A&A 388, 1048-1061 (2002)
DOI: 10.1051/0004-6361:20020542
A. Tritschler - W. Schmidt
Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104 Freiburg, Germany
Received 12 November 2001 / Accepted 8 April 2002
Abstract
We investigate the thermal and morphological fine structure of
a small sunspot, which includes the determination of
brightness temperatures and characteristic spatial scales
as well as their distribution inside the sunspot. The
identification and isolation of sunspot fine structure is
accomplished by means of a feature-finding algorithm applied
to a high-resolution time sequence taken simultaneously
in three continuum bands of the solar spectrum.
In order to compensate for seeing and instrumental
effects, we apply the phase-diversity technique combined
with a deconvolution method. The findings can
be summarized as follows: (1) umbral dots
are found to be on average 760K
cooler than the immediate
surrounding photosphere outside the spot.
(2) Some exceptional hot penumbral grains exceed the
average temperature of the brightest granules of the spots
surroundings by typically 150K. (3) The size
distribution of umbral dots and penumbral grains
support the idea that the smallest structures are still spatially
unresolved. (4) The distribution function of umbral dot peak
intensities points to the existence of
two umbral dot "populations'' indicating different
efficiency of energy transport. (5) The classification
of penumbral filaments into "dark'' and "bright'' depends
on the immediate surroundings.
Key words: Sun: sunspots - techniques: image processing
Umbral dots (UDs) are point-like bright features inside the umbra.
They are irregularly distributed inside the umbral regions, but
this distribution is far from random. UDs cover from 6-18% of the umbral area in dependence on identification criteria, data
quality and data type (spectra or filtergram) and sunspot brightness
(Adjabshirzadeh & Koutchmy 1983; Pahlke & Wiehr 1990;
Sobotka et al. 1993, 1997a).
The appearance and brightness of UDs
is closely related to the properties of the local umbral background
(Sobotka et al. 1992a,b, 1993). Measured UD-brightnesses
vary on a broad range between 0.08 -0.90
(Sobotka et al. 1993, 1997b)
and UD-temperatures are found typically to lie below the
temperature of the mean quiet sun (Grossmann-Doerth et al. 1986;
Bumba et al. 1990; Pahlke & Wiehr 1990; Aballe-Villero 1991;
Sütterlin & Wiehr 1998). However, exceptional high UD temperatures,
close to the mean quiet sun, derived from
a two-colour photometry are reported by Beckers & Schröter (1968)
and Koutchmy & Adjabshirzadeh (1981).
Observed UD sizes vary between 0.2-1.4 arcsec
(Beckers & Schröter 1968; Adjabshirzadeh & Koutchmy
1978, 1980, 1983; Koutchmy & Adjabshirzadeh 1981;
Grossmann-Doerth et al. 1986; Aballe-Villero 1991; Sobotka et al. 1993, 1997a).
There is evidence that UDs are not spatially resolved, so that
the size distribution function increases towards the diffraction
limit given by the telescope (Sobotka et al. 1997a).
Thus, the smaller the UDs, the more numerous they are, or, in other
words, no characteristic size of UDs can be given, because
the size seems to concentrate below the diffraction limit of the
respective measurement.
Although UDs are conspicious in brightness and temperature, they are rather unspectacular in other physical parameters. Observationally, magnetic fields in UDs are not substantially weaker than those of the umbral surroundings. Measured magnetic field reductions on spatial scales corresponding to the observed UDs are in the range 1-20% (Adjabshirzadeh & Koutchmy 1983; Pahlke & Wiehr 1990; Lites et al. 1991; Wiehr & Degenhardt 1993; Schmidt & Balthasar 1994; Tritschler & Schmidt 1997). There is no evidence of a significant velocity difference between UDs and the surrounding umbral regions (Pahlke & Wiehr 1990; Lites et al. 1991; Wiehr 1994; Schmidt & Balthasar 1994). Thus, UDs are magnetically and dynamically (vertical flows) "invisible'' and nearly undistinguishable from the properties of the umbral surroundings. This is in contradiction to common ideas of sunspot umbral structure, which cover mainly the cluster model or follow magnetoconvective beliefs.
Within the framework of the cluster model UDs are identified with intrusions of field free hotter fluid channeled between the flux tubes that are darkened due to the suppression of convection by the magnetic field (Obridko 1975; Parker 1979a,b; Choudhuri 1986). Before reaching the photosphere the magnetic field lines are closed at the top of the intrusions. The hot plasma inside the field free regions piles up and leads to a local increase of temperature and pressure. Thus the column rises vertically as a whole until the photosphere is reached where the pressure drops drastically and the magnetic field lines are no longer able to close at the top of the column. The plasma leaks out through a "magnetic valve''. The UD decays when the pressure inside decreases sufficiently so that the surrounding magnetic field closes again.
In the context of a magnetoconvective model UDs are interpreted
as the top of convective cells present in a homogeneous vertical
magnetic field (Knobloch & Weiss 1984). The presence of a strong magnetic
field modifies the nature of convection substantially. What
convection looks like depends critically on the parameter ,
the ratio of magnetic to thermal diffusivity.
For
oscillatory convection occurs in the upper sunspot
layers, while steady convection evolves only in the deeper sunspot
layers. Numerical simulations of two-dimensional magnetoconvection
show that oscillatory and steady convection couple in such a way
that one gets hot regions of upwelling gas with somewhat dispersed
magnetic field (Weiss et al. 1990).
