SSC/Virial/Polytropes - Revision history

Jet53man: /* See Also */

2022-04-25T17:04:15Z

Jet53man: /* Relationship to Detailed Force Balance Solution */

2021-10-13T18:49:10Z

Relationship to Detailed Force Balance Solution

← Older revision		Revision as of 14:49, 13 October 2021
Line 4,124:		Line 4,124:
	</div>		</div>

	Notice that the bracketed term that is to be evaluated at <math>~\xi_1</math> is identical in both expressions. After [[SSC/Virial/Polytropes#Renormalization\|renormalization, as ~~drived~~ above]], the statement of virial equilibrium for embedded polytropes is,		Notice that the bracketed term that is to be evaluated at <math>\xi_1</math> is identical in both expressions. After [[SSC/Virial/Polytropes#Renormalization\|renormalization, as derived above]], the statement of virial equilibrium for embedded polytropes is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Grunt Work */

2021-10-13T18:46:17Z

Grunt Work

← Older revision		Revision as of 14:46, 13 October 2021
Line 3,158:		Line 3,158:
	</div>		</div>

	Inserting the functional ~~expressions~~ for <math>~r_a</math> from {{ Horedt70 }}, and our structural form factors, <math>\tilde\mathfrak{f}_i</math>, gives,		Inserting the functional expression for <math>r_a</math> from {{ Horedt70 }}, and our structural form factors, <math>\tilde\mathfrak{f}_i</math>, gives,

	<div align="center">		<div align="center">

Jet53man: /* Contrast with Detailed Force-Balanced Solution */

2021-10-13T18:40:15Z

Contrast with Detailed Force-Balanced Solution

← Older revision		Revision as of 14:40, 13 October 2021
Line 1,779:		Line 1,779:
	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

	Inserting ~~Horedt's~~ expressions for <math>~r_a</math> and <math>~p_a</math> into the ~~viral~~ equilibrium expression, we have,		Inserting the expressions for <math>r_a</math> and <math>p_a</math>, as provided by {{ Horedt70 }}, into the virial equilibrium expression, we have,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Solution Expressed in Terms of K and M (Whitworth's 1981 Relation) */

2021-10-13T18:32:40Z

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

← Older revision		Revision as of 14:32, 13 October 2021
Line 1,304:		Line 1,304:
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">
	Equation and accompanying sentence ~~extracted without modification~~ from p. 970 of<br />		Equation and accompanying sentence drawn directly from p. 970 of<br />
	{{ Whitworth81figure }}<br />		{{ Whitworth81figure }}<br />
	© Royal Astronomical Society<br />		© Royal Astronomical Society
			</td></tr>
			<tr>
			<td align="left">
	<!--		<!--
	[~~http~~:~~//adsabs~~.~~harvard.edu/abs/1981MNRAS.195..967W~~ Whitworth (1981)]~~<br />~~		[[File:Whitworth1981Eq5.jpg\|500px\|center\|Whitworth (1981, MNRAS, 195, 967)]]
	~~"''Global Gravitational Stability for One-dimensional Polytropes''"<br />~~		[[Image:AAAwaiting01.png\|400px\|center\|Whitworth (1981) Eq. 5]]
	~~Monthly Notices of the Royal Astronomical Society, vol~~. ~~195, pp~~. ~~967-977 <br />~~
	-->		-->
	</td><~~/tr~~>		The general equilibrium condition, <math>(d\mathcal{U}/dR)_{R_0} = 0</math>, reduces to
			<table border="0" align="center" width="100%" cellpadding="5">
	<tr>		<tr>
	<td align="center">		<td align="right" width="25%"><math>R_0 \rightarrow R_\mathrm{eq}\, ,</math></td>
	<!-~~- [[File:Whitworth1981Eq5.jpg\|500px\|center\|Whitworth~~ (~~1981, MNRAS, 195, 967~~)~~]] --~~>		<td align="center" width="5%"> </td>
	~~[[Image:AAAwaiting01.png\|400px\|center\|Whitworth~~ (~~1981~~) Eq. ~~5]]~~		<td align="left">
			<math>P_\mathrm{ex} = K(3M_0/4\pi R^3_\mathrm{eq})^\eta - 3GM_0^2/20\pi R^4_\mathrm{eq}</math>
			</td>
			<td align="right" width="8%">(5)</td>
			</tr>
			</table>
			(subscript 'eq' for equilibrium).
	</td>		</td>
	</tr>		</tr>
Line 1,322:		Line 1,331:
	</div>		</div>

