8.4 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Equation of time | 6/9 | https://en.wikipedia.org/wiki/Equation_of_time | reference | science, encyclopedia | 2026-05-05T11:12:39.793727+00:00 | kb-cron |
E, the Sun's eccentric anomaly (note that this is different from M); ν, the Sun's true anomaly; λ = ν + λp, the Sun's true longitude on the ecliptic.
All these angles are shown in the figure on the right, which shows the celestial sphere and the Sun's elliptical orbit seen from the Earth (the same as the Earth's orbit seen from the Sun). In this figure ε is the obliquity, while e = √1 − (b/a)2 is the eccentricity of the ellipse. Now given a value of 0 ≤ M ≤ 2π, one can calculate α(M) by means of the following well-known procedure: First, given M, calculate E from Kepler's equation:
M
=
E
−
e
sin
E
{\displaystyle M=E-e\sin {E}}
Although this equation cannot be solved exactly in closed form, values of E(M) can be obtained from infinite (power or trigonometric) series, graphical, or numerical methods. Alternatively, note that for e = 0, E = M, and by iteration:
E
≈
M
+
e
sin
M
{\displaystyle E\approx M+e\sin {M}}
This approximation can be improved, for small e, by iterating again:
E
≈
M
+
e
sin
M
+
1
2
e
2
sin
2
M
{\displaystyle E\approx M+e\sin {M}+{\frac {1}{2}}e^{2}\sin {2M}}
, and continued iteration produces successively higher order terms of the power series expansion in e. For small values of e (much less than 1) two or three terms of the series give a good approximation for E; the smaller e, the better the approximation. Next, knowing E, calculate the true anomaly ν from an elliptical orbit relation
ν
=
2
arctan
(
1
+
e
1
−
e
tan
1
2
E
)
{\displaystyle \nu =2\arctan \left({\sqrt {\frac {1+e}{1-e}}}\tan {\tfrac {1}{2}}E\right)}
The correct branch of the multiple valued function arctan x to use is the one that makes ν a continuous function of E(M) starting from νE=0 = 0. Thus for 0 ≤ E < π use arctan x = arctan x, and for π < E ≤ 2π use arctan x = arctan x + π. At the specific value E = π for which the argument of tan is infinite, use ν = E. Here arctan x is the principal branch, |arctan x| < π/2; the function that is returned by calculators and computer applications. Alternatively, this function can be expressed in terms of its Taylor series in e, the first three terms of which are:
ν
≈
E
+
e
sin
E
+
1
4
e
2
sin
2
E
{\displaystyle \nu \approx E+e\sin {E}+{\frac {1}{4}}e^{2}\sin {2E}}
. For small e this approximation (or even just the first two terms) is a good one. Combining the approximation for E(M) with this one for ν(E) produces:
ν
≈
M
+
2
e
sin
M
+
5
4
e
2
sin
2
M
{\displaystyle \nu \approx M+2e\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}}
. The relation ν(M) is called the equation of the center; the expression written here is a second-order approximation in e. For the small value of e that characterises the Earth's orbit this gives a very good approximation for ν(M). Next, knowing ν, calculate λ from its definition:
λ
=
ν
+
λ
p
{\displaystyle \lambda =\nu +\lambda _{p}}
The value of λ varies non-linearly with M because the orbit is elliptical and not circular. From the approximation for ν:
λ
≈
M
+
λ
p
+
2
e
sin
M
+
5
4
e
2
sin
2
M
{\displaystyle \lambda \approx M+\lambda _{p}+2e\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}}
. Finally, knowing λ calculate α from a relation for the right triangle on the celestial sphere shown above
α
=
arctan
(
cos
ε
tan
λ
)
{\displaystyle \alpha =\arctan \left(\cos {\varepsilon }\tan {\lambda }\right)}
Note that the quadrant of α is the same as that of λ, therefore reduce λ to the range 0 to 2π and write
α
=
arctan
(
cos
ε
tan
λ
+
k
π
)
{\displaystyle \alpha =\arctan \left(\cos {\varepsilon }\tan {\lambda }+k\pi \right)}
, where k is 0 if λ is in quadrant 1, it is 1 if λ is in quadrants 2 or 3 and it is 2 if λ is in quadrant 4. For the values at which tan is infinite, α = λ. Although approximate values for α can be obtained from truncated Taylor series like those for ν, it is more efficacious to use the equation
α
=
λ
−
arcsin
(
y
sin
(
α
+
λ
)
)
{\displaystyle \alpha =\lambda -\arcsin \left(y\sin \left(\alpha +\lambda \right)\right)}
where y = tan2(ε/2). Note that for ε = y = 0, α = λ and iterating twice:
α
≈
λ
−
y
sin
2
λ
+
1
2
y
2
sin
4
λ
{\displaystyle \alpha \approx \lambda -y\sin {2\lambda }+{\frac {1}{2}}y^{2}\sin {4\lambda }}
.
=== Final calculation === The equation of time is obtained by substituting the result of the right ascension calculation into an equation of time formula. Here Δt(M) = M + λp − α[λ(M)] is used; in part because small corrections (of the order of 1 second), that would justify using E, are not included, and in part because the goal is to obtain a simple analytical expression. Using two-term approximations for λ(M) and α(λ) allows Δt to be written as an explicit expression of two terms, which is designated Δtey because it is a first order approximation in e and in y.