Kepler's laws of planetary motion

Three rules from the sky

For years, Tycho Brahe measured where the planets were, night after night. His assistant, Johannes Kepler, found three rules hidden in the numbers.

The first law: ellipses

Each planet moves in an ellipse, with the Sun at one focus (not the centre). Most planets' ellipses are nearly circles; comets' are long and thin.

The second law: equal areas

The line from the Sun to a planet sweeps out equal areas in equal times. So a planet moves fastest when it is nearest the Sun, and slowest when farthest. At the two ends, $v_{\text{near}}r_{\text{near}} = v_{\text{far}}r_{\text{far}}$.

This is the conservation of angular momentum: the Sun's pull points straight at the Sun and has no torque about it.

The third law: periods

The square of a planet's year is proportional to the cube of its mean distance:

With $a$ in astronomical units (the Earth's distance) and $T$ in years, simply $T^2 = a^3$. Try it.

You should find all six planets on one curve, and that a planet nine times as far out as the Earth has a year $27$ times as long.

Watch out. The third law compares bodies orbiting the same central body. Moons of Jupiter follow it among themselves, with a different constant from the planets.

A worked example

Io circles Jupiter at $4.22 \times 10^{5}\ \text{km}$ every $1.77$ days. Europa orbits at $6.71 \times 10^{5}\ \text{km}$. Find Europa's period.

1. Ratio of distances. $\dfrac{6.71}{4.22} = 1.59$.
2. Third law. $T = 1.77\ \text{days} \times 1.59^{3/2} = 1.77\ \text{days} \times 2.005 = 3.55\ \text{days}$.
3. Check. The measured value is $3.55$ days.

Think first. Use ratios: the constant in $T^2 \propto a^3$ cancels when you compare two bodies around the same centre.

Practise with new numbers

Try it in Python

The program checks the third law with six real planets.

Remember these

• First law: ellipses, with the Sun at a focus.
• Second law: equal areas in equal times; $v_{\text{near}}r_{\text{near}} = v_{\text{far}}r_{\text{far}}$.
• Third law: $T^2 \propto a^3$ for bodies around the same centre.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A planet's year is $8$ Earth years. How far is it from its Sun-like star?

2. In which season of its orbit does the Earth move fastest?

3. Why is the Sun at a focus of each orbit, and not at the centre?

4. A comet's farthest distance is $35$ times its nearest. How many times faster does it move when nearest?