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Part I: Wave Equation

Part II: Boundary Conditions

2024-01-22T03:24:08Z

Created page with "FORCETOC  =Radial Oscillations of Polytropic Spheres= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part I: Wave Equation<br /> </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part II: Boundary Conditions<br /> </td> <td align="..."

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FORCETOC

=Radial Oscillations of Polytropic Spheres=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Stability/Polytropes|Part I:   Wave Equation]]<br /> </td>
<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Stability/Polytropes/Pt2|Part II:  Boundary Conditions]]<br /> </td>
<td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Stability/Polytropes/Pt3|III:  Tables]]<br /> </td>
</tr>
</table>

==Tables==
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="5">
Quantitative Information Regarding Eigenvectors of Oscillating Polytropes

<math>~(\Gamma_1 = 5/3)</math>
</th>
</tr>
<tr>
<td align="center">
{{Math/MP_PolytropicIndex}}
</td>
<td align="center">
<math>\frac{\rho_c}{\bar\rho}</math>
</td>
<td align="center">
Excerpts from Table 1 of

{{ HRW66 }}

<math>~s^2 (n+1)/(4\pi G\rho_c)</math>
</td>
<td align="center">
Excerpts from Table 3 of

[http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974)]

<math>~\sigma_0^2 R^3/(GM)</math>
</td>
<td align="center">
<math>\frac{(n+1) \mathrm{Cox74}}{3 \mathrm{HRW66}} \cdot \frac{\bar\rho}{\rho_c}</math>
</td>
</tr>
<tr>
<td align="center">
<math>~0</math>
</td>
<td align="center">
<math>~1</math>
</td>
<td align="center">
<math>~1/3</math>
</td>
<td align="center">
<math>~1</math>
</td>
<td align="center">
<math>~1</math>
</td>
</tr>

<tr>
<td align="center">
<math>~1</math>
</td>
<td align="center">
<math>~3.30</math>
</td>
<td align="center">
<math>~0.38331</math>
</td>
<td align="center">
<math>~1.892</math>
</td>
<td align="center">
<math>~0.997</math>
</td>
</tr>

<tr>
<td align="center">
<math>~1.5</math>
</td>
<td align="center">
<math>~5.99</math>
</td>
<td align="center">
<math>~0.37640</math>
</td>
<td align="center">
<math>~2.712</math>
</td>
<td align="center">
<math>~1.002</math>
</td>
</tr>

<tr>
<td align="center">
<math>~2</math>
</td>
<td align="center">
<math>~11.4</math>
</td>
<td align="center">
<math>~0.35087</math>
</td>
<td align="center">
<math>~4.00</math>
</td>
<td align="center">
<math>~1.000</math>
</td>
</tr>

<tr>
<td align="center">
<math>~3</math>
</td>
<td align="center">
<math>~54.2</math>
</td>
<td align="center">
<math>~0.22774</math>
</td>
<td align="center">
<math>~9.261</math>
</td>
<td align="center">
<math>~1.000</math>
</td>
</tr>

<tr>
<td align="center">
<math>~3.5</math>
</td>
<td align="center">
<math>~153</math>
</td>
<td align="center">
<math>~0.12404</math>
</td>
<td align="center">
<math>~12.69</math>
</td>
<td align="center">
<math>~1.003</math>
</td>
</tr>

<tr>
<td align="center">
<math>~4.0</math>
</td>
<td align="center">
<math>~632</math>
</td>
<td align="center">
<math>~0.04056</math>
</td>
<td align="center">
<math>~15.38</math>
</td>
<td align="center">
<math>~1.000</math>
</td>
</tr>
</table>

=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 [[#Groundwork|above discussion]], the LAWE for any polytrope of index, <math>~n</math>, may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<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>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<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>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\sigma_c^2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{3\omega^2}{2\pi G\rho_c} \, .</math>
</td>
</tr>
</table>
</div>

Following a [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#Integrating_Outward_Through_the_Core|parallel discussion]], we begin by multiplying the LAWE through by <math>~\theta</math>, obtaining a 2<sup>nd</sup>-order ODE that is relevant at every individual coordinate location, <math>~\xi_i</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta_i {x_i''}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[4\theta_i - (n+1)\xi_i (- \theta^')_i\biggr] \frac{x_i'}{\xi_i}
- (n+1)\biggl[ \frac{\sigma_c^2}{6\gamma_g} -
\frac{\alpha}{\xi_i } (- \theta^')_i\biggr] x_i </math>
</td>
</tr>
</table>
</div>

