9.0 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Equation of time | 7/9 | https://en.wikipedia.org/wiki/Equation_of_time | reference | science, encyclopedia | 2026-05-05T11:12:39.793727+00:00 | kb-cron |
Δ
t
e
y
=
−
2
e
sin
M
+
y
sin
(
2
M
+
2
λ
p
)
=
−
7.659
sin
M
+
9.863
sin
(
2
M
+
3.5932
)
{\displaystyle \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)}
minutes This equation was first derived by Milne, who wrote it in terms of λ = M + λp. The numerical values written here result from using the orbital parameter values, e = 0.016709, ε = 23.4393° = 0.409093 radians, and λp = 282.9381° = 4.938201 radians that correspond to the epoch 1 January 2000 at 12 noon UT1. When evaluating the numerical expression for Δtey as given above, a calculator must be in radian mode to obtain correct values because the value of 2λp − 2π in the argument of the second term is written there in radians. Higher order approximations can also be written, but they necessarily have more terms. For example, the second order approximation in both e and y consists of five terms
Δ
t
e
2
y
2
=
Δ
t
e
y
−
5
4
e
2
sin
2
M
+
4
e
y
sin
M
cos
(
2
M
+
2
λ
p
)
−
1
2
y
2
sin
(
4
M
+
4
λ
p
)
{\displaystyle \Delta t_{e^{2}y^{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)}
This approximation has the potential for high accuracy, however, in order to achieve it over a wide range of years, the parameters e, ε, and λp must be allowed to vary with time. This creates additional calculational complications. Other approximations have been proposed, for example, Δte which uses the first order equation of the center but no other approximation to determine α, and Δte2 which uses the second order equation of the center. The time variable, M, can be written either in terms of n, the number of days past perihelion, or D, the number of days past a specific date and time (epoch):
M
=
2
π
t
Y
n
{\displaystyle M={\frac {2\pi }{t_{Y}}}n}
days
=
M
D
+
2
π
t
Y
D
{\displaystyle =M_{D}+{\frac {2\pi }{t_{Y}}}D}
days
=
6.240
040
77
+
0.017
201
97
D
{\displaystyle =6.240\,040\,77+0.017\,201\,97D}
M
=
6.240
040
77
+
0.017
201
97
D
{\displaystyle M=6.240\,040\,77+0.017\,201\,97D}
Here MD is the value of M at the chosen date and time. For the values given here, in radians, MD is that measured for the actual Sun at the epoch, 1 January 2000 at 12 noon UT1, and D is the number of days past that epoch. At periapsis M = 2π, so solving gives D = Dp = 2.508109. This puts the periapsis on 4 January 2000 at 00:11:41 while the actual periapsis is, according to results from the Multiyear Interactive Computer Almanac (abbreviated as MICA), on 3 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 n and adding the actual periapsis date for a given year. However, high accuracy can be achieved by using the formulation in terms of D. When D > Dp, M is greater than 2π and one must subtract a multiple of 2π (that depends on the year) from it to bring it into the range 0 to 2π. Likewise for years prior to 2000 one must add multiples of 2π. For example, for the year 2010, D varies from 3653 on 1 January at noon to 4017 on 31 December at noon; the corresponding M values are 69.0789468 and 75.3404748 and are reduced to the range 0 to 2π by subtracting 10 and 11 times 2π respectively. One can always write: 5) D = nY + d where:
nY = number of days from the epoch to noon on 1 January of the desired year 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:
a
=
−
7.659
sin
(
6.240
040
77
+
0.017
201
97
(
365.25
(
y
−
2000
)
+
d
)
)
{\displaystyle a=-7.659\sin(6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d))}
b
=
9.863
sin
(
2
(
6.240
040
77
+
0.017
201
97
(
365.25
(
y
−
2000
)
+
d
)
)
+
3.5932
)
{\displaystyle b=9.863\sin \left(2(6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d))+3.5932\right)}
Δ
t
e
y
=
a
+
b
{\displaystyle \Delta t_{ey}=a+b}
[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 d. A comparison of plots of Δt, Δtey, and results from MICA all for the year 2000 is shown in the figure. The plot of Δtey is seen to be close to the results produced by MICA, the absolute error, Err = |Δtey − MICA2000|, is less than 1 minute throughout the year; its largest value is 43.2 seconds and occurs on day 276 (3 October). The plot of Δt is indistinguishable from the results of MICA, the largest absolute error between the two is 2.46 s on day 324 (20 November).
==== Continuity ==== For the choice of the appropriate branch of the 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: