Nonthermal Electron Energy Derived from OVSA Radio Burst Spectra
Jeongwoo Lee1 , Gelu M. Nita1 , and Dale E. Gary1
1 Physics
Department, New Jersey Institute of Technology, Newark, NJ 07102
ABSTRACT
We calculate parameters of nonthermal electrons in solar flares from the microwave
flux spectra of 412 flares measured with the Owens Valley Solar Array (OVSA) in
2001-2002. The parameters are: power-law index δ of the electron energy distribution
and total number and energy of the electrons integrated over energy and space. In
the database, we have selected the events with spectral/temporal behaviors indicative
of nonthermal gyrosynchrotron radiation so that the simplified expressions of gyrosynchrotron formulae presented by Dulk and Marsh can be used. We use the measured
spectral index for determining the power-law index, the peak frequency ν ∗ for column
density of nonthermal electrons, and finally the peak flux Sν∗ for the total number of
the electrons. We also employ an empirical relationship between area and flux that
we found from the Nobeyama Radioheliograph (NoRH) data to complement the nonimaging data. The total energy of the nonthermal electrons was computed for two
different minimum energies, 10 keV and 100 keV, in order to see a possible difference
in total energy content between events of soft, abundant electrons and the other case of
hard, but fewer electrons.
The resulting total energy from 588 peaks forms a power-law distribution with index
around −1.2, which is insensitive to the minimum energy adopted for integration. The
distribution reaches ∼1033 erg if integrated above 10 keV and ∼2×1031 erg if above 100
keV. The corresponding total numbers for largest flares are ∼1041 and ∼1038 , respectively. There is an apparent trend that softer events show larger numbers of electrons
concentrated at low energy interval (10-100 keV) and thus total energy content exceeding that of harder events. This trend, however, vanishes or reverses if we restrict the
number integration to higher energies (≥100 keV). We therefore conclude that a plausible acceleration mechanism should have no preferred value of δ in terms of efficiency
of nonthermal electron production. The derived flare parameters are best correlated
with the observed peak flux divided by the square of the peak frequency, so that the
total nonthermal energy integrated above 100 keV is well approximated by ∼(2×1024
erg)(Sν∗ [sfu])2.7 (ν ∗ [GHz])−5.2 . The implications of the derived values of these parameters
for electron acceleration is briefly discussed.
Subject headings: acceleration of particles — radiation mechanisms: nonthermal — Sun:
corona — Sun: flares — Sun: radio radiation — Sun: X-rays
–2–
1.
Introduction
Since nonthermal electrons in solar flares were detected through hard X ray spectral observations (Kane & Anderson 1970), their energetics has been one of the major interests in studies
of particle acceleration. The observed hard X ray (HXR) flux is related to the energy deposition
rate under thick target bremsstrahlung model (Brown 1982), and should be integrated over time to
yield the energy content. Lin $ Hudson (1971, 1976)analyzed HXR observations of the August 1972
series of large flares from ESRO TD-1 and OSO-7 spacecraft to give estimate of total energy in ≥20
keV electrons ranging from 1.5 × 1030 to 1.1 × 1032 ergs. Brown & Hoyng (1975) have analyzed the
large X-ray burst of 1972 August 4, to deduce number and energy as large as 3.5 × 1039 electrons
and 2.0 × 1032 erg above 25 keV, with which they discussed problems of diagnosing flare particle
acceleration mechanism from hard X-ray bursts. Most recently, the X4.8 flare of 2002 July 23 has
been carefully analyzed with the very high resolution spectra obtained with the NASA Reuven
Ramaty High Energy Solar Spectroscopic Imager (RHESSI). Lin et al. (2003) found the energy
deposited by ≥ 10 keV electrons in the rise phase reaches greater than ∼4×1032 ergs without assuming a thermal component, and Lin et al. (2003) estimated ∼2×1031 ergs taking into account the
presence of a hot thermal plasma. Since these energies amount to a significant fraction of the total
energy that are believed to be released by large solar flares ≥ 1032 − 1033 ergs, the acceleration of
nonthermal electrons should be an important form of flare energy dissipation Lin $ Hudson (1971,
1976). The large number of nonthermal electrons also places a strong constraint on the particle
acceleration theory, because the inferred total number even exceeds that of background thermal
particles in a typical flare loop (Miller et al. 1997).
While the HXR flux gives the energy deposition rate, microwave observations provide instantaneous number of electrons emitting gyrosynchrotron and thus play a complementary role to the
HXR in this diagnostic. Gary (1985) analyzed 13 solar flares with microwave flux greater than
500 sfu that were observed mostly with the Berne radiometer at 3–35 GHz, to find the number of
nonthermal electrons in the range from 4 × 1033 to 1 × 1039 . When these numbers are compared
with those obtained with HXR data (obtained with the Hard X-Ray Burst Spectrometer (HXRBS)
experiment on the Solar Maximum Mission (SMM) spacecraft) after proper conversion to the electron production rate, they could be made to agree to each other so long as proper field strengths
are chosen. Lee, Gary $ Zirin (1994) performed modelling of magnetic loops mainly constrained to
reproduce unusual spectra from four X-class flares in 1990-1991, which leads to total number of ≥10
keV electrons ranging 7 × 1035 − 3 × 1038 . Although neither authors presented the corresponding
energy, a rough estimate gives 6 × 1025 to 2 × 1031 ergs if we assume the mean energy of individual
electrons is ∼10 keV. These microwave-based results suggest that the nonthermal electron energy
may distribute in a much wider range than previously found with HXR observations.
