Latitude from Polaris and Solar Altitude
A student team at an unknown northern-hemisphere site observes Polaris at an altitude of 43°. On June 21 at local solar noon, a 1.00-m vertical stick casts a shadow 0.35 m long, and the Sun's altitude is measured as 70.5°. On December 21 at solar noon, the same stick casts a shadow 2.35 m long, and the Sun's altitude is 23.5°. The noon shadow points north on both dates. The team claims the site is at 43°N and that the Sun will be directly overhead there on the equinoxes. A classmate argues that the June and December data cannot both come from the same latitude.
1. Determine the latitude from the Polaris observation and verify it with the June 21 solar altitude. Show the relationship you used.
Read the worked answer
In the Northern Hemisphere, the altitude of Polaris above the horizon equals the observer's latitude. Polaris is observed at 43°, so the latitude is 43°N. At local solar noon, the Sun's altitude is related to latitude and solar declination by altitude = 90° - latitude + declination. On June 21 the declination is about +23.5°. Substituting gives 90° - 43° + 23.5° = 70.5°, which matches the measured altitude. The December observation also fits: 90° - 43° - 23.5° = 23.5°. The two seasonal measurements are therefore consistent with the same latitude, so the classmate's argument is incorrect.
2. Explain why the noon shadow points north on both dates and why its length changes so much between June and December.
Read the worked answer
At 43°N, the Sun at local solar noon is always south of the zenith, even on June 21 when it is highest. A vertical stick's shadow points opposite the Sun, so the shadow points north on both dates. The shadow length depends on the Sun's altitude: a higher Sun produces a shorter shadow. On June 21 the noon altitude is 70.5°, so the shadow is only 0.35 m for a 1.00-m stick. On December 21 the noon altitude is 23.5°, so the same stick casts a much longer 2.35-m shadow. The difference is caused by the change in solar declination as Earth's axis remains tilted during its orbit.
3. Evaluate the team's claim that the Sun will be directly overhead at their site on the equinoxes.
Read the worked answer
The claim is false. On an equinox the solar declination is 0°, so the noon Sun altitude at 43°N is 90° - 43° + 0° = 47°. That is well short of 90°, which would be directly overhead. The Sun can be directly overhead only at latitudes between the Tropic of Cancer and the Tropic of Capricorn, and on the equinoxes it is overhead at the equator. At 43°N the noon Sun remains south of the zenith on the equinoxes and casts a north-pointing shadow. The team has confused a mid-latitude location with a tropical or equatorial one.