User:Tohline/SSC/Structure/Polytropes/VirialSummary - Revision history

Tohline: /* Fix External Pressure While Varying Mass */

2019-02-11T22:12:25Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 22:12, 11 February 2019
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	</tr>		</tr>
	</table>		</table>

			----

			Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set,
			<div align="center">
			<math>~\mathcal{A} = \frac{4\pi n}{3(n+1)}</math>       and       <math>~\mathcal{B} = \biggl( \frac{4\pi}{3} \biggr) \, ,</math>
			</div>
			and display, in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes. Specifically, each curve in the left-hand panel of Figure 1 results from the mass-radius expression,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
			- \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
			+ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
			</td>
			</tr>
			</table>

			(As has been demonstrated in an [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.)

			<table border="1" align="center" cellpadding="8">
			<tr>
			<th align="center" colspan="2">Figure 1:   Virial-Analysis-Based Equilibrium Sequences</th>
			</tr>
			<tr>
			<td align="center" rowspan="2">
			[[File:MassRadiusVirialLabeled.png\|350px\|Virial Theorem Mass-Radius Relation]]
			</td>
			<td align="center">
			[[File:Stahler1983TitlePage0.png\|300px\|center\|Stahler (1983) Title Page]]
			</td>
			</tr>
			<tr>
			<td align="center" bgcolor="white">
			[[File:Stahler_MRdiagram1.png\|300px\|center\|Stahler (1983) Figure 17 (edited)]]
			</td>
			</tr>
			</table>

			For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]. 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.


	----		----

Tohline: /* Fix External Pressure While Varying Mass */

2019-02-10T22:09:06Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 22:09, 10 February 2019
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	----		----

	Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set		Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set all three structural form factors to unity, in which case the pair of coefficients,
	<div align="center">		<div align="center">
	<math>~\mathcal{A} = 4\~~pi n/[3(n+~~1)]</math>       and       <math>~\mathcal{B} = 4\pi /3</math>,		<math>~\mathcal{A} = \frac{1}{5}</math>       and       <math>~\mathcal{B} = \biggl( \frac{3}{4\pi} \biggr)^{1/n}</math>,
	</div>		</div>
	and ~~display, in~~ the ~~left~~-~~hand panel of Figure 1, the behaviors~~ of equilibrium sequences as governed by the algebraic mass-radius relation,		and the resulting mass-radius relation of equilibrium sequences is governed by the algebraic mass-radius relation,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<math>~		<math>~
	\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4		\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
	-\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}		- \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{M}{M_\mathrm{SWS}}\biggr]^{(n+1)/n}
	+ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, ,</math>		+ ~\frac{3}{20\pi} \biggl( \frac{n+1}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	for seven different values of the polytropic index, as labeled. (As has been demonstrated in an [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#~~modNormalizations~~\|accompanying discussion]], ~~instead of choosing these specific values of~~ <math>~~~\mathcal{A}~~</math> ~~and~~ <math>~~~\mathcal{B}~~</math>~~, we could alternatively have obtained this same algebraic~~ relation ~~by shifting to a different pair of mass- and radius-normalizations — namely~~, <math>~~~M_\mathrm{mod}~~</math> and <math>~R_\~~mathrm{mod}~~</math>.)		The left-hand panel of Figure 1 displays this behavior for equilibrium sequences having seven different values of the polytropic index, as labeled. As has been demonstrated in an [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation\|accompanying discussion]], this governing mass-radius relation can be analytically manipulated into a form that provides either an explicit <math>~M(R)</math> or <math>~R(M)</math> relation for the cases labeled, <math>~n = 1, 3,</math> and <math>~\infty</math>; a generic root-finding technique has been used to generate [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#TabulatedValues\|points along each of the other depicted equilibrium sequences]].

