A&A 377, 297-311 (2001)
DOI: 10.1051/0004-6361:20011056
S. Zieba - J. Masowski - A. Michalec - A. Kuak^{}
Astronomical Observatory, Jagiellonian University, ul Orla 171, 30-244 Kraków, Poland
Received 18 July 2000 / Accepted 11 July 2001
Abstract
Three types of observations: the daily values of the solar radio flux
at 7 frequencies, the daily international sunspot number and the daily Stanford
mean solar magnetic field were processed in order to find all the periodicities
hidden in the data. Using a new approach to the radio data, two time series were
obtained for each frequency examined, one more sensitive to spot magnetic fields,
the other to large magnetic structures not connected with sunspots. Power
spectrum analysis of the data was carried out separately for the minimum (540 days
from 1 March 1996 to 22 August 1997) and for the rising phase (708 days from
23 August 1997 to 31 July 1999) of the solar cycle 23. The Scargle periodograms obtained,
normalized for the effect of autocorrelation, show the majority of known periods
and reveal a clear difference between the periodicities found in the minimum and
the rising phase. We determined the rotation rate of the "active longitudes"
in the rising phase as equal to 444.4
4 nHz (
).
The results indicate that appropriate and careful analysis of daily radio data at
several frequencies allows the investigation of solar periodicities generated in
different layers of the solar atmosphere by various phenomena related to the periodic
emergence of diverse magnetic structures.
Key words: Sun: activity - Sun: radio radiation - Sun: magnetic fields - Sun: rotation
The study of periodicities in different solar data is important for the understanding of solar magnetic activity. For over a decade many authors have reported various periods other than those at 11 yr and 27 days, of which the one near 154 days is the best known (Rieger et al. 1984; Dennis 1985; Bogart & Bai 1985; Lean & Brueckner 1989; Bai & Cliver 1990; Dröge et al. 1990; Pap et al. 1990; Carbonell & Ballester 1990, 1992; Bai & Sturrock 1991, 1993; Kile & Cliver 1991; Bouver 1992; Oliver et al. 1998; Ballester et al. 1999). This investigation was designed to show that accurate daily radio observations can be very useful for studying the solar periodicities generated in different layers of the solar atmosphere, and in different phases of the solar cycle. We used a new approach to the daily measured radio fluxes which allow us to separate to some extent the radio emission generated in the strong magnetic fields of active regions from that emitted by large but weaker magnetic structures. All the solar data from the minimum and the rising phase of solar cycle 23 were processed separately using the Scargle periodogram technique (Scargle 1982). The statistical significance levels of all the periods found were estimated through the FAP (false alarm probability - the probability that the given periodogram value z is generated by noise), as well as by a Monte Carlo approach to the periodograms obtained (Horne & Baliunas 1986; Bai 1992b; Özgüç & Ataç 1994; Oliver & Ballester 1995).
Papers published by Das & Chatterjee (1996); Das & Nag (1998, 1999) presented some periods in radio data without any discussion about their significance level. Our analysis of radio observations however, gives the majority of the known periods, reveals a clear difference between periodicities observed in two phases of the solar cycle and shows that daily measured radio fluxes at various frequencies are very useful for the systematic study of solar periodicities observed in the different layers of the solar atmosphere.
Linear model F_{i} = B + h(ISN)_{i} | |||||||
frequency in MHz | 405 | 810 | 1215 | 1620 | 2800 | 4995 | 8800 |
B- basic component [su] | |||||||
h- radio flux production | |||||||
% of the data variance | 71.0 | 78.6 | 81.9 | 85.7 | 87.5 | 83.5 | 61.6 |
explained by the model | |||||||
Boltzman formula model | |||||||
frequency in MHz | 405 | 810 | 1215 | 1620 | 2800 | 4995 | 8800 |
parameters of A_{1} [su] | |||||||
the Boltzman A_{2} [su] | |||||||
sigmoidal [days] | |||||||
formula [days] | |||||||
- radio flux production | |||||||
% of the data variance | 81.0 | 91.8 | 91.4 | 91.5 | 92.5 | 87.5 | 79.3 |
explained by the model |
It is trivial to form the time series from the daily values of the sunspot numbers and the mean magnetic field, but if we want to use the radio data for the study of magnetic activity, we must first eliminate from the observed daily flux (free from bursts) the thermal emission which comprises the majority of the daily measured value. This can be done through a procedure in which the flux observed on day t_{i}, F_{i} is divided between the thermal, almost constant component (often called the basic component - B) and the slowly varying component (SVC), whose value changes every day and is generated by mechanisms dependent on the magnetic field. It is usually assumed that this component is proportional to a certain daily index of activity, for example (ISN)_{i} and then the daily flux F_{i} can be described by the following linear formula: Fi = B + (SVC)_{i} = B + h (ISN)_{i}, where h is the production of the radio flux from the spot with ISN = 1 (Krüger 1979).
The assumption that B is constant over the large time intervals is rather strong
and cannot be accepted, especially at a time when the level of radio flux rises
systematically. Here we propose a new approach in which the basic component
B is not constant over the given time interval but changes every day according to
the Boltzman sigmoidal formula:
Figure 1: Daily values of the radio flux at 810 MHz observed from Cracow. The horizontal solid line shows the constant value of the basic component B = 39.8 su resulting from the linear formula, while the dashed curve shows values of the basic component calculated from the best fitted parameters, A_{1}=38.9 su, A_{2}=62.5 su, days, days according to the Boltzman formula . The division into the minimum and rising phase is also indicated. | |
Open with DEXTER |
Figure 2: a) The cumulative distribution function of the Scargle power for the original, minimum ISN time series. The vertical axis is the number of frequencies whose power exceeds z. The straight line is the best fit to the points for values of power lower than 5. b) The normalized periodogram of the original, minimum ISN time series with FAP significance levels indicated. | |
Open with DEXTER |
Then the observed daily flux , and has a similar interpretation to h. To determine the above parameters we used the observed radio data and daily sunspot numbers over the whole time interval investigated, 1 March 1996-31 July 1999 (1248 days). The best fit values of these parameters are shown in Table 1. The difference between the two models is clearly seen, especially for four frequencies 405, 810, 1215, 8800 MHz. To demonstrate this we present in Fig. 1, as an example, the daily values of the radio flux at 810 MHz observed from Cracow as well as the calculated values of the basic component B and B_{i}. The data in this figure also explain our division into the minimum and the rising phase.
