User:Tohline/SSC/Structure/Other Analytic Models
Other Analytically Definable, Spherical Equilibrium Models
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Linear Density Distribution
R. F. Stein (1966) defines the "Linear Stellar Model" as a star whose density "varies linearly from the center to the surface," that is,
<math>\rho(r) = \rho_c\biggl( 1 - \frac{r}{R} \biggr) \, ,</math>
where, <math>~\rho_c</math> is the central density and, <math>~R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically,
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<math>~M_r(r)</math> |
<math>~=</math> |
<math>~\int_0^r 4\pi r^2 \rho(r) dr</math> |
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<math>~=</math> |
<math>~\frac{4\pi\rho_c r^3}{3} \biggl[1 - \frac{3}{4} \biggl( \frac{r}{R} \biggr)\biggr] \, ,</math> |
in which case we can write,
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<math>~g_0 \equiv \frac{G M_r(r) \rho(r)}{r^2} </math> |
<math>~=</math> |
<math>~\frac{4\pi\rho_c^2 r}{3} \biggl( 1 - \frac{r}{R} \biggr) \biggl[1 - \frac{3}{4} \biggl( \frac{r}{R} \biggr)\biggr] \, .</math> |
Hence,
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<math>~P(r)</math> |
<math>~=</math> |
<math>~P_c - \int_0^r g_0(r) dr</math> |
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<math>~=</math> |
<math>~\frac{\pi G\rho_c^2 R^2}{36} \biggl[5 - 24 \biggl( \frac{r}{R} \biggr)^2 + 28 \biggl( \frac{r}{R} \biggr)^3 - 9 \biggl( \frac{r}{R} \biggr)^4 \biggr] \, ,</math> |
where, we have enforced the surface boundary condition, <math>~P(R) = 0</math>, and the central pressure is,
<math>~P_c = \frac{5\pi}{36} G\rho_c^2 R^2 \, .</math>
Parabolic Density Distribution
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