Celestial Programming : Equation Of Time - Simplified

The Equation Of Time is a correction applied to Solar Time. It is mainly used today in celestial navigation, but was also used as a correction applied to sundials. Sundials would have a table or graph of the equation of time printed on them, so the observer could adjust the observed time to the true time.

The equation below is a simplified version, published by W. M. Smart in "Textbook On Spherical Astronomy". A more accurate version would require a nutation series, and precise RA and Dec of the Sun. But this version is accurate to within a few seconds over 60 centuries.

An even more simplified version is produced below. It differs by Smart's version by a few minutes at most, and is based only on the day of the year, so no correction for precession or other factors are made. The Astronomical Almanac publishes a simplefied version for each year. In the code, I have included versions from 1982 and 2016 purely to illustrate how it changes with time.

W. M. Smart

$$ \begin{align*} T = &\frac{jd-2415020.0}{36525} \\ \epsilon = &23.452294 - 0.0130125T - 0.00000164T^2 + 0.000000503T^3 \\ y = &\tan^2(\epsilon/2) \\ L = &279.69668 + 36000.76892T + 0.0003025T^2 \\ e = &0.01675104 - 0.0000418T - 0.000000126T^2 \\ M = &358.47583 + 35999.04975T - 0.000150T^2 - 0.0000033T^3 \\ E = & y\sin(2L) - 2e\sin(M) + 4ey\sin(M)\cos(2L) \\& - \frac{1}{2}y^2\sin(4L) - \frac{5}{4}e^2\sin(2M) \\ \end{align*} $$ \(jd\) is the Julian Date.
\(E\) is the Equation Of Time, and the result is in radians.
Multiply by \(\frac{60 \pi}{180 \cdot 15}\) to convert to minutes.


$$ -7.655 \sin(d) + 9.873 \sin(2d + 3.588) $$

Where \( d \) is the day of the year.
Result is in minutes

White is the Smart version, red is the Basic version.