However, the obvious resemblance between the two models is, that UDs in either picture tend to weaken the magnetic fields and are connected with an upward motion. Both properties are not observed. How to reconcile observations with the predictions of the models? Lites et al. (1991) first pointed out that the contradiction with existing observations may be due to some point to the fact that in both models radiative transfer is neglected. Radiative transfer is important in the upper layers of a sunspot, where the opacity becomes small. Upward motions are probably damped out in these layers due to considerable amount of energy being lost by radiative transfer. The lack of upward flow excludes the UD to be interpreted as Choudhuris magnetic valve, but the UD can still be the upper part of the tapering column, from which the energy leaks out due to efficient radiative transfer. Thus, the reduction of the magnetic field inside the UD is not observable, because the measurements are not done in a line formed below the vertex. The brightness of an UD is due to energy diffusing radiatively from the vertex, resulting in UD temperatures lower than that of the photosphere outside the spot.
In contrast to the still puzzling nature of UDs the penumbra and its fine structure has lost some of its mysteriousness. The recent progress in two-dimensional spectroscopy and spectropolarimetry together with time sequences of high spatial resolution and promising numerical simulations seem to give a concise picture of the penumbral phenomena and its fine structure, which is dominated by the bright penumbral filaments with their comet-like shaped heads - the penumbral grains (PGs) (Muller 1973). Observations of PGs give an average width of 0.35 arcsec and length scales in the range 0.5-2.0 arcsec (Krat et al. 1972; Bruzek 1977; Muller 1973; Moore 1981; Bonet et al. 1982). Grossmann-Doerth & Schmidt (1981) state an upper limit of 0.55 arcsec for bright and dark filaments, whereas analysis of penumbral power spectra indicate significant signal close to the diffraction limit (Harvey & Breckinridge 1973; Stachnik et al. 1983; Lites et al. 1990; Denker et al. 1995). Sanchez Almeida & Bonet (1998a,b) estimate that the real size of penumbral filaments is below the achievable resolution varying on scales of a few kilometers. The brightness of PGs reaches that of the mean surrounding photosphere outside the spot. Individual PGs can even exceed the brightness of the brightest granules (Sütterlin & Wiehr 1998).
The morphological appearance of the penumbra suggests
the existence of two components: a bright
and a dark component covering 43-49%
and
51-57%
of the penumbral region (Muller 1973; Collados et al. 1988).
The average brightness
of the bright and the dark component is in the
range 0.78
-1.07
and
0.61
-0.79
,
respectively
(Krat et al. 1972; Grossmann-Doerth & Schmidt 1981;
Collados et al. 1988). Contrary to the visual glimpse, the corresponding
penumbral intensity distribution reveals a single-peaked shape
(Grossmann-Doerth & Schmidt 1981; Collados et al. 1988; Denker 1998). Thereafter
the term bright and dark filament is of only local meaning and
penumbral filaments are defined by their immediate surroundings.
However, Collados et al. (1988) show with a simple two-component
model that this is not in direct contradiction
to the existence of two components.
In accordance with magneto-hydrodynamical simulations, the penumbral structure is stamped by the dynamcis of ascending flux tubes (Schlichenmaier et al. 1998). In this picture, the PGs are interpreted as the foot points of flux tubes. The flux tubes, heated by a systematic flow (Evershed flow), establish temperatures even higher than the hottest granules in the surrounding photosphere outside the spot. The optically thick bright filaments correspond to the radiative signature of the horizontally outflowing hot plasma channeled by the flux tube. During the outflow the plasma cools and the original bright filament thins, until it becomes optical thin and cannot be distinguished from the dark background (Schlichenmaier et al. 1999). This gives a natural explanation of the characteristic tail of the bright filaments.
In this investigation we analyze the morphologic properties and brightness of a small sunspot as described in Tritschler & Schmidt (2002) (hereafter: TS). Section 3 introduces the algorithm used to identify and isolate the sunspots small-scale structures. Section 4 presents the results, like umbral and penumbral brightness and size distributions, filling factors, brightness-background relations and brightness-size relations. In Sect. 5 the results are discussed and compared to already existing observations. A summary of our findings is given in Sect. 5.
Observations were made on May 19, 1995, with the German Vacuum Tower Telescope (VTT) located at the Observatorio del Teide on Tenerife. In order to observe simultaneously in three different continuum bands (402.1 nm, 569.5 nm and 709.1 nm) of the solar spectrum, the observations also employed the Multichannel Filter System of the VTT (Kentischer 1995). To apply the phase-diversity technique, a defocused image was taken additionally in one of the wavelength channels (569.5 nm, phase-diversity channel).
The analyzed data set covers a 90 min sequence of a well
developed sunspot (NOAA 7871, f-spot, 25 arcsec in diameter),
observed out of disc center (
).
First steps of the analysis included gain correction and subtraction of dark current. For more details of the observation, the calibration process and the image reconstruction and processing methods see TS.
The study of sunspot fine structure covers the consideration of size, brightness and area fraction as well as the distribution of these quantities. This implies the identification and isolation of the relevant features in the umbra and penumbra, which is accomplished by means of a feature-finding algorithm (FFA) working in the following way.
