SSC/Stability/Polytropes

Radial Oscillations of Polytropic Spheres

Part I:   Wave Equation

Part II:  Boundary Conditions

III:  Tables

Polytropes

Groundwork

 
 
 
 

Adiabatic (Polytropic) Wave Equation

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. If the initial, unperturbed equilibrium configuration is a polytropic sphere whose internal structure is defined by the function, $\theta (\xi )$, then

$r_0$	$=$	$a_n\xi ,$
${\rho }_0$	$=$	${\rho }_c{\theta }^n,$
$P_0$	$=$	$K{\rho }_0^{(n+1)/n}=K{\rho }_c^{(n+1)/n}{\theta }^{n+1},$
$g_0$	$=$	$\frac{GM(r_0)}{r_0^2}=\frac{G}{r_0^2}\left[4\pi a_n^3{\rho }_c\right(-{\xi }^2\frac{d\theta }{d\xi }\left)\right],$

where,

$a_n$

$=$

$[\frac{(n+1)K}{4\pi G}\cdot {\rho }_c^{(1-n)/n}{]}^{1/2}.$

Hence, after multiplying through by $a_n^2$, the above adiabatic wave equation can be rewritten in the form,

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

$=$

$0.$

In addition, given that,

$\frac{g_0}{a_n}$

$=$

$4\pi G{\rho }_c(-\frac{d\theta }{d\xi }),$

and,

$\frac{a_n^2{\rho }_0}{P_0}$

$=$

$\frac{(n+1)}{(4\pi G{\rho }_c)\theta }=\frac{a_n^2{\rho }_c}{P_c}\cdot \frac{{\theta }_c}{\theta },$

we can write,

$\frac{d^{2}x}{d{\xi }^2}+\left[\frac{4-(n+1)V(\xi )}{\xi }\right]\frac{dx}{d\xi }+\left[{\omega }^2\right(\frac{a_n^2{\rho }_c}{{\gamma }_g{P}_c})\frac{{\theta }_c}{\theta }-(3-\frac{4}{{\gamma }_g})\cdot \frac{(n+1)V(x)}{{\xi }^2}]x$

$=$

$0,$

where we have adopted the function notation,

$V(\xi )$

$\equiv$

$-\frac{\xi }{\theta }\frac{d\theta }{d\xi }.$

As can be seen in the following set of retyped expressions, this is the form of the polytropic wave equation published by 📚 J. O. Murphy & R. Fiedler (1985b, Proc. Astron. Soc. Australia, Vol. 6, no. 2, pp. 222 - 226), at the beginning of their discussion.

Polytropic Wave Equation extracted† from J. O. Murphy & R. Fiedler (1985) Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models Proceedings of the Astronomical Society of Australia, Vol. 6, no. 2, pp. 222 - 226 © Astronomical Society of Australia

Comment by J. E. Tohline: There appears to be a sign error in the numerator of the second term of the polytropic wave equation as published by Murphy & Fiedler and as retyped here; there also appears to be an error in the definition of the coefficient, α*, as given in the text of their paper. $\frac{d^2\eta }{d{\zeta }^2}+\left(\frac{4+(n+1)V}{\zeta }\right)\frac{d\eta }{d\zeta }+(\frac{{\omega }_k^2{\theta }_c}{\theta }-\frac{{\alpha }^{*}(n+1)V}{{\zeta }^2})\eta$ $=$ $0,$ where:    ${\omega }_k^2=\frac{{\sigma }_k^2{\alpha }_n^2{\rho }_c}{\gamma P_c}.$

†As displayed here, the layout of the equations has been modified from the original publication.

It is also the same as the radial pulsation equation for polytropic configurations that appears as equation (56) in 📚 M. Hurley, P. H. Roberts, & K. Wright (1966, ApJ, Vol. 143, pp. 535 - 551); hereafter, HRW66. The relevant set of equations from HRW66 is retyped here.

