User:Tohline/SSC/IsothermalCollapse

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Collapse of Isothermal Spheres

Whitworth's (1981) Isothermal Free-Energy Surface
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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. 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


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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