Kepler's Laws of Planetary Motion

The order of the universe in three laws

Johannes Kepler (1571–1630) was a German astronomer with poor eyesight. Having caught smallpox as a child, he could not observe the stars well with his own eyes. To do his work, he needed someone else's.

In 1600 he went to Prague as an assistant to Tycho Brahe, the greatest observer of the age. For 20 years Tycho had recorded the position of Mars in the sky with the naked eye, building up the most precise observations possible without a telescope. Tycho died suddenly in 1601, and Kepler took over his data.

For the next 8 years he tried to explain the orbit of Mars. First he tried the 2,000-year-old dogma: the circle. It did not work: there was a discrepancy of 8 arcminutes (2/15 of a degree) with the observations. Most astronomers would have considered such a small difference "negligible". Kepler refused. He later wrote: "If I had believed that we could ignore these eight minutes, I would have patched up my hypothesis accordingly. But since it was not permissible to ignore them, those eight minutes pointed the road to a complete reformation in astronomy."

He abandoned the circle. For years he tried other orbital shapes: egg shapes, ovals. Finally he tried the ellipse. It worked. In 1609 he published his first two laws. Ten years later, in 1619, the third followed.

First law: the ellipse and the focus

Every planet traces an ellipse around the Sun; the Sun lies at one of the two foci of the ellipse.

An ellipse is like a "stretched circle". It has two foci (for a circle the two foci coincide at the centre). Its shape is set by the eccentricity (e):

• e = 0 → a perfect circle
• e = 0.5 → a noticeable ellipse
• e → 1 → a very long, squashed ellipse

The eccentricities of the planets are generally small:

• Earth: e ≈ 0.017 (almost a perfect circle, which is why nobody caught the difference for 2,000 years)
• Mars: e ≈ 0.093 (thanks to Mars, Kepler found the rule)
• Mercury: e ≈ 0.21 (the most eccentric planet in the Solar System)
• Halley's Comet: e ≈ 0.97 (a very long orbit; it comes back to the Sun every 76 years)

The point closest to the Sun is called perihelion and the farthest aphelion. Earth passes perihelion in early January (around 4 January), so northern hemisphere winter is exactly when we are closest to the Sun. The seasons are caused not by distance but by the tilt of the Earth's axis (a common misconception).

Second law: equal areas (the secret of speed)

The line joining the Sun and a planet sweeps out equal areas in equal times.

At first sight this law looks abstract. In fact it has a beautiful physical meaning: a planet moves FAST when it is near the Sun and SLOW when it is far away. Near the Sun the line to the planet is short, so to sweep the same area it has to turn through a wide angle quickly. Far away the line is long, so it can sweep the same area through a smaller angle, and therefore more slowly.

In modern physics, this is conservation of angular momentum:

$L = m \cdot r^2 \cdot \frac{d\theta}{dt} = \text{constant}$

So as $r$ decreases, $d\theta/dt$ (the angular speed) increases, as Newton later proved.

The example of Halley's Comet: at perihelion it flies at 54 km/s; at aphelion (beyond Neptune) it slows to 0.9 km/s, a difference of 60 times!

Third law: period and distance (T² = a³)

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

In astronomy this is the most practical of the laws. If you measure the semi-major axis in AU (astronomical units, the Earth-Sun distance, 150 million km) and the period in years, the constant of proportionality is exactly 1:

$T^2 = a^3$

Some examples:

As you can see, T² and a³ match almost perfectly. In 1687 Newton showed that this law follows naturally from the law of universal gravitation, F = G·M·m / r². So Kepler's laws are not just descriptions but logical consequences of the law of gravity.

Kepler's legacy to science

Kepler's discovery was not just three laws; it was a demonstration of the modern scientific method:

• Trust the data: he did not ignore an 8-arcminute discrepancy
• Break with dogma: circular orbits may have seemed "aesthetically beautiful", but when they did not match reality he chose the ellipse
• Mathematics is the language of nature: his three laws are purely mathematical; physics is not philosophy but measurable reality

This approach lies at the foundation of Galileo, Newton and all of modern science.

One step further

In Newton's Principia Mathematica (1687), Kepler's laws appear as proved theorems:

• First law: under any central force with F ∝ 1/r², orbits are conic sections (ellipses, parabolas, hyperbolas)
• Second law: a consequence of conservation of angular momentum
• Third law: follows exactly from the relation G·M = 4π²·a³/T², which lets us measure the mass of the Sun

All modern space missions are guided by Kepler's mathematics: Cassini on its way to Saturn, James Webb settling at its Lagrange point, SpaceX rockets reaching orbit; all of them obey these three laws.

Sources

• NASA: Orbits and Kepler's Laws
• Stanford Encyclopedia of Philosophy: Kepler
• Johannes Kepler, Astronomia Nova (1609)