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The gamma-law equation of state is very useful in studies of astrophysics. It is basically an equation that connects specific internal energy (internal energy per unit mass), pressure and mass density given by

$$ P = \left(\gamma-1\right)\rho u, $$

where $u$ is the internal energy per mass, $\gamma$ is the adiabatic index. I want to know under what conditions do equations of state of this form apply? Any other physical understandings regarding this kind of EQS is also welcomed. I just want to have more physics insights into it.

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