Sunrise on the shortest day
- Round
- Theory
- Points
- 2
- Level
- Bronze
- Taught in
- Section 2.3
Adapted from IOAA 2007 (Chiang Mai), theory problem 1.1. We have restated it in our own words and kept the original data. The solution was last checked on 17 October 2026.
An observer is at latitude 42.5° N and longitude 71° W, where the clocks show UT−5. Estimate the time of sunrise by the clock on 21 December. Ignore refraction and the size of the Sun's disc.
- On 21 December the Sun's declination is −23.44°. By (1.15),
- On 21 December the equation of time is +1.9 min, so by (2.3) the local mean time is
- The clocks keep the mean time of the meridian at 75° W. The observer is 4° east of it, so the clocks are
AnswerThe Sun rises at about 7:16 by the clock, or 7:18 if the equation of time is ignored. The first official solution treats the small triangle near the horizon as if it were flat and gets 7:10, which is 8 minutes out at this latitude. The alternative solution printed with it uses the exact formula and gets 7:18.
This problem also appears among the olympiad problems of Section 2.3, Longitude, time zones and universal time, which teaches the ideas it uses. Equation and figure numbers refer to the course.