The equation of time resulting from the tilt of the axis is slightly more difficult to comprehend than is that resulting from the elliptical orbit. The diagram to the right shows both a top view and a side view of the earth and the "orbit" of the sun. Again, since the sun obviously does not orbit the earth, this concept is used to simplify, and perhaps it would be more accurate to say the "path that the sun takes as we view it from the earth".
As earlier we found the difference between a, the angle we considered for a circular orbit, and v, the angled we considered for an elliptical orbit, something similar is done here. The image in the top view shows two angles of the sun as viewed from earth; e, the angle of the sun in a horizontal orbit, and b, the angle of the sun in a tilted orbit. The horizontal orbit here equates to a circular orbit of the earth around the sun, and the tilted orbit takes into account the tilt of the axis of the earth. We are therefore trying to find (ε - β)
Using the diagram as a reference, most of this can be done with trigonometric identities.
α = 23.5 degrees // tilt of axis
d = cos α
b = sin ε
c = cos ε
When we were dealing with the ellipse, we labeled N as the number of days since perihelion in the earth's orbit. Here, we will label N as the number of days since vernal equinox, occurring on March 21, since this is a marking point in the orbit that is a result of the tilted axis and not of the elliptical motion. We can therefore define ε:
ε = 2π/365.24 = .017203 N // angle of mean sun after N days since vernal equinox
Note that we are dealing with radians here, so the proper conversions must be used. Using one more simple trigonometric identity, we can easily find β:
tan β = d/c
β = arctan (d/c)
The arctan function returns a value on the interval (-π/2,π/2), and we want an angle between 0 and 2π to equate to the 360 degree orbit of the earth. We can use the value of ε to convert β to the proper value according to the following rules:
; add π to β
; add 2π to β
After finding β and ε, we can use the same method to calculate the equation of time resulting from the tilt of the axis, keeping in mind that the angles are in radians and need to be converted:
EOT(tilt) = (ε - β) * 180/π * 3.989
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