Vis-Viva Equation — Orbital Speed Calculator
Calculate orbital speed at any point in an elliptical orbit using the vis-viva equation.
Compare to circular orbit speed.
Orbital Speed
The vis-viva equation gives the speed of an object at any point in its orbit around a central body.
General vis-viva equation:
v = √(GM × (2/r - 1/a))
Where:
- v = orbital speed at current position
- G = 6.674 × 10⁻¹¹ N·m²/kg²
- M = mass of the central body
- r = current distance from the center of the central body
- a = semi-major axis of the orbit
For a circular orbit (r = a):
v_circular = √(GM/r)
Key insights:
- At periapsis (closest point): v is maximum
- At apoapsis (farthest point): v is minimum
- A circular orbit has constant speed
- “Vis-viva” means “living force” in Latin — it relates kinetic and potential energy
Solar system examples (circular orbit speeds):
- Mercury: 47.4 km/s
- Earth: 29.78 km/s
- Mars: 24.1 km/s
- Jupiter: 13.1 km/s
- Pluto: 4.7 km/s
Hohmann transfer orbit: To go from one circular orbit to another, spacecraft use a Hohmann transfer: apply two impulses at periapsis and apoapsis of an elliptical transfer orbit. This is the most fuel-efficient two-burn maneuver for changing orbital altitude.
The vis-viva equation is fundamental to planning all interplanetary missions.