User:Tohline/SSC/IsothermalSimilaritySolution

Similarity Solution

Much of the material in this chapter has been drawn from §4.1 of a review article by Tohline (1982) titled, Hydrodynamic Collapse.

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

Drawing from an accompanying chapter's introductory discussion, we begin with the set of governing equations that describe the collapse of isothermal spheres from an Eulerian frame of reference.

Eulerian Frame
<math>~\frac{\partial M_r}{\partial r} </math> <math>~=</math> <math>~4\pi r^2 \rho \, , </math> <math>~\frac{\partial M_r}{\partial t} </math> <math>~=</math> <math>~- 4\pi r^2 \rho v_r \, ,</math> <math>~\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r}</math> <math>~=</math> <math>~- c_s^2 \biggl( \frac{\partial \ln \rho}{\partial r}\biggr) - \frac{GM_r}{r^2} \, .</math>

Notice that, following Larson's (1969) lead, we have replaced the standard continuity equation with 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>

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,

<math>~M_r(r,t)</math>	<math>~=</math>	<math>~\biggl(\frac{c_s^2 t}{G}\biggr) m(\zeta) \, ,</math>
<math>~\rho(r,t)</math>	<math>~=</math>	<math>~\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) P(\zeta) \, ,</math>
<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,

<math>~\frac{dU}{d\zeta}</math>	<math>~=</math>	<math>~ \frac{}{[ (\zeta U +1)^2 - \zeta^2]} \, , </math>
<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>,

<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

© 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