Find the circular orbital velocity, orbital period, and escape velocity for a satellite at any altitude around Earth, the Moon, Mars, or the Sun. Choose a central body to set its gravitational parameter μ and radius, then enter altitude — the calculator works in SI units throughout.
A satellite in a circular orbit is in perpetual free fall: gravity bends its straight-line motion into a closed loop, and the speed at which that exactly balances curvature is the circular orbital velocity. This calculator solves the three fundamental quantities of a circular orbit — speed, period, and escape velocity — directly from the central body's gravitational parameter μ = GM and the orbital radius r, the standard two-body relations used in every mission-design tool.
For a circular orbit of radius r about a body with gravitational parameter μ = GM, setting gravity equal to the required centripetal force gives the circular speed:
v = √(μ / r)
The period — the time for one full revolution — follows from circumference over speed and reduces to Kepler's third law:
T = 2π · √(r³ / μ)
Here r is measured from the centre of the body, so r = R_body + h where h is the altitude above the surface. Using μ instead of separate G and M values avoids the large uncertainty in G and is the convention in astrodynamics.
It is counter-intuitive, but a satellite in a higher orbit moves more slowly. Because v = √(μ/r), speed falls as the square root of radius: double the orbital radius and the speed drops by a factor of √2. At the same time the orbit is longer, so the period grows even faster — T scales as r^1.5. The International Space Station at ~400 km orbits in about 92 minutes at roughly 7.7 km/s, while a satellite far higher up takes much longer and travels more slowly. This trade is the basis of orbital phasing and rendezvous: to catch up with a target ahead of you, you drop to a lower, faster orbit, then climb back.
Orbits are grouped by altitude. Low Earth Orbit (LEO, roughly 160–2000 km) hosts the ISS, most Earth-observation satellites, and large communications constellations; periods are around 90–130 minutes. Medium Earth Orbit (MEO, ~20,000 km) is home to GPS and other navigation systems with ~12-hour periods. Geostationary orbit (GEO) sits at about 35,786 km altitude (r ≈ 42,164 km), where the period is exactly one sidereal day so the satellite appears fixed over a point on the equator — ideal for communications and weather satellites.
Escape velocity is the speed at which an object's kinetic energy exactly cancels its gravitational potential energy, letting it coast to infinity with no further thrust:
v_e = √(2μ / r) = √2 · v_circular
So escape velocity is always √2 ≈ 1.414 times the local circular speed. From Earth's surface it is about 11.2 km/s; from the Moon only about 2.4 km/s, which is why launching from the Moon is so much cheaper. Escape velocity is a speed, not a direction — any trajectory at or above v_e (in the absence of other forces) is unbound.
μ = GM is the product of the universal gravitational constant G and the mass M of the central body. It is used instead of G and M separately because μ is known to far higher precision than either factor (G is one of the least precisely measured constants). Values used here are Earth 3.986×10¹⁴, Moon 4.903×10¹², Mars 4.283×10¹³, and Sun 1.327×10²⁰ m³/s².
At a ~400 km altitude the orbital radius is about 6,771 km. Plugging into v = √(μ/r) with Earth's μ gives roughly 7.67 km/s, and a period near 92 minutes — exactly what this calculator returns and what is observed. The Station completes about 16 orbits per day.
Yes. The formulas v = √(μ/r) and T = 2π√(r³/μ) are exact only for circular orbits. Real orbits are usually slightly elliptical, where speed varies between perigee and apogee. For an ellipse you use the vis-viva equation v = √(μ(2/r − 1/a)) with semi-major axis a; the circular case is the special instance where r = a everywhere.
A circular orbit has kinetic energy equal in magnitude to half the potential energy. To escape, total energy must reach zero, which requires twice the kinetic energy, and since kinetic energy goes as v², the speed only needs to increase by √2. Hence v_e = √2 · v_circular at the same radius.
Yes — select the central body and the calculator swaps in that body's μ and surface radius. So you can size a lunar orbit, a Mars science orbit, or even a heliocentric orbit around the Sun. Just remember the altitude is measured above that body's surface, and these are two-body results that ignore perturbations from other bodies.