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__FORCETOC__


=Virial Equilibrium of Isothermal Spheres=

==Review==
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">
<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 appears as Eq. (1.2) in {{ Bonnor56full }}:

<table border="1" align="center" cellpadding="8" width="60%">
<tr>
<td align="center" colspan="1">
Reprint of the opening (introductory) paragraph from …<br />
{{ Bonnor56figure }}
</td>
</tr>
<tr>
<td align="left" colspan="1">
<font color="darkgreen">"It has recently been suggested by Terletsky<sup>&dagger;</sup> that for a large mass <math>M</math> of gas, of volume <math>V</math> and temperature {{ Math/VAR_Temperature01 }}, containing <math>N</math> molecules under boundary pressure <math>p</math>, the equation of state should be not

<table border="0" align="center" cellpadding="3" width="100%">
<tr>
<td align="right" width="35%">
<math>PV</math>
</td>
<td align="center" width="5%"><math>=</math></td>
<td align="left">
<math>NkT</math>
</td>
<td align="right" width="8%">(1.1)</td>
</tr>
</table>

but
<table border="0" align="center" cellpadding="3" width="100%">
<tr>
<td align="right" width="35%">
<math>PV</math>
</td>
<td align="center" width="5%"><math>=</math></td>
<td align="left">
<math>NkT - \alpha G M^2 V^{-1 / 3} \, ,</math>
</td>
<td align="right" width="8%">(1.2)</td>
</tr>
</table>
where {{ Math/C_BoltzmannConstant }} is Boltzmann's constant, {{ Math/C_GravitationalConstant }} is Newton's constant of gravitation, and <math>\alpha</math> is a constant depending on the shape of the mass. The proposed correction of Boyle's Law arises because, for a large mass, one has to take account of the gravitational interactions between the molecules."</font>
</td>
</tr>
<tr><td align="left">
<sup>&dagger;</sup>Y. P. Terletsky (1952, Zh. Eksper. Teor. Fiz., Vol. 22, p. 506)<br />
----
Notes from J. E. Tohline regarding this referenced article:<br />
<ul>
<li>The full title of this (Russian language) journal is, ''Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki'', sometimes abbreviated as, ''ZhETF''.</li>
<li>English translations of ''ZhETF'' articles dating back to 1967 can be found in the [http://jetp.ras.ru ''Journal of Experimental and Theoretical Physics''] ''(JETP)''; I have been unable to find an English translation (or even the original Russian-language version) of Terletsky's 1952 article.</li>
<li>A more accessible article by [https://ui.adsabs.harvard.edu/abs/1966AZh....43...96G/abstract I. L. Genkin (1966, Astronomicheskii Zhurnal, Vol. 43, p. 96)] heavily references Terletsky's work.</li>
</ul>
</td></tr>
</table>

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 the one derived by {{ Bonnor56 }} if we set the prefactor on his last term, <math>\alpha = (4\pi/3)^{1/3}/5</math>. (Indeed, later on the first page of his paper, {{ Bonnor56 }} 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.

[[SSCpt1/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">
<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>

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

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