  Context. Increasing our knowledge of the interior structure, composition and
density distribution of exoplanets is crucial to make progress in the
understanding of exoplanetary formation, migration and habitability. However,
the directly measurable mass and radius values offer little constraint on
interior structure, because the inverse problem is highly degenerate. Therefore
there is a clear need for a third observable of exoplanet interiors. This third
observable can be the $k_2$ fluid Love number which measures the central mass
concentration of an exoplanet. Aims. The aims of this paper are (i) to develop
a basic model to fit the long-term radial velocity and TTV variations caused by
tidal interactions, (ii) to apply the model to the WASP-18Ab system, and (iii)
to estimate the Love number of the planet. Methods. Archival radial velocity,
transit and occultation timing data are collected and fitted via the model
introduced here. Results. The best model fit to the archival radial velocity
and timing data of WASP-18Ab was obtained with a Love number of the massive
($\sim 10 M_\mathrm{Jup}$) hot Jupiter WASP-18Ab: $k_{2,Love} =
0.62^{+0.55}_{-0.19}$. This causes apsidal motion in the system, at a rate of
$\sim0.0087\pm0.0033^\circ / \mathrm{days} \approxeq 31.3\pm11.8$
arcseconds/day. When checking possible causes of periastron precession, other
than the relativistic term or the non-spherical shape of the components, we
found a companion star to the WASP-18 system, named WASP-18B which is a
probable M6.5V dwarf with $\sim 0.1~M_\odot$ at 3519 AU distance from the
transit host star. We also find that small orbital eccentricities may be real,
rather than an apparent effect caused by the non spherical stellar shape.

Earth and Planetary Astrophysics