Kepler's laws

Around 1600, Johannes Kepler spent years trying to make the planets move in perfect circles. The observations of Mars refused to fit. What he found instead are three rules that still steer every spacecraft we launch.

The numbers refused to fit the circles. So Kepler changed the circles.

Look

Around 1600, Johannes Kepler worked with years of very precise observations of Mars made by Tycho Brahe. Everyone assumed planets moved in perfect circles. The numbers refused to fit. After years of work, Kepler found three rules that do fit.

1. Orbits are ellipses (stretched circles), with the Sun a little off-centre at a point called a focus.
2. A planet speeds up near the Sun and slows down far from it. Draw a line from the Sun to the planet: it sweeps out equal areas in equal times.
3. Bigger orbits take much longer. Not just longer because the path is longer, but disproportionately longer.

In the sandbox above, Halley’s Comet is loaded. Watch it whip around the Sun and then crawl through the outer part of its orbit. The coloured slices are all swept in the same amount of time: thin and long far away, short and wide close in. Same area, same time. That is the second law in action.

Understand

An ellipse is described by two numbers:

• The semi-major axis a: half the longest width. It sets the size.
• The eccentricity e: how stretched it is. 0 is a circle; close to 1 is very long and thin.

The closest point to the Sun is the periapsis (perihelion for the Sun), at distance a(1 − e). The farthest point is the apoapsis (aphelion), at a(1 + e).

Object	a (au)	e
Earth	1.000	0.017
Mars	1.524	0.093
Halley’s Comet	17.9	0.967

Third law. For everything orbiting the Sun, the square of the period is proportional to the cube of the semi-major axis. In years and au:

T² = a³

Jupiter: a = 5.20 au, so T = 5.20^1.5 = 11.9 years. Halley: a = 17.9 au gives T = 75.9 years, close to its famous 75 to 76-year return.

The sandbox shows T²/a³ in years² per au³. Change the orbit around the Sun as much as you like: it stays 1.

Master

Newton derived all three laws from his law of gravity. The third law in full form is:

T = 2π · √(a³ / μ),   μ = G(M + m)

So T²/a³ = 4π²/μ is the same for every orbit around one central body, but different for different bodies. Switch the sandbox’s central body to Earth and the constant becomes about 9.9 × 10⁻⁵ s² km⁻³ instead of 1 yr² au⁻³. The m in μ is why Kepler’s version is slightly off for Jupiter, whose mass is about 0.1% of the Sun’s.

The second law is conservation of angular momentum. The areal velocity is dA/dt = h/2, where h = r²·dν/dt is the specific angular momentum, constant for any central force. At periapsis and apoapsis the velocity is perpendicular to the radius, so r_p·v_p = r_a·v_a: the speed ratio equals the inverse distance ratio. For Halley, r_a/r_p = (1 + e)/(1 − e) ≈ 60, so it moves about 60 times faster at perihelion than at aphelion.

To place a body on its orbit at a given time you need the mean anomaly M = n(t − t_p), with mean motion n = 2π/T, and then Kepler’s equation, M = E − e sin E. It has no closed-form solution; the tools on this site solve it by Newton iteration, which converges in a few steps even for Halley’s e = 0.967.