User:Tohline/SSC/Virial/Isothermal - Revision history

Tohline: /* Isolated, Nonrotating Configuration */

2016-08-13T22:56:49Z

Isolated, Nonrotating Configuration

← Older revision		Revision as of 22:56, 13 August 2016
Line 91:		Line 91:

	==Isolated, Nonrotating Configuration==		==Isolated, Nonrotating Configuration==
	For a nonrotating configuration <math>(C=J=0)</math> that is not influenced by the effects of a bounding external medium <math>(D=P_e = 0)</math>, the statement of virial equilibrium is,		For a nonrotating configuration <math>~(C=J=0)</math> that is not influenced by the effects of a bounding external medium <math>~(D=P_e = 0)</math>, the statement of virial equilibrium is,
	<div align="center">		<div align="center">
	<math>		<math>~
	(1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1} = 0 \, .		(1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1} = 0 \, .
	</math>		</math>
Line 99:		Line 99:

	===Isothermal Evolutions===		===Isothermal Evolutions===
	For isothermal configurations <math>(\delta_{1\gamma_g} = 1)</math>, one and only one equilibrium state arises where,		For isothermal configurations <math>~(\delta_{1\gamma_g} = 1)</math>, one and only one equilibrium state arises where,
	<div align="center">		<div align="center">
	<math>		<math>~
	B_I = A\chi^{-1} \, ,		B_I = A\chi^{-1} \, ,
	</math>		</math>
Line 107:		Line 107:
	that is,		that is,
	<div align="center">		<div align="center">
	<math>		<math>~
	R_\mathrm{eq} = R_0 \chi_\mathrm{eq} = \frac{A}{B_I}\cdot R_0 = \frac{GM}{5c_s^2} \, .		R_\mathrm{eq} = R_0 \chi_\mathrm{eq} = \frac{A}{B_I}\cdot R_0 = \frac{GM}{5c_s^2} \, .
	</math>		</math>

Tohline: /* Quartic Solution */ Clean up font size of (especially) in-line equations

2016-08-13T22:52:17Z

Quartic Solution: Clean up font size of (especially) in-line equations

Show changes

Tohline: /* P-V Diagram */ Clean up font size of in-line mathematical expressions

2016-08-13T22:39:05Z

P-V Diagram: Clean up font size of in-line mathematical expressions

← Older revision		Revision as of 22:39, 13 August 2016
Line 153:		Line 153:

	====P-V Diagram====		====P-V Diagram====
	Returning to the dimensionless form of this expression and multiplying through by <math>[-\chi/(3D)]</math>, we obtain,		Returning to the dimensionless form of this expression and multiplying through by <math>~[-\chi/(3D)]</math>, we obtain,
	<div align="center">		<div align="center">
	<math>		<math>
Line 159:		Line 159:
	</math>		</math>
	</div>		</div>
	Now, taking a cue from the solution presented above for an isolated isothermal configuration, we choose to set the previously unspecified scale factor, <math>R_0</math>, to,		Now, taking a cue from the solution presented above for an isolated isothermal configuration, we choose to set the previously unspecified scale factor, <math>~R_0</math>, to,
	<div align="center">		<div align="center">
	<math>		<math>
Line 165:		Line 165:
	</math>		</math>
	</div>		</div>
	in which case <math>B_I = A</math>, and the quartic equation governing the radii of equilibrium states becomes, simply,		in which case <math>~B_I = A</math>, and the quartic equation governing the radii of equilibrium states becomes, simply,
	<div align="center">		<div align="center">
	<math>		<math>
Line 177:		Line 177:
	</math>		</math>
	</div>		</div>
	For a given choice of <math>P_e</math> and <math>c_s</math>, <math>\Pi^{1/2}</math> can represent a dimensionless mass, in which case,		For a given choice of <math>~P_e</math> and <math>~c_s</math>, <math>~\Pi^{1/2}</math> can represent a dimensionless mass, in which case,
	<div align="center">		<div align="center">
	<math>		<math>
Line 183:		Line 183:
	</math>		</math>
	</div>		</div>
	Alternatively, for a given choice of configuration mass and sound speed, this parameter, <math>\Pi</math>, can be viewed as a dimensionless external pressure; or, for a given choice of <math>M</math> and <math>P_e</math>, <math>\Pi^{-1/8}</math> can represent a dimensionless sound speed. In most of what follows we will view <math>\Pi</math> as a dimensionless external pressure.		Alternatively, for a given choice of configuration mass and sound speed, this parameter, <math>~\Pi</math>, can be viewed as a dimensionless external pressure; or, for a given choice of <math>~M</math> and <math>~P_e</math>, <math>~\Pi^{-1/8}</math> can represent a dimensionless sound speed. In most of what follows we will view <math>~\Pi</math> as a dimensionless external pressure.

