User:Tohline/SSC/IsothermalSimilaritySolution
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 | |||||||||
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Notice that, in place of the standard continuity equation, we will use the following equivalent statement of mass conservation:
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<math>~\frac{dM_r}{dt}</math> |
<math>~=</math> |
<math>~0 </math> |
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<math>~\Rightarrow ~~~ 0</math> |
<math>~=</math> |
<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math> |
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<math>~=</math> |
<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math> |
Solution
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,
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<math>~M_r(r,t)</math> |
<math>~=</math> |
<math>~\biggl(\frac{c_s^2 t}{G}\biggr) m(\zeta) \, ,</math> |
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<math>~\rho(r,t)</math> |
<math>~=</math> |
<math>~\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) P(\zeta) \, ,</math> |
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<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>~P(\zeta)</math> and <math>~U(\zeta)</math>, namely,
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<math>~\frac{dU}{d\zeta}</math> |
<math>~=</math> |
<math>~ \frac{}{[ (\zeta U +1)^2 - \zeta^2]} \, , </math> |
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<math>~\frac{dP}{d\zeta}</math> |
<math>~=</math> |
<math>~\frac{}{[ (\zeta U +1)^2 - \zeta^2]} \, ,</math> |
and a single equation defining <math>~m(\zeta)</math>,
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<math>~m(\zeta)</math> |
<math>~=</math> |
<math>~P \biggl[ U + \frac{1}{\zeta} \biggr] \, .</math> |
See Especially
- M. V. Penston (1969, MNRAS, 144, 425): Dynamics of Self-Gravitating Gaseous Sphers - III. Analytic Results in the Free-Fall of Isothermal Cases
- Richard B. Larson (1969, MNRAS, 145, 271): Numerical Calculations of the Dynamics of Collapsing Proto-Star
- F. H. Shu (1977, ApJ, 214, 488-497): Self-Similar Collapse of Isothermal Spheres and Star Formation
- C. Hunter (1977, ApJ, 218, 834-845): The Collapse of Unstable Isothermal Spheres
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© 2014 - 2021 by Joel E. Tohline |