SSC/Stability/Polytropes/Pt2

Radial Oscillations of Polytropic Spheres

Part I:   Wave Equation

Part II:  Boundary Conditions

III:  Tables

Boundary Conditions

As we have pointed out in the context of a general discussion of boundary conditions associated with the adiabatic wave equation, the eigenfunction, $x$, will be suitably well behaved at the center of the configuration if,

$\frac{dx}{d{r}_0}=0$        at         $r_0=0,$

which, in the context of our present discussion of polytropic configurations, leads to the inner boundary condition,

$\frac{dx}{d\xi }=0$        at         $\xi =0.$

This is precisely the inner boundary condition specified by HRW66 — see their equation (57), which has been reproduced in the above excerpt from HWR66.

As we have also shown in the context of this separate, general discussion of boundary conditions associated with the adiabatic wave equation, the pressure fluctuation will be finite at the surface — even if the equilibrium pressure and/or the pressure scale height go to zero at the surface — if the radial eigenfunction, $x$, obeys the relation,

$r_0\frac{dx}{d{r}_0}$

$=$

$(4-3{\gamma }_g+\frac{{\omega }^2{R}^3}{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}})\frac{x}{{\gamma }_g}$        at         $r_0=R.$

Or, given that, in polytropic configurations, $r_0=a_n\xi$,

$\xi \frac{dx}{d\xi }$

$=$

$\frac{x}{{\gamma }_g}[4-3{\gamma }_g+\frac{{\omega }^2(a_n{\xi }_1{)}^3}{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}]$        at         $\xi ={\xi }_1,$

where, the subscript "1" denotes equilibrium, surface values. As can be deduced from our above summary of the properties of polytropic configurations,

$G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$=$

$4\pi G{a}_n^3{\rho }_c(-{\xi }_1^2{\theta }_1^{\text{'}}).$

Hence, for spherically symmetric polytropic configurations, the surface boundary condition becomes,

$\frac{dx}{d\xi }$	$=$	$\frac{x}{{\gamma }_g\xi }[4-3{\gamma }_g+{\omega }^2(\frac{1}{4\pi G{\rho }_c}\left)\frac{\xi }{(-{\theta }^{\text{'}})}\right]$         at         $\xi ={\xi }_1,$
$\Rightarrow (n+1)\frac{dx}{d\xi }$	$=$	$\frac{x}{{\gamma }_g\xi }[(n+1)(4-3{\gamma }_g)+{\omega }^2(\frac{1+n}{4\pi G{\rho }_c}\left)\frac{\xi }{(-{\theta }^{\text{'}})}\right]$
	$=$	$-\frac{x}{{\gamma }_g\xi }[(n+1)(3{\gamma }_g-4)-{\omega }^2(\frac{1+n}{4\pi G{\rho }_c}\left)\frac{\xi }{(-{\theta }^{\text{'}})}\right]$         at         $\xi ={\xi }_1.$

Adopting notation used by HRW66, specifically, as demonstrated above,

$-{\omega }^2\left(\frac{1+n}{4\pi G{\rho }_c}\right)\to (s^{\text{'}}{)}^2,$

and, from equation (50) of HRW66,

$-{\theta }^{\text{'}}\to q$         at         $\xi ={\xi }_1,$

this outer boundary condition becomes,

$(n+1)\frac{dx}{d\xi }$

$=$

$-\frac{x}{{\gamma }_g\xi }[(n+1)(3{\gamma }_g-4)+\frac{\xi (s^{\text{'}}{)}^2}{q}]$         at         $\xi ={\xi }_1.$

With the exception of the leading negative sign on the right-hand side, this expression is identical to the outer boundary condition identified by equation (58) of HRW66 — see the excerpt reproduced above.

Overview

The eigenvector associated with radial oscillations in isolated polytropes has been determined numerically and the results have been presented in a variety of key publications:

• P. LeDoux & Th. Walraven (1958, Handbuch der Physik, 51, 353) —
• R. F. Christy (1966, Annual Reviews of Astronomy & Astrophysics, 4, 353) — Pulsation Theory
• 📚 M. Hurley, P. H. Roberts, & K. Wright (1966, ApJ, Vol. 143, pp. 535 - 551) — The Oscillations of Gas Spheres
• J. P. Cox (1974, Reports on Progress in Physics, 37, 563) — Pulsating Stars

See Also