Orbital Velocity Calculator
Compute the orbital velocity around a central body using v = √(GM/r) × √(1+e), an approximation that scales the circular-orbit speed by the eccentricity. Useful for first-order estimates of satellite velocities, planetary orbital speeds, and binary star systems.
Last updated: May 2026
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About this calculator
For a circular orbit, Newton's second law balanced against gravity gives v = √(G·M/r), where G = 6.674×10⁻¹¹ N·m²/kg² is the gravitational constant, M is the central body's mass in kg, and r is the orbital radius in m. This calculator applies an eccentricity factor √(1 + e) — a simplified scaling that approximates the velocity at perihelion (closest approach) of an elliptical orbit relative to the equivalent circular orbit at the semi-major axis. The exact vis-viva equation for elliptical orbits is v = √(G·M·(2/r − 1/a)), where a is the semi-major axis; at perihelion (r = a(1−e)) this gives v_peri = √(G·M/a) · √((1+e)/(1−e)). The simplified formula here approximates this for small to moderate eccentricities. Variables: centralMass in kg (Earth = 5.972×10²⁴, Sun = 1.989×10³⁰, Moon = 7.342×10²²); orbitRadius in m (low Earth orbit ~ 6.78×10⁶, geostationary ~ 4.22×10⁷, lunar orbit ~ 3.84×10⁸); eccentricity (0 = perfect circle, 0.99 = highly elliptical like Halley's Comet); bodyType is informational and doesn't affect the calculation. Edge cases: orbitRadius must be > 0 and ideally > body radius (else you're inside the body); eccentricity ≥ 1 means parabolic (escape) or hyperbolic trajectory, not a closed orbit. Reference velocities: ISS at ~7.66 km/s; geostationary satellites at ~3.07 km/s; Moon at ~1.02 km/s; Earth around Sun at ~29.78 km/s; Mercury at perihelion ~58.98 km/s. For comparison to the velocity-needed-to-escape, the escape velocity at radius r is v_esc = √(2GM/r) = √2 × v_circular ≈ 1.414 × circular velocity at the same radius.
How to use
Example 1 — Low Earth orbit (ISS altitude). Earth mass = 5.972×10²⁴ kg; orbital radius from Earth's centre = 6.78×10⁶ m (about 400 km above the surface, ISS altitude). Circular orbit, e = 0. Enter Central Mass = 5.972e24, Orbital Radius = 6.78e6, Eccentricity = 0, Body Type = Earth. v = √(6.674×10⁻¹¹ × 5.972×10²⁴ / 6.78×10⁶) × √1 ≈ √(5.876×10⁷) ≈ 7666 m/s ≈ 7.67 km/s. ✓ The ISS actually orbits at 7.66 km/s — the formula gets the right answer for circular LEO. One complete orbit takes 2π × 6.78×10⁶ / 7666 ≈ 5560 s ≈ 93 minutes, matching observed ISS orbital period. Example 2 — Earth's orbit around the Sun (slightly elliptical). Sun mass = 1.989×10³⁰ kg; Earth's mean orbital radius ≈ 1.496×10¹¹ m (1 AU); Earth orbit eccentricity ≈ 0.017. Enter 1.989e30, 1.496e11, 0.017, Sun. v ≈ √(6.674×10⁻¹¹ × 1.989×10³⁰ / 1.496×10¹¹) × √1.017 ≈ √(8.87×10⁸) × 1.0085 ≈ 29790 × 1.0085 ≈ 30,043 m/s. ✓ Earth's actual orbital velocity averages 29.78 km/s — the formula's eccentricity correction gives about 30 km/s (perihelion value), close to the observed perihelion speed of ~30.29 km/s.
Frequently asked questions
What's the difference between orbital velocity and escape velocity?
