SSCpt1/Virial/FormFactors - Revision history

Jet53man: /* Gravitational Potential Energy (n = 5) */

2021-10-26T20:25:58Z

Gravitational Potential Energy (n = 5)

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	This exactly matches the normalized gravitational potential energy derived independently in the context of our [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#Expression_for_Free_Energy\|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>W_\mathrm{core}^*</math>.		This exactly matches the normalized gravitational potential energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy\|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>W_\mathrm{core}^*</math>.

	Hence, also, as defined in the [[SSCpt1/Virial#Gathering_it_All_Together\|accompanying introductory discussion]], the constant, <math>\mathcal{A}</math>, that appears in our general free-energy equation is (for <math>n=5</math> polytropic configurations),		Hence, also, as defined in the [[SSCpt1/Virial#Gathering_it_All_Together\|accompanying introductory discussion]], the constant, <math>\mathcal{A}</math>, that appears in our general free-energy equation is (for <math>n=5</math> polytropic configurations),

Jet53man: /* Thermal Energy (n = 5) */

2021-10-26T20:23:44Z

Thermal Energy (n = 5)

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	This exactly matches the normalized thermal energy derived independently in the context of our [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#Expression_for_Free_Energy\|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>S_\mathrm{core}^*</math>. Its similarity to the expression for the gravitational potential energy — which is relevant to the virial theorem — is more apparent if it is rewritten in the following form:		This exactly matches the normalized thermal energy derived independently in the context of our [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy\|exploration of <math>(n_c, n_e) = (5,1)</math> bipolytropes]], referred to in that discussion as <math>S_\mathrm{core}^*</math>. Its similarity to the expression for the gravitational potential energy — which is relevant to the virial theorem — is more apparent if it is rewritten in the following form:
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Jet53man: /* Fiddling Around */

2021-10-14T17:00:34Z

Fiddling Around

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	==Fiddling Around==		==Fiddling Around==
	NOTE (from Tohline on 17 March 2015): Chronologically, this "Fiddling Around" subsection was developed before our discovery of the ~~VH74~~ derivations. It put us on track toward the correct development of general expressions for the structural form factors that are applicable to pressure-truncated polytropic spheres. But this subsection's conclusions are superseded by the ~~VH74~~ work.		NOTE (from Tohline on 17 March 2015): Chronologically, this "Fiddling Around" subsection was developed before our discovery of the {{ VH74hereafter }} derivations. It put us on track toward the correct development of general expressions for the structural form factors that are applicable to pressure-truncated polytropic spheres. But this subsection's conclusions are superseded by the {{ VH74hereafter }} work.

	In this subsection, for simplicity, we will omit the "tilde" over the variable <math>\xi</math>. In the case of <math>n=1</math> structures,		In this subsection, for simplicity, we will omit the "tilde" over the variable <math>\xi</math>. In the case of <math>n=1</math> structures,

Jet53man: /* Compare With General Expressions Based on VH74 Work */

2021-10-14T16:58:47Z

Compare With General Expressions Based on VH74 Work

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	====Compare With General Expressions Based on VH74 Work====		====Compare With General Expressions Based on VH74 Work====
	Based on the general expressions [[#PTtable\|derived above]] in the context of ~~VH74's work~~, for the specific case of <math>n=1</math> polytropic configurations, the three structural form factor should be,		Based on the general expressions [[#PTtable\|derived above]] in the context of {{ VH74hereafter }}, for the specific case of <math>n=1</math> polytropic configurations, the three structural form factor should be,

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	which also matches the expression presented in the [[#Summary_.28n.3D1.29\|summary table, above]]. So this adds support to the deduction, above, that ~~VH74~~ have provided us with the information necessary to develop general expressions for the three structural form factors.		which also matches the expression presented in the [[#Summary_.28n.3D1.29\|summary table, above]]. So this adds support to the deduction, above, that {{ VH74hereafter }} have provided us with the information necessary to develop general expressions for the three structural form factors.

