Statistical properties of radio flux densities of solar flares

The present study is based on a list of radio bursts recorded with the NoRP from 2000 to 2010 at multiple frequencies: 1, 2, 3.75, 9.4, and 17 GHz (Nakajima et al. 1985). The raw data of the NoRP have a time resolution of 0.1 s. For each day of the recorded bursts, we selected two quiescent periods before and after each flare and calculated their mean flux densities and their standard deviations for each frequency. We redefine the start and end times of a flare at each frequency as the first and last five consecutive points whose flux densities exceed the mean quiescent flux densities by more than five times the corresponding standard deviation, respectively. To ensure that the signal is strong enough for spectral analysis, we only consider flares with a duration greater than 200 s and a peak flux density at least 1.1 times the corresponding averaged quiescent flux densities at one of these frequencies. Then 209 flares were selected. For our analyses, we also requires that the peak flux density at each frequency exceeds the corresponding averaged quiescent flux densities by more than 10%, and some frequencies are dropped due to calibration of the flux density during the flare. Then at each frequency the number of bursts is less than 209. Overall, we have 182, 195, 200, 197, and 166 events for the 1, 2, 3.75, 9.4, and 17 GHz frequency channels, respectively.

To quantify the significance of flux variation on different timescales, we use a running smooth technique. To subtract contribution from the background properly, we selected background periods at least 5 minutes away from the flare's start and end points at each frequency. The upper curves (a) in the top panels of1 Figure 1 show light curves of two flares at 2 GHz. The lower curves (b) in these panels show the background subtracted light curves running smoothed over a duration of 0.9 s. The curves (c) in the middle panels show the difference between (b) and (a) with the background subtraction. One can define the power of the flux density variation at 0.9 s as the sum of the absolute values of (c). To subtract contributions to the power spectrum from the background, the background power spectra are also calculated and rescaled before the subtraction. Then we may have the power at T = t × 0.1 s:

$$\begin{eqnarray}&&F(T)=\displaystyle \sum _{i}|{f}_{i}-{f}_{i}(t)|-\displaystyle \sum _{j}|{b}_{j}-{b}_{j}(t)|\times \displaystyle \frac{l}{{l}_{b}},\end{eqnarray} \tag{ 1 }$$

where f_i and f_i (t) are the original and smoothed flare flux densities, respectively, b_j and b_j (t) are for the background, and l and l_b represent the duration of the flare and the background, respectively.

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In general, the power spectra should always be positive. However, due to the background subtraction, negative power may be obtained at some timescales. Figure 2 shows the scatter plot of the power at 0.3 s indicated as fluence v.s. the standard deviation of the background (panel (a)) and their normalized distributions (panels (b) and (c)). With the increase of frequency, there are more fluctuations in the background, and more events have a negative power at 0.3 s.

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To avoid uncertainties close to the temporal resolution of 0.1 s, one may set the power at 0.3 s to be zero with the background power spectrum normalized at T = 0.3 s. Then we may have the power at T = t × 0.1 s:

$$\begin{eqnarray}\begin{array}{ll}F(T)= & \displaystyle \sum _{i}|{f}_{i}-{f}_{i}(t)|-\displaystyle \sum _{i}|{f}_{i}-{f}_{i}(3)|\\ & \times \left[\displaystyle \sum _{j}|{b}_{j}-{b}_{j}(t)|/\displaystyle \sum _{j}|{b}_{j}-{b}_{j}(3)|\right].\end{array}\end{eqnarray} \tag{ 2 }$$

The solid lines in the bottom panels of Figure 1 show these power spectra. Comparing the two flares in Figure 1, one can see that the flare on 2002 April 9 shown in the left panel has prominent impulsive emission, and its power spectrum starts to rise at the shortest time scale. While there is no evident impulsive emission for the flare on 2001 March 25 shown in the right panel and its power spectrum starts to rise after 10 s.

