Editing SSCpt2/SolutionStrategies (section)

=Spherically Symmetric Configurations (Part II)=
<!--  [[Image:LSU_Structure_still.gif|74px|left]] -->
Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [[SSCpt1/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, 
&nbsp;<br />
&nbsp;<br />
<div align="center">
<math>~\vec{v} = \hat{e}_r v_r = 0 \, .</math>
</div>

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,

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

<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} </math> ,<br />
</div>
and,
<div align="center">
<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 />
</div>


(We recognize the first of these expressions as being the statement of [[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; {{Math/VAR_Pressure01}}<math>(r)</math>, {{Math/VAR_Density01}}<math>(r)</math>,  and {{Math/VAR_NewtonianPotential01}}<math>(r)</math>. As has been outlined in our discussion of [[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 {{Math/VAR_Pressure01}} and {{Math/VAR_Density01}}.

==Solution Strategies==
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Solution<br />Strategies</b>]]</font>
|}
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/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,

<div align="center">
<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> ,
</div>

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,

<div align="center">
<math>~M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .
</div>

We can also state that,

<div align="center">
{{Math/EQ_SSmassConservation01}}
</div>

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,

<div align="center">
<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>
</div>

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,

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

Substituting this expression into the hydrostatic balance equation gives,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>

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

<div align="center">
<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,
<div align="center">
<math>\frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) \, ,</math>
</div>

which can be used to replace the left-hand-side of the Poisson equation and give,

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

that is, a single second-order governing differential equation which depends only on the two unknown functions, {{Math/VAR_Enthalpy01}} and {{Math/VAR_Density01}}.

<b>Numerical integration examples:</b>
* Isothermal sphere &#8212; 
** [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Emden's (1907) tabulation]].
** Tabulation by [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar &amp; Wares (1949, ApJ, 109, 551)].
** Outline of [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|our numerical integration scheme]].
* Spherical polytropes &#8212;
** [[SSC/Structure/Polytropes#Tabulated_Properties|Some tabulated global properties]].
** Outline of [[SSC/Structure/Polytropes#Straight_Numerical_Integration|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,

<div align="center">
<math>~\frac{d}{dr}(H+\Phi) = 0 \, .</math>
</div>

This means that, throughout our configuration, the functions {{Math/VAR_Enthalpy01}}<math>(\rho)~</math> and {{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,

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

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

<div align="center">
<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, {{Math/VAR_Enthalpy01}}, {{Math/VAR_Density01}}, and {{Math/VAR_NewtonianPotential01}} to one another.


<b>Self-Consistent Field (SCF) Technique:</b>
* Spherical polytropes &#8212;
** [[SSC/Structure/Polytropes#Tabulated_Properties|Some tabulated global properties]].
** Outline of [[SSC/Structure/Polytropes#HSCF_Technique|our implementation of the HSCF scheme]].