Orbital Period Calculator

Enter any 2 values (semi-major axis / orbital radius, mass of the central object, or orbital period) to calculate the missing variable.

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Orbital Period Formula

The orbital period is the time required for one complete revolution around a central body. For a two-body system, the exact relation is:

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

For most practical cases, the orbiting object is much less massive than the central object, so the common approximation is:

T \approx 2\pi\sqrt{\frac{a^3}{GM}}

This equation shows that orbital period depends primarily on the size of the orbit and the mass being orbited. Larger orbits take longer, while more massive central bodies produce shorter periods for the same orbital size.

Variable Meanings

How to Use the Calculator

1. Enter the orbital size as a radius for circular motion or a semi-major axis for elliptical motion.
2. Enter the mass of the central object being orbited.
3. Select the units that match your inputs.
4. Calculate the missing value. The tool can solve for orbital period, orbital radius/semi-major axis, or central mass when the other two are known.

Important Orbit Geometry Note

For a circular orbit, the orbital radius and semi-major axis are the same:

a = r

For an elliptical orbit, use the semi-major axis, not the distance at periapsis, apoapsis, or the average of the major and minor axes. The orbital period is determined by the semi-major axis of the ellipse.

Rearranged Forms

If you need to solve for a different variable, the same relationship can be rearranged.

Solve for semi-major axis:

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

Solve for total system mass:

M + m = \frac{4\pi^2 a^3}{GT^2}

If the orbiting mass is negligible:

M \approx \frac{4\pi^2 a^3}{GT^2}

Kepler-Style Comparison Between Two Orbits

When comparing two objects orbiting the same central body, the period-to-distance relationship simplifies to:

T^2 \propto a^3

So for two orbits around the same mass:

\frac{T_1^2}{T_2^2} = \frac{a_1^3}{a_2^3}

This is especially useful for comparing satellites, planets, or moons without directly solving the full equation each time.

What the Result Means

• Short period: the object is close to the central mass, the central mass is very large, or both.
• Long period: the orbit is large, the central mass is smaller, or both.
• Doubling orbital size: the period increases by more than double because period scales with the 3/2 power of orbital size.

Common Input Mistakes

• Using altitude instead of orbital radius: if an object is orbiting above a planet, radius must be measured from the planet’s center. Add the planet’s radius to the altitude.
• Mixing units: the equation assumes consistent units. The calculator handles conversions, but conceptually the standard form uses meters, kilograms, and seconds.
• Using instantaneous distance in an elliptical orbit: use the semi-major axis instead.
• Ignoring the orbiting mass in a binary system: if both objects are massive, use M + m rather than only the central mass.

Where This Formula Is Used

• Satellite mission design
• Planetary motion analysis
• Moon and binary system calculations
• Astrophysics and celestial mechanics
• Estimating central mass from observed orbital motion

FAQ

Does the mass of the orbiting object matter?

Yes, in the full two-body equation the total mass is M + m. However, when the orbiting object is much smaller than the central object, its contribution is usually so small that it can be ignored.

Why is the orbital period based on semi-major axis?

In elliptical motion, orbital speed changes continuously throughout the orbit, but the total orbital period depends on the size of the ellipse, represented by the semi-major axis.

Can this calculator be used for circular and elliptical orbits?

Yes. For circular orbits, enter the orbital radius. For elliptical orbits, enter the semi-major axis.

Can I estimate the mass of a planet or star from an orbit?

Yes. If you know the orbital size and period, you can rearrange the equation to solve for the central mass.