In a first step, for each frame a binary mask is worked out,
which defines the umbra and the penumbra (see TS).
In a second step, from each frame a smoothed version (boxcar average
with size 0.570.57 arcsec2) is subtracted
to create a difference image with contrast enhanced fine structure. On the
basis of this difference image a binary mask is generated by
setting all pixels above a certain intensity threshold (
)
equal to
1 and all other pixels equal to 0.
This mask is applied to the actual frame. A single structure
is thus defined by a contour line - depending on
-
in the difference frame. For each identified structure the following
parameters are determined: peak intensity, P, average intensity, A,
local background intensity, B, and size.
The local background intensity, B, results from an intensity average
over the adjacent intensity minima next to the structure under
consideration. These minimum-contour intensities are denoted
with D. The structure size is measured by means of the
spatial extent at half of the intensity difference P-B (FWHM-area) and is
specified in form of an effective diameter
:
the diameter of a circle with the same area.
The diffraction limit in the three continua is about
0.144, 0.205 and 0.244 arcsec, respectively, corresponding to
a size of
2 pixel for the given
image scale of 0.0823 arcsec pixel-1.
Therefore all structures with a FWHM-area < 4, 7, 10 pixel, respectively, are excluded.
Identified structures inside the umbra or penumbra are labeled as UDs or PGs, respectively. At this point we want to make some further comments on the algorithm:
Figure 1 visualizes the relation between the peak
intensity P and the mean local background intensity B of the identified
features inside the umbra (upper row) and
the penumbra (lower row) in the three observed continuum bands.
![]() |
Figure 1:
P-B relation of identified structures in the three
observed continuum bands at 402.1 nm,
569.5 nm and 709.1 nm.
Upper panel: umbral structures (UDs). Lower panel: penumbral
structures (PGs). ![]() |
Open with DEXTER |
The penumbra does not show the sector-like distribution in
the P-B plane. The individual peak intensities, P, of the PGs and the
corresponding background intensities, B, are less correlated
than for structures in the umbra. PGs are on average
only 3% (50 K) darker than the mean quiet sun.
The peak brightness P of individual PGs is in the range 0.71-1.35
(5590-6490 K). The local background brightness is on
average
(
K) with individual
values between 0.42-1.03 (5000-6080 K). The
average ratio
amounts to
,
whereas
exceptional bright PGs reveal a P/B ratio up to 2.77.
A summary of all relevant values is given in Table 2
at the end of Sect. 6.
![]() |
Figure 2: Intensity distributions in the observed continuum bands inside the umbra (upper panel) and penumbra (lower panel). Bin size = 0.02. P: peak intensity (thick full line). A: average intensity (dotted line). B: average background intensity at the location of the structure (thin line). D: minimum contour intensities in the vicinity of the identified structure (dashed line). Given values refer to the maximum of the distribution function. All distributions are normalized with respect to the total number of identified structures except for the D-distribution, which is normalized with respect to the total number of minimum intensity values. |
Open with DEXTER |
Alternatively, we can take another point of view. For each specified background intensity, B, a range of structure brightness (UD or PG) is observed. Thus, there exist bright as well as faint structures at the same background intensity. Therefore, each background intensity can be assigned a mean intensity, that results from an average over the corresponding range of observed UD or PG brightness. These mean intensities are marked in Fig. 1 with the +-symbol. For reasons of representation binned +-values are plotted.
In Fig. 2 the distribution of the peak intensities, P, the averaged intensities, A, the local background intensities, B, and the individual minimum-contour intensities, D, are displayed for the umbra (upper row) and the penumbra (lower row). The distribution functions refer to the whole sequence and are normalized to the total number of structures found in the corresponding sunspot region. The distribution of the UD-intensities P and A are asymmetric and broader compared to the background intensities. Most pronounced in the blue continuum band, the shape of the P-distribution shows the contribution of two components. The two peaks are located at 0.34 (5230 K) and 0.46 (5480 K). The green continuum shows two peaks located at 0.44 (5060 K) and 0.63 (5460 K), respectively. The red continuum shows a single peak located at 0.59 (5030 K). The distribution of the individual background intensities, D, and the mean local background intensity, B, are single-peaked and asymmetric. The asymmetry is due to UDs that occur in regions with strong intensity gradients. The maximum of B is located at 0.38 (4890 K), that of D is somewhat lower at 0.36 (4910 K).
![]() |
Figure 3: Penumbral intensity distribution of one of the best images for the three observed continuum bands 402.1 nm (thick line), 569.5 nm (thin line) and 709.1 nm (dotted line). Bin size = 0.01. |
Open with DEXTER |
The penumbral intensity distributions of P, A, B and D (Fig. 2, lower row) are single-peaked and symmetric.
Characteristic values for P and A,
are located at 0.98 (5990 K) and 0.94 (5960 K),
respectively. Background intensities, B and D, peak at 0.79
(5730 K) and 0.75 (5670 K), respectively.
The D-distribution reveals a second component
located at very low intensities. This contribution is formed
by intensities that belong rather to the umbra than to the penumbra
and can be explained as follows: the FFA searches for the minimum
contour line around the selected structure, which is done by
following the intensity gradient until the slope
changes the sign from negative to positive.
Especially for PGs, this situation does not occur until
umbral regions are reached.
