= Surface sound speed of a polytropic star
{title2=$c_s^2\simeq\gamma gz/(m+1)$}
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.
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