	Notice that, when <math>~P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math> (see the left-hand column of our [[SSC/Virial/Polytropes#TwoPointsOfView\|table titled "Two Points of View"]]).		Notice that, when <math>P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>K</math> and <math>M_\mathrm{limit}</math> (see the left-hand column of our [[SSC/Virial/Polytropes#TwoPointsOfView\|table titled "Two Points of View"]]).

	====Solution Expressed in Terms of M and Central Pressure====		====Solution Expressed in Terms of M and Central Pressure====

Jet53man at 22:44, 10 October 2021

2021-10-10T22:44:35Z

← Older revision		Revision as of 18:44, 10 October 2021
Line 1,482:		Line 1,482:
	</div>		</div>

	It is important to appreciate that, in the expressions for <math>~r_a</math> and <math>~p_a</math>, the tilde indicates that the Lane-Emden function and its derivative are to be evaluated, not at the radial coordinate, <math>~\xi_1</math>, that is traditionally associated with the "first zero" of the Lane-Emden function and therefore with the surface of the ''isolated polytrope,'' but at the radial coordinate, <math>~\tilde\xi</math>, where the internal pressure of the isolated polytrope equals <math>~P_e</math> and at which the ''embedded'' polytrope is to be truncated. The coordinate, <math>~\tilde\xi</math>, therefore identifies the surface of the embedded — or, pressure-truncated — polytrope. We also have taken the liberty of attaching the subscript "limit" to <math>~M</math> in both defining relations because it is clear that {{ Horedt70 }} intended for the normalization mass to be the mass of the pressure-truncated object, not the mass of the associated ''isolated'' (and untruncated) polytrope. In anticipation of further derivations, below, we note here the ratio of ~~Hoerdt's~~ normalization parameters to ours, assuming <math>~\gamma = (n+1)/n</math>:		It is important to appreciate that, in the expressions for <math>~r_a</math> and <math>~p_a</math>, the tilde indicates that the Lane-Emden function and its derivative are to be evaluated, not at the radial coordinate, <math>~\xi_1</math>, that is traditionally associated with the "first zero" of the Lane-Emden function and therefore with the surface of the ''isolated polytrope,'' but at the radial coordinate, <math>~\tilde\xi</math>, where the internal pressure of the isolated polytrope equals <math>~P_e</math> and at which the ''embedded'' polytrope is to be truncated. The coordinate, <math>~\tilde\xi</math>, therefore identifies the surface of the embedded — or, pressure-truncated — polytrope. We also have taken the liberty of attaching the subscript "limit" to <math>~M</math> in both defining relations because it is clear that {{ Horedt70 }} intended for the normalization mass to be the mass of the pressure-truncated object, not the mass of the associated ''isolated'' (and untruncated) polytrope. In anticipation of further derivations, below, we note here the ratio of the {{ Horedt70 }} normalization parameters to ours, assuming <math>~\gamma = (n+1)/n</math>:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,488:		Line 1,488:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\biggl( \frac{R_\mathrm{~~Hoerdt~~}}{R_\mathrm{norm}} \biggr)^{n-3}</math>		<math>~\biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)^{n-3}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,517:		Line 1,517:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\biggl( \frac{P_\mathrm{~~Hoerdt~~}}{P_\mathrm{norm}} \biggr)^{n-3}</math>		<math>~\biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)^{n-3}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,545:		Line 1,545:
	</div>		</div>

