SSC/Stability/n1PolytropeLAWE - Revision history

Jet53man: /* Setup */

2022-05-25T22:41:30Z

Setup

← Older revision		Revision as of 18:41, 25 May 2022
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	</table>		</table>
	</div>		</div>


	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

			<span id="WorkInProgress"> </span>

	<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">		<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

Jet53man: /* See Also */

2021-10-26T17:05:40Z

See Also

Jet53man: /* First Guess (n1) */

2021-10-26T17:02:17Z

First Guess (n1)

← Older revision		Revision as of 13:02, 26 October 2021
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\Rightarrow ~~~~~		<math>\Rightarrow ~~~
	\biggl[\frac{d\ln[ (\xi^2 \sin^2\xi)x^']^{1/(2\alpha)}}{d\ln\xi} +1 \biggr]		\biggl[\frac{d\ln[ (\xi^2 \sin^2\xi)x^']^{1/(2\alpha)}}{d\ln\xi} +1 \biggr]
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\Rightarrow ~~~~~		<math>\Rightarrow ~~~
	\frac{d\ln[ (\xi^2 \sin^2\xi)x^']^{-1/(2\alpha)}}{d\ln\xi}		\frac{d\ln[ (\xi^2 \sin^2\xi)x^']^{-1/(2\alpha)}}{d\ln\xi}
	</math>		</math>

Jet53man: /* Attempt at Deriving an Analytic Eigenvector Solution */

2021-10-26T17:00:23Z

Attempt at Deriving an Analytic Eigenvector Solution

← Older revision		Revision as of 13:00, 26 October 2021
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\sigma^2</math>		<math>\sigma^2</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\equiv</math>		<math>\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{\omega^2}{2\pi G\rho_c \gamma_g} \, ,		\frac{\omega^2}{2\pi G\rho_c \gamma_g} \, ,
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\alpha</math>		<math>\alpha</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\equiv</math>		<math>\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~3-\frac{4}{\gamma_g}		3-\frac{4}{\gamma_g}
	\, .		\, .
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~ \frac{1}{\xi^2 \sin^2\xi} \cdot \frac{d}{d\xi}\biggl[ \xi^2 \sin^2\xi \frac{dx}{d\xi} \biggr]		<math>\frac{1}{\xi^2 \sin^2\xi} \cdot \frac{d}{d\xi}\biggl[ \xi^2 \sin^2\xi \frac{dx}{d\xi} \biggr]
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{1}{\xi^2 \sin\xi} \biggl[ 2\alpha ( \sin\xi - \xi \cos \xi ) - \sigma^2 \xi^3 \biggr] x		\frac{1}{\xi^2 \sin\xi} \biggl[ 2\alpha ( \sin\xi - \xi \cos \xi ) - \sigma^2 \xi^3 \biggr] x
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~- \biggl[ \frac{2\alpha}{\xi^2} \biggl( \frac{\xi \cos \xi}{\sin\xi} -1\biggr) + \sigma^2 \biggl( \frac{\xi}{\sin\xi}\biggr) \biggr] x		- \biggl[ \frac{2\alpha}{\xi^2} \biggl( \frac{\xi \cos \xi}{\sin\xi} -1\biggr) + \sigma^2 \biggl( \frac{\xi}{\sin\xi}\biggr) \biggr] x
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~- \biggl[ \frac{2\alpha}{\xi^2} \frac{\xi^2}{\sin\xi} \cdot \frac{d}{d\xi} \biggl( \frac{\sin\xi}{\xi} \biggr)		- \biggl[ \frac{2\alpha}{\xi^2} \frac{\xi^2}{\sin\xi} \cdot \frac{d}{d\xi} \biggl( \frac{\sin\xi}{\xi} \biggr)
	+ \sigma^2 \biggl( \frac{\xi}{\sin\xi}\biggr) \biggr] x		+ \sigma^2 \biggl( \frac{\xi}{\sin\xi}\biggr) \biggr] x
	</math>		</math>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~- \biggl[ \frac{2\alpha}{\xi} \frac{d}{d\xi}\biggl( \frac{\sin\xi}{\xi} \biggr) + \sigma^2 \biggr] \biggl( \frac{\xi}{\sin\xi}\biggr) x \, ,		- \biggl[ \frac{2\alpha}{\xi} \frac{d}{d\xi}\biggl( \frac{\sin\xi}{\xi} \biggr) + \sigma^2 \biggr] \biggl( \frac{\xi}{\sin\xi}\biggr) x \, ,
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~ \frac{d}{d\xi} \biggl( \frac{\sin\xi}{\xi} \biggr)		<math>\frac{d}{d\xi} \biggl( \frac{\sin\xi}{\xi} \biggr)
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{\sin\xi}{\xi^2} \biggl[ \frac{\xi \cos\xi}{\sin\xi} - 1 \biggr] \, .		\frac{\sin\xi}{\xi^2} \biggl[ \frac{\xi \cos\xi}{\sin\xi} - 1 \biggr] \, .
	</math>		</math>
	</td>		</td>
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	</div>		</div>

