An observable consequence of the solar rotation is that the azimuthal degeneracy of global modes of oscillation is raised. The rotational splitting is defined by :

where denotes the frequency of the mode of degree l, radial order n and azimuthal order m.

This figure gives some recent results obtained by various groups for the low l degree rotational splitting frequencies. The value of the splitting corresponds to an average over the given radial order range assuming that all the splittings for a given degree l are measuring the same quantity. The weighting coefficients used to compute the mean splitting values have been used in the inversion process too.

For low rotation the frequency shifts are given by:

where is the unknown rotation rate versus depth and latitude, , is the colatitude and r the normalized solar radius. The kernels depend on the model and the eigenfunctions of the mode which are assumed to be known exactly (e.g. Christensen-Dalsgaard & Berthomieu 1991).

We search for the unknown rotation as a linear combination of piecewise polynomials, projected on a tensorial product of B-spline basis (Corbard et al., 1996). We take into account the observational errors (with standard deviation ) and a regularization term to avoid the large spurious variations of the solution induced by the ill-conditioned inversion problem. Thus we search coefficients that represent the projection of the rotation rate on the chosen B-splines basis by minimizing:

The regularization parameters and and functions and are used to define the trade off between the resolution and the propagation of the input errors. We can choose any derivative of the rotation rate. This leads to different behavior of the rotation especially at locations where the observed modes are not enough to constrain the solution. For the results discussed here we choose , , i=2 and , , j=1 .

The inversion depends on the number of piecewise polynomials, on the order of the spline functions and on the distribution of the break points of these polynomials. In what follows, the distribution of these points along the radius has been chosen according to the density of the turning points of the considered p-modes for a given dataset.

The radial and latitudinal full widths at mid height ( and ) of these kernels provide an estimation of the spatial resolution of the inversion at a point . It increases with lower regularizing term, at the expense of larger errors on the solution.

The radial and latitudinal resolution reached at at the equator are respectively and . Nevertheless, we obtain a lot of additional peaks localized at the surface. Generally, a well peaked kernel can be obtained only if a large amount of cancelations eliminates the dominant contribution of the rotational kernels near the surface. In this inversion the cancelation is not completely done showing that a better inversion should require observation of high degree modes which are able to give informations about zones near the solar surface. We note however that the peaks shown near the surface remain unresolved with the grid used for the plot and that their contribution to the whole kernel is very small.

Although the two kernels at are well localized, the two ones at are not distinguishable and are both localized at the equator and . This result shows that our inversion is not able to detect a latitudinal dependency of the rotation rate even if one exists at . The solution obtained at all latitudes below 0.3 is an estimation of the rotation rate in the equatorial plane. A more important problem arise from the fact that the averaging kernels calculated at and below this depth are still localized between 0.25 and 0.28. This indicates that the true rotation rate at a target location below 0.2 contributes only for a few part in the calculation of the inferred rotation rate at this location.

The solutions we obtain show that the different values given by observers for the low l splittings lead to different behaviours of the inferred rotation rate in the core. Therefore the inversion is sensitive to the actual observational differences. Nevertheless the shape of the averaging kernels corresponding to solar core layers makes the interpretation of results in term of the true rotation rate very difficult. For example, the averaging kernel calculated at presents important oscillations up to . Therefore the interpretation of the value of the rotation rate obtained at depends strongly on the behavior of the rotation rate in the radiative interior. We note however that the integral of the averaging kernel between and is null. Thus, if we suppose that the rotation rate is constant in the radiative interior and down to , we can interpret the value obtained at as a weighted averaged of the true rotation rate between 0.1 and the weighting function being the corresponding part of the averaging kernel. Nevertheless although the solid rotation found between and is thought to be reliable, its extension down to 0.25 is more speculative and some little gradients in these zones can modify significantly the interpretation of the value of the rotation rate found at and deeper.

A better understanding of the rotation of the core should require a better agreement between the different analysis of the different observations. However, this work shows some limits in the utilization of a 2D RLS code to probe the rotation of the core. Some other global inversion techniques as SOLA should be helpful by searching explicitly averaging kernels without any negative part. Such kernels, even larger than those obtained in this work should be useful to complete the interpretation of our results in the core.