Editing Equation of time (section)

=== Final calculation ===

The equation of time is obtained by substituting the result of the right ascension calculation into an equation of time formula. Here {{math|1=Δ''t''(''M'') = ''M'' + ''λ''<sub>p</sub> − ''α''[''λ''(''M'')]}} is used; in part because small corrections (of the order of 1 second), that would justify using {{math|''E''}}, are not included, and in part because the goal is to obtain a simple analytical expression. Using two-term approximations for {{math|''λ''(''M'')}} and {{math|''α''(''λ'')}} allows {{math|Δ''t''}} to be written as an explicit expression of two terms, which is designated {{math|Δ''t''<sub>''ey''</sub>}} because it is a first order approximation in {{math|''e''}} and in {{math|''y''}}.
: 1) <math>\Delta t_{ey} = -2e\sin{M} + y\sin \left( 2M + 2\lambda_p \right) = -7.659\sin{M} + 9.863\sin \left( 2M + 3.5932 \right)</math> minutes
This equation was first derived by Milne,{{r|Milne|p=375}} who wrote it in terms of {{math|1=''λ'' = ''M'' + ''λ''<sub>p</sub>}}. The numerical values written here result from using the orbital parameter values, {{math|''e''}} = {{val|0.016709}}, {{math|''ε''}} = {{val|23.4393}}° = {{val|0.409093}}&nbsp;radians, and {{math|''λ''<sub>p</sub>}} = {{val|282.9381}}° = {{val|4.938201}}&nbsp;radians that correspond to the epoch 1&nbsp;January 2000 at 12 noon [[UT1]]. When evaluating the numerical expression for {{math|Δ''t''<sub>''ey''</sub>}} as given above, a calculator must be in radian mode to obtain correct values because the value of {{math|2''λ''<sub>p</sub> − 2π}} in the argument of the second term is written there in radians. Higher order approximations can also be written,{{r|Muller|p=Eqs (45) and (46)}} but they necessarily have more terms. For example, the second order approximation in both {{math|''e''}} and {{math|''y''}} consists of five terms{{r|Hughes+|p=1535}}
: 2) <math>\Delta t_{e^2y^2} = \Delta t_{ey} - \frac{5}{4}e^2\sin{2M} + 4ey\sin{M}\cos \left( 2M + 2\lambda_p \right) - \frac{1}{2}y^2\sin \left( 4M + 4\lambda_p \right)</math>
This approximation has the potential for high accuracy, however, in order to achieve it over a wide range of years, the parameters {{math|''e''}}, {{math|''ε''}}, and {{math|''λ''<sub>p</sub>}} must be allowed to vary with time.{{r|Duffett-Smith|p=86}}{{r|Hughes+|p=1531,1535}} This creates additional calculational complications. Other approximations have been proposed, for example, {{math|Δ''t''<sub>''e''</sub>}}{{r|Duffett-Smith|p=86}}<ref name="Williams"/> which uses the first order equation of the center but no other approximation to determine {{math|''α''}}, and {{math|Δ''t''<sub>''e''<sup>2</sup></sub>}}<ref name="ApproxSolCoord"/> which uses the second order equation of the center.

The time variable, {{math|''M''}}, can be written either in terms of {{math|''n''}}, the number of days past perihelion, or {{math|''D''}}, the number of days past a specific date and time (epoch):
: 3) <math>M = \frac{2\pi}{t_Y} n</math> days <math>= M_D + \frac{2\pi}{t_Y} D</math> days <math>= 6.240\, 040\, 77 + 0.017\, 201\, 97D</math>
: 4)  <math>M = 6.240\, 040\, 77 + 0.017\, 201\, 97D</math>

