Difference between revisions of "User:Tohline/SSC/IsothermalSimilaritySolution"
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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.  Several authors ([[#See_Especiallyreferences 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==  ==Establishing Set of Governing Equations== 
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Similarity Solution
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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
Accompanying chapter discussion …
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,
Eulerian Frame  


Notice that, in place of the standard continuity equation, we will use 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> 
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
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