The Equations of Stellar structure

Transcription

1 PART IV: STARS I

2 Section 1 The Equations of Stellar structure 1.1 Hyostatic Equilibrium da P P + dp r Consider a volume element, density ρ, thickness and area da, at distance r from a central point. The gravitational force on the element is Gm(r) r 2 ρ(r) da where m(r) is the total mass contained within radius r. In hyostatic equilibrium the volume element is supported by the (dierenc in the) pressure force, da; i.e., da = Gm(r) r 2 ρ(r) da 81

3 or = Gm(r)ρ(r) r 2 = ρ(r) g(r) (1.1) or + ρ(r) g(r) = which is the equation of hyostatic equilibrium. (Note the minus sign, which arises because increasing r goes with decreasing P (r).) More generally, the gravitational and pressure forces may not be in equilibrium, in which case there will be a nett acceleration. The resulting equation of motion is simply + ρ(r) g(r) = ρ(r) a (1.2) where a is the acceleration, d 2 r/ dt 2. Note that we have not specied the nature of the supporting pressure; in stellar interiors, it is typically a combination of gas pressure, P = nkt, and radiation pressure P R = 4σ 3c T 4 (1.31) (Section 1.8). Turbulent pressure may be important under some circumstances. 1.2 Mass Continuity r R 82

4 The quantities m(r) and ρ(r) in eqtn. (1.1) are clearly not independent, but we can easily derive the relationship between them. The mass in a spherical shell of thickness is = 4π r 2 ρ(r). Thus = 4πr2 ρ(r); the Equation of Mass Continuity. 1 (1.3) 1 Spherical symmetry Implicit in eqtn. (1.3) is the assumption of spherical symmetry. This assumption is reasonable as long as the centrifugal force arising from rotation is small compared to Newtonian gravity The centrifugal force is a maximum at the equator, so for a star of mass M and equatorial radius R e our condition for spherical symmetry is mv 2 e R e = mω 2 R e GM m R 2 e for a test particle of mass m at the equator, where the rotational velocity is v e (equatorial angular velocity ω); that is, ω 2 GM, or, equivalently, (1.4) Re r 3 GM v e giving (1.5) R e ffir GM P rot 2π (1.6) R 3 e This condition is true for most stars; e.g., for the Sun, P rot 25.4 d (at the equator); ffir GM 2π = 1 4 s.1 d. R 3 e However, a few stars (such as the emission-line B stars) do rotate so fast that their shapes are signicantly distorted. The limiting case is when the centrifugal acceleration equals the gravitational acceleration at the equator: GM m R 2 e = mv2 max R e, or r GM v max = R e Equating the potential at the pole and the equator, GM R p = GM R e + v2 e 2 whence R e = 1.5R p in this limiting case. Note that this returns the right answer for the total mass: M = Z R = Z R 4πr 2 ρ(r) = 4 3 πr3 ρ(r) 83

5 Moving media Equation 1.3 applies when hyostatic equilibrium holds. If there is a ow (as in the case of stellar winds for example), the equation of mass continuity reects dierent physics. In spherical symmetry, the amount of material owing through a spherical shell of thickness at some radius r must be the same for all r. Since the ow at the surface, 4πR ρ(r 2 )v(r ) is the stellar mass-loss rate Ṁ, we have Ṁ = 4πr 2 ρ(r)v(r) 1.3 Energy continuity The increase in luminosity in going from r to r + is just the energy generation within the corresponding shell: i.e, dl = L(r + ) L(r) = 4πr 2 ρ(r)ɛ(r); dl = 4πr2 ρ(r)ɛ(r) (1.7) where ρ(r) is the mass density and ɛ(r) is the energy generation rate per unit mass at radius r. Stars don't have to be in energy equilibrium moment by moment (e.g., if the gas is getting hotter or colder), but equilibrium must be maintained on longer timescales. Evidently, the stellar luminosity is given by integrating eqtn. (1.7): L = R 4πr 2 ρ(r)ɛ(r) if we make the approximation that the luminosity (measured at the surface) equals the instantaneous energy-generation rate (in the interior). This assumption holds for most of a star's lifetime, since the energy diusion timescale is short compared to the nuclear timescale (see discussions in Sections 12.3 and 12.4). The equations of hyostatic equilibrium (dp/), mass continuity (/), and energy continuity (dl/), can be used (together with suitable boundary conditions) as a basis for the calculation of stellar structures. 1.4 Virial Theorem The virial theorem expresses the relationship between gravitational and thermal energies in a `virialised system' (which may be a star, or a cluster of galaxies). There are several ways to 84

