2. Forward problem and hypothesis
2.1. Basic equations
The Sun is oscillating simultaneously in many thousands of global acoustic modes. The observation of these modes at the solar surface and the knowledge of their sub-surface properties are the basis by which helioseismology can sound the interior of the Sun.
Each mode can be described by three integers: the degree l, the azimuthal order m and the radial order n. In a spherically symmetric non-rotating star the eigenfrequencies of the modes are independent of m. The rotation of the Sun induces a preferred axis of symmetry and the frequency difference between westward and eastward propagating waves on the solar surface contains the signature of the global rotation of the Sun.
The rotation period for the Sun (about 1 month) is very long compared to the periods of the observed p-modes (about 5 minutes), thus we can use a linear perturbation theory to predict the effect of rotation on the p-modes. According to this theory and under the assumption that the effect of the magnetic field is negligible, the difference between the frequency of a mode with azimuthal order m and the frequency that this mode would have in a non-rotating (but otherwise identical) star is given in terms of the eigenfunctions of the non rotating star (e.g. Hansen et al. 1977; Christensen-Dalsgaard & Berthomieu 1991).
If we denote by and the radial and horizontal displacement of the fluid from its equilibrium position, the displacement has the form:
where are the spherical polar coordinates defined from the solar rotation axis and 's are spherical harmonics. The so-called frequency splitting can be written as a weighted average of the unknown rotation rate :
with and where the full kernels derived from the first order perturbation theory are given by:
where are normalized Legendre polynomials, is the density and .
In first approximation, we can neglect terms with first and second derivatives of the rotation rate (by partial integration with respect to the colatitude of Eq.(2)) assuming that one has smooth variation in latitude and the so-called rotational kernel reduces to (Cuypers 1980):
The radial part of the rotational kernel is the same as in the 1D inversion problem where the rotation is supposed to be latitudinal independent i.e. (e.g. Gough 1981). The kernel Eq. (3) is symmetric about the equator and the factor of two is introduced by the assumption that the rotation rate has a similar symmetry property i.e. . The functions , are determined by solving the differential equations describing the motion of a self-gravitating fluid body in a standard solar model (Unno et al. 1989).
We note that the approximation Eq. (5) of the rotational kernel includes a term which does not appear in Sekii's approximation (Sekii 1993) and which becomes of significant importance compared to the term only for the low l. For higher degree modes, this kernel reduces to Sekii's approximation and the terms of Eq. (3) that are neglected have been shown by Pijpers and Thompson (1996) to be small compared to except near the inner turning point of the modes. Therefore their contribution to the integral Eq. (2) is negligible for the observed modes. Nevertheless, while this approximation of the rotational kernel simplifies the problem and decreases the number of calculations, our work takes into account the full kernel and this should have a significant effect especially if f- or even g-modes become available. We note however that, for the present data, using the full or the approximate kernel leads to the same solution in the zones that are sounded by the observed modes.
The object of all the 2D inversion codes is to infer the rotation rate versus depth and latitude from the observed splittings by inverting the integral relation Eq.(2).
2.2. Boundary conditions
2.2.1. At the surface
Some direct observations of the rotation at the solar surface are available and one may want to force the inferred rotation to match the observed surface rotation. The sidereal rotational frequencies are obtained as a function of latitude at the solar surface by different techniques such as the Doppler shift of photospheric spectral lines or by tracking sunspots, small magnetic features or supergranulation cells (see the review by Schröter (1985)). The values which are derived are within a few percent but they lead to a different variation of the solar rotation as a function of latitude during the solar cycle. These differences could be explained by the different depths where indicators are anchored but a complete interpretation of these observations is strongly related to a better theoretical understanding of the interaction between rotation, convection and magnetic fields.
The rotation of surface layers has been determined spectroscopically from standard techniques used at Mount Wilson by Snodgrass (1984). In this work we use the rotation results from Doppler velocity measurements made at the Mount Wilson 150 foot tower telescope between 1967 and 1984 and related by Snodgrass & Ulrich (1990). The sidereal plasma rotation rate averaged over the entire period is given by:
All the magnetic tracers are believed to represent the rotation of deeper layers. The observation of small magnetic features leads to a rotation rate slower than the rotation rate of the supergranular pattern but faster than the rotation rate of sunspot groups or the plasma (Komm & Howard 1993). Thus it gives a mean value of the rotation rates estimated by the different indicators and we also use these data to study the sensitivity of the inversion to different surface constraints. The fit of the main sidereal rotational rate of small magnetic features that we use is given by Komm & Howard (1993) from the analysis of magnetograms taken with the NSO Vacuum Telescope on Kitt peak between 1975 and 1991:
The surface rates (Eqs. (6), (7)) are averaged over a long period and are not contemporaneous with the LOWL observations. Moreover they describe the rotation of layers that are not necessarily strictly the solar surface. This might introduce spurious effects in the inversion results if these observations were used as strong constraints. For this reason, we choose to take into account these data in a more flexible way by introducing a parameter as explained in Appendix B and discussed in Sect. 4.2.3.
2.2.2. At the center
At the limit the rotation rate has no latitudinal dependence. Thus the functional space where we search the rotation rate must generate only functions which are in agreement with the physical condition:
This condition insures the regularity of the solution at the center and is easy to insert in the inversion process as discussed in Appendix A.
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998