# A burst with double radio spectrum observed up to 212 GHz

C.G. Giménez de Castro1,2G.D. Cristiani2,3P.J.A. Simões6C.H. Mandrini2,3E.
Correia1,4P. Kaufmann1,5 1 CRAAM, Universidade Presbiteriana Mackenzie,
01302-907, São Paulo, Brasil; email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
2 Instituto de Astronomía y Física del Espacio, CONICET-UBA, CC. 67 Suc. 28,
1428, Buenos Aires, Argentina; email<EMAIL_ADDRESS><EMAIL_ADDRESS>
3 Facultad de Ciencias Exactas y Naturales, FCEN-UBA, Buenos Aires, Argentina
4 Instituto Nacional de Pesquisas Espaciais, São José dos Campos, Brazil
5 Centro de Componentes Semicondutores, Universidade Estadual de Campinas,
Campinas, Brasil
6 School of Physics & Astronomy, University of Glasgow, Glasgow, Scotland

###### Abstract

We study a solar flare that occurred on September 10, 2002, in active region
NOAA 10105 starting around 14:52 UT and lasting approximately 5 minutes in the
radio range. The event was classified as M2.9 in X-rays and 1N in H$\alpha$.
Solar Submillimeter Telescope observations, in addition to microwave data give
us a good spectral coverage between 1.415 and 212 GHz. We combine these data
with ultraviolet images, hard and soft X-rays observations and full-disk
magnetograms. Images obtained from Ramaty High Energy Solar Spectroscopic
Imaging data are used to identify the locations of X-ray sources at different
energies and to determine the X-ray spectrum, while ultra violet images allow
us to characterize the coronal flaring region. The magnetic field evolution of
the active region is analyzed using Michelson Doppler Imager magnetograms. The
burst is detected at all available radio-frequencies. X-ray images (between 12
keV and 300 keV) reveal two compact sources and 212 GHz data, used to estimate
the radio source position, show a single compact source displaced by 25′′ from
one of the hard X-ray footpoints. We model the radio spectra using two
homogeneous sources, and combine this analysis with that of hard X-rays to
understand the dynamics of the particles. Relativistic particles, observed at
radio wavelengths above 50 GHz, have an electron index evolving with the
typical soft–hard–soft behaviour.

###### keywords:

Radio Bursts, Association with Flares; Radio Bursts, Microwave; X-Ray Bursts,
Association with Flares; Flares, Relation to Magnetic Field; Chromosphere,
Active

##  1 Introduction

High frequency radio observations, above 50 GHz, bring information about
relativistic particles (see e.g. Ramatyetal:1994 and Trottetetal:1998).
Moreover, the efficiency of synchrotron emission, responsible for the radio
radiation, increases as the electron energy increases, contrary to the
bremsstrahlung mechanism which is the origin of the Hard X-ray (HXR) emission
[White et al. (2011)]. This makes observations at high frequencies very
attractive for the analysis of high energy particles. For typical magnetic
fields on the Chromosphere and mildly relativistic electrons, gyrosynchrotron
theory expects a peak frequency at approximately 10 GHz. Therefore the caveat
of submillimeter observations is that flare emission becomes weaker as the
observing frequency increases. At the same time, at high frequencies, earth
atmosphere becomes brighter and absorbs much of the incoming radiation.
Notwithstanding some X-class flares have shown a second spectrum besides the
microwaves spectrum, with an optically thick emission at submillimeter
frequencies, sometimes described as an upturn (see e.g. Kaufmannetal:2004,
Silvaetal:2007, Luthietal:2004b). Nonetheless, Cristianietal:2008 found, in a
medium size flare, a second radio component peaking around 200 GHz. We call
these cases double radio spectrum events.

Although different mechanisms were proposed to explain the double radio
spectrum events [Kaufmann and Raulin (2006), Fleishman and Kontar (2010)], the
conservative approach of two distinct synchrotron sources can fit reasonably
well to the observations [Silva et al. (2007), Trottet et al. (2008)]. We
note, however, that observations at higher frequencies are needed to
completely determine the radiation mechanism of those events that only show
the optically thick emission of the second component, like in
Kaufmannetal:2004 and, because of their strong fluxes, have much stringent
requirements.

Therefore, the double radio spectrum bursts may represent a kind of events
whose low frequency component is the classical gyrosynchrotron from mildly
relativistic particles peaking around 10 GHz, and the high frequency component
is also synchrotron emission with peak frequency around or above 50 GHz,
depending on the flare characteristics (in some cases above 400 GHz).

In this work we present a detailed analysis of a double radio spectrum burst
occurred during a GOES M class event on September 10, 2002 and observed in
radio from 1.415 to 212 GHz. We’ll show that the low frequency component is
well correlated with the HXR observed with RHESSI up to 300 keV, hence we can
study the dynamics of the mildly relativistic electrons inside the coronal
loop. On the other hand, the high frequency component, which is also well
represented by an electron synchrotron source, should be produced by a
different particle population and, likely, in a different place.

