User:Tohline/SSC/Stability/n3PolytropeLAWE

Radial Oscillations of n = 3 Polytropic Spheres

Groundwork

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

Adiabatic Wave (or Radial Pulsation) Equation

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

<math>~=</math>

<math>0 \, ,</math>

where we have adopted the function notation,

<math>~V(\xi)</math>

<math>~\equiv</math>

<math>~- \frac{\xi}{\theta} \frac{d \theta}{d\xi} \, .</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."

Numerical Integration

It can be shown straightforwardly that this matches the LAWE used by Schwarzschild (1941), if <math>~n</math> is set to 3. 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>

The algorithm is as follows:

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

Related Discussions

• 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)
  • n = 5:  Attempt at Formulating an Analytic Solution