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

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1. Introduction

Solar oblateness is a fundamental property of the Sun and its value is a direct input for some astrophysical computations. This is mainly the case for precise determination of planetary orbits and specially for the motion of Mercury and other minor planets, such as Icarus. The solar oblateness, if one is able to measure its value with a high accuracy, could be used to determine the Eddington-Robertson parameter ([FORMULA]) in the Parametrized-Post-Newtonian theories of gravity. This oblateness, which is linked to the internal structure, via its gravitational and rotational potentials, permits one to access certain physical properties of the layers below the surface. This is certainly the most interesting feature which has not yet been too much approached. Such features may concern at first the borderlines of the different layers such as the tachocline or other subsurface sections. Moreover, the oblateness of the Sun is directly linked to the quadrupole moment, a quantity which gives us information on the solar potential distortion. In this paper, the behavior of the oblateness [FORMULA], from the core to the surface, is carefully investigated, to understand how each layer of the Sun is affected by the differential rotation, and how each of these layers reacts to the perturbation of the potential. The shape of the profiles of [FORMULA], computed as a function of the radius and of the latitude, leads us to determine its theoretical value at the surface.

Several studies and observations have been undertaken since 1966 to evaluate the solar oblateness. The measurements began with the Princeton Solar Distortion Telescope, and yielded the first evaluation of the ellipticity of the Sun. Dicke and Goldenberg found a value for [FORMULA] as large as [FORMULA], a revisited value, deduced from the original data (Dicke and Goldenberg, 1967), taking into account all corrections for the seeing effects (Dicke and Goldenberg, 1974). In 1968, Goldreich and Schubert showed that the theoretical maximum solar oblateness, consistent with the stability of the Sun, is [FORMULA]. They emphasized the fact that this value was consistent with the gradients of mean molecular weight, as calculated in standard solar models at that time. In 1975, Hill and Stebbins gave an intrinsic visual value of the difference between equatorial and polar diameters: [FORMULA] milli arcsecond (mas) or [FORMULA]. They extracted this value from observations influenced by an excess brightness, monitored at the same time to evaluate the necessary corrections. Then, in 1983, Kislik considered what effect the oblateness of the Sun might have on the astrometric (radar, optical) determinations of planetary orbits. He looked for the residuals of parameters determined from various kinds of observations, when the equations of planetary motion do not allow for the solar oblateness. In this study, he estimated that [FORMULA] should be in the range [FORMULA] (Kislik, 1983). This same year, Dicke, Kuhn and Libbrecht made new measurements of the Sun's shape at Mount Wilson, with an improved version of the Princeton Solar Oblateness Telescope, which has been used in 1966. The measurements of the solar oblateness yielded a value [FORMULA] mas or [FORMULA] (Dicke et al., 1985), the upper bound being only half of that observed in 1966: [FORMULA] mas. The data obtained in 1984 by Dicke et al. (1986) lead to significantly lesser values than those obtained in 1983: [FORMULA] mas or [FORMULA]. The 1985 data yielded a value [FORMULA] mas or [FORMULA], which comes in between the 1984 and 1983 values of the solar oblateness (the 1984 value being the lowest). The authors concluded (Dicke et al., 1987) that the quantity [FORMULA] may vary with the 11.14 yr period of the solar cycle. In 1986, assuming that the Sun is in hydrostatic equilibrium, Bursa (1986) gave a new estimation of the Sun's oblateness range: [FORMULA], where the upper limit requires a heavy core and the lower one corresponds to a nearly homogeneous body. In 1990, on the basis of radar observations, Afanas'eva et al. determined the quadrupole moment of the Sun by methods of celestial mechanics and deduced a range for the oblateness of the Sun from the theory of the figures of celestial bodies, assuming that the Sun rotates as a rigid body. They found: [FORMULA]. Motivated by the suggestion of Dicke, Kuhn and Libbrecht, that the magnitude of the oblateness might be a function of the solar cycle, Maier, Twigg and Sofia gave, in 1992, their preliminary results of the solar diameter from a balloon flight of the Solar Disk Sextant (SDS) experiment. They found [FORMULA] for the solar oblateness, but [FORMULA]30[FORMULA] offset from the polar-equator position. Additional studies, based upon flights in 1992 and 1994 (Lydon and Sofia, 1996), lead to a measured oblateness of respectively [FORMULA] and [FORMULA], indicating little or no variation. At the same time, from July 1993 to July 1994, Rösch et al. conducted observations at the Pic-du-Midi observatory by means of a scanning heliometer, which operates by fast photoelectric scans of opposite limbs of the Sun. Over this period, Rozelot & Rösch (1997) reported [FORMULA] mas or [FORMULA], a value averaged from observations made in 1993 and 1994. New campaigns, conducted in 1995 and 1996, lead respectively to [FORMULA] mas and [FORMULA] mas (Rozelot, 1997). At last, Kuhn et al. (1998), found with the data obtained on board of the SOHO spacecraft, that the Sun's shape and temperature vary with the latitude in an inexpectedly complex way. The autors concluded also that the solar oblateness itself presents no strong evidence of varying with the solar cycle.

From the above reported data, two conclusions can be drawn, one concerning the observational values and the other one, the theoretical values. As far as the first one is concerned, a general agreement seems to exist on the fact that [FORMULA] does not exceed [FORMULA], the averaged value being a bit below, around [FORMULA]. It is not yet clear if a solar cycle dependence is real or not; in such a case the amount of variation would be no more than [FORMULA]. All these values are of importance to constrain the solar models and it is not yet obvious if the global shape of the Sun follows a perfect ellipsoid or not. This point will be discussed later on (Sect. 4.4). As far as the theoretical values are concerned, one could accept from the above quoted data that [FORMULA] would lie around [FORMULA]. It is then of high interest to determine [FORMULA] using one of the most recent solar models of mass and density combined with an up-to-date rotational model, which depends both on the latitude and on the distance to the rotation axis. This rotational model derives from helioseismological observations of p-mode rotational frequency splitting, deduced from measurements made on board the SOHO spacecraft. Observational data of high quality obtained from this satellite allowed Pijpers (1998) to determine an excess of the equatorial diameter on the polar diameter by some 0.017 arcsecond (Note, 1998).

In the following sections, we present the theoretical method of calculation and the models used to obtain the oblateness [FORMULA] (Sect. 2). Next, we give the results and their interpretations (Sect. 3). Finally, we discuss the relation between the behavior of [FORMULA] and the particular aspects of the solar interior (Sect. 4).

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Online publication: March 17, 2000