Since the Earth orbits the Sun in an ellipse, there is a point where the Earth is closest to the Sun (the Perihelion) and a point where the Earth is farthest from the Sun (the Aphelion). These events occur on January 3rd and July 4th, respectively. Another important term to define before deriving the Equation of Time due to the Earth’s Elliptical Orbit is eccentricity. Eccentricity is the measure of the degree to which the shape of the orbit is elliptical. The eccentricity can range anywhere from zero to one, in which zero is a perfect circle. We will need to compute the eccentricity of Earth’s orbit to complete this equation.
Let a = the angle of the mean Sun measured from its perihelion date
Let v = the angle of the true Sun measured from its perihelion date
We want to find the difference between the two angles (a – v).
Let N = the number of days since the perihelion (Jan 3rd)
a = (360º / 365.24 days) * N = 0.986N
Now we need to find the eccentricity (e) letting a = the semi-major axis of the ellipse and b = the semi-minor axis.
e = [(a2 + b2)1/2] / a
The eccentricity of the Earth’s elliptical orbit is calculated to be 0.016713.
v = a + (360 / π) * esin(a)
= 0.986N + (360 / π) * 0.016713*sin(0.986N)
We also need to find the number of minutes it takes the Earth to rotate one degree, which we already calculated for the effect of the tilt of the Earth’s axis.
= 1440 (minutes in one day) / 361º (rotated by Earth in one day) ≈ 3.989 min/degree
The Equation of Time due to the Earth’s elliptical orbit is:
(a – v) * 3.989 min/degree
= [(0.986N) – [(0.986N + (360 / π) * (0.016713)*sin(0.986N)]] * 3.989
= [(-360 / π) * (0.016713sin(0.986N))] * 3.989
To show an example using this equation we can return to the previous example and ask the question: What is the equation of time (due to the Earth’s elliptical orbit) on May 25th? This is 143 days after the perihelion.
= [(-360 / π) * (0.016713sin(0.986*143))] * 3.989 = -4.8007 = -4 minutes and 48 minutes.