Mathematical Appendix 3

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Mathematical appendix 3 The Moon–Earth system This discussion relates to ideas in Section 3.3. The motions of the Moon are more difficult to define in ecliptic or equatorial coordinates than those of the Sun, because the plane of the motion is inclined at a mean angle of 5◦ 09 to the plane of the ecliptic. Moreover, this plane rotates slowly over a period of 18.61 years. The ascending node, at which the Moon crosses the ecliptic from south to north, moves backwards along the ecliptic at a nearly uniform rate of 0.0022 deg per mean solar hour (see Table 3.2). This regression goes through a complete cycle in 18.61 years. Viewed in equatorial coordinates, when the ascending node corresponds to the First Point of Aries (ϒ) the Moon’s maximum declination north and south of the equator in the next month will be (23◦ 27 + 5◦ 09) = 28◦ 36. Later, after 9.3 years, when the descending node is at the vernal equinox the maximum lunar declination is only (23◦ 27 − 5◦ 09) = 18◦ 18. Clearly the terms in the Equilibrium Tide that depend on the lunar declination will have a pronounced 18.61-year modulation. The eccentricity of the Moon’s orbit has a mean value of 0.0549, more than three times the eccentricity of the Earth–Sun orbit. Because of the effect of the Sun’s gravitational attraction the obliquity or inclination varies between 4◦ 58 and 5◦ 19, and the eccentricity varies from 0.044 to 0.067. The lunar distance r2,toafirst approximation (Roy, 1978) is: r¯ 2 = (1 + e cos(s − p) + solar perturbations) r2 where r¯2 is the mean lunar distance, s is the Moon’s geocentric mean ecliptic ◦ longitude (which increases by ω2 = 0.5490 per mean solar hour) and p is the longitude of lunar perigee, which rotates with an 8.85-year period. The true ecliptic longitude of the Moon also increases at a slightly irregular rate through the orbit (as does the solar longitude): λ2 = s + 2e sin(s − p) + solar perturbations where λ2 is in radians. The right ascension of the Moon is calculated from its ecliptic longitude and ecliptic latitude: 2 A2 = λ2 − tan (l/2) sin 2λ2 6 The Moon–Earth system 7 Relative to the ecliptic the Moon’s latitude is given by: ◦ sin(ecliptic latitude) = sin(λ2 − N) sin(5 09 ) where λ2 is the ecliptic longitude and N is the mean longitude of the ascending node which regresses over an 18.61 year cycle. The lunar declination can be calculated from this ecliptic latitude using similar formulae. The declination and right ascension of the Moon and of the Sun may all be represented as series of harmonics with different amplitudes and angular speeds. The Equilibrium Tide may also be represented as the sum of several harmonics by entering these astronomical terms into the equation in the associated appendix..

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