Vis-Viva Equation - Orbital Speed Calculator

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 (International Space Station, 408 km): 7,660 m/s
• GPS (Global Positioning System, 20,200 km): 3,870 m/s
• Geostationary (35,786 km): 3,070 m/s
• Moon (384,400 km): 1,022 m/s

A note on the units. Both distance fields take AU (Astronomical Units), one AU being the mean Earth-Sun distance of 149.6 million km. That suits solar-system work, but it makes Earth orbits awkward to enter, because they are minute by comparison. The distances in the list above work out to 0.0000453 AU for the space station, 0.000178 AU for GPS, 0.000282 AU for geostationary and 0.00257 AU for the Moon. Remember these are measured from the centre of the Earth, not from the surface, so add 6,371 km to any altitude before converting. If you pick Earth as the central body, expect to type very small numbers.

Reading the answer. Compare your speed against the circular-orbit speed the panel also shows. Faster than circular means you are inside your own semi-major axis, heading outward, somewhere between periapsis and apoapsis. Slower means you are beyond it and falling back in. Equal means the orbit is circular. That single comparison tells you which half of the orbit you are on without any further calculation.