Orbital Period Calculator Formula

Understand the math behind the orbital period calculator. Each variable explained with a worked example.

Use the Orbital Period Calculator

Formulas Used

Orbital Period

period_seconds = 2 * pi * sqrt(pow(semi_major_axis, 3) / (6.674e-11 * central_mass))

Period in Days

period_days = 2 * pi * sqrt(pow(semi_major_axis, 3) / (6.674e-11 * central_mass)) / 86400

Period in Years

period_years = 2 * pi * sqrt(pow(semi_major_axis, 3) / (6.674e-11 * central_mass)) / 31557600

Variables

VariableDescriptionDefault
semi_major_axisSemi-Major Axis(m)149597870700
central_massCentral Body Mass(kg)1.989e+30

How It Works

How the Orbital Period Is Calculated

The orbital period is the time a body takes to complete one full orbit around a central mass.

Kepler's Third Law (Generalized)

T = 2π √(a³ / (G M))

  • *a* is the semi-major axis of the orbit (metres)
  • *G* is the gravitational constant, 6.674 × 10⁻¹¹ N m² kg⁻²
  • *M* is the mass of the central body (kg)

    • This assumes the orbiting body's mass is negligible compared to the central body.

    Worked Example

    Find the orbital period of Earth around the Sun (a = 1.496e11 m, M = 1.989e30 kg).

    semi_major_axis = 149597870700central_mass = 1.989e+30
    1. 01T = 2π √(a³ / (G M))
    2. 02a³ = (1.496e11)³ = 3.348e33 m³
    3. 03G M = 6.674e-11 × 1.989e30 = 1.327e20
    4. 04a³ / (G M) = 2.524e13
    5. 05T = 2π × √(2.524e13) ≈ 3.156e7 s ≈ 365.25 days

    Ready to run the numbers?

    Open Orbital Period Calculator