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=C_{V}dT+P\,dV.

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

dQ=C_{P}dT-V\,dP.

For an adiabatic process $dQ=0\,$ and recalling $\gamma ={\frac {C_{P}}{C_{V}}}\,$ , one finds

{\frac {V\,dP=C_{P}dT}{P\,dV=-C_{V}dT}}

{\frac {dP}{P}}=-{\frac {dV}{V}}\gamma .

One can solve this simple differential equation to find

PV^{\gamma }={\text{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={\frac {\rho k_{B}T}{\mu m_{H}}},

where $k_{B}\,$ is Boltzmann's constant. Substituting this into the above equation along with $V=[grams]/\rho \,$ and $\gamma =5/3\,$ for an ideal monatomic gas one finds

K={\frac {k_{B}T}{\mu m_{H}\rho ^{2/3}}},

where $\mu \,$ is the mean molecular weight of the gas or plasma; and $m_{H}\,$ is the mass of the Hydrogen atom, which is extremely close to the mass of the proton, $m_{p}\,$ , 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

S=k_{B}\ln \Omega +S_{0}\,

where $\Omega \,$ , the density of states in statistical theory, takes on the value of K as defined above.

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