Now, using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the substitutions,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x_i'</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
\frac{x_+ - x_-}{2 \Delta_\xi} \, ;
</math>
</td>
</tr>
</table>
</div>
and,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
x_i''
</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} \, ,</math>
</td>
</tr>
</table>
</div>
which will provide an approximate expression for <math>~x_+ \equiv x_{i+1}</math>, given the values of <math>~x_- \equiv x_{i-1}</math> and <math>~x_i</math>. Specifically, if the center of the configuration is denoted by the grid index, <math>~i=1</math>, then for zones, <math>~i = 3 \rightarrow N</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta_i \biggl[ \frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[4\theta_i - (n+1)\xi_i (- \theta^')_i\biggr] \biggl[ \frac{x_+ - x_-}{2 \xi_i \Delta_\xi} \biggr]
- (n+1)\biggl[ \frac{\sigma_c^2}{6\gamma_g} -
\frac{\alpha}{\xi_i } (- \theta^')_i\biggr] x_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \theta_i \biggl[ \frac{x_+ }{\Delta_\xi^2} \biggr] + \biggl[4\theta_i - (n+1)\xi_i (- \theta^')_i\biggr] \biggl[ \frac{x_+ }{2 \xi_i\Delta_\xi} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\theta_i \biggl[ \frac{- 2x_i + x_-}{\Delta_\xi^2} \biggr]
- \biggl[4\theta_i - (n+1)\xi_i (- \theta^')_i\biggr] \biggl[ \frac{- x_-}{2 \xi_i \Delta_\xi} \biggr]
- (n+1)\biggl[ \frac{\sigma_c^2}{6\gamma_g} -
\frac{\alpha}{\xi_i } (- \theta^')_i\biggr] x_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ x_+ \biggl[2\theta_i +\frac{4\Delta_\xi \theta_i}{\xi_i} - \Delta_\xi (n+1)(- \theta^')_i\biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x_- \biggl[\frac{4\Delta_\xi \theta_i}{\xi_i} - \Delta_\xi (n+1)(- \theta^')_i - 2\theta_i\biggr]
+ x_i\biggl\{4\theta_i - 2\Delta_\xi^2(n+1)\biggl[ \frac{\sigma_c^2}{6\gamma_g} -
\frac{\alpha}{\xi_i } (- \theta^')_i\biggr] \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x_- \biggl[\frac{4\Delta_\xi \theta_i}{\xi_i} - \Delta_\xi (n+1)(- \theta^')_i - 2\theta_i\biggr]
+ x_i\biggl\{4\theta_i - \frac{\Delta_\xi^2(n+1)}{3}\biggl[ \frac{\sigma_c^2}{\gamma_g} -
2\alpha \biggl(- \frac{3\theta^'}{\xi}\biggr)_i\biggr] \biggr\} \, .</math>
</td>
</tr>
</table>
</div>

<span id="KickStart">In order to kick-start the integration</span>, we will set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then we will draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \Delta_\xi</math>, away from the center. Specifically, we will set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
x_2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F} \Delta_\xi^2}{60} \biggr] \, ,</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\mathfrak{F}
</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{\sigma_c^2}{\gamma_g} - 2\alpha\biggr]\, .</math>
</td>
</tr>
</table>
</div>

=See Also=

* Radial Oscillations of [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density sphere]]
* Radial Oscillations of Isolated Polytropes
[[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|Setup]]
n = 1:  [[SSC/Stability/n1PolytropeLAWE|Attempt at Formulating an Analytic Solution]]
n = 3:  [[SSC/Stability/n3PolytropeLAWE|Numerical Solution]] to compare with {{ Schwarzschild41full }}
n = 5:  [[SSC/Stability/n5PolytropeLAWE|Attempt at Formulating an Analytic Solution]]

* In an accompanying [[Appendix/Ramblings/SphericalWaveEquation#Playing_With_Spherical_Wave_Equation|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 [[Apps/ImamuraHadleyCollaboration#Papaloizou-Pringle_Tori|stability of Papaloizou-Pringle tori]].

* <math>n=3</math> …
{{ Eddington18full }}, ''On the Pulsations of a Gaseous Star and the Problem of the Cepheid Variables.   Part I.''
{{ Schwarzschild41full }}, ''Overtone Pulsations of the Standard Model'': This work is referenced in §38.3 of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>]. It contains an analysis of the radial modes of oscillation of <math>~n=3</math> polytropes, assuming various values of the adiabatic exponent.

* <math>n=2</math> …
{{ Miller29full }}, ''The Effect of Distribution of Density on the Period of Pulsation of a Star''
{{ PG61full }}, ''Radial Pulsations of the Polytrope, n = 2''

* <math>n=\tfrac{3}{2}</math> … D. Lucas (1953, Bul. Soc. Roy. Sci. Liege, 25, 585) … Citation obtained from the Prasad & Gurm (1961) article.

* <math>n=1</math> … Citation also appears at the beginning of this chapter, and in the Prasad & Gurm (1961) article.
{{ Chatterji51full }}, ''Radial Oscillations of a Gaseous Star of Polytropic Index I''
{{ Chatterji52full }}, ''Anharmonic Pulsations of a Polytropic Model of Index Unity''

* Composite Polytropes … [http://adsabs.harvard.edu/abs/1968MNRAS.140..235S M. Singh (1968, MNRAS, 140, 235-240)], ''Effect of Central Condensation on the Pulsation Characteristics''

* Summary of Known Analytic Solutions … [http://adsabs.harvard.edu/abs/1981MNRAS.197..351S R. Stothers (1981, MNRAS, 197, 351-361)], ''Analytic Solutions of the Radial Pulsation Equation for Rotating and Magnetic Star Models''

* Interesting Composite! … [http://adsabs.harvard.edu/abs/1948MNRAS.108..414P C. Prasad (1948, MNRAS, 108, 414-416)], ''Radial Oscillations of a Particular Stellar Model''

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