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Orbital Period Calculator — Kepler's Third Law

Q: How long is a year on Jupiter?

About **11.86 Earth years** (4,333 days). Jupiter's semi-major axis is 5.2 AU from the Sun.

Q: What is an Astronomical Unit (AU)?

An AU is the average Earth-Sun distance: **1 AU = 1.496 × 10¹¹ m** ≈ 93 million miles or 149.6 million kilometers.

Find any orbit's period with Kepler's Third Law: T² = 4π²a³/(GM), using NIST's G = 6.674×10⁻¹¹. Enter semi-major axis and central mass — free, instant.

NASA Goddard Space Flight Center data · July 2026
Edited by Martín Rodríguez
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Per the NASA Planetary Fact Sheet (GSFC/NSSDC) and NIST's CODATA values for the gravitational constant G = 6.674 × 10⁻¹¹ N·m²/kg², Kepler's Third Law relates orbital period to semi-major axis: T² = 4π²a³ / (GM), where a is the semi-major axis, G is the gravitational constant, and M is the mass of the central body. Used in US classrooms following Next Generation Science Standards (NGSS HS-ESS1-4), in NASA mission planning at JPL, and by amateur astronomers tracking exoplanets discovered by Kepler/TESS missions.

When to use this calculator

Calculate orbital periods of planets and satellites.
Solve orbital mechanics problems.
Verify Kepler's Third Law with real data.
Calculate exoplanet orbits.
Understand why distant planets have longer years.

Orbital Periods of Solar System Planets (NASA Planetary Fact Sheet)

Planet	Semi-major Axis (AU)	Orbital Period (Years)
Mercury	0.387	0.24
Venus	0.723	0.62
Earth	1.000	1.00
Mars	1.524	1.88
Jupiter	5.203	11.86
Saturn	9.537	29.46
Uranus	19.19	84.01
Neptune	30.07	164.8

Fuente: NASA Goddard Space Flight Center, NSSDC Planetary Fact Sheet (nssdc.gsfc.nasa.gov/planetary/factsheet/). 1 AU = 1.496 × 10¹¹ m; masa del Sol = 1.989 × 10³⁰ kg; G = 6.674 × 10⁻¹¹ N·m²/kg² (NIST CODATA).

How it works

Kepler's Third Law

Kepler's Third Law in its Newtonian form relates the orbital period to the semi-major axis:

T² = (4π² / GM) · a³

where:

G = 6.674 × 10⁻¹¹ N·m²/kg² (universal gravitational constant)

M = mass of the central body (kg)

a = semi-major axis of the orbit (m)

T = orbital period (s)

The original form (Kepler, 1619) stated only that T² ∝ a³ for planets around the Sun, without knowing why. Newton's derivation (1687) added the physical cause—gravity—and introduced M, making the law applicable to any two-body system, not just the solar planets.

How It's Calculated

Starting from T² = (4π² / GM) · a³, you solve for whichever variable is unknown:

Period from semi-major axis: T = 2π · √(a³ / GM)

Semi-major axis from period: a = ∛(GM · T² / 4π²)

Central body mass from T and a: M = 4π² · a³ / (G · T²)

That third rearrangement is scientifically powerful: it's how astronomers measure the mass of planets (from moon orbits), stars (from exoplanet transits), and even black holes (from stellar orbits around the galactic center). The mass of Sagittarius A*, the Milky Way's central black hole (~4 × 10⁶ solar masses), was determined this way using stellar orbits.

Practical shortcut in the solar system: if you use AU for distance and Earth years for time, the formula simplifies to T² = a³ (because all constants cancel out when normalized to Earth's orbit). This lets you compute planetary periods mentally.

Orbital Periods of the Planets (Around the Sun)

1 AU = 1.496 × 10¹¹ m · Mass of the Sun = 1.989 × 10³⁰ kg

Notice the non-linear scaling: Neptune is ~30× farther than Earth but its year is ~165× longer, consistent with T ∝ a^(3/2).

Real-World Applications

Geostationary satellites: at exactly T = 24 hours, the required orbital radius is ~42,164 km from Earth's center (~35,786 km altitude). This is a direct application of the formula using Earth's mass (5.972 × 10²⁴ kg).

ISS: orbits at ~408 km altitude with a period of ~92 minutes — also verifiable with this calculator.

Exoplanet detection: when a planet transits its star repeatedly, astronomers measure T; combined with estimated stellar mass M, they calculate a and infer whether the planet lies in the habitable zone.

Moon of Jupiter: Galileo used the orbital periods of Io, Europa, Ganymede, and Callisto (1610) as the earliest observational confirmation of bodies not orbiting Earth — a key argument for heliocentrism.

