On the theory of the oblateness of the sun

In this paper we first emphasize why it is important to know the successive zonal harmonics of the Sun's figure with high accuracy: mainly fundamental astrometry, helioseismology. planetary motions and relativistic effects. Then we briefly comment why the Sun appears oblate, going back to primitive definitions in order to underline some discrepancies in theories and to emphasize again the relevant hypotheses. We propose a new theoretical approach entirely based on an expansion in terms of Legendre's functions, including the differential rotation of the Sun at the surface. This permits linking the two first spherical harmonic coefficients (J(2) and J(4)) with the geometric parameters that can be measured on the Sun (equatorial and polar radii). We emphasize the difficulties in inferring gravitational oblateness from visual measurements of the geometric oblateness, and more generally a dynamical flattening. Results are given for different observed rotational laws. It is shown that the surface oblateness is surely upper bounded by 11 milliarcsecond. As a consequence of the observed surface and sub-surface differential rotation laws, we deduce a measure of the two first gravitational harmonics, the quadrupole and the octopole moment of the Sun: J(2) = -(6.13+/-2.52) x 10(-7) if all observed data are taken into account, and respectively, J(2) = -(6.84 +/- 3.75) x 10-7 if only sunspot data are considered, and J(2) = -(3.49 +/- 1.86) x 10(-7) in the case of helioseismic data alone. The value deduced from all available data for the octopole is: J(4) = (2.8 +/- 2.1) x 10(-12). These values are compared to some others found in the literature.