# Declination of the Sun

## Words to Begin With

• Horizon is the horizontal plane that extends from the point where the observer is standing, to infinity, straight through space. Since we're only working with relatively short distances (compared to the Universe), a line extending N-S will be quite sufficient.
• Altitude (A) is the angle of the sun over the horizon. In this problem, we will be working with the sun at noon, so it will either be over the N or S horizon.
• Zenith (Z) is the angle that the sun is from directly overhead, and it is equal to 90-A. It, too, can be over the S or N horizon, but there is little need to state it.
• Declination (D) is the latitude at which the sun is directly overhead. It is always between 23.5 N and 23.5 S latitude, those occurring on the Solstices.
• Latitude (L) is the location N or S of the equator at which the observer is located. (It is determined by radii from the center of Earth at different angles to the equator. If such an angle is swept along the surface of the planet, it draws a circle.)

## A Quick Lesson in Basic Geometry ### Diagram A

In Diagram A above, CD and AB are parallel, intersecting EF at G and H, respectively. Congruent (same) sets of angles are: 1, 3, 5, and 7; and 2, 4, 6, and 8. This is because opposite angles (ie: 1 and 3) are congruent (which is true for any 2 intersecting lines), and because same-side angles of parallel lines (ie: 3 and 7) are congruent.

I will refer to this as the SS (Same-Side) theorem.

## And the Rest is Simple Mathematics...

Honest, I'm not kidding about this! I couldn't do these for the longest time, but I finally figured out how and where I went wrong: I was trying to make these problems a lot harder than they were! Just relax; they aren't that hard. Take it from someone who knows.

### Three Possible Cases

Important Note: If you are given information about D, and you know either L or A, you can get whichever of them you do not have. #### Case 1.

Case 1.
• Z as the sum of D and L, from the SS Theorem.
• Z will therefore be larger than both D and L.
• D and the observer are in different hemispheres. #### Case 2.

Case 2.
• L is the sum of D and Z, from the SS theorem.
• L is greater than each D and Z.
• D and the observer are in the same hemisphere.
• The observer is farther from the equator than D is. #### Case 3.

Case 3.
• D is the sum of L and Z, from the SS theorem.
• D is greater than each L and Z.
• D and the observer are in the same hemisphere.
• The observer is closer to the equator than D is.

## How 'Bout an Example?

Suppose it is the first day of summer, and the sun is over 23.5 N, and you've just observed the sun at an altitude of 50 degrees over the S horizon. What's your latitude?

First, let's examine what we have as information.

• Declination = 23.5 N.
• Altitude = 50 over S horizon.

Next, what can we infer?

• Zenith = 40.
• Since we have to look S to see the sun with a declination over a line of N latitude, we must be further to the N than the declination, and we are in the same hemisphere as the declination.
• This is, therefore, a Case 2 situation.

Now, we solve for our latitude.

• D + Z = L; 23.5 + 40 = 63.5 N (remember we said it is in the same hemisphere as the declination.)