Quasi-Periodic Pulsations: Fermi/GBM Results

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Introduction

Over the past 40 years, quasi-periodic pulsations (QPP) in solar flares have been reported from observations across the electromagnetic spectrum on timescales from sub-second intervals up to several minutes (see e.g. [1], [2], [3]). While there seems to be an overwhelming amount of observational data, the underlying physical mechanism that could generate such QPPs still remains unclear.

In general, the analysis and interpretation of the power spectral density (PSD) of solar flares is difficult. This is because solar flares, together with many other astrophysical sources, have very steep power spectra in the low-frequency region. This type of variability is known as red-noise (e.g. [4]). When determining the significance of possible periodicities in the PSD, this red-noise component has to be accounted for in order not to severely overestimate the significance of identified frequencies [5].

Here, we present time series and periodogram analyses of four solar flares which were observed by the Fermi Gamma-Ray Burst Monitor (GBM) and display compelling quasi-periodic behavior in their light curves. A more comprehensive analysis on this subject can be found here

The Gamma-Ray Burst Monitor

The GBM [6] is one of the instruments onboard the Fermi Gamma-Ray Space Telescope launched on June 11, 2008. Specifically designed for gamma-ray burst (GRB) studies, GBM observes the whole unocculted sky with a total of 12 thallium-activated sodium iodide (NaI(Tl)) scintillation detectors covering the energy range from 8 keV to 1 MeV and two bismuth germanate scintillation detectors (BGO) sensitive to energies between 150 keV and 40 MeV. Therefore, GBM offers superb capabilities for the analyses of not only GRBs but solar flares as well.

What is red-noise?

Red-noise has nothing to do with measurement errors or the systematics of the detectors, which are also called noise. Red-noise is an intrinsic property of the observed source and is due to erratic, aperiodic brightness changes. In contrast to white noise, which displays a flat spectrum in a PSD, i.e. is power independent of frequency, red-noise is characterized by a power law of the form of P=N^-α and is found not only in solar flares but also in e.g. X-ray binaries, Seyfert galaxies and gamma-ray bursts. As a first order approximation, red-noise is the realization of a linear stochastic and weakly non-stationary process.

In time-series analysis of solar flares, it is common practice to artificially suppress this low-frequency component by de-trending the light curves of solar flares [8]. This can be achieved for example by smoothing the light curve with a moving average or by applying a Gaussian filter and performing the periodogram analysis on the residual emission, i.e. the smoothed version is subtracted from the original data-set. Unfortunately, this procedure can give rise to misleading results as we show in the following.

By randomizing the phase as well as the amplitudes, one can generate a purely random time series that displays a 1/f^&alpha: (red-noise) dependence in the PSD. We created a time series with an f^-1.8 periodogram shape, subtracted a simple moving average and then calculated a Lomb-Scargle diagram of the residuals. The result is remarkable. Although we started with a purely random, red-noise dominated time-series, we obtain a PSD with three frequencies whose power exceeds the 3σ confidence limit. This approach in signal processing clearly returns false-positive frequencies.

Figure 1: Upper panel: Synthetic red-noise time series with a power law index of α=-1.8. Simple (boxcar) moving average is overplotted (dashed line). The inset presents the residual signal. Middle panel: Lomb-Scargle Periodogram of the residual light curve. The 3σ CL has been denoted by a dashed line

Shown by the example above, we strongly discourage subtracting smoothed versions of raw light curves when looking for intrinsic frequencies in the red-noise dominated PSD.

One way to estimate the significance of induced frequencies on top of an underlying red-noise continuum in a PSD was put forward by Vaughan 2005 [8]. In short, one simply converts the periodogram into log-space in both frequency and power density. For such a log-periodogram, one can then clearly identify the power-law component in the low-frequency range and a "cutoff" where white noise or another, additional noise component takes over. One can easily determine the power-law parameters by fitting a linear function to the low-frequency periodogram bins using the least squares method.

Results

We calculated two periodograms of a RHESSI solar flare for which QPPs have been reported (see also [7]). Firstly, a Lomb-Scargle-periodogram was calculated on the residual emission after a simple moving average was subtracted from the raw data. With this method, several peaks were found above the 3 σ threshold (see Fig. 2). The peak with the highest value of normalized power is located at f ~ 0.025 Hz. Another peak worth mentioning is located at f ~ 0.24 Hz, which is the expected rotational frequency of RHESSI around its spin axis and thus, an instrumental artifact.

As we now demonstrate, the significance of the peak at f ~ 0.025 Hz is highly overestimated by the latter method. In Fig. 2 a PSD is shown which was calculated using the raw and undetrended light curve applying the technique by Vaughan 2005. A significant spectral feature at ~0.24 Hz is found which, once again, is the expected rotation period of the RHESSI instrument. However, this PSD is lacking any other frequency above the 3 σ confidence limit. The discrepancy between the two methods is easily explained. Firstly, the whole raw light curve was used without artificially detrending it beforehand. Secondly, the method of Lomb and Scargle assumes a white noise continuum and does not take into consideration the red-noise component of the solar flare.

Figure 2: Upper panel: Summed and background subtracted light curve of the solar flare observed by RHESSI on January 1st, 2005. The inset shows the residual emission after subtracting a simple (boxcar) moving average. Middle panel: Periodogram analysis performed on the residual emission. Lower panel: PSD with best fit broken power-law (solid green line) and the 3σ significance level (dotted line) indicated. The arrow points to the significant frequency at f ~ 0.244 Hz.

In conclusion, we could not confirm the reported QPP in the above RHESSI solar flare.

We then applied the aforementioned method to four solar flares which were observed by GBM. All of these solar flares show a promising quasi-periodic behavior in their light curves, but none of these QPPs were statistically significant. (For clarity, only 2 out of 4 solar flares are shown here. For the detailed analysis we refer the reader again to [7]).

Figure 3: Left panels, solar flare observed by GBM on March 14th, 2011. The panels are the same as Fig. 2. Right panels, solar flare observed by GBM on March 15th, 2011. The panels are the same as Fig. 2.

Conclusions

Power spectra that are dominated by red-noise ensure that a periodogram analysis is not trivial, the interpretation of a peak in a PSD is more complex than usual, and that the estimate of the peak's significance is important. Again, red-noise is an intrinsic source property. In other words, having shown that the variations in the corresponding solar flares are not quasi-periodic at the 3σ level does not mean that these variations are not real.

While we consider it unnecessary to invoke oscillatory regimes of plasma instabilities, we note that the physics of these variations is certainly interesting and worth further study.

References

[1] "Sixteen-Second Periodic Pulsations Observed in the Correlated Microwave and Energetic X-Ray Emission from a Solar Flare"

[2] "Oscillations of Hard X-Ray Flare Emission Observed by RHESSI: Effects of Super-Alfvénic Beams?"

[3] "Observational Studies of the X-Ray Quasi-Periodic Oscillations of a Solar Flare"

[4] "Timing of the Crab Pular III. The Slowing Down and the Nature of the Random Process"

[5] "A ~4.6 h quasi-periodic oscillation in the BL Lacertae PKS 2155-304?"

[6] "The Fermi Gamma-ray Burst Monitor"

[7] "Quasi-Periodic Pulsations in the Gamma-Ray Emission of a Solar Flare"

[8] "A simple test for periodic signals in red noise"