Will we see the pulsar?
- Round
- Theory
- Points
- 10
- Level
- Silver
- Taught in
- Section 10.3
Adapted from IOAA 2012 (Rio de Janeiro), theory problem S8. We have restated it in our own words and kept the original data. The official solution reaches −3.24, which corresponds to taking the Sun's absolute bolometric magnitude as 4.80, and with the course's 4.74 it is −3.30. The solution was last checked on 23 October 2026.
A pulsar 1000 pc from Earth has a luminosity of 10 000 times the Sun's, all of it sent out in two opposite beams. Each beam is a uniform cone with a full opening angle of 4°. The beams are tilted by 30° from the spin axis, and the spin axis points in a random direction. What is the probability that we see its pulses? If we see it, what is its apparent bolometric magnitude?
- Each beam has a half-width of 2°, so by (10.7) the probability is
- Each cone covers a solid angle of
- The Sun at 10 pc gives the flux
AnswerThe probability is 3.5%, and the pulsar would have an apparent bolometric magnitude of −3.30, brighter than any star in the sky.
This problem also appears among the olympiad problems of Section 10.3, Pulsars, which teaches the ideas it uses. Equation and figure numbers refer to the course.