Entropy (astrophysics)

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

dQ=dU−dW.

For an ideal gas in this special case, the internal energy, U, is only a function of 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

dQ=CVdT+PdV.

Further manipulation using the differential version of the ideal gas law, the previous equation, and assuming constant pressure, one finds

dQ=CPdT−VdP.

For an adiabatic process dQ=0 and recalling γ=CPCV, one finds

VdP=CPdTPdV=−CVdT

dPP=−dVVγ.

One can solve this simple differential equation to find

PVγ=constant=K

This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows

P=ρkBTμmH,

where kB is Boltzmann's constant. Substituting this into the above equation along with V=[g]/ρ and γ=5/3 for an ideal monatomic gas one finds

K=kBT(ρ/μmH)2/3,

where μ is the mean molecular weight of the gas or plasma; and mH is the mass of the Hydrogen atom, which is extremely close to the mass of the proton, mp, 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

ΔS=3/2ln⁡K.

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