In some studies the HXR and microwave parameters gathered for a large number of flares
are binned into each parameter space to obtain frequency distributions of the parameters. As
well documented in a comprehensive study (Crosby, Aschwanden & Dennis 1993), those analyzes
of both HXR and microwave observations have revealed that the frequency distribution of each
–3–
flare parameter commonly appears in a form of power-law with index typically lying between −1.5
and −2.0. While most of the power-law distributions have been obtained with directly observed
quantities, Crosby, Aschwanden & Dennis (1993) showed that the total energy, as derived from
the HXR fluxes and their durations, also forms a power-law distribution, but with a slightly lower
index, −1.53. The result was interpreted based on both stochastic exponential energy build up
model (Rosner & Vaiana 1978) and the avalanche model under the principle of self-organized
criticality (Lu et al. 1993). In the former case, the slope of the frequency distribution can tell the
ratio of the energy build up time to flare interval, while in the latter case, it may constrain the
threshold for magnetic reconnection used in the simulation of avalanche models.
In this paper we derive the flare parameters using microwave spectral data obtained recently
with the OVSA (Nita, Gary & Lee submited, hereafter Paper I). We note that microwave diagnostics
on total electron energy has been utilized less frequently than the HXR diagnostics for a couple of
difficulties including lack of simultaneous imaging and spectral capability. We address the latter
problem to some extent in this paper. First, we utilize our unique ability to tell which flux is
optically thick or thin so that we can choose an observed flux right for the derivation of the desired
physical quantity under the gyrosynchrotron radiation mechanism. This allows us to use the directly
measured the peak frequency and the optically thin spectral slope in deriving the column density
and the electron power-law index, respectively. Second, we utilize an independent estimate of
the source area inferred from NoRH data, to complement the non-imaging spectral data from the
above-mentioned OVSA database.
In §2 we give an overview of the OVSA data and outline the analysis strategy. In §3 we
use NoRH data to give the statistical relationship between the flux density distribution and the
brightness temperature distribution. In §4 we combine the OVSA and NoRH results to deduce the
statistical distribution of physical parameters, and we discuss related issues in §5.
2.
Data and Strategy
The main data set used in this study was obtained from the 412 flares observed with OVSA
during 2001-2002. OVSA is a solar-dedicated interferometer array consisting of two 27-m dishes
and four 2-m dishes, taking daily observations in 16:00-24:00 UT (Hurford, Read & Zirin 1984).
Although the main purpose of the instrument is to provide imaging data, in order to study such a
large number of bursts we restrict ourselves to total power (integrated flux density) data without
spatial resolution. Because of difficulties with the 27-m dishes due to various effects arising from
their restricted field of view, we further restrict our analysis to data from the 2-m antennas, which
measures only total intensity (Stokes I). Details of the instrument, data calibration, and description
of the dataset are given in Paper I.
One of the puzzling findings in Paper I was that the cm radio emissions show an apparent frequency-dependent flux density limit that increases with the square of the frequency on a
–4–
frequency-flux density plot, whereas the dm emissions (see Paper I) do not follow the behavior. On one hand, this characteristic has been used as a criterion for distinguishing the radiation
mechanism. Namely the events with spectral maximum above 3 GHz and showing the frequencydependent limit share similar spectral and temporal behaviors and are due to a common mechanism,
gyrosynchrotron radiation. On the other hand, this finding also brings to our attention what this
frequency-dependent flux limit implies to the flare parameter distribution. We reproduce this
property in Figure 1 where the peak frequencies are plotted against peak flux scaled by ν 2 . It is
obvious that the cm events do not cross the vertical limit (dotted line), which we empirically set at
∼ 300 sfu/GHz2 . The dotted horizontal line represents the dm frequency limit of 2.6 GHz. No event
has been recorded in OVSA data outside of these two empirical limits. We do not consider this
frequency-independent limit in Sν /ν 2 as an artifact, because no instrumental sensitivity threshold
with such particular frequency dependence is known. We also note that such limit is found only
for the peak fluxes and peak frequencies. If we had included, in Figure 1, all fluxes other than the
peak time values, the quantity Sν /ν 2 would have more randomly distributed instead of showing a
systematic trend as presently shown in the figure.
