SSC/Stability/Polytropes/Pt2 - Revision history

Joel2: Created page with "FORCETOC =Radial Oscillations of Polytropic Spheres=

Part I: Wave Equation

Part II: Boundary Conditions

2024-01-22T03:23:02Z

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>

===Boundary Conditions===

As we have pointed out in the context of [[SSC/Perturbations#Boundary_Conditions|a general discussion of boundary conditions associated with the adiabatic wave equation]], the eigenfunction, <math>~x</math>, will be suitably well behaved at the center of the configuration if,
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<math>~\frac{dx}{dr_0} = 0</math>        at         <math>~r_0 = 0 \, ,</math>
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which, in the context of our present discussion of polytropic configurations, leads to the inner boundary condition,
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<math>~\frac{dx}{d\xi} = 0</math>        at         <math>~\xi = 0 \, .</math>
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This is precisely the inner boundary condition specified by {{ HRW66hereafter }} — see their equation (57), which has been reproduced in the above excerpt from HWR66.

As we have also shown in the context of this separate, [[SSC/Perturbations#Boundary_Conditions|general discussion of boundary conditions associated with the adiabatic wave equation]], the pressure fluctuation will be finite at the surface — even if the equilibrium pressure and/or the pressure scale height go to zero at the surface — if the radial eigenfunction, <math>~x</math>, obeys the relation,

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<math>~r_0 \frac{dx}{dr_0}</math>
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<math>~=</math>
</td>
<td align="left">
<math>~\biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \frac{x}{\gamma_g}</math>        at         <math>~r_0 = R \, .</math>
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Or, given that, in polytropic configurations, <math>~r_0 = a_n\xi</math>,

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<td align="right">
<math>~\xi \frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{x}{\gamma_g} \biggl[ 4 - 3\gamma_g + \frac{\omega^2 (a_n \xi_1)^3}{GM_\mathrm{tot}}\biggr] </math>        at         <math>~\xi = \xi_1 \, ,</math>
</td>
</tr>
</table>
</div>

where, the subscript "1" denotes equilibrium, surface values. As can be deduced from our above summary of the properties of polytropic configurations,
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<math>~GM_\mathrm{tot}</math>
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<math>~=</math>
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<math>~4\pi G a_n^3 \rho_c (-\xi_1^2 \theta_1^') \, .</math>
</td>
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</table>
</div>
Hence, for spherically symmetric polytropic configurations, the surface boundary condition becomes,

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

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<td align="right">
<math>~\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{x}{\gamma_g \xi} \biggl[ 4 - 3\gamma_g + \omega^2 \biggl( \frac{1}{4\pi G \rho_c } \biggr) \frac{\xi}{(-\theta^')}\biggr] </math>
        at         <math>~\xi = \xi_1 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~~(n+1)\frac{dx}{d\xi}</math>
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<td align="center">
<math>~=</math>
</td>
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<math>~\frac{x}{\gamma_g \xi} \biggl[ (n+1)(4 - 3\gamma_g) + \omega^2 \biggl( \frac{1+n}{4\pi G \rho_c } \biggr) \frac{\xi}{(-\theta^')}\biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-\frac{x}{\gamma_g \xi} \biggl[ (n+1)(3\gamma_g-4) - \omega^2 \biggl( \frac{1+n}{4\pi G \rho_c } \biggr) \frac{\xi}{(-\theta^')}\biggr] </math>
        at         <math>~\xi = \xi_1 \, .</math>
</td>
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</table>
</div>

Adopting notation used by {{ HRW66hereafter }}, specifically, as demonstrated above,
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<math>~-\omega^2 \biggl( \frac{1+n}{4\pi G \rho_c } \biggr) \rightarrow (s^')^2 \, , </math>
</div>
and, from equation (50) of {{ HRW66hereafter }},
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<math>~-\theta^' \rightarrow q </math>
        at         <math>~\xi = \xi_1 \, ,</math>
</div>
this outer boundary condition becomes,

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

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<td align="right">
<math>~(n+1)\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-\frac{x}{\gamma_g \xi} \biggl[ (n+1)(3\gamma_g-4) + \frac{\xi (s^')^2}{q}\biggr] </math>
        at         <math>~\xi = \xi_1 \, .</math>
</td>
</tr>
</table>
</div>
With the exception of the leading negative sign on the right-hand side, this expression is identical to the outer boundary condition identified by equation (58) of {{ HRW66hereafter }} — see the [[SSC/Stability/Polytropes#HRW66excerpt|excerpt reproduced above]].

==Overview==
The eigenvector associated with radial oscillations in isolated polytropes has been determined numerically and the results have been presented in a variety of key publications:
* P. LeDoux & Th. Walraven (1958, Handbuch der Physik, 51, 353) —
* [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C R. F. Christy (1966, Annual Reviews of Astronomy & Astrophysics, 4, 353)] — ''Pulsation Theory''
* {{ HRW66full }} — ''The Oscillations of Gas Spheres''
* [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974, Reports on Progress in Physics, 37, 563)] — ''Pulsating Stars''

=See Also=

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