Statistical properties of soft X-ray solar flares
F. Lepreti † , V. Carbone † , P. Veltri † and P. Giuliani
Dipartimento di Fisica, Università della Calabria, I-87036 Rende (CS), Italy
†
Istituto Nazionale per la Fisica della Materia, Unità di Cosenza, Italy
Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel
Abstract. We investigate some statistical properties of soft X-ray bursts produced by solar flares. The Probability Density
Functions (PDFS) of soft X-ray intensity fluctuations are shown to display wide, non-gaussian tails. The shape of the PDFs is
nearly unchanged as the timelag, used to calculate fluctuations, varies. A very similar behavior is found for PDFs of energy
dissipation fluctuations in a shell model of Magnetohydrodynamic (MHD) turbulence. Recalling also that both flare soft X-ray
bursts and dissipative bursts in the MHD shell model are characterized by a power law distribution for waiting times between
successive bursts, we suggest that the results shown in this paper support the idea that solar flares could represent bursty
dissipative events of MHD turbulence.
INTRODUCTION
the observed WTD is well described by a Lévy function, which asimptotically displays a power law behavior.
These results indicate that waiting times are statistically
self-similar (see also Fig. 3 in ref. [11]) and suggest the
presence of long-range correlations in the flaring process.
Boffetta et al. [10] showed that energy dissipation
bursts in a shell model of magnetohydrodynamic (MHD)
turbulence are characterized by power law distributions,
including the WTD, as in the case of solar flare observations. On this basis, they suggested that the presence of
long-range correlations could be related to the nonlinear
dynamics occurring in fully developed MHD turbulence.
To the aim of investigating in more detail the physical
origin of solar flare statistics, in this work we will analyze, besides the waiting time distribution, the scaling
behavior of the Probability Density Functions (PDFs) of
SXR intensity fluctuations. As a comparison, the same
analysis will be performed on fluctuations of energy dissipation rate in a shell model of MHD turbulence.
Bursts of X-ray emission are one of the main signatures
of solar flares. Statistical properties of solar X-ray bursts
have been extensively studied in order to investigate the
physical mechanisms underlying flares. Several authors
showed that probability distributions of flare peak flux,
fluence and duration are well represented by power laws
[1, 2, 3, 4, 5, 6]. An explanation of these results was
proposed by Lu and Hamilton [7] by using “avalanche
models” (also called “sandpile models”) based on the
idea that the coronal magnetic field is in a state of selforganized criticality (SOC) [8]. These models are able
to reproduce the power law behavior of solar flare size
distributions.
However, it has recently been pointed out that the
statistics of time intervals ∆t between two successive
bursts (also called waiting times) is very important
for solar flare modeling [9, 10]. In sandpile models,
avalanches are independent on each other and follow
a Poisson statistics. This gives rise to an exponential
waiting time distribution (WTD) [9, 10]. On the other
hand, recent analyses of hard X-ray (HXR) and soft Xray (SXR) observations showed that the solar flare WTD
clearly deviates from the exponential behavior expected
from avalanche models [9, 10].
In particular, Boffetta et al. [10], by using SXR data
acquired by the Geostationary Operational Environmental Satellites (GOES), found that the WTD follows a
power law P ∆t ∝ ∆t β , with β 2 4, for waiting times
greater than a few hours. Lepreti et al. [11] extended this
analysis, by showing that the sequence of SXR bursts
is not consistent with a local Poisson process and that
ANALYSIS OF GOES SOFT X-RAY DATA
The SXR data used in this paper were acquired by the
GOES 10 satellite in the 1-8 Å band, during the time
interval between 1998 August 1 and 2000 July 29. The
SXR flux f t (shown in Fig. 1) was measured with a
sampling time of 1 minute.
In order to calculate waiting times, bursts are defined
as the time intervals during which the condition f t fth is satisfied. The threshold is defined as f th f t 3σ , where the average and the standard deviation are
calculated through an iterative procedure, excluding the
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FIGURE 1. Soft X-ray flux measured by the GOES 10 satellite in the 1-8 Å band, in the interval between 1998 August 1
and 2000 July 29.
FIGURE 3. PDFs of soft X-ray intensity fluctuations at different timelags τ in the interval 0.03 hrs τ 8 7 103 hrs
ANALYSIS OF AN MHD SHELL MODEL
In this section, the same analysis tools described above
will be applied to a shell model of magnetohydrodynamic turbulence. This model is a dynamical system
which aims to reproduce the main features of nonlinear
dynamics occurring in MHD turbulence [10, 12].
