User:Tohline/SSC/Structure/Other Analytic Models

Other Analytically Definable, Spherical Equilibrium Models

Linear Density Distribution

In an article titled, "Stellar Evolution: A Survey with Analytic Models," R. F. Stein (1966, in Stellar Evolution, Proceedings of an International Conference held at the Goddard Space Flight Center, Greenbelt, MD, U.S.A., edited by R. F. Stein & A. G. W. Cameron, pp. 1-105) defines the "Linear Stellar Model" as a star whose density "varies linearly from the center to the surface," that is (see his equation 3.1),

<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, following our general solution strategy for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,

<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(r) \equiv \frac{G M_r(r) \rho(r)}{r^2} </math>

<math>~=</math>

<math>~\frac{4\pi G \rho_c^2 r}{3} \biggl( 1 - \frac{r}{R} \biggr) \biggl[1 - \frac{3}{4} \biggl( \frac{r}{R} \biggr)\biggr] \, .</math>

Hence, proceeding via what we have labeled as "Technique 1", and enforcing the surface boundary condition, <math>~P(R) = 0</math>, Stein (1966) determines that (see his equation 3.5),

<math>~P(r)</math>	<math>~=</math>	<math>~- \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, it can readily be deduced, as well, that the central pressure is,

<math>~P_c = \frac{5\pi}{36} G\rho_c^2 R^2 \, .</math>

Parabolic Density Distribution

In an article titled, "Radial Oscillations of a Stellar Model," C. Prasad (1949, MNRAS, 109, 103) investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,

<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \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, following our general solution strategy for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,

<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}{5} \biggl( \frac{r}{R} \biggr)^2 \biggr] \, ,</math>

in which case we can write,

<math>~g_0(r) \equiv \frac{G M_r(r) \rho(r)}{r^2} </math>	<math>~=</math>	<math>~\frac{4\pi G \rho_c^2 r}{3} \biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2\biggr] \biggl[1 - \frac{3}{5} \biggl( \frac{r}{R} \biggr)^2\biggr] </math>
	<math>~=</math>	<math>~\frac{4\pi G \rho_c^2 r}{15} \biggl[ 5 - 8\biggl(\frac{r}{R} \biggr)^2 + 3\biggl( \frac{r}{R} \biggr)^4\biggr] \, .</math>

Hence, proceeding via what we have labeled as "Technique 1", and enforcing the surface boundary condition, <math>~P(R) = 0</math>, Prasad (1949) determines that,

<math>~P(r)</math>	<math>~=</math>	<math>~- \int_0^r g_0(r) dr</math>
	<math>~=</math>	<math>~\frac{2\pi G\rho_c^2 R^2}{15} \biggl[2 - 5 \biggl( \frac{r}{R} \biggr)^2 + 4 \biggl( \frac{r}{R} \biggr)^4 - \biggl( \frac{r}{R} \biggr)^6 \biggr] </math>
	<math>~=</math>	<math>~\frac{4\pi G\rho_c^2 R^2}{15} \biggl[1-\biggl(\frac{r}{R}\biggr)^2\biggr]^2 \biggl[1-\frac{1}{2}\biggl(\frac{r}{R}\biggr)^2\biggr] \, ,</math>

where, it can readily be deduced, as well, that the central pressure is,

<math>~P_c = \frac{4\pi}{15} G\rho_c^2 R^2 \, .</math>

© 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