Thus, in our approach to the radio data we create two time series from the
observation at each frequency. The first, the SVC (slowly varying component) time
series, consists of diurnal values calculated as the difference between the daily
observed flux and the daily value of the basic component computed from our model,
(SVC)_{i} = F_{i} - B_{i}. The second, the RRE (radio residual emission) time
series describes the every day difference between the radio observations
and our model of the daily radio flux,
.
Taking the time series
SVC and RRE, we can analyse cyclic variations of those magnetic structures
which modified the observed radio emission. However, the SVC series are more
sensitive to spot magnetic fields, while the RRE series are
sensitive to large magnetic structures not connected with sunspots.
Figure 3: a) The normalised periodogram of the minimum, original SVC 810/0 time series. b), c), ... same as a) but recalculated after successively removing from the original data one, two, and more sine curves having periods with peaks whose FAP values are smaller than 0.5%. In each graph the removed periods are indicated at the top. The dashed lines show FAP significance levels. | |
Open with DEXTER |
The search for periodicities in all the time series was performed by calculating the Scargle normalized periodograms (Scargle 1982).This technique (see Horne & Baliunas 1986) has several advantages over the conventional fast Fourier transformation method and provides, through FAP, a simple estimate of the significance of the height of a peak in the power spectrum. However, the FAP value is easy to calculate only for that time series for which the successive data are independent. In our case, the all analysed time series were prepared from the daily values of the different solar indices which are not independent but correlated with a characteristic correlation time of a week (Oliver & Ballester 1995). Therefore, for all our FAP calculations used to estimate the statistical significance of a peak z, in the Scargle power spectrum, we applied the formula , where N is the number of independent frequencies, k is the normalization factor due to data correlation and z_{m} = z/k the normalized power (Bai & Cliver 1990; Bai 1992b). To determine the normalization factor k for the given time series we followed the procedure described by Bai & Cliver (1990). The key step of this procedure is the choice of a spectral window as well as the number of independent frequencies. Since we would like to analyse naturally limited time series connected with two different phases of the solar cycle over the largest possible range of periods, we took the interval 43 to 1447 nHz (8-270 days) as the spectral window for all the investigated time series. The shortest period, 8 days, is connected with a possible range of correlations of the analysed data (see Tables 2 and 3 where the autocorrelation coeficients with the lag = 1 day and lag = 7 days are given). The longest periods, 270 days, results from the actual length of the minimum time series equal to 540 days. The number of totally independent frequencies inside the chosen window is given by the value of the independent Fourier spacing, , where T is the time span of the data (Scargle 1982). In the case of our time series, days and nHz for the minimum, while for the rising phase days and nHz. However, de Jager (1987) has shown by Monte-Carlo simulations that the Fourier powers taken at intervals of one-third of the independent Fourier spacing are still statistically independent. Thus, we accepted the numbers 198 and 259 as the numbers of independent frequencies in the chosen window for the minimum and the rising phase time series, respectively. To illustrate the method used for determination of the normalization factor k, we will process one of our time series step by step; as an example we take the ISN time series in the minimum phase. First, we calculate the Scargle power normalized by the variance of the data, (Horne & Baliunas 1986) for all 198 independent frequencies which allows us to construct the graph presented in Fig. 2a. This shows the cumulative number of frequencies for which the Scargle power exceeds a certain value z. Then we fit all values of power z < 5 to the equation , which gives the value of the normalization factor k equal 2.59, since when the formula for FAP reduces to . Finally, we normalize the power spectrum once more by dividing the Scargle power by 2.59 to obtain the normalized periodogram for which FAP values are easily calculated. Therefore, if we substitute z = 30.54, k =2.59, and N = 198 in the FAP formula we get the normalized power and FAP = 0.0015 for the highest peak in the ISN minimum time series periodogram. Figure 2b presents this normalized periodogram of the minimum ISN series together with FAP significance levels obtained from FAP formula with N = 198 and k = 2.59.
