The radius vector from the Sun to the planet sweeps out equal areas, in equal periods of time.|
(Based on a diagram by John P. Oliver)
Above, each area is equal, but the arcs that sweep out the areas are not; closer to the Sun, the arcs are longer, because the planet moves faster (see below); further from the Sun, the arcs are shorter, because the planet moves slower.
Close to the Sun, a given arc sweeps out a small area; further away, a larger area. This means that it takes less time to cover a given arc close to the Sun, so the planet must be moving faster; and more time to cover a given arc further from the Sun, so the planet must be moving slower. In other words, the Law of Areas implies that as the planets move away from the Sun, they slow down; and as they move toward it, they speed up. For small changes in distance (low-eccentricity orbits), the change in speed is about the same, in percentage terms, as the change in distance; that is, for each percent that a planet moves closer to the Sun it speeds up by about one percent, and for each percent that it moves away from the Sun it slows down by about one percent. As an example, as the Earth moves 1.7% closer to the Sun, in January, it speeds up by 1.7%, and as it moves 1.7% further from the Sun, in July, it slows down by 1.7% (see Cassini Measures the Motion of the Sun for a diagram). For larger changes in distance, things are not so simple, except at perihelion and aphelion, where the motion is perpendicular to the radius vector; at those points, the speed is inversely proportional to the distance; so if an object were 10 times further from the Sun at aphelion than at perihelion, it would be going 10 times slower. For some comets, aphelion is as much as 100,000 times further out than perihelion, and although the comet moves nearly 200 miles per second at perihelion, at aphelion, going 100,000 times slower, it only moves about 10 feet per second, and takes 300 years to move as far as it would in one day, at perihelion.
It is possible to calculate the speed at any point in the orbit, using the principle of conservation of energy, in which the kinetic energy of the planet, calculated from its speed and mass, plus its potential energy, calculated from its mass and distance from the Sun, are added together, to get a constant value. Solving the resulting equation yields a formula relating the speed to the orbital size and eccentricity, and the current distance from the Sun. That formula is called the Vis Viva Equation. (this paragraph will be expanded at a later date, as the mathematical and physical concepts are beyond the introductory level of the course, and other topics need attention, as well).
As a planet moves from aphelion to perihelion, it sweeps out half the area of the orbit; and the same area is swept out, as it moves from perihelion to aphelion; so the times required to go from one point to the other are equal. This is because on the way in, the planet is speeding up, and on the way out, it is slowing down; and for every point on the inward journey where the planet has a particular speed, there is a symmetrical point on the outward journey where the planet has the same speed; so it covers the equal arcs at different speeds at different times, but with the same average speed, over the half-orbit shown above.
In the diagram above, the planet moves from one end of the minor axis to the other end, sweeping out a small area as it moves through perihelion. Covering the same distance (half the perimeter of the ellipse) as it moves through aphelion, it sweeps out a much larger area. This is because every moment the planet spends on the perihelion half-ellipse, it is closer to the Sun, moving faster; and at every moment it spends on the aphelion half-ellipse, it is further from the Sun, moving slower.
The Force Behind The Law: A Brief Note of Caution
The reason that the planets move this way is that the force which keeps them in their orbits points at the Sun. When they are moving away from the Sun, its pull is backwards and sideways relative to their direction of motion, so they slow down and curve toward the Sun. When they are moving toward the Sun, its pull is forward and sideways relative to their direction of motion, so they speed up and curve toward the Sun. Neither the nature of the force, its strength, or the way that its strength changes with distance can alter that statement. The Law of Areas would be true no matter what kind of force the Sun exerted on the planets, as long as that force pointed directly toward the Sun.
The force which the Sun exerts on the planets is the same as the gravitational force which the Earth exerts on us; and is, as discovered by Isaac Newton, an inverse-square law, meaning that it gets weaker as the planet moves away from the Sun, and stronger as the planet moves toward the Sun. Students often mistakenly presume or state that planets slow down as they move away from the Sun, because the Sun's force is getting weaker, with the implication that if they didn't go slower, its force couldn't hold onto them; and that planets speed up as they move toward the Sun, because the Sun's force is getting stronger, and a stronger force makes them go faster than a weaker one would. This sounds reasonable, but is a totally incorrect analysis of the Law of Areas. No matter how the Sun's force changed with distance -- getting weaker as the inverse square of the distance, or in any other way; not changing; or even getting stronger -- only the shape of the orbital motion would be affected. The Law of Areas would be completely unaffected, because all that depends on, as already stated, is that the force is backwards, slowing the planet down, as it moves outward; and is forwards, speeding the planet up, as it moves inward.