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

Tohline: /* Contrast with Detailed Force-Balanced Solution */

2016-08-12T16:52:13Z

Contrast with Detailed Force-Balanced Solution

← Older revision		Revision as of 16:52, 12 August 2016
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	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr><th align="center" colspan="1">		<tr><th align="center" colspan="1">
	<font size="+1" color="red"><b>WRONG!!</b></font>  For the correct form-factor expressions, go [[User:Tohline/SSC/~~Virial~~/~~FormFactors~~#~~PTtable~~\|here]].		<font size="+1" color="red"><b>WRONG!!</b></font>  For the correct form-factor expressions, go [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Free-Energy_of_Truncated_Polytropes\|here]].
	</th></tr>		</th></tr>
	<tr><th align="center" colspan="1">		<tr><th align="center" colspan="1">
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	</div>		</div>

	<font size="+1" color="red"><b>WRONG!!</b></font>  For the correct form-factor expressions, go [[User:Tohline/SSC/~~Virial~~/~~FormFactors~~#~~PTtable~~\|here]].		<font size="+1" color="red"><b>WRONG!!</b></font>  For the correct form-factor expressions, go [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Free-Energy_of_Truncated_Polytropes\|here]].

	Notice that, in an effort to differentiate them from their counterparts developed earlier for "isolated" polytropes, we have affixed a tilde to each of these three form-factors, <math>~\mathfrak{f}_i</math>.		Notice that, in an effort to differentiate them from their counterparts developed earlier for "isolated" polytropes, we have affixed a tilde to each of these three form-factors, <math>~\mathfrak{f}_i</math>.

Tohline: /* Contrast with Detailed Force-Balanced Solution */ Insert warning regarding this incorrect set of form-factor expressions

2016-08-12T16:44:53Z

Contrast with Detailed Force-Balanced Solution: Insert warning regarding this incorrect set of form-factor expressions

← Older revision		Revision as of 16:44, 12 August 2016
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	<div align="center" id="PTtable">		<div align="center" id="PTtable">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
			<tr><th align="center" colspan="1">
			<font size="+1" color="red"><b>WRONG!!</b></font>  For the correct form-factor expressions, go [[User:Tohline/SSC/Virial/FormFactors#PTtable\|here]].
			</th></tr>
	<tr><th align="center" colspan="1">		<tr><th align="center" colspan="1">
	Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes		Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
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	</table>		</table>
	</div>		</div>

			<font size="+1" color="red"><b>WRONG!!</b></font>  For the correct form-factor expressions, go [[User:Tohline/SSC/Virial/FormFactors#PTtable\|here]].

	Notice that, in an effort to differentiate them from their counterparts developed earlier for "isolated" polytropes, we have affixed a tilde to each of these three form-factors, <math>~\mathfrak{f}_i</math>.		Notice that, in an effort to differentiate them from their counterparts developed earlier for "isolated" polytropes, we have affixed a tilde to each of these three form-factors, <math>~\mathfrak{f}_i</math>.

Tohline at 23:16, 10 July 2016

2016-07-10T23:16:35Z

← Older revision		Revision as of 23:16, 10 July 2016
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	</div>		</div>
	This needs to be checked, perhaps with specific applications to the cases <math>~n=1</math> and <math>~n=5</math>.		This needs to be checked, perhaps with specific applications to the cases <math>~n=1</math> and <math>~n=5</math>.


			=See Also=
			<ul>
			<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
			</ul>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Solution Expressed in Terms of M and Central Pressure */

2015-08-07T17:57:06Z

Solution Expressed in Terms of M and Central Pressure

← Older revision		Revision as of 17:57, 7 August 2015
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	====Solution Expressed in Terms of M and Central Pressure====		====Solution Expressed in Terms of M and Central Pressure====
	Beginning again with the [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|relevant statement of virial ~~equilibirum~~]], namely,		Beginning again with the [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|relevant statement of virial equilibrium]], namely,
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: /* Solution Expressed in Terms of K and M (Whitworth's 1981 Relation) */

2015-08-07T16:52:49Z

Solution Expressed in Terms of K and M (Whitworth's 1981 Relation)

← Older revision		Revision as of 16:52, 7 August 2015
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	If the structural form factors are set equal to unity, this exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:		If the structural form factors are set equal to unity, this exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:

