  Of all the solar fundamental parameters (mass, diameter, gravity at the
surface,...), the gravitational moments have been quite often ignored in the
past, mainly due to the great difficulty to get a reliable estimate. Even
though the order of magnitude of the solar quadrupole moment $J_2$ is now known
to be $10^{-7}$, its accurate value is still discussed. Indeed, the expansion
in multipoles $J_{(l,~ l = 2, ...)}$ of the gravitational potential of a
rotating body affects the orbital motion of planets at a relativistic level. We
will recall here the recent progresses made in testing General Relativity
through the contribution of the first solar quadrupole moment. Using the
Eddington-Robertson parameters, we recall the constraints both on a theoretical
and experimental point of view. Together with $\gamma$, which encodes the
amount of curvature of space-time per unit rest-mass, the Post--Newtonian
Parameter $\beta$ contributes to the relativistic precession of planets. The
latter parameter encodes the amount of non-linearity in the superposition law
of gravitation. Even though in principle, it would be possible to extract $J_2$
from planetary ephemerides, we observe that it is significantly correlated with
other solution parameters (semi-major axis of planets, mass of asteroids...).
Focusing on the $J_2$ correlations, we show that in general, when ~$\beta$ and
~$\gamma$ are freed, the correlations ~[$\beta, J_2$] and ~[$\gamma, J_2$] are
$\approx$ 45\% and $\approx$ 55\% respectively. Moreover, all the planetary
dynamics-based values are biased by the Lense--Thiring effect, which has never
been modeled and solved for so far, but can be estimated to $\approx$ 7\%. It
is thus possible to get a good estimate of the solar quadrupole
moment:$1.66\times10^{-7}$$\leq$$J_2$$\leq$$2.32\times10^{-7}$.