6 obtain the theorem (e.g., by integrating the equation of hyostatic equilibrium; see Section 1.4); the approach adopted here, appropriate for stellar interiors, is particularly direct. The total thermal energy of a star is ( ) 3 U = kt (r) n(r) dv (1.8) 2 V (mean energy per particle, times number of particles per unit volume, integrated over the volume of the star); but and so V = 4 3 πr3 whence dv = 4πr 2 P = nkt U = R Integrating by parts, U = πr2 P (r). ] [4πP (r) r3 R 3 = 3 2 PS P C PS P C 4π r3 3 dp 2πr 3 dp where P S, P C are the surface and central pressures, and the rst term vanishes because P S ; but, from hyostatic equilibrium and mass continuity, so dp = Gm(r)ρ(r) r 2 4πr 2 ρ(r) (1.1, 1.3) U = = 1 2 M M 2πr 3 Gm(r) r 2 4πr 2 Gm(r) r. (1.9) However, the total gravitational potential energy (dened to be zero for a particle at innity) is M Gm(r) Ω = ; (1.1) r comparing eqtns. (1.9) and (1.1) we see that 2U + Ω = the Virial Theorem. (1.11) 85

7 An alternative derivation From hyostatic equilibrium and mass conservation, we have that = Gm(r). 4πr 4 = Gm(r)ρ(r), r 2 (1.1) = 4πr2 ρ(r) (1.3) Multiplying both sides by 4 /3πr 3 V gives us V dp = 4 3 πr3 Gm 4πr 4 = 1 Gmr. 3 Integrating over the entire star, Z R V dp = P V R but P as r R, so 3 Z V (R ) P dv + Ω =. Z V (R ) P dv, = 1 3 Ω We can write the equation of state in the form P = (γ 1)ρu where γ is the ratio of specic heats and u is the specic energy (i.e., internal [thermal] energy per unit mass); 3 Z V (R ) (γ 1)ρu dv + Ω =. Now ρu is the internal energy per unit volume, so integrating over volume gives us the total thermal energy, U; and 3(γ 1)U + Ω =. For an ideal monatomic gas, γ = 5 /3, and we recover eqtn. (1.11) Implications Suppose a star contracts, resulting in a more negative Ω (from eqtn. (1.1); smaller r gives bigger Ω ). The virial theorem relates the change in Ω to a change in U: U = Ω 2. Thus U becomes more positive; i.e., the star gets hotter as a result of the contraction. This is as one might intuitively expect, but note that only half the change in Ω has been accounted for; the remaining energy is `lost' in the form of radiation. Overall, then, the eects of gravitational contraction are threefold (in addition to the trivial fact that the system gets smaller): 86

8 (i) The star gets hotter (ii) Energy is radiated away (iii) The total energy of the system, T, decreases (formally, it becomes more negative that is, more tightly bound); T = U + Ω = Ω/2 + Ω = 1 2 Ω These results have an obvious implication in star formation; a contracting gas cloud heats up until, eventually, thermonuclear burning may start. This generates heat, hence gas pressure, which opposes further collapse as the star comes in to hyostatic equilibrium Red Giants The evolution of solar-type main-sequence stars is a one of the most amatic features of stellar evolution. All numerical stellar-evolution models predict this transition, and yet we lack a simple, didactic physical explanation: Why do some stars evolve into red giants though some do not? This is a classic question that we consider to have been answered only unsatisfactorily. This question is related, in a more general context, to the formation of core-halo structure in self-gravitating systems. D. Sugimoto & M. Fujimoto (ApJ, 538, 837, 2) Nevertheless, semi-phenomenological descriptions aord some insight; these can be presented in various degrees of detail, of which the most straighforward argument is as follows. We simplify the stellar structure into an inner core and an outer envelope, with masses and radii M c, M e and R c, R e (= R ) respectively. At the end of core hyogen burning, we suppose that core contraction happens quickly faster than the Kelvin-Helmholtz timescale, so that the Virial Theorem holds, and thermal and gravitational potential energy are conserved to a satisfactory degree of approximation. We formalize this supposition by writing Ω + 2U = constant Ω + U = constant (Virial theorem) (Energy conservation) These two equalities can only hold simultaneously if both Ω and U are individually constant, summed over the whole star. In particular, the total gravitational potential energy is constant 87

