Editing Equation of time (section)

=== Alternative calculation ===

Another procedure for calculating the equation of time can be done as follows.<ref name="Williams"/> Angles are in degrees; the conventional [[order of operations]] applies.
:{{mvar|n}} = {{sfrac|360°|365.24&nbsp;days,}}
where {{mvar|n}} is the Earth's mean angular orbital velocity in degrees per day, a.k.a. ''"the mean daily motion"''.

: <math>A = \left( D + 9 \right) n</math>
where {{mvar|D}} is the date, counted in days starting at 1 on 1&nbsp;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&nbsp;December. {{mvar|A}} is the angle the Earth ''would'' move on its orbit at its average speed from the December solstice to date {{mvar|D}}.

: <math>B = A + 0.0167\cdot\frac{360^{\circ}}{\pi}\sin \left( \left( D - 3 \right) n \right)</math>
{{mvar|B}} is the angle the Earth moves from the solstice to date {{mvar|D}}, including a first-order correction for the Earth's orbital eccentricity, 0.0167&nbsp;. The number&nbsp;3 is the approximate number of days from 31&nbsp;December to the current date of the Earth's [[perihelion]]. This expression for {{mvar|B}} can be simplified by combining constants to:
: <math> B = A + 1.914^{\circ}\cdot\sin\left( \left( D - 3 \right) n \right)</math>.

: <math>C=\frac{A-\arctan\frac{\tan B}{\cos 23.44^\circ}}{180^\circ}</math>

Here, {{mvar|C}} 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 "[[Turn (geometry)|half-turns]]". The value 23.44° is the [[obliquity of the ecliptic|tilt of the Earth's axis ("obliquity")]]. The subtraction gives the conventional sign to the equation of time. For any given value of {{mvar|x}}, {{math|[[Inverse trigonometric functions#arctan|arctan]] ''x''}} (sometimes written as {{math|tan{{sup|−1}} ''x''}}) 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 {{mvar|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:

: <math>\mathrm{EOT} = 720\left( C - \operatorname{nint}{C} \right)</math>&nbsp;minutes

The expression {{math|[[Nearest integer function|nint(''C'')]]}} means the ''nearest integer'' to {{mvar|C}}. On a computer, it can be programmed, for example, as {{nowrap|{{mono|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&nbsp;hours) that the Earth takes to rotate one half turn relative to the Sun, to get the equation of time.

Compared with published values,<ref name="Waugh"/> this calculation has a [[root mean square]] error of only 3.7&nbsp;s. The greatest error is 6.0&nbsp;s. This is much more accurate than the approximation described above, but not as accurate as the elaborate calculation.

==== Solar declination ====
{{Main article|Declination|Position of the Sun}}

The value of {{mvar|B}} in the above calculation is an accurate value for the Sun's ecliptic longitude (shifted by 90°), so the solar declination {{mvar|δ}} becomes readily available:

: <math>\delta = -\arcsin\left( \sin 23.44^{\circ}\cdot\cos B \right)</math>

which is accurate to within a fraction of a degree.