The highest | Auto | k | NP | FAP | % | The highest | Auto | k | NP | FAP | % | ||||||
peak at | correlation | [%] | of | peak at | correlation | [%] | of | ||||||||||
Freq. | Per. | lag | var. | Freq. | Per. | lag | var. | ||||||||||
[nHz] | [day] | 1 | 7 | exp. | [nHz] | [day] | 1 | 7 | exp. | ||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||||
ISN | MMF | ||||||||||||||||
0 | 84a | 138 | .839 | .126 | 2.59 | 11.8 | 0.15 | 17.4 | 0 | 858a | 13.5 | .523 | -.187 | 1.56 | 19.0 | 0.00 | 12.6 |
-1 | 419a | 27.6 | .476 | -.079 | 1.54 | 19.0 | 0.00 | 23.8 | |||||||||
-2 | 400a | 28.9 | .390 | -.095 | 1.65 | 13.7 | 0.02 | 30.0 | |||||||||
SVC 405 | RRE 405 | ||||||||||||||||
0 | 418a | 27.7 | .801 | .007 | 2.50 | 19.8 | 0.00 | 21.0 | 0 | 416a | 27.8 | .714 | .014 | 2.25 | 23.4 | 0.00 | 22.6 |
SVC 810 | RRE 810 | ||||||||||||||||
0 | 423a | 27.4 | .905 | .177 | 1.56 | 49.9 | 0.00 | 35.9 | 0 | 418a | 27.7 | .801 | .218 | 1.46 | 42.9 | 0.00 | 26.4 |
-1 | 396a | 29.2 | .895 | .285 | 1.71 | 18.7 | 0.00 | 45.2 | -1 | 152a | 76.2 | .765 | .346 | 1.70 | 27.7 | 0.00 | 40.6 |
-2 | 150b | 77.2 | .883 | .321 | 1.89 | 15.6 | 0.00 | 52.1 | -2 | 381a | 30.4 | .707 | .216 | 2.06 | 15.3 | 0.00 | 48.8 |
-3 | 76b | 152 | .874 | .228 | 2.01 | 14.1 | 0.01 | 57.7 | -3 | 268b | 43.2 | .667 | .242 | 2.07 | 11.8 | 0.14 | 54.2 |
-4 | 529b | 21.9 | .852 | .105 | 2.10 | 11.2 | 0.28 | 61.9 | |||||||||
-5 | 238b | 48.6 | .836 | .164 | 2.11 | 11.5 | 0.20 | 66.0 | |||||||||
SVC 1215 | RRE 1215 | ||||||||||||||||
0 | 424a | 27.3 | .912 | .219 | 1.60 | 34.6 | 0.00 | 26.7 | 0 | 157a | 73.7 | .757 | .271 | 1.72 | 18.8 | 0.00 | 13.7 |
-1 | 80b | 145 | .905 | .297 | 1.72 | 14.8 | 0.01 | 34.5 | -1 | 420a | 27.6 | .718 | .160 | 1.91 | 18.7 | 0.00 | 26.7 |
-2 | 151b | 76.6 | .893 | .207 | 1.79 | 15.0 | 0.01 | 41.8 | -2 | 383b | 30.2 | .683 | .221 | 1.93 | 12.2 | 0.10 | 34.3 |
-3 | 528b | 21.9 | .886 | .112 | 1.90 | 12.1 | 0.11 | 47.5 | -3 | 267b | 43.4 | .649 | .233 | 1.89 | 11.8 | 0.14 | 40.6 |
-4 | 113b | 102 | .873 | .176 | 1.94 | 11.5 | 0.20 | 52.9 | |||||||||
-5 | 498b | 23.2 | .855 | .084 | 1.95 | 11.8 | 0.15 | 57.8 | |||||||||
-6 | 242b | 47.8 | .839 | .129 | 2.18 | 11.8 | 0.15 | 62.5 | |||||||||
SVC 1620 | RRE 1620 | ||||||||||||||||
0 | 425a | 27.2 | .898 | .111 | 2.15 | 23.1 | 0.00 | 25.0 | 0 | 155b | 74.7 | .687 | .197 | 2.00 | 11.7 | 0.16 | 10.6 |
-1 | 420b | 27.6 | .653 | .123 | 2.07 | 11.2 | 0.28 | 19.3 | |||||||||
-2 | 382b | 30.3 | .627 | .159 | 2.12 | 10.8 | 0.40 | 27.2 | |||||||||
SVC 2800 | RRE 2800 | ||||||||||||||||
0 | 85b | 136 | .931 | .134 | 2.38 | 13.0 | 0.05 | 17.4 | 0 | 383b | 30.2 | .647 | .044 | 2.32 | 7.8 | 7.78 | 7.6 |
-1 | 424a | 27.3 | .921 | .020 | 2.57 | 13.5 | 0.03 | 28.0 | |||||||||
SVC 4995 | RRE 4995 | ||||||||||||||||
0 | 90a | 129 | .862 | .286 | 2.00 | 18.3 | 0.00 | 14.3 | 0 | 53a | 218 | .710 | .347 | 1.64 | 23.5 | 0.00 | 19.4 |
-1 | 124a | 93.3 | .832 | .166 | 2.20 | 13.9 | 0.02 | 24.6 | -1 | 158a | 73.2 | .625 | .224 | 1.83 | 19.5 | 0.00 | 30.3 |
SVC 8800 | RRE 8800 | ||||||||||||||||
0 | 122b | 94.9 | .635 | .355 | 1.63 | 15.8 | 0.00 | 10.3 | 0 | 156b | 74.2 | .638 | .396 | 1.52 | 16.2 | 0.00 | 10.0 |
-1 | 154b | 75.2 | .593 | .289 | 1.71 | 14.9 | 0.01 | 19.9 | -1 | 121b | 95.6 | .598 | .339 | 1.54 | 14.3 | 0.01 | 18.1 |
-2 | 83b | 139 | .558 | .285 | 1.55 | 11.8 | 0.15 | 24.3 |
The columns of the table show: 1: time series ("-1" indicates that the one sine curve with
period given a row above was removed from the original data, "-2" indicates that the two sine curves with periods given in two rows above were removed from the original data, and so on), 2: frequency and period of
the highest peak in a given time series, a small letter following a frequency value indicates to which
interval of probability ("a": <0.1%, "b": 0.1-1%, "c": 1-5% ) this period belongs to
after the randomising procedure, 3: the autocorrelation coefficients calculated
with two lags equal 1 and 7 days respectively, 4: the normalization factor k computed
according to the procedure described in Sect.3, 5: the normalized power equal to
the Scargle power divided by k, 6: the FAP value resulting from the normalized power,
7: in successive rows a percentage of the original data variance explained by the
prominent sinusoidal signals found in a given type of data.
