Orbital Velocity Calculator

Find the Speed Needed for a Circular Orbit

Orbital Velocity Formula

Formula

\[ v = \sqrt{\frac{GM}{r}} \qquad \text{with} \qquad r = R + h \]

Where:

\(v\) — orbital velocity, the constant speed of the object along the circular orbit (m/s)
\(G\) — gravitational constant, \(G = 6.67430 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}\)
\(M\) — mass of the central body (kg)
\(r\) — orbital radius, the distance from the centre of the body to the orbit (m)
\(R\) — mean radius of the central body (m)
\(h\) — altitude of the orbit above the body's surface (m)

The formula states that in a circular orbit, gravity provides exactly the centripetal force needed to keep the object moving in a circle: setting \(GMm/r^2 = mv^2/r\) and solving for \(v\) gives the result. The calculator also computes the orbital period from \(T = 2\pi\sqrt{r^3/GM}\).

Orbital Velocity – Calculation Example

Example: the International Space Station flies in a roughly circular orbit about 400 km above Earth's surface. Find its orbital speed and period.

Earth's standard gravitational parameter is \(GM = 3.986004 \times 10^{14}\ \text{m}^3/\text{s}^2\) and its mean radius is \(R = 6371\ \text{km}\). The orbital radius is:

\[ r = R + h = 6371\ \text{km} + 400\ \text{km} = 6771\ \text{km} = 6.771 \times 10^{6}\ \text{m} \]

\[ v = \sqrt{\frac{3.986004 \times 10^{14}}{6.771 \times 10^{6}}} \approx 7673\ \text{m/s} \approx 7.67\ \text{km/s} \]

\[ T = 2\pi\sqrt{\frac{(6.771 \times 10^{6})^3}{3.986004 \times 10^{14}}} \approx 5545\ \text{s} \approx 92.4\ \text{min} \]

Result: the ISS must travel at about 7.67 km/s (roughly 27,600 km/h or 17,150 mph) to hold its 400 km circular orbit, completing one lap around Earth — a distance of about 42,543 km — in about 92 minutes.

How to Use the Calculator

Select the Central body — Earth, Moon, Mars, Jupiter, the Sun, or Custom body. A preset body automatically uses its known mass and radius.
Choose the Distance type: either Altitude above the surface (typical for spacecraft orbits) or Orbital radius from the centre.
Enter the Orbit altitude (h) (or the orbital radius, depending on your choice) and pick its unit — km, m, or miles.
If you chose Custom body, fill in the Mass of the body (M) in Earth masses, kilograms, or solar masses. When entering an altitude, also provide the Radius of the body (R) so the calculator can convert altitude to orbital radius.
Read the results: the orbital speed in six different units, plus the orbital period and the distance travelled per revolution.

What Is Orbital Velocity?

Orbital velocity is the speed an object must maintain to travel in a circular path around a celestial body without falling into it or drifting away. It is not a matter of escaping gravity — gravity is fully present in orbit — but of moving sideways fast enough that the curved fall toward the body exactly matches the curvature of the orbit. An astronaut aboard the ISS is in continuous free fall; the station simply moves forward so fast that it keeps missing the Earth.

The key insight from the formula \(v = \sqrt{GM/r}\) is that orbital speed depends only on the mass of the central body and the orbital radius — not on the mass of the orbiting object. A bowling ball and a feather at the same altitude require exactly the same speed. Notice also the inverse relationship with distance: the higher the orbit, the slower the required speed. At 400 km the ISS races along at 7.67 km/s, while a satellite in geostationary orbit at 35,786 km needs only about 3.07 km/s.

The orbital period \(T = 2\pi\sqrt{r^3/GM}\) tells you how long one full revolution takes. It grows rapidly with altitude because the circumference grows while the speed shrinks. Kepler's third law hides inside this expression: the square of the period is proportional to the cube of the orbital radius. For Earth, an orbit whose period equals one sidereal day (23 h 56 min) lies at 35,786 km altitude — this is the geostationary belt used by communications satellites.

