Editing SSC/Dynamics/IsothermalCollapse (section)

===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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<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>
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<math>~=</math>
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<math>~0  \, ,</math>
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<math>~\frac{\partial m}{\partial t} + 4\pi r^2\rho u</math>
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<math>~=</math>
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<math>~0 \, ,</math>
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<math>~\frac{\partial m}{\partial r} - 4\pi r^2 \rho</math>
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<math>~=</math>
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<math>~ 0 \, .  </math>
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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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<math>~\frac{dM_r}{dt}</math>
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<math>~=</math>
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<math>~0 </math>
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<math>~\Rightarrow ~~~ 0</math>
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<math>~=</math>
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<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math>
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