The highest | Auto | k | NP | FAP | % | The highest | Auto | k | NP | FAP | % | ||||||
peak at | correlation | [%] | of | peak at | correlation | [%] | of | ||||||||||
Freq. | Per. | lag | var. | Freq. | Per. | lag | var. | ||||||||||
[nHz] | [day] | 1 | 7 | exp. | [nHz] | [day] | 1 | 7 | exp. | ||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||||
ISN | MMF | ||||||||||||||||
0 | 442a | 26.2 | .906 | .150 | 2.19 | 19.2 | 0.00 | 44.6 | 0 | 799a | 14.5 | .650 | -.415 | 1.54 | 25.8 | 0.00 | 15.5 |
-1 | 127a | 91.1 | .897 | .179 | 2.37 | 13.9 | 0.02 | 49.7 | -1 | 844a | 13.7 | .634 | -.340 | 1.70 | 26.9 | 0.00 | 30.1 |
-2 | 876a | 13.2 | .548 | -.187 | 1.93 | 12.2 | 0.14 | 36.4 | |||||||||
SVC 405 | RRE 405 | ||||||||||||||||
0 | 418a | 27.7 | .731 | .086 | 1.90 | 24.5 | 0.00 | 21.9 | 0 | 418a | 27.7 | .641 | .064 | 2.15 | 15.2 | 0.01 | 9.8 |
-1 | 55a | 210 | .695 | .102 | 2.10 | 14.3 | 0.02 | 28.8 | -1 | 52a | 223 | .609 | .072 | 2.23 | 13.2 | 0.05 | 17.9 |
-2 | 444a | 26.1 | .663 | .017 | 2.15 | 11.1 | 0.39 | 33.9 | |||||||||
SVC 810 | RRE 810 | ||||||||||||||||
0 | 443a | 26.1 | .935 | .215 | 1.01 | 74.0 | 0.00 | 33.0 | 0 | 417a | 27.8 | .863 | .206 | 1.30 | 52.0 | 0.00 | 20.2 |
-1 | 416a | 27.8 | .924 | .300 | 1.21 | 50.4 | 0.00 | 45.4 | -1 | 450a | 25.7 | .842 | .275 | 1.58 | 27.2 | 0.00 | 30.2 |
-2 | 75a | 154 | .918 | .385 | 1.32 | 38.2 | 0.00 | 53.5 | -2 | 56a | 207 | .824 | .323 | 1.74 | 20.3 | 0.00 | 37.4 |
-3 | 456a | 25.4 | .904 | .275 | 1.51 | 20.1 | 0.00 | 58.0 | -3 | 71b | 163 | .806 | .245 | 1.91 | 13.2 | 0.05 | 42.0 |
-4 | 62a | 187 | .893 | .318 | 1.59 | 21.5 | 0.00 | 62.3 | |||||||||
-5 | 103a | 112 | .878 | .239 | 1.67 | 21.4 | 0.00 | 66.4 | |||||||||
-6 | 477a | 24.3 | .861 | .147 | 1.80 | 14.2 | 0.02 | 69.0 | |||||||||
-7 | 385a | 30.1 | .853 | .193 | 1.74 | 15.4 | 0.01 | 71.5 | |||||||||
-8 | 127b | 91.1 | .843 | .205 | 1.76 | 15.2 | 0.01 | 73.9 | |||||||||
-9 | 562b | 20.6 | .829 | .143 | 1.77 | 12.8 | 0.07 | 75.6 | |||||||||
-10 | 498b | 23.2 | .819 | .184 | 1.84 | 12.3 | 0.12 | 77.3 | |||||||||
SVC 1215 | RRE 1215 | ||||||||||||||||
0 | 444a | 26.1 | .957 | .240 | 1.14 | 68.5 | 0.00 | 36.4 | 0 | 418a | 27.7 | .872 | .253 | 1.37 | 33.4 | 0.00 | 14.4 |
-1 | 77a | 150 | .948 | .333 | 1.41 | 38.7 | 0.00 | 46.6 | -1 | 449a | 25.8 | .861 | .303 | 1.52 | 27.6 | 0.00 | 24.9 |
-2 | 416a | 27.8 | .938 | .206 | 1.61 | 35.2 | 0.00 | 55.6 | -2 | 76a | 152 | .845 | .349 | 1.74 | 24.9 | 0.00 | 34.3 |
-3 | 104a | 111 | .935 | .261 | 1.74 | 21.8 | 0.00 | 60.6 | -3 | 59a | 196 | .823 | .261 | 1.88 | 15.7 | 0.00 | 40.0 |
-4 | 457a | 25.3 | .922 | .159 | 1.93 | 14.3 | 0.02 | 64.1 | |||||||||
-5 | 64b | 181 | .912 | .188 | 1.93 | 14.7 | 0.01 | 67.4 | |||||||||
-6 | 129b | 89.7 | .900 | .109 | 2.00 | 14.3 | 0.02 | 70.3 | |||||||||
-7 | 480a | 24.1 | .893 | .035 | 1.94 | 15.2 | 0.01 | 73.0 | |||||||||
-8 | 563b | 20.6 | .888 | .077 | 1.96 | 12.7 | 0.08 | 75.0 | |||||||||
-9 | 499b | 23.2 | .882 | .118 | 2.02 | 13.2 | 0.05 | 77.0 | |||||||||
-10 | 385b | 30.1 | .874 | .158 | 2.04 | 11.9 | 0.18 | 78.7 | |||||||||
-11 | 49b | 236 | .869 | .161 | 2.03 | 10.2 | 0.31 | 80.3 | |||||||||
-12 | 361b | 32.1 | .858 | .116 | 1.92 | 12.6 | 0.09 | 81.9 | |||||||||
-13 | 148b | 78.2 | .850 | .128 | 1.84 | 12.5 | 0.10 | 83.1 | |||||||||
SVC 1620 | RRE 1620 | ||||||||||||||||
0 | 444a | 26.1 | .955 | .250 | 1.44 | 47.7 | 0.00 | 38.6 | 0 | 419a | 27.6 | .831 | .149 | 1.89 | 21.8 | 0.00 | 12.2 |
-1 | 77a | 150 | .946 | .329 | 1.68 | 33.6 | 0.00 | 48.9 | -1 | 78a | 148 | .816 | .181 | 2.03 | 22.6 | 0.00 | 23.8 |
-2 | 416a | 27.8 | .934 | .196 | 1.93 | 25.0 | 0.00 | 56.3 | -2 | 450a | 25.7 | .787 | .061 | 2.26 | 16.1 | 0.00 | 31.9 |
-3 | 128a | 90.4 | .930 | .239 | 2.03 | 17.8 | 0.00 | 61.0 | |||||||||
-4 | 105b | 110 | .920 | .064 | 2.17 | 12.3 | 0.12 | 64.2 | |||||||||
SVC 2800 | RRE 2800 | ||||||||||||||||
0 | 443a | 26.1 | .927 | .251 | 1.47 | 41.0 | 0.00 | 37.9 | 0 | 78a | 148 | .775 | .178 | 1.94 | 21.5 | 0.00 | 11.8 |
-1 | 77a | 150 | .917 | .315 | 1.65 | 34.3 | 0.00 | 47.8 | -1 | 421a | 27.5 | .744 | .066 | 2.13 | 16.7 | 0.00 | 20.7 |
-2 | 416a | 27.8 | .900 | .180 | 1.93 | 19.2 | 0.00 | 53.5 | |||||||||
-3 | 128b | 90.4 | .893 | .207 | 2.09 | 13.4 | 0.04 | 57.2 | |||||||||
-4 | 430b | 26.9 | .883 | .146 | 2.08 | 12.2 | 0.12 | 60.6 | |||||||||
-5 | 487a | 23.8 | .876 | .159 | 1.97 | 15.2 | 0.01 | 64.1 | |||||||||
-6 | 66b | 175 | .868 | .200 | 2.07 | 14.0 | 0.02 | 67.7 | |||||||||
-7 | 52a | 223 | .854 | .112 | 2.16 | 15.6 | 0.00 | 71.2 | |||||||||
SVC 4995 | RRE 4995 | ||||||||||||||||
0 | 442a | 26.2 | .891 | .182 | 1.75 | 31.6 | 0.00 | 37.5 | 0 | 79a | 147 | .755 | .009 | 2.66 | 9.1 | 2.78 | 7.1 |
-1 | 77a | 150 | .870 | .221 | 2.00 | 21.3 | 0.00 | 45.3 | |||||||||
-2 | 420b | 27.6 | .859 | .114 | 2.20 | 11.3 | 0.33 | 49.4 | |||||||||
