Kepler's Third Law Calculator

Compare two orbits using T₁²/T₂² = r₁³/r₂³ — no need to know central body mass.

🌍 Orbits📐 T²∝r³🔭 Kepler

Enter three of the four values (T₁, r₁, T₂, r₂) — the calculator finds the fourth.

T₁ (period of body 1)
r₁ (orbital radius of body 1)
T₂ (leave blank to find)
r₂ (leave blank to find)

Units must be consistent (both periods in years + both radii in AU, or both in SI units)

⚠️ Enter exactly three values — leave one blank to solve for it.

What Is Kepler's Third Law?

Kepler's Third Law in ratio form: T₁²/T₂² = r₁³/r₂³, or equivalently T₁²/r₁³ = T₂²/r₂³ = 4π²/GM. The power of this ratio form is that the central body mass M cancels — you can compare any two objects orbiting the same body using only period and radius ratios. Using Earth as reference (T=1 yr, r=1 AU): for any other planet T² = r³ (in years and AU).

Kepler discovered this empirically in 1619 from Tycho Brahe's planetary observations — without knowing why it worked. Newton derived it in 1687 from the inverse-square law of gravitation and centripetal force. The derivation: GMm/r² = mω²r → GM = ω²r³ = (2π/T)²r³ → T² = 4π²r³/GM → T² ∝ r³.

In our solar system with Earth as reference: Mars (r = 1.524 AU) → T = 1.524^(3/2) = 1.88 years ✓. Jupiter (r = 5.20 AU) → T = 5.20^(3/2) = 11.86 years ✓. The law works with any consistent unit system — years and AU are particularly convenient for the solar system.

The ratio form also applies to moons orbiting planets, stars in binary systems, and exoplanet systems. If two exoplanets orbit the same star and one period is known, the other can be derived from transit timing, without knowing the stellar mass.

Formula Reference Table

Solve ForFormulaNotes
Kepler's Third LawT² ∝ r³For orbits around same body
Ratio formT₁²/T₂² = r₁³/r₂³Eliminates central mass M
Find T₂T₂ = T₁·(r₂/r₁)^(3/2)Using known T₁, r₁, r₂
Find r₂r₂ = r₁·(T₂/T₁)^(2/3)Using known T₁, r₁, T₂
Solar system formT(yr)² = r(AU)³Earth = 1 yr, 1 AU reference
Physical lawT² = 4π²r³/(GM)Newton's derivation

3 Worked Examples

Example 1
Mars Orbital Period

Earth: T₁=1 yr, r₁=1 AU. Mars: r₂ = 1.524 AU.

  • T₂ = T₁ × (r₂/r₁)^(3/2) = 1 × (1.524)^(3/2)
  • T₂ = 1 × 1.881 = 1.881 years
  • Actual: 1.881 years ✓ — perfect agreement
✓ Mars period = 1.881 years
Example 2
Find Neptune's Distance

Neptune: T₂ = 164.8 yr. Earth reference.

  • r₂ = r₁ × (T₂/T₁)^(2/3) = 1 × (164.8)^(2/3)
  • r₂ = (164.8)^(2/3) = 30.07 AU
  • Actual Neptune: 30.07 AU ✓
✓ Neptune distance = 30.07 AU
Example 3
Moon Comparison

Earth's Moon: T₁ = 27.3 days, r₁ = 384,400 km. Hypothetical moon at r₂ = 100,000 km.

  • T₂ = 27.3 × (100,000/384,400)^(3/2)
  • T₂ = 27.3 × (0.2601)^(3/2) = 27.3 × 0.1327 = 3.62 days
✓ Period at 100,000 km = 3.62 days

Real-World Applications

☀️
Solar System Mapping
Kepler derived all planetary distances relative to Earth using only period ratios, measured by timing planetary positions from Earth. The AU was the fundamental unit; absolute distances came later from parallax measurements.
🔭
Exoplanet Systems
Multiple transiting planets in the same system allow period ratios to determine radius ratios, checking orbital stability and potentially revealing unseen companions through transit timing variations.
🌙
Moon Systems
Jupiter's 4 Galilean moons (Io, Europa, Ganymede, Callisto) follow Kepler's Third Law perfectly. Their period ratios (1:2:4:8 approximately) reveal the gravitational resonance that drives Io's volcanism.
⚖️
Binary Stars
Two stars in a binary system: T is the orbital period of both (same orbit around common center of mass). Combining T and r gives M₁ + M₂. Further analysis of the center of mass gives individual masses.
🔬
Gravitational Research
Kepler's Law is used to determine planetary and stellar masses indirectly, without landing on the objects. Any orbiting body reveals the central mass via M = 4π²r³/(GT²).

