SSC/Stability/BiPolytrope00Details - Revision history

Jet53man: /* Related Discussions */

2022-10-24T19:16:58Z

Related Discussions

← Older revision		Revision as of 15:16, 24 October 2022
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	=Related Discussions=		=Related Discussions=
	* [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|Summary of Above, Detailed Derivation]]		* [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|Summary of Above, Detailed Derivation]]
	* [[Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes\|Searching for Additional, Analytically Specified Eigenvectors of Zero-Zero Bipolytropes]]		* [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes\|Searching for Additional, Analytically Specified Eigenvectors of Zero-Zero Bipolytropes]]


	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: /* Match Prasad-like Envelope Eigenvector to the Core Eigenvector */

2022-10-24T19:09:28Z

Match Prasad-like Envelope Eigenvector to the Core Eigenvector

← Older revision		Revision as of 15:09, 24 October 2022
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	</table>		</table>
	</div>		</div>
	Not surprisingly, this is identical in form to the eigenvalue problem first presented by [[SSC/Stability/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|Sterne (1937)]] in connection with an examination of radial oscillations in uniform-density spheres. For the core of our zero-zero bipolytrope, we can therefore adopt any one of the [[SSC/Stability/UniformDensity#Sterne.27s_General_Solution\|polynomial eigenfunctions and corresponding eigenfrequencies]] derived by Sterne. We will insist that the eigenfrequency of the envelope match the eigenfrequency of the core; and, following ~~[http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b)]~~ (see the top paragraph of the right-hand column on p. 223 of their article), we seek solutions for which there is continuity in both the eigenfunction and its first derivative at the interface <math>~(\xi = 1)</math>.		Not surprisingly, this is identical in form to the eigenvalue problem first presented by [[SSC/Stability/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|Sterne (1937)]] in connection with an examination of radial oscillations in uniform-density spheres. For the core of our zero-zero bipolytrope, we can therefore adopt any one of the [[SSC/Stability/UniformDensity#Sterne.27s_General_Solution\|polynomial eigenfunctions and corresponding eigenfrequencies]] derived by Sterne. We will insist that the eigenfrequency of the envelope match the eigenfrequency of the core; and, following {{ MF85bfull }} (see the top paragraph of the right-hand column on p. 223 of their article), we seek solutions for which there is continuity in both the eigenfunction and its first derivative at the interface <math>~(\xi = 1)</math>.

	===Eigenfrequencies===		===Eigenfrequencies===

Jet53man: /* Envelope Solution Outline */

2022-10-11T22:36:01Z

Envelope Solution Outline

← Older revision		Revision as of 18:36, 11 October 2022
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	==Eureka Regarding Prasad's 1948 Paper==		==Eureka Regarding Prasad's 1948 Paper==
	===Envelope Solution Outline===		===Envelope Solution Outline===
	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 5 December 2016: Yesterday, I stumbled on this key paper by Prasad (1948) while I was looking back through the published literature to catalog who has solved the polytropic wave equation numerically.]]~~[http://adsabs.harvard.edu/abs/1948MNRAS.108..414P C. Prasad (1948, MNRAS, 108, 414-416)]~~ has examined a closely related problem and, as it turns out, the mathematical approach that he used to solve that problem analytically is gratifyingly useful to me here. If, [[#More_General_Solution\|as above]], we restrict our investigation to configurations for which,		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 5 December 2016: Yesterday, I stumbled on this key paper by Prasad (1948) while I was looking back through the published literature to catalog who has solved the polytropic wave equation numerically.]]{{ Prasad48full }} has examined a closely related problem and, as it turns out, the mathematical approach that he used to solve that problem analytically is gratifyingly useful to us here. If, [[#More_General_Solution\|as above]], we restrict our investigation to configurations for which,
	<div align="center">		<div align="center">
	<math>g^2 = \mathcal{B} \, ,</math>		<math>g^2 = \mathcal{B} \, ,</math>
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	= \biggl[2 \biggl(\frac{\rho_c}{\rho_e}-1 \biggr)\biggr]^{-1} \, .</math>		= \biggl[2 \biggl(\frac{\rho_c}{\rho_e}-1 \biggr)\biggr]^{-1} \, .</math>
	</div>		</div>
	This wave equation is very similar to equation (2) of ~~[http://adsabs.harvard.edu/abs/1948MNRAS.108..414P Prasad (1948)]~~. If, following ~~Prasad~~'s guidance, we then assume a series solution of the form,		This wave equation is very similar to equation (2) of {{ Prasad48 }}. If, following {{ Prasad48hereafter }}'s guidance, we then assume a series solution of the form,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<math>c_0 = -1 \pm \sqrt{1+\alpha} \, .</math>		<math>c_0 = -1 \pm \sqrt{1+\alpha} \, .</math>
	</div>		</div>
	This is precisely the value of the exponent, <math>~c_0</math>, that we derived — in a more stumbling fashion — [[#More_General_Solution\|above]] and, as is shown by the following ~~framed image~~, it is identical to the exponent derived by ~~[http://adsabs.harvard.edu/abs/1948MNRAS.108..414P Prasad (1948)]~~.		This is precisely the value of the exponent, <math>c_0</math>, that we derived — in a more stumbling fashion — [[#More_General_Solution\|above]] and, as is shown by the following text excerpt, it is identical to the exponent derived by {{ Prasad48 }}.

