# What is the orbital period?

Orbital period calculator: Kepler’s third law T = 2π√(a³ ÷ GM) for an orbit around the Earth, Sun, Moon, a planet or any mass, or the semi-major axis for a period, with the orbit speed.

- Page: https://www.acalculator.org/physics/orbital-period-calculator
- JSON spec: https://www.acalculator.org/physics/orbital-period-calculator.json
- Version: a3efd80c08f0

## Default answer

Example with the default inputs (Find Orbital period, Orbiting around Earth, Size of orbit as Height above surface, Height above the surface (h) 248.548476894934 mi): T = 1.54024 h, with a circular orbit speed of 7.67259 km/s.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| find | Find | The period from the size of the orbit, or the size from the period. |
| body | Orbiting around | The central body, or your own mass. |
| M | Central mass (M) | The mass of the body being orbited. |
| by | Size of orbit as | The semi-major axis (from the centre), or the height of a circular orbit above the surface: typed for a period, shown for a period you type. |
| h | Height above the surface (h) | The height of a circular orbit above the mean surface. |
| R | Body radius (R) | The radius of your body, to add to the height. |
| a | Semi-major axis (a) | Half the longest width of the orbit, from the centre: the radius of a circular orbit. |
| au | Axis or height unit | The unit of the semi-major axis you type, or of the axis or height found from a period. |
| T | Orbital period (T) | The time for one orbit, in the unit chosen below. |
| Tu | Period unit | The unit of the orbital period. |
| m | Orbiting mass (m) | Optional: the mass of the orbiting body, for two bodies of similar size. Leave empty when it is tiny. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Answer | The period, or the semi-major axis, to 6 significant figures. |
| seconds | Orbital period (s) | T in seconds. |
| minutes | Orbital period (min) | T in minutes. |
| hours | Orbital period (h) | T in hours. |
| days | Orbital period (days) | T in days of 86,400 s. |
| years | Orbital period (years) | T in Julian years of 365.25 days. |
| axisKm | Semi-major axis (km) | a, from the centre of the central body, in kilometres. |
| axisAu | Semi-major axis (au) | a in astronomical units (149,597,870,700 m). |
| height | Height above the surface (km) | a − R: how high a circular orbit of radius a is above the mean surface. |
| speed | Circular orbit speed (km/s) | √(GM ÷ a): the speed on a circular orbit of radius a. |

## Method

T = 2π √(a³ ÷ μ) and a = ∛(μT² ÷ 4π²), with μ = GM of the central body (+ G × m, the orbiting mass, when given); a = R + h for a circular orbit at height h; circular orbit speed √(μ ÷ a).

## Assumptions

- Two bodies only: no pull from other planets or moons, and the central body acts as a point mass (a sphere).
- An elliptical orbit has the period of a circular orbit whose radius is its semi-major axis; a height gives a circular orbit.

## Worked examples

1. find = period, body = earth, by = height, h = 400,000 gives seconds = 5,544.865458, minutes = 92.414424, speed = 7.672594, result = T = 1.54024 h. Source: OpenStax, University Physics Volume 1, §13.4 Satellite Orbits and Energy (Equations 13.7 and 13.8; Example 13.9: the ISS at 400 km, 7.67 × 10³ m/s and 5.55 × 10³ s), https://openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-energy (retrieved 2026-10-19) (with M_E = 5.96 × 10²⁴ kg); NASA JPL Solar System Dynamics, Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440; 1 au = 149,597,870,700 m; Julian year 365.25 d; mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19).
2. find = period, body = earth, by = axis, a = 384,400, au = km, m = 73,460,000,000,000,000,000,000 gives days = 27.284601. Source: OpenStax, University Physics Volume 1, §13.5 Kepler’s Laws of Planetary Motion (Equation 13.11, T² = 4π²a³ ÷ GM; Example 13.13: Halley’s comet, T = 75.3 years, a = 17.8 AU), https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion (retrieved 2026-10-19); NASA JPL Solar System Dynamics, Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440; 1 au = 149,597,870,700 m; Julian year 365.25 d; mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19).
3. find = period, body = sun, by = axis, a = 1, au = AU gives days = 365.256898, years = 1.000019, result = T = 1.00002 years. Source: OpenStax, University Physics Volume 1, §13.5 Kepler’s Laws of Planetary Motion (Equation 13.11, T² = 4π²a³ ÷ GM; Example 13.13: Halley’s comet, T = 75.3 years, a = 17.8 AU), https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion (retrieved 2026-10-19); NASA JPL Solar System Dynamics, Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440; 1 au = 149,597,870,700 m; Julian year 365.25 d; mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19) (the sidereal year is 365.25636 d).
4. find = axis, body = earth, by = axis, T = 86,164.09054, Tu = s, au = km gives axisKm = 42,164.169415, result = a = 42,164.2 km. Source: OpenStax, University Physics Volume 1, §13.5 Kepler’s Laws of Planetary Motion (Equation 13.11, T² = 4π²a³ ÷ GM; Example 13.13: Halley’s comet, T = 75.3 years, a = 17.8 AU), https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion (retrieved 2026-10-19); NASA JPL Solar System Dynamics, Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440; 1 au = 149,597,870,700 m; Julian year 365.25 d; mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19).
5. find = axis, body = sun, by = axis, T = 75.3, Tu = yr, au = AU gives axisAu = 17.831636, result = a = 17.8316 au. Source: OpenStax, University Physics Volume 1, §13.5 Kepler’s Laws of Planetary Motion (Equation 13.11, T² = 4π²a³ ÷ GM; Example 13.13: Halley’s comet, T = 75.3 years, a = 17.8 AU), https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion (retrieved 2026-10-19) (17.8 AU); NASA JPL Solar System Dynamics, Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440; 1 au = 149,597,870,700 m; Julian year 365.25 d; mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19).
6. find = axis, body = earth, by = height, T = 86,164.09054, Tu = s, au = km gives h = 35,793.161015, result = h = 35,793.2 km. Source: OpenStax, University Physics Volume 1, §13.5 Kepler’s Laws of Planetary Motion (Equation 13.11, T² = 4π²a³ ÷ GM; Example 13.13: Halley’s comet, T = 75.3 years, a = 17.8 AU), https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion (retrieved 2026-10-19); NASA JPL Solar System Dynamics, Astrodynamic Parameters (GM of the Sun, Earth and Moon, DE440; 1 au = 149,597,870,700 m; Julian year 365.25 d; mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19).
7. find = period, body = custom, M = 5,960,000,000,000,000,000,000,000, by = axis, a = 384,000, au = km gives seconds = 2,370,558.696621. Source: OpenStax, University Physics Volume 1, §13.4 Satellite Orbits and Energy (Equations 13.7 and 13.8; Example 13.9: the ISS at 400 km, 7.67 × 10³ m/s and 5.55 × 10³ s), https://openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-energy (retrieved 2026-10-19), Example 13.10 (T = 2.36 × 10⁶ s at 3.84 × 10⁸ m gives M_E = 6.01 × 10²⁴ kg).

