Editing SSC/Dynamics/IsothermalCollapse

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=Collapse of Isothermal Spheres=
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! style="height: 150px; width: 150px; background-color:#ffff99;;" |[[H_BookTiledMenu#Nonlinear_Dynamical_Evolution|<b>Collapse of<br />Isothermal<br />Spheres</b>]]
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==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,

<div align="center">
<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 />
</div>


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,
<div align="center">

<span id="EOS:Isothermal"><font color="#770000">'''Isothermal Equation of State'''</font></span><p></p>

<math>~P = c_s^2 \rho \, ,</math>
</div>
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,
<div align="center">
<math>~c_s^2 = \frac{\mathfrak{R}T}{\bar\mu} \, ,</math>
</div>
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,
<div align="center">
<math>~dm = 4\pi r^2 \rho dr \, ,</math>
</div>
we see that the mass enclosed within radius, <math>~r</math>, is,
<div align="center">
<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,
<div align="center">
<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 />
</div>

<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>

==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, 
<div align="center">
<math>~v_r = \frac{\partial r}{\partial t} \, ,</math>
</div>
and the combined Euler + Poisson equation may be rewritten as (see their equation 1),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>
</div>

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),
<div align="center">
<table border="0" cellpadding="5" align="center">

<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>
</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),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>

<tr>
  <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>
</div>
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,

<div align="center">
<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>

<tr>
  <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>

=See Also=

* [http://adsabs.harvard.edu/abs/1967LIACo..14..305P  M. V. Penston (1967, &hellip;)]: &nbsp; ''A Numerical Integration of the Nonlinear Equations of Spherical Collapse''
* [http://adsabs.harvard.edu/abs/1966RGOB..117..299P M. V. Penston (1966, Royal Observatory Bulletins. Series E, No. 117)]: &nbsp; ''Dynamics of Self-Gravitating Gaseous Spheres (Part I)'' [Not available digitally]
* [http://adsabs.harvard.edu/abs/1968ApJ...152..515B P. Bodenheimer &amp; A. Sweigart (1968, ApJ, 152, 515)]: &nbsp; ''Dynamic Collapse of the Isothermal Sphere''
* [http://adsabs.harvard.edu/abs/1969MNRAS.145..457P M. V. Penston (1969, MNRAS, 145, 457)]: &nbsp; ''Dynamics of Self-Gravitating Gaseous Sphers - II. Collapses of Gas Spheres With Cooling and the Behavior of Polytropic Gas Spheres''
* [http://adsabs.harvard.edu/abs/1969MNRAS.144..425P M. V. Penston (1969, MNRAS, 144, 425)]: &nbsp; ''Dynamics of Self-Gravitating Gaseous Sphers - III. Analytic Results in the Free-Fall of Isothermal Cases''
* [http://adsabs.harvard.edu/abs/1969MNRAS.145..271L Richard B. Larson (1969, MNRAS, 145, 271)]: &nbsp; ''Numerical Calculations of the Dynamics of Collapsing Proto-Star''
* [http://adsabs.harvard.edu/abs/1977ApJ...214..488S F. H. Shu (1977, ApJ, 214, 488-497)]: &nbsp; ''Self-Similar Collapse of Isothermal Spheres and Star Formation''
* [http://adsabs.harvard.edu/abs/1977ApJ...218..834H C. Hunter (1977, ApJ, 218, 834-845)]: &nbsp; ''The Collapse of Unstable Isothermal Spheres''
* [http://adsabs.harvard.edu/abs/1979ApJ...230..839B J. Buff, H. Gerola, &amp; R. F. Stellingwerf (1979, ApJ, 230, 839-846)]: &nbsp; ''A Predictability Limit for Collapsing Isothermal Spheres''  &#8212; they numerically study properties of eigenvectors near the Bonnor-Ebert limit.
* [http://adsabs.harvard.edu/abs/1985MNRAS.214....1W A. Whitworth &amp; D. Summers (1985, MNRAS, 214, 1 - 25)]:&nbsp; ''Self-Similar Condensation of Spherically Symmetric Self-Gravitting Isothermal Gas Clouds''
* [http://adsabs.harvard.edu/abs/1993ApJ...416..303F Prudence N. Foster &amp; Roger A. Chevalier (1993, ApJ, 416, 303)]: &nbsp; ''Gravitational Collapse of an Isothermal Sphere''
* [http://adsabs.harvard.edu/abs/2013RMxAA..49..127R A. C. Raga, J. C. Rodr&iacute;guez-Ram&iacute;rez, A. Rodr&iacute;guez-Gonz&aacute;lez, V. Lora, &amp; A. Esquivel (2013, Revista Mexicana de Astronom&iacute;a y Astrof&iacute;sica, 49, 127-135)]: &nbsp; ''Analytic and Numerical Calculations of the Radial Stability of the Isothermal Spheres''
* [http://adsabs.harvard.edu/abs/2017ApJ...835…40C E. Coughlin (2017, ApJ, 835, article id. 40)]: &nbsp; ''Spherically Symmetric Cold Collapse:  The Exact Solutions and a Comparison with Self-Similar Solutions''


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