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)×10⁻⁷ if all observed data are taken into account, and respectively, J ₂=−(6.84±3.75)×10⁻⁷ if only sunspot data are considered, and J ₂=−(3.49±1.86)×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)×10⁻¹². These values are compared to some others found in the literature. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1005238718479     

A new outlook on the `Differential Theory' of the solar Quadrupole Moment and Oblateness

The modeling of the quadrupole moment J ₂ and of the oblateness , two key solar parameters, derives from the development in successive spherical harmonics of the gravitational potential. These harmonics are representative of the shape of the Sun, by studying the local distortion of the internal layers, under their distribution of mass and velocity. The first aim of this paper is to study, over the radius r and the colatitude , the structure of the internal layers of the Sun through a geometrical approach, considering J ₂ and under a differential form. The second aim is to determine their theoretical values, after integration over r and , taking the best available models of density and rotation into account constrained by helioseismic data. The novelty of our approach lies in the analysis of the profiles of the two above-mentioned solar parameters, under differential form, from the core to the surface. This analysis allows us to comply with the physical processes located in the transition regions, namely the tachocline and maybe a new subsurface layer which could be called the leptocline. The profiles of tildeJ ₂ show two sharp decreases, which are directly connected to the shear layers located at 0.7 R and beneath the surface. The profiles of tilde show five changes of curvature, which seem to be connected to solar processes, such as the matter circulation flows, seismic events or the storage of the magnetic field, phenomena taking place in the transition regions. These sets of profiles allow us to propose a configuration scenario composed of a double layer. Moreover, as a result of the theoretical determination of tildeJ ₂ and tilde, the values at the surface of the quadrupole moment and of the oblateness can be deduced, which are 1.60×10^–7 and 8.77×10^–6, respectively. As a result of an analysis of available data, we may admit J ₂=(2.0±0.4)×10^–7. The theoretical computations of J ₂ and at the surface will be confronted in the near future with the values measured in space by means of the Picard microsatellite. This mission should permit one to measure at the same time both the total solar irradiance and the latitudinal diameters in any position angle (after removing the passing spots or faculae at the limb).