pc. Thus 1 parsec is the distance from which the radius of Earth's orbit, 1 AU, subtends an angle of 1~~. Another unit of distance often encountered is the light-year (abbreviated ly), the distance traveled by light through a vacuum in one year: 1 ly = 9.461 x 1017 cm. One parsec is equivalent to 3.262 ly. Even Proxima Centauri, the nearest star other than the Sun, has a parallax angle of less than 1". (Proxima Centauri is a member of the triple star system 3.2 The Magnitude Scale 65 Earth  Figure 3.2 Stellar parallax: d = 1/p" pc. a Centauri, and has a parallax angle of 0.77". If Earth's orbit around the Sun were represented by a dime, then Proxima Centauri would be located 1.5 miles away!) In fact, this cyclic change in a star's position is so difficult to detect that it was not until 1838 that it was first measured, by Friedrich Wilhelm Bessel (1784-1846), a German mathematician and astronomer.' Using spacecraft high above Earth's distorting atmosphere, parallax angles approaching 0.001" have been measured, corresponding to a distance of 1000 pc - 1 kiloparsec (kpc). This distance is still quite small compared to the 8 kpc distance to the center of our Milky Way Galaxy, so stellar parallax is useful only for surveying the local neighborhood of the Sun. Example 3.1 In 1838, after 4 years of observing 61 Cygni, Bessel announced his measurement of a parallax angle of 0.316" for that star. This corresponds to a distance of 1 1 d= ~~ pc= pc=3.16 pc= 10.31y, p 0.316 within 10% of the modern value 11.1 ly. 61 Cygni is one of the Sun's nearest neighbors. 3.2 The Magnitude Scale Nearly all of the information astronomers have received about the universe beyond our solar system has come from the careful study of the light emit-ted by stars, galaxies, and interstellar clouds of gas and dust. Our modern 1Tycho Brahe had searched for stellar parallax 250 years earlier, but his instruments were too imprecise to find it. Tycho concluded that Earth does not move through space, and he was thus unable to accept Copernicus's model of a heliocentric solar system. 66 Chapter 3 The Continuous Spectrum of Light understanding of the universe has been made possible by the quantitative measurement of the intensity and polarization of light in every part of the electromagnetic spectrum. The Greek astronomer Hipparchus was one of the first skywatchers to cat-alog the stars that he saw. In addition to compiling a list of the positions of some 850 stars, Hipparchus invented a numerical scale to describe how bright each star appeared in the sky. He assigned an apparent magnitude m = 1 to the brightest stars in the sky, and he gave the dimmest stars visible to the naked eye an apparent magnitude of m = 6. Note that a smaller apparent magnitude means a brighter-appearing star. Since Hipparchus's time, astronomers have extended and refined his ap-parent magnitude scale. In the nineteenth century, it was thought that the human eye responded to the difference in the logarithms of the brightness of two luminous objects. This theory led to a scale in which a difference of one magnitude between two stars implies a constant ratio between their brightness. By the modern definition, a difference of 5 magnitudes corresponds exactly to a factor of 100 in brightness, so a difference of one magnitude corresponds exactly to a brightness ratio of 1001/5 -_ 2.512. Thus a first magnitude star appears 2.512 times brighter than a second magnitude star, 2.5122 = 6.310 times brighter than a third magnitude star, and 100 times brighter than a sixth magnitude star. Using sensitive instruments called photometers, astronomers can measure the apparent magnitude of an object with an accuracy of f0.01 magnitude, and differences in magnitudes with an accuracy of f0.002 magnitude. Hipparchus's scale has been extended in both directions, from m = -26.81 for the Sun to approximately m = 29 for the faintest object detectable. The tqtal range of more than 55 magnitudes corresponds to over 10055/5 = (102)11 = 1022 for the ratio of the apparent brightness of the Sun to that of the faintest star or galaxy yet observed. The "brightness" of a star is actually measured in terms of the radiant flux F received from the star. The radiant flux is the total amount of light energy of all wavelengths that crosses a unit area oriented perpendicular to the direction of the light's travel in unit time; that is, it is the number of ergs of starlight energy arriving per second at one square centimeter of a detector aimed at the star.2 Of course, the radiant flux received from an object depends on both its intrinsic luminosity (energy emitted per second) and its distance from the observer. The same star, if located farther from Earth, would appear less bright in the sky. 21 erg = 10-7 joule. 3.2 The Magnitude Scale 67   Imagine a star of luminosity L surrounded by a huge spherical shell of radius r. Then, assuming that no light is absorbed during its journey out to the shell, the radiant flux, F, measured at distance r is related to the star's luminosity by  Since L does not depend on r, the radiant flux is inversely proportional to the square of the distance from the star. This is the well-known inverse square law for light.3 Example 3.2 The luminosity of the Sun is Lo = 3.826 x 1033 erg s-1. At a distance of 1 AU = 1.496 x 1013 cm, Earth receives a radiant flux above its absorbing atmosphere of F = 4 rz = 1.360 x 106 erg s-1 cm-z. This value of the solar flux is known as the solar constant. At a distance of 10 pc = 2.063 x 106 AU, an observer would measure the radiant flux to be only (1/2.063 x 106)z as large. That is, the radiant flux from the Sun would be 3.196 x 10-7 erg s-1 cm-z at a distance of 10 pc. Using the inverse square law, astronomers can assign an absolute magni-tude, M, to each star.4 This is defined to be the apparent magnitude a star would have if it were located at a distance of 10 pc. Recall that a difference of 5 magnitudes between the apparent magnitudes of two stars corresponds to the smaller-magnitude star being 100 times brighter than the larger-magnitude star. This allows us to specify their flux ratio as  Fz = 100~'"''1-"'.2)/5 . . (3.3) Taking the logarithm of both sides leads to the alternate form: F ml - mz = -2.51ogio C F ) . (3.4) z The connection between a star's apparent and absolute magnitudes and its distance may be found by combining Eqs. (3.2) and (3.3): z lOO~'"''-n~t>/s = lo F (10 pc) '  3If the star is moving with a speed near that of light, the inverse square law must be modified slightly. 4The magnitudes discussed hereafter are actuahy bolometric magnitudes, measured over all wavelengths of light; see page 82. 68 Chapter 3 The Continuous Spectrum of Light where Flo is the flux that would be received if the star were at a distance of 10 PC, and d is the star's distance, measured in parsecs. Solving for d gives d = lO(m-M+5)/5 PC (3.5) The quantity m - M is therefore a measure of the distance to a star and is called the star's distance modulus: m - M = 5loglo(d) - 5 = 5loglo ( 10 PC). (3.6) Example 3.3 The apparent magnitude of the Sun is mg _ -26.81, and its distance is 1 AU = 4.848 x 10-6 PC. Equation (3.6) shows that the absolute magnitude of the Sun is MSun = mSun - 5loglo(d) + 5 = 4.76, as already given. The Sun's distance modulus is thus ms,, - MSn = -31.57.5 For two stars at the same distance, Eq. (3.2) shows that the ratio of their radiant fluxes is equal to the ratio of their luminosities. Thus Eq. (3.3) for absolute magnitudes becomes 100(M1-M2)/5 = L2 L1. (3.7) Letting one of these stars be the Sun reveals the direct relation between a star's absolute magnitude and its luminosity: M = Msun - 2.51og1o ( L (3.8) o where the absolute magnitude and luminosity of the Sun are Mgun = 4.76 and Lp = 3.826 x 1033 erg s-1, respectively. It is left as an exercise for the reader to show that a star's apparent magnitude m is related to the radiant flux F received from the star by m = Ms. - 2.51og1o F , (3.9) ( F'io,o ) S The magnitudes m and M for the Sun have a "Sun" subscript (instead of "O") to avoid confusion with Mo, the standard symbol for the Sun's mass. 3.3 The Wave Nature of Light - 69 where Flo,o is the radiant flux received from the Sun at a distance of 10 pc (see Example 3.2). The inverse square law for light, Eq. (3.2), relates the intrinsic properties of a star (luminosity L and absolute magnitude M) to the quantities measured at a distance from that star (radiant flux F and apparent magnitude m). At first glance, it may seem that astronomers must start with the measurable quan-tities F and m and then use the distance to the star (if known) to determine the star's intrinsic properties. However, if the star belongs to an important class of objects known as pulsating variable stars, its intrinsic luminosity L and absolute magnitude M can be determined without any knowledge of its distance. Equation (3.5) then gives the distance to the variable star. As will be discussed in Section 14.1, these stars act as beacons that illuminate the fundamental distance scale of the universe. 3.3 The Wave Nature of Light Much of the history of physics is concerned with the evolution of our ideas about the nature of light. The speed of light was first measured with some accuracy in 1675, by the Danish astronomer Ole Roemer (1644-1710). Roemer observed the moons of Jupiter as they passed into the giant planet's shadow, and he was able to calculate when future eclipses of the moons should occur by using Kepler's laws. However, Roemer discovered that when Earth was moving closer to Jupiter, the eclipses occurred earlier than expected. Similarly, when Earth was moving away from Jupiter, the eclipses occurred behind schedule. Roemer realized that the discrepancy was caused by the differing amounts of time it took for light to travel the changing distance between the two planets, and he concluded that 22 minutes was required for light to cross the diameter of Earth's orbit.6 The resulting value of 2.2 x lOlo cm s-1 was close to the modern value of the speed of light. In 1983 the speed of light in vacuo was recognized as a fundamental constant of nature whose value is, by definition, c = 2.99792458 x 1010 cm s-1. Even the fundamental nature of light has long been debated. Isaac Newton, for example, believed that light must consist of a rectilinear stream of particles, because only such a stream could account for the sharpness of shadows. Chris tian Huygens (1629-1695), a contemporary of Newton, advanced the idea that light must consist of waves. According to Huygens, light is described by the usual quantities appropriate for a wave. The distance between two successive wave crests is the wavelength A, and thenumber of waves per second that 6 We now know that it takes light about 16.5 minutes to travel 2 AU. 70 , Chapter 3 The Continuous Spectrum of Light  Figure 3.3 Double-slit experiment. pass a point in space is the frequency v of the wave. Then the speed of the light wave is given by  Both the particle and wave models could explain the familiar phenomena of the reflection and refraction of light. However, the particle model of light prevailed, primarily on the strength of Newton's reputation, until its wave nature was conclusively demonstrated by Thomas Young's (1773-1829) famous double-slit experiment. In a double-slit experiment, monochromatic light of wavelength A from a single source passes through two narrow, parallel slits that are separated by a distance d. The light then falls upon a screen a distance L beyond the two slits (see Fig. 3.3). The series of light and dark interference fringes that Young observed on the screen could be explained only by a wave model of light. As the light waves pass through the narrow slits,7 they spread out (diffract) radially in a succession of crests and troughs. Light obeys a superposition principle, so when two waves meet, they add algebraically; see Fig. 3.4. At the screen, if a wave crest from one slit meets a wave crest from the other slit, a bright fringe or maximum is produced by the resulting constructive interference. But if a wave crest from one slit meets a wave trough from the other slit, they cancel each other, and a dark fringe or minimum results from this destructive interference. 7 Actually, Young used pinholes in his original experiment.  3.3 The Wave Nature of Light 73 Region Wavelength  Gamma ray A < 0.1 A X-ray 0.1 A < A < 100 A Ultraviolet 100 A < A < 4000 A Visible 4000 A < A < 7000 A Infrared 7000 A < A < 1 mm Microwave 1 mm < A < 10 cm Radio 10 cm < Table 3.1 The Electromagnetic Spectrum.  knowledge of the geometrical conditions of the motion is complete. A doubt about these things is no longer possible; a refutation of these views is inconceivable to the physicist. The wave theory of light is, from the point of view of human beings, certainty. Today, astronomers utilize light from every part of the electromagnetic spectrum. The total spectrum of light consists of electromagnetic waves of all wavelengths, ranging from very short wavelength gamma rays to very long wavelength radio waves. Table 3.1 shows how the electromagnetic spectrum has been arbitrarily divided into various wavelength regions. Like all waves, electromagnetic waves carry both energy and momentum in the direction of propagation. The amount of energy carried by a light wave is described by the Poynting vector, S. The Poynting vectorll points in the direction of the electromagnetic wave's propagation and has a magnitude equal to the amount of energy per unit time that crosses a unit area oriented perpendicular to the direction of the propagation of the wave. Because the magnitudes of the fields E and B vary harmonically with time, the quantity of practical interest is the time-averaged value of the Poynting vector over one cycle of the electromagnetic wave. In a vacuum the magnitude of the time-averaged Poynting vector, (S), is (S) = 8~EoBo (cgs units of erg s-1 cm-2) (3.12) = 1 EoBo (SI units of watt m-2), 2y0   where Eo and Bo are the maximum magnitudes (amplitudes) of the electric "The Poynting vector is named after John Henry Poynting (1852-1914), the physicist who first described it. In cgs units S = cE x B/4-7r, and in SI units S = E x B/po. 74 Chapter 3 The Continuous Spectrum of Light  Frad ~reflection) Figure 3.6 Radiation pressure force. and magnetic fields.l2 The time-averaged Poynting vector thus provides a description of the radiant flux in terms of the electric and magnetic fields of the light waves. However, it should be remembered that the radiant flux discussed in Section 3.2 involves the amount of energy received at all wavelengths from a star, whereas Eo and Bo describe an electromagnetic wave of a specified wavelength. Because an electromagnetic wave carries momentum, it can exert a force on a surface hit by the light. The resulting radiation pressure depends on whether the light is reflected from or absorbed by the surface. If the light is completely absorbed, then the force due to radiation pressure is in the direction of the light's propagation and has magnitude Fr~ - (S)A cos B (absorption), (3.13) c where B is the angle of incidence of the light as measured from the direction perpendicular to the surface of area A (see Fig. 3.6). Alternatively, if the light is completely reflected, then the radiation pressure force must act in a direction perpendicular to the surface; the reflected light cannot exert a force parallel to the surface. Then the magnitude of the force is ~,r~ - 2(S)A cos2 B (reflection). (3.14) c R,a,diation pressure has a negligible effect on physical systems under ev-eryday conditions. However, radiation pressure may play a dominant role in 12For an electromagnetic wave in a vacuum, Eo and Bo are related by Eo = Bo (cgs units) or Eo = cBo (SI units).  3.4 Blackbody Radiation 75 lo [ ._ - . g Betelgeuse 6 4 ou 'd 2 a 0 0  Q .U -.:. \ . 2 . d -4 ~ , . _6 .:: . . -1B Rigel' I......~. .:.......'.... ....... 6 5 Right ascension (hr) Figure 3.7 The constellation of Orion.  determining some aspects of the behavior of extremely luminous objects such as early main-sequence stars, red supergiants, or accreting compact stars. It may also have a significant effect on the small particles of dust found through-out the interstellar medium. 3.4 Blackbody Radiation Anyone who has looked at the constellation of Orion on a clear winter night has noticed the strikingly different colors of red Betelgeuse (in Orion's north-east shoulder) and blue-white Rigel (in the southwest leg); see Fig. 3.7. These colors betray the difference in the surface temperatures of the two stars. Betel-geuse has a surface temperature of about 3400 K, significantly cooler than the 10,100 K surface of Rige1.13 The connection between the color of light emitted by a hot object and its temperature was first noticed in 1792 by the English maker of fine porcelain, Thomas Wedgewood. All of his ovens became red-hot at the same temperature, independent of their size, shape, and construction. Subsequent investigations by many physicists revealed that any object with a temperature above abso-lute zero emits light of all wavelengths with varying degrees of efficiency; an ideal emitter is an object that absorbs all of the light energy incident upon it, -8 _ 13Both of these stars are pulsating variables (Chapter 14), so the values quoted are average temperatures. 76 Chapter 3 The Continuous Spectrum of Light  4 3 0 C 2   6  1000 3000 5000 7000 9000 11,000 13,000 15,000 Wavelength ~ (~) Figure 3.8 Blackbody spectrum [Planck function Ba(T)~. and reradiates this energy with the characteristic spectrum shown in Fig. 3.8. Because an ideal emitter reflects no light, it is known as a blackbody, and the radiation it emits is called blackbody radiation. Stars and planets are blackbodies, at least to a rough first approximation. Figure 3.8 shows that a blackbody of temperature T emits a continu-ous spectrum with some energy at all wavelengths and that this blackbody spectrum peaks at a wavelength ~ma,~, which becomes shorter with increasing temperature. The relation between ~m~ and T is known as Wien's displace-ment law:14 .~n,~T = 0.290 cm K. (3.15) Example 3.4 Betelgeuse has a surface temperature of 3400 K. If we treat Betelgeuse as a blackbody, Wien's displacement law shows that its continuous spectrum peaks at a wavelength of _ 0.290 cm K 3400 K = 8'S3 x 10-5 cm = 8530 ~r, which is in the infrared region of the electromagnetic spectrum. Rigel, with a surface temperature of 10,100 K, has a continuous spectrum that peaks at a 14In 1911, the German physicist Wilhelm Wien received the Nobel Prize for his theoretical contributions to understanding the blackbody spectrum. 3.6 The Color Index 85  Figure 3.10 Color-color diagram for main-sequence stars. The dashed line is for a blackbody. The color indices U - B and B - V are immediately seen to be    FSU d~ U - B = -2.51ogio F~sB d~) + CU-B, (3.29) G where Cp_B - CU - CB. A similar relation holds for B - V. From Eq. (3.26), note that although the apparent magnitudes depend on the radius R of the model star and its distance r, the color indices do not, because the factor of (R/r)2 cancels in Eq. (3.29). Thus the color index is a measure solely of the temperature of a model blackbody star. Figure 3.10 is a color-color diagram showing the relation between the (U - B) and (B - V) color indices for main-sequence stars.24 Astronomers face the difficult task of connecting a star's position on a color-color diagram with the physical properties of the star itself. If stars actually behaved as blackbodies, the color-color diagram would be the straight dashed line shown in Fig. 3.10. However, stars are not true blackbodies. As will be discussed in detail in Chapter 9, some light is absorbed as it travels through a star's atmosphere, and the amount of light absorbed depends on both the wavelength of the light and the temperature of the star. Other factors also play a role, causing the color indices of main sequence and supergiant stars of the same 24 As will be discussed in Section 10.6, main-sequence stars are powered by the nuclear fusion of hydrogen nuclei in their centers. Approximately 80% to 90% of all stars are main-sequence stars. The letter labels in Fig. 3.10 are spectral types; see Section 8.1. 86 Chapter 3 The Continuous Spectrum of Light U ~ o.s 0 ~c o.6 w  0.2 0.0~ ~ ~ w~_ 3000 4000 5000 6000 7000 Wavelength (f1) Figure 3.11 Sensitivity functions S(.~) for U, B, and V filters. temperature to be slightly different. The color-color diagram in Fig. 3.10 shows that the agreement between actual stars and model blackbody stars is best for very hot stars. Example 3.7 A star of spectral type 05 (to be defined in Section 8.1) has a surface temperature of 44,500 K and color indices U - B = -1.19 and B - V = -0.33. The large negative value of U - B indicates that this star appears brightest at ultraviolet wavelengths, as can be confirmed using Wien's displacement law, Eq. (3.19). The spectrum of a 44,500 K blackbody peaks at (5000 1~ ) (5800 K) ~ma" 44, 500 K - 652 1~, in the ultraviolet region of the electromagnetic spectrum. This wavelength is much shorter than the wavelengths transmitted by the U, B, and V filters (see Fig. 3.11), so we will be dealing with the smoothly declining long wavelength "tail" of the Planck function Ba(T). We can use the values of the color indices to estimate the constant CU-B in Eq. (3.29), and CB_v in a similar equation for the color index B - V. In this estimate, we will use a step function to represent the sensitivity function: S(~) = 1 inside the filter's bandwidth, and S(~) = 0 otherwise. The integrals in Eq. (3.29) may then be approximated by the value of the Planck function Ba at the center of the filter bandwidth, multiplied by that bandwidth. Thus, Chapter 4 THE THEORY OF SPECIAL RELATIVITY 4.1 The Failure of the Galilean Transformations A wave is a disturbance that travels through a medium. Water waves are dis-turbances traveling through water, and sound waves are disturbances traveling through air. James Clerk Maxwell predicted that light consists of "modulations of the same medium which is the cause of electric and magnetic phenomena," but what was the medium through which light waves traveled? At the time, physicists believed that light waves moved through a medium called the lu-miniferous ether. This idea of an all-pervading ether had its roots in the science of early Greece. In addition to the four earthly elements of Earth, Air, Fire, and Water, the Greeks believed that the heavens were composed of a fifth perfect element: the ether. Maxwell echoed their ancient belief when he wrote: There can be no doubt that the interplanetary and interstellar spaces are not empty, but are occupied by a material substance or body, which is certainly the largest, and probably the most uni-form body of which we have any knowledge. This modern reincarnation of the ether had been proposed for the sole purpose of transporting light waves; an object moving through the ether would experi-ence no mechanical resistance, so Earth's velocity through the ether could not be directly measured. In fact, no mechanical experiment is capable of determining the absolute velocity of an observer. It is impossible to tell whether you are at rest or in uniform motion (not accelerating). This general principle was recognized very 93 94 Chapter 4 The Theory of Special Relativity 0   clock Figure 4.1 Inertial reference frame. early. Galileo described a laboratory completely enclosed below the deck of a smoothly sailing ship and argued that no experiment done in this uniformly moving laboratory could measure the ship's velocity. To see why, consider two inertial reference frames, S and S'. As discussed in Section 2.2, an inertial reference frame may be thought of as a laboratory in which Newton's first law is valid: An object at rest will remain at rest and an object in motion will remain in motion in a straight line at constant speed unless acted upon by an unbalanced force. As shown in Fig. 4.1, the laboratory consists of (in principle) an infinite collection of meter sticks and synchronized clocks that can record the position and time of any event that occurs in the laboratory, at the location of that event; this removes the time delay involved in relaying information about an event to a distant recording device. With no loss of generality, the frame S' can be taken as moving in the positive x-direction (relative to the frame S) with constant velocity u, as shown in Fig. 4.2.1 Furthermore, the clocks in the two frames can be started when the origins of the coordinate systems, O and O~, coincide at time t = t~ = 0. Observers in the two frames S and S' measure the same moving object, recording its position (x, y, z) and (x~, y', z') at time t and t', respectively. An appeal to common sense and intuition shows that these measurements are 1This does not imply that the frame S is at rest, and that S' is moving. S' could be at rest while S moves in the negative x'-direction, or both frames may be moving. The point of the following argument is that there is no way to tell; only the relative velocity u is meaningful. 