Free Kepler's Third Law Calculator

Orbital Period Calculator and Semi-Major Axis Calculator – This tool is built around Kepler's third law equation, enabling the determination of orbital periods, semi-major axes, or central masses for any gravitationally bound two-body system. It functions as a planetary motion calculator for studying planets, moons, and exoplanets, and can double as a star mass calculator when orbital data from a planet or moon is available.

Definition of Kepler's Third Law

In the early 17th century, Johannes Kepler discovered three empirical rules governing the orbits of the planets. The third law, often summarized as the law of harmonies, establishes a relationship between the orbital period $T$ (the time to complete one full revolution) and the semi-major axis $a$ (half the longest diameter of the elliptical orbit):

$$T^{2} \propto a^{3}$$

This proportionality holds for all objects orbiting the same central body. Expressed in words: the square of the orbital period is directly proportional to the cube of the semi-major axis.

Mathematical Formulation

The precise form of Kepler's third law emerges from Newtonian mechanics. For a simplified circular orbit, the gravitational force between the central mass $M$ and the orbiting mass $m$ provides the required centripetal force:

$$m \frac{v^{2}}{r} = \frac{G M m}{r^{2}}$$

Here $v$ is the orbital speed, $r$ the orbital radius, and $G = 6.67408 \times 10^{-11}~\text{m}^{3}\,\text{kg}^{-1}\,\text{s}^{-2}$ the gravitational constant. Replacing $v$ with $\frac{2\pi r}{T}$ and simplifying gives:

$$\frac{r^{3}}{T^{2}} = \frac{G M}{4\pi^{2}}$$

Hence the ratio $r^{3}/T^{2}$ is constant for all planets around a given star. When elliptical orbits are taken into account, the radius $r$ is replaced by the semi-major axis $a$, and the planet's own mass $m$ must be included:

$$\frac{a^{3}}{T^{2}} = \frac{G (M + m)}{4\pi^{2}}$$

This is the complete Kepler's third law equation used by the calculator. When $M \gg m$, the term $m$ becomes negligible, and the simple proportionality $T^{2} \propto a^{3}$ is an excellent approximation.

Using the Calculator

The interface is straightforward: supply any two of the three parameters – orbital period, semi-major axis, or central mass – and the missing value is computed automatically.

• Orbital Period: Enter the time taken for one orbit.
• Semi-Major Axis: Specify the average distance from the center of mass.
• Central Mass: Provide the mass of the star or central body.

For greater accuracy, especially in systems where the orbiting body has a substantial mass relative to the central body, enable the "planet mass" option and enter the mass of the satellite. This activates the full equation $a^{3}/T^{2} = G(M+m)/(4\pi^{2})$. The calculator supports several unit systems, including metric (meters, seconds, kilograms) and astronomical units (AU, years, solar masses), making it adaptable to different contexts.

Quick test: With $M = 1$ solar mass and $T = 1$ year, the computed semi-major axis is exactly 1 AU – confirming Earth's orbit and verifying the tool's correct implementation of Kepler's third law.

Solar System Verification

The following table lists the orbital parameters of the eight planets and the dwarf planet Pluto. Each row satisfies the relationship $a^{3}/T^{2} = \text{constant}$ (with minor deviations due to the planets' own masses), serving as a direct validation of the law.

Planet	Orbital period (yr)	Semi-major axis (AU)	Mass (Earth masses)
Mercury	0.240	0.387	0.0553
Venus	0.615	0.723	0.815
Earth	1.000	1.000	1.000
Mars	1.881	1.524	0.107
Jupiter	11.862	5.203	317.83
Saturn	29.457	9.537	95.159
Uranus	84.021	19.191	14.536
Neptune	164.8	30.069	17.147
Pluto	247.94	39.482	0.0021

A quick check for Mercury: $(0.387)^{3} / (0.240)^{2} \approx 1.007$, while Earth's ratio is exactly 1. The small variations highlight the need for the full equation that includes each planet's mass.

Estimating Central Mass

This orbital period calculator and semi-major axis calculator also serves as a star mass calculator. To find the mass of a central object, input the orbital period and semi-major axis of a satellite that orbits it, then select "central mass" as the desired output. For cases where the satellite's mass is much smaller than the central body, the result approximates $M \approx a^{3} / T^{2}$ when using AU and years. Earth's own values ($T=1$ yr, $a=1$ AU) yield $M \approx 1$ solar mass – precisely the mass of the Sun.

Broader Applications

Although originally derived for the Solar System, Kepler's third law applies universally. Astronomers use it to determine binary star masses, to predict spacecraft trajectories, and to characterize exoplanetary systems. With this free online tool, anyone can explore these cosmic relationships without manual calculation – simply enter two known quantities and let the Kepler third law equation do the rest.