Binary mass function ($F$)

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 $a_1=a{M}_2/(M_1+M_2)$ and Kepler's third law $4{\pi }^2{a}^3=G(M_1+M_2)P^2$ eliminate the unobserved separation:It has units of mass. The right side is at most $M_2$, so $f(M)\le M_2$ 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 $f(M)=P{K}_1^3(1-e^2{)}^{3/2}/(2\pi G)$. 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.

Randomly oriented binary angular-momentum vectors are uniform in $\cos i$. After identifying $i$ with $\pi -i$, their normalized inclination density on $0\le i\le \pi /2$ is $p(i)=\sin i$.