SSCpt1/Virial - Revision history

Jet53man: /* Volume Integrals */

2023-12-14T19:47:36Z

Volume Integrals

← Older revision		Revision as of 15:47, 14 December 2023
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	<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|configuration embedded in an external medium]] of pressure, <math>P_e</math>, will have a (pressure-truncated) surface whose radius, <math>R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>P_e</math>. In this case as well, one might choose to refer to <math>M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>M_\mathrm{limit}</math> from <math>M_\mathrm{tot}</math>, continuing to rely on <math>M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.		<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|configuration embedded in an external medium]] of pressure, <math>P_e</math>, will have a (pressure-truncated) surface whose radius, <math>R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>P_e</math>. In this case as well, one might choose to refer to <math>M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>M_\mathrm{limit}</math> from <math>M_\mathrm{tot}</math>, continuing to rely on <math>M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.

	<font color="maroon"><b>BiPolytropes</b></font>: When discussing [[SSC/~~BipolytropeGeneralization_Version2~~#Bipolytrope_Generalization\|bipolytropes]], the limit of integration, <math>R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.		<font color="maroon"><b>BiPolytropes</b></font>: When discussing [[SSC/BipolytropeGeneralizationVersion2#Bipolytrope_Generalization\|bipolytropes]], the limit of integration, <math>R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.
	</td></tr>		</td></tr>
	</table>		</table>

Jet53man: /* Some Detailed Examples */

2023-12-14T19:31:26Z

Some Detailed Examples

← Older revision		Revision as of 15:31, 14 December 2023
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	====Some Detailed Examples====		====Some Detailed Examples====

	In an [[SSCpt1/Virial/FormFactors#Structural_Form_Factors\|accompanying discussion]], we derive detailed expressions for a selected subset of the above structural form factors and corresponding energy terms in the case of spherically symmetric configurations that obey an {{ Template:Math/MP_PolytropicIndex }} = 5 or an {{ Template:Math/MP_PolytropicIndex }} = 1 polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should help with the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases, which cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures\|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#Free_Energy\|cores of bipolytropes]].		In an [[SSCpt1/Virial/FormFactors#Structural_Form_Factors\|accompanying discussion]], we derive detailed expressions for a selected subset of the above structural form factors and corresponding energy terms in the case of spherically symmetric configurations that obey an {{ Template:Math/MP_PolytropicIndex }} = 5 or an {{ Template:Math/MP_PolytropicIndex }} = 1 polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should help with the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases, which cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures\|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic51#Free_Energy\|cores of bipolytropes]].

	===Gathering it All Together===		===Gathering it All Together===

Jet53man: /* Energy Extrema */

2021-10-09T21:20:16Z

Energy Extrema

← Older revision		Revision as of 17:20, 9 October 2021
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	<table border="1" align="center" width="90%" cellpadding="20">		<table border="1" align="center" width="90%" cellpadding="20">
	<tr><td align="left">		<tr><td align="left">
	<b><font color="purple">ASIDE:</font></b> When we discuss the equilibrium of isothermal, rotating configurations that are immersed in an external medium, we will draw on the work of {{ Weber76full }} and the work of {{ Tohline85full }} which, in turn draws upon ~~[http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981, ApJ, 248, 717)]~~. In preparation for that discussion, we will go ahead and show how the {{ Tohline85 }} statement of virial equilibrium — his equation (9) — is the same as the equation that defines free energy extrema that has been derived here; and we will show how the {{ Weber76 }} "energy integral" — his equation (B3) — relates to our dimensionless free-energy function.		<b><font color="purple">ASIDE:</font></b> When we discuss the equilibrium of isothermal, rotating configurations that are immersed in an external medium, we will draw on the work of {{ Weber76full }} and the work of {{ Tohline85full }} which, in turn, draws upon {{ Tohline81full }}. In preparation for that discussion, we will go ahead and show how the {{ Tohline85 }} statement of virial equilibrium — his equation (9) — is the same as the equation that defines free energy extrema that has been derived here; and we will show how the {{ Weber76 }} "energy integral" — his equation (B3) — relates to our dimensionless free-energy function.

