Orbital Velocity Formula
Reference for orbital velocity v = sqrt(GM/r).
Calculate the speed for a stable circular orbit around Earth or Moon from orbital radius and central mass.
The Formula
Orbital velocity is the speed needed for an object to stay in a stable circular orbit. Going slower causes the object to fall inward. Going faster moves it to a higher orbit.
Variables
| Symbol | Meaning |
|---|---|
| v | Orbital velocity (m/s) |
| G | Gravitational constant (6.674 × 10⁻¹¹ N⋅m²/kg²) |
| M | Mass of the central body (kg) |
| r | Orbital radius — distance from center of the central body (meters) |
Example 1
Find the orbital velocity of the ISS (altitude 408 km above Earth)
M = 5.972 × 10²⁴ kg
r = 6,371 km + 408 km = 6,779 km = 6.779 × 10⁶ m
v = √(6.674 × 10⁻¹¹ × 5.972 × 10²⁴ / 6.779 × 10⁶)
v ≈ 7,661 m/s ≈ 27,580 km/h
Example 2
Find the orbital velocity for a geostationary orbit (35,786 km altitude)
r = 6,371 + 35,786 = 42,157 km = 4.216 × 10⁷ m
v = √(6.674 × 10⁻¹¹ × 5.972 × 10²⁴ / 4.216 × 10⁷)
v ≈ 3,075 m/s ≈ 11,070 km/h
When to Use It
Use the orbital velocity formula when:
- Designing satellite orbits at specific altitudes
- Understanding why higher orbits have slower speeds
- Calculating launch requirements for space missions
- Comparing orbital speeds around different planets
Key Notes
- Formula for circular orbits: v = √(GM/r): G is Newton's gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), M is the mass of the central body, and r is the orbital radius (measured from the center, not the surface).
- Higher orbit means slower speed: Orbital velocity decreases as the orbital radius increases. The Moon (384,400 km) orbits at ~1 km/s; the ISS (408 km altitude) orbits at ~7.66 km/s.
- Only applies to circular orbits: For elliptical orbits, use the vis-viva equation: v² = GM(2/r − 1/a), where a is the semi-major axis. Circular orbits are a special case where r = a.
- Geostationary orbit: At ~42,164 km from Earth's center (~35,786 km altitude), orbital velocity equals Earth's rotation rate. Satellites at this altitude appear stationary — used for TV and weather satellites.
- Independent of satellite mass: The orbital velocity formula has no term for the satellite's own mass, because in the limit M_satellite ≪ M_central body, the mass of the orbiting object cancels out.