Kepler's Laws of Planetary Motion: Three Rules Wrung Out of Twenty Years of Data

Eight Arcminutes That Would Not Go Away

In 1600, Johannes Kepler took a job as assistant to Tycho Brahe, the Danish nobleman who had spent two decades measuring the positions of the planets with instruments the size of a room and no telescope at all — the telescope had not been invented yet. Brahe’s naked-eye positions were accurate to about one arcminute, a sixtieth of a degree, roughly the width of a coin seen from sixty metres. Nobody had ever measured the sky that well.

Brahe died in 1601 and Kepler inherited the data. He spent the next five years trying to fit the orbit of Mars to a circle, the shape that everyone from Aristotle to Copernicus had assumed was compulsory for heavenly bodies. He got close. His best circular model reproduced Brahe’s Mars positions to within eight arcminutes. Any earlier astronomer would have declared victory and gone home, because eight arcminutes was smaller than the error in every catalogue that had come before. Kepler knew Brahe’s data was better than that, and eight arcminutes was therefore not noise — it was a message. He threw the circles away.

The First Law: The Sun Sits Off-Centre

What came out of that wreckage, published in Astronomia Nova in 1609, was the first law: each planet moves in an ellipse, with the Sun at one focus. Not at the centre. At a focus, one of two special points inside the ellipse, and the other focus is empty space.

How stretched an ellipse is gets measured by its eccentricity, e, running from 0 for a perfect circle up toward 1 for something nearly flat. If the semi-major axis — half the long diameter — is a, then the closest approach to the Sun is a(1 − e) and the farthest is a(1 + e). These points are called perihelion and aphelion.

The surprising part is how gentle real planetary ellipses are. Earth’s eccentricity is 0.0167. Drawn accurately on paper, our orbit is indistinguishable from a circle by eye; what gives it away is that the Sun is noticeably off-centre, sitting about 2.5 million kilometres from the middle. We are 147.1 million kilometres out in early January and 152.1 million in early July, which is why, counterintuitively for the northern hemisphere, we are closest to the Sun in the depths of winter. Earth’s seasons come from axial tilt, not from distance. Mercury is the outlier among planets at e = 0.206, and comets routinely run above 0.9.

The Second Law: Equal Areas in Equal Times

Kepler’s second law is stranger and, in some ways, deeper. Draw a line from the Sun to the planet. As the planet moves, that line sweeps out area like a windscreen wiper. Kepler found that the line sweeps equal areas in equal times — always, for every planet, everywhere on the orbit.

Because the line is short near perihelion and long near aphelion, the only way the swept areas can match is if the planet covers far more angle when it is close in. In plain terms: planets speed up as they fall toward the Sun and slow down as they climb away. Earth manages about 30.3 km/s in January and 29.3 km/s in July. Halley’s Comet swings from roughly 54 km/s at perihelion down to under 1 km/s out past Neptune.

Kepler had no idea why this was true. Newton later showed it is simply conservation of angular momentum: gravity pulls straight along the Sun–planet line, so it produces no torque about the Sun, so the quantity r × v cannot change. Equal areas in equal times is that conservation law wearing a geometric disguise.

Eccentricity e 0.50

Semi-major axis a (AU) 1.00

Pause Orbital period 1.00 years

The Third Law: The One He Was Really Looking For

Kepler spent his life convinced that the solar system was built on musical and geometric harmonies, and he kept hunting for a rule linking a planet’s distance to its year. He found it on 15 May 1618 and published it in Harmonices Mundi in 1619: the square of the orbital period is proportional to the cube of the semi-major axis, T² ∝ a³.

Choose the right units and the proportionality becomes an equality. Measure T in Earth years and a in astronomical units — one AU being Earth’s average distance, 149.6 million kilometres — and the law reads simply T² = a³. Mars sits at a = 1.524 AU, so its year is 1.524^1.5 = 1.88 years. Jupiter at 5.204 AU takes 11.87 years. Neptune at 30.07 AU takes 164.8. The rule held for every planet Kepler knew and for every one discovered since, including Uranus in 1781 and Neptune in 1846, neither of which he could have imagined.

Newton Shows They Were Gravity All Along

Kepler’s laws are empirical. They describe; they do not explain. The explanation arrived in 1687, when Isaac Newton published the Principia and derived all three from a single assumption: that every mass attracts every other with a force F = Gm₁m₂/r², combined with his three laws of motion.

The first law falls out of the mathematics of an inverse-square force, which permits only conic sections as trajectories. The second law is angular momentum conservation. The third emerges by balancing gravity against the centripetal requirement of curved motion, giving T² = 4π²a³/GM, where M is the mass of the central body. That formula is Kepler’s third law with the constant made explicit, and it shows what Kepler could not have known: the constant depends on the mass at the centre. Swap the Sun for Earth and the same equation hands you a geostationary radius of 42,164 kilometres for a 24-hour orbit, which is why every television satellite hangs at that exact altitude. The deeper reason orbiting objects never fall in is covered in our piece on why the Moon does not fall, and the modern picture of what gravity actually is appears in our guide to gravity from Newton to Einstein.

Where the Rules Bend

Kepler’s laws are exact only for two point masses alone in the universe. Reality is messier. Planets tug on each other, so orbits slowly precess and wobble; it was these gravitational nudges from an unseen body that led to Neptune being predicted on paper before anyone pointed a telescope at it. General relativity adds a further correction, tiny for most planets but measurable for Mercury, whose perihelion advances an extra 43 arcseconds per century — a discrepancy that stood unexplained for half a century until Einstein accounted for it in 1915.

Even so, the three laws remain the working tools of the field. Every exoplanet mass and orbit derived from the transit and radial-velocity methods runs through Kepler’s third law. So does every estimate of a galaxy’s mass, and every flight plan that threads a spacecraft to Mars.

A Lesson in Taking Errors Seriously

The most instructive thing about Kepler is not the ellipse. It is the eight arcminutes. He had a model that was better than anything before it, a residual smaller than the measurement error of every rival dataset, and a professional life’s worth of reasons to call it finished. Instead he trusted that Brahe’s numbers were good enough that a small disagreement had to mean something.

Two thousand years of assuming that circles were the only shape fit for the heavens ended because one man refused to round away a discrepancy the width of a fingernail at arm’s length.