JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96,NO. B2,PAGES 2431-2440, FEBRUARY 10,1991 The Rotation of LAGEOS B. BERTOTTI Dipartirnento di Fisica Nucleare e Teorica, Universitd di Pavia, Pavia, Italy L. I•ss Istituto di Fisica dello Spazio Interplanetario, CNR, Frascati, Italy In view of the need of an accurate modelling of nongravitational forces on laser-tracked satellites, it is important to understand their rotational dynamics, which determinesthe temperature anisotropy and the enõuingradiation recoil effects. We propose a model of the torques acting on LAGEOS due to eddy currents and gravity gradient. The electromotive forces induced in the spacecraft by its rotation in the magnetic field of the Earth dissipate angularmomentumand producea precession of the spinaxis; the oblatespacecraft will precess in the gravitational field of the Earth at a rate proportional to the rotation period. Therefore the gravitational torques become more and more important with time and eventually may produce a chaotic dynamics. The predicted evolution of the spin period agrees very well with the few experimental dataavailable and corresponds to an approximately exponential growth rate of about 3 years. 1. INTRODUCTION In the 1960s, space physicists devoted a good deal of attention to the natural torques acting on spacecraft, with the hope of controlling their attitude without active propulsion systems; for example, by activating magnetsinside a space- craft orbiting in the magnetic field of the Earth one could control its attitude [Wilson, 1959, 1960, 1964;Hecht and Manger, 1964]. Colombo investigated the rotation of Ex- plorer XI and studied the torques induced by the gravita- tional gradient [Colombo, 1964] and by permanent magneti- zation with hysteresis effects [Colombo, 1967]. Colombo [1967] also give a short and semiempirical discussionon the loss of angular momentum due to eddy currents, one of the main topics of the present work. With the advent of active and precise attitude control systems, the problem of natural torqueson spacecraft lost its main practical interest and laid neglectedfor some time, but now thereis a strong, new motivation from the need for a full understanding of the very weak forces acting on laser- tracked satellites like LAGEOS [Cohen et al., 1985]. It is now possible to measure its long-term accelerationto an accuracy of •10 -•ø cm/s2; indeed, this very high accuracy, and a good theoretical model to match it, is required by the exceedingly preciserange measurements available (down to 1 cm) and the need for a good orbital solution valid for a long time. A very accurate knowledge of the orbital elements of LAGEOS is important for geodetic applications. Indeed, at present, the anomalous, along-track acceleration, which is the main causefor the uncertaintyin the positionof LA- GEOS, is evaluated with a least squares fit which uses as unknown parametersits averages over an interval from 1 to 2 weeks. However, this time interval will become shorter as the accuracy in the distance measurementsincreases, lead- ing to an unpleasant proliferation of unknown parameters;in fact, a new type of data analysis in which each pass is independently fitted has been proposed [Milani and Mel- Copyright 1991 by the American Geophysical Union. Paper number 90JB01949. 0148-0227/91/90JB-01949505.00 chioni, 1989]. A better understandingof the nongravitational forces will become more and more important in the future. Recently, Ciufolini [1986] proposed using two LAGEOS satellites with supplementary inclinations to measure the gravitomagnetic force due to the angular momentum of the Earth; the final error of this relativistic experiment is limited by our "poor" knowledge of the nongravitational forces acting on the two spacecraft [Ciufolini, 1987]. At this very high accuracy, several complex and little known effects come into play: one could say, the nongravi- tational forces acting on LAGEOS are an interesting prob- lem of space physics in itself [Rubincam, 1982, 1987; Rubin- cam et al., !987; Afonso et al., 1980, 1985, 1990; Anselmo et al., 1983; Barlief et al., 1986; Farinella et al., 1990; Ciufolini et al., 1990]. We have a drag due to the interaction with the charged particles; the radiation pressure from the Sun, as modified by the eclipses; the radiation pressure from the Earth, bothin the optical andthe infrared band; and, finally and crucially, the reaction due to the anisotropic radiation emitted by the spacecraftwith an inhomogeneous tempera- ture distribution. In particular, Rubincam [1987] and Rubin- cam et al. [1987] has shown that the observed, long-term change in the semimajor axis of LAGEOS could be ex- plained by the "radiation rocket" effect due to the terrestrial infrared radiation on a spinning spacecraft, although the direct Earth albedo radiation pressure could also give a similar effect [Anselmo et al., 1983]. Since the temperature anisotropyis determined by the vectoria! angular velocity, a theoretical model for its time evolution is required. The "radiation rocket" effect may play a role for a laser-trackedsatellite on three counts: (1)the solar heating, especially as a consequence of eclipses, which produce long-term effects on the orbital elements; (2) the infrared radiation of the Earth [Rubincam, 1987]; and (3) for slowly rotating satellites, the "daily" Yarkowsky effect due to a lack of temperature uniformity along their equator, produc- ing an additional force orthogonal to the spin axis. The knowledgeof the spin vector of the spacecraftis essentialfor an understanding of these effects. For a passive, nonferromagnetic spacecraft like LA- 2431