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. $$