We first present the data analyzed, explain the reduction methods and give the
clues that justify the interpretations in Section sec:observations. The
spectral analysis, both at radio and Hard X-rays is the kernel of our work,
thus it deserves the entire Section sec:analysis. We divide the interpretation
of the event in two wide energy bands for the mildly relativistic and the
relativistic particles in Section sec:discusion. The consequences of our
interpretations are presented as our final remarks in Section sec:fin.

##  2 Observations

sec:observations

###  2.1 The data

sec:data

The impulsive phase of the solar burst SOL2002-09-10T14:53 started at 14:52:30
UT, in active region (AR) NOAA 10105 (S10E43) and lasted a few minutes. The
flare is classified as GOES M2.9 in soft X-Rays and 1N in H$\alpha$. The burst
was observed by various radiotelescopes around the world: (1) the United
States Air Force (USAF) Radio Solar Telescope Network (RSTN, Guidiceetal:1981)
at 1.415, 2.695, 4.995, 8.8 and 15.4 GHz with 1 s time resolution; (2) the
solar polarimeter at 7 GHz of the Itapetinga Observatory with 12 ms time
resolution [Kaufmann (1971), Correia, Kaufmann, and Melnikov (1999)]; (3) the
solar patrol radiotelescopes of the Bern University at 11.8, 19.6, 35 and 50
GHz with 100 ms time resolution (unfortunately, at that time, the channel at
8.4 GHz was not working); (4) the null interferometer at 89.4 GHz of the
University of Bern with 15 ms time resolution [Lüthi, Magun, and Miller
(2004)] and (5) the Solar Submillimeter Telescope (SST) at 212 and 405 GHz and
40 ms time resolution [Kaufmann et al. (2008)]. All telescopes, except the
SST, have beams greater than the solar angular size. The null interferometer
can be used to remove the Quiet Sun contribution. The SST beam sizes are 4′
and 2′ at 212 and 405 GHz respectively, and form a focal array that may locate
the centroid position of the emitting source and correct the flux density for
mispointing. Below we comment more on this. Since there are no overlapping
frequencies between the different instruments, we have to rely on the own
telescope calibration procedures, which were successfully verified along their
long operation history. All of them, except the SST, claim to determine the
flux density with an accuracy of the order of 10%. When applying the multi-
beam method [Giménez de Castro et al. (1999)] to SST data, the accuracy is of
the order of 20%. We have used these figures in the present work.

HXR data for this event were obtained with the Reuven Ramaty High Energy Solar
Spectroscopic Imager (RHESSI, Linetal:2002). RHESSI provides imaging and
spectral observations with high spatial (0.5′′) and spectral (1 keV)
resolution in the 3 keV–17 MeV energy range. We also used complementary data:
extreme–ultraviolet (EUV) images from the Extreme ultraviolet Imaging
Telescope (SOHO/EIT, Delaboudiniereetal:1995), and full disk magnetograms from
the Michelson Doppler Imager (SOHO/MDI, Scherreretal:1995). During September
10, 2002, EIT was working in CME watch mode. Observations were taken at half
spatial resolution and with a temporal cadence of 12 minutes, which is too low
compared to the flare duration to detect any kind of flare temporal evolution.
Early during the day, and up to $\sim$14:48 UT, data were taken at 195 Å. At
around the flare time, from 15:00 UT, and later, images were obtained at 304
Å. There is only one image at this wavelength displaying the flare
brightening. The Transition Region and Coronal Explorer (TRACE) was not
observing this AR.

###  2.2 The photospheric evolution around the flare time

sec:photosphere

The active region, where the flare occurred, arrived at the eastern solar limb
on September 7, 2002, as an already mature region. It appeared formed by an
unusually strong leading negative polarity and a weaker dispersed positive
following polarity. AR 10105 is the recurrence of AR 10069 seen on the disk
during the previous solar rotation.

Figure 1.: Magnetic field evolution seen in MDI line of sight magnetograms
during September 9 and 10, 2002. All images have been rotated to solar Sun
center and saturated above (below) 300 G (-300 G). Notice that the field
intensity in the negative spot is such that saturation is very clear in MDI
data in Figure fig:MDI-HXR-sst. The arrows in panel (a) point to a chain of
small bipoles located in the moat region surrounding the spot. The green box
in panel (c) represents the field of view of the Figure fig:MDI-HXR-
sst.fig:field-evolution

