Spherically Symmetric Configurations

Principal Governing Equations

If the self-gravitating configuration that we wish to construct is spherically symmetric, then the coupled set of multidimensional, partial differential equations that serve as our principal governing equations can be simplified to a coupled set of one-dimensional, ordinary differential equations. This is accomplished by expressing each of the multidimensional spatial operators — gradient (<math>\nabla</math>), divergence (<math>\nabla\cdot</math>), and Laplacian (<math>\nabla^2</math>) — in spherical coordinates (<math>r, \theta, \varphi</math>) (see, for example, the Wikipedia discussion of integration and differentiation in spherical coordinates) then setting to zero all derivatives that are taken with respect to the angular coordinates <math>\theta</math> and <math>\varphi</math>. After making this simplification, our governing equations become,

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

Adiabatic Form of the
First Law of Thermodynamics

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</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>

Structure

Equilibrium, spherically symmetric structures are obtained by searching for time-independent solutions to the above set of simplified governing equations. The steady-state flow field that must be adopted to satisfy both a spherically symmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_r v_r = 0</math>. After setting the radial velocity, <math>v_r</math>, and all time-derivatives to zero, we see that the <math>1^\mathrm{st}</math> (continuity) and <math>3^\mathrm{rd}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> (Euler) and <math>4^\mathrm{th}</math> give, respectively,

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

and,

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

(We recognize the first of these expressions as being the statement of hydrostatic balance for spherically symmetric configurations.)

We need one supplemental relation to close this set of equations because there are two equations, but three unknown functions — <math>~P</math>(r), <math>~\rho</math>(r), and <math>~\Phi</math>(r). As has been outlined in our discussion of supplemental relations for time-independent problems, in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between <math>~P</math> and <math>~\rho</math>.

Solution Strategies

When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where <math>~\rho</math> is not a function of <math>\theta</math> or <math>\varphi</math>), the differential mass <math>dm_r</math> that is enclosed within a spherical shell of thickness <math>dr</math> is,

<math>dm_r = \rho dr \oint dS = r^2 \rho dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} d\varphi = 4\pi r^2 \rho dr</math> ,

where we have pulled from the Wikipedia discussion of integration and differentiation in spherical coordinates to define the spherical surface element <math>dS</math>. Integrating from the center of the spherical configuration (<math>r=0</math>) out to some finite radius <math>r</math> that is still inside the configuration gives the mass enclosed within that radius, <math>M_r</math>; specifically,

<math>M_r \equiv \int_0^r dm_r = \int_0^4 4\pi r^2 \rho dr</math> .

We can also state that,

<math>\frac{dm_r}{dr} = 4\pi r^2 \rho </math> ,

which is often referred to as the differential relation that replaces the equation of continuity as a statement of mass conservation in spherically symmetric, static equilibrium structures.

Technique #1

Integrating the

Stability & Dynamics

Stability

SUMMARY: The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically.

Dynamics

SUMMARY: The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically.

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