As to why there could be such a limit in microwave flux, we propose a simple idea that Sν /ν 2
is somehow related to a sort of maximum energy that a solar flare can afford. According to the
Rayleigh-Jeans law, the observed flux is given by the source solid angle, Ω, or its area, A and
effective temperature Teff as:
2k
2k
Sν
= 2 Teff Ω = 2 2 Teff A
(1)
ν2
c
r c
where k is the Boltzmann constant, c is the speed of light, and r = 1 AU is the distance from the
source to the observer. The empirical limit we found now becomes a limit of the product of A and
Teff . Note that for optically thin emission Teff should be replaced by the brightness temperature,
Tb = Teff [1 − exp(−τ )], but since we are talking about upper limits, we tacitly assume that the
emission is optically thick. Note that the quantity on the right hand side without some constants,
kTeff A, has the dimension of an energy per column density. We may even assume its association
with the column density, because these quantities are measured at the peak frequency and the peak
frequency is mostly determined by the column density (see §4.2). In this sense, we may naively
interpret Sν /ν 2 as a quantity representing an energy content, and it is presumable that the energy
content per flare has a maximum limit.
Since the source area has an obvious practical upper limit, namely, the largest active region
size, we tentatively estimate this limit from the maximum value of Sν /ν 2 assuming a constant
effective temperature in all flares. For convenience, we introduce dimensionless parameters such as,
ν
S
Teff
, S4 = 4 , T9 = 9
10GHz
10 sfu
10 K
into (1), and then find that the angular source diameter, D = 2 Ω/π, be expressed as
S4
1
D = 0.7
ν10 T9
ν10 =
(2)
–5–
and the empirical limit we found may be mapped into a limit of the radio source size, scaled by an
effective temperature factor:
D T9 1.2 .
(3)
max
This result suggests that in the case of an effective temperature of 109 K, a reasonable temperature
for gyrosynchrotron emission (Dulk & Marsh 1982), the associated source size becomes 1.2 , which
is also a reasonable value compared with the typical size of an active region. A decrease of the
effective temperature of one order of magnitude would require an increase of the source size by a
factor of about 3 in order to maintain the same limit. This compares well with the largest source
size, 1.37 , directly measured by Nobeyama Radio Heliograph, (NoRH), at 17 GHz, according to
the NoRH flare list.
Although the above assumption of the constant effective temperature provides a means for
simple estimate for the limiting source size, we should consider, in a more realistic calculation,
the variations of A and Teff from flare to flare. Under the gyrosynchrotron emission, Teff is a
complicated function of the nonthermal electrons in phase space and magnetic field as well as
frequency (Ramaty 1969). In the optically thick regime, Teff is related to the mean energy of the
nonthermal electrons, alternatively called effective temperature by Dulk & Marsh (1982), and we
can explore the corresponding electron energy and magnetic field. In the optically-thin regime, we
must instead consider the brightness temperature Tb , which is determined by the total emissivity
and cannot be identified with a mean temperature. The spectral slope becomes important in the
optically thin regime, and can be directly related to the electron distribution in phase space.
The strategy adopted in this paper is as follows: (1) as typical, we simplify the problem
in some aspects: the electrons are assumed to have an isotropic pitch angle distribution and a
nonthermal energy distribution in a single power law form. Under this assumption the spectral
slope in the optically thin frequencies can be used to determine the electron power law index directly
without other assumptions. (2) We make use of NoRH data in the next section, which provides
flux density, area, and brightness temperature through imaging observations, to determine the
relationship between brightness temperature and flux density, and apply this empirical relationship
to the OVSA dataset in order to separate Tb from the observed Sν . (3) Knowing the brightness
temperature and the electron power-law index, we can determine the magnetic field contribution
to the observed microwave flux. This allows us to determine the entire set of parameters involved
with gyrosynchrotron radiation.
3.
The Relationship between Flux and Brightness Temperature Inferred from
NoRH Data
We also use the flare list obtained with the Nobeyama Radioheliograph (NoRH) containing
383 flares with peak fluxes at 17 GHz above 10 sfu, observed by NoRH since 1992. The flare list
gives flux densities, source sizes as a ratio of flare area to beam area, and equivalent brightness
–6–
temperatures. To convert the area ratio to area, we assumed a beam size of 20 at 17 GHz.
Although the database provides such quantities as the flux density, brightness temperature and
area for events observed at 17 or 34 GHz, it is, however, not just as simple as finding a correlation
between any two desired quantities. While the quantity presented in Figure 1 is the spectrally
determined peak flux density (i.e. at optical depth τ 1), the 17 and 34 GHz fluxes in NoRH
data are in many cases due to optically thin radiation. As we noted earlier, the optically thin Tb is
highly sensitive to frequency and its magnitude does not imply a mean energy. On the other hand
the area of the optically thin source is independent of frequency, so the diagnostics are valid even
though the burst may consist of multiple sources and may be inhomogeneous to some extent. For
this reason we use the NoRH data to define the distribution of source size A vs. Sν and take the
peak Tb as a quantity to be derived.
We look for power law relationships among the three relevant quantities in Equation (1):
y≡
Sν
,
ν2
A ∼ yα ,
T ∼ y 1−α .
(4)
Thus we want to determine the value of α, which can be determined in the case of the NoRH
database by simply correlating the flux density and the area listed in that database. However, in
order to apply it to the OVSA dataset, we must first check for consistency between the two.
