User:Tohline/SSC/Virial/PolytropesSummary - Revision history

Tohline: /* Mapping from Above Discussion */ Modified display of Figure 17 from Stahler (1983)

2015-08-28T17:07:04Z

Mapping from Above Discussion: Modified display of Figure 17 from Stahler (1983)

← Older revision		Revision as of 17:07, 28 August 2015
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	This analytic function is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following table.		This analytic function is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following table.

			<!-- COMMENT OUT ORIGINAL FIGURE until reprint permission received

	<table border="1" align="center" cellpadding="3">		<table border="1" align="center" cellpadding="3">
	<tr>		<tr>
	<td align="center" rowspan="2">		<td align="center" rowspan="3">
	[[File:MassRadiusVirialLabeled.png\|350px\|Virial Theorem Mass-Radius Relation]]		[[File:MassRadiusVirialLabeled.png\|350px\|Virial Theorem Mass-Radius Relation]]
	</td>		</td>
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	</td>		</td>
	</tr>		</tr>
			</table>

			-->

			<table border="1" align="center" cellpadding="3" width="80%">
			<tr>
			<td align="center" rowspan="3">
			[[File:MassRadiusVirialLabeled.png\|350px\|Virial Theorem Mass-Radius Relation]]
			</td>
			<td align="center">
			Figure 17 extracted<sup>&dagger;</sup>from p. 184 of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S S. W. Stahler (1983)]<p></p>
			"''The Equilibria of Rotating Isothermal Clouds.     II. Structure and Dynamical Stability''"<p></p>
			ApJ, vol. 268, pp. 165-184 © [http://aas.org/ American Astronomical Society]
			</td>
			</tr>
			<tr>
			<td align="center">
			<!-- [[File:Stahler_MRdiagram1.png\|300px\|center\|Stahler (1983) Figure 17 (edited)]] -->
			[[Image:AAAwaiting01.png\|400px\|center\|Stahler (1983) Figure 17 (edited)]]
			</td>
			</tr>
			<tr><td align="left"><sup>&dagger;</sup>Figure displayed here, as a digital image, has been modified from the original publication only via the addition of the word "<font color="red">SCHEMATIC</font>".</td></tr>
	</table>		</table>

Tohline: /* Structural Form Factors */ Inserted, along with dated comment, "work in progress" warning sign

2015-01-14T03:05:25Z

Structural Form Factors: Inserted, along with dated comment, "work in progress" warning sign

← Older revision		Revision as of 03:05, 14 January 2015
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	</div>		</div>
	and that, specifically in the context of spherically symmetric, pressure-truncated polytropes, we can write,		and that, specifically in the context of spherically symmetric, pressure-truncated polytropes, we can write …

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	</div>		</div>


			<font color="red"><b>
			January 13, 2015:
			</b></font>
			As is noted in our [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Third_Effort\|accompanying outline of work]], I no longer believe that <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math> have the same expressions as in isolated polytropes. Hence, all of the material that follows is suspect and needs to be reworked.

			{{LSU_WorkInProgress}}

	After plugging these nontrivial expressions for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> into the righthand sides of the above equations for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> and, simultaneously, using Horedt's detailed force-balanced expressions for <math>~r_a</math> and <math>~p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>~P_e/P_\mathrm{norm}</math> in these same equations, we find that,		After plugging these nontrivial expressions for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> into the righthand sides of the above equations for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> and, simultaneously, using Horedt's detailed force-balanced expressions for <math>~r_a</math> and <math>~p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>~P_e/P_\mathrm{norm}</math> in these same equations, we find that,
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	</div>		</div>

	It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial\|virial theorem presented above]].		It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[User:Tohline/SSC/Virial/PolytropesSummary#ConciseVirial\|virial theorem presented above]].

	===Physical Meaning of Parameter <math>~\eta_\mathrm{ad}</math>===		===Physical Meaning of Parameter <math>~\eta_\mathrm{ad}</math>===

Tohline: /* Determining Expressions for Free-Energy Coefficients */ Finished deriving expression for the free-energy coefficient, A

2015-01-07T03:29:52Z

Determining Expressions for Free-Energy Coefficients: Finished deriving expression for the free-energy coefficient, A

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Tohline: /* Borrowing from Bipolytrope Discussion */

2015-01-07T02:11:44Z

Borrowing from Bipolytrope Discussion

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	</math>		</math>
	</div>		</div>
	As has [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|also been discussed in the accompanying chapter]], we can deduce from Stahler's detailed force-balanced models that the equilibrium radius of embedded, <math>~n=5</math> polytropes is given in terms of the dimensionless, ''truncated'' Lane-Emden radius, <math>~\tilde\xi</math>, by the expression,		As has [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|also been discussed in the accompanying chapter]], we can deduce from Stahler's detailed force-balanced models that the equilibrium radius of embedded, <math>~n=5</math> polytropes is given in terms of the dimensionless, ''truncated'' Lane-Emden radius, <math>~\tilde\xi</math> — and our corresponding variable, <math>\ell</math> — by the expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">

Tohline: /* Further Evaluation of n = 5 Polytropic Structures */

2015-01-07T00:08:46Z

Further Evaluation of n = 5 Polytropic Structures

Show changes

Tohline: /* Take Care Comparing Gravitational Potential Energies */

2015-01-01T23:28:18Z

Take Care Comparing Gravitational Potential Energies

← Older revision		Revision as of 23:28, 1 January 2015
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	</div>		</div>
			So the energy integral becomes,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm} }</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			- 3 \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr) \biggl( \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2}
			\int_0^{\tilde\xi} \biggl[\xi^3 (3 + \xi^2)^{-3/2} \biggr] \xi \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} d\xi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			- \biggl( \frac{2^3 \cdot 3^{6}}{\pi } \biggr)^{1/2}
			\int_0^{\tilde\xi} \xi^4 \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-4} d\xi \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			This needs to be compared with the <math>~W_\mathrm{grav}^*</math> integral that we previously have handled in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy\|the chapter discussing bipolytrope models]].