The width of the individual B- and D-distributions is
about 15% and 17% of the mean quiet sun
intensity, whereas the P- and A-distributions are about
20% and 17% broad.
![]() |
Figure 4:
Distribution of size
![]() |
Open with DEXTER |
For comparison, Fig. 3 displays the total penumbral intensity distribution for the observed continuum bands of one of the best images in our sequence. The distributions are all single-peaked and symmetric.
The size distribution of identified structures
(full thick line) in the umbra and the
penumbra is shown in Fig. 4.
Also displayed are
the distribution functions that result from the uncorrected
data (indicated by the shaded area).
All distribution functions reveal, independent from the wavelength and
the sunspot region, the same behaviour: the distributions show no peak
but an increase with decreasing size
up to the diffraction
limit. The shape of the distributions changes only little with the
threshold intensity
.
The reconstructions lead to the
following result: the average diameter
and deviations
(
)
of UDs and PGs amounts to
arcsec and
arcsec, respectively.
The average size changes slightly with
:
an increase
of
leads to a decrease of the average size of the
structures. With regard to PGs the derived sizes must be taken
as upper limits since
tends to overestimate the
size of an elongated structure.
Umbra | Penumbra | |||
![]() ![]() |
![]() |
f |
![]() |
f |
0.35 ![]() |
0.11 ![]() |
0.41 ![]() |
0.15 ![]() |
|
402.1 | 0.35 ![]() |
0.11 ![]() |
0.39 ![]() |
0.14 ![]() |
0.41 ![]() |
0.16 ![]() |
0.44 ![]() |
0.14 ![]() |
|
569.5 | 0.39 ![]() |
0.13 ![]() |
0.43 ![]() |
0.12 ![]() |
0.47 ![]() |
0.16 ![]() |
0.49 ![]() |
0.14 ![]() |
|
709.1 | 0.46 ![]() |
0.16 ![]() |
0.48 ![]() |
0.13 ![]() |
![]() |
Figure 5:
Relation between peak intensity P and size
![]() |
Open with DEXTER |
From the FWHM-sizes, the area fraction f occupied by the structures can be
calculated by the ratio of the summed area of all identified structures
to the total area of the sunspot region under consideration.
This quantity is very
sensitive to
and less sensitive to the minimum size.
With a decrease of
the area fraction increases, since the
number of identified structures increases. Within the umbra, the area
fraction triples, while within the penumbra it doubles. In particular
for the penumbra these values must be regarded as lower limits. The
search algorithm does not identify the whole bright filament, but only
the bright head. The total area of the filament is probably three times
as big. Thus, the area fraction is underestimated.
The relation between the peak intensity, P, and the
effective diameter,
,
is illustrated in Fig. 5
for the reconstructed data.
For each identified structure of specified
size, there exists a range of observed brightnesses. Thus, there exist
bright as well as faint structures of equal equivalent size
.
We follow the same procedure as described before with respect to
the relation between peak intensity and mean local
background intensity of identified structures
(see Sect. 4.1, Fig. 1).
Each size
can be assigned a mean intensity, that
results from an average over the
corresponding range of observed brightness. These mean intensities are
marked in Fig. 5 with the +-symbol. As in
Fig. 1 binned +-values are displayed. In all three
continuum bands, these mean intensities show a slight increase with
size. For
arcsec the mean peak intensity
fluctuates because large structures
are less frequent than small ones. Thus, only a few points
contribute to the mean peak intensity. The increase of the mean peak
intensity with size is most conspicious for the penumbra.
This behaviour is ascribed to the contribution of unresolved
conglomerats of structures. The FFA interprets them as a large single
structure. As a result, for each size there is a deficit of
bright intensity structures.
![]() |
Figure 6: Intensity distribution of UD-peak intensities for the observed continuum bands on the bases of the reconstructed (full line) and the uncorrected (dotted line) data. Upper panel: default values for the minimum size of selected UDs. Lower panel: reduced minimum size for the green and red continuum band. Bin size = 0.02. |
Open with DEXTER |
As already mentioned before, the shape of the distribution of peak
intensities P and averaged intensities A for UDs is very sensitive
to the choice of
and the minimum FWHM-size.
The variation of the threshold intensity (
)
and
the minimum size (4, 7 and 10 pixel) reveals that
the two-component structure is clearly evident
only for a combination of the lowest threshold intensity and a size
near the diffraction limit. The peaks are
located at 0.30 and 0.45, with the saddle point at 0.40.
Since the size of UDs is uncoupled
to their brightness, the variation of
does not lead to
a shift towards higher or lower intensities but to a change
in UD frequency because low-contrast UDs are rejected. As a consequence
the separation of the two components smears.
Figure 6 shows the distribution of UD peak
intensities P as a function of wavelength and minimum size
based on the reconstructed and the uncorrected data set.
For the default values of the minimum size, the following can be stated:
the uncorrected data set leads to a double-peaked distribution of UD
intensities in all three continua. In the blue and green continuum, the
components (or "populations'') are clearly separated.
Bright and dark UDs are almost
equally frequent. In the red continuum, dark UDs
are more frequent than bright ones by factor of 1.5. The corresponding
reconstructed intensity distributions also indicate a two-component
distribution, at least for the green and blue continuum band.
The image restoration dilutes the double-peak distribution, particularly
in the red wavelength band.
