Kepler’s laws

Planets move in ellipses

Narrated video. Transcript:

For thousands of years people watched the planets wander across the night sky. Here is the sky over India, with Mars drifting slowly against the stars.

Tycho Brahe measured these positions very carefully, night after night, without a telescope. His assistant Johannes Kepler spent years looking for a pattern in the numbers.

Kepler found that the orbits are not circles. Watch the path of Mars drawn out from the data: it is a slightly squashed circle, an ellipse.

Here is how to draw an ellipse. Fix two pins, the foci, loop a string around them and trace a curve with a pencil, keeping the string tight.

The longest diameter is the major axis. Half of it is the semi-major axis, a. The two points on the ellipse closest to and farthest from a focus are on this axis.

Kepler’s first law, the law of orbits: every planet moves in an ellipse with the Sun at one focus. The other focus is just an empty point in space.

The point nearest the Sun is the perihelion, P. The point farthest away is the aphelion, A. For the Earth the ellipse is so close to a circle that you could not tell by eye.

A circle is simply an ellipse whose two foci sit on top of each other. Next, let us see how fast a planet moves along its ellipse.

Equal areas, and the law of periods

Narrated video. Transcript:

Draw a line from the Sun to the planet. As the planet moves, this line sweeps out an area, like a windscreen wiper. Watch the shaded wedge grow.

Kepler’s second law, the law of areas: the line from the Sun to the planet sweeps out equal areas in equal intervals of time.

Near the Sun the wedge is short and wide. Far from the Sun it is long and thin. To have the same area, the planet must move faster near the Sun and slower far away.

Why? The pull of the Sun always points straight at the Sun. A force like this produces no torque about the Sun, so the planet’s angular momentum L stays constant.

The area swept per second works out to be L divided by 2m. If L is constant, the rate of sweeping area is constant. That is the second law.

Kepler’s third law, the law of periods, connects the size of an orbit to the time it takes. Here are the planets with their orbits and their years.

The square of the period is proportional to the cube of the semi-major axis: T squared is proportional to a cubed. Watch the column of T²/a³ come out almost the same for every planet.

So a planet four times farther out takes eight times longer to go round, because four cubed is sixty-four, and the square root of sixty-four is eight.

Try it: launch a planet

Interactive simulation. Things to try:

• Press Launch (or Space). The planet starts 1 AU from the Sun, moving sideways.
• Each coloured wedge takes the same time, 1/12 of an orbit. Check the area readout.
• Try speeds from 20 to 40 km/s and watch T²/a³ stay at 1 yr²/AU³.
• At the escape speed the path no longer closes.

Quick check

A planet moves around the Sun in an elliptical orbit. Where is its speed greatest?

1. At the aphelion, farthest from the Sun
2. At the perihelion, closest to the Sun
3. Halfway between, at the ends of the minor axis
4. The speed is the same everywhere

Law of periods

The Earth takes 1 year to go round the Sun. A planet has an orbit with a semi-major axis 4 times as large as the Earth’s. How long is its year?

1. 4 years
2. 8 years
3. 16 years
4. 64 years

Summary

• Law of areas (from conservation of L): ΔA/Δt = L / 2m = constant
• Law of periods: T² ∝ a³
• Two planets: T₁² / T₂² = a₁³ / a₂³

• Law of orbits: every planet moves in an ellipse with the Sun at one focus.
• Law of areas: the line from the Sun to a planet sweeps equal areas in equal times, so a planet is fastest at the perihelion.
• The law of areas holds because gravity points towards the Sun, so it exerts no torque and angular momentum is conserved.
• Law of periods: T² ∝ a³, where a is the semi-major axis. Bigger orbits take much longer.