Orbital Velocity: How to Calculate Orbit Speed
A satellite stays up by moving sideways fast enough that, as gravity pulls it toward the planet, the curved surface falls away beneath it at the same rate. The speed that makes this work is the orbital velocity. For a circular orbit it depends on only two numbers, the mass of the central body and the distance from its center, and the Orbital Velocity Calculator computes it from exactly those inputs, plus an optional eccentricity for elliptical orbits. This guide explains the formula, works the numbers for the International Space Station, and shows how to read the result.
What orbital velocity is
Orbital velocity is the speed an object needs to hold a stable orbit at a given distance from a central body. Slower, and the orbit dips lower on the far side (or, for a satellite already at low altitude, into the atmosphere); faster, and the orbit stretches outward; at √2 times the circular speed the object escapes entirely.
The speed is set by a balance between gravity and inertia. A more massive central body pulls harder, so a higher speed is needed to stay in orbit. A larger orbit feels weaker gravity and needs a lower speed. Astronomers run the same relationship in reverse: measure how fast and how far out a moon or planet orbits, and you can solve for the mass of whatever it circles, which is how the masses of planets, stars and the Milky Way's central black hole are measured.
The formula
For a circular orbit, setting the gravitational pull equal to the centripetal force needed to bend the path gives:
v = √(G·M ÷ r)
where G is the gravitational constant, 6.674 × 10⁻¹¹ m³·kg⁻¹·s⁻² (CODATA 2022 gives 6.67430 × 10⁻¹¹), M is the central body's mass in kilograms, and r is the orbital radius in meters, measured from the center of the central body. The result is in meters per second.
The calculator multiplies this by √(1 + e), where e is the orbital eccentricity (0 for a circle). That factor is not a fudge: if you enter the periapsis distance (the closest approach) as r, then √(G·M·(1 + e) ÷ r) is exactly the speed at periapsis given by the vis-viva equation, v² = G·M·(2/r − 1/a). Leave e at 0 for a circular orbit. For an elliptical orbit the result is the maximum speed, reached at closest approach, not the average speed around the orbit.
Worked example. Estimate the speed of a satellite in a near-circular orbit 400 km above Earth's surface, about where the space station flies.
1. Earth's mass is 5.972 × 10²⁴ kg and its mean radius 6,371 km (NASA Earth fact sheet). Orbital radius = 6,371 + 400 = 6,771 km = 6.771 × 10⁶ m.
2. G·M = 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ = 3.986 × 10¹⁴ m³/s².
3. Divide by r: 3.986 × 10¹⁴ ÷ 6.771 × 10⁶ = 5.886 × 10⁷ m²/s².
4. Take the square root: √(5.886 × 10⁷) ≈ 7,672 m/s, about 7.7 km/s (values rounded to four significant figures at each step; the calculator rounds the final answer to the nearest m/s).
5. With e = 0, √(1 + 0) = 1 and the result is unchanged.
The ISS orbits at roughly 7.66–7.67 km/s and completes an orbit in about 93 minutes: one circumference, 2π × 6.771 × 10⁶ m ≈ 4.25 × 10⁷ m, divided by 7,672 m/s is about 5,550 s. The same arithmetic with the Moon's orbital radius of 3.844 × 10⁸ m gives 1,018 m/s, close to the Moon's actual mean orbital speed of about 1.02 km/s.
Reading the result
It scales predictably. Double the orbital radius and the speed falls by a factor of √2; quadruple the central mass and the speed doubles. That makes sanity checks easy: a satellite that the formula says moves faster at a higher altitude is a sign of a unit error.
It weighs distant objects. Rearranging gives M = v²·r ÷ G. The Planetary Mass Calculator does the equivalent from an orbital period and semi-major axis using Kepler's third law.
It sets the bar for launch and escape. The circular speed at a target altitude is the sideways velocity a launch vehicle must deliver, before the losses to gravity and drag that mission planners add on top. Escape speed at the same radius is √2 times higher: for the 400 km orbit above, about 10,850 m/s. The Escape Velocity Calculator computes √(2·G·M ÷ r) directly from the same mass and radius.
Common mistakes
Using altitude instead of radius. r is measured from the center of the central body, not its surface. For an Earth orbit add Earth's radius, about 6,371 km, to the altitude. Entering 400 km as r would give a speed of about 31.6 km/s, four times too high.
Mixing units. The constant assumes SI units: kilograms and meters in, meters per second out. Convert kilometers, astronomical units or solar masses before entering them (1 AU = 1.496 × 10¹¹ m; the Sun's mass is 1.989 × 10³⁰ kg).
Misreading the eccentricity term. With e > 0 the calculator returns the periapsis speed, and only if the radius you entered is the periapsis distance. Speed elsewhere on an ellipse needs the full vis-viva equation.
Expecting precision navigation. The formula is the idealized two-body result. Atmospheric drag, the gravity of other bodies and Earth's non-spherical shape perturb real orbits; the ISS, for example, needs periodic reboosts because drag slowly lowers it.
Key takeaways
- Formula: v = √(G·M ÷ r), with M in kilograms and r in meters from the center of the central body, gives speed in m/s; the calculator multiplies by √(1 + e) for the periapsis speed of an elliptical orbit.
- Example: a 400 km Earth orbit (r = 6.771 × 10⁶ m) gives about 7,672 m/s, matching the ISS's roughly 7.66 km/s.
- Radius, not altitude: add the body's radius to the altitude before entering r.
- Escape is √2 higher: about 10.85 km/s at the same 400 km orbit.