Binary mass function

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.