9 Stars are centrally condensed, so we make the approximation that M c M e ; then, adding the core and envelope, Ω GM 2 c R c + GM cm e R We are interested in evolution, so we take the derivative with respect to time, d Ω dt = = GM 2 c R 2 c dr c dt GM cm e R 2 dr dt i.e., dr dr c = M c M e ( R R c ) 2. The negative sign demonstrates that as the core contracts, the envelope must expand a good rule of thumb throughout stellar evolution, and, in particular, what happens at the end of the main sequence for solar-type stars. 1.5 Mean Molecular Weight The `mean molecular weight', µ, is 2 simply the average mass of particles in a gas, expressed in units of the hyogen mass, m(h). That is, the mean mass is µm(h); since the number density of particles n is just the mass density ρ divided by the mean mass we have n = ρ µm(h) and P = nkt = ρ kt (1.12) µm(h) In order to evaluate µ we adopt the standard astronomical nomenclature of X = mass fraction of hyogen Y = mass fraction of helium Z = mass fraction of metals. (where implicitly X + Y + Z 1). For a fully ionized gas of mass density ρ we can infer number densities: No. of nuclei Element: H He Metals No. of electrons Xρ m(h) Xρ m(h) Y ρ 4m(H) 2Y ρ 4m(H) Zρ Am(H) (A/2)Zρ Am(H) 2 Elsewhere we've used µ to mean cos θ. Unfortunately, both uses of µ are completely standard; but fortunately, the context rarely permits any ambiguity about which `µ' is meant. And why `molecular' weight for a potentially molecule-free gas? Don't ask me... 88

10 where A is the average atomic weight of metals ( 16 for solar abundances), and for the nal entry we assume that A i 2Z i for species i with atomic number Z i. The total number density is the sum of the numbers of nuclei and electrons, n ρ m(h) (2X + 3Y /4 + Z /2) ρ µm(h) (1.13) (where we have set Z /2 + Z /A Z /2, since A 2). We see that µ (2X + 3Y /4 + Z /2) 1. (1.14) We can op the approximations to obtain a more general (but less commonly used) denition, µ 1 = i Z i + 1 f i A i where f i is the mass fraction of element i with atomic weight A i and atomic number Z i. For a fully ionized pure-hyogen gas (X = 1, Y = Z = ), µ = 1 /2; for a fully ionized pure-helium gas (Y = 1, X = Z = ), µ = 4 /3; for a fully ionized gas of solar abundances (X =.71, Y =.27, Z =.2), µ = Pressure and temperature in the cores of stars Solar values Section 1.1 outlines how the equations of stellar structure can be used to construct a stellar model. In the context of studying the solar neutrino problem, many very detailed models of the Sun's structure have been constructed; for reference, we give the results one of these detailed models, namely Bahcall's standard model bp2stodel.dat. This has T c = K ρ c = kg m 3 P c = N m 2 with 5% (95%) of the solar luminosity generated in the inner.1 (.2) R Central pressure (1) We can use the equation of hyostatic equilibrium to get order-of-magnitude estimates of conditions in stellar interiors; e.g., letting = R (for an approximate solution!) then = Gm(r)ρ(r) r 2 (1.1) 89

