13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Equation of time | 8/9 | https://en.wikipedia.org/wiki/Equation_of_time | reference | science, encyclopedia | 2026-05-05T11:12:39.793727+00:00 | kb-cron |
arctan
η
x
=
arctan
x
+
π
round
(
η
−
arctan
x
π
)
{\displaystyle \arctan _{\eta }x=\arctan x+\pi \operatorname {round} {\left({\frac {\eta -\arctan x}{\pi }}\right)}}
. It produces a value that is as close to η as possible. The function round rounds to the nearest integer. Applying this yields:
Δ
t
(
M
)
=
M
+
λ
p
−
arctan
M
+
λ
p
(
cos
ε
tan
λ
)
{\displaystyle \Delta t(M)=M+\lambda _{p}-\arctan _{M+\lambda _{p}}\left(\cos {\varepsilon }\tan {\lambda }\right)}
. The parameter M + λp arranges here to set Δt to the zero nearest value which is the desired one.
=== Secular change === The difference between the MICA and Δt results was checked every 5 years over the range from 1960 to 2040. In every instance the maximum absolute error was less than 3 s; the largest difference, 2.91 s, occurred on 22 May 1965 (day 141). However, in order to achieve this level of accuracy over this range of years it is necessary to account for the secular change in the orbital parameters with time. The equations that describe this variation are:
e
=
1.6709
×
10
−
2
−
4.193
×
10
−
5
(
D
36
525
)
−
1.26
×
10
−
7
(
D
36525
)
2
ε
=
23.4393
−
0.013
(
D
36
525
)
−
2
×
10
−
7
(
D
36
525
)
2
+
5
×
10
−
7
(
D
36
525
)
3
degrees
λ
p
=
282.938
07
+
1.7195
(
D
36
525
)
+
3.025
×
10
−
4
(
D
36
525
)
2
degrees
{\displaystyle {\begin{aligned}e&=1.6709\times 10^{-2}-4.193\times 10^{-5}\left({\frac {D}{36\,525}}\right)-1.26\times 10^{-7}\left({\frac {D}{36525}}\right)^{2}\\\varepsilon &=23.4393-0.013\left({\frac {D}{36\,525}}\right)-2\times 10^{-7}\left({\frac {D}{36\,525}}\right)^{2}+5\times 10^{-7}\left({\frac {D}{36\,525}}\right)^{3}{\mbox{ degrees}}\\\lambda _{\mathrm {p} }&=282.938\,07+1.7195\left({\frac {D}{36\,525}}\right)+3.025\times 10^{-4}\left({\frac {D}{36\,525}}\right)^{2}{\mbox{ degrees}}\end{aligned}}}
According to these relations, in 100 years (D = 36525), λp increases by about 0.5% (1.7°), e decreases by about 0.25%, and ε decreases by about 0.05%. As a result, the number of calculations required for any of the higher-order approximations of the equation of time requires a computer to complete them, if one wants to achieve their inherent accuracy over a wide range of time. In this event it is no more difficult to evaluate Δt using a computer than any of its approximations. In all this note that Δtey as written above is easy to evaluate, even with a calculator, is accurate enough (better than 1 minute over the 80-year range) for correcting sundials, and has the nice physical explanation as the sum of two terms, one due to obliquity and the other to eccentricity that was used previously in the article. This is not true either for Δt considered as a function of M or for any of its higher-order approximations.
=== Alternative calculation === Another procedure for calculating the equation of time can be done as follows. Angles are in degrees; the conventional order of operations applies.
n = 360°/365.24 days, where n is the Earth's mean angular orbital velocity in degrees per day, a.k.a. "the mean daily motion".
A
=
(
D
+
9
)
n
{\displaystyle A=\left(D+9\right)n}
where D 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. A is the angle the Earth would move on its orbit at its average speed from the December solstice to date D.
B
=
A
+
0.0167
⋅
360
∘
π
sin
(
(
D
−
3
)
n
)
{\displaystyle B=A+0.0167\cdot {\frac {360^{\circ }}{\pi }}\sin \left(\left(D-3\right)n\right)}
B 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:
B
=
A
+
1.914
∘
⋅
sin
(
(
D
−
3
)
n
)
{\displaystyle B=A+1.914^{\circ }\cdot \sin \left(\left(D-3\right)n\right)}
.
C
=
A
−
arctan
tan
B
cos
23.44
∘
180
∘
{\displaystyle C={\frac {A-\arctan {\frac {\tan B}{\cos 23.44^{\circ }}}}{180^{\circ }}}}
Here, 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 "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 x, arctan x (sometimes written as tan−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 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:
E
O
T
=
720
(
C
−
nint
C
)
{\displaystyle \mathrm {EOT} =720\left(C-\operatorname {nint} {C}\right)}
minutes The expression nint(C) means the nearest integer to C. 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. Compared with published values, 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.
==== Solar declination ====