1 THE MOTION OF CELESTIAL BODIES Kaae Aksnes Institute of Theoetical Astophysics Univesity of Oslo Keywods: celestial mechanics, two-body obits, thee-body obits, petubations, tides, non-gavitational foces, planets, satellites, space pobes Contents 1. Intoduction 2. Two-body poblem 2.1. Obit in Space 2.2. Detemination of Obits fom Obsevations. Thee-body poblem 4. Petubations of planets and satellites 5. Dynamics of asteoids and planetay ings 6. Dynamics of atificial satellites and space pobes 7. Tides 8. Non-gavitational petubations 9. Dynamical evolution and stability of the Sola System 10. Conclusion Acknowledgement Glossay Bibliogaphy Biogaphical Sketch Summay The histoy of celestial mechanics is fist biefly suveyed, identifying the majo contibutos and thei contibutions. The Ptolemaic and Copenican wold models, Keple s laws of planetay motion and Newton s laws of univesal gavity ae pesented. It is shown that the obit of a body moving unde the gavitational attaction of anothe body can be epesented by a conic section. The six obital elements ae defined, and it is indicated how they can be detemined fom obseved positions of the body on the sky. Some special cases, pemitting exact solutions of the motion of thee gavitating bodies, ae also teated. With two-body motion as a fist appoximation, the petubing effects of othe bodies ae next deived and applied to the motions of planets, satellites, asteoids and ing paticles. The main effects of the Eath s oblateness on the motions of atificial satellites ae explained, and tajectoies fo sending a space pobe fom one planet to anothe ae shown. The influences of gavitational tides and nongavitational foces due to sola adiation and gas dag ae also teated. Finally, the long-tem evolution and stability of the Sola System ae biefly discussed. 1. Intoduction Befoe tuning to moe technical aspects of celestial motions, a shot histoical suvey
2 will be offeed. The celestial bodies and thei motions must always have attacted the attention of obsevant people. The Sun, Moon and planets wee in many ealy cultues associated with gods. The names of ou planets all come fom Geco-Roman mythology. Comets wee often viewed as bad omens of wa, pestilence and disaste. The egula inteval between new moons povided a basis fo the fist luna calendas, while sola calendas wee based on the appaent motion of the Sun among the stas, as manifested by changes in the times of ising and setting of these objects thoughout the yea. Aleady aound 700 B.C. the Babylonians had ecoded on stone tablets the motions of the Sun, Moon and the planets against the sta backgound, and had pedicted luna and sola eclipses with emakable pecision. But they don t seem to have had a geometical pictue of the motions in tems of obits. The concept of obits, o athe a device to compute obits, was intoduced by the Geek, notably Claudius Ptolemy. He developed in the Almagest aound 140 A.D. the Ptolemaic system in which the Sun, Moon and planets each move in a cicle (epicycle) whose cente moves on the peiphey of anothe cicle (defeent) which is in tun centeed on a point slightly displaced fom the Eath s cente. This geocentic wold pictue stood the gound fo 1400 yeas until Nicolaus Copenicus on his death bed in 154 intoduced the heliocentic system with the Sun in the middle, but it took moe than 100 yeas befoe the Copenican system was geneally accepted. The next fundamental pogess was made by Johannes Keple who in 1609 and 1619 published his thee laws of planetay motion: 1. Each planetay obit is an ellipse with the Sun situated at one of the foci. 2. The line joining the Sun and a planet will sweep out equal aeas in equal times.. The cubes of the semi-majo axes of the obits ae popotional to the squaes of the planets peiods of evolution. These laws and also Galileo Galilei s studies of the motions of falling objects, at about the same time, povided the basis fo Isaac Newton s laws of motion and univesal gavity, published in his famous Philosophiae Natualis Pincipia Mathematica in 1687: 1. Evey body emains at est o unifom motion unless acted upon by an extenal foce. 2. Acceleation is popotional to the impessed foce and takes place in the diection in which the foce acts.. To evey action thee is an equal and opposite eaction. 4. Two point masses will attact each othe with a foce which is popotional to the poduct of the masses divided by the squae of the distance between them, the popotionality facto being called the univesal constant of gavitation. 5. The gavitational attaction of an extended body of spheical shape will be as if all its mass had been concentated at the cente of the body.
