  We provide evidence of the existence of KAM quasi-periodic attractors for a
dissipative model in Celestial Mechanics. We compute the attractors extremely
close to the breakdown threshold.
  We consider the spin-orbit problem describing the motion of a triaxial
satellite around a central planet under the simplifying assumption that the
center of mass of the satellite moves on a Keplerian orbit, the spin-axis is
perpendicular to the orbit plane and coincides with the shortest physical axis.
We also assume that the satellite is non-rigid; as a consequence, the problem
is affected by a dissipative tidal torque that can be modeled as a
time-dependent friction, which depends linearly upon the velocity.
  Our goal is to fix a frequency and compute the embedding of a smooth
attractor with this frequency. This task requires to adjust a drift parameter.
  The goal of this paper is to provide numerical calculations of the condition
numbers and verify that, when they are applied to the numerical solutions, they
will lead to the existence of the torus for values of the parameters extremely
close to the parameters of breakdown. Computing reliably close to the breakdown
allows to discover several interesting phenomena, which we will report in
[CCGdlL20a].
  The numerical calculations of the condition numbers presented here are not
completely rigorous, since we do not use interval arithmetic to estimate the
round off error and we do not estimate rigorously the truncation error, but we
implement the usual standards in numerical analysis (using extended precision,
checking that the results are not affected by the level of precision,
truncation, etc.). Hence, we do not claim a computer-assisted proof, but the
verification is more convincing that standard numerics. We hope that our work
could stimulate a computer-assisted proof.
