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) who, as we shall see, justified these two indexes on physical grounds. 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]. 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.

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Milne's (1930) Choice of Equations 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>

Envelope

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</math>

<math>~=</math>

<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>

Now, returning to the definition of <math>~\beta</math>, we recognize that the total pressure in the envelope can be written in the form of a modified ideal gas relation, namely,

<math>~P = \frac{1}{\beta}\biggl(\frac{\Re}{\mu_e}\biggr) \rho T \, ,</math>

with the specific <math>~T(\rho)</math> behavior just derived. This allows us to write the envelope's total pressure as,

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

which can be immediately associated with a polytropic relation of the form,

<math>~P = K_e \rho^{1 + 1/n_e} \, ,</math>

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^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \, .</math>

So, from the solution, <math>~\phi(\eta)</math>, to the Lane-Emden equation of index <math>~n=3</math>, we will be able to determine that,

<math>~\rho</math>

<math>~=</math>

<math>~\rho_e \phi^3 \, ,</math>

and,

<math>~r</math>

<math>~=</math>

<math>~a_3 \eta \, ,</math>

where — see our general introduction to the Lane-Emden equation —

<math>~a_3^2</math>

<math>~=</math>

<math>~\biggl( \frac{K_e}{\pi G}\biggr) \rho_e^{-2/3} \, .</math>

This is the envelope structure that will be incorporated into our derivation of the bipolytrope's properties, below.

In contrast to this approach, Milne (1930) chose to relate the solution to the envelope's <math>~n=3</math> Lane-Emden equation directly to the temperature via the expression,

<math>T = \lambda \phi \, ,</math>

and deduced that the corresponding radial scale-factor is (see Milne's equation 27),

<math>~a^2_\mathrm{Milne}</math>

<math>~=</math>

<math>~\frac{1}{\lambda^2} \biggl(\frac{\Re}{\mu_e}\biggr)^{2} \frac{(1-\beta)}{\beta^2} \biggl(\frac{3}{\pi a_\mathrm{rad}G}\biggr)\, .</math>

In order to demonstrate the relationship between our radial scale-factor <math>~(a_3)</math> and Milne's, we note that,

<math>~\phi^3</math>	<math>~=</math>	<math>~\biggl(\frac{T}{\lambda}\biggr)^3 = \frac{\rho}{\rho_e}</math>
<math>~\Rightarrow~~~~~\lambda^3</math>	<math>~=</math>	<math>~\rho_e \biggl(\frac{T^3}{\rho}\biggr)</math>
	<math>~=</math>	<math>~\rho_e \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]</math>
<math>~\Rightarrow~~~~~\lambda^{-2}</math>	<math>~=</math>	<math>~\rho_e^{-2/3} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{-2/3} \, .</math>

Hence,

<math>~a^2_\mathrm{Milne}</math>	<math>~=</math>	<math>~\rho_e^{-2/3} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{-2/3}\biggl(\frac{\Re}{\mu_e}\biggr)^{2} \frac{(1-\beta)}{\beta^2} \biggl(\frac{3}{\pi a_\mathrm{rad}G}\biggr)</math>
	<math>~=</math>	<math>~\rho_e^{-2/3} \biggl( \frac{1}{\pi G} \biggr) \biggl[ \biggl(\frac{\Re}{\mu_e}\biggr)^{4} \frac{(1-\beta)}{\beta^{4}} \biggl(\frac{3}{a_\mathrm{rad}}\biggr) \biggr]^{1/3}</math>
	<math>~=</math>	<math>~\rho_e^{-2/3} \biggl( \frac{K_e}{\pi G} \biggr) \, .</math>

It is clear, therefore, that the two radial scale-factors are the same.

Core

In contrast to the envelope, Milne (1930) assumed that the (non-relativistic) electron degeneracy pressure dominates over the other two pressure contributions in the core. As we have demonstrated elsewhere, the non-relativistic (NR) expression for the degeneracy pressure is,

<math>~P_\mathrm{deg}\biggr|_\mathrm{NR}</math>

<math>~=</math>

<math>~\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl[ \frac{\rho}{(\mu_{e^-}) m_p}\biggr]^{5/3} \, ,</math>

which can be associated with a polytropic relation of the form,

<math>~P = K_c \rho^{1 + 1/n_c} \, ,</math>

with,

<math>~n_c</math>	<math>~=</math>	<math>~\frac{3}{2} \, ,</math>
<math>~K_c</math>	<math>~=</math>	<math>~\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl[ \frac{1}{(\mu_{e^-}) m_p}\biggr]^{5/3} \, .</math>

(Note that, here only, we have used the parameter, <math>~\mu_{e^-}</math>, to denote the molecular weight of electrons, instead of just <math>~\mu_e</math>, in order not to confuse it with the mean molecular weight assigned, below, to the envelope material.) So, from the solution, <math>~\theta(\xi)</math>, to the Lane-Emden equation of index <math>~n=\tfrac{3}{2}</math>, we will be able to determine that,

<math>~\rho</math>

<math>~=</math>

<math>~\rho_0 \theta^{3/2} \, ,</math>

and,

<math>~r</math>

<math>~=</math>

<math>~a_{3/2} \xi \, ,</math>

where — see our general introduction to the Lane-Emden equation —

<math>~(a_{3/2})^2</math>

<math>~=</math>

<math>~\biggl( \frac{5K_c}{2^3\pi G}\biggr) \rho_0^{-1/3} \, .</math>

This is the core structure that will be incorporated into our derivation of the bipolytrope's properties, below.

This is precisely the approach taken by Milne (1930). Just before his equation (43), Milne states that the equation of state that is relevant for the core is,

<math>~p</math>

<math>~=</math>

<math>~\frac{1}{5}\biggl( \frac{3}{4\pi}\biggr)^{2/3} \frac{h^2}{(2m_H)^{5/3} m_e q_e^{2/3} }~ \rho^{5/3} \, ,</math>

which, upon regrouping terms gives,

<math>~p</math>

<math>~=</math>

<math>~\frac{1}{5}\biggl( \frac{3}{\pi}\biggr)^{2/3}\biggl( \frac{1}{2^4 q_e^2 }\biggr)^{1/3} \frac{h^2}{m_e }\biggl( \frac{\rho}{2 m_H} \biggr)^{5/3} \, .</math>

Recognizing that Milne set <math>~q_e = 2</math>, as "the statistical weight of an electron," and that he adopted a molecular weight of the electrons, <math>~\mu_{e^-}=2</math>, this expression for the equation of state of the core exactly matches our expression for <math>~P_\mathrm{deg}|_\mathrm{NR}</math>.

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>

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