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.

Our Derivation

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>

Milne's (1930) Presentation

Envelope's Equation of State

As has been detailed in our introductory discussion of analytically expressible equations of state and as is summarized in the following table, often the total gas pressure can be expressed as the sum of three separate components: a component of ideal gas pressure, a component of radiation pressure, and a component due to a degenerate electron gas.

Ideal Gas

Degenerate Electron Gas

Radiation

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

	<math>~P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math>
where: <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math>
and:	<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math>

<math>~P_\mathrm{rad} = \frac{1}{3} a_\mathrm{rad} T^4</math>

Milne (1930) considered that the effects of electron degeneracy pressure could be ignored in the envelope of his composite polytrope and, accordingly, employed an expression for the total pressure of the form,

<math>~P_\mathrm{gas} + P_\mathrm{rad} \, .</math>

Milne also introduced the parameter, <math>~\beta</math>, to define the ratio of gas pressure to total pressure in the envelope, that is,

<math>\beta \equiv \frac{P_\mathrm{gas}}{P} \biggr|_\mathrm{env} \, ,</math>

in which case, also,

<math>\frac{P_\mathrm{rad}}{P} \biggr|_\mathrm{env} = 1-\beta </math> and <math>\frac{P_\mathrm{gas}}{P_\mathrm{rad}} \biggr|_\mathrm{env} = \frac{\beta}{1-\beta} \, , </math>

or (see Milne's equation 24),

<math>\biggl( \frac{\mathfrak{\Re}}{\mu_e}\biggr) \rho = \frac{1}{3}a_\mathrm{rad}T^3 \biggl( \frac{\beta}{1-\beta} \biggr) \, .</math>

If the parameter, <math>~\beta</math>, is constant throughout the envelope — which Milne assumes — then this last expression can be interpreted as defining a <math>~T(\rho)</math> function throughout the envelope of the form,

<math>T = \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \rho^{1/3} \, .</math>

Viewing the total pressure in the envelope as having an "ideal gas" form, namely,

with this specific <math>~T(\rho)</math> behavior, allows us to express the envelope's total pressure in terms of the polytropic relation,

with,

<math>~n_e</math>	<math>~=</math>	<math>~3 \, ,</math>
<math>~K_e</math>	<math>~=</math>	<math>~\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \, .</math>

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