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 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

$dQ=C_{\text{v}}dT+P\,dV.$

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

$dQ=C_{\text{p}}dT-V\,dP.$

For an adiabatic process $dQ=0\,$ and recalling $\gamma ={C_{\text{p}}}/{C_{\text{v}}}\,$, one finds

${\frac {V\,dP=C_{\text{p}}dT}{P\,dV=-C_{\text{v}}dT}}$

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

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

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

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

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

where $\mu \,$ is the mean molecular weight of the gas or plasma; and $m_{\text{H}}$ 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

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