REMEMBERING MICH ELE MOONS

The new year begins very sadly for our community. Michele Moons died on January8 1998, at the age of 46, after having struggled for more than a year against herillness. She has left her husband Yves, her sons Christophe, Olivier and Nicolas, anda large number of friends and colleagues all over the world.

She was assistant editor of this journal, a member of the organizing committeeof commission 7 and a member of commission 20 of the International AstronomicalUnion, a member of the Belgian committee of the European Southern Observatoryand of the American Astronomical Society of the Pacific. But, above all, she was abrilliant researcher in celestial mechanics whose work has left a major mark on thediscipline. Since she loved research so much that she spoke with me about futureresearch projects just a few days before her passing, let me celebrate her memory bybriefly recalling the history of her unfortunately short but excellent research activity.

Michele began her career in 1981 with a Ph.D. concerning physical libration ofthe Moon, under the direction of Prof. J. Henrard at the University of Namur. Atthat time, the construction of high precision analytic ephemerides was undergoinga revival due to the systematic use of Lie’s series and algebraic manipulators oncomputers, thus this was one of the major research topics in celestial mechanics.

Celestial Mechanics and Dynamical Astronomy68: 199–204, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

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The first version of Michele’s theory of theforcedlibrations of the Moon (Moons,1982a) included the effects of a spherical Earth and of the Sun on an eccentricorbit, and attained, or even surpassed, the level of precision of the existing theoriesby Migus (1977) and Eckhardt (1981). It was then improved by allowing a fullanalytical dependence on the physical parameters of the Moon’s gravity field – a veryimportant feature to allow the parameters’ determination by fitting the observations –and including an accurate description of the Moon’sfreelibrations (Moons, 1982b).Furthermore, the indirect perturbations of the planets (Moons, 1984) and the effectsof the non-sphericity of the Earth (Moons, 1986) were also taken into account. Thismade Michele’s theory of lunar librations the most accurate and complete available,attaining a remarkable accuracy of about 10 centimeters. To compare, the accuracyof experimental measurements, realized by reflecting lasers off mirrors positionedon the Moon by the Apollo astronauts, is about 3–5 cm.

To achieve this high accuracy, several thousand terms had to be taken into accountin the perturbing series. This required the design of a suitable algebraic manipulator;Michele collaborated with J. Henrard and H. Claes in programming the MS algebraicmanipulator, a Fortran code adapted to VAX computers, and then designed a moresimplified and exportable version, the MINIMS, which, in spite of its simplicity,is still very suitable for implementing analytic perturbation algorithms in celestialmechanics.

Michele’s theory on the Moon’s librations is still in active use. Although thephysical parameters of the Moon’s potential are now basically known, her theory isstill used to determine the amplitudes, phases, and frequencies of the Moon’s freelibrations (Chapront and Chapront–Touze, 1997; Williams, private communicationand papers in preparation). The study of the Moon’s free librations is of great impor-tance; such librations should be damped by tides on a timescales ranging from 104 to106 years (Newhall and Williams, 1997). Therefore, the existence of these free libra-tions implies the existence of some sort of excitation mechanism. A few mechanismshave been proposed (impacts on the lunar surface, a recent passage through a spin-orbit resonance, the existence of a liquid core), but a quantitative model that accountsfor the librations “measured” using Michele’s theory has still to be developed.

After having completed her lunar theory, Michele lived a transition period evolv-ing, in phase with the changing goals of celestial mechanics, from high-accuracyanalytic ephemerides to dynamical studies of interesting phenomena, most of whichwere of chaotic character. She first studied the phenomenon of shepherding of plan-etary rings in the framework of the elliptic Hill problem (Delhaise and Moons, 1988;Moons et al., 1988), and then she examined (with J. Henrard) the problem of theanomalous inclination of the Uranian satellite Miranda.

It had been suggested (Tittemore and Wisdom, 1989; Malhotra and Dermott,1990) that Miranda’s inclination could have been pumped while passing, during tidalevolution of its orbit, through the 3/1 mean-motion resonance with Umbriel. Henrardand Moons (1992) computed the probability of capture in the various secondaryresonances which characterize the dynamics of the Miranda–Umbriel 3/1 resonance,

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improving the estimates provided by Malhotra (1990) and leading to a quantitativeunderstanding of the inclination pumping phenomenon. In a second contemporarypaper, but published only two years later (Moons and Henrard, 1994), they studiedthe global structure of the phase-space around and inside the Miranda–Umbriel 3/1resonance, achieving an understanding of the origin of the chaotic layers. Thesepapers also mark a technical transition in Michele’s research, passing from pureanalytical studies on classical series expansions to the use of action-angle variableswith a semi-numerical method (Henrard, 1990).

I started to work with Michele during the last year that I spent in Namur formy doctorate. At first the reasons for which we decided to study the mean-motioncommensurabilities of the asteroid belt were entirely academic. Up to that time,indeed, mean motion commensurabilities had been investigated in the framework ofthe elliptic restricted three-body model, while for the study of secular resonancesand the computation of proper elements a more accurate model was commonly used,taking into account the precessions and deformations of Jupiter’s orbit forced by theother planets’ perturbations. Our aim was therefore to upgrade the theory of mean-motion resonant dynamics, using the same model adopted for the description of thesecular dynamics of the asteroid belt.

We expected to find theν5 secular resonance inside most of the main mean motioncommensurabilities because previous work (Yoshikawa, 1989, 1990, 1991; Ferraz-Mello et al., 1993) had pointed out the existence of the so-called “corotation reso-nance”, which is nothing but the ghost of theν5 resonance in the approximation whereJupiter is assumed to evolve on an elliptic fixed orbit. In contrast, we were rathersurprised when we discovered that the other secular resonances – mainlyν6, ν16 andKozai’s – are also commonly present inside mean-motion commensurabilities andstructure their long-term dynamics.

