What is the vis-viva equation?

The vis-viva equation is v² = GM × (2/r − 1/a), where G is the gravitational constant, M is the central body's mass, r is the current distance from the central body, and a is the semi-major axis of the orbit. It gives the orbital speed at any point without needing the full trajectory solution.

What is the semi-major axis?

The semi-major axis (a) is half the longest diameter of an elliptical orbit. It equals the average of the periapsis distance (closest point) and apoapsis distance (farthest point): a = (r_peri + r_apo) / 2. For a circular orbit, a equals the constant orbital radius.

Why is orbital speed higher at periapsis than apoapsis?

By conservation of energy, the sum of kinetic and potential energy is constant. Closer to the central body, potential energy is lower, so kinetic energy — and hence speed — must be higher. This is a direct consequence of the vis-viva equation: a smaller r increases the 2/r term, raising v.

What is GM (standard gravitational parameter)?

GM is the product of the gravitational constant G (6.674 × 10⁻¹¹ N·m²/kg²) and the central body's mass M. It is often known more precisely as a single value than G and M separately, and is directly what appears in orbital mechanics equations.

How does eccentricity relate to this equation?

Eccentricity (e) shapes the orbit but does not appear explicitly in the vis-viva equation. It determines periapsis r_peri = a(1−e) and apoapsis r_apo = a(1+e). You can plug those radii into the vis-viva equation to get the periapsis and apoapsis speeds.

Does this apply to hyperbolic trajectories?

Yes — for a hyperbolic escape trajectory a is negative (by convention), and the vis-viva equation still holds. The result gives the hyperbolic excess speed at any given radius.

Vis-Viva Calculator

Name: Vis-Viva Calculator
Availability: InStock
Author: Nham Vu

Enter a central body, semi-major axis, and current distance to instantly compute orbital speed using the vis-viva equation.

Orbital Parameters

Central Body

Semi-Major Axis a (km)

Average of periapsis & apoapsis distances

Current Radius r (km)

Distance from central body center

Eccentricity e (optional, 0–1)

Quick Presets

Formula: v = sqrt(GM × (2/r − 1/a))

Select a preset or enter orbital parameters, then click Calculate.

v = sqrt(GM(2/r − 1/a))

Orbital speed at current radius

vs. circular speed at same radius

km / second

m / second

miles per hour

Values Used

GM:

Circular Orbital Speed Reference

Body	Low orbit speed	Bar

Low orbit = altitude 200 km above mean surface radius.

Summary

Enter a central body, semi-major axis, and current distance to instantly compute orbital speed using the vis-viva equation.

How it works

Select a central body (Sun, Earth, Mars, Jupiter, etc.) or enter a custom gravitational parameter (GM).
Enter the semi-major axis of the orbit (the average of periapsis and apoapsis distances).
Enter the current orbital radius — the distance from the central body at the point you want to evaluate.
The calculator applies the vis-viva equation: v = sqrt(GM × (2/r − 1/a)).
Optionally enter eccentricity to auto-compute periapsis and apoapsis speeds for comparison.
Results are shown in km/s, m/s, and mph with a speed comparison bar.

Use cases

Find how fast a satellite moves at different points in its elliptical orbit.
Compute perihelion and aphelion speeds for planets and comets.
Verify delta-v budgets for Hohmann transfer orbit burns.
Teach orbital mechanics and Kepler's laws in physics or astronomy classes.
Model hypothetical orbits around custom or exoplanet central bodies.
Calculate ISS orbital speed at its current altitude.
Understand why spacecraft speed up near periapsis and slow near apoapsis.
Prepare for astrodynamics coursework or engineering competitions.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu