Figure 1 (click here) shows a white-light image and a magnetogram of AR 5417 taken
at Big Bear Solar Observatory (BBSO). The east limb can be seen on the
left side of the images, and lines of solar longitude (in steps of
) and latitude (south 
and 
) are shown
superposed. The leading sunspot shows positive polarity and the
following sunspot shows negative polarity. The active region is
surrounded by areas of negative polarity. The radio image at 3.6 GHz
covering 256'' (cf. Fig. 4), which was reconstructed from the
observations taken between 21:05 UT and 23:58 UT, is superposed for
comparison. Note that the radio signal shows two sources, one of them
coincides with the active region, while the westward source coincides
with the location of an earlier M3.1 flare which erupted at 20:38 UT,
about half an hour before our observing time, reached its maximum in
soft X-rays at 20:52 UT (about 10 min before our observations began),
but took several hours to decay. The Big Bear images were taken at
20:01 UT, before the flare occcured.
Figures 2 (click here) and 3 (click here) show two image reconstructions of
the left-hand circular polarization using conventional methods where
spectral information is not used. The contours in the figures are in
brightness temperature, 
, expressed in a logarithmic scale (contour 4:
, contour 5: 
, contour 6: 
). Figure 2 (click here) shows a reconstruction using the CLEAN algorithm,
while Fig. 3 (click here) shows a reconstruction using only the spatial part of
our MEM algorithm (
). The two reconstructions show a single radio
source extending on the spatial axis scale from about 0'' to +100'' at
frequencies below about 2.5 GHz. Both show that the active region centered at
about +29'' is visible at all higher frequencies, and they both show the
flare-related source to the right of the active region. Other features are
sidelobes, which are more pronounced in the corresponding dirty map not
shown here. The largest sidelobe
temperatures are about one order of magnitude smaller than the
temperatures of the active region; both reconstructions suppress
sidelobes by about one order of magnitude.
Figure 4 (click here) shows the reconstruction of the same radio data using the additional spectral information. The spatial/spectral MEM algorithm reconstruction shows the same sources seen in the two previous figures, but the sidelobes are suppressed by almost two orders of magnitude, which shows that including the spectral information greatly improves the dynamic range. This improvement is best illustrated by directly comparing brightness temperature maps at a given frequency as in Fig. 5 (click here) which shows maps at frequencies (a) 2.0 GHz, (b) 5.0 GHz, (c) 7.0 GHz, and (d) 9.0 GHz derived from the spatial/spectral MEM reconstruction (filled gray areas), the spatial MEM reconstruction (solid line) and the CLEAN reconstruction (dashed line). Note that the amplitude scale is logarithmic. Even where the spatial/spectral MEM map shows relatively strong sidelobes, as in Fig. 5 (click here)b, the sidelobes are about one order of magnitude smaller than in the other two reconstructions.
The reconstructed image of the right-hand circular polarization data is very similar to that in Fig. 4 (click here) indicating that the degree of polarization is low. This is to be expected for observations near the limb (cf. Lee et al. 1993), where the magnetic field lines are nearly perpendicular to the line of sight. Spectra from the spatial/spectral MEM reconstruction in both right- and left-hand polarization are used to derive coronal temperatures and magnetic field strengths in the next section.
Now, we briefly discuss the flare-related source located to the west of
the active region. In H
images from BBSO, we found that the
flare event comprised two ribbons aligned along the interferometer
fringes. Although the nonthermal radio emission was over by the time
of our observations, the H
ribbons remained quite bright and
separated slowly in classic post-flare evolution. We know from events
seen with Yohkoh that a large, spatially-static soft X-ray loop was likely
responsible for the continued radio emission from this location.
Due to temporal evolution of this source, however, which can be
different at different frequencies, we might expect inconsistencies
in the reconstructed image averaged over the entire period as in
Fig. 4 (click here), where the flare-related source is located at different
spatial positions at different frequencies from about 80'' at 3 GHz to
about 70'' at 6 GHz and shows two separate sources at higher frequencies
centered at about 55'' and 80''. Because the unwanted temporal
evolution undoubtedly affects the spectra of this flare-related source,
we do not attempt to interpret it here. Instead, we focus on the
active-region-related source. To see to what extent the flare-related
source influences the active-region-related source, we divided the data
set into six subsets of equal length of about 
hour duration and
reconstructed maps from these subsets. Although the reconstructions
were based on fewer visibilities, we found that the
active-region-related source gave a consistently reconstructed
morphology in all subsets except the first. In this first subset, an
apparent splitting of the active-region source above 6 GHz was seen,
likely an inaccurate reconstruction due to temporal variations in the
shape of the flare source during this first half-hour period. In the
other five time periods, the shape of the flare-related source was
quite stable while its brightness decreased slowly. The ratio of flare
to active region maximum temperature, averaged over 
, is
1.6 in the first period, decreases to 1.2 in the second, 0.8 in the
third and remains 0.7 in the last three subsets. Thus, the emission of
the active-region-related source integrated over the entire period
should be only slightly affected by the flare-related source.
We assume that the shape of the active-region-related source is well
represented in the synthesis map of Fig. 4 (click here). In any case,
our main purpose here is to show the relative improvement of the MEM
algorithm over other reconstruction methods.
Figure 6 (click here) shows two examples of brightness temperature spectra
derived from the spatial/spectral MEM reconstruction (
), the spatial
MEM reconstruction (
), and the CLEAN reconstruction (+).
