3 Dynamical stability of polytropic gas spheres

   
3.1 The equation of radial pulsations

We shall now investigate the dynamical stability of polytropic gas spheres against radial perturbations. We use a method and presentation similar to that adopted for isothermal spheres in Chavanis (2002a). The equations describing the motions of a gaseous star are the equation of continuity, the Euler equation and the Poisson equation

$${\partial \rho\over \partial t}+\nabla (\rho {\vec v})=0,$$ (34)

 

$${\partial {\vec v}\over \partial t}+({\vec v}\nabla){\vec v}=-{1\over\rho}\nabla p-\nabla\Phi,$$ (35)

$$\Delta\Phi=4\pi G\rho.$$ (36)

It is possible to show that viscosity does not change the onset of instability. Therefore, for simplicity, we have directly used the Euler equation instead of the Navier-Stokes equation. We assume furthermore that the pressure and the density are related to each other by the polytropic equation of state (1). We write small perturbations around equilibrium in the form

$${\vec v}=\delta {\vec v}({\vec r},t),\qquad \rho=\overline{\rho}+\delta \rho({\vec r},t),$$ (37)

$$p=\overline{p}+\delta p({\vec r},t),\qquad \Phi=\overline{\Phi}+\delta \Phi({\vec r},t),$$ (38)

where the bar refers to the stationary solution (in the following we shall drop the bar). The linearized equations for the perturbations are

$$\rho{\partial {\delta{\vec v}}\over \partial t}=-K\gamma \nab... ...gamma-1}\delta\rho)-\delta\rho\nabla\Phi-\rho\nabla\delta\Phi,$$ (39)

$${\partial \delta \rho\over \partial t}+\nabla (\rho {\delta{\bf v}})=0,$$ (40)

$$\Delta\delta\Phi=4\pi G\delta\rho.$$ (41)

Restricting ourselves to radial perturbations and writing the time dependance of the perturbation in the form $\delta v\sim {\rm e}^{\lambda t}$, $\delta\rho\sim {\rm e}^{\lambda t}$,..., the equations of the problem become

$$\lambda\rho\delta v=-K\gamma{{\rm d}\over {\rm d}r}(\rho^{\ga... ...m d}\Phi\over {\rm d}r}-\rho{{\rm d}\delta\Phi\over {\rm d}r},$$ (42)

$$\lambda \delta \rho+ {1\over r^{2}}{{\rm d}\over {\rm d}r}(\rho r^{2}\delta{v})=0,$$ (43)

$${1\over r^{2}}{{\rm d}\over {\rm d}r}\biggl ( r^{2}{{\rm d}\delta\Phi\over {\rm d}r}\biggr )=4\pi G\delta\rho.$$ (44)

We introduce the function q(r) by the relation

$$\delta\rho={1\over 4\pi r^{2}}{{\rm d}q\over {\rm d}r}\cdot$$ (45)

Physically, q(r) represents the mass perturbation $\delta M(r)=\int_{0}^{r}\delta\rho 4\pi r^{2}{\rm d}r$ within the sphere of radius r. Thus, by definition, q(0)=0. Then, it is readily seen that the Poisson equation (44) is equivalent to

$${{\rm d}\delta\Phi\over {\rm d}r}={Gq\over r^{2}},$$ (46)

which is just the perturbed Gauss theorem (4). On the other hand, the continuity Eq. (43) leads to the relation

$$\delta v=-{\lambda\over 4\pi \rho r^{2}}q.$$ (47)

Substituting these results back into Eq. (42), we obtain

 

$${\lambda^{2}\over 4\pi r^{2}}q=K\gamma{{\rm d}\over {\rm d}r}\big... ... r^{2}}{{\rm d}q\over {\rm d}r}{{\rm d}\Phi\over {\rm d}r}+{G\rho\over r^{2}}q.$$ (48)

Using the condition of hydrostatic equilibrium (3), the foregoing equation can be rewritten

$$K\gamma {{\rm d}\over {\rm d}r}\biggl ({\rho^{\gamma-2}\over ... ...}\biggr )+{Gq\over r^{2}}={\lambda^{2}\over 4\pi \rho r^{2}}q,$$ (49)

which is the required equation of radial pulsations for a polytrope.

