SSC/Structure/LimitingMasses - Revision history

Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2024-01-15T23:23:23Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 19:23, 15 January 2024
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	==Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses==		==Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses==
	[[SSC/Structure/BiPolytropes/~~Analytic15~~#Limit_when_m3_~~.3D_0~~\|As we have shown elsewhere]], in the limit <math>m_3 \rightarrow 0</math>, the physically relevant root of the above analytic relation is <math>\xi_i = 3</math> and the mass contained in the core is,		[[SSC/Structure/BiPolytropes/Analytic51#Limit_when_m3_=_0\|As we have shown elsewhere]], in the limit <math>m_3 \rightarrow 0</math>, the physically relevant root of the above analytic relation is <math>\xi_i = 3</math> and the mass contained in the core is,
	<div align="center">		<div align="center">
	<math>		<math>

Joel2: /* Schönberg-Chandrasekhar Mass */

2024-01-15T23:20:13Z

Schönberg-Chandrasekhar Mass

← Older revision		Revision as of 19:20, 15 January 2024
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	In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, {{ Beech88full }} matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see {{ BTZ2012 }}]. Beech's results were not significantly different from those reported by {{ SC42 }}; in particular, the value of <math>\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.		In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, {{ Beech88full }} matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see {{ BTZ2012 }}]. Beech's results were not significantly different from those reported by {{ SC42 }}; in particular, the value of <math>\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.

	[[SSC/Structure/BiPolytropes/~~Analytic15~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|In an accompanying derivation]] [see, also, {{ EFC98full }}] we have gone one step farther, matching an analytically prescribable, <math>n_e = 1</math> polytropic envelope to an analytically prescribable, <math>n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>\mu_c/\mu_e > 3</math> and that the interface location, <math>\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:		[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|In an accompanying derivation]] [see, also, {{ EFC98full }}] we have gone one step farther, matching an analytically prescribable, <math>n_e = 1</math> polytropic envelope to an analytically prescribable, <math>n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>\mu_c/\mu_e > 3</math> and that the interface location, <math>\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:

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Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2023-12-31T01:15:37Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 21:15, 30 December 2023
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	</td>		</td>
	<td align="center" colspan="1" bgcolor="yellow">		<td align="center" colspan="1" bgcolor="yellow">
	Equilibrium Sequences for <br /><math>(n_c, n_e) = (5, 1)</math> Bipolytropes		Equilibrium Sequences for <br /><math>(n_c, n_e) = (5, 1)</math> Bipolytropes<br />having various values of <math>\mu_e/\mu_c</math>
	</td>		</td>
	</tr>		</tr>

Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2023-12-31T01:12:57Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 21:12, 30 December 2023
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	Virial-Determined Mass-Radius Sequences for Embedded Polytropes, as detailed in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Concise_Mass-Radius_Relation\|accompanying discussion]]		Virial-Determined Mass-Radius Sequences for Embedded Polytropes, as detailed in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Concise_Mass-Radius_Relation\|accompanying discussion]]
	</td>		</td>
	<td align="~~center~~" colspan="1" width="25%" bgcolor="pink">  </td>		<td align="left" colspan="1" width="25%" bgcolor="pink">
	<td align="center" colspan="1" bgcolor="yellow">~~ ~~ </td>		<div align="center">Detailed Force-Balanced Determination of Mass-Radius Sequences for Embedded Polytropes</div>
			----

			''Top:''  Sequences with n = 1, 2.5, 3, 3.05, 3.5, 5, 6. <br />
			''Bottom:''   Same as ''top'' without n = 1, 2.5 while adding isothermal (n = ∞).
			</td>
			<td align="center" colspan="1" bgcolor="yellow">
			Equilibrium Sequences for <br /><math>(n_c, n_e) = (5, 1)</math> Bipolytropes
			</td>
	</tr>		</tr>

Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2023-12-31T01:01:18Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 21:01, 30 December 2023
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	Figure 17 from <br />{{ Stahler83figure }}		Figure 17 from <br />{{ Stahler83figure }}
	</td>		</td>
	<td align="center" colspan="1" width="25%" bgcolor="lightgreen">~~ ~~ </td>		<td align="center" colspan="1" width="25%" bgcolor="lightgreen">
			Virial-Determined Mass-Radius Sequences for Embedded Polytropes, as detailed in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Concise_Mass-Radius_Relation\|accompanying discussion]]
			</td>
	<td align="center" colspan="1" width="25%" bgcolor="pink">  </td>		<td align="center" colspan="1" width="25%" bgcolor="pink">  </td>
	<td align="center" colspan="1" bgcolor="yellow">  </td>		<td align="center" colspan="1" bgcolor="yellow">  </td>
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	<!-- [[Image:MassRadiusVirialLabeled.png\|250px\|center]] -->		<!-- [[Image:MassRadiusVirialLabeled.png\|250px\|center]] -->
	[[~~Image~~:~~MassVsRadiusCombined02~~.png\|~~250px~~\|center]]		[[File:VirialDeterminedMRsequencesLabeled.png\|300px\|center]]
	</td>		</td>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
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	</td>		</td>
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			<tr>
			<td align="center" bgcolor="white">

			</td>
			<td align="center" bgcolor="white">

			</td>
			<td align="center" rowspan="1" bgcolor="white">
			<!-- [[Image:MassRadiusVirialLabeled.png\|250px\|center]] -->
			[[Image:MassVsRadiusCombined02.png\|250px\|center]]
			</td>
			<td align="center" bgcolor="white">

			</td>
	</tr>		</tr>

	</table>		</table>
	</div>		</div>

Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2023-12-28T01:01:43Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 21:01, 27 December 2023
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	</td>		</td>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
	[[File:Stahler1983Comparison.png\|250px\|center\|To be compared with Stahler (1983)]]		<!-- [[File:Stahler1983Comparison.png\|250px\|center\|To be compared with Stahler (1983)]] -->
			[[File:DFBsequenceB.png\|250px\|center]]
	</td>		</td>
	<td align="center" rowspan="1" bgcolor="white">		<td align="center" rowspan="1" bgcolor="white">

Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2023-12-28T00:57:15Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 20:57, 27 December 2023
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	</td>		</td>
	<td align="center" rowspan="1" bgcolor="white">		<td align="center" rowspan="1" bgcolor="white">
	[[Image:PlotSequencesBest02.png\|250px\|center]]		<!-- [[Image:PlotSequencesBest02.png\|250px\|center]] -->
			[[File:TurningPoints51Bipolytropes.png\|250px\|center]]
	</td>		</td>
	</tr>		</tr>

Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2023-12-28T00:50:43Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 20:50, 27 December 2023
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	</td>		</td>
	<td align="center" rowspan="1" bgcolor="white">		<td align="center" rowspan="1" bgcolor="white">
	[[Image:MassRadiusVirialLabeled.png\|250px\|center]]		<!-- [[Image:MassRadiusVirialLabeled.png\|250px\|center]] -->
			[[Image:MassVsRadiusCombined02.png\|250px\|center]]
	</td>		</td>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">

Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2023-12-28T00:38:43Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 20:38, 27 December 2023
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	<!-- [[File:Stahler1983TitlePage0.png\|300px\|center\|Stahler (1983) Title Page]] -->		<!-- [[File:Stahler1983TitlePage0.png\|300px\|center\|Stahler (1983) Title Page]] -->
	<!-- [[Image:AAAwaiting01.png\|300px\|center\|Stahler (1983) Title Page]] -->		<!-- [[Image:AAAwaiting01.png\|300px\|center\|Stahler (1983) Title Page]] -->
	{{ Stahler83figure }}		Figure 17 from <br />{{ Stahler83figure }}
	</td>		</td>
	<td align="center" colspan="1" width="25%" bgcolor="lightgreen">  </td>		<td align="center" colspan="1" width="25%" bgcolor="lightgreen">  </td>
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	<tr>		<tr>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
	[[File:Stahler_MRdiagram1.png\|~~200px~~\|center\|Stahler (1983) Figure 17 ~~(edited)~~]]		[[File:Stahler_MRdiagram1.png\|250px\|center\|Stahler (1983) Figure 17]]
	<!-- [[Image:AAAwaiting01.png\|300px\|center\|Stahler (1983) Figure 17 (edited)]] -->		<!-- [[Image:AAAwaiting01.png\|300px\|center\|Stahler (1983) Figure 17 (edited)]] -->
	</td>		</td>
	<td align="center" rowspan="1" bgcolor="white">		<td align="center" rowspan="1" bgcolor="white">
	[[Image:MassRadiusVirialLabeled.png\|~~200px~~\|center]]		[[Image:MassRadiusVirialLabeled.png\|250px\|center]]
	</td>		</td>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
	[[File:Stahler1983Comparison.png\|~~200px~~\|center\|To be compared with Stahler (1983)]]		[[File:Stahler1983Comparison.png\|250px\|center\|To be compared with Stahler (1983)]]
	</td>		</td>
	<td align="center" rowspan="1" bgcolor="white">		<td align="center" rowspan="1" bgcolor="white">
	[[Image:PlotSequencesBest02.png\|~~200px~~\|center]]		[[Image:PlotSequencesBest02.png\|250px\|center]]
	</td>		</td>
	</tr>		</tr>

