Editing SSC/Perturbations (section)

===Consistent Lagrangian Formulation===

The (Lagrangian) time derivatives in these equations define how a given physical parameter &#8212; for example, {{Math/VAR_Density01}}, <math>v_r</math>, or {{Math/VAR_SpecificInternalEnergy01}} &#8212; should vary with time in a (Lagrangian) fluid element that is not fixed in space but, rather, is moving along with the flow.  However, the radial derivatives describe the spatial variation of various physical parameters as measured at fixed locations in space; that is, as written, the radial derivatives do not track conditions as viewed by a (Lagrangian) fluid element that is moving along with the flow because the position <math>r</math> of each (Lagrangian) fluid element is itself changing with time.  A proper Lagrangian representation of the spatial derivatives can be formulated in the case of one-dimensional, spherically symmetric flows by using <math>M_r</math> (or, equivalently, <math>m_r</math>) instead of <math>r</math> as the independent variable. Making the substitution,
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<math>\frac{d}{dr} = \frac{dM_r}{dr}\frac{d}{dM_r} = 4\pi \rho r^2 \frac{d}{dM_r} \, ,</math> 
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in the first two equations above gives, respectively,

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<math>\frac{d\rho}{dt} = - 4\pi \rho^2 r^2  \frac{dv_r}{dM_r} - \frac{2\rho v_r}{r}  \, ,</math> 
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and,

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<math>\frac{dv_r}{dt} = - 4\pi r^2 \frac{dP}{dM_r} - \frac{GM_r}{r^2} \, .</math>
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