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 \(\frac{60 * 180}{15 \pi}\) to convert to minutes.
Hughes, Yallop & Hohenkerk
$$
\begin{align*}
t = &\frac{jd + UT^h/24-2451545.0}{36525} \\
T = &t + \Delta T(t) \\
\Delta T(t) = &\begin{cases}
0 &\text{ from AD 1650 to 1900 } \\
[-3.36 + 1.35(t + 2.33)^2] \times 10^{-8} & \text{ elsewhere }
\end{cases} \\
\\
ST\degree = &100.4606 - 36000.77005t + 0.000388t^2 - 3 \times 10^{-8}t^3 \\
\\
L\degree = &280.46607 + 36000.76980T + 0.0003025T^2 \\
G\degree = &357.528 + 35999.0503T \\
\epsilon\degree = &23.4393 - 0.01300T - 0.0000002T^2 + 0.0000005T^3 \\
C\degree = & (1.9146 - 0.00484T - 0.000014T^2)\sin G + (0.01999 - 0.00008T)\sin{2G} \\
\\
L_\odot\degree = &L + C - 0.0057\\
y = &\tan^2\frac{\epsilon}{2} \\
f = &180/\pi \\
\alpha\degree = &L_\odot - yf \sin 2L_\odot + 0.5y^2f \sin 4L_\odot \\
\\
E\degree = &(ST - \alpha) - (15UT^h - 180\degree) \\
&\text{if } E > 10\degree \text{ then } E = E -360\degree \\
\end{align*}
$$
Basic
$$ -7.655 \sin(d) + 9.873 \sin(2d + 3.588) $$
Where \( d \) is the day of the year.
Result is in minutes
White is the Smart version, red is the Basic version.