I Where does a polytropic EoS come from?

[Vrbic]

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.

 

[George Jones]

Try "Black Holes, White Dwarfs, and Neutron stars: The Physics of Compact Objects" by Shapiro and Teukolsky.

 

[Ken G]

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.)