4.3 Time and Space in Special Relativity 105 the length of the rod as measured in the frame S' is L' = x2' - xl'. What is the length of the rod measured from S? Because the rod is moving relative to S, care must be taken to measure the x-coordinates xl and x2 of the ends of the rod at the same time. Then Eq. (4.16), with tl = t2, shows that the length L = x2 - xl measured in S may be found from or x21 - x1 = (x2 - xi) _ u(tz _ ti)  L' _ 1 - U2 /C2 (4.28) Because the rod is at rest relative to S', L' will be called Lrest. Similarly, be-cause the rod is moving relative to S, L will be called La,oving. Thus Eq. (4.28) becomes Lmoving = Lrest0 - U2 /C2. (4.29) This equation shows the effect of length contraction on a moving rod. It says that length or distance is measured differently by two observers in relative motion. If a rod is moving relative to an observer, that observer will measure a shorter rod than will an observer at rest relative to it. The longest length, called the rod's proper length, is measured in the rod's rest frame. Only lengths or distances parallel to the direction of the relative motion are affected by length contraction; distances perpendicular to the direction of the relative motion are unchanged (c.f. Eqs. 4.17-4.18). Example 4.1 Cosmic rays from space collide with the nuclei of atoms in Earth's upper atmosphere, producing elementary particles called muons. Muons are unstable and decay after an average lifetime T = 2.20 x 10-6 s, as measured in a laboratory where the muons are at rest. That is, the number of muons in a given sample should decrease with time according to N(t) _ No e-tlT, where No is the number of muons originally in the sample at time t = 0. At the top of Mt. Washington in New Hampshire, a detector counted 563 muons hr-1 moving downward at a speed u = 0.9952c. At sea level, 1907 m below the first detector, another detector counted 408 muons hr-1.6 The muons take (1.907 x 105 cm)/(0.9952c) = 6.39 x 10-6 s to travel from the top of Mt. Washington to sea level. Thus it might be expected that the number of muons detected per hour at sea level would have been N - No e-t/7 - 563 e-(6.39"l0-6 s)/(2.20x10-6 s) _ 31 muons hr-1. s Details of this experiment can be found in Frisch and Smith (1963). 106 Chapter 4 The Theory of Special Relativity Figure 4.8 Muons moving downward past Mt. Washington. (a) Mountain frame. (b) Muon frame. This is much less than the 408 muons hr-1 actually measured at sea level! How did the muons live long enough to reach the lower detector? The problem with the preceding calculation is that the lifetime of 2.20 x 10-6 s is measured in the muon's rest frame as Otrest, but the experimenter's clocks on Mt. Washington and below are moving relative to the muons. They measure the muon's lifetime to be 'Wrest 2.20 x 10-6 s Otmoving = - = 2.25 X 10-5 S, 1 - U2 /C2 1 - (0.9952)z more than ten times a muon's lifetime when measured in its own rest frame. The moving muons' clocks run slower, so more of them survive long enough to reach sea level. Repeating the preceding calculation using the muon lifetime as measured by the experimenters gives N - No e-t/T - 563 e-(6'3s"lo-6 5)/(z.z5xlo-5 S) _ 424 muons hr-1. When the effects of time dilation are included, the theoretical prediction is in excellent agreement with the experimental result. From a muon's rest frame, its lifetime is only 2.20 x 10-6 s. How would an observer riding along with the muons, as shown in Fig. 4.8, explain their ability to reach sea level? The observer would measure a severely length-contracted Mt. Washington (in the direction of the relative motion only). The distance traveled by the muons would not be L,.est = 1907 m, but rather Lmoving = Lrest 1 - u2/c2 = 1907 m 1 - (0.9952)z = 186.6 m. Thus it would take (1.866 x 104 cm)/(0.9952c) = 6.25 x 10-7 s for the muons to travel the length-contracted distance to the detector at sea level, as measured Chapter 5 The Interaction of Light and Matter 128 Chapter 5 The Interaction of Light and Matter Wavelength Equivalent (A) Name Atom Width (A) 3859.922 Fe I 1.554 3886.294 Fe I 0.920 3905.532 Si I 0.816 3933.682 K Ca II 20.253 3968.492 H Ca II 15.467 4045.825 Fe I 1.174 4101.748 h, H6 H I 3.133 4226.740 g Ca I 1.476 4340.475 G', H.y H I 2.855 4383.557 d Fe I 1.008 4404.761 Fe I 0.898 4861.342 F, Hp H I 3.680 5167.327 b4 Mg I 0.935 5172.698 b2 Mg I 1.259 5183.619 bl Mg I 1.584 5889.973 D2 Na I 0.752 5895.940 D1 Na I 0.564 6562.808 C, H, H I 4.020 Table 5.1 Wavelengths of the Strong Fraunhofer Lines. The atomic no-tation is explained in Section 8.1, and the equivalent width of a spectral line is defined in Section 9.4. (Data from Lang, Astrophysical Formulae, Second Edition, Springer-Verlag, New York, 1980.) measured to be 6562.50 A. Equation (5.1) shows that the radial velocity of Vega is v,. _ c (Abs - Arest) _ -1.4 x 106 cm s-1 = -14 km s-1; Arest the minus sign means that Vega is approaching the Sun. Recall from Sec-tion 1.3, however, that stars also have a proper motion, p, perpendicular to the line of sight. Vega's angular position in the sky changes by It = 0.345" yr-1. At a distance of r = 8.0 pc, this proper motion is related to the star's trans-verse velocity, ve, by Eq. (1.4). Expressing r in cm and Ec in rad s-1 results in ve = r/-C = 1.3 x 106 cm,s-l. This transverse velocity of 13 km s-1 is comparable to Vega's radial velocity. 5.1 Spectral Lines 129 Light from telescope  Diffraction grating   Camera mirror Figure 5.2 Spectrograph. Vega's speed through space relative to the Sun is thus v=w,?+ve=l9kms-1. The average speed of stars in the solar neighborhood is about 25 km s-1. In reality, the measurement of a star's radial velocity is complicated by the 29.8 km s-1 motion of Earth around the Sun, which causes the observed wave-length ~obs of a spectral line to vary sinusoidally over the course of a year. This effect of Earth's speed may be easily compensated for by subtracting the component of Earth's orbital velocity along the line of sight from the star's measured radial velocity. Modern methods can measure radial velocities with an accuracy of nearly f10 m s-1! Today astronomers use spectrographs to measure the spectra of stars and galaxies; see Fig. 5.2.3 After passing through a narrow slit, the starlight is collimated by a mirror and directed onto a diffraction grating. A diffraction grating is a piece of glass onto which narrow, closely spaced lines have been evenly ruled (typically several thousand lines per millimeter); the grating may be made to transmit the light (a transmission grating) or reflect the light (a reflection grating). In either case, the grating acts like a long series of neighboring double slits. Different wavelengths of light have their maxima 3As will be discussed in Chapter 7, measuring the radial velocities of stars in binary star systems allows the masses of the stars to be determined. 130 Chapter 5 The Interaction of Light and Matter occurring at different angles 9 given by Eq. (3.11): d sin B = nA (n = 0, 1, 2, ... ), where d is the distance between adjacent lines of the grating, n is the order of the spectrum, and B is measured from the line normal (or perpendicular) to the grating. (n = 0 corresponds to B = 0 for all wavelengths, so the light is not dispersed into a spectrum in this case.) The spectrum is then focused onto a photographic plate or electronic detector for recording. The ability of a spectrograph to resolve two closely spaced wavelengths separated by an amount 0A depends on the order of the spectrum, n, and the total number of lines of the grating that are illuminated, N. The smallest difference in wavelength that the grating can resolve is  where A is either of the closely spaced wavelengths being measured. The ratio A/0A is the resolving power of the grating.4 Astronomers recognized the great potential for uncovering the secrets of the stars in the empirical rules that had been obtained for the spectrum of light: Wien's law, the Stefan-Boltzmann equation, Kirchhoff's laws, and the new science of spectroscopy. By 1880 Gustav Wiedemann found that a detailed investigation of the Fraunhofer lines could reveal the temperature, pressure, and density of the layer of the Sun's atmosphere that produces the lines. The splitting of spectral lines by a magnetic field was discovered by Pieter Zeeman of the Netherlands in 1897, raising the possibility of measuring stellar magnetic fields. But a serious problem blocked further progress: However impressive, these results lacked the solid theoretical foundation required for the interpreta-tion of stellar spectra. For example, the absorption lines produced by hydrogen are much stronger for Vega than for the Sun. Does this mean that Vega's com-position contains significantly more hydrogen than the Sun's? The answer is no, but how can this information be gleaned from the dark absorption lines of a stellar spectrum recorded on a photographic plate? The answer required a new understanding of the nature of light itself. 5.2 Photons Despite Heinrich Hertz's absolute certainty in the wave nature of light, the solution to the riddle of the continuous spectrum of blackbody radiation led 4 In some cases, the resolving power of a spectrograph may be determined by other factors, for example, the slit width.  5.3 The Bohr Model of the Atom 141 atom? The energy lost by the electron is carried away by the photon, so Ephoton = Ehigh - I!' low hc__13.6eV 1 13.6eV 1 n2 high - ( nl w 12400 eV ~ 1 1\ _ -13.6 eV C32 - 22 I . Solving for the wavelength gives A = 6565 A, within 0.03% of the measured value of the Ha spectral line. The reverse process may also occur. If a photon has an energy equal to the difference in energy between two orbits (with the electron in the lower orbit), the photon may be absorbed by the atom. The electron uses the photon's energy to make an upward transition from the lower orbit to the higher orbit. The relation between the photon's wavelength and the quantum numbers of the two orbits is again given by Eq. (5.15). After the quantum revolution, the physical processes responsible for Kirch-hoff's laws (discussed in Section 5.1) finally became clear. A hot, dense gas or hot solid object produces a continuous spectrum with no dark spectral lines. This is the continuous spectrum of blackbody radiation, described by the Planck functions B,\(T) and B(T), emitted at any temperature above absolute zero. The wavelength Am,,., at which the Planck function B,\ (T) obtains its maximum value is given by Wien's displacement law, Eq. (3.15). A hot, diffuse gas produces bright emission lines. Emission lines are produced when an electron makes a downward transition from a higher to a lower orbit. The energy lost by the electron is carried away by a single photon. For example, the hydrogen Balmer emission lines are produced by electrons "falling" from higher orbits down to the n = 2 orbit; see Fig. 5.6. A cool, diffuse gas in front of a source of continuous spectrum produces dark absorption lines in the continuous spectrum. Absorption lines are produced when an electron makes a transition from a lower to a higher orbit. If an incident photon in the continuous spectrum has exactly the right amount of energy, equal to the difference in energy between a higher orbit and the electron's initial orbit, the photon is absorbed by Chapter 6 TELESCOPES 6.1 Basic Optics From the beginning, astronomy has been an observational science. In com-parison with what was previously possible with the naked eye, Galileo's use of the new optical device known as the telescope greatly enhanced our ability to observe the universe (see Section 2.2). Today we are still increasing our ability to "see" faint objects and to resolve them in greater detail. As a result, modern observational astronomy continues to supply scientists with more clues to the physical nature of our universe. Although observational astronomy now covers the entire range of the elec-tromagnetic spectrum, along with many areas of particle physics, the most familiar part of the field remains in the optical regime of the human eye (ap proximately 4000 A to 7000 A). Consequently, telescopes and detectors de-signed to investigate optical-wavelength radiation will be discussed in some detail. Galileo's telescope was a refracting telescope that made use of lenses through which light would pass, ultimately forming an image. Later, New-ton designed and built a reflecting telescope that made use of mirrors as the principal optical component. Both refractors and reflectors remain in use today. To understand the effects of an optical system on the light coming from an astronomical object, we will focus first on refracting telescopes. The path of a light ray through a lens can be understood using Snell's law of refraction. Recall that as a light ray travels from one transparent medium to another, its e~ path is bent at the interface. The amount that the ray is bent depends on the ratio of the wavelength-dependent indices of refraction na - c/va of each 159 160 Chapter 6 Telescopes n12,  Figure 6.1 Snell's law of refraction.   Figure 6.2 (a) A converging lens, fa > 0. (b) A diverging lens, f,\ < 0. material, where va represents the speed of light within the specific medium., if Bl is the angle of incidence, measured with respect to the normal to the interface between the two media, and 02 is the angle of refraction, also measured relative to the normal to the interface (see Fig. 6.1), then Snell's law is given by nla sin Bl = n2a sin B2. (6.1) If the surfaces of the lens are shaped properly, a beam of light rays of a given wavelength, originally traveling parallel to the axis of symmetry of the lens, called the optical axis of the system, can be brought to a focus at a point along that axis by a converging lens (Fig. 6.2a). Alternatively, the light can be made to diverge by a diverging lens and the light rays will appear to originate from a single point along the axis (Fig. 6.2b). The unique point in either case is referred to as the focal point of the lens, and the distance to that point from the center of the lens is known as the focal length, f. For a converging lens the focal length is taken to be positive, and for a diverging lens the focal length is negative. For an extended object, the image will also necessarily be extended. If a photographic plate or some other detector is to record this image, the detector 'It is only in a vacuum that va - c, independent of wavelength. The speed of light is wavelength-dependent in other environments. 6.3 Radio Telescopes 183 considered to be constant over that interval, then the integral simplifies to give P = SAOv, where A is the effective area of the aperture. A typical radio source has a spectral flux density S(v) on the order of one Jansky (Jy), 1 Jy = 10-26 W m-2 Hz-1 = 10-23 erg s-1 cm-2 Hz-1. Spectral flux density measurements of several mJy are not uncommon. With such weak sources, a large aperture is needed to collect enough photons to be measurable. Example 6.3 The second strongest radio source in the sky, after the Sun, is the galaxy Cygnus A. At 400 MHz (a wavelength of 75 cm), its spectral flux density is 4500 Jy. Assuming that a 25-m-diameter radio telescope is 100% efficient and is used to collect the radio energy of this source over a frequency bandwidth of 5 MHz, the total power detected by the receiver would be \2 P = S(v)~r C ~ I Ov = 1.1 x 10-6 erg s-1. One problem that radio telescopes share with optical telescopes is the need for greater resolution. Rayleigh's criterion (Eq. 6.6) applies to radio telescopes just as it does in the visible regime, except that radio wavelengths are much longer than those involved in optical work. Therefore, to obtain a level of resolution comparable to what is reached in the visible, much larger diameters are needed. Example 6.4 To obtain a resolution of 1" at a wavelength of 21 cm using a single aperture, the dish diameter must be D = 1.22 ~ = 1.22 0~21 m 52.8 km. B ( 4.85 x 10-6 rad )  For comparison, the largest single-dish radio telescope in the world is the fixed 300 m (1000 foot) diameter dish at Arecibo Observatory, Puerto Rico (see Fig. 6.20). One advantage of working at such long wavelengths is that small deviations from an ideal parabolic shape are not nearly as crucial. Since the relevant criterion is to be within some small fraction of a wavelength (say ~/20) of what is considered to be a perfect shape, variations of 1 cm are tolerable when observing at 21 cm. 184 Chapter 6 Telescopes  Figure 6.20 The 300-m radio telescope at Arecibo Observatory, Puerto Rico. (Courtesy of the NAIC-Arecibo Observatory, which is operated by Cornell University for the National Science Foundation.) Although it is clearly prohibitive to build individual dishes of sufficient size to produce the resolution at radio wavelengths that is anything like what is obtainable from the ground in the visible regime, astronomers have nevertheless been able to resolve radio images to better than 0.0015". This remarkable resolution is accomplished using a process not unlike the interference technique used in the Young double-slit experiment. Figure 6.21 shows two radio telescopes separated by a baseline of distance d. Since the distance from telescope B to the source is greater than the distance from telescope A to the source by an amount L, a specific wavefront will arrive at B after it has reached A. The two signals will be in phase and their superposition will result in a maximum if L is equal to an integral number of wavelengths (L = nA, where n = 0, 1, 2, ... for constructive interference). Similarly, if L is an odd integral number of half-wavelengths, then the signals will be exactly out of phase and a superposition of signals will result in a minimum in the signal strength [L = (n- 2 )A, where n = l, 2.... for destructive interference]. Since the pointing angle B is related to d and L by sin B = ~ , (6.11) it is then possible to determine accurately the position of the source using the interference pattern that is produced by combining the signals of the two antennas. Equation (6.11) is completely analogous to Eq. (3.11) describing the Young double-slit experiment. 188 Chapter 6 Telescopes Besides atmospheric absorption, the situation in the infrared is complicated still further because steps must be taken to cool the detector, if not the entire telescope. Using Wien's displacement law (Eq. 3.15) the peak wavelength of a blackbody of temperature 300 K is found to be nearly 10 ym. Thus the telescope and its detectors can produce radiation in just the wavelength region the observer might be interested in. Of course, the atmosphere can radiate in the infrared as well, including the production of molecular IR emission lines. In 1983 the Infrared Astronomy Satellite (IRAS) was placed in a 560-mile-high orbit, well above Earth's obscuring atmosphere. The 0.6-m imag-ing telescope was cooled to liquid helium temperatures, and its detectors were designed to observe at a variety of wavelengths, from 12 pm to 100 lim. Before its coolant was exhausted, IRAS proved to be very successful. Among its many accomplishments was the detection of dust in orbit around young stars, possi-bly indicating the formation of planetary systems. IRAS was also responsible for many important observations concerning the nature of galaxies. Based upon the success of IRAS, the European Space Agency plans to launch the Infrared Space Observatory (ISO) in 1995. The observatory will be cooled, just as IRAS was, but to obtain nearly 1000 times the resolution of IRAS, ISO will be able to point toward a target for a much longer period of time, allowing it to collect a greater number of photons.lo ISO will make use of the revolutionary new detectors now becoming avail-able for infrared work. Since infrared photons are not generally energetic enough to activate a photographic emulsion or eject electrons from most stan dard metals, detector technology differs from that used in the optical. Tradi-tionally, bolometers have been the detectors of choice for infrared observations. These devices, whose properties (e.g., electrical resistance) change as a result of being heated by infrared radiation, are usually composed of only one ele-ment (pixel). To form an image of a source, the detector must scan the source one section at a time, a very time-consuming and inaccurate process. Today, semiconductor technology has begun to supply arrays of detectors that are much like a silicon CCD, differing only in their use of a variety of hybrid (or "doped") materials, each sensitive in a specific wavelength range. Designed to investigate the electromagnetic spectrum at the longer wave-lengths of the microwave regime, the Cosmic Background Explorer (COBE; Fig. 6.25a) was launched in 1989 and finally switched off in 1993. COBE made a number of important observations, including very precise mea- loNASA has been developing plans to build its own successor to IRAS, called the Space Infrared Telescope Facility (SIRTF). However, at the time this book was written, SIRTF had not yet received funding from Congress.  6.4 Infrared, Ultraviolet, and X-Ray Astronomy 187 - X-rays -*I Visible a. Ultraviolet 0 ~F- Infrared ~ Microwave a. Optical 4 - Radio waves ->  window ~= 1.0 m I~ Molecular--, absorption Molecular *-Tabsorption Ionospheric ~ Radio window r- reflection 0.0 1 1 1 1 Illllllt 1 lug u i i i i I 1 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 logjoA (cm) Figure 6.24 The transparency of Earth's atmosphere as a function of wavelength. an effective collecting area that is 27 times greater than that of an individual telescope. To produce even higher resolution maps, the National Radio Astronomy Observatory operates the Very Long Baseline Array, composed of a series of telescopes scattered throughout the continental United States, Hawaii, and St. Croix in the U. S. Virgin Islands. 