The Laws of Planetary Motion

Core Concepts

The geometry of ellipses

An ellipse is an oval defined by two special points, its foci (singular: focus): for any point on the ellipse, the sum of the distances to the two foci is constant. You can draw one with two pins and a loop of string. Two numbers describe it:

• Semi-major axis (a): half the longest diameter — the average distance of the planet from the Sun. This sets the size of the orbit and appears in Kepler's third law.
• Eccentricity (e): how "stretched" the ellipse is, from 0 to just under 1. An eccentricity of 0 is a perfect circle; values near 1 are long, thin ovals.

In our solar system, Earth's orbit is nearly circular (e ≈ 0.017), Mars's is more noticeable (e ≈ 0.09), and Mercury's is the most stretched of the major planets (e ≈ 0.21). Comets can reach eccentricities of 0.9 or higher — extreme ellipses that bring them close to the Sun and fling them far away.

Kepler's first law: the law of ellipses

Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus.

The Sun occupies one focus; the other is empty. Because of this, every planet's distance from the Sun changes over the year: the closest point is perihelion and the farthest is aphelion. (For orbits around Earth — the Moon or satellites — the terms are perigee and apogee.)

Kepler's second law: the law of equal areas

A line from the Sun to a planet sweeps out equal areas in equal intervals of time.

When the planet is near perihelion, it covers a short arc but a wide "triangle" (the Sun is close); near aphelion, a long arc but a narrow one. For the areas to stay equal, the planet must move faster near perihelion and slower near aphelion. Earth, for example, moves slightly faster in January (near perihelion) than in July. This law is really a statement of conservation of angular momentum: with no external torque, mass × speed × distance from the Sun stays constant, so speed must rise when distance falls.

Kepler's third law: the harmonic law

The square of a planet's orbital period is proportional to the cube of its semi-major axis.

Written for the solar system with P in Earth years and a in astronomical units (AU; 1 AU = Earth–Sun distance, a commonly taught reference value):

P² = a³

This one compact equation lets you compute a planet's year from its distance, or its distance from its year:

• Earth: a = 1 AU → P² = 1³ = 1 → P = 1 year. ✓
• Jupiter: a = 5.2 AU → P² = 5.2³ ≈ 140.6 → P ≈ 11.9 years. ✓
• Mercury: a = 0.387 AU → P² ≈ 0.058 → P ≈ 0.24 year (~88 days). ✓

The third law is a proportion, not a fixed constant — it only works when the orbiting masses are similar (planets around the same star). Newton later generalized it to include both masses, which is how astronomers measure the masses of stars, black holes, and exoplanets.

Why the laws work together

The three laws are not separate curiosities; they describe one physical picture. The first law sets the track (ellipse, Sun at one focus). The second sets the timing (equal areas → variable speed). The third sets the global scale (larger orbit ↔ longer year). Any model of the solar system must satisfy all three simultaneously — and Newton's gravity, as the next topics show, produces all three from a single force law.

How It Works / Step-by-Step Process

Using Kepler's third law to find a planet's year:

1. Get the semi-major axis in AU. For Mars, a ≈ 1.52 AU (a commonly taught value).
2. Cube it: a³ = 1.52³ ≈ 3.51.
3. Set equal to P²: P² = 3.51.
4. Take the square root: P ≈ √3.51 ≈ 1.87 years (about 687 days).
5. Check the direction of the relationship: a larger a gives a larger P — outer planets always have longer years, which is why Jupiter (a = 5.2 AU) needs ~11.9 years and Neptune (a ≈ 30 AU) needs ~165 years.
6. Remember the units: the shortcut only works with years and AU. Using kilometers and seconds requires the full Newtonian version with G and masses.

Common Confusions