User:Tohline/SSC/IsothermalCollapse
Collapse of Isothermal Spheres
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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 1st 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.
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
- 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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