SSC/Stability/Polytropes/Pt3

Radial Oscillations of Polytropic Spheres

Part I:   Wave Equation

Part II:  Boundary Conditions

III:  Tables

Tables

Quantitative Information Regarding Eigenvectors of Oscillating Polytropes $({\mathrm{\Gamma }}_1=5/3)$
$n$	$\frac{{\rho }_c}{\overline{\rho }}$	Excerpts from Table 1 of 📚 Hurley, Roberts, & Wright (1966) $s^2(n+1)/(4\pi G{\rho }_c)$	Excerpts from Table 3 of J. P. Cox (1974) ${\sigma }_0^2{R}^3/(GM)$	$\frac{(n+1)*\mathrm{C}\mathrm{o}\mathrm{x}\mathrm{7}\mathrm{4}}{3*\mathrm{H}\mathrm{R}\mathrm{W}\mathrm{6}\mathrm{6}}\cdot \frac{\overline{\rho }}{{\rho }_c}$
$0$	$1$	$1/3$	$1$	$1$
$1$	$3.30$	$0.38331$	$1.892$	$0.997$
$1.5$	$5.99$	$0.37640$	$2.712$	$1.002$
$2$	$11.4$	$0.35087$	$4.00$	$1.000$
$3$	$54.2$	$0.22774$	$9.261$	$1.000$
$3.5$	$153$	$0.12404$	$12.69$	$1.003$
$4.0$	$632$	$0.04056$	$15.38$	$1.000$

Numerical Integration from the Center, Outward

Here we show how a relatively simple, finite-difference algorithm can be developed to numerically integrate the governing LAWE from the center of a polytropic configuration, outward to its surface.

Drawing from our above discussion, the LAWE for any polytrope of index, $n$, may be written as,

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

where,

${\sigma }_c^2$

$\equiv$

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

Following a parallel discussion, we begin by multiplying the LAWE through by $\theta$, obtaining a 2^nd-order ODE that is relevant at every individual coordinate location, ${\xi }_i$, namely,

${\theta }_i{x_i}^{″}$

$=$

$-[4{\theta }_i-(n+1){\xi }_i(-{\theta }^{\text{'}}{)}_i]\frac{{x_i}^{\prime }}{{\xi }_i}-(n+1)[\frac{{\sigma }_c^2}{6{\gamma }_g}-\frac{\alpha }{{\xi }_i}(-{\theta }^{\text{'}}{)}_i]x_i$

Now, using the general finite-difference approach described separately, we make the substitutions,

${x_i}^{\prime }$

$\approx$

$\frac{x_{+}-x_{-}}{2{\mathrm{\Delta }}_{\xi }};$

and,

${x_i}^{″}$

$\approx$

$\frac{x_{+}-2x_i+x_{-}}{{\mathrm{\Delta }}_{\xi }^2},$

which will provide an approximate expression for $x_{+}\equiv x_{i+1}$, given the values of $x_{-}\equiv x_{i-1}$ and $x_i$. Specifically, if the center of the configuration is denoted by the grid index, $i=1$, then for zones, $i=3\to N$,

${\theta }_i\left[\frac{x_{+}-2x_i+x_{-}}{{\mathrm{\Delta }}_{\xi }^2}\right]$	$=$	$-[4{\theta }_i-(n+1){\xi }_i(-{\theta }^{\text{'}}{)}_i\left]\right[\frac{x_{+}-x_{-}}{2{\xi }_i{\mathrm{\Delta }}_{\xi }}]-(n+1)[\frac{{\sigma }_c^2}{6{\gamma }_g}-\frac{\alpha }{{\xi }_i}(-{\theta }^{\text{'}}{)}_i]x_i$
$\Rightarrow {\theta }_i\left[\frac{x_{+}}{{\mathrm{\Delta }}_{\xi }^2}\right]+[4{\theta }_i-(n+1){\xi }_i(-{\theta }^{\text{'}}{)}_i\left]\right[\frac{x_{+}}{2{\xi }_i{\mathrm{\Delta }}_{\xi }}]$	$=$	$-{\theta }_i\left[\frac{-2x_i+x_{-}}{{\mathrm{\Delta }}_{\xi }^2}\right]-[4{\theta }_i-(n+1){\xi }_i(-{\theta }^{\text{'}}{)}_i\left]\right[\frac{-x_{-}}{2{\xi }_i{\mathrm{\Delta }}_{\xi }}]-(n+1)[\frac{{\sigma }_c^2}{6{\gamma }_g}-\frac{\alpha }{{\xi }_i}(-{\theta }^{\text{'}}{)}_i]x_i$
$\Rightarrow x_{+}[2{\theta }_i+\frac{4{\mathrm{\Delta }}_{\xi }{\theta }_i}{{\xi }_i}-{\mathrm{\Delta }}_{\xi }(n+1)(-{\theta }^{\text{'}}{)}_i]$	$=$	$x_{-}[\frac{4{\mathrm{\Delta }}_{\xi }{\theta }_i}{{\xi }_i}-{\mathrm{\Delta }}_{\xi }(n+1)(-{\theta }^{\text{'}}{)}_i-2{\theta }_i]+x_i\{4{\theta }_i-2{\mathrm{\Delta }}_{\xi }^2(n+1)[\frac{{\sigma }_c^2}{6{\gamma }_g}-\frac{\alpha }{{\xi }_i}(-{\theta }^{\text{'}}{)}_i\left]\right\}$
	$=$	$x_{-}[\frac{4{\mathrm{\Delta }}_{\xi }{\theta }_i}{{\xi }_i}-{\mathrm{\Delta }}_{\xi }(n+1)(-{\theta }^{\text{'}}{)}_i-2{\theta }_i]+x_i\{4{\theta }_i-\frac{{\mathrm{\Delta }}_{\xi }^2(n+1)}{3}[\frac{{\sigma }_c^2}{{\gamma }_g}-2\alpha (-\frac{3{\theta }^{\text{'}}}{\xi }{)}_i\left]\right\}.$