We postulate that the flux density, area, and brightness temperature all separately follow power
law distributions and define the distribution functions of these quantities as
Φy (y) ∼ y −γ ,
φA (A) ∼ A−β ,
ψT (T ) ∼ T −l ,
(5)
respectively, where γ, β, and l are completely general indexes to be related to α. From the conservation of total numbers, using equations 4 and 5 we get
dy γ−1
Φy (y) ∼ A−(1+ α )
(6)
φA (A) =
dA
and
ψT (T ) =
dy γ−α
Φy (y) ∼ T − 1−α .
dT
Comparing Equations (5)–(7), we get the following relations for the indexes:
α = (γ − 1)/(β − 1) and l = (γ − α)/(1 − α).
(7)
(8)
These relations allow us to compute two of the above indexes, given the other two. One of the
indexes, γ can be obtained directly from the OVSA dataset by fitting a power law to the number
2
, as shown in Figure 2. From the figure we see that the flux distridistribution versus Speak /νpeak
bution is indeed fit well with a single power law, of index γ = 1.56 ± 0.04. However, at least one
other index is needed, and for that we require spatially resolved observations.
It is straight-forward to check that the NoRH database of 383 flares with peak fluxes at 17
2
distribution of
GHz above 10 sfu agrees with this distribution. Figure 3a displays the Speak /νpeak
–7–
this data set. Although all observations were at the same, generally optically thin frequency, 17
GHz, the power-law index of the distribution, 1.63 ± 0.07 is in excellent agreement with the value
of γ determined from OVSA data.
We also estimate the power-law index of the source size distribution, available from the same
dataset. Although this dataset gives the distribution of the source sizes at 17 GHz, we assume that
only the normalization constant, not the slope itself, may be affected by this restriction. This is
reasonable because, as we mentioned earlier, the source size becomes independent of frequency for
optically thin emission. From Figure 3b we estimate that the power-law index of the source size
distribution is β = 2.57 ± 0.23. Plugging these results into equation 8, we get α = 0.40 ± 0.07
and l = 2.04 ± 0.22. To test the consistency of our assumptions, we plot in 3c the distribution
of brightness temperatures. The power-law index of this distribution, 1.65 ± 0.09, agrees with the
above-estimated l = 2.04 ± 0.22 within the 2σ limit. Figure 3d provides a direct estimate of the
2
dependence. Its value, 0.32 ± 0.2, although with large
power law index of the area-Speak /νpeak
uncertainty, is again close to the estimated α = 0.40 ± 0.07. We conclude that it is statistically
valid to consider the distribution of bursts versus flux density as a product of two other power law
distributions of source area A and effective temperature Teff .
4.
Analysis of the OVSA Data
In this section we use the above-found relationship between flux and effective temperature to
derive parameters of flare electrons such as electron energy power law, total number and energy of
nonthermal electrons. We have already selected the events likely due to gyrosynchrotron emission.
For convenience, we use Dulk & Marsh’s (1982) simplified expressions for gyrosynchrotron radiation
which assumes the electrons are accelerated above 10 keV in a single power-law electron energy
distribution with isotropic pitch angle distribution in the following form:
n(E) = N (δ − 1)E0δ−1 E −δ .
(9)
Dulk & Marsh (1982) give simplified expressions for nonthermal gyrosynchrotron radiation for the
x-mode and for 2 ≤ δ ≤ 7, θ ≥ 20◦ and ν/νB ≥ 10 where the eigenfrequency νB = 2.8 × 106 B is
in Hz, magnetic field B, in gauss, and N is the electron number per unit volume in cm−3 . Within
the validity range of the Dulk & Marsh formulae, we will determine the volume and thus the total
volume integrated number Ntot = N V and the corresponding total energy Etot .
4.1.
Electron Power-law Index
The electron power law-index δ can be directly determined from the observed spectral index
of gyrosynchrotron radiation, since the spectral index of the optically thin microwave radiation is
determined by the emissivity, ην . According to Dulk & Marsh’s simplified expression, the microwave
–8–
emissivity is given by
Sνthin ∼ ην ∼ ν 1.22−0.90δ .
(10)
We can thus determine the electron power-law index δ from the measured optically thin spectral
index q by δ = (q + 1.22)/0.9 without any further assumption.
In Figure 4 we plot the resulting distribution of the electron power-law index for three groups
of microwave bursts. In the left panel, the groups are divided according to the peak flux. The
darker grey, grey, and lighter grey areas show all data with Sνpeak ≥ 10 sfu, ≥300 sfu, and ≥3000
sfu, respectively. The maximum population is found near δ ≈ 4.0 with a trend that weaker events
show a wider distribution extending to higher index and stronger events show a more confined
distribution at 2 < δ < 4. The trend of stronger events having harder index is more obvious when
we plot the distributions in groups divided according to the scaled flux, y, as shown in the right
panel of Figure 4. The shape of distribution shows little variation with groups.