	The above quartic equation can be rearranged immediately to give the external pressure that is required to obtain a particular configuration radius, namely,		The above quartic equation can be rearranged immediately to give the external pressure that is required to obtain a particular configuration radius, namely,
Line 204:		Line 204:
	The black curve traces out the function,		The black curve traces out the function,
	<div align="center">		<div align="center">
	<math>		<math>~
	\Pi = (\chi - 1)/\chi^4 \, ,		\Pi = (\chi - 1)/\chi^4 \, ,
	</math>		</math>
	</div>		</div>
	and shows the dimensionless external pressure, <math>\Pi</math>, that is required to construct a nonrotating, self-gravitating, isothermal sphere with an equilibrium radius <math>\chi</math>. The pressure becomes negative at radii <math>\chi < 1</math>, hence the solution in this regime is unphysical.		and shows the dimensionless external pressure, <math>~\Pi</math>, that is required to construct a nonrotating, self-gravitating, isothermal sphere with an equilibrium radius <math>~\chi</math>. The pressure becomes negative at radii <math>~\chi < 1</math>, hence the solution in this regime is unphysical.

	[[User:Tohline/SphericallySymmetricConfigurations/Virial#Visual_Representation\|Figure 1]] displays the free energy surface that "lies above" the two-dimensional parameter space (<math>1.2 < \chi < 1.51</math>; <math>0.103 < \Pi < 0.104</math>) that is identified here by the thin, red rectangle.		[[User:Tohline/SphericallySymmetricConfigurations/Virial#Visual_Representation\|Figure 1]] displays the free energy surface that "lies above" the two-dimensional parameter space <math>~(1.2 < \chi < 1.51</math>; <math>~0.103 < \Pi < 0.104)</math> that is identified here by the thin, red rectangle.
	</td>		</td>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
Line 219:		Line 219:
	</table>		</table>
	</div>		</div>
	In the absence of self-gravity (''i.e.,'' <math>A=0</math>), the product of the external pressure and the volume should be constant. The corresponding relation, <math>\Pi = \chi^{-3}</math>, is shown by the blue dashed curve in the figure. As the figure illustrates, when gravity is included the P-V relationship pulls away from the PV = constant curve at sufficiently small volumes. Indeed, the curve turns over at a finite pressure, <math>\Pi_\mathrm{max}</math>, and for every value of <math>\Pi < \Pi_\mathrm{max}</math> a second, more compact equilibrium configuration appears. The location of <math>\Pi_\mathrm{max}</math> along the curve is identified by setting <math>\partial\Pi/\partial\chi = 0</math>, that is, it occurs where,		In the absence of self-gravity (''i.e.,'' <math>~A=0</math>), the product of the external pressure and the volume should be constant. The corresponding relation, <math>~\Pi = \chi^{-3}</math>, is shown by the blue dashed curve in the figure. As the figure illustrates, when gravity is included the P-V relationship pulls away from the PV = constant curve at sufficiently small volumes. Indeed, the curve turns over at a finite pressure, <math>~\Pi_\mathrm{max}</math>, and for every value of <math>~\Pi < \Pi_\mathrm{max}</math> a second, more compact equilibrium configuration appears. The location of <math>~\Pi_\mathrm{max}</math> along the curve is identified by setting <math>~\partial\Pi/\partial\chi = 0</math>, that is, it occurs where,
	<div align="center">		<div align="center">
	<math>		<math>
Line 231:		Line 231:
	<span id="BonnorEbertMass">Hence,</span>		<span id="BonnorEbertMass">Hence,</span>
	<div align="center">		<div align="center">
	<math>\Pi_\mathrm{max} = \biggl( \frac{2^2}{3} \biggr)^{-4} \biggl( \frac{2^2}{3}-1 \biggr) = \frac{3^3}{2^8} \approx 0.105469\, ;</math>		<math>~\Pi_\mathrm{max} = \biggl( \frac{2^2}{3} \biggr)^{-4} \biggl( \frac{2^2}{3}-1 \biggr) = \frac{3^3}{2^8} \approx 0.105469\, ;</math>
	</div>		</div>
	therefore, from above,		therefore, from above,
	<div align="center">		<div align="center">
	<math>		<math>~
	M_\mathrm{max} = \biggl( \frac{3^4\cdot 5^3}{2^{10}\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2}		M_\mathrm{max} = \biggl( \frac{3^4\cdot 5^3}{2^{10}\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2}
	\approx 1.77408 \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .		\approx 1.77408 \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .

Tohline: /* Bonnor's (1956) Equivalent Relation */ Clean up font size of in-line equations

2016-08-13T22:28:17Z

Bonnor's (1956) Equivalent Relation: Clean up font size of in-line equations

← Older revision		Revision as of 22:28, 13 August 2016
Line 129:		Line 129:

	====Bonnor's (1956) Equivalent Relation====		====Bonnor's (1956) Equivalent Relation====
	Inserting the expressions for the coefficients <math>B_I</math>, <math>A</math>, and <math>D</math> gives,		Inserting the expressions for the coefficients <math>~B_I</math>, <math>~A</math>, and <math>~D</math> gives,
	<div align="center">		<div align="center">
	<math>		<math>~
	3Mc_s^2 ~- \frac{3}{5} \frac{GM^2}{R} = 3 P_e \biggl( \frac{4\pi}{3} R^3\biggr) \, ,		3Mc_s^2 ~- \frac{3}{5} \frac{GM^2}{R} = 3 P_e \biggl( \frac{4\pi}{3} R^3\biggr) \, ,
	</math>		</math>
	</div>		</div>
	or, because the volume <math>V = (4\pi R^3/3)</math> for a spherical configuration, we can write,		or, because the volume <math>~V = (4\pi R^3/3)</math> for a spherical configuration, we can write,
	<div align="center">		<div align="center">
	<math>		<math>
	3P_e V = 3Mc_s^2 ~- \frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/3} \frac{GM^2}{V^{1/3}} \, .		~3P_e V = 3Mc_s^2 ~- \frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/3} \frac{GM^2}{V^{1/3}} \, .
	</math>		</math>
	</div>		</div>
Line 145:		Line 145:
	<table border="2">		<table border="2">
	<tr><td>		<tr><td>
	~~<!--~~ [[File:Bonnor1951Eq1.2.jpg\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]] -->		[[File:Bonnor1951Eq1.2.jpg\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]]
	[[Image:AAAwaiting01.png\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]]		<!--[[Image:AAAwaiting01.png\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]]-->
	</td></tr>		</td></tr>
	</table>		</table>
	</div>		</div>
	Once we realize that, for an isothermal configuration, twice the thermal energy content, <math>2S</math>, can be written as <math>(3NkT)</math> just as well as via the product, <math>(3Mc_s^2)</math>, we see that our expression is identical to Bonnor's if we set the prefactor on Bonnor's last term, <math>\alpha = (4\pi/3)^{1/3}/5</math>. (Indeed, later on the first page of his paper, Bonnor points out that this is the appropriate value for <math>\alpha</math> when considering a uniform density sphere.)		Once we realize that, for an isothermal configuration, twice the thermal energy content, <math>~2S</math>, can be written as <math>~(3NkT)</math> just as well as via the product, <math>~(3Mc_s^2)</math>, we see that our expression is identical to Bonnor's if we set the prefactor on Bonnor's last term, <math>~\alpha = (4\pi/3)^{1/3}/5</math>. (Indeed, later on the first page of his paper, Bonnor points out that this is the appropriate value for <math>~\alpha</math> when considering a uniform density sphere.)

