We will now describe our final results of the fitting of the low degree () p-mode parameters using the GONG data and demonstrate that their determination is better than others found in the literature. In summary, we fitted the Fourier spectra instead of the power spectra for all l, except for . The leakage and the noise covariance matrices were calculated as described in Sects. 2.2 and 2.3, respectively. The variance of the spectrum of a single mode, , was represented by a Lorentzian profile (Eq. 3). In our preliminary results (Rabello-Soares & Appourchaux 1998), we used the `cleaned' spectra for all modes with , 4, 5 and 6. However, here we are going to use the `cleaned' spectra only for modes inside the leakage frequency range, as discussed in Sect. 3.3.
By estimating accurately the low degree p-mode parameters using high spatial resolution data, we enabled the creation in the future of a homogeneous set of frequencies and rotational splittings for estimating the solar structure and rotation profile.
The frequency results are shown in Table 1. We have identified 139 p modes with (81 with ) with an estimate of the uncertainties on the fitted frequency of less than 0.2 µHz. Frequencies in parenthesis were positively identified using the collapsogramme technique described by Appourchaux (1998) but could not be fitted properly due to their narrow linewidths; the error bars are rough estimates taking into account the frequency resolution and the number of modes.
Table 1. Frequencies and errors in µHz as measured by GONG. Frequencies in parenthesis were positively identified but could not be fitted properly.
We have compared our frequencies with those of the GONG project for a three-month time series starting on 6 June 1996, which is essentially contained within the period used here. In that analysis, the power spectra was fitted instead of the Fourier spectra and the leakage between the elements of a multiplet was not taken into account (Hill et al. 1996). Its number of identified modes for is very small. Out of the 55 frequency determinations in common with ours for , 93 coincide within 3 (using combined errors), 82 within 2 and 55 within 1.
We have compared our frequencies with those of the GOLF instrument on board the SOHO satellite (Lazrek et al. 1997) which performs full-disk integrated light velocity measurements. They used an eight-month observational period, of which four overlap with the GONG observations used here. Their frequencies are in good agreement with ours. Out of the 72 frequency determinations in common with ours for , 100 coincide within 3 (using combined errors), 93 within 2 and 71 within 1, meaning that the scatter is due to a normally distributed noise. For , the estimate of the uncertainties on the fitted GOLF frequency is similar to ours for the same modes but for , 2 and 3, they are significantly larger than ours, approximately times larger. This is due to a combination of different signal-to-noise ratios, modes detected and length of observing time. In addition, the GOLF frequency errors are the average of statistical errors in the fittings done for different authors. Last but not least, the time series for in the GONG data analysis is calculated in a different way than the other degrees; it is obtained through the spatial average of each solar image, in such a way that, for , GONG simulates an integrated light instrument. In the fitting, the modes were fitted simultaneously. This is necessary not only for modes with µHz where the modes overlap the modes but also for modes with µHz. Otherwise, the background noise level will be overestimated due to the modes, causing a decrease in linewidths. The fitted parameters are consistent with those found using spherical harmonic decomposition; however, they were not used in this work.
We also compared our results with the LOWL data, which is a one-site instrument. The data we used was velocity measurements (125 square pixels), starting in February 1995 and it is one-year long, thus having a six-month overlap with the GONG data used here (Tomczyk et al. 1995). Out of the 87 LOWL frequency determinations for , 93 coincide within 3 (using combined errors), 72 within 2 and 45 within 1.
Despite the good agreement between the frequency determination, the GONG frequencies are systematically lower than GOLF and LOWL frequencies. The weighted average of the frequency differences between our work and LOWL is µHz (87 observations) and between our work and GOLF µHz (72 observations). The LOWL observational period is slightly further away from the solar minimum, which occurred at the beginning of 1996, than the GONG observations used here. Its frequency difference is in agreement with a decrease in frequency of µHz between solar maximum and minimum found by several authors (Hill et al. 1991). Even though the GOLF observational period is inside the solar minimum, its frequency difference is barely significant.
4.2. Splitting coefficients
Fig. 6b shows the splitting coefficients results found for up to 6 as a function of the lower turning point position. The coefficients for and high-order modes are very scattered and have larger error bars than the others. The small number of elements in the multiplets and the very small value of the splitting, of the order of 10 times the uncertainty in the mode frequency, impose difficulties on the splitting determination. This scattering in the splitting coefficients determination is also present in the analysis of a very low spatial resolution instrument, LOI/SOHO (Appourchaux et al. 1998c), and in integrated light instruments.
The splitting for the three-month GONG series is poorly determined. This is probably due to the short period of observation and the incorrect treatment of the mode leakage. However, it agrees with our results towards higher degree modes. The weighted averages over n of sidereal coefficients are: nHz for (3 observations), nHz for (4 observations), nHz for (6 observations), nHz for (9 observations), nHz for (7 observations) and nHz for (7 observations), which could be compared with the results found here (see Table 2).
Table 2. Weighted averages over sidereal splitting coefficient values of the modes with the same l and their errors (in nHz).
When compared with GOLF determinations (Lazrek et al. 1997), our error bars are smaller and our measurements are similarly scattered for modes. However, for and 3 modes, our data are less scattered and the error bars are much smaller.
Finally, the estimated linewidth (Eq. 3) of the modes obtained from the fit are shown on Fig. 9. The power-law behaviour of the linewidths is clearly visible for all degrees at the low and high frequency limits: (28 observations) for µHz and (41 observations) for µHz from an error-weighted fit to the data. At intermediate frequencies, there is a small depression in the linewidth around 2900 µHz which is predicted by theory (Balmforth 1992) and first detected by Fröhlich et al. (1997).
© European Southern Observatory (ESO) 1999
Online publication: April 28, 1999