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

Tohline: /* Virial Equilibrium of Pressure-Truncated Polytropes */

2019-02-10T21:32:46Z

Virial Equilibrium of Pressure-Truncated Polytropes

← Older revision		Revision as of 21:32, 10 February 2019
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	</table>		</table>

	Below, we will examine the behavior of individual ~~viral~~-equilibrium sequences, using various physical arguments to ~~specify appropriate values~~ of the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set <math>~\mathcal{A} = 4\pi n/[3(n+1)]</math> and <math>~\mathcal{B} = 4\pi /3</math> and display, ~~in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes. That is, each curve~~ in the left-hand panel of Figure 1 ~~results from~~ the mass-radius ~~expression~~,		----

			Below, we will examine the behavior of individual virial-equilibrium sequences, using various physical arguments to justify our choice of specific expressions for the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set
			<div align="center">
			<math>~\mathcal{A} = 4\pi n/[3(n+1)]</math>       and       <math>~\mathcal{B} = 4\pi /3</math>,
			</div>
			and display, in the left-hand panel of Figure 1, the behaviors of equilibrium sequences as governed by the algebraic mass-radius relation,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4		\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
	-\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}		-\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}
	+ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>		+ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	(As has been demonstrated in an [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod~~}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B~~}</math>.)		for seven different values of the polytropic index, as labeled. (As has been demonstrated in an [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#modNormalizations\|accompanying discussion]], instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math>.)

	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 [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]. 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" align="center" cellpadding="8">
			<tr>
			<th align="center" colspan="2">Figure 1:   Virial-Analysis-Based Equilibrium Sequences</th>
	<table border="1" align="center" cellpadding="3">		</tr>
	<tr>		<tr>
	<td align="center" rowspan="2">		<td align="center" rowspan="2">
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	</tr>		</tr>
	</table>		</table>

			For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]. 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.

	=See Also=		=See Also=

Tohline: /* Fix External Pressure While Varying Mass */

2019-02-09T20:17:07Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 20:17, 9 February 2019
Line 552:		Line 552:
	</table>		</table>
	(As has been demonstrated in an [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.)		(As has been demonstrated in an [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.)

			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 [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]. 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" align="center" cellpadding="3">
			<tr>
			<td align="center" rowspan="2">
			[[File:MassRadiusVirialLabeled.png\|350px\|Virial Theorem Mass-Radius Relation]]
			</td>
			<td align="center">
			[[File:Stahler1983TitlePage0.png\|300px\|center\|Stahler (1983) Title Page]]
			</td>
			</tr>
			<tr>
			<td align="center" bgcolor="white">
			[[File:Stahler_MRdiagram1.png\|300px\|center\|Stahler (1983) Figure 17 (edited)]]
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* Fix External Pressure While Varying Mass */

2019-02-09T20:12:44Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 20:12, 9 February 2019
Line 531:		Line 531:
	</tr>		</tr>
	</table>		</table>

			Below, we will examine the behavior of individual viral-equilibrium sequences, using various physical arguments to specify appropriate values of the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>. Here, in an effort to provide a broad overview, we set <math>~\mathcal{A} = 4\pi n/[3(n+1)]</math> and <math>~\mathcal{B} = 4\pi /3</math> and display, in a single diagram, the behaviors of equilibrium sequences having seven different polytropic indexes. That is, each curve in the left-hand panel of Figure 1 results from the mass-radius expression,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
			-\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}
			+ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
			</td>
			</tr>
			</table>
			(As has been demonstrated in an [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#modNormalizations\|accompanying discussion]], we could alternatively have obtained this same algebraic relation by shifting to a different pair of mass- and radius-normalizations — namely, <math>~M_\mathrm{mod}</math> and <math>~R_\mathrm{mod}</math> — instead of choosing these specific values of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.)

	=See Also=		=See Also=

Tohline: /* Virial Equilibrium of Pressure-Truncated Polytropes */

2019-02-09T01:52:20Z

Virial Equilibrium of Pressure-Truncated Polytropes

← Older revision		Revision as of 01:52, 9 February 2019
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	=Virial Equilibrium of Pressure-Truncated Polytropes=		=Virial Equilibrium of Pressure-Truncated Polytropes=

	Here we will draw heavily from an [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytropes\|~~accompanying~~ ''Free Energy Synopsis'']].		Here we will draw heavily from …
			* An accompanying [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytropes\|''Free Energy Synopsis'']];
			* A detailed analysis of the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation\|Virial Equilibrium of Adiabatic Spheres]].

Tohline: /* Fix External Pressure While Varying Mass */

2019-02-09T00:38:36Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 00:38, 9 February 2019
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	<td align="right">		<td align="right">
	<math>~		<math>~
	\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)_\mathrm{crit}^4		\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)_\mathrm{crit}^4
	</math>		</math>
	</td>		</td>

Tohline: /* Fix External Pressure While Varying Mass */

2019-02-09T00:21:09Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 00:21, 9 February 2019
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	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr>		<tr>
	<th align="center" colspan="1">''Case P'' Free-Energy ~~Surface~~</th>		<th align="center" colspan="1">''Case P'':   Free-Energy Surfaces</th>
	</tr>		</tr>
	<tr><td align="center">		<tr><td align="center">
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	Across this '''Case P''' free-energy surface, extrema — and, hence, equilibrium configurations — will arise wherever the virial-equilibrium condition is met, that is, wherever		Across this '''Case P''' free-energy surface, extrema — and, hence, equilibrium configurations — will arise wherever the virial-equilibrium condition is met, that is, wherever

			<table border="1" cellpadding="8" align="center">
			<tr>
			<th align="center" colspan="1">''Case P'':   Virial Equilibrium Sequences</th>
			</tr>
			<tr><td align="center">

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</td>		</td>
	</tr>		</tr>
			</table>
			</td></tr>
	</table>		</table>
	An equilibrium model ''sequence'' is thereby defined for each chosen polytropic index in the range, <math>~0 \le n < \infty</math>.		An equilibrium model ''sequence'' is thereby defined for each chosen polytropic index in the range, <math>~0 \le n < \infty</math>.

Tohline: /* Virial Equilibrium of Pressure-Truncated Polytropes */

2019-02-09T00:16:42Z

Virial Equilibrium of Pressure-Truncated Polytropes

← Older revision		Revision as of 00:16, 9 February 2019
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	</td></tr>		</td></tr>
			</table>

			Across this '''Case P''' free-energy surface, extrema — and, hence, equilibrium configurations — will arise wherever the virial-equilibrium condition is met, that is, wherever

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^4
			-\frac{3\mathcal{B}}{4\pi} \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^{(n-3)/n}
			+ \frac{3 \mathcal{A}}{4\pi} \biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .</math>
			</td>
			</tr>
			</table>
			An equilibrium model ''sequence'' is thereby defined for each chosen polytropic index in the range, <math>~0 \le n < \infty</math>.

			Along each equilibrium sequence for which <math>~3 \le n < \infty</math>, there is one ''specific'' equilibrium state that marks a transition from dynamically stable to dynamically unstable configurations. For a given mass, the radius of this ''critical'' configuration is identified by the relation,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			\biggl( \frac{R}{R_\mathrm{SWS}}\biggr)_\mathrm{crit}^4
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\mathcal{A}}{4\pi}\biggl(\frac{n - 3}{n}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \, .
			</math>
			</td>
			</tr>
	</table>		</table>

Tohline: /* Fix External Pressure While Varying Mass */

2019-02-08T22:11:43Z

Fix External Pressure While Varying Mass

← Older revision		Revision as of 22:11, 8 February 2019
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	where the pair of constants, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, have the same definitions in terms of the ''[[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Free-Energy_of_Truncated_Polytropes\|structural form factors]]'' as provided above. The relevant dimensionless free-energy surface is, then, given by the expression,		where the pair of constants, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, have the same definitions in terms of the ''[[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Free-Energy_of_Truncated_Polytropes\|structural form factors]]'' as provided above. The relevant dimensionless free-energy surface is, then, given by the expression,
	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="8" align="center">
			<tr>
			<th align="center" colspan="1">''Case P'' Free-Energy Surface</th>
			</tr>
	<tr><td align="center">		<tr><td align="center">