Orbital Period Calculator (Kepler III)

Compute orbital period with Kepler's third law T = 2π·√(a³/GM). Presets for Sun, Earth, Mars, Jupiter. Inverse modes for semi-major axis and central mass.

Calculation parameters
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Usato nelle modalità preset. La massa tabulata è quella di NASA Planetary Fact Sheet.
Massa del corpo attrattore. Default: massa del Sole 1,98892·10³⁰ kg.
Raggio orbitale medio o semiasse maggiore dell'ellisse. 1 AU = 1,495978707·10¹¹ m (per l'orbita terrestre).
Periodo di rivoluzione in secondi. 1 anno siderale ≈ 31 557 600 s (365,25 d).
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Kepler's third law

Kepler's third law, formulated by Johannes Kepler in 1619 in his Harmonices Mundi, is one of the three laws governing planetary orbital motion. In its classical form (pre-Newton) it states that «the squares of the orbital periods are proportional to the cubes of the semi-major axes of the orbits»:

T² ∝ a³

Isaac Newton, in the Principia (1687), derived this from universal gravitation and found the explicit constant:

T = 2π · √(a³ / (G·M))

where T is the orbital period in seconds, a the semi-major axis in metres, G = 6.67430·10⁻¹¹ m³/(kg·s²) the gravitational constant, and M the central mass.

The four modes

  1. From M and a → T: classic mode. Given central mass and semi-major axis, returns period.
  2. Preset central body → T: Sun, Earth, Moon, Mars, Jupiter, Saturn or Proxima Centauri; mass from NASA fact sheets.
  3. From T and M → a: inverse for finding orbital radius from period. For geostationary satellites (T = 86 164 s) gives a ≈ 42 164 km (35 786 km altitude).
  4. From T and a → M: the astronomer's scale. Measuring T and a of a binary or exoplanet system yields the central star's mass.

Modern applications

  • Artificial satellites: ISS (a ≈ 6 770 km, T ≈ 92 min), GPS (a ≈ 26 600 km, T = 12 h), geostationary (T = 23 h 56 min 4 s).
  • Interplanetary missions: Hohmann transfer windows and cruise times. A Mars-bound probe takes ≈ 8.5 months.
  • Exoplanet discovery: since 1992, thousands of exoplanets have been found via transits (Kepler, TESS) or radial velocity (HARPS, ESPRESSO), using Kepler's third law to extract semi-major axes from observed periods and stellar masses.
  • Dark matter: galactic rotation curves show that Kepler III applied to outer stars implies far more mass than luminous (Zwicky 1933, Rubin 1970).
  • Supermassive black holes: the S1-S10 stars orbiting Sagittarius A* enabled a mass measurement of ≈ 4.15·10⁶ solar masses for the Milky Way's central black hole (Nobel 2020, Ghez and Genzel).

Harmonic form

In AU, sidereal years and solar masses the law becomes:

T²(yr) = a³(AU) / (M/M☉)

Mars: a = 1.524 AU → T = 1.88 yr. Venus: a = 0.723 AU → T = 0.615 yr. Applies to Jupiter-Io with M = M_Jupiter ≈ 1/1047 M_Sun.

Limits

  • Closed orbits (bound, negative total energy).
  • Orbiting mass ≪ M; for comparable binaries use M_total = M₁ + M₂.
  • Central field, no significant third-body perturbations.
  • Newtonian regime; near black holes use Schwarzschild/Kerr.

Parabolic (e = 1) or hyperbolic (e > 1) trajectories have no period: they are open and described by flyby time instead.

From Kepler's law to Newton's gravitation

The third law was published by Johannes Kepler in 1619 in his work Harmonices Mundi, crowning years of analysis of Tycho Brahe's extremely precise observations. Kepler discovered empirically that the square of the period is proportional to the cube of the semi-major axis (T² ∝ a³), but he did not know its physical cause. It was Isaac Newton, in 1687, who proved that this relation follows directly from the law of universal gravitation: by equating the gravitational force to the centripetal force needed for the orbit one obtains exactly T = 2π·√(a³/(G·M)). The proportionality constant, which Kepler regarded as a harmonic mystery, turned out to depend only on the mass of the central body. It was one of the triumphs of Newtonian physics and the proof that the same laws govern the fall of an apple and the motion of the planets.