Adjacent to AR 10105, on its west, there is a dispersed bipolar facular region
and a new AR to its SE (AR 10108, see Figure fig:field-evolutiona). Vigorous
moving magnetic feature (MMF, see HarveyHarvey:1973) activity is seen (compare
Figure fig:field-evolutiona to Figure fig:field-evolutionb and see the movie
mag-field-evol.mpg that accompanies this paper) around the big leading spot
and a moat boundary is visible in Figure fig:field-evolutiona mainly in a E-SE
segment, where magnetic field aggregates at what appears the confluence of
three supergranular cells. In particular, we have indicated within arrows in
Figure fig:field-evolutiona a chain of small bipoles. Though this image is
affected by projection effects, these bipolar regions persist in MDI
magnetograms. Figure fig:field-evolutionb shows that the eastern negative
polarity of the eastermost bipole rotates counter-clockwise around the
positive one. This motion should increase the magnetic field shear in the
region and, at the same time, favour the interaction with nearby bipoles in
the MMFs to its north. Figure fig:field-evolutionc (see a zoom of this image
in Figure fig:MDI-HXR-sst) shows the magnetic field at 14:27 UT on September
10, this is the closest in time magnetogram to the flare occurrence at 14:52
UT. In this magnetogram the negative polarity to the east (which is along the
moat boundary) has decreased in size as flux cancellation proceeds with the
positive bipole polarity. Along this period, flux cancellation also proceeds
between this positive polarity and the nearby negative polarity of the bipole
to the west. Finally, this small bipole is hardly observable by the end of the
day (see Figure fig:field-evolutiond). The magnetic field evolution just
discussed lets us infer the origin of the flare eruption and the location of
the pre-reconnected set of loops and the reconnected loops of which only one
is visible in the XR and EUV data (see Figure fig:MDI-HXR-sst and Section
fig:MDI-HXR-sst).

###  2.3 Time profiles in radio and X-rays

sec:profiles

Figure 2.: Radio data at selected frequencies. The vertical bar on the 2.695
GHz plot indicates the short pulse. We include also 75–300 keV HXR from RHESSI
and 1-8 Å SXR from GOES.fig:profiles

The time profiles are shown in Fig fig:profiles. The flare appears as a single
impulsive peak in GOES data with a start at around 14:50 UT, peak time at
14:56 UT and a total duration of about 20 minutes. The soft-X rays (SXR)
emission has a maximum flux $2.9\times 10^{-5}\ \mathrm{Wm}^{-2}$ (M2.9). We
have derived a temperature $T_{goes}=14.4$ MK and emission measure
$EM_{goes}=3.4\times 10^{48}\ \mathrm{cm}^{-3}$ using the Chianti 6.0.1
Coronal Model, during the HXR impulsive peak interval (14:52:50 - 14:53:00).

The impulsive phase is clearly visible at all radio frequencies covering
almost two decades from 1 to 200 GHz, starting at approximately 14:52:30 UT. A
short pulse is observed at the beginning of the event between 14:52:50 and
14:53 UT (indicated by a vertical bar on the 2.695 GHz panel) with a flux in
excess of the main emission well defined between 1.415 and 4.995 GHz, above
this range it is hardly distinguishable. The short duration and the narrow
spectrum of this pulse reminds us of a similar one that also occurred during
the rising phase of an event (SOL2002-08-30T13:28, GimenezdeCastroetal:2006).
At 212 GHz, the flux density time profile is composed of a single peak, with
maximum at 14:53:20 UT with a total duration of around 2 minutes. There is no
clear evidence of an extended phase like in Luthietal:2004a and
Trottetetal:2011. Besides, the HXR emission above 50 keV ends around the peak
time of the radio emission, although the start time of both are similar, below
30 keV the emission extends longer. We note that this is one of the weakest
GOES events that has a clear submillimeter counterpart. Indeed, as a
comparison, the flare SOL2002-08-30T13:28 (X1.5) had a 212 GHz peak density
flux of 150 s.f.u. [Giménez de Castro et al. (2009)]

We observe a frequency dependent time delay in the radio data, which in the
four second integrated HXR data is not evident. Normalized time profiles
integrated in one second bins are shown in Figure fig:delays (top) at each
frequency with different colours. This figure gives the impression that each
different frequency starts at slightly different times. To further investigate
this, we analyzed the rising times at different frequencies. Since the start
time is subjected to big uncertainties, we compared the time elapsed between a
reference time and the time at which the emission reaches a certain normalized
flux density. Moreover, we find more accurate to measure delays during the
rising phase than during the, almost flat, maximum. We used two different
normalized levels, 10% and 50%. For each frequency $\nu$ we found
$t_{10}(\nu)$ and $t_{50}(\nu)$, the times when the flux density reaches the
10% and 50% relative level respectively. The reference is the 75–300 keV
emission, with $t_{10}^{*}$ and $t_{50}^{*}$ defined when the HXR flux is 10%
and 50% of the peak flux respectively. Delays are then defined as
$\Delta_{10}(\nu)=t_{10}(\nu)-t_{10}^{*}$ and
$\Delta_{50}(\nu)=t_{50}(\nu)-t_{50}^{*}$. To some extent $t_{10}^{*}$ and
$t_{50}^{*}$ are arbitrarily defined, but in doing so we simultaneously have
an indication of the relation between radio and HXR. Delays are determined
with 0.5 s accuracy, derived from the worst time resolution we have in the
radio data. The result is shown in Figure fig:delays (bottom). Above 10 GHz
there is a continuous shift, which is, within the uncertainties, independent
of the level (10% or 50%). This is qualitatively observed in the time profiles
(Fig. fig:delays, top). Previous works have shown that
millimeter/submillimeter emission is delayed from microwaves (see e.g.
Limetal:1992,Trottetetal:2002, Luthietal:2004a), but this is the first time to
our knowledge that a spectrum of the delay is presented. Below 7 GHz the
presence of the short pulse distorts this trend.

