Minimum companion mass from radial velocity ($M_p\sin i$)

For a companion of negligible mass compared with its host, $M_p\sin i=K[P/(2\pi G){]}^{1/3}{M}_{\star }^{2/3}\sqrt{1-e^2}$. The observable supplies a projected mass; the minimum at a specified eccentricity occurs for an edge-on orbit. A exoplanet transit can constrain inclination, while low-inclination stars or brown dwarfs can otherwise mimic a lower-mass companion. The exact relation retains $(M_{\star }+M_p{)}^{2/3}$ in the denominator.

At fixed observed period and semi-amplitude, $M_p\sin i\propto M_{\star }^{2/3}$ and $a\propto M_{\star }^{1/3}$. At fixed planet mass and amplitude threshold, the largest circular orbit instead scales as $M_{\star }^{-1}$. A revised stellar radius affects transit-derived planet radii but not these leading dynamical relations.

For a circular low-mass companion and a semi-amplitude threshold $K_{\min }$, $a_{\max }=G{M}_p^2{\sin }^{2}i/(M_{\star }{K}_{\min }^2)$. The edge-on case is the largest geometrically accessible distance. Adequate baseline and cadence are separate requirements, and a full peak-to-peak detection threshold is twice the corresponding semi-amplitude.