SSC/Stability/GammaVariation - Revision history

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T23:26:01Z

Example Fundamental Modes for Isolated Configurations

@@ Line 1,636: / Line 1,636: @@
    <td align="left"><sup>b</sup>{{ HRW66 }}</td>
    <td align="center">[[SSC/Stability/Polytropes/Pt3#Tables|here]]</td>
 </tr>
@@ Line 1,644: / Line 1,662: @@
    <td align="center"><math>0.0</math></td>
    <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
-   <td align="center">[[SSC/Stability/n3PolytropeLAWE#Schwarzschild_(1941)|here]]</td>
+   <td align="center">n/a</td>
 </tr>

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T23:06:30Z

Example Fundamental Modes for Isolated Configurations

← Older revision		Revision as of 19:06, 23 January 2024
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	<td align="center"><math>0</math></td>		<td align="center"><math>0</math></td>
	<td align="center">1</td>		<td align="center">1</td>
	<td align="center"><math>\~~frac{5}{3}~~</math></td>		<td align="center">Any <math>\gamma</math></td>
	<td align="center"><math>2</math></td>		<td align="center"><math>6(\gamma - 4/3)</math></td>
	<td align="left"><sup>c</sup>{{ Sterne37 }}</td>		<td align="left"><sup>c</sup>{{ Sterne37 }}</td>
	<td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|here]]</td>		<td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|here]]</td>

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T22:56:10Z

Example Fundamental Modes for Isolated Configurations

@@ Line 1,571: / Line 1,571: @@
    <td align="center" bgcolor="lightgrey" colspan="6">&nbsp; </td>
 </tr>
 <tr>
    <td align="center"><math>1</math></td>
@@ Line 1,595: / Line 1,596: @@
    <td align="center"><math>1.1492896</math></td>
    <td align="center" colspan="2">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|Our imposed surface B.C.]]</td>
 </tr>

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T22:43:56Z

Example Fundamental Modes for Isolated Configurations

← Older revision		Revision as of 18:43, 23 January 2024
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	</tr>		</tr>

			<tr>
			<td align="center" bgcolor="lightgrey" colspan="6">  </td>
			</tr>
	<tr>		<tr>
	<td align="center"><math>1</math></td>		<td align="center"><math>1</math></td>
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	<tr>		<tr>
	<td align="center"~~><math>3</math></td>~~		<td align="center" bgcolor="lightgrey" colspan="6">  </td>
	~~<td align="right">54.18248</td>~~
	~~<td align~~="~~center~~"~~><math>\frac{4}{3}</math></td>~~
	~~<td align="center"><math>0.0</math></td>~~
	~~<td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>~~
	~~<td align~~="~~center~~">~~[[SSC/Stability/n3PolytropeLAWE#Schwarzschild_(1941)\|here]]~~</td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="center">~~ ~~</td>		<td align="center"><math>3</math></td>
	<td align="right">~~ ~~</td>		<td align="right">54.18248</td>
	<td align="center"><math>\frac{5}{3}</math></td>		<td align="center"><math>\frac{5}{3}</math></td>
	<td align="center"><math>0.34175</math></td>		<td align="center"><math>0.34175</math></td>
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	<td align="left"><sup>b</sup>{{ HRW66 }}</td>		<td align="left"><sup>b</sup>{{ HRW66 }}</td>
	<td align="center">[[SSC/Stability/Polytropes/Pt3#Tables\|here]]</td>		<td align="center">[[SSC/Stability/Polytropes/Pt3#Tables\|here]]</td>
			</tr>

			<tr>
			<td align="center"> </td>
			<td align="right"> </td>
			<td align="center"><math>\frac{4}{3}</math></td>
			<td align="center"><math>0.0</math></td>
			<td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
			<td align="center">[[SSC/Stability/n3PolytropeLAWE#Schwarzschild_(1941)\|here]]</td>
	</tr>		</tr>

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T19:42:01Z

Example Fundamental Modes for Isolated Configurations

← Older revision		Revision as of 15:42, 23 January 2024
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	As an example, consider specifying <math>\gamma = \tfrac{5}{3}</math> for an isolated, <math>n = 1</math> polytrope. The following table records the value of the square of the eigenfrequency that has been independently determined by three different research groups: 1.155 by {{ Chatterji51 }}; 1.1499 by {{ HRW66 }}; and 1.1492896 [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|herein]]. Also for comparison, the corresponding ''eigenfunction'' obtained from two of these investigations has been displayed graphically [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|herein]].		As an example, consider specifying <math>\gamma = \tfrac{5}{3}</math> for an isolated, <math>n = 1</math> polytrope. The following table records the value of the square of the eigenfrequency that has been independently determined by three different research groups: 1.155 by {{ Chatterji51 }}; 1.1499 by {{ HRW66 }}; and 1.1492896 [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|herein]]. Also for comparison, the corresponding ''eigenfunction'' obtained from two of these investigations has been displayed graphically [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|herein]].