We compare this result (Fig. 4) with studies of HXR spectral index. Dennis (1985) shows that
the distribution of HXR spectral index at a given peak count rate gets narrower with increasing
count rate with a trend of harder index at higher count rates, which is similar to the present result
for microwave spectrum. McTiernan & Petrosian (1991) showed that spectral indexes from hard
X rays (HXRBS/SMM) form a broader distribution than those obtained from gamma ray bursts
(GRS/SMM). If we assume that the GRS data represent higher energy electrons than the HXRBS
data do, the flux-dependent distribution of the electron power-law index shown in Figure 4 could be
qualitatively consistent with their result. In case of thick target bremsstrahlung, the spectral index
of hard X ray spectrum is smaller than the electron power law index by about unity (Brown 1971).
Then the typical HXR spectral index ≈ 3.5 − 4.0 (as read from McTiernan & Petrosian’s Fig. 1)
would correspond to δ ≈ 4.5 − 5.0. This is within the range of our result that microwave spectral
index mostly implies 2.5 < δ < 5 with a weak trend of the indexes from microwave spectra being
smaller than those from HXR spectra. This is also consistent with earlier result that microwave
emitting electrons appear harder than HXR electrons (e.g. Silva, Wang & Gary 2000).
From this point on, we restrict our investigation to the events with y > 0.5 so that we can
proceed with those events obeying the power-law behavior (Fig. 2). We also have to exclude the
events with δ ≤ 2 for which the mean energy will diverge (see below, eq.[14]). We further limit to
the events with δ < 6, so that we can safely fit the observed spectrum to nonthermal single powerlaw distribution as we assumed. If a spectrum otherwise has a too steep slope, confusion with
thermal component may occur. Originally we have 770 bursts from 412 flares, and after applying
these criteria, the final dataset consists of 588 data points.
4.2.
Total Number and Total Energy
To determine the total number or energy, we should be able to separate the contributions
of brightness temperature and area to the observed peak flux and peak frequency. For this, our
–9–
analysis will rely on our earlier result that, statistically, there is a power law relationship between
the flux and Teff and the flux and area, so that we can separate these two contributions. Although
such a relationship may not apply to individual events, we expect that it can serve as a means to
derive statistical properties. Based on the result given in the previous section, we assume
TB,peak = 7 × 108 K
S
4, peak
2
ν10, peak
0.60
.
(11)
This assumed brightness temperatures are then compared with theoretical effective temperature
(the brightness temperature in the optically thick regime), which is given by Dulk & Marsh (1982)
as
ν 0.50+0.085δ
Teff ≈ 2.2 × 109 10−0.31δ | sin θ|−0.36−0.06δ
.
(12)
νB
By equating (12) with (11), we obtain the magnetic field B. The viewing angle θ is another
unknown, but we propose to use the angle-averaged quantities over some range (e.g. 20◦ –70◦ ). This
is of course not a good approximation for other studies like the study of gamma ray directivity
(McTiernan & Petrosian 1991), where gamma rays are due to more deeply penetrating and therefore
sensitive to the viewing angle. Our justification is that microwaves (as far as optically thick part is
concerned) are emitted from electrons occupying the whole loop in which case magnetic field vector
spans a wide range of viewing angle.
We finally make use of the peak frequency which is largely determined by the column density
N L along the line of sight. Dulk & Marsh (1982) give a simplified expression for the peak frequency
as
(13)
νpeak ≈ 2.72 × 103 100.27δ | sin θ|0.41+0.03δ B 0.68+0.03δ (N L)0.32−0.03δ .
We derive N L from this equation with the measured νpeak and the above-determined magnetic field
B. We then combine the N L with the source area Ω = A/1AU2 , to determine the total number of
nonthermal electrons by
Ntot = (N L)A
(14)
The total energy of the nonthermal electrons would be determined by multiplying the total
number N by the mean energy distribution E:
Etot ≡ Ntot E.
(15)
Since we assume a power law electron energy distribution, the mean (over energy space per volume)
energy of individual electrons is defined as:
∞
E n(E)dE
δ−1
=
E = E0∞
E0
(16)
δ
−2
n(E)dE
E0
where E0 is the low cutoff energy. Because of the power-law nature, the mean energy E from all
flares appear in a distribution steeply decreasing from E0 . Thus the total energy, in spite of its
– 10 –
dependence on δ, is largely determined by the total number. (Lin et al. (2003)claimed that 10-100
keV takes bulk (10-50%) of the total flare energy.)
We plot the frequency distribution of the total number and energy in Figure 5. The top left
panel shows the total number of electrons above 10 keV, which obeys a power law starting from
∼1033 and ending at ∼6×1042 with index −1.15 ± 0.02. The top right panel is a result of redoing
this procedure but counting only the electrons above 100 keV mainly to see the effect of the low
energy bound in counting the nonthermal energy content. In the latter case, the total number of
flare electrons appears in a power law as steep as −1.23±0.02 reaching ∼3×1034 . The slope changes
only a little with an order of magnitude change of E0 . The bottom panels show the total energies
of nonthermal electrons again for two the cases of E0 . They also form a power law distribution
with index of −1.18 ± 0.026 extending from ∼ 1025 erg to ∼ 1033 erg for E0 = 10 keV, and a little
steeper distribution reaching 3 × 1031 erg with index −1.24 ± 0.02 for E0 = 100 keV.
