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Distance from orbits

Space and time · level 6, lesson 1 of 6

A question to start

Earth is 1 au from the Sun and Mars is about 1.52 au. If, seen from the Sun, the two planets are 90° apart, how far is it from Earth to Mars?

The idea

Treat the two orbits as circles round the Sun. The Sun, Earth and Mars make a triangle. If the angle at the Sun is \(\theta\), the cosine rule gives the third side:

\[d^2 = r_E^2 + r_M^2 - 2\,r_E\,r_M\cos\theta,\]

where \(r_E = 1\) au and \(r_M = 1.52\) au.

  • \(\theta = 0°\): Mars is straight out from Earth (opposition). \(d = 1.52 - 1 = 0.52\) au.
  • \(\theta = 180°\): Mars is behind the Sun (conjunction). \(d = 1.52 + 1 = 2.52\) au.

The real orbits are slightly oval, so the true range is a bit wider: from about 54.6 to about 401 million km.

Distance from the Sun Mars Earth
Closest 206.7 million km 147.1 million km
Average 228.0 million km 149.6 million km
Farthest 249.3 million km 152.1 million km

The very closest approach happens when Mars is near its closest to the Sun and Earth near its farthest: \(206.7 - 152.1 = 54.6\) million km.

θdMars: 1.52 auEarth: 1 auθ = 90° here

Worked examples

1. At 90°. \(\cos 90° = 0\), so \(d^2 = 1 + 1.52^2 = 3.31\). \(d \approx 1.82\) au, about 272 million km.

2. At 60°. \(\cos 60° = 0.5\). \(d^2 = 1 + 2.31 - 2 \times 1.52 \times 0.5 = 1.79\). \(d \approx 1.34\) au, about 200 million km.

3. Into light time. 1.82 au is about \(1.82 \times 8.3 \approx 15\) light-minutes.

Try it

  1. What shape do the Sun, Earth and Mars make?
  2. What is the distance at opposition, using circular orbits?
  3. Find the distance when \(\theta = 120°\). (\(\cos 120° = -0.5\).)
  4. Why is the real closest distance smaller than 0.52 au?
Answers
  1. A triangle, with the Sun at one corner.
  2. \(1.52 - 1 = 0.52\) au, about 78 million km.
  3. \(d^2 = 1 + 2.31 + 1.52 = 4.83\), so \(d \approx 2.2\) au.
  4. The orbits are slightly oval. When Mars is near its closest to the Sun and Earth near its farthest, the gap is only 54.6 million km.

Common mistake

Taking \(\theta\) as the angle seen from Earth. The cosine rule here needs the angle at the Sun, between the lines to each planet.

Where it leads