Evection

The evection (Latin evehere , evectum : to lead out, to drive out) describes in the celestial mechanical moon theory a periodic disturbance of the lunar orbit .

discovery

It was already known in ancient times that the moon did not move through its orbit at a constant angular velocity . With a period of 27.55 days, the anomalistic month , the position of the moon fluctuates by approximately ± 6.3 degrees from the mean position. This difference caused by the elliptical shape of the lunar orbit is called great inequality or the equation of the center point and was later explained by Kepler using Kepler's second law in the context of the two-body problem . The Greek astronomer Ptolemy noted in his famous work Almagest (and in turn quotes Hipparchus) that there is a further deviation from the uniform movement, which is significantly smaller at ± 1.27 degrees and has a period of 31.8 days. This second deviation is called evection .

The name goes back to Ismael Boulliau , who in his "Astronomia Philolaica" (1645) tried to describe the second deviation of the lunar movement by a periodic movement of the free focus of the lunar ellipse called "evectio".

calculation

If the earth-moon system were an isolated two-body system, the great inequality would already explain the position of the moon with high accuracy. However, this assumption is by no means justified; The sun in particular influences the earth-moon system and leads to deviations from the elliptical lunar orbit, as follows from Kepler's laws. Within the framework of a perturbation theory, these deviations can be calculated by assuming that the orbital elements of the moon are subject to changes over time due to the influence of the sun. While the position of the perigee and the ascending node "wander" linearly in time due to the disturbance (so-called secular disturbances ), all other orbital elements and in particular the semi-major axis , numerical eccentricity and orbital inclination are subject to periodic disturbances, which are caused by the ecliptical length of the moon λ _m and the sun λ _s . A special disturbance term shows that the numerical eccentricity of the lunar orbit changes by a summand that is proportional to , where Π denotes the mean secular ecliptical length of the perigee. Another term changes the position of the perigee proportionally . These perturbations lead to a change in the ecliptical length of the moon in a first approximation around the summand: ${\ displaystyle \ cos (\ lambda _ {s} -2 \ lambda _ {m} + \ Pi)}$ ${\ displaystyle \ sin (\ lambda _ {s} -2 \ lambda _ {m} + \ Pi)}$

{\ displaystyle \ Delta \ lambda _ {m} = {\ frac {15} {4}} \ mu e_ {m} \ sin (\ lambda _ {s} -2 \ lambda _ {m} + \ Pi), }

where e _m ≈ 0.0549 is the mean numerical eccentricity of the lunar orbit and μ = ω _s / ω _m ≈ 0.075, the ratio of the sidereal month to the sidereal year. With an amplitude of only about 0.88 degrees, this first approximation only provides a rough estimate. Closer analysis shows that the amplitude

{\ displaystyle \ Delta \ lambda _ {m} ^ {\ textit {max}} = e_ {m} {\ Bigl (} {\ frac {15} {4}} \ mu + {\ frac {263} {16 }} \ mu ^ {2} + {\ frac {48217} {708}} \ mu ^ {3} + {\ frac {1880537} {9216}} \ mu ^ {4} {\ Bigr)} + ​​{\ mathcal {O}} (\ mu ^ {5}) \ approx 1 {,} 27 ^ {\ circ}}

amounts.

The period of the disturbance results from

{\ displaystyle P = {\ frac {2 \ pi} {(2-c) \ omega _ {m} -2 \ omega _ {s}}} \ approx 31 {,} 8 {\ text {days}}, }

where c ≈0.992 is a correction factor that depends on μ.

In comparison to other disturbances in the lunar orbit ( variation , annual and parallactic inequality, etc.), the proportionality to the numerical eccentricity of the lunar orbit, which evection has in common with the great inequality, is important. Since the calculation presented here is in principle also valid for the moons of other planets, it becomes relevant for moons with a large eccentricity and a large frequency ratio μ. However, one quickly sees that μ is much smaller in all other large moons of the solar system than in the earth's moon (≈ 1/13). The Jupitermond Kallisto has a ratio of ≈ 1/260, by the lesser eccentricity of about 0.007 but the effect is less than 1% of the size at the Earth's moon. At Saturn's moon Iapetus μ ≈ 1/135 and the eccentricity is about 0.3, so that the effect on this moon is about half as big as on the earth's moon. However, in the large moons of the gas planets, disturbances caused by the flattening of the central planet and disturbances from neighboring planets are far more relevant.

Individual evidence

^ Otto Neugebauer : A History of Ancient Mathematical Astronomy, Vol. 3 . (= Studies in the history of mathematics and physical sciences; 1). Springer, Berlin 1975, ISBN 3-540-06995-X , p. 1109.
↑ Manfred Schneider : Himmelsmechanik, Vol. 2: System models . BI Science Verlag, Mannheim 1993, ISBN 3-411-15981-2 , chap. 26, p. 543.
↑ ^a ^b Ibid., P. 552

[1] Otto Neugebauer : A History of Ancient Mathematical Astronomy, Vol. 3 . (= Studies in the history of mathematics and physical sciences; 1). Springer, Berlin 1975, ISBN 3-540-06995-X , p. 1109.

[schneider1-2] Manfred Schneider : Himmelsmechanik, Vol. 2: System models . BI Science Verlag, Mannheim 1993, ISBN 3-411-15981-2 , chap. 26, p. 543.

[schneider2-3] Ibid., P. 552