SSC/VirialEquilibrium/UniformDensity - Revision history

Jet53man at 21:13, 14 October 2021

2021-10-14T21:13:09Z

← Older revision		Revision as of 17:13, 14 October 2021
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	==Review==		==Review==
	In an [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Virial_Equilibrium\|introductory discussion]] of the virial equilibrium structure of spherically symmetric configurations — see especially the section titled, [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Energy_Extrema\|''Energy Extrema'']] — we deduced that a system's equilibrium radius, <math>~R_\mathrm{eq}</math>, measured relative to a reference length scale, <math>~R_0</math>, ''i.e.,'' the dimensionless equilibrium radius,		In an [[SSCpt1/Virial#Virial_Equilibrium\|introductory discussion]] of the virial equilibrium structure of spherically symmetric configurations — see especially the section titled, [[SSCpt1/Virial#Energy_Extrema\|''Energy Extrema'']] — we deduced that a system's equilibrium radius, <math>~R_\mathrm{eq}</math>, measured relative to a reference length scale, <math>~R_0</math>, ''i.e.,'' the dimensionless equilibrium radius,
	<div align="center">		<div align="center">
	<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_0} \, ,</math>		<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_0} \, ,</math>
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	</table>		</table>
	</div>		</div>
	This should be compared with our [[~~User:Tohline/~~SSC/Structure/UniformDensity#Summary\|detailed force-balance solution]] of the interior structure of an isolated, nonrotating, uniform-density sphere, which gives the precise expression,		This should be compared with our [[SSC/Structure/UniformDensity#Summary\|detailed force-balance solution]] of the interior structure of an isolated, nonrotating, uniform-density sphere, which gives the precise expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	The expression for <math>~P_c</math> derived from our identification of an extremum in the free energy is identical to the expression derived from the more precise, detailed force-balance analysis, except that the leading numerical coefficients differ by a factor of <math>~(5\mathfrak{f}_A/2\mathfrak{f}_W)</math>.		The expression for <math>~P_c</math> derived from our identification of an extremum in the free energy is identical to the expression derived from the more precise, detailed force-balance analysis, except that the leading numerical coefficients differ by a factor of <math>~(5\mathfrak{f}_A/2\mathfrak{f}_W)</math>.

	From a free-energy analysis alone, the best we can do is assume that both structural form factors, <math>~\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, are of order unity. But knowing the detailed force-balance solution allows us to evaluate both form factors. From our [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#FormFactors\|introductory discussion of the free energy function]], their respective definitions are,		From a free-energy analysis alone, the best we can do is assume that both structural form factors, <math>~\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, are of order unity. But knowing the detailed force-balance solution allows us to evaluate both form factors. From our [[SSCpt1/Virial#FormFactors\|introductory discussion of the free energy function]], their respective definitions are,
	<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>
	Now, because the configuration under discussion has a uniform density, we should set <math>~\rho(x)/\rho_c = 1</math> in the definition of <math>~\mathfrak{f}_W</math> which, after evaluation of the nested integrals, gives <math>~\mathfrak{f}_W = 1</math>. But, instead of being uniform throughout the configuration, in the [[~~User:Tohline/~~SSC/Structure/UniformDensity#Summary\|detailed force-balance model]], the pressure drops from the center to the surface according to the relation,		Now, because the configuration under discussion has a uniform density, we should set <math>~\rho(x)/\rho_c = 1</math> in the definition of <math>~\mathfrak{f}_W</math> which, after evaluation of the nested integrals, gives <math>~\mathfrak{f}_W = 1</math>. But, instead of being uniform throughout the configuration, in the [[SSC/Structure/UniformDensity#Summary\|detailed force-balance model]], the pressure drops from the center to the surface according to the relation,
	<div align="center">		<div align="center">
	<math>\frac{P(x)}{P_c} = 1 - x^2 \, .</math>		<math>\frac{P(x)}{P_c} = 1 - x^2 \, .</math>
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	</table>		</table>
	</div>		</div>
	where, in the last step as was [[~~User:Tohline/~~SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model\|recognized above]], we have set,		where, in the last step as was [[SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model\|recognized above]], we have set,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	===Comparison with Detailed Force-Balance Model===		===Comparison with Detailed Force-Balance Model===
	It is reasonable to ask how close this virial expression for the equilibrium radius is to the [[~~User:Tohline/~~SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium\|exact result]]. As before, from a free-energy analysis alone, the best we can do is assume that both structural form factors, <math>~\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, are of order unity. But we can do better than this. To begin with, because <math>~\rho</math> is uniform throughout the configuration, <math>~\mathfrak{f}_W = 1</math>, even though the configuration is truncated by the imposed external pressure. We need to reassess how <math>~\mathfrak{f}_A</math> is evaluated, however, because the pressure does not drop to zero at the surface of the configuration.		It is reasonable to ask how close this virial expression for the equilibrium radius is to the [[SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium\|exact result]]. As before, from a free-energy analysis alone, the best we can do is assume that both structural form factors, <math>~\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, are of order unity. But we can do better than this. To begin with, because <math>~\rho</math> is uniform throughout the configuration, <math>~\mathfrak{f}_W = 1</math>, even though the configuration is truncated by the imposed external pressure. We need to reassess how <math>~\mathfrak{f}_A</math> is evaluated, however, because the pressure does not drop to zero at the surface of the configuration.

