The rising Sun through the air

Adapted from IOAA 2018 (Beijing), theory problem 3. We have restated it in our own words and kept the original data. The original statement of (a) is unclear about where the distances are measured from, and we measure them from the centre of the disc as seen, the reading for which the official answer holds. The solution was last checked on 17 October 2026.

Consider sunrise at Beijing ($\phi =40°$ N) on the day of the March equinox.

(a) Just after sunrise, let $r_{\text{l}}$, $r_{\text{d}}$, $r_{\text{r}}$ and $r_{\text{u}}$ be the distances from the centre of the Sun's disc as seen through the air, where refraction places it, to the edge of the disc towards the left, down, right and up. Put the four in order, using less than, equal to and more than. [4 points]

(b) By how much does the air make the top edge of the Sun appear earlier than it would without an atmosphere? Take the refraction near the horizon as 35′, and consider only the daily motion of the Sun. [6 points]

Answer(a) $r_{\text{d}}<r_{\text{u}}<r_{\text{l}}=r_{\text{r}}$. (b) The top edge appears about 3.05 minutes earlier. The official solution uses the stars' rate of 15.04′ per minute and gets 3.04 minutes, which makes no real difference.