An opposition of Neptune at the equinox
- Round
- Theory
- Points
- 5
- Level
- Bronze
- Taught in
- Section 6.1
Adapted from IOAA 2023 (Katowice), theory problem 1. We have restated it in our own words and kept the original data. The solution was last checked on 23 October 2026.
Neptune was at opposition on 21 September 2024. In which year was Neptune last at opposition near the time of the March equinox? Take the orbits of Earth and Neptune as circles, and the semi-major axis of Neptune's orbit as 30.070 au.
- By Kepler's third law Neptune's sidereal period is
- On 21 September the Sun is close to the September equinox, which under the problem's assumptions puts it at ecliptic longitude 180°, so at the opposition of 2024 Neptune was at longitude about 0°.
- At an opposition in March the Sun is at longitude 0°, so Neptune must be at 180°. Neptune was last there half a turn earlier,
- As a check, by (6.2)
AnswerThe opposition came about 82.4 years before late September 2024, near year 1942.3, so it was the opposition in the spring of 1942. The official solution accepts 1941 or 1942 under these assumptions. With the real, slightly eccentric orbits the closest opposition to the equinox was in 1943.
This problem also appears among the olympiad problems of Section 6.1, Configurations and synodic periods, which teaches the ideas it uses. Equation and figure numbers refer to the course.