	==Fiddling Around==		==Fiddling Around==

Jet53man: /* Foundation (n = 5) */

2021-10-14T16:53:42Z

Foundation (n = 5)

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	As is detailed in our [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#Profile\|accompanying discussion of bipolytropes]] — see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope\|discussion of the properties of ''isolated'' polytropes]] — in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,		As is detailed in our [[SSC/Structure/BiPolytropes/Analytic51#Profile\|accompanying discussion of bipolytropes]] — see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope\|discussion of the properties of ''isolated'' polytropes]] — in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,
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Jet53man: /* Implication for Structural Form Factors */

2021-10-14T16:51:17Z

Implication for Structural Form Factors

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	If we now adopt the ~~VH74~~ expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,		If we now adopt the {{ VH74hereafter }} expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,
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Jet53man: /* Implication for Structural Form Factors */

2021-10-14T16:41:48Z

Implication for Structural Form Factors

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	We should point out that ~~[http://adsabs.harvard.edu/abs/1993ApJS...88..205L Lai, Rasio, & Shapiro (1993b, ApJS, 88, 205)]~~ define a different set of dimensionless structure factors for ''isolated'' polytropic spheres — <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10) is used in the determination of the gravitational potential energy.		We should point out that {{ LRS93bfull }} define a different set of dimensionless structure factors for ''isolated'' polytropic spheres — <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10) is used in the determination of the gravitational potential energy.
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Jet53man: /* Implication for Structural Form Factors */

2021-10-14T16:21:53Z

Implication for Structural Form Factors

← Older revision		Revision as of 12:21, 14 October 2021
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	Building on the work of ~~VH74~~, we have, quite generally,		Building on the work of {{ VH74hereafter }}, we have, quite generally,

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Jet53man at 22:34, 13 October 2021

2021-10-13T22:34:15Z

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	===Presentation===		===Presentation===

	~~[http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)]~~ have provided analytic expressions for the gravitational potential energy and the internal energy — which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively — that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres. [The same expression for <math>~\Omega</math> is also effectively provided in §1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]		{{ VH74full }} have provided analytic expressions for the gravitational potential energy and the internal energy — which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively — that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres. [The same expression for <math>~\Omega</math> is also effectively provided in §1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]
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	<!-- [[Image:VialaHoredt1974.png\|500px\|center]] ~~-->~~		<!--
			[[Image:VialaHoredt1974.png\|500px\|center]]
	Astronomy & Astrophysics, 33: 195-202, (1974)<br />		Astronomy & Astrophysics, 33: 195-202, (1974)<br />
	POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />		POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />
	Y. P. Viala & Gp. Horedt<br />		Y. P. Viala & Gp. Horedt<br />
			-->
			{{ VH74figure }}
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	A couple of key equations drawn directly from ~~[http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala & Horedt (1974)]~~ have been shown here. As its title indicates, the paper includes discussion of — and accompanying equation derivations for — equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries: planar sheets, axisymmetric cylinders, and spheres. We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only. These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.		A couple of key equations drawn directly from {{ VH74 }} have been shown here. As its title indicates, the paper includes discussion of — and accompanying equation derivations for — equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries: planar sheets, axisymmetric cylinders, and spheres. We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only. These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.
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	Rewriting these two expressions to accommodate our parameter notations — recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation\|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization\|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> — we have from ~~Viala & Horedt's (~~VH74~~) work~~,		Rewriting these two expressions to accommodate our parameter notations — recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation\|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization\|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> — we have from {{ VH74 }},

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	===Renormalization===		===Renormalization===
	Both of the energy-term expressions derived by ~~Viala & Horedt~~ are written in terms of <math>\rho_c</math> and		Both of the energy-term expressions derived by {{ VH74 }} are written in terms of <math>\rho_c</math> and
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	So, employing our preferred normalization, the VH74 expressions become,		So, employing our preferred normalization, the {{ VH74 }} expressions become,

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174.64.8.247: /* See Also */

2021-08-01T20:35:13Z

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	=See Also=		=See Also=
	<ul>		<ul>
			<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
	<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]</li>		<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]</li>
	</ul>		</ul>