As a comparison, Crosby et al. (1993) showed that total energies in electrons above 25 keV as
derived from HXRBS data appear in a power law with index −1.53 ± 0.02 within the range: 1028
erg Etot 1032 erg. Our result, if converted to those for electrons above 25 keV, is a power law
with index −1.21 ± 0.02 within the range: 1023 erg Etot 1033 erg. Our microwave-based results
are therefore similar to theirs based on HXRs. Especially, the upper limit of the energy distribution
µ
X 1032 erg). As a few
1033 erg vs. Etot
agrees to each other within an order of magnitude (Etot
distinctions, the power law distribution of total energy is less steeper (Γµ ≈ −1.21 vs. ΓX ≈ −1.53),
µ
X 1028
1025 erg vs. Etot
and the power-law distribution extends to much lower limit (Etot
erg) by apparently three orders of magnitude. The latter could be due to the greater sensitivity
of microwave observations to nonthermal electrons than HXR observations. Our distributions
shown in Figure 5 may have already included sub-flares (1029 ergs), which were not detected in
previous HXR observations. The origin of the differing power law index is of new concern. As
a preliminary speculation, we consider that it originating either from intrinsic difference between
HXR and microwave emitting electrons (i.e., precipitating vs. trapped populations, respectively,
Lee 2003) or due to an influence of the greater sensitivity of microwaves to weaker events on the
overall distribution of the flare parameters.
4.3.
Derived Total Energy vs. Directly Measured Parameters
In Figure 6 we correlate the derived total energy with the directly observed parameters: the
spectral index, peak frequency and peak flux, in order to know which parameter best represents
the energy content of nonthermal electrons. We plot the data points as dots again for two cases
of E0 assumed: E0 = 10 keV in the top panels and E0 = 100 keV in the bottom panels. The line
in each panel represents a linear fit to the data, and the number written in the upper part of each
panel is the slope of the fit.
In the upper first panel, we see a trend that the derived total energy has in fact correlates with
– 11 –
δ in a way that softer event have higher derived total energy. This means that a softer event has
a large number of electrons (mostly confined within low energy interval, say, 10–100 keV) and can
outnumber the harder events in terms of the total energy content. This happens as a feature of
single power-law approximation. and thus we show, in the lower first panel, the other case where
energies only in the ≥100 keV electrons are integrated to give the total energy. The dependence on
δ is largely removed in this case. If we increase E0 further to 200 keV, the trend is almost unseen,
and above 300 keV the trend even reverses in a way that harder events show larger total energies.
We therefore conclude that, in terms of total content of the high energy (≥200 keV) electrons, there
is no particular value of δ preferred during flare acceleration.
The second column shows the correlation between the total energy and the peak frequency. The
peak frequency generally increases with the column density of nonthermal electrons and magnetic
field. Absence of good correlation in the second columns regardless of E0 implies that both the
column density and magnetic field strength may not be an important factor in governing the total
nonthermal energy produced in solar flares.
Finally, in the third column, we find a good correlation of the derived total energy with the
peak flux, and actually a little better correlation is with the scaled flux y = Sν /ν 2 , as shown. As
we raise E0 to higher values, the correlation gets even stronger. In the last panel, for instance, we
find Etot integrated above 100 keV is well approximated by (24.29 ± 0.05 erg)y 2.69±0.06 . To keep
our interpretation of the scaled flux (§2), it is likely that the strength of a flare is not simply a
matter of efficiency of particle acceleration per volume, but how wide area the particle acceleration
takes place is also important.
5.
5.1.
Discussion
Our Diagnostic Procedure
One of key assumptions for this microwave diagnostics is that the peak brightness temperature
should have a power-law relationship with the scaled flux, Sν /ν 2 or, equivalently, with area, (eq.11)
as partly demonstrated by the empirical relationship from NoRH. The real reason we needed this
assumption is not that the are should be known in deriving the total number, but that we need to
know the unmeasured magnetic field, one of the two major factors determining the gyrosynchrotron
radiation. We used the empirical relationship from NoRH data, from power-law distributions.
Such approach may not be adequate for analyzing an individual event, but here used to derive a
statistical property. Our assumption Teff (Sν /ν 2 )0.6 means that the rest dependence comes from
the area variation with flux is allowed to that extent. Use of y ≡ Sν /ν 2 as a central parameter
here is motivated by several factors that it has many distinct statistical properties (maximum
limit independent of frequency, obeying a well developed power-law and a good criterion giving
organization of δ distribution) and it could be interpreted as some sort of effective energy. We
limited application of this assumption to only flares with y < 0.5 where all the statistical properties
– 12 –
seem well observed.