-3 | 103b | 112 | .856 | .132 | 2.34 | 11.0 | 0.39 | 53.4 | |||||||||
SVC 8800 | RE 8800 | ||||||||||||||||
0 | 79a | 147 | .756 | .365 | 1.74 | 42.7 | 0.00 | 26.9 | 0 | 81a | 143 | .726 | .338 | 1.89 | 33.5 | 0.00 | 19.6 |
-1 | 66a | 175 | .696 | .189 | 2.20 | 25.9 | 0.00 | 40.5 | -1 | 67a | 173 | .668 | .185 | 2.28 | 20.9 | 0.00 | 31.5 |
-2 | 51a | 227 | .617 | .073 | 2.55 | 11.3 | 0.32 | 37.8 |
Since peaks in a periodogram may arise from aliasing or other phenomena not present in Gaussian noise (e.g., spectral leakage arising from the spacing of the data and from the finite length of the time series), the FAP values alone, are insufficient for establishing whether or not strong peaks in a periodogram are indeed real periodicities in the time series. Also, some small peaks present in the original periodogram can be real in the case when the normalized factor k would be too large in consequence of treating real periods as noise. We test for genuine peaks by recomputing the periodogram after randomising the data on the time grid. This procedure (Delache et al. 1985; Özgüç & Ataç 1994) maintains the noise characteristic of the time series but destroys all coherent signals, especially those with periods longer than the chosen cut interval of the data. In our time series, we cut the data with a seven day interval. It preserves to some extent the correlation characteristics of the data, so if a period results from a strong correlation inside the data, the number of cases in which it is observed should be rather large. We repeated this simulation 10000 times, every time computing the number of cases in which the recalculated power values for the periods having peaks in the original spectrum are equal to or larger than the peaks power of the real data. The results of these calculations are presented in Tables 4 and 5 as small letters situated after the frequency of the peaks found in the original periodograms. The letter "a" indicates that, for 10000 simulations, in less then 10 cases the peak value at the given frequency exceeded the corresponding peak power of the real data. Successive letters mark the intervals for which the probability (calculated from 10000 simulations) of obtaining as high a peak as in the original periodogram by chance are as follows: "b": 0.1-1%, "c": 1-5%, "d": 5-15%.
When more than one periodic signal is present in the data, multiple significant peaks appear in the periodogram. Alternatively, a true signal at frequency can cause peaks in the periodogram at frequencies other than because of the finite length of the data and irregularities in the data spacing. A useful procedure for determining whether any additional peaks with significant false alarm probability are physically real is an iterative peak removal technique (Delache & Scherrer 1983; Horne & Baliunas 1986). The highest peak in the original data periodogram provides the frequency corresponding to the strongest sinusoidal signal present in the data. Using the method of least squares the phase and amplitude of this sinusoidal signal are fitted from the original data. This allows subtraction of the best fitted sine curve from the time series and then recalculation of a new periodogram. This procedure is repeated as long as the FAP of the peaks connected with subtracted sine curves are smaller than 0.5%, producing the main periods in each of the analysed time series. To see the effect of removing these peaks from the original data, we present in Fig. 3 the six consecutive periodograms, calculated according to this procedure for the case of the minimum SVC 810 time series. We see that the original periodogram is dominated by a strong peak at frequency 423 nHz ( ), so the structure of the periodogram near this frequency is difficult to recognise. When the sine curve with this period is removed from the data, the new period at begins to be visible (Fig. 3b). Subtracting the next sine curves from the data we come to Fig. 3f which presents the periodogram of the last descendant time series having period whose peak value gives FAP smaller than 0.5%. Comparing all the periodograms seen in this figure, we can conclude that the period values indicated by peak positions do not change much during the removed procedure, which confirms the significance of the periods found. The results of the removal procedure are summarised in Table 2 for the minimum and in Table 3 for the rising phase. Both tables show for the original time series (marked by 0), and those with successively removed sine curves (denoted by -1, -2, -3) such characteristic parameters as: frequency and period of the highest peak, the autocorrelation coefficients, the normalization factor k, the normalized power equal to the Scargle power divided by k, the FAP value resulting from the normalized power.
