# GEK1506 Heavenly Mathematics & Cultural Astronomy, Tutorial 5, Solutions

1. The vernal equinox is the intersection point of the ecliptic and the celestial equator where the declination is increasing. The vernal equinox is the moment in time when the Sun is at the vernal equinox.
• Heliocentric orbit: axis points along the rays or perpendicular to the rays.
• Points along the ecliptic on the celestial sphere.
• Length of the day.
• Rising position of the Sun.
• Length of the noon shadow.
2. The vernal equinox used to be the first point of Aries but is now in Pisces and will gradually move to Aquarius because of the precession of the equinox.
3. The Sun is now in Gemini at the time of the summer solstice and in Sagittarius at the time of the winter solstice.
4. The maximal altitude of the Sun on a given day depends on the latitude of the observer and the time of the year. The celestial equator intersects the meridian at an altitude equal to the colatitude. This is equal to the maximal altitude of the Sun at the equinoxes. To get the the maximal altitude of the Sun we must add the declination if we are in the northern hemisphere or subtract the declination if we are in the southern hemisphere. But if the answer comes out bigger than 90, we must instead subtract the answer from 180.

In order to do this in a uniform way, we can instead use the following convention. We will measure altitude along the meridian from the south point on the horizon. So an altitude of 80 would mean 80 degrees in the south, while an altitude of 100 would mean 80 degrees in the north. In that case, the formula becomes 90 - latitude + declination.

5. 4min = 24h/365.
6. 48min=24h/30.

Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com