6 Analysis of the l-dependence of the frequency shifts

The LOWL instrument (Tomczyk et al. 1995), located in Mauna Loa, Hawaï, is a Doppler imager based on a Potassium Magneto-Optical Filter, and it has been collecting solar observations for more than six years. With the installation of a similar experiment at the Observatorio del Teide, a new network called ECHO (Experiment for Coordinated Helioseismic Observations) intends to continue the solar observations for a complete solar cycle (Tomczyk et al. 2000) with an increased duty cycle. Recently, six years of data have been re-analyzed through a new pipeline producing mode parameters for low and intermediate degrees (Jiménez-Reyes 2001). We have used the mode frequencies given by this analysis to compare the $\ell$- and frequency-dependence of the frequency shift at low and intermediate degrees.

Figure 6 shows the normalized frequency shift using LOWL observations for $\ell = 1$ up to 99. It has been performed using those modes between 2.5 and 3.7 mHz, as we did for low degree. The inverse mode mass, which was calculated as well for each one of the modes fitted and then averaged in the same way, follows remarkably well the results. The figure shows as well the $\ell$-dependence of low degree p-mode frequency shifts for Mark-I found in Sect. 3. The general trend confirms the $\ell$-dependence of the frequency shift, the sensitivity at high degree ( $\ell = 100$) being almost twice to that of low-degree. We notice that the LOWL $\ell = 2$and Mark-I $\ell = 0$, 2 are in good agreement and significantly lower than the inverse mode mass curve. However the LOWL data error bars in $\delta\nu$ for $\ell = 1$ and $\ell = 2$ overlap significantly and the small value for $\ell = 0$, 2 could either be due to particularly small $\delta\nu$ in $\ell = 2$ but also in $\ell = 0$. It will therefore be important to check this results with independent observations in the future. This is important because if a geometrical effect purely related to the integrated disk measurements affect the even modes measured by Mark-I, this should not be seen for the LOWL resolved measurements. If confirmed, this lower value of the frequency shift for $\ell = 2$ may therefore have a physical origin and be the signature of a localized perturbation.

Figure 7: Normalized frequency dependence of LOWL data averaged for modes from $\ell = 1$ up to 99 in intervals of 150 $\mu$Hz. The results for very low degree shown in Fig. 5 are also plotted. The solid lines represent the best fit of the inverse mode mass to both data sets.

Figure 7 reproduces the frequency dependence for low degree shown in Fig. 5 together with the frequency shift averaged for $\ell$'s between 1 up to 99 in intervals of 150 $\mu$Hz obtained from LOWL data. Again, the best fit to the inverse mode mass is shown. The ratio between the two slopes in the inverse mode mass fits is 0.74 and is nearly equal to the ratio between the mean mode mass calculated for both mode sets (0.75) showing that the $\ell$-dependence of the frequency shift is again well described by the $\ell$-dependence of the inverse mode mass. However, although the sensitivity is higher for higher degree modes (LOWL data), the fit is worse than for low degree (Mark-I data); moreover, in this later case, while roughly consistent with the sizes of the error bars, the scatter seems to be organized as an oscillation on top of the fitted line. The higher sensitivity of the frequency shift at high degree seems to be in contradiction with what Régulo et al. (1994) pointed out, but in agreement with recent analysis of Chaplin et al. (1998). While all results point towards the existence of a perturbation confined close to the surface, there is still no convincing evidence of another cause that could exist deeper down as suggested by Régulo et al. (1994). On the other hand, the oscillation, also pointed out by Anguera Gubau et al. (1992), that seems to be present in the results for low degree, has the same period ($\approx$400-450 $\mu$Hz) that the one that is clearly found at higher frequencies (see Fig. 5 bottom) and deserves further attention and confirmation.

Finally, we notice that the frequency dependence of the frequency shift obtained from the low degree modes of LOWL data is in very good agreement with the result plotted here using Mark-I data. This, added to the fact that the 6 years of LOWL data were covered also by Mark-I observations, make us very confident in the validity of our analysis of the $\ell$-dependence using both instruments.