Lowering the minimum size to 4 pixel in the green and red continuum results in an increase of the
number of the bright UDs. For the uncorrected data, two separated peaks
with similar frequency are visible in the red continuum.
From a morphological point of view, the observed penumbra is
two-component structured: bright filaments with a mean intensity
of 0.92 (5910 K) cover 14%
of the total penumbral area and are separated by dark lanes of a mean
intensity
of 0.76 (5670 K)
(see Table 2 at the end of Sect. 6).
Although the FFA was not applied to identify dark structures, we
consider the average minimum-contour intensity
as a good approximation to the mean "brightness'' of the dark filaments.
However, we can make no statement about the size and area
fraction of the dark filaments. Thus, the area fraction of
the dark filaments is set to 86%, which corresponds
to the area that is not occupied by bright filaments.
Mean brightness and area fraction of the two components
lead to a penumbral brightness of 0.78.
Krat et al. (1972) find that bright filaments typically show intensities around 0.78%, whereas dark filaments are around 0.62. Their analysis leads to the following picture: bright filaments are built up of bright chained grains, the PGs. In the inner (outer) part of the penumbra, the PGs are on average 0.90 (0.95) bright and cover approximately 43% of the penumbral area (Muller 1973). The dark filaments are on average 0.52 (0.60) bright and cover an area fraction of 57%.
The investigations of Grossmann-Doerth & Schmidt (1981) show that rather quiet penumbral regions exhibit mean intensities in the range 0.80-0.97 and 0.62-0.77 for bright and dark structures, respectively. Penumbral intensity fluctuations vary between 9.2% and 18.3%.
A photometric study of Collados et al. (1988) in the two continua at 500 nm and 550 nm yields a mean brightness of 0.85 and 0.83 for bright structures and 0.61 and 0.66 for dark structures, respectively. The corresponding area fractions amount to 49% and 51%. The average brightness of the whole penumbra therefore amounts to 0.73 and 0.74, while mean intensity fluctuations amount to 11.5% (500 nm) and 7.9% (550 nm).
Compared to former findings, in our work the area
fraction occupied by the bright penumbral structures amounts to only 1/3
of the value adopted by Muller (1973). As mentioned before
(Sect. 4.2) this is due to the
fact that the FFA selects primarily the bright heads of the filaments,
while the extended tail of the filament is sorted out.
Also, the average intensities of the dark filaments based on our analysis
deviate significantly from former results. As already discussed in TS
this is only partially ascribed to the influence of stray light.
At first, there is growing evidence that small sunspots are
brighter and hotter than large sunspots. At second, it cannot be
excluded that also for small sunspots the umbral and penumbral
brightness is correlated. Furthermore, the observations have been
carried out during the late phase of the solar cycle, where
sunspots are believed to be brighter than during the early phase of the
solar cycle. This may explain the enhanced
intensities of dark penumbral structures and the reduced
penumbral rms-fluctuations, which amount on average to
11.3%, 9.5% and 8.8%
for the blue, green and red continuum band (see TS).
In opposition to the morphological appearance, the intensity distribution of the whole penumbra at no time shows a split and a completely separated second component, apparently in contradiction to the picture of the penumbra given by Muller (1973). Our single-peaked penumbral intensity distributions, as shown in Fig. 3, rather indicate the following: what is a bright and dark filament is defined locally by the immediate penumbral vicinity. The terms bright and dark filament have only a local meaning. This is consistent with Grossmann-Doerth & Schmidt (1981) and Collados et al. (1988).
By means of a simple model of bright and dark penumbral structures,
Collados et al. (1988) show that the observed single-peaked
distributions are reproducable, if the mean intensity of bright and dark
filaments is in the range 0.85-0.90 and 0.60-0.64, respectively, and
if the individual distributions of the two components have a width
10-15% of the average quiet sun intensity.
In dependence of the choice of the free parameters (width, area
fraction, mean intensity) of the single components, double-peaked
distributions or asymmetries appear. This result is
supported by our results. Our distributions show widths that are on
average
16% (bright component) and
17%
(dark component). Therefore, the observed single-peaked distributions
do not directly contradict the results given by Muller (1973).
The observed mean intensity ratios P/B for UDs are in good agreement
with already existing results. Sobotka et al. (1993) find that
the distribution function of the intensity ratio P/B accumulates
at 1.35 with an average value and deviation of
.
Our observations lead to an average value and deviation of
,
with a peak of the distribution function at 1.23, which
is slightly below the values of Sobotka et al. (1993). This can be
ascribed to the choice of the threshold intensity
.
Already a
leads to higher average values of
for P/B.
The sector-like shape of the relation between P and B for umbral
structures is well established in all observed continuum bands and
confirms former studies (Sobotka et al. 1992a, 1993).
The upper edge of the sector is below a straight line of slope
3, which is assumed to be approximately the true ratio
of P/B (Sobotka et al. 1992a, 1993). The established correlation
bewteen P and B presents evidence that the peak intensity of UDs
is related to the corresponding local umbral background.
Otherwise, the UD-intensities would be distributed
irregular in the halfplane above the P/B=1 line.
Penumbral features are slighty less correlated to the background than umbral features. For the inner penumbra, the determination of the local background intensity is more insecure compared to the umbra, because of the steep intensity gradient. The algorithm moves too far into the umbra and therefore underestimates the average background intensity. As a result, the P/B values are overestimated. Points in the P-B diagram slide horizontally to the left side. This degrades the correlation and falsifies the relation between P and B. For the penumbra we therefore cannot exclude a dependence between P and B, similar to that of the umbra.
