Editing SSC/Stability/BiPolytropes (section)

=Equilibrium Models=

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The equilibrium sequences displayed in Figure 1 &#8212; for pressure-truncated polytropes &#8212; and in Figure 2 &#8212; for <math>(n_c, n_e) = (5, 1)</math> bipolytropes &#8212; were constructed by solving the second-order ODE that governs detailed force-balance throughout spherically symmetric configurations that obey polytropic equations of state.  In Figure 1, the sequences labeled <math>n = 1</math> and <math>n = 5</math> can be specified entirely through analytical expressions; these are well-known analytic solutions to the Lane-Emden equation.  In Figure 2, ''all seven equilibrium sequences'' are specifiable analytically; the location(s) of ''turning points'' along each sequence (when they exist) are also completely specifiable analytically.

In this chapter we examine the relative stability of various equilibrium <math>(n_c, n_e) = (5, 1)</math> bipolytropes.  A model is uniquely selected once we specify the radial location, <math>\xi_i</math>, of the core/envelope interface, and specify the ratio <math>\mu_e/\mu_c \le 1</math> of the mean-molecular weights of the gas in the envelope to the gas in the core.  Once this pair of parameter values has been specified, the model's radius <math>R_\mathrm{eq}</math> and total mass <math>M_\mathrm{tot}</math> are known analytically, as is the relative size of the core &#8212; both in terms of its fractional radius, <math>q \equiv r_i/R_\mathrm{eq}</math>, and its fractional mass, <math>\nu \equiv m_\mathrm{core}/M_\mathrm{tot}</math>.  Knowledge of the <math>(q, \nu)</math> parameter pair precisely identifies the selected model's location in the Figure 2 diagram.

In line with the '''<font color="red">principal question</font>''' stated above, we are particularly interested in examining the relative stability of models that are associated with the "maximum-mass turning points" along bipolytropic sequences for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math> &#8212; see the solid green circular markers in Figure 2.
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