Joel2: /* Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses */

2023-12-28T00:36:04Z

Relationship Between the Bonnor-Ebert and Schönberg-Chandrasekhar Critical Masses

← Older revision		Revision as of 20:36, 27 December 2023
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	<table border="2" cellpadding="8">		<table border="2" cellpadding="8">
	<tr>		<tr>
	<td align="center" colspan="2">		<td align="center" colspan="4">
	<font color="darkblue">Equilibrium Mass-Radius ~~Diagram~~ </font>		<font color="darkblue">Equilibrium Mass-Radius Diagrams </font>
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center" bgcolor="white">		<td align="center" width="25%" bgcolor="white">
	<!-- [[File:Stahler1983TitlePage0.png\|300px\|center\|Stahler (1983) Title Page]] -->		<!-- [[File:Stahler1983TitlePage0.png\|300px\|center\|Stahler (1983) Title Page]] -->
	<!-- [[Image:AAAwaiting01.png\|300px\|center\|Stahler (1983) Title Page]] -->		<!-- [[Image:AAAwaiting01.png\|300px\|center\|Stahler (1983) Title Page]] -->
	{{ Stahler83figure }}		{{ Stahler83figure }}
	</td>		</td>
	<td align="center" colspan="1" bgcolor="~~white~~">		<td align="center" colspan="1" width="25%" bgcolor="lightgreen">  </td>
			<td align="center" colspan="1" width="25%" bgcolor="pink">  </td>
	</td>		<td align="center" colspan="1" bgcolor="yellow">  </td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
	[[File:Stahler_MRdiagram1.png\|~~300px~~\|center\|Stahler (1983) Figure 17 (edited)]]		[[File:Stahler_MRdiagram1.png\|200px\|center\|Stahler (1983) Figure 17 (edited)]]
	<!-- [[Image:AAAwaiting01.png\|300px\|center\|Stahler (1983) Figure 17 (edited)]] -->		<!-- [[Image:AAAwaiting01.png\|300px\|center\|Stahler (1983) Figure 17 (edited)]] -->
	</td>		</td>
	<td align="center" rowspan="1" bgcolor="white">		<td align="center" rowspan="1" bgcolor="white">
	[[Image:MassRadiusVirialLabeled.png\|~~300px~~\|center]]		[[Image:MassRadiusVirialLabeled.png\|200px\|center]]
	</td>		</td>
	~~</tr>~~

	~~<tr>~~
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
	[[File:Stahler1983Comparison.png\|~~300px~~\|center\|To be compared with Stahler (1983)]]		[[File:Stahler1983Comparison.png\|200px\|center\|To be compared with Stahler (1983)]]
	</td>		</td>
	<td align="center" rowspan="1" bgcolor="white">		<td align="center" rowspan="1" bgcolor="white">
	[[Image:PlotSequencesBest02.png\|~~300px~~\|center]]		[[Image:PlotSequencesBest02.png\|200px\|center]]
	</td>		</td>
			</tr>
	</tr>		</tr>
	</table>		</table>