User:Tohline/SSC/IsothermalSimilaritySolution
Similarity Solution
Much of the material in this chapter has been drawn from §4.1 of a review article by Tohline (1982) titled, Hydrodynamic Collapse.
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Several authors (references given, below) have shown that when isothermal pressure gradients are important during a gas cloud's collapse, the equations governing the collapse admit a set of similarity solutions. Certain properties of these solutions can be described analytically and are instructive models for comparison with more detailed, numerical collapse calculations.
Establishing Set of Governing Equations
Drawing from an accompanying chapter's introductory discussion, we begin with the set of governing equations that describe the collapse of isothermal spheres from an Eulerian frame of reference.
Eulerian Frame  


Notice that, following Larson's (1969) lead, we have replaced the standard continuity equation with the following equivalent statement of mass conservation:
<math>~\frac{dM_r}{dt}</math> 
<math>~=</math> 
<math>~0 </math> 
<math>~\Rightarrow ~~~ 0</math> 
<math>~=</math> 
<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math> 

<math>~=</math> 
<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math> 
Solution
Summary
A similarity solution becomes possible for these equations when the single independent variable,
<math>~\zeta = \frac{c_s t}{r} \, ,</math>
is used to replace both <math>~r</math> and <math>~t</math>. Then, if <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math> assume the following forms,
<math>~M_r(r,t)</math> 
<math>~=</math> 
<math>~\biggl(\frac{c_s^2 t}{G}\biggr) m(\zeta) \, ,</math> 
<math>~\rho(r,t)</math> 
<math>~=</math> 
<math>~\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \, ,</math> 
<math>~v_r(r,t)</math> 
<math>~=</math> 
<math>~ c_s U(\zeta) \, ,</math> 
the three coupled partial differential equations reduce to two coupled ordinary differential equations for the functions, <math>~\Rho (\zeta)</math> and <math>~U(\zeta)</math>, namely,
<math>~\frac{dU}{d\zeta}</math> 
<math>~=</math> 
<math>~ \frac{(\zeta U +1) [\Rho (\zeta U +1) 2)]}{[ (\zeta U +1)^2  \zeta^2]} \, , </math> 
<math>~\frac{dP}{d\zeta}</math> 
<math>~=</math> 
<math>~\frac{\zeta \Rho [2\Rho (\zeta U +1)]}{[ (\zeta U +1)^2  \zeta^2]} \, ,</math> 
and a single equation defining <math>~m(\zeta)</math>,
<math>~m(\zeta)</math> 
<math>~=</math> 
<math>~\Rho \biggl[ U + \frac{1}{\zeta} \biggr] \, .</math> 
The parameters <math>~\zeta, m, \Rho</math>, and <math>~U</math>, and this summary set of equations are exactly those used by Hunter (1977) in his analysis of this problem. But they differ in form from the relations used by Penston (1969), Larson (1969), and Shu (1977) primarily because these authors chose to use a similarity variable,
<math>~x = \pm \frac{1}{\zeta} \, ,</math>
instead of <math>~\zeta</math>. Hunter's analysis is the most complete and his relations will be used here, but a transformation between his presentation and those of the other authors can be easily obtained from Table 1 of Hunter (1977).
Proof
See Especially
 M. V. Penston (1969, MNRAS, 144, 425): Dynamics of SelfGravitating Gaseous Sphers  III. Analytic Results in the FreeFall of Isothermal Cases
 Richard B. Larson (1969, MNRAS, 145, 271): Numerical Calculations of the Dynamics of Collapsing ProtoStar
 F. H. Shu (1977, ApJ, 214, 488497): SelfSimilar Collapse of Isothermal Spheres and Star Formation
 C. Hunter (1977, ApJ, 218, 834845): The Collapse of Unstable Isothermal Spheres
© 2014  2021 by Joel E. Tohline 