	Next, we demonstrate that this pair of parametric relations satisfies the virial theorem and, in so doing, demonstrate how <math>~r_a</math> and <math>~p_a</math> may be ''directly'' related to each other. Given that ~~Hoerdt's chosen~~ normalization radius and normalization pressure are defined in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math>, we begin with the [[SSC/Virial/Polytropes#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.2\|virial theorem derived above]] in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math>, setting <math>~\gamma_g = (n+1)/n</math>.		Next, we demonstrate that this pair of parametric relations satisfies the virial theorem and, in so doing, demonstrate how <math>~r_a</math> and <math>~p_a</math> may be ''directly'' related to each other. Given that the normalization radius and normalization pressure chosen by {{ Horedt70 }} are defined in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math>, we begin with the [[SSC/Virial/Polytropes#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.2\|virial theorem derived above]] in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math>, setting <math>~\gamma_g = (n+1)/n</math>.

	<div align="center">		<div align="center">
Line 1,615:		Line 1,615:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~P_e = p_a P_\mathrm{~~Hoerdt~~}</math>		<math>~P_e = p_a P_\mathrm{Horedt}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,708:		Line 1,708:
	</div>		</div>

	Now, ~~Hoerdt~~ has given analytic expressions for <math>~r_a</math> and <math>~p_a</math> in terms of the Lane-Emden function and its first derivative. The question is, what should the expressions for our structural form factors be in order for this virial expression to hold true for all pressure-truncated polytropic structures? As has been [[SSC/Virial/Polytropes#Summary\|summarized above]], in the case of an ''isolated'' polytrope, whose surface is located at <math>~\xi_1</math> and whose global properties are defined by evaluation of the Lane-Emden function at <math>~\xi_1</math>, we know that (see the [[SSC/Virial/Polytropes#Summary\|above summary]]),		Now, {{ Horedt70 }} has given analytic expressions for <math>~r_a</math> and <math>~p_a</math> in terms of the Lane-Emden function and its first derivative. The question is, what should the expressions for our structural form factors be in order for this virial expression to hold true for all pressure-truncated polytropic structures? As has been [[SSC/Virial/Polytropes#Summary\|summarized above]], in the case of an ''isolated'' polytrope, whose surface is located at <math>~\xi_1</math> and whose global properties are defined by evaluation of the Lane-Emden function at <math>~\xi_1</math>, we know that (see the [[SSC/Virial/Polytropes#Summary\|above summary]]),

	<div align="center">		<div align="center">
Line 1,770:		Line 1,770:
	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

	Inserting ~~Hoerdt~~'s expressions for <math>~r_a</math> and <math>~p_a</math> into the viral equilibrium expression, we have,		Inserting Horedt's expressions for <math>~r_a</math> and <math>~p_a</math> into the viral equilibrium expression, we have,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Relationship to Detailed Force Balance Solution */

2021-10-10T22:40:39Z

Relationship to Detailed Force Balance Solution

← Older revision		Revision as of 18:40, 10 October 2021
Line 3,862:		Line 3,862:
	==Relationship to Detailed Force Balance Solution==		==Relationship to Detailed Force Balance Solution==

