2.2 Differential rotation and rotational splitting
The Sun’s rotation lifts the degeneracy between modes of the same and different ,
resulting in “rotational splitting” of the frequencies as waves propagating with and against
the direction of rotation (prograde and retrograde) have higher and lower frequencies. To first
order, the splitting is proportional to the rotation rate multiplied by
Figure 7 shows a typical acoustic spectrum of GONG data at . The effect of the
rotation causes the ridges at each to slant away from the axis; closer examination reveals that
the ridges have an S-curve shape that arises from the differential rotation, and also shows the ridge
structure due to leakage, which will be discussed below in Section 2.3.
Figure 7: Spectrum for in the plane (top) and detail (bottom) of a single ridge
(radial order) showing the curvature due to differential rotation and the multiple-ridge structure
arising from spherical harmonic leakage.
Because modes of different values sample different latitude ranges, with the sectoral ()
modes confined to a belt around the equator and the zonal or modes reaching to the poles, as
illustrated in Figure 8, we can measure the rotation as a function of latitude.
Figure 8: Spherical harmonic patterns for : left, ; center, ; right, .
A given () multiplet consists of frequency measurements if each () spectrum is
analyzed separately, though some fraction of these frequencies may be missing in any given data set. This
amount of data was somewhat unwieldy in the early days of helioseismology. It is therefore common to
express as a polynomial expansion, for example,
where the basis functions are polynomials related to the Clebsch–Gordan coefficients by
(Ritzwoller and Lavely, 1991). Indeed, in many analysis schemes coefficients of the expansion are derived by
fitting directly to the acoustic spectrum and the individual frequencies are not measured. This approach can
improve the stability of the fits, perhaps at the cost of imposing systematic errors. Early work used
Legendre polynomials; however, most modern work uses either Clebsch–Gordan coefficients or the
Ritzwoller–Lavely formulation, which come closer to being truly orthogonal for the solar rotation problem.
Only the odd-order coefficients encode the rotational asymmetry, while the even-order coefficients contain
information about the structural asphericity. Roughly speaking, the coefficient describes the rotation
rate averaged over all latitudes, and the and higher coefficients describe the differential