= Minimum companion mass from radial velocity
{title2=$M_p\sin i$}
<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 is
$$
K=\left(\frac{2\pi G}{P}\right)^{1/3}
\frac{M_p\sin i}{(M_\star+M_p)^{2/3}\sqrt{1-e^2}}.
$$
Cubing and rearranging defines the observed <binary mass function>
$$
f=\frac{PK^3(1-e^2)^{3/2}}{2\pi G}
=\frac{M_p^3\sin^3i}{(M_\star+M_p)^2}.
$$
For 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 $\sin i\leq1$, the smallest compatible companion <mass> occurs for an <edge-on orbit>, $i=\pi/2$.
Only when $M_p\ll M_\star$ does this simplify to
$$
M_p\sin i\simeq K\left(\frac{P}{2\pi G}\right)^{1/3}M_\star^{2/3}\sqrt{1-e^2}.
$$
A 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.
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