Figure 3.: Top: normalized flux densities from 1.415 to 212 GHz. Normalization
fluxes are taken in the interval 14:5310 – 14:5330 UT. The dashed curve is the
normalized 75–300 keV flux. Dotted horizontal lines, represent 10% and 50%
levels. Bottom: Measured delays between radio and HXR. The horizontal gray
band represents the uncertainty at 75–300 keV rising levels.fig:delays

###  2.4 X-Ray imaging and radio-source positions

sec:images

We produced X-ray images using RHESSI data. These images are constructed with
the PIXON algorithm [Hurford et al. (2002)] with an accumulation time of four
seconds (from 14:52:52 UT to 14:52:56 UT) for lower energy bands (below 80
keV), and twelve seconds (from 14:52:52 UT to 14:53:04 UT) for higher energy
bands (above 100 keV). We used collimators 1–6 and a pixel size of 0.5′′. We
considered the following energy bands: 12–25 keV, 40–80 keV, 100–250 keV,
150–300 keV and 250–300 keV. The images show two sources clearly defined (see
Fig. fig:MDI-HXR-sst). There is a loop or arcade connecting the two footpoints
visible in the 12-25 keV energy band image (see Fig. fig:MDI-HXR-sst).

Figure 4.: MDI magnetogram closest in time to the event. The black lines
denote magnetic polarity inversion lines. Blue contours are RHESSI 12–25 keV
for the interval 14:52:52 to 14:52:56 UT, magenta lines are RHESSI 40–80 keV
contour levels at 20%, 50% and 80% of the image maximum for the same time
interval. A contour corresponding to 60% of the maximum intensity of the
closest in time EIT image in 304 Å is also shown (red dashed contour). Green
points indicate the centroid positions of the submillimeter emission from
14:52:40 to 14:54:40 UT (impulsive peak in radio) averaged every 0.4 s. The
yellow dot–dashed circle denotes the absolute uncertainty in the determination
of the radio–source positions ($\sim$30′′), mainly limited by pointing
accuracy.fig:MDI-HXR-sst

Fig. fig:MDI-HXR-sst shows the magnetogram closest in time ($\sim$14:27 UT) to
the flare overlaid by RHESSI contours in the 12–25 (blue contours) and 40–80
(magenta contours) keV energy bands. Also included is a 60% of the maximum
intensity contour of the closest in time 304 Å EIT image (red lines). Black
thin contours correspond to the magnetic polarity inversion line, i.e. they
separate positive from negative line of sight magnetic field. It is evident
that both RHESSI footpoints overlay opposite sign polarities, with one of them
located on the positive polarity, the evolution of which we discussed in
Section sec:photosphere, and the other one on the negative polarity, north of
it. Whithin the uncertainties of the reconstruction method, we did not observe
any displacement in the HXR sources.

We determined the centroid of the sources emitting at 212 GHz every 40 ms
during the impulsive phase of the event, assuming that the source size is
small compared to the SST beam sizes (For a review of the multi–beam method
see GimenezdeCastroetal:1999). At the same time, we corrected the flux density
for mispointing. We note that since beam sizes are of the order of arcminutes,
when they are not aligned with the emitting source the flux obtained from a
single beam may be wrong.

In Figure fig:MDI-HXR-sst we have superimposed the positions of the 212 GHz
burst emission centroids averaged every 0.4 s. They seem to be separated by
25′′ from one of the HXR footpoints. The dot–dashed yellow circle represents
the absolute uncertainty in the determination of the radio source position
($\sim$30”); this uncertainty is mainly due to the radiotelescope pointing
accuracy. Because of this large absolute uncertainty, it is not possible to
determine where the submillimeter source is located inside the loop, at the
footpoints or at the loop top. The two X-ray sources in the flare seem to be
associated with the bright ultraviolet enhancement seen by EIT encircling
them. This brightening would correspond to an upper chromospheric loop also
traced by the lowest energy RHESSI isocontours. Timing analysis does not
reveal a displacement of the 212 GHz emission centroids in a privileged
direction during the impulsive phase of the event. We can infer the
compactness of the submillimeter source or sources because of the low level of
spread, lower than 10′′.