The wavevector space is divided in discrete shells
of radius kn 2n k0 . Two complex dynamical variables
un t and bn t , representing, respectively, velocity and
magnetic field increments on an eddy of scale l k n 1 ,
are assigned to each shell. The evolution equations for
un t and bn t are obtained by: a) introducing general
quadratic couplings between neighbouring shells; b) imposing the conservation of MHD ideal invariants. In this
way, the following set of nonlinear Ordinary Differential
Equations can be obtained:
FIGURE 2. Distribution of waiting times between successive soft X-ray flares. The solid line represent a power law with
an exponent β 2 33.
dun
2
ν kn un fn Tn un bn (1)
dt
dbn
2
µ k n b n Gn u n b n (2)
dt
where where ν and µ are, respectively, the kinematic
viscosity and the resistivity, f n is an external forcing
term, Tn and Gn are the nonlinear quadratic terms [10,
12].
In this work, we are interested in analyzing the statistical features of the energy dissipation rate ε t , given by
bursts [10]. The waiting time distribution obtained from
our analysis is similar to WTDs shown in previous works
[10, 11], that is, it displays a power law tail, for ∆t 5 hrs, with an exponent β 2 33 0 16 (see Fig. 2),
despite the fact we used a dataset covering a shorter
period and we selected flares with a different criterion.
In addition to the WTD, we are interested in analyzing
the scaling bahavior of SXR intensity fluctuations δ f τ f t τ f t . This is done by calculating the PDFs of
standardized fluctuations δ Fτ δ fτ δ fτ δ fτ2 1 2
(where brackets represent time averages) at different lagtimes τ . In Fig. 3, we report the PDFs of δ Fτ for 0.5 hrs
τ 8 7 103 hrs. It can be seen that the PDFs don’t
change their shape significantly, as τ varies, and display
strong, non gaussian tails.
ε t ν ∑ kn2 un 2 η ∑ kn2 bn 2
n
(3)
n
Time series ε t (see Fig. 4) can be obtained from numerical simulatons of the model [10], and dissipation bursts
can be found through the condition ε t ε th , where εth
is a suitable threshold value for ε t .
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FIGURE 4. Time series of energy dissipation ε t for the
MHD shell model.
FIGURE 6. PDFs of energy dissipation fluctuations, for the
MHD shell model at different timelags τ .
SXR data.
CONCLUSION
In this paper, we performed a statistical analysis of the
soft X-ray bursts produced by solar flares and compared
the results with a shell model of MHD turbulence. As already evidenced in previous papers[10, 11], we point out
that the waiting time distribution displays a power law
tail, both for flare SXR bursts and for energy dissipation
bursts in the MHD shell model.
Besides the waiting time distribution, we investigated
the scaling behavior of SXR intensity fluctuations. We
showed that the PDFs of these fluctuations are characerized by the presence of wide, non-gaussian tails and
have the same shape at different timescales. A very similar behavior was found for PDFs of energy dissipation
fluctuations in the MHD shell model.
In our opinion, these results suggest that solar flares
could represent bursty dissipative events of MHD turbulence, as proposed by Boffetta et al. [10]. Following this idea, the power law behavior of the WTD can
be attributed to long-range correlations arising from the
nonlinear dynamics acting in the system, while the nongaussian PDFs can be interpreted as the result of strong
events accumulating on the dissipative scale through the
occurrence of nonlinear interactions.
FIGURE 5. Distribution of waiting times between successive dissipative bursts in the MHD shell model. The solid line
represent a power law with an exponent β 2 07.
Boffetta et al. [10] already showed that dissipative
bursts in the MHD shell model reproduce the statistical
properties of X-ray solar flares, that is, the power law
distributions for peak flux, total energy, durations and
waiting times. The threshold chosen in the present paper
to select dissipation bursts and calculate waiting times
is given by εth ε t 3σ where ε t and σ are
calculated in the same way as for GOES SXR data. The
waiting time distribution (see Fig. 5) is characterized by
the presence of a power law tail with a scaling exponent
2 07 0 10.
As in the case of GOES SXR data, we also calculated the PDFs of standardized fluctuations of energy dissipation at different lagtimes τ , that is, δ E τ δ ετ δ ετ δ ετ2 1 2, where δ ετ ε t τ ε t . Fig. 6
shows that the PDFs of δ Eτ have strong non-gaussian
tails and don’t change their shape significantly as τ
varies, in good agreement with the behavior observed for
ACKNOWLEDGMENTS
We thank Luca Sorriso-Valvo for useful discussions.
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