	Going back to our [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Energy_Content_for_a_System_of_a_Given_Size_and_Internal_Structure\|original definition of the thermodynamic energy reservoir for spherically symmetric adiabatic systems]],		Going back to our [[SSCpt1/Virial#Energy_Content_for_a_System_of_a_Given_Size_and_Internal_Structure\|original definition of the thermodynamic energy reservoir for spherically symmetric adiabatic systems]],
	<div align="center">		<div align="center">
	<math>\mathfrak{W}_A = \frac{1}{({\gamma_g}-1)} \int_0^R 4\pi r^2 P dr		<math>\mathfrak{W}_A = \frac{1}{({\gamma_g}-1)} \int_0^R 4\pi r^2 P dr
	\, ,</math>		\, ,</math>
	</div>		</div>
	we begin by normalizing the radial coordinate to <math>~R_0</math>, the radius of the isolated (''i.e.,'' not truncated) sphere, because we know [[~~User:Tohline/~~SSC/Structure/UniformDensity#Summary\|from the detailed force-balanced solution]] that, structurally, the pressure varies with <math>~r</math> inside the configuration as,		we begin by normalizing the radial coordinate to <math>~R_0</math>, the radius of the isolated (''i.e.,'' not truncated) sphere, because we know [[SSC/Structure/UniformDensity#Summary\|from the detailed force-balanced solution]] that, structurally, the pressure varies with <math>~r</math> inside the configuration as,
	<div align="center">		<div align="center">
	<math>\frac{P(x)}{P_c} = 1 - x^2 \, ,</math>		<math>\frac{P(x)}{P_c} = 1 - x^2 \, ,</math>
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	</math>		</math>
	</div>		</div>
	Plugging this expression for <math>~\mathfrak{f}_A</math> along with <math>~\mathfrak{f}_W = 1</math> into the [[~~User:Tohline/~~SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model_2\|just-derived virial equilibrium solution]] gives,		Plugging this expression for <math>~\mathfrak{f}_A</math> along with <math>~\mathfrak{f}_W = 1</math> into the [[SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model_2\|just-derived virial equilibrium solution]] gives,

	<div align="center">		<div align="center">
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	</table>		</table>
	</div>		</div>
	This result exactly matches the solution for the equilibrium radius of a pressure-truncated, uniform-density sphere that has been [[~~User:Tohline/~~SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium\|derived elsewhere]].		This result exactly matches the solution for the equilibrium radius of a pressure-truncated, uniform-density sphere that has been [[SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium\|derived elsewhere]].


	=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 }}

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2021-10-14T21:10:22Z

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__FORCETOC__ 

=Uniform-Density Sphere=

==Review==
In an [[User:Tohline/SphericallySymmetricConfigurations/Virial#Virial_Equilibrium|introductory discussion]] of the virial equilibrium structure of spherically symmetric configurations — see especially the section titled, [[User:Tohline/SphericallySymmetricConfigurations/Virial#Energy_Extrema|''Energy Extrema'']] — we deduced that a system's equilibrium radius, <math>~R_\mathrm{eq}</math>, measured relative to a reference length scale, <math>~R_0</math>, ''i.e.,'' the dimensionless equilibrium radius,
<div align="center">
<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_0} \, ,</math>
</div>
is given by the root(s) of the following equation:
<div align="center">
<math>
2C \chi^{-2} + ~ (1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1} -~ 3D\chi^3 = 0 \, ,
</math>
</div>
where the definitions of the various coefficients are,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right">
<math>~A</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>\frac{3}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \mathfrak{f}_W \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~B</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\frac{K M_\mathrm{tot} }{(\gamma_g-1)} \biggl( \frac{3M_\mathrm{tot} }{4\pi R_0^3} \biggr)^{\gamma_g - 1} \cdot \mathfrak{f}_A
= \frac{\bar{c_s}^2 M_\mathrm{tot} }{(\gamma_g - 1)} \cdot \mathfrak{f}_A \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~B_I</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
3c_s^2 M_\mathrm{tot} \cdot \mathfrak{f}_M \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~C</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\frac{5J^2}{4M_\mathrm{tot} R_0^2} \cdot \mathfrak{f}_T \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~D</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>
\frac{4}{3} \pi R_0^3 P_e \, .
</math>
</td>
</tr>
</table>
</div>