##  3 Spectral Analysis

sec:analysis

Figure 5.: Radio spectra ad selected times. The data, integrated in 1 s bins,
are represented with their error bars. Dashed curves correspond to the low
frequency fitting and dot-dashed curves to the high frequencies fitting.
Continuous lines represent the total solution (low frequency + high
frequency). fig:rspec-fit

Figure fig:rspec-fit shows integrated radio spectra at selected one second
time intervals. The existence of composed spectra is evident after 14:52:50
UT. We distinguish two components that we call the low frequency and high
frequency components. The low frequency one has a peak frequency around 10
GHz, while the high frequency is maximum at around 35 GHz. Because only the
high frequency component shows clearly its optically thin part, we have used
frequencies 50, 89.4 and 212 GHz to compute the spectral index $\alpha_{hf}$
along the event. Figure fig:delta bottom shows $\alpha_{hf}$ in function of
time together with error bars. During the maximum of the event the spectral
index remains stable, but at the beginning and the end a softening is
observed, i.e. the index has an SHS behaviour. To get insight into the
characteristics of electron populations that produced this emission, we have
fitted the data to two homogeneous gyrosynchrotron sources using the
traditional Ramaty:1969 procedure. The suprathermal electron distributions are
represented by a power law with electron indices $\delta_{l\\!f}$ and
$\delta_{h\\!f}$ for the low frequency and high frequency sources
respectively. At each time interval, a model was fitted to the low frequency
and another model to the high frequency, the sum of both was compared to the
data until the best solution was obtained. In general we fixed the source
size, the magnetic intensity, medium density and the electron energy cutoffs
(see Table tbl:parametros). We allowed changes only in the electron indices
$\delta_{l\\!f}$, $\delta_{h\\!f}$ and the total number of accelerated
electrons. The fittings are shown in the Figure fig:rspec-fit. It is evident
that below 9 GHz the fittings are rather poor, which is an indication that the
source is not totally homogeneous [Klein, Trottet, and Magun (1986)],
nonetheless we may be confident on the electron index and number, which depend
on the optically thin part, and on the magnetic field which is defined by the
peak frequency. Moreover, the good agreement with similar parameters derived
from HXR is another indication of the goodness of the fittings (see below).

| Low Frequency | High Frequency |
---|---|---|---
Parameter | Component | Component | Unit
Mag. Field | 380 | 2000 | G
Diameter | 18 | 5 | arc sec
Height | $10^{9}$ | $10^{8}$ | cm
Low En. Cutoff | 20 | 20 | keV
High En. Cutoff | 10 | 10 | MeV
Maximum Total Number of $e^{-}$ | $4.7\times 10^{37}$ | $7.6\times 10^{34}$ |
Table 1.: Fixed parameters used to fit the spectral data and maximum total
electron number derived from the computations.tbl:parametros

Figure 6.:  HXR spectra at selected interval times. Data are integrated in 4 s
bins and represented with error bars. Blue curves represent the thermal
solution, green curves are the broken power law solution and red curves are
the total solution (Thernal+broken power law). Magenta curves represent the
expected emission from the relativistic electrons used to fit the radio high
frequency component. Dashed curves are the background.fig:xspec-fit

HXR spectra were taken from 14:52:00UT to 14:54:00UT in 4 second intervals,
using front detectors 1, 3, 4, 5, 6, 8 and 9, for energies between 3 and 290
keV. We excluded the time bin when there was a change in attenuator state,
from 0 to 1. Due to the high flare activity during this RHESSI observation
interval, we selected the background emission from the subsequent night period
(15:18:48 – 15:22:12 UT). Figure fig:xspec-fit shows the photon flux spectra
at selected time intervals along with the fitting used to calibrate the data.
Using standard OSPEX111see ’OSPEX, Reference Guide’, Kim Tolbert, at
http://hesperia.gsfc.nasa.gov/ssw/packages/spex/doc/ospex˙explanation.htm
procedures we applied a model of a thermal source, including continuum and
lines, and a double power law. We found significant counts up to 300 keV,
during peak time (14:52:59 UT). The thermal component is well represented by
an isothermal source with a mean temperature $T_{hsi}=20.7$ MK and emission
meassure $EM_{hsi}=1.2\times 10^{48}\ \mbox{cm}^{-3}$ during the HXR impulsive
peak interval (14:52:50 - 14:53:00). The break energy remains always below 80
keV, and the electron index below it is always harder than the index above.
The later was used to compare with radio data because electrons with energies
$<100$ keV affect the gyroemission only in the optically thick part of the
spectrum [White et al. (2011)] which we did not try to fit as noted before. We
also have to take into account that HXR emission depends on the electron flux,
therefore for non relativistic particles we should add 0.5 to $\delta_{X}$ to
compare with radio [Holman et al. (2011)], and, since the HXR spectra are
computed below 300 keV, we should compare $\delta_{X}$ with $\delta_{l\\!f}$,
because relativistic electrons are needed to produce gyrosynchrotron emission
for frequencies above 50 GHz. In the bottom panel of Figure fig:delta we can
see the evolution of electron indices. We observe that $\delta_{X}$ (dot-
dashed curve) remains stable at the beginning and increases at the end. The
low frequency index $\delta_{l\\!f}$ remains stable until 14:53:25 UT and is
comparable to $\delta_{X}$, then a sudden change takes place making it harder.
On the other hand, the high frequency index (continuous line) $\delta_{h\\!f}$
is much harder than $\delta_{l\\!f}$ and $\delta_{X}$ as it was observed in
previous works (see e.g. GimenezdeCastroetal:2009, Trottetetal:1998) and it
evolves in the same way as the spectral index $\alpha_{hf}$.

