The definition of entropy in astrophysics is often actually just the adiabatic constant $K$ relating $P=K\rho^\gamma$ where $\gamma$ is the usual adiabatic index, $P$ is the pressure and $\rho$ is the mass density. This is explained in the short Wikipedia article https://en.wikipedia.org/wiki/Entropy_(astrophysics)
However on that Wikipedia page I don't understand how to get the last two equations.
First, since we also have $P=\frac{\rho}{\mu m_p} k_B T$, we can plug this into the entropy equation above and solve for $K$ assuming $\gamma=5/3$ for a monatomic ideal gas. When I do this, I get
$$ K = P/\rho^{5/3} = \frac{\rho k_B T}{\mu m_p \rho^{5/3}} = \frac{k_B T}{\mu m_p \rho^{2/3}} $$
whereas that Wikipedia page has $(\frac{\rho}{\mu m_p})^{2/3}$ in the denominator which is convenient since that's just the number density $n^{2/3}$. But I cannot figure out the algebraic manipulation to get to that form -- am I missing something obvious?
Second question is how to derive the relation between that adiabatic entropy and the standard thermodynamic entropy given as the last equation on that Wikipedia page:
$$ \Delta S = 3/2 \ln K $$
What is $\Delta S$ and where does the natural log come from?
Edit: I answered my own first question: the trick is to start with $K=PV^\gamma$ where $V$ is the volume, and to substitute $V=(\mu m_p)/\rho$. Then it works out and you get $K=k_B T / (\frac{\rho}{\mu m_p})^{2/3}$. That Wikipedia page has a typo (or lack of clarity) when it says $V=$[g]$/\rho$...
I would still appreciate a brief pedagogical review of where $ \Delta S = 3/2 \ln K $ comes from (I know this is intro thermodynamics but I'm blanking.)