Once the pressure exerted by the external medium (<math>~P_e</math>), and the configuration's mass (<math>~M_\mathrm{tot}</math>), angular momentum (<math>~J</math>), and specific entropy (via <math>~K</math>) — or, in the isothermal case, sound speed (<math>~c_s</math>) — have been specified, the values of all of the coefficients are known and <math>~\chi_\mathrm{eq}</math> can be determined.

==Adiabatic Evolution of an Isolated Sphere==
Here we seek to determine the equilibrium radius of a non-rotating configuration (<math>~J = 0</math>) that undergoes adiabatic compression/expansion (<math>\delta_{1\gamma_g} =~0</math>) and that is not confined by an external medium (<math>P_e = 0~</math>).
===Solution===
In this case, the statement of virial equilibrium is simplified considerably. Specifically, <math>~\chi_\mathrm{eq}</math> is given by the root(s) of the equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~A\chi_\mathrm{eq}^{-1} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3(\gamma_g-1) B\chi_\mathrm{eq}^{3 -3\gamma_g} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~~\chi_\mathrm{eq}^{3\gamma_g-4} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3(\gamma_g-1) B}{A}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl[ 3K M_\mathrm{tot} \biggl( \frac{3M_\mathrm{tot} }{4\pi R_0^3} \biggr)^{\gamma_g - 1} \cdot \mathfrak{f}_A \biggr]
\biggl[ \frac{3}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \mathfrak{f}_W \biggr] ^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl( \frac{1}{R_0} \biggr)^{3\gamma_g-4} \biggl[ 5\biggl( \frac{3}{4\pi} \biggr)^{\gamma_g-1} \biggr(\frac{K}{G}\biggr)
M^{(\gamma_g-2)} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_W} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

In other words,
<div align="center">
<math>
R_\mathrm{eq} = \biggl[ 5\biggl( \frac{3}{4\pi} \biggr)^{\gamma_g-1} \biggr(\frac{K}{G}\biggr)
M^{(\gamma_g-2)} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_W} \biggr]^{1/(3\gamma_g-4)} \, .
</math>
</div>

===Comparison with Detailed Force-Balance Model===
This derived solution will look more familiar if, instead of <math>~K</math>, we express the solution in terms of the central pressure,
<div align="center">
<math>P_c = K\rho_0^{\gamma_g} \, ,</math>
</div>
where, for this uniform-density sphere, <math>~\rho_0 = 3M_\mathrm{tot}/(4\pi R_\mathrm{eq}^3)</math>. Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~K</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>P_c \biggl( \frac{4\pi R_\mathrm{eq}^3}{3M_\mathrm{tot}} \biggr)^{\gamma_g} \, ,</math>
</td>
</tr>
</table>
</div>
and the solution takes the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>R_\mathrm{eq}^{3\gamma_g - 4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>5\biggl( \frac{4\pi}{3} \biggr) \biggr(\frac{P_c R_\mathrm{eq}^{3\gamma_g}}{GM^2_\mathrm{tot}}\biggr)
\cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_W} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ R_\mathrm{eq}^{4} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\biggl( \frac{3}{20\pi} \biggr) \biggr(\frac{GM^2_\mathrm{tot}}{P_c}\biggr)
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \, .</math>
</td>
</tr>
</table>
</div>
Or, solving for the central pressure,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~P_c</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\biggl( \frac{3}{20\pi} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr) \frac{GM^2_\mathrm{tot}}{R_\mathrm{eq}^{4} }
\, .</math>
</td>
</tr>
</table>
</div>
This should be compared with our [[User:Tohline/SSC/Structure/UniformDensity#Summary|detailed force-balance solution]] of the interior structure of an isolated, nonrotating, uniform-density sphere, which gives the precise expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~P_c</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\biggl( \frac{3}{8\pi} \biggr) \frac{GM^2_\mathrm{tot}}{R_\mathrm{eq}^{4} }
\, .</math>
</td>
</tr>
</table>
</div>
The expression for <math>~P_c</math> derived from our identification of an extremum in the free energy is identical to the expression derived from the more precise, detailed force-balance analysis, except that the leading numerical coefficients differ by a factor of <math>~(5\mathfrak{f}_A/2\mathfrak{f}_W)</math>.