	----		----
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	</table>		</table>
	</div>		</div>
	Next, because {{ Tohline85 }} considered only uniform-density configurations, all of the dimensionless filling factors can be set to unity in the definitions of the leading coefficients of all of our energy terms; but, following ~~[http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981)]~~, the leading coefficients of two of our energy terms should be modified to include a factor involving the configuration's eccentricity,		Next, because {{ Tohline85 }} considered only uniform-density configurations, all of the dimensionless filling factors can be set to unity in the definitions of the leading coefficients of all of our energy terms; but, following {{ Tohline81 }}, the leading coefficients of two of our energy terms should be modified to include a factor involving the configuration's eccentricity,
	<div align="center">		<div align="center">
	<math>e \equiv \biggl( 1 - \frac{Z_\mathrm{eq}^2}{R_\mathrm{eq}^2} \biggr)^{1/2} \, ,</math>		<math>e \equiv \biggl( 1 - \frac{Z_\mathrm{eq}^2}{R_\mathrm{eq}^2} \biggr)^{1/2} \, ,</math>
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	<math>c_s^2 = \biggl( \frac{3}{4\pi} \biggr) c_\mathrm{norm}^2 \, ,</math>		<math>c_s^2 = \biggl( \frac{3}{4\pi} \biggr) c_\mathrm{norm}^2 \, ,</math>
	</div>		</div>
	it is clear that this last form of our "free energy extrema" expression is identical to {{ Tohline85 }} virial equilibrium equation (9), which appears in print in a simpler but also more cryptic form as,		it is clear that this last form of our "free energy extrema" expression is identical to the {{ Tohline85 }} virial equilibrium equation (9), which appears in print in a simpler but also more cryptic form as,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Energy Extrema */

2021-10-09T02:24:13Z

Energy Extrema

← Older revision		Revision as of 22:24, 8 October 2021
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	<table border="1" align="center" width="90%" cellpadding="20">		<table border="1" align="center" width="90%" cellpadding="20">
	<tr><td align="left">		<tr><td align="left">
	<b><font color="purple">ASIDE:</font></b> When we discuss the equilibrium of isothermal, rotating configurations that are immersed in an external medium, we will draw on the work of ~~[http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber (1976, ApJ, 208, 113)] — ''Oscillation and Collapse of Interstellar Clouds'' —~~ and the work of ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985, ApJ, 292, 181)] — ''Star Formation: Phase Transition, not Jeans Instability'' —~~ which, in turn draws upon [http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981, ApJ, 248, 717)]. In preparation for that discussion, we will go ahead and show how ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline's (1985)]~~ statement of virial equilibrium — his equation (9) — is the same as the equation that defines free energy extrema that has been derived here; and we will show how ~~[http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)]~~ "energy integral" — his equation (B3) — relates to our dimensionless free-energy function.		<b><font color="purple">ASIDE:</font></b> When we discuss the equilibrium of isothermal, rotating configurations that are immersed in an external medium, we will draw on the work of {{ Weber76full }} and the work of {{ Tohline85full }} which, in turn draws upon [http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981, ApJ, 248, 717)]. In preparation for that discussion, we will go ahead and show how the {{ Tohline85 }} statement of virial equilibrium — his equation (9) — is the same as the equation that defines free energy extrema that has been derived here; and we will show how the {{ Weber76 }} "energy integral" — his equation (B3) — relates to our dimensionless free-energy function.

