Photons from the faintest stars

Adapted from IOAA 2009 (Tehran), theory problem 2. We have restated it in our own words and kept the original data, adding the Sun's apparent magnitude and the solar constant from the IOAA 2025 data sheet. The solution was last checked on 16 October 2026.

A main-sequence star like the Sun has an apparent magnitude of 6, about the faintest that the naked eye can see. Estimate how many photons from it enter the eye each second. Assume that the pupil of the eye is 6 mm across and that all the star's light has a wavelength of 550 nm. The Sun has an apparent magnitude of −26.74, and its light brings 1361 W m⁻² to Earth. A larger magnitude means a fainter star, and 5 magnitudes are a factor of 100 in brightness.

AnswerAbout 8500 photons a second, or roughly ${10}^4$, enter the eye from a star at the limit of naked-eye visibility.