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Dynamics of the Sun-Earth-Moon System

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Dynamics of the Sun-Earth-Moon System GENERAL I ARTICLE Dynamics of the Sun-Earth-Moon System S M Alladin and G M Ballabh The dynamics of the Sun-Earth-Moon system is discussed with special attention to the effects of Sun's perturbations on the Moon's orbit around the Earth. Important secular effects are the re­ gression of the nodes, the advance of the perigee S M Alladin is a retired and the increase in the Moon's mean longitude. professor of Astronomy, We discuss the relationship of the first with the Osmania University, Hyderabad. He specialized occurrence of eclipses, the second with the fluc­ in dynamical astronomy tuations in the moon's synodic period, and the and did research on the third with the slowing down of the Earth's rota­ dynamics of colliding and tion due to tidal friction. The Sun-Earth-Moon merging galaxies. He is system is compared with other triple celestial interested in the interac­ tion between religion and systems with regard to the intensity of tidal ef­ science. fects. 1. Introduction The Sun-Earth-Moon system (Figure 1) is of vital im­ portance for several reasons. It is our native system and is the system with which we are most intimately G M Ballabh is a retired associated. It has stimulated the intellect of man from professor of Astronomy) Osmania University, time immemorial and continues to do so. It was the Hyderabad. His field of first three-body problem to be studied with Newtonian research includes mechanics. It is now a very well studied system and dynamical astronomy and hence it can be used as a standard system with which positional astronomy. At other triple systems may be compared. Abhyankar [1] present he is the Chair- man of the Standing has given a good account of the Indian contributions to Advisory Committee for this problem. the Positional Astronomy Centre, Kolkata The Holy Quran has drawn attention to the Sun-Earth­ Moon system. In one place (Chapter 11, verse 6) it says: Keywords "He (Allah) it is Who made the Sun radiate a brilliant Sun-Earth-Moon system, peri­ odic and secular variations, 3- light and the Moon reflect a lustre and ordained for it body systems, tidal effects. stages that you might know the number of years and -6------------------------------~~----------------------------- RESONANCE I August 2005 GENERAL I ARTICLE Figure 1. The Sun-Earth­ Moon system. SUN mathematics. Allah has not created this but with truth. He details the signs for a people who have knowledge." As indicated in the above verse, the study of t.he Moon has been particularly useful in the development of math­ ematics. The motion of the Moon is very complicated. Sir Isaac Newton is supposed to have told his friend Hal­ ley that lunar theory "made his head ache and kept him awake so often that he would think of it no more" [2]. The Earth and the Moon are sufficiently close to the Sun The motion of the and hence the Moon's orbit around the Earth is appre­ Moon is very ciably perturbed by the gravitational attraction of the complicated. Sir Sun. Smaller perturbations arise due to the attraction Isaac Newton is of other planets and also due to the deviations of the supposed to have shape of the Earth and the Moon from perfect spheri­ told his friend Halley cal symmetry. Non-gravitational effects also affect the that lunar theory motion. "made his head ache The Moon moves around the Earth in an approximately and kept him awake elliptic orbit which can be described by the six orbital so often that he elements shown in Figure 2, namely: semi-major axis 'a' would think of it no which gives the size of the orbit, eccentricity 'e' which more" -RE-S-O-N-A-N-CE--I-A-U9-U-st--2-00-5------------~-----------------------------7 GENERAL I ARTICLE Figure 2. Orbital elements. Z i M y Ecliptic Plane x Orbital Elements describes the shape of the orbit, inclination 'i' which gives the orientation of the orbit with respect to the ecliptic, longitude of the node '0' which gives the posi­ tion of the intersection point of the ascending node of the lunar orbit and the ecliptic, longitude of perigee 'w' which defines the point in the orbit at which the Moon is closest to the Earth and the time of perigee passage 'T' which is the time at which the Moon is at perigee. On account of the