The Moon from two sides of Earth
- Round
- Theory
- Points
- 20
- Level
- Bronze
- Taught in
- Section 1.7
Adapted from IOAA 2008 (Bandung), theory problem 1. We have restated it in our own words and kept the original data, adding values for Earth's radius and the Moon's distance. The solution was last checked on 15 October 2026.
Two observers on Earth's equator, nearly 180° apart in longitude, observe the position of the Moon against the background stars at the same moment. The Moon's declination is zero. Sketch the situation, and find the difference between the right ascensions of the Moon that the two observers measure. Take Earth's radius to be 6371 km and the Moon's distance to be 384 400 km.
- The Moon is in the plane of the equator. Both observers can see it at once only if it is on both horizons, rising for one observer and setting for the other. Each line of sight then just grazes Earth, so it is at right angles to the radius through the observer, and this is why the observers are nearly, but not exactly, 180° apart in longitude.
- In the right-angled triangle formed by Earth's centre, one observer and the Moon, the side opposite the angle at the Moon is Earth's radius, 6371 km, and the hypotenuse is the Moon's distance, 384 400 km. Each observer therefore sees the Moon displaced against the stars by the angle
- The two directions differ by twice this angle, 1.899°, or 113.96′. Along the celestial equator 15′ of arc is 1 minute of right ascension, so the difference is
AnswerThe two right ascensions differ by about 7.6 minutes of time, which is 1.90° on the sky.
This problem also appears among the olympiad problems of Section 1.7, Problems for Chapter 1, which teaches the ideas it uses. Equation and figure numbers refer to the course.