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

$dQ=C_{\text{v}}dT+PdV.$

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

$dQ=C_{\text{p}}dT-VdP.$

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

$\frac{VdP=C_{\text{p}}dT}{PdV=-C_{\text{v}}dT}$

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

One can solve this simple differential equation to find

$P{V}^{\gamma }=\text{constant}=K$

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

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