User:Tohline/SSC/Structure/PolytropesEmbedded
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
Embedded Polytropic Spheres
In a separate discussion we showed how to determine the structure of isolated polytropic spheres. These are rather idealized stellar structures in which the pressure and density both drop to zero at the surface of the star. Here we consider how the equilibrium radius of a polytropic configuration of a given <math>M</math> and <math>~K_\mathrm{n}</math> is modified when it is embedded in an external medium of pressure <math>P_e</math>. We will begin by focusing on polytropes of index <math>~n</math> = 1 and <math>~n</math> = 5 because their structures can be described by analytic mathematical expressions.
<math>~n</math> = 1 Polytrope
Drawing from the earlier discussion of isolated polytropes, we will reference various radial locations within the spherical configuration by the dimensionless radius,
<math> \xi \equiv \frac{r}{a_\mathrm{n=1}} , </math>
where,
<math> a_\mathrm{n=1} \equiv \biggl[\frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr)_{n=1}\biggr]^{1/2} = \biggl[\frac{K}{2\pi G} \biggr]^{1/2} \, . </math>
Summary
Again, from the earlier discussion, we can describe the properties of an isolated, spherical <math>~n</math> = 1 polytrope as follows:
- Mass:
- In terms of the central density, <math>\rho_c</math>, and <math>~K_\mathrm{n}</math>, the total mass is,
<math>M = \frac{4}{\pi} \rho_c (\pi a_{n=1})^3 = 4\pi^2 \rho_c \biggl[\frac{K}{2\pi G} \biggr]^{3/2} = \rho_c \biggl[\frac{2\pi K^3}{G^3} \biggr]^{1/2}</math> ;
- and, expressed as a function of <math>M</math>, the mass that lies interior to the dimensionless radius <math>\xi</math> is,
<math>\frac{M_\xi}{M} = \frac{1}{\pi} \biggl[ \sin\xi - \xi\cos\xi \biggr] \, ,~~~~~~\mathrm{for}~\pi \ge \xi \ge 0 \, .</math>
- Pressure:
- The central pressure of the configuration is,
<math>P_c = \biggl[ \frac{G^3}{2\pi} \rho_c^4 M^2 \biggr]^{1/3} = \biggl[ \frac{G^3}{2\pi} \rho_c^6 \biggl(\frac{2\pi K^3}{G^3} \biggr) \biggr]^{1/3} = K\rho_c^2</math> ;
- and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<math>P_\xi= P_c \biggl[ \frac{\sin\xi}{\xi} \biggr]^2</math> .
<math>~n</math> = 5 Polytrope
To derive the radial distribution of the Lane-Emden function <math>\Theta_H(r)</math> for an <math>~n</math> = 5 polytrope, we must solve,
<math>\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - (\Theta_H)^5</math> ,
subject to the above-specified boundary conditions. Following Emden (1907), C67 (pp. 93-94) shows that by making the substitutions,
<math> \xi = \frac{1}{x} = e^{-t} \, ; ~~~~~\Theta_H = \biggl(\frac{x}{2}\biggr)^{1/2} z = \biggl(\frac{1}{2}e^t\biggr)^{1/2}z \, , </math>
the differential equation can be rewritten as,
<math> \frac{d^2 z}{dt^2} = \frac{1}{4}z (1 - z^4) \, . </math>
This equation has the solution,
<math> z = \pm \biggl[ \frac{12 C e^{-2t}}{(1 + C e^{-2t})^2} \biggr]^{1/4} \, , </math>
that is,
<math> \Theta_H = \biggl[ \frac{3 C }{(1 + C \xi^2)^2} \biggr]^{1/4} \, . </math>
where <math>C</math> is an integration constant. Because <math>\Theta_H</math> must go to unity when <math>\xi = 0</math>, we see that <math>C=1/3</math>. Hence,
<math> \Theta_H = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1/2} \, . </math>
Related Wikipedia Discussions
|
|
|---|
|
© 2014 - 2021 by Joel E. Tohline |