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Astron. Astrophys. 345, 1027-1037 (1999)

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Appendix A: calculation of the a1 coefficient as a function of the leakage matrix for l = 1 fitting the Fourier spectra

According to Eq. (1), the observed Fourier spectra for [FORMULA], taking into account only the leakage within the multiplet, is given by:

[EQUATION]

where [FORMULA] is the leakage between [FORMULA] and [FORMULA] present in the observed spectra. The index that indicate the mode degree and the order n are omitted. By analogy, the assumed leakage matrix for [FORMULA] will be:

[EQUATION]

and, neglecting the noise covariance, the expected covariance matrix (Eq. 2) is equal to:

[EQUATION]

The variances [FORMULA] will be represented by a Lorentzian profile (Eq. 3):

[EQUATION]

where [FORMULA] is the power amplitude, [FORMULA] is the full linewidth at half maximum, [FORMULA] is frequency subtracting the `unperturbed' mode frequency and [FORMULA] is the rotational splitting coefficient as defined by Eq. (4). Assuming the probability density function of the observed Fourier spectra to be a multivariate normal distribution, the mode parameters will be found minimizing the logarithm of the likelihood of Eq. (32) in Paper I. Substituting the above equations for [FORMULA] into the logarithm of Eq. (32):

[EQUATION]

The sum will be over a frequency interval large enough to characterize the Lorentzians (Eq. A4) and centered at the central frequency of the multiplet. The process of finding the minimum of S means that any partial derivative of S with respect to any mode parameter must be 0. It is impossible to find an analytical solution that solves the set of equations involving the partial derivatives. Nevertheless, we have assumed that all other parameters are much less sensitive to the variation in [FORMULA] than the splitting [FORMULA]. The variation of the amplitudes, linewidth and central frequency using different values for [FORMULA] are proved to be very small in the performed Fourier fittings (Sect. 3.2) and are neglected. The only functions of [FORMULA] in Eq. (A5) are [FORMULA] and:

[EQUATION]

The partial derivative of S becomes:

[EQUATION]

If the discrete summation over the frequency bins is replaced by an integral, the fitting is made at frequency intervals much larger than the true splitting ([FORMULA]) and the observed power spectrum ([FORMULA]) is replaced by its mean over several realizations (i.e. its Lorentzian profile which has the correct rotational splitting [FORMULA]), it is easy to show that:

[EQUATION]

Substituting these equations into Eq. (A7) and making it equal to zero:

[EQUATION]

An expression of [FORMULA] as a function of [FORMULA] is given by:

[EQUATION]

Note that when [FORMULA], [FORMULA] will be equal to [FORMULA] which is the maximum of the function. The function is symmetric around [FORMULA] only when [FORMULA] and could be approximated by a parabola. Otherwise, it is highly asymmetric (Fig. A1).

[FIGURE] Fig. A1. Splitting coefficient [FORMULA] as a function of [FORMULA] given by Eq. (A10) for [FORMULA] nHz. The continuous line is for [FORMULA] and the dashed line is for [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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