# 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:

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

## Solution

A similarity solution becomes possible for these equations when the single independent variable,

$~\zeta = \frac{c_s t}{r} \, ,$

is used to replace both $~r$ and $~t$. Then, if $~M_r$, $~\rho$, and $~v_r$ assume the following forms,

 $~M_r(r,t)$ $~=$ $~\biggl(\frac{c_s^2 t}{G}\biggr) m(\zeta) \, ,$ $~\rho(r,t)$ $~=$ $~\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) P(\zeta) \, ,$ $~v_r(r,t)$ $~=$ $~- c_s U(\zeta) \, ,$

the three coupled partial differential equations reduce to two coupled ordinary differential equations for the functions, $~P(\zeta)$ and $~U(\zeta)$, namely,

 $~\frac{dU}{d\zeta}$ $~=$ $~ \frac{}{[ (\zeta U +1)^2 - \zeta^2]} \, ,$ $~\frac{dP}{d\zeta}$ $~=$ $~\frac{}{[ (\zeta U +1)^2 - \zeta^2]} \, ,$

and a single equation defining $~m(\zeta)$,

 $~m(\zeta)$ $~=$ $~P \biggl[ U + \frac{1}{\zeta} \biggr] \, .$

# 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