[[File:EquationOfTime612.png|thumb|upright=2.2|right|Curves of {{math|Δ''t''}} and {{math|Δ''t''<sub>''ey''</sub>}} along with symbols locating the daily values at noon (at 10-day intervals) obtained from the ''Multiyear Interactive Computer Almanac'' vs {{math|''d''}} (day) for the year 2000]]
[[File:Length of solar day.png|thumb|450px|Derivative of −Δt. The axis on the right shows the length of the [[solar day]].]]
Here {{math|''M<sub>D</sub>''}} is the value of {{math|''M''}} at the chosen date and time. For the values given here, in radians, {{math|''M<sub>D</sub>''}} is that measured for the actual Sun at the epoch, 1&nbsp;January 2000 at 12&nbsp;noon UT1, and {{math|''D''}} is the number of days past that epoch. At periapsis {{math|1=''M'' = 2π}}, so solving gives {{math|1=''D'' = ''D''<sub>p</sub>}} = {{val|2.508109}}. This puts the periapsis on 4&nbsp;January 2000 at 00:11:41 while the actual periapsis is, according to results from the ''Multiyear Interactive Computer Almanac''<ref name="MICA"/> (abbreviated as MICA), on 3&nbsp;January 2000 at 05:17:30. This large discrepancy happens because the difference between the orbital radius at the two locations is only 1 part in a million; in other words, radius is a very weak function of time near periapsis. As a practical matter this means that one cannot get a highly accurate result for the equation of time by using {{math|''n''}} and adding the actual periapsis date for a given year. However, high accuracy can be achieved by using the formulation in terms of {{math|''D''}}.

When {{math|''D'' > ''D''<sub>p</sub>}}, ''M'' is greater than 2{{pi}} and one must subtract a multiple of 2{{pi}} (that depends on the year) from it to bring it into the range 0 to 2{{pi}}. Likewise for years prior to 2000 one must add multiples of 2{{pi}}. For example, for the year 2010, {{math|''D''}} varies from {{val|3653}} on 1&nbsp;January at noon to {{val|4017}} on 31&nbsp;December at noon; the corresponding {{math|''M''}} values are {{val|69.0789468}} and {{val|75.3404748}} and are reduced to the range 0 to 2{{pi}} by subtracting 10 and 11 times 2{{pi}} respectively.

One can always write:

5)  {{math|1=''D'' = ''n''<sub>Y</sub> + ''d''}}

where:

* {{math|''n''<sub>Y</sub>}} =  number of days from the epoch to noon on 1&nbsp;January of the desired year
* {{math|0 ≤ ''d'' ≤ 364}} (365 if the calculation is for a leap year).

The resulting equation for years after 2000, written as a sum of two terms, given 1), 4) and 5), is:

<math>a = -7.659\sin(6.240\, 040\, 77 + 0.017\, 201\, 97(365.25(y-2000) + d))</math>

<math>b = 
9.863\sin \left( 2 (6.240\, 040\, 77 + 0.017\, 201\, 97 (365.25(y-2000)+ d)) + 3.5932 \right)</math>

6) <math>\Delta t_{ey} =  a + b</math> [minutes]

In plain text format:

7) EoT =  -7.659sin(6.24004077 + 0.01720197(365*(y-2000) + d)) + 9.863sin( 2 (6.24004077 + 0.01720197 (365*(y-2000) + d)) + 3.5932 ) [minutes]

Term "a" represents the contribution of eccentricity, term "b" represents contribution of obliquity.

The result of the computations is usually given as either a set of tabular values, or a graph of the equation of time as a function of {{math|''d''}}.  A comparison of plots of {{math|Δ''t''}}, {{math|Δ''t''<sub>''ey''</sub>}}, and results from MICA all for the year 2000 is shown in the figure. The plot of {{math|Δ''t''<sub>''ey''</sub>}} is seen to be close to the results produced by MICA, the absolute error, {{nowrap|1=Err = {{abs|{{math|Δ''t''<sub>''ey''</sub>}} − MICA2000}}}}, is less than 1&nbsp;minute throughout the year; its largest value is 43.2&nbsp;seconds and occurs on day&nbsp;276 (3&nbsp;October). The plot of {{math|Δ''t''}} is indistinguishable from the results of MICA, the largest absolute error between the two is 2.46&nbsp;s on day&nbsp;324 (20&nbsp;November).

==== Continuity ====

For the choice of the appropriate branch of the {{math|arctan}} relation with respect to function continuity a modified version of the arctangent function is helpful. It brings in previous knowledge about the expected value by a parameter. The modified arctangent function is defined as:
: <math>\arctan_\eta x = \arctan x + \pi\operatorname{round}{\left( \frac{\eta - \arctan x}{\pi} \right)}</math>.
It produces a value that is as close to {{math|''η''}} as possible. The function {{math|round}} rounds to the nearest integer.

Applying this yields:
: <math>\Delta t(M) = M + \lambda_p - \arctan_{M + \lambda_p} \left( \cos{\varepsilon}\tan{\lambda} \right)</math>.
The parameter {{math|''M'' + ''λ''<sub>p</sub>}} arranges here to set {{math|Δ''t''}} to the zero nearest value which is the desired one.