Editing SSC/Structure/UniformDensity (section)

=Uniform-Density Sphere Embedded in an External Medium=
For the ''isolated'' uniform-density sphere, discussed above, the surface of the configuration was identified as the radial location where the pressure drops to zero.  Here we embed the sphere in a hot, tenuous medium that exerts a confining "external" pressure, <math>~P_e</math>, and ask how the configuration's equilibrium radius, <math>~R_e</math>, changes in response to this applied external pressure, for a given (fixed) total mass and central pressure, <math>~P_c</math>. 

Following [[SSC/Structure/UniformDensity#Solution_Technique_1|solution technique #1]], the derivation remains the same up through the integration of the hydrostatic balance equation to obtain the relation,
<div align="center">
<math>P  = P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</div>
Now we set <math>~P = P_e</math> at the surface of our spherical configuration &#8212; that is, at <math>~r=R_e</math> &#8212; so we can write,
<div align="center">
<math>P_c - P_e = \frac{2\pi G}{3} \rho_c^2 R_e^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr)</math>

<math>\Rightarrow ~~~~~ P_c \biggl( 1 - \frac{P_e}{P_c} \biggr) = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr) \, ,</math>
</div>
where <math>M</math> is the total mass of the configuration.  Solving for the equilibrium radius, we have,
<div align="center">
<math> R_e = \biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, .</math>
</div>
As it should, when the ratio <math>~P_e/P_c \rightarrow 0</math>, this relation reduces to the one obtained, above, for the isolated uniform-density sphere, namely,
<div align="center">
<math> R_e^4 = \biggl( \frac{3}{8\pi} \biggr) \frac{G M^2}{P_c}  \, .</math>
</div>

{{ SGFfooter }}