We develop a self--consistent model for the evolution
of the spin axis of LAGEOS and the related long--term semimajor axis
perturbations, due to asymmetric emission/reflection of radiation from
the satellite's surface. We show that the theory developed by
Bertotti and Iess (1990) for the evolution of LAGEOS' rotation under
magnetic and gravitational torques, which we have somewhat generalized
here, can lead to a successful fit of the observed semimajor axis
residuals, provided the correct initial conditions for the direction
of the spin axis are chosen. The remaining residuals have a r.m.s.
dispersion of $0.50\times 10^{-12}$~m/s$^2$, comparable to that of
previous solutions, based on purely empirical fits of the spin axis
direction as a function of time. The spectrum of the residuals
indicates that they are probably due to unmodelled radiation forces
(e.g., from Earth albedo and/or penumbra passages). Our solution
allows us to predict the future evolution of LAGEOS' rotation for
about another decade in the future, until the spin rate will become so
slow that some basic assumptions of the theory will fail. A similar
model can also be used to model/predict the coupled spin--orbit
evolution of the LAGEOS~II satellite, launched in late 1992, although
the available data still cover a span of time too short for reaching
significant quantitative conclusions concerning this satellite.
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