Vis-Viva Equation — Orbital Speed Calculator
Calculate orbital velocity at any point using the vis-viva equation v^2=GM(2/r-1/a).
Find speeds at periapsis, apoapsis, and circular orbit for comparison.
Orbital Speed
How Orbital Velocity Is Calculated Using the Vis-Viva Equation
The vis-viva equation gives the orbital speed of a body at any point in its orbit around a central mass, regardless of orbit shape (circular or elliptical).
Vis-Viva Equation:
v² = GM × (2/r − 1/a)
Where:
- v = orbital speed at the given point (m/s)
- G = gravitational constant = 6.674 × 10⁻¹¹ N·m²/kg²
- M = mass of central body (kg)
- r = current distance from central body (m)
- a = semi-major axis of orbit (m) — for circular orbits, a = r
Simplification for Circular Orbits (a = r):
v = √(GM / r)
Worked Example — Earth’s orbital speed around the Sun:
- G = 6.674 × 10⁻¹¹, M_sun = 1.989 × 10³⁰ kg
- r = 1.496 × 10¹¹ m (1 AU)
- GM = 1.327 × 10²⁰
- v = √(1.327×10²⁰ / 1.496×10¹¹) = √(8.87×10⁸) = 29,784 m/s ≈ 29.8 km/s
Hohmann Transfer Orbit (to change orbits efficiently): Burn at periapsis to raise apoapsis, then burn at apoapsis to circularize. The vis-viva equation calculates the required ΔV at each point.
Speed Comparison at Different Altitudes (Earth orbit):
- ISS (408 km): 7,660 m/s
- GPS (20,200 km): 3,870 m/s
- Geostationary (35,786 km): 3,070 m/s
- Moon (384,400 km): 1,022 m/s