Figure 7.: Top panel: time profiles at 15.8 GHz (dark gray), 75–300 keV (light
gray) and 212 GHz (continuous line). The bottom panel shows the time evolution
of the electron indices together with the values and error bars of the high
frequency optically thin spectral index $\alpha_{h\\!f}$. fig:delta

##  4 Discussion

sec:discusion

The optically thin emission above 50 GHz is produced by relativistic
particles, while the optically thick emission at microwaves and the HXR
observed by RHESSI are emitted by mildly relativistic particles. We roughly
divide the analysis in these two energy bands.

###  4.1 Dynamics of mildly relativistic electrons

sec:mild-relativistic

In order to understand the dynamics of the mildly relativistic electrons we
compare the HXR emission, which it was observed up to 300 keV with the low
frequency radio data. The comparison of the temporal evolution of both sets of
data (see Figures fig:profiles and fig:delta) supports the existence of
trapped electrons because: 1) the duration of the impulsive phase in HXR is
shorter than in radio and 2) the peak time in HXR occurs before the radio
peak, even at low frequencies. Therefore, the HXR time profile is not
necessarily the representation of the injected electrons, since there are
transport effects along the loop, or at least, HXR may represent the injected
electrons that precipitate directly, without being subject to trapping.

We can use the spectral analysis to derive the rate of the injected electrons
in the emitting area in function of time. To do so, we write a simplified
continuity equation that depends only on time, since we are not interested on
how the electron distribution changes in terms of energy, pitch angle, or
depth. In this simplified model, we are interested only in the total
instantaneous number of electrons, $N(t)$, inside the magnetic loop,
incremented by a source, $Q(t)$, from the acceleration site and decremented by
the precipitated electrons, $P(t)$. Therefore, the continuity equation should
be

$\frac{dN(t)}{dt}=Q(t)-P(t)\ .\ilabel{eq:cont}$ (1)

Integrating the above equation in the interval $(t_{i},t_{i+1})$ (with
$i=0,1,2\dots$) and solving for $Q(t_{i})$ yields

$Q(t_{i})\Delta t_{i+1}=N(t_{i+1})-N(t_{i})+P(t_{i})\Delta t_{i+1}\
,\ilabel{eq:injection}$ (2)

with $\Delta t_{i+1}=t_{i+1}-t_{i}$. Since we are comparing $<300$ keV
emission with gyrosynchrotron, we can identify the instantaneous number of
electrons inside the loop $N(t)$ with the trapped particles emitting the lower
frequency component. On the other hand, the particles leaving the volume
$P(t)$ produce the HXR emission observed by RHESSI. In our picture, the low
frequency component is produced all along a loop with a length of $10^{9}$ cm,
while the HXR emission is produced in a narrow slab (see e.g. Holmanetal:2011)
with a very small surface (see Figure fig:MDI-HXR-sst); therefore, we can
neglect the gyroemission produced within this small volume. Furthermore, no
change would be appreciated if one includes the particles responsible for the
high frequency component since they are two orders of magnitude less than
those that produce the low frequency component. (See Table tbl:parametros)

We divided the event in 10 second intervals and assumed that within these
intervals the conditions do not change. The computation of Equation
eq:injection is straightforward and the result is presented in Figure
fig:injection. Since $>100$ keV data have good S/N ratio only between 14:52:50
UT and 14:53:30 UT, (see Figure fig:xspec-fit) the analysis is restricted to
this interval, although is clear from the time profiles that there are
emitting particles before and after. We observe a continuous injection with
two peaks, one at the beginning and the second during the decay of the
impulsive phase. To verify our results, we sum the precipitated electrons,
$\sum_{i}P(t_{i})\Delta t=3.4\times 10^{37}$, and we compare this number with
the maximum number of electrons existing instantaneously inside the loop,
$\max[N(t_{i})]=4.7\times 10^{37}$. The difference between these two numbers
maybe due to the fact that we are limited to the time interval in which the
HXR data are statistically meaningful; therefore, we cannot track the
precipitation until the end of the gyrosynchrotron emission.

Figure 8.: Top: time profiles at 15.4 (blue) and 75–300 keV (red). Bottom: the
precipitated electrons $P(t)\Delta t$ (green); the instantaneous number of
electrons, $N(t)$ (blue), and the injected electrons $Q(t)\Delta t$
(red).fig:injection

We observe that changes in $\delta_{l\\!f}$ (Fig. fig:delta) can be related to
the injections occurred at around 14:52:51 UT and 14:53:15 UT (Figure
fig:injection). The electron index $\delta_{X}$ lies around 5, as
$\delta_{l\\!f}$ for the same period, although, it increases slowly until
14:53:20 UT when it suddenly softens by around 0.5. The difference in time
evolution of $\delta_{X}$ and $\delta_{l\\!f}$ is an indication that the
softening of the former is a consequence of the trapping. Since
$\delta_{l\\!f}$ has two constant values, we conclude that it is not affected
by the medium. In the above analysis we rely upon the parameters derived from
the radio and HXR data fittings. Although we do not claim that the obtained
solutions are unique, the fact that two independent fittings give very
comparable results, give us confidence on them.

The progressive delay of the radio emission observed in Figure fig:delays must
be interpreted differently depending on the frequency range. Between 1 and 5
GHz, the short pulse dominates the emission during the rising phase;
therefore, its contribution should be removed to asses the delay of the main
emission. This may lead to ambiguous results, hence, we preferred not to
analyze delays in this range. In the range between 9 and 30 GHz lies the peak
frequency, i.e. the optical opacity $\tau$ is approximately 1, hence, the
gyrosynchrotron self absorption is critical. The peak frequency ($\tau\simeq
1$) of the low frequency component shifts from approximately 7 GHz to 15 GHz
from 14:52:40 UT to the peak time around 14:5259 UT (Fig. fig:rspec-fit).
Since there is no change in the magnetic field, this can be interpreted by the
accumulation of accelerated electrons inside the loop due to the trapping that
increases its density, which is the dominant factor of the self-absorption
mechanism. The shift makes the low frequencies more absorbed and thus
increases the relative importance of higher frequencies. Therefore, even with
a rather constant $\delta_{l\\!f}$ (Fig. fig:delta) the progressive delay
should be observed. In our fittings at 14:52:40 UT we have $\tau(\nu=7\
\mbox{GHz})=0.7$ and $\tau(\nu=10\ \mbox{GHz})=0.15$. Later on, during peak
time $\tau(\nu=7\ \mbox{GHz})=60$ and $\tau(\nu=10\ \mbox{GHz})=3.7$. We note
that the emission is proportional to $1-e^{-\tau}$, hence, we have a relative
amplification of $(1-e^{-60})/(1-e^{-0.7})\simeq 1.4$ at 7 GHz, while it is
$(1-e^{-3.7})/(1-e^{-0.15})\simeq 7.5$ at 10 GHz. The relative amplification
changes the rate at which the emission rises and, therefore, the time when the
signal reaches a certain level with respect to its maximum.

###  4.2 Relativistic Electrons

sec:relativistic

The centroid position of the emitting source at 212 GHz remains quite stable
during the flare which may imply that the source is compact. Moreover, it is
placed 25′′ far from one of the HXR footpoints (Figure fig:MDI-HXR-sst). A
similar result was obtained by Trottetetal:2008 for the SOL2003-10-28T11:10
flare; during the impulsive phase (interval B in their work) the centroid
positions of the 210 GHz emission lie at approximately 10′′ from the center of
one of the HXR footpoints (250–300 keV), but are coincident with the location
of precipitating high energy protons with energies above 30 MeV seen in
$\gamma$-ray imaging of the 2.2 MeV line emission. Since for our work we do
not have $\gamma$-ray imaging to compare with, we should be cautious because
the uncertainty in position is of the order of the position shift.

Since we observe the optically thin part of the high frequency spectra, the
obtained $\delta_{h\\!f}$ is not affected by the medium and, within data
uncertainties, it must be correct. On the other hand, we assumed a standard
viewing angle $\theta=45^{\circ}$ which gives us a mean value of the magnetic
field $B$ and total number of electrons $N$. Increasing (reducing) $\theta$
results in smaller (larger) $B$ and $N$. Although we cannot rightly evaluate
$\theta$ with our data, we do not expect an extreme value for it since the AR
is located not far from Sun center (E43). Furthermore we did consider an
isotropic electron distribution. The total number of accelerated electrons is
$8\times 10^{34}$ during peak time, and they should not produce enough
bremsstrahlung flux to be detected by RHESSI detectors. We confirmed this by
computing its HXR emission using the bremthick222Developed by G. Holman, last
revision May 2002. Obtained from the RHESSI site:
http://hesperia.usfc.nasa.gov/hessi/modelware.htm program (magenta curves in
Figure fig:xspec-fit) which is orders of magnitude smaller than the mildly
relativistic electron emission and remains below (except during one time
interval) the background.

