The nearby star Fomalhaut harbours a cold, moderately eccentric dust belt
with a sharp inner edge near 133 au. A low-mass, common proper motion companion
(Fom b), was discovered near the inner edge and was identified as a planet
candidate that could account for the belt morphology. However, the most recent
orbit determination based on four epochs of astrometry over eight years reveals
a highly eccentric orbit that appears to cross the belt in the sky plane
projection. We perform here a full orbital determination based on the available
astrometric data to independently validate the orbit estimates previously
presented. Adopting our values for the orbital elements and their associated
uncertainties, we then study the dynamical interaction between the planet and
the dust ring, to check whether the proposed disk sculpting scenario by Fom b
is plausible. We used a dedicated MCMC code to derive the statistical
distributions of the orbital elements of Fom b. Then we used symplectic N-body
integration to investigate the dynamics of the dust belt, as perturbed by a
single planet. Different attempts were made assuming different masses for Fom
b. We also performed a semi-analytical study to explain our results. Our
results are in good agreement with others regarding the orbit of Fom b. We find
that the orbit is highly eccentric, is close to apsidally aligned with the
belt, and has a moderate mutual inclination relative to the belt plane of. If
coplanar, this orbit crosses the disk. Our dynamical study then reveals that
the observed planet could sculpt a transient belt configuration with a similar
eccentricity to what is observed, but it would not be simultaneously apsidally
aligned with the planet. This transient configuration only occurs a short time
after the planet is placed on such an orbit (assuming an initially circular
disk), a time that is inversely proportional to the planet's mass, and that is
in any case much less than the 440 Myr age of the star. We constrain how long
the observed dust belt could have survived with Fom b on its current orbit, as
a function of its possible mass. This analysis leads us to conclude that Fom b
is likely to have low mass, that it is unlikely to be responsible for the
sculpting of the belt, and that it supports the hypothesis of a more massive,
less eccentric planet companion Fom c.Comment: 17 pages, 15 figures, accepted for publication in Astronomy \&
Astrophysic
