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=PGE for Spherically Symmetric Configurations=
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[[H_BookTiledMenu#Spherically_Symmetric_Configurations|One-Dimensional PGEs]]
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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 — gradient, divergence, and Laplacian — in spherical coordinates&dagger; <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,
  
  
  

<div align="center">
'''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> 

Adiabatic Form of the 
'''First Law of Thermodynamics''' 

{{ Template:Math/EQ_FirstLaw02 }}

'''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> 
</div>

=Footnotes=

&dagger;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 — [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Solution Strategies]]
</li>
<li>
Part 2 of ''Spherically Symmetric Configurations'': Stability — [[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>

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