What is the semimajor axis of an orbit?

For an elliptical orbit, the semimajor axis (a) is half the length of the longest diameter of the ellipse. For a circular orbit it equals the orbital radius. It is the standard measure of orbit size in celestial mechanics and appears directly in Kepler's Third Law.

What formula does this calculator use?

It rearranges Kepler's Third Law — T² = 4π²a³ / (GM) — to isolate a: a = ∛(G × M × T² / 4π²). G is the gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), M is the central body mass, and T is the orbital period in seconds.

What is an astronomical unit (AU)?

One AU is the mean distance from Earth to the Sun, defined as exactly 149,597,870.7 km. It is the standard unit for measuring distances within the Solar System. For example, Mars orbits at about 1.52 AU and Jupiter at about 5.20 AU.

How do I find the semimajor axis of a geostationary satellite?

A geostationary satellite has a period of one sidereal day (86,164 seconds). Plugging this and Earth's mass (5.972 × 10²⁴ kg) into the formula gives a ≈ 42,164 km from Earth's center, or roughly 35,786 km above the surface.

Does this work for moons and satellites?

Yes. Replace M with the mass of the planet the moon or satellite orbits. For the ISS orbiting Earth, use Earth's mass (~5.972 × 10²⁴ kg) and the ISS period (~92 minutes) to get an orbital radius of roughly 6,787 km.

What are the limitations of this calculator?

The formula assumes a two-body system where the orbiting body is much less massive than the central body. Perturbations from other bodies, relativistic effects, and atmospheric drag are not modeled. For high-precision mission planning, specialized software is required.

Semimajor Axis Calculator

Name: Semimajor Axis Calculator
Availability: InStock
Author: Nham Vu

Enter an orbital period and central body mass to calculate the semimajor axis of the orbit using Kepler's Third Law.

Orbital Parameters

Solar System Preset

Orbital Period

Time for one complete orbit

Central Body Mass (kg)

Use scientific notation: 1.989e30

Formula: a = ∛(G·M·T² / 4π²)
G = 6.674 × 10⁻¹¹ N·m²/kg²

Select a preset or enter custom values, then click Calculate.

Earth

a = ∛(GM·T² / 4π²)

— km

Semimajor Axis

vs. Earth orbit (1 AU = 149,597,871 km)

Kilometers

Astronomical Units

Light-Minutes

Input Values Used

Orbital period:

Central mass:

Solar System Comparison (planets around Sun)

Planet	Axis (AU)	Scale

Summary

Enter an orbital period and central body mass to calculate the semimajor axis of the orbit using Kepler's Third Law.

How it works

Select a Solar System preset to load known orbital data, or choose Custom to enter your own values.
For custom input, provide the orbital period (in seconds, minutes, hours, days, or years) and the central body mass in kilograms.
The calculator applies the inverted form of Kepler's Third Law: a = ∛(G × M × T² / 4π²).
Results are displayed in kilometers, astronomical units (AU), and light-minutes for easy cross-referencing.
A comparison bar shows how the computed orbit compares in size to Earth's orbit around the Sun (1 AU).

Use cases

Determine the orbital radius of an exoplanet from its measured period.
Verify Solar System orbital data using Kepler's Third Law.
Calculate the altitude of a satellite given its orbital period.
Find the semimajor axis of a hypothetical orbit for science fair or homework problems.
Explore how orbital period and orbit size are related for different central bodies.
Compute geostationary orbit altitude from a 24-hour period and Earth's mass.
Illustrate that shorter periods correspond to tighter, faster orbits.
Prepare inputs for n-body simulation software that requires semimajor axis values.

Frequently Asked Questions

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