In the bottom panels, we also show the maximum difference between the original and smoothed light curves f_t at different timescales, which can be interpreted as the maximum contribution to the flux density from the impulsive component. The size of the emission region has an upper limit of the light travel distance L_t ≤ t × 0.1 s×c, where c is the speed of light. Then the brightness temperature on different timescales has a lower limit of:

$$\begin{eqnarray}\begin{array}{ll}{T}_{b}(T) & \ge {{f}_{t}}\left(\displaystyle \frac{{D}^{2}}{{L}_{t}^{2}}\right)\left(\displaystyle \frac{{c}^{2}}{2{\nu }^{2}{k}_{{\rm{B}}}}\right)\\ & \simeq 0.82{\left(\displaystyle \frac{0.1t\nu }{1{\rm{GHz}}}\right)}^{-2}\left(\displaystyle \frac{{f}_{t}}{10\,{\rm{sfu}}}\right){\rm{MK}},\end{array}\end{eqnarray} \tag{ 3 }$$

where k_B is the Boltzmann constant, D = 1.5 × 10¹³ cm is the distance to the Sun, ν is the observation frequency, and 1 sfu = 1.0 × 10⁻¹⁹ erg s⁻¹ cm⁻² Hz⁻¹. When the impulsive emission is prominent, this figure shows that the brightness temperature has a characteristic value higher than the coronal temperature and this temperature increases with the decrease of timescale T. The brightness temperature of the gradual emission is at least two orders of magnitude lower.

Since phenomena below 1 s are closely related to the electron acceleration and transport processes, one may use T = 0.9 s to separate the light curves into an impulsive ((c) of Fig. 1) and a gradual ((b) of Fig. 1) component. Panel (a) of Figure 3 shows the correlation between the peak of the gradual component, i.e. the peak flux density of the background subtracted smoothed light curves, and the peak flux density of the impulsive component, the maximum difference (positively defined) between the original and the smoothed light curve. Panel (b) of Figure 3 shows that the slope of this correlation can be divided into two groups, one for the two low-frequency channels and the other for the three high-frequency channels. As shown in the left panel of Figure 1, the peaks of the gradual components at the two low frequencies can have contributions from many simultaneous short timescale pulses and do not necessarily reflect physical processes on large scales. This may explain the good correlation between the peak flux densities of the impulsive and gradual components at 1 and 2 GHz. The slope decreases with the increase of frequency at high frequencies, where the impulsive component is less prominent.

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Panels (c) and (d) of Figure 3 show the normalized distributions of the peak flux densities of the gradual and impulsive components, respectively. At 1 GHz, the distributions of the gradual and the impulsive component are very similar, implying contamination of the gradual component by many impulsive pulses, and both peak at a few tens of sfu. At other frequencies, while the gradual components peak near 100 sfu, the impulsive components peak near the level of the background fluctuations, which is defined as 5 times of the averaged standard deviation of the impulsive component of the corresponding backgrounds, implying more flares with lower impulsive peak flux densities.

Panels (e) and (f) of Figure 3 show the scatter plots of the peak flux density distributions of the gradual and impulsive components, respectively. Here again one can see that these distributions at 1 GHz are similar, but there are quantitative differences. For most flares, the peak flux density of the gradual component is higher than that of the impulsive component (see panel (a)), and the distribution of the gradual component appears to shift toward a slightly higher flux density. In particular, the medium flux density is above 100 sfu for the gradual component but less than 100 sfu for the impulsive component. These results suggest that the gradual component at 1 GHz is usually not as strong as the impulsive component, and the gradual component that we derived above can have significant contributions from pileups of many impulsive pulses. If this pileup effect does not depend on the peak flux density, as implied by self-similarity of X-ray fluxes of solar flares (Zhang & Liu 2015), one expects an identical peak flux density distribution of the impulsive and gradual components defined above.