	Let's plug these form-factors into our expressions for the dimensionless equilibrium radius, <math>~\chi_\mathrm{ad}</math>, and dimensionless surface pressure, <math>~\Pi_\mathrm{ad}</math>, that have been derived from the identification of extrema in the free-energy function and see how they compare to the dimensionless radius, <math>~r_a \equiv R_\mathrm{eq}/R_\mathrm{Horedt}</math>, and dimensionless pressure, <math>~p_a \equiv P_e/P_\mathrm{Horedt}</math>, given by ~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt's (1970)]~~ detailed force-balance models. Note that expressions for <math>~r_a</math> and <math>~p_a</math> are given in our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|accompanying discussion of embedded polytropic spheres]] and that the conversion from the {{ Horedt70 }} scaling parameters to our normalization parameters, <math>~R_\mathrm{Horedt}/R_\mathrm{norm}</math> and <math>~P_\mathrm{Horedt}/P_\mathrm{norm}</math>, can be found in our [[SSCpt1/Virial#Normalizations\|introductory discussion of the virial equilibrium of spherical configurations]].		Let's plug these form-factors into our expressions for the dimensionless equilibrium radius, <math>~\chi_\mathrm{ad}</math>, and dimensionless surface pressure, <math>~\Pi_\mathrm{ad}</math>, that have been derived from the identification of extrema in the free-energy function and see how they compare to the dimensionless radius, <math>~r_a \equiv R_\mathrm{eq}/R_\mathrm{Horedt}</math>, and dimensionless pressure, <math>~p_a \equiv P_e/P_\mathrm{Horedt}</math>, given by the detailed force-balance models provided by {{ Horedt70 }}. Note that expressions for <math>~r_a</math> and <math>~p_a</math> are given in our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|accompanying discussion of embedded polytropic spheres]] and that the conversion from the {{ Horedt70 }} scaling parameters to our normalization parameters, <math>~R_\mathrm{Horedt}/R_\mathrm{norm}</math> and <math>~P_\mathrm{Horedt}/P_\mathrm{norm}</math>, can be found in our [[SSCpt1/Virial#Normalizations\|introductory discussion of the virial equilibrium of spherical configurations]].

	<div align="center">		<div align="center">
Line 4,147:		Line 4,147:
	</table>		</table>
	</div>		</div>
	This relation can now be written in terms of ~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt's (1970)]~~ dimensionless radius and pressure.		This relation can now be written in terms of the {{ Horedt70 }} dimensionless radius and pressure.
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Relationship to Detailed Force Balance Solution */

2021-10-10T22:36:46Z

Relationship to Detailed Force Balance Solution

← Older revision		Revision as of 18:36, 10 October 2021
Line 3,862:		Line 3,862:
	==Relationship to Detailed Force Balance Solution==		==Relationship to Detailed Force Balance Solution==

	Let's plug these form-factors into our expressions for the dimensionless equilibrium radius, <math>~\chi_\mathrm{ad}</math>, and dimensionless surface pressure, <math>~\Pi_\mathrm{ad}</math>, that have been derived from the identification of extrema in the free-energy function and see how they compare to the dimensionless radius, <math>~r_a \equiv R_\mathrm{eq}/R_\mathrm{~~Hoerdt~~}</math>, and dimensionless pressure, <math>~p_a \equiv P_e/P_\mathrm{~~Hoerdt~~}</math>, given by [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt's (1970)] detailed force-balance models. Note that expressions for <math>~r_a</math> and <math>~p_a</math> are given in our [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|accompanying discussion of embedded polytropic spheres]] and that the conversion from ~~Hoerdt's~~ scaling parameters to our normalization parameters, <math>~R_\mathrm{~~Hoerdt~~}/R_\mathrm{norm}</math> and <math>~P_\mathrm{~~Hoerdt~~}/P_\mathrm{norm}</math>, can be found in our [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Normalizations\|introductory discussion of the virial equilibrium of spherical configurations]].		Let's plug these form-factors into our expressions for the dimensionless equilibrium radius, <math>~\chi_\mathrm{ad}</math>, and dimensionless surface pressure, <math>~\Pi_\mathrm{ad}</math>, that have been derived from the identification of extrema in the free-energy function and see how they compare to the dimensionless radius, <math>~r_a \equiv R_\mathrm{eq}/R_\mathrm{Horedt}</math>, and dimensionless pressure, <math>~p_a \equiv P_e/P_\mathrm{Horedt}</math>, given by [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt's (1970)] detailed force-balance models. Note that expressions for <math>~r_a</math> and <math>~p_a</math> are given in our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|accompanying discussion of embedded polytropic spheres]] and that the conversion from the {{ Horedt70 }} scaling parameters to our normalization parameters, <math>~R_\mathrm{Horedt}/R_\mathrm{norm}</math> and <math>~P_\mathrm{Horedt}/P_\mathrm{norm}</math>, can be found in our [[SSCpt1/Virial#Normalizations\|introductory discussion of the virial equilibrium of spherical configurations]].