MMF | SVC | SVC | SVC | SVC | SVC | SVC | SVC | ISN | Line | M | SVC | ISN | Time series | ||
RRE | RRE | RRE | RRE | RRE | RRE | RRE | [nHz] | [day] | RRE | Imp. | NP | FAP | |||
405 | 810 | 1215 | 1620 | 2800 | 4995 | 8800 | [% ] | ||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
858a | 851c | 846d | 845d | 854.5 | 13.5 | a | c | - | MMF/0 | ||||||
851d | 13 | + | 6 | 19.0 | 0.00 | ||||||||||
673c | 679c | 679d | 675c | 674b | 672b | 672b | 673.0 | 17.2 | - | 2b3c | b | ISN/-1 | |||
662c | 665c | 685c | 23 | 3c | 15 | 6.7 | 20.93 | ||||||||
588d | 585c | 580b | 584c | 584b | 585d | 583c | 586.0 | 19.8 | - | 2b2c | c | SVC 1215/-7 | |||
603d | 594d | 600c | 23 | c | 10 | 8.4 | 4.27 | ||||||||
534b | 529b | 528b | 526b | 526b | 525c | 529b | 528.1 | 21.9 | - | 5bc | b | SVC 1215/-3 | |||
548d | 23 | + | 19 | 12.1 | 0.11 | ||||||||||
491c | 498b | 496.8 | 23.3 | - | bc | - | SVC 1215/-5 | ||||||||
512d | 503d | 504c | 496b | 495b | 497d | 21 | 2bc | 11 | 11.8 | 0.15 | |||||
419a | 418a | 423a | 424a | 425a | 424a | 424b | 432a | 423.5 | 27.3 | a | 5ab | a | SVC 810/0 | ||
416a | 418a | 420a | 420b | 434c | 432c | 18 | 3ab2c | 58 | 49.9 | 0.00 | |||||
400a | 396a | 397b | 398c | 398c | 397.8 | 29.1 | a | ab2c | - | SVC 810/-1 | |||||
4 | - | 15 | 18.7 | 0.00 | |||||||||||
390a | 374c | 372d | 382.6 | 30.3 | - | ac | - | RRE 810/-2 | |||||||
382a | 381a | 383b | 382b | 383b | 386b | 18 | 2a4b | 28 | 15.3 | 0.00 | |||||
307c | 306c | 308d | 299.5 | 39 | - | 2c | - | SVC 810/-6 | |||||||
295d | 293c | 292b | 16 | bc | 6 | 8.2 | 5.14 | ||||||||
286d | 272c | 265d | 275c | 281d | 265d | 270.6 | 43 | + | 2c | - | RRE 810/-3 | ||||
268b | 267b | 266b | 21 | 3b | 11 | 11.8 | 0.14 | ||||||||
238b | 242b | 238d | 240.0 | 48 | - | 2b | - | SVC 1215/-6 | |||||||
4 | - | 6 | 11.8 | 0.14 | |||||||||||
197d | 195c | 197c | 195c | 198c | 217d | 194c | 199.0 | 58 | - | 4c | c | RRE 8800/-3 | |||
217d | 204d | 215c | 23 | c | 6 | 8.9 | 2.55 | ||||||||
150b | 151b | 150c | 147d | 156c | 154b | 154.4 | 75 | - | 3b2c | - | RRE 810/-1 | ||||
143d | 152a | 157a | 155b | 159b | 158a | 156b | 16 | 3a3b | 35 | 27.7 | 0.00 | ||||
125d | 121b | 124a | 122b | 126c | 126.4 | 92 | - | a2b | c | SVC 8800/0 | |||||
136c | 135c | 139d | 121b | 18 | b2c | 17 | 15.8 | 0.00 | |||||||
107c | 113b | 112.7 | 103 | - | bc | - | SVC 1215/-4 | ||||||||
118d | 116d | 118c | 11 | c | 5 | 11.5 | 0.20 | ||||||||
96c | 85b | 90a | 84a | 89.0 | 130 | c | ab | a | SVC 4995/0 | ||||||
88d | 96b | 83b | 13 | 2b | 20 | 18.3 | 0.00 | ||||||||
75b | 76b | 80b | 80b | 74d | 80c | 77.4 | 150 | - | 4bc | - | SVC 1215/-1 | ||||
69c | 82c | 67d | 13 | 2c | 15 | 14.8 | 0.01 | ||||||||
60d | 55c | 56c | 58c | 47c | 54c | 53.0 | 218 | + | 5c | - | RRE 4995/0 | ||||
52c | 52c | 50b | 53a | 13 | ab2c | 15 | 23.5 | 0.00 |
The first nine columns give frequencies of the lines found in the indicated type of data.
The small letter following the frequency denotes the interval to which belongs
the probability to obtain by chance as high a peak at this frequency as in the original
periodogram. The intervals are as follows: "a": <0.1% "b": 0.1-1%, "c": 1-5%,
"d": 5-15%. The bold letters indicate frequencies of the
large peaks having FAP values smaller than 0.5%.
Columns 10 and 11 give the mean frequency and mean period of the line. They are
calculated only from the periods marked by letters "a", "b", "c". The number
under the mean frequency is the difference between the highest and the lowest frequency
from all the periods found.
The next three Cols. 12-14 describe in a shorter way in what type of data
(magnetic, radio, spot numbers) the given line is visible. We mark this using
the letters "a", "b", "c", and two symbols "+" and "-". The symbol "-" means that
the given line is not noticed in the respective type of data.