Verifying the law in the Solar System

The third law holds with extraordinary precision when observing the planets. Mercury, at 0.39 astronomical units from the Sun, has a period of 88 days; Earth, at 1 AU, of one year; Mars, at 1.52 AU, of 1.88 years; Jupiter, at 5.2 AU, of nearly 12 years; Neptune, at 30 AU, of as much as 165 years. For all of them the ratio T²/a³ stays constant: expressed in years and astronomical units, it equals exactly 1 for the Sun. This constancy makes it possible, by measuring period and distance, to derive the mass of the central body: this is how astronomers weigh stars, planets and even galaxies, and with the transit method — used by the Kepler and TESS space telescopes — discover and characterise thousands of exoplanets around other stars.

Updated July 2026

Frequently Asked Questions

Why doesn't the period depend on eccentricity?

Because Kepler's third law contains the semi-major axis a, not the shape of the orbit. Two orbits with the same semi-major axis have the same period, whether nearly circular or highly elliptical. The reason is that the total orbital energy depends only on the semi-major axis: a highly elongated comet and a circular planet with the same 'mean distance' take the same time to complete one revolution.

How is stellar mass measured via exoplanets?

By measuring the period T and semi-major axis a of the planet's orbit, one inverts Kepler's third law: M = 4π²·a³/(G·T²). Since the planet's mass is negligible compared with the star's, the result gives the stellar mass directly. This is the method by which astronomers determine the masses of stars by observing their planets or companion stars in binary systems.

What is the geostationary-satellite period?

A geostationary satellite has a period of exactly one sidereal day, about 23 hours 56 minutes, so it stays fixed above the same point on the equator while the Earth rotates. Applying the third law with Earth's mass gives the corresponding orbital radius: about 42,164 km from the Earth's centre, that is about 35,786 km of altitude above the surface.

Why sidereal rather than tropical year?

The sidereal year is the time the Earth takes to complete one full orbit around the Sun relative to the fixed stars, about 365.256 days: it is the true orbital period. The tropical year (365.242 days) instead measures the return of the seasons and is slightly shorter because of the precession of the equinoxes. For orbital calculations the sidereal year must be used, consistent with the third law.

How does it generalise to binary systems?

When the two masses are comparable, both orbit the common centre of mass and the third law contains the total mass: T² = 4π²·a³/(G·(M₁+M₂)), where a is the semi-major axis of the relative orbit. Measuring period and distance gives the sum of the masses; with further data on each star's motion the two individual masses can be separated. It is the key method for weighing double stars.

Does it work for parabolic or hyperbolic trajectories?

No, Kepler's third law holds only for closed elliptical (or circular) orbits, the only periodic ones. Parabolic and hyperbolic trajectories are open: the body passes once and does not return, so it has no period. They are typical of non-periodic comets and of probes escaping after a gravity assist, for which other energy relations are used instead.

What about orbits near a black hole?

Far from a black hole, Kepler's third law remains an excellent approximation, because Newtonian gravity describes the field well. Close to the event horizon, however, general relativity is needed: orbits undergo relativistic precession (as for Mercury around the Sun, but amplified) and there is a minimum stable circular orbit, beyond which matter inevitably plunges into the black hole.

Who formulated Kepler's third law?

The law was stated by Johannes Kepler in 1619 in his work Harmonices Mundi, based on Tycho Brahe's observations. Kepler discovered it empirically as a harmonic relation among the planets, without explaining its cause. It was Isaac Newton, in 1687, who derived it from the law of universal gravitation, showing that the constant depends on the central body's mass. It is the third of Kepler's three laws of planetary motion.

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