Assumptions, Limitations and Practical Tips

Circular orbits only. The formula gives the exact speed for a circular orbit. Real orbits are usually ellipses, where the speed varies along the path — fastest at the lowest point (perigee), slowest at the highest point (apogee).
Point-mass assumption. The body is treated as if all its mass were concentrated at its centre, which is excellent for planets and stars but less accurate close to irregular bodies such as small asteroids.
No atmosphere and no perturbations. Drag from an atmosphere, atmospheric braking, the gravity of moons or other planets, and relativistic effects near very massive bodies are all ignored. Low orbits decay over time precisely because of drag.
Two-body problem only. The calculation assumes the orbiting object's mass is negligible compared with the central body. This holds for satellites, spacecraft, and even planets orbiting the Sun.
Altitude of zero means an orbit skimming the surface — physically impossible for bodies with an atmosphere, but useful for comparing theoretical surface-orbit speeds (about 7.91 km/s for Earth).
When using a Custom body, make sure the orbital radius you enter is larger than the body's radius; otherwise the calculator will warn you that the orbit would lie inside the body.

Related Physics

Orbital motion is a classic application of centripetal force — gravity plays the role of the force pulling the object toward the centre. The same balance also follows from the conservation of energy: in a circular orbit, kinetic and gravitational potential energy stay constant, and trading altitude for speed in elliptical orbits is a direct energy exchange.

Frequently Asked Questions

What is the difference between orbital velocity and escape velocity?

Orbital velocity is the speed needed to stay in a circular orbit, while escape velocity is the speed needed to break free of the body's gravity entirely. For any given radius, escape velocity equals the circular orbital speed multiplied by \(\sqrt{2}\), i.e. about 1.41 times larger. Around Earth's surface that means 7.91 km/s to orbit but 11.19 km/s to escape. A satellite traveling exactly at orbital speed has just enough energy to keep circling forever, but not enough to leave.

Why do higher orbits require slower speeds?

Gravity weakens with distance, following the inverse-square law, so a satellite farther from the body needs less sideways speed to keep missing the surface. The formula \(v = \sqrt{GM/r}\) shows the speed shrinking as the radius grows. Near Earth the ISS flies at 7.67 km/s at 400 km, while the Moon, at about 384,400 km, moves at only about 1.02 km/s. Counterintuitively, though, higher orbits still take longer to complete because the circumference grows and the speed falls.

How fast does a satellite travel in low Earth orbit?

Typical low Earth orbit (LEO) satellites travel between about 7.5 and 7.8 km/s, or roughly 27,000–28,000 km/h. The ISS at 400 km moves at 7.67 km/s, and most LEO satellites sit between 200 and 2,000 km altitude, where speeds vary only slightly because the altitude is small compared with Earth's 6,371 km radius. At these speeds a satellite circles the entire planet in about 90–130 minutes.

What altitude gives a geostationary orbit around Earth?

A geostationary orbit lies at 35,786 km above the equator, where the orbital period equals one sidereal day (23 h 56 min 4 s). At that altitude the satellite moves at about 3.07 km/s and appears fixed over one spot on the ground, which is why TV and weather satellites use this belt. The same logic applies to other bodies: a geostationary orbit around Mars, for example, lies at about 17,000 km altitude.

Can I use this calculator for elliptical orbits?

No — the formula \(v = \sqrt{GM/r}\) applies only to circular orbits. In an elliptical orbit the speed constantly changes: the satellite moves fastest at perigee (its closest point) and slowest at apogee (its farthest point). For ellipses you can use the vis-viva equation \(v = \sqrt{GM(2/r - 1/a)}\), where \(a\) is the semi-major axis. The circular result equals the vis-viva speed when the ellipse becomes a circle, i.e. when \(r = a\).

Why do astronauts float inside the ISS if it travels so fast?

Astronauts float not because gravity is absent but because they are in continuous free fall together with the spacecraft. Gravity at 400 km altitude is still about 90% as strong as on the surface. The station and everything in it accelerate toward Earth at the same rate, so there is no contact force between them — this creates the sensation of weightlessness. If the station suddenly stopped, it would fall straight down like any other object.

How do I calculate orbital velocity by hand for another planet?

Convert the orbital radius to metres, look up the planet's standard gravitational parameter \(GM\) (or multiply its mass by \(G = 6.67430 \times 10^{-11}\)), then take the square root of \(GM/r\). For example, Mars has \(GM = 4.2828 \times 10^{13}\ \text{m}^3/\text{s}^2\); at 300 km altitude the radius is about \(3.69 \times 10^6\) m, giving \(v \approx 3.41\) km/s. Always use metres, kilograms and seconds to avoid unit errors.