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Orbital Period Calculator

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.

When to use

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.

Formula

T = 2π × √(a³ / GM); v_circular = √(GM / a); GM(custom) = 6.67430 × 10⁻¹¹ × central_mass_kg

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.

Inputs

Parameter	Type	Unit	Required	Description
`semi_major_axis`	number		yes	Orbit size a, measured from the centre of the central body (radius of a circular orbit), in the unit given by axis_unit. Range: > 0, ≤ 1000000000000000000
`axis_unit`	enum: km \| au \| m		default km	Unit of semi_major_axis.
`central_body`	enum: sun \| earth \| moon \| mars \| jupiter \| custom		default earth	Body being orbited; its GM is taken from the JPL astrodynamic constants.
`central_mass_kg`	number	kg	no	Mass of the central body, only used when central_body is custom (GM = G × M with G = 6.67430 × 10⁻¹¹). Range: > 0, ≤ 1e+42

Outputs

Output	Type	Unit	Description
`period_seconds`	number	s	T = 2π √(a³ / GM).
`period_minutes`	number	min	T / 60.
`period_hours`	number	h	T / 3600.
`period_days`	number	days	T / 86,400.
`period_years`	number	years	T / (365.25 × 86,400) (Julian years).
`orbital_speed_km_s`	number	km/s	√(GM / a), the speed of a circular orbit of radius a.
`semi_major_axis_km`	number	km	The orbit size in kilometres.
`gm_m3_s2`	number	m³/s²	Standard gravitational parameter of the central body.

Example

Earth around the Sun (1 AU): {"semi_major_axis":1,"axis_unit":"au","central_body":"sun"} → {"period_days":365.2569,"period_years":1.00002,"orbital_speed_km_s":29.785,"semi_major_axis_km":149597870.7}

ISS: 6,791 km from Earth's centre (about 420 km altitude): {"semi_major_axis":6791,"axis_unit":"km","central_body":"earth"} → {"period_seconds":5569.44,"period_minutes":92.824,"orbital_speed_km_s":7.661}

GET https://tttkmbb.com/api/v1/calculate/orbital-period?semi_major_axis=1&axis_unit=au&central_body=sun

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/orbital-period (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/orbital-period · Markdown: https://tttkmbb.com/science/orbital-period.md · JSON definition: https://tttkmbb.com/science/orbital-period.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="orbital-period"
OpenAPI operationId: calculate_orbital_period
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Orbital period (reference)
Wikipedia – Standard gravitational parameter (reference)
NASA JPL Solar System Dynamics – Astrodynamic parameters (GM of the Sun and planets) (government)

FAQ

Is the semi-major axis the altitude?

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.

Does the mass of the orbiting body matter?

Only when it is not negligible compared with the central body (e.g. binary stars, Pluto–Charon): then GM should be G × (M + m).

Why is the year 365.26 days rather than 365.24?

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.

Related calculators

Escape Velocity Calculator — Escape speed and circular orbital speed at the surface of the same bodies.
Centripetal Force Calculator — Centripetal force and acceleration of the circular orbit.
Gravitational Force Calculator — The gravitational force providing the centripetal acceleration.