Expert: Courtney Seligman - 10/14/2008

Question
What is the angle created by the sun at sun rise and sun set on June 21 and what is it on December 21 at 43 degrees north latitude?  How is that measure obtained?

Answer
Your question was apparently rejected and placed in the general question pool, which is where I just found it. Not knowing why it was placed there (the person you asked may have thought it sounded like a homework question, which we are not supposed to answer), I can't give you an exact numerical answer, but I can explain the concepts involved.

Given the dates, I believe you are supposed to calculate the change in the Sun's azimuth when it rises and sets on the first day of summer, when it is as far north of the Celestial Equator as possible, and on the first day of winter, when it is as far south of the Celestial Equator as possible. (The actual date of the first day of winter is a day or two later than in your question, but I think that's what you meant).

Azimuth is the angle that an object has along the horizon, measured from the North point on the horizon. On the first day of Spring and Fall, when the Sun is on the Celestial Equator, it rises due East and sets due West, with an azimuth of 90 degrees East when it rises, and an azimuth of 90 degrees West when it sets. At the Summer Solstice, when it is as far north as possible (23 1/2 degrees north of the Celestial Equator), it would rise and set at least 23 1/2 degrees north of East and West, and at the Winter Solstice, when it is as far south as possible (23 1/2 degrees south of the Celestial Equator), it would rise and set at least 23 1/2 degrees south of East and West.

Exactly how far north or south it would rise and set depends upon your latitude. At the Equator, the angle is exactly the same as the 23 1/2 degree distance between the Celestial Equator and the Sun; but as you move toward the Pole, the sky seems to tilt the other way, causing the angle between the East/West points and the Sun's rising/setting azimuth to increase, slowly at low latitudes, then faster, at mid to high latitudes.

To calculate the angle, you have to solve a spherical triangle, called the PZS or Astronomical Triangle, in which one side is the distance between the Pole and the Zenith (90 degrees minus your latitude), one side is the distance between the Pole and the Sun (90 minus the Sun's angle north of the Celestial Equator), and the other side is the distance between the Zenith and the horizon (90 degrees, exactly, for sunrise and sunset). The angle at the Zenith, between the arc from the Zenith to the Pole, and the arc from the Zenith to the Sun, is the azimuth of the Sun. Finding that angle requires spherical trigonometry, but if you are familiar with that, it is fairly straightforward. If you aren't familiar with that, you need someone else to calculate the angle for you.

The easiest way to do this, if you don't know spherical trigonometry, is to use Internet planetarium software. There are many sites which allow you to view the sky at various times by entering your longitude and latitude and the date and time you want to view. Not all of them show the azimuth of an object, but some do, and if you use one of those, and adjust the time until the Sun is on the horizon, you'll find the answer you need.

Courtney Seligman
Professor of Astronomy
Long Beach City College