As was explained, and as can be seen in the diagram, the angle we are interested in is the difference between a and v, specifically (a-v). Each angle must first be calculated. Lets first calculate the average angle the earth travels in its orbit over the course of a single day.
The angles are measured from perihelion. Call N the number of days since perihelion, occurring on earth on January 3rd. We can therefore easily calculate a:
a = .986 N // angle in a circular orbit
The angle v depends on the eccentricity of the orbit. Eccentricity is a measure of how elliptical the shape of the orbit is. If a is the semi-major axis and b is the semi-minor axis, then we can calculate the eccentricity by the following formula:
// eccentricity of elliptical orbit
An ellipse can range anywhere from a perfect circle to a line, and its eccentricity can range anywhere from 0 (a=b) to 1(b=0). The eccentricities of most planets are extremely small, and that of the earth is approximately .016713.
Once the eccentricity is found, the angle v follows a simple formula:
// angle in an elliptical orbit
Now the difference in the angles can be determined. To convert this to minutes, it is necessary to determine how many minutes pass during each degree of rotation of the earth. Note this is not the same as calculating how many days pass during each degree of revolution of the earth's orbit, as we did earlier. We use the simple formula:
This is easy to understand. There are 1440 minutes in a day (60 * 24), and the approximation is used that the earth rotates 361 degrees in a single day. This is because the earth has traveled about another degree in its orbit, and so it must rotate a degree more for the sun location in the sky to be consistent. Now we can calculate the equation of time for the elliptical orbit:
EOT(ellipse) = (a - v) * 3.989 minutes.
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