perturbations mentioned earlier all these elements change with time. Lunar theory has been developed analytically by very eminent mathematicians such as Newton, Euler, Laplace, At present when Poisson, Hansen, Delauny, Hill and Brown. At present powerful computers when powerful computers are available the motion of are available the the Moon is determined accurately by numerical inte­ motion of the Moon is gration of its equations of motion. A good textbook on determined accurately the motion of the Moon has been written by Cook [3]. by numerical In this article we shall briefly review the salient features integration of its of the lunar theory arising out of the Sun's perturba­ equations of motion. tion on the lunar orbit. We shall also mention non- -8----------------------------~------------R-ES-O-N-A-N-C-E-I-A-U-9-Us-t--20-0-5 GENERAL I ARTICLE gravitational affects arising from the slowing down of the Earth's rotation due to tidal effects. 2. Sun's Perturbations on the Moon's Orbit If we consider the Earth, the Moon and the Sun as a system of three mass points, then the equation of motion of the Moon with respect to the Earth as origin is given by (see [2]): .~ G(E+m)r=GM (E_rl) r 3 0 A 3 13' (1) + r u r where f is the radius vector of the Moon, r' is the radius vector of the Sun and Li == r 1 - r is the radius vector of the Sun from the Moon (Figure 3); E, m and M0 are the masses of the Earth, Moon and Sun respectively and G is the gravitational constant. The second term represents the principal force on the Moon due to the attraction of the Earth while the term on the right hand side represents the tidal force on the Moon due to the Sun. It is the difference between the force with which the Sun attracts the Moon and the force with which the Sun attracts the Earth. The Sun is about 3,30,000 times more massive than the Earth while its distance is about 400 times the distance of the Moon. If we put these numeral values we find that the principal force is about 70 times the maximum tidal force. s r' E~-------r------~M Figure 3. -RE-S-O-N-AN-C-E-I-A-U-g-US-1-2-0-05-----------~---------------------------9 GENERAL I ARTICLE The right hand side of (1) can be expressed as the gra­ dient of the disturbing function, R, given by: 1 1'.1") R = GM0 ( t1 - r'3 (2) Since (r Ir') ~ 1/400 we obtain an expansion for R in terms of r Ir'. We then express R in terms of the orbital elements. Some of the important terms are as follows: k2 Ma2 ) 1 3 2 1 1 2 R = moon [- + _e - -ecosM - -e cos2M+ ( a~un 4 8 2 8 ~ e2cos{2(Q + w - 'IjJ')} - ~e2COS{2(Q + w - 'IjJ') + M}+ 3 15 4cOS {2 (f2+w-'ljJ') +2M} - Se2cos{ 2 (f2+w-'ljJ') +2M}+ 3 3 4ecos{2(O+w-'ljJ')+3M}+4e2cos{2(n + w - 'ljJ') + 4M}]. I The mean anomaly M is the angle described by a radius (3) vector with constant veloc­ ity. At any time t it is given In the above expansion terms of higher order than two by: in e have been ignored. M=n(t-T), where n(=2n-/period of revo­ Here a, e, 0, ware the elements of the Moon's orbit and lution of the body) is the con­ M is its mean anomaly!. The angle 'ljJ' is the mean stant angular velocity and T longitude2 of the Sun. is the time of perihelion pas­ sage. We can make estimates for the changes in the orbital elements with time by using Lagrange's planetary equa­ 2 The mean Sun is a fictitious body which is supposed to tions which relate the time derivatives of the elements move along the celestial with the derivatives of the disturbing function with re­ equator with the Sun's mean spect to the elements. We get: angular velocity n. Then the mean longitude, yI, of the Sun . 2 BR a=-­ at any time equals the right na ascension of the mean Sun BO" and is given by: 2 yI=n(t-the time when the Sun e = (~) BR _ (v'I- e ) BR was at the vernal equinox). na2 e BO' na2 e Bw' -10-----------------------------~------------R-ES-O-N-A-N-C-E-I-A-U-9-U-St--2-00-S GENERAL I ARTICLE 2 (J" = ( 1 - e ) 8R 2 8R - na2e 8e - na 8a ' n=. ( 1 ) -.,8R na2 sin iJ1 - e2 8", 2 -cosi ) 8R ) 8R W= (J1-e ( na2 sin iJ1 - e2 8i + na2e 8e ' di ( cos i ) 8R ( 1 ) 8R dt = na2 sin iJ1 - e2 8w - na2 sin iJ1 - e2 8n' It may be convenient to use the mean anomaly M = n( t - T) = nt + (J" instead of a itself, which is the co­ efficient of the secular acceleration. This implies that the perturbation function can be written in the form R(a, e, i, 0, w, M).
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