Equation of Time (Fourier Series Approximation)

A sundial and a clock do not keep the same time. The equation of time $E$ is the difference between the time a sundial reads (apparent solar time) and the time a clock keeps (mean solar time). It is positive when the Sun is ahead of the clock, about +16 minutes in early November, and negative when it is behind, about -14 minutes in mid February. Two effects add up to it: Earth's orbit is an ellipse, so the Sun's apparent speed through the year changes, and the Sun travels along the ecliptic, which is tilted against the celestial equator.

This page does not compute those effects one by one. It uses the Fourier series in the fractional year that NOAA's Global Monitoring Laboratory prints in its General Solar Position Calculations note. The fractional year $\gamma$, in radians, is $2\pi/Y$ times $n - 1 + (h-12)/24$, where $Y$ is the days in the year (365, or 366 in a leap year), $n$ the day of the year and $h$ the hour of the day. Then $E$, in minutes, is 229.18 times a sum of five terms in $\cos\gamma$, $\sin\gamma$, $\cos 2\gamma$ and $\sin 2\gamma$. The 229.18 is $1440/(2\pi)$, the minutes in a day per radian, rounded as NOAA prints it. The hour moves the answer by at most about 30 seconds a day, so any clock will do for it.

This is a Fourier approximation, not a physical law and not an almanac. Measured here against an independent solar position calculation across 2003 and 2004, it stays within about 0.7 minute (about 40 seconds), and no worse than 0.75 minute in the 366-day year. At the instant of the worked example in NREL's Solar Position Algorithm report (17 October 2003, day 290, 19:30:30 universal time), it gives 14.88 minutes against the report's 14.6415, which is 14 seconds high. That is good enough to correct a sundial or to place solar noon to the minute, and not good enough for celestial navigation.

Two quick checks. At noon on 3 November (day 307) the series gives 16.37 minutes, against 16 min 25 s (16.417 minutes) in a rough published table of the extremes; at noon on 11 February (day 42) it gives -14.20 against -14 min 15 s. Both are well inside 0.75 minute. Solar noon on a mean-time clock at the middle meridian of a time zone falls at 12:00 less $E$; at other longitudes the longitude page adds 4 minutes for each degree.