User:Tohline/SSC/IsothermalCollapse

Collapse of Isothermal Spheres

Establishing Set of Governing Equations

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>

Governing Relations Adopted in Various Research Publications

Each of the papers referenced below presents results from an investigation into the development of nonlinear structure throughout the volume of a dynamically collapsing isothermal sphere. Here we demonstrate that, in each of these published studies, the coupled set of governing equations is the same as (either the Lagrangian or the Eulerian) set of equations that we have identified, above. The published results differ from study to study, either because …

• the adopted initial model configurations and/or adopted boundary conditions differ; or
• instead of seeking the solution of a particular initial value problem, the authors choose to examine in broad terms how the structure of the flow should develop, given the nature of the governing equations.

Bodenheimer & Sweigart (1968)

As is detailed at the beginning of §II of their paper, Bodenheimer & Sweigart (1968, ApJ, 152, 515) adopt a Lagrangian Frame of reference. The coordinate, <math>~r(r_0, t)</math>, varies with time and is used to track the location of each <math>~M_r(r_0)</math> mass shell whose initial radial location is, <math>~r_0</math>. Hence,

<math>~v_r = \frac{\partial r}{\partial t} \, ,</math>

and the Euler equation may be rewritten as (see their equation 1),

<math>~\rho \frac{\partial^2 r}{\partial t^2} </math>

<math>~=</math>

<math>~- \frac{dP}{dr} - \frac{GM_r \rho}{r^2} \, .</math>

Formally, they adopt an ideal-gas equation of state (see their equation 3) and implement the isothermality condition by holding the gas temperature constant, that is to say, they effectively set,

<math>~c_s^2 = \frac{\mathfrak{R}T}{\mu} \,.</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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