	----		----
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	<tr><td align="center">		<tr><td align="center">
	<!-- [[Image:Tohline1985_Eq9.png\|500px\|center]] -->		<!-- [[Image:Tohline1985_Eq9.png\|500px\|center]] -->
			<!--
	The Astrophysical Journal, 292: 181-187, 1985 May 1<br />		The Astrophysical Journal, 292: 181-187, 1985 May 1<br />
	STAR FORMATION: PHASE TRANSITION, NOT JEANS INSTABILITY<br />		STAR FORMATION: PHASE TRANSITION, NOT JEANS INSTABILITY<br />
	Joel E. Tohline<br />		Joel E. Tohline<br />
			-->
			{{ Tohline85figure }}<br /> <br />
	<math>\beta(\sin^{-1}e/e)^2 [\tfrac{1}{2} - \beta ] + kV^* - F_s^* = 0 \, .</math>       (9)		<math>\beta(\sin^{-1}e/e)^2 [\tfrac{1}{2} - \beta ] + kV^* - F_s^* = 0 \, .</math>       (9)
	</td></tr>		</td></tr>
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	First, in order to match sign conventions, we need to multiply our "free energy extrema" equation through by minus one; second, we should set <math>\delta_{1\gamma_g} = 1</math> because ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)]~~ was only concerned with isothermal systems; then, because ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)]~~ normalizes each energy term by		First, in order to match sign conventions, we need to multiply our "free energy extrema" equation through by minus one; second, we should set <math>\delta_{1\gamma_g} = 1</math> because {{ Tohline85 }} was only concerned with isothermal systems; then, because {{ Tohline85 }} normalizes each energy term by
	<div align="center">		<div align="center">
	<math>E^* \equiv \biggl( \frac{2^2 \cdot 3^2}{5^3} \biggr) \frac{G^2 M_\mathrm{tot}^5}{J^2} \, ,</math>		<math>E^* \equiv \biggl( \frac{2^2 \cdot 3^2}{5^3} \biggr) \frac{G^2 M_\mathrm{tot}^5}{J^2} \, ,</math>
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	</table>		</table>
	</div>		</div>
	Next, because ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)]~~ considered only uniform-density configurations, all of the dimensionless filling factors can be set to unity in the definitions of the leading coefficients of all of our energy terms; but, following [http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981)], the leading coefficients of two of our energy terms should be modified to include a factor involving the configuration's eccentricity,		Next, because {{ Tohline85 }} considered only uniform-density configurations, all of the dimensionless filling factors can be set to unity in the definitions of the leading coefficients of all of our energy terms; but, following [http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981)], the leading coefficients of two of our energy terms should be modified to include a factor involving the configuration's eccentricity,
	<div align="center">		<div align="center">
	<math>e \equiv \biggl( 1 - \frac{Z_\mathrm{eq}^2}{R_\mathrm{eq}^2} \biggr)^{1/2} \, ,</math>		<math>e \equiv \biggl( 1 - \frac{Z_\mathrm{eq}^2}{R_\mathrm{eq}^2} \biggr)^{1/2} \, ,</math>
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	</table>		</table>
	</div>		</div>
	Next we need to appreciate that ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)]~~ adopted the dimensionless parameter, <math>\beta \equiv T_\mathrm{rot}/\|W_\mathrm{grav}\|</math>, instead of the normalized radius, <math>\chi</math>, as the order parameter that is varied when searching for extrema in the free-energy function. So, in our equation that defines "free energy extrema" we need to replace <math>\chi_\mathrm{eq}</math> with <math>\beta_\mathrm{eq}</math>, using the relation,		Next we need to appreciate that {{ Tohline85 }} adopted the dimensionless parameter, <math>\beta \equiv T_\mathrm{rot}/\|W_\mathrm{grav}\|</math>, instead of the normalized radius, <math>\chi</math>, as the order parameter that is varied when searching for extrema in the free-energy function. So, in our equation that defines "free energy extrema" we need to replace <math>\chi_\mathrm{eq}</math> with <math>\beta_\mathrm{eq}</math>, using the relation,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	which is the definition of the coefficient "<math>k</math>" that is provided as equation (7) of ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)]~~. Hence, dropping the factor of two out front, our expression for "free energy extrema" becomes,		which