From a free-energy analysis alone, the best we can do is assume that both structural form factors, <math>~\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, are of order unity. But knowing the detailed force-balance solution allows us to evaluate both form factors. From our [[User:Tohline/SphericallySymmetricConfigurations/Virial#FormFactors|introductory discussion of the free energy function]], their respective definitions are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_W</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\mathfrak{f}_A</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\int_0^1 \biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math>
</td>
</tr>
</table>
</div>
Now, because the configuration under discussion has a uniform density, we should set <math>~\rho(x)/\rho_c = 1</math> in the definition of <math>~\mathfrak{f}_W</math> which, after evaluation of the nested integrals, gives <math>~\mathfrak{f}_W = 1</math>. But, instead of being uniform throughout the configuration, in the [[User:Tohline/SSC/Structure/UniformDensity#Summary|detailed force-balance model]], the pressure drops from the center to the surface according to the relation,
<div align="center">
<math>\frac{P(x)}{P_c} = 1 - x^2 \, .</math>
</div>
Integrating over this function, in accordance with the definition of <math>~\mathfrak{f}_A</math>, gives,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\mathfrak{f}_A</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ 3\int_0^1 (1-x^2) x^2 dx = 3 \biggl[ \frac{x^3}{3} - \frac{x^5}{5} \biggr]_0^1 = \frac{2}{5} \, .</math>
</td>
</tr>
</table>
</div>
Hence, the ratio,
<div align="center">
<math>
\frac{5\mathfrak{f}_A}{2\mathfrak{f}_W} = 1 \, ,
</math>
</div>
which brings into perfect agreement the two separate determinations of the equilibrium expressions for <math>~R_\mathrm{eq}</math> and <math>~P_c</math> in terms of one another and the total mass.

This demonstrates that the free-energy approach to determining the equilibrium radius of a spherical configuration is only handicapped by its inability to precisely nail down values of the structural form factors. But this is not a severe limitation as the (dimensionless) form factors are generally of order unity. In contrast, the free-energy analysis brings with it a capability to readily evaluate the global stability of equilibrium configurations.

==Adiabatic Evolution of Pressure-truncated Sphere==
Here we seek to determine the equilibrium radius of a non-rotating configuration (<math>~J = 0</math>) that undergoes adiabatic compression/expansion (<math>\delta_{1\gamma_g} =~0</math>) and that is embedded in a hot, tenuous external medium whose confining pressure, <math>~P_e</math>, truncates the configuration.
===Solution===
In this case, virial equilibrium implies that <math>~\chi_\mathrm{eq}</math> is given by the root(s) of the equation,
<div align="center">
<math>
3(\gamma_g-1) B\chi^{3 -3\gamma_g} ~ -~A\chi^{-1} -~ 3D\chi^3 = 0 \, .
</math>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~D</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
(\gamma_g-1) B\chi_\mathrm{eq}^{-3\gamma_g} - \frac{A}{3}\chi_\mathrm{eq}^{-4}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~\frac{4\pi}{3} R_0^3 P_e</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl[ K M_\mathrm{tot} \biggl( \frac{3M_\mathrm{tot} }{4\pi R_0^3} \biggr)^{\gamma_g - 1} \cdot \mathfrak{f}_A \biggr] \chi_\mathrm{eq}^{-3\gamma_g}
- \biggl[ \frac{1}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \mathfrak{f}_W \biggr] \chi_\mathrm{eq}^{-4}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ P_e</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl[ K \biggl( \frac{3M_\mathrm{tot} }{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} \cdot \mathfrak{f}_A \biggr]
- \biggl[ \biggl(\frac{3}{20\pi} \biggr) \frac{GM_\mathrm{tot} ^2}{R_\mathrm{eq}^4} \cdot \mathfrak{f}_W \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
P_c \cdot \mathfrak{f}_A - \biggl(\frac{3}{20\pi} \biggr) \frac{GM_\mathrm{tot} ^2}{R_\mathrm{eq}^4} \cdot \mathfrak{f}_W \, ,
</math>
</td>
</tr>

</table>
</div>
where, in the last step as was [[User:Tohline/SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model|recognized above]], we have set,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~K</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>P_c \biggl( \frac{4\pi R_\mathrm{eq}^3}{3M_\mathrm{tot}} \biggr)^{\gamma_g} \, .</math>
</td>
</tr>
</table>
</div>
Hence, for any external pressure, <math>~P_e < P_c</math>, the pressure-confined equilibrium radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
~R_\mathrm{eq}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl[ \biggl(\frac{3}{20\pi} \biggr) \frac{GM_\mathrm{tot} ^2}{P_c} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A}
\biggl( 1 - \frac{P_e}{P_c} \cdot \frac{1}{\mathfrak{f}_A} \biggr)^{-1} \biggr]^{1/4} \, .
</math>
</td>
</tr>
</table>
</div>

