# Similarity Solution

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

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
 $~\frac{\partial M_r}{\partial r}$ $~=$ $~4\pi r^2 \rho \, ,$ $~\frac{\partial M_r}{\partial t}$ $~=$ $~- 4\pi r^2 \rho v_r \, ,$ $~\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r}$ $~=$ $~- c_s^2 \biggl( \frac{\partial \ln \rho}{\partial r}\biggr) - \frac{GM_r}{r^2} \, .$

Notice that, in place of the standard continuity equation, we will use the following equivalent statement of mass conservation:

 $~\frac{dM_r}{dt}$ $~=$ $~0$ $~\Rightarrow ~~~ 0$ $~=$ $~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r}$ $~=$ $~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .$

# See Especially

 © 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation