  We consider a Celestial Mechanics model: the spin-orbit problem with a
dissipative tidal torque, which is a singular perturbation of a conservative
system. The goal of this paper is to show that it is possible to compute
quasi-periodic attractors accurately and reliably for parameter values
extremely close to the breakdown. Therefore, it is possible to obtain
information on mathematical phenomena at breakdown. The method we use
incorporates 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 mathematical
advantages; (ii) very accurate integration of the ODE (high order Taylor
methods implemented with extended precision) for the map at its jets; (iii) a
very efficient KAM method for maps which computes the attractor and its tangent
spaces ( quadratically convergent step with low storage requirements, and low
operation count); (iv) the algorithms are backed by a rigorous a-posteriori KAM
Theorem, which establishes that if the algorithm, produces a very approximate
solution of functional equation with reasonable condition numbers. then there
is a true solution nearby; and (v) the continuation algorithm is guaranteed to
reach arbitrarily close to the border of existence if it is given enough
computer resources. As a byproduct of the accuracy that we maintain till
breakdown, we study several scale invariant observables of the tori used in the
renormalization group of infinite dimensional spaces. In contrast with
previously studied simple models, the behavior at breakdown of the spin-orbit
problem does not satisfy standard scaling relations which implies that the
spin-orbit problem is not described by a hyperbolic fixed point of a
renormalization operator.
