Editing SSC/Structure/BiPolytropes (section)

==Setup==

Here we lay the mathematical foundation for building a spherically symmetric structure in which the core and the envelope are described by different barotropic equations of state.  The <math>2^\mathrm{nd}</math> and <math>3^\mathrm{rd}</math> columns of Table 1 provide the relevant set of relations for a structure in which the core and envelope obey polytropic equations of state that have, respectively, polytropic indexes <math>n_c</math> and <math>n_e</math>.  Drawing on our [[SSC/Structure/Polytropes#Lane-Emden_Equation|related discussion of isolated polytropes]], it is clear that the structural profile in both regions should be given by a solution of the
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}
</div>
If either region is governed by an isothermal, rather than polytropic, equation of state then, as we have discussed in the context of both [[SSC/Structure/IsothermalSphere#Isothermal_Sphere|isolated]] and [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|pressure-bounded isothermal spheres]] and as is shown by equation (374) of [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]], the structural profile should be given instead by a solution of the related equation,
<div align="center">
<math>
\frac{1}{\chi^2} \frac{d}{d\chi} \biggl( \chi^2 \frac{d\psi}{d\chi} \biggr) = e^{-\psi} \, .
</math>
</div>
The <math>1^\mathrm{st}</math> column of Table 1 provides the relevant set of associated structural relations for an isothermal core.

<div align="center" id="TableSetup">
<table border="1" cellpadding="5" width="80%">
<tr>
  <td align="center" colspan="3">
<font size="+1"><b>Table 1:</b> Setup</font>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
<font size="+1" color="darkblue">
'''Core''':
</font>
Choose isothermal equation of state or polytropic index <math>n_c</math>
  </td>
  <td align="center">
<font size="+1" color="darkblue">
'''Envelope'''
</font>
  </td>
</tr>

<tr>
  <td align="center">
Isothermal <math>(n_c = \infty)</math>
  </td>
  <td align="center">
<math>n = n_c</math>
  </td>
  <td align="center">
<math>n = n_e</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{1}{\chi^2} \frac{d}{d\chi} \biggl( \chi^2 \frac{d\psi}{d\chi} \biggr) = e^{-\psi}
</math>

sol'n:
<math>
\psi(\chi)
</math>
  </td>
  <td align="center">
<math>
\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\theta}{d\xi} \biggr) = - \theta^{n_c}
</math>

sol'n:
<math>
\theta(\xi)
</math>
  </td>
  <td align="center">
<math>
\frac{1}{\eta^2} \frac{d}{d\eta} \biggl( \eta^2 \frac{d\phi}{d\eta} \biggr) = - \phi^{n_e}
</math>

sol'n:
<math>
\phi(\eta)
</math>
  </td>
</tr>

<tr>
  <td align="center">

<!-- BEGIN LEFT BLOCK details -->
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Specify:  <math>c_s^2</math> and <math>\rho_0 ~\Rightarrow</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 e^{-\psi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_s^2 \rho_0 e^{-\psi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{c_s^2}{4\pi G\rho_0} \biggr]^{1/2} \chi</math>
  </td>
</tr>

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

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

<!-- BEGIN CENTER BLOCK details -->
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Specify:  <math>K_c</math> and <math>\rho_0 ~\Rightarrow</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \theta^{n_c}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
<!-- END CENTER BLOCK details -->

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

<!-- BEGIN RIGHT BLOCK details -->
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Knowing:  <math>K_e</math> and <math>\rho_e ~\Rightarrow</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_e \phi^{n_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</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</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r</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)</math>
  </td>
</tr>
</table>
<!-- END RIGHT BLOCK details -->

  </td>
</tr>
<tr>
  <td align="left" colspan="3">
<font color="red">'''NOTE typed on 13 May 2019:'''</font>&nbsp; Prior to this date, in both the middle and right-hand colums, the RHS of the ''polytropic'' version of the Lane-Emden equation incorrectly had a ''positive'' sign.  This was a type-setting error that has now been corrected.
  </td>
</tr>
</table>
</div>