	====P-V Diagram====		====P-V Diagram====

Tohline at 23:17, 10 July 2016

2016-07-10T23:17:45Z

← Older revision		Revision as of 23:17, 10 July 2016
Line 411:		Line 411:
	</table>		</table>
	</div>		</div>



			=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>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Quartic Solution */ Comment-out one Whitworth image

2015-05-24T20:52:24Z

Quartic Solution: Comment-out one Whitworth image

← Older revision		Revision as of 20:52, 24 May 2015
Line 404:		Line 404:
	<tr>		<tr>
	<td align="center" bgcolor="white">		<td align="center" bgcolor="white">
	[[File:WhitworthFig1aCopy.jpg\|450px\|center\|Whitworth (1981) Figure 1a]]		<!-- [[File:WhitworthFig1aCopy.jpg\|450px\|center\|Whitworth (1981) Figure 1a]] -->
			[[Image:AAAwaiting01.png\|450px\|center\|Whitworth (1981) Figure 1a]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Bonnor's (1956) Equivalent Relation */ Comment-out Bonnor 1951 image

2015-05-24T20:49:35Z

Bonnor's (1956) Equivalent Relation: Comment-out Bonnor 1951 image

← Older revision		Revision as of 20:49, 24 May 2015
Line 145:		Line 145:
	<table border="2">		<table border="2">
	<tr><td>		<tr><td>
	[[File:Bonnor1951Eq1.2.jpg\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]]		<!-- [[File:Bonnor1951Eq1.2.jpg\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]] -->
			[[Image:AAAwaiting01.png\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]]
	</td></tr>		</td></tr>
	</table>		</table>

Tohline: Create new page to discuss SSC/Virial/Isothermal

2014-07-06T00:55:10Z

Create new page to discuss SSC/Virial/Isothermal

New page

=Virial Equilibrium of Isothermal Spheres=
{{LSU_HBook_header}}

==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 B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} 3B_I ~-~3A\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{1}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,</math>
</td>
</tr>

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

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

==Isolated, Nonrotating Configuration==
For a nonrotating configuration <math>(C=J=0)</math> that is not influenced by the effects of a bounding external medium <math>(D=P_e = 0)</math>, the statement of virial equilibrium is,
<div align="center">
<math>
(1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1} = 0 \, .
</math>
</div>

===Isothermal Evolutions===
For isothermal configurations <math>(\delta_{1\gamma_g} = 1)</math>, one and only one equilibrium state arises where,
<div align="center">
<math>
B_I = A\chi^{-1} \, ,
</math>
</div>
that is,
<div align="center">
<math>
R_\mathrm{eq} = R_0 \chi_\mathrm{eq} = \frac{A}{B_I}\cdot R_0 = \frac{GM}{5c_s^2} \, .
</math>
</div>

==Nonrotating Configuration Embedded in an External Medium==
For a nonrotating configuration <math>(C=J=0)</math> that is embedded in, and is influenced by the pressure <math>P_e</math> of, an external medium, the statement of virial equilibrium is,
<div align="center">
<math>
(1-\delta_{1\gamma_g})~3 B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} 3B_I ~-~3A\chi^{-1} -~ 3D\chi^3 = 0 \, .
</math>
</div>

===Bounded Isothermal===
For isothermal configurations <math>(\delta_{1\gamma_g} = 1)</math>, we deduce that equilibrium states exist at radii given by the roots of the equation,
<div align="center">
<math>
3B_I ~-~3A\chi^{-1} -~ 3D\chi^3 = 0 \, .
</math>
</div>

