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(→‎Step 8: Throughout the envelope ~(\eta_i \le \eta \le \eta_s): Fix a few mistakes/incompletions in Step 8 expressions)
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Here we lay out the procedure for constructing a [[User:Tohline/SSC/Structure/BiPolytropes#BiPolytropes|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 [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930, MNRAS, 91, 4)].  While this system cannot be described by closed-form, analytic expressions, it is of particular interest because &#8212; as far as we have been able to determine &#8212; 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 [[User:Tohline/SSC/Structure/BiPolytropes#Solution_Steps|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, &sect;28 of Chandrasekhar's book titled, "An Introduction to the Study of Stellar Structure" [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]], and at the end of that chapter (specifically, p. 182), Chandrasekhar acknowledges that Milne's "method is largely used in &sect; 28."  It seems fitting, therefore, that we highlight the features of the ''specific'' bipolytropic configuration that [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)] chose to build.
Here we lay out the procedure for constructing a [[User:Tohline/SSC/Structure/BiPolytropes#BiPolytropes|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 [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930, MNRAS, 91, 4)].  While this system cannot be described by closed-form, analytic expressions, it is of particular interest because &#8212; as far as we have been able to determine &#8212; 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 [[User:Tohline/SSC/Structure/BiPolytropes#Solution_Steps|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, &sect;28 of Chandrasekhar's book titled, "An Introduction to the Study of Stellar Structure" [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]], and at the end of that chapter (specifically, p. 182), Chandrasekhar acknowledges that Milne's "method is largely used in &sect; 28."  It seems fitting, therefore, that we highlight the features of the ''specific'' bipolytropic configuration that [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)] chose to build.


==Steps 2 &amp; 3==
==Our Derivation==
===Steps 2 &amp; 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<sup>nd</sup>-order ODE,
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<sup>nd</sup>-order ODE,
<div align="center">
<div align="center">
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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 [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]]).  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>.
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 [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]]).  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>==
===Step 4:  Throughout the core <math>~(0 \le \xi \le \xi_i)</math>===
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<div align="center">
<table border="0" cellpadding="3">
<table border="0" cellpadding="3">
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</div>
</div>


==Step 5: Interface Conditions==
===Step 5: Interface Conditions===


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<div align="center">
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</div>


==Step 8:  Throughout the envelope <math>~(\eta_i \le \eta \le \eta_s)</math>==
===Step 8:  Throughout the envelope <math>~(\eta_i \le \eta \le \eta_s)</math>===
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<div align="center">
<table border="0" cellpadding="3">
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</table>


</div>
==Milne's (1930) Presentation==
===Envelope's Equation of State===
As has been detailed in our [[User:Tohline/SR#Equation_of_State|introductory discussion of equations of state]] and as is summarized in the following table, often the total gas pressure, <math>~p_\mathrm{env}</math>, can be expressed as being 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.
<table width="95%" align="center" border=1 cellpadding=5>
<tr>
<td align="center" width="25%"><font color="darkblue">Ideal Gas</font></td>
<td align="center"><font color="darkblue">Degenerate Electron Gas</font></td>
<td align="center" width="25%"><font color="darkblue">Radiation</font></td>
</tr>
<tr>
<td align="center">
{{User:Tohline/Math/EQ_EOSideal0A}}
</td>
<td align="center">
{{User:Tohline/Math/EQ_ZTFG01}}
</td>
<td align="center">
{{User:Tohline/Math/EQ_EOSradiation01}}
</td>
</tr>
</table>
[http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)] considered that the effects of electron degeneracy pressure could be ignored in the envelope of his composite polytrope and, accordingly, the expression he used for the total pressure in the envelope was,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~p_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_\mathrm{gas} + P_\mathrm{rad} \, .</math>
  </td>
</tr>
</table>
</div>
</div>



Revision as of 16:14, 27 May 2015

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

Whitworth's (1981) Isothermal Free-Energy Surface
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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 2nd-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 equations of state and as is summarized in the following table, often the total gas pressure, <math>~p_\mathrm{env}</math>, can be expressed as being 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

LSU Key.png

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

LSU Key.png

<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>

LSU Key.png

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

E. A. Milne (1930) considered that the effects of electron degeneracy pressure could be ignored in the envelope of his composite polytrope and, accordingly, the expression he used for the total pressure in the envelope was,

<math>~p_\mathrm{env}</math>

<math>~=</math>

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


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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