2 Properties of polytropic gas spheres

   
2.1 The Lane-Emden equation

Polytropic stars are characterized by an equation of state of the form

$$p=K\rho^{\gamma},$$ (1)

where K and $\gamma$ are constants. The index n of the polytrope is defined by the relation

$$\gamma=1+{1\over n}\cdot$$ (2)

The equation of state (1) corresponds to an adiabatic equilibrium in regions where convection keeps the star stirred up and produces a uniform entropy distribution ( $s={\rm const.}$). In that case, $\gamma$ is the ratio of specific heats c_p/c_V at constant pressure and volume. For a monoatomic gas, $\gamma=5/3$. The equation of state (1) also describes a polytropic equilibrium characterized by a uniform specific heat $c\equiv \delta Q/{\rm d}T$. In this more general situation $\gamma=(c_{\rm p}-c)/(c_{V}-c)$. Convective equilibrium is recovered for c=0 and isothermal equilibrium is obtained in the limit of infinite specific heat $c\rightarrow +\infty$. A power law relation between the pressure and the density is also the limiting form of the equation of state describing a gas of cold degenerate fermions (Chandrasekhar 1932). In that case, the constant K can be expressed in terms of fundamental constants. In the classical limit $\gamma=5/3$, n=3/2 and $K={1\over 20}(3/\pi)^{2/3}h^{2}/ m$ (where h is the Planck constant) and in the relativistic limit $\gamma=4/3$, n=3 and $K={1\over 8}(3/\pi)^{1/3}hc$ (where c is the speed of light). Historically, the index $\gamma=5/3$ appears in the classical theory of white dwarf stars initiated by Fowler (1926) and the index $\gamma=4/3$ is related to the limiting mass of Chandrasekhar (1931).

The condition of hydrostatic equilibrium for a spherically symmetrical distribution of matter reads

$${{\rm d}p\over {\rm d}r}=-\rho{{\rm d}\Phi\over {\rm d}r},$$ (3)

where $\Phi$ is the gravitational potential. Using the Gauss theorem

$${{\rm d}\Phi\over {\rm d}r}={GM(r)\over r^{2}},$$ (4)

where $M(r)=\int_{0}^{r}\rho 4\pi r^{2}{\rm d}r$ is the mass contained within the sphere of radius r, we can derive the fundamental equation of equilibrium (Chandrasekhar 1932)

$${1\over r^{2}}{{\rm d}\over {\rm d}r}\biggl ({r^{2}\over\rho}{{\rm d}p\over {\rm d}r}\biggr )=-4\pi G\rho.$$ (5)

Equations (1), (5) fully determine the structure of polytropic gas spheres. Letting

$$\rho=\rho_{0}\theta^{n},\qquad \xi=\biggl\lbrack {4\pi G\rho_{0}^{1-1/n}\over K(n+1)}\biggr\rbrack^{1/2}r,$$ (6)

where $\rho_{0}$ is the central density, we can reduce the condition of hydrostatic equilibrium to the Lane-Emden equation (Chandrasekhar 1932)

$${1\over\xi^{2}}{{\rm d}\over {\rm d}\xi}\biggl (\xi^{2}{{\rm d}\theta\over {\rm d}\xi}\biggr )=-\theta^{n},$$ (7)

with boundary conditions

$$\theta(0)=1,\qquad \theta'(0)=0.$$ (8)

For n>3, the Lane-Emden equation admits an analytical solution which is singular at the origin:

$$\theta_{s}=\biggl\lbrack {2(n-3)\over (n-1)^{2}}\biggr \rbrack^{1\over n-1}{1\over \xi^{2\over n-1}}\cdot$$ (9)

Regular solutions of the Lane-Emden equation must in general be computed numerically. For $\xi\rightarrow 0$, we can expand the function $\theta$ in Taylor series. The first terms in this expansion are given by

$$\theta=1-{1\over 6}\xi^{2}+{n\over 120}\xi^{4}+...$$ (10)