Figure 7 displays the distribution functions of the intensity ratio P/B for structures inside the umbra (upper panel) and the penumbra (lower panel). The distributions are single-peaked and asymmetric: they decrease rapidly to lower and slowly to higher intensities. Typical P/B values are found around the peak at 1.22 (1.17) for umbral (penumbral) features.
The identified UDs do not reach the temperature
of the surroundings outside the spot. Typically, UDs are
680 K hotter than the umbral minimum and
760 K colder than the mean surroundings outside the
spot. Individual UD temperatures vary between 4860-5950 K.
The application of image reconstruction techniques combined with the determination of
brightness temperatures provides evidence that UDs do not reach quiet sun
temperatures (Bumba et al. 1990; Grossmann-Doerth et al. 1986;
Sütterlin & Wiehr 1998). Pahlke & Wiehr (1990) derive a two-component model
of an umbra, which reproduces spatially unresolved Stokes-V-profiles of
photospheric lines and find that the UD-temperature is much less than that
of the surrounding quiet sun. Thus, our results are in agreement with
former findings.
![]() |
Figure 7: Distribution of the intensity ratio P/B based on the reconstructed (full line) and the uncorrected (dotted line) data in the observed continuum bands. Upper panel: umbra. Lower panel: penumbra. All distributions are normalized with respect to the total number of identified structures. Bin size = 0.01. |
Open with DEXTER |
In contrast hereto results from a two-color photometry and the use of color temperatures (Beckers & Schröter 1968; Koutchmy & Adjabshirzadeh 1981; Aballe-Villero 1991) show that UDs reach or exceed the temperature of the surrounding quiet sun. The difference between the use of color and brightness temperatures and the consequences have been already discussed in TS. Here we just want to remind the reader that observed color temperatures above quiet sun values do not imply that UDs have brightness temperatures higher than the photosphere outside the sunspot.
Some of the PGs show exceptionally high brightness temperatures:
they exhibit temperatures in the range
K, close to the photospheric conditions
of the surroundings outside the spot. Maximum temperatures are reached
at 6490 K, which is 450 K above the mean
quiet sun temperature and as hot as the brightest granule
(6500 K) in our field of view.
This confirms recent results (Sütterlin & Wiehr 1998) and
predictions of numerical simulations (Schlichenmaier et al. 1998).
In accordance with magneto-hydrodynamical simulations, the penumbral
structure is dominated by the dynamcis of ascending flux tubes. In this
picture, the PGs are interpreted as the intersection of these flux
tubes with the
-level. The flux tubes, heated by a
systematic flow (Evershed flow), establish
temperatures even higher than the hottest granules.
The optically thick bright filaments correspond
to the radiative signature of the horizontally outflowing hot plasma
channeled by the flux tube. During the outflow the plasma cools
until the filament becomes optical thin and
cannot be distinguished from the dark background.
This leads to the characteristic tail of the bright filaments.
Since the hot flux tube is embedded in a cool surrounding it cools
by radiative heat exchange (Schlichenmaier et al. 1999) and leads
to a slight heating of the background. To which extent
this heating or cooling of the flux tubes can account for
the observed brightness of the dark filaments - and the single-peaked
penumbral intensity distributions - needs further
investigation. In principle, it is conceivable that
the brightness of the penumbral background is a consequence
of this heating mechanism due to the cooling of the flux
tubes. However, the cooling must account for a mean flux difference
between the umbral and penumbral background of approximately
40%.
The area fraction occupied by UDs can be determined
directly on the basis of the observed size and number of UDs per umbra,
but the dependence from data quality and the identification criteria is
obvious. The observed area fractions lie in the range of 6-18%. Rather dark umbrae show a somewhat lower percentage
(6-8%) than brighter umbrae
(10-15%) (Sobotka et al. 1993).
Sobotka et al. (1997a) obtain a similar result (
%).
Therefore, our results based on the reconstructed
(
13.1%) and the
uncorrected data (
15.8%) fit well into already
existing results, if we keep in mind that the observed sunspot is rather
small and bright (see TS). An untypical high area fraction
of 30% for a small umbra, found by
Sobotka et al. (1988), was due to the moderate resolution of the data set.
Spectroscopic investigations lead to somehow lower
area fractions of
5%
(Adjabshirzadeh & Koutchmy 1983; Pahlke & Wiehr 1990).
The determination of UD sizes is still a challenge, which is reflected in the huge amount of different methods to measure the UD size or to infer the true size of UDs. Among these are image reconstruction techniques and stray light corrections (Adjabshirzadeh & Koutchmy 1978, 1980; Grossmann-Doerth et al. 1986), the application of identification algorithms, the two-color photometry (Beckers & Schröter 1968; Koutchmy & Adjabshirzadeh 1981; Aballe-Villero 1991) or a combination of the individual methods (Sobotka et al. 1993, 1997a). Accordingly, the results vary on a wide range (0.3-0.6 arcsec Aballe-Villero 1991, 0.3-1.4 arcsec Grossmann-Doerth et al. 1986). The present work confirms and supplements the work of Sobotka et al. (1997a).