11 becomes P C P S R GM ρ R 2 where P C, P S are the central and surface pressures; and since P C P S, P C GM ρ = 3 4π R GM 2 R Pa (=N m 2 ) (1.15) (or about 1 1 atmospheres) for the Sun. Because of our crude approximation in integrating eqtn. (1.1) we expect this to be an underestimate (more nearly an average than a central value), and reference to the detailed shows it falls short by about an order of magnitude. Nevertheless, it does serve to demonstrate high core values, and the M 2 /R 4 dependence of P C. Central pressure (2) Another estimate can be obtained by dividing the equation of hyostatic equilibrium (1.1) by the equation of mass continuity (1.3): ffi = Gm(r) 4πr 4 Integrating over the entire star, Z M = PC PS = Z M Gm(r) 4πr 4 Evidently, because R r, it must be the case that Z M Gm(r) 4πr 4 Z M Gm(r) 4πR 4 {z }. (1.16) that is, = GM2 8πR 4 ; P C > GM2 8πR 4 > N m 2 (+P S, but P S ) for the Sun. This is a weaker estimate than, but is consistent with, eqtn. (1.15), and shows the same overall scaling of P C M 2 /R Central temperature For a perfect gas, P = nkt = ρkt µm(h) (1.12) 9

12 but so P C GM ρ R T C µm(h) GM k R ρ ρ C (1.15) K (1.17) for the Sun which is quite close to the results of detailed calculations. (Note that at these temperatures the gas is fully ionized and the perfect gas equation is an excellent approximation.) Mean temperature We can use the Virial Theorem to obtain a limit on the mean temperature of a star. We have 3 U = kt (r) n(r) dv (1.8) V 2 3 = 2 kt (r) ρ(r) µm(h) dv and = Ω = V M > M M > GM 2 2R 3 ρ(r) kt (r) 2 µm(h) Gm(r) r Gm(r) R ρ(r) (1.1) From the Virial Theorem, 2U = Ω (eqtn. 1.11), so 3k µm(h) M T > GM 2 2R The integral represents the sum of the temperatures of the innitesimal mass elements contributing to the integral; the mass-weighted average temperature is M T T = M M T = M > GM µm(h) 2R 3k 91

13 For the Sun, this evaluates to T ( ) > K (using µ =.61), i.e., kt 2 ev comfortably in excess of the ionization potentials of hyogen and helium (and enough to substantially ionize the most abundant metals), justifying the assumption of complete ionization in evaluating µ. 1.7 MassLuminosity Relationship We can put together our basic stellar-structure relationships to demonstrate a scaling between stellar mass and luminosity. From hyostatic equilibrium, = Gm(r)ρ(r) r 2 P M R ρ (1.1) but our equation of state is P = (ρkt )/(µm(h)), so T µm R. For stars in which the dominant energy transport is radiative, we have so that L(r) r2 dt k R T 3 r2 dt κ R ρ(r) T 3 (3.11) L RT 4 κ R ρ. From mass continuity (or by inspection) ρ M/R 3, giving i.e., L R4 T 4 κ R M R4 κ R M ( ) µm 4 ; R L µ4 κ R M 3. This simple dimensional analysis yields a dependency which is in quite good agreement with observations; for solar-type main-sequence stars, the empirical massluminosity relationship is L M

14 1.8 The role of radiation pressure So far we have, for the most part, considered only gas pressure. What about radiation pressure? From Section 1.8, the magnitude of the pressure of isotropic radiation is P R = 4σ 3c T 4. Thus the ratio of radiation pressure to gas pressure is P R/P G = 4σ / kρt 3c T 4 µm(h) [kg m 3 K 3 ] T 3 ρ (1.31) (1.18) For ρ = kg m 3 and T C ( ) 1 7 K, P R /P G.1. That is, radiation pressure is relatively unimportant in the Sun, even in the core. However, radiation pressure is is important in stars more massive than the Sun (which have hotter central temperatures). Eqtn (1.18) tells us that P R/P G T 3 ρ T 3 R 3 M but T M /R, from eqtn. (1.17); that is, P R/P G M The Eddington limit Radiation pressure plays a particular role at the surfaces of luminous stars, where the radiation eld is, evidently, no longer close to isotropic (the radiation is escaping from the surface). As we saw in Section 1.8, the momentum of a photon is hν/c. If we consider some spherical surface at distance r from the energy-generating centre of a star, where all photons are owing outwards (i.e., the surface of the star), the total photon momentum ux, per unit area per unit time, is / L (4πr 2 ) [J m 3 = kg m 1 s 2 ] c The Thomson-scattering cross-section (for electron scattering of photons) is [ σ T = 8π ( ) e 2 2 ] 3 m e c 2 = m 2. 93

