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.
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
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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