Celestial & Orbital Mechanics

Planets Sweep Equal Areas on an Ellipse

A planet traces an ellipse with the Sun at one focus. It sweeps equal areas in equal times, so it moves faster at perihelion. Period and semi-major axis lock together as T² ∝ a³.

Kepler stared at years of Mars data until he gave up the circle for an ellipse. Three laws tie the shape, the speed, and the period of an orbit into one picture.

First law: the orbit is an ellipse and the Sun sits at one of its two foci. When the eccentricity e is zero the shape is a perfect circle; as e grows it stretches. The nearest point is perihelion at r = a(1−e), the farthest is aphelion at r = a(1+e).

Second law: the line from the Sun to the planet sweeps out equal areas in equal times. So the planet races near perihelion and dawdles near aphelion. This area rule is exactly the statement that angular momentum is conserved.

Third law: the square of the period is proportional to the cube of the semi-major axis. Written as T² = a³ (with T in years and a in astronomical units), it says a planet farther out takes far longer to go around once.

Read the equal-area law as a speed and the planet moves fastest closest to the Sun. The ratio of perihelion speed to aphelion speed is (1+e)/(1−e). The equation linking position to speed is the vis-viva relation v² = μ(2/r − 1/a), which the next lesson unpacks.

Now watch all three laws at once. As the planet rounds the ellipse, the wedge from the Sun fills at a steady rate. The percentage of the period that has passed always equals the percentage of the area that has been swept.

In PracticeKnowing the eccentricity gives the perihelion and aphelion distances at once (r = a(1∓e)), and the semi-major axis alone fixes the period as T = a¹.5. Thanks to the equal-area law you can gauge the relative speed anywhere on the orbit from the ratio (1+e)/(1−e).

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