Polytrope

A polytrope obeys $p=K{\rho }^{1+1/n}$ for constants $K$ and polytropic index $n$.

A polytropic equation of state relates pressure and density by $p=K{\rho }^{\Gamma }$ with constant $K$ and exponent $\Gamma$. Writing $\Gamma =1+1/n$ defines the polytropic index $n$.

A polytropic atmosphere obeys $p=K{\rho }^{1+1/m}$ for polytropic index $m$. Under uniform vertical gravity and a free surface, its density and pressure are powers of depth below that surface.

In a plane-parallel polytropic atmosphere of index $m$, let $z$ measure depth from a zero-pressure boundary and take constant gravitational acceleration $g$. The stellar hydrostatic equation and $p=K{\rho }^{1+1/m}$ integrate to $(m+1)p/\rho =gz$. Consequently the adiabatic sound speed satisfies $c_s^2=\gamma p/\rho =\gamma gz/(m+1)$, so $c_s\propto z^{1/2}$. This local relation does not justify extrapolating a surface model to the stellar core.

A polytropic atmosphere is neutrally stratified when $1+1/m=\gamma$, so its equilibrium pressure-density relation matches the adiabatic relation and its buoyancy frequency vanishes.

A surface gravito-inertial wave combines free-surface gravity restoration with Coriolis restoration. In a uniformly rotating deep atmosphere its incompressible dispersion relation can be written

In a neutrally stratified polytropic atmosphere, vertically trapped polynomial solutions of degree $n\ge 1$ describe acoustic pressure modes. The degree-zero member is the surface-gravity mode.