6.4 Infrared, Ultraviolet, and X-Ray Astronomy Given the enormous amount of data supplied by optical and radio observations, it is natural to consider studies in other wavelength regions. Unfortunately, such observations are either difficult or impossible to perform from the ground due to Earth's atmosphere being opaque to most wavelength regions outside of the visible and radio bands. Figure 6.24 shows the transparency of the atmosphere as a function of wavelength. Long wavelength ultraviolet radiation and some regions in the infrared are able to traverse the atmosphere with limited success but other wavelength regimes are completely blocked. For this reason, special measures must be taken to gather information at many photon energies. The primary contributor to infrared absorption is water vapor. As a result, if an observatory can be placed above most of the atmospheric water vapor, some observations can be made from the ground. To this end, NASA operates a 3-m infrared telescope on Mauna Kea, where the humidity is quite low. Even at 14,000 feet, however, the problem is not completely solved. To get above more of the atmosphere, balloon and aircraft observations have also been used. 188 Chapter 6 Telescopes Besides atmospheric absorption, the situation in the infrared is complicated still further because steps must be taken to cool the detector, if not the entire telescope. Using Wien's displacement law (Eq. 3.15) the peak wavelength of a blackbody of temperature 300 K is found to be nearly 10 pm. Thus the telescope and its detectors can produce radiation in just the wavelength region the observer might be interested in. Of course, the atmosphere can radiate in the infrared as well, including the production of molecular IR emission lines. In 1983 the Infrared Astronomy Satellite (IRAS) was placed in a 560-mile-high orbit, well above Earth's obscuring atmosphere. The 0.6-m imag-ing telescope was cooled to liquid helium temperatures, and its detectors were designed to observe at a variety of wavelengths, from 12 ym to 100 ym. Before its coolant was exhausted, IRAS proved to be very successful. Among its many accomplishments was the detection of dust in orbit around young stars, possi-bly indicating the formation of planetary systems. IRAS was also responsible for many important observations concerning the nature of galaxies. Based upon the success of IRAS, the European Space Agency plans to launch the Infrared Space Observatory (ISO) in 1995. The observatory will be cooled, just as IRAS was, but to obtain nearly 1000 times the resolution of IRAS, ISO will be able to point toward a target for a much longer period of time, allowing it to collect a greater number of photons.lo ISO will make use of the revolutionary new detectors now becoming avail-able for infrared work. Since infrared photons are not generally energetic enough to activate a photographic emulsion or eject electrons from most stan dard metals, detector technology differs from that used in the optical. Tradi-tionally, bolometers have been the detectors of choice for infrared observations. These devices, whose properties (e.g., electrical resistance) change as a result of being heated by infrared radiation, are usually composed of only one ele-ment (pixel). To form an image of a source, the detector must scan the source one section at a time, a very time-consuming and inaccurate process. Today, semiconductor technology has begun to supply arrays of detectors that are much like a silicon CCD, differing only in their use of a variety of hybrid (or "doped") materials, each sensitive in a specific wavelength range. . Designed to investigate the electromagnetic spectrum at the longer wave-lengths of the microwave regime, the Cosmic Background Explorer (COBE; Fig. 6.25a) was launched in 1989 and finally switched off in 1993. COBE made a number of important observations, including very precise mea- IoNASA has been developing plans to build its own successor to IRAS, called the Space Infrared Telescope Facility (SIRTF). However, at the time this book was written, SIRTF had not yet received funding from Congress.  BINARY STARS AND STELLAR PARAMETERS  7.1 The Classification of Binary Stars A detailed understanding of the structure and evolution of stars (the goal of Part II) requires knowledge about their physical characteristics. We have seen that knowledge of blackbody radiation curves, spectra, and parallax enables us to determine a star's effective temperature, luminosity, radius, composition, and other parameters. However, the only direct way to determine the mass of a star is by studying its gravitational interaction with other objects. In Chapter 2 Kepler's laws were used to calculate the masses of members of our solar system. However, the universality of the gravitational force allows Kepler's laws to be generalized to include the orbits of stars about one another or even the orbital interactions of galaxies, as long as proper care is taken to refer all orbits to the center of mass of the system. Fortunately, nature has provided ample opportunity for astronomers to observe binary star systems. At least half of all "stars" in the sky are actually multiple systems, two or more stars in orbit about a common center of mass. Analysis of the orbital parameters of these systems provides vital information about a variety of stellar characteristics, including mass. The methods used to analyze the orbital data vary somewhat depending on the geometry of the system, its distance from the observer, and the relative masses and luminosities of each component. Consequently, binary star systems are classified according to their specific observational characteristics. , . _ .. ~ ,~.d~+'.~P` 202 Chapter 7 Binary Stars and Stellar Parameters  Figure 7.1 An astrometric binary, which contains one visible member. The unseen component is implied by the oscillatory motion of the observed element of the system. Optical double. These systems are not true binaries but simply lie along the same line of sight (i.e., they have similar right ascensions and declinations). As a consequence of their large physical separations, they are not gravitationally bound and are not useful in determining stellar masses. Visual binary. Both stars in the binary can be resolved independently and, assuming that the orbital period is not prohibitively long, it is possible to monitor the motion of each member of the system. These systems provide important information about the angular separation of the stars from their mutual center of mass. If the distance to the binary is also known, the linear separations of the stars can then be calculated. Astrometric binary. If one member of a binary is significantly brighter than the other, it may not be possible to observe both members directly. In such a case the existence of the unseen member may be deduced by observing the oscillatory motion of the visible component. Since New-ton's first law requires that a constant velocity be maintained by a mass unless a force is acting upon it, such an oscillatory behavior requires that another mass be present (see Fig. 7.1). Eclipsing binary. For binaries that have orbital planes oriented ap-proximately along the line of sight of the observer, one star may pe-   Chapter 9 STELLAR ATMOSPHERES 9.1 The Description of the Radiation Field The light that astronomers receive from a star comes from the star's atmo-sphere, the transparent layers of gas overlying the opaque interior. A flood of photons pours from these layers, releasing the energy produced by the ther-monuclear reactions in the star's center. The temperature of the atmospheric layers from which these photons escape determines the features of the star's spectrum. To interpret the observed spectral lines properly, we must describe how light travels through the gas that makes up a star. Figure 9.1 shows rays of light with a wavelength between A and A + dA passing through a surface of area dA at an angle B into a cone of solid angle dSZ.l The angle B is measured from the direction perpendicular to the surface, so dA cos B is the area dA projected onto a plane perpendicular to the direction in which the radiation is traveling. If Ea dA is the amount of energy that these rays carry into the cone in a time interval dt, then the average intensity of the rays is defined as _ Ea dA dA dt dA cos 8 d52 ' The energy Ea dA in the numerator becomes vanishingly small as the quantities in the denominator go to zero, but their ratio approaches a limiting value of IA, called the specific intensity, usually referred to simply as the intensity. Thus, in spherical coordinates, Ea dA = Ia dA dt dA cos B dS2 = Ia dA dt dA cos B sin B dB do 'The surface is a mathematical location in space and is not necessarily a real physical surface. The concept of a solid angle and its units of steradians (sr) was discussed in Sec-tion 6.1: 255 256 Chapter 9 Stellar Atmospheres  Figure 9.1 Intensity IA. is the amount of electromagnetic radiation energy having a wavelength between A and A+dA that passes in time dt through the area dA into a solid angle d52 = sin B dB do. The specific intensity therefore has units of erg s-1 cm-3 sr-1.2 The Planck function BA, Eq. (3.20), is an example of the specific intensity for the special case of blackbody radiation. In general, however, the energy of the light need not vary with wavelength in the same way as it does for blackbody radiation. Later we will see under what circumstances we may set IA = B,\. Imagine a light ray of intensity IA as it propagates through a vacuum. Because IA is defined in the limit dSZ --+ 0, the energy of the ray does not spread out (or diverge). The intensity is therefore constant along any ray traveling through empty space. In general, the specific intensity IA varies with direction. The mean in-tensity of the radiation is found by integrating the specific intensity over all directions and dividing the result by 47r sr, the solid angle enclosed by a sphere, to obtain an average value of Ia. In spherical coordinates, this average value 1S3 (Ia> --_ 1 IIa dSt = 27r 7r I,\ sin e de d^. (9.1) 47r 1 47r 10-o 10=0 For an isotropic radiation field (one with the same intensity in all directions), (IA) = h. Blackbody radiation is isotropic and has JA) = BA- To determine how much energy is contained within the radiation field, we can use a "trap" consisting of a small cylinder of length dL, open at both ends, with perfectly reflecting walls inside; see Fig. 9.2. Light entering the trap at Z Recall from Section 3.5 that erg cm-3 indicates an energy per unit area per unit wave-length interval, erg cm-2 cm-1, not an energy per unit volume. 3 Many texts refer to the average intensity as Ja instead of (Ia). 248 Chapter 8 The Classification of Stellar Spectra Class Type of Star Ia-O Extreme, luminous supergiants Ia Luminous supergiants Ib Less luminous supergiants II Bright giants III Normal giants IV Subgiants V Main-sequence (dwarf) stars VI, sd Subdwarfs D White dwarfs Table 8.3 Morgan-Keenan Luminosity Classes. two-dimensional Morgan-Keenan (M-K) system of spectral classification.21 A luminosity class, designated by a Roman numeral, is appended to a star's Harvard spectral type. The numeral "I" (subdivided into classes Ia and Ib) is reserved for the supergiant stars, and "V" denotes a main-sequence star. The ratio of the strengths of two closely spaced lines is often employed to place a star in the appropriate luminosity class. In general, for stars of the same spectral type, narrower lines are usually produced by more luminous stars.22 The Sun is a G2 V star, and Betelgeuse is classified as M2 Ia.z3 The series of Roman numerals extends below the main sequence; the subdwarfs (class VI or "sd" ) reside slightly below the main sequence because they are deficient in metals (elements heavier than helium).24 The M-K system does not ex-tend to the white dwarfs, which are classified by the letter D. Table 8.3 lists the luminosity classes, and Fig. 8.15 shows the corresponding divisions on the H-R diagram and the location of a selection of specific stars. The two-dimensional M-K classification scheme enables astronomers to locate a star's position on the Hertzsprung-Russell diagram based entirely on the appearance of its spectrum. Once the star's absolute magnitude, M, has been read from the vertical axis of the H-R diagram, the distance to the star 21Edith Kellman of Yerkes printed the 55 spectra and was a co-author of the atlas; hence the additional "K" in MKK Atlas. 22In Section 9.4, we will find that because the atmospheres of more luminous stars are less dense, there are fewer collisions between atoms to distort the energy of their orbitals and so broaden the spectral lines. Z`~Betelgeuse, a pulsating variable star, is sometimes given the intermediate classification M2 Iab. 24Astronomers simply refer to all elements heavier than helium as metals. 262 Chapter 9 Stellar Atmospheres S  4 a wE 3 3 W 2 Hs  I I ~ ~ I I I ~ ~ I I I I I A.'SCO 4~00 4.700 1000 5100 E+000 6~ ~LC~. %.100 BL'CO BJ.CO 'l00~ .'15W I~.aCO NbMIQlIy~11 (A) Figure 9.5 The spectrum of the Sun. The dashed line is the curve of an ideal blackbody having the Sun's effective temperature. (Figure from Aller, Atoms, Stars, and Nebnlae, Revised Edition, Harvaxd University Press, Cambridge, MA, 1971.) effort is required to obtain a more accurate value of the surface temperature.7 Figure 9.5 shows that the Sun's spectrum deviates substantially from the shape of the blackbody Planck function, Ba, because solar absorption lines remove light from the Sun's continuous spectrum. The decrease in intensity produced by the dense series of metallic absorption lines in the solar spectrum is espe-cially effective; this effect is called line blanketing. In fact, there are many measures of a star's temperature. In addition to the effective temperature obtained from the Stefan-Boltzmann law, stellar temperature scales include the following: ~ The excitation temperature, defined by the Boltzmann equation (8.4). ~ The ionization temperature, defined by the Saha equation (8.6). ~ The kinetic temperature, contained in the Maxwell-Boltzmann dis-tribution, Eq. (8.1). ~ The color temperature, obtained by fitting the shape of a star's con-tinuous spectrum to the Planck function, Eq. (3.20). These temperatures are the same for the simple case of a gas confined within a box. The confined gas particles and blackbody radiation will come into 7See Bohm-Vitense (1981) for more details concerning the determination of temperatures.  9.2 Stellar Opacity 261 Just as the pressure of a gas exists throughout the volume of the gas and not just at the container walls, the radiation pressure of a "photon gas" exists everywhere in the radiation field. Imagine removing the reflecting surface dA in Fig. 9.4 and replacing it with a mathematical surface. The incident photons will now keep on going through dA; instead of reflected photons, photons will be streaming through dA from the other side. Thus, for an isotropic radiation field, there will be no change in the expression for the radiation pressure if the leading factor of 2 (which originated in the change in momentum upon reflection of the photons) is removed and the angular integration is extended over all solid angles: Prad,a dA = 1 Ia dA cos2 9 dS2 (9.7) e sphere 1 2' ~~r h dA cos2 B sin B dB dO c ~-o Je=o . = 3~ Ia da. (9.8) However, it may be that the radiation field is not isotropic. In that case, Eq. (9.7) for the radiation pressure is still valid but the pressure depends on the orientation of the mathematical surface dA. The total radiation pressure produced by photons of all wavelengths is found by integrating Eq. (9.8): ~'00 Prad = J Prad,a dA. 0 For blackbody radiation, it is left as a problem to show that _ 47r 1000 4o-T4 _ 1 4 _ 1 Prad 3c Ba (T ) d'~ = 3c 3 aT 3 u. (9.9) Thus the blackbody radiation pressure is one-third of the energy density. (For comparison, the pressure of an ideal monatomic gas is two-thirds of its energy density.) 9.2 Stellar Opacity  The classification of stellar spectra is an ongoing process. Even the most basic task, such as finding the surface temperature of a particular star, is complicated by the fact that stars are not actually blackbodies. The Stefan-Boltzmann re-lation, in the form of Eq. (3.17), defines a star's effective temperature, but some 9.4 The Structure of Spectral Lines 293 0.9 ~ 0.8 ~-ILo i m 0.7 0.6 0.5 - Eddington approximation Observed limb darkening 0.4 r 0 10 20 30 40 50 60 70 80 90 Angle 9 (deg) Figure 9.17 Observed and theoretical solar limb darkening for light inte-grated over all wavelengths. 9.4 The Structure of Spectral Lines We now have a formidable theoretical arsenal to bring to bear on the analysis of spectral lines. The shape of an individual spectral line contains a wealth of information about the environment in which it was formed. Figure 9.18 shows a graph of the radiant flux, Fa, as a function of wavelength for a typical absorption line. In the figure, Fa is expressed as a fraction of F~, the value of the flux from the continuous spectrum outside the spectral line. Near the central wavelength, ~o, is the core of the line, and the sides sweeping upward to the continuum are the line's wings. Individual lines may be narrow or broad, shallow or deep. The quantity (F~ - Fa)/F~ is referred to as the depth of the line. The strength of a spectral line is measured in terms of its equivalent width. The equivalent width W of a spectral line is defined as the width in angstroms of a box (shaded in Fig. 9.18) reaching up to the continuum that has the same area as the spectral line. That is, W = ~ F~ F, F~ d~, (9.54) where the integral is taken from one side of the line to the other. The equivalent width of a line in the visible spectrum, shaded in Fig. 9.18, is usually on the order of 0.1 1~. Another measure of the width of a spectral line is the distance in angstroms from one side of the line to the other, where its depth 294 Chapter 9 Stellar Atmospheres  Wavelength Figure 9.18 The shape of a typical spectral line. (F, - FA)/(F, - F,\o) = 1/2; this is called the full width at half-maximum and will be denoted by (0A)1/2. The spectral line shown in Fig. 9.18 is termed optically thin because there is no wavelength at which the radiant flux has been completely blocked. The opacity r1a of the stellar material is greatest at the wavelength Ao at the line's center and decreases moving into the wings. From the discussion on page 278, this means that the center of the line is formed at higher (and cooler) regions of the stellar atmosphere. Moving into the wings from Ao, the line formation occurs at progressively deeper (and hotter) layers of the atmosphere. In Sec-tion 11.2 this idea will be applied to the absorption lines produced in the solar photosphere. Three main processes are responsible for the broadening of spectral lines. Each of these mechanisms produces its own distinctive line shape or line profile. 1. Natural broadening. Spectral lines cannot be infinitely sharp, even for motionless, isolated atoms. According to Heisenberg's uncertainty principle, Eq. (5.19), as the time available for an energy measurement decreases, the inherent uncertainty of the result increases. Because an electron in an excited state occupies its orbital for only a brief instant, At, the orbital's energy, E, cannot have a precise value. Thus the uncertainty in the energy, AE, of the orbital is DE ~ At* (The electron's lifetime in the ground state may be taken as infinite,  L  Chapter 10 THE INTERIORS OF STARS 10.1 Hydrostatic Equilibrium In the last two chapters many of the observational details of stellar spectra were discussed along with the basic physical principles behind the production of the observed lines. Analysis of that light, collected by ground-based and orbital telescopes, allows astronomers to determine a variety of quantities re-lated to the outer layers of stars, such as effective temperature, luminosity, and composition. However, with the exceptions of the detection of neutrinos from the Sun (which will be discussed later in this chapter and in Chapter 11) and from Supernova 1987A (Section 13.3), no direct way exists to observe the central regions of stars. To deduce the detailed internal structure of stars requires the generation of computer models that are consistent with all known physical laws and that ultimately agree with observable surface features. Such calculations require very large and sophisticated software programs (often referred to as codes) and powerful computers. Although much of the theoretical foundation of stellar structure was understood in the first half of this century, not until the 1960s were sufficiently fast computing machines available to carry out all of the nec-essary calculations. Arguably the greatest success of theoretical astrophysics to date has been the computer modeling of stellar structure and evolution. Despite all of the successes of such calculations, however, numerous questions remain unanswered. The solution to many of these problems requires a more detailed theoretical understanding of the physical processes in operation in the interiors of stars, combined with significant computational power.  