Editing SSC/Structure/UniformDensity (section)

==Summary==
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From the above derivations, we can describe the properties of a uniform-density, self-gravitating sphere as follows:
* <font color="red">Mass</font>:  
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
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<math>M = \frac{4\pi}{3} \rho_c R^3 </math> ;
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: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
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<math>\frac{M_r}{M} = \biggl(\frac{r}{R} \biggr)^3</math> .
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* <font color="red">Pressure</font>: 
: Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
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<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{\pi}{6} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;<br /> 
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), &sect;5.13, p. 45, Eq. (5.13.4)
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: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
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<math>P(r) = P_c \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
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* <span id="UniformSphereEnthalpy"><font color="red">Enthalpy</font>:</span> 
: Throughout the configuration, the enthalpy is given by the relation,
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<math>H(r) = \frac{P(r)}{ \rho_c} = \frac{GM}{2R} \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
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* <span id="UniformSpherePotential"><font color="red">Gravitational potential</font>: </span>
: Throughout the configuration &#8212; that is, for all <math>r \leq R</math> &#8212; the gravitational potential is given by the relation,
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<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{G M}{2R} \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
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: Outside of this spherical configuration&#8212; that is, for all <math>r \geq R</math> &#8212;  the potential should behave like a point mass potential, that is,
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<math>\Phi(r) = - \frac{GM}{r} </math> .
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: Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a uniform-density, spherically symmetric configuration:
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<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \frac{1}{2}\biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] \biggr\} = - \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .<br />
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), &sect;5.8.9, p. 36, Eq. (5.8.23)
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* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
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<math>M \propto R^3  ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
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* <font color="red">Central- to Mean-Density Ratio</font>:
: Because this is a uniform-density structure, the ratio of its central density to its mean density is unity, that is,
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<math>\frac{\rho_c}{\bar{\rho}} = 1 </math> .
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