Expert: Courtney Seligman - 11/26/2008

Question
Thanks for taking my question professor Seligman.

I found this site and your name while searching for information on my question. You answered a question regarding azimuth of sunrise and sunset - your answer, although not exactly addressing my question will give me an opportunity to practice spherical trig!

My question was asked by my wife - and of course I have to come up with an answer:
How can one calculate the angle with the horizon that the path of the sun makes when setting, given latitude and day of year?
I may have figured this out, being able to obtain the declination of the sun for any day. But before giving my wife an answer I would like your confirmation!
Is the path exactly 90 degrees on the equinox and at the equator?

Many thanks
Bill

Answer
You are correct in saying that at the Equator, the Sun would rise vertically (at a 90 degree angle to the horizon). This is true, regardless of the Sun's declination.

At other latitudes, however, the angle that it makes with the horizon as it rises (and the exactly equal angle it makes as it sets) depends upon its declination. If it is on the Celestial Equator, as it is at one of the Equinoxes, it rises at an angle of (90 - your latitude), which is called the complement of the latitude, or the co-latitude. E.g., at 35 degrees latitude, it would rise at 90 - 35 = 55 degrees, relative to the horizon.

As the Sun moves north or south relative to the Celestial Equator, you have to use, as you presumed, spherical trigonometry to calculate the angle. I can't draw a diagram here, but you need to solve for one of the angles in the spherical triangle made by the Celestial Pole, the Zenith, and the rising Sun. This PZS triangle has as its sides:

(1) the arc from the Zenith to the Celestial Pole, which is equal to 90 - the latitude

(2) the arc from the Celestial Pole to the Sun, which is equal to 90 - the Sun's declination

(3) the arc from the Zenith to the rising Sun, which is 90 degrees.

You need to solve for the angle made at the Sun, between the vertical circle from the Sun to the zenith, and the hour circle from the Sun to the Celestial Pole. If we call that angle S, then the angle the Sun rises at, relative to the horizon, is 90 - S.

It's not clear to me whether you wanted to go over the equations involved, or just to verify the concepts; so if you would like to see the actual equations, just let me know.

Courtney Seligman
Professor of Astronomy
Long Beach City College