SSC/Stability/n1PolytropeLAWE - Revision history

Joel2: /* Radial Oscillations of n = 1 Polytropic Spheres (Pt 4) */

2025-07-15T17:57:15Z

Radial Oscillations of n = 1 Polytropic Spheres (Pt 4)

← Older revision		Revision as of 13:57, 15 July 2025
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	=Radial Oscillations of n = 1 Polytropic Spheres (Pt 4)=		=Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)=
	<table border="1" align="center" width="100%" colspan="8">		<table border="1" align="center" width="100%" colspan="8">
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Joel2: /* Radial Oscillations of n = 1 Polytropic Spheres (Pt 1) */

2025-07-15T17:51:32Z

Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)

← Older revision		Revision as of 13:51, 15 July 2025
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	=Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)=		=Radial Oscillations of n = 1 Polytropic Spheres (Pt 4)=
	<table border="1" align="center" width="100%" colspan="8">		<table border="1" align="center" width="100%" colspan="8">
	<tr>		<tr>
	<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Stability/n1PolytropeLAWE\|Part I:   Search for Analytic Solutions]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/n1PolytropeLAWE\|Part I:   Search for Analytic Solutions]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Stability/n1PolytropeLAWE/Pt2\|Part II:  New Ideas]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/n1PolytropeLAWE/Pt2\|Part II:  New Ideas]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Stability/n1PolytropeLAWE/Pt3\|III:  What About Bipolytropes?]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/n1PolytropeLAWE/Pt3\|Part III:  What About Bipolytropes?]]<br /> </td>
			<td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Stability/n1PolytropeLAWE/Pt4\|Part IV:  Most General Structural Solution]]<br /> </td>
	</tr>		</tr>
	</table>		</table>

Joel2: /* Graphical Reassessment */

2024-01-22T02:57:31Z

Graphical Reassessment

← Older revision		Revision as of 22:57, 21 January 2024
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	This expression clearly is not zero, so our trial eigenfunction is not a good one. However, the terms in the wave equation did combine somewhat to give a fairly compact — ''albeit nonzero'' — expression. So we may be on the right track!		This expression clearly is not zero, so our trial eigenfunction is not a good one. However, the terms in the wave equation did combine somewhat to give a fairly compact — ''albeit nonzero'' — expression. So we may be on the right track!


	=See Also=		=See Also=

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Joel2: /* Setup */

2024-01-21T19:29:34Z

Setup

← Older revision		Revision as of 15:29, 21 January 2024
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	<br />		<br />

	This is identical to the formulation of the wave equation that is relevant to the (n = 1) core of the composite polytrope studied by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b)]; for comparison, their expression is displayed~~, here~~, in the following boxed-in image.		This is identical to the formulation of the wave equation that is relevant to the (n = 1) core of the composite polytrope studied by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b)]; for comparison, their expression is displayed, in the following boxed-in image.

	<div align="center">		<div align="center">
	<table border="2" cellpadding="10" id="MurphyFiedler1985b">		<table border="2" cellpadding="10" id="MurphyFiedler1985b">
	<tr>		<tr>
	<th align="center">		<td align="center">
	n = 1 Polytropic Formulation of ~~Wave Equation~~ as Presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)]		n = 1 Polytropic Formulation of the LAWE as Presented by …
	</th>
			{{ MF85bfigure }}
			<!--[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] -->
			</td>
	<tr>		<tr>
	<td>		<td>
	[[File:MurphyFiedlerN1formulation.png\|700px\|center\|Murphy & Fiedler (1985b)]]		<!-- [[File:MurphyFiedlerN1formulation.png\|700px\|center\|Murphy & Fiedler (1985b)]] -->
			<table border="0" align="center" cellpadding="5">
			<tr>
			<td align="center">
			<math>
			\frac{d^2\eta}{d\zeta^2}
			+ \frac{1}{\zeta}\biggl[ 4 + \frac{2(\zeta \cos\zeta - \sin\zeta)}{\sin\zeta}\biggr]\frac{d\eta}{d\zeta}
			+ \biggl[ \frac{\omega_k^2 \zeta}{\sin\zeta} + \frac{2\alpha^* (\zeta \cos\zeta - \sin\zeta)}{\zeta^2\sin\zeta} \biggr]\eta
			= 0
			</math>
			</td>
			</tr>
			</table>
	</td>		</td>
	</tr>		</tr>

Joel2: /* Radial Oscillations of n = 1 Polytropic Spheres (Pt 1) */

2024-01-21T18:12:26Z

Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)

← Older revision		Revision as of 14:12, 21 January 2024
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	As far as we have been able to ascertain, the first technical examination of radial oscillation modes in <math>n=1</math> polytropes was performed — using numerical techniques — in 1951 by L. D. Chatterji; at the time, he was in the Mathematics Department of Allahabad University. His two papers on this topic were published in, what is now referred to as, the ''Proceedings of the Indian National Science Academy'' (PINSA). The citations that immediately follow this opening paragraph provide inks to both of these papers by Chatterji, but the links may be insecure. (Citations/links to articles that provide analyses of models having other polytropic indexes are provided at the [[#See_Also\|bottom of this chapter]].) Apparently [https://www.springer.com/journal/43538 Springer] is archiving recent PINSA volumes, but their holdings do not date back as early as 1951.		As far as we have been able to ascertain, the first technical examination of radial oscillation modes in <math>n=1</math> polytropes was performed — using numerical techniques — in 1951 by L. D. Chatterji; at the time, he was in the Mathematics Department of Allahabad University. His two papers on this topic were published in, what is now referred to as, the ''Proceedings of the Indian National Science Academy'' (PINSA). The citations that immediately follow this opening paragraph provide inks to both of these papers by Chatterji, but the links may be insecure. (Citations/links to articles that provide analyses of models having other polytropic indexes are provided at the [[#See_Also\|bottom of this chapter]].) Apparently [https://www.springer.com/journal/43538 Springer] is archiving recent PINSA volumes, but their holdings do not date back as early as 1951.

	* {{ ~~Chatterji51~~ }}, ''Radial Oscillations of a Gaseous Star of Polytropic Index I''		* {{ Chatterji51full }}, ''Radial Oscillations of a Gaseous Star of Polytropic Index I''
	* {{ ~~Chatterji52~~ }}, ''Anharmonic Pulsations of a Polytropic Model of Index Unity''		* {{ Chatterji52full }}, ''Anharmonic Pulsations of a Polytropic Model of Index Unity''

			A detailed review of {{ Chatterji51hereafter }} is provided in [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Setup\|an accompanying discussion]].

	==Groundwork==		==Groundwork==

Joel2 at 02:11, 21 January 2024

2024-01-21T02:11:07Z

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Joel2: Created page with "FORCETOC =Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)=

Part I: Search for Analytic Solutions

Part II: New Ideas

2024-01-21T01:52:02Z

Created page with "FORCETOC =Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part I: Search for Analytic Solutions<br /> </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part II: New Ideas<br /> </td> <td align="center" ro..."

New page
FORCETOC
=Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Stability/n1PolytropeLAWE|Part I:   Search for Analytic Solutions]]<br /> </td>
<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Stability/n1PolytropeLAWE/Pt2|Part II:  New Ideas]]<br /> </td>
<td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Stability/n1PolytropeLAWE/Pt3|III:  What About Bipolytropes?]]<br /> </td>
</tr>
</table>

=See Also=

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