====Bonnor's (1956) Equivalent Relation====
Inserting the expressions for the coefficients <math>B_I</math>, <math>A</math>, and <math>D</math> gives,
<div align="center">
<math>
3Mc_s^2 ~- \frac{3}{5} \frac{GM^2}{R} = 3 P_e \biggl( \frac{4\pi}{3} R^3\biggr) \, ,
</math>
</div>
or, because the volume <math>V = (4\pi R^3/3)</math> for a spherical configuration, we can write,
<div align="center">
<math>
3P_e V = 3Mc_s^2 ~- \frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/3} \frac{GM^2}{V^{1/3}} \, .
</math>
</div>
It is instructive to compare this expression for a self-gravitating, isothermal equilibrium sphere to the one that was presented in 1956 by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) as equation (1.2) in a paper titled, "Boyle's Law and Gravitational Instability":
<div align="center">
<table border="2">
<tr><td>
[[File:Bonnor1951Eq1.2.jpg|600px|center|Bonnor (1956, MNRAS, 116, 351)]]
</td></tr>
</table>
</div>
Once we realize that, for an isothermal configuration, twice the thermal energy content, <math>2S</math>, can be written as <math>(3NkT)</math> just as well as via the product, <math>(3Mc_s^2)</math>, we see that our expression is identical to Bonnor's if we set the prefactor on Bonnor's last term, <math>\alpha = (4\pi/3)^{1/3}/5</math>. (Indeed, later on the first page of his paper, Bonnor points out that this is the appropriate value for <math>\alpha</math> when considering a uniform density sphere.)

====P-V Diagram====
Returning to the dimensionless form of this expression and multiplying through by <math>[-\chi/(3D)]</math>, we obtain,
<div align="center">
<math>
\chi^4 - \frac{B_I}{D} \chi + \frac{A}{D} = 0 \, .
</math>
</div>
Now, taking a cue from the solution presented above for an isolated isothermal configuration, we choose to set the previously unspecified scale factor, <math>R_0</math>, to,
<div align="center">
<math>
R_0 = \frac{GM}{5c_s^2} \, ,
</math>
</div>
in which case <math>B_I = A</math>, and the quartic equation governing the radii of equilibrium states becomes, simply,
<div align="center">
<math>
\chi^4 - \frac{\chi}{\Pi} + \frac{1}{\Pi} = 0 \, ,
</math>
</div>
where,
<div align="center">
<math>
\Pi \equiv \frac{D}{B_I} = \frac{4\pi R_0^3 P_e}{3Mc_s^2} = \frac{4\pi P_e G^3 M^2}{3\cdot 5^3 c_s^8} \, .
</math>
</div>
For a given choice of <math>P_e</math> and <math>c_s</math>, <math>\Pi^{1/2}</math> can represent a dimensionless mass, in which case,
<div align="center">
<math>
M = \Pi^{1/2} \biggl( \frac{3\cdot 5^3}{2^2\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .
</math>
</div>
Alternatively, for a given choice of configuration mass and sound speed, this parameter, <math>\Pi</math>, can be viewed as a dimensionless external pressure; or, for a given choice of <math>M</math> and <math>P_e</math>, <math>\Pi^{-1/8}</math> can represent a dimensionless sound speed. In most of what follows we will view <math>\Pi</math> as a dimensionless external pressure.

The above quartic equation can be rearranged immediately to give the external pressure that is required to obtain a particular configuration radius, namely,
<div align="center">
<math>
\Pi = \frac{(\chi - 1)}{\chi^4} \, .
</math>
</div>
The resulting behavior is shown by the black curve in Figure 2.

<div align="center">
<table border="2" cellpadding="8">
<tr>
<td align="center" colspan="2">
'''Figure 2:''' <font color="darkblue">Equilibrium Isothermal P-V Diagram </font>
</td>
</tr>
<tr>
<td valign="top" width=450 rowspan="1">
The black curve traces out the function,
<div align="center">
<math>
\Pi = (\chi - 1)/\chi^4 \, ,
</math>
</div>
and shows the dimensionless external pressure, <math>\Pi</math>, that is required to construct a nonrotating, self-gravitating, isothermal sphere with an equilibrium radius <math>\chi</math>. The pressure becomes negative at radii <math>\chi < 1</math>, hence the solution in this regime is unphysical.

