Kepler’s Third Law
Kepler's Third Law, sometimes called the 'Harmonic Law', states that the square of a planet's orbital period is directly proportional to the cube of its average orbital radius. In plain terms: planets that orbit farther from the Sun take disproportionately longer to complete one orbit. Earth takes one year to orbit at roughly 150 million kilometres, whereas Neptune, nearly 30 times farther out, takes about 165 years. This elegant mathematical relationship was discovered by the German astronomer Johannes Kepler and published in 1619, decades before Newton explained why it is true. The law applies not just to the planets of our Solar System but to any two bodies orbiting the same central mass — moons orbiting a planet, artificial satellites, or even stars orbiting a black hole. This universality makes it one of the most powerful tools in astronomy. By measuring how long a moon takes to complete one orbit and how far it sits from its planet, scientists can calculate the planet's mass without ever visiting it. The same principle was used to 'weigh' Jupiter using its Galilean moons long before space probes were invented. Kepler derived his three laws purely from careful observation of Tycho Brahe's planetary data — without any underlying theory of gravity. It was Isaac Newton who later showed that Kepler's Third Law follows mathematically from his own Universal Law of Gravitation combined with his second law of motion, giving the relationship a firm theoretical foundation rooted in the physics of centripetal force and gravitational attraction.
- The Proportionality Statement
- Turning the Proportionality into an Equation
- Deriving k from Newton's Laws
- Using the Law to Find Masses
The Proportionality Statement
For any planet orbiting the Sun, if T is the orbital period (how long one full orbit takes) and r is the semi-major axis (average orbital radius), then T² ∝ r³. This means doubling the orbital radius does not merely double the period — it increases it by a factor of 2^(3/2) ≈ 2.83. A planet twice as far from the Sun takes nearly three times as long to complete an orbit.
Turning the Proportionality into an Equation
We introduce a constant k so that T² = k·r³. For planets in our Solar System, the constant k depends only on the Sun's mass. When we use units of years for T and astronomical units (AU) for r (where 1 AU = Earth's orbital radius), the constant k equals exactly 1, giving the simple form T² = r³. For example, Mars has r ≈ 1.52 AU, so T² = 1.52³ ≈ 3.51, giving T ≈ 1.87 years — very close to the actual 1.88 years.
Deriving k from Newton's Laws
Consider a planet of mass m in a circular orbit of radius r around a star of mass M. The gravitational force provides the centripetal force: GMm/r² = mω²r, where ω = 2π/T is the angular velocity. Rearranging gives T² = (4π²/GM)·r³. Therefore the constant k = 4π²/(GM). This derivation reveals that k depends only on the central mass M, not on the orbiting body's mass. Two different planets around the same star share the same k.
Using the Law to Find Masses
Rearranging the Newtonian form gives M = 4π²r³/(GT²). This is enormously useful: by observing the orbital period and radius of any natural or artificial satellite, we can calculate the mass of the central body. Astronomers used this method to determine the mass of the Milky Way's central black hole by tracking stars orbiting it. The star S2 has T ≈ 15.9 years and r ≈ 1000 AU, yielding a central mass of roughly 4 million solar masses.
The Key Equation
- T — Orbital period (seconds (s)): The time taken for one complete orbit around the central body.
- G — Universal gravitational constant (N m² kg⁻²): G = 6.674 × 10⁻¹¹ N m² kg⁻²; the fundamental constant of gravitational attraction.
- M — Mass of the central body (kilograms (kg)): The mass of the star, planet, or other body being orbited. The orbiting body's mass is negligible in comparison.
- r — Orbital radius (semi-major axis) (metres (m)): The average distance between the orbiting body and the centre of the central body. For a circular orbit this is simply the radius.
Worked Example
Problem. Jupiter's moon Io has an orbital period of T = 1.769 days and orbits at a radius of r = 4.22 × 10⁸ m from Jupiter's centre. Use Kepler's Third Law to calculate the mass of Jupiter. (G = 6.674 × 10⁻¹¹ N m² kg⁻²)
Answer. Mass of Jupiter ≈ 1.90 × 10²⁷ kg
Origins · 1 minJohannes Kepler and how this was discovered
Kepler published his three laws of planetary motion between 1609 and 1619. The Third Law appeared in 'Harmonices Mundi' (The Harmony of the World, 1619), where he showed that T² ∝ r³ holds for all planets. He derived these laws entirely from the meticulous observational data of Danish astronomer Tycho Brahe, for whom he had worked as an assistant in Prague. Kepler's laws provided the first precise mathematical description of planetary motion and laid the groundwork for Newton's theory of universal gravitation. Kepler believed the orbital periods of the planets corresponded to musical harmonies — he literally tried to write out the 'music of the spheres', assigning each planet a melodic line based on the ratio of its fastest to slowest orbital speed.
Related Concepts
- Newton's Law of Universal Gravitation — Newton's gravitational law provides the theoretical derivation of Kepler's Third Law, showing why T² ∝ r³ and what the constant of proportionality physically means.
- Gravitational Field and Orbital Motion — Understanding how gravitational fields create centripetal acceleration is essential for deriving and applying Kepler's Third Law to real orbits.
- Kepler's First and Second Laws — Kepler's First Law (elliptical orbits) and Second Law (equal areas in equal times) are the companion laws to the Third Law, together forming a complete description of planetary motion.
- Circular Motion and Centripetal Force — The derivation of Kepler's Third Law from Newton's gravity requires equating gravitational force with centripetal force, making circular motion a key prerequisite concept.