	<div align="center">		<div align="center">
	<table border="2">		<table border="1" cellpadding="5" width="80%">
	<tr><td>		<tr><td align="center">
	[~~[File~~:~~Whitworth1981Eq5~~.~~jpg\|500px\|center\|~~Whitworth (1981, ~~MNRAS,~~ 195, 967~~)]]~~		Equation and accompanying sentence extracted without modification from p. 970 of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]<p></p>
			"''Global Gravitational Stability for One-dimensional Polytropes''"<p></p>
			Monthly Notices of the Royal Astronomical Society, vol. 195, pp. 967-977 © Royal Astronomical Society
	</td></tr>		</td></tr>
			<tr>
			<td align="center">
			<!-- [[File:Whitworth1981Eq5.jpg\|500px\|center\|Whitworth (1981, MNRAS, 195, 967)]] -->
			[[Image:AAAwaiting01.png\|400px\|center\|Whitworth (1981) Eq. 5]]
			</td>
			</tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Alternate Derivation of Gravitational Potential Energy */

2015-08-07T16:24:09Z

Alternate Derivation of Gravitational Potential Energy

← Older revision		Revision as of 16:24, 7 August 2015
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	</table>		</table>
	</div>		</div>
	We now recall two earlier expressions that show the role that our structural form factors play in the evaluation of <math>~W</math> and <math>~P_c</math>, namely,		We now recall two earlier expressions that show the role that our structural form factors play in the evaluation of <math>~W_\mathrm{grav}</math> and <math>~P_c</math>, namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Central and Mean Pressure */Get rid of reference to mean pressure

2015-08-07T16:21:45Z

Central and Mean Pressure: Get rid of reference to mean pressure

← Older revision		Revision as of 16:21, 7 August 2015
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	</div>		</div>

	===Central ~~and Mean~~ Pressure===		===Central Pressure===
	It is also worth pointing out that [[User:Tohline/Appendix/References\|Chandrasekhar [C67]]] — see his Equations (80) & (81), p. 99 — introduces a dimensionless structural form factor, <math>~W_n</math>, for the central pressure via the expression,		It is also worth pointing out that [[User:Tohline/Appendix/References\|Chandrasekhar [C67]]] — see his Equations (80) & (81), p. 99 — introduces a dimensionless structural form factor, <math>~W_n</math>, for the central pressure via the expression,
	<div align="center">		<div align="center">
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	</div>		</div>

			<!-- (August 2015) REMOVE NEXT FEW LINES, AS CONNECTION WITH MEAN PRESSURE IS UNJUSTIFIED ...
	Looking back at our [[User:Tohline/SphericallySymmetricConfigurations/Virial#FormFactors\|original definition of the structural form factors]], we note that,		Looking back at our [[User:Tohline/SphericallySymmetricConfigurations/Virial#FormFactors\|original definition of the structural form factors]], we note that,
	<div align="center">		<div align="center">
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	<math> \frac{\bar{P} R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} = \frac{3}{4\pi (5-n)} \, .</math>		<math> \frac{\bar{P} R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} = \frac{3}{4\pi (5-n)} \, .</math>
	</div>		</div>
			END DELETION -->

	===Alternate Derivation of Gravitational Potential Energy===		===Alternate Derivation of Gravitational Potential Energy===

Tohline: /* Contrast with Detailed Force-Balanced Solution */ Inserted dated note along with "work in progress" warning sign because I no longer think the assumptions are valid

2015-01-14T02:50:39Z

Contrast with Detailed Force-Balanced Solution: Inserted dated note along with "work in progress" warning sign because I no longer think the assumptions are valid

← Older revision		Revision as of 02:50, 14 January 2015
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	These same expressions may or may not work for pressure-truncated polytropes, even if the evaluation radius is shifted from <math>~\xi_1</math> to <math>~\tilde\xi</math>. Let's see …		These same expressions may or may not work for pressure-truncated polytropes, even if the evaluation radius is shifted from <math>~\xi_1</math> to <math>~\tilde\xi</math>. Let's see …

