User:Tohline/SSC/VirialEquilibrium/UniformDensity - Revision history

Tohline at 23:15, 10 July 2016

2016-07-10T23:15:25Z

← Older revision		Revision as of 23:15, 10 July 2016
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	</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 [[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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Tohline: /* Comparison with Detailed Force-Balance Model */ Add link to exact solution

2014-05-09T20:33:00Z

Comparison with Detailed Force-Balance Model: Add link to exact solution

← Older revision		Revision as of 20:33, 9 May 2014
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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 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 [[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]],		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]],

Tohline: /* Uniform-Density Sphere */ Finish derivation of pressure-truncated sphere

2014-05-09T00:08:52Z

Uniform-Density Sphere: Finish derivation of pressure-truncated sphere

← Older revision		Revision as of 00:08, 9 May 2014
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	</div>		</div>

			===Comparison with Detailed Force-Balance Model===
			It is reasonable to ask how close this virial expression for the equilibrium radius is to the 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]].

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Tohline: /* Uniform-Density Sphere */

2014-05-08T16:29:00Z

Uniform-Density Sphere

← Older revision		Revision as of 16:29, 8 May 2014
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	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.		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>


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Tohline: /* Adiabatic Evolution of an Isolated Sphere */ Adjust subsection headings

2014-05-07T22:48:54Z

Adiabatic Evolution of an Isolated Sphere: Adjust subsection headings

← Older revision		Revision as of 22:48, 7 May 2014
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	</div>		</div>

	~~===Discussion===~~		===Comparison with Detailed Force-Balance Model===

	====Comparison with Detailed Force-Balance ~~Solution=~~===
	This derived solution will look more familiar if, instead of <math>~K</math>, we express the solution in terms of the central pressure,		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">		<div align="center">

Tohline: /* Discussion */ Demonstration that two solutions give the same detailed result, after accounting for the structural form factors

2014-05-07T22:47:43Z

Discussion: Demonstration that two solutions give the same detailed result, after accounting for the structural form factors

← Older revision		Revision as of 22:47, 7 May 2014
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	===Discussion===		===Discussion===

			====Comparison with Detailed Force-Balance Solution====
	This derived solution will look more familiar if, instead of <math>~K</math>, we express the solution in terms of the central pressure,		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">		<div align="center">
Line 210:		Line 211:
	<td align="left">		<td align="left">
	<math>\biggl( \frac{3}{20\pi} \biggr) \biggr(\frac{GM^2_\mathrm{tot}}{P_c}\biggr)		<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>		\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	or, solving for the central pressure,		Or, solving for the central pressure,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 251:		Line 252:
	</table>		</table>
	</div>		</div>
	The ~~expressions are~~ identical, 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>.

	~~Accordingly~~, the ~~equilibrium mass~~-~~radius relationship for adiabatic configurations~~ of ~~a given specific entropy is~~,		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">		<div align="center">
	<math>		<table border="0" cellpadding="5" align="center">
	M^{(\~~gamma_g -~~ 2)} \~~propto R_~~\~~mathrm~~{eq}^{(3\~~gamma_g~~ -4)} \, .
	</math>		<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>		</div>
	~~Notice that, for <math>\gamma_g=2</math>, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass. Conversely, notice that~~, for <math>\gamma_g = 4/3</math>, the mass of the configuration is independent of the radius. For <math>\gamma_g</math> > <math> 2</math> or <math>\gamma_g </math>< <math>4/3</math>, configurations with larger mass (but the same specific entropy) have larger equilibrium radii. However, for <math>\gamma_g</math> in ~~the range, <math>2</math> > <math>\gamma_g </math> > <math>4/3</math>, configurations~~ with ~~larger mass have smaller equilibrium radii. Note that~~ the ~~result obtained for the isothermal configuration could have been obtained by setting~~ <math>\~~gamma_g = 1~~</math> ~~in this adiabatic solution~~, ~~because <math>K = c_s^2</math> when <math>\gamma_g = 1</math>.~~		Integrating over this function, in accordance with the definition of <math>~\mathfrak{f}_A</math>, gives,