What This Formula Does NOT Include

General relativity corrections: for Mercury's perihelion precession (43 arcseconds/century), Newtonian gravity gives a slightly wrong result. For most engineering and astronomy purposes this is negligible, but it matters for GPS satellites and high-precision astrometry.

Third-body perturbations: the Moon's orbit is noticeably perturbed by the Sun; this formula treats it as a clean two-body problem.

Non-gravitational forces: radiation pressure, atmospheric drag (for low Earth orbits), and tidal forces are not accounted for.

Binary systems with comparable masses: when both bodies have significant mass, the correct form is T² = 4π² · a³ / G(M₁ + M₂), where a is the total separation. Using only one mass introduces proportional error.

Common Mistakes

Mixing unit systems: the most frequent error. If a is in AU and M is in kg, the formula breaks. Use SI throughout (meters, kilograms, seconds), or use the normalized AU/year shortcut only when M = M☉.

Confusing semi-major axis with orbital radius or apoapsis: for elliptical orbits, a = (periapsis + apoapsis) / 2. Using the maximum distance from the central body overestimates a and therefore T.

Assuming circular orbits: Kepler's Third Law applies to ellipses. Circular orbits are a special case (eccentricity = 0) where a equals the constant radius.

Ignoring the mass of the orbiting body: valid only when M_orbiter ≪ M_central. Earth's mass relative to the Sun is ~3 × 10⁻⁶ — negligible. But for double stars or comparable-mass systems, omitting M_orbiter causes real error.

Related Calculators

Simple Pendulum Period — oscillatory mechanics with a similar T = 2π · √(L/g) structure

Escape velocity and circular orbital velocity calculators share the same GM/r foundation

Real example: Earth orbiting the Sun

Data: a = 1.496 × 10¹¹ m (1 AU), M = 1.989 × 10³⁰ kg.

Formula: T² = (4π² / GM) · a³ = (4π² / (6.674e-11 × 1.989e30)) · (1.496e11)³.

Result: T ≈ 3.156 × 10⁷ s ≈ 365.25 days ≈ 1 year.

Interpretation: matches the length of an Earth year, as expected.

Earth completes its orbit in 365.25 days — matches the NASA Planetary Fact Sheet and validates Kepler's Third Law.

Frequently asked questions

What is Kepler's Third Law?

The square of a planet's orbital period is proportional to the cube of its semi-major axis: T² ∝ a³. Planets farther from their star have longer orbital periods.

What is a semi-major axis?

The semi-major axis is half the longest diameter of an elliptical orbit. For nearly circular orbits, it's approximately the orbital radius.

How long is a year on Jupiter?

About 11.86 Earth years (4,333 days). Jupiter's semi-major axis is 5.2 AU from the Sun.

Does Kepler's Third Law work for artificial satellites?

Yes. Replace M with Earth's mass. The ISS (a ≈ 6,771 km from Earth's center) has an orbital period of about 92 minutes.

What is an Astronomical Unit (AU)?

An AU is the average Earth-Sun distance: 1 AU = 1.496 × 10¹¹ m ≈ 93 million miles or 149.6 million kilometers.

Can I use Kepler's law to study exoplanets?

Yes. If you know the star's mass and measure the planet's orbital period (via transit detection), you can calculate the exoplanet's orbital distance.

Does the planet's mass matter in the calculation?

For planets much less massive than their star, the planet's mass is negligible. For binary systems with comparable masses, use M₁ + M₂.

What units should I use for Kepler's Third Law?

Always use SI units: meters for distance, kilograms for mass, and seconds for time. Mixing units will give incorrect results.

Sources & references

NASA / GSFC NSSDC - Planetary Fact Sheet — NASA Goddard Space Flight Center
NASA JPL - Solar System Dynamics & HORIZONS Ephemeris — NASA Jet Propulsion Laboratory
NIST - CODATA Value: Newtonian Constant of Gravitation — National Institute of Standards and Technology

Methodology & trust

Editorial

ciencia calculator with its formula verified automatically against NASA / GSFC NSSDC - Planetary Fact Sheet, per our editorial policy and methodology.

Updates

Updated: July 2026. Parameters are verified periodically against the cited sources.

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Limitations

Indicative results. For critical decisions, consult a professional.

📌 How to cite this calculator

APA format

Rodríguez, M. (2026). Orbital Period Calculator — Kepler's Third Law. Hacé Cuentas. https://hacecuentas.com/en/orbital-period-planet

BibTeX

@misc{hacecuentas_orbital_period_planet_2026,
  author       = {Rodríguez, Martín},
  title        = {{Orbital Period Calculator — Kepler's Third Law}},
  year         = {2026},
  howpublished = {\url{https://hacecuentas.com/en/orbital-period-planet}},
  note         = {Hacé Cuentas}
}

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