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.