Expert: Philip Stahl - 12/12/2006

Question
at the equator does the sunrise directly east  

Answer
Hello.

First, let's set the reference bearing on the horizon. Imagine a circle of 360 degrees, and mark off the four cardinal directions of the compass at 90 degree intervals.




W x-------------------------------x (E)

If due north is set at 0 degrees, then due east will be at 90 degrees, due south at 180 deg, and due west at 270 degrees. These are azimuth points on the horizon.

The above diameter represents the ideal sunrise-sunset line which is perpendicular to the N-S line.

Now, the equator is at latitude zero degrees.

The equation to find the Sun's azimuth at sunrise is given by:

cos (A_sr)  = sin (d)/  cos (L)

where 'd' denotes the Sun's declination, and L the latitude of the place.

Since the equator has L = 0 and cos (L) = cos (0) = 1 we may write:

cos (A_sr) =  sin (d)

(Bearing in mind this applies ONLY at th equator!)

As it turns out, sin (d) = sin (0) = 0 ONLY on the date of the equinoxes (Mar. 21 and Sept. 23).

If you work out the Sun's azimuth for either or both these dates, you get:

A _sr =  cos^-1 (0) =  90 degrees.

which is due east or directly east. As per our reference frame for the horizon.

HOWEVER, this does NOT hold at any other time! Why? Because the value of the Sun's declination is not constant, but changes from (-23.5) degrees (at the winter solstice) to +23.5 degrees at the summer solstice.

To illustrate, let's take the date of the winter solstice or Dec. 22, for which d = -23.5 degrees.

then

sin (d) = sin (-23.5) =  -0.3987

Then the Sun's azimuth on the date is:

A_sr =  cos^-1 (-0.3987)  =  113.5 degrees

Going by our reference circle, one can see that this would be at an angle of (113.5 - 90) degrees SOUTH of the E-W line. In other words, it would be 23.5 degrees south of due east.

If you go on the web you may be able to get a free (or share ware) program like 'Astronomy Lab' that does all these computations for you.

From the input values, such as I illustrated here, you can actually see (in an output graphic) how the Sun's azimuth changes through the year at the equator, and that it is actually 'due east' on only two dates - the equinoxes.

Hope this info helps!