{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"orbital-period"},"result":{"entity_type":"calculator","id":"orbital-period","calculator_id":"orbital-period","canonical_url":"https://tttkmbb.com/science/orbital-period","name":"Orbital Period Calculator","title":"Orbital Period Calculator – Kepler's Third Law Period and Circular Orbital Speed around the Sun, Earth or Any Mass","category":"science","category_name":"Biology, Earth & Space","tool_name":"calculate_orbital_period","featured_mcp_tool":false,"description":"Computes the orbital period from the semi-major axis and the central body's gravitational parameter GM with Kepler's third law T = 2π√(a³/GM), plus the circular orbital speed √(GM/a), for the Sun, Earth, Moon, Mars, Jupiter or a custom mass.","use_when":"You need the period or speed of a satellite, moon or planet from its orbit size (e.g. the ISS at 6,791 km from Earth's centre, or 1 AU around the Sun).","do_not_use_when":"You need the speed to leave a body (use escape-velocity), a transfer orbit or the position along an elliptical orbit at a given time, or relativistic corrections.","inputs":[{"name":"semi_major_axis","label":"Semi-major axis","type":"number","required":true,"max":1000000000000000000,"exclusive_min":0,"description":"Orbit size a, measured from the centre of the central body (radius of a circular orbit), in the unit given by axis_unit.","example":6791},{"name":"axis_unit","label":"Axis unit","type":"enum","required":false,"default":"km","values":[{"value":"km","label":"Kilometres"},{"value":"au","label":"Astronomical units (1 AU = 149,597,870.7 km)","aliases":["astronomical_unit"]},{"value":"m","label":"Metres"}],"description":"Unit of semi_major_axis.","example":"km"},{"name":"central_body","label":"Central body","type":"enum","required":false,"default":"earth","values":[{"value":"sun","label":"Sun"},{"value":"earth","label":"Earth"},{"value":"moon","label":"Moon"},{"value":"mars","label":"Mars"},{"value":"jupiter","label":"Jupiter"},{"value":"custom","label":"Custom mass (give central_mass_kg)"}],"description":"Body being orbited; its GM is taken from the JPL astrodynamic constants.","example":"earth"},{"name":"central_mass_kg","label":"Central mass (custom)","type":"number","unit":"kg","required":false,"max":1e+42,"exclusive_min":0,"description":"Mass of the central body, only used when central_body is custom (GM = G × M with G = 6.67430 × 10⁻¹¹).","example":5.972e+24}],"outputs":[{"name":"period_seconds","label":"Period","type":"number","unit":"s","decimals":2,"description":"T = 2π √(a³ / GM)."},{"name":"period_minutes","label":"Period (minutes)","type":"number","unit":"min","decimals":3,"description":"T / 60."},{"name":"period_hours","label":"Period (hours)","type":"number","unit":"h","decimals":4,"description":"T / 3600."},{"name":"period_days","label":"Period (days)","type":"number","unit":"days","decimals":4,"description":"T / 86,400."},{"name":"period_years","label":"Period (years)","type":"number","unit":"years","decimals":5,"description":"T / (365.25 × 86,400) (Julian years)."},{"name":"orbital_speed_km_s","label":"Circular orbital speed","type":"number","unit":"km/s","decimals":3,"description":"√(GM / a), the speed of a circular orbit of radius a."},{"name":"semi_major_axis_km","label":"Semi-major axis","type":"number","unit":"km","decimals":3,"description":"The orbit size in kilometres."},{"name":"gm_m3_s2","label":"GM used","type":"number","unit":"m³/s²","decimals":0,"description":"Standard gravitational parameter of the central body."}],"input_schema":{"type":"object","properties":{"semi_major_axis":{"description":"Orbit size a, measured from the centre of the central body (radius of a circular orbit), in the unit given by axis_unit.","type":"number","maximum":1000000000000000000,"exclusiveMinimum":0,"examples":[6791]},"axis_unit":{"description":"Unit of semi_major_axis.","type":"string","enum":["km","au","m"],"default":"km","examples":["km"]},"central_body":{"description":"Body being orbited; its GM is taken from the JPL astrodynamic constants.","type":"string","enum":["sun","earth","moon","mars","jupiter","custom"],"default":"earth","examples":["earth"]},"central_mass_kg":{"description":"Mass of the central body, only used when central_body