Our observed sizes on the basis of an effective diameter
vary over a wide range, depending on the selection criteria. The
choice of a threshold intensity just above the noise level
results in effective diameters of 0.25-0.98 arcsec. Most probably, the
largest UD proxies correspond to conglomerates of UDs
that cannot be separated by the FFA. However, the existence
of very large UDs cannot be excluded. Grossmann-Doerth et al. (1986) report about specimen with diameters up to 1.4 arcsec. The distribution of the UD size,
,
features an increase
towards the diffraction limit: the smaller the structures, the more
numerous they are. This means, that the UDs cannot be assigned
to a typical size, since the spatial resolution is not sufficient to
resolve the smallest UDs. From the steep increase towards
the diffraction limit can be inferred, that the size
of an UD is not far below the diffraction limit (Sobotka et al. 1997a).
The authors find an average UD-size of
arcsec.
Our analysis leads to UD-sizes
that are in agreement with the aforementioned results: UDs show average diameters
of
arcsec for the reconstructed data and
arcsec for the uncorrected data.
In a similar investigation Sobotka et al. (1993) find a mean UD-diameter of
arcsec. Their corresponding size distribution of
the observed UDs is symmetric and single-peaked with a maximum at
0.4 arcsec which is above the spatial resolution in the data
(0.3 arcsec, 240 km). Adjabshirzadeh & Koutchmy
(1983) obtain a one-component size distribution with a maximum at
0.225 arcsec (165 km) for a single spot with only
the brightest UDs considered. The shape of the size distribtuion
function obviously depends on the volume of the UD-sample, the noise
level, the spatial resolution of the data set and of the
UD-identification method (by visual inspection or automatic).
The comet-like shaped heads of the bright
filaments, the PGs (Muller 1973), are on average 0.35 arcsec wide and 0.5-2.0 arcsec long (Krat et al. 1972;
Muller 1973; Moore 1981; Bonet et al. 1982). Based on a
statistical analysis of the penumbral fine structure Grossmann-Doerth et al. (1986) give 0.55 arcsec as an upper limit for the size of
the bright and dark filaments. Altogether, penumbral spatial scales are
close to the diffraction limit. Our analysis leads to somewhat bigger penumbral
scales: on average the bright filaments show effective diameters of
0.15 arcsec (reconstructed) and
arcsec (uncorrected). As stated before,
the effective diameter overestimates the size of an elongated
structure. The size distribution of the PGs shows the same
behaviour as that for the UDs: an increase towards the diffraction
limit, which suggests that the smallest
penumbral structures are still unresolved.
The distribution function of UD peak intensities P and the average UD
intensities A is double-peaked with two well-separated maxima in all
observed continua for the uncorrected data set. The appearance of
the double peak is very sensitive to the choice of the threshold
intensity
and the minimum size. Only for values of
and the minimum size near the noise level and the diffraction
limit respectively, the two UD-populations are distinguishable.
The intensities accumulate approximately at 0.32 and 0.45. In the
reconstructed data set for the red continuum, the double-peaked
distribution does not appear until the minimum size is set
below the diffraction limit for the
red wavelength. If the positions are checked, it can be shown
that the members of the hot population are settled at the edge of the
umbra and on a streak combining the lightbridge with the penumbra
on the opposite side. Members of the cool population are located
in the central regions of the umbra. However, the overall
distribution of UDs inside the umbra is irregular, but not random.
There are regions which are void of UDs. In these regions, UDs neither
pop up nor existing UDs manoeuvre into. An example is the darkest part of the
umbra. Figure 8
illustrates this behaviour for the reconstructed data
set.
![]() |
Figure 8:
UD-population inside the umbra
(image scale: 1 minor tickmark
![]() ![]() |
Open with DEXTER |
A similar study was carried out by Sobotka et al. (1997b). The authors consider the distribution function of time-averaged UD peak intensities. This distribution function shows two separated maxima located at 0.34 and 0.48, which differs only slightly from our results for the blue continuum.
The relation between the UD peak intensity and the local background suggests the following: the brightest UDs are located where the background is high or marked by an intensity gradient (lightbridge, near penumbra). The brightness or temperature of UDs appears to be controlled or at least influenced by the background intensity or another physical quantity which characterizes the background. Thus, in central located UDs energy is transported less efficient compared to peripheral located UDs. The existence of two different populations was a priori assumed by Grossmann-Doerth et al. (1986), who differentiated between peripheral UDs (PUDs) and central UDs (CUDs) according to their location inside the umbra. The close proximity to the penumbra raises the question whether the pre-defined PUDs and the members of the bright population - at least those which are located at the boundary to the penumbra - are remnants of former penumbral filaments. In order to verify this, sufficient long time sequences of high spatial resolution covering all evolutionary states of sunspot penumbral filaments are needed.
