The Moon at full and at first quarter

Adapted from IOAA 2014 (Suceava), theory problem P14. We have restated it in our own words and kept the original data. The original does not state the law of brightness with phase, and the official solution uses the matte-sphere law, normalised to 2/3 at full phase, which we state. The official solution rounds the factor 2/(3π) to 0.2 and then slips in the arithmetic, getting −10.5 at first quarter instead of −11.0. The official answer at full Moon is −12.25, from a distance of 0.00256 au. With $H$ defined at full phase, as in this section, the full Moon comes out at −12.70, close to the real −12.74. The real Moon at first quarter is near −10, fainter than a matte sphere, because of the shadows of its rough surface. The book gives no marks for this problem. The solution was last checked on 23 October 2026.

The absolute magnitude of the Moon is $H=0.25$. Find the Moon's apparent magnitude at full Moon and at first quarter. Take the distance from the Moon to Earth as 385 000 km and the distances from the Sun to Earth and to the Moon as 1 au. Assume the Moon scatters light as a matte sphere, so that the light it sends towards an observer at phase angle $i$ is proportional to $\frac{2}{3}[(1-i/\pi )\cos i+(\sin i)/\pi ]$, with $i$ in radians, Take $H$ as the magnitude the Moon would have at 1 au from both the Sun and Earth if this expression were equal to 1.

AnswerThe full Moon has magnitude −12.26 and the first-quarter Moon −11.0 in this model.