Editing SSC/FreeEnergy/EquilibriumSequenceInstabilities (section)

==Equation of State Considerations==

Spherically symmetric, self-gravitating, equilibrium configurations can be constructed from gases exhibiting a wide variety of degrees of compressibility.  When examining how the internal structure of such configurations varies with compressibility, or when examining the relative stability of such structures, it can be instructive to construct models using a [[SR#Time-Independent_Problems|polytropic equation of state]], 

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{{ Math/EQ_Polytrope01 }}
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because the degree of compressibility can be adjusted by simply changing the value of the polytropic index, {{Math/MP_PolytropicIndex}}, across the range, <math>0 \le n \le \infty</math>.  (Alternatively, one can vary the effective adiabatic exponent of the gas, <math>\Gamma = 1 + 1/n</math>.)  In particular, <math>n = 0 ~~ (\Gamma = \infty)</math> represents a ''hard'' equation of state and describes an incompressible configuration, while <math>n = \infty ~~(\Gamma = 1)</math> represents an isothermal and extremely ''soft'' equation of state.


As has been detailed in an [[SSC/Structure/Polytropes#Polytropic_Spheres|accompanying discussion]], the structural properties of spherical polytropes can be described entirely in terms of a dimensionless radial coordinate, <math>\xi</math>, and by the radial dependence of the dimensionless enthalpy function, <math>\theta_n(\xi)</math>, and its first radial derivative, <math>\theta^'_n(\xi)</math>.  At the center of each configuration <math>~(\xi=0)</math>, <math>\theta_n = 1</math> and <math>\theta^'_n  = 0</math>.  The surface of each 
[[SSC/Structure/Polytropes#Polytropic_Spheres|''isolated'' polytrope]] is identified by the radial coordinate, <math>\xi_1</math>, at which <math>~\theta_n</math> first drops to zero.  As a class, isolated polytropes exhibit three attributes that are especially key in the context of our present discussion:
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For example, the radial mass-density distribution and the Lagrangian mass coordinate are given, respectively, but the expressions,
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<math>~\frac{\rho(\xi)}{\rho_c}</math>
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<math>~=</math>
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<math>~[\theta_n(\xi)]^n \, ,</math>
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<math>~m_r(\xi)</math>
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<math>~=</math>
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<math>~4\pi a_n^3 \rho_c \biggl[ - \xi^2 \theta^'_n(\xi)\biggr] \, .</math>
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<ol>
  <li>The equilibrium structure is dynamically stable if <math>n < 3</math>.
  <li>The equilibrium structure has a finite radius if <math>n < 5</math>.
  <li>The equilibrium structure can be described in terms of closed-form analytic expressions 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>]].
</ol>


Isothermal spheres are discussed in a wide variety of astrophysical contexts because it is not uncommon for physical conditions to conspire to create an extended volume throughout which a configuration exhibits uniform temperature.  But, as can be surmised from our list of three key polytrope attributes and recognition that equilibrium isothermal configurations are polytropes with index <math>n=\infty</math>, mathematical models of [[SSC/Structure/IsothermalSphere#Isothermal_Sphere|isolated isothermal spheres]] are relatively cumbersome to analyze because they extend to infinity, they are dynamically unstable, and they are not describable in terms of analytic functions.  In such astrophysical contexts, we have sometimes found it advantageous to employ an <math>n=5</math> polytrope instead of an isothermal sphere.  An [[SSC/Structure/Polytropes#n_.3D_5_Polytrope|isolated <math>~n=5</math> polytrope]] can serve as an effective surrogate for an isothermal sphere because it is both infinite in extent and dynamically unstable, but it is less cumbersome to analyze because its structure can be described by closed-form analytic expressions.