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Tohline: /* Virial Chapter */

2015-01-01T23:04:22Z

Virial Chapter

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	<math>~		<math>~
	\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi \, .		\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Also,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\rho_\mathrm{norm}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} M_\mathrm{tot}^{-5}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl[ \biggl( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}\biggr]^{-5}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl( \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)^{5/2} \rho_c
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl(\frac{3^2}{2^4\pi^2} \biggr)^{1/2} \biggl( \frac{\pi^5 }{2^5 \cdot 3^{20}} \biggr)^{1/2} \rho_c
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{\pi^3 }{2^9 \cdot 3^{18}} \biggr)^{1/2} \rho_c \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Hence,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \frac{\rho}{\rho_c}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Virial Chapter */

2015-01-01T22:13:21Z

Virial Chapter

← Older revision		Revision as of 22:13, 1 January 2015
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	</table>		</table>
	</div>		</div>
			We can replace <math>~r^* \equiv r/R_\mathrm{norm}</math> with <math>~\xi \equiv r/a_5</math> by recognizing that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~M_\mathrm{tot}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \rho_c^{-2/5} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr) M_\mathrm{tot}^2 \, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~a_5</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{3K}{2\pi G} \biggr)^{1/2} \rho_c^{-2/5}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{3K}{2\pi G} \biggr)^{1/2} \biggl( \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr) M_\mathrm{tot}^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^2 \, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~R_\mathrm{norm}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~ \frac{a_5}{R_\mathrm{norm}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Hence,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~r^*</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi \, .
			</math>
			</td>
			</tr>
			</table>
			</div>



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Tohline: /* Serious Concern */

2014-12-31T18:25:14Z

Serious Concern

← Older revision		Revision as of 18:25, 31 December 2014
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	</div>		</div>
			According to the line of reasoning presented above, the coefficient, <math>~A</math>, is related to the gravitational potential energy via the expression,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~W_\mathrm{grav}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-3A\chi^{-1} E_\mathrm{norm} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-3A \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \frac{1}{R}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-3A \biggl(\frac{GM_\mathrm{tot}^2}{R} \biggr) \, .</math>
			</td>
			</tr>
			</table>
			</div>
			From our [[User:Tohline/SphericallySymmetricConfigurations/Virial#Expressions_for_Various_Energy_Terms\|introductory layout of the free-energy function for polytropes]] — see, also, p. 64, Equation (12) of [[User:Tohline/Appendix/References\|Chandrasekhar [C67]]] — the gravitational potential energy is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~W_\mathrm{grav}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ - \int_0^{R_\mathrm{limit}} \biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~M_r</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\int_0^r 4\pi r^2 \rho dr \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Now, independent of the chosen normalization, if we use <math>~M_\mathrm{tot}</math> to represent the total mass of an ''isolated'' <math>~n=5</math> polytrope, then from [[User:Tohline/SSC/Structure/PolytropesEmbedded#Review_2\|an earlier review]], we have,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~M_\mathrm{tot}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_c^{-1/5} \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			and we can write, in terms of the Lane-Emden dimensionless radius, <math>~\xi</math>,
			<div align="center">
			<math>
			\frac{M_r}{M_\mathrm{tot}} = \xi^3 (3 + \xi^2)^{-3/2} \, .
			</math>
			</div>
			===Virial Chapter===
			Now, in our discussion of the virial equilibrium of embedded polytropes, we used the normalizations specified above and wrote,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~W_\mathrm{grav}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			- 4\pi GM_\mathrm{tot} R_\mathrm{norm}^2 \rho_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} \biggl[\frac{M_r}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^*
			</math>
			</td>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			- E_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} 3\biggl[\frac{M_r}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^* \, .
			</math>
			</td>
			</tr>
			</table>
			</div>





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Tohline: /* Take Care Comparing Gravitational Potential Energies */

2014-12-31T17:09:57Z

Take Care Comparing Gravitational Potential Energies

← Older revision		Revision as of 17:09, 31 December 2014
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	==Take Care Comparing Gravitational Potential Energies==		==Take Care Comparing Gravitational Potential Energies==
	Does this derived relation for the coefficient, <math>~A</math>, make sense? Well, we've derived the relation by comparing two separate expressions for the ~~normalized~~ gravitational potential energy that were normalized in slightly different ways, so the leading numerical coefficient may not be correct. We need to repeat the derivation, checking the relative normalizations carefully. But before doing this, let's determine what we ''expected'' the relation to be, based on the expressions for the structural form factors that we have be using.		Does this derived relation for the coefficient, <math>~A</math>, make sense? Well, we've derived the relation by comparing two separate expressions for the gravitational potential energy that were normalized in slightly different ways, so the leading numerical coefficient may not be correct. We need to repeat the derivation, checking the relative normalizations carefully. But before doing this, let's determine what we ''expected'' the relation to be, based on the expressions for the structural form factors that we have been using.

	From the [[User:Tohline/SSC/Virial/PolytropesSummary#Serious_Concern\|lead-in paragraphs of this subsection]], we have previously assumed that,		From the [[User:Tohline/SSC/Virial/PolytropesSummary#Serious_Concern\|lead-in paragraphs of this subsection]], we have previously assumed that,