User:Tohline/SSC/IsothermalCollapse

Collapse of Isothermal Spheres

We begin with the set of time-dependent governing equations for spherically symmetric systems, 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 for the sound speed, then integrating forward in time, in a self-consistent fashion, the following coupled set of equations:

<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>

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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