Radial Pulsation Equation as Presented† by M. Hurley, P. H. Roberts, & K. Wright (1966) The Oscillations of Gas Spheres The Astrophysical Journal, Vol. 143, pp. 535 - 551 © American Astronomical Society

The radial pulsation equation is $\frac{d^{2}X}{d{x}^2}$ $=$ $-\frac{1}{x}\frac{dX}{dx}[4+(n+1)x\frac{{\theta }^{\prime }}{\theta }]-\frac{{\theta }^{\prime }X}{\gamma \theta x}[(n+1)(3\gamma -4)-\frac{x{s}^2}{{\theta }^{\prime }}],$ (56) with end-point conditions $X^{\prime }=0,$    at     $x=0,$ (57) Comment by J. E. Tohline: As is shown in the subsection on "Boundary Conditions," below, it appears as though the term on the right-hand-side of HRW66's equation (58) is incorrect, as published and as retyped here; it should be preceded with a negative sign. and $(n+1)\frac{dX}{dx}$ $=$ $\frac{X}{\gamma x}[(n+1)(3\gamma -4)+\frac{x{s}^2}{q}],$     at     $x=x_0.$ (58)

†Set of equations and accompanying text displayed here, as a single digital image, exactly as they appear in the original publication.

In order to make clearer the correspondence between our derived expression and the one published by HRW66, we will rewrite the HRW66 radial pulsation equation: (1) Gathering all terms on the same side of the equation; (2) making the substitution,

${\theta }^{\text{'}}\to -\frac{\theta V}{x};$

and (3) reattaching a "prime" to the quantity, $s$, to emphasize that it is a dimensionless frequency.

ASIDE: In their equation (46), HRW66 convert the eigenfrequency, $s$ — which has units of inverse time — to a dimensionless eigenfrequency, $s^{\text{'}}$, via the relation, $s=(\frac{4\pi G{\rho }_c}{1+n}{)}^{1/2}{s}^{\text{'}}\cdots \cdots (46)$ Then, immediately following equation (46), they state that they will "omit the prime on $s$ henceforward." As a result, the dimensionless eigenfrequency that appears in their equations (56) and (58) is unprimed. This is unfortunate as it somewhat muddies our efforts, here, to demonstrate the correspondence between the HRW66 polytropic radial pulsation equation and ours. In our subsequent manipulation of equation (56) from HRW66 we reattach a prime to the quantity, $s$, to emphasize that it is a dimensionless frequency. But this prime on $s$ should not be confused with the prime on $\theta$ (HRW66 equation 56) or with the prime on $X$ (HRW66 equation 57), both of which denote differentiation with respect to the radial coordinate.

With these modifications, the HRW66 radial pulsation equation becomes,

$0$	$=$	$\frac{d^{2}X}{d{x}^2}+\left[\frac{4-(n+1)V}{x}\right]\frac{dX}{dx}-\frac{V}{\gamma x^2}[\frac{x^2(s^{\text{'}}{)}^2}{\theta V}+(3\gamma -4)(n+1)]X$
	$=$	$\frac{d^{2}X}{d{x}^2}+\left[\frac{4-(n+1)V}{x}\right]\frac{dX}{dx}+[-\frac{(s^{\text{'}}{)}^2}{\gamma \theta }-(3-\frac{4}{\gamma }\left)\frac{(n+1)V}{x^2}\right]X.$

The correspondence with our derived expression is complete, assuming that,

$(s^{\text{'}}{)}^2$	$=$	$-{\omega }^2\left(\frac{a_n^2{\rho }_c}{P_c}\right){\theta }_c$
	$=$	$-{\omega }^2\left[\frac{n+1}{4\pi G{\rho }_c}\right].$

As has been explained in the above "ASIDE," this is exactly the factor that HRW66 use to normalize their eigenfrequency, $s$, and make it dimensionless $(s^{\text{'}})$. It is clear, as well, that HRW66 have adopted a sign convention for the square of their eigenfrequency that is the opposite of the sign convention that we have adopted for ${\omega }^2$. That is, it is clear that,

$s^2\leftrightarrow -{\omega }^2.$

See Also