Orbital Velocity

v = \sqrt{\frac{GM}{r}}

v =

M =

r =

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Constant used — built into this formula, no need to enter

G = 6.6743 \times 10^{-11}\ \text{m}^{3}\text{/(kg}{\cdot}\text{s}^{2}\text{)}

Newtonian constant of gravitation

Learning zone

A satellite stays in a circular orbit when gravity supplies exactly the centripetal force it needs — setting GMm/r² equal to mv²/r and cancelling the satellite's mass gives v = √(GM/r). The cancellation is the famous part: orbital speed depends only on the central body's mass and the orbit's radius, never on what is orbiting. The International Space Station, circling Earth (M ≈ 5.97 × 10²⁴ kg) at r ≈ 6.79 × 10⁶ m, moves at √(6.674 × 10⁻¹¹ × 5.97 × 10²⁴ / 6.79 × 10⁶) ≈ 7 660 m/s — about 7.7 km/s, one lap of the planet every 92 minutes.

The inverse square root also explains a counter-intuitive fact: higher orbits are slower. GPS satellites at r ≈ 26 600 km amble along at 3.9 km/s, half the ISS's speed. Astronomers run the formula in reverse constantly — measure a moon's or a star's orbital speed and radius, and M = v²r/G weighs the central body, which is how we know the mass of everything from Jupiter to the black hole at the centre of the Milky Way.

Orbital Velocity

v = \sqrt{\frac{GM}{r}}

Where

$v$ = Orbital velocity
$M$ = Central mass
$r$ = Orbital radius

Missing one of these? Work it out first, then come back

Orbital velocity — Linear Momentum (p = mv), Power from Force and Velocity (P = Fv)
Central mass — Orbital Period, Escape Velocity
Orbital radius — Orbital Period, Centripetal Acceleration (a = v²/r)

Part of this formula set

Gravitation and orbital mechanics

8 formulasNewton's inverse-square law and what falls out of it in orbit — field strength, orbital speed and period, Kepler's third law, and escape velocity.