An outlook on the estimate of the solar quadrupole moment from relativistic gravitation contributions

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}$.