is the definition of the coefficient "<math>k</math>" that is provided as equation (7) of {{ Tohline85 }}. Hence, dropping the factor of two out front, our expression for "free energy extrema" becomes,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	Finally, realizing that the square of the sound speed is related to our <math>c_\mathrm{norm}^2</math> via the relation [note that ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)]~~ uses <math>a^2</math> in place of <math>c_s^2</math>],		Finally, realizing that the square of the sound speed is related to our <math>c_\mathrm{norm}^2</math> via the relation [note that {{ Tohline85 }} uses <math>a^2</math> in place of <math>c_s^2</math>],
	<div align="center">		<div align="center">
	<math>c_s^2 = \biggl( \frac{3}{4\pi} \biggr) c_\mathrm{norm}^2 \, ,</math>		<math>c_s^2 = \biggl( \frac{3}{4\pi} \biggr) c_\mathrm{norm}^2 \, ,</math>
	</div>		</div>
	it is clear that this last form of our "free energy extrema" expression is identical to ~~[http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline's (1985)]~~ virial equilibrium equation (9), which appears in print in a simpler but also more cryptic form as,		it is clear that this last form of our "free energy extrema" expression is identical to {{ Tohline85 }} virial equilibrium equation (9), which appears in print in a simpler but also more cryptic form as,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr><td align="center">		<tr><td align="center">
	<!-- [[Image:Weber1976_EqB3.png\|500px\|center]] -->		<!-- [[Image:Weber1976_EqB3.png\|500px\|center]] -->
			<!--
	The Astrophysical Journal, 208: 113-126, 1976 August 15<br />		The Astrophysical Journal, 208: 113-126, 1976 August 15<br />
	OSCILLATION AND COLLAPSE OF INTERSTELLAR CLOUDS<br />		OSCILLATION AND COLLAPSE OF INTERSTELLAR CLOUDS<br />
	Stephen V. Weber<br />		Stephen V. Weber<br />
			-->
			{{ Weber76figure }}<br /> <br />
	<math>E = \dot{a}^2 + \dot\gamma^2/2 + J^2/\alpha^2		<math>E = \dot{a}^2 + \dot\gamma^2/2 + J^2/\alpha^2
	- \tfrac{4}{3} \log \alpha^2 \gamma + \tfrac{1}{3}\alpha^2 \gamma P_\mathrm{ext} -		- \tfrac{4}{3} \log \alpha^2 \gamma + \tfrac{1}{3}\alpha^2 \gamma P_\mathrm{ext} -
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	</table>		</table>
	</div>		</div>
	where, for axisymmetric configurations (set <math>\beta=\alpha</math> in ~~[http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)]~~ equation 12),		where, for axisymmetric configurations (set <math>\beta=\alpha</math> in equation 12 of {{ Weber76 }}),
	<div align="center">		<div align="center">
	<math>J_\mathrm{Weber} \equiv \alpha^2 \Omega = \biggl( \frac{R}{R_0} \biggr)^2 (\dot\varphi_c t_0)^2 \, .</math>		<math>J_\mathrm{Weber} \equiv \alpha^2 \Omega = \biggl( \frac{R}{R_0} \biggr)^2 (\dot\varphi_c t_0)^2 \, .</math>
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	</table>		</table>
	</div>		</div>
	The right-hand-side of this expression exactly matches ~~[http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)]~~ "energy integral" for oblate-spheroidal configurations — see his equation (B3) for the case, <math>e > 0</math> — except that Weber's energy integral includes an additional pair of terms (<math>{\dot\alpha}^2 + {\dot\gamma}^2/2</math>) to account for kinetic energy associated with the overall collapse or expansion of the configuration. [NOTE: The logarithmic term ultimately needs to be <math>\ln \alpha^2\gamma</math> instead of <math>\ln\chi^3</math> in order to reflect an oblate-spheroidal, rather than spherical, volume. This term also needs to be fixed in the above discussion of Tohline's work.]		The right-hand-side of this expression exactly matches the {{ Weber76 }} "energy integral" for oblate-spheroidal configurations — see his equation (B3) for the case, <math>e > 0</math> — except that Weber's energy integral includes an additional pair of terms (<math>{\dot\alpha}^2 + {\dot\gamma}^2/2</math>) to account for kinetic energy associated with the overall collapse or expansion of the configuration. [NOTE: The logarithmic term ultimately needs to be <math>\ln \alpha^2\gamma</math> instead of <math>\ln\chi^3</math> in order to reflect an oblate-spheroidal, rather than spherical, volume. This term also needs to be fixed in the above discussion of Tohline's work.]
	</td></tr>		</td></tr>
	</table>		</table>