			Not unexpectedly, when a different value of <math>\gamma</math> is specified, the result is a different radial oscillation eigenfrequency along with a different eigenfunction. However, as was first demonstrated by {{ Sterne37 }}, for an <math>n = 0</math> (uniform-density) polytrope, even though the eigenfrequency varies with the choice of <math>\gamma</math>, the radial displacement ''eigenfunction'' is identically the same for all chosen <math>\gamma</math>.

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T19:20:55Z

Example Fundamental Modes for Isolated Configurations

← Older revision		Revision as of 15:20, 23 January 2024
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	==Example Fundamental Modes for Isolated Configurations==		==Example Fundamental Modes for Isolated Configurations==
	For an isolated polytrope whose surface does not extend to infinity — that is, for <math>0 < n < 5</math> — the eigenvector for the fundamental mode of radial oscillation depends on the specification of a single parameter: <math>\gamma</math>. Then, for virtually any choice of the square of the radial oscillation frequency, <math>\sigma_c^2</math>, the governing polytropic LAWE can be integrated (usually, numerically) to obtain the radial-displacement, <math>x(\xi)</math>, that is consistent with that choice of <math>\sigma_c^2</math>. While this function, <math>x(\xi)</math>, satisfies the LAWE, its slope at the surface of the polytrope usually will not satisfy the physically relevant boundary condition. ~~So other~~		For an isolated polytrope whose surface does not extend to infinity — that is, for <math>0 < n < 5</math> — the eigenvector for the fundamental mode of radial oscillation depends on the specification of a single parameter: <math>\gamma</math>. Then, for virtually any choice of the square of the radial oscillation frequency, <math>\sigma_c^2</math>, the governing polytropic LAWE can be integrated (usually, numerically) to obtain the radial-displacement, <math>x(\xi)</math>, that is consistent with that choice of <math>\sigma_c^2</math>. While this function, <math>x(\xi)</math>, satisfies the LAWE, its slope at the surface of the polytrope usually will not satisfy the physically relevant boundary condition. Other "guesses" for <math>\sigma_c^2</math> must be made until the <math>x(\xi)</math> function satisfies the proper boundary condition; the result provides the eigenfrequency and eigenfunction (together, the eigenvector) that are associated with the specified value of <math>\gamma</math>.

			As an example, consider specifying <math>\gamma = \tfrac{5}{3}</math> for an isolated, <math>n = 1</math> polytrope. The following table records the value of the square of the eigenfrequency that has been independently determined by three different research groups: 1.155 by {{ Chatterji51 }}; 1.1499 by {{ HRW66 }}; and 1.1492896 [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|herein]]. Also for comparison, the corresponding ''eigenfunction'' obtained from two of these investigations has been displayed graphically [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|herein]].

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T18:52:02Z

Example Fundamental Modes for Isolated Configurations

← Older revision		Revision as of 14:52, 23 January 2024
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	==Example Fundamental Modes for Isolated Configurations==		==Example Fundamental Modes for Isolated Configurations==
			For an isolated polytrope whose surface does not extend to infinity — that is, for <math>0 < n < 5</math> — the eigenvector for the fundamental mode of radial oscillation depends on the specification of a single parameter: <math>\gamma</math>. Then, for virtually any choice of the square of the radial oscillation frequency, <math>\sigma_c^2</math>, the governing polytropic LAWE can be integrated (usually, numerically) to obtain the radial-displacement, <math>x(\xi)</math>, that is consistent with that choice of <math>\sigma_c^2</math>. While this function, <math>x(\xi)</math>, satisfies the LAWE, its slope at the surface of the polytrope usually will not satisfy the physically relevant boundary condition. So other

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
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	<td align="center"><math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math>		<td align="center"><math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math>
	<td align="center">Publication</td>		<td align="center">Publication</td>
	<td align="center">Relevant<br />~~H_Book~~ Chapter</td>		<td align="center">Relevant<br />JETohlineWiki<br />Chapter</td>
	</tr>		</tr>

