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

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

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

$\Delta S=3/2\ln K.$