315 316 Chapter 10 The Interiors of Stars  dr  A FP,b Figure 10.1 In a static star the gravitational force on a mass element is exactly canceled by the outward force due to a pressure gradient in the star. A cylinder of mass dm is located at a distance r from the center of the star. The height of the cylinder is dr, and the areas of the top and bottom are both A. The density of the gas is assumed to be p at that position. The theoretical study of stellar structure, coupled with observational data, clearly shows that stars are dynamic objects, usually changing at an impercep-tibly slow rate by human standards, although at other times changing in very rapid and dramatic ways, such as during a supernova explosion. That such changes must occur can be seen by simply considering the observed energy output of a star. In the Sun, 3.826 x 1033 ergs of energy are emitted every second. This rate of energy output would be sufficient to melt a 0C block of ice measuring 1 AU x 1 mile x 1 mile in only 0.3 s, assuming that the absorption of the energy were 100% efficient. Because stars do not have infinite supplies of energy, they must eventually use up their reserves and die. Stellar evolution is the result of a constant fight against the relentless pull of gravity. Because the gravitational force is always attractive, an opposing force must exist if a star is to avoid collapse. This force is provided by pressure. To calculate how the pressure must vary with depth, consider a cylinder of mass dm whose base is located a distance r from the center of a spherical star (see Fig. 10.1). The areas of the top and bottom of the cylinder are each A and the cylinder's height is dr. Furthermore, assume that the only forces acting on the cylinder are gravity and the pressure force, which is always normal to the surface and may vary with distance from the center of the star. Using Chapter I1 THE SUN  11.1 The Solar Interior Over the last few chapters we have investigated the theoretical foundations of stellar structure, treating the star as being comprised of an atmosphere and an interior. The distinction between the two regions is fairly nebulous. Loosely, the atmosphere is considered to be that region where the optical depth is less than unity and the simple approximation of photons diffusing through optically thick material is not justified (see Eq. 9.25). Instead, atomic line absorption and emission must be considered in detail in the stellar atmosphere. On the other hand, nuclear reaction processes deep in the stellar interior play a crucial role in the star's energy output and its inevitable evolution. Due to its proximity to us, the star for which we have the greatest amount of observational data is our Sun. This wealth of information provides us with an excellent test of the theory of stellar atmospheres and interiors. Although most of the fundamental predictions of standard computer models of the Sun are in excellent agreement with observations, some discrepancies still remain. The resolution of these differences will certainly shed new light on a number of important astrophysical problems and could very well have a significant impact on other more fundamental areas of physics. Based on its observed luminosity and effective temperature, our Sun is classified as a typical main-sequence star of spectral class G2 with a surface composition of X = 0.73 and Z = 0.02 (the mass fractions of hydrogen and metals, respectively). To understand how it has evolved to this point, recall that according to the Vogt-Russell theorem the mass and composition of a star dictate its internal structure. Neglecting any mass loss that may have occurred in the past, our Sun has been converting hydrogen to helium via the 381 382 Chapter 11 The Sun  Figure 11.1 A schematic diagram of the Sun's interior. pp chain during most of its lifetime, thereby changing its composition and its structure. By comparing the results of radioactive dating tests of Moon rocks and meteorites with stellar evolution calculations, the current age of the Sun is determined to be approximately 4.52 x 109 yr. From this information, a standard solar model may be constructed for the present-day Sun using the physical principles discussed in preceding chap-ters. A schematic diagram of such a model is shown in Fig. 11.1.1 According to one evolutionary sequence leading to a present-day standard model, during its lifetime, the mass fraction of hydrogen (X) in the Sun's center has de-creased from its initial value of 0.71 to 0.34 while the central mass fraction of helium (Y) has increased accordingly, from 0.27 to 0.64.2 Furthermore, due to diffusive settling of elements heavier than hydrogen, the mass fraction of hydrogen near the surface may have increased by approximately 0.03 while the mass fraction of helium has decreased by a similar amount. The change in the Sun's central composition since it began nuclear burning has had a direct influence on its observable luminosity and radius, as shown in Fig. 11.2; the lu-minosity has increased by 40% while the size of the star has increased by more than 10%. 1 Nonstandard solar models invoke exotic or hypothetical physical mechanisms in an at-tempt to explain lingering discrepancies between the standard models and certain observa-tions, such as the flux of solar neutrinos; see below. 2The data quoted here and in the following discussion are from the standard solar model of Joyce Guzik, private communication, 1994. 11.1 The Solar Interior page 383 ssoo 5750 15700 [r` 15650 0 i I .0 p .0 1.0 2.0 3.0 4.0 5.0 6~ Time (109 yr) Figure 11.2 The evolution of the Sun from its birth to the present. As a result of changes in its internal composition, the Sun has become larger and brighter. The solid lines indicate its radius and luminosity while the dashed line represents its effective temperature. The radius and luminos-ity curves are relative to present-day values. (Data from Guzik, private communication, 1994.) From a theoretical standpoint, it is not at all clear how this change in solar energy output has altered Earth during its history, primarily because of un-certainties in the behavior of the terrestrial environment. Understanding the complex interaction between the Sun and Earth involves the detailed calcula-tion of convection in Earth's atmosphere, as well as the effects of the atmo-sphere's time-varying composition and the nature of the continually changing reflectivity of Earth's surface.3 Table 11.1 gives the values of the central temperature, pressure, density, and composition for a standard solar model of Guzik.4 Because of the Sun's past evolution, its composition is no longer homogeneous (constant through out) but instead shows the influence of ongoing nucleosynthesis, surface convec-tion, and elemental diffusion (settling of heavier elements). The composition structure of the Sun is shown in Fig. 11.3 for iH, 2 He, and 2 He. Since the 3 The ratio of the amount of reflected sunlight to incident sunlight is known as the albedo. Earth's albedo is affected by, among other things, the amount of surface water and ice. 4 Various researchers find slightly different values for central parameters, depending on assumptions about composition, opacities, convection, and so on. For instance, typical values of the central density range from approximately 150 g cm-3 to 160 g cm-3 while central temperatures range from 1.56 x 107 K to 1.58 x 107 K. 384 Chapter 11 The Sun Temperature 1.58 x 107 K Pressure 2.50 x 1017 dynes cm-2 Density 1.62 x 102 g cm-3 X 0.336 Y 0.643 Table 11.1 Central Conditions in the Sun. (Data from Guzik, private communication, 1994.) Sun's primary energy production mechanism is the pp chain, 2 He is an inter-mediate species in the reaction sequence. During the conversion of hydrogen to helium, 2 He is produced and then destroyed again [see Eqs. (10.46), (10.47), and (10.48)]. At the top of the hydrogen burning region, where the temper-ature is lower, 2He is relatively more abundant because it is produced more easily than it is destroyed.s At greater depths, the higher temperatures allow the helium-helium interaction to proceed more rapidly and the 2 He abundance again decreases (the temperature structure of the Sun is shown in Fig. 11.4). The slight ramp in the iH and 2 He curves near 0.7 Ro reflects evolutionary changes in the position of the base of the surface convection zone, combined with the effects of elemental diffusion. Within the convection zone, turbulence results in essentially complete mixing and a homogeneous composition. The largest contribution to the energy production in the Sun occurs at ap-proximately one-tenth of the solar radius, as can be seen in the luminosity curve (Fig. 11.5) and the curve of its derivative with respect to radius (Fig. 11.6). If this result seems unexpected, consider that the mass conservation equation (Eq. 10.8) dMT = 41rr2P dr gives dM,. = 4,7rr2p dr = p dV, indicating that the amount of mass within a certain radius interval increases with radius simply because the volume of a spherical shell, dV = 4,7rr2 dr, increases with r for a fixed choice of dr. Of course, the mass contained in the shell also depends on the density of the gas. Consequently, even if the amount of energy liberated per gram of material (E) decreases steadily from the center outward, the largest contribution to the total luminosity will occur, not at the S Recall that much higher temperatures are required for helium-helium compared to proton-proton interactions. Chapte~ 12 THE PROCESS OF STAR FORMATION 12.1 Interstellar Dust and Gas When we look into the heavens, it appears as though the stars are unchanging, pointlike sources of light that shine steadily. On casual inspection, even our own Sun appears constant. But, as we have seen in the last chapter, this is not the case; sunspots come and go, flares erupt, the corona changes shape, and even the Sun's luminosity appears to be fluctuating very slightly over long periods of time. In fact all stars change. Usually the changes are so gradual and over such long time intervals when measured in human terms that we do not notice them. Occasionally, however, the changes are extremely rapid and dramatic, as in the case of a supernova explosion. By invoking our understanding of the physics of stellar interiors and atmospheres developed thus far, we can now examine this process of how stars evolve during their lives. In some sense the evolution of a star is cyclic. It is born out of gas and dust that exists between the stars, known as the interstellar medium (ISM). During its lifetime, depending on the star's total mass, much of that material may be returned to the ISM through stellar winds and explosive events. Sub-sequent generations of stars can then form from this processed material. As a result, to understand the evolution of a star, it is important to study the nature of the ISM. On a dark night some of the dust clouds that populate our Galaxy can be seen in the band of stars that is the disk of the Milky Way Galaxy (see Fig. 12.1). It is not that these dark regions are devoid of stars, but. rather 437 438 Chapter 12 The Process of Star Formation  Figure 12.1 Dust clouds obscure the stars located behind them in the disk of the Milky Way. (Courtesy of Palomar/ Caltech.) that the stars located behind intervening dust clouds are obscured. This ob-scuration, referred to as interstellar extinction, is due to the scattering and absorption of the starlight (see Fig. 12.2). Given the effect that extinction can have on the apparent magnitude of a star, the distance modulus equation (Eq. 3.6) must be modified appropriately. In a given wavelength band centered on A, we now have ma = Ma + 5loglo d - 5 + aa, (12.1) where d is in pc and aA represents the number of magnitudes of absorption or scattering present along the line of sight. If aa is large enough, a star that would otherwise be visible to the naked eye or through a telescope could no longer be detected. This is the reason for the dark bands running through the Milky Way. Clearly a,\ must be related to the optical depth of the material, measured back along the line of sight. From Eq. (9.16), the fractional change in the intensity of the light is given by Ia/ h o = e-Ta. , Combining this with Eq. (3.4), we can now relate the optical depth to the Chapter 13  J POST-MAIN-SEQUENCE STELLAR EVOLUTION 13.1 Evolution on the Main Sequence In Section 10.6 we learned that the existence of the main sequence is due to the nuclear reactions that convert hydrogen into helium in the cores of stars. The evolutionary process of protostellar collapse to the zero-age main sequence was discussed in Chapter 12. In this chapter we will follow the lives of stars as they age, beginning on the main sequence. This evolutionary process is an inevitable consequence of the relentless force of gravity and the change in chemical composition due to nuclear reactions. To maintain their luminosities, stars must tap sources of energy contained within, either nuclear or gravitational.' Pre-main-sequence evolution is char-acterized by two basic time scales, the free-fall time scale (Eq. 12.16) and the Kelvin-Helmholtz (or thermal readjustment) time scale (Eq. 10.30). Main-se-quence and post-main-sequence evolution are also governed by a third time scale, the time scale of nuclear reactions (Eq. 10.31). As we saw in Exam-ple 10.4, the nuclear time scale is on the order of 1010 years for the Sun, much longer than the Kelvin-Helmholtz time scale of roughly 107 years, estimated in Example 10.3. It is the difference in time scales for the various phases of evolution of individual stars that explains why approximately 80% to 90% of all stars in the solar neighborhood are observed to be main-sequence stars (see Section 8.2); we are more likely to find stars on the main sequence simply because that stage of evolution requires the most time, while later stages of 'We have already seen in Problem 10.3 that chemical energy cannot play a significant role in the energy budgets of stars. 483 484 Chapter 13 Post-Main-Sequence Stellar Evolution evolution proceed more rapidly. However, as a star switches from one nu-clear source to the next, gravitational energy can play a major role and the Kelvin-Helmholtz time scale will again become important. Careful study of the main sequence of an observational H-R diagram such as Fig. 8.11 or the observational mass-luminosity relation (Fig. 7.7) reveals that these curves are not simply thin lines but have finite widths. The widths of the main sequence and the mass-luminosity relation are due to a number of factors, including observational errors, differing chemical compositions of the individual stars in the study, and varying stages of evolution on the main sequence. In this section, the evolution of stars on the main sequence will be consid-ered. Although all stars on the main sequence are converting hydrogen into helium and, as a result, share similar evolutionary characteristics, differences do exist. For instance, as was mentioned in Section 10.6, ZAMS stars with masses less than about 1.2 Mo have radiative cores while more massive stars have convective cores. First consider a star typical of those at the low-mass end of the main sequence, such as the Sun. As was mentioned in the discussion of Fig. 11.2, the Sun's luminosity, radius, and temperature have all increased steadily since it reached the ZAN1S some 4.5 billion years ago. This evolution occurs because, as the pp chain converts hydrogen into helium, the mean molecular weight /t of the core increases (Eq. 10.21). According to the ideal gas law (Eq. 10.14), unless the density and/or temperature of the core also increases, there will be insufficient gas pressure to support the overlying layers of the star. As a result, the core must be compressed. While the density of the core increases, gravitational potential energy is released, and, as required by the virial theorem (Section 2.4), half of the energy is radiated away and half of the energy goes into increasing the thermal energy and hence the temperature of the gas.2 Now, since the pp chain nuclear reaction rate goes as pX2T6 (see Eq. 10.50), the increased temperature and density more than offset the decrease in the mass fraction of hydrogen, and the luminosity of the star slowly increases, along with its radius and effective temperature. Main-sequence and post-main-sequence evolutionary tracks of stars of vari-ous masses, as computed in the pioneering study by Icko Iben, Jr., are shown in Fig. 13.1, and the amount of time required to evolve between points indicated on the figure are given in Table 13.1.3 The locus of points labeled 1 represents 2 This temperature increase means that the region of the star that is hot enough to undergo nuclear reactions increases slightly during the main-sequence phase of evolution. 3 Note that in these models, the solar luminosity was taken to be Lo = 3.86 x 1033 ergs s-1, rather than the value adopted elsewhere in this book. 13.4 Stellar Clusters 529 13.4 Stellar Clusters    Over the past two chapters we have seen a story develop that depicts the lives of stars. They are formed from the ISM, only to return most of that material to the ISM through stellar winds, by the ejection of planetary nebulae, or via supernova explosions. The matter that is given back, however, has been enriched with heavier elements that were produced through the various sequences of nuclear reactions governing a star's life. As a result, when the next generation of stars is formed, it possesses higher concentrations of these heavy elements than did its ancestors. This cyclic process of star formation, death, and rebirth is evident in the variations in composition between stars. It is generally believed that the universe began with the Big Bang some 10 to 20 billion years ago and that hydrogen and helium were essentially the only elements produced by the nucleosynthesis that occurred during the initial fireball. Consequently, the first stars to form were extremely metal poor, having very low values for Z. Each succeeding generation of star production resulted in higher and higher proportions of heavier elements, leading to metal-rich stars for which Z - 0.03. Metal-poor stars are referred to as Population II while metal-rich stars are called Population I. The classifications of Population II and Population I are due originally to their identifications with kinematically distinct groups of stars within our Galaxy. Population I stars have velocities relative to the Sun that are low compared to Population II stars. Furthermore, Population I stars are found predominantly in the disk of the Milky Way, while Population II stars can be found well above or below the disk. It was only later that astronomers real-ized that these two groups of stars differed chemically as well. Not only do populations tell us something about evolution, but the kinematic characteris-tics, positions, and compositions of Population I and Population II stars also provide us with a great deal of information about the formation and evolution of the Milky Way Galaxy. Recall from Section 12.2 that during the collapse of a molecular cloud, a process of cascading fragmentation results. This leads to the creation of stellar clusters, ranging in size from tens of stars to hundreds of thousands of stars. Every member of a given cluster is formed from the same cloud, at the same time, and all with essentially identical compositions. Thus, excluding such effects as rotation, magnetic fields, or membership in a binary star system, the Vogt-Russell theorem suggests that the differences in evolutionary states between the various stars in the cluster are due solely to their initial masses. Extreme Population 11 clusters formed when the Galaxy was very young, Chapter 14 STELLAR PULSATION 14.1 Observations of Pulsating Stars In August of 1595, a Lutheran pastor and amateur astronomer named David Fabricius observed the star o Ceti. As he watched over a period of months, the brightness of this second magnitude star in the constellation Cetus (the Sea Monster) slowly faded. By October, the star had vanished from the sky. Several more months passed as the star eventually recovered and returned to its former brilliance. In honor of this miraculous event, o Ceti was named Mira, meaning "wonderful." Mira continued its rhythmic dimming and brightening, and by 1660 the 11-month period of its cycle was established. The regular changes in brightness were mistakenly attributed to dark "blotches" on the surface of a rotating star. Supposedly, Mira would appear fainter when these dark areas were turned toward Earth. Figure 14.1 shows the light curve of Mira for a 42-year interval. Today astronomers recognize that the changes in Mira's brightness are due not to dark spots on its surface, but to the fact that Mira is a pulsating star, a star that dims and brightens as its surface expands and contracts. Mira is the prototype of the long-period variables, stars that have somewha.t irregular light curves and pulsation periods between 100 and 700 days. Nearly two centuries elapsed before another pulsating star was discovered. In 1784 John Goodricke of York, England, found that the brightness of the star S Cephei varies regularly with a period of 5 days, 8 hours, 48 minutes. This discovery cost Goodricke his life; he contracted pneumonia while observing S Cephei and died at the age of 21. The light curve of S Cephei, shown in Fig. 14.5, is less spectacular than that of o Ceti. It varies by less than one 541 542 Chapter 14 Stellar Pulsation   '8 : i . ' . 4 :.t .: . . . . . . . t 2  10000 11000 12000 13000 14000 15000 Julian day - 2435000 Figure 14.1 The light curve of Mira from 1953 to March 1995. The time is measured in Julian days; Julian day 0 started at noon (UT) on January 1, 4713 s.c., and JD 2435000 is September 14, 1954. (Courtesy of Janet A. Mattei, AAVSO Director.) magnitude in brightness and never fades from view. Nevertheless, pulsating stars similar to S Cephei, called classical Cepheids, are vitally important to astronomy. Today, some 20,000 pulsating stars have been cataloged by astronomers. One woman, Henrietta Swan Leavitt (1868-1921; see Fig. 14.2), discovered more than 10% of these stars while working as a "computer" for Edward Charles Pickering at Harvard University. Her tedious task was to compare two photographs of the same field of stars taken at different times, and detect any star that va      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ried in brightness. Eventually she discovered 2400 classical Cepheids, most of them located in the Small Magellanic Cloud, with periods between 1 and 50 days. Leavitt took advantage of this opportunity to in-vestigate the nature of the classical Cepheids in the Small Magellanic Cloud. Noticing that the more luminous Cepheids took longer to go through their pul-sation cycles, she plotted the apparent magnitudes of these stars against their pulsation periods. The resulting graph, shown in Fig. 14.3 demonstrated that the apparent magnitudes of classical Cepheids are closely correlated with their periods, with an intrinsic uncertainty of only Am ,z:~ 0.5 at a given period. Because all of the stars in the Small Magellanic Cloud are roughly the same distance from us (about 60 kpc), the differences in their apparent magnitudes must be the same as the differences in their absolute magnitudes [c.f. Eq. (3.6) for the distance modulus]. Thus the observed differences in these staxs' Chapte~ 1 S THE DEGENERATE REMNANTS OF STARS 15.1 The Discovery of Sirius B In 1838 Friedrich Wilhelm Bessel used the technique of stellar parallax to find the distance to the star 61 Cygni. Following this first successful measurement of a stellar distance, Bessel applied his talents to another likely candidate: Sirius, the brightest appearing star in the sky. Its parallax angle of p~~ = 0.377~~ corresponds to a distance of only 2.65 pc, or 8.65 ly. Sirius's brilliance in the night sky is in part due to its proximity to Earth. As he followed the star's path through the heavens, Bessel found that it deviated slightly from a straight line. After ten years of precise observations, Bessel concluded in 1844 that Sirius is actually a binary star system. Although unable to detect the companion of the brighter star, he deduced that its orbital period was about 50 years (the modern value is 49.9 years) and predicted its position. The search was on for the unseen "Pup," the faint companion of the luminous "Dog Star." The telescopes of Bessel's time were incapable of finding the Pup so close to the glare of its bright counterpart, and following Bessel's death in 1846 the en-thusiasm for the quest waned. Finally in 1862, the son of the prominent Amer ican lensmaker Alvan Clark tested his father's new 18-inch refractor (3 inches larger than any previous instrument) on Sirius, and he promptly discovered the Pup at its predicted position. The dominant Sirius A was found to be nearly one thousand times brighter than the Pup, now called Sirius B; see Fig. 15.1. The details of their orbits about their center of mass (see Fig. 15.2 and Prob-lem 7.4) revealed that Sirius A ~and Siriusre~: }aave masses of about 2.3 Mo and 577 page 578 Chapter 15 The Degenerate Remnants of Stars Figure 15.1 The white dwarf, Sirius B, beside the overexposed image of Sirius A. (Courtesy of Lick Observatory.) 