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T01:15:45Z

Example Fundamental Modes for Isolated Configurations

← Older revision		Revision as of 21:15, 22 January 2024
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	<tr>		<tr>
	<td align="center"><math>0</math></td>		<td align="center"><math>0</math></td>
	<td align="~~right~~">1</td>		<td align="center">1</td>
	<td align="center"><math>\frac{5}{3}</math></td>		<td align="center"><math>\frac{5}{3}</math></td>
	<td align="center"><math>2</math></td>		<td align="center"><math>2</math></td>
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	<tr>		<tr>
	<td align="center"><math>1</math></td>		<td align="center"><math>1</math></td>
	<td align="~~right~~">3~~.30~~</td>		<td align="center"><math>\frac{\pi^2}{3}</math></td>
	<td align="center"><math>\frac{5}{3}</math></td>		<td align="center"><math>\frac{5}{3}</math></td>
	<td align="center"><math>1.155</math></td>		<td align="center"><math>1.155</math></td>
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	<td align="center"><math>\frac{5}{3}</math></td>		<td align="center"><math>\frac{5}{3}</math></td>
	<td align="center"><math>1.1492896</math></td>		<td align="center"><math>1.1492896</math></td>
	<td align="~~left~~"~~>Our imposed surface B.C.</td>~~		<td align="center" colspan="2">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|Our imposed surface B.C.]]</td>
	~~<td align~~="~~center~~">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|~~here~~]]</td>
	</tr>		</tr>

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T01:10:56Z

Example Fundamental Modes for Isolated Configurations

← Older revision		Revision as of 21:10, 22 January 2024
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	<td align="center"><math>1</math></td>		<td align="center"><math>1</math></td>
	<td align="right">3.30</td>		<td align="right">3.30</td>
			<td align="center"><math>\frac{5}{3}</math></td>
			<td align="center"><math>1.155</math></td>
			<td align="left"><sup>d</sup>{{ Chatterji51 }}</td>
			<td align="center">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|here]]</td>
			</tr>

			<tr>
			<td align="center"> </td>
			<td align="right"> </td>
	<td align="center"><math>\frac{5}{3}</math></td>		<td align="center"><math>\frac{5}{3}</math></td>
	<td align="center"><math>1.1499</math></td>		<td align="center"><math>1.1499</math></td>
	<td align="left"><sup>b</sup>{{ HRW66 }}</td>		<td align="left"><sup>b</sup>{{ HRW66 }}</td>
	<td align="center">[[SSC/Stability/Polytropes/Pt3#Tables\|here]]</td>		<td align="center">[[SSC/Stability/Polytropes/Pt3#Tables\|here]]</td>
			</tr>

			<tr>
			<td align="center"> </td>
			<td align="right"> </td>
			<td align="center"><math>\frac{5}{3}</math></td>
			<td align="center"><math>1.1492896</math></td>
			<td align="left">Our imposed surface B.C.</td>
			<td align="center">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope\|here]]</td>
	</tr>		</tr>

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	<li> <math>\omega^2 = s^2_\mathrm{HRW66}</math></li>		<li> <math>\omega^2 = s^2_\mathrm{HRW66}</math></li>
	<li> <math>\omega^2 = n^2_\mathrm{Sterne37}</math></li>		<li> <math>\omega^2 = n^2_\mathrm{Sterne37}</math></li>
			<li> <math>\sigma_c^2 = 3\gamma \omega^2_\mathrm{Chatterji}</math></li>
	</ol>		</ol>
	</td>		</td>

Joel2: /* Example Fundamental Modes for Isolated Configurations */

2024-01-23T00:57:02Z

Example Fundamental Modes for Isolated Configurations

← Older revision		Revision as of 20:57, 22 January 2024
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	<td align="center"><math>0</math></td>		<td align="center"><math>0</math></td>
	<td align="right">1</td>		<td align="right">1</td>
			<td align="center"><math>\frac{5}{3}</math></td>
			<td align="center"><math>2</math></td>
			<td align="left"><sup>c</sup>{{ Sterne37 }}</td>
			<td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|here]]</td>
			</tr>

			<tr>
			<td align="center"> </td>
			<td align="right"> </td>
	<td align="center"><math>\frac{5}{3}</math></td>		<td align="center"><math>\frac{5}{3}</math></td>
	<td align="center"><math>2</math></td>		<td align="center"><math>2</math></td>
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	<li> <math>\sigma_c^2 = \tfrac{3}{2}\gamma \omega^2_\mathrm{Sch}</math></li>		<li> <math>\sigma_c^2 = \tfrac{3}{2}\gamma \omega^2_\mathrm{Sch}</math></li>
	<li> <math>\omega^2 = s^2_\mathrm{HRW66}</math></li>		<li> <math>\omega^2 = s^2_\mathrm{HRW66}</math></li>
			<li> <math>\omega^2 = n^2_\mathrm{Sterne37}</math></li>
	</ol>		</ol>
	</td>		</td>