[[User:Tohline/SphericallySymmetricConfigurations/Virial#Visual_Representation|Figure 1]] displays the free energy surface that "lies above" the two-dimensional parameter space (<math>1.2 < \chi < 1.51</math>; <math>0.103 < \Pi < 0.104</math>) that is identified here by the thin, red rectangle.
</td>
<td align="center" bgcolor="white">
[[File:Bonnor1956Fig1.jpg|450px|center|Equilibrium P-R Diagram]]
</td>
</tr>

</table>
</div>
In the absence of self-gravity (''i.e.,'' <math>A=0</math>), the product of the external pressure and the volume should be constant. The corresponding relation, <math>\Pi = \chi^{-3}</math>, is shown by the blue dashed curve in the figure. As the figure illustrates, when gravity is included the P-V relationship pulls away from the PV = constant curve at sufficiently small volumes. Indeed, the curve turns over at a finite pressure, <math>\Pi_\mathrm{max}</math>, and for every value of <math>\Pi < \Pi_\mathrm{max}</math> a second, more compact equilibrium configuration appears. The location of <math>\Pi_\mathrm{max}</math> along the curve is identified by setting <math>\partial\Pi/\partial\chi = 0</math>, that is, it occurs where,
<div align="center">
<math>
\frac{\partial\Pi}{\partial\chi} = -4 \chi^{-5}(\chi - 1) + \chi^{-4} = 0 \, ,
</math>

<math>
\Rightarrow ~~~~~ \chi = \frac{2^2}{3} \approx 1.333333 \, .
</math>
</div>
<span id="BonnorEbertMass">Hence,</span>
<div align="center">
<math>\Pi_\mathrm{max} = \biggl( \frac{2^2}{3} \biggr)^{-4} \biggl( \frac{2^2}{3}-1 \biggr) = \frac{3^3}{2^8} \approx 0.105469\, ;</math>
</div>
therefore, from above,
<div align="center">
<math>
M_\mathrm{max} = \biggl( \frac{3^4\cdot 5^3}{2^{10}\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2}
\approx 1.77408 \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .
</math>
</div>

====Quartic Solution====
In the above <math>P-V</math> diagram discussion, we rearranged the quartic equation governing equilibrium configurations to give <math>\Pi</math> for any chosen value of <math>\chi</math>. Alternatively, the four roots of the quartic equation — <math>\chi_1</math>, <math>\chi_2</math>, <math>\chi_3</math> and <math>\chi_4</math> in the presentation that follows — will identify the radii at which a spherical configuration will be in equilibrium for any choice of the external pressure, <math>\Pi</math>, assuming the roots are real.
<div align="center">
<table border="1" cellpadding="10" bgcolor="darkblue">
<tr>
<td align="center" bgcolor="lightblue">
Roots of the quartic equation: <math>\chi^4 - \chi \Pi^{-1}+ \Pi^{-1} = 0 </math>
</td>
</tr>

<tr><td>

<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\chi_1</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
+\frac{1}{2} y_r^{1/2} + \frac{1}{2} D_q \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\chi_2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
+\frac{1}{2} y_r^{1/2} - \frac{1}{2} D_q \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\chi_3</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-\frac{1}{2} y_r^{1/2} + \frac{1}{2} E_q \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\chi_4</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-\frac{1}{2} y_r^{1/2} - \frac{1}{2} E_q \, ,
</math>
</td>
</tr>

</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>D_q</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
y_r^{1/2} \biggl[ \frac{2}{\Pi} y_r^{-3/2} - 1 \biggr]^{1/2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>E_q</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
y_r^{1/2} \biggl[ - \frac{2}{\Pi} y_r^{-3/2} - 1 \biggr]^{1/2} \, ,
</math>
</td>
</tr>

</table>
</div>
and,
<div align="center">
<math>
y_r \equiv \biggl( \frac{1}{2\Pi^2} \biggr)^{1/3} \biggl\{ \biggl[ 1 + \sqrt{1-\frac{2^8}{3^3}\Pi} \biggr]^{1/3} + \biggl[ 1 - \sqrt{1-\frac{2^8}{3^3}\Pi} \biggr]^{1/3} \biggr\} \, ,
</math>
</div>
is the real root of the cubic equation,
<div align="center">
<math>
y^3 - \frac{4y}{\Pi} - \frac{1}{\Pi^{2}} = 0 \, .
</math>
</div>