Common Mistakes to Avoid

⚠️
Mixing units

T and r must use consistent units. If T₁ is in years and T₂ is in months, convert one. The ratio form is unit-agnostic as long as T₁ and T₂ use the same unit, and r₁ and r₂ use the same unit.

⚠️
Applying to different central bodies

T₁²/T₂² = r₁³/r₂³ ONLY applies if both bodies orbit the SAME central mass. You can't compare Earth's orbital period around the Sun with the Moon's orbital period around Earth.

⚠️
Using diameter instead of orbital radius

r is the orbital radius (center-to-center distance), not altitude above surface. For Earth, orbital radius = 6,371 km + altitude.

⚠️
Forgetting the exponents

Find T₂: T₂ = T₁ × (r₂/r₁)^(3/2). Find r₂: r₂ = r₁ × (T₂/T₁)^(2/3). Mixing up 3/2 and 2/3 gives wrong answers.

⚠️
Applying to non-gravitational orbits

Kepler's Law is derived from gravity. It doesn't apply to charged-particle orbits in magnetic fields or other non-gravitational circular motions.

Frequently Asked Questions

What did Kepler discover? ▾
Kepler's 3 laws (1609–1619): (1) Orbits are ellipses with the Sun at one focus. (2) Planets sweep equal areas in equal time (angular momentum conservation). (3) T² ∝ r³ (period-radius relationship). He found these empirically from Tycho Brahe's 20 years of precise planetary data — without calculus or Newton's gravity.
How did Newton derive Kepler's Third Law? ▾
Setting gravitational force = centripetal force: GMm/r² = mv²/r = m(2πr/T)²/r. Simplifying: GM = (2π)²r³/T² → T² = 4π²r³/GM. The inverse-square law gravity directly predicts Kepler's T² ∝ r³.
Why do the planets orbit in the same direction? ▾
The solar system formed from a collapsing nebula with net angular momentum. Conservation of angular momentum preserved the net rotation direction throughout collapse and planet formation. All planets orbit counter-clockwise as viewed from Earth's north pole.
What are orbital resonances? ▾
Resonances occur when period ratios are simple fractions (1:2, 1:3, 2:3). Jupiter-Saturn: 5:2 resonance. Io-Europa-Ganymede: 1:2:4. These aren't coincidences — gravitational interactions over billions of years push bodies into resonances. Resonances can be stable (Trojan asteroids) or unstable (Kirkwood gaps in asteroid belt).
How do astronomers use Kepler's law with exoplanets? ▾
Transit photometry measures T. Radial velocity gives v_star = 2πr_star/T. Mass ratio: M_planet/M_star = r_planet/r_star. Combined with T: r_planet = (GM_star × T²/(4π²))^(1/3) (needs M_star from stellar type). Period ratios between multiple planets in the same system are directly measurable with no M needed.
What is the asteroid belt and Kirkwood gaps? ▾
Kirkwood gaps are gaps in the asteroid belt at orbital resonances with Jupiter (1:3, 2:5, 3:7, 1:2). Asteroids near these resonances are repeatedly pulled by Jupiter at the same orbital point — leading to orbital instability and ejection. Kepler's law predicts the exact resonant period from the gap radius.
Does Kepler's law work for elliptical orbits? ▾
Yes — using the semi-major axis a instead of radius r: T² = 4π²a³/GM. For a circular orbit, a = r. Comets have highly elliptical orbits with enormous a; Halley's comet (period 75.3 years) has a = (75.3)^(2/3) AU = 17.8 AU despite approaching within 0.59 AU at perihelion.
What is the 'third law' constant? ▾
For solar system planets: T²/r³ = 4π²/(GM_sun) = 2.974×10⁻¹⁹ s²/m³ = 1 yr²/AU³. This constant is the same for every planet — a direct measure of solar mass. For Earth-orbiting satellites: same law with M_Earth. Different constant for each central body.

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Formula Explorer connections

Interpretation: This relationship connects mass, distance, orbit or spacetime behavior through gravitation and astrophysical scaling. Assumption: Many calculations assume spherical bodies, point masses, circular orbits, weak fields or Newtonian gravity; relativistic regimes require the stated correction.

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