Orbital velocity and period, escape velocity, surface gravity, your weight on the Moon or Mars, and the gravitational force between two masses.
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A circular orbit 400 km above the Earth needs 7.6726 km/s and takes 92.414 minutes per lap. Mass of the satellite doesn't matter.
Newton's law of gravitation, F = Gm₁m₂/r², gives the pull between two masses. For a body of mass M the same law sets the speed of a circular orbit, v = √(GM/r), its period from Kepler's third law, T = 2π√(r³/GM), the escape velocity √(2GM/r) and the surface gravity GM/R². Pick the Earth, Moon, Mars, Jupiter or the Sun, or enter any mass and radius.
The default, an orbit 400 km above Earth like the International Space Station's, gives 7.67 km/s (about 17,200 mph) and 92.4 minutes per lap. Entering one sidereal day, 23.934 hours, as the period returns the geostationary radius of 42,164 km.
Radii are volumetric means from NASA's fact sheets and rotation is ignored, so Earth's surface gravity comes out at 9.82 m/s², between the 9.78 m/s² measured at the equator and 9.83 m/s² at the poles. Orbits are circular and drag-free.
A circular orbit's speed is v = √(GM/r), with r measured from the planet's center. At 400 km above Earth, r = 6,371 + 400 = 6,771 km and GM = 3.986 × 10¹⁴ m³/s², so v = 7,672.62 m/s, about 7.67 km/s, and one lap takes 92.41 minutes.
Add the body's radius to the altitude before using the formula: r = R + h. Plugging in the 400 km altitude alone gives 31,567 m/s, more than four times the real orbital speed.
The satellite's own mass cancels out, so a space station and a loose bolt at the same height move at the same speed.
Lower orbits are faster. Skimming Earth's mean surface would take 7,909.81 m/s and 84.35 minutes per lap, while the geostationary radius needs only 3,074.67 m/s.
The period of a circular orbit is T = 2π√(r³/GM). For the 400 km orbit, T = 5,544.8 s, or 92.41 minutes.
Solved for the radius, r = (GMT²/4π²)^(1/3). One sidereal day, 23.9344696 hours, gives r = 42,164.2 km, the geostationary radius. A satellite in an equatorial orbit there keeps pace with Earth's rotation.
Subtracting Earth's mean radius of 6,371 km leaves an altitude of 35,793.2 km. The usual quoted figure, 35,786 km, subtracts the 6,378 km equatorial radius instead.
The OpenStax chapter on satellite orbits derives both formulas from the inward force a circular path needs.
Escape velocity is v_esc = √(2GM/r), √2 ≈ 1.414 times the circular-orbit speed at the same radius. From Earth's surface it is 11,186 m/s; from 400 km up it is 10,851 m/s.
It is the speed at which an object coasting with no further thrust and no air drag never falls back. A real launch needs more, because drag, gravity losses and steering add to the requirement.
Surface gravity is g = GM/R², and weight is mass times g. With the masses and mean radii in NASA's planetary fact sheets:
The mass stays 70 kg everywhere; only the weight changes. The model treats each body as a non-rotating sphere, so Earth comes out at 9.820 m/s², a little above standard gravity, 9.80665 m/s².
Newton's law of gravitation is F = Gm₁m₂/r², with r measured center to center. Two 1 kg masses 1 m apart attract with 6.674 × 10⁻¹¹ N. Earth and the Moon, 384,400 km apart, pull on each other with 1.98 × 10²⁰ N.
G = 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² is the CODATA 2022 value. It is known to about 2 parts in 100,000, so the last one or two significant figures of any result are uncertain.
Default is the 400 km Earth orbit. Sidereal period finds the altitude for a 23.9344696-hour period, escape surface is escape velocity from Earth's surface, and Moon weight is a 70 kg mass on the Moon.
| Scenario | Selected result |
|---|---|
| Default | 7,672.62 m/s |
| Sidereal period | 35,793.2 km altitude |
| Escape surface | 11,186.2 m/s |
| Moon weight | 113.699 N |
For the inward acceleration behind any circular path, use the circular motion calculator. Short flights near the ground, where g is constant, belong in the projectile motion calculator.
Checked against: GM = 6.6743e-11 × 5.9722e24, r = 6771 km, v = √(GM/r) = 7672.619, T = 2π√(r³/GM) = 1.5402335 h
Checked against: r = (GM T²/4π²)^⅓ = 42164.2 km (the published GEO radius); minus the 6371 km mean radius (35,786 km is quoted above the 6378 km equatorial radius)
Checked against: √(GM/R) with R = 6371.0 km = 7909.813
Checked against: √(2GM/R) = 11186.165 m/s (NASA fact sheet lists 11.186 km/s)
About 7.67 km/s at 400 km altitude, which is 27,600 km/h or roughly 17,200 mph, and one lap takes 92.4 minutes. The speed depends only on Earth's GM and the orbit radius (6,771 km here), not on the station's mass. Raising the orbit to 410 km slows it by 5.7 m/s and lengthens the lap by 12 seconds.
11.186 km/s from the surface, about 40,270 km/h, the figure on NASA's Earth fact sheet. It is the speed at which an unpowered object never falls back, ignoring air drag, and it is the same for a pebble and a rocket. The Moon's is 2.38 km/s, Mars's 5.03 km/s and Jupiter's about 60 km/s. From 400 km up it drops to 10.85 km/s.
Escaping takes twice the kinetic energy of a circular orbit at the same height. In a circular orbit the kinetic energy is GMm/2r; reaching infinity needs GMm/r, the full depth of the potential well. Doubling kinetic energy multiplies speed by √2 ≈ 1.414, so the 7.67 km/s orbital speed at 400 km above Earth becomes an escape speed of 10.85 km/s.
35,786 km above the equator, a radius of 42,164 km from Earth's centre. At that radius Kepler's third law gives a period of one sidereal day, 23 h 56 min 4 s, so the satellite keeps pace with Earth's rotation. Entering that period here gives the same radius and an altitude of 35,793 km, because the calculator subtracts the 6,371 km mean radius rather than the 6,378 km equatorial radius.
About 16.6% of your Earth weight on the Moon and 38% on Mars. Surface gravity GM/R² is 1.62 m/s² on the Moon and 3.73 m/s² on Mars with NASA's mean radii, against the 9.80665 m/s² standard gravity defined for Earth. A 70 kg person weighs 113.7 N on the Moon, which a scale calibrated on Earth would show as 11.6 kg.
This calculator includes 8 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.