The Sun Asphericities: Astrophysical Relevance
  Of all the fundamental parameters of the Sun (diameter, mass, temperature...), the gravitational multipole moments (of degree l and order m) that determine the solar moments of inertia, are still poorly known. However, at the first order (l=2), the quadrupole moment is relevant to many astrophysical applications. It indeed contributes to the relativistic perihelion advance of planets, together with the post-Newtonian (PN) parameters; or to the precession of the orbital plane about the Sun polar axis, the latter being unaffected by the purely relativistic PN contribution. Hence, a precise knowledge of the quadrupole moment is necessary for accurate orbit determination, and alternatively, to obtain constraints on the PN parameters. Moreover, the successive gravitational multipole moments have a physical meaning: they describe deviations from a purely spherical mass distribution. Thus, their precise determination gives indications on the solar internal structure. Here, we explain why it is difficult to compute these parameters, how to derive the best values, and how they will be determined in a near future by means of space experiments. 
J-P. Rozelot, S. Pireaux, S. Lefebvre and T. Corbard