Decades before Newton wrote down a law of gravity, Johannes Kepler stared at Tycho Brahe's planetary observations and pulled three patterns out of the data. They are still the cleanest description of how planets move that anyone has ever produced.
In the late 1500s, the Danish astronomer Tycho Brahe built the most precise pre-telescope observatory in the world and spent twenty years recording the positions of planets — Mars in particular — to roughly one arcminute, or 1/60th of a degree. When Tycho died in 1601, his assistant Johannes Kepler inherited the data and spent the next eight years trying to fit a circular orbit to it. He couldn't. The data wouldn't bend.
Kepler eventually gave up on circles, and in giving up, he discovered three rules so general that they apply not just to the planets in our solar system but to satellites, comets, exoplanets, and binary stars. He had no idea why the rules worked — that wouldn't come until Newton's Principia in 1687. He just saw that they did.
Each planet moves in an ellipse, with the Sun at one focus.
This is the result we took for granted in the previous module. Kepler arrived at it empirically, by trying every shape he could think of until ellipses fit the Mars data. The reason ellipses (and not, say, ovals or egg-shapes) is the inverse-square nature of gravity — but proving that requires calculus Kepler didn't have.
A line drawn from the planet to the Sun sweeps out equal areas in equal times.
This is the strangest of the three at first glance, and the most beautiful once you see it. It says that even though a planet's distance to the Sun and its speed both change throughout its orbit, the area swept out per unit time is constant. Near perihelion, where the planet is close to the Sun, it moves fast; near aphelion, where it's far, it crawls. The two effects exactly compensate.
Modern physics recognizes Kepler's second law as a statement of angular momentum conservation. The areal velocity $\frac{dA}{dt}$ equals $\frac{L}{2m}$, where $L$ is angular momentum and $m$ is the planet's mass. Since gravity is a central force — it always points along the line between the two bodies — there is no torque on the orbit, so $L$ is constant, and so is the rate at which area is swept.
The square of a planet's orbital period is proportional to the cube of its semi-major axis.
In equation form:
$$ T^2 = \frac{4\pi^2}{GM} a^3 $$where $T$ is the orbital period, $a$ is the semi-major axis, $G$ is Newton's gravitational constant, and $M$ is the mass of the central body. The constant of proportionality $4\pi^2 / GM$ depends only on what you're orbiting around — the Sun, Earth, Jupiter, whatever — so every body orbiting that same central mass obeys the same $T^2 \propto a^3$ relationship.
We can derive the third law for the special case of a circular orbit using only what we already know. Setting gravity equal to centripetal force gives $v = \sqrt{GM/a}$. The period of a circular orbit is the circumference divided by the speed: $T = 2\pi a / v$. Squaring and substituting yields exactly the equation above. It is one of those satisfying moments where you realize a 17th-century empirical law falls out of high-school physics.
Of the three laws, the second is the hardest to grasp from words alone and the most rewarding to watch. The interactive below shows a planet on an elliptical orbit, with a wedge sweeping out the area between the planet and the central body. The wedge's color stays bold for a fixed time interval and then fades — so you can compare the shapes of wedges drawn at different points in the orbit and see, with your own eyes, that they all have the same area.
Crank up the eccentricity and watch what happens at perihelion: the planet whips around fast, and the wedge becomes short and fat. Out near aphelion it crawls, and the wedge stretches long and thin. Different shapes, same area. The areal rate readout in green stays constant — that's the conservation law in action.
To draw the planet at a specific moment in time, we need to know where it is at that moment — and this is where the curriculum has to admit a small piece of unavoidable trickery. The relationship between time and position on an elliptical orbit isn't simple. It's governed by Kepler's equation, a transcendental relation that has to be solved numerically.
The full machinery is in the appendix for the curious; the short version is that we step the mean anomaly $M$ forward at a constant rate (because Kepler's second law guarantees angular momentum is conserved, $M$ ticks linearly with time), then iteratively solve $M = E - e \sin E$ for the eccentric anomaly $E$, then convert $E$ to a position. Newton-Raphson converges in about four iterations. It runs every animation frame and you'd never know.
We've now seen that orbits are ellipses (first law), that the planet's motion along that ellipse follows a strict areal-velocity rule (second law), and that the size of the orbit fixes the period (third law). The next module zooms in on the geometry of the ellipse itself — what the parameters $a$ and $e$ really mean, and how to read off perihelion and aphelion distances from them.