===Polytropic Core===
Using the notation established in Table 1, if the core obeys a polytropic equation of state, the variable <math>\xi</math> will denote the dimensionless radial coordinate through the core and the relevant solution is a function, <math>\theta(\xi)</math>, whose value goes to <math>1</math> and whose first derivative, <math>d\theta/d\xi</math>, goes to <math>0</math> at <math>\xi = 0</math>.  Then, given a value of the central density, <math>\rho_0</math>, the density throughout the core is,
<div align="center">
<math>\rho(\xi) = \rho_0 \theta(\xi)^{n_c}</math> ;
</div>
and, given a value of the polytropic constant in the core, <math>K_c</math>, the pressure throughout the core is,
<div align="center">
<math>P(\xi) = K_c \rho_0^{1+1/n_c} \theta(\xi)^{n_c + 1}</math>.
</div>
Likewise, given <math>\rho_0</math> and <math>K_c</math>, the radial coordinate <math>r</math> (in dimensional rather than dimensionless units) and the mass enclosed within this radius, <math>M_r</math>, are given by the bottom two expressions shown in the <math>2^\mathrm{nd}</math> column of Table 1.  

The structure of an [[SSC/Structure/Polytropes#Lane-Emden_Equation|isolated polytrope]] would be described by following the function <math>\theta(\xi)</math> all the way out to the surface, that is, to the radial location <math>\xi_\mathrm{surf}</math> where <math>\theta(\xi)</math> first drops to zero.  (Analytic solutions of this type are presented elsewhere for [[SSC/Structure/Polytropes#n_.3D_0_Polytrope|<math>n = 0</math>]], [[SSC/Structure/Polytropes#n_.3D_1_Polytrope|<math>n = 1</math>]], and [[SSC/Structure/Polytropes#n_.3D_5_Polytrope|<math>n = 5</math>]].)  In constructing a bipolytrope, we will instead follow <math>\theta(\xi)</math> out to a radius <math>\xi_i < \xi_\mathrm{surf}</math>, then build an envelope whose inner radius &#8212; or ''base'' &#8212; is at the ''interface'' radius, <math>r_i</math>, that corresponds to <math>\xi_i</math>.  For any choice of the pair of polytropic indexes, <math>n_c</math> and <math>n_e</math>, a series of bipolytropes can then be constructed by choosing a variety of different interface radii.

===Polytropic Envelope===
Again following the notation of Table 1, we will use <math>\eta</math> to identify the dimensionless radial coordinate through the envelope, and the relevant solution of the Lane-Emden equation will be the function, <math>\phi(\eta)</math>.  The ''particular'' solution that we seek for the envelope will differ from the solution obtained in the core not only because the governing polytropic index is different but also because the envelope solution will be constrained by different boundary conditions:  The value of the function <math>\phi(\eta)</math> as well as its first derivative, <math>d\phi/d\eta</math>, must be specified at some radial location within the envelope.  Because the envelope does not extend all the way to the center of the structure, it makes more sense to choose a solution in which <math>\phi</math> is set to unity at the inner edge of the envelope &#8212; that is, at the radial location <math>\eta_i</math> that corresponds to <math>r_i</math> &#8212; rather than at <math>\eta = 0</math>.  By setting <math>\phi = 1</math> at the base of the envelope, it should be clear from the expression, 
<div align="center">
<math>
\rho = \rho_e \phi^{n_e} \, ,
</math>
</div>
(taken from the <math>3^\mathrm{rd}</math> column of Table 1) that <math>\rho_e</math> represents the value of the gas density at the base of the envelope. The value of <math>\rho_e</math> can be obtained from knowledge of the gas density at the outer edge of the core (''i.e.'', at <math>r_i</math>) combined with the specified molecular-weight jump condition at the interface. Looking ahead at the first interface condition for a polytropic core catalogued in Table 2, we see more specifically that, after setting <math>\phi_i = 1</math>,
<div align="center">
<math>
\rho_e = \rho_0 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \, .
</math>
</div> 

As we will show in connection with Table 3, the value of <math>d\phi/d\eta</math> at the base of the envelope (''i.e.'', at <math>r_i</math>) also will be set by the properties of the core  at the interface &#8212; specifically, by the values of <math>(d\theta/d\xi)_i</math> and <math>\theta_i</math>.