1.0 Mp, respectively. A more recent determination for the mass of Sirius B is 1.053 0.028 Mo, and it is this value that will be used below. Clark's discovery of Sirius B was made near the opportune time of apastron, when the two stars were most widely separated (by just 10"). The great difference in their luminosities (LA = 23.5 Lo and LB = 0.03 L(D) makes observations at other times much more difficult. When the next apastron arrived 50 years later, spectroscopists had developed the tools to measure the stars' surface temperatures. From the Pup's faint appearance, astronomers expected it to be cool and red. They were startled when Walter Adams, working at Mt. Wilson Observatory in 1915, discovered that to the contrary, Sirius B is a hot, blue-white star that emits much of its energy in the ultraviolet. A modern value of the temperature of Sirius B is 27,000 K, much hotter than Sirius A's 9910 K. The implications for the star's physical characteristics were astounding. Using the Stefan-Boltzmann law, Eq. (3.17), to calculate the size of Sirius B results in a radius of only 5.5 x 108 cm ~ 0.008 Ro. Sirius B has the mass of the Sun confined within a volume smaller than Earth! The average density of Sirius B is 3.0 x 106 g cm-3, and the acceleration due to gravity at its surface is about 4.6 x 108 cm s-2. On Earth, the pull of gravity on a teaspoon of white-dwarf material would be 14.5 billion dynes (over 16 tons), and on the surface of the white dwarf it would weigh 470,000 times more. This fierce  15.2 White Dwarfs 579 A 12 6 -12 ~ -12 -6 0 6 12 ll8 24 Distance (AU)   4 Figure 15.2 The orbits of Sirius A and Sirius B. The center of mass of the system is marked with an "x ." gravity reveals itself in the spectrum of Sirius B; it produces an immense pressure near the surface that results in very broad hydrogen absorption lines; see Fig. 8.14.1 Other than these lines, its spectrum is a featureless continuum. Astronomers first reacted to the discovery of Sirius B by dismissing the results, calling them "absurd." However, the calculations are so simple and straightforward that this attitude soon changed to the one expressed by Ed dington in 1922: "Strange objects, which persist in showing a type of spectrum entirely out of keeping with their luminosity, may ultimately teach us more than a host which radiate according to rule." Like all sciences, astronomy advances most rapidly when confronted with exceptions to its theories. 15.2 White Dwarfs Obviously Sirius B is not a normal star. It is a white dwarf, a class of stars that have approximately the mass of the Sun and the size of Earth. Although as many as one-quarter of the stars in the vicinity of the Sun may be white dwarfs, the average characteristics of these faint stars have been difficult to determine because a complete sample has been obtained only within 10 pc of the Sun. Figures 8.13 and 8.15 show that the white dwarfs occupy a narrow sliver of the H-R diagram that is roughly parallel to and below the main sequence. 1Reca11 the discussion of pressure broadening in Section 9.4. N 580 Chapter 15 The Degenerate Remnants of Stars Although white dwarfs are typically whiter than normal stars, the name itself is something of a misnomer since they come in all colors, with surface tempera-tures ranging from less than 5000 K to more than 80,000 K. Their spectral type, D (for "dwarf"), has several subdivisions. The largest group (about two-thirds of the total number, including Sirius B), called DA white dwarfs, display only pressure-broadened hydrogen absorption lines in their spectra. Hydrogen lines are absent from the DB white dwarfs (8%), which show only helium absorption lines, and the DC white dwarfs (14%) show no lines at all--only a continuum devoid of features. It is instructive to estimate the conditions at the center of a white dwarf of mass MWd and radius RWd, using the values for Sirius B given in the preceding section. Equation (14.3) with r = 0 shows that the central pressure is roughly2 P, ti 3 7rGp2R2 d ~ 3.8 x 1023 dyne cm-2, about 1.5 million times larger than the pressure at the center of the Sun. A crude estimate of the central temperature may be obtained from Eq. (10.61) for the radiative temperature gradient 3  or Assuming that the surface temperature, TWd, is much smaller than the central temperature and using -9 = 0.2 cm 2 g-1 for electron scattering [Eq. (9.21) with X = 0] gives _ 3Tp LWd 1~ T~ ~ [ 4ac 47rRWd ] ^ 7.6 x 10 K. Thus the central temperature of a white dwarf is several times 107 K. These estimated values for a white dwarf lead directly to a surprising con-clusion. Although hydrogen makes up roughly 70% of the visible mass of the universe, it cannot be present in appreciable amounts below the surface layers of a white dwarf. Otherwise, the dependence of the nuclear energy genera-tion rates on density and temperature [see Eq. (10.49) for the pp chain and 2 Remember that Eq. (14.3) was obtained for the unrealistic assumption of constant density. 3 As will be discussed later in Section 15.5, the assumption of a radiative temperature gradient is incorrect because the energy is actually carried outward by electron conduction. However, Eq. (10.61) is sufficient for the purpose of this estimation. dT _ 3 Tp L,. dr 4ac T3 41rr2 TWd - Tc _ 3 -kp LWa RWd - 0 4ac TWd 4~RWd  15.2 White Dwarfs 581 3 2 B7 ' ; B8 : B9 ~ AO_   -3 40,000 30,000 20,000 10,000 Te ~K) Figure 15.3 DA white dwarfs on an H-R diagram. A line marks the location of the 0.50 Mo white dwarfs, and a portion of the main sequence is at the upper right. (Data from Bergeron, Saffer, and Liebert, Ap. J., 39l, 228, 1992.) Eq. (10.53) for the CNO cycle] would produce white dwarf luminosities several orders of magnitude larger than those actually observed. Similar reasoning ` applied to other reaction sequences implies that thermonuclear reactions are not involved in producing the energy radiated by white dwarfs and that their centers must therefore consist of particles that are incapable of fusion at these densities and temperatures. As was discussed in Section 13.2, white dwarfs are manufactured in the cores of low and intermediate-mass stars (those with an initial mass below 8 or 9 Mo on the main sequence) near the end of their lives on the asymptotic giant branch of the H-R diagram. Because any star with a helium core mass exceeding about 0.5 Mo will undergo fusion, most white dwarfs consist pri-marily of completely ionized carbon and oxygen nuclei.4 As the aging giant expels its surface layers as a planetary nebula, the core is exposed as a white dwarf progenitor. The distribution of DA white dwarf masses is sharply peaked at 0.56 Mp, with some 80 percent lying between 0.42 Mo and 0.70 Mo; see Fig. 15.3. The much larger main-sequence masses quoted earlier imply high mass-loss rates while on the asymptotic giant branch, involving thermal pulses and perhaps a superwind phase as well. 4Low-mass helium white dwarfs may also exist, and rare oxygen-neon-magnesium white dwarfs have been detected in a few novae. 582 Chapter 15 The Degenerate Remnants of Stars The exceptionally strong pull of the white dwarf's gravity is responsible for the characteristic hydrogen spectrum of DA white dwarfs. Heavier nuclei are pulled below as the lighter hydrogen rises to the surface, resulting in a thin outer layer of hydrogen covering a layer of helium on top of the carbon-oxygen core.' This vertical stratification of nuclei according to their mass takes only 100 years or so in the hot atmosphere of the star. The origin of the non-DA (e.g., DB and DC) white dwarfs is not yet clear. Efficient mass-loss may occur on the asymptotic giant branch associated with the thermal pulse or superwind phases, stripping the white dwarf of nearly all of its hydrogen. Alternately, a single white dwarf may be transformed between the DA and non-DA spectral types by convective mixing in its surface layers .6 For example, the helium convection zone's penetration into a thin hydrogen layer above could change a DA into a DB white dwarf. White dwarfs with surface temperatures of Te ;zz~ 12, 000 K lie within the instability strip of the H-R diagram and pulsate with periods between 100 and 1000 s; see Fig. 8.15 and Table 14.1. These ZZ Ceti variables, named after the prototype discovered in 1968 by Arlo Landolt, are variable DA white dwarfs; hence they are also known as DAV stars. The pulsation periods correspond to nonradial g-modes that resonate within the white dwarf's surface layers of hydrogen and helium.7 Because these g-modes involve almost perfectly horizontal displacements, the radii of these compact pulsators hardly change. Their brightness variations (typically a few tenths of a magnitude) are due to temperature variations on the stars' surfaces. Since most stars will end their lives as white dwarfs, these must be the most common type of variable star in the universe, although only about thirty have been detected. Successful numerical calculations of pulsating white dwarf models were car-ried out by American astronomer Don Winget and others. They were able to demonstrate that it is the hydrogen partial ionization zone that is responsible for driving the oscillations of the ZZ Ceti stars, as mentioned in Section 14.2. These computations also confirmed the elemental stratification of white dwarf envelopes. Winget and his colleagues went on to predict that hotter DB white dwarfs should also exhibit g-mode oscillations driven by the helium partial ion-ization zone. Within a year's time, this prediction was confirmed when the first S Estimates of the relative masses of the hydrogen and helium layers range from m(H)/m(He) ~ 10-2 to 10-11 for DA white dwarfs. 6 As will be seen in Section 15.5, steep temperature gradients produce convection zones in the white dwarf's surface layers. 7 The nonradial pulsation of stars was discussed in Section 14.4. Unlike the g-modes of normal stars, shown in Fig. 14.16, the g-modes of white dwarfs are confined to their surface layers. 15.4 The Chandrasekhar Limit 591 _ N   14  2F  0.0 0:2 0.4 0.6 0.8 1.0 1.2 1.4 Mass (MlMo) Figure 15.7 Radii of white dwarfs of M~d < Mch at T = 0 K. It is important to emphasize that neither the nonrelativistic or relativis-tic formulas for the electron degeneracy pressure [Eqs. (15.11~ and (15.14~, respectively] contains the temperature. Unlike the gas pressure of the ideal gas law and the expression for radiation pressure, the pressure of a completely degenerate electron gas is independent of its temperature. This has the effect of decoupling the mechanical structure of the star from its thermal properties. We have already seen one implication of this decoupling in Section 13.2, where the helium core flash was described as the result of the independence of the mechanical and thermal behavior of the degenerate helium core of a low-mass star. When helium burning begins in the core, it proceeds with-out an accompanying increase in pressure that would normally expand the core and therefore restrain the rising temperature. The resulting rapid rise in temperature leads to a runaway production of nuclear energy-the helium flash-which lasts until the temperature becomes sufficiently high to remove the degeneracy of the core, allowing it to expand. On the other hand, a star may have so little mass that its core temperature never becomes high enough to initiate the helium burning. The result in this case is the formation of a helium white dwarf. adding just a bit more mass to white dwarf with very nearly 1.44 Mo. This will be considered in Section 17.4, where Type I supernovae are discussed. A Type I supernova occurs when a white dwarf pulls gas from a giant companion star in a binary system. 592 Chapter 15 The Degenerate Remnants of Stars 15.5 The Cooling of White Dwarfs Most stars end their lives as white dwarfs. These glowing embers scattered throughout space are a galaxy's memory of its past glory. Because no fusion occurs in their interiors, white dwarfs simply cool off at an essentially constant radius as they slowly deplete their supply of thermal energy (recall Fig. 15.3). Much effort has been directed at understanding the rate at which a white dwarf cools so its lifetime and the time of its birth may be calculated. Just as pale-ontologists can read the history of Earth's life in the fossil record, astronomers may be able to recover the history of star formation in our Galaxy by studying the statistics of white-dwarf temperatures. This section will be devoted to a discussion of the principles involved in this stellar archaeology. First we must ask how energy is transported outward from the interior of a white dwarf. In an ordinary star, photons travel much farther than atoms do before suffering a collision that robs them of energy (recall Examples 9.1 and 9.2). As a result, photons are normally a more efficient carrier of energy to the stellar surface. In a white dwarf, however, the degenerate electrons can travel long distances before losing energy in a collision with a nucleus, since the vast majority of the lower-energy electron states are already occupied. Thus, in a white dwarf, energy is carried by electron conduction rather than by radiation. This is so efficient that the interior of a white dwarf is nearly isothermal, with the temperature dropping significantly only in the nondegenerate surface layers. Figure 15.8 shows that a white dwarf consists of a nearly constant-temperature interior surrounded by a thin nondegenerate envelope that transfers heat less efficiently, allowing the energy to leak out slowly. The steep temperature gradient near the surface creates convection zones that may alter the appearance of the white dwarf's spectrum as it cools (as described in Section 15.2). The structure of the (nondegenerate) surface layers of a star is described at the beginning of Appendix H. For a white dwarf of surface luminosity Lwd and mass Mwd, Eq. (H.1) for the pressure P as a function of the temperature T in the envelope is" P - 4 167ac GMwd k _ l / (17 3 Lwd ro%~mx l where Ko [called "A" in Eq. (H.1)] is the coefficient of the bound-free Kramers 14 Equation (15.15) assumes that the envelope is in radiative equilibrium, with the energy carried outward by photons. Even when convection occurs in the surface layers of a white dwarf, it is not expected to have a large effect on the cooling. Chapte~ 16 BLACK HOLES 16.1 The General Theory of Relativity Gravity, the weakest of the four forces of nature, is nonetheless the dominant force in sculpting the universe on the largest scale. Newton's law of universal gravitation, F = GM2 , (16.1) r remained an unquestioned cornerstone of astronomers' understanding of heav-enly motion until the beginning of the twentieth century. Its application had explained the motions of the known planets and had accurately predicted the existence and position of the planet Neptune in 1846. The sole blemish on New-tonian gravitation was the inexplicably large rate of shift in the orientation of Mercury's orbit. The gravitational influences of the other planets cause the major axis of Mercury's elliptical orbit to slowly swing around the Sun in a counterclockwise direction relative to the fixed stars; see Fig. 16.1. The angular position at which perihelion occurs shifts at a rate of 574~~ per century.l However, Newton's law of gravity was unable to explain 43" per century of this shift, an inconsistency that led some mid-nineteenth century physicists to suggest that Eq. (16.1) should be modified from an exact inverse-square law. Others thought that an unseen planet, nicknamed Vulcan, might occupy an orbit inside Mercury's. Between the years 1907 and 1915 Albert Einstein developed a new theory of gravity, his general theory of relativity. In addition to resolving the 1The value of 1.5 per century encountered in some texts includes the very large effect of the precession of Earth's rotation axis on tl~lestial coordinate system, described in Section 1.3. 633 634 Chapter 16 Black Holes  Figure 16.1 Perihelion shift of Mercury's orbit; the shift per orbit has been exaggerated by a factor of 105. mystery of Mercury's orbit, it predicted many new phenomena that were later confirmed by experiment. In this and the next section we will describe just enough of the physical content of general relativity to provide the background needed for future discussions of black holes and cosmology. Einstein's view of the universe provides an exhilarating challenge to the imaginations of all students of astrophysics. Before embarking on our study of general relativity, it will be helpful to take an advanced look at this new gravitational landscape. The general theory of relativity is fundamentally a geometric description of how distances (intervals) in spacetime are measured in the presence of mass. For the moment, the effects on space and time will be considered separately, although the reader should always keep in mind that relativity deals with a unified spacetime. Near a massive object, both space and time must be described in a new way. Distances between points in the space surrounding a massive object are altered in a way that can be interpreted as space becoming curved through a fourth spatial dimension perpendicular to all of the usual three directions. The human mind balks at picturing this situation, but an analogy is easily found. Imagine four people holding the corners of a rubber sheet, stretching it tight and flat. This represents the flatness of empty space that exists in the absence of mass. Also imagine that a polar coordinate system has been painted on the sheet, with evenly spaced concentric circles spreading out from its center. Now lay a heavy bowling ball (representing the Sun) at the center of the sheet, and watch the indentation of the sheet as it curves down and stretches in response 16.3 Black Holes 667 horizon of a nonrotating black hole, all worldlines converge at the singularity. Even photons are pulled in toward the center. This means that the astronomer never has an opportunity to glimpse the singularity because no photons can reach her from there. She can, however, see the light that falls in behind her from events in the outside universe, but she does not see the entire history of the universe as it unfolds. Although the elapsed coordinate time in the outside world does become infinite, the light from all of these events does not have time to reach the astronomer. Instead, these events occur in her "elsewhere." Just 6.6 x 10-5 s of proper time after passing the event horizon, she is inexorably drawn to the singularity.21 Black holes may be formed in several ways. The collapse of the center of a sufficiently massive supergiant star may result in the formation of a central black hole. A neutron star in a close binary system may gravitationally strip enough mass from its companion that the neutron star's self-gravity exceeds the ability of the degeneracy pressure to support it, again resulting in a black hole. Black holes may also have been manufactured in the earliest instants of the universe. Presumably, these primordial black holes would have been formed with a wide range of masses, both much greater (_ 105 M(D) and much less (_ 10-5 g) than the lower limit of 3 Mp for black holes formed by stellar collapse. The only criterion for a black hole is that its entire mass must lie within the Schwarzschild radius, so the Schwarzschild metric is valid at the event horizon. Example 16.3 If Earth could somehow (miraculously) be compressed suffi-ciently to become a black hole, its radius would only be RS = 2GM/c2 = 0.887 cm. Although a primordial black hole could be this size, it is almost impossible to imagine packing Earth's entire mass into so small a ball. In re-ality, the cubic centimeter of material at Earth's center contains only 13 g, so nothing special happens 0.887 cm from the center.22 Whatever the formation process, it is certain to be very complicated. The collapse of a star will rarely be symmetrical. Detailed calculations have demon-strated, however, that any irregularities are radiated away by gravitational waves (see Section 17.5). As a result, once the surface of the collapsing star reaches the event horizon, the exterior spacetime horizon is spherically sym-metric and described by the Schwarzschild metric. Z1A thorough description of the final view of the falling astronomer may be found in Rothman et al. (1985). 22 The reader is reminded that the SchwarzscW_ metric is only valid outside matter. It does not describe the spacetime inside Earth. 668 Chapter 16 Black Holes Another complication is the fact that all stars rotate, and therefore so will the resulting black hole. Remarkably, however, any black hole can be completely described by just three numbers: its mass, angular momentum, and electric charge.23 Black holes have no other attributes or adornments, a condition commonly expressed by saying that "a black hole has no hair." 24 There is a firm upper limit for a rotating black hole's angular momentum. The maximum value of the angular momentum for a black hole of mass M is Lmax = GM2 (16.25) C If the angular momentum of a rotating black hole were to exceed this limit, there would be no event horizon and a naked singularity would appear, in violation of the Law of Cosmic Censorship. Example 16.4 The maximum angular momentum for a solar-mass black hole is 2 L,r,a,x - GMo = 8.81 x 1048 g cm2 s-1. c By comparison, the angular momentum of the Sun (assuming uniform rotation) is 1.63 x 1048 g cm 2 s-1, about 18% of Ln,ax. We should expect that many stars will have angular momenta that are comparable to L,Y,a.,, and so vigorous (if not maximal) rotation ought to be common for stellar-mass black holes. The structure of a maximally rotating black hole is shown in Fig. 16.22.25 The rotation has distorted the central singularity from a point into a flat ring, and the event horizon has assumed the shape of an ellipsoid. The figure also shows additional features caused by the rotation. As a massive object spins, it induces a rotation in the surrounding spacetime, a phenomenon known as frame dragging. To gain some insight into this effect, recall the behavior of a pendulum swinging at the north pole of Earth. As Earth rotates, the plane of the pendulum's swing remains fixed with respect to the distant stars. The stars define a nonrotating frame of reference for the universe, and it is relative to this frame that the pendulum's swing remains planar. However, the rotating 23If magnetic monopoles exist, the "magnetic charge" would also be required for a complete specification. However, both magnetic and electric charge can be safely ignored because stars should be very nearly neutral. 24 The "no hair" theorem actually applies only to the universe outside the event horizon. Inside, the spacetime geometry is complicated by the mass distribution of the collapsed star. 25The Kerr metric for a rotating black hole was derived from Einstein's field equations by a New Zealand mathematician, Roy Kerr, in 1963. 