In a first paper (Morbidelli and Moons, 1993) we showed that theν5 andν6 secularresonances cause strong chaos at large eccentricity in the 2/1 and 3/2 mean-motioncommensurabilities; in the 3/2 commensurability the Hildas fill the regular regionat moderate eccentricity, bounded by secular resonances, while in the 2/1 commen-surability the corresponding region is mysteriously empty of asteroids. In a secondpaper (Moons and Morbidelli, 1995) we showed that the 3/1, 5/2, 4/1 and 7/3 mean-motion commensurabilities are globally chaotic due to the overlapping of secularresonances, so that the eccentricity, whatever its initial value, evolves chaoticallyand may become equal to unity on a timescale of about 1 million years, forcingthe resonant asteroid to collide with the Sun, as observed in the numerical simula-tions by Farinella et al. (1994). Although the existence of Kirkwood gaps associatedwith these resonances had been explained on the basis of simpler models (Wisdom,1983, 1985; Yoshikawa, 1990, 1991), only this model of overlapping secular reso-nances allows one to understand the orbital evolution of resonant asteroids observedin numerical simulations (Morbidelli and Moons, 1995) and their extremely shortdynamical lifetimes (Gladman et al., 1997).

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Our studies of mean-motion resonant dynamics could not have been done usingthe classical series expansions, since we had to avoid all restrictions on the maximalvalues of eccentricity and inclination. Therefore Michele extended the approach byFerraz-Mello and Sato (1989) for the accurate local evaluation of the Hamiltonian andits derivatives (Moons, 1993). Using her formulæ, she also designed an integratorof the averaged resonant equations of motion which extends the original work ofSchubart (1978) by introducing the secular theory of Jupiter’s motion (Moons, 1994).Her integrator is very suitable to explore in detail the dynamics associated with meanmotion commensurabilities.

Michele used her integrator to achieve two results. In (Morbidelli et al., 1995a)we computed the distance of real non-resonant asteroids from the chaotic bordersof mean motion resonances, showing that several asteroid families are truncated bysuch borders. This result implies that families must have injected many large asteroidsinto the resonances, and therefore it has been the starting point of the recent paperby Cellino et al. (1997) on the existence of “asteroid showers” characterizing theimpact history of the Earth. In a second paper (Henrard et al., 1995), our results onthe 2/1 resonance were extended, showing the existence of a dynamical path formedby several secondary and secular resonances which connects small eccentricities tolarge eccentricities, passing through high inclinations. This result explained for thefirst time the strange evolution of one test particle numerically followed by Wisdom(1987).

In the meantime, the discovery of several objects beyond Neptune, in the so-called Kuiper belt, and the numerical simulations by Holman and Wisdom (1993)and Levison and Duncan (1993) motivated us to apply our tools to understand theglobal dynamical structure of the Kuiper Belt (Morbidelli et al., 1995b).

In recognition of her important scientific production, Michele was invited to givea review lecture on the Kirkwood gaps at the IVth von Humboldt Colloquium heldin Ramsau (Austria) in 1996. It was at that meeting she first discovered somethingstrange within her: the precursor of her medical odyssey. Despite that she gave agreat talk, an extremely complete and balanced review of all the efforts made since1866 to explain the strange gaps that Kirkwood first pointed out to be associated tomean-motion resonances. Already weakened by her illness, it was terribly hard andpainful for her to transfer her review lecture to paper. It took more than a year, but shefinally succeeded in writing an excellent paper, even better than her oral presentation,which certainly deserves to be carefully read (Moons, 1997).

But Michele was not yet satisfied as one more scientific result had still to beachieved for her: Nesvorny and Ferraz-Mello (1997) had recently made a majoradvancement in the understanding of the origin of the gap associated with the 2/1resonance, but a complete clear analytical description of the structure of the seculardynamics within the 2/1 resonance still remained to be done. I remember Michele– during one of her frequent visits to Nice to rest under the spring sun after hertherapies – saying to me: “we can do it, so we must do it”. And so we did what I believeis the most extended application of the semi-numerical perturbation techniques to

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describe highly non-linear dynamics. The paper has just been submitted toIcarus, andalso includes the important contribution provided by Fabio Migliorini who tragicallydied in a mountain accident last November, on the connection between the fewexisting resonant bodies and the Themis asteroid family (Moons et al., 1998).

Michele loved science; she loved the sun and the sea. She loved life. After a coupleof meetings in the western United States, I introduced her to the joys of backpackingand wild camping. I remember her, laying under the stars in the Havasu canyon orstaring at the moon –ma copine, as she used to call it – in the Arches National Park.When she was in her hospital bed, one of the last things she said to me was: “I wouldrather be in the desert, alone under the stars”. I know what it meant for her.

Adieu, my friend.

Alessandro MorbidelliCNRS, Observatoire de la Cote d’AzurB.P. 229, 06304 Nice Cedex 4, France

(email: morby@obs-nice.fr)

References

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Chapront, J. and Chapront-Touze, M.: 1997, ‘Lunar motion: theory and observations’,Celest. Mech.,66, 31–38.

Delhaise, F. and Moons, M.: 1988, ‘Effects of a non-circular shepherd upon a planetary ring’,Long-term dynamical behaviour of natural and artificial N-body systems(A.E. Roy ed.), Kluwer,Dordrecht, The Netherlands, 173–180.

Eckhardt, D.H.: 1981, ‘Theory of the libration of the Moon’,The Moon and the Planets, 25, 3.Farinella, P., Froeschle, Ch., Froeschle, C. Gonczi, R., Hahn, G., Morbidelli, A. and Valsecchi,

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