Editing SSC/Dynamics/IsothermalCollapse (section)

==Establishing Set of Governing Equations==
We begin with the [[SSCpt1/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|set of time-dependent governing equations for spherically symmetric systems]] &#8212; as viewed from a ''Lagrangian'' frame of reference &#8212; namely,

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<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr}  \biggr] = 0 </math><br />


<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr} </math><br />


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho \, ,</math><br />
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but, in place of the adiabatic form of the 1<sup>st</sup> Law of Thermodynamics, we enforce isothermality both in space and time by adopting the,
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<span id="EOS:Isothermal"><font color="#770000">'''Isothermal Equation of State'''</font></span><p></p>

<math>~P = c_s^2 \rho \, ,</math>
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where, <math>~c_s</math>, is the isothermal sound speed.  This is equivalent to adopting an [[SR#Equation_of_State|ideal-gas equation of state]] and setting,
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<math>~c_s^2 = \frac{\mathfrak{R}T}{\bar\mu} \, ,</math>
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then implementing the isothermality condition by holding the gas temperature constant.  Recognizing that the differential element of mass that is contained within a spherical shell of width, <math>~dr</math>, at radial location, <math>~r</math>, is,
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<math>~dm = 4\pi r^2 \rho dr \, ,</math>
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we see that the mass enclosed within radius, <math>~r</math>, is,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \int_0^r r^2 \rho dr \, .</math>
  </td>
</tr>
</table>
</div>
Hence, we find from the Poisson equation that,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\Phi}{dr}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{G}{r^2} \biggl[ 4\pi \int_0^4 r^2 \rho dr \biggr] = \frac{GM_r}{r^2} \, ,</math>
  </td>
</tr>
</table>
</div>
which, when combined with the Euler equation gives the,

<div align="center" id="EulerPoisson">
<span id="PGE:Euler"><font color="#770000">'''Combined Euler + Poisson Equation'''</font></span><br />

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{GM_r}{r^2} \, . </math><br />
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<span id="IsothermalLagrangianFrame">In summary, then, the collapse of an isothermal gas cloud can be modeled by appropriately specifying an initial state and value of the sound speed, then integrating forward in time the following coupled set of equations:</span>
<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">Lagrangian Frame</th>
</tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_s^2 \rho </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{dM_r}{dr} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi r^2 \rho \, ,  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d\rho}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{dv_r}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho}\frac{dP}{dr} - \frac{GM_r}{r^2}  \, .</math>
  </td>
</tr>
</table>

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

<span id="IsothermalEulerianFrame">The lefthand side of the two equations containing time derivatives will take a different form if, alternatively, the choice is made to view the evolution from an ''Eulerian'' frame of reference.  In this case the set of governing equations becomes,</span>
<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">Eulerian Frame</th>
</tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_s^2 \rho </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial M_r}{\partial r} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi r^2 \rho \, ,  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \rho}{\partial t} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[\frac{1}{r^2}\frac{\partial (r^2  \rho v_r)}{\partial r}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho}\frac{\partial P}{\partial r} - \frac{GM_r}{r^2}  \, .</math>
  </td>
</tr>
</table>

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

In either case, we can reduce the number of dependent variables by one &#8212; and, in conjunction, reduce the set of equations by one &#8212; by explicitly replacing <math>~P</math> by the term on the righthand side of the isothermal equation of state, in which case the first term on the righthand side of the Euler + Poisson equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\rho} \frac{dP}{dr}</math>
  </td>
  <td align="center">
<math>~~\rightarrow~~</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^2}{\rho} \biggr) \frac{d\rho}{dr} = c_s^2 \biggl( \frac{d\ln\rho}{dr} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>