  The surface rotations of some red giants are so fast that they must have been
spun up by tidal interaction with a close companion, either another star, a
brown dwarf, or a planet. We focus here on the case of red giants that are spun
up by tidal interaction with a planet. When the distance between the planet and
the star decreases, the spin period of the star decreases, the orbital period
of the planet decreases, and the reflex motion of the star increases. We study
the change rate of these three quantities when the circular orbit of a planet
of 15 M$_{J}$ that initially orbits a 2 M$_\odot$ star at 1 au shrinks under
the action of tidal forces during the red giant phase. We use stellar evolution
models coupled with computations of the orbital evolution of the planet, which
allows us to follow the exchanges of angular momentum between the star and the
orbit in a consistent way. We obtain that the reflex motion of the red giant
star increases by more than 1 m s$^{-1}$ per year in the last $\sim$40 years
before the planet engulfment. During this phase, the reflex motion of the star
is between 660 and 710 m s$^{-1}$. The spin period of the star increases by
more than about 10 minutes per year in the last 3000 y before engulfment.
During this period, the spin period of the star is shorter than 0.7 year.
During this same period, the variation in orbital period, which is shorter than
0.18 year, is on the same order of magnitude. Changes in reflex-motion and spin
velocities are very small and thus most likely out of reach of being observed.
The most promising way of detecting this effect is through observations of
transiting planets, that is, through{\it } changes of the beginning or end of
the transit. A space mission like PLATO might be of great interest for
detecting planets that are on the verge of being engulfed by red giants.