15 This is a major opacity source in ionized atmospheres, so the force is exerted mostly on electrons; the radiation force per electron is F R = σ TL 4πr 2 c [J m 1 N] which is then transmitted to positive ions by electrostatic interactions. For stability, this outward force must be no greater than the inward gravitational force; for equality (and assuming fully ionized hyogen) σ T L 4πr 2 c = GM (m(h) + m e) r 2 GMm(H) r 2. This equality gives a limit on the maximum luminosity as a function of mass for a stable star the Eddington Luminosity, L Edd = 4πGMc m(h) σ T M M [J s 1 ], or L Edd L M M Since luminosity is proportional to mass to some power (roughly, L M 3 4 on the upper main sequence; cf. Section 1.7), it must be the case that the Eddington Luminosity imposes an upper limit to the mass of stable stars. (In practice, instabilities cause a super-eddington atmosphere to become clumpy, or `porous', and radiation is able to escape through paths of lowered optical depth between the clumps. Nevertheless, the Eddington limit represents a good approximation to the upper limit to stellar luminosity. We see this limit as an upper bound to the Hertzsprung-Russell diagram, the so-called `Humphreys-Davidson limit'.) 94

16 How to make a star (for reference only) 1.1 Introduction We've assembled a set of equations that embody the basic principles governing stellar structure; renumbering for convenience, these are (r) = 4πr 2 ρ(r) Mass continuity (1.3=1.19) = Gm(r)ρ(r) r 2 Hyostatic equilibrium (1.1=1.2) dl(r) = 4πr 2 ρ(r)ε(r) Energy continuity (1.7=1.21) and radiative energy transport is described by L(r) = 16π 3 r 2 k R (r) dt act 3 Radiative energy transport. (3.11=1.22) The quantities P, ε, and k R (pressure, energy-generation rate, and Rosseland mean opacity) are each functions of density, temperature, and composition; those functionalities can be computed separately from the stellar structure problem. For analytical work we can reasonably adopt power-law dependences, k R (r) = κ(r) = κ ρ a (r)t b (r) (2.6=1.23) and (anticipating Section 13) ε(r) ε ρ(r)t α (r), (13.9=1.24) together with an equation of state; for a perfect gas P (r) = n(r)kt (r). (1.12=1.25) Mass is a more fundamental physical property than radius (the radius of a solar-mass star will change by orders of magnitude over its lifetime, while its mass remains more or less constant), 95

17 so for practical purposes it is customary to reformulate the structure equations in terms of mass as the independent variable. We do this simply by dividing the last three of the foregoing equations by the rst, giving (r) = 1 4πr 2 ρ(r) (1.26) (r) = Gm(r) 4πr 4 (1.27) dl(r) (r) = ε(r) = ε ρ(r)t α (r) (1.28) dt (r) (r) = 3k RL(r) 16π 2 r 4 act 3 (r) (where all the radial dependences have been shown explicitly). (1.29) To solve this set of equations we require boundary conditions. At m = (i.e., r = ), L(r) = ; at m = M (r = R ) we set P =, ρ =. In practice, the modern approach is to integrate these equations numerically. 3 However, much progress was made before the advent of electronic computers by using simplied models, and these models still arguably provide more physical insight than simply running a computer program Homologous models Homologous stellar models are dened such that their properties scale in the same way with fractional mass m m(r)/m. That is, for some property X (which might be temperature, or density, etc.), a plot of X vs. m is the same for all homologous models. Our aim in constructing such models is to formulate the stellar-structure equations so that they are independent of absolute mass, but depend only on relative mass. We therefore recast the variables of interest as functions of fractional mass, with the dependency on absolute mass assumed to be a power law: r = M x 1 r (m) = M x 1 ρ(r) = M x 2 ρ (m) dρ(r) = M x 2 dρ T (r) = M x 3 T (m) dt (r) = M x 3 dt P (r) = M x 4 P (m) = M x 4 dp L(r) = M x 5 L (m) dl(r) = M x 5 dl 3 The interested student can run his own models using the EZ (`Evolve ZAMS') code; W. Paxton, PASP, 116, 699,

