In astrophysics, what is referred to as "entropy" is actually the adiabatic constant derived as follows.
Using the first law of thermodynamics for a quasi-static, infinitesimal process for a hydrostatic system
 formula_1 
For an ideal gas in this special case, the internal energy, "U", is a function of only the temperature "T"; therefore the partial derivative of heat capacity with respect to "T" is identically the same as the full derivative, yielding through some manipulation
 formula_2
Further manipulation using the differential version of the ideal gas law, the previous equation, and assuming constant pressure, one finds
 formula_3
For an adiabatic process formula_4 and recalling formula_5, one finds
One can solve this simple differential equation to find
 formula_6
This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows
 formula_7
where formula_8 is the Boltzmann constant.
Substituting this into the above equation along with formula_9 and formula_10 for an ideal monatomic gas one finds
 formula_11
where formula_12 is the mean molecular weight of the gas or plasma; and formula_13 is the mass of the hydrogen atom, which is extremely close to the mass of the proton, formula_14, the quantity more often used in astrophysical theory of galaxy clusters.
This is what astrophysicists refer to as "entropy" and has units of [keV⋅cm2]. This quantity relates to the thermodynamic entropy as
 formula_15
References.
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