Polytrope

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

For a symmetric self-gravitating slab with polytropic equation of state $p=K{\rho }^2$ and constant $K>0$, hydrostatic equilibrium gives $2K{\rho }^{\prime }=-{\psi }^{\prime }$. The plane-parallel Poisson equation for Newtonian gravity then yields ${\rho }^{\prime \prime }+(2\pi G/K)\rho =0$. Reflection symmetry at the midplane fixes the title profile. For a vacuum surface, retain the nonnegative profile only on $|z|\le z_s=\pi /(2k)$ and put $\rho =0$ outside. The half-thickness is independent of central density in this ideal index-one model, while the surface mass density is $\Sigma =2{\rho }_c/k$.

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 $n$.

The coefficient $K$ in a polytropic equation of state $p=K{\rho }^{\Gamma }$. It is spatially constant for one simple polytrope, but need not stay constant between different equilibrium models. For a fixed-composition ideal gas with $\Gamma$ equal to its adiabatic index, $K$ parametrizes specific entropy: $s=c_v\ln K+\text{constant}$ with a suitable dimensional reference inside the logarithm.

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 gravitational restoration with the Coriolis force. For a uniform incompressible fluid occupying $z<0$, take gravity downward, rotation $\mathbf{\Omega }=\Omega {\mathbf{e}}_z$ normal to the surface and $f=2\Omega$. A mode proportional to $e^{i(kx-\omega t)+qz}$ must decay downward, so $q>0$.

The linear rotating momentum equations and incompressibility giveThe kinematic boundary condition is $w=-i\omega \eta$ at the surface. The dynamic condition is $p^{\prime }=\rho g\eta$, so vertical momentum gives ${\omega }^2=gq$. Eliminating $q$ yieldsFor $f=2\Omega$, this isFor nonzero horizontal wavenumber, this branch has ${\omega }^2>f^2$ and an evanescent vertical structure. With no rotation it reduces to the deep-water gravity wave relation ${\omega }^2=g|k|$.

The orientation and incompressible deep-fluid assumptions matter. A tilted rotation axis, finite depth or compressible polytropic atmosphere has a different eigenvalue problem; the displayed formula is the stated incompressible model.

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.