Some assumptions are necessary for deducing the brightness temperature
from observations with one-dimensional spatial resolution. At each
iteration of the algorithm, the model brightness temperature map must
be converted to solar flux units to compare with the measured amplitudes.
This is done assuming an area for each pixel of 1'' square. Thus, the
flux in each pixel of the 1-d model is imagined to be
condensed into a single 1'' pixel along the fringe direction. The
algorithm thus produces a map whose brightness is too high by the amount
of the extent of the source along the fringes, in arcseconds. Of course,
the true source extent is unknown, but we make the assumption that the
sources are the same size along the fringes as they are across the fringes
(that is, they are circular in shape), and have corrected the brightness
temperatures accordingly. This modification was made for Figs. 3
and 4, and also in the spectra that are discussed in this section.
Figure 6: The brightness temperature spectra at spatial positions
a) +33'' and b) +39''. Each plot shows spectra
derived from the spatial/spectral MEM reconstruction (
),
the spatial MEM reconstruction (
),
and the CLEAN reconstruction (+)
At position +33'' (Fig. 6 (click here)a), which corresponds to the center of the active region source, the spectra stay more or less flat below about 5 GHz and decrease at higher frequencies with a steep slope of -5, which indicates that the emission is optically thick below 5 GHz and that thermal gyroresonance is the emission mechanism (e.g. Gary & Hurford 1994). For an isothermal corona, the optically thick part of the spectrum should be perfectly flat. The appearance of apparently real deviations from a flat spectrum in Fig. 6 (click here) could be due to real variations of temperature with height in the active region, or could be artificial due to our assumption of a circularly shaped source at all frequencies.
The spatial MEM leads generally to slightly higher temperatures than the spatial/spectral MEM, while CLEAN produces smaller temperatures. This may be due to a redistribution of the flux into sidelobes in the latter case. The transition from optically thick to optically thin occurs at almost the same frequency and the high-frequency slope is also more or less the same for the three different reconstructions. The spatial/spectral MEM reconstruction shows the smoothest spectrum of the three due to the added frequency constraint, but all three methods do a reasonable job for this position of the strongest source in the maps.
At position +39'' (Fig. 6 (click here)b), which is farther from the center of the active region, the spectra are flat for frequencies below 3.4 GHz and the high-frequency slope is -4 which is still too steep for free-free emission, and so is likely to be produced by gyroresonance. Again, the temperatures of the spatial MEM reconstruction are, in general, slightly higher than the ones of the spatial/spectral MEM resconstruction, while the CLEAN reconstruction produces smaller temperatures. The high-frequency slope is easily determined in the spatial/spectral MEM reconstruction, while it is rather difficult in the other two reconstructions. This clearly shows the advantage gained by including the spectral smoothness criterion.
The transition between the optically thick and optically thin parts of
the spectrum is a sharp ``kink" in the spatial MEM reconstruction, while
it rolls off more smoothly in the spatial/spectral MEM
reconstruction. The smooth transition is perhaps an undesired effect
of using the spectral smoothness criterion, so one must recognize the
potential for artificially smoothing sharp variations in the spectrum.
For larger 
, this becomes more pronounced and can, in extreme
cases, significantly alter the spectrum. As discussed in Sect. 2.1,
we used this behavior to determine a value of 
that minimizes the
unwanted smoothing.
The spectra can be interpreted using knowledge about radio emission
mechanisms such as thermal bremsstrahlung (free-free) emission and
thermal gyroresonance present in the solar corona (e.g.,
Dulk 1985; Gary & Hurford 1994). The average brightness
temperature of the flat part of a thermal spectrum gives directly the
electron temperature, 
, independent of the emission mechanism.
Figure 7 (click here) shows the electron temperature, 
, derived
from the spectra as a function of spatial position. The solid line
represents 
of the active region centered at about +29'', while
the dotted line gives 
of the low-frequencies (below 2 GHz)
source. The data are smoothed by a 5'' running mean. The temperature of
the active region is about 
with a slight maximum at
+37'', while the temperature of the broad single source is everywhere
somewhat higher with a broad maximum at +38''. At spatial positions >
80'' and at positions < 14'', the emission is optically thin, the
presented values are thus lower limits to the true temperature.
Figure 7: The electron temperature measured from the flat parts of the
brightness temperature spectra (
: solid
line, 
GHz: dotted line)
Figure 8: The total magnetic field strength at the base of the corona
assuming s = 3 everywhere measured from the gyroresonance
spectra (
: solid line, 
GHz:
dotted line)
From the shape of the spectra, we have already noted that thermal
gyroresonance is the dominant emission mechanism throughout the active
region. This allows us to derive the magnetic field strength in
the solar corona following the procedure described by Gary
& Hudford (1994). The highest magnetic field strength is given by
where s is the harmonic number.
The frequency at which the optical depth is unity (
) is
just above the frequency where the spectrum turns over, so we determine
at 
. Since 
is continous
with spatial position, we assume s = 3 at all positions following
Gary & Hurford (1994) and Lee et al. (1993).
Figure 8 (click here) shows the total magnetic field strength at the base
of the corona. The derived value depends slightly on the value of 
, so we plot two curves. The solid line shows the magnetic field
strength derived with 
given by the solid line curve in
Fig. 7 (click here), while the dotted line uses 
from the dotted
line in Fig. 7 (click here). Both temperatures give the same result,
that the magnetic field strength of the active region is about 800 G with a
maximum of 870 G at spatial position +28'' (which is 9'' to the east
of the position of maximum temperature).