Editing SSCpt1/Virial (section)

=====Looking Ahead to Bipolytropes=====
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<b><font color="purple">ASIDE:</font></b> When we discuss the free energy of bipolytropic configurations, we will need to divide the expression for <math>\mathfrak{S}_A/E_\mathrm{norm}</math> into two parts &#8212; one accounting for the reservoir of thermodynamic energy in the bipolytrope's "core" and one accounting for the reservoir of thermodynamic energy in the bipolytrope's "envelope."  It is useful to develop this two-part expression here, while the definition of <math>\mathfrak{S}_A</math> is fresh in our minds and to show how the two-part expression reduces to the simpler expression for <math>\mathfrak{S}_A/E_\mathrm{norm}</math>, just derived, when there is no distinction drawn between the properties of the core and the envelope. 

In what follows, we will use the subscript ''core'' (or "c") when referencing physical properties of the bipolytrope's core and the subscript ''env'' (or "e") for the envelope; and, as above, we will use <math>x \equiv r/R_\mathrm{edge}</math> to denote the dimensionless radial location within a configuration of radius, <math>R_\mathrm{edge}</math>.  The dimensionless radial coordinate, <math>q \equiv x_i = r_i/R_\mathrm{edge}</math>, will identify the radial ''interface'' where the core meets the envelope; that is, <math>q</math> will identify both the outer edge of the core and the inner edge of the envelope.  In general, separate expressions will define the run of pressure through the core and through the envelope.  We can assume that, for the core, the pressure drops monotonically from a value of <math>P_0</math> at the center of the configuration according to an expression of the form,
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<math>P_\mathrm{core}(x) = P_0 [1 - p_c(x)]</math> &nbsp; &nbsp;&nbsp; for &nbsp; &nbsp;&nbsp; <math>0 \leq x \leq q \, ,</math>
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and that, for the envelope, the pressure drops monotonically from a value of <math>P_{ie}</math> at the interface according to an expression of the form,
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<math>P_\mathrm{env}(x) = P_{ie} [1 - p_e(x)]</math> &nbsp; &nbsp;&nbsp; for &nbsp; &nbsp;&nbsp; <math>q \leq x \leq 1 \, ,</math>
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where <math>p_c(x)</math> and <math>p_e(x)</math> are both dimensionless functions that will depend on the equations of state that are chosen for the core and envelope, respectively.  By prescription, the pressure in the envelope must drop to zero at the surface of the bipolytropic configuration, hence, we should expect that <math>p_e(1) = 1</math>.  Furthermore, by prescription, the pressure in the core will drop to a value, <math>P_{ic}</math>, at the interface, so we can write,
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<math>P_{ic} = P_0 [1 - p_c(q)] \, .</math>
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In equilibrium &#8212; that is, when <math>R_\mathrm{edge} = R_\mathrm{eq}</math> &#8212; we will demand that the pressure at the interface be the same, whether it is referenced in the core or in the envelope, that is, we will demand that <math>P_{ic} = P_{ie} \, .</math>  It will therefore prove to be strategically advantageous to rewrite the expression for the run of pressure through the core in terms of the pressure at the interface rather than in terms of the central pressure; specifically,
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<math>P_\mathrm{core}(x) = P_{ic} \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] \, .</math> 
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Referencing these prescriptions for <math>P_\mathrm{core}(x)</math> and <math>P_\mathrm{env}(x)</math>, the two-part expression for the reservoir of thermodynamic energy is,
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<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
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<math>=</math>
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<math>
\frac{1}{({\gamma_c}-1)}  \int_0^{r_i/R_\mathrm{norm}}  4\pi (r^*)^2 P^*_\mathrm{core} dr^*
+ \frac{1}{({\gamma_e}-1)}  \int_{r_i/R_\mathrm{norm}}^\chi  4\pi (r^*)^2 P^*_\mathrm{env} dr^* 
</math>
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&nbsp;
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<math>=</math>
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<math>
\frac{4\pi \chi^3 }{({\gamma_c}-1)} \biggl[ \frac{P_{ic}}{P_\mathrm{norm}} \biggr] \int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx
+ \frac{4\pi \chi^3 }{({\gamma_e}-1)} \biggl[ \frac{P_{ie}}{P_\mathrm{norm}} \biggr]  \int_q^1  \biggl[1 - p_e(x) \biggr]  x^2 dx \, .
</math>
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As is implied by the subscripts on the adiabatic exponents that appear in the leading factor of each of the two terms, we are assuming that, as the bipolytropic system expands or contracts, the thermodynamic properties of the material in the envelope will vary as prescribed by an adiabat of index, <math>\gamma_e</math>, while the thermodynamic properties of material in the core will vary as prescribed by a, generally different, adiabat of index, <math>\gamma_c</math>.  Therefore, as the radius of the bipolytropic configuration, <math>R_\mathrm{edge}</math>, is varied, the density of each fluid element will vary and, in the core, the pressure of each fluid element will vary as <math>P \propto \rho^{\gamma_c}</math> while, in the envelope, the pressure of each fluid element will vary as <math>P \propto \rho^{\gamma_e}</math>.  If we furthermore assume that the mass in the core and the mass in the envelope remain constant during a phase of contraction or expansion, the density of each fluid element will vary as <math>R_\mathrm{edge}^{-3}</math>, whether the material is associated with the core or with the envelope.  Therefore, using the subscript, "eq," to identify the value of thermodynamic quantities when the system is in an equilibrium state and, accordingly, <math>R_\mathrm{edge} = R_\mathrm{eq}</math>, we can write,