In order to kick-start the integration, we will set the displacement function value to $x_1=1$ at the center of the configuration $({\xi }_1=0)$, then we will draw on the derived power-series expression to determine the value of the displacement function at the first radial grid line, ${\xi }_2={\mathrm{\Delta }}_{\xi }$, away from the center. Specifically, we will set,

$x_2$

$=$

$x_1[1-\frac{(n+1)\mathfrak{𝔉}{\mathrm{\Delta }}_{\xi }^2}{60}],$

where,

$\mathfrak{𝔉}$

$\equiv$

$[\frac{{\sigma }_c^2}{{\gamma }_g}-2\alpha ].$

See Also

• Radial Oscillations of Uniform-density sphere
• Radial Oscillations of Isolated Polytropes
  • Setup
  • n = 1:  Attempt at Formulating an Analytic Solution
  • n = 3:  Numerical Solution to compare with 📚 M. Schwarzschild (1941, ApJ, Vol. 94, pp. 245 - 252)
  • n = 5:  Attempt at Formulating an Analytic Solution

• In an accompanying Chapter within our "Ramblings" Appendix, we have played with the adiabatic wave equation for polytropes, examining its form when the primary perturbation variable is an enthalpy-like quantity, rather than the radial displacement of a spherical mass shell. This was done in an effort to mimic the approach that has been taken in studies of the stability of Papaloizou-Pringle tori.

• $n=3$ …
  • 📚 A. S. Eddington (1918, MNRAS, Vol. 79, pp. 2 - 22), On the Pulsations of a Gaseous Star and the Problem of the Cepheid Variables.   Part I.
  • 📚 M. Schwarzschild (1941, ApJ, Vol. 94, pp. 245 - 252), Overtone Pulsations of the Standard Model: This work is referenced in §38.3 of [KW94]. It contains an analysis of the radial modes of oscillation of $n=3$ polytropes, assuming various values of the adiabatic exponent.

• $n=2$ …
  • 📚 J. C. P. Miller (1929, MNRAS, Vol. 90, pp. 59 - 64), The Effect of Distribution of Density on the Period of Pulsation of a Star
  • 📚 C. Prasad & H. S. Gurm (1961, MNRAS, Vol. 122, pp. 409 - 411), Radial Pulsations of the Polytrope, n = 2

• $n=\frac{3}{2}$ … D. Lucas (1953, Bul. Soc. Roy. Sci. Liege, 25, 585) … Citation obtained from the Prasad & Gurm (1961) article.

• $n=1$ … Citation also appears at the beginning of this chapter, and in the Prasad & Gurm (1961) article.
  • 📚 L. D. Chatterji (1951, PINSA, Vol. 17, No. 6, pp. 467 - 470), Radial Oscillations of a Gaseous Star of Polytropic Index I
  • 📚 L. D. Chatterji (1952, PINSA, Vol. 18, No. 3, pp. 187 - 191), Anharmonic Pulsations of a Polytropic Model of Index Unity

• Composite Polytropes … M. Singh (1968, MNRAS, 140, 235-240), Effect of Central Condensation on the Pulsation Characteristics

• Summary of Known Analytic Solutions … R. Stothers (1981, MNRAS, 197, 351-361), Analytic Solutions of the Radial Pulsation Equation for Rotating and Magnetic Star Models

• Interesting Composite! … C. Prasad (1948, MNRAS, 108, 414-416), Radial Oscillations of a Particular Stellar Model