SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2 - Revision history

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-31T05:19:15Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 01:19, 31 December 2023
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	is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure. For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of {{ Stahler83 }}. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.		is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure. For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of {{ Stahler83 }}. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.
	<table border="1" cellpadding="3" align="center">		<table border="1" cellpadding="3" align="center" width="70%">
	<tr>		<tr>
	<td align="center" colspan="2">		<td align="center" colspan="2">
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	<td align="center">		<td align="center">
	[[File:Stahler_MRdiagram1.png\|300px\|center\|Stahler (1983) Figure 17]]		[[File:Stahler_MRdiagram1.png\|300px\|center\|Stahler (1983) Figure 17]]
			</td>
			</tr>

			<tr>
			<td align="left" colspan="2">
			''Left-hand Panel'':   As detailed below, the three orange-dashed sequences — n = 1, n = 3, and isothermal — are analytically prescribed while the others have been determined via an iterative procedure. Also as detailed below, a solid-yellow circular marker identifies where along each sequence (n > 1) the model with the largest radius resides; for n = 1, the equilibrium sequence asymptotically approaches the maximum radius, <math>R/R_\mathrm{SWS} = [15/(8\pi)]^{1 / 2}</math>, where the mass climbs to infinity. The solid-green circular marker identifies where along each sequence (n ≥ 3) the maximum-mass model resides.
	</td>		</td>
	</tr>		</tr>

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-31T04:47:52Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 00:47, 31 December 2023
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	<td align="left">0.58847</td>		<td align="left">0.58847</td>
	</tr>		</tr>
	<tr><td align="~~center~~" ~~colspan~~="3"> </td></tr>		<tr>
			<td align="right">0.0100</td>
			<td align="center"> </td>
			<td align="left">0.4161145</td>
			</tr>
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-31T04:43:40Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 00:43, 31 December 2023
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	<td align="left">5.69164</td>		<td align="left">5.69164</td>
	</tr>		</tr>
	~~<tr><td align="center" colspan="3"> </td></tr>~~
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>
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	<td align="left">5.47056</td>		<td align="left">5.47056</td>
	</tr>		</tr>
	~~<tr><td align="center" colspan="3"> </td></tr>~~
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>
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	<td align="left">1.502865</td>		<td align="left">1.502865</td>
	</tr>		</tr>
	~~<tr><td align="center" colspan="3"> </td></tr>~~
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>
	</table>		</table>
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	<td align="left">0.848625</td>		<td align="left">0.848625</td>
	</tr>		</tr>
	~~<tr><td align="center" colspan="3"> </td></tr>~~
	</table>		</table>
	</td>		</td>
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	<td align="left">0.58847</td>		<td align="left">0.58847</td>
	</tr>		</tr>
	~~<tr><td align="center" colspan="3"> </td></tr>~~
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-31T04:38:53Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 00:38, 31 December 2023
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	<td align="left">0.58847</td>		<td align="left">0.58847</td>
	</tr>		</tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
	<tr><td align="center" colspan="3"> </td></tr>		<tr><td align="center" colspan="3"> </td></tr>
	</table>		</table>
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	<tr>		<tr>
	<td align="left" colspan="5">		<td align="left" colspan="5">
	NOTE:   Along each sequence (fixed value of "n"), the coordinates, <math>(R/R_\mathrm{SWS}, M/M_\mathrm{SWS}),</math> of the model with the largest radius are typed in a dark green font; the identified coordinate values not only satisfy the ~~viral~~-balance equation but also the relation,		NOTE:   Along each sequence (fixed value of "n"), the coordinates, <math>(R/R_\mathrm{SWS}, M/M_\mathrm{SWS}),</math> of the model with the largest radius are typed in a dark green font; the identified coordinate values not only satisfy the virial-balance equation but also the relation,
	<div align="center">		<div align="center">
	<math>\biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^4</math>		<math>\biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^4 = \frac{3}{20\pi}\biggl(\frac{n-1}{n}\biggr)\biggl(\frac{M}{M_\mathrm{SWS}} \biggr)^2 \, .</math>
			</div>
			Similarly, for sequences having n > 3, the coordinates of the model with the maximum mass are highlighted by a yellow background color; the coordinate values not only satisfy the virial-balance equation but also the relation,
			<div align="center">
			<math>\biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^4 = \frac{1}{20\pi}\biggl(\frac{n-3}{n}\biggr)\biggl(\frac{M}{M_\mathrm{SWS}} \biggr)^2 \, .</math>
	</div>		</div>
	</td>		</td>

