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Gravitation and circular orbit calculator

Orbital speed, period, escape velocity and local gravity of a circular orbit, for any central body, ISS included.

System

The central body
kgExample value: 5.972 × 10²⁴ kg, the mass of the Earth (Sun ≈ 1.989 × 10³⁰ kg, Moon ≈ 7.35 × 10²² kg). Enter as 5.972e24.
kmUsed for display only (disc to scale, altitude h = r − R). Example value: 6,371 km.
The orbit
kmMeasured from the centre of the body, not from its surface: body radius + altitude. Example value: 6,771 km, the ISS orbit (6,371 + 400).

Orbit

the plan and the climb
M 5.97 × 10²⁴ kg · r 6771 km · R 6371 kmvgcentral bodysatellite1 px = 0.137 km/s1 px = 0.229 m/s²orbital radius r 6771 kmbody radius R 6371 kmaltitude h 400 km0510speed (km/s)v 7.67 · vesc 10.90100200period (min)T 92.4 min0500010 0000510radius from the centre (km)gravity (m/s²)g 8.69
Speed
v = √(G·M/r), G = 6.6743 × 10⁻¹¹ m³/(kg·s²) → v = 7.6725 km/s
Period
T = 2π·r/v = 5544.9 s
Escape
vesc = √(2·G·M/r) = √2·v = 10.851 km/s
Note

The light band in the three panels is the inside of the body, where these laws do not apply.

the three laws of the radiusthe escape speed, √2 times the orbital onethe values at the entered orbit

Result

continuous computation
At this orbit
v7.6725km/scompleted in 1 h 32 min
Orbital period
5544.9 s
Escape speed
10.851 km/s
Gravity at this radius
8.694 m/s²

Escape speed from this radius: 10.851 km/s, that is √2 times the 7.6725 km/s of the orbit.

What the mass fixes
Kepler constant
10 096 km³/s²

It depends only on the central mass.

Calculation, assumptions and sources

OverviewThe mass of the central body and the orbital radius give the orbital speed, the period of revolution (Kepler’s 3rd law), the escape speed and the local gravity.

The mass of the central body and the orbital radius give the orbital speed, the period of revolution (Kepler’s 3rd law), the escape speed and the local gravity.

Method & formulasv = √(G·M / r)
v = √(G·M / r)T = 2π·√(r³ / G·M), Kepler: r³/T² = G·M/4π²vesc = √(2·G·M / r) = √2 · vg(r) = G·M / r²G = 6.67430 × 10⁻¹¹ m³/(kg·s²), codata 2018

Gravitation supplies the centripetal acceleration (G·M·m/r² = m·v²/r) and the mass of the satellite cancels out. Kepler’s third law follows. The escape speed comes from energy (½mv² = G·M·m/r) and is √2 times the circular orbital speed, at any altitude. G is known to 2.2 × 10⁻⁵ in relative terms; the results inherit this precision.

Orbital radius (r)
the distance to the centre of the central body: body radius plus altitude.
Escape velocity
the minimum speed to recede to infinity with no further propulsion, from distance r. It does not depend on the direction of launch; 11.2 km/s from the Earth’s surface.
Body radius (R)
used for display only: the central body to the scale of the orbit and the altitude h = r − R. The calculation uses only M and r.
Kepler constant (r³/T²)
the same for all satellites of one central body, the Moon as well as the ISS.
Validity domainCircular orbits around a spherically symmetric body, satellite mass negligible compared with M.

Circular orbits around a spherically symmetric body, satellite mass negligible compared with M. Elliptical orbits, perturbations (J2 oblateness, atmospheric drag, third body) and relativistic effects are outside the model. A geostationary orbit targets the sidereal day (86,164 s), not the 86,400 s solar day. The body masses given as hints are known far better than G; the uncertainty comes from G.

Orbital radius and altitudeFor the ISS, r = 6,771 km.

For the ISS, r = 6,771 km. Using the 400 km altitude instead of the radius would give 31.6 km/s instead of 7.67 km/s and a period of 80 seconds instead of 92 minutes. The orbital speed varies as 1/√r: the ISS goes round in 92 minutes, the Moon in 27 days. A braked satellite descends, then speeds up.