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Of the seven periods found, the 2.71-2.44, 1.86-1.73, and 1.37-1.29-hr pairs are each separated by 1 day~^ in frequency, and are a combination of a true period and an alias not resolved by the CLEAN algorithm; the 2-D periodogram of Figure 2.20 shows that the members of an alias/true period pair never occur together at any single wavelength sample. The alias detection is a result of the overlapping of aliasing from similar periods which artificially raise an alias above the true period peak. Thus CLEAN locks onto this false peak giving an erroneous result. Inspection of Figure 2.20 shows that the 2.44-, 1.86- and 1.37-hr periods were more frequently found by the c l e a n algorithm (particularly in the Si ill lines). The integrated power spectrum (Figure 2.21) shows that the 3.34- and 2.44-hr periods are dominant. The 2.44-hr peak has twice the power of the 2.71-hr period, and the 1.86- and 1.37-hr periods have somewhat more power than the 1.72- and 1.29-hr periods respectively. However, on their own. Figures 2.20 and 2.21 are not conclusive in resolving the true/alias pairs. Thus, in spite of the extensive observations and the use of the

CLEAN algorithm, the aliasing is still so severe that there is ambiguity in determining

the true periods and their associated aliases.

In an attem pt to resolve this, a ‘pre-whitening’ procedure was used to remove sequentially the variation responsible for each period (and the associated aliases) in the temporal domain. For the time series at each wavelength sample, the amplitude and phase of the unambiguous 3.34-hr period was determined from the power spec­ trum. Corresponding sinusoids were then subtracted from the time series. Power spectra of the pre-whitened time series confirmed th at the 3.34-hr period was suc­ cessfully removed, together with its associated aliases, but, as expected, showed the 2.71- or 2.44-hr periods as the stronger in different parts of the line profiles.

To gain an insight into this problem, four-period fits to a number of representative time series in the line profiles of He I A4471Â and S im A4552Â, both close to the line centre and line wings, were produced, by first adopting the 2.71-hr period, and then the 2.44-hr period (together with the 3.34-hr period and whichever other two periods were found in the c l e a n spectrum). The fits were used to generate noise-free artificial data, which were time-sampled using the observational sampling. For the line-centre time series, power spectra of the artificial data showed that the 2.71-hr

period had the greatest power, regardless of whether the input sinusoid had a 2.44- or 2.71-hr period. For time-series spectra towards the wings of the line profiles, c l e a n correctly identified either the 2.44- or 2.71-hr period specified. It was therefore concluded th at the 2.71-hr period is most probably an alias onto which power leaks from the true 3.34- and 2.44-hr periods. Using the same test, it was concluded that the 1.73-hr period is an alias of the 1.86-hr period. Since the amplitude of the 1.37- and 1.29-hr period were so weak, this procedure was unable to distinguish between them with any confidence, and these two periods have been carried through the rest of the analysis. An example of the aliasing of the 2.44-hr period is displayed in Figure 2.22(a) and (b).

2 .4.4 A m p litu d e and phase o f th e d e te c te d p eriod s

Testing the CLEAN technique using groups of sinusoids and the observational time samphng showed th at CLEAN removed power from some of the input frequencies (see Table 2.7 and Appendix B). This lead to a significant underestimate of the amplitude of the input signal based on measuring the power peaks in the final

CLEANed periodograms. This was also the conclusion of a recent study of Fourier

techniques on irregularly sampled data by Carbonell et al. (1992). Estimation of the amplitude of individual frequencies from the dirty periodogram proved more accurate. However, in a similar study of e Per, Hahula (1992) has suggested that the amplitudes of multiple-input frequencies are more consistently derived from the

CLEANed rather than the dirty periodogram.

