= Solution
<Kepler's third law> is $P^2=4\pi^2a^3/[G(M_S+M_P)]$, with $a$ the semimajor axis of the relative <orbit>. For a circular edge-on <orbit>, the star's speed is $V_S=2\pi a_S/P$, and the <centre of mass> relation gives $a_S=aM_P/(M_S+M_P)$. Cubing the speed expression and eliminating $a$ yields
$$
\frac{M_P^3}{(M_S+M_P)^2}=\frac{PV_S^3}{2\pi G}.
$$
For $M_P\ll M_S$, this reduces to
$$
\boxed{M_P^3\simeq\frac{V_S^3PM_S^2}{2\pi G}.}
$$
Without the edge-on assumption, a factor $\sin^3i$ multiplies the numerator, explaining the minimum-mass degeneracy of the <radial-velocity method>.
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