Orbital velocity is the speed needed to maintain a stable orbit at a given altitude — fast enough to balance gravity's pull, but not so fast that the object escapes. Escape velocity is the minimum speed needed to leave the central body's gravitational influence entirely (with kinetic energy at infinity equal to zero). The two are related by v_escape = √2 × v_circular at the same radius. For low Earth orbit, v_circ ≈ 7.8 km/s and v_escape ≈ 11.0 km/s. Any speed between v_circ and v_escape gives an elliptical orbit that returns; speeds equal to v_escape give a parabolic escape; speeds above v_escape give a hyperbolic trajectory that recedes forever. To go from low Earth orbit to the Moon you don't need v_escape — you need a transfer-orbit speed of about 10.8 km/s at perigee — but to leave the Earth-Moon system entirely requires v_escape from Earth's orbit, about 16.5 km/s relative to the Sun (the heliocentric escape from 1 AU).
Why does Kepler's third law follow from orbital velocity?
Kepler's third law states T² ∝ a³, where T is the orbital period and a is the semi-major axis. Derive it from circular-orbit dynamics: orbital period T = 2πa/v, and v = √(GM/a). Squaring: T² = 4π²a²/v² = 4π²a²/(GM/a) = 4π²a³/(GM). So T² = (4π²/GM)·a³ — exactly Kepler's third law, with the proportionality constant determined by the central body's mass. For elliptical orbits the same law holds with a being the semi-major axis (not the instantaneous radius), proven by integrating the vis-viva equation. This is why distant planets take so long to orbit the Sun — Neptune (a ≈ 30 AU) has T = 30^(3/2) ≈ 165 years, versus Earth's 1 year at 1 AU.
How does eccentricity change the orbital speed?
For an elliptical orbit, speed varies along the orbit. At perihelion (closest approach), r is smallest and v is largest; at aphelion (farthest), r is largest and v is smallest. The exact relation is the vis-viva equation: v² = GM(2/r − 1/a). At perihelion r = a(1−e), giving v_peri = √(GM/a) · √((1+e)/(1−e)). At aphelion r = a(1+e), giving v_apo = √(GM/a) · √((1−e)/(1+e)). So v_peri/v_apo = (1+e)/(1−e); for Halley's Comet (e ≈ 0.97), perihelion speed is over 50× aphelion speed. Earth's orbit has e ≈ 0.017, so perihelion (early January) and aphelion (early July) speeds differ by only about 3% (30.29 vs 29.29 km/s). This calculator's √(1+e) scaling is a rough approximation that gives the perihelion velocity for small e; for accurate elliptical velocity at arbitrary points, use the full vis-viva equation.
What are the most common mistakes computing orbital velocity?
The first is mixing units — using km instead of m for radius, or g instead of kg for mass, throws the result off by orders of magnitude. The G constant is in SI units (m, kg, s) and demands consistent SI inputs. The second is using altitude above the surface instead of orbital radius from the body's centre; ISS altitude is ~400 km but its orbital radius is 6,378 + 400 = 6,778 km. The third is forgetting that "orbital velocity" of a planet around the Sun and "orbital velocity" of a satellite around the planet are completely different — the central mass must be the body you're orbiting, not Earth or the Sun by default. The fourth is using the circular-orbit formula on highly elliptical orbits without applying the vis-viva correction; the simple v = √(GM/r) understates speeds at perihelion of comets and high-eccentricity satellites. The fifth is forgetting that orbits are 3-dimensional; the speed is a scalar but the velocity vector has direction along the tangent to the orbit, which matters for rendezvous and orbital insertion calculations.
When should I not use this calculator?
Skip it for highly eccentric orbits (e > 0.3) where the √(1+e) approximation breaks down badly; use the full vis-viva equation. Don't use it for multi-body problems (Lagrange points, three-body resonances, satellite mission design) where gravity from multiple sources matters; specialised n-body software is required. It's the wrong tool for relativistic orbits (Mercury's perihelion precession, orbits near black holes) where general relativity adjusts the Newtonian result by terms in v²/c²; for those, use post-Newtonian or full GR formulas. Avoid it for orbits with significant atmospheric drag (low Earth orbit below ~600 km, especially during solar maximum) — drag changes the orbit secularly and a static velocity isn't meaningful. Finally, for trajectory planning of actual missions, use mission-design tools (GMAT, STK, FreeFlyer) that account for J2 oblateness, third-body perturbations, and other effects; this calculator gives a first-order estimate, not a flight-quality answer.