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<math>~- \frac{3\omega^2}{2\pi G \rho_c}  \, .</math>
<math>~\frac{3\omega^2}{2\pi G \rho_c}  \, .</math>
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Revision as of 05:28, 18 February 2017

Radial Oscillations of n = 3 Polytropic Spheres

Whitworth's (1981) Isothermal Free-Energy Surface
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Background

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

Adiabatic Wave (or Radial Pulsation) Equation

LSU Key.png

<math>~ \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x = 0 </math>

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:

<math> \frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr] x = 0 , </math>

where,

<math>~g_\mathrm{SSC} \equiv \frac{P_c}{R\rho_c} \, ,</math>       and       <math>~\tau_\mathrm{SSC} \equiv \biggl[\frac{R^2 \rho_c}{P_c}\biggr]^{1/2} \, .</math>

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

<math>~0 </math>

<math>~=</math>

<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + \biggl[\frac{\omega^2}{\gamma_g \theta} \biggl(\frac{n+1 }{4\pi G \rho_c} \biggr) - \biggl(3-\frac{4}{\gamma_g}\biggr) \cdot \frac{(n+1)V(x)}{\xi^2} \biggr] x </math>

 

<math>~=</math>

<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{(n+1)}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} + \frac{(n+1)}{\theta}\biggl[\frac{\sigma_c^2}{6\gamma_g } - \frac{\alpha}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] x \, ,</math>

where we have adopted the dimensionless frequency notation,

<math>~\sigma_c^2</math>

<math>~\equiv</math>

<math>~\frac{3\omega^2}{2\pi G \rho_c} \, .</math>

Here we perform a numerical integration of the governing LAWE for <math>~n=3</math> polytropes. We can directly compare our results with Schwarzschild's (1941) published work on "Overtone Pulsations for the Standard [Stellar] Model." To begin with, it is straightforward to demonstrate that the last form of the LAWE, provided above, matches equation (2) from Schwarzschild (1941), if <math>~n</math> is set to 3; note as well that Schwarzschild's dimensionless oscillation frequency — defined in his equation (1) and which we will label, <math>~\omega_\mathrm{Sch}</math> — is related to our dimensionless frequency via the expression,

<math>~\sigma_c^2</math>

<math>~~\leftrightarrow~~</math>

<math>~\biggl( \frac{3\gamma_g}{2} \biggr) \omega_\mathrm{Sch}^2 \, .</math>

Paragraph extracted from M. Schwarzschild (1941)

"Overtone Pulsations for the Standard Model"

ApJ, vol. 94, pp. 245 - 252 © American Astronomical Society

Schwarzschild (1941, ApJ, 94, 245)

      3A. S. Eddington (1930), The Internal Constitution of the Stars, pp. 188 and 192.

M. Schwarzschild (1941) numerically integrated the LAWE for <math>~n=3</math> polytropic spheres to find eigenvectors (i.e., the spatially discrete eigenfunction and corresponding eigenfrequency) for five separate oscillation modes (the fundamental mode, plus the 1st, 2nd, 3rd, and 4th overtones) for models having four different adopted adiabatic indexes <math>~\gamma_g = \tfrac{4}{3}, \tfrac{10}{7}, \tfrac{20}{13}, \tfrac{5}{3})</math>. Table 1 catalogs the eigenfrequencies that Schwarzschild determined for these twenty different models/modes (see his Table 1).

Table 1:   From Table 1 of M. Schwarzschild (1941)

Mode <math>~\alpha = 0.0</math>

<math>~(\gamma_g = 4/3)</math>
<math>~\alpha = 0.2</math>

<math>~(\gamma_g = 10/7)</math>
<math>~\alpha = 0.4</math>

<math>~(\gamma_g = 20/13)</math>
<math>~\alpha = 0.6</math>

<math>~(\gamma_g = 5/3)</math>
<math>~\omega_\mathrm{Sch}^2</math> <math>~\omega_\mathrm{Sch}^2</math> <math>~\omega_\mathrm{Sch}^2</math> <math>~\mathfrak{F} = \biggl[\frac{3\omega_\mathrm{Sch}^2}{2} - 2\alpha \biggr]</math> <math>~\omega_\mathrm{Sch}^2</math>
0 0.00000 0.05882 0.10391 -0.64414 0.13670
1 0.16643 0.19139 0.21998 -0.47003 0.25090
2 0.3392 0.3648 0.3920 -0.2120 0.4209
3 0.5600 0.5863 0.6136 +0.1204 0.6420
4 0.8283 0.8554 0.8832 +0.5248 0.9117

Numerical Integration

Here we use the finite-difference algorithm described separately to integrate the discretized LAWE from the center of the polytropic configuration, outward to its surface, which in this case — see, for example, p. 77 of Horedt (2004) — is located at the polytropic-coordinate location,

<math>~\xi_\mathrm{max} = 6.89684862 \, .</math>

It is assumed, at the outset, that we have in hand an appropriately discretized description of the unperturbed, equilibrium properties of an <math>~n=3</math> polytrope; specifically, at each radial grid line, we have tabulated values of the radial coordinate, <math>~0 \le \xi_i \le \xi_\mathrm{max}</math>, the Lane-Emden function, <math>~\theta_i</math>, and its first radial derivative, <math>~\theta_i'</math>.