Jet53man: /* Choices Made by Other Researchers */

2021-10-08T22:01:12Z

Choices Made by Other Researchers

← Older revision		Revision as of 18:01, 8 October 2021
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	=====Choices Made by Other Researchers=====		=====Choices Made by Other Researchers=====

	As is detailed in a [[SSC/Structure/PolytropesEmbedded#General_Properties\|related discussion]], our definitions of <math>R_\mathrm{norm}</math> and <math>P_\mathrm{norm}</math> are close, but not identical, to the scalings adopted by ~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970, MNRAS, 151, 81)]~~ and by ~~[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981, MNRAS, 195, 967)]~~. The following relations can be used to switch from our normalizations to theirs:		As is detailed in a [[SSC/Structure/PolytropesEmbedded#General_Properties\|related discussion]], our definitions of <math>R_\mathrm{norm}</math> and <math>P_\mathrm{norm}</math> are close, but not identical, to the scalings adopted by {{ Horedt70 }} and by {{ Whitworth81 }}. The following relations can be used to switch from our normalizations to theirs:
	<div align="center">		<div align="center">

Jet53man: /* Our Choices */

2021-10-08T21:58:45Z

Our Choices

← Older revision		Revision as of 17:58, 8 October 2021
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	=====Our Choices=====		=====Our Choices=====
	It is appropriate for us to define some characteristic scales against which various physical parameters can be normalized — and, hence, their relative significance can be specified or measured — as the free energy of various systems is examined. As the system size is varied in search of extrema in the free energy, we generally will hold constant the total system mass and the specific entropy of each fluid element. (When isothermal rather than adiabatic variations are considered, the sound speed rather than the specific entropy will be held constant.) Hence, following the lead of both ~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970, MNRAS, 151, 81)]~~ and ~~[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981, MNRAS, 195, 967)]~~, we will express the various characteristic scales in terms of the constants, <math>G, M_\mathrm{tot},</math> and the polytropic constant, <math>K.</math> Specifically, we will normalize all length scales, pressures, energies, mass densities, and the square of the speed of sound by, respectively,		It is appropriate for us to define some characteristic scales against which various physical parameters can be normalized — and, hence, their relative significance can be specified or measured — as the free energy of various systems is examined. As the system size is varied in search of extrema in the free energy, we generally will hold constant the total system mass and the specific entropy of each fluid element. (When isothermal rather than adiabatic variations are considered, the sound speed rather than the specific entropy will be held constant.) Hence, following the lead of both {{ Horedt70full }} and {{ Whitworth81full }}, we will express the various characteristic scales in terms of the constants, <math>G, M_\mathrm{tot},</math> and the polytropic constant, <math>K.</math> Specifically, we will normalize all length scales, pressures, energies, mass densities, and the square of the speed of sound by, respectively,
	<div align="center">		<div align="center">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">

Jet53man: /* Volume Integrals */

2021-10-08T21:44:25Z

Volume Integrals

← Older revision		Revision as of 17:44, 8 October 2021
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	<div>		<div>
	<font color="red">Rotational Kinetic Energy:</font> We will also consider a system that is rotating with a specified [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|''simple'' angular velocity profile]], <math>\dot\varphi(\varpi)</math>, in which case, from our discussion of the [[VE#Scalar_Virial_Theorem\|scalar virial theorem]] — see, specifically, the reference to Equation (8), on p. 16 of [[Appendix/References#EFE\|EFE]] — the (ordered) kinetic energy,		<font color="red">Rotational Kinetic Energy:</font> We will also consider a system that is rotating with a specified [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|''simple'' angular velocity profile]], <math>\dot\varphi(\varpi)</math>, in which case, from our discussion of the [[VE#Scalar_Virial_Theorem\|scalar virial theorem]] — see, specifically, the reference to Equation (8), on p. 16 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] — the (ordered) kinetic energy,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

174.64.8.247: /* Virial Equilibrium of Spherically Symmetric Configurations */

2021-07-28T16:03:53Z

Virial Equilibrium of Spherically Symmetric Configurations

← Older revision		Revision as of 12:03, 28 July 2021
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	=Virial Equilibrium of Spherically Symmetric Configurations=		=Virial Equilibrium of Spherically Symmetric Configurations=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:#9390DB;" \|
			<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations\|<b>Free-Energy<br />of<br />Spherical<br />Systems</b>]]</font>
			\|}
	==Free Energy Expression==		==Free Energy Expression==
	===Review===		===Review===

174.64.8.247: /* Some Detailed Examples */

2021-07-16T21:44:34Z

Some Detailed Examples

← Older revision		Revision as of 17:44, 16 July 2021
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	====Some Detailed Examples====		====Some Detailed Examples====

	In an [[~~SSC~~/Virial/FormFactors#Structural_Form_Factors\|accompanying discussion]], we derive detailed expressions for a selected subset of the above structural form factors and corresponding energy terms in the case of spherically symmetric configurations that obey an {{ Template:Math/MP_PolytropicIndex }} = 5 or an {{ Template:Math/MP_PolytropicIndex }} = 1 polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should help with the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases, which cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures\|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy\|cores of bipolytropes]].		In an [[SSCpt1/Virial/FormFactors#Structural_Form_Factors\|accompanying discussion]], we derive detailed expressions for a selected subset of the above structural form factors and corresponding energy terms in the case of spherically symmetric configurations that obey an {{ Template:Math/MP_PolytropicIndex }} = 5 or an {{ Template:Math/MP_PolytropicIndex }} = 1 polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should help with the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases, which cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures\|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy\|cores of bipolytropes]].

	===Gathering it All Together===		===Gathering it All Together===

Jet53man: /* See Also */

2021-07-16T17:20:08Z

← Older revision		Revision as of 13:20, 16 July 2021
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	<ul>		<ul>
	<li>[[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>
			<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]</li>
	</ul>		</ul>


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