6 Linear combinations

Winget et al. (1994) and Vuille et al. (2000) show that most of the periodicities are in fact linear combination peaks of the main peaks (eigenmodes). Combination peaks are what we call peaks in the FT whose frequencies are equal to the sum or difference of two (or more) the the $\ell=1$ or 2 mode frequencies. The criteria for selection of the combination peaks was that the frequency difference between the combination peak and the sum of the "parent mode'' frequencies must be smaller than our resolution, which is typically around 1 $\mu$Hz. The last column of Table 11 shows the frequency difference.

Figure 14: Pre-whitened peaks in the 2000 transform.

For example, only 28 of the more than 180 peaks in Winget et al. (1994) are $\ell=1$ modes; the rest are combination peaks up to third order (i.e., three modes are involved). The $\ell=1$ modes lie in the region 1000 to 2500 $\mu$Hz, and are identified as modes k=18 to 8. In the 1994 data set analyzed by Vuille et al., combination peaks up to 4th order were detected. In the 2000 data set we identify combination peaks up to 6th order, and most if not all remaining peaks are in fact linear combination peaks, as demonstrated in Table 11 and is shown in the pre-whitened FT of the 2000 data (see Fig. 14). Here too, we use the nomenclature k^a, for example 15^-, to represent a subcomponent with m=-1 of the k=15 mode.

The so-called $\ell=2$ mode at $1255~\mu$Hz, as well as k=17 and k=15 modes, have subcomponents, but probably they are not different m value components, and are caused, most likely, by amplitude modulation. We say this because the frequency splittings are drastically different than in previous data, and for the $\ell=2$ mode, there are more than 5 possible subcomponent peaks present. We did not do an exhaustive search for all of the possible combination peaks up to 6th order in the Fourier transform, as we only took into account the peaks that had a probability smaller than 1/1000 of being due to noise, and studied if they could be explained as combination peaks.

Brickhill's (1992) pulsation-convection interaction model predicts, and the observations reported by Winget et al. and Vuille et al. agree, that a combination peak involving two different modes always has a larger relative amplitude than a combination involving twice the frequency of a given mode (also called a harmonic peak). Wu's (2001) analytical expression leads to a factor of 1/2 difference between a combination peak with two modes versus a harmonic peak, assuming that the amplitudes of both eigenmodes are the same. Vuille et al. claim that the relatively small amplitude of the k=13 mode in 1994 is affected by destructive beating of the nonlinear peak ( $2\times 15-18$) and that the k=16 mode amplitude is affected by the (15+18-17) combination peak. It is noteworthy that the peak at 1423.62 $\mu$Hz is only 3.52 $\mu$Hz from k=15, so it might be the 15^- mode. However, the previously identified 15^- was 6.7 $\mu$Hz from it, and we consider the $1423.62~\mu$Hz peak to be either a result of amplitude modulation of the k=15 mode or yet another combination peak.

We note that the wealth of combination peaks and their relative amplitude offers insight into the amplitude limiting mechanism and would be worthy of the considerable theoretical and numerical effort required to understand it.