	<div align="center">		<div align="center">
	<table border="2" cellpadding="10" width="75%">		<table border="2" cellpadding="10" width="75%">
	<tr>		<tr>
	<th align="center">		<td align="center">
	Equation and accompanying text extracted~~<sup>&dagger;</sup>~~ from ~~[http://adsabs.harvard.edu/abs/1948MNRAS.108~~.~~.414P C. Prasad (1948)]<p></p>~~		Equation and accompanying text extracted from the middle of p. 415 of …<br />
	~~"''Radial Oscillations~~ of ~~a Particular Stellar Model''"<p>~~</p>		{{ Prasad48figure }}
	~~Monthly Notices of the Royal Astronomical Society, vol. 108, pp. 414-416 © Royal Astronomical Society~~		</td>
	</th>
	<tr>		<tr>
	<td>		<td>
	[[File:Prasad1948Eq4.png\|400px\|left\|Prasad (1948)]]		<!-- [[File:Prasad1948Eq4.png\|400px\|left\|Prasad (1948)]] -->
			<font color="darkgreen">
			Assuming a solution in series
			</font>
			<table border="0" align="center" width="100%" cellpadding="5"><tr>
			<td align="right">
			<math>\xi_1 = x^q \sum_0^\infty a_k x^k \, ,</math>
			</td>
			<td align="right" width="40%">
			(4)
			</td>
			</tr></table>
			<font color="darkgreen">
			the indicial equation gives <math>q = -1 \pm \sqrt{1 + \alpha} \, .</math>
			</font>
	</td>		</td>
	</tr>		</tr>
	~~<tr><td align="left"><sup>&dagger;</sup>Displayed here exactly as presented in the original publication.</td></tr>~~
	</table>		</table>
	</div>		</div>

	Using equation (7) from ~~[http://adsabs.harvard.edu/abs/1948MNRAS.108..414P Prasad (1948)]~~ as a guide, we hypothesize that the eigenfrequency of the ''j''<sup>th</sup> mode in the envelope is given by the relation,		Using equation (7) from {{ Prasad48 }} as a guide, we hypothesize that the eigenfrequency of the ''j''<sup>th</sup> mode in the envelope is given by the relation,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	where, the last expression results from recognizing that <math>~\alpha_e = c_0^2 + 2c_0 \, ,</math> and we have adopted the notation,		where, the last expression results from recognizing that <math>\alpha_e = c_0^2 + 2c_0 \, ,</math> and we have adopted the notation,
	<div align="center">		<div align="center">
	<math>		<math>
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	</div>		</div>

	And guided by equation (6) from ~~[http://adsabs.harvard.edu/abs/1948MNRAS.108..414P Prasad (1948)]~~, we hypothesize that successive coefficients in the (truncated) series that defines the radial structure of each mode is governed by the recurrence relation,		And guided by equation (6) from {{ Prasad48 }}, we hypothesize that successive coefficients in the (truncated) series that defines the radial structure of each mode is governed by the recurrence relation,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Try Quartic Core Eigenfunction */

2022-10-11T22:09:58Z

Try Quartic Core Eigenfunction

← Older revision		Revision as of 18:09, 11 October 2022
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	===Try Quartic Core Eigenfunction===		===Try Quartic Core Eigenfunction===
	Let's ~~begin with Sterne~~'s quartic function ~~and see if we can match it to~~ the ~~envelope's power~~-~~law eigenfunction~~. ~~From~~ Sterne's ~~presentation~~, ~~we have~~,
			Let's focus on the analytically specified ''quartic'' displacement function (for which the quantum number, <math>j=2</math>) derived by {{ Sterne37full }} for uniform-density spheres. As has been detailed in [[SSC/Stability/UniformDensity#Sterne's_Presentation\|our accompanying discussion]] of {{ Sterne37hereafter }}'s work, the relevant eigenfunction expression is,

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

Jet53man: /* Attempt to Find Eigenfunction for the Envelope */

2022-10-11T21:53:08Z

Attempt to Find Eigenfunction for the Envelope

← Older revision		Revision as of 17:53, 11 October 2022
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	==Attempt to Find Eigenfunction for the Envelope==		==Attempt to Find Eigenfunction for the Envelope==

	Adopting some of the notation used by ~~[http://adsabs.harvard.edu/abs/1937MNRAS..97..582S T. E. Sterne (1937)]~~ and enunciated in our [[SSC/Stability/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|accompanying discussion of the uniform-density sphere]], we'll define,		Adopting some of the notation used by {{ Sterne37full }} and enunciated in our [[SSC/Stability/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|accompanying discussion of the uniform-density sphere]], we'll define,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Idea Involving Logarithmic Derivatives */

2022-04-12T17:26:12Z

Idea Involving Logarithmic Derivatives

Show changes

Jet53man: /* A Specific Choice of the Density Ratio */

2022-04-12T17:23:43Z

A Specific Choice of the Density Ratio

← Older revision		Revision as of 13:23, 12 April 2022
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	===A Specific Choice of the Density Ratio===		===A Specific Choice of the Density Ratio===
	Now, let's focus on the ''specific'' model for which <math>~\rho_e/\rho_c = 1/2</math>. In this case,		Now, let's focus on the ''specific'' model for which <math>\rho_e/\rho_c = 1/2</math>. In this case,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	Note that this last expression goes to zero at the surface of the bipolytrope, that is, at <math>~\xi = 1/q</math>. For this ''specific'' case, the wave equation for the envelope becomes,		Note that this last expression goes to zero at the surface of the bipolytrope, that is, at <math>\xi = 1/q</math>. For this ''specific'' case, the wave equation for the envelope becomes,

	<div align="center">		<div align="center">

Jet53man: /* Example11 */

2022-04-12T17:20:07Z

Example11

Show changes

Jet53man: /* Match Prasad-like Envelope Eigenvector to the Core Eigenvector */

2022-04-12T17:17:41Z

Match Prasad-like Envelope Eigenvector to the Core Eigenvector

← Older revision		Revision as of 13:17, 12 April 2022
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~0</math>		<math>0</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	(1 - \eta^2)\frac{d^2x}{d\eta^2} +		(1 - \eta^2)\frac{d^2x}{d\eta^2} +
	( 4 - 6\eta^2 ) \frac{1}{\eta} \cdot \frac{dx}{d\eta}		( 4 - 6\eta^2 ) \frac{1}{\eta} \cdot \frac{dx}{d\eta}
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	</table>		</table>
	</div>		</div>
	This means that, demanding that the two ''dimensional'' frequencies <math>~(\omega)</math> be the same requires that the ratio of the ''dimensionless'' frequencies <math>(\sigma)</math> be,		This means that, demanding that the two ''dimensional'' frequencies <math>(\omega)</math> be the same requires that the ratio of the ''dimensionless'' frequencies <math>(\sigma)</math> be,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Match Prasad-like Envelope Eigenvector to the Core Eigenvector */

2022-04-12T17:12:53Z

Match Prasad-like Envelope Eigenvector to the Core Eigenvector

← Older revision		Revision as of 13:12, 12 April 2022
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	</table>		</table>
	</div>		</div>
	Not surprisingly, this is identical in form to the eigenvalue problem first presented by [[SSC/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|Sterne (1937)]] in connection with an examination of radial oscillations in uniform-density spheres. For the core of our zero-zero bipolytrope, we can therefore adopt any one of the [[SSC/UniformDensity#Sterne.27s_General_Solution\|polynomial eigenfunctions and corresponding eigenfrequencies]] derived by Sterne. We will insist that the eigenfrequency of the envelope match the eigenfrequency of the core; and, following [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b)] (see the top paragraph of the right-hand column on p. 223 of their article), we seek solutions for which there is continuity in both the eigenfunction and its first derivative at the interface <math>~(\xi = 1)</math>.		Not surprisingly, this is identical in form to the eigenvalue problem first presented by [[SSC/Stability/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|Sterne (1937)]] in connection with an examination of radial oscillations in uniform-density spheres. For the core of our zero-zero bipolytrope, we can therefore adopt any one of the [[SSC/Stability/UniformDensity#Sterne.27s_General_Solution\|polynomial eigenfunctions and corresponding eigenfrequencies]] derived by Sterne. We will insist that the eigenfrequency of the envelope match the eigenfrequency of the core; and, following [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b)] (see the top paragraph of the right-hand column on p. 223 of their article), we seek solutions for which there is continuity in both the eigenfunction and its first derivative at the interface <math>~(\xi = 1)</math>.

	===Eigenfrequencies===		===Eigenfrequencies===
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\sigma^2\|_\mathrm{core} \equiv \frac{3\omega^2_\mathrm{core}}{2\pi \gamma_g G\rho_c} \, ,</math>		<math>\sigma^2\|_\mathrm{core} \equiv \frac{3\omega^2_\mathrm{core}}{2\pi \gamma_g G\rho_c} \, ,</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\sigma^2\|_\mathrm{env} \equiv \frac{3\omega^2_\mathrm{env}}{2\pi \gamma_g G\rho_e} \, .</math>		<math>\sigma^2\|_\mathrm{env} \equiv \frac{3\omega^2_\mathrm{env}}{2\pi \gamma_g G\rho_e} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	This means that, demanding that the two ''dimensional'' frequencies <math>~(\omega)</math> be the same requires that the ratio of the ''dimensionless'' frequencies <math>~(\sigma)</math> be,		This means that, demanding that the two ''dimensional'' frequencies <math>~(\omega)</math> be the same requires that the ratio of the ''dimensionless'' frequencies <math>(\sigma)</math> be,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{ \sigma^2\|_\mathrm{core}}{\sigma^2\|_\mathrm{env} }</math>		<math>\frac{ \sigma^2\|_\mathrm{core}}{\sigma^2\|_\mathrm{env} }</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{\rho_e}{\rho_c} \, .</math>		<math>\frac{\rho_e}{\rho_c} \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</div>		</div>

	Now, according to [[SSC/UniformDensity#Sterne.27s_General_Solution\|Sterne's derivation]], the dimensionless eigenfrequency associated with the <math>~j^\mathrm{th}</math> mode in the core is,		Now, according to [[SSC/Stability/UniformDensity#Sterne.27s_General_Solution\|Sterne's derivation]], the dimensionless eigenfrequency associated with the <math>~j^\mathrm{th}</math> mode in the core is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="center" colspan="4">		<td align="center" colspan="4">
	[[File:~~C0Plus2~~.png\|350px\|C0 Plus]]		[[File:C0Plus3.png\|350px\|C0 Plus]]
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="center" colspan="4">		<td align="center" colspan="4">
	[[File:~~C0Minus2~~.png\|350px\|C0 Minus]]		[[File:C0Plus2.png\|350px\|C0 Minus]]
	</td>		</td>
	</tr>		</tr>