Editing SSC/Structure/OtherAnalyticRamblings (section)

====Compare LAWE to Hydrostatic Balance Condition====

Returning to the [[SSC/Structure/OtherAnalyticModels#Generic|generic formulation derived earlier]], we have,

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<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) x </math>
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<math>~=</math>
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<math>~\biggl(\frac{P_0}{P_c}\biggr)\frac{1}{\chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
+ \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{1}{\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, .
</math>
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Dividing this entire expression through by <math>~(P_0/P_c)x</math> gives,

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<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} </math>
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<math>~=</math>
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<math>~\frac{1}{x \chi_0^4} \frac{d}{d\chi_0}\biggl( \chi_0^4 x^' \biggr) 
+ \frac{d}{d\chi_0}\biggl[ \ln \biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{1}{x\chi_0^\alpha}\frac{d}{d\chi_0}\biggl(\chi_0^\alpha x\biggr) \, .
</math>
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&nbsp;
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<math>~=</math>
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<math>~\biggl( \frac{x^'}{x} \biggr) \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \chi_0^4 x^' \biggr) \biggr]
+ \frac{d}{d\chi_0}\biggl[ \ln \biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl(\chi_0^\alpha x\biggr)\biggr] \, .
</math>
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Now, let's step aside from the LAWE and look directly at the differential relationship between the mass-density and the pressure, as dictated by combining the [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|two principal governing relations]], the

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

<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} </math> ,<br />
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and,
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<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 />
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In combination, we have,
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<math>~-4\pi G \rho_0 </math>
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<math>~=</math>
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<math>~\frac{1}{r_0^2}\frac{d}{dr_0}\biggl[ \frac{r_0^2}{\rho_0} \frac{dP_0}{dr_0}\biggr] </math>
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<math>~\Rightarrow ~~~~ -4\pi G \rho_0 \biggl(\frac{R^2\rho_c}{P_c}\biggr)</math>
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<math>~=</math>
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<math>~\frac{1}{\chi_0^2}\frac{d}{d\chi_0}\biggl[ \frac{\chi_0^2}{(\rho_0/\rho_c)}\cdot \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr] </math>
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<math>~\Rightarrow ~~~~ -[4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)</math>
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<math>~=</math>
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<math>~
\biggl( \frac{\rho_0}{\rho_c} \biggr)^{-1}\frac{1}{\chi_0^2}\frac{d}{d\chi_0}\biggl[ \chi_0^2 \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr] 
+ \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \cdot \frac{d}{d\chi_0}\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}
</math>
  </td>
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<math>~\Rightarrow ~~~~ -[4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)^2\biggl(\frac{P_0}{P_c}\biggr)^{-1}</math>
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<math>~=</math>
  </td>
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<math>~
\frac{1}{\chi_0^2 (P_0/P_c)}\frac{d}{d\chi_0}\biggl[ \chi_0^2 \frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr)\biggr] 
+ \frac{d}{d\chi_0}\biggl[\ln\biggl(\frac{P_0}{P_c}\biggr)\biggr] \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr]
</math>
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&nbsp;
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<math>~=</math>
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<math>~\biggl(\frac{p^'}{p}\biggr)
\frac{1}{\chi_0^2 p^'}\frac{d}{d\chi_0}\biggl[ \chi_0^2 p^'\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr]
</math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{d\ln(p)}{d\chi_0}\cdot
\frac{d}{d\chi_0}\biggl[ \ln(\chi_0^2 p^')\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr] \, ,
</math>
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where,
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<math>~p^'</math>
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<math>~\equiv</math>
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<math>~\frac{d}{d\chi_0}\biggl(\frac{P_0}{P_c}\biggr) \, .</math>
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Let's compare the form of this "equilibrium" relation with the form of the LAWE just constructed:
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<math>~- [4\pi G \rho_c \tau_\mathrm{SSC}^2] \biggl( \frac{\rho_0}{\rho_c} \biggr)^2\biggl(\frac{P_0}{P_c}\biggr)^{-1}</math>
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<math>~=</math>
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<math>~\biggl( \frac{p^'}{p} \biggr) \cdot 
\frac{d}{d\chi_0}\biggl[ \ln(\chi_0^2 p^')\biggr] + \frac{d\ln(p)}{d\chi_0}\cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \frac{\rho_0}{\rho_c}\biggr)^{-1}\biggr] 
</math>
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&nbsp;
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<font size="+1">''versus''</font>
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&nbsp;
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<math>~- \sigma^2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} </math>
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<math>~=</math>
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<math>~\biggl( \frac{x^'}{x} \biggr) \cdot \frac{d}{d\chi_0}\biggl[\ln\biggl( \chi_0^4 x^' \biggr) \biggr]
+ \frac{d\ln(p)}{d\chi_0}\cdot  \frac{d}{d\chi_0}\biggl[\ln\biggl(\chi_0^\alpha x\biggr)\biggr] 
</math>
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</td></tr>
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I like this layout because it unveils similarities in the way the differential operators interact with the functions that describe the radial profiles of variables &#8212; specifically, the mass-density, the pressure, and the fractional radial displacement, <math>~x</math>, during pulsations.  However, it is not yet obvious how best to translate between the two differential equations in order to aid in solving for the unknown variable, <math>~x(\chi_0)</math>.