Choose orbital period, semi-major axis, or central mass, then enter the other two quantities. Common distance, mass, and time units are supported; the optional orbiting mass enables the full ideal two-body model.

Ideal two-body orbit. Use center-to-center radius for a circle, or the relative orbit’s semi-major axis for an ellipse.

Orbiting mass (optional)

Blank treats this mass as negligible.

Orbital Period Formula

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

T = 2π√((a³) / (G(M + m)))

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

T ≈ 2π√((a³) / (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

Symbol Meaning Typical SI Unit Practical Note
T Orbital period seconds (s) The total time for one full orbit
a Semi-major axis meters (m) For a circular orbit, this is the orbital radius
G Universal gravitational constant N·m2/kg2 Connects mass and gravitational attraction
M Mass of the central object kilograms (kg) Examples: planet, moon, star, or black hole
m Mass of the orbiting object kilograms (kg) Often negligible compared with M

How to Use the Calculator

  1. Select the quantity to solve for, then enter the two required quantities. Orbital size is center-to-center radius for circular motion or the relative orbit’s semi-major axis for elliptical motion.
  2. If the orbiting mass is significant, enter it in the optional section. Blank assumes negligible orbiting mass.
  3. Select the units that match your inputs.
  4. Select the result unit and Calculate. Expand Calculation details for the equation and SI values.

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](G(M + m)T²) / (4π²)

Solve for total system mass:

M + m = (4π² a³) / (GT²)

If the orbiting mass is negligible:

M ≈ (4π² a³) / (GT²)

Kepler-Style Comparison Between Two Orbits

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

T² ∝ a³

So for two orbits around the same mass:

(T₁²) / (T₂²) = (a₁³) / (a₂³)

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, enter the optional orbiting mass and use the relative separation’s semi-major axis; the model uses M + m.

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.