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The astronomical unit

Space and time · level 5, lesson 4 of 6

A question to start

Long before anyone knew how far the Sun was, astronomers knew Mars was about one and a half times farther from it than Earth. How?

The idea

Johannes Kepler found a rule linking a planet's year to its distance from the Sun. If you measure the year in Earth years and the distance in units of Earth's distance from the Sun, then

\[\text{year}^2 = \text{distance}^3.\]

That unit, Earth's average distance from the Sun, is the astronomical unit (au). Timing a planet's year is easy, so Kepler's rule gave every planet's distance in au. Measuring just one distance in kilometres, as Cassini did with Mars, then turned the whole map into kilometres.

Today the au is fixed by agreement. Since 2012 it is exactly 149,597,870,700 m, about 149.6 million km.

Mars Earth
Year, in Earth years about 1.88 1
Distance from the Sun, in au about 1.52 1
Distance from the Sun, in km about 228 million about 149.6 million
Earth1 auMars1.52Jupiter5.2 au

Worked examples

1. Mars from its year. Year \(= 1.88\). \(1.88^2 \approx 3.53\). The distance is the number whose cube is 3.53: about 1.52 au.

2. Into kilometres. \(1.52 \times 149.6 \approx 227\) million km.

3. Jupiter. Its year is about 11.86 Earth years. \(11.86^2 \approx 141\), and \(5.2^3 \approx 141\). So Jupiter is about 5.2 au from the Sun.

Try it

  1. What is an astronomical unit?
  2. What does Kepler's rule link together?
  3. A planet is 4 au from the Sun. How long is its year?
  4. Why was measuring one distance in kilometres enough for the whole solar system?
Answers
  1. Earth's average distance from the Sun: about 149.6 million km, now fixed exactly.
  2. A planet's year and its distance from the Sun.
  3. \(4^3 = 64\), and \(8^2 = 64\), so 8 Earth years.
  4. Kepler's rule already gave every distance in au; one distance in km fixed the size of the au.

Common mistake

Using the rule with days and kilometres. In this simple form it only works with Earth years and au.

Where it leads