Kepler's Laws of Planetary Motion — Ellipses, Areas and Periods

High School⏱ 9 min readClassical Mechanics

Kepler's three laws of planetary motion — elliptical orbits, equal areas in equal times, and the T² ∝ a³ harmony — and how Newton explained them.

Quick answer

Kepler's laws describe planetary motion: (1) orbits are ellipses with the Sun at one focus, (2) a planet sweeps equal areas in equal times, (3) the square of the orbital period is proportional to the cube of the average distance, T² ∝ a³.

$T^2 = \frac{4\pi^2}{GM}a^3$

Unit: T in seconds or years, a in meters or AU (astronomical units)

Example: Mars orbits at 1.524 times Earth's distance, so its year is 1.524^1.5 ≈ 1.88 Earth years — matching its observed 687-day period.

What you will learn

State all three of Kepler's laws in plain language
Explain what orbital eccentricity and the semi-major axis mean
Use T² ∝ a³ to compare orbital periods
Describe how Newton derived Kepler's laws from gravitation

From Tycho's data to three laws

In the early 1600s, Johannes Kepler inherited decades of naked-eye planetary observations from the Danish astronomer Tycho Brahe — the most precise data in existence. Kepler spent years trying to fit Mars's orbit to circles, and it would not fit. His breakthrough was abandoning circles entirely.

The result was three empirical laws, published in 1609 and 1619, that describe how planets move — found from data alone, with no theory of why. Newton supplied the why seventy years later with universal gravitation.

The three laws

Law 1 — The law of ellipses. Every planet orbits the Sun in an ellipse, with the Sun at one focus. An ellipse has two foci; a circle is the special case where they coincide.

Law 2 — The law of equal areas. A line from the Sun to a planet sweeps out equal areas in equal times. Concretely: the planet moves faster when near the Sun (perihelion) and slower when far (aphelion).

Law 3 — The law of harmonies. The square of a planet's orbital period is proportional to the cube of its average distance from the Sun: $T^2 \propto a^3$ .

T^2 = \frac{4\pi^2}{GM}\,a^3

The third law with its constant — which encodes the Sun's mass M.

Ellipses, foci and eccentricity

An ellipse is defined by two numbers. The semi-major axis $a$ is half the longest diameter — effectively the planet's average distance from the Sun. The eccentricity $e$ measures how stretched the ellipse is: $e = 0$ is a perfect circle, and values approaching 1 are long and thin.

Most planetary orbits are nearly circular: Earth's eccentricity is just $0.017$ , Venus's $0.007$ . Mercury's is $0.206$ — noticeably egg-shaped — and Pluto's is $0.249$ , which is part of why its distance from the Sun varies so much. Comets are the extreme case: Halley's comet has $e = 0.967$ , swooping from near the Sun out past Neptune.

Focus, not center

An ellipse has two focal points. The Sun occupies one; the other is empty. When e is small the foci nearly coincide and the orbit looks like a circle — but the offset is always there.

Using the third law

The third law is the most useful quantitatively. For any two bodies orbiting the same star:

$\frac{T_1^2}{a_1^3} = \frac{T_2^2}{a_2^3}$

Astronomers measure distances in astronomical units (AU) — 1 AU is Earth's average distance — and periods in years, which makes the arithmetic clean: for the solar system, $T^2 = a^3$ with $T$ in years and $a$ in AU.

Example: Mars sits at $a = 1.524\ \mathrm{AU}$ . Its period is $T = \sqrt{1.524^3} = \sqrt{3.540} \approx 1.881\ \mathrm{yr}$ , or about 687 days — the observed Martian year. Jupiter at $5.203\ \mathrm{AU}$ gives $T = \sqrt{5.203^3} \approx 11.86\ \mathrm{yr}$ , matching its ~12-year orbit.

Newton's explanation

Newton showed that all three laws fall out of $F = Gm_1m_2/r^2$ plus his laws of motion:

Ellipses are the natural trajectories of motion under an inverse-square central force (for bound, negative-energy orbits).
Equal areas follow because a central force exerts no torque about the Sun, so angular momentum — and hence the areal sweep rate — is conserved.
$T^2 \propto a^3$ comes from equating gravity to centripetal force for a circular orbit: $GMm/r^2 = m(2\pi r/T)^2/r$ , which rearranges to $T^2 = (4\pi^2/GM)r^3$ .

Newton's derivation turned Kepler's empirical patterns into consequences of one universal force — arguably the first great unification in physics, connecting earthly falling bodies to celestial orbits.

Beyond planets: moons, satellites and weighing stars

Kepler's laws apply to any system dominated by one central mass: moons around planets, satellites around Earth, and binary stars around each other. The constant in the third law, $4\pi^2/GM$ , changes with the central mass — for Jupiter's moons, $M$ is Jupiter's mass.

They break down where gravity is not the whole story: orbits very close to massive bodies need general relativity (Mercury's perihelion precession was the famous clue), and multi-planet systems show small mutual perturbations. But as a first description of orbital motion, Kepler's three laws remain the starting point of celestial mechanics.

A weighing machine

Rearrange the third law as M = 4π²a³/GT²: measure a moon's orbit and you can weigh its planet. This is how the masses of the outer planets — and of distant stars with planets — were first determined.

Variables and units

Symbol	Name	Unit	Meaning
$T$	Period	s (or yr)	Orbital period — time for one full orbit
$a$	Semi-major axis	m (or AU)	Half the longest diameter of the ellipse; the planet's average distance
$e$	Eccentricity	dimensionless	How stretched the ellipse is: 0 = circle, near 1 = very elongated

Common mistakes

✗ Planets move at constant speed around the Sun.

By the second law, a planet speeds up near perihelion and slows down near aphelion. Only a perfectly circular orbit has constant speed.

✗ The Sun sits at the center of each orbit.

The Sun sits at one focus of the ellipse — off-center. The other focus is empty space. Earth's orbit is nearly circular, so the offset is small, but it is real.

✗ Kepler's laws were derived from Newton's gravity.

Historically it was the reverse: Kepler found the laws empirically from Tycho Brahe's observations (1609–1619), and Newton later showed they follow from universal gravitation.

Real-world applications

Mission planners use the third law to compute orbital periods of satellites from their altitudes.
Astronomers detect exoplanets by the periodic wobble their orbits impose on the host star.
Kepler's laws underpin the Hohmann transfer orbits used to travel between planets.
The third law lets astronomers weigh a planet from the orbit of its moon alone.

Frequently asked questions

Are Kepler's laws exact?

Yes — they are approximations for one light planet orbiting one heavy star. When the two bodies have comparable mass, both orbit their common center of mass. Mutual perturbations from other planets also cause small deviations, which is how astronomers discovered Neptune — from unexplained wobbles in Uranus's orbit.

Why ellipses rather than some other shape?

The ellipse comes from the inverse-square nature of gravity — it is the shape traced by an object moving under a central 1/r² force with the right energy and angular momentum. A circle is simply the special ellipse with zero eccentricity.

What does the second law have to do with angular momentum?

Equal areas in equal times is the geometric face of angular momentum conservation. Because gravity pulls straight toward the Sun (a central force), it exerts no torque, so the planet's angular momentum cannot change — and constant angular momentum is exactly what sweeping equal areas means.

Do the laws apply to moons and satellites too?

Yes, to excellent precision. Each moon's period and orbital radius satisfy T² ∝ a³ around its planet, and each satellite satisfies it around Earth. The proportionality constant differs (it encodes the central body's mass), but the pattern holds everywhere gravity dominates.