I Where does a polytropic EoS come from?

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The discussion addresses the origins and derivation of the polytropic equation of state (EoS) used in modeling compact objects like neutron stars and white dwarfs. It highlights two types of polytropic EoS: one that accurately reflects the density-dependent pressure due to degeneracy pressure in fermions, and another that simplifies temperature dependence into density dependence. The first type is particularly relevant for white dwarfs and neutron stars, where low temperatures allow for a close approximation to zero-temperature conditions. This approximation is significant because it simplifies calculations by focusing on density rather than temperature. Understanding these concepts is crucial for studying the physics of compact objects.
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I have read few texts about compact objects (neutron stars, white dwarfs) and the easiest approximation of equation of state is taken as a polytrope. But nowhere is written why, how can I derive it or from what (or I missed it). Can someone explain me it or refer me to some text?
Thank you.
 
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Try "Black Holes, White Dwarfs, and Neutron stars: The Physics of Compact Objects" by Shapiro and Teukolsky.
 
There are two flavors of polytropic equations of state that see common usage. One is where the equation of state really is polytropic (i.e., the pressure really is determined by the density, which is a low-temperature approximation called degeneracy pressure that applies to fermions approaching their Pauli-exclusion-principle controlled ground state), and the other is where the temperature dependence is important to the pressure but is subsumed into the density dependence (so that latter type is not a true polytrope, but can be treated as such because it simplifies things, because it allows you to ignore the explicit temperature structure). You seem to be interested in the first type, where degeneracy has driven the temperature down so low that you are close to reaching the zero-temperature approximation for the fermionic pressure. That's a decent approximation in white dwarfs and neutron stars, and then the pressure depends primarily only on the density, you don't need to know the temperature because the Pauli exclusion principle has made it so low. (Of course, "low temperature" is a relative term, these stars are pretty hot by stellar standards but their temperature is low in the sense that kT is way lower than the kinetic energy per degenerate particle.)
 
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