is custom (GM = G × M with G = 6.67430 × 10⁻¹¹). Unit: kg.","type":"number","maximum":1e+42,"exclusiveMinimum":0,"examples":[5.972e+24],"x-unit":"kg"}},"additionalProperties":false,"required":["semi_major_axis"]},"output_schema":{"type":"object","properties":{"period_seconds":{"description":"T = 2π √(a³ / GM). Unit: s.","type":"number","x-unit":"s"},"period_minutes":{"description":"T / 60. Unit: min.","type":"number","x-unit":"min"},"period_hours":{"description":"T / 3600. Unit: h.","type":"number","x-unit":"h"},"period_days":{"description":"T / 86,400. Unit: days.","type":"number","x-unit":"days"},"period_years":{"description":"T / (365.25 × 86,400) (Julian years). Unit: years.","type":"number","x-unit":"years"},"orbital_speed_km_s":{"description":"√(GM / a), the speed of a circular orbit of radius a. Unit: km/s.","type":"number","x-unit":"km/s"},"semi_major_axis_km":{"description":"The orbit size in kilometres. Unit: km.","type":"number","x-unit":"km"},"gm_m3_s2":{"description":"Standard gravitational parameter of the central body. Unit: m³/s².","type":"number","x-unit":"m³/s²"}}},"formula":"T = 2π × √(a³ / GM); v_circular = √(GM / a); GM(custom) = 6.67430 × 10⁻¹¹ × central_mass_kg","method":"Kepler's third law for a body of negligible mass in a two-body orbit. GM values (m³/s²): Sun 1.32712440018 × 10²⁰, Earth 3.986004418 × 10¹⁴, Moon 4.9048695 × 10¹², Mars 4.282837 × 10¹³, Jupiter 1.26686534 × 10¹⁷. For an elliptical orbit the speed output is the circular value at radius a, not the varying true speed.","sources":[{"name":"Wikipedia – Orbital period","url":"https://en.wikipedia.org/wiki/Orbital_period","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Standard gravitational parameter","url":"https://en.wikipedia.org/wiki/Standard_gravitational_parameter","type":"reference","retrieved_at":"2026-09-24"},{"name":"NASA JPL Solar System Dynamics – Astrodynamic parameters (GM of the Sun and planets)","url":"https://ssd.jpl.nasa.gov/astro_par.html","type":"government","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"Earth around the Sun (1 AU)","inputs":{"semi_major_axis":1,"axis_unit":"au","central_body":"sun"},"expected":{"period_days":365.2569,"period_years":1.00002,"orbital_speed_km_s":29.785,"semi_major_axis_km":149597870.7},"url":"https://tttkmbb.com/api/v1/calculate/orbital-period?semi_major_axis=1&axis_unit=au&central_body=sun"},{"name":"ISS: 6,791 km from Earth's centre (about 420 km altitude)","inputs":{"semi_major_axis":6791,"axis_unit":"km","central_body":"earth"},"expected":{"period_seconds":5569.44,"period_minutes":92.824,"orbital_speed_km_s":7.661},"url":"https://tttkmbb.com/api/v1/calculate/orbital-period?semi_major_axis=6791&axis_unit=km&central_body=earth"}],"faq":[{"q":"Is the semi-major axis the altitude?","a":"No: it is measured from the centre of the central body, so add the body's radius to the altitude (ISS: 6,371 km + 420 km ≈ 6,791 km). A note is added when a is smaller than the body's radius."},{"q":"Does the mass of the orbiting body matter?","a":"Only when it is not negligible compared with the central body (e.g. binary stars, Pluto–Charon): then GM should be G × (M + m)."},{"q":"Why is the year 365.26 days rather than 365.24?","a":"Kepler's law gives the sidereal year (365.256 days, one orbit relative to the stars); the calendar (tropical) year of 365.242 days is shorter because of the precession of the equinoxes."}],"tags":["orbital period","kepler third law","orbital speed","satellite period","semi major axis","gravitational parameter"],"related":[{"calculator_id":"escape-velocity","reason":"Escape speed and circular orbital speed at the surface of the same bodies."},{"calculator_id":"centripetal-force","reason":"Centripetal force and acceleration of the circular orbit."},{"calculator_id":"gravitational-force","reason":"The gravitational force providing the centripetal acceleration."}],"links":{"html":"https://tttkmbb.com/science/orbital-period","markdown":"https://tttkmbb.com/science/orbital-period.md","json":"https://tttkmbb.com/science/orbital-period.json","api":"https://tttkmbb.com/api/v1/calculate/orbital-period","schema":"https://tttkmbb.com/api/v1/calculators/orbital-period","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T03:45:46Z"}