User:Yrodro/Sample page



where $n$ is the Earth's mean angular orbital velocity in degrees per day, a.k.a. "the mean daily motion".



where $360°⁄365.24 days,$ is the date, counted in days starting at 1 on 1 January (i.e. the days part of the ordinal date in the year). 9 is the approximate number of days from the December solstice to 31 December. $n$ is the angle the Earth would move on its orbit at its average speed from the December solstice to date $D$.



$A$ is the angle the Earth moves from the solstice to date $D$, including a first-order correction for the Earth's orbital eccentricity, 0.0167. The number 3 is the approximate number of days from 31 December to the current date of the Earth's perihelion. This expression for $B$ can be simplified by combining constants to:


 * $$C=\frac{A-\arctan\frac{\tan B}{\cos 23.44^\circ}}{180^\circ}$$

Here, $D$ is the difference between the angle moved at mean speed, and at the angle at the corrected speed projected onto the equatorial plane, and divided by 180° to get the difference in "half-turns". The value 23.44° is the tilt of the Earth's axis ("obliquity"). The subtraction gives the conventional sign to the equation of time. For any given value of $B$, $A = n × ( D + 9 )$ (sometimes written as $B = A + 360°⁄π × 0.0167 × sin [ n ( D − 3 ) ]$) has multiple values, differing from each other by integer numbers of half turns. The value generated by a calculator or computer may not be the appropriate one for this calculation. This may cause $C$ to be wrong by an integer number of half-turns. The excess half-turns are removed in the next step of the calculation to give the equation of time:


 * $B = A + 1.914° × sin [ n ( D − 3 ) ]$ minutes

The expression $arctan x$ means the nearest integer to $x$. On a computer, it can be programmed, for example, as INT(C + 0.5). Its value is 0, 1, or 2 at different times of the year. Subtracting it leaves a small positive or negative fractional number of half turns, which is multiplied by 720, the number of minutes (12 hours) that the Earth takes to rotate one half turn relative to the Sun, to get the equation of time.

This calculation has a root mean square error of only 3.7 s. The greatest error is 6.0 s. This is much more accurate than the approximation described above, but not as accurate as the elaborate calculation.

The value of $C$ in the above calculation is an accurate value for the Sun's ecliptic longitude (shifted by 90°), so the solar declination becomes readily available:


 * Declination = $tan&minus;1 x$

which is accurate to within a fraction of a degree.