	It would seem that the eigenfunction, <math>~x(\xi)</math>, should be expressible in terms of trigonometric functions and powers of <math>~\xi</math>; indeed, it appears as though the expression governing this eigenfunction would simplify considerably if <math>~x \propto \sin\xi/\xi</math>. With this in mind, we have made some attempts to ''guess'' the exact form of the eigenfunction. Here is one such attempt.		It would seem that the eigenfunction, <math>x(\xi)</math>, should be expressible in terms of trigonometric functions and powers of <math>~\xi</math>; indeed, it appears as though the expression governing this eigenfunction would simplify considerably if <math>x \propto \sin\xi/\xi</math>. With this in mind, we have made some attempts to ''guess'' the exact form of the eigenfunction. Here is one such attempt.

	===First Guess (n1)===		===First Guess (n1)===

Jet53man: /* First Guess (n1) */

2021-10-26T16:57:37Z

First Guess (n1)

← Older revision		Revision as of 12:57, 26 October 2021
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">

Jet53man: /* First Guess (n1) */

2021-10-26T16:55:43Z

First Guess (n1)

← Older revision		Revision as of 12:55, 26 October 2021
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	Let's try,		Let's try,
	<div align="center">		<div align="center">
	<math>~x = \frac{\sin\xi}{\xi} \, ,</math>		<math>x = \frac{\sin\xi}{\xi} \, ,</math>
	</div>		</div>
	which means,		which means,
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~x^' \equiv \frac{dx}{d\xi}</math>		<math>x^' \equiv \frac{dx}{d\xi}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{\sin\xi}{\xi^2} \biggl[ \frac{\xi \cos\xi}{\sin\xi} - 1 \biggr]		\frac{\sin\xi}{\xi^2} \biggl[ \frac{\xi \cos\xi}{\sin\xi} - 1 \biggr]
	\, .		\, .
	</math>		</math>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~- \biggl[ \frac{2\alpha}{\xi} \frac{d}{d\xi}\biggl( \frac{\sin\xi}{\xi} \biggr) + \sigma^2 \biggr] \biggl( \frac{\xi}{\sin\xi}\biggr) x		- \biggl[ \frac{2\alpha}{\xi} \frac{d}{d\xi}\biggl( \frac{\sin\xi}{\xi} \biggr) + \sigma^2 \biggr] \biggl( \frac{\xi}{\sin\xi}\biggr) x
	= - \biggl[ \frac{2\alpha x^'}{\xi} + \sigma^2 \biggr] \, .		= - \biggl[ \frac{2\alpha x^'}{\xi} + \sigma^2 \biggr] \, .
	</math>		</math>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ \frac{x^'}{\xi}		<math>\frac{x^'}{\xi}
	\biggl\{ \frac{\xi}{(\xi^2 \sin^2\xi)x^'} \cdot \frac{d[ (\xi^2 \sin^2\xi)x^']}{d\xi} \biggr\}		\biggl\{ \frac{\xi}{(\xi^2 \sin^2\xi)x^'} \cdot \frac{d[ (\xi^2 \sin^2\xi)x^']}{d\xi} \biggr\}
	</math>		</math>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ \frac{x^'}{\xi}		<math>\frac{x^'}{\xi}
	\biggl[\frac{d\ln[ (\xi^2 \sin^2\xi)x^']}{d\ln\xi} \biggr] \, .		\biggl[\frac{d\ln[ (\xi^2 \sin^2\xi)x^']}{d\ln\xi} \biggr] \, .
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~ \frac{x^'}{\xi}		<math>\frac{x^'}{\xi}
	\biggl[\frac{d\ln[ (\xi^2 \sin^2\xi)x^']}{d\ln\xi} +2\alpha \biggr]		\biggl[\frac{d\ln[ (\xi^2 \sin^2\xi)x^']}{d\ln\xi} +2\alpha \biggr]
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>		<math>
	~-\sigma^2		-\sigma^2
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~ \Rightarrow ~~~~~		<math>\Rightarrow ~~~~~
	\biggl[\frac{d\ln[ (\xi^2 \sin^2\xi)x^']^{1/(2\alpha)}}{d\ln\xi} +1 \biggr]		\biggl[\frac{d\ln[ (\xi^2 \sin^2\xi)x^']^{1/(2\alpha)}}{d\ln\xi} +1 \biggr]
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~- \frac{\sigma^2}{2\alpha } \biggl( \frac{\xi}{x^'} \biggr)		- \frac{\sigma^2}{2\alpha } \biggl( \frac{\xi}{x^'} \biggr)
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~ \Rightarrow ~~~~~		<math>\Rightarrow ~~~~~
	\frac{d\ln[ (\xi^2 \sin^2\xi)x^']^{-1/(2\alpha)}}{d\ln\xi}		\frac{d\ln[ (\xi^2 \sin^2\xi)x^']^{-1/(2\alpha)}}{d\ln\xi}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~1 + \frac{\sigma^2}{2\alpha } \biggl( \frac{\xi}{x^'} \biggr) \, .		1 + \frac{\sigma^2}{2\alpha } \biggl( \frac{\xi}{x^'} \biggr) \, .
	</math>		</math>
	</td>		</td>