The algorithm is as follows (substitute <math>~n=3</math> everywhere):

  • Establish an equally spaced radial-coordinate grid containing <math>~N</math> grid zones (and, accordingly, <math>~N+1</math> grid lines), in which case the grid-spacing parameter, <math>~\Delta_\xi \equiv \xi_\mathrm{max}/N</math>.
  • Specify a value of the adiabatic exponent, <math>~\gamma</math>, which, in turn, determines the value of the parameter, <math>~\alpha \equiv (3-4/\gamma) \, .</math>
  • Choose a value for the (square of the) dimensionless oscillation frequency, <math>~\sigma_c^2</math>, which we will accomplish by assigning a value to the parameter,

    <math>~\mathfrak{F} \equiv \frac{\sigma_c^2}{\gamma} - 2\alpha \, .</math>

  • Set the eigenfunction to unity at the center <math>~(\xi_0 = 0)</math> of the configuration, that is, set <math>~x_0 = 1</math>.
  • Determine the value of the eigenfunction at the first grid line away from the center — having coordinate location, <math>~\xi_1 = \Delta_\xi </math> — via the expression,

    <math>~ x_1 </math>

    <math>~=</math>

    <math>~ x_0 \biggl[ 1 - \frac{\Delta_\xi^2 (n+1) \mathfrak{F}}{12} \biggr] \, .</math>

  • At all other grid lines, <math>~i=2,N</math>, determine the value of the eigenfunction, <math>~x_i</math>, via the expression,

    <math>~x_i \biggl[2\theta_{i-1} +\frac{4\Delta_\xi \theta_{i-1}}{\xi_{i-1}} - \Delta_\xi (n+1)(- \theta^')_{i-1}\biggr] </math>

    <math>~=</math>

    <math>~ x_{i-1}\biggl\{4\theta_{i-1} - \frac{\Delta_\xi^2(n+1)}{3}\biggl[ \mathfrak{F}+2\alpha - 2\alpha \biggl(- \frac{3\theta^'}{\xi}\biggr)_{i-1} \biggr] \biggr\} + x_{i-2} \biggl[\frac{4\Delta_\xi \theta_{i-1}}{\xi_{i-1}} - \Delta_\xi (n+1)(- \theta^')_{i-1} - 2\theta_{i-1}\biggr] \, .</math>

Now, in searching for an appropriate boundary condition at the surface of the configuration, it will be useful to tabulate, not only the value of the eigenfunction at the surface, <math>~x_N</math>, but also its logarithmic derivative. A finite-difference expression of this logarithmic derivative that is consistent with the above-described finite-difference algorithm, is,

<math>~\frac{d\ln x}{d\ln \xi} \biggr|_\mathrm{surface}</math>

<math>~\approx</math>

<math>~\frac{\xi_\mathrm{max}}{x_N} \biggl[ \frac{x_{N+1}-x_{N-1}}{2\Delta_\xi} \biggr] \, .</math>

Everything is known here, except for the quantity, <math>~x_{N+1}</math>, which can be evaluated using the last expression in our algorithm one more time to, in effect, evaluate the eigenfunction just outside the surface. That is, we obtain <math>~x_{N+1}</math> and, in turn, obtain a value for the logarithmic derivative at the surface, via the expression,

<math>~x_{N+1} \biggl[2\theta_{N} +\frac{4\Delta_\xi \theta_{N}}{\xi_\mathrm{max}} - \Delta_\xi (n+1)(- \theta^')_{N}\biggr] </math>

<math>~=</math>

<math>~ x_{N}\biggl\{4\theta_{N} - \frac{\Delta_\xi^2(n+1)}{3}\biggl[ \mathfrak{F}+2\alpha - 2\alpha \biggl(- \frac{3\theta^'}{\xi}\biggr)_{N} \biggr] \biggr\} + x_{N-1} \biggl[\frac{4\Delta_\xi \theta_{N}}{\xi_\mathrm{max}} - \Delta_\xi (n+1)(- \theta^')_{N} - 2\theta_{N}\biggr] \, .</math>

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