User:Tohline/SSC/IsothermalCollapse

Collapse of Isothermal Spheres

We begin with the set of time-dependent governing equations for spherically symmetric systems — as viewed from a Lagrangian frame of reference — namely,

Equation of Continuity

<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr} \biggr] = 0 </math>

Euler Equation

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr} </math>

Poisson Equation

<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho \, ,</math>

but, in place of the adiabatic form of the 1^st Law of Thermodynamics, we enforce isothermality both in space and time by adopting the,

Isothermal Equation of State

<math>~P = c_s^2 \rho \, ,</math>

where, <math>~c_s</math>, is the isothermal sound speed. Recognizing that the differential element of mass that is contained within a spherical shell of width, <math>~dr</math>, at radial location, <math>~r</math>, is,

<math>~dm = 4\pi r^2 \rho dr \, ,</math>

we see that the mass enclosed within radius, <math>~r</math>, is,

<math>~M_r</math>

<math>~=</math>

<math>~4\pi \int_0^4 r^2 \rho dr \, .</math>

Hence, we find from the Poisson equation that,

<math>~\frac{d\Phi}{dr}</math>

<math>~=</math>

<math>~\frac{G}{r^2} \biggl[ 4\pi \int_0^4 r^2 \rho dr \biggr] = \frac{GM_r}{r^2} \, ,</math>

which, when combined with the Euler equation gives the,

Combined Euler + Poisson Equation

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{GM_r}{r^2} \, . </math>

In summary, then, the collapse of an isothermal gas cloud can be modeled by appropriately specifying an initial state and value of the sound speed, then integrating forward in time the following coupled set of equations:

Lagrangian Frame
<math>~P</math> <math>~=</math> <math>~c_s^2 \rho </math> <math>~\frac{dM_r}{dr} </math> <math>~=</math> <math>~4\pi r^2 \rho \, , </math> <math>~\frac{d\rho}{dt} </math> <math>~=</math> <math>~- \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr} \biggr] \, ,</math> <math>~\frac{dv_r}{dt} </math> <math>~=</math> <math>~- \frac{1}{\rho}\frac{dP}{dr} - \frac{GM_r}{r^2} \, .</math>

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

<math>~P</math> <math>~=</math> <math>~c_s^2 \rho </math> <math>~\frac{dM_r}{dr} </math> <math>~=</math> <math>~4\pi r^2 \rho \, , </math> <math>~\frac{\partial \rho}{\partial t} </math> <math>~=</math> <math>~- \biggl[\frac{1}{r^2}\frac{\partial (r^2 \rho v_r)}{\partial r} \biggr] \, ,</math> <math>~\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r}</math> <math>~=</math> <math>~- \frac{1}{\rho}\frac{\partial P}{\partial r} - \frac{GM_r}{r^2} \, .</math>

See Especially

• P. Bodenheimer & A. Sweigart (1968, ApJ, 152, 515):   Dynamic Collapse of the Isothermal Sphere
• 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
• A. Whitworth & D. Summers (1985, MNRAS, 214, 1 - 25):  Self-Similar Condensation of Spherically Symmetric Self-Gravitting Isothermal Gas Clouds
• Prudence N. Foster & Roger A. Chevalier (1993, ApJ, 416, 303):   Gravitational Collapse of an Isothermal Sphere
• A. C. Raga, J. C. Rodríguez-Ramírez, A. Rodríguez-González, V. Lora, & A. Esquivel (2013, Revista Mexicana de Astronomía y Astrofísica, 49, 127-135):   Analytic and Numerical Calculations of the Radial Stability of the Isothermal Spheres

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