For the five periods, the peak amplitudes, in a range ±0.004 h r“ ^ centred on the peak frequency in each of the 435 dirty periodograms, were identified and recorded together with the phases (given by arctan(Aim/ARe); where Aim and Arc are the

imaginary and real amplitudes of the dirty power spectrum) of the best-fit sinusoids with respect to the mean time of observation. These quantities, along with the mean spectrum, are plotted in Figure 2.23. Two significance levels have also been plotted. These were computed by using a Monte-Carlo simulation technique. For a single iteration, each wavelength-sample time series was randomised and a power spectrum computed. As in the real data, the highest peak in the dirty spectrum was found in the range ±0.004 h r“ ^ centred on the frequency of the candidate period

o X k (U

o

(b)(i)

d

d

F req u en cy (hr

Figure 2.22: (a) and (b). An example of the aliasing in the line-centre time series of Hel X4471Â. The dirty periodogram of the data time series (a)(i), when C L E A N e d (a)(ii) gives the periods identified using Table 2.5. P2 is an alias of the 2.44-hr period. Using the input sinusoid data of Table 2.7 (fit to observed data), results in the dirty (b)(1) and C L E A N e d (b)(ii) periodograms. Despite the explicit input of the 2.44-hr period, the 2.71-hr period is recovered

Table 2.6: Comparison of significance levels Power (xlO - ') Power (xlO - ') (100 iterations) (1000 iterations) Period (hrs) 68% 95% 99% 68% 95% 99% 3.339 3.49 4.84 4.93 3.69 5.15 6.15 2.425 3.99 5.18 5.53 3.84 5.34 6.12 1.859 3.92 5.43 5.81 3.76 5.14 6.06 1.366 3.67 5.84 6.62 3.78 5.06 6.62 1.292 3.81 4.88 5.84 3.77 5.12 5.93

(Table 2.5); the 2nd and 33rd highest peaks from the amplitude spread of 100 such iterations were chosen as the 99% and 68% confidence levels for each period at the given wavelength sample respectively.

A comparison of a 100- hnd 1000-iteration solution is given in Table 2.6. The 99% amplitude of 1000-iterations is slightly higher than that of the 100-iteration case, although the 68% confidence levels for both cases are comparable. This increase is accredited to the randomising process failing to remove enough of the periodicity during an iteration, with the result that increasing the number of iterations raises the 99% confidence level. This 99% level is only raised by an average of 3% of the 100-iteration value for each of the periods. Therefore, the 100 iteration solution represented a good approximation of that of the additional magnitude solution and so justifies the adoption of the smaller sample confidence levels.

Figure 2.23 shows the dominance of the 3.34-, 2.44-, and 1.86-hr periods in the He I A4471Â, and Sim AA4552, 4567, 4574Â lines, and weak variability in Mgii A4481Â. Weak variability is also detected for unidentified lines at A4530Â for both the 3.34- and 2.44-hr periods, and at AA4506, 4538Â for the 3.34-hr period only. The 1.37- and 1.29-hr periods are very weak, but present for the Hel and Sim lines. For aU the periods there is virtually no variability present in the H ell A4541Â, or N III AA4511, 4515, 4518Â. This confirms the TVS results (Figures 2.6 and 2.7) and the 2-D periodogram in Figure 2.20.

Period = 3.339—houra P e rio d - 2 .4 3 & - h o u n P e rio d - 1.8 5 9 - h o u r # P e rio d = 1.3 6 6 - h o u r s P e rio d = 1 2 9 2 - h o u r # 00 g O) 4460 4460 4500 4520 4540 4560 4580

H eliocentric Wavelength (Â)

F igure 2.23: Sinusoidal semiamplitudes (SSA) and phases (P) with respect to the mean

tim e o f observation. The SSA are 10~^ those shown. The significance levels are plotted: 68% (dashed line); and 99% (dotted line)

tred around AA4530Â and 4538Â. For the former wavelength, it is suspected that either A1 III A4529Â or (more likely) NII A4530Â is responsible, but no obvious can­ didate can be identified for the signal at A4538Â. It is unlikely to be Hell A4541Â, since other lines seem to show symmetric amplitude profiles about the line centre. It is therefore concluded th at there is no significant signal in N HI AA4511, 4515, 4518Â, or in Hell A4541Â.

The variation of the amplitude across a line profile for each period for the 3.34- , 1.86-, and 1.37/1.29-hr periods show ‘boxy’ amplitude profiles; however, for the 2.44-hr period, the amplitude strength is concentrated in the line wings, possibly indicating a larger horizontal velocity component. This would fit easily in an NRP interpretation, since the pulsation can be subject to both horizontal and vertical motion in the corotating frame. However, for a RM interpretation there would have to be a mechanism which allows one of the sequences of ‘spots’ to possess a sizable horizontal velocity component. A possible explanation could be differential rotation, or th at the origin of the 2.44-hr Ipv is above the photosphere orbiting close to the star.