{
  "id": "orbital-period",
  "version": "a3efd80c08f0",
  "status": "published",
  "name": "Orbital Period Calculator",
  "question": "What is the orbital period?",
  "summary": "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.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/orbital-period-calculator",
  "markdown": "https://www.acalculator.org/physics/orbital-period-calculator.md",
  "kind": "function",
  "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."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "The period from the size of the orbit, or the size from the period.",
        "type": "string",
        "enum": [
          "period",
          "axis"
        ]
      },
      "body": {
        "title": "Orbiting around",
        "description": "The central body, or your own mass.",
        "type": "string",
        "enum": [
          "earth",
          "sun",
          "moon",
          "mercury",
          "venus",
          "mars",
          "jupiter",
          "saturn",
          "uranus",
          "neptune",
          "pluto",
          "custom"
        ]
      },
      "M": {
        "title": "Central mass (M)",
        "description": "The mass of the body being orbited.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0.000001,
        "maximum": 1e+40
      },
      "by": {
        "title": "Size of orbit as",
        "description": "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.",
        "type": "string",
        "enum": [
          "height",
          "axis"
        ]
      },
      "h": {
        "title": "Height above the surface (h)",
        "description": "The height of a circular orbit above the mean surface.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0,
        "maximum": 1000000000000
      },
      "R": {
        "title": "Body radius (R)",
        "description": "The radius of your body, to add to the height.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.001,
        "maximum": 1000000000000000
      },
      "a": {
        "title": "Semi-major axis (a)",
        "description": "Half the longest width of the orbit, from the centre: the radius of a circular orbit.",
        "type": "number",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "au": {
        "title": "Axis or height unit",
        "description": "The unit of the semi-major axis you type, or of the axis or height found from a period.",
        "type": "string",
        "enum": [
          "km",
          "m",
          "mi",
          "AU"
        ]
      },
      "T": {
        "title": "Orbital period (T)",
        "description": "The time for one orbit, in the unit chosen below.",
        "type": "number",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "Tu": {
        "title": "Period unit",
        "description": "The unit of the orbital period.",
        "type": "string",
        "enum": [
          "s",
          "min",
          "h",
          "d",
          "yr"
        ]
      },
      "m": {
        "title": "Orbiting mass (m)",
        "description": "Optional: the mass of the orbiting body, for two bodies of similar size. Leave empty when it is tiny.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0,
        "maximum": 1e+40
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The period, or the semi-major axis, to 6 significant figures.",
      "format": "text"
    },
    "seconds": {
      "label": "Orbital period (s)",
      "description": "T in seconds.",
      "format": "number"
    },
    "minutes": {
      "label": "Orbital period (min)",
      "description": "T in minutes.",
      "format": "number"
    },
    "hours": {
      "label": "Orbital period (h)",
      "description": "T in hours.",
      "format": "number"
    },
    "days": {
      "label": "Orbital period (days)",
      "description": "T in days of 86,400 s.",
      "format": "number"
    },
    "years": {
      "label": "Orbital period (years)",
      "description": "T in Julian years of 365.25 days.",
      "format": "number"
    },
    "axisKm": {
      "label": "Semi-major axis (km)",
      "description": "a, from the centre of the central body, in kilometres.",
      "format": "number"
    },
    "axisAu": {
      "label": "Semi-major axis (au)",
      "description": "a in astronomical units (149,597,870,700 m).",
      "format": "number"
    },
    "height": {
      "label": "Height above the surface (km)",
      "description": "a − R: how high a circular orbit of radius a is above the mean surface.",
      "format": "number"
    },
    "speed": {
      "label": "Circular orbit speed (km/s)",
      "description": "√(GM ÷ a): the speed on a circular orbit of radius a.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "period",
      "body": "earth",
      "M": "1.989e30 kg",
      "by": "height",
      "h": "400 km",
      "R": "6371 km",
      "a": 384400,
      "au": "km",
      "T": 1,
      "Tu": "d"
    },
    "outputs": {
      "result": "T = 1.54024 h",
      "seconds": 5544.865458049257,
      "minutes": 92.41442430082094,
      "hours": 1.5402404050136824,
      "days": 0.06417668354223677,
      "years": 0.0001757061835516407,
      "axisKm": 6771.0084,
      "axisAu": 0.00004526139555541147,
      "speed": 7.672593828568173
    },
    "text": "T = 1.54024 h, with a circular orbit speed of 7.67259 km/s."
  },
  "examples": [
    {
      "given": {
        "find": "period",
        "body": "earth",
        "by": "height",
        "h": 400000
      },
      "expect": {
        "seconds": 5544.865458049257,
        "minutes": 92.41442430082094,
        "speed": 7.672593828568173,
        "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); hand calculation in content.mdx: 2π √(6,771,008.4³ ÷ 3.98600435507 × 10¹⁴)"
    },
    {
      "given": {
        "find": "period",
        "body": "earth",
        "by": "axis",
        "a": 384400,
        "au": "km",
        "m": 7.346e+22
      },
      "expect": {
        "days": 27.28460104210842
      },
      "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); OpenStax Astronomy 2e §3.3 (a³ = (M₁ + M₂)P²); 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); hand calculation in content.mdx"
    },
    {
      "given": {
        "find": "period",
        "body": "sun",
        "by": "axis",
        "a": 1,
        "au": "AU"
      },
      "expect": {
        "days": 365.2568983272364,
        "years": 1.000018886590654,
        "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); hand calculation in content.mdx: 2π √(1 au³ ÷ GM_Sun)"
    },
    {
      "given": {
        "find": "axis",
        "body": "earth",
        "by": "axis",
        "T": 86164.09054,
        "Tu": "s",
        "au": "km"
      },
      "expect": {
        "axisKm": 42164.16941524308,
        "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); hand calculation in content.mdx: ∛(GM_E × 86,164.09054² ÷ 4π²)"
    },
    {
      "given": {
        "find": "axis",
        "body": "sun",
        "by": "axis",
        "T": 75.3,
        "Tu": "yr",
        "au": "AU"
      },
      "expect": {
        "axisAu": 17.83163568726182,
        "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); hand calculation in content.mdx with Julian years"
    },
    {
      "given": {
        "find": "axis",
        "body": "earth",
        "by": "height",
        "T": 86164.09054,
        "Tu": "s",
        "au": "km"
      },
      "expect": {
        "h": 35793.16101524308,
        "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); hand calculation in content.mdx: 42,164.169 − 6,371.0084 km"
    },
    {
      "given": {
        "find": "period",
        "body": "custom",
        "M": 5.96e+24,
        "by": "axis",
        "a": 384000,
        "au": "km"
      },
      "expect": {
        "seconds": 2370558.6966213584
      },
      "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); hand calculation in content.mdx with G = 6.67430 × 10⁻¹¹"
    }
  ],
  "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"
  ],
  "related": [
    "escape-velocity",
    "gravitational-force",
    "angular-velocity",
    "centripetal-force"
  ],
  "changelog": []
}