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<math>\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{core}</math>
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<math>=</math>
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<math>\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_c} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c} \, ,</math>
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and,
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<math>\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{env}</math>
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<math>=</math>
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<math>\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_e} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e} \, .</math>
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In particular, for any <math>R_\mathrm{edge}</math>, material associated with the core that lies at the interface will have a pressure given by the relation,
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<math>P_{ic}</math>
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<math>=</math>
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<math>
(P_{ic})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c} 
= (P_{ic})_\mathrm{eq} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{+3\gamma_c}\biggl( \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)^{-3\gamma_c} 
= (P_{ic})_\mathrm{eq} \chi_\mathrm{eq}^{+3\gamma_c} \chi^{-3\gamma_c} 
\, ,</math>
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while material associated with the envelope that lies at the interface will have a pressure given by the relation,
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<math>P_{ie}</math>
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<math>=</math>
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<math>
(P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e} 
= (P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{+3\gamma_e}\biggl( \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)^{-3\gamma_e} 
= (P_{ie})_\mathrm{eq} \chi_\mathrm{eq}^{+3\gamma_e} \chi^{-3\gamma_e} 
\, .</math>
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Hence,
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<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
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<math>=</math>
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<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx
</math>
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&nbsp;
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&nbsp;
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<math>
+ ~\frac{4\pi }{({\gamma_e}-1)}  \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_e} 
\int_q^1  \biggl[1 - p_e(x) \biggr]  x^2 dx \, .
</math>
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----


Now, let's see how this expression simplifies if <math>P_{ie} = P_{ic}</math> and <math>\gamma_e = \gamma_c</math> and, hence, the properties of the envelope are indistinguishable from the properties of the core.  We note, first, that in this limit, <math>P_\mathrm{core}(x)</math> and <math>P_\mathrm{env}(x)</math> must be identical functions of <math>x</math>, that is, it must be the case that <math>p_e(x)</math> is related to <math>p_c(x)</math> via the relation,
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<math>1 - p_e(x) </math>
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<math>=</math>
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<math>\frac{1 - p_c(x)}{1-p_c(q)} \, .</math>
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We therefore obtain,
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<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
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<math>=</math>
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<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\biggl\{ \int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx + \int_q^1  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx \biggr\}
</math>
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&nbsp;
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<math>=</math>
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<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_0 \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\biggl\{ \int_0^1  \biggl[1 - p_c(x)\biggr]  x^2 dx  \biggr\}
</math>
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&nbsp;
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<math>=</math>
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<math>
\frac{4\pi }{({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma}
\int_0^{1}  \biggl[ \frac{P(x)}{P_c} \biggr] x^2 dx \biggr\} \, ,
</math>
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as desired.

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