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-31T04:21:18Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 00:21, 31 December 2023
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	<span id="TabulatedValues">First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.</span>		<span id="TabulatedValues">First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.</span>
	<div align="center">		<div align="center">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5" width="80%">
	<tr>		<tr>
	<td align="center" colspan="5">For Various Values of <math>n</math>, Numerically Determined Solutions to the Virial-Equilibrium Relation …		<td align="center" colspan="5">For Various Values of <math>n</math>, Numerically Determined Solutions to the Virial-Equilibrium Relation …
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	</td>		</td>

			</tr>

			<tr>
			<td align="left" colspan="5">
			NOTE:   Along each sequence (fixed value of "n"), the coordinates, <math>(R/R_\mathrm{SWS}, M/M_\mathrm{SWS}),</math> of the model with the largest radius are typed in a dark green font; the identified coordinate values not only satisfy the viral-balance equation but also the relation,
			<div align="center">
			<math>\biggl(\frac{R}{R_\mathrm{SWS}} \biggr)^4</math>
			</div>
			</td>
	</tr>		</tr>
	</table>		</table>

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-31T00:44:57Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 20:44, 30 December 2023
Line 992:		Line 992:

	is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure. For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of {{ Stahler83 }}. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.		is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure. For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of {{ Stahler83 }}. It seems that our derived, analytically prescribable, mass-radius relationship — which is, in essence, a statement of the scalar virial theorem — embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.
	<table border="1~~" width="100%~~" cellpadding="3" align="center">		<table border="1" cellpadding="3" align="center">
	<tr>		<tr>
	<td align="center" colspan="2">		<td align="center" colspan="2">
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	<tr>		<tr>
	<td align="center" rowspan="2">		<td align="center" rowspan="2">
	[[File:MassRadiusVirialLabeled.png\|350px\|Virial Theorem Mass-Radius Relation]]		<!-- [[File:MassRadiusVirialLabeled.png\|350px\|Virial Theorem Mass-Radius Relation]] -->
			[[File:VirialDeterminedMRsequencesLabeled.png\|350px\|Virial-Determined MR Sequences]]
	</td>		</td>
	<td align="center">		<td align="center">

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-30T21:07:33Z

Plotting Concise Mass-Radius Relation

@@ Line 1,075: / Line 1,075: @@
 <div align="center">
 <table border="1" align="center" cellpadding="5">
 <tr>
    <td align="center" colspan="1"><math>~n =2</math></td>

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-30T20:55:54Z

Plotting Concise Mass-Radius Relation

@@ Line 1,604: / Line 1,604: @@
    <td align="left">1.82708</td>
 </tr>
 </table>
 </td>

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-30T19:16:56Z

Plotting Concise Mass-Radius Relation

@@ Line 1,178: / Line 1,178: @@
    <td align="left">5.69164</td>
 </tr>
 <tr><td align="center" colspan="3">&nbsp;</td></tr>
 <tr><td align="center" colspan="3">&nbsp;</td></tr>
@@ Line 1,311: / Line 1,316: @@
    <td align="left">5.47056</td>
 </tr>
 </table>
 </td>
@@ Line 1,427: / Line 1,437: @@
    <td align="center">&nbsp;</td>
    <td align="left">2.578605</td>
 </tr>
 <tr><td align="center" colspan="3">&nbsp;</td></tr>

Joel2: /* Plotting Concise Mass-Radius Relation */

2023-12-30T18:18:03Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 14:18, 30 December 2023
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	<td align="right">0.5000</td>		<td align="right">0.5000</td>
	<td align="center"> </td>		<td align="center"> </td>
	<td align="left">2.~~75557~~</td>		<td align="left">2.7555746</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="right">0.4500</td>		<td align="right">0.4500</td>
	<td align="center"> </td>		<td align="center"> </td>
	<td align="left">2.~~543602~~</td>		<td align="left">2.8890287</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="right">0.3800</td>		<td align="right">0.3800</td>
	<td align="center"> </td>		<td align="center"> </td>
	<td align="left">2.~~94829~~</td>		<td align="left">2.9482952</td>
	</tr>		</tr>
	<tr>		<tr>