   
3.2 The condition of marginal stability

Considering the case of marginal stability $\lambda=0$ and introducing the dimensionless variables defined in Sect. 2, we can reduce the equation of radial pulsations to the form

$${{\rm d}\over {\rm d}\xi}\biggl ({\theta^{1-n}\over\xi^{2}}{{\rm d}F\over {\rm d}\xi}\biggr )+{nF\over\xi^{2}}=0$$ (50)

with F(0)=0. Denoting by ${\cal L}$ the differential operator appearing in Eq. (50) and using the Lane-Emden Eq. (7), we easily establish that

 

$${\cal L}(\xi^{2}\theta') {{\rm d}\over {\rm d}\xi}\biggl ({\theta^{1-n}\over\xi^{2}}{{\rm d}\over {\rm d}\xi}(\xi^{2}\theta')\biggr )+n\theta'\qquad\qquad\qquad$$ =

$$-{{\rm d}\over {\rm d}\xi}(\theta^{1-n}\times \theta^{n})+n\theta'=(n-1)\theta',$$ = (51)

 

$${\cal L}(\xi^{3}\theta^{n}) {{\rm d}\over {\rm d}\xi}\biggl ({\theta^{1-n}\over\xi^{2}}{{\rm d}\over {\rm d}\xi}(\xi^{3}\theta^{n})\biggr )+n\xi\theta^{n} \qquad$$ =

$$(3+n)\theta'+\xi n\theta''+n\xi\theta^{n} =(3-n)\theta'.$$ = (52)

Therefore, the general solution of Eq. (50) is

$$F(\xi)=c_{1}\biggl (\xi^{3}\theta^{n}+{n-3\over n-1}\xi^{2}\theta'\biggr ),$$ (53)

where c₁ is an arbitrary constant. The connexion with isothermal configurations (see Chavanis 2002a) is particularly obvious if we make the correspondance $\theta'\leftrightarrow \psi'$ and $\theta^{n}\leftrightarrow {\rm e}^{-\psi}$.

   
3.3 Boundary conditions

The equation of pulsations (49) must be supplemented by appropriate boundary conditions. For self-confined polytropes (1<n<5), we require that the Lagrangian derivative of the pressure vanishes at the surface of the configuration, i.e.

$${{\rm d}\over {\rm d}t}\delta p+\delta v{{\rm d}p\over {\rm d}r}=0, \qquad {\rm at}\ r=R.$$ (54)

Using Eqs. (45), (47), this condition can be rewritten

$${p\over\rho}\biggl ({{\rm d}q\over {\rm d}r}-{q\over\rho}{{\rm d}\rho\over {\rm d}r}\biggr )=0, \qquad {\rm at}\ r=R,$$ (55)

or, in dimensionless form,

$$\theta{{\rm d}F\over {\rm d}\xi}-n{{\rm d}\theta\over {\rm d}\xi}F=0, \qquad {\rm at}\ \xi=\xi_{1}.$$ (56)

The index of the marginally stable polytrope is obtained by substituting the solution (53) in the boundary condition (56). Since $\theta(\xi_{1})=0$, this yields n=3. Therefore, the transition from stability to instability occurs for a polytropic index n=3 or, equivalently, for an adiabatic index $\gamma=4/3$. Of course, this result is well-known and could have been obtained from the general theorems of stellar pulsation (see, e.g., Cox 1980). However, we provide here an alternative method based on the explicit resolution of the equation of pulsations for polytropes.

According to Eq. (45), the perturbation profile at the point of marginal stability is given by

$${\delta\rho\over\rho_{0}}={1\over 4\pi\xi^{2}}{{\rm d}F\over {\rm d}\xi},$$ (57)

with the expression (53) for $F(\xi)$. Simplifying the derivative with the aid of the Lane-Emden Eq. (7), we obtain

$${\delta\rho\over\rho}={3\over 4\pi}c_{1}\biggl ({2\over n-1}-v\biggr ),$$ (58)

where v is the Milne variable (15). For n=3, it reduces to

$$\delta\rho={3\over 4\pi}c_{1}\rho_{0}\theta^{3}(1-v).$$ (59)

