Orbital Period Calculator

What the orbital period calculator does

The orbital period is the time an object takes to complete one orbit. This calculator finds it in two ways: from the size of an orbit and the masses of the two bodies involved, using Kepler's third law, or from the density of the central body alone for a close orbit skimming its surface.

Below is what an orbital period is, the equations behind it, and a worked example.

How to use it

1. For an orbit by size, enter the semi-major axis and the masses of the two bodies, and the calculator gives the period.
2. For a close surface orbit, enter just the central body's density, and the calculator gives the minimum orbital period.
3. Press Calculate for the period, or Reset to clear it.

What an orbital period is

The orbital period is simply how long an object takes to go once around its orbit. For the Earth around the Sun, it is one year; for the Moon around the Earth, about 27 days; for a low satellite, around 90 minutes. The period depends on how large the orbit is and on the gravity holding it together, and it is one of the most basic facts you can know about any orbit, from a spacecraft to a planet to a distant binary star.

What makes the orbital period so useful is its tight, predictable link to the size of the orbit. Bigger orbits take longer, and not just in proportion: the relationship follows a precise mathematical law discovered by Johannes Kepler. This means that knowing an orbit's size lets you predict its period, and measuring a period lets you deduce the orbit's size, or even the mass of the body at the centre. This calculator captures that relationship from both directions.

The equation it uses

For an orbit of a given size, the period follows Kepler's third law:

T = 2π √(a³ ÷ (G (M + m)))

Here T is the orbital period, a is the semi-major axis, which is the average orbital radius, G is the gravitational constant, and M and m are the masses of the two bodies. The period grows with the orbit's size, raised to the three-halves power, so a wider orbit takes disproportionately longer. It shrinks with the combined mass of the bodies, since stronger gravity whips an orbit around faster. The calculator evaluates this relationship to give the period.

Orbits where both bodies matter

For a small object orbiting a huge one, like a satellite around the Earth or a planet around the Sun, only the big body's mass really counts, because the small one's gravity is negligible by comparison. But when the two bodies are more comparable, such as two stars orbiting each other in a binary system, both masses contribute to the gravity that sets the period, and both must be included.

This is why the formula uses the sum of the two masses rather than just one. In a binary star system, the two stars each pull on the other, and the orbit responds to their combined gravity, circling faster than it would around either star alone. The calculator includes both masses precisely so it can handle these cases, from a feather-light satellite around a planet to two massive stars locked in mutual orbit. For the lopsided case, simply entering a tiny second mass recovers the familiar single-body result.

The period that depends only on density

The calculator offers a second, more surprising route to an orbital period. For an object orbiting in a very low orbit, skimming just above the surface of a body, the period turns out to depend only on the average density of that body, not on its size at all. A small dense world and a large world of the same density give the same close-orbit period.

This elegant result falls out of Kepler's law when the orbital radius equals the body's own radius, and the size cancels away, leaving only the density. For the Earth, it gives a minimum orbital period of about 84 to 90 minutes, which is indeed roughly how long the lowest satellites take to circle the planet, and no satellite around Earth can orbit faster than this. The same period would apply to any rocky body of Earth's density, whatever its size. The calculator computes this density-only period as a neat shortcut for the fastest possible orbit around a body.

Units and precision

The calculator takes the semi-major axis in units from metres to parsecs, including astronomical units, and the masses in kilograms, Earth masses, or Sun masses, which suits everything from satellites to star systems. Density is taken in units like grams per cubic centimetre. It returns the period in seconds, hours, days, or years. The calculations are exact applications of Kepler's law, and results carry several significant figures.

A worked example

Take the Earth orbiting the Sun, with a semi-major axis of one astronomical unit, the Sun's mass, and the Earth's much smaller mass.

Kepler's third law gives a period of T = 2π √(a³ ÷ (G (M + m))) ≈ 1.000 year, exactly as it should, since the astronomical unit is defined as Earth's orbital distance. Using the second method, entering Earth's average density of about 5.5 grams per cubic centimetre gives a close-orbit period of about 84 minutes, the fastest anything can orbit Earth, matching the periods of the lowest satellites.

Questions people ask

How do you calculate orbital period?

Use Kepler's third law, T = 2π√(a³/(G(M+m))), where a is the semi-major axis and M and m are the two masses. Bigger orbits and lighter bodies give longer periods.

Why include both bodies' masses?

Because both contribute to the gravity that sets the period. For a satellite around a planet the smaller mass is negligible, but for two comparable bodies like binary stars, both matter.

How can a period depend only on density?

For an orbit skimming a body's surface, the size cancels out of Kepler's law, leaving only the density. Any body of the same density gives the same close-orbit period, whatever its size.

What is the shortest orbit around Earth?

About 84 to 90 minutes, set by Earth's density. No satellite can circle Earth faster than this, which is why the lowest orbits all have periods near an hour and a half.

References

A quick note on where the physics comes from. Kepler's third law and the orbital period are standard gravitation, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. NASA describes orbital periods for planets and satellites. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 1, Section 13.5, Kepler's Laws of Planetary Motion. https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion
2. HyperPhysics, Kepler's Laws. http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html
3. NASA, Orbits and Kepler's Laws. https://science.nasa.gov/resource/orbits-and-keplers-laws/