Jet53man: /* Fourth Guess (n1) */

2021-10-26T16:52:32Z

Fourth Guess (n1)

Show changes

Jet53man: /* Fifth Guess (n1) */

2021-10-26T16:47:20Z

Fifth Guess (n1)

Show changes

Jet53man: /* Graphical Reassessment */

2021-10-26T16:42:14Z

Graphical Reassessment

Show changes

Jet53man: /* Graphical Reassessment */

2021-10-26T16:39:07Z

Graphical Reassessment

← Older revision		Revision as of 12:39, 26 October 2021
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ \frac{1}{\sin^3\xi} \biggl[		<math>\frac{1}{\sin^3\xi} \biggl[
	\xi^2 \sin\xi (1-\cos^2\xi) + \xi^2\sin\xi \cos^2\xi - \xi^3 \cos\xi		\xi^2 \sin\xi (1-\cos^2\xi) + \xi^2\sin\xi \cos^2\xi - \xi^3 \cos\xi
	+ \xi^2\sin\xi - \xi \sin^2\xi \cos\xi		+ \xi^2\sin\xi - \xi \sin^2\xi \cos\xi
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ - \alpha + \frac{1}{\sin^3\xi} \biggl[		<math>- \alpha + \frac{1}{\sin^3\xi} \biggl[
	2\xi^2 \sin\xi - \xi^3 \cos\xi - \xi \sin^2\xi \cos\xi + \alpha \xi \cos\xi \sin^2\xi		2\xi^2 \sin\xi - \xi^3 \cos\xi - \xi \sin^2\xi \cos\xi + \alpha \xi \cos\xi \sin^2\xi
	\biggr] \, ;		\biggr] \, ;

← Older revision		Revision as of 13:05, 26 October 2021
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	=See Also=		=See Also=


	* Radial Oscillations of [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density sphere]]		* Radial Oscillations of [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density sphere]]
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	* 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]].		* 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> …		* <math>n=3</math> …
	** {{ Eddington18full }}, ''On the Pulsations of a Gaseous Star and the Problem of the Cepheid Variables.   Part I.''		** {{ 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.		** {{ 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> …		* <math>n=2</math> …
	** {{ Miller29full }}, ''The Effect of Distribution of Density on the Period of Pulsation of a Star''		** {{ Miller29full }}, ''The Effect of Distribution of Density on the Period of Pulsation of a Star''
	** {{ PG61full }}, ''Radial Pulsations of the Polytrope, n = 2''		** {{ 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=\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.		* <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''		** {{ Chatterji51full }}, ''Radial Oscillations of a Gaseous Star of Polytropic Index I''
	** {{ Chatterji52full }}, ''Anharmonic Pulsations of a Polytropic Model of Index Unity''		** {{ Chatterji52full }}, ''Anharmonic Pulsations of a Polytropic Model of Index Unity''