	~~It is also instructive to write the coefficient <math>B</math> in terms of the average sound speed as defined above. In this case~~,
	<div align="center">		<div align="center">
	<math>		<table border="0" cellpadding="5" align="center">
	R_\~~mathrm~~{eq} = ~~R_0~~ \biggl[ \frac{GM}{5 \~~bar~~{~~c_s~~}~~^2 R_0~~} \biggr]^{1~~/(4- 3~~\~~gamma_g)~~} \, ,		<tr>
	</~~math~~>		<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>		</div>
	so the ~~equilibrium radius of an isolated, nonrotating, uniform density, adiabatic sphere is~~,		Hence, the ratio,
	<div align="center">		<div align="center">
	<math>		<math>
	R_\~~mathrm~~{~~eq} = R_0 =~~ \~~frac~~{GM}{5 \~~bar~~{~~c_s~~}^2 } \, .		\frac{5\mathfrak{f}_A}{2\mathfrak{f}_W} = 1 \, ,
	</math>		</math>
	</div>		</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.


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Tohline: /* Adiabatic Evolution of an Isolated Sphere */ Begin comparison with detailed force-balance solution

2014-05-07T21:28:39Z

Adiabatic Evolution of an Isolated Sphere: Begin comparison with detailed force-balance solution

← Older revision		Revision as of 21:28, 7 May 2014
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	===Discussion===		===Discussion===

			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 expressions are identical, except that the leading numerical coefficients differ by a factor of <math>~(5\mathfrak{f}_A/2\mathfrak{f}_W)</math>.

	Accordingly, the equilibrium mass-radius relationship for adiabatic configurations of a given specific entropy is,		Accordingly, the equilibrium mass-radius relationship for adiabatic configurations of a given specific entropy is,
	<div align="center">		<div align="center">

Tohline: /* Uniform-Density Sphere */ Clarify solution and add to discussion

2014-05-07T20:30:50Z

Uniform-Density Sphere: Clarify solution and add to discussion

← Older revision		Revision as of 20:30, 7 May 2014
Line 91:		Line 91:

	==Adiabatic Evolution of an Isolated Sphere==		==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>). 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,		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">		<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>		<math>
	3(\~~gamma_g-~~1) ~~B\chi~~^{3 -3\gamma_g} ~~~-~A~~\~~chi~~^{-1} ~~= 0~~ \, .		\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>		</math>
			</td>
			</tr>
			</table>
	</div>		</div>

	In other words,		In other words,
	<div align="center">		<div align="center">
	<math>		<math>
	R_\mathrm{eq} = ~~R_0~~ \~~chi_\mathrm{eq} =~~ \biggl[ \frac{3~~(\gamma_g-1) B~~}{A} \~~cdot R_0^{(3\gamma_g-4)~~} \biggr]^{~~1/(3~~\gamma_g-4)} = \~~biggl[ 5\biggl~~( \frac{3}{~~4\pi~~} \biggr)^{\gamma_g-1} \cdot \frac{~~KM^~~{~~(\gamma_g-2)~~}}{G} \biggr]^{1/(3\gamma_g-4)} \, .		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>		</math>
	</div>		</div>

			===Discussion===
	Accordingly, the equilibrium mass-radius relationship for adiabatic configurations of a given specific entropy is,		Accordingly, the equilibrium mass-radius relationship for adiabatic configurations of a given specific entropy is,
	<div align="center">		<div align="center">

Tohline: /* Isolated Uniform-Density Sphere */ Modified section and subsection titles

2014-05-07T19:57:19Z

Isolated Uniform-Density Sphere: Modified section and subsection titles

← Older revision		Revision as of 19:57, 7 May 2014
Line 1:		Line 1:
	__FORCETOC__ <!-- will force the creation of a Table of Contents -->		__FORCETOC__ <!-- will force the creation of a Table of Contents -->
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=~~Isolated~~ Uniform-Density Sphere=		=Uniform-Density Sphere=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	==Review==		==Review==
Line 90:		Line 90:
	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.		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.

	==~~Specific Case Being Considered Here~~==		==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>). 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,		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>). 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">		<div align="center">

Tohline: Force table of contents

2014-05-07T02:12:46Z

Force table of contents

← Older revision		Revision as of 02:12, 7 May 2014
Line 1:		Line 1:
	<!-- ~~__FORCETOC__~~ will force the creation of a Table of Contents -->		__FORCETOC__ <!-- will force the creation of a Table of Contents -->
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Isolated Uniform-Density Sphere=		=Isolated Uniform-Density Sphere=