Binary Star Orbit

This calculator applies Kepler's Third Law in its solar-system-units form, M_total = a^3 / P^2, where Semi-Major Axis (a) is in astronomical units, Orbital Period (P) is in years, and the result is total system mass in solar masses. This is Newton's reformulation of Kepler's original relation -- Kepler found only that P^2 is proportional to a^3, and it was Newton's law of gravitation that revealed the proportionality constant is the total system mass when a, P, and M are expressed in these particular units (see OpenStax's Astronomy 2e derivation of Newton's form of Kepler's Third Law, cited below). Because mass scales with the cube of Semi-Major Axis but only the inverse square of Orbital Period, Semi-Major Axis moves Total System Mass more strongly for the same percentage change in either input -- doubling the separation roughly octuples the implied mass, while doubling the period only quarters it. Each Star (if equal) is simply half of Total System Mass, a convenience figure that only applies when the pair is a genuine equal-mass binary; for an unequal pair you would need the individual mass ratio from radial-velocity or astrometric data, which this calculator does not model. Orbital Velocity and Separation are computed independently from the same two inputs using basic circular-orbit geometry (v = 2*pi*a/P and separation converted to light-seconds) rather than from the derived mass, so they respond only to the input whose units they actually depend on: Separation moves only with Semi-Major Axis, and Period (days) moves only with Orbital Period.

The calculator treats the relative orbit as circular; a genuinely eccentric binary's true mass, from the full vis-viva form of Kepler's law, would differ from this simplified circular estimate, so treat Total System Mass as a first-pass approximation rather than a precise fit to a measured eccentric orbit. Orbital Energy is the system's specific orbital energy (energy per unit reduced mass), computed as -mu / (2a) where mu is the standard gravitational parameter (G times Total System Mass). It comes out negative at every valid input combination -- a negative specific orbital energy is the standard signature of a bound, closed orbit, since it takes positive energy input to unbind the pair to infinity. A larger magnitude (a more negative value) means a more tightly bound system, driven the same way Total System Mass is: strongly by Semi-Major Axis and Orbital Period through the Kepler relationship.