Entropy (astrophysics) explained

In astrophysics, what is referred to as "entropy" is actually the adiabatic constant derived as follows.^[1]

Using the first law of thermodynamics for a quasi-static, infinitesimal process for a hydrostatic system

dQ=dU-dW.

^[2]

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_vdT+PdV.

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

dQ=C_pdT-VdP.

dQ=0

and recalling

\gamma={C_p

}/\,, ^[3] one finds

	VdP=C_pdT
	PdV=-C_vdT

	dP
	P

	dV
	V

\gamma.

One can solve this simple differential equation to find

PV^\gamma=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=	\rhok_BT
	\mum_H

where

k_B

is the Boltzmann constant.Substituting this into the above equation along with

V=[g]/\rho

and

\gamma=5/3

for an ideal monatomic gas one finds

k_BT

(\rho/\mu

	2/3
m
	H)

where

\mu

is the mean molecular weight of the gas or plasma; ^[4] 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⋅cm<sup>2</sup>]. This quantity relates to the thermodynamic entropy as

\DeltaS=3/2lnK.

Notes and References

Web site: Adiabatic Condition Development . 2024-11-03 . hyperphysics.phy-astr.gsu.edu.
Web site: m300l5 . 2024-11-03 . personal.ems.psu.edu.
Web site: THERMAL PROPERTIES OF MATTER . 2024-11-03 . www.sciencedirect.com.
Web site: Mean molecular weight .