The letters "a", "b", and "c" indicate that the line is recognised
and the periods found have the probability level described by these
letters, while the symbol "+" indicates that in the given type of data are periods
having the probability level described by the letter "d".
The second row in Col. 14 gives the importance number which indirectly measure
how large is support for this line in all the analysed time series (see text).
The last two columns in the first row, determine the time series in which the highest peak
for the given line have been observed. The second row gives the normalized power
of this peak as well as the resulting FAP value.
MMF | SVC | SVC | SVC | SVC | SVC | SVC | SVC | ISN | Line | M | SVC | ISN | Time series | ||
RRE | RRE | RRE | RRE | RRE | RRE | RRE | [nHz] | [day] | RRE | Imp. | NP | FAP | |||
405 | 810 | 1215 | 1620 | 2800 | 4995 | 8800 | [% ] | ||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
876a | 887b | 887d | 886d | 886c | 883.4 | 13.1 | a | b | c | MMF/-2 | |||||
886d | 886d | 887c | 881c | 11 | 2c | 11 | 12.2 | 0.14 | |||||||
844a | 863c | 864c | 853.2 | 13.6 | a | 2c | - | MMF/-1 | |||||||
841c | 23 | c | 8 | 26.9 | 0.00 | ||||||||||
814a | 823d | 822d | 806.5* | 14.4 | a | + | - | MMF/0 | |||||||
799a | 820d | 24 | a | + | 10 | 25.8 | 0.00 | ||||||||
756a | 762.4 | 15.2 | a | - | - | MMF/-3 | |||||||||
765c | 767c | 763c | 761c | 11 | 4c | 9 | 7.9 | 8.86 | |||||||
565b | 562b | 563b | 566c | 571b | 572b | 572d | 571c | 569.3 | 20.3 | - | 5bc | c | SVC 810/-9 | ||
566b | 572c | 574b | 574b | 579d | 576c | 575d | 17 | 3b2c | 28 | 12.8 | 0.07 | ||||
502c | 498b | 499b | 504b | 497a | 499b | 507d | 499.3 | 23.2 | - | a4bc | + | SVC 1215/-10 | |||
504c | 498d | 496c | 498c | 11 | 3c | 21 | 13.2 | 0.05 | |||||||
477a | 480a | 483b | 487a | 486b | 482.6 | 24.0 | - | 3ab | b | SVC 1215/-7 | |||||
10 | - | 21 | 15.2 | 0.01 | |||||||||||
456a | 457a | 457b | 464d | 453c | 455d | 459.5 | 25.2 | - | 2abc | + | SVC 810/-3 | ||||
467b | 466d | 467c | 14 | bc | 18 | 20.1 | 0.00 | ||||||||
444d | 444a | 443a | 444a | 444a | 443a | 442a | 439b | 442a | 444.4 | 26.0 | + | 6ab | a | SVC 810/0 | |
450a | 449a | 450a | 443b | 11 | 3ab | 56 | 74.0 | 0.00 | |||||||
429b | 429b | 432b | 430b | 431d | 431.7 | 26.8 | - | 4b | - | SVC 2800/-4 | |||||
434d | 434c | 436d | 436b | 7 | bc | 16 | 12.2 | 0.12 | |||||||
418a | 416a | 416a | 416a | 416a | 420b | 415c | 414c | 417.4 | 27.7 | - | 5abc | c | RRE 810/-1 | ||
418a | 417a | 418a | 419a | 421a | 420a | 7 | 6a | 60 | 50.4 | 0.00 | |||||
399a | 399.0 | 29.0 | a | - | - | MMF/-3 | |||||||||
399d | 400c | 398c | 2 | 2c | 7 | 10.8 | 0.54 | ||||||||
377a | 382c | 385a | 385b | 383c | 376d | 382d | 381.1 | 30.4 | a | ab2c | + | SVC 810/-7 | |||
387d | 381b | 381c | 381c | 383c | 383d | 373c | 14 | b4c | 22 | 15.4 | 0.01 | ||||
362b | 361b | 366d | 366c | 363.8 | 31.8 | - | 2bc | - | SVC 1215/-12 | ||||||
365d | 363c | 5 | c | 8 | 12.6 | 0.09 | |||||||||
285d | 284c | 284d | 284c | 283d | 292c | 285c | 287.0 | 40 | - | 3c | c | SVC 1215/-14 | |||
281d | 284d | 290b | 11 | b | 7 | 10.3 | 0.85 | ||||||||
157d | 149d | 146c | 148a | 151c | 152d | 150.0 | 77 | + | b2c | b | SVC 1215/-13 | ||||
141d | 155c | 16 | c | 6 | 12.5 | 0.10 | |||||||||
128b | 127b | 129b | 128a | 128b | 127b | 117d | 127a | 126.8 | 91 | - | a5b | a | SVC 1620/-3 | ||
120c | 12 | c | 26 | 17.8 | 0.00 | ||||||||||
103a | 104a | 105b | 102c | 103b | 103.6 | 112 | - | 2a2bc | - | SVC 1215/-3 | |||||
104c | 104c | 102d | 3 | 2c | 19 | 21.8 | 0.00 | ||||||||
85a | 98c | 88d | 88d | 93c | 91.6 | 126 | - | a2c | - | SVC 405/-3 | |||||
86c | 96b | 13 | bc | 11 | 9.6 | 1.70 | |||||||||
75a | 77a | 77a | 77a | 77a | 79a | 74b | 76.8 | 151 | - | 6a | b | SVC 8800/0 | |||
71b | 76a | 78a | 78a | 79a | 81a | 10 | 5ab | 61 | 42.7 | 0.00 | |||||
62a | 64b | 65c | 66b | 67d | 66a | 64.1 | 181 | - | 2a2bc | - | SVC 8800/-1 | ||||
59a | 65d | 64b | 67a | 8 | 2ab | 30 | 25.9 | 0.00 | |||||||
55a | 56a | 49b | 52a | 51b | 52.3 | 221 | - | 3a2b | - | RRE 810/-2 | |||||
52a | 54d | 51a | 7 | 2a | 31 | 20.3 | 0.00 |
* This is the mean value from two close magnetic lines.
