Entropy (astrophysics)

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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.