User:Tohline/SSC/Structure/Other Analytic Models

Other Analytically Definable, Spherical Equilibrium Models

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,

<math>~M_r(r)</math>	<math>~=</math>	<math>~\int_0^r 4\pi r^2 \rho(r) dr</math>
	<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,

<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,

<math>~P(r)</math>	<math>~=</math>	<math>~P_c - \int_0^r g_0(r) dr</math>
	<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

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation