Skip to content

Seeing Mars from Earth

Waves, sound and light · level 7, lesson 5 of 6

A question to start

Mars is half as wide as Earth, but even at its closest it looks like a bright dot in the sky. How big does it look through a telescope, and when is the best time to look?

The idea

How big something looks depends on its angular size: its width divided by its distance, in radians:

\[\theta = \frac{D}{d}.\]

Astronomers often use seconds of arc: \(1 \text{ rad} = 206{,}265''\).

How Mars and the Moon look from Earth Mars the Moon
Width 6,779 km 3,474 km
Distance 54.6 to 401.4 million km about 384,000 km
Angular size 3.5'' to 25.6'' about 1,860'' (31')

Mars looks biggest at opposition, when Earth passes between Mars and the Sun. That happens about every 780 days, a little over two years. Even then, Mars is about 70 times smaller in angle than the Moon.

Moon, eyeMars, 100×Mars, eye

Worked examples

1. Mars at its closest. \(\theta = 6779/54.6\times10^6 = 1.24\times10^{-4}\) rad \(= 1.24\times10^{-4} \times 206{,}265 = 25.6''\).

2. Through a telescope. At \(M = 100\), Mars looks \(25.6 \times 100 = 2{,}560''\), about 43', bigger than the Moon looks to the eye.

3. At its farthest, \(6779/401.4\times10^6 \times 206{,}265 = 3.5''\): seven times smaller than at its closest.

Try it

  1. What is the angular size of an object?
  2. Find the Moon's angular size in seconds of arc.
  3. What magnification makes Mars at 25.6'' look as big as the Moon to the eye?
  4. Why is opposition the best time to look at Mars?
Answers
  1. Its width divided by its distance, the angle it covers.
  2. \(3474/384{,}400 \times 206{,}265 \approx 1{,}860''\).
  3. \(1860/25.6 \approx 73\).
  4. Mars is then at its closest to Earth, so it looks biggest and brightest.

Common mistake

Forgetting to match units. Width and distance must be in the same unit, both in km here, before dividing.

Where it leads