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

Doppler spectroscopy measures the line-of-sight component of a host star's stellar reflex motion, rather than the full companion mass. Let the measured orbital period, orbital eccentricity and stellar radial-velocity semi-amplitude be $P,e,K$, with host mass $M_{\star }$ and sky-plane orbital inclination $i$. The exact two-body relation isCubing and rearranging defines the observed binary mass functionFor fixed $M_{\star }$, the function $M_p^3/(M_{\star }+M_p{)}^2$ is strictly increasing: its logarithmic derivative is $3/M_p-2/(M_{\star }+M_p)>0$. Since $\sin i\le 1$, the smallest compatible companion mass occurs for an edge-on orbit, $i=\pi /2$.

Only when $M_p\ll M_{\star }$ does this simplify toA small inferred projected mass alone therefore does not exclude a more massive companion at small orbital inclination. An exoplanet transit constrains orbital inclination and helps remove this degeneracy. When the companion mass is appreciable, solve the exact binary mass function instead of inserting a total-mass correction on the wrong side of the approximate formula.

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.