</td></tr>
</table>
</div>

Because <math>\Pi</math> must be positive in physically realistic solutions, we conclude that the two roots involving <math>E_q</math> — that is, <math>\chi_3</math> and <math>\chi_4</math> — are imaginary and, hence, unphysical. The other two roots — <math>\chi_1</math> and <math>\chi_2</math> — will be real only if the arguments inside the radicals in the expression for <math>y_r</math> are positive. That is, <math>\chi_1</math> and <math>\chi_2</math> will be real only for values of the dimensionless external pressure,
<div align="center">
<math>\Pi \leq \Pi_\mathrm{max} \equiv \frac{3^3}{2^8} \, .</math>
</div>
This is the same upper limit on the external pressure that was derived above, via a different approach, and translates into a maximum mass for a pressure-bounded isothermal configuration of,
<div align="center">
<math>M_\mathrm{max} = \Pi_\mathrm{max}^{1/2} \biggl(\frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2}
= \biggl(\frac{3^4\cdot 5^3}{2^{10}\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2} \, .</math>
</div>

When combined, a plot of <math>\chi_1</math> versus <math>\Pi</math> and <math>\chi_2</math> versus <math>\Pi</math> will reproduce the solid black curve shown in Figure 2, but with the axes flipped. The top-right quadrant of Figure 3 presents such a plot, but in logarithmic units along both axes; also <math>\Pi</math> is normalized to <math>\Pi_\mathrm{max}</math> and <math>\chi</math> is normalized to the equilibrium radius <math>(4/3)</math> at that pressure. This is the manner in which [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) chose to present this result for uniform-density, spherical isothermal <math>(\gamma_\mathrm{g}=1)</math> configurations. Our solid and dashed curve segments — identifying, respectively, the <math>\chi_1(\Pi)</math> and <math>\chi_2(\Pi)</math> solutions to the above quadratic equation — precisely match the solid and dashed curve segments labeled "1" in Whitworth's Figure 1a (replicated here in the bottom-right quadrant of Figure 3).

<div align="center">
<table border="2" cellpadding="8">
<tr>
<td align="center" colspan="2">
'''Figure 3:''' <font color="darkblue">Equilibrium R-P Diagram </font>
</td>
</tr>
<tr>
<td valign="top" width=450 rowspan="2">
''Top:'' The solid curve traces the function <math>\chi_1(\Pi)</math> and the dashed curve traces the function <math>\chi_2(\Pi)</math>, where <math>\chi_1</math> and <math>\chi_2</math> are the two real roots of the quartic equation,
<div align="center">
<math>
\chi^4 - \frac{\chi}{\Pi} + \frac{1}{\Pi} = 0 \, .
</math>
</div>
Logarithmic units are used along both axes; <math>\Pi</math> is normalized to <math>\Pi_\mathrm{max}</math>; and <math>\chi</math> is normalized to the equilibrium radius <math>(4/3)</math> at <math>\Pi_\mathrm{max}</math>.

''Bottom:'' A reproduction of Figure 1a from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967). The solid and dashed segments of the curve labeled "1" identify the equilibrium radii, <math>R_\mathrm{eq}</math>, that result from embedding a uniform-density, isothermal <math>(\gamma_\mathrm{g} = 1)</math> gas cloud in an external medium of pressure <math>P_\mathrm{ex}</math>.

''Comparison:'' The curve shown above that traces out <math>\chi_1(\Pi)</math> and <math>\chi_2(\Pi)</math> should be identical to the "Whitworth" curve labeled "1".
</td>
<td align="center" bgcolor="white">
[[File:WhitworthLogFig1a_norm.jpg|450px|center|To be compared with Whitworth (1981)]]
</td>
</tr>
<tr>
<td align="center" bgcolor="white">
[[File:WhitworthFig1aCopy.jpg|450px|center|Whitworth (1981) Figure 1a]]
</td>
</tr>

</table>
</div>

{{LSU_HBook_footer}}