Editing SSCpt1/PGE

__FORCETOC__ 
<!-- __NOTOC__ will force TOC off -->

=PGE for Spherically Symmetric Configurations=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:lightgreen;" |
<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|One-Dimensional<br />PGEs]]</font>
|}
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 [[PGE#Principal_Governing_Equations|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 &#8212; gradient, divergence, and Laplacian &#8212; in spherical coordinates<sup>&dagger;</sup> <math>(r, \theta, \varphi)</math> 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,
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr}  \biggr] = 0 </math><br />


<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />

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



<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{ Template:Math/EQ_FirstLaw02 }}


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

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

=Footnotes=

<sup>&dagger;</sup>See, for example, the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates].

=See Also=
<ul>
  <li>
Part 2 of ''Spherically Symmetric Configurations'':  Structure &#8212; [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Solution Strategies]]
  </li>
  <li>
Part 2 of ''Spherically Symmetric Configurations'':  Stability &#8212; [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Linearization of Governing Equations]]
  </li>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>


{{ SGFfooter }}