2 Observations

The radio fluxes at the four lowest frequencies 405, 810, 1215, 1620 MHz are from our new accurate observations, which we began on 1 March 1996 (http://www.oa.uj.edu.pl.). Every day we measure the total solar radio flux at ten frequencies within the decimeter range of wavelengths using the 8-m radiotelescope built in 1995 at the Cracow Astronomical Observatory (Zieba et al. 1996). The international sunspot numbers (ISN), the Stanford mean solar magnetic field (MMF) and radio fluxes at the frequencies 2800, 4995 and 8800 MHz were taken from Solar Geophysical Data bulletins (1996-1999). We added to our radio observations those at higher frequencies for two reasons. Firstly, we wanted to examine periodicities of the radio emission in a wide enough radio band to analyse the radio flux coming from a large part of the solar atmosphere and secondly to compare our observations with those of others.

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:

$$B_i = A_2 + (A_1 - A_2) / (1 + \exp ( (t_i -t_{\circ}) / \Delta t) )$$

where A₁, A₂, $t_{\circ}$, $\Delta t$ are the parameters and t_i is the day number.

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, A1=38.9 su, A2=62.5 su, $t_{\circ }=855$ days, $\Delta t=111$ days according to the Boltzman formula $B_i=A_2+(A_1-A_2) / (1+\exp ((t_i-t_{\circ}) / (\Delta t))$. The division into the minimum and rising phase is also indicated.

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.

Then the observed daily flux $F_i = B_i + h_{\rm B} (ISN)_i$, and $h_{\rm B}$ 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, $(RRE)_i = F_i - B_i - h_{\rm B} (ISN)_i$. 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.