# 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

${\displaystyle dQ=dU-dW.\,}$

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

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

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

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

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

 ${\displaystyle {\frac {V\,dP=C_{P}dT}{P\,dV=-C_{V}dT}}}$ ${\displaystyle {\frac {dP}{P}}=-{\frac {dV}{V}}\gamma .}$

One can solve this simple differential equation to find

${\displaystyle PV^{\gamma }={\text{constant}}=K\,}$

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

${\displaystyle P={\frac {\rho k_{B}T}{\mu m_{H}}},}$

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

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

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

${\displaystyle \Delta S=3/2\ln K.}$