= Low circular-orbit period measures mean density
{title2=$P^2=3\pi/(G\bar\rho)$}
For a spherical body of radius $R$, mass $M$ and <mean density> $\bar\rho=3M/(4\pi R^3)$, a low <circular orbit> has period $P\simeq2\pi\sqrt{R^3/(GM)}$, by equating the gravitational acceleration from <Newton's law of universal gravitation> to the <centripetal acceleration>. Substitution proves the displayed relation. Thus similar low-orbit periods indicate similar mean densities, even for bodies with very different masses and radii. At altitude $h$, the right side is multiplied by $(1+h/R)^3$; the simple density inference requires $h\ll R$ and neglects rotation, nonsphericity and other gravitating bodies.
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