Editing SSC/Dynamics/FreeFall (section)

==Assembling the Key Relations==

We begin with the [[SSCpt1/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|set of time-dependent governing equations for spherically symmetric systems]], namely,

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<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 />

{{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 />
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By definition, an element of fluid is in "free fall" if its motion in a gravitational field is unimpeded by pressure gradients.  The most straightforward way to illustrate how such a system evolves is to set <math>~P = 0</math> in all of the governing equations.  In doing this, the continuity equation and the Poisson equation are unchanged; the equation formulated by the first law of thermodynamics becomes irrelevant; and the Euler equation becomes, 
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<math>~\frac{dv_r}{dt} =  - \frac{d\Phi}{dr} \, ,</math>
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or, recognizing that <math>~v_r = dr/dt</math>,
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<math>~\frac{d^2r}{dt^2} =  - \frac{d\Phi}{dr} \, .</math>
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