Editing SSC/Structure/BiPolytropes (section)

==Solution Steps==

* Step 1:  Choose <math>n_c</math> and <math>n_e</math>.
* Step 2:  Adopt boundary conditions at the center of the core (<math>\theta = 1</math> and <math>d\theta/d\xi = 0</math> at <math>\xi=0</math>), then solve the Lane-Emden equation to obtain the solution, <math>\theta(\xi)</math>, and its first derivative, <math>d\theta/d\xi</math> throughout the core; the radial location, <math>\xi = \xi_s</math>, at which <math>\theta(\xi)</math> first goes to zero identifies the natural surface of an [[SSC/Structure/Polytropes#Lane-Emden_Equation|isolated polytrope]] that has a polytropic index <math>n_c</math>.
* Step 3  Choose the desired location, <math>0 < \xi_i < \xi_s</math>, of the outer edge of the core. 
* Step 4:  Specify <math>K_c</math> and <math>\rho_0</math>; the structural profile of, for example, <math>\rho(r)</math>, <math>P(r)</math>, and <math>M_r(r)</math> is then obtained throughout the core &#8212; over the radial range, <math>0 \le \xi \le \xi_i</math> and <math>0 \le r \le r_i</math> &#8212; via the relations shown in the <math>2^\mathrm{nd}</math> column of Table 1.
* Step 5:  Specify the ratio <math>\mu_e/\mu_c</math> and adopt the boundary condition, <math>\phi_i = 1</math>; then use the interface conditions as rearranged and presented in Table 3 to determine, respectively:
** The gas density at the base of the envelope, <math>\rho_e</math>;
** The polytropic constant of the envelope, <math>K_e</math>, relative to the polytropic constant of the core, <math>K_c</math>;
** The ratio of the two dimensionless radial parameters at the interface, <math>\eta_i/\xi_i</math>;
** The radial derivative of the envelope solution at the interface, <math>(d\phi/d\eta)_i</math>.
* Step 6:  The last sub-step of solution step 5 provides the boundary condition that is needed &#8212; in addition to our earlier specification that <math>\phi_i = 1</math> &#8212; to derive the desired ''particular'' solution, <math>\phi(\eta)</math>, of the Lane-Emden equation that is relevant throughout the envelope; knowing <math>\phi(\eta)</math> also provides the relevant structural first derivative, <math>d\phi/d\eta</math>, throughout the envelope.
* Step 7:  The surface of the bipolytrope will be located at the radial location, <math>\eta = \eta_s</math> and <math>r=R</math>, at which <math>\phi(\eta)</math> first drops to zero.
* Step 8:  The structural profile of, for example, <math>\rho(r)</math>, <math>P(r)</math>, and <math>M_r(r)</math> is then obtained throughout the envelope &#8212; over the radial range, <math>\eta_i \le \eta \le \eta_s</math> and <math>r_i \le r \le R</math> &#8212; via the relations provided in the <math>3^\mathrm{rd}</math> column of Table 1.


<div align="center" id="Table3">
<table border="1" cellpadding="5">
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<font size="+1"><b>Table 3:</b> Sub-steps of Solution Step 5</font> <br>
(derived from the relations in Table 2)
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<font size="+1" color="darkblue">
'''Polytropic Core'''
</font>
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<font size="+1" color="darkblue">
'''Isothermal Core'''
</font>
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<math>\frac{\rho_e}{\rho_0}</math>
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<math>=</math>
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<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \phi_i^{-n_e}</math>
  </td>
</tr>

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<math>\biggl( \frac{K_e}{K_c} \biggr) \rho_0^{1/n_e - 1/n_c}</math>
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<math>=</math>
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<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} \theta^{1 - n_c/n_e}_i</math>
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<math>\frac{\eta_i}{\xi_i}</math>
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<math>=</math>
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<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>
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<math>\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
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<math>=</math>
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<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>
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</table>
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  </td>
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<math>\biggl( \frac{\rho_e}{\rho_0} \biggr)</math>
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<math>=</math>
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<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) e^{-\psi_i} \phi_i^{-n_e}</math>
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<math>\frac{K_e \rho_0^{1/n_e} }{c_s^2} </math>
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<math>=</math>
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<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} e^{+\psi_i/n_e} </math>
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<math>\frac{\eta_i}{\chi_i}</math>
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<math>=</math>
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<math>(n_e + 1)^{-1/2}\biggl( \frac{\mu_e}{\mu_c} \biggr) e^{-\psi_i/2} \phi_i^{(1-n_e)/2}</math>
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<math>- \biggl(\frac{d\phi}{d\eta} \biggr)_i</math>
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<math>=</math>
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<math>(n_e + 1)^{-1/2} e^{+\psi_i/2} \phi_i^{(n_e+1)/2} \biggl(\frac{d\psi}{d\chi} \biggr)_i</math>
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<span id="UVplane">
Taking the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions on the left-hand side of Table 3 produces,</span>
<div align="center">
<math>
\frac{\eta_i \phi_i^{n_e}}{(d\phi/d\eta)_i} = \frac{\xi_i \theta_i^{n_c}}{(d\theta/d\xi)_i} \biggl( \frac{\mu_e}{\mu_c}\biggr) \, .
</math>
</div>
''Multiplying'' the <math>3^\mathrm{rd}</math> expression by the <math>4^\mathrm{th}</math> expression generates,
<div align="center">
<math>
(n_e+1)\frac{\eta_i (d\phi/d\eta)_i}{ \phi_i } = (n_c+1)\frac{\xi_i (d\theta/d\xi)_i}{ \theta_i } \biggl( \frac{\mu_e}{\mu_c}\biggr) \, .
</math>
</div>
These are two relations that Chandrasekhar found to be useful in his analysis of "composite [polytropic] configurations" &#8212; after setting <math>\mu_e=\mu_c</math> they match, respectively, equations 486 &amp; 489 in &sect; 28 of [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]].