16.3 Black Holes 667   horizon of a nonrotating black hole, all worldlines converge at the singularity. Even photons are pulled in toward the center. This means that the astronomer never has an opportunity to glimpse the singularity because no photons can reach her from there. She can, however, see the light that falls in behind her from events in the outside universe, but she does not see the entire history of the universe as it unfolds. Although the elapsed coordinate time in the outside world does become infinite, the light from all of these events does not have time to reach the astronomer. Instead, these events occur in her "elsewhere." Just 6.6 x 10-5 s of proper time after passing the event horizon, she is inexorably drawn to the singularity.21 Black holes may be formed in several ways. The collapse of the center of a sufficiently massive supergiant star may result in the formation of a central black hole. A neutron star in a close binary system may gravitationally strip enough mass from its companion that the neutron star's self-gravity exceeds the ability of the degeneracy pressure to support it, again resulting in a black hole. Black holes may also have been manufactured in the earliest instants of the universe. Presumably, these primordial black holes would have been formed with a wide range of masses, both much greater (~ 105 Mo) and much less (~ 10-5 g) than the lower limit of 3 Mo for black holes formed by stellar collapse. The only criterion for a black hole is that its entire mass must lie within the Schwarzschild radius, so the Schwarzschild metric is valid at the event horizon. Example 16.3 If Earth could somehow (miraculously) be compressed suffi-ciently to become a black hole, its radius would only be RS = 2GM/c2 = 0.887 cm. Although a primordial black hole could be this size, it is almost impossible to imagine packing Earth's entire mass into so small a ball. In re-ality, the cubic centimeter of material at Earth's center contains only 13 g, so nothing special happens 0.887 cm from the center.22 Whatever the formation process, it is certain to be very complicated. The collapse of a star will rarely be symmetrical. Detailed calculations have demon-strated, however, that any irregularities are radiated away by gravitational waves (see Section 17.5). As a result, once the surface of the collapsing star reaches the event horizon, the exterior spacetime horizon is spherically sym-metric and described by the Schwarzschild metric. Z1A thorough description of the final view of the falling astronomer may be found in Rothman et al. (1985). 22The reader is reminded that the Schwarzschild metric is only valid outside matter. It does not describe the spacetime inside Earth. 16.3 Black Holes _ 669  Figure 16.22 The structure of a maximally rotating black hole, with the ring singularity seen edge-on. The location of the event horizon at the equator is r = 2 RS = GM/c2. spacetime close to a massive spinning object produces a local deviation from the nonrotating frame that describes the universe at large. Near a rotating black hole, frame dragging is so severe that there is a nonspherical region outside the event horizon called the ergosphere where any particle must move in the same direction that the black hole rotates; see Fig. 16.22. Spacetime within the ergosphere is rotating so rapidly that a particle would have to travel faster than the speed of light to remain at the same angular coordinate (e.g., at the same value of 0 in the coordinate system used by a distant observer). The outer boundary of the ergosphere is called the static limit, so named because once beyond this boundary a particle can remain at the same coordinate as the effect of frame dragging diminishes. Even Earth's rotation produces very weak frame dragging. Scheduled to be launched into a polar orbit in 1999, the Stanford Gravity Probe B experiment will attempt to detect Earth's frame dragging by measuring the precession of four gyroscopes made of precisely shaped spheres of fused quartz 3.8 cm in diameter. Although the expected precession rate is only 0.042" yr-1, the effect is cumulative. By comparing the changes that occur in the gyroscopes' different initial orientations, the frame dragging should be measurable. At this point, the reader should be warned that the previous descriptions of a black hole's structure inside the event horizon, such as Fig. 16.22, are based on vacuum solutions to Einstein's field equations. These solutions were obtained by ignoring the effects of the mass of the collapsing star, so the vac-uum. solutions do not describe the interior of a real black hole. Furthermore, 670 Chapter 16 Black Holes  \:.-~---~-_~ \ , ,~ i    Figure 16.23 Depiction of a Schwarzschild throat connecting two different regions of spacetime. Any attempted passage of matter or energy through the throat would cause it to collapse. the present laws of physics, including general relativity, break down under the extreme conditions found very near the center. The details of the singularity cannot be fully described until a theory of quantum gravity is found. The pres-ence of a singularity is assured, however. In 1965 an English mathematician, Roger Penrose, proved conclusively that every complete gravitational collapse must form a singularity. The possibility of using a black hole as a tunnel connecting one location in spacetime with another (perhaps in a different universe) has inspired both physicists and science fiction writers. Most conjectures of spacetime tunnels are based on vacuum solutions to Einstein's field equations and as such don't apply to the interiors of real black holes. Still, they have become part of the popular culture and we will consider them briefly here. Figure 16.23 depicts a spacetime tunnel called a Schwarzschild throat (also known as an Einstein-Rosen bridge), which uses the Schwarzschild geometry of a nonrotating black hole to connect two regions of spacetime. The width of the throat is a minimum at the event horizon, and the "mouths" may be interpreted as opening onto two different locations in spacetime. It is tempting to imagine this as a tunnel, and writers of speculative fiction have dreamed of white holes pouring out mass or serving as passageways for starships and cute Disney robots. However, it appears that any attempt to send a tiny amount of matter or energy (even a stray photon) through the throat would cause it to collapse. For a real nonrotating black hole, all worldlines end at the inescapable singularity, where Chapter 17 CLOSE BINARY STAR SYSTEMS 17.1 Gravity in a Close Binary Star System As explained in Chapter 7, at least half of all "stars" in the sky are actually multiple systems, consisting of two (or more) stars in orbit about their common center of mass. In most of these systems the stars are sufficiently far apart that they have a negligible impact on one another. They evolve essentially independently, living out their lives in isolation except for the gentle grip of gravity that binds them together. If the stars are very close, with a separation roughly equal to the diameter of the larger star, then one or both stars may have their outer layers gravita-tionally deformed into a teardrop shape. As a star rotates through the tidal bulge raised by its partner's gravitational pull, it is forced to pulsate. These oscillations are damped by the mechanisms discussed in Section 14.2. Orbital and rotational energy are dissipated in this way until the system reaches the state of minimum energy for its (constant) angular momentum, resulting in synchronous rotation and circular orbits. Thereafter the same side of each star always faces the other as the system rotates rigidly in space and no fur-ther energy can be lost by tidally driven oscillations.' The distorted star may even lose some of its photospheric gases to its companion. The spilling of gas from one star onto another can lead to some spectacular celestial fireworks, the subject of this chapter. To understand how gravity operates in a close binary star system, consider 'If one of the stars is a compact object such:~ ,,, , white dwarf or a neutron star, its spin may not be synchronized. ,- .- 683 684 Chapter 17 Close Binary Star Systems two stars in a circular orbit in the x-y plane with angular velocity w = vl/rl = V2/r2. Here, vi and rl are the orbital speed of star 1 and its distance from the center of mass of the system, and similarly for star 2. It is useful to choose a corotating coordinate system that follows the rotation of the two stars about their center of mass. If the center of mass is at the origin, then the stars will be at rest in this rotating reference frame, with their mutual gravitational attraction balanced by the outward "push" of a centrifugal force. 2 The centrifugal force vector on a mass m in this frame a distance r from the origin is then Fc = mw 2r "r, (17.1) in the outward radial direction. It is usually easier to work with the gravitational potential energy, given by Eq. (2.14), U9=-G Mm , r instead of the gravitational force. 3 To do this in a rotating coordinate system, a fictitious "centrifugal potential energy" must be included in the potential energy through the use of Eq. (2.13), rf Uf-U2=0Uc=- f F, - dr. Here, F, is the centrifugal force vector, ri and r f are the initial and final position vectors, respectively, and dr is the infinitesimal change in the position vector (recall Fig. 2.9). The change in centrifugal potential energy is thus  AU,=-Jrf mw2rdr=-2mw2(rf-r?). rs Realizing that only changes in potential energy are physically meaningful, U, _ 0 at r = 0 can arbitrarily be chosen to give the final result for the centrifugal potential energy, Uc _ -~~2r2, (17.2) Figure 17.1 shows a corotating coordinate system in which two stars with masses Ml and M2 are separated by a distance a. The stars are located on 1 2 The centrifugal force is an inertial force (as opposed to a physical force) that must be included when describing motion in a rotating coordinate system. There is another inertial force, called the Coriolis force, that will be neglected in what follows. 3Most stars can be treated as point masses in what follows because the mass is concen-trated at their centers, allowing their teardrop shapes to be neglected.  Appendix A ASTRONOMICAL AND PHYSICAL CONSTANTS Astronomical Constants  Solar mass 1 Mp = 1.989 x 1033 g Solar luminosity 1 Lo = 3.826 x 1033 ergs s-1 Solar radius 1 Ro = 6.9599 x 101 cm Solar effective temperature To = 5770 K Earth mass 1 M = 5.974 x 1027 g Earth radius 1 R = 6.378 x 10$ cm Light year 1 ly = 9.4605 x 1017 cm Parsec 1 pc = 3.0857 x 101g cm = 3.2616 ly Astronomical unit 1 AU = 1.4960 x 1013 cm Sidereal day = 23h 56I" 04.09054s Solar day = 86400 s Sidereal year = 3.155815 x 107 s 'Iropical year = 3.155693 x 107 s A-2 Appendix A Astronomical and Physical Constants  Gravitational constant G = 6.67259 x 10-8 dyne cm 2 g-2 Speed of light (exact) c = 2.99792458 x 101 cm s-1 Planck's constant h = 6.6260755 x 10-27 erg s h - h/27r = 1.05457266 x 10-27 erg s Boltzmann's constant k = 1.380658 x 10-16 erg K-1 Stefan-Boltzmann constant Q = 5.67051 x 10-5 erg cm-2 s-1 K-4 Radiation constant a = 4Q/c = 7.56591 x 10-15 erg cm-3 K-4 Proton mass MP = 1.6726231 x 10-24 g Neutron mass m,, = 1.674929 x 10-24 g Electron mass me = 9.1093897 x 10-2$ g Hydrogen mass mH = 1.673534 x 10-z4 g Atomic mass unit 1 u = 1.6605402 x 10-24 g = 931.49432 MeV/c2 Coulomb law constant (cgs) kc - 1 (SI) = 8.9875518 x 109 N m2 C-2 Electric charge (cgs) e = 4.803206 x 10-1 esu (SI) = 1.60217733 x 10-19 C Electron volt 1 eV = 1.60217733 x 10-12 erg Avagadro's number NA = 6.0221367 x 1023 mole-1 Gas constant R = 8.314510 x 107 ergs mole-1 K-1 Bohr radius ao = h2/m,e2 = 5.29177249 x 10-9 cm Rydberg constant RH = Pe4/47rh3c = 1.09677585 x 105 cm-1 Suggested Readings TECHNICAL Cohen, E. Richard, and Taylor, Barry N., "The 1986 Adjustment of the Fun-damental Physical Constants," Reviews of Modern Physics, 59, 1121, 1987. Lang, Kenneth R., Astrophysical Data: Planets and Stars, Springer-Verlag, New York, 1992.  Physical Constants  Appendix B SOLAR SYSTEM DATA Planetary Physical Data Equatorial Average Sidereal Mass Radius Density Rotation Planet (M) (R) (g cm-3) Period (d) Oblateness Albedo Mercury 0.0553 0.382 5.43 58.65 0.0 0.06 Venus 0.8150 0.949 5.25 243.01 0.0 0.77 Earth 1.0000 1.000 5.52 0.997 0.0034 0.30 Mars 0.1074 0.533 3.93 1.026 0.0052 0.15 Jupiter 317.894 11.19 1.33 0.414 0.0648 0.51 Saturn 95.184 9.46 0.71 0.444 0.1076 0.50 Uranus 14.537 4.01 1.24 0.718 0.030 0.66 Neptune 17.132 3.81 1.67 0.671 0.022 0.62 Pluto 0.0022 0.182 2.1 6.387 0.0 0.6 Planetary Orbital Data Sidereal Orbital Equatorial Semimajor Orbital Orbital Inclination Inclination Planet Axis (AU) Eccentricity Period (yr) to Ecliptic () to Orbit () Mercury 0.3871 0.2056 0.2408 7.004 7.0 Venus 0.7233 0.0068 0.6152 3.394 177.4 Earth 1.0000 0.0167 1.0000 0.000 23.45 Mars 1.5237 0.0934 1.8809 1.850 23.98 Jupiter 5.2028 0.0483 11.8622 1.308 3.08 Saturn 9.5388 0.0560 29.4577 2.488' 26.73 Uranus 19.1914 0.0461 84.0139 0.774 97.92 Neptune 30.0611 0.0097 164.793 1.774 28.8 Pluto 39.5294 0.2482 248.54 17.148 122.46 A-4 Appendix B Solar System Data Data of Selected Major Satellites Orbital Orbital Parent Mass Radius Density Period Distance Satellite Planet (1025 g) (103 km) (g cm-3) (d) (103 km) Moon Earth 7.35 1.738 3.34 27.322 384.4 10 Jupiter 8.92 1.815 3.55 1.769 421.6 Europa Jupiter 4.87 1.569 3.04 3.551 670.9 Ganymede Jupiter 14.9 2.631 1.93 7.155 1070 Callisto Jupiter 10.8 2.400 1.83 16.689 1880 Titan Saturn 13.5 2.575 1.88 15.945 1222 Triton Neptune 2.14 1.355 2.05 5.877 354.8 Appendix C THE CONSTELLATIONS  Latin Name R. A. Dec. 'Ranslation Genitive Abbrev. h Andromeda Andromedae And 1 +40 Princess of Ethiopia Antlia Antliae Ant 10 -35 Air Pump Apus Apodis Aps 16 -75 Bird of Paradise Aquarius Aquarii Aqr 23 -15 Water Bearer Aquila Aquilae Aql 20 + 5 Eagle Ara Arae Ara 17 -55 Altar Aries Arietis Ari 3 +20 Ram Auriga Aurigae Aur 6 +40 Charioteer Bootes Bootis Boo 15 +30 Herdsman Caelum Caeli Cae 5 -40 Chisel Camelopardalis Camelopardis Cam 6 +70 Giraffe Cancer Cancri Cnc 9 +20 Crab A-5 A-6 Appendix C The Constellations Latin Name R. A. Dec. Translation Genitive Abbrev. h Canes Venatici Canum Venaticorum CVn 13 +40 Hunting Dogs Canis Major Canis Majoris CMa 7 -20 Big Dog Canis Minor Canis Minoris CMi 8 + 5 Little Dog Capricornus Capricorni Cap 21 -20 Goat Carina Carinae Car 9 -60 Ship's Keel Cassiopeia Cassiopeiae Cas 1 +60 Queen of Ethiopia Centaurus Centauri Cen 13 -50 Centaur Cepheus Cephei Cep 22 +70 King of Ethiopia Cetus Ceti . Cet 2 -10 Sea Monster (whale) Chamaeleon Chamaeleontis Cha 11 -80 Chameleon Circinus Circini Cir 15 -60 Compass Columba Columbae Col 6 -35 Dove Coma Berenices Comas Berenices Com 13 +20 Berenice's Hair Corona Australis Corona,e Australis CrA 19 -40 Southern Crown Corona Borealis Coronae Borealis CrB 16 +30 Northern Crown Corvus Corvi Crv 12 -20 C row Crater Crateris Crt 11 -15 Cup Crux Crucis Cru 12 -60 Southern Cross r r _ Appendix C The Constellations A-7 Latin Name R. A. Dec. Translation Genitive Abbrev. h Cygnus Swan Delphinus Dolphin, Porpoise Dorado Swordfish Draco Dragon Equuleus Little Horse Eridanus River Eridanus Fornax Furnace Gemini Twins Grus Crane Hercules Son of Zeus Horologium Clock Hydra Water Snake Hydrus Sea Serpent Indus Indian Lacerta Lizard Leo Lion Leo Minor Little Lion Lepus Hare Cygni Cyg 21 +40 Delphini Del 21 +10 Doradus Dor 5 -65 Dra,conis Dra 17 +65 Equulei Equ 21 +10 Eridani Eri 3 -20 Fornacis For 3 -30 Geminorum Gem 7 +20 Gruis Gru 22 -45 Herculis Her 17 +30 Horologii Hor 3 -60 Hydrae Hya 10 -20 Hydri Hyi 2 -75 Indi Ind 21 -55 Lacertae Lac 22 +45 Leonis Leo 11 +15 Leonis Minoris LMi 10 +35 Leporis Lep 6 -20 A_g Appendix C The Constellations Latin Name R. A. Dec. Translation Genitive Abbrev. h Libra Librae Lib 15 -15 Balance, Scales Lupus Lupi Lup 15 -45 Wolf Lynx Lyncis Lyn 8 +45 Lynx Lyra Lyrae Lyr 19 +40 Lyre, Harp Mensa Mensae Men 5 -80 Table, Mountain Microscopium Microscopii Mic 21 -35 Microscope Monoceros Monocerotis Mon 7 - 5 Unicorn Musca Muscae Mus 12 -70 Fly Norma Norma,e Nor 16 -50 Square, Level Octans Octantis . Oct 22 -85 Octant Ophiuchus Ophiuchi Oph 17 0 Serpent-bearer Orion Orionis Ori 5 + 5 Hunter Pavo Pavonis Pav 20 -65 Peacock Pegasus Pegasi Peg 22 +20 Winged Horse Perseus Persei Per 3 +45 Rescuer of Andromeda Phoenix Phoenicis Phe 1 -50 Phoenix Pictor Pictoris Pic 6 -55 Painter, Easel Pisces Piscium - Psc 1 +15 Fish Appendix C The Constellations A-9 Latin Name R. A. Dec. Translation Genitive Abbrev. h Piscis Austrinus Piscis Austrini PsA 22 -30 Southern Fish Puppis Puppis Pup 8 -40 Ship's Stern Pyxis Pyxidis Pyx 9 -30 Ship's Compass Reticulum Reticuli Ret 4 -60 Net Sagitta Sagittas Sge 20 +10 Arrow Sagittarius Sagittarii Sgr 19 -25 Archer Scorpius Scorpii Sco 17 -40 Scorpion Sculptor Sculptoris Scl 0 -30 Sculptor Scutum Scuti Sct 19 -10 Shield Serpens Serpentis Ser 17 0 Serpent Sextans Sextantis Sex 10 0 Sextant Taurus Tauri Tau 4 +15 Bull Telescopium Telescopii Tel 19 -50 Telescope Triangulum Trianguli Tri 2 +30 Triangle Triangulum Australe Trianguli Australis RA 16 -65 Southern Triangle Tucana Tucanae Tuc 0 -65 Toucan Ursa Major Ursae Majoris UMa 11 +50 Big Bear Ursa Minor Ursae Minoris UMi 15 +70 Little Bear A-10 Appendix C The Constellations Latin Name R. A. Dec. Translation Genitive Abbrev. h Vela Velorum Vel 9 -50 Ship's Sail Virgo Virginis Vir 13 0 Maiden, Virgin Volans Volantis Vol 8 -70 Flying Fish Vulpecula - Vulpeculae Vul 20 +25 Little Fox Appendix D THE BRIGHTEST STARS Spectral Class V M~ Name Star A B A B A B Sirius c~ CMa A1V wdb -1.46 + 8.7 +1.4 +11.6 Canopus a Car FOlb-II -0.72 -3.1 Rigel Kentaurus a Cen G2V KOV -0.01 + 1.3 +4.4 + 5.7 Arcturus a Boo K2IIIp -0.06 -0.3 Vega a Lyr AOV +0.04 +0.5 Capella a Aur GIII M1V +0.05 +10.2 -0.7 + 9.5 Rigel ,C3 Ori B8Ia B9 +0.14 + 6.6 -6.8 - 0.4 Procyon a CMi F5IV-V wdb +0.37 +10.7 +2.6 +13.0 Betelgeuse a Ori M2Iab +0.41v -5.5 Achernar a Eri B5V -~-0.51 -1.0 - Hadar ,l3 Cen B1III ? +0.63 + 4 -4.1 - 0.8 Altair a Aql A7IV-V +0.77 +2.2 Acrux a Cru B1IV B3 +1.39 + 1.9 -4.0 - 3.5 Aldebaran a Tau K5III M2V +0.86 +13 -0.2 +12 Spica a Vir B1V +0.91v -3.6 Antares a Sco MIIb B4eV +0.92v + 5.1 -4.5 - 0.3 Pollux ,Q Gem KOIII +1.16 +0.8 Fomalhaut a PsA A3V K4V +1.19 + 6.5 +2.0 + 7.3 Deneb a Cyg A2Ia +1.26 -6.9 Mimosa ,Q Cru B0.5IV +1.28v -4.6 a Values labeled v designate variable stars. b wd represents a white dwarf star. ~ Capella has a third member of spectral class M5V, V = -F13.7, and Mv = +13. A-12 Appendix D The Brightest Stars Proper Radial R. A.a Dec.' Distance Motion Velocity Name h m ' (PC) (" yr-1) (km s-1) Sirius 6 42.9 -16 39 2.6 1.33 - 7.6 Canopus 6 22.8 -52 40 30 0.02 +20.5 Rigel Kentaurus 14 36.2 -60 38 1.3 3.68 -24.6 Arcturus 14 13.4 +19 27 11 2.28 - 5.2 Vega 18 35.2 +38 44 8.0 0.34 -13.9 Capella 5 13.0 +45 57 14 0.44 +30.2 Rigel 5 12.1 -08 15 250 0.00 +20.7 Procyon 7 36.7 +05 21 3.5 1.25 - 3.2 Betelgeuse 5 52.5 +07 24 150 0.03 +21.0 Achernar 1 35.9 -57 29 20 0.10 +19 Hadar 14 00.3 -60 08 90 0.04 -12 Altair 19 48.3 +08 44 5.1 0.66 _ -26.3 Acrux 12 23.8 -62 49 120 0.04 -11.2 Aldebaran 4 33.0 +16 25 16 0.20 +54.1 Spica 13 22.6 -10 54 80 0.05 + 1.0 Antares 16 26.3 -26 19 120 0.03 - 3.2 Pollux 7 42.3 +28 09 12 0.62 + 3.3 Fomalhaut 22 54.9 -29 53 7.0 0.37 -+- 6.5 Deneb 20 39.7 +45 06 430 0.00 - 4.6 Mimosa 12 44.8 -59 24 150 0.05 a Right ascension and declination are given in epoch 1950.0. Suggested Readings TECHNICAL HofHeit, Dorrit, and Warren, Wayne H. Jr., The Bright Star Catalogue, Fifth Edition, Yale University Observatory, New Haven, 1991. Lang, Kenneth R., Astrophysical Data: Planets and Stars, Springer-Verlag, New York, 1992. Appendix E STELLAR DATA   Main-Sequence Stars (Luminosity Class V) Sp. Te Type (K) L/Lp R/Ro M/Mo Mbot BC Mv U - B B - V 05 44500 790000 15 60 -10.1 -4.40 -5.7 -1.19 -0.33 06 41000 420000 13 37 -9.4 -3.93 -5.5 -1.17 -0.33 07 38000 260000 - - -8.9 -3.68 -5.2 -1.15 -0.32 08 35800 170000 11 23 -8.4 -3.54 -4.9 -1.14 -0.32 09 33000 97000 - - -7.8 -3.33 -4.5 -1.12 -0.31 BO 30000 52000 8.4 17.5 -7.1 -3.16 -4.0 -1.08 -0.30 B1 25400 16000 - - -5.9 -2.70 -3.2 -0.95 -0.26 B2 22000 5700 - - -4.7 -2.35 -2.4 -0.84 -0.24 B3 18700 1900 4.2 7.6 -3.5 -1.94 -1.6 -0.71 -0.20 B5 15400 830 4.1 5.9 -2.7 -1.46 -1.2 -0.58 -0.17 B6 14000 500 - - -2.1 -1.21 -0.9 -0.50 -0.15 B7 13000 320 - - -1.6 -1.02 -0.6 -0.43 -0.13 B8 11900 180 3.2 3.8 -1.0 -0.80 -0.2 -0.34 -0.11 B9 10500 95 - - -0.3 -0.51 +0.2 -0.20 -0.07 AO 9520 54 2.7 2.9 +0.3 -0.30 +0.6 -0.02 -0.02 A1 9230 35 - - +0.8 -0.23 +1.0 +0.02 +0.01 A2 8970 26 - - +1.1 -0.20 +1.3 +0.05 +0.05 A3 8720 21 - - +1.3 -0.17 +1.5 +0.08 +0.08 A5 8200 14 1.9 2.0 +1.7 -0.15 +1.9 +0.10 +0.15 A7 7850 10.5 - - +2.1 -0.12 +2.2 +0.10 +0.20 A8 7580 8.6 - - +2.3 -0.10 +2.4 +0.09 +0.25 FO 7200 6.5 1.6 1.6 +2.6 -0.09 +2.7 +0.03 +0.30 F2 6890 3.2 - - +3.4 -0.11 +3.5 0.00 +0.35 F5 6440 2.9 1.4 1.4 +3.5 -0.14 +3.6 -0.02 +0.44 F8 6200 2.1 - - +3.8 -0.16 +4.0 +0.02 +0.52 A-13 A-14 Appendix E Stellar Data Main-Sequence Stars (Luminosity Class V) Sp. Te Type (K) L/Lo R/Ro M/ILio 1L1,,., BC Mv U - B B - V GO 6030 1.5 1.1 1.05 +4.2 -0.18 +4.4 +0.06 +0.58 G2 5860 1.1 - - +4.5 -0.20 +4.7 +0.12 +0.63 Sun' 5780 1.00 1.00 1.00 +4.64 -0.19 +4.83 +0.17 +0.68 Sun b 5770 1.00 1.00 1.00 +4.76 -0.07 +4.83 +0.16 +0.64 G5 5770 0.79 0.89 0.92 +4.9 -0.21 +5.1 +0.20 +0.68 G8 5570 0.66 - - +5.1 -0.40 +5.5 +0.30 +0.74 KO 5250 0.42 0.79 0.79 +5.6 -0.31 +5.9 +0.45 +0.81 Kl 5080 0.37 - - +5.7 -0.37 +6.1 +0.54 +0.86 K2 4900 0.29 - - +6.0 -0.42 +6.4 +0.64 +0.91 K3 4730 0.26 - - +6.1 -0.50 +6.6 +0.80 +0.96 K4 4590 0.19 - - +6.4 -0.55 +7.0 - +1.05 K5 4350 0.15 0.68 0.67 +6.7 -0.72 +7.4 +0.98 +1.15 K7 4060 0.10 - - +7.1 -1.01 +8.1 +1.21 +1.33 MO 3850 0.077 0.63 0.51 +7.4 -1.38 +8.8 +1.22 +1.40 Ml 3720 0.061 - - +7.7 -1.62 +9.3 +1.21 +1.46 M2 3580 0.045 0.55 0.40 +8.0 -1.89 +9.9 +1.18 +1.49 M3 3470 0.036 - - +8.2 -2.15 +10.4 +1.16 +1.51 M4 3370 0.019 - - +8.9 -2.38 +11.3 +1.15 +1.54 M5 3240 0.011 0.33 0.21 +9.6 -2.73 +12.3 +1.24 +1.64 M6 3050 0.0053 - - +10.3 -3.21 +13.5 +1.32 +1.73 M7 2940 0.0034 - - +10.8 -3.46 +14.3 +1.40 +1.80 M8 2640 0.0012 0.17 0.06` +11.9 -4.1 +16.0 +1.53 +1.93 'Values adopted by Schmidt-Kaler (1982). b Values adopted in this book. 'Schmidt-Kaler (1982) uses a value of 0.06 Me for the M8 star, which is below the lower limit of 0.085 Me for stable hydrogen burning on the main sequence. However, stars in the range of 0.06-0.08 Me may undergo an extended period (109-101 years) of hydrogen fusion before failing to stabilize as main-sequence stars; see Liebert and Probst (1987).  ,yYJr      Appendix E Stellar Data A-15 Giant Stars (Luminosity Class III) Sp. Te Type (K) L/Lo R/Ro M/Mo Mboi BC Mv U - B B - V 05 42500 990000 18 - -10.3 -4.05 -6.3 -1.18 -0.32 06 39500 650000 - - -9.9 -3.80 -6.1 -1.17 -0.32 07 37000 440000 - - -9.5 -3.58 -5.9 -1.14 -0.32 08 34700 340000 - - -9.2 -3.39 -5.8 -1.13 -0.31 09 32000 220000 - - -8.7 -3.13 -5.6 -1.12 -0.31 BO 29000 110000 13 20 -8.0 -2.88 -5.1 -1.08 -0.29 B1 24000 39000 - - -6.8 -2.43 -4.4 -0.97 -0.26 B2 20300 17000 - - -5.9 -2.02 -3.9 -0.91 -0.24 B3 17100 5000 - - -4.6 -1.60 -3.0 -0.74 -0.20 B5 15000 1800 6.3 7 -3.5 -1.30 -2.2 -0.58 -0.17 B6 14100 1100 - - -2.9 -1.13 -1.8 -0.51 -0.15 B7 13200 700 - - -2.5 -0.97 -1.5 -0.44 -0.13 BS 12400 460 - - -2.0 -0.82 -1.2 -0.37 -0.11 B9 11000 240 - - -1.3 -0.71 -0.6 -0.20 -0.07 AO 10100 106 3.4 4 -0.4 -0.42 +0.0 -0.07 -0.03 A1 9480 78 - - -0.1 -0.29 +0.2 +0.07 +0.01 A2 9000 65 - - +0.1 -0.20 +0.3 +0.06 +0.05 A3 8600 53 - - -f-0.3 -0.17 +0.5 +0.10 +0.08 A5 8100 43 3.3 - +0.6 -0.14 +0.7 +0.11 +0.15 A7 7650 29 - - +1.0 -0.10 +1.1 +0.11 +0.22 A8 7450 26 - - +1.1 -0.10 +1.2 +0.10 +0.25 FO 7150 20 2.9 - +1.4 -0.11 +1.5 +0.08 +0.30 F2 6870 17 - - +1.6 -0.11 +1.7 +0.08 +0.35 F5 6470 17 3.3 - +1.6 -0.14 +1.6 +0.09 +0.43 F8 6150 - - - - -0.16 - +0.10 +0.54 GO 5850 34 5.7 1.0 +0.8 -0.20 +1.0 +0.21 +0.65 G2 5450 40 - - +0.6 -0.27 +0.9 +0.39 +0.77 G5 5150 43 8.3 1.1 +0.6 -0.34 +0.9 +0.56 +0.86 G8 4900 51 - - +0.4 -0.42 +0.8 +0.70 +0.94 KO 4750 60 11 1.1 +0.2 -0.50 +0.7 +0.84 +1.00 K1 4600 69 - - +0.1 -0.55 +0.6 +1.01 +1.07 K2 4420 79 - - -0.1 -0.61 +0.5 +1.16 +1.16 K3 4200 110 - - -0.5 -0.76 +0.3 +1.39 +1.27 K4 4000 170 - - -0.9 -0.94 0.0 - +1.38 K5 3950 220 32 1.2 -1.2 -1.02 -0.2 +1.81 +1.50 K? 3850 280 - - -1.5 -1.17 -0.3 +1.83 +1.53  A-16 Appendix E Stellar Data Giant Stars (Luminosity Class III) Sp. Te Type (K) L/Lo R/Ro M/Mo Mbol BC M~ U - B B - V MO 3800 330 42 1.2 -1.6 -1.25 -0.4 +1.87 +1.56 Ml 3720 430 - - -1.9 -1.44 -0.5 +1.88 +1.58 M2 3620 550 60 1.3 -2.2 -1.62 -0.6 +1.89 +1.60 M3 3530 700 - - -2.5 -1.87 -0.6 +1.88 +1.61 M4 3430 880 - - -2.7 -2.22 -0.5 +1.73 +1.62 M5 3330 930 92 - -2.8 -2.48 -0.3 +1.58 +1.63 M6 3240 1070 - - -2.9 -2.73 -0.2 +1.16 +1.52 Appendix E Stellar Data A-1'7 Supergiant Stars (Luminosity Class Approximately lab) Sp. Te Type (K) L/Lo R/Ro M/Mo Mbol BC Mv U - B B - V 05 40300 1100000 22 70 -10.5 -3.87 -6.6 -1.17 -0.31 06 39000 900000 21 40 -10.2 -3.74 -6.5 -1.16 -0.31 07 35700 710000 - - -10.0 -3.48 -6.5 -1.14 -0.31 08 34200 620000 22 28 -9.8 -3.35 -6.5 -1.13 -0.29 09 32600 530000 - - -9.7 -3.18 -6.5 -1.13 -0.27 BO 26000 260000 25 25 -8.9 -2.49 -6.4 -1.06 -0.23 B1 20800 150000 - - -8.3 -1.87 -6.4 -1.00 -0.19 B2 18500 110000 - - -8.0 -1.58 -6.4 -0.94 -0.17 B3 16200 76000 - - -7.6 -1.26 -6.3 -0.83 -0.13 B5 13600 52000 41 20 -7.2 -0.95 -6.2 -0.72 -0.10 B6 13000 49000 - - -7.1 -0.88 -6.2 -0.69 -0.08 B7 12200 44000 - - -7.0 -0.78 -6.2 -0.64 -0.05 B8 11200 40000 - - -6.9 -0.66 -6.2 -0.56 -0.03 B9 10300 35000 - - -6.7 -0.52 -6.2 -0.50 -0.02 AO 9730 35000 66 16 -6.7 -0.41 -6.3 -0.38 -0.01 A1 9230 35000 - - -6.7 -0.32 -6.4 -0.29 +0.02 A2 9080 36000 - - -6.7 -0.28 -6.5 -0.25 +0.03 A3 8770 35000 - - -6.7 -0.21 -6.5 -0.14 +0.06 A5 8510 35000 86 13 -6.7 -0.13 -6.6 -0.07 +0.09 A7 8150 33000 - - -6.7 -0.06 -6.6 0.00 +0.12 A8 7950 32000 - - -6.6 -0.03 -6.6 +0.11 +0.14 FO 7700 32000 100 12 -6.6 -0.01 -6.6 +0.15 +0.17 F2 7350 31000 - - -6.6 0.00 -6.6 +0.18 +0.23 F5 6900 32000 130 10 -6.6 -0.03 -6.6 +0.27 +0.32 F8 6100 31000 - - -6.6 -0.09 -6.5 +0.41 +0.56 GO 5550 30000 190 10 -6.6 -0.15 -6.4 +0.52 +0.76 G2 5200 29000 - - -6.5 -0.21 -6.3 +0.63 +0.87 G5 4850 29000 240 12 -6.5 -0.33 -6.2 +0.83 +1.02 G8 4600 29000 - - -6.5 -0.42 -6.1 +1.07 +1.15 KO 4420 29000 290 13 -6.5 -0.50 -6.0 +1.17 +1.24 K1 4330 30000 - - -6.6 -0.56 -6.0 +1.28 +1.30 K2 4250 29000 - - -6.5 -0.61 -5.9 +1.32 +1.35 K3 4080 33000 - - -6.6 -0.75 -5.9 -F1.60 +1.46 K4 3950 34000 - - -6.7 -0.90 -5.8 - +1.53 K5 3850 38000 440 13 -6.8 -1.01 -5.8 +1.80 +1.60 K7 3700 41000 - - -6.9 -1.20 -5.7 +1.84 +1.63 A-18 Appendix E Stellar Data Supergiant Stars (Luminosity Class Approximately Iab) Sp. Te Type (K) L/Lo R/Rp M/Mp Mb~1 BC M~r U - B B - V MO 3650 41000 510 13 -6.9 -1.29 -5.6 +1.90 +1.67 MI1 3550 44000 - - -7.0 -1.38 -5.6 +1.90 +1.69 M2 3450 55000 660 19 -7.2 -1.62 -5.6 +1.95 +1.71 M3 3200 56000 - - -7.7 -2.13 -5.6 +1.95 +1.69 M4 2980 160000 - - -8.3 -2.75 -5.6 +2.00 +1.76 M5 2800 300000 2300 24 -9.1 -3.47 -5.6 +1.60 +1.80 M6 2600 450000 - - -9.5 -3.90 -5.6 - - Except for the stellar radii, the data in the foregoing tables were taken from Schmidt-Kaler (1982). The values of the stellar radii were calculated using \2 ~ Ro=C7,/ ,/Lo. Suggested Readings TECHNICAL Liebert, James, and Probst, Ronald G., "Very Low Mass Stars," Annv.al Re-view of Astronomy and Astrophysics, ~5, 473, 1987. Schmidt-Kaler, Th., "Physical Parameters of the Stars," Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology, New Series, Group VI, Volume 2b, Springer-Verlag, Berlin, 1982.  Appendix F THE MESSIER CATALOG  t   R. A.6 Dec.b M NGC Name Const. mva h m ' Typec 1 1952 Crab Tau 8.4: 5 34.5 +22 01 SNR 2 7089 Aqr 6.5 21 33.5 -0 49 GC 3 5272 CVn 6.4 13 42.2 +28 23 GC 4 6121 Sco 5.9 16 23.6 -26 32 GC 5 5904 Ser 5.8 15 18.6 + 2 05 GC 6 6405 Sco 4.2 17 40.1 -32 13 OC 7 6475 Sco 3.3 17 53.9 -34 49 OC 8 6523 Lagoon Sgr 5.8: 18 03.8 -24 23 N 9 6333 Oph 7.9: 17 19.2 -18 31 GC 10 6254 Oph 6.6 16 57.1 - 4 06 GC 11 6705 Set 5.8 18 51.1 - 6 16 OC 12 6218 Oph 6.6 16 47.2 - 1 57 GC 13 6205 Her 5.9 16 41.7 +36 28 GC 14 6402 Oph 7.6 17 37.6 - 3 15 GC 15 7078 Peg 6.4 21 30.0 +12 10 GC 16 6611 Ser 6.0 18 18.8 -13 47 OC 17 6618 Swan d Sgr 7: 18 20.8 -16 11 N 18 6613 Sgr 6.9 18 19.9 -17 08 OC 19 6273 Oph 7.2 17 02.6 -26 16 GC 20 6514 Trifid Sgr 8.5: 18 02.6 -23 02 N 21 6531 Sgr 5.9 18 04.6 -22 30 OC 22 6656 Sgr 5.1 18 36.4 -23 54 GC 23 6494 Sgr 5.5 17 56.8 -19 01 OC 24 6603 Sgr 4.5: 18 16.9 -18 29 OC 25 Sgr 4.6 18 31.6 -19 15 OC 26 6694 Set 8.0 18 45.2 - 9 24 OC 27 6853 Dumbbell Vul 8.1: 19 59.6 +22 43 PN  A-19 A-20 Appendix F The Messier Catalog  R. A. b Dec. b M NGC Name Const. mva h m ' Type'  28 6626 29 6913 30 7099 31 224 Andromeda And 32 221 And 33 598 Triangulum Tri 34 1039 Per 35 2168 Gem 36 1960 Aur 37 2099 Aur 38 1912 Aur 39 7092 Cyg 40 UMa 41 2287 CMa 42 1976 Orione Ori 43 1982 Ori 44 2632 Praesepe Cnc 45 Pleiades Tau 46 2437 Pup 47 2422 Pup 48 2548 Hya 49 4472 Vir 50 2323 Mon 51 5194 Whirlpoolf CVn 52 7654 Cas 53 5024 Corn 54 6715 Sgr 55 6809 Sgr 56 6779 Lyr 57 6720 Ring Lyr 58 4579 Vir 59 4621 Vir 60 4649 Vir 61 4303 Vir 62 6266 Oph 63 5055 Sunflower CVn 64 4826 Evil Eye Corn 65 3623 Leo 66 3627 Leo 67 2682 Cnc 68 4590 Hya 69 6637 Sgr Sgr 6.9: 18 24.5 -24 52 GC Cyg 6.6 20 23.9 +38 32 OC Cap 7.5 21 40.4 -23 11 GC 3.4 0 42.7 +41 16 SbI-II 8.2 0 42.7 +40 52 cE2 5.7 1 33.9 +30 39 Sc(s)II-III 5.2 2 42.0 +42 47 OC 5.1 6 08.9 +24 20 OC 6.0 5 36.1 +34 08 OC 5.6 5 52.4 +32 33 OC 6.4 5 28.7 +35 50 OC 4.6 21 32.2 +48 26 OC 8: 12 22.4 +58 05 DS 4.5 6 47.0 -20 44 OC 4: 5 35.3 - 5 23 N 9: 5 35.6 - 5 16 N 3.1 8 40.1 +19 59 OC 1.2 3 47.0 +24 07 OC 6.1 7 41.8 -14 49 OC 4.4 7 36.6 -14 30 OC 5.8 8 13.8 - 5 48 OC 8.4 12 29.8 + 8 00 E2 5.9 7 03.2 - 8 20 OC 8.1 13 29.9 +47 12 Sbc(s)I-II 6.9 23 24.2 +61 35 OC 7.7 13 12.9 +18 10 GC 7.7 18 55.1 -30 29 GC 7.0 19 40.0 -30 58 GC 8.2 19 16.6 +30 11 GC 9.0: 18 53.6 +33 02 PN 9.8 12 37.7 +11 49 Sab(s)II 9.8 12 42.0 +11 39 E5 8.8 12 43.7 +11 33 E2 9.7 12 21.9 + 4 28 Sc(s)I 6.6 17 01.2 -30 07 GC 8.6 13 15.8 +42 02 Sbc(s)II-III 8.5 12 56.7 +21 41 Sab(s)II 9.3 11 18.9 +13 05 Sa(s)I 9.0 11 20.2 +12 59 Sb(s)II 6.9 8 50.4 +11 49 OC 8.2 12 39,5 -26 45 GC 7.7 18 31.4 -32 21 GC   Appendix F The Messier Catalog A-21 R. A.b Dec.b M NGC Name Const. m~a h m ~ Type~ 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 L  70 6681 Sgr 8.1 18 43.2 -32 18 GC 71 6838 Sge 8.3 19 53.8 +18 47 GC 72 6981 Aqr 9.4 20 53.5 -12 32 GC 73 6994 Aqr 9.1 20 58.9 -12 38 OC 74 628 Psc 9.2 1 36.7 +15 47 Sc(s)I 75 6864 Sgr 8.6 20 06.1 -21 55 GC 76 650/651 Per 11.5: 1 42.3 +51 34 PN 77 1068 Cet 8.8 2 42.7 - 0 O1 Sb(rs)II 78 2068 Ori 8: 5 46.7 + 0 03 N 1904 Lep 8.0 5 24.5 -24 33 GC 6093 Sco 7.2 16 17.0 -22 59 GC 3031 UMa 6.8 9 55.6 +69 04 Sb(r)I-II 3034 UMa 8.4 9 55.8 +69 41 Amorph 5236 Hya 7.6: 13 37.0 -29 52 SBc(s)II 4374 Vir 9.3 12 25.1 +12 53 E1 4382 Com 9.2 12 25.4 +18 11 SO pec 4406 Vir 9.2 12 26.2 +12 57 SO/E3 4486 Virgo A Vir 8.6 12 30.8 +12 24 EO 4501 Com 9.5 12 32.0 +14 25 Sbc(s)II 4552 Vir 9.8 12 35.7 +12 33 SO 4569 Vir 9.5 12 36.8 +13 10 Sab(s)I-II 4548 Com 10.2 12 35.4 +14 30 SBb(rs)I-II 6341 Her 6.5 17 17.1 +43 08 GC 2447 Pup 6.2: 7 44.6 -23 52 OC 4736 CVn 8.1 12 50.9 +41 07 RSab(s) 3351 Leo 9.7 10 44.0 +11 42 SBb(r)II 3368 Leo 9.2 10 46.8 +11 49 Sab(s)II 3587 Owl UMa 11.2: 11 14.8 +55 Ol PN 4192 Com 10.1 12 13.8 +14 54 SbII 4254 Com 9.8 12 18.8 +14 25 Sc(s)I 4321 Com 9.4 12 22.9 +15 49 Sc(s)I 5457 Pinwheel UMa 7.7 14 03.2 +54 21 Sc(s)I 5866 UMa 10.5 15 06.5 +55 46 SO 581 Cas 7.4: 1 33.2 +60 42 OC 4594 Sombrero Vir 8.3 12 40.0 -11 37 Sa/Sb 3379 Leo 9.3 10 47.8 +12 35 EO 4258 CVn 8.3 12 19.0 +47 18 Sb(s)II 6171 Oph 8.1 16 32.5 -13 03 GC 3556 UMa 10.0 11 11.5 +55 40 Sc(s)III 3992 UMa 9.8 11 57.6 +53 23 SBb(rs)I 205 And 8.0 0 40.4 +41 41 SO/E pec A-22 Appendix F The Messier Catalog a : indicates approximate apparent visual magnitude. b Right ascension and declination are given in epoch 2000.0. ~ Type abbreviations correspond to: SNR = supernova remnant, GC = globular cluster, OC = open cluster, N = diffuse nebula, PN = planetary nebula, DS = double star. Galaxies are indicated by their morphological Hubble types. d M17, the Swan nebula, is also known as the Omega nebula. e M42 also corresponds to the Trapezium H II region. f M51 also includes NGC 5195, the satellite to the Whirlpool galaxy. Suggested Readings TECHNICAL Hirshfeld, Alan, Sinnott, Roger W., and Ochsenbein, Francois, Sky Catalogue 2000.0, Second Edition, Cambridge University Press and Sky Publishing Cor-poration, New York, 1991. Sandage, Allan, and Bedke, John, The Carnegie Atlas of Galaxies, Carnegie Institution of Washington, Washington, D.C., 1994. Appendix G A PLANETARY ORBIT CODE Program ORBIT real*8 a, aau, e, time, tyears, dt, LoM, period, Pyears real*8 Mstrsun, Mstar, Msun, theta, dtheta, r, x, y real*8 G, AU, spyr integer*4 n, k, kmax, i C define basic constants data G, Msun, AU, spyr 1 / 6.67259d-08, 1.989d33, 1.4960d13, 3.155815d7 / c open output file for orbital parameters open(unit=l0,file='orbit.dat',form='formatted',status='unknown') c enter physical parameters for the system write(*,*) ' Enter the mass of the parent star (in solar masses):' read(*,*) Mstrsun write(*,*) ' Enter the semimajor axis of the orbit (in AU):' read(*,*) aau write(*,*) ' Enter the eccentricity:' read(*.*) e c calculate orbital period using Kepler's third law (Eq. 2.35) Pyears = sqrt(aau**3/Mstrsun) write(*,*) ' ' write(*,100) Pyears c enter the number of time steps and the time interval to be printed write(*,*) ' ' write(*,*) ' '  A-23 A-24 Appendix G A Planetary Orbit Code write(*,*) ' You may now enter the number of time steps to ', 1 'be calculated and the' write(*,*) ' frequency with which you want time steps printed.' write(*,*) ' Note that taking too large a time step during the', 1 ' calculation will' write(*,*) ' produce inaccurate results.' write(*.*) ' ' write(*,*) ' Enter the number of time steps desired for the ', 1 'calculation: ' read(*.*) n write(*,*) ' ' write(*,*) ' How often do you want time steps printed?' write(*,*) ' 1 = every time step' -. write(*,*) ' 2 = every second time step' write(*,*) ' 3 = every third time step' write(*,*) ' etc.' read(*,*) kmax c convert to cgs units period = Pyears*spyr dt = period/float(n-1) a = aau*AU Mstar = Mstrsun*Msun c initialize print counter, angle, and elapsed time theta = O.OdO time = 0.Od0 c print output header write(10,200) Mstrsun, aau, Pyears, e c start main time step loop do 10 i = 1,n c increment print counter k = k+i c use Eq. (2.3) to find r r = a*(1.Od0-e**2)/(1.Od0+e*cos(theta)) c convert to cartesian coordinates and print time and position c also print last point to close ellipse if (k.eq.l or. i.eq.n) then x = r*cos(theta)/AU ,, y = r*sin(theta)/AU " tyears = time/spyr Appendix G A Planetary Orbit Code A-25 write(10,300) tyears, x, y end if c calculate the angular momentum per unit mass L/m (Eq. 2.32) LoM = sqrt(G*Mstar*a*(1.Od0-e**2)) c calculate the next value of theta by combining Eqs. (2.27) c and (2.29) (Kepler's second law) dtheta = LoM/r**2*dt theta = theta + dtheta c update the elapsed time time = time + dt c check print counter then return to the top of the loop if (k.eq.kmax) k = 0 10 continue write(*,*) ' ' write(*,*) ' The calculation is finished and listed in ', 1 'orbit.dat' stop c formats 100 format(ix,' The period of this orbit is ',f10.3,' years') 200 format(1x,25x,'Elliptical Orbit',//, 1 26x,'Mstar = ',f10.3,' Mo',/, 2 26x,'a = ',f10.3,' AU',/, 3 26x,'P = ',f10.3,' yr',/, 4 26x,'e = ',f10.3,///, 5 lix,'t (yr)',14x,'x (AU)',12x,'y (AU)') 300 format(7x,f10.3,1Ox,f10.3,8x,f10.3) end Appendix H STATSTAR, A STELLAR STRUCTURE CODE r  The FORTRAN computer program listed here (STATSTAR) is based on the equations of stellar structure and the constitutive relations developed in Chap-ters 9 and 10. An example of the output generated by STATSTAR is given in Appendix I. STATSTAR is designed to illustrate as clearly as possible many of the most important aspects of numerical stellar astrophysics. To accomplish this goal, STATSTAR models are restricted to a fixed composition throughout (i.e., homogeneous main-sequence models). The four basic stellar structure equations are computed in the functions dPdr, dMdr, dLdr, and dTdr. These functions calculate the derivatives dP/dr (Eq. 10.7), dM,./dr (Eq. 10.8), dLT/dr (Eq. 10.45), and dT/dr [Eqs. (10.61) and (10.81)]. The density [p(r) = rho] is calculated directly from the ideal gas law and the radiation pressure equation (Eq. 10.26), given local values for the pressure [P (r) = P ( i ) ] , temperature [T (r) = T ( i ) ] , and the mean molecular weight (~, = mu, assumed here to be for a completely ionized gas only); note that i is the number of the mass shell currently being calculated (i - 1 at the sur-face, see Fig. 10.11). Once the density is determined, both the opacity [~(r) _ kappa] and the nuclear energy generation rate [E(r) = epslon] may be calcu-lated. The opacity is determined using the bound-bound and bound-free opac-ity formulae [Eqs. (9.19) and (9.20), respectively], together with electron scat-tering (Eq. 9.21). The energy generation rate is computed from the equation  A-27 A-28 Appendix H STATSTAR, A Stellar Structure Code for the total pp chain (Eq. 10.49) and for the CNO cycle (Eq. 10.53). Each of these calculations is carried out in the equation-of-state subroutine, EOS.1 The program begins by asking the user to supply the desired stellar mass (Msolar, in solar units), the trial luminosity (Lsolar, also in solar units), the trial effective temperature (Te, in kelvin), and the mass fractions of hydrogen (X) and metals (Z). Using the stellar structure equations, the program proceeds to integrate from the surface of the star toward the center, stopping when a problem is detected or when a satisfactory solution is obtained. If the inward integration is not successful, a new trial luminosity and/or effective temper-ature must be chosen. Recall that the Vogt-Russell theorem states that a unique stellar structure exists for a given mass and composition. Satisfying the central boundary conditions therefore requires specific surface boundary conditions. It is for this reason that a well-defined main sequence exists. ff~~i~f~ the central boundary conditions exactly by the crude shooting method employed by STATSTAR, the calculation is terminated when the core is approached. The stopping criteria used here are that the interior mass MT < O.O1MS and the interior luminosity LT < 0.1L3, when the radius r < 0.02RS, where MS, LS, and Rs are the surface mass, luminosity, and radius, respectively. [Within STATSTAR, Mr = M_r (i), Ms = Ms, LT = L_r( i ) , LS = Ls, r = r ( i ) , and Rs = Rs.] Once the criteria for halting the integration are detected, the conditions at the center of the star are estimated by an extrapolation procedure. Since the pressure, temperature, and density are all assumed to be zero at the surface of the star, it is necessary to begin the calculation with approxima-tions to the basic stellar structure equations. This can be seen by noting that the mass, pressure, luminosity, and temperature gradients are all proportional to the density and are therefore exactly zero at the surface. It would appear that applying these gradients in their usual form implies that the fundamental physical parameters cannot change from their initial values since the density would remain zero at each step! One way to overcome this problem is to assume that the interior mass and luminosity are both constant through a number of surface zones. In the case of the luminosity, this is clearly a valid assumption since temperatures are not sufficient to produce nuclear reactions near the surfaces of main-sequence stars. For the interior mass, the assumption is not quite as obvious. However, we will see that in realistic stellar models, the density is so low near the surface that 'State-of-the-art research codes use much more sophisticated prescriptions for the equa-tions of state.     Appendix H STATSTAR, A Stellar Structure Code A-29 the approximation is indeed valid. Of course, it is important to verify that the assumption is not being violated to within some specified limit. Assuming that the surface zone is radiative, and given the surface values M,. = MS and LT = LS, dividing Eq. (10.7) by Eq. (10.61) leads to dP _ 167rac GM, T3 dT 3 LS Since relatively few free electrons exist in the atmospheres of stars, electron scattering can be neglected and 7~ may be replaced by the bound-free and free-free Kramers' opacity laws [Eqs. (9.19) and (9.20)] expressed in the forms 74f = Ab fp/T3.5 and -9ff = Aff p/T3*5, respectively. Defining A - Ab f -1- Aff and using Eq. (10.14) to express the density in terms of the pressure and temperature through the ideal gas law (assuming that radiation pressure may be neglected), dP _ 167r GM, ack T7.5 dT 3 Ls Ap,mH P Integrating with respect to temperature and solving for the pressure, we find that __ C 1 167r GM, ack ~ 1/2 4.25 P 4.25 3 LS ApmH T ' It is now possible to write T in terms of the independent variable r through Eq. (10.61), again using the ideal gas law and Kramers' law, along with Eq. (H.1) to eliminate the dependence on pressure. Integrating, T = GMS (4 25k) (r RS) ' (H.2) Equation (H.2) is first used to obtain a value for T(r), then Eq. (H.1) gives P(r). At this point it is possible to calculate p, -9, and E from the usual equation-of-state routine, EOS. A very similar procedure is used in the case that the surface is convective. In this situation Eq. (10.81) may be integrated directly if y is constant. This gives T = GMS 11k H ) ( r RS ) ' (H.3) Now, since convection is assumed to be adiabatic in the interior of our simple model, the pressure may be found from Eq. (10.75). Subroutine STARTMDL computes Eqs. (H.1), (H.2), and (H.3). The conditions at the center of the star are estimated by extrapolating from the last zone that was calculated by direct numerical integration. Beginning  A-30 Appendix H STATSTAR, A Stellar Structure Code with Eq. (10.7), and identifying MT = 47rpor3/3, where po is taken to be the average density of the central ball (the region inside the last zone calculated by the usual procedure),2 dP - -G M,.Po = - 47r G r. - Po dr r2 3 Integrating, JP P dP = - 4 GP o J T r dr o 3 0 and solving for the central pressure results in PO = P + 3 Gpor2. (H.4) Other central quantities can now be found more directly. Specifically, the central density is estimated to be po = Mr/(47rr3/3), where Mr and r are the values of the last zone calculated. Neglecting radiation pressure, To may be determined from the ideal gas law. Finally, the central value for the nuclear energy generation rate is computed using eo = Lr/MT. The numerical integration technique employed here is a fourth-order Runge-Kutta algorithm, which is accurate through fourth order in the step size Or = deltar. This means that if Or/r = 0.01, the solutions for P, MT, LT, and T are accurate to approximately a few parts in 0.014 = 10-$, assuming that the results of the previous zone were exact. To accomplish this accuracy, the Runge-Kutta algorithm evaluates derivatives at several intermediate points between mass shell boundaries. Details of the Runge-Kutta method are given in many numerical analysis texts and will not be discussed further here. STATSTAR execution times vary depending on the machine being used. For instance, on PCs with 486 33MHz chips or higher, a model can be com-pleted in a few seconds; on faster machines only a fraction of a second is required. It should be pointed out that if STATSTAR is compiled on a VAX computer running VMS, the \G-FLOAT option should be invoked. This option provides the large exponent range required of most astrophysical calculations. ZYou might notice that dPIdr goes to zero as the center is approached. This behavior is indicative of the smooth nature of the solution. Close inspection of the graphs in Section 11.1 showing the detailed interior structure of the Sun illustrates that the first derivatives of many physical quantities go to zero at the center. , Appendix H STATSTAR, A Stellar Structure Code A-31 Program STATSTAR - c c This program will calculate a static stellar model using the c equations developed in the text. The user is expected to supply the c star's mass, luminosity, effective temperature, and composition c (X and Z). If the choices for these quantities are not consistent c with the central boundary conditions, an error message will be c generated and the user will then need to supply a different set of c initial values. c real*8 r(999), P(999), M_r(999), L-r(999), T(999), deltar, Te real*8 rho(999), kappa(999), epslon(999), d1Pd1T(999) real*8 X, Y, Z, XCNO, mu real*8 Ms, Ls, Rs, T0, PO real*8 Pcore, Tcore, rhocor, epscor, rhomax real*8 Rsolar, Msolar, Lsolar, Qm, Rcrat, Mcrat, Lcrat real*8 deltam, dlPlim real*8 Rsun, Msun, Lsun : real*8 f_iml(4), f_i(4), dfdr(4) real*8 dMdr, dPdr, dLdr, dTdr real*8 sigma, c, a, G, k-B, m-H, pi, gamma, gamrat, kPad, tog_bf, 1 g_ff character clim, rcf common /cnstnt/ sigma, c, a, G, k-B, m-H, pi, gamma, gamrat, kPad, 1 g-f f c c deltar = radius integration step c idrflg = set size flag c = 0 (initial surface step size of Rs/1000.) c = 1 (standard step size of Rs/100.) c = 2 (core step size of Rs/5000.) c Nstart = number of steps for which starting equations are used c (the outermost zone is assumed to be radiative) c Nstop = maximum number of allowed zones in the star c Igoof = final model condition flag c = -1 (number of zones exceeded; also the initial value) c = 0 (good model) c = 1 (core density was extreme) c ~ = 2 (core luminosity was extreme) _ - A-32 Appendix H STATSTAR, A Stellar Structure Code c = 3 (extrapolated core temperature is too low) c = 4 (mass became negative before center was reached) c = 5 (luminosity became negative before center was reached) c X, Y, Z = mass fractions of hydrogen, helium, and metals c T0, PO = surface temperature and pressure (TO = PO = 0 is assumed) c Ms, Ls, Rs = mass, luminosity, and radius of the star (cgs units) c data Nstart, Nstop, Igoof, ierr / 10, 999, -1, 0 / data P0, T0, dlPlim / O.OdO, O.OdO, 99.9d0 / data Rsun, Msun, Lsun / 6.9599d+10, 1.989d+33, 3.826d+33 / data sigma, c, a, G, k~, m~t, pi, gamma, tog_bf, g~f 1 / 5.67051d-5, 2.99792458d+10, 7.56591d-15, 6.67259d-8, 2 1.380658d-i6, 1.673534d-24, 3.141592654d0, 1.6666667d0, O.O1d0, 3 1.Od0 / c c Assign values to constants (cgs units) c Rsun = radius of the Sua c Msun = mass of the Sun c Lsun = luminosity of the Sun c sigma = Stefan-Boltzmann constaat c c = speed of light in vacuum c a = 4*sigma/c (radiation pressure constant) c G = universal gravitational coastaat c k ~ = Boltzmann constant c m1i = mass of hydrogen atom c pi = 3.141592654 c gamma = 5/3 (adiabatic gamma for a monatomic gas) c gamrat = gamma/(gamma-1) c kPad = P/T**(gamma/(gamma-1)) *) Msolar   Appendix H STATSTAR, A Stellar Structure Code A-33 write(*,*) ' Enter the luminosity of the star (in solar units):' read(*,*) Lsolar write(*,*) ' Enter the effective temperature of the star (in K):' read(*.*) Te 10 continue write(*,*) ' Enter the mass fraction of hydrogen (X):' read(*,*) X write(*,*) ' Enter the mass fraction of metals (Z):' read(*.*) Z Y= 1.d0-X-Z if (Y.1t.0.Od0) then write(*,100) go to 10 end if c c Select the mass fraction CNO to be 50% of Z. XCNO = Z/2.Od0 c c Calculate the mass, luminosity, and radius of the star. c The radius is calculated from Eq. (3.17).    