Annular eclipses against total ones
- Round
- Theory
- Points
- 10
- Level
- Silver
- Taught in
- Section 6.4
Adapted from IOAA 2010 (Beijing), theory problem S15. We have restated it in our own words and kept the original data. Like the official solution, we take the Moon's distance to change steadily with time between perigee and apogee. Real counts over five thousand years give a ratio nearer 1.4 when hybrid eclipses are counted as annular. The solution was last checked on 23 October 2026.
The Moon's apparent diameter is on average slightly less than the Sun's, so annular eclipses are a little more common than total ones. The longest total eclipse lasts about 7.5 minutes and the longest annular eclipse about 12.5 minutes, both measured from second to third contact. Estimate the ratio of the number of annular eclipses to the number of total ones over a long time. Take Earth's orbit as a circle and the eccentricity of the Moon's orbit as 0.0549, and count hybrid eclipses as annular.
- Let the Sun's angular radius be
- By Kepler's second law the angular speed goes as
- If the Moon's distance, and so its angular radius, changes steadily with time between perigee and apogee, the share of the time when the Moon looks smaller than the Sun is proportional to
- A central eclipse is equally likely at any point of the orbit, so the ratio of annular to total eclipses is
AnswerThere are about 1.34 annular eclipses for every total one, roughly four to three.
This problem also appears among the olympiad problems of Section 6.4, The Moon's orbit and eclipses, which teaches the ideas it uses. Equation and figure numbers refer to the course.