SSC/Stability/BiPolytropes - Revision history

Joel2: /* Supplemental Chapters */

2025-12-08T18:26:42Z

Supplemental Chapters

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	==Supplemental Chapters==		==Supplemental Chapters==
	<ol>		<ol>
	<li>~~Containes~~ [[SSC/Stability/InstabilityOnsetOverview#Displacement_Functions_Summary\|Displacement Functions Summary]]</li>		<li>Contains [[SSC/Stability/InstabilityOnsetOverview#Displacement_Functions_Summary\|Displacement Functions Summary]]</li>
	<li>[[SSC/Stability/BiPolytropes/PlannedApproach\|Earlier Planned Approach]]</li>		<li>[[SSC/Stability/BiPolytropes/PlannedApproach\|Earlier Planned Approach]]</li>
	<li>[[SSC/Stability/BiPolytropes/HeadScratching\|Headscratching]]</li>		<li>[[SSC/Stability/BiPolytropes/HeadScratching\|Headscratching]]</li>

Joel2: /* Overview */

2025-11-22T23:06:30Z

Overview

← Older revision		Revision as of 19:06, 22 November 2025
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	When modeling bipolytropes, the default expectation is that an increase in <math>\xi_i</math> along a given sequence will correspond to an increase in the relative size — both the radius and the mass — of the core. This expectation is realized along the Figure 2 sequences that have the largest mean-molecular weight ratios: <math>\mu_e/\mu_c</math> = 1 and ½. But the behavior is different along the other five illustrated sequences. For sufficiently large <math>\xi_i</math>, the relative radius of the core begins to decrease; along each sequence, a solid purple circular marker identifies the location of this ''turning point'' in radius. Furthermore, along sequences for which <math>\mu_e/\mu_c < \tfrac{1}{3}</math>, eventually the fractional mass of the core reaches a maximum and, thereafter, decreases even as the value of <math>\xi_i</math> continues to increase. In Figure 2, a solid green circular marker identifies the location of this ''maximum mass turning point'' along each of these sequences; the analytically determined values of <math>\xi_i, q </math> and <math>\nu</math> that are associated with each of these ''turning points'' are provided in the table adjacent to Figure 2. (Additional properties of these equilibrium sequences are discussed in [[SSC/FreeEnergy/PolytropesEmbedded#Behavior_of_Equilibrium_Sequence\|yet another accompanying chapter]].)		When modeling bipolytropes, the default expectation is that an increase in <math>\xi_i</math> along a given sequence will correspond to an increase in the relative size — both the radius and the mass — of the core. This expectation is realized along the Figure 2 sequences that have the largest mean-molecular weight ratios: <math>\mu_e/\mu_c</math> = 1 and ½. But the behavior is different along the other five illustrated sequences. For sufficiently large <math>\xi_i</math>, the relative radius of the core begins to decrease; along each sequence, a solid purple circular marker identifies the location of this ''turning point'' in radius. Furthermore, along sequences for which <math>\mu_e/\mu_c < \tfrac{1}{3}</math>, eventually the fractional mass of the core reaches a maximum and, thereafter, decreases even as the value of <math>\xi_i</math> continues to increase. In Figure 2, a solid green circular marker identifies the location of this ''maximum mass turning point'' along each of these sequences; the analytically determined values of <math>\xi_i, q </math> and <math>\nu</math> that are associated with each of these ''turning points'' are provided in the table adjacent to Figure 2. (Additional properties of these equilibrium sequences are discussed in [[SSC/FreeEnergy/PolytropesEmbedded#Behavior_of_Equilibrium_Sequence\|yet another accompanying chapter]].)

	<font color="red">'''The principal question is:'''</font> ''Along bipolytropic sequences, are maximum-mass models associated with the onset of dynamical instabilities?''		<span id="PrincipalQ">
			<font color="red">'''The principal question is:'''</font> ''Along bipolytropic sequences, are maximum-mass models associated with the onset of dynamical instabilities?''</span>

Joel2: /* Organizational Index */

2024-07-21T18:00:22Z

Organizational Index

← Older revision		Revision as of 14:00, 21 July 2024
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	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  (5,1) Radial Oscillations]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  (5,1) Radial Oscillations]]<br /> </td>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>		<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>
			</tr>
			<tr>
			<td align="left" width="100%" colspan="4">
			These four chapters, labeled Parts I - IV, are segments of the much longer chapter titled, [[SSC/Stability/BiPolytropes/PlannedApproach\|SSC/Stability/BiPolytropes/PlannedApproach]]. An [[SSC/Stability/BiPolytropes/Index\|accompanying organizational index]] has helped us write this chapter succinctly.
			</td>
	</tr>		</tr>
	</table>		</table>