Another important factor in the result would be the assumption that the observed gyrosynchrotron emission is entirely contributed by nonthermal electrons in single power-law energy distribution starting from 10 keV extending to infinite energy. This is a built-in parameter in the
simplified expressions for nonthermal gyrosynchrotron radiation by Dulk & Marsh (1982). Real
flares would have a more complicated than single power-law energy distribution, which nevertheless was employed here to enable statistical study of a large database. As a result the total number
and thus total energy in our calculations is very sensitive to the electron power law index. As an
example, for a typical large flare with peak flux, 104 sfu, and peak frequency, 10 GHz, the total
energy comes out as Etot ≈ 2.5 × 1032 , 3.2 × 1033 , 1.0 × 1035 ergs for δ = 2.5, 3.0, 3.5, respectively.
(cf. White et al. 2003, analysis in 2002 July 23 flare). A differing result can be found in a more realistic case of either a double power-law distribution (Brandenbrug et al. ) or thermal/nonthermal
combination (Benka 1991), in which case we are using some intermediate power-law index (see
below). The limitation of single power-law assumption is common to both HXR and microwaves.
For instance, the energy of nonthermal electrons during the rise phase of the 2002 July 23 flare has
been estimated to differ by a factor of 20 depending on whether or not the thermal component is
assumed to co-exist (Lin et al. 2003). To reduce this ambiguity associated with the existence of
thermal population, we have confined our investigation to the events with 2 < δ < 6, i.e. the more
obvious nonthermal events.
5.2.
Upper Limit in the Nonthermal Energy Distribution
The present result for the upper limit in the distribution of total energy ∼1033 erg) is almost
comparable to or exceeds by an order of magnitude the previously known from HXRs: ∼ 1032
erg above 25 keV for the 1972 August series flares (Lin & Hudson 1971, 1976) or ∼4×1032 erg
for the 2002 July 23 flare (Lin et al. 2003). This is somewhat unexpected considering that the
energy is obtained by integrated the instant energy deposition rate over the whole flare period, and
then the microwave prediction should be lower than the HXR total energy in general. What we
could expect though is that the peak time spectra and the derived energy could be comparable to
the time integrated energy deposition rate if energy deposition rate rapidly rises to the peak time
and decays afterward. In this case both quantities could be comparable in an order-of-magnitude
calculation. Another context that should be considered is (Even though τa > τd ) is that microwaves
and HXRs may represent the electrons trapped and the precipitating electrons, respectively. Thus
a ratio comes in depending on the mirror ratio and pitch angles. If electrons are accelerated in the
corona and precipitate into the chromosphere by 10% to emit thick target bremsstrahlung at the
footpoints, then the agreement between the present finding of maximum electron energy ∼ 1033
erg and the previously known HXR result ∼ 1032 erg may be excellent. In any case, for the various
reasons mentioned above the numbers derived here should be taken as rough estimates, Considering
this nature, the agreement should be rough basis only. and the agreement should be understood
– 13 –
only that level. Important future improvement would be individual event, a more detained study
More accurate estimate (total number derived from HXR to actually accelerated total number)
would include the ratio of acceleration time vs. loss time scale (Lin & Hudson 1971). Alternatively
convert the microwave flux to particle per time, like Gary (1985).
5.3.
Statistical Properties of Flare Parameters
The derived total nonthermal energy per flare show a power law like other flare parameters
do (see review in §1), but it extends over a wider range and with a lower index ∼ −1.2 than any
previously known power law distribution. The power-law index only weakly depends on the low
energy cut-off, E0 . Therefore the power law itself is robust. As to why it makes a difference with
HXR distribution (Crosby et al. 1993), we may consider two opposing possibilities. One is that
microwave diagnostics lead to an overestimate of energy for large flares as biased to longer trapping
time, which would then result in a more extended power-law distribution to higher energies at a
lower index. It should however be noted that the total energy from HXRs is obtained by time
integration of the whole lightcurves, and therefore can also be affected by a longer duration of
flares. Alternatively we suspect that that microwave has a greater sensitivity to weaker nonthermal
events, to show a wider distribution extending to a much lower energy with a lower index. Either
the detector threshold or data selection criterion can indeed affect the slope of the frequency
distribution. If we proceed with the current value of the index ∼ −1.2 and the stochastic energy
build-up model (Rosner & Vaiana 1978), we find that the flaring time interval should be longer
than the build-up time by a factor of five, instead of two as given by Crosby et al. (1993).
We also note that the the power-law with index -1.8 was found mostly for directly observed
quantities. At radio wavelengths, monochromatic flux at a fixed frequency (Akabane 1956) or
the maximum flux distribution at the peak frequency (Paper I) show such a power law with an
index around 1.7, and the scaled flux, Sν /ν 2 , in the present study also shows a power-law with
index around −1.56 ± 0.04. A series of flare parameter studies based on the same HXR data set
(HXRBS/SMM) presented slightly differing slopes depending on the selection criterion (Crosby et
al. 1993). Also a relatively low slope was found for a derived quantity such as total energy in HXR
emitting electrons, which is derived after multiplication of energy deposition rate and duration,
each of which has its own distribution. In the present case, the mean energy and area showed
much steeper distributions with index −3.6 ± 0.34 and −2.57 ± 0.23, respectively. Presumably the
distributions of these auxiliary parameters such as mean energy and area play a role in shaping the
resulting distribution of total electron energy, although they show steep distribution. Therefore, a
more careful determination of these parameters could be important to establish the universality of
the power-law distribution of major flare parameters (cf. Wheatland & Sturrock (1996)).
