Kepler's laws
Johannes Kepler discovered three laws describing how planets move. They remain accurate descriptions of orbits today.
Kepler's first law
Planets orbit the Sun in ellipses, with the Sun at one focus. An ellipse is a stretched circle defined by two foci; the sum of distances from any point on it to the two foci is constant. Most planetary orbits are nearly circular, but comets often have very elongated ellipses.
Eccentricity
Eccentricity (e) measures how stretched an ellipse is: 0 is a perfect circle, and values approaching 1 are very elongated. Earth's orbital eccentricity is about 0.017; Mars's is about 0.09; Halley's Comet's is about 0.97. The semi-major axis (a), half the longest diameter, is the planet's average distance from the Sun.
Kepler's second law
A line from the Sun to a planet sweeps out equal areas in equal times. This means a planet moves faster when closer to the Sun (at perihelion) and slower when farther (at aphelion). Earth moves about 3% faster in January than in July.
Kepler's third law
The square of a planet's orbital period is proportional to the cube of its semi-major axis: \(P^2 = a^3\), when P is in years and a in astronomical units (AU), where 1 AU is Earth's average distance from the Sun, about 150 million km. Planets farther from the Sun take much longer to orbit.
P^2 = a^3Using the third law
Jupiter orbits at about 5.2 AU. Then \(P^2 = 5.2^3 \approx 140.6\), so \(P \approx 11.9\) years, matching its observed period of about 12 years. Conversely, if an asteroid takes 8 years to orbit, \(a^3 = 64\), so \(a = 4\) AU.
What Kepler's laws mean
Kepler's laws describe how planets move but not why. They are empirical laws, discovered from data. Isaac Newton later showed that they follow from his laws of motion and universal gravitation, explaining them in terms of underlying physics.
Beyond the solar system
Kepler's laws apply to any object orbiting another under gravity: moons around planets, artificial satellites, and planets around other stars. Astronomers use them to find the masses of stars and planets from orbital periods and distances, and the Kepler space telescope (2009-2018) was named in his honour.
Worked example
Mars has a semi-major axis of about 1.52 AU. Its period is P = square root of (1.52 cubed) = square root of 3.51, about 1.88 years, or roughly 687 days. Saturn at 9.5 AU has P = square root of 857, about 29.3 years, so Saturn returns to the same place in our sky only about every 29 years.
Key takeaways
Kepler's first law: orbits are ellipses with the Sun at a focus. Eccentricity measures elongation; Earth's is about 0.017. Second law: equal areas in equal times, so planets move faster near the Sun. Third law: P^2 = a^3 in years and AU; 1 AU is about 150 million km. Kepler's laws are empirical; Newton explained them. They apply to moons, satellites, and exoplanets.
Flashcard
Orbits are ellipses with the Sun at a focus
Check your understanding
An asteroid orbits the Sun at a semi-major axis of 9 AU. What is its orbital period?