Roche-lobe-filling period-density relation
= Roche-lobe-filling period-density relation
{c}
{title2=$P\propto(R_2^3/M_2)^{1/2}$}
If $R_2=R_L\simeq Ca(M_2/M)^{1/3}$, <Kepler's third law> gives
$$
P=\frac{2\pi}{\sqrt{GC^3}}\left(\frac{R_2^3}{M_2}\right)^{1/2}.
$$
Thus a <Roche lobe>-filling star's period measures its inverse square-root mean <mass density>, largely independently of the companion mass in this approximation. A local <mass-radius relation> $R_2\propto M_2^\zeta$ gives $P\propto M_2^{(3\zeta-1)/2}$.