Figure 4: The normalised periodograms of the minimum ISN and SVC time series after removing from the original data all the sine curves having periods with FAP values smaller than 0.5%. The removed periods are shown at the top of each graph. The dashed lines indicate FAP significance levels. | |
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Figure 5: Same as Fig. 4. but for the rising phase ISN and SVC time series after removing from the original data all the sine curves having periods with FAP values smaller than 0.5%. | |
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Figure 6: Same as Fig. 4. but for the minimum MMF and RRE time series after removing from the original data all the sine curves having periods with FAP values smaller than 0.5%. | |
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Figure 7: Same as Fig. 4. but for the rising phase MMF and RRE time series after removing from the original data all the sine curves having periods with FAP values smaller than 0.5%. | |
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Most of the calculated periodograms have been obtained from radio observations (SVC and RRE time series calculated for seven different frequencies), but the ISN and MMF data were also analysed. The final periodograms obtained when all the main sinusoidal signals given in Tables 2 and 3 are subtracted from the original data are presented in Figs. 4-7. Careful examination of these periodograms shows that many periods having a formal significance level near 30% are seen in time series coming from different observational data. This fact allow us to suggest that even those periods could be real.
Taking into account all the information about the periods found obtained with the three different approaches to the data described in Sect.3, we prepare Table 4 for the minimum, and Table 5 for the rising phase, which bring together information about characteristic periodicities (lines) observed in these phases of the solar cycle 23 (we use the spectroscopic term "line" for the mean period calculated from the periods observed in various time series, which, as we suppose, represent the same characteristic periodicity). As two time series were created from the radio data at each frequency, we use two successive rows in the radio data columns to separate periods observed in SVC and RRE radio time series. The exact frequencies of the detected periods were found with a 1 nHz resolution in the vicinity of peaks seen in periodograms constructed from normalized power values on the grid of the independent frequencies appropriate to the given window. In all the figures, the points marked in the presented periodograms are situated in the places resulting from the grid of the independent frequencies, while the solid lines are drawn from the power values computed every 1 nHz. To discribe in some measure a "strength" of the line we introduced so called the importance number of the line. This number shows how large is support for this line from all the analysed time series. It is calculated according to the following rule. The each letter "a" in the line description columns (Cols. 12-14) gives for the importance number "5", the letter "b" gives "3" , and the letter "c" only "1". It is only one line with the importance number "5" in Tables 4 and 5. There are a few lines with a smaller importance number than "5" recognised in the analysed data, but we do not include them in Tables 4 and 5. However, it is important to notice that some of the lines included in Tables 4 and 5 can be unreal in a sense that they are created from periods which in fact belong to two different but neighboring lines. For such a case the calculated mean period is somewhere between the periods of these two neighboring, unknown lines. The probability of such a situation increses for lines having long periods and large values of (Col. 10, second row).
To aid in further discussion of the lines we have prepared Fig. 8, which shows all the lines present in Tables 4 and 5. The level of darkness in this 3D graph illustrate the importance number of the lines. A close look at Fig. 8 shows a clear difference among the 18 lines observed in the minimum and the 22 lines found in the rising phase of solar sunspot activity cycle. Although the 9 lines have almost the same periods in both the phases, the strength (measured indirectly by the importance number) of nearly all the 18 lines change, indicating that perhaps the physical mechanisms responsible for them also change with the solar phase. From Fig. 8 it is evident that the lines in the rising phase gather into three groups:
Many previous studies by a number of authors have resulted in a wide range of solar periodicities, which are not easy to explain. This indicates that the problem of solar periodicities is still open and more systematic efforts should be undertaken. Here, we do not want to discuss all possible causes of the observed periods, but we want to present a suggestion which may be of help in further investigations.
Recently Oliver et al. (1998) proposed that the periodic emergence of magnetic flux, manifested as sunspots, triggers the near 158 day periodicity in high-energy solar flares. As different magnetic features have different rates of rotation (Gilman 1974; van Tend & Zwaan 1976; Erofeev 1999) we think that a periodic emergence and a constant conversion of various magnetic structures explain the origin of the observed lines and their transformation with the phase of solar cycle 23. The main arguments supporting this idea are as follows:
Figure 8: Characteristic lines generated by solar phenomena in a) the minimum and b) the rising phase of solar cycle 23. The three group of lines are inicated: the level of darkness qualitatively illustrate the importance of lines. | |
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To summarise, our analysis of radio observations gives the majority of the known periods, reveals a clear difference among periodicities observed in the two phase of the solar cycle examined and shows that daily measured radio fluxes at various frequencies are very useful for the systematic study of solar periodicities observed in the different layers of the solar atmosphere. We are preparing a similar analysis for the next phases of solar cycle 23: the maximum and the declining phase.
We found the rotation rate of the "active longitudes" in the rising phase as equal to 444.4 4 nHz (26 0 0 3). We suppose that this period can be identified with a fundamental period of unknow Sun's clock as a lot of the known periodicities are subharmonics of it. We think that in the minimum the lines observed are conected with the small and large-scale magnetic fields which then dominate, while in the rising phase most lines are generated by new magnetic structures connected with long lived active regions formed within "active longitudes".
To understand the cause of all the observed periodicities, we still need new observational data, but more careful analyse of old radio data can also be useful (in preparation). However, our investigation, indicates that the solution of the observed solar periodicities should be sought in a complicated Sun's magnetic system which generate in the different solar data the compound set of solar periodicities.
Acknowledgements
First of all, we would like to thank our anonymous referee for very valuable remarks and suggestions which helped us to significantly improve the current version of the paper. The autors are grateful to Dr. K. Chyzy for assistance in preparation of the manuscript. This work was supported in partly by KBN grant No. 158/E - 338/SPUB - 204/93.