Binary Star System Orbital Speed - 2D Centre-of-Mass Map and Radial Velocity Curves

Two bodies, one ellipse

Two stars of masses m₁ and m₂ orbit their common centre of mass. Their separation follows a single Kepler ellipse of semi-major axis a with period P² = 4π² a³ / (G (m₁+m₂)); in solar masses, years and AU this is P = √(a³/M). Each star moves on a similar ellipse with size scaled by the other star's mass fraction, so v₁ = v_rel m₂/M and v₂ = v_rel m₁/M.

Speed and the radial-velocity curve

The relative speed obeys the vis-viva equation v² = G M (2/r - 1/a): fastest at periapsis, slowest at apoapsis. A spectrograph only sees the line-of-sight component, v_r = K [cos(ν+ω) + e cos ω], with semi-amplitude K = (m_other/M) √(G M / (a (1-e²))) sin i. That is how masses of real binaries are measured.

Honest simplifications: point masses, Newtonian gravity, no tides, mass transfer or relativity; the top-down view is face-on and the inclination slider only affects the velocity curves.

• Choose Equal twins and then Giant + dwarf: the lighter star always has the bigger orbit and speed.
• Choose Highly eccentric and watch the stars rush through periapsis while the shaded wedge keeps the same area.
• Lower the inclination: the radial-velocity amplitude shrinks with sin i, the real orbit does not.
• Double the semi-major axis: the period grows by 2^1.5.