# Question regarding the path of sun crossing zenith before the summer solstice

I stay in Bangalore, India (12.9716° N, 77.5946° E). For my latitude, the Sun crosses zenith (the highest point in the sky) on 24th of April every Summer, which happens to be about 2 months prior to the June solstice (summer solstice to us here in the Northern Hemisphere).

Shouldn't the Sun be reaching its highest point on the longest day, i.e. on June 21st, instead of April 24th?
Doesn't the Sun take the longest arc across the sky on the day it passes through zenith?

For reference, the angle at which the Sun rises/sets in my location is about 13 degrees, which happens to be almost vertical to the horizon.

• The sun is overhead when it's declination matches your latitude. The sun actually passes its longest arc on the equinoxes, since the equator is the longest line of declination. The two are unrelated. Between the Tropics of Capricorn and Cancer (where you are), the sun will be overhead twice a year (for you, the other one will be about 2 months after the June solstice). Outside this area, the sun is never overhead, but is closest to overhead on the solstices.
– user21
Nov 30, 2017 at 15:13
• Interesting question. Although I agree with @barrycarter that longest line of declination is at the equator, I am not sure that the longest line of declination visible above the horizon is the equator. If I did my math correctly, a star that is just circumpolar for me (latitude 40 N) has a longer arc above the horizon than the equator does, even though the "radius" is smaller. The question is when is H*cos(dec) the maximum, where H is the hour angle between the horizon and meridian for an object at declination of "dec" (assuming my formula is correct). That answer will have to wait. Nov 30, 2017 at 18:53

Question 2: Doesn't the Sun take the longest arc across the sky on the day it passes through zenith?

Answer: There are two effects. (In the remaining description, I will be referring to an observer in the northern latitude.) As the Sun moves northward from 0 degrees declination, the radius of the circle that it travels is smaller. In other words, the total length of the 20 degree declination circle is smaller than the 10 degree declination circle which is smaller than the 0 degree declination circle (the celestial equator). But at the same time, the fraction of the declination circle that is visible above the horizon is greater. At some combination of declination, these two effects result in the maximum length of the declination circle above the horizon. If the declination is north or south of that optimized solution, the length of the arc is smaller.

Skipping the calculations for now and going right to the solution, here is a graph showing the length of the arc for various latitudes and declinations. This graph plots the results relative to the length of the arc along the celestial equator which is 180 degrees regardless of your latitude (ignoring refraction and the Sun's diameter).

If your latitude is between 0 and 23.4 degrees, the longest arc is before the Sun reaches the zenith. For example, an observer at 20 degrees latitude witnesses the longest arc when the Sun is at about 13 degrees declination which is 7 degrees before the Sun reaches the zenith. If you live north of 23.4 degrees, the Sun is never overhead. North of 30 degrees, the longest arc is when the Sun is highest in the sky.

I can go into more details and the math, but I do not know if I should do it here or maybe in a new question.

Question 1: Shouldn't the Sun be reaching its highest point on the longest day, i.e. on June 21st, instead of April 24th?

Answer: No. The Sun (or any celestial object) is highest when its declination equals your latitude. That is when the object is at the zenith. If your latitude is between 23.4 S and 23.4 N, the Sun's declination equals your latitude on some day other than the longest day. (The Sun's declination varies between -23.4 and +23.4, approximately.)

If you live outside of that range of those latitudes, then the Sun is closest to the zenith on the longest day of the year, and therefore is highest in the sky on the longest day of the year.