Over a period of a few hundred years the shape of the analemma will slowly change and thus the Equation-of-Time will also change. For now we will assume that perihelion (when the Earth is closest to the sun) will occur around January 2nd. We need to find the angle v the Earth makes in relation with the sun after perihelion and compare it to the angle a the Earth would make with the sun if the orbit were circular.
view comparative diagram
We can easily calculate this angle on an "average" day - that is, if the Earth were in a circular orbit - as follows:

average angle = 360° / 365.24 days in a year
average angle = .985653° per day.

Since the Earth is moving faster on some days and slower on other days, we want to find the difference on those days and compare it to an average day.

The formula for eccentricity of orbit was derived from the book, "Practical Astronomy With Your Calculator", by Peter Duffett Smith, Cambridge University Press, ISBN 0-521-35699-7. This is a wonderful book full of all kinds of astronomy formulas that can be done on a hand-held calculator. The original formulas in this book were simplified somewhat for this demonstration.

This is the method for finding the angle of the Earth in relation with the sun with the Earth in an elliptical orbit:

Let the average angle = .985653° per day and N = day number starting January 1 (i.e. Feb. 2 = 33), then
where
e = .016713 (a measure of the shape of the ellipse):

This formula is only an approximation. There is a formula described in the Smith book that will give a more accurate angle, but this formula is close enough for this graphic demonstration.

To convert the angle to time is easily done. The Earth rotates approximately 361° in 24 hours, and there are 1,440 minutes in 24 hours, so:

1,440/361 = 3.98892 minutes per degree of Earth's rotation.

Example: How much does the sun's position differ from what our watch reads on January 10th?

While this doesn’t seem like much, don't forget that the time difference is accumulative. As the following graph shows, over the course of 3 months the error adds up to almost 8 minutes. The following graph was created with the above formulas and a spreadsheet application.
The following table is for the first 10 days after perihelion. The "Day" column is the number of days after January 2nd. (please note the difference in the day number). The "Angle" column is the angle that the Earth makes in relation to the sun for an elliptical orbit. The "Avg Angle per Day" column is what the average angle would be if the orbit were circular. You can see from the chart that after 10 days, the sun has drifted to the east and would take over 1 minute to reach its highest point in the sky even though according to your watch, it should be straight overhead.

An ellipse is a unique figure in astronomy as it is the path of any orbiting body around another. An ellipse is a flattened circle. Two fixed points inside the ellipse, F1 and F2 are called the foci. For the Earth–sun system, F1 is the position of the sun, F2 is an imaginary point in space, while the Earth follows the path of the ellipse.

An ellipse has a special property. The sum of the distances from P1 to the foci is the same as the sum of the distances from P2 to the foci. This is true for any point P on the ellipse. The distance a is the semimajor axis, while the distance b is the semiminor axis.

The eccentricity e of an ellipse is a measure of the asymmetry of the ellipse. It is the ratio of:

distance from center to a focus : semimajor axis

The eccentricity e can be calculated as follows:
The eccentricity of the ellipse in the figure is .661. The eccentricity of the Earth’s orbit around the sun is .017. While this does not seem like much, it amounts to this:
semimajor axis of Earth’s orbit = 92,956,198 miles
eccentricity = .017
r = ea = 1,553,570 miles
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