			<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}}

	Inserting Hoerdt's expressions for <math>~r_a</math> and <math>~p_a</math> into the viral equilibrium expression, we have,		Inserting Hoerdt's expressions for <math>~r_a</math> and <math>~p_a</math> into the viral equilibrium expression, we have,

Tohline: /* Contrast with Detailed Force-Balanced Solution */

2014-12-26T20:57:39Z

Contrast with Detailed Force-Balanced Solution

← Older revision		Revision as of 20:57, 26 December 2014
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	This all seems to make a great deal of sense. Only the structural parameter that is derived from an integral over the pressure distribution, <math>~\mathfrak{f}_A</math>, gets modified when the polytropic configuration is truncated. Notice, as well, that the term that has been added to the definition of <math>~\mathfrak{f}_A</math> naturally goes to zero in the limit of <math>~\tilde\xi \rightarrow \xi_1</math>, that is, for an isolated polytrope. We should definitely go back to the original definitions of all three structural parameters and prove that this is the case. But, in the meantime, here is the summary:		This all seems to make a great deal of sense. Only the structural parameter that is derived from an integral over the pressure distribution, <math>~\mathfrak{f}_A</math>, gets modified when the polytropic configuration is truncated. Notice, as well, that the term that has been added to the definition of <math>~\mathfrak{f}_A</math> naturally goes to zero in the limit of <math>~\tilde\xi \rightarrow \xi_1</math>, that is, for an isolated polytrope. We should definitely go back to the original definitions of all three structural parameters and prove that this is the case. But, in the meantime, here is the summary:

	<div align="center">		<div align="center" id="PTtable">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr><th align="center" colspan="1">		<tr><th align="center" colspan="1">

Tohline: /* Strategy2 */ Finish inserting correct factor of M_limit/M_tot in "strategy2" discussion

2014-10-12T19:34:45Z

Strategy2: Finish inserting correct factor of M_limit/M_tot in "strategy2" discussion

← Older revision		Revision as of 19:34, 12 October 2014
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	<td align="left">		<td align="left">
	<math>		<math>
	\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggl[ \frac{4\pi}{(n+1)^n}\biggr]^{1/(n-3)} \, ,		\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{n-1} \biggr]^{1/(n-3)} \, ,
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{(n+1)/(n-3)} \, .		\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^{(n+1)/(n-3)} \, .
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{1}{5} \~~cdot~~ \frac{~~(5-n~~)\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} = 1 \, ,</math>		<math>\frac{(5-n)}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2}
			= \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	<math>		<math>
	\biggl( \frac{3}{4\pi} \biggr)^{1/n}		\biggl( \frac{3}{4\pi} \biggr)^{1/n}
	\biggl[ \frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}		\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
			\cdot [(5-n)\tilde\mathfrak{f}_A]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>
			\frac{4\pi}{3} \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi_\mathrm{eq}^{3(n+1)/n}
	\cdot [(5-n)\tilde\mathfrak{f}_A]		\cdot [(5-n)\tilde\mathfrak{f}_A]
	</math>		</math>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~-3\chi^{-1} +n\mathcal{B}_\mathrm{mod} \chi^{-3/n} +~ \mathcal{D}_\mathrm{mod}\chi^3 \, .</math>		<math>~-3\mathcal{A}_\mathrm{mod} \chi^{-1} +n\mathcal{B}_\mathrm{mod} \chi^{-3/n} +~ \mathcal{D}_\mathrm{mod}\chi^3 \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1 \, .</math>		<math>~\mathcal{A}_\mathrm{mod} \, .</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~1</math>		<math>~\mathcal{A}_\mathrm{mod} </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{3}{4\pi}\chi_\mathrm{eq}^{-4} </math>		<math>~\frac{3}{4\pi}\chi_\mathrm{eq}^{-4} \mathcal{A}_\mathrm{mod} </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~\frac{P_c}{P_\mathrm{norm}}		<math>~\Rightarrow~~~\frac{P_c}{P_\mathrm{norm}} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[ 4\pi(n+1) (\tilde\theta^')^2 ~~\chi_\mathrm{eq}^{4}~~ \biggr]^{-1}		<math>~\biggl[ 4\pi(n+1) (\tilde\theta^')^2 \biggr]^{-1}
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{~~tot~~}^2}		<math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{limit}^2}
	</math>		</math>
	</td>		</td>