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We consider a Celestial Mechanics model: the spin-orbit problem with adissipative tidal torque, which is a singular perturbation of a conservativesystem. The goal of this paper is to show that it is possible to computequasi-periodic attractors accurately and reliably for parameter valuesextremely close to the breakdown. Therefore, it is possible to obtaininformation on mathematical phenomena at breakdown. The method we useincorporates the same time numerical and rigorous improvements. Among them (i)the formalism is based on studying the time-one map of the spin-orbit problem(which reduces the dimensionality of the problem) and has mathematicaladvantages; (ii) very accurate integration of the ODE (high order Taylormethods implemented with extended precision) for the map at its jets; (iii) avery efficient KAM method for maps which computes the attractor and its tangentspaces ( quadratically convergent step with low storage requirements, and lowoperation count); (iv) the algorithms are backed by a rigorous a-posteriori KAMTheorem, which establishes that if the algorithm, produces a very approximatesolution of functional equation with reasonable condition numbers. then thereis a true solution nearby; and (v) the continuation algorithm is guaranteed toreach arbitrarily close to the border of existence if it is given enoughcomputer resources. As a byproduct of the accuracy that we maintain tillbreakdown, we study several scale invariant observables of the tori used in therenormalization group of infinite dimensional spaces. In contrast withpreviously studied simple models, the behavior at breakdown of the spin-orbitproblem does not satisfy standard scaling relations which implies that thespin-orbit problem is not described by a hyperbolic fixed point of arenormalization operator.

Accurate computations up to break-down of quasi-periodic attractors in the dissipative spin-orbit problem