The behavior of $\theta(\xi)$ at large distances deserves a more specific discussion. For 1<n<5, the density falls off to zero at a finite radius R, identified as the radius of the star. If we denote by $\xi_{1}$ the value of the normalized distance at which $\theta=0$then, for $\xi\rightarrow \xi_{1}$, we have

$$\theta=-\xi_{1}\theta'_{1}\biggl \lbrack {\xi_{1}-\xi\over\xi... ...biggl ({\xi_{1}-\xi\over\xi_{1}}\biggr )^{3}+...\biggr\rbrack.$$ (11)

For n>5, the polytropes extend to infinity, like the isothermal configurations recovered in the limit $n= \infty$. For $\xi\rightarrow +\infty$,

 

$$\theta=\biggl\lbrack {2(n-3)\over (n-1)^{2}}\biggr \rbrack^{1\ove... ...k {\sqrt{7n^{2}-22n-1}\over 2(n-1)}\ln\xi+\delta\biggr\rbrack\biggr\rbrace\cdot$$ (12)

The curve (12) intersects the singular solution (9) infinitely often at points that asymptotically increase geometrically in the ratio 1: ${\rm exp}\lbrace 2(n-1)\pi/\sqrt{7n^{2}-22n-1}\rbrace$. Since $\theta^{n}\sim \xi^{-2n\over n-1}$ at large distances, these configurations have an "infinite mass'', which is clearly unphysical. In the following, we shall restrict these configurations to a "box'' of radius R, as in the classical Antonov problem. Therefore, Eq. (7) must be solved for $\xi\le \alpha$ with

$$\alpha=\biggl\lbrack {4\pi G\rho_{0}^{1-1/n}\over K(n+1)}\biggr\rbrack^{1/2}R.$$ (13)

Note that, for a fixed box radius R, $\alpha$ is a measure of the central density $\rho_{0}$. The case n=5 is special. For this index, the Lane-Emden equation can be solved analytically and yields the result:

$$\theta={1\over (1+{1\over 3}\xi^{2})^{1/2}}\cdot$$ (14)

The total mass of this configuration is finite but its potential energy diverges. Therefore, this polytrope must also be confined within a box. In Fig. 1, we have plotted different density profiles with index n=3,5,6.

Figure 1: Density profiles of polytropes with index n=3,5,6. The dashed line corresponds to the singular solution (9).

   
2.2 The Milne variables

As in the analysis of isothermal gas spheres, it will be convenient in the following to introduce the Milne variables (u,v) defined by (Chandrasekhar 1932):

$$u=-{\xi\theta^{n}\over\theta'},\qquad v=-{\xi\theta'\over\theta}\cdot$$ (15)

Taking the logarithmic derivative of u and v with respect to $\xi$ and using Eq. (7), we get

$${1\over u}{{\rm d}u\over {\rm d}\xi}={1\over \xi}(3-nv-u),$$ (16)

$${1\over v}{{\rm d}v\over {\rm d}\xi}={1\over\xi}(u+v-1).$$ (17)

Due to the homology invariance of the polytropic configuations (see Chandrasekhar 1932), the Milne variables u and v satisfy a first order differential equation

$${u\over v}{{\rm d}v\over {\rm d}u}=-{u+v-1\over u+nv-3}\cdot$$ (18)

The solution curve in the (u,v) plane (see Figs. 2-4) is parametrized by $\xi$. It starts from the point (u,v)=(3,0) with a slope $({\rm d}v/{\rm d}u)_{0}=-{5\over 3n}$ as $\xi\rightarrow 0$. For 1<n<5, the curve is monotonous and tends to $(u,v)=(0,+\infty)$ as $\xi\rightarrow \xi_{1}$. More precisely, using Eq. (11), we have

$$uv^{n}\sim \omega_{n}^{n-1},\qquad \omega_{n}=-\xi_{1}^{n+1\over n-1}\theta'_{1} \qquad (\xi\rightarrow \xi_{1}).$$ (19)

For n>5, the solution curve spirals indefinitely around the fixed point $(u_{{\rm s}},v_{{\rm s}})=({n-3\over n-1}, {2\over n-1})$, corresponding to the singular solution (9), as $\xi$ tends to infinity. All polytropic spheres must necessarily lie on this curve. For bounded polytropic spheres, $\xi$ must be terminated at the box radius $\alpha$. For n=5, the Milne variables are related according to

 

u+3v=3. (20)

Figure 2: The (u,v) plane for polytropes with index 1<n<5. The construction is made explicitly for n=4.

Figure 3: The (u,v) plane for polytropes with index n=5.

Figure 4: The (u,v) plane for polytropes with index n>5. The construction is made explicitly for n=6.

   
2.3 The maximum mass and minimum temperature of confined polytropes

For polytropes confined within a box of radius R, there exists a well-defined relation between the mass M of the configuration and the central density $\rho_{0}$ (through the parameter $\alpha$). Starting from the relation

 

$$M=\int_{0}^{R}\rho 4\pi r^{2}{\rm d}r =4\pi\rho_{0}\biggl\lbrack ... ...1}\over 4\pi G}\biggr\rbrack^{3/2}\int_{0}^{\alpha}\theta^{n}\xi^{2}{\rm d}\xi,$$ (21)

and using the Lane-Emden Eq. (7), we get

$$M=-4\pi \rho_{0}\biggl\lbrack {K(1+n)\rho_{0}^{1/n-1}\over 4\pi G}\biggr\rbrack^{3/2}\alpha^{2}\theta'(\alpha).$$ (22)

Expressing the central density in terms of $\alpha$, using Eq. (13), we obtain after some rearrangements

$$M=-4\pi\biggl\lbrack {K(1+n)\over 4\pi G}\biggr\rbrack^{n\over n-1} R^{n-3\over n-1}\alpha^{n+1\over n-1}\theta'(\alpha).$$ (23)

Introducing the parameter

$$\eta\equiv {M\over 4\pi}\biggl\lbrack { 4\pi G\over K(1+n)}\biggr\rbrack^{n\over n-1}{1\over R^{n-3\over n-1} },$$ (24)

the foregoing relation can be rewritten

$$\eta=-\alpha^{n+1\over n-1}\theta'(\alpha).$$ (25)

For n<5, the normalized box radius $\alpha$ in necessarily restricted by the inequality $\alpha\le\xi_{1}$. For the limiting value $\alpha=\xi_{1}$, corresponding to an isolated polytrope satisfying $\rho(R)=0$, we have

$$\eta(\xi_{1})=\omega_{n}.$$ (26)

The quantity $\omega_{n}$, defined by Eq. (19), has been tabulated by Chandrasekhar (1932). The definition (24) of $\eta$ and the relation (25) between $\eta$ and $\alpha$ are consistent with the formulae derived in the case of an isothermal gas (see Chavanis 2002a). This connexion is particularly relevant if we interpret the constant K as a polytropic temperature $\Theta_{\gamma}$ (see Chandrasekhar 1932, p. 86). For $n\rightarrow +\infty$, $\Theta_{\gamma}=T$ and the parameter $\eta$ reduces to the corresponding one for isothermal spheres ( $\eta\sim \eta_{\infty}/n$, $\eta_{\infty}={\beta GMm\over R}$, $\beta=1/kT$).

Figure 5: Mass-density profiles for polytropic configurations with index n=2,3,4,5,6. A mass peak appears for the first time for the critical index n=3.

The function $\eta(\alpha)$ is represented in Fig. 5 for different values of the polytropic index n. Instead of $\alpha$, we could have used the density contrast

$${\cal R}\equiv {\rho_{0}\over \rho(R)}=\theta(\alpha)^{-n},$$ (27)

which also provides a relevant parametrization of the solutions. Using the Lane-Emden Eq. (7) and the definition of the Milne variables (15), it is straightforward to check that the condition of extremum ${\rm d}\eta/{\rm d}\alpha=0$ is equivalent to

$$u_{0}={n-3\over n-1}\equiv u_{{\rm s}},$$ (28)

