User:Tohline/SSC/Structure/BiPolytropes/Analytic1.5 3

BiPolytrope with <math>n_c = \tfrac{3}{2}</math> and <math>n_e=3</math>

Here we lay out the procedure for constructing a bipolytrope in which the core has an <math>~n_c=\tfrac{3}{2}</math> polytropic index and the envelope has an <math>~n_e=3</math> polytropic index. We will build our discussion around the work of E. A. Milne (1930, MNRAS, 91, 4). While this system cannot be described by closed-form, analytic expressions, it is of particular interest because — as far as we have been able to determine — its examination by Milne represents the first "composite polytrope" to be discussed in the astrophysics literature. In deriving the properties of this model, we will follow the general solution steps for constructing a bipolytrope that are outlined in a separate chapter of this H_Book. That group of general solution steps was drawn largely from chapter IV, §28 of Chandrasekhar's book titled, "An Introduction to the Study of Stellar Structure" [C67], and at the end of that chapter (specifically, p. 182), Chandrasekhar acknowledges that Milne's "method is largely used in § 28." It seems fitting, therefore, that we highlight the features of the specific bipolytropic configuration that E. A. Milne (1930) chose to build.

Steps 2 & 3

Throughout the core, the properties of this bipolytrope can be expressed in terms of the Lane-Emden function, <math>~\theta(\xi)</math>, which derives from a solution of the 2^nd-order ODE,

<math> \frac{1}{\xi^2} \frac{d}{d\xi} \biggl[ \xi^2 \frac{d\theta}{d\xi}\biggr] = - \theta^{3/2} \, , </math>

subject to the boundary conditions,

<math>~\theta = 1</math>       and       <math>~\frac{d\theta}{d\xi} = 0</math>       at       <math>~\xi = 0</math>.

The first zero of the function <math>~\theta(\xi)</math> and, hence, the surface of the corresponding isolated <math>~n=\tfrac{3}{2}</math> polytrope is located at <math>~\xi_s = 3.65375</math> (see Table 4 in chapter IV on p. 96 of [C67]). Hence, the interface between the core and the envelope can be positioned anywhere within the range, <math>~0 < \xi_i < \xi_s = 3.65375</math>.

Step 4: Throughout the core <math>~(0 \le \xi \le \xi_i)</math>

Specify: <math>~K_c</math> and <math>~\rho_0 ~\Rightarrow</math>
<math>~\rho</math>	<math>~=</math>	<math>~\rho_0 \theta^{n_c}</math>	<math>~=</math>	<math>~\rho_0 \theta^{3/2}</math>
<math>~P</math>	<math>~=</math>	<math>~K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}</math>	<math>~=</math>	<math>~K_c \rho_0^{5/3} \theta^{5/2}</math>
<math>~r</math>	<math>~=</math>	<math>~\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>	<math>~=</math>	<math>~\biggl[ \frac{5K_c}{8\pi G} \biggr]^{1/2} \rho_0^{-1/6} \xi</math>
<math>~M_r</math>	<math>~=</math>	<math>~4\pi \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>	<math>~=</math>	<math>~\biggl[ \frac{5^3K_c^3}{2^5 \pi G^3} \biggr]^{1/2} \rho_0^{1/2} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>

Step 5: Interface Conditions

			Setting <math>~n_c=\tfrac{3}{2}</math>, <math>~n_e=3</math>, and <math>~\phi_i = 1 ~~~~\Rightarrow</math>
<math>\frac{\rho_e}{\rho_0}</math>	<math>~=</math>	<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \phi_i^{-n_e}</math>	<math>~=</math>	<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{3/2}_i </math>
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>	<math>~=</math>	<math>\rho_0^{1/n_c - 1/n_e}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} \theta^{1 - n_c/n_e}_i</math>	<math>~=</math>	<math>\rho_0^{1/3}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-4/3} \theta^{1/2}_i</math>
<math>\frac{\eta_i}{\xi_i}</math>	<math>~=</math>	<math>\biggl[ \frac{n_c + 1}{n_e+1} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{(n_c-1)/2} \phi_i^{(1-n_e)/2}</math>	<math>~=</math>	<math>\biggl(\frac{5}{8}\biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{1/4}</math>
<math>\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>	<math>~=</math>	<math>\biggl[ \frac{n_c + 1}{n_e + 1} \biggr]^{1/2} \theta_i^{- (n_c + 1)/2} \phi_i^{(n_e+1)/2} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>	<math>~=</math>	<math>\biggl(\frac{5}{8}\biggr)^{1/2} \theta_i^{- 5/4} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>

Step 8: Throughout the envelope <math>~(\eta_i \le \eta \le \eta_s)</math>

					Knowing: <math>~K_e/K_c</math> and <math>~\rho_e/\rho_0</math> from Step 5   <math>\Rightarrow</math>
<math>~\rho</math>	<math>~=</math>	<math>~\rho_e \phi^{n_e}</math>	<math>~=</math>	<math>~\rho_0 \biggl(\frac{\rho_e}{\rho_0}\biggr) \phi^3</math>	<math>~=</math>	<math>~\rho_0 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{3/2}_i \phi^3</math>
<math>~P</math>	<math>~=</math>	<math>~K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}</math>	<math>~=</math>	<math>K_c \rho_0^{4/3} \biggl(\frac{K_e }{K_c}\biggr) \biggl(\frac{\rho_e}{\rho_0}\biggr)^{4/3} \phi^{4}</math>	<math>=</math>	<math>K_c \rho_0^{5/3} \theta^{5/2}_i \phi^{4}</math>
<math>~r</math>	<math>~=</math>	<math>~\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta</math>	<math>~=</math>	<math>~\biggl[ \frac{K_c}{\pi G} \biggr]^{1/2} \rho_0^{-1/3} \biggl( \frac{K_e}{K_c}\biggr)^{1/2} \biggl( \frac{\rho_e}{\rho_0} \biggr)^{-1/3} \eta</math>	<math>~=</math>	<math>~\biggl[ \frac{K_c}{\pi G} \biggr]^{1/2} \rho_0^{-1/6} \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-1/4} \eta</math>
<math>~M_r</math>	<math>~=</math>	<math>~4\pi \biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{(3-n_e)/(2n_e)} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>	<math>~=</math>	<math>~4\pi \biggl[ \frac{K_c}{\pi G} \biggr]^{3/2} \biggl( \frac{K_e}{K_c}\biggr)^{3/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>	<math>~=</math>	<math>~\biggl( \frac{2^4}{\pi} \biggr)^{1/2} \biggl[ \frac{K_c}{G} \biggr]^{3/2} \rho_0^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-2} \theta_i^{3/4} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>

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