Astron. Astrophys. 355, 743-750 (2000)

6. Bias introduced when fitting a Lorentzian profile to an asymmetric resonance

6.1. Determination of a bias uncertainty

The uncertainty on a bias (frequency, line width or splitting) cannot be calculated from those corresponding to the quantities delivered by the two kinds of fits, because these uncertainties are not dependent, the stochastic excitation being the same in the two cases. To compute a bias uncertainty, we perform Monte Carlo simulations: for each radial order and mode pair, we simulate 200 stochastic realisations of a Nigam's profile. For each realisation, we do a Nigam and a Lorentzian fit. Then, for each mode parameter, the histogram of the difference between the two results gives the uncertainty we require. Note that the uncertainty on a bias can be significantly smaller than the uncertainty on the absolute value of the mode parameters (see Figs. 7 to 12).

 Fig. 7. The frequency difference between asymmetric and Lorentzian fittings as a function of frequency and mode degree. The errors bars are computed by Monte Carlo simulations as described in the text.

 Fig. 8. The linewidth difference between asymmetric and Lorentzian fittings as a function of frequency and mode degree.

 Fig. 9. Determination of the linewidth using Lorentzian profiles.

 Fig. 10. Determination of the linewidth using the Nigam formula.

 Fig. 11. The splitting difference between asymmetric fitting and Lorentzian fitting, as a function of frequency and mode degree.

 Fig. 12. Determination of the splitting (synodic) using the Nigam formula. Results for higher frequencies have larger errors bars and are not shown here.

6.2. Bias on the frequency

The frequency difference between symmetric and asymmetric fits is plotted versus the frequency in Fig. 7 for the different modes. There is a systematic shift for all the modes: the frequency infered from fitting a Lorentzian profile is lower than those using an asymmetric profile. The mean value of the bias is about 42 nHz for =0, 85 nHz for =1, 54 nHz for =2 and 24 nHz for =3. Not only the bias but also its variation with frequency is an important input for the inversion because the difference between frequencies has more impact on the deduced solar structure than frequencies themselves. The difference between the behaviour of =1 and the other degree modes can be understood and reproduced in simulations (Lorentzian fitting on pure Nigam profiles). =1 behaves as an isolated mode (see Sect. 5.1) and is influenced not only by B but predominantly by . For 1, the influence of the neighbouring mode dominates. The frequencies inferred from asymmetric fitting are given in Table 1.

Table 1. Frequency and errors in µHz derived from asymmetric fitting for modes =0, =1, =2 and =3 with radial orders from n=12 to n=25. The observation duration is T=805 days.

6.3. Bias on the linewidth

The width difference between symmetric and asymmetric fitting depends on the mode degree and on the frequency (see Fig. 8): the widths inferred from a Lorentzian profile are lower than those from an asymmetric profile for degree =1 and =0 and inversely for =2. For =3 the sign is frequency dependent. The same simulations than those mentioned in the previous subsection show that the influence of the components on each other in a multiplet is dominant for the sign of the bias. The evolution with the radial order is, as for the frequencies, influenced by both the line width and the neighbouring mode.

The importance of this is such that when the asymmetry is taken into account, the dispersion of the width with the degree is considerably reduced as shown by Figs. 9 and 10.

6.4. Bias on the splitting

The difference between determinations of the rotational splitting based on Lorentzian or asymmetric profiles depends on the degree (see Fig. 11) and the difference increases with increasing frequency for modes =1 and =2. As for the width, the sign of the bias depends on the degree, which we also confirmed by the simulations. The Fig. 12 shows the results obtained and Table 2 gives mean values over frequency for the two kinds of fits. Note from Table 2 that, as we would expect, the uncertainty induced by the stochastic excitation is not reduced when using an asymmetric fit, but the dispersion with the degree is somewhat reduced by a more valid fit. In particular, the relatively low value for the splitting of the =2, observed here in the Lorentzian case and reported in some publications (Lazrek et al. 1997, Gizon et al. 1996), is no longer obtained in the asymmetric case. Moreover, the use of the Nigam formula permits a good determination of the splitting farther in the high frequency range than with a Lorentzian formula.

Table 2. Average and standard error in nHz of the synodic splitting over n when using asymmetric or Lorentzian fitting.

© European Southern Observatory (ESO) 2000

Online publication: March 9, 2000