Editing SSC/Structure/BiPolytropes (section)

==Interface Conditions==
===The Sch&ouml;nberg-Chandrasekhar Condition===
The <math>2^\mathrm{nd}</math> and <math>3^\mathrm{rd}</math> columns of Table 1 provide the mathematical relations that are needed to implement the interface matching conditions specified by equation (1) of {{ SC42 }} (see the above article excerpt) when the core obeys a polytropic equation of state.  For example, in order to ensure that,
<div align="center">
<math>
P(r_i)|_c = P(r_i)|_e \, ,
</math>
</div>
the expression for <math>P</math> given in the <math>2^\mathrm{nd}</math> column and ''evaluated at the interface'' is set equal to the expression for <math>P</math> given in the <math>3^\mathrm{rd}</math> column and ''evaluated at the interface''.  This results in the <math>2^\mathrm{nd}</math> relation shown in the left-hand column of Table 2, namely,
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<math>
K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}_i = K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}_i \, .
</math>
</div>
Likewise, in order to ensure that,
<div align="center">
<math>
M_r(r_i)|_c = M_r(r_i)|_e \, ,
</math>
</div>
the expression for <math>M_r</math> given in the <math>2^\mathrm{nd}</math> column of Table 1 and ''evaluated at the interface'' is set equal to the expression for <math>M_r</math> given in the <math>3^\mathrm{rd}</math> column of Table 1 and ''evaluated at the interface''.  This results in the <math>4^\mathrm{th}</math> relation shown in the left-hand column of Table 2.  The <math>3^\mathrm{rd}</math> relation shown in the left-hand column of Table 2 ensures that the dimensionless coordinate used to identify the surface of the core, <math>\xi_i</math>, and the dimensionless coordinate used to identify the base of the envelope, <math>\eta_i</math>, refer to exactly the same physical (dimensional) radial location, <math>r_i</math>.  It is obtained by setting the expression for <math>r</math> given in the <math>2^\mathrm{nd}</math> column of Table 1 and ''evaluated at the interface'' equal to the expression for <math>r</math> given in the <math>3^\mathrm{rd}</math> column of Table 1 and ''evaluated at the interface''. 

Finally, drawing on the expressions for <math>\rho</math> that are given in the <math>2^\mathrm{nd}</math> and <math>3^\mathrm{rd}</math> columns of Table 1, we note that the <math>1^\mathrm{st}</math> relation shown in the left-hand column of Table 2 ensures that,
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<math>
\frac{\rho(r_i)}{\mu}\biggr|_c = \frac{\rho(r_i)}{\mu}\biggr|_e \, .
</math>
</div>
This relation also ensures that, as desired,
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<math>
T(r_i)|_c = T(r_i)|_e \, ,
</math>
</div>
if, as assumed by {{ SC42 }} (again, see the above article excerpt), <math>T</math> is related to <math>P</math> and <math>\rho</math> through the ideal-gas equation of state.  More specifically, according to the ideal-gas equation of state, 
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<math>
\frac{1}{T} = \biggl[ \frac{H}{kP} \biggr] \frac{\rho}{\mu} \, .
</math>
</div>
Hence, if <math>P</math> is continuous across the interface, as has already been assured, then the continuity of <math>T</math> across the interface will be ensured by guaranteeing the continuity of <math>\rho/\mu</math> across the interface.


<div align="center" id="Table2">
<table border="1" cellpadding="5">
<tr>
  <td align="center" colspan="2">
<font size="+1"><b>Table 2:</b> Interface Conditions</font>
  </td>
</tr>
<tr>
  <td align="center" colspan="1">
<font size="+1" color="darkblue">
'''Polytropic Core'''
</font>
  </td>
  <td align="center">
<font size="+1" color="darkblue">
'''Isothermal Core'''
</font>
  </td>
</tr>

<tr>
  <td align="center">

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<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\biggl( \frac{\rho_0}{\mu_c} \biggr) \theta^{n_c}_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_e}{\mu_e} \biggr) \phi^{n_e}_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>[ (n_c + 1)K_c ]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(\xi^2 \frac{d\theta}{d\xi} \biggr)_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>[ (n_e + 1)K_e ]^{3/2} \rho_e^{(3-n_e)/(2n_e)} \biggl(\eta^2 \frac{d\phi}{d\eta} \biggr)_i</math>
  </td>
</tr>
</table>
<!-- END LEFT BLOCK details -->

  </td>
  <td align="center">

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<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\biggl( \frac{\rho_0}{\mu_c} \biggr) e^{-\psi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_e}{\mu_e} \biggr) \phi^{n_e}_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c_s^2 \rho_0 e^{-\psi_i}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl[ \frac{c_s^2}{4\pi G\rho_0} \biggr]^{1/2} \chi_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl[ \frac{c_s^6}{4\pi G^3\rho_0} \biggr]^{1/2} \biggl( \chi^2 \frac{d\psi}{d\chi} \biggr)_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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)_i</math>
  </td>
</tr>
</table>
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  </td>
</tr>

<tr>
  <td align="left">
After setting <math>\mu_c=\mu_e=1</math>, these four relations become identical to equations 482-485 (p. 172) of [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]].
  </td>
  <td align="left">
After setting <math>n_e = 1</math>, these four relations are essentially identical to, respectively, equations (13), (10), (12), &amp; (11) of {{ Beech88full }}; adopting <math>n_e=3</math> instead, they serve as the foundation of earlier work by {{ HC41full }}.
  </td>
</tr>
</table>
</div>

<span id="TohlineGeneralization">

===The Tohline Generalization===
</span>
[Introduced '''<font color="red">10 June 2013</font>'''] It should be pointed out that, while the interface conditions shown in Table 2 and the solution steps that follow do ensure that the gas pressure is continuous across the interface and allow for a discontinuity in the mass-density across the interface, they do not actually force the temperature to be continuous across the interface.  More generally, pressure continuity is ensured if,
<div align="center">
<math>
\frac{\rho(r_i)}{\mu/T(r_i)}\biggr|_c = \frac{\rho(r_i)}{\mu/T(r_i)}\biggr|_e \, .
</math>
</div>
So a discontinuity across the interface will arise if the ratio of the molecular weights, <math>\mu_c/\mu_e</math>, is not unity, or if there is a discontinuity in the temperature across the interface &#8212; that is, if <math>T(r_i)|_e \ne T(r_i)|_c</math>, or both.  Because they were using bipolytropes to model optically thick stellar interiors, {{ SC42 }} argued that the temperature should also be continuous across the interface and, hence, that a discontinuity in the density would be introduced at the interface from a discontinuity in the molecular weight.  [[SSC/Structure/LimitingMasses#Relationship_Between_the_Bonnor-Ebert_and_Sch.C3.B6nberg-Chandrasekhar_Critical_Masses|In a separate chapter where we discuss the relationship between the Sch&ouml;nberg-Chandrasekhar critical mass and the Bonnor-Ebert critical mass]], we will argue that a discontinuous drop in the density is introduced by a substantial jump in the gas temperature at the interface.  In making this alternate assumption, the structural equations describing the bipolytropic model will remain unchanged; we will only need to replace the ratio <math>\mu_c/\mu_e</math> by the ratio <math>T(r_i)|_e/T(r_i)|_c</math>.