## FAQ

### How do you calculate the orbital period?

Use Kepler’s third law: T = 2π √(a³ ÷ GM), with a the semi-major axis (the radius of a circular orbit) and GM the gravitational parameter of the body orbited. 400 km above the Earth, a = 6,771 km and T = 5,545 s, about 92.4 minutes.

### What is Kepler’s third law?

The square of the period is proportional to the cube of the semi-major axis: T² = 4π²a³ ÷ GM. In years, astronomical units and solar masses it becomes a³ = (M₁ + M₂)P², so a planet at 1 au from the Sun takes 1 year.

### How high is a geostationary orbit?

It turns once per sidereal day, 86,164.09 s, so a = ∛(GM T² ÷ 4π²) = 42,164 km from the Earth’s centre, 35,793 km above its mean surface.

### Does the mass of the satellite matter?

Only when it is not tiny next to the central body. Newton’s form uses both masses, T = 2π √(a³ ÷ G(M + m)). For the Moon this matters: with its mass added, 384,400 km gives 27.28 days, against 27.45 days without it.

### How do you find the semi-major axis from the period?

Turn the law around: a = ∛(GM T² ÷ 4π²). Halley’s comet returns every 75.3 years, so its semi-major axis is about 17.83 au, and its far point is 2 × 17.83 − 0.586 = 35.07 au from the Sun.

### What is the semi-major axis?

Half the longest width of an elliptical orbit, measured through the centre; for a circle it is the radius. It is also the average of the closest and farthest distances from the body orbited.

## Sources

- OpenStax, University Physics Volume 1, §13.5 Kepler’s Laws of Planetary Motion: Equation 13.11 (T² = 4π²a³ ÷ GM); a is half the sum of the perihelion and aphelion distances; Example 13.13 (Halley’s comet: T = 75.3 years, a = 17.8 AU, aphelion 35.0 AU), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion
- OpenStax, University Physics Volume 1, §13.4 Satellite Orbits and Energy: Equation 13.7 (v = √(GM ÷ r)) and 13.8 (T = 2π √(r³ ÷ GM)); Example 13.9 (the ISS at 400 km: 7.67 × 10³ m/s, 5.55 × 10³ s); Example 13.10 (the Moon: T = 2.36 × 10⁶ s at 3.84 × 10⁸ m gives M_E = 6.01 × 10²⁴ kg), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-energy
- OpenStax, Astronomy 2e, §3.3 Newton’s Universal Law of Gravitation: Kepler’s third law with both masses, a³ = (M₁ + M₂) × P² (au, years and solar masses), retrieved 2026-10-19. https://openstax.org/books/astronomy-2e/pages/3-3-newtons-universal-law-of-gravitation
- NASA JPL Solar System Dynamics, Astrodynamic Parameters: GM of the Sun 1.32712440041279419 × 10²⁰ m³/s², of the Earth 398,600.435507 km³/s², of the Moon 4,902.800118 km³/s², of Mercury 22,031.868551 km³/s² and of Venus 324,858.592 km³/s² (DE440); 1 au = 149,597,870,700 m; Julian year 365.25 d; mean sidereal day 86,164.09054 s, retrieved 2026-10-19. https://ssd.jpl.nasa.gov/astro_par.html
- NASA JPL Solar System Dynamics, Planetary Satellite Physical Parameters: planet GM values (Mars 42,828.37362; Jupiter 126,686,531.9; Saturn 37,931,206.23; Uranus 5,793,951.3; Neptune 6,835,099.97; Pluto 869.3 km³/s²) and the Moon’s mean radius 1,737.4 km, retrieved 2026-10-19. https://ssd.jpl.nasa.gov/sats/phys_par/
- NASA JPL Solar System Dynamics, Planetary Physical Parameters: mean radii of the planets, retrieved 2026-10-19. https://ssd.jpl.nasa.gov/planets/phys_par.html
- NASA NSSDCA, Sun Fact Sheet: volumetric mean radius 695,700 km, retrieved 2026-10-19. https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
- NIST, CODATA 2022: Newtonian constant of gravitation G = 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻², retrieved 2026-10-19. https://physics.nist.gov/cgi-bin/cuu/Value?bg