At the three high frequency channels, most of the impulsive components have a peak flux density comparable to the background fluctuations (Panels (d) and (f)), while the peak flux density distribution of the gradual components may be approximated by log-normal distributions (Panels (c) and (e)). At 2 GHz the distribution of the peak flux density of the gradual component is contaminated by the impulsive emission (Panel (a) and the left panel of Fig. 1), while the distribution of the peak flux density of the impulsive component is affected by the background (Panels (d) and (f)). Using Equation (3), one can estimate the lower limit to the maximum brightness temperature from the impulsive peak flux densities at different frequencies. At 1 GHz, the brightness temperature can reach a few tens of billion Kelvin. This temperature decreases by more than four orders of magnitude above 10 GHz, implying distinct emission processes.

We also extracted the intervals with a flux density above the background fluctuations defined as five times the standard derivation of the impulsive component of the corresponding backgrounds. The durations of the impulsive and gradual components are defined as the total duration of these corresponding intervals. Figure 4 shows statistical properties of these durations. The duration of the impulsive component is always lower than that of the corresponding gradual component and decreases with the increase of frequency (Panels (a) and (d) of Fig. 4). However the duration of the impulsive component increases quickly with the increase of that of the gradual component and these two become comparable at 1 and 2 GHz for flares with long impulsive durations (Panels (a) and (b) of Fig. 4). This again shows that the gradual components at low frequencies are contaminated by the impulsive emission.

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It is interesting to note that the distributions of durations of the gradual components are nearly identical at all frequencies (Panel (c) of Fig. 4), which implies a physical correlation between the impulsive and the gradual emission. As mentioned above, the gradual component at 1 GHz has significant contributions from pileups of many impulsive pulses. A physical correlation between the impulsive and the gradual emission can naturally explains that it has the same duration distribution as the other frequencies. We also note that radio emission is well-correlated with the impulsive phase of X-ray flares (Neupert 1968) and the gradual component studied here is mostly emitted in the impulsive phase of solar flares. However, the distribution of durations of the impulsive components at 1 GHz peaks at a few tens of seconds, which is one order of magnitude longer than peak locations of the distributions of durations at the three high frequencies (Panel (d) of Fig. 4). Again, the durations of the impulsive components at these high frequencies are likely affected by background fluctuations and do not reflect the intrinsic properties of the impulsive emission. The distribution of the durations of the impulsive component at 2 GHz is broader than that at other frequencies (Panel (d) of Fig. 4), reminiscent of the distribution of the corresponding peak flux density (Panel (d) of Fig. 3).

We also integrated flux densities of the impulsive and gradual components over the duration to obtain the corresponding energy densities (i.e., fluences). Figure 5 shows their statistical properties. The fluence of the impulsive component is always lower than that of the gradual component (Panel (a)) and the two components appear to be well correlated except for the 17 GHz channel (Panels (a) and (b)), where the background fluctuation is the strongest. Panel (b) shows that the slopes of these correlations are always less 1, implying relatively less impulsive emission in more powerful flares. Interestingly, peak locations of the fluence distributions of the gradual components appear to increase monotonically with the increase of frequency (Panel (c)), while those of the impulsive components decrease from 1 GHz to 2 GHz first and then increase monotonically with the frequency (Panel (d)), implying distinct emission mechanisms at 1 GHz. In general, most of the energy, which can be estimated with the product of the energy density and the frequency, is radiated via the gradual component at high frequencies.

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To further draw distinctions between the impulsive and gradual components, in Figure 6 we show the correlations between the fluence and the product of duration and peak flux density of the impulsive (top two rows) and gradual components (bottom two rows). In general, these quantities are well correlated. However, the slopes for the gradual component are always close to 1 (Panel (f) of the bottom two rows), as expected for the relatively smooth and seemingly energy independent light curves. The slopes for the impulsive components, however, are always less than 1 (Panel (f) of the top two rows), implying shorter durations or less frequent appearance of stronger pulses. These two quantities become comparable for small events and the slope appears to be higher at the two high frequencies, implying different origin of the impulsive emission at different frequencies. Nevertheless, the power-law scaling of these two quantities still implies an energy (or scale) independent behaviour of the impulsive component. In summary, these correlations imply that most energy of the gradual components is released near the peak of the flux density, while for the impulsive component, most energy is released by the more numerous pulses with a lower flux density.

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