where, by definition, $u_{0}=u(\alpha)$ and $u_{{\rm s}}$ refers to the singular solution (9). For $n\rightarrow \infty$, we recover the condition u₀=1 previously derived for isothermal configurations (Chavanis 2002a). The values of $\alpha$ for which $\eta$ is extremum are given by the intersection(s) between the solution curve in the (u,v) plane and the line u=u_s (see Figs. 2-4). For 1<n<3, $u_{{\rm s}}<0$ and there is no intersection. The mass-density relation is therefore monotonous. For $3\le n\le 5$, the curve $\eta(\alpha)$ presents a single maximum at $\alpha_{1}$. For n=3, this maximum is reached at the extremity of the curve ( $\alpha_{1}=\xi_{1}$). For n=5, $\xi_{1}\rightarrow +\infty$ and the function $\eta(\alpha)$ is explicitly given by

$$\eta={\alpha^{5/2}\over 3(1+{1\over 3}\alpha^{2})^{3/2}}\cdot$$ (29)

The maximum of $\eta$ is located at $\alpha_{1}=\sqrt{15}$. Finally, for n>5, the mass-density relation presents an infinite number of damped oscillations since the line defined by Eq. (28) passes through the center of the spiral in the (u,v) plane. If we denote by $\alpha_{k}$ the locii of the extrema of $\eta(\alpha)$, these values asymptotically follow the geometric progression

$$\alpha_{k}\sim \biggl\lbrack {\rm e}^{2(n-1)\pi\over \sqrt{7n... ...biggr\rbrack^{k}\qquad (k\rightarrow \infty, \ {\rm integer}),$$ (30)

obtained by substituting the asymptotic expansion (12) in Eq. (28). For $n\rightarrow \infty$, we recover the ratio ${\rm e}^{2\pi/\sqrt{7}}=10.74...$ corresponding to classical isothermal configurations (Semelin et al. 1999; Chavanis 2002a).

From the above results, it is clear that restricted polytropic spheres with index $n\ge 3$ can exist only for

$$\eta\le -\alpha_{1}^{n+1\over n-1}\theta'(\alpha_{1})\equiv\eta_{{\rm max}}.$$ (31)

This implies in particular the existence of a limiting mass (for a given confining radius R) such that

$$M\le M_{{\rm max}}=4\pi\eta_{{\rm max}}\biggl\lbrack { K(1+n)\over 4\pi G}\biggr\rbrack^{n\over n-1} R^{n-3\over n-1}.$$ (32)

For $n= \infty$, we recover the limiting mass $M_{{\rm c}}=2.52{RkT\over Gm}$ above which an isothermal sphere cannot sustain self-gravity (see, e.g., Chavanis 2002b). Alternatively, for a given mass M and radius R, the inequality (31) implies the existence of a minimum value of the polytropic temperature $\Theta_{\gamma}=Km/k$. Indeed, $\Theta_{\gamma}$ is restricted by the inequality

$$\Theta_{\gamma}\ge {4\pi Gm\over k(1+n)}\biggl ({M\over 4\pi ... ..._{{\rm max}}}\biggr )^{n-1\over n}{1\over R^{n-3\over n}}\cdot$$ (33)

In the limit $n= \infty$, we recover the critical temperature $kT_{{\rm c}}={GMm\over 2.52R}$ below which an isothermal sphere is expected to collapse (Lynden-Bell & Wood 1968). This critical point, corresponding to ${\rm d}\eta/{\rm d}\alpha=0$, appears for the first time for the index n=3. This observation will take a deeper physical significance in the stability analysis performed in the following section. For $n\le 3$, the mass-density profile is monotonous and $\eta\le \omega_{n}$. The total mass (resp. temperature) of confined polytropes is always smaller (resp. larger) than the corresponding one for isolated polytropes. However, this bound does not correspond to a condition of extremum ${\rm d}\eta/{\rm d}\alpha=0$ but rather to the impossibility of constructing polytropes with $\alpha>\xi_{1}$ (since $\theta$ can become negative).