Better mirror coatings

Adapted from IOAA 2019 (Keszthely), theory problem 4. We have restated parts (a) and (b) in our own words and kept the original data. The solution was last checked on 20 October 2026.

A student has a Cassegrain telescope whose two mirrors have aluminium coatings that reflect ${\epsilon }_1=91\mathrm{\%}$ of the light.

(a) By how much does the limiting magnitude change if both coatings are replaced by premium ones that reflect ${\epsilon }_2=98\mathrm{\%}$? (5 points)

(b) The student also uses a star diagonal, a flat mirror at 45° that turns the light towards the eye, with the same 91% coating. By how much does the limiting magnitude change if, together with the premium coatings on the two main mirrors, the diagonal is replaced by one that reflects ${\epsilon }_3=99\mathrm{\%}$? (3 points)

Ignore any dependence on wavelength and any other losses.

AnswerThe premium mirror coatings raise the limiting magnitude by 0.16, and with the better diagonal as well by 0.25. The original problem also asked whether the change is obvious to the eye, and the official answer says that it is. We have left that part out, since it is a matter of the observer's experience more than of calculation.