Johannes Kepler was able to test several rival hypotheses for how the Sun and the planets

are arranged in the Solar System, eventually leading to his three laws of planetary motion.

In 1609, he published the first two laws in a book called Astronomia Nova, which focused

on the movements of the planet Mars. Mars was something of a conundrum - its observed

motions didn't match any of the proposed models of the solar system,

which involved circular orbits.

Kepler's First Law states simply that Mars travels in an elliptical orbit, with the Sun

at one focus of the ellipse. Although he chose to list it first, Kepler only came to this

conclusion after figuring out his “second” law, which says that if you draw a line from

the Sun to Mars, and wait a fixed amount of time, that line will sweep out a certain area

as Mars moves along its orbit. What Kepler noticed was that this area is exactly the

same no matter where in the orbit you are.

This is often phrased as Kepler's “equal area in equal time” law, and this law works

because Mars doesn't move at a constant velocity - it speeds up the closer it gets

to the Sun. So if Mars is approaching perihelion, the point in the orbit nearest to the Sun,

it's traveling faster than if it's at aphelion, the point that's farthest away.

In the first case, the line connecting Mars to the Sun is very short, but because the

planet is moving faster, it covers a lot of distance. In the second case, the line segment

is much longer, but Mars also moves more slowly. Either way, the area swept out in a fixed

amount of time is the same.

Kepler and his contemporaries could see that Mars doesn't move at a constant rate, but

they didn't know why. The inverse relationship that Kepler proposed between distance from

the Sun and orbital velocity could explain the puzzling observations of Mars' movements,

but only if the orbit is an ellipse. A circular orbit would mean no change in distance from

the Sun with time, and thus the velocity would be constant as well.

These two statements--that

Mars travels in an elliptical orbit and that its speed varies so that the Mars-Sun line

sweeps out equal areas in equal time--were generalized to include all planets in 1621,

and they constitute Kepler's first and second laws of planetary motion.

The 2nd Law, it turns out, is also a consequence of the conservation of angular momentum (which

was not a concept known to Kepler in the seventeenth century). Angular momentum is a measure of

the amount of rotational motion in a body or system of bodies, like Mars and the Sun,

and in the absence of outside forces, it's a fixed quantity. This implies a tradeoff

between the distance at which Mars orbits and its velocity -- like Kepler noticed. Just

as an ice skater spins faster after pulling her arms close to her body, Mars has to move

faster when it gets closer to the Sun. Kepler's statement that the area swept out by the Mars-Sun

line is constant is equivalent to the statement that angular momentum is a constant as well

-- that is to say, that it's conserved.