3 This healded the beginning of celestial mechanics which made it possible to accuately pedict the motions of the celestial bodies. Newton showed that the gavitational attaction between two point masses, o finite bodies of spheical shape, leads to motions in obits which ae conic sections (see Figue 1). Duing the next 200 yeas this field was dominated by mathematicians and astonomes like L. Eule ( ), J. L. Lagange ( ), P. S. Laplace ( ), U. J. Leveie ( ), F. Tisseand ( ) and H. Poincaé ( ). Laplace had a vey deteministic view of the Sola System, believing that if the positions and velocities of the celestial bodies could be specified at some instant, the motions at any time befoe o afte this instant could be calculated with high accuacy. Lagange showed that thee ae five possible equilibium solutions of the thee-body poblem. In a failed attempt to find a geneal solution to this poblem, Poincaé poved that the motions of thee o moe gavitating bodies ae in geneal chaotic, meaning that the motions ae unpedictable ove vey long time spans, thus negating Laplace s optimistic view. As we will comment on late, the consequences of chaotic motion have not been fully appeciated until ecently. Figue1. A cicle, ellipse, paabola o hypebola can be constucted as intesections of planes with a cone. The intoduction of the theoy of special and geneal elativity by A. Einstein ( ) has not affected planetay obit theoies appeciably, except fo Mecuy which moves close enough to the Sun, so that the deviation fom Newtonian mechanics becomes measuable and was, in fact, the fist confimation of geneal elativity. Relativistic effects must also sometimes be taken into account fo atificial satellites and space pobes tacked with vey high accuacy, and elativistic effects can be vey ponounced fo tight and massive binay stas.
4 In the fist half of the twentieth centuy celestial mechanics went into a decline fo lack of new and inteesting poblems. The launch of the fist atificial satellite on 4 Octobe 1957 maked the beginning of a new and exciting ea in celestial mechanics. The advent of satellites, and late space pobes, bought with it a demand fo not only obit calculation of unpecedented accuacy, but also contol of a vaiety of obit types not manifested by the natual bodies. A plethoa of new satellites, asteoids, comets and planetay ings have been discoveed with planetay pobes and poweful moden telescopes on the gound and in space. The dynamics of planetay ings and the evolution of the Sola System ae pesenting special challenges to celestial mechanics. 2. Two-Body Poblem Let m 1 and m 2 be two point masses (o extended bodies of spheical shape with adial distibution of mass), a distance apat, which attact each othe with the foce mm F = G and 2 ae the position vectos whee G is the univesal constant of gavitation. If 1 of the two masses measued fom an abitay but unacceleated point, then accoding to Newton s second law m 11= Gmm 1 2, m 22 = Gmm 1 2 whee = 2 1 is the position vecto fom m 1 to m 2, and a dot above a symbol means diffeentiation with espect to time. This gives the diffeential equation called the equation of motion of the two-body poblem: + μ = 0 whee Gm1 m2 μ = ( + ). Newton showed that the esulting motion, i.e., the solution of, f as this equation, can be expessed in pola coodinates ( ) p = 1 + e cos f which is the equation of a conic section. As illustated on Figue 1, the intesection of a cone with a plane paallel to the base of the cone is a cicle( e = 0). By tilting the plane the cicle tuns into ellipses (0 < e < 1) of inceasing eccenticities until the plane is e = is cut out. If the tilt is futhe paallel to the side of the cone when a paabola ( 1)
