Kepler's Third Law Calculator

No. Kepler's third law uses the semi-major axis, which equals the radius for a circular orbit but also works for ellipses of any eccentricity. An ellipse with semi-major axis a has the same orbital period as a circle of radius a around the same central mass. This is a remarkable result: eccentricity does not affect period.

By measuring the orbital velocity and distance of stars and gas clouds at various radii from the Galactic center, astronomers use M = v^2 r / G (from circular orbit balance) or equivalently T^2 = a^3/M. The observation that rotation velocity remains flat at large radii (instead of decreasing) implies mass continues to increase — evidence for an extended dark matter halo.

Yes. For a binary system of total mass M1+M2, T^2 = a^3/(M1+M2) in solar units, where a is the semi-major axis of the relative orbit. If additional information (velocity ratio from spectroscopy, or astrometric position ratio) gives the mass ratio, individual masses M1 and M2 follow. Binary stars are the only direct source of stellar mass measurements.

Near strong gravitational fields (like neutron stars and black holes), GR modifies the orbit. For stars orbiting Sgr A*, the GR correction causes the periapsis to precess (advance) with each orbit — the same effect (at much smaller scale) that explains Mercury's perihelion precession of 43 arcseconds per century. The GR correction for S2 orbiting Sgr A* was first measured in 2020, confirming GR in the strong-field regime.

Indirectly, Kepler's third law gives us the masses and orbital parameters of Solar System bodies, which when combined with radioactive dating of meteorites gives the age of the Solar System (~4.568 billion years). More directly, the orbital period constrains the distance from the Sun, which constrains where a body formed in the protoplanetary disk — informing models of Solar System formation.

The exact form of Kepler's third law is T^2 = 4pi^2 a^3 / (G(M1+M2)), where both masses enter. For planetary satellites (planet mass >> satellite mass), the satellite mass is negligible. For binary systems where both masses are comparable, both must be included. The Earth-Moon system uses M_earth + M_moon as the effective central mass when computing the Moon's orbital period precisely.

Yes — this is how Sgr A*'s mass was measured. Stars in tight orbits around Sgr A* (studied by the UCLA and MPE groups for 30+ years) follow Keplerian orbits. From the measured periods and semi-major axes, the enclosed mass is directly calculated as M = a^3/T^2. This earned Andrea Ghez and Reinhard Genzel the 2020 Nobel Prize in Physics.

The Titius-Bode law is an empirical rule (not a physical law) that predicted the spacing of planet semi-major axes: a = 0.4 + 0.3 x 2^n AU for integer n. It correctly predicted the location of the asteroid belt before Ceres was discovered. Unlike Kepler's third law (which has a rigorous derivation from Newtonian gravity), the Titius-Bode law has no theoretical foundation and fails for Neptune and beyond.

In a perfect two-body gravitational system, orbits are exact Keplerian ellipses. In the real Solar System, gravitational perturbations from other planets cause: precession of the perihelion (orbit slowly rotates), changes in orbital eccentricity and inclination, and in resonant cases, significant energy exchange that can either stabilize or destabilize orbits over millions of years. These effects are studied through perturbation theory and numerical integration.