  Love numbers measure the reaction of a celestial body to perturbing forces,
such as the centrifugal force caused by rotation, or tidal forces resulting
from the interaction with a companion body. These parameters are related to the
interior density profile. The non-point mass nature of the host star and a
planet orbiting around each other contributes to the periastron precession. The
rate of this precession is characterized mainly by the second-order Love
number, which offers an opportunity to determine its value. We collected all
available radial velocity (RV) data, along with the transit and occultation
times from the previous investigations of the system. We supplemented the data
set with 19 new RV data points of the host star WASP-19A obtained by HARPS.
Here, we summarize the technique for modeling the RV observations and the
photometric transit timing variations (TTVs) to determine the rate of
periastron precession in this system for the first time. We excluded the
presence of a second possible planet up to a period of ~4200 d and with a
radial velocity amplitude bigger than ~1 m/s. We show that a constant period is
not able to reproduce the observed radial velocities. We also investigated and
excluded the possibility of tidal decay and long-term acceleration in the
system. However, the inclusion of a small periastron precession term did indeed
improve the quality of the fit. We measured the periastron precession rate to
be 233 $^{+25}_{-35}$''/. By assuming synchronous rotation for the planet, it
indicates a k2 Love number of 0.20 $^{+0.02}_{-0.03}$ for WASP-19Ab. The
derived k2 value of the planet has the same order of magnitude as the estimated
fluid Love number of other Jupiter-sized exoplanets (WASP-18Ab, WASP-103b, and
WASP-121b). A low value of k2 indicates a higher concentration of mass toward
the planetary nucleus.