5 inceased hypebolas ( e > 1) esult. The paamete p (semilatus ectum) is the length of a nomal to the symmety axis of the ellipse, paabola o hypebola though the focus of each of these cuves. The angle f is called the tue anomaly and is the angle between the adius vecto and the diection to the neaest point on the cuve (peicente), see Figue 2. Fo an ellipse, p is elated to the semimajo axis a and the eccenticity e though the equation 2 p = a(1 e ). Figue 2. The position of m 2 elative to m 1 in an ellipse. With these definitions, Keple s second and thid law can be expessed as 2 f = μ p, a P μ = (2 ) 2 2 Π. μ = ( + ) whee 1 Fo a planet Gm1 m2 m is the mass of the Sun and m2 is the much smalle mass of the planet. Thus, μ will vay slightly fom planet to planet, wheeas Keple assumed that the ight-hand side of the last equation is the same constant fo all the planets. But even fo massive Jupite the deviation is only 1%, which was fa too small to be detected in the obsevations that Keple had at hand. An impotant application of Keple s thid law has been to weigh the planets which have satellites. Let the peiod, semi-majo axis and mass be Pam,, fo a planet and P, a, m fo its satellite. By applying the law, fist to the planet in its motion about the Sun of mass M, and then to the satellite in its motion about the planet, we find afte dividing the latte equation by the fome: 2 Pa' m+ m' m = 2 P' a M + m M, whee the last appoximation is based on the fact that the Sun s mass is much lage
6 than the planet s mass, which is in tun much lage than the satellite s mass. Since all the paametes on the left of this equation can be measued, this gives the planet s mass in tems of the Sun s mass. The velocity V in an elliptic, paabolic o hypebolic obit is given by the enegy integal 1 2 μ μ V =, 2 2a whee a is positive fo an ellipse, negative fo a hypebola and infinite fo a paabola. The thee tems of this equation epesent, espectively, the kinetic enegy, the potential enegy and the total enegy pe unit mass. Fom this equation we obtain the velocity of escape V E fom the cental body at the distance, i.e. the paabolic velocity when a =, V E 2μ = = 2VC, whee V C is the velocity in a cicula obit of adius. In ode to calculate the tue anomaly f fo a given instant t, we need to intoduce two auxiliay angles, which fo objects in elliptic obits ae called the mean anomaly M and the eccentic anomaly E, elated though Keple s equation, M = nt ( τ ) = E esine, n = μ a whee τ is the time when the object passes though the peicente and n is called the mean motion. Afte obtaining E fom this equation (by iteation), f can be calculated fom the equation 1 tan f + e tan E = 2 1 e Obit in Space Two-body motion can be specified though six constant obital elements a, e, τ, I, Ω, ω of which the fist thee, as we have shown, define the size and shape of the obit, and the last thee oient the obital plane in space (see Figue ) with espect to some efeence plane. These obital elements ae called a the semimajo axis
7 e the eccenticity τ the time of peicente passage I the inclination (angle between obital plane and efeence plane) Ω the longitude of the ascending node (measued fom a fixed diection in the efeence plane) ω the agument of peicente (angle fom ascending node to peicente) Figue. The oientation of the obit of a celestial body by means of the angles I, Ω, ω Detemination of Obits fom Obsevations So fa we have discussed how to compute positions and velocities of celestial bodies at specific times given thei obital elements. The invese poblem is consideably moe complex and can only be solved by iteation. Hee we will limit ouselves to a sketch of how to poceed. The classical methods fo obit detemination assume that at least thee positions on the sky at diffeent times have been obseved of the body whose obit is sought. Moden obsevations of natual and atificial celestial bodies also include distance and adial velocity measuements with, fo instance, adas and lases. In the classical obit detemination poblem, we assume that thee sky positions ae available. The intevals between the obsevations need to be neithe too shot no too long fo the methods to convege. The body s distance is unknown, so each obsevation yields only two coodinates. Thee positions then supply 6 coodinates which ae usually sufficient fo detemining the 6 obital elements of the body. If we ae dealing with a body in obit about the Sun, the iteation stats off by guessing the distance of the body fom the obseve at the time of the middle obsevation. The heliocentic position of the Eath and the obseve s position on the Eath ae assumed known. By means of elations fo
