Kepler's Third Law Calculator Formula

Understand the math behind the kepler's third law calculator. Each variable explained with a worked example.

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Formulas Used

Semi-Major Axis

semi_major_au = pow(pow(period_years, 2), 1/3)

Semi-Major Axis (km)

semi_major_km = pow(pow(period_years, 2), 1/3) * 149597870.7

Variables

VariableDescriptionDefault
period_yearsOrbital Period(yr)1

How It Works

Kepler's Third Law

For bodies orbiting the same central mass, the square of the period is proportional to the cube of the semi-major axis.

Simplified Form (Solar System)

T² = a³ when T is in Earth years and a is in AU.

Rearranging: a = T^(2/3)

Worked Example

Mars has an orbital period of 1.881 years. Find its semi-major axis.

period_years = 1.881
  1. 01a = T^(2/3)
  2. 02T² = 1.881² = 3.538
  3. 03a = 3.538^(1/3) = 1.524 AU
  4. 04In km: 1.524 × 149 597 870.7 ≈ 228 million km

Frequently Asked Questions

Does this work for moons of other planets?

Yes, but the constant changes because it depends on the central body's mass. The simplified form uses Solar mass implicitly.

Who was Johannes Kepler?

Kepler (1571-1630) was a German astronomer who discovered three empirical laws of planetary motion from Tycho Brahe's observations.

How precise is T² = a³?

Exact for a massless test particle orbiting the Sun. For real planets the correction factor (1 + m_planet/m_Sun) is typically less than 0.001.

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