	<div align="center">		<div align="center">
Line 3,875:		Line 3,875:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{R_\mathrm{eq}}{R_\mathrm{~~Hoerdt~~}} \biggl( \frac{R_\mathrm{~~Hoerdt~~}}{R_\mathrm{norm}} \biggr)		<math>~\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)
	\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)</math>		\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)</math>
	</td>		</td>
Line 3,931:		Line 3,931:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{P_\mathrm{e}}{P_\mathrm{~~Hoerdt~~}} \biggl( \frac{P_\mathrm{~~Hoerdt~~}}{P_\mathrm{norm}} \biggr)		<math>~\frac{P_\mathrm{e}}{P_\mathrm{Horedt}} \biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)
	\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr)</math>		\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr)</math>
	</td>		</td>
Line 3,980:		Line 3,980:
	</div>		</div>

	Now we insert the form-factor expressions [[~~User:Tohline/~~SSC/Virial/Polytropes#Summary\|from above]], to obtain,		Now we insert the form-factor expressions [[SSC/Virial/Polytropes#Summary\|from above]], to obtain,

	<div align="center">		<div align="center">
Line 4,115:		Line 4,115:
	</div>		</div>

	Notice that the bracketed term that is to be evaluated at <math>~\xi_1</math> is identical in both expressions. After [[~~User:Tohline/~~SSC/Virial/Polytropes#Renormalization\|renormalization, as drived above]], the statement of virial equilibrium for embedded polytropes is,		Notice that the bracketed term that is to be evaluated at <math>~\xi_1</math> is identical in both expressions. After [[SSC/Virial/Polytropes#Renormalization\|renormalization, as drived above]], the statement of virial equilibrium for embedded polytropes is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 4,186:		Line 4,186:
	</table>		</table>
	</div>		</div>
	where, from our separate [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|summary of ~~Hoerdt~~'s presentation]],		where, from our separate [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation\|summary of Horedt's presentation]],
	<div align="center">		<div align="center">
	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">

Jet53man: /* More Precise Form Factors */

2021-10-10T22:31:17Z

More Precise Form Factors

← Older revision		Revision as of 18:31, 10 October 2021
Line 3,525:		Line 3,525:

	====More Precise Form Factors====		====More Precise Form Factors====
	Here we attempt to determine proper expressions for several form factors such that the equilibrium configurations determined from virial analysis will precisely match the [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|pressure-truncated polytropic configurations]] that have been determined from detailed force-balanced models that have been derived and published separately by ~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)]~~, by ~~[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]~~ and by ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]~~. It seems simplest to begin with the [[SSC/~~BipolytropeGeneralization_Version2~~#Generalized_Free-Energy_Expression\|free-energy expressions that we have already generalized in the context of bipolytropic configurations]], properly modified to embed the "core" in an external medium of pressure, <math>~P_e</math>, rather then inside an envelope that has a different polytropic index. Specifically,		Here we attempt to determine proper expressions for several form factors such that the equilibrium configurations determined from virial analysis will precisely match the [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|pressure-truncated polytropic configurations]] that have been determined from detailed force-balanced models that have been derived and published separately by {{ Horedt70full }}, by {{ Whitworth81full }} and by {{ Stahler83full }}. It seems simplest to begin with the [[SSC/BipolytropeGeneralizationVersion2#Generalized_Free-Energy_Expression\|free-energy expressions that we have already generalized in the context of bipolytropic configurations]], properly modified to embed the "core" in an external medium of pressure, <math>~P_e</math>, rather then inside an envelope that has a different polytropic index. Specifically,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 3,631:		Line 3,631:
	=====n = 5 Polytropic=====		=====n = 5 Polytropic=====

