Orbital Period Calculator

The semi-major axis is half the longest diameter of an ellipse. For a circular orbit, it equals the radius. For an elliptical orbit (like most planetary orbits), it represents the average of the periapsis and apoapsis distances. In Kepler's third law, the semi-major axis determines the orbital period regardless of the orbital shape (eccentricity).

From T^2 = a^3/M in solar units (T in years, a in AU): M = a^3/T^2 = 1^3/1^2 = 1 solar mass — trivially by definition. To get the actual mass in kg from observational data: M_sun = 4 pi^2 a^3 / (G T^2) = 4 pi^2 x (1.496x10^11 m)^3 / (6.674x10^-11 x (3.156x10^7 s)^2) = 1.989x10^30 kg.

When a transiting exoplanet is discovered, its period is measured directly from the transit interval. The host star's mass is estimated from spectroscopy. Then a = (M_star x T^2)^(1/3) in AU (T in years) gives the semi-major axis. This is the primary method for determining exoplanet orbital distances in transit surveys like Kepler and TESS.

Yes, with the central mass being the planet rather than the Sun. For Earth's Moon: T^2 = a^3/M_earth in units where M_earth replaces the Sun and distances are in Earth-Moon units. More practically, for any central mass M and orbital radius r, T = 2 pi x sqrt(r^3 / (GM)) seconds. This is used to design satellite constellations and predict satellite overflight times.

Kepler's third law shows that the period depends only on the semi-major axis (not the eccentricity) for a given central mass. This is a consequence of energy conservation: two orbits with the same semi-major axis but different eccentricities have the same total energy (potential + kinetic), and therefore the same period. Higher eccentricity means moving faster at periapsis and slower at apoapsis, which exactly compensates.

The International Space Station orbits at about 400 km altitude, giving an orbital radius of about 6,771 km from Earth's center. The period is about 92.68 minutes. This means the ISS orbits Earth approximately 15-16 times per day. Astronauts aboard see about 15-16 sunrises and sunsets per day.

A galactic year (cosmic year) is the time for the Sun to complete one orbit around the Milky Way's center. The Sun is located about 8.5 kpc from the center and orbits at about 220 km/s. One galactic year is approximately 225-250 million years (Earth years). In the Sun's lifetime of 4.6 billion years, it has completed about 18-20 galactic orbits.

The sidereal period is the true orbital period measured with respect to distant stars (the inertial background). The synodic period is the period measured with respect to the Sun as seen from Earth — the time between successive conjunctions with the Sun (e.g., between successive oppositions for outer planets). The relationship is: 1/P_syn = 1/P_earth - 1/P_planet for outer planets.

From Kepler's third law, T = sqrt(a^3/M), so T is inversely proportional to sqrt(M). Doubling the central mass reduces the period by sqrt(2) at the same orbital radius — the orbit is faster. This makes sense: stronger gravity requires faster orbital speed to maintain the same circular orbit radius. In binary star systems, both masses enter the denominator: M_total = M1 + M2.