===Comparison with Detailed Force-Balance Model===
It is reasonable to ask how close this virial expression for the equilibrium radius is to the [[User:Tohline/SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium|exact result]]. As before, from a free-energy analysis alone, the best we can do is assume that both structural form factors, <math>~\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, are of order unity. But we can do better than this. To begin with, because <math>~\rho</math> is uniform throughout the configuration, <math>~\mathfrak{f}_W = 1</math>, even though the configuration is truncated by the imposed external pressure. We need to reassess how <math>~\mathfrak{f}_A</math> is evaluated, however, because the pressure does not drop to zero at the surface of the configuration.

Going back to our [[User:Tohline/SphericallySymmetricConfigurations/Virial#Energy_Content_for_a_System_of_a_Given_Size_and_Internal_Structure|original definition of the thermodynamic energy reservoir for spherically symmetric adiabatic systems]],
<div align="center">
<math>\mathfrak{W}_A = \frac{1}{({\gamma_g}-1)} \int_0^R 4\pi r^2 P dr
\, ,</math>
</div>
we begin by normalizing the radial coordinate to <math>~R_0</math>, the radius of the isolated (''i.e.,'' not truncated) sphere, because we know [[User:Tohline/SSC/Structure/UniformDensity#Summary|from the detailed force-balanced solution]] that, structurally, the pressure varies with <math>~r</math> inside the configuration as,
<div align="center">
<math>\frac{P(x)}{P_c} = 1 - x^2 \, ,</math>
</div>
where, <math>~x \equiv r/R_0</math>. Integrating only out to the edge of the ''truncated'' sphere, which we will identify as <math>~R_e</math> and, correspondingly,
<div align="center">
<math>~x_e \equiv \frac{R_e}{R_0} = \biggl( 1 - \frac{P_e}{P_c} \biggr)^{1/2} \, ,</math>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{W}_A</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\frac{4\pi P_c R_0^3}{({\gamma_g}-1)} \int_0^{x_e} ( 1-x^2 ) x^2 dx</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{4\pi P_c R_0^3}{({\gamma_g}-1)} \biggl[ \frac{x^3}{3}-\frac{x^5}{5} \biggr]_0^{x_e}
= \frac{P_c }{({\gamma_g}-1)} \biggl( \frac{4\pi R_e^3}{3} \biggr) \biggl[ 1-\frac{3}{5}x_e^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\frac{M_\mathrm{tot} }{({\gamma_g}-1)} \biggl( \frac{P_c}{\rho_c} \biggr) \biggl[ 1-\frac{3}{5}\biggl(1 - \frac{P_e}{P_c}\biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Hence, in the case of a pressure-truncated, uniform-density sphere, we surmise that the relevant structural form factor is,
<div align="center">
<math>
\mathfrak{f}_A = 1-\frac{3}{5}\biggl(1 - \frac{P_e}{P_c}\biggr) = \frac{2}{5} + \frac{3}{5}\frac{P_e}{P_c} \, .
</math>
</div>
Plugging this expression for <math>~\mathfrak{f}_A</math> along with <math>~\mathfrak{f}_W = 1</math> into the [[User:Tohline/SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model_2|just-derived virial equilibrium solution]] gives,

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

<tr>
<td align="right">
<math>~\biggl(\frac{3}{20\pi} \biggr) \frac{GM_\mathrm{tot} ^2}{R_\mathrm{eq}^4} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~P_c \cdot \biggl[ \frac{2}{5} + \frac{3}{5}\frac{P_e}{P_c} \biggr] - P_e
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{2}{5} P_c \biggl( 1 - \frac{P_e}{P_c} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~~ R_\mathrm{eq}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
\biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, .
</math>
</td>
</tr>

</table>
</div>
This result exactly matches the solution for the equilibrium radius of a pressure-truncated, uniform-density sphere that has been [[User:Tohline/SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium|derived elsewhere]].

=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>

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SSC/VirialEquilibrium/UniformDensity - Revision history

Jet53man at 21:13, 14 October 2021

Jet53man: Created page with "__FORCETOC__ =Uniform-Density Sphere= ==Review== In an User:Tohline/Sphericall..."

Jet53man: Created page with "FORCETOC =Uniform-Density Sphere= ==Review== In an User:Tohline/Sphericall..."