User:Tohline/SSC/Structure/BiPolytropes/Analytic0 0

BiPolytrope with <math>n_c = 0</math> and <math>n_e=0</math>

Here we construct a bipolytrope in which both the core and the envelope have uniform densities, that is, the structure of both the core and the envelope will be modeled using an <math>n = 0</math> polytropic index. It should be possible for the entire structure to be described by closed-form, analytic expressions. Generally, we will follow the general solution steps for constructing a bipolytrope that we have outlined elsewhere. [On 1 February 2014, J. E. Tohline wrote: This particular system became of interest to me during discussions with Kundan Kadam about the relative stability of bipolytropes.]

Step 4: Throughout the core (<math>0 \le \chi \le \chi_i</math>)

Specify: <math>~P_0</math> and <math>\rho_0 ~\Rightarrow</math>
<math>~\rho</math>	<math>~=</math>	<math>~\rho_0</math>
<math>~P</math>	<math>~=</math>	<math>P_0 - \frac{2}{3} \pi G \rho_0^2 r^2</math>	<math>~=</math>	<math>P_0 \biggl( 1 - \frac{2\pi}{3}\chi^2 \biggr)</math>
<math>~r</math>	<math>~=</math>	<math>\biggl[ \frac{P_0}{G \rho_0^2} \biggr]^{1/2} \chi</math>	<math>~=</math>	<math>\biggl[ \frac{P_0}{G \rho_0^2} \biggr]^{1/2} \chi</math>
<math>~M_r</math>	<math>~=</math>	<math>\frac{4\pi}{3} \rho_0 r^3</math>	<math>~=</math>	<math>\frac{4\pi}{3} \rho_0 \biggl[ \frac{P_0}{G \rho_0^2} \biggr]^{3/2} \chi^3 = \frac{4\pi}{3} \biggl[ \frac{P_0^3}{G^3 \rho_0^4} \biggr]^{1/2} \chi^3</math>

Step 5: Interface Conditions

Specify: <math>~\chi_i</math> and <math>~\rho_e/\rho_0</math>, and demand …
<math>~P_{ei}</math>	<math>~=</math>	<math>~P_{ci}</math>	<math>~=</math>	<math>P_0 \biggl( 1 - \frac{2\pi}{3}\chi_i^2 \biggr)</math>

Step 6: Envelope Solution (<math>~\chi > \chi_i</math>)

<math>~\rho</math>	<math>~=</math>	<math>~\rho_e</math>
<math>~P</math>	<math>~=</math>	<math>P_{ei} + \biggl(\frac{2}{3} \pi G \rho_e\biggr) \biggl[ 2(\rho_0 - \rho_e) r_i^3\biggl( \frac{1}{r} - \frac{1}{r_i}\biggr) - \rho_e(r^2 - r_i^2) \biggr]</math>
	<math>~=</math>	<math>P_{ei} + \frac{2\pi}{3} \biggl(\frac{\rho_e}{\rho_0}\biggr) P_0 \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \chi_i^3\biggl( \frac{1}{\chi} - \frac{1}{\chi_i}\biggr) - \frac{\rho_e}{\rho_0} (\chi^2 - \chi_i^2) \biggr]</math>
<math>~\frac{P}{P_0}</math>	<math>~=</math>	<math>1 - \frac{2\pi}{3}\chi_i^2 + \frac{2\pi}{3} \biggl(\frac{\rho_e}{\rho_0}\biggr) \chi_i^2 \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( \frac{1}{\xi} - 1\biggr) - \frac{\rho_e}{\rho_0} (\xi^2 - 1) \biggr]</math>
<math>~M_r</math>	<math>~=</math>	<math>\frac{4\pi}{3} \biggl[ \rho_0 r_i^3 + \rho_e(r^3 - r_i^3) \biggr]</math>
	<math>~=</math>	<math>\frac{4\pi}{3} \biggl[ \frac{P_0^3}{G^3 \rho_0^4} \biggr]^{1/2} \biggl[\chi_i^3 +\frac{\rho_e}{\rho_0} \biggl( \chi^3 - \chi_i^3 \biggr) \biggr]</math>
	<math>~=</math>	<math>\frac{4\pi}{3} \biggl[ \frac{P_0^3}{G^3 \rho_0^4} \biggr]^{1/2} \chi_i^3\biggl[1 +\frac{\rho_e}{\rho_0} \biggl( \xi^3 - 1\biggr) \biggr]</math>

Related Discussions

• Analytic solution with <math>n_c = 5</math> and <math>n_e=1</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