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=Spherically Symmetric Configurations (Part II)=
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Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [[User:Tohline/SphericallySymmetricConfigurations/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|identified 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, 
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<math>~\vec{v} = \hat{e}_r v_r = 0 \, .</math>
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After setting the radial velocity, <math>~v_r</math>, and all time-derivatives to zero, we see that the 1<sup>st</sup> (continuity) and 3<sup>rd</sup> (first law of thermodynamics) equations are trivially satisfied while the 2<sup>nd</sup> (Euler) and 4<sup>th</sup> give, respectively,

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<span id="HydrostaticBalance"><font color="#770000">'''Hydrostatic Balance'''</font></span><br />

<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} </math> ,<br />
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and,
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<span id="Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr)  = 4\pi G \rho </math> .<br />
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(We recognize the first of these expressions as being the statement of [[User:Tohline/PGE/ConservingMomentum#Time-independent_Behavior|hydrostatic balance]] appropriate for spherically symmetric configurations.)  

We need one supplemental relation to close this set of equations because there are two equations, but three unknown functions &#8212; {{User:Tohline/Math/VAR_Pressure01}}<math>(r)~</math>, {{User:Tohline/Math/VAR_Density01}}<math>(r)~</math>,  and {{User:Tohline/Math/VAR_NewtonianPotential01}}<math>(r)~</math>. As has been outlined in our discussion of [[User:Tohline/SR#Time-Independent_Problems|supplemental relations for time-independent problems]] &#8212; and as is discussed further, below &#8212; in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}.

==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 {{User:Tohline/Math/VAR_Density01}} 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,

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<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> ,
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where we have pulled from the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates 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,

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<math>~M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .
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We can also state that,

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{{User:Tohline/Math/EQ_SSmassConservation01}}
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This differential relation is often identified as a statement of mass conservation that replaces the equation of continuity for spherically symmetric, static equilibrium structures.  

===Technique 1===
Integrating the Poisson equation once, from the center of the configuration <math>~(r=0)</math> out to some finite radius, <math>~r</math>, that is still inside the configuration, gives,

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<math> 
~\int_0^r d\biggl( r^2 \frac{d \Phi}{dr} \biggr)  = \int_0^r 4\pi G r^2 \rho dr 
</math><br />

<math> 
\Rightarrow ~~~~~ r^2 \frac{d \Phi}{dr} \biggr|_0^r  = GM_r \, .
</math>
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Now, as long as <math>~d\Phi/dr</math> increases less steeply than <math>~r^{-2}</math> as we move toward the center of the configuration &#8212; indeed, we will find that <math>~d\Phi/dr</math> usually goes smoothly to zero at the center &#8212; the term on the left-hand-side of this last expression will go to zero at <math>~r=0</math>.  Hence, this first integration of the Poisson equation gives,

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<math> 
~\frac{d \Phi}{dr}  = \frac{G M_r}{r^2} \, .
</math>
</div>

Substituting this expression into the hydrostatic balance equation gives,

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{{User:Tohline/Math/EQ_SShydrostaticBalance01}}
</div>

that is, a single governing integro-differential equation which depends only on the two unknown functions, {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}} .


===Technique 2===
As long as we are examining only barotropic structures, we can replace <math>~dP/\rho</math> by <math>~dH</math> in the hydrostatic balance relation to obtain,

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<math>~\frac{dH}{dr} =- \frac{d\Phi}{dr} \, .</math>
</div>

If we multiply this expression through by <math>~r^2</math> then differentiate it with respect to <math>~r</math>, we obtain,
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<math>\frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) \, ,</math>
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which can be used to replace the left-hand-side of the Poisson equation and give,

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<math>~\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =-  4\pi G \rho \, ,</math>
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that is, a single second-order governing differential equation which depends only on the two unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}} and {{User:Tohline/Math/VAR_Density01}}.

<b>Examples:</b>
* [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration|Isothermal sphere]] &#8212; outline of our numerical integration scheme.

===Technique 3===
As in Technique #2, we replace <math>~dP/\rho</math> by <math>~dH</math> in the hydrostatic balance relation, but this time we realize that the resulting expression can be written in the form,

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<math>~\frac{d}{dr}(H+\Phi) = 0 \, .</math>
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This means that, throughout our configuration, the functions {{User:Tohline/Math/VAR_Enthalpy01}}<math>(\rho)~</math> and {{User:Tohline/Math/VAR_NewtonianPotential01}}<math>(\rho)~</math> must sum to a constant value, call it <math>~C_\mathrm{B}</math>.  That is to say, the statement of hydrostatic balance reduces to the ''algebraic'' expression,

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<math>H + \Phi = C_\mathrm{B} \, .</math>
</div>

This relation must be solved in conjunction with the Poisson equation,

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

giving us two equations (one algebraic and the other a <math>2^\mathrm{nd}</math>-order ODE) that relate the three unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_NewtonianPotential01}} to one another.

=See Also=

* Part I of ''Spherically Symmetric Configurations'': [[User:Tohline/SphericallySymmetricConfigurations/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|Simplified Governing Equations]]

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