Astron. Astrophys. 355, 365-374 (2000)

3. Determination of the oblateness

Differential Eq. 13 was solved to obtain the function of the potential and to deduce through Eq. 22 the behavior of , shell by shell, for different latitudes: 0, 15, 30, 45, 60, 75 and 90 degrees. The numerical integration step is determined by the model. The number of steps has been taken as 220; to test the robustness of the solution, this sample has been extended up to 2100 steps and no significant difference has been found. For the numerical applications, the adopted values for the solar parameters are those given by Allen (1976): and .

3.1. Study of the successive layers considered as thin shells

In this study, we consider simultaneously two types of rotation for the Sun: a global rigid rotation and a differential rotation computed with the above mentioned rotation law. The corresponding curves for a Sun splitted in successive shells, are shown in Fig. 1.

Fig. 1. Profile of the oblateness of the Sun splitted in successive shells and numerically computed both for the rigid rotation case and the differential rotation case for latitudes 0, 15, 30, 45, 60, 75 and 90 degrees. Around , the passage of the transition zone is clearly visible.

The profiles, which are computed for the rigid rotation and for the differential rotation , globally evolve identically up to , then diverge up to the surface. Looking in more detail, the curve corresponding to is shifted from the others as early as . The curves present a first variation around and diverge. The curves, corresponding to 0, 15, 30, , 45 and 60 (in decreasing order of the latitudes) increase and present a second very slight variation near . The curves corresponding to 75 and 90 decrease from up to , then increase until a variation of the gradient around , and those corresponding to 75 and 90 decrease up to , then increase. From , each curve increases and reaches a maximum around (Fig. 2). In their decreasing, they all present two changes of curvature, located around and . Only the curve corresponding to w=1 is globally increasing (Fig. 1), but also presents two changes of curvature located around and .

Fig. 2. Zoom of the right side of the previous global profiles (Fig. 1) for the differential rotation case at 0, 15, 30 (Left Axis) and 45 degrees (Right Axis) to show two of the main changes of curvature located around and .

Above 0.999, a change of curvature is visible with a minimum near 0.99991 at latitudes 0, 15, 30, 45, and 60 degrees. For latitudes 75 and 90, the curves decrease without any clearly visible variation. Nevertheless, these ones are not completely linear around , where a minimum appears. In the same range, the curve corresponding to increases presenting a very slight minimum near . The oblateness being directly linked to the solar potential, it will be modified in a region where this potential is affected by flows or events.

These detected variations can be linked to the borderline of the shear layer which governs the transitions between the different zones within the Sun. In the following discussion, we present a synthesis of actual results and we propose a scenario for the structure of the solar layers.

3.2. The radial integration case

This computed oblateness behavior is consistent with the differential rotation law expressing that the higher the rotation rate of a shell, the higher the amount of . The integration over each shell yields the behavior of for a body of radius that can be expressed at each given latitude (Fig. 3).

Fig. 3. Profiles of the oblateness of the Sun integrated on successive shells and depending upon the latitudes.

where x' is associated to the radius of the shells.

These integrated curves diverge near and present for some of them a slight variation located approximately at . Then, they increase to reach a stable value near the surface.

To compare this theoretical model with available observations, we also present the profiles (Fig. 4) which illustrate the latitudinal variation of the solar semi diameter observed by means of the solar astrolabe at CERGA in France (Laclare et al., 1996) and in Santiago, Chili (No"el, 1999). These data are compared with our results. The observational results of F. Laclare show that the largest ellipticity is obtained for 0 (by extrapolation), 30, 45 and 90, and the more oblate regions are located at 60 and 75. The observational results of E. Noïl show that the largest ellipticity is obtained for 0 (by extrapolation), 15, 45 and 60, and the more oblate regions are located at 30, 75 and 90. The latitudinal order of our curves presents the order in which the solar regions are the most elliptic (large curvature) towards the most oblate regions. The largest ellipticity is obtained for 0, then 15 and 30. The Sun is less elliptic at 75 and 90, which indicates a possible bump around the royal zones centered on 60.

Fig. 4. Deviations from the mean solar semi diameter, deduced from the observations obtained by means of a solar astrolabe by F. Laclare (1996) at CERGA in France and by E. Noïl (1999) in Santiago, Chili.

3.3. The latitudinal integration case

The sum of the radial integrated oblatenesses over all heliographic latitudes yields the total behavior of (Fig. 5).

Fig. 5. Profiles of the integrated oblatenesses of the Sun. The plain curve is computed, taken into account the differential rotation by means of helioseismic data. The dashed curve is obtained in the case of rigid rotation .

The model used stops at , but we can extrapolate the curve of the integrated profile, without a too large error on the final result. So, the oblateness of the Sun can be determined as

Comparing the profile of for the differential rotation case with the profile of in the rigid rotation case, where we find , we see that the oblateness of the Sun is increased by the differential rotation of a quantity equal to (Fig. 6).

Fig. 6. Zoom of the right side of the previous profiles (Fig. 5) in the differential rotation case and in the rigid rotation case, to determine the value of oblateness at the surface in these two cases. The plain curve permits us to determine for the oblateness in the differential rotation case and in the rigid rotation case. Note the break which appears at .

© European Southern Observatory (ESO) 2000

Online publication: March 17, 2000
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