Mars's distance in the manner of Copernicus
- Round
- Theory
- Points
- 10
- Level
- Bronze
- Taught in
- Section 6.2
Adapted from IOAA 2009 (Tehran), theory problem 13. We have restated it in our own words and kept the original data. The worked example of 6.1 runs the same geometry the other way, from the radius to the time. The solution was last checked on 23 October 2026.
Imagine living in the time of Copernicus, before Kepler's laws were known, but accepting that the planets go round the Sun. Mars's orbital period is 687 days, and Mars is seen at quadrature 106 days after an opposition. Find the distance of Mars from the Sun in astronomical units, taking both orbits as circles.
- At opposition the Sun, Earth and Mars lie on one line. In 106 days Earth moves round the Sun by
- The angle at the Sun between Earth and Mars is therefore
- At quadrature the triangle formed by the Sun, Earth and Mars has a right angle at Earth, so
- The radius of Mars's orbit is
AnswerMars is 1.52 au from the Sun, close to its real mean distance of 1.524 au.
This problem also appears among the olympiad problems of Section 6.2, Elongation, retrograde motion and visibility, which teaches the ideas it uses. Equation and figure numbers refer to the course.