Vis-Viva Equation

\[v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)}\]

Variables

vorbital velocity (m/s)

mu[μ]standard gravitational parameter (m³/s²)

rinstantaneous distance from the central body (m)

asemi-major axis of the orbit (m)

Description

What is this formula?

The Vis-Viva Equation calculates the orbital velocity of a spacecraft or celestial body at any point along an orbit.

Unlike the circular orbit equation, the Vis-Viva Equation is valid for elliptical, circular, parabolic, and hyperbolic trajectories.

It is one of the most important equations in astrodynamics.

When to use it

Use this equation when determining spacecraft velocity at a specific orbital position, analyzing transfer orbits, or studying planetary and satellite motion.

It is widely used in mission design, orbital maneuvers, and trajectory analysis.

Example

A spacecraft is in an elliptical Earth orbit with:

μ = 3.986×10^14 m³/s²

Distance from Earth center = 8.0×10^6 m

Semi-major axis = 1.0×10^7 m

Formula:

v = √(μ(2/r − 1/a))

Substitution:

v = √(3.986×10^14 × (2/(8.0×10^6) − 1/(1.0×10^7)))

v = 8151 m/s

Result:

Orbital velocity = 8.15 km/s

Applications

Orbital maneuver analysis

Transfer orbit calculations

Satellite mission planning

Interplanetary trajectory design

Astrodynamics

Space mission engineering