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What Is Kepler's Third Law?
Kepler's Third Law, also called the Law of Harmonies, states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Published by Johannes Kepler in 1619, this law applies to any two bodies in gravitational orbit around each other, from artificial satellites to binary star systems.
Newton later showed that Kepler's Third Law is a natural consequence of the law of universal gravitation. The Newtonian form includes the masses of both bodies, making it a powerful tool for determining the mass of celestial objects. For most planetary systems, the mass of the orbiting body is negligible compared to the central body, simplifying the formula.
The Formula
Where T is the orbital period, G is the gravitational constant (6.674 x 10^-11 N m^2/kg^2), M is the central body mass, and a is the semi-major axis of the orbit.
Solar System Data
| Planet | a (AU) | T (years) | T²/a³ |
|---|---|---|---|
| Mercury | 0.387 | 0.241 | 1.002 |
| Venus | 0.723 | 0.615 | 1.001 |
| Earth | 1.000 | 1.000 | 1.000 |
| Mars | 1.524 | 1.881 | 0.999 |
| Jupiter | 5.203 | 11.86 | 0.999 |
| Saturn | 9.537 | 29.46 | 1.000 |
Applications
- Determining stellar masses: By observing the period and orbital radius of a planet or companion star, astronomers calculate the mass of the central star.
- Satellite orbits: Engineers use Kepler's Third Law to determine the required altitude for a desired orbital period, such as geostationary orbit at 35,786 km.
- Exoplanet discovery: Transit timing and radial velocity measurements yield orbital parameters from which planet properties are derived using Kepler's law.
- Galaxy mass estimation: Orbital velocities of stars at known radii reveal the mass distribution within galaxies.
Frequently Asked Questions
Does Kepler's Third Law work for elliptical orbits?
Yes, Kepler's Third Law applies to all elliptical orbits. The semi-major axis "a" is the average of the closest (periapsis) and farthest (apoapsis) distances from the central body. The law works regardless of eccentricity.
Why does T squared divided by a cubed equal 1 for the Solar System?
When using years for time and AU for distance, the ratio T^2/a^3 equals approximately 1 for all planets because Earth defines both units: 1 year and 1 AU give T^2/a^3 = 1. This convenient simplification only works when orbiting the Sun with these specific units.
How accurate is Kepler's Third Law?
For single-planet systems or when the orbiting body mass is negligible, Kepler's Law is extremely accurate. Deviations arise from gravitational perturbations by other planets, relativistic effects near massive objects, and non-negligible satellite mass. For the Solar System planets, the law holds to better than 0.1% accuracy.