= Mass of an exponential spherical atmosphere
{title2=$M=4\pi\rho_0h_0(R^2+2Rh_0+2h_0^2)$}
For a spherical surface of radius $R$ with exterior <mass density> $\rho(r)=\rho_0e^{-(r-R)/h_0}$, integrate the spherical <volume element> to obtain $M=4\pi\rho_0\int_0^\infty(R+h)^2e^{-h/h_0}\,dh$. The moments $\int_0^\infty h^ne^{-h/h_0}\,dh=n!h_0^{n+1}$ give the displayed expression. Its leading thin-atmosphere approximation is surface area times column mass, $4\pi R^2\rho_0h_0$; the two positive correction terms account for the growth of shell area with altitude.
Back to article page