18 where the x i exponents are constants to be determined, and r, ρ etc. depend only on the fractional mass m. We also have m(r) = mm (r) = M and, from eqtns. (2.6=1.23, 13.9=1.24) κ(r) = κ ρ a (r)t b (r) b = κ ρ a (m)t b (m)m ax 2 M bx 3 ε(r) ε ρ(r)t α (r), = ε ρ (m)t α (m)m x 2 M αx 3, Transformed equations We can now substitute these into our structure equations to express them in terms of dimensionless mass m in place of actual mass m(r): Mass continuity: M (x 1 1) (r) = 1 4πr 2 ρ(r) (m) = becomes (1.26) 1 4πr 2 (m)ρ (m) M (2x 1+x 2 ) (1.3) The condition of homology requires that the scaling be independent of actual mass, so we can equate the exponents of M on either side of the equation to nd 3x 1 + x 2 = 1 (1.31) Hyostatic equlibrium: M (x 4 1) (r) = Gm(r) 4πr 4 becomes (1.27) dp (m) = Gm 4πr 4 M (1 4x 1) (1.32) whence 4x 1 + x 4 = 2 (1.33) Energy continuity: dl(r) (r) = ε(r) = ε ρ(r)t α (r) becomes (1.28) M (x 5 1) dl (m) = ε ρ (m)t α (m)m x 2+αx 3 (1.34) whence x 2 + αx = x 5 (1.35) 97

19 Radiative transport: dt (r) (r) = 3k RL(r) 16π 2 r 4 act 3 (r) becomes (1.29) M (x 3 1) dt (m) = 3(κ ρ a T b)l 5+(b 3)x 3 +ax 2 4x 1 ) 16π 2 r 4acT 3(m)M(x (1.36) whence 4x 1 + (4 b)x 3 = ax 2 + x (1.37) Equation of state: P (r) = n(r)kt (r) = ρ(r)kt (r) µm(h) (1.38) (neglecting any radial dependence of µ) M x 4 P (m) = ρ (m)kt (m) M (x 2+x 3 ) (1.39) µm(h) whence x 2 + x 3 = x 4 (1.4) We now have ve equations for the ve exponents x i, which can be solved for given values of a, b, and α, the exponents in eqtns. (2.6=1.23) and (13.9=1.24). This is quite tedious in general, but a simple solution is aorded by the (reasonable) set of parameters a = 1 b = 3.5 α = 4 (xxx see Tayler); the solutions include x 1 = 1 /13 x 5 = 71 /13( 5.5) which we will use in the next section. 98

20 Results Collecting the set structure equations that describe a sequence of homologous models: (m) dp (m) dl (m) = 1 4πr 2 (m)ρ (m) = Gm 4πr 4 (1.3) (1.32) = ε ρ (m)t α (m) (1.34) dt (m) = 3(κ ρ a T b)l 16π 2 r 4acT 3(m) (1.36) P (m) = ρ (m)kt (m) µm(h) These can be solved numerically using the boundary conditions r = L = at m =, P = ρ = at m = 1. (1.39) However, we can aw some useful conclusions analytically. First, it's implicit in our denition of homologous models that there must exist a mass-luminosity relation for them; since L = M x 5 L (1) it follows immediately that L M x 5 ( M 5.5 ) which is not too bad compared to the actual main-sequence massluminosity relationship (L M 3 4, especially considering the crudity of the modelling. Secondly, since it follows that L R 2 T 4 eff R = M x 1 r (1) L = M x 5 L (1), or M x 5 L (1) M 2x 1 r(1)t 2 eff 4 M T 4 /(x 5 2x 1 ) eff T 52 /69 eff. 99

