Kepler's laws: how the planets move

Rules hidden in the sky

For many years, the astronomer Tycho Brahe measured where the planets were, night after night. Johannes Kepler studied his numbers and found three rules. Later, Newton showed that his law of gravitation explains all three.

The first law: ovals, not circles

Each planet goes round the Sun on an oval path, called an ellipse. The Sun is not at the centre of the oval. It sits to one side, at a point called a focus. Most planets have paths that are very nearly circles. A comet's path is long and thin.

The second law: equal areas in equal times

Join the Sun to the planet with a line. As the planet moves, the line sweeps out an area. In equal times it sweeps out equal areas.

Near the Sun the line is short. So the planet must move fast to sweep the same area. Far from the Sun the line is long, and the planet moves slowly.

The third law: a longer year farther out

The time a planet takes to go once round the Sun is its year, $T$. Let $a$ be its average distance from the Sun. Kepler found:

Measure $a$ in astronomical units (AU). One AU is the Earth's distance from the Sun, about $1.5 \times 10^{8}\ \text{km}$. Measure $T$ in Earth years. Then simply $T^2 = a^3$.

You should find that the planet speeds up each time it passes close to the Sun. A planet twice as far out as the Earth needs almost three years for one trip.

Watch out. $T^2 = a^3$ needs $a$ in AU and $T$ in Earth years. It works for bodies going round the Sun. Moons going round a planet follow the same pattern, but with a different number.

How Newton used the third law

Newton asked what pull keeps a planet on its path. Take a planet moving on a circle of radius $r$, to keep it simple.

1. To move round a circle at speed $v$, a body needs a pull towards the centre. For each kilogram, this pull goes as $\dfrac{v^2}{r}$. (You will meet this again in Class 11.)
2. In one year the planet goes once round, a distance $2\pi r$. So $v = \dfrac{2\pi r}{T}$, and the pull goes as $\dfrac{r}{T^2}$.
3. Kepler's third law says $T^2 \propto r^3$. So the pull goes as $\dfrac{r}{r^3} = \dfrac{1}{r^2}$.

That is the square of the distance in the law of gravitation, from the first lesson of this course. Newton checked it with the Moon. The Moon is about $60$ Earth radii from the Earth's centre, so the Earth's pull on each kilogram there should be $60^2$ times weaker than at the ground. The Moon's motion agreed.

A worked example

Ceres, the largest body in the asteroid belt, is about $2.77\ \text{AU}$ from the Sun. How long is its year? Saturn takes $29.5$ Earth years to go round. How far is it from the Sun?

1. Ceres. $T^2 = a^3 = 2.77^3 = 21.3$, so $T = \sqrt{21.3} = 4.61\ \text{years}$.
2. Saturn. $a^3 = T^2 = 29.5^2 = 870$. The number that, multiplied by itself three times, gives $870$ is $9.55$. So $a = 9.55\ \text{AU}$.
3. So Saturn is about ten times as far out as the Earth, but its year is about thirty times as long.

Practise with new numbers

Try it in Python

The program checks the third law with six planets.

Remember these

• First law: each planet moves on an ellipse, with the Sun at a focus, not at the centre.
• Second law: 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.
• Third law: $T^2 \propto a^3$. With $a$ in AU and $T$ in Earth years, $T^2 = a^3$.
• From the third law, Newton found that the Sun's pull falls as $\dfrac{1}{r^2}$.

Check yourself

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

1. Where in its path does a planet move fastest?

2. One planet is $9$ times as far from the Sun as another. How many times as long is its year?

3. Why do the outer planets take so long to go round the Sun?

4. What did Newton learn from Kepler's third law?