Entropy (astrophysics)

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In astrophysics, what is referred to as "entropy" is actually the adiabatic constant derived as follows.{{Cite web |title=Adiabatic Condition Development |url=http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiabc.html |access-date=2024-11-03 |website=hyperphysics.phy-astr.gsu.edu}}

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

: $dQ\; =\; dU-dW.$ {{Cite web |title=m300l5 |url=https://personal.ems.psu.edu/~brune/m300f99/m300l5.html#:~:text=The%20First%20Law%20of%20Thermodynamics,m2s-2). |access-date=2024-11-03 |website=personal.ems.psu.edu}}

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 $dQ=0\backslash ,$ and recalling $\backslash gamma\; =\; \{C\_\backslash text\{p\}\}/\{C\_\backslash text\{v\}\}\backslash ,$, {{Cite web |title=THERMAL PROPERTIES OF MATTER |url=https://www.sciencedirect.com/science/article/abs/pii/B9780120598601500278 |access-date=2024-11-03 |website=www.sciencedirect.com}} one finds

:

$\backslash frac\{V\backslash ,dP\; =\; C\_\backslash text\{p\}\; dT\}\{P\backslash ,dV\; =\; -C\_\backslash text\{v\}\; dT\}$

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

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 $k\_\backslash text\{B\}$ is the Boltzmann constant.

Substituting this into the above equation along with $V=[\backslash mathrm\{g\}]/\backslash rho\backslash ,$ and $\backslash gamma\; =\; 5/3\backslash ,$ for an ideal monatomic gas one finds

: $$

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

where $\backslash mu\backslash ,$ is the mean molecular weight of the gas or plasma; {{Cite web |title=Mean molecular weight |url=http://astronomy.nmsu.edu/jasonj/565/docs/09_03.pdf }}and $m\_\backslash text\{H\}$ 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 .