Ms = Msolar*Msun Ls = Lsolar*Lsun Rs = sqrt(Ls/(4.d0*pi*sigma))/Te**2 Rsolar = Rs/Rsun c Begin with a very small step size since surface conditions vary c rapidly. c deltar = -Rs/1000.Od0 idrflg = 0 c c Calculate mean molecular weight mu assuming complete ionization c (see Eq. 10.21).' mu = 1.Od0/(2.Od0*X + 0.75d0*Y + 0.5d0*Z)  A-34 Appendix H STATSTAR, A Stellar Structure Code  I c Calculate the delimiter between adiabatic convection and radiation i c (see Eq. 10.87). I c gamrat = gamma/(gamma - i.OdO) c c Initialize values of r, P, Ms, Ls, T, rho, kappa, and epslon at c the surface. The outermost zone is assumed to be zone 1. The zone c number increases toward the center. c r(i) = Rs MS(1) = Ms LS(1) = Ls T(1) = TO P(1) = PO if (PO.le.0.Od0 .or. TO.le.0.Od0) then rho(1) = O.OdO kappa(1) = O.OdO epslon(1) = O.OdO else call EOS(X, Z, XCNO, mu, P(1), T(1), rho(i), kappa(1), 1 epslon(1), tog_bf, 1 ,ierr) if (ierr.ne.0) stop end if c c Apply approximate surface solutions to begin the integration, c assuming radiation transport in the outermost zone (do 20 loop). c irc = 0 for radiation, irc = 1 for convection. c Assume arbitrary initial values for kPad, and d1Pd1T. c d1Pd1T = d1nP/d1nT (see Eq. 10.87) c kPad = 0.3d0 irc = 0 d1Pd1T(1) = 4.25d0 do 20 i = 1, Nstart _. ip1=i+1 call STARTMDL(deltar, X, Z, mu, Rs, r(i), Mx(i), Lx(i), 1 r(ipl), P(ipi), MS(ipi), L~(ipl), T(ip1), tog_bf, irc) call EOS(X, Z, XCNO, mu, P(ipl), T(ipi), rho(ipi), kappa(ipl), 1 epslon(ipi), tog_bf, ipl, ierr)  Appendix H STATSTAR, A Stellar Structure Code A-35 if (ierr.ne.0) then write(*,400) r(i)/Rs, rho(i), Ms(i)/Ms, kappa(i), T(i), 1 epslon(i), P(i), Ls(i)/Ls stop end if c c Determine whether convection will be operating in the next zone by c calculating d1nP/d1nT numerically between zones i and i+l (ipl). c Update the adiabatic gas constant if necessary. if (i.gt.l) then d1Pd1T(ipl) = log(P(ipl)/P(i))/log(T(ipl)/T(i)) else d1Pd1T(ipl) = d1Pd1T(i) end if : if (d1Pd1T(ipi).lt.gamrat) then irc = 1 else irc=0 kPad = P(ipi)/T(ipi)**gamrat end if c c Test to see whether the surface assumption of constant mass is still c valid. c deltaM = deltar*dMdr(r(ipi), rho(ipl)) MS(ipi) = MS(i) + deltaM if (abs(deltaM).gt.0.001d0*Ms) then write(*,200) if (ipi.gt.2) ipl = ipi - 1 go to 30 end if 20 continue - .- , c c This is the main integration loop. The assumptions of constant c interior mass and luminosity are no longer applied.  30 Nsrtpl = ipl + 1 do 40 i = Nsrtpi, Nstop im1=i-1 A-36 Appendix H STATSTAR, A Stellar Structure Code c Initialize the Runge-Kutta routine with zone i-1 quantities c and their derivatives. Note that the pressure, mass, luminosity, c and temperature are stored in the memory locations f_iml(1), c f_imi(2), f_iml(3), and f_imi(4), respectively. The derivatives of c those quantities with respect to radius are stored in dfdr(1), c dfdr(2), dfdr(3), and dfdr(4). Finally, the resulting values for c P, Ms, Ls, and T are returned from the Runge-Kutta routine in c f_i(1), f_i(2), f_i(3), and f_i(4), respectively. c c The stellar structure equations dPdr (Eq. 10.7), dMdr (Eq. 10.8), c dLdr (Eq. 10.45), and dTdr (Eq. 10.61 or Eq. 10.81) are calculated c in function calls, defined later in the code. c f_iml(1) = P(iml) f_iml(2) = MS (imi) f_imi(3) = L-r(iml) f_iml(4) = T(iml) dfdr(1) = dPdr(r(imi), Ms (iml), rho(iml)) dfdr(2) = dMdr(r(imi), rho(imi)) dfdr(3) = dLdr(r(iml), rho(imi), epslon(iml)) dfdr(4) = dTdr(r(iml), Ms(iml), L-r(imi), T(imi), rho(iml), 1 kappa(iml), mu, irc) call RUNGE(f_imi, dfdr, f_i, r(imi), deltar, irc, X, Z, XCNO, 1 mu, i, ierr) if (ierr.ne.0) then write(*,300) write(*,400) r(iml)/Rs, rho(iml), M-r(imi)/Ms, kappa(imi), 1 T(imi), epslon(iml), P(iml), L-r(iml)/Ls stop end if c c Update stellar parameters for the next zone, including adding c dr to the old radius (note that dr < p since the integration is c inward). r(i) = r(imi) + deltar P(i) = f_i(1) Ms(i) = f_i(2) L-r (i) = f_i(3) T(i) = f_i(4)   Appendix H STATSTAR, A Stellar Structure Code A-37 c Calculate the density, opacity, and energy generation rate for c this zone. c call EOS(X, Z, XCNO, mu, P(i), T(i), rho(i), kappa(i), 1 epslon(i), tog_bf, i, ierr) if (ierr.ne.0) then write(*,400) r(iml)/Rs, rho(iml), M~(imi)/Ms, kappa(iml), 1 T(iml), epslon(iml), P(iml), L~(imi)/Ls stop end if c c Determine whether convection will be operating in the next zone by c calculating d1nP/d1nT and comparing it to gamma/(gamma-1) c (see Eq. 10.87). Set the convection flag appropriately. d1Pd1T(i) = log(P(i)/P(im1))/log(T(i)/T(iml)) if (d1Pd1T(i).lt.gamrat) then irc = 1 else irc = 0 end if c c Check to see whether the center has been reached. If so, set Igoof and c estimate the central conditions rhocor, epscor, Pcore, and Tcore. c The central density is estimated to be the average density of the c remaining central ball, the central pressure is determined by c applying Eq. (H.4), and the central value for the energy c generation rate is calculated to be the remaining interior c luminosity divided by the mass of the central ball. Finally, the c central temperature is computed by applying the ideal gas law c (where radiation pressure is neglected).  if (r(i).le.abs(deltar) .and. 1 (L~(i).ge.0.id0*Ls .or. Ms(i).ge.0.O1d0*Ms)) then Igoof = 6 else if (LS(i).1e.0.Od0) then ' Igoof = 5 rhocor = Ms(i)/(4.Od0/3.Od0*pi*r(i)**3) if (M_r(i).ne.0.Od0) then a epscor = Ls(i)/Ms(i) A-38 Appendix H STATSTAR, A Stellar Structure Code else epscor = O.OdO end if Pcore = P(i) + 2.Od0/3.Od0*pi*G*rhocor**2*r(i)**2 Tcore = Pcore*mu*m1i/(rhocor*k.B) else if (MS (i).1e.0.Od0) then Igoof = 4 Rhocor = O.OdO epscor = O.OdO Pcore = O.OdO Tcore = O.OdO else if (r(i).1t.0.02d0*Rs and. M-r(i).1t.0.01d0*Ms apd. 1 Ls(i).1t.0.id0*Ls) then rhocor = ms(i)/(4.Od0/3.Od0*pi*r(i)**3) _ rhomax = 10.Od0*(rho(i)/rho(im1))*rho(i) epscor = Lx(i)/Ms(i) Pcore = P(i) + 2.Od0/3.Od0*pi*G*rhocor**2*r(i)**2 Tcore = Pcore*mu*m-H/(rhocor*k-B) if (rhocor.lt.rho(i) or. rhocor.gt.rhomax) then Igoof = 1 else if (epscor.lt.epslon(i)) then Igoof = 2 else if (Tcore.lt.T(i)) then Igoof = 3 else Igoof = 0 end if end if if (Igoof.ne.-1) then istop = i go to 50 . . . , end if c c Is it time to change the,step size? c if (idrflg.eq.0 and. Ms(i).1t.0:99d0*Ms) then deltar = -Rs/100.Od0 idrflg = 1 end if  Appendix H STATSTAR, A Stellar Structure Code A-39  if (idrflg.eq.l .and. deltar.ge.0.5*r(i)) then deltar = -Rs/5000.Od0 idrflg = 2 end if istop = i 40 continue -- c c Generate warning messages for the central conditions. c rhocor = M_r(istop)/(4.Od0/3.Od0*pi*r(istop)**3) epscor = LS(istop)/Ms(istop) Pcore = P(istop) + 2.Od0/3.Od0*pi*G*rhocor**2*r(istop)**2 Tcore = Pcore*mu*m~I/(rhocor*k~) 50 continue if (Igoof.ne.0) then if (Igoof.eq.-i) then write(*,5000) write(*,5100) else if (Igoof.eq.l) then write(*,6000) write(*,5200) rho(istop) if (rhocor.gt.l.Od10) write(*,5300) else if (Igoof.eq.2) then write(*,6000) -- , write(*,5400) epslon(istop) else if (Igoof.eq.3) then write(*,6000) write(*,5500) T(istop) else if (Igoof.eq.4) then write(*,5000) write(*,5600) else if (Igoof.eq.5) then write(*,5000) write(*,5700) else if (Igoof.eq.6) then write(*,5000) write(*,5800) end if  A-40 Appendix H STATSTAR, A Stellar Structure Code else write(*,7000) end if c c Print the central conditions. If necessary, set limits for the c central radius, mass, and luminosity to avoid format field overflows. c Rcrat = r(istop)/Rs _. if (Rcrat.lt.-9.999d0) Rcrat = -9.999d0 Mcrat = M-r(istop)/Ms if (Mcrat.lt.-9.999d0) Mcrat = -9.999d0 Lcrat = Ls (istop)/Ls if (Lcrat.lt.-9.999d0) Lcrat = -9.999d0 write( *,2000) Msolar, Mcrat, Rsolar, Rcrat, Lsolar, Lcrat, Te, 1 rhocor, X, Tcore, Y, Pcore, Z, epscor, d1Pd1T(istop) write(20,1000) write(20,2000) Msolar, Mcrat, Rsolar, Rcrat, Lsolar, Lcrat, Te, 1 rhocor, X, Tcore, Y, Pcore, Z, epscor, d1Pd1T(istop) write(20,2500) Ms c c Print data from the center of the star outward, labeling convective c or radiative zones by c or r, respectively. If abs(d1nP/d1nT) c exceeds 99.9, set a print warning flag (*) and set the output limit c to +99.9 or -99.9 as appropriate to avoid format field overflows. c write(20,3000) do 60 ic = 1, istop i=istop-ic+i Qm = 1.0d0 - M_r(i)/Ms if (d1Pd1T(i).lt.gamrat) then rcf = 'c' else rcf = 'r' end if if (abs(d1Pd1T(i)).gt.dlPlim) then d1Pd1T(i) = sign(d1Plim,dlPd1T(i)) clim = '*' else Appendix H STATSTAR, A Stellar Structure Code A-41 write(20,4000) r(i), Qm, L_r(i), T(i), P(i), rho(i), kappa(i), 1 epslon(i), clim, rcf, d1Pd1T(i) 60 continue write(*,9000) c Format statements c 100 format(' ',/,' You must have X + Z <= 1',/, 1 ' please reenter composition',/) 200 format(' ',/, 1 ' The variation in mass has become larger than 0.001*Ms',/, 2 ' leaving the approximation loop before Nstart was reached',/) 300 format(' ',/,' The problem occurred in the Runge-Kutta routine',/) 400 format(' Values from the previous zone are:',/, 1 10x,'r/Rs = ',1pe12.5,' ', 2 12x,'rho = ',1pe12.5,' g/cm**3',/, 3 lOx,'M_r/Ms = ',ipe12.5,' ', 4 12x,'kappa = ',1pe12.5,' cm**2/g',/, 5 lOx,'T = ',1pe12.5,' K', 6 12x,'epsilon = ',ipe12.5,' ergs/g/s',/, 7 lOx,'P = ',ipe12.5,' dynes/cm**2',/, 8 lOx,'LS/Ls = ',1pe12.5) 1000 format(' ',15x,'A Homogeneous Main-Sequence Model',/) 2000 format(' ',/, 1 ' The surface conditions are:',10x,'The central conditions are:', 2 //, 3 ' Mtot = ',Opf13.6,' Msun',12x,'Mc/Mtot = ',1pe12.5,/, 4 ' Rtot = ',Opf13.6,' Rsun',12x,'Rc/Rtot = ',1pe12.5,/, 5 ' Ltot = ',Opf13.6,' Lsun',12x,'Lc/Ltot = ',ipe12.5,/, 6 ' Teff = ',Opf13.6,' K ',12x,'Density = ',ipe12.5, 7 ' g/cm**3',/, 8 ' X = ',Opf13.6,' ',12x,'Temperature = ',1pe12.5,' K',/, g ' Y = ',Opf13.6,' ',12x,'Pressure = ',Spe12.5, 1 ' dynes/cm**2',/, 2 ' Z = ',Opf13.6,' ',12x,'epsilon = ',ipe12.5, 3 ' ergs/s/g'>/, 4 ' ', 13x ,' ',12x,'d1nP/d1nT = ',ipe12.5,//) 2500 format(' Notes: ',/, _. 1 ' (1) Mass is listed as Qm = 1.0 - M-T/Mtot, where Mtot 2 ipe13.6,' g',/, -" A-42 Appendix H STATSTAR, A Stellar Structure Code 3 ' (2) Convective zones are indicated by c, radiative zones by r', 4 /~ 5 ' (3) d1nP/d1nT may be limited to +99.9 or -99.9; if so it is', 6 ' labeled by *',//) 3000 format(' ',5x,'r',7x,'qm',7x,'LS',7x,'T',8x,'P',7x, 1 'rho',6x,'kap',6x,'eps',3x,'d1Pd1T') 4000 format(' ',ip8e9.2,2a1,Opf5.1) 5000 format(' ',/,15x,'Sorry to be the bearer of bad news, but...',/, 1 15x,' Your model has some problems',/) 5100 format(' ', 8x,'The number of allowed shells has been exceeded' 1 ./) 5200 format(' ', 14x,'The core density seems a bit off,'/, 1 5x,' density should increase smoothly toward the center.',/, 2 5x,' The density of the last zone calculated was rho = ', 3 1pe10.3,' gm/cm**3',/) 5300 format(' ', ix,'It looks like you will need a degenerate', 1 ' neutron gas and general relativity',/, 2 ' to solve this core. Who do you think I am, Einstein?',/) 5400 format(' ', 14x,'The core epsilon seems a bit off,',/, 1 9x,' epsilon should vary smoothly near the center.',/, 2 9x,' The value calculated for the last zoae was eps =', 3 1pe10.3,' ergs/g/s',/) 5500 format(' ',8x,' Your extrapolated central temperature is too low' 1 ,/,8x,' a little more fine tuning ought to do it.',/, 2 8x,' The value calculated for the last zone was T = ', .. 3 1pe10.3,' K',/) 5600 format(' ', lOx,'You created a star with a hole in the center!', 1 /) 5700 format(' ', 30x,'This star has a negative central luminosity!',/) 5800 format(' ', 5x,'You hit the center before the mass and/or ', 1 'luminosity were depleted!',/) 6000 format(' ',///,15x,'It looks like you are getting close,',/, 1 12x,'however, there are still a few minor errors',/) 7000 format(' ',///,15x,'CONGRATULATIONS, I THINK YOU FOUND IT!',/, 1 9x,'However, be sure to look at your model carefully.',//) 9000 format(' ',10x,'***** The integration has been completed *****', 1 /,lOx,' The model has been stored in starmodl.dat', 2 /) stop end A-44 Appendix H STATSTAR, A Stellar Structure Code c Subroutine EOS calculates the values of density, opacity, the c guillotine-to-gaunt factor ratio, and the energy generation rate at c the radius r. G Subroutine EOS(X, Z, XCNO, mu, P, T, rho, kappa, epslon, tog_bf, 1 izone, ierr) real*8 X, Z, mu, P, T, rho, kappa, epslon, tog_bf, Prad, Pgas real*8 k_bf, , k-ff, k_e real*8 T6, fx, fpp, epspp, epsCNO, XCNO, oneo3, twoo3, psipp, Cpp real*8 sigma, c, a, G, k-B, m-H, pi, gamma, gamrat, kPad, g-ff common /cnstnt/ sigma, c, a, G, k-B, m-H, pi, gamma, gamrat, kPad, 1 g_ff data oneo3, tvoo3 / 0.333333333d0, 0.666666667d0 / c c Solve for density from the ideal gas law (remove radiation c pressure); see Eq. (10.26). c if (T.le.0.Od0 or. P.le.0.Od0) then ierr = 1 write(*,100) izone, T, P return end if Prad = a*T**4/3.Od0 Pgas = P - Prad rho=(mu*m-i/k-B) * (Pgas/T) if (rho.lt.0.Od0) then ierr = 1 write(*,200) izone, T, P, Prad, Pgas, rho return end if c c Calculate opacity, including the guillotine-to-gaunt factor ratio; c see Novotny (1973), p. 469. k_bf, k_ff, and k_e are the bound-free, c free-free, and electron scattering opacities, given by Eqs. (9.19), c (9.20), and (9.21), respectively. c   tog_bf = 2.82d0*(rho*(1.Od0 + X))**0.2d0 k_bf = 4.34d25/tog_bf*Z*(i.OdO + X)*rho/T**3.5d0 k_ff = 3.68d22*g-ff*(1.Od0 - Z)*(i.OdO + X)*rha1T**3:5d0 k_e = 0.2d0*(1.Od0 + X)   Appendix H STATSTAR, A Stellar Structure Code A-45  kappa = k-bf + k_ff + k_e c c Compute energy generation by the pp chain and the CNO cycle. These c are calculated using Eqs. (10.49) and (10.53), which come from c Fowler, Caughlan, and Zimmerman (1975). The screening factor for c the pp chain is calculated as fpp; see Clayton (1968), p. 359ff. c T6 = T*1.Od-06 fx = 0.133d0*X*sqrt((3.Od0 + X)*rho)/T6**1.5d0 fpp = 1.Od0 + fx*X psipp = 1.Od0 + 1.412d8*(1.Od0/X - 1.Od0)*exp(-49.98*T6**(- oneo3)) Cpp = 1.Od0 + 0.0123d0*T6**oneo3 + 0.0109d0*T6**twoo3 1 + 0.000938d0*T6 epspp = 2.38d6*rho*X*X*fpp*psipp*Cpp*T6**(-twoo3) 1 *exp(-33.80d0*T6**(-oneo3)) CCNO = 1.Od0 + 0.0027d0*T6**oneo3 - 0.00778d0*T6**twoo3 1 - 0.000149d0*T6 epsCNO = 8.67d27*rho*X*XCNO*CCNO*T6**(-twoo3) 1 *exp(-152.28d0*T6**(-oneo3)) epslon = epspp + epsCNO c c Formats c 100 format(' ',/,' Something is a little wrong here.', 1 /,' You are asking me to deal with either a negative temperature' 2 /,' or a negative pressure. I am sorry but that is not in my' 3 ' contract!',/,' You will have to try again with different', 4 ' initial conditions.',/, 5 ' In case it helps, I detected the problem in zone ',i3, 6 ' with the following',/,' conditions:',/, 7 10x,'T = ',ipe10.3,' K',/, 8 lOx,'P = ',1pe10.3,' dynes/cm**2') 200 format(' ',/,' I am sorry, but a negative density was detected.', 1 /,' my equation-of-state routine is a bit baffled by this new', 2 /,' physical system you have created. The radiation pressure', 3 /,' is probably too great, implying that the star is unstable.' 4 /,' Please try something a little less radical next time.',/, 5 ' In case it helps, I detected the problem in zone ',i3, 6 ' with the following',/,' conditions:',/, 7 lOx,'T = ',ipe10.3,' K',/,  A-46 Appendix H STATSTAR, A Stellar Structure Code 8 lOx,'P_total = ',1pe10.3,' dynes/cm**2',/, 9 lOx,'Psad = ',ipe10.3,' dynes/cm**2',/, 1 lOx,'P_gas = ',ipe10.3,' dynes/cm**2',/, 2 lOx,'rho = ',Spe10.3,' g/cm**3') return end c c The following four function subprograms calculate the gradients of c pressure, mass, luminosity, and temperature at r. real*8 Function dPdr(r, Ms, rho) real*8 r, Ms, rho real*8 sigma, c, a, G, k-B, m1i, pi, gamma, gamrat, kPad, g-ff common /cnstnt/ sigma, c, a, G, k-B, m-H, pi, gamma, gamrat, kPad, 1 g_ff c Eq. (10.7) dPdr = -G*rho*Ms'/r**2 return end c real*8 Function dMdr(r, rho) real*8 r, rho real*8 sigma, c, a, G, k-B, m-H, pi, gamma, gamrat, kPad, g-ff common /cnstnt/ sigma, c, a, G, k-B, m-H, pi, gamma, gamrat, kPad, 1 gff c Eq. (10.8) dMdr = 4.Od0*pi*rho*r**2 return end c real*8 Function dLdr(r, rho, epslon) real*8 r, rho, epslon real*8 sigma, c, a, G, k-B, m.Ii, pi, gamma, gamrat, kPad, g-ff common /cnstnt/ sigma, c, a, G, k-B, m1i, pi, gamma, gam at, kPad, 1 g_ff c Eq. (10.45) dLdr = 4.Od0*pi*rho*epslon*r**2 return end _ ., < Appendix H STATSTAR, A Stellar Structure Code A-47  real*8 Function dTdr(r, Ms, L_r, T, rho, kappa, mu, irc) real*8 r, Ms, Ls, T, rho, kappa, mu real*8 sigma, c, a, G, k-B, m1-I, pi, gamma, gamrat, kPad, g_ff common /cnstnt/ sigma, c, a, G, k-B, m-H, pi, gamma, gamrat, kPad, 1 g_ff c This is the radiative temperature gradient (Eq. 10.61). if (irc.eq.0) then dTdr = - (3.Od0/(16.Od0*pi*a*c))*kappa*rho/T**3*L-r/r**2 c This is the adiabatic convective temperature gradient (Eq. 10.81). else dTdr = -1.Od0/gamrat*G*M-r/r**2*mu*m..H/k-B end if return end c c This is a fourth-order Runge-Kutta integration routine. c Subroutine RUNGE(f_imi, dfdr, f_i, r_imi, deltar, irc, X, Z, XCNO, 1 mu, izone, ierr) real*8 f_iml(4), dfdr(4), f_i(4), df1(4), df2(4), df3(4), 1 f_temp(4) real*8 r_imi, r_i, deltar, dr12, dr16, r12, X, Z, XCNO, mu c _ dr12 = deltar/2.Od0 dr16 = deltar/6.Od0 . _-r12 = r_iml + dr12 r.i = r_imi + deltar c c Calculate intermediate derivatives from the fundamental stellar c structure equations found in Subroutine FUNDEQ. c  c do S0i=1, 4 f_temp(i) = f_iml(i) + dr12*dfdr(i) 10 continue call FUNDEQ(r12, f_temp, dfl, irc, X, Z, XCNO, mu, izone, ierr) if (ierr.ne.0) return do 20 i = 1, 4 f_temp(i) = f_imi(i) + dr12*dfi(i) 20 continue A-48 Appendix H STATSTAR, A Stellar Structure Code call FUNDEQ(r12, f_temp, df2, irc, X, Z, XCNO, mu, izone, ierr) if (ierr.ne.0) return do 30 i = 1, 4 f_temp(i) = f_iml(i) + deltar*df2(i) 30 continue call FUNDEQ(r_i, f_temp, df3, irc, X, Z, XCNO, mu, izone, ierr) if (ierr.ne.0) return c c Calculate the variables at the next shell (i + 1). c do 40 i = 1, 4 f_i(i) = f_imi(i) + dri6*(dfdr(i) + 2.Od0*dfi(i) 1 + 2.Od0*df2(i) + df3(i)) 40 continue return end c c This subroutine returns the required derivatives for RUNGE, the c Runge-Kutta integration routine. c Subroutine FUNDEQ(r, f, dfdr, irc, X, Z, XCNO, mu, izone, ierr) real*8 f(4), dfdr(4), X, Z, XCNO, r, Ms, P, T, Ls, rho, kappa, 1 epslon, tog_bf, mu real*8 dPdr, dMdr, dLdr, dTdr P = f(1) MS = f (2) Ls = f (3) T = f (4) call EOS(X, Z, XCNO, mu, P, T, rho, kappa, epslon, tog_bf, izone, 1 ierr) dfdr(1) = dPdr(r, M~, rho) dfdr(2) = dMdr(r, rho) dfdr(3) = dLdr(r, rho, epslon) dfdr(4) = dTdr(r, Ms, L~, T, rho, kappa, mu, irc) return end  Appendix I    STATSTAR STELLaR MODELS STATSTAR (see Appendix H) can be used to generate a theoretical main sequence once the composition is chosen. Although the results difFer somewhat from those of more sophisticated codes, many of the important features of stellar interiors are present.l Problems 10.18, 10.19, and 10.20 are designed to investigate the changes in stellar structure on the main sequence caused by variations in mass and/or composition. The STATSTAR main sequence for the composition X = 0.7, Y = 0.292, and Z = 0.008 is shown in Fig. L 1. Furthermore, Table L 1 shows the depen-dence of effective temperature on mass for the same composition. A sample STATSTAR session and output file are also presented. The user is asked to input the mass of the star, an estimate of its luminosity, the desired effective temperature, and the mass fractions of hydrogen and metals. STATSTAR will then compute a model stellar interior, integrating from the surface toward the center, as described in Appendix H. 1The careful reader may notice, for instance, that the 1 Mp model presented here has a convective core and a radiative envelope, which is actually the reverse of a real main-sequence star like the Sun (see the extended discussion of the solar interior in Section 11.1). The reasons for these discrepancies reside largely in our approximate treatments of the opacity (particu-larly at lower temperatures) and thermodynamics (our assumption of adiabatic convection). Both of these approximations are poor ones near the surfaces of cooler stars. A-51 A-52 Appendix I STATSTAR Stellar Models 5 4F 3 a 2 0 0  -2 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4 o Te (K) Figure 1.1 The STATSTAR main sequence for X = 0.7, Y = 0.292, and Z = 0.008.  o Te (K) M/Mo Te (K) M/Mo Te (K) -A1/M@ T 0.50 2321.4 2.25 12260.0 4.75 19730.0 9.00 27061.2 0.60 2910.8 2.50 13240.0 5.00 20302.0 9.50 27712.0 0.70 3523.0 2.75 14170.8 5.50 21354.0 10.00 28263.6 0.80 4163.3 3.00 15007.3 6.00 22310.0 10.50 28845.2 0.90 4832.8 3.25 15790.8 6.50 23217.0 11.00 29414.6 1.00 5500.2 3.50 16525.0 7.00 24074.0 11.50 29964.8 1.25 7203.6 3.75 17252.0 7.50 24880.0 12.00 30496.5 1.50 8726.4 4.00 17904.0 8.00 25613.6 12.50 31009.0 1.75 10090.0 4.25 18546.8 8.50 26332.0 13.00 31493.0 2.00 11218.4 4.50 19153.6 Table 1.1 The Variation of Effective Temperature with Mass Along the STATSTAR Main Sequence, Assuming X = 0.7, Y = 0.292, and Z = 0.008. Appendix I STATSTAR Stellar Models A-53 Here is an example of the required input, together with STATSTAR's response:  Enter the mass of the star (in solar units): 1.0 Enter the luminosity of the star (in solar units): 0.86071 Enter the effective temperature of the star (in K): 5500.2 Enter the mass fraction of hydrogen (X): 0.70 Enter the mass fraction of metals (Y): 0.008 , CONGRATULATIONS, I THINK YOU FOUND IT! However, be sure to look at your model carefully. The surface conditions are: The central conditions are: Mtot = 1.000000 Msun Rtot = 1.020998 Rsun Ltot = .860710 Lsun Teff = 5500.200000 K X = .700000 Y = .292000 Z = .008000 Mc/Mtot = 4.00418E-04 Rc/Rtot = 1.90000E-02 Lc/Ltot = 7.67225E-02 Density = 7.72529E+01 g/cm**3 Temperature = 1.41421E+07 K Pressure = 1.46284E+17 dynes/cm**2 epsilon = 3.17232E+02 ergs/s/g d1nP/d1nT = 2.49808E+00   ***** The integration has been completed ***** The model has been stored in starmodl.dat RST]^ab78;<?@ÿ~zvrnjf]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]cV]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c*]c*]c]c]]';<[ \ Q R U V Y Z + , / 0 3 4 ÿ{wsokg]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c& < =  [ \ ] ^ t 5678Lÿ{wsnjfb]c]c]cV]c]c]c]c]c]c UV]c]c]c]c]c]cV]c]c]c]c]c]cV]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c&Kx  !uyz|}~zuplhd`]c]c]c]cV]cV]c]c]c]c]c]c]c]cV]c]cV]c]cV]cV]cV]cV]cV]c]c]c]c]c]c]c]c]cV]c]c]c]c]c]cV]c%  7;hkyz}~KNRUVYZ¾|xtplhd]c]c]c]c]c]c]c]c]c]c]c]]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]cV]c]c]c]c]c]c]c]c]c]c'   bdrtwxzÿ{wsqomki]]]]]]c]c]c]c]c]c]c]c]c]c(]c]c]c]c]c]c]c]c]c]c]cV]c]cV]c]c]cV]c]c]c]c]c]cV]c]c'ghkl  #$()8Iÿ~zuqmie]c]c]c]cV]c]c]c]c]c]c]c]c]c]cV]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c& ;<@CEFHITUY[\]^`a~B C þ{wsokg]c]c]c]c]c]c]c]c]]]]V]cV]cV]c]]V]cV]c V]^c]c]c]cV]c]cV]c]c]c]c]c]cV]cV]c]]c]V]c]c]c'C F G z { ~  !!!!!!!!!!""!")"""####'#-###$$$$ÿ{wsokg]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c&$$$$%%%%% %H%N%%%%%%%%%&&&''''I'\'''''\(](^(_(g(h(¾|xtplhd]c]c]c]c]c]c]c]c]c]cU]c]c]c]c]c]cU]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]cV] c]c]c&h(i(j(m(n())) )**!*"***(+)+,+-+r+s+++++++++,,,,C,D,G,H,--¾~zvrnjf]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]cV]c]c]c]c]c]c]c]c]c]c]c]c]c]c&---,---A-B-C-D-H-R-S-T-U-S.T.W.X.}.~..........G/H/K/L/ 0 0 00000|xtplhd]c]c]c]c]c]c]c]c]c]c]cV]c]c]c]c]c]c]c]c]c]c]c]c]c]c]cV]V]V]V]]c]^c]^c]c]c]c]c]c]c'0000V1W1222222333 33344 4 4*4+4.4/4M4N4Q4R4c4d4g4h444444ÿ{wsokg]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c&444444444V5W5X5Y5Z5]5^5552636y6|6666677777777888888}yuplgV]c]cV]c]c]c]c] ]c]c]c]c]cV]c]cV]c]cV]c]c] ]c]c]cV]c]c]c]c] ] ]]c]c]c]c]c]c]c]c]c]c'8 9!9c9h99999999999999999*:0:x:y:}:~::::::;;; ; ;Ŀ|wsniea]c]cV]cV]c]cV]cV]c]c]c]c]c]c]c]cV]c]c]c]c]c]cV]c]cV]cV]cV]cV]c]cV]cV]c]cV]c]cV]c]c]c]c$ ; ; ;;;;;;;;;;;;;;;;;;;;;<< < <4<5<9<:<w<x<{<|<<<<þ{wsokgc]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]c]cV]c]c]^c]c]^c]^c]c]c]cV]c]c]c]c]c]c]c]cV]cV]cV]c%<<<<=== =x=y=}=~===>>>>>>?? 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