The path of a planet is not a circle. It is an ellipse. And the Sun does not sit at the centre of that ellipse. It sits at one of two points called foci.
An ellipse has a defining rule. From any point on it, the distances to the two foci add to the same total. That total is the length of the long axis, , and is the semi-major axis.
How far the Sun is displaced from the centre is the eccentricity . At the shape is a circle, and it grows thinner as approaches . For the Earth it is , which no eye could tell from a circle.
This is the first law. Kepler drew it out of the observations of Mars in .
Draw a line from the Sun to the planet. As the planet moves, that line sweeps out a wedge.
Let it sweep for equal intervals of time and the areas come out equal. Near the Sun the wedge is long and thin, far from it short and broad. The shapes differ entirely; the areas do not.
Which means the planet moves quickly when it is close and slowly when it is far. The Earth is nearest in January, and that is when it is moving fastest.
This is the second law, published in the same year, . It was later recognised as saying exactly that angular momentum is conserved.
So far everything has concerned a single planet. The third law ties the planets to one another.
Write the orbital period as and the semi-major axis as , and comes out the same for every planet. It holds for anything going round the Sun: planets, asteroids, comets alike.
Put in numbers. For the Earth and . For Mars , so , and years, so . They match.
This is the third law, which came in , nine years after the other two. Kepler had spent much of that time trying to explain the spacing of the planets with regular solids, and that attempt came to nothing.
None of the three was derived from theory. What Kepler had was the record of observations Tycho Brahe had gathered over years, and nothing else.
These were observations made before the telescope. Even so, the record of Tycho was extraordinarily accurate for its day, good to about arcminutes. An arcminute is a sixtieth of a degree.
Kepler first tried to fit the path of Mars with a circle. It very nearly worked, but a discrepancy of arcminutes refused to go away. With anyone else observations, that gap could have been dismissed as error.
With the record of Tycho it could not. For the sake of those arcminutes, Kepler gave up the circle and arrived at the ellipse.
The three laws state precisely how a planet moves. Four centuries later they still place the planets of the solar system almost exactly.
Why a planet moves that way, though, is nowhere in them. No reason is given for the ellipse, for the equal areas, or for the link between period and axis.
Only the direction can be read off: the planet is pulled toward the Sun. That much the three laws show. How strongly it is pulled, they do not.
Newton filled that gap. In the next article we derive, from these three laws, a force falling off as the square of the distance.