8 two-body motion, the heliocentic and geocentic positions at all thee times ae computed. The diffeence between the guessed and computed values of the distance, o of some elated quantity, fo the middle obsevation will make it possible to impove on the guess and epeat the iteation until convegence. Nomally moe than thee obsevations will be available. The obit deived fom thee of the obsevations can then be impoved by means of the method of least-squaes in which the obital elements ae adjusted in such a way that the sum of squaes of the esiduals between obseved and computed positions is minimized Bibliogaphy TO ACCESS ALL THE 27 PAGES OF THIS CHAPTER, Visit: Goldeich P. & Temaine S. (1979a). The fomation of the Cassini Division in Satun s ings, Icaus 4, [This studies how Mimas emoves ing paticles fom the location of the 2:1 obital esonance with Mimas]. Goldeich P. & Temaine S. (1979b). Pecession of the epsilon ing of Uanus, Astonomical Jounal 84, [It is shown that Uanus elliptical epsilon ing of paticles otates as one body]. Laska J. (1989) A numeical expeiment on the chaotic behavio of the Sola System. Natue 8, [Laska shows that despite the chaotic natue of the motion of the planets, thei obits ae stable fo seveal hunded million yeas]. Malhota, R. (199). The oigin of Pluto s peculia obit. Natue, 65, [This discusses why Pluto neve collides with Neptune although thei obits intesect]. Mobidelli, A. (2002). Moden integations of sola system dynamics. Annual Review of Eath and Planetay Sciences, 0, [The long-tem development of planetay obits is studied by means of new numeical integation techniques]. Muay C.D. and Demott S.F. (1999). Sola System Dynamics, 592 pp, Cambidge Univesity Pess. [Moden textbook on celestial mechanics]. Peale, S.J. (199). Pluto s stange obit. Natue 65, [The consequences of Pluto s 2: obital efeence with Neptune is discussed]. Peale S.J., Cassen P. & Reynolds R.T. (1979) Melting of Io by tidal dissipation, Science 20, [this pape pedicted Io s volcanism befoe it was discoveed by the Voyage 1 spacecaft]. Roy A.E. (2005). Obital Motion, 526 pp, Institute of Physics Publishing, ISBN ISBN [Textbook on celestial mechanics]. Tsiganis K., Gomes R., Mobidelli A. & Levison H.F. (2005). Oigin of the obital achitectue of the giant planets of the Sola System. Natue 45, [This simulation shows that Satun, Uanus and Neptune fomed much close to the Sun than thei cuent locations]. Wisdom J. (1982). The oigin of the Kikwood gaps: A mapping fo asteoidal motion nea the /1 commensuability. Astonomical Jounal 87, [This pape explains the absence of asteoids with
9 peiods 1/ of Jupite s peiod]. Wisdom J. and Holman M.J. (1991). Symplectic maps fo the n-body poblem, Astonomical Jounal 102, [This pape intoduces a numeical integation method allowing vey long time steps]. Biogaphical Sketch Kaae Aksnes, bon at Kvam, Noway 25 Mach 198, Pofesso of Astonomy, Univesity of Oslo, Noway, Mastes Degee in astonomy, Univesity of Oslo, Noway 196, Ph.D. in astonomy, Yale Univesity, New Haven, USA He has fo ove 40 yeas woked on poblems of dynamics of atificial and natual satellites, comets and asteoids, and has been involved with seveal US and Euopean space missions (Maine 9, Voyage 1&2, GPS, ERS, Rosetta). He headed fo 16 yeas a Woking Goup fo Planetay System Nomenclatue fo the Intenational Astonomical Union. He was employed fo 11 yeas in USA at the Smithsonian Astophysical Obsevatoy and Jet Populsion Laboatoy, and 0 yeas in Noway at the Nowegian Defence Reseach Establishment, Univesity of Tomso and Univesity of Oslo. He has published moe than 10 scientific papes. Pof. Aksnes is a membe of Ameican Astonomical Society, Intenational Astonomical Union and Nowegian Academy of Science and Lettes.