SSC/Stability/n1PolytropeLAWE

Radial Oscillations of n = 1 Polytropic Spheres

Groundwork

In an accompanying discussion, we derived the so-called,

Adiabatic Wave (or Radial Pulsation) Equation

$\frac{d^{2}x}{d{r}_0^2}+[\frac{4}{r_0}-(\frac{g_0{\rho }_0}{P_0}\left)\right]\frac{dx}{d{r}_0}+\left(\frac{{\rho }_0}{{\gamma }_{\mathrm{g}}{P}_0}\right)[{\omega }^2+(4-3{\gamma }_{\mathrm{g}})\frac{g_0}{r_0}]x=0$

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Because this widely used form of the radial pulsation equation is not dimensionless but, rather, has units of inverse length-squared, we have found it useful to also recast it in the following dimensionless form:

$\frac{d^{2}x}{d{\chi }_0^2}+[\frac{4}{{\chi }_0}-(\frac{{\rho }_0}{{\rho }_c}\left)\right(\frac{P_0}{P_c}{)}^{-1}\left(\frac{g_0}{g_{\mathrm{S}\mathrm{S}\mathrm{C}}}\right)]\frac{dx}{d{\chi }_0}+(\frac{{\rho }_0}{{\rho }_c}\left)\right(\frac{P_0}{P_c}{)}^{-1}\left(\frac{1}{{\gamma }_{\mathrm{g}}}\right)[{\tau }_{\mathrm{S}\mathrm{S}\mathrm{C}}^2{\omega }^2+(4-3{\gamma }_{\mathrm{g}})(\frac{g_0}{g_{\mathrm{S}\mathrm{S}\mathrm{C}}}\left)\frac{1}{{\chi }_0}\right]x=0,$

where,

$g_{\mathrm{S}\mathrm{S}\mathrm{C}}\equiv \frac{P_c}{R{\rho }_c},$       and       ${\tau }_{\mathrm{S}\mathrm{S}\mathrm{C}}\equiv [\frac{R^2{\rho }_c}{P_c}{]}^{1/2}.$

In a separate discussion, we showed that specifically for isolated, polytropic configurations, this linear adiabatic wave equation (LAWE) can be rewritten as,

$0$	$=$	$\frac{d^{2}x}{d{\xi }^2}+\left[\frac{4-(n+1)V(\xi )}{\xi }\right]\frac{dx}{d\xi }+\left[\frac{{\omega }^2}{{\gamma }_g\theta }\right(\frac{n+1}{4\pi G{\rho }_c})-(3-\frac{4}{{\gamma }_g})\cdot \frac{(n+1)V(x)}{{\xi }^2}]x$
	$=$	$\frac{d^{2}x}{d{\xi }^2}+[\frac{4}{\xi }-\frac{(n+1)}{\theta }(-\frac{d\theta }{d\xi }\left)\right]\frac{dx}{d\xi }+\frac{(n+1)}{\theta }[\frac{{\sigma }_c^2}{6{\gamma }_g}-\frac{\alpha }{\xi }(-\frac{d\theta }{d\xi }\left)\right]x,$

where we have adopted the dimensionless frequency notation,

${\sigma }_c^2$

$\equiv$

$\frac{3{\omega }^2}{2\pi G{\rho }_c}.$

Here we focus on an analysis of the specific case of isolated, $n=1$ polytropic configurations, whose unperturbed equilibrium structure can be prescribed in terms of analytic functions. Our hope — as yet unfulfilled — is that we can discover an analytically prescribed eigenvector solution to the governing LAWE.

See Also