The spectral index of the high frequency component $\alpha_{hf}$ shows a SHS
behaviour, but in this case we cannot conclude whether its origin comes from
the acceleration mechanism or from the interaction with the medium as before.
We tend to think that the former should be the cause, since these are
relativistic particles and their interaction with the medium should be less
effective. The emission must come from a compact region with a strong magnetic
field and the electron index $\delta_{h\\!f}$ should be harder than
$\delta_{X}$ and $\delta_{l\\!f}$, as is the case. These arguments support the
evidence of the existence of a separated source where relativistic electrons
are the responsible for the emission.

The progressive delay of the radio emission above 50 GHz (Figure fig:delays)
can be interpreted considering the initial hardening of the spectral index
$\alpha_{hf}$. If it is due to the acceleration mechanism that accelerates
first the low energy particles and later the high energy particles, then, the
progressive delay is a consequence.

##  5 Final Remarks

sec:fin

From our simple continuity model, we inferred the evolution of injected number
of electrons, appearing as a continuous injection with two distinctive pulses
separated by approximately 30 seconds. From the timing of those pulses, the
first one produces the HXR and initiates the radio low frequency emission. The
second pulse, while slightly stronger than the first, builds up into the radio
emission but do not contributes to generate more HXR. This description
suggests that the two injection might have different initial pitch angle
distributions, because our fittings do not show magnetic field changes during
the burst. The first injection might be formed by an electron beam aligned
with the magnetic field direction, a fraction of the population is trapped by
magnetic mirroring, while the other fraction enter the loss cone and
precipitates, producing HXR. The second pulse should then be formed by a beam
with a wider pitch angle distribution, or even isotropic, keeping most of the
electrons trapped, producing radio emission, with little precipitation (no
significant increase in HXR).

From our observations we conclude that the radio spectra cannot be explained
by an homogeneous gyrosynchrotron source, therefore we adopted a model based
on two homogeneous sources since it is the simplest hypothesis that fits
reasonably the data. Although we don’t have images to support this assumption,
it is plausible that it is the case. As shown by the magnetic field evolution
discussed in Section 2.2, we can conclude that the flare originates by the
interaction of magnetic loops anchored in the MMF polarities. Magnetic energy
is probably increased in the configuration by shearing motions, in particular,
the rotation of the negative polarity around the positive one. After magnetic
reconnection occurs, two sets of reconnected loops should be present. In the
example we have analyzed, we observe a set of loops in SXR (and also in EUV)
with two HXR footpoints. Considering our description in Section
sec:photosphere, we speculate that the second set of reconnected loops (not
visible) could be anchored in the higher magnetic field positive polarity
located north of the northern HXR footpoint and the negative polarity that
rotates around the positive bipole polarity. Such a set of reconnected loops
would have a larger volume than the ones that are observed; therefore,
considering that the same amount of energy is injected in both sets, the
emission in the second one could be less intense and, then, not visible. This
second magnetic structure is also suggested by the 25′′ displacement of the
emitting source at 212 GHz respect to one of the HXR footpoints (Figure
fig:MDI-HXR-sst). While it has been shown that the reconnection of many
different magnetic loops is more efficient to accelerate high energy electrons
(see Trottetetal:2008 and references therein), we might conclude that this
complex mechanism operates even in medium size events as the one we analyzed
in this work.

Another possible scenario is a loop structure where the low frequency source
represents the coronal part of the loop (with a lower effective magnetic field
strength), and the high frequency source represents the low coronal or
chromospheric footpoints of the loop (with a higher effective magnetic field
strength). Melnikovetal:2011, using simulations of electron dynamics and
gyrosynchrotron emission in a loop structure, demonstrated that two spectral
components can be produced from one single loop, where mildly-relativistic
electrons produce microwave emission in the loop, while relativistic electrons
produce higher frequency (reaching sub-THz frequencies) emission from the
footpoints.

#### Acknowledgements

CGGC is grateful to FAPESP (Proc. 2009/18386-7). CHM acknowledge financial
support from the Argentinean grants UBACyT 20020100100733, PIP 2009-100766
(CONICET), and PICT 2007-1790 (ANPCyT). PJAS is grateful to FAPESP (Proc.
2008/09339-2) and to the European Commission (project HESPE
FP7-2010-SPACE-1-263086). GDC and CHM are members of the Carrera del
Investigador Científico (CONICET), CGGC is level 2 fellow of CNPq and
Investigador Correspondiente (CONICET).

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