21 This is a bit o the mark for real stars, which show a slower increase in temperature with mass; but combining this with the mass-luminosity relationship gives L T 4x 5/(x 5 2x 1 ) eff T 284 /69 eff which again is not too bad for the lower main sequence indeed, the qualitative result that there is a luminositytemperature relationship is, in eect, a prediction that a main sequence exists in the HR diagram Polytropes and the Lane-Emden Equation Another approach to simple stellar-structure models is to assume a `polytropic' formalism. We can again start with hyostatic equilibrium and mass continuity, = Gm(r)ρ(r) r 2 = 4πr2 ρ(r). Dierentiating eqtn. (1.1) gives us ( ) d r 2 dp = G (r) ρ (1.1) (1.3) whence, from eqtn. (1.3), ( ) 1 d r 2 dp r 2 = 4πGρ (1.41) ρ This is, essentially, already the `Lane-Emden equation' (named for the American astronomer Jonathan Lane and the Swiss Jacob Emden). It appears that we can't solve hyostatic equilibrium without knowing something about the pressure, i.e., the temperature, which in turn suggests needing to know about energy generation processes, opacities, and other complexities. Surprisingly, however, we can solve for temperature T without these details, under some not-too-restrictive assumptions about the equation of state. Specically, we adopt a polytropic law of adiabatic expansion, P = Kρ γ = Kρ 1 /n+1 (1.42) where K is a constant, γ is the ratio of specic heats, 4 and in this context n is the adopted polytropic index. 4 This relation need not necessarily be taken to be an equation of state it simply expresses an assumption regarding the evolution of pressure with radius, in terms of the evolution of density with radius. The Lane-Emden equation has applicability outside stellar structures. 1

22 Introducing eqtn. (1.42) into (1.41) gives ( K d r 2 dρ γ ) r 2 = 4πGρ. (1.43) ρ We now apply the so-called Emden transformation, ξ = r α, θ = T T c (1.44) (where α is a constant and T c is the core temperature), to eqtn. (1.42), giving ( ρ ρ c ) 1 1/γ ( ) ρ P 1/γ ( ) ρt 1/γ = = ; i.e., ρ c P c ρ c T c ( ) T 1/γ = = θ 1/γ, or T c ρ = ρ c θ 1 /γ 1 (where we have also used P = nkt, ρt ), whence eqtn. (1.43) becomes ( ) K d (αξ)2 d {ρ c θ n } (n+1) /n (αξ) 2 d(αξ) ρ c θ n = 4πGρ c θ n. d(αξ) Since we have d dξ (θn+1 ) = (n + 1)θ n dθ dξ (n + 1)Kρ 1 /n 1 c 4πGα 2 1 ξ 2 d dξ ( ξ 2 dθ ) = θ n. dξ Finally, letting the constant α (which is freely selectable, provided its dimensionality length is preserved) be α [ ] (n + 1)Kρ 1 /n 1 1/2 c, 4πG we obtain the Lane-Emden equation, relating (scaled) radius to (scaled) temperature: ( 1 d ξ 2 ξ 2 dθ ) = θ n, dξ dξ or, equivalently, 1 ξ 2 ( 2ξ dθ dξ + ξ2 d2 θ dξ 2 ) + θ n = d2 θ dξ dθ ξ dξ + θn =. 11

23 We can solve this equation with the boundary conditions θ = 1, dθ dξ = at ξ = (recalling that ξ and θ are linear functions of r and T ). Numerical solutions are fairly straightforward to compute; analytical solutions are possible for polytropic indexes n =, 1 and 5 (i.e., ratios of specic heats γ =, 2, and 1.2); these are, respectively, θ(ξ) = 1 ξ 2 /6 ζ = 6, = sin ξ/ξ = π, and = ( 1 + ξ 2 /3 ) 1/2 = A polytrope with index n = has a uniform density, while a polytrope with index n = 5 has an innite radius. A polytrope with index n = corresponds to a so-called `isothermal sphere', a self-gravitating, isothermal gas sphere, used to analyse collisionless systems of stars (in particular, globular clusters). In general, the larger the polytropic index, the more centrally condensed the density distribution. We've done a lot of algebra; what about the physical interpretation of all this? What about stars? Neutron stars are well modeled by polytropes with index about in the range n =.51. A polytrope with index n = 3 /2 provides a reasonable model for degenerate stellar cores (like those of red giants), white dwarfs, brown dwarfs, gas giants, and even rocky planets. Main-sequence stars like our Sun are usually modelled by a polytrope with index n = 3, corresponding to the Eddington standard model of stellar structure. 12