Editing SSC/Dynamics/IsothermalCollapse (section)

==Governing Relations Adopted in Various Research Publications==

Each of the papers [[#See_Also|referenced below]] presents results from an investigation into the development of nonlinear structure throughout the volume of a dynamically collapsing isothermal sphere.  Here we demonstrate that, in each of these published studies, the coupled set of governing equations is the same as (either the Lagrangian or the Eulerian) set of equations that we have identified, above.  The published results differ from study to study, either because &hellip;
* the adopted initial model configurations and/or adopted boundary conditions differ; or 
* instead of seeking the solution of a ''particular'' initial value problem, the authors choose to examine in broad terms how the structure of the flow ''should'' develop, given the nature of the governing equations.

===Direct Numerical Simulation of the Initial Value Problem===

====Bodenheimer &amp; Sweigart (1968)====
As is detailed at the beginning of &sect;II of their paper, [http://adsabs.harvard.edu/abs/1968ApJ...152..515B Bodenheimer &amp; Sweigart (1968, ApJ, 152, 515)] adopt a ''Lagrangian Frame'' of reference.  The coordinate, <math>~r(r_0, t)</math>, is used to track over time the location of each <math>~M_r(r_0)</math> mass shell whose ''initial'' radial location is, <math>~r_0</math>.  Hence, 
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<math>~v_r = \frac{\partial r}{\partial t} \, ,</math>
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and the combined Euler + Poisson equation may be rewritten as (see their equation 1),
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\rho \frac{\partial^2 r}{\partial t^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{dP}{dr} - \frac{GM_r \rho}{r^2}  \, .</math>
  </td>
</tr>
</table>
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Bodenheimer &amp; Sweigart (1968) recognize that, when viewed from this Lagrangian perspective, integration of the continuity equation would be redundant as conservation of mass is guaranteed by using the differential expression (see their equation 2),
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<table border="0" cellpadding="5" align="center">

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  <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>
</table>
</div>

to track how the mass density varies as the coordinate location of each mass shell, <math>~r</math>, varies.

===Similarity Solution===

====Larson (1969)====
In Appendix C of a seminal paper, [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)] presents an ''asymptotic similarity solution for the isothermal collapse of a sphere.''  He begins by stating that, <font color="darkgreen">in Eulerian form, the equations governing the isothermal collapse may be written</font> (see his equation ''set'' C1),
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + \frac{Gm}{r^2}  + \mathfrak{R} T ~\frac{d\ln\rho}{d r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\frac{\partial m}{\partial t} + 4\pi r^2\rho u</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial m}{\partial r} - 4\pi r^2 \rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0 \, .  </math>
  </td>
</tr>
</table>
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Apart from the different adopted variable ''names'', it is clear that the first and third of these equations have exact counterparts in the set of [[#IsothermalEulerianFrame|"Eulerian Frame" governing equations]] that we have identified, above.  The second equation replaces the continuity equation, providing a different but equally valid statement of mass conservation.  It is most straightforwardly derived by recognizing that the quantity, <math>~m</math>, can be used as a Lagrangian tracer whose (Lagrangian) time-derivative is zero throughout an evolution, then switching to an Eulerian frame of reference.  That is,

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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dM_r}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
</tr>

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  <td align="right">
<math>~\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math>
  </td>
</tr>
</table>
</div>