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

[math]\displaystyle{ dQ = dU-dW.\, }[/math]

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

[math]\displaystyle{ dQ = C_{V} dT+P\,dV. }[/math]

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

[math]\displaystyle{ dQ = C_{P} dT-V\,dP. }[/math]

For an adiabatic process [math]\displaystyle{ dQ=0\, }[/math] and recalling [math]\displaystyle{ \gamma = \frac{C_{P}}{C_{V}}\, }[/math], one finds

[math]\displaystyle{ \frac{V\,dP = C_{P} dT}{P\,dV = -C_{V} dT} }[/math]

[math]\displaystyle{ \frac{dP}{P} = -\frac{dV}{V}\gamma. }[/math]

One can solve this simple differential equation to find

[math]\displaystyle{ PV^{\gamma} = \text{constant} = K\, }[/math]

This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows

[math]\displaystyle{ P=\frac{\rho k_{B}T}{\mu m_{H}}, }[/math]

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

[math]\displaystyle{ K = \frac{k_{B}T}{(\rho/\mu m_{H})^{2/3}}, }[/math]

where [math]\displaystyle{ \mu\, }[/math] is the mean molecular weight of the gas or plasma; and [math]\displaystyle{ m_{H}\, }[/math] is the mass of the Hydrogen atom, which is extremely close to the mass of the proton, [math]\displaystyle{ m_{p}\, }[/math], 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

[math]\displaystyle{ \Delta S = 3/2 \ln K . }[/math]

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