3. Fitting the GOLF data
The GOLF series used covers the 805 days between April 11, 1996 and June 25, 1998, with more than 99% continuity. The GOLF raw data have been converted to velocities using the calibration techniques described by García (private communication) prior to carrying out an FFT transform on the entire series, with a sampling time of 80 seconds. Only the GOLF detector PM2 has been used in this analysis.
The classical fitting technique in this domain consists of fitting a simple Fourier power spectrum, often using codes derived from those offered by Appourchaux et al. (1998). For such a spectrum, statistics are applicable and the fitting is then by the maximum likelihood method .
The entire analysis was limited to the spectral range 2000 to 3600 . Our objective requires very high precision, which we cannot obtain reliably outside of this range. Above 3600 the modes overlap seriously. Below 2000 , the signal to solar background ratio is smaller and, more importantly, the number of bins which contribute to the line is small due to the smaller linewidth. These limitations are particularly critical for the asymmetry parameter.
A critical point is the choice of the spectral window used for fitting the p-modes profiles, i.e. how much of the spectrum we fit at a time. Some workers choose to fit the entire p-mode spectrum, arguing that this is the only way to correctly take account of the effect of the far wings from distant modes (Roca Cortes et al. 1998). This can be done only if the model profile assumed for the modes is valid everywhere in the window or if the residual errors are negligible compared with errors due to the stochastic noise. It also assumes that the overlapping far wings are additive on a scale of spectral power, which is not evident. As the Nigam formula used in this paper is obtained by a Taylor expansion around the eigenfrequency of the mode (Nigam & Kosovichev 1998), we adopted the approach of fitting only those close multiplets that have an evident overlap, i.e. fitting was carried out over limited spectral ranges with the neighbouring =0 and =2 grouped together, as were the =1 and =3. The choice of a precise window is discussed in Sect. 5 and corresponds to a trade off between the effects of the stochastic noise and of the validity of the formula. This alternative to a fit of the whole spectrum has the cost of considering the contribution of other resonances as a flat "background". In all cases, there exists an unknown continuous component due to solar noise (plus perhaps instrumental noise) for which arbitrary parameters are used. In the present work, we performed tests and simulations to quantify the errors due to our assumptions. These include that the background is represented by a constant value, the slope being assumed to be zero over the limited range of window. All of the components in the fitted range were assumed to have the same asymmetry factor, but the linewidth was allowed to vary with the degree. The relative amplitude of the mode components was taken as a fixed empirical function of m, which was derived by taking a mean over observations from different orders. For =2, we use (m=2)/(m=0)=1.7 and for =3, (m=3)/(m=1)=1.7 (Lazrek, private comunication).
Here, the splitting is defined as the mean for each multiplet of the separation between the modes m=1.
The uncertainty quoted here for each parameter determined is the statistical error obtained from the fitting program, using appropriately the covariance matrix of the fit. We performed Monte Carlo simulations (based on a method developed by Fierry Fraillon et al. 1997) which confirm the reality of these uncertainties.
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000