= Binary mass function
{title2=$F$}
{wiki}
A <binary mass function> combines the observed <orbital period> $P$ and <radial velocity> semi-amplitude $K_1$ of a visible member of a <spectroscopic binary>. For a circular <Kepler orbit>, $K_1=2\pi a_1\sin i/P$, where $a_1$ is the visible <star>'s orbital semimajor axis about the <centre of mass> and $i$ the <orbital inclination>. The <centre of mass> relation $a_1=aM_2/(M_1+M_2)$ and <Kepler's third law> $4\pi^2a^3=G(M_1+M_2)P^2$ eliminate the unobserved separation:
$$
f(M)=\frac{PK_1^3}{2\pi G}=\frac{M_2^3\sin^3i}{(M_1+M_2)^2}.
$$
It has units of <mass>. The right side is at most $M_2$, so $f(M)\leq M_2$ is a strict observational lower bound on the unseen companion's <mass> within this model. If $M_1$ is independently estimated, setting $\sin i=1$ and solving the equation gives a stronger minimum companion <mass>. An unknown <orbital inclination> generally prevents a unique pair of stellar <masses> from being inferred from one <velocity> curve alone.
For an eccentric <Kepler orbit> with measured <orbital eccentricity> $e$, the radial-velocity semi-amplitude is $K_1=2\pi a_1\sin i/[P\sqrt{1-e^2}]$, so the observable expression becomes $f(M)=PK_1^3(1-e^2)^{3/2}/(2\pi G)$. A fitted semi-amplitude, rather than an arbitrary instantaneous <radial velocity>, is required. This is why an unseen companion can be constrained dynamically even when its own <spectral lines> are absent.
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