– 14 –
5.4.
Electron Acceleration Mechanism
We naturally consider the electron power-law index δ as a key parameter in the electron
acceleration problem. The distribution of δ we found for three ranges of y shows a weak trend
that strong (y > 30) events are mostly hard distributions (Fig. 4), which seemed to suggest that
acceleration efficiency goes with δ. However the trend almost vanishes in our investigation confined
to relatively harder events 2 < δ < 6. it came to our attention that moderate (y < 30) and weaker
(y < 3) events have both soft and hard electrons, namely, they show δ in a wide range, and the
number of strong (y > 30) events is really small. A next surprising result is that the derived total
energy Etot lacks a correlation with the electron index δ as shown in Figure 6. As a main reason,
in a softer event, abundant number of electrons reside in a relatively low energy range (say 10–100
keV), which offsets the less contribution from high energy (≥100 keV) electrons in small number.
With the present result alone, there seems to be no preferred value of δ for strong flares in terms of
total nonthermal electron energy. Both an event with abundant softer electrons and another with
harder and fewer electrons can produce strong flares.
We compare this result with the stochastic electron acceleration model, in particular, that
of Hamilton & Petrosian (1992), which demonstrated how the characteristic shape of electron
energy distribution, i.e. n(E) in our notation, depends on the relative importance of acceleration
and Coulomb collisions. They showed that in a collisionless environment, an efficient stochastic
acceleration produces abundant electrons below a high energy cut-off (determined by the lowest
frequency of plasma turbulence) above which the electron number density drops rather rapidly. We
note that this kind of acceleration scenario may produce the events that we identified as large Etot
with (apparently) softer electrons. There is no difficulty in explaining the opposing case of large
Etot derived from harder events in terms of any acceleration mechanism efficient at higher energies.
We therefore conclude that our result of Etot being insensitive to δ could imply the interplay of the
acceleration and Coulomb collisions in various environment or multiple acceleration mechanisms
producing electron energy distribution in differing shapes.
JL has been supported by NASA grants NAG5-10891 and NAG5-11875. The OVSA is supported by NSF grants ATM-0077273 and AST-0307670 to New Jersey Institute of Technology. We
gratefully acknowledge the open data policy of NoRH.
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This preprint was prepared with the AAS LATEX macros v5.0.
– 17 –
Fig. 1.— The distribution of peak flux density divided by the square of peak frequency. All cm bursts,
(C-diamonds, CD-squares), and almost all D-type bursts (triangles), have the same limit of this ratio that
we empirically set at ∼ 300 sfu/Ghz2 ,represented by the dotted vertical line. The dotted horizontal line
represents the dm frequency limit of 2.6 GHz. No event has been recorded in OVSA data above these two
empirical limits. The lack of events in the left-down corner of the plot is just an artifact due to the 1 sfu
frequency-independent sensitivity limit of OVSA.
– 18 –
2
Fig. 2.— The Speak /νpeak
distribution from OVSA data above 2.6 GHz. The power-law index of this
distribution is 1.56 ± 0.04 the same as, within the statistical limits, the one derived from NORH observations
at a single frequency.
– 19 –
2
Fig. 3.— (a)- The Speak /νpeak
distribution from NORH data at 17 GHz. A power-law index of 1.63 ± 0.07
is found. (b)-Source area distribution from NORH. Area is expressed in units of NORH beam area, which
is about 20” at 17 GHz. The tail of the distribution obeys a power-law with an index of 2.57 ± 0.23. (c)Brightness temperature distribution from NORH. The power-law index of this distribution is 1.65 ± 0.09.
2
(d)-The correlation plot of source area and Speak /νpeak
at 17 GHz from NORH data. The distribution is
best fitted by a power-law with an index of 0.32 ± 0.02.
– 20 –
Fig. 4.— Distribution of the electron power-law index derived from the optically-thin spectral slopes of
the OVSA spectra. These index distributions are shown at three different groups according to the flux (left
panel) and according to the scaled flux, y ≡ Sν /ν 2 (right panel).
– 21 –
Fig. 5.— The total number (upper panel) and total energy (lower panel) distributions of nonthermal
electrons derived from the OVSA microwave spectra. The inset in the bottom panel shows mean energy E
of the nonthermal electrons.
– 22 –
Fig. 6.— Derived total energy vs. directly observed flare parameters. Top panels are energy integrated
from E0 = 10 keV and the bottom panels, energy integrated above E0 = 100 keV. From the left to right
columns the scatter plots show correlation of the total energy with the (a)spectral index, (b) peak frequency,
(c) the scaled peaked flux, y, respectively. The number put in each panel represents the slope of the linear
best fit.