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Astron. Astrophys. 317, 919-924 (1997)

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3. The near surface magnetic perturbation determined from BBSO data

Libbrecht and Woodard (1990) extracted the a -coefficients of Eq. (1) for l =5-60 from their observations at the Big Bear Solar Observatory. They noted that the coefficients changed significantly between the activity minimum in 1986 and 1988 when the Sun was in the rising phase of solar activity. They also found a corresponding sizeable shift in the centroid frequencies (m =0). They argued that the source of these changes must reside very near the solar surface because the size of the effect scaled with mode inertia. They also noted that the source of the perturbation is strongest in the active latitudes. We (Dziembowski and Goode 1991) used these data in a search for the signature of a strong, internal magnetic field. We found the signature of a buried quadrupole toroidal field near the base of the convection zone. We emphasized that in the 1988 data the dominant contributor to the frequencies, by far, was the near surface magnetic field. Our analysis allowed us to localize the source of this effect to within some 1 Mm of the base of the photosphere. A similar conclusion was reached by Goldreich et al. (1991) from their analysis of the centroid shifts.

The data of Libbrecht and Woodard went through several preliminary iterations of which they generously provided us the results. The BBSO data we use in this paper are from their final iteration. The four sets of observations were made in the middle of 1986, 1988, 1989 and 1990, respectively. Each of these data sets spans l from 5 to 140 and covers a three month period. From these data, we find no evidence for an internal magnetic field. Of the aforementioned conclusions arising from use of earlier releases of the BBSO data, only this one is changed.

Our purpose here is to derive information about the near surface perturbation, and use it to evaluate its effect on low-l mode frequencies. To that end, we use Eq. (7) of Dziembowski and Goode (1991), where we drop the term due to internal magnetic field. This equation may be written in the following form,

[EQUATION]

where the [FORMULA] describe the contribution of the second order effect of rotation to the fine structure and next term describes the near surface perturbation. The I 's are the mode inertia and L is again [FORMULA], and the [FORMULA] are what we want. We assume a common normalization of the eigenfunctions fixing their amplitude at the base of the photosphere. The contribution of the second order effect of rotation is calculated using the seismically determined rotation rate in the interior(which is determined from the odd-a coefficients). For the p-modes, our ignorance of the rotation in the core is irrelevent here because the second order effect of rotation is dominated by the distortion it causes which arises mostly in the outer layers. Furthermore, its overall effect is small. Only for the lowest-l values and for [FORMULA] does the second order effect of rotation play any role, and even there it is within the errors.

The form of the near surface contribution to even-a coefficients given in Eq. (2) is easy to justify. At this point, we do not specify the perturbation, but we assume that it is located well above the lower turning point of the modes considered. For the set of modes we consider, the lower turning point is located below 0.95 [FORMULA]. Therefore, the radial eigenfunctions are l -independent. Second, we assume that the perturbing force may be expanded in Legendre polynomials, [FORMULA]. The expansion starts with k =1 because the spherically symmetric part does not contribute to the a -coefficients, and we truncate it at k =3 because we have data up through [FORMULA]. With these assumptions, the variational expression for frequency perturbation [FORMULA] -mode (e.g. Eq. (2) of Dziembowski and Goode 1991) leads to

[EQUATION]

The [FORMULA] -coefficients are [FORMULA] -dependent because the radial eigenfunctions are strong functions of the mode frequency. The angular integral defining the term Q is given by

[EQUATION]

The l -dependence of the perturbation enters through this coefficient and I. Modes with higher l 's are trapped in the outer layers and naturally have lower inertia. The m -dependence arises only from the Q -coefficients which are polynomials of order [FORMULA] in m. In the limit of [FORMULA], we have

[EQUATION]

where

[EQUATION]

In this limit, [FORMULA] contributes only to [FORMULA] coefficient, Goode and Kuhn(1990). This approximation is applicable to the set of modes we consider. It doesn't work perfectly for the lowest l -values in the BBSO data, however, its accuracy is well within the quoted errors for those modes. The approximation given by Eq. (5) implies that the centroid shifts have the same form as a spherically symmetric perturbation. Therefore, if the structural inversion uses only modes for which this approximation is valid, then the near surface perturbation does not corrupt the inversions. However, for probing the core, we need low l modes for which Eq. (5) is invalid.

Our objective is to find the functions, [FORMULA]. To this end, we discretize each [FORMULA] function by a three term Legendre expansion of argument [FORMULA] where [FORMULA] and [FORMULA] are the maximum and minimum frequencies in the data and [FORMULA]. We determine the coefficients by a least squares fitting. The resulting [FORMULA] 's are shown in Fig. 3. In the uppermost panel, we show [FORMULA] describing the centroid shift relative to 1986. The quantity [FORMULA] -the spherically symmetric part of the magnetic perturbation-cannot be observationally determined. We emphasize that only the [FORMULA] coefficient differs significantly from zero in the 1986 data, and that this term corresponds to a quadrupole toroidal field geometry for the near surface perturbation.

[FIGURE] Fig. 3. From top panel to bottom [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are shown as a function of mode frequency. [FORMULA] is calculated with respect to the 1986 BBSO data. The other [FORMULA] 's are calculated for 1986, 1988, 1989 and 1990 BBSO data. The normalization of the radial displacement amplitude at the base of the photosphere was chosen to 3 [FORMULA]. With such a normalization, the mode inertia in Eq. (2) is of order unity for modes of frequency near 3 mHz.

One may compare the [FORMULA] 's for 1986 and 1988 in Fig. 3 to their counterparts in Fig. 1 of Dziembowski and Goode (1991). In doing so, we stress that the [FORMULA] 's defined in the latter paper incorporated the [FORMULA] factor. Beyond this, there are only slight differences between the two sets of results.

The changes in frequency are linked to the activity cycle. Woodard et al. (1991) have observed that the frequency changes are correlated with the surface magnetic activity on a timescale of months. That leaves no room for doubt that the frequencies reflect magnetic changes occuring in the outer layers. Precisely how this occurs is unclear. Whether it is due to a changing fibril structure or whether there is a role for a subsurface velocity field remains to be seen. Henceforth, we denote the near surface perturbation varying with activity as NSPA. We make this distinction not only to emphasize the fact that we are studying this effect, but also that the NSPA is different from any other near surface effect, like that due to vigorous convection. The [FORMULA] -dependence of [FORMULA] results from the fact that the radial eigenfunctions in the outer layers vary significantly with frequency. The fact that the frequency dependence in all the [FORMULA] 's is as weak as it is in Fig. 3 argues for a localization of the perturbing agent very close to the photosphere where the radial eigenfunctions have been uniformly normalized.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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