	From our [[SSC/Structure/BiPolytropes/~~FreeEnergy5_1~~#Free_Energy_of_BiPolytrope_with\|analysis of the free energy of <math>~(n_c, n_e) = (5, 0)</math> bipolytropes]], we deduce that the coefficient, <math>~\mathcal{A}</math>, that quantifies the [[SSC/Structure/BiPolytropes/~~FreeEnergy5_1~~#The_Core_2\|gravitational potential energy of a pressure-truncated <math>~n = 5</math> polytrope]] is,		From our [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with\|analysis of the free energy of <math>(n_c, n_e) = (5, 0)</math> bipolytropes]], we deduce that the coefficient, <math>\mathcal{A}</math>, that quantifies the [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_2\|gravitational potential energy of a pressure-truncated <math>n = 5</math> polytrope]] is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 3,653:		Line 3,653:
	</table>		</table>
	</div>		</div>
	the coefficient, <math>~\mathcal{B}_\mathrm{core}</math>, that quantifies the [[SSC/Structure/BiPolytropes/~~FreeEnergy5_1~~#The_Core_3\|thermal energy content of a pressure-truncated <math>~n = 5</math> polytrope]] is,		the coefficient, <math>\mathcal{B}_\mathrm{core}</math>, that quantifies the [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_3\|thermal energy content of a pressure-truncated <math>n = 5</math> polytrope]] is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 3,674:		Line 3,674:
	</table>		</table>
	</div>		</div>
	and the coefficient, <math>~\mathcal{D}</math> — which can be obtained by setting <math>~P_{ic} \rightarrow P_e</math> in expressions drawn from out analysis of the [[SSC/Structure/BiPolytropes/~~FreeEnergy5_1~~#The_Core_3\|thermal energy content of an <math>~(5, 0)</math>bipolytrope]] — is,		and the coefficient, <math>\mathcal{D}</math> — which can be obtained by setting <math>P_{ic} \rightarrow P_e</math> in expressions drawn from out analysis of the [[SSC/Structure/BiPolytropes/FreeEnergy51#The_Core_3\|thermal energy content of an <math>~(5, 0)</math>bipolytrope]] — is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Renormalization */

2021-10-10T22:25:57Z

Renormalization

← Older revision		Revision as of 18:25, 10 October 2021
Line 3,111:		Line 3,111:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{R_\mathrm{eq}}{R_\mathrm{~~Hoerdt~~}} \cdot \frac{R_\mathrm{~~Hoerdt~~}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{norm}}{R_\mathrm{ad}}</math>		<math>~\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \cdot \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{norm}}{R_\mathrm{ad}}</math>
	</td>		</td>
	</tr>		</tr>
Line 3,149:		Line 3,149:
	</div>		</div>

	Inserting the functional expressions for ~~Hoerdt's~~ <math>~r_a</math> and our structural form factors, <math>\tilde\mathfrak{f}_i</math>, gives,		Inserting the functional expressions for <math>~r_a</math> from {{ Horedt70 }}, and our structural form factors, <math>\tilde\mathfrak{f}_i</math>, gives,

	<div align="center">		<div align="center">
Line 3,224:		Line 3,224:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{P_e}{P_\mathrm{~~Hoerdt~~}} \cdot \frac{P_\mathrm{~~Hoerdt~~}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{norm}}{P_\mathrm{ad}}</math>		<math>~\frac{P_e}{P_\mathrm{Horedt}} \cdot \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{norm}}{P_\mathrm{ad}}</math>
	</td>		</td>
	</tr>		</tr>
Line 3,263:		Line 3,263:
	</div>		</div>

	Inserting the functional expressions for ~~Hoerdt's~~ <math>~p_a</math> and our structural form factors, <math>\tilde\mathfrak{f}_i</math>, gives,		Inserting the functional expressions for <math>~p_a</math> from {{ Horedt70 }}, and our structural form factors, <math>\tilde\mathfrak{f}_i</math>, gives,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

← Older revision		Revision as of 13:04, 25 April 2022
Line 4,289:		Line 4,289:
	=See Also=		=See Also=
	<ul>		<ul>
	<li>[[~~User:Tohline/~~SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>		<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
	</ul>		</ul>


	{{ SGFfooter }}		{{ SGFfooter }}