Editing SSC/Virial/Isothermal (section)

====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 &hellip;<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.)