Variation (astronomy)

In astronomy , variation in the celestial mechanical moon theory describes a periodic disturbance of the lunar orbit .

discovery

The Greek astronomer Ptolemy already describes in his famous work Almagest that the moon does not follow its orbit with a constant angular velocity , but with a period of 27.55 days, the anomalistic month , fluctuates by about ± 6.3 degrees compared to the mean position. This difference is called the midpoint equation and its maximum amount is called Great Inequality . Ptolemy describes another 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 .

It was not until 1590 that the Danish astronomer Tycho Brahe noticed that there is another periodic fluctuation of about 0.66 degrees. It is called variation and has a period of half a synodic month with 14.8 days . In contrast to the Great Inequality , it (like evection ) is not caused by Kepler's second law and the Kepler equation resulting from it , but represents a periodic orbital disruption. It was first found in the context of Newton's theory of gravity by analyzing the three-body system Earth-Moon -Sun a satisfactory explanation.

calculation

Illustration of the contraction of the lunar orbit towards the sun, which leads to the variation . The compression is shown greatly exaggerated.

The earth-moon system is not an isolated two-body system, so that the calculation of the position of the moon requires a correction that goes beyond the great inequality , which is due in particular to the gravitational influences of the sun. In the context of a perturbation theory, one can calculate that the Kepler's orbit elements of the moon are subject to temporal changes due to the influence of the sun: The position of the perigee and the ascending node "move" linearly in time due to the perturbation (so-called secular perturbations ), all orbit elements and in particular the semi-major axis , numerical eccentricity and orbital inclination of periodic disturbances which depend on the ecliptical length of the moon λ _m and the sun λ _s . Some terms have periodic dependencies on the double angle between the sun and the moon , including a term that affects the semi-major axis. This term can be understood as the compression of the lunar orbit towards the sun. These perturbations lead to a change in the ecliptical length of the moon in a first approximation around the summand: ${\displaystyle 2(\lambda _{s}-\lambda _{m})}$

${\displaystyle \Delta \lambda _{m}={\frac {11}{8}}\mu ^{2}\sin 2(\lambda _{s}-\lambda _{m}),}$

where μ = ω _s / ω _m ≈0.075, the ratio of the sidereal month to the sidereal year. This first approximation provides only a rough estimate with an amplitude of only about 0.44 degrees. Closer analysis shows that the total amplitude is 39.5 arc minutes, i.e. H. 0.66 degrees. The first links

${\displaystyle \Delta \lambda _{m}^{\textit {max}}={\frac {11}{4}}\mu ^{2}+{\frac {59}{12}}\mu ^{3}+{\frac {893}{72}}\mu ^{4}+{\frac {2855}{108}}\mu ^{5}+{\mathcal {O}}(\mu ^{6})\approx 35'=0{,}583^{\circ }}$

do not depend on the numerical eccentricity in contrast to the large deviation and evection. The remaining 5 arc minutes, however, result from terms that depend on both the eccentricity of the lunar and earth orbits. The period of the disturbance results from

${\displaystyle P={\frac {2\pi }{2(\omega _{m}-\omega _{s})}}\approx 29{,}6{\text{Tage}},}$

d. H. exactly one synodic month.

The calculation presented here is in principle also valid for the moons of other planets. Since it practically only depends on the frequency ratio μ, one quickly sees that it is much smaller for all other large moons of the solar system than for the earth's moon (μ≈1 / 13). In relation to μ, the Saturn moon Iapetus with μ≈1 / 135 is in second place before the Jupiter moon Callisto with μ≈1 / 260. However, due to the quadratic dependence of μ, the size of the effect at Iapetus is only 1% or 0.25% of the size at the Earth's moon. In addition, as in the case of evection, disturbances due to the flattening of the central planet and neighboring planets are far more relevant in the large moons of the gas planets .

Individual evidence

1. ^ Life and work of Claudius Ptolemy ( Memento from September 28, 2007 in the Internet Archive )
2. ↑ Ptolemy . encyclopedia2.thefreedictionary.com. Columbia Electronic Encyclopedia. Columbia University Press. Retrieved March 5, 2018
3. ↑ ^{a b c} M. Schneider: Himmelsmechanik , chap. 26, Vol. 2, BI Wiss. Verlag, Mannheim (1993), pp. 551-552
4. ↑ Ibid., P. 543

See also

• Evection
• Three body problem