	~~The [[SSC/Stability/BiPolytropes/Index\|accompanying organizational index]] has helped us write this chapter succinctly.~~

	=In Search of Marginally Unstable (n<sub>c</sub>,n<sub>e</sub>) = (5,1) Bipolytropes=		=In Search of Marginally Unstable (n<sub>c</sub>,n<sub>e</sub>) = (5,1) Bipolytropes=

Joel2: /* Organizational Index */

2024-01-08T20:17:50Z

Organizational Index

← Older revision		Revision as of 16:17, 8 January 2024
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	<tr>		<tr>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   The Search]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   The Search]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  ~~to be named~~]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  Review of MF85b]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  (5,1) Radial Oscillations]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  (5,1) Radial Oscillations]]<br /> </td>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>		<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>

Joel2: /* Review of the Analysis by Murphy & Fiedler (1985b) */

2024-01-08T20:17:05Z

Review of the Analysis by Murphy & Fiedler (1985b)

Show changes

Joel2: /* Radial Oscillations of (nc, ne) = (5, 1) Models */

2024-01-08T20:07:54Z

Radial Oscillations of (nc, ne) = (5, 1) Models

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Joel2: /* Organizational Index */

2024-01-08T20:07:21Z

Organizational Index

← Older revision		Revision as of 16:07, 8 January 2024
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	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   The Search]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   The Search]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  to be named]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  to be named]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  ~~to be named~~]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  (5,1) Radial Oscillations]]<br /> </td>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>		<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>
	</tr>		</tr>

Joel2 at 20:00, 8 January 2024

2024-01-08T20:00:26Z

Show changes

Joel2: /* Organizational Index */

2024-01-08T19:44:46Z

Organizational Index

← Older revision		Revision as of 15:44, 8 January 2024
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	=Organizational Index=		=Organizational Index=
			<table border="1" align="center" width="100%" colspan="8">
			<tr>
			<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   to be named]]<br /> </td>
			<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  to be named]]<br /> </td>
			<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  to be named]]<br /> </td>
			<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  to be named]]<br /> </td>
			</tr>
			</table>

	The [[SSC/Stability/BiPolytropes/Index\|accompanying organizational index]] has helped us write this chapter succinctly.		The [[SSC/Stability/BiPolytropes/Index\|accompanying organizational index]] has helped us write this chapter succinctly.

Joel2: Created page with "FORCETOC =Organizational Index= The accompanying organizational index has helped us write this chapter succinctly. =In Search of Marginally Unstable (n_c,n_e) = (5,1) Bipolytropes=

← Older revision		Revision as of 16:17, 8 January 2024
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	<tr>		<tr>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   The Search]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   The Search]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  ~~to be named~~]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  Review of MF85b]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  (5,1) Radial Oscillations]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  (5,1) Radial Oscillations]]<br /> </td>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>		<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>

← Older revision		Revision as of 16:07, 8 January 2024
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	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   The Search]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes\|Part I:   The Search]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  to be named]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2\|Part II:  to be named]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  ~~to be named~~]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3\|III:  (5,1) Radial Oscillations]]<br /> </td>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>		<td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4\|IV:  Reconciliation]]<br /> </td>
	</tr>		</tr>

SSC/Stability/BiPolytropes - Revision history

Joel2: /* Supplemental Chapters */

Joel2: /* Overview */

Joel2: /* Organizational Index */

Joel2: /* Organizational Index */

Joel2: /* Review of the Analysis by Murphy & Fiedler (1985b) */

Joel2: /* Radial Oscillations of (nc, ne) = (5, 1) Models */

Joel2: /* Organizational Index */

Joel2 at 20:00, 8 January 2024

Joel2: /* Organizational Index */

Joel2: Created page with "__FORCETOC__ =Organizational Index= The accompanying organizational index has helped us write this chapter succinctly. =In Search of Marginally Unstable (nc,ne) = (5,1) Bipolytropes=

Joel2: Created page with "FORCETOC =Organizational Index= The accompanying organizational index has helped us write this chapter succinctly. =In Search of Marginally Unstable (n_c,n_e) = (5,1) Bipolytropes=