This perturbation profile is plotted in Fig. 6. It has one node at $\xi_{(1)}=2.16...$ On the other hand, the velocity profile is given by Eq. (47). Normalizing with a typical velocity $c_{{\rm S}}$, we find that

$${\delta v\over c_{{\rm S}}}=-{\lambda'\over 4\pi}c_{1}{\theta'\over\theta^{n}}\biggl ({n-3\over n-1}-u\biggr ).$$ (60)

We assume that we are just at the onset of the instability $(\lambda'=0^{+})$ so that Eq. (60) is applicable with $\lambda'>0$. Substituting explicitly for n=3, we get

$${\delta v\over c_{{\rm S}}}=-{\lambda'\over 4\pi}c_{1}\xi,$$ (61)

and we observe that the velocity is proportional to the radial distance (see also Cox 1980).

Figure 6: Density perturbation profile for the critical polytrope of index n=3.

For polytropes confined within a box, the boundary condition consistent with the conservation of mass is $F(\alpha)=0$. With the expression (53) for $F(\xi)$, this yields

$$u_{0}={n-3\over n-1},$$ (62)

which is precisely Eq. (28). Therefore, the point of marginal stability coincides with the point of maximum mass (or minimum temperature) in the series of equilibrium. For n<3, restricted polytropes are always stable since the function $\eta(\alpha)$ is monotonous. For $3\le n\le 5$, the configurations with $\alpha> \alpha_{1}$ are unstable but there are no secondary instabilities. For n>5, a first mode of stability is lost for $\alpha=\alpha_{1}$. Subsequent oscillations in the mass-density profile are associated with secondary instabilities. A summary of the stability limits for polytropic gas spheres is given in Table 1. In Fig. 7 we have plotted the density contrast at the critical point $\alpha=\alpha_{1}$ as a function of the index of the polytrope. For $n\le 3$, the polytropes are stable and ${\cal R}_{{\rm c}}=+\infty$. For n>3, only configurations with a density contrast ${\cal R}\le {\cal R}_{{\rm c}}$ are stable. For $n\rightarrow +\infty$, the critical density contrast tends to its classical value ${\cal R}_{{\rm c}}=32.1$ (see Chavanis 2002a).

Figure 7: Critical density contrast as a function of the polytropic index n.

Using Eqs. (58), (60), the profiles of density and velocity that trigger the instabilities at the critical points can be written

$${\delta\rho\over\rho}={3\over 4\pi}c_{1}(v_{{\rm s}}-v ),$$ (63)

$${\delta v\over c_{{\rm S}}}=-{\lambda'\over 4\pi}c_{1}{\theta'\over\theta^{n}}(u_{{\rm s}}-u).$$ (64)

where we recall that $\xi\le\alpha_{k}$ for the mode of instability of order k. We can determine the qualitative behavior of these profiles graphically by considering the intersections between the solution curve in the (u,v) plane and the lines $v=v_{{\rm s}}$ and $u=u_{{\rm s}}$ (see Figs. 2-4). These intersections determine the nodes of the perturbation profile. For $3\le n\le 5$, there is only one intersection since the (u,v) curve is monotonous. For n=5, we obtain the analytical solutions

$${\delta\rho\over\rho}={3\over 8\pi}c_{1}{3-\xi^{2}\over 3+\xi^{2}},$$ (65)

$${\delta v\over c_{{\rm S}}}={\lambda'\over 72\pi}c_{1}\xi(\xi^{2}-15).$$ (66)

The perturbation $\delta\rho$ has its node at $\xi_{(1)}=\sqrt{3}$. For n>5, there are several intersections with the spiral since the lines $v=v_{{\rm s}}$ and $u=u_{{\rm s}}$ pass through the fixed point. The description of the perturbation profiles is similar to the one given for isothermal spheres ($n= \infty$). The density profile of the fundamental mode of instability (at $\alpha=\alpha_{1}$) has only one node. High order modes of instability (at $\alpha=\alpha_{k}$, k>1) present numerous oscillations whose nodes asymptotically follow a geometric progression. We refer to our previous paper (Chavanis 2002a) for a more precise description of these results.

  

Table 1: Summary of the stability limits for polytropic gas spheres.