We summarize our findings as follows:
Umbra | Penumbra | |||||||
![]() ![]() |
(P)- | (P)+ |
![]() |
![]() |
(P)- | (P)+ |
![]() |
![]() |
0.23 | 0.69 | 0.39 ![]() |
0.34 | 0.55 | 1.45 | 0.87 ![]() |
0.84 | |
402.1 | 4980 | 5880 | 5350 ![]() |
5270 | 5670 | 6690 | 6090 ![]() |
6100 |
0.36 | 0.94 | 0.55 ![]() |
0.44 | 0.71 | 1.35 | 0.97 ![]() |
0.98 | |
569.5 | 4860 | 5950 | 5280 ![]() |
5220 | 5590 | 6490 | 5990 ![]() |
6000 |
0.42 | 0.95 | 0.60 ![]() |
0.54 | 0.76 | 1.34 | 0.99 ![]() |
0.96 | |
709.1 | 4730 | 5830 | 5170 ![]() |
5040 | 5490 | 6450 | 5900 ![]() |
5860 |
![]() ![]() |
(A)- | (A)+ |
![]() |
![]() |
(A)- | (A)+ |
![]() |
![]() |
0.22 | 0.59 | 0.36 ![]() |
0.28 | 0.53 | 1.23 | 0.83 ![]() |
0.83 | |
402.1 | 4950 | 5720 | 5300 ![]() |
5460 | 5640 | 6450 | 6040 ![]() |
6070 |
0.35 | 0.80 | 0.51 ![]() |
0.47 | 0.70 | 1.21 | 0.92 ![]() |
0.94 | |
569.5 | 4830 | 5670 | 5200 ![]() |
5110 | 5500 | 6310 | 5910 ![]() |
5930 |
0.40 | 0.82 | 0.56 ![]() |
0.53 | 0.74 | 1.18 | 0.94 ![]() |
0.93 | |
709.1 | 4690 | 5550 | 5080 ![]() |
4990 | 5450 | 6190 | 5820 ![]() |
5800 |
![]() ![]() |
(B)- | (B)+ |
![]() |
![]() |
(B)- | (B)+ |
![]() |
![]() |
0.18 | 0.43 | 0.28 ![]() |
0.24 | 0.26 | 1.01 | 0.67 ![]() |
0.72 | |
402.1 | 4800 | 5450 | 5100 ![]() |
5030 | 5040 | 6260 | 5830 ![]() |
5910 |
0.29 | 0.55 | 0.40 ![]() |
0.38 | 0.42 | 1.03 | 0.76 ![]() |
0.80 | |
569.5 | 4640 | 5290 | 4950 ![]() |
4900 | 5000 | 6080 | 5660 ![]() |
5700 |
0.35 | 0.59 | 0.44 ![]() |
0.44 | 0.43 | 1.03 | 0.77 ![]() |
0.79 | |
709.1 | 4550 | 5140 | 4800 ![]() |
4790 | 4760 | 5950 | 5510 ![]() |
5570 |
![]() ![]() |
(D)- | (D)+ |
![]() |
![]() |
(D)- | (D)+ |
![]() |
![]() |
0.15 | 0.64 | 0.28 ![]() |
0.21 | 0.16 | 1.27 | 0.68 ![]() |
0.66 | |
402.1 | 4710 | 5830 | 5110 ![]() |
4950 | 4820 | 6390 | 5840 ![]() |
5830 |
0.24 | 0.80 | 0.40 ![]() |
0.36 | 0.27 | 1.27 | 0.76 ![]() |
0.76 | |
569.5 | 4510 | 5760 | 4960 ![]() |
4930 | 4670 | 6220 | 5670 ![]() |
5670 |
0.30 | 0.85 | 0.44 ![]() |
0.40 | 0.32 | 1.24 | 0.78 ![]() |
0.80 | |
709.1 | 4390 | 5670 | 4800 ![]() |
4720 | 4570 | 6140 | 5540 ![]() |
5500 |
![]() ![]() |
(P/B)- | (P/B)+ |
![]() |
![]() |
(P/B)- | (P/B)+ |
![]() |
![]() |
1.03 | 2.82 | 1.39 ![]() |
1.23 | 1.02 | 2.94 | 1.31 ![]() |
1.16 | |
402.1 | 1.00 | 1.17 | 1.05 ![]() |
1.02 | 1.00 | 1.19 | 1.05 ![]() |
1.02 |
1.06 | 2.46 | 1.38 ![]() |
1.22 | 1.03 | 2.77 | 1.30 ![]() |
1.17 | |
569.5 | 1.01 | 1.21 | 1.07 ![]() |
1.04 | 1.01 | 1.26 | 1.06 ![]() |
1.04 |
1.08 | 2.48 | 1.36 ![]() |
1.18 | 1.05 | 2.73 | 1.29 ![]() |
1.19 | |
709.1 | 1.02 | 1.25 | 1.08 ![]() |
1.04 | 1.02 | 1.30 | 1.07 ![]() |
1.05 |
Despite the numerous observational results for umbral and penumbral fine structure, the need for further investigations is obvious. Some of the key questions are: (a) what leads to the different UD-populations? (b) What is the regulating mechanism responsible for the appearence and disappearance of UDs? (c) What is the real size of UDs and PGs? (d) What is the fundamental relation between the intensity of UDs and the local background? Over and above the morphological studies presented herein, spectroscopic and polarimetric measurements are needed to determine the magnetic field configuration and the flows at the scale of umbral and penumbral structures. With the availibity of adaptive optics and suitable postfocus instruments, these questions could be answered in a near future.
Acknowledgements
The authors wish to thank Rolf Schlichenmaier for his inspiring help in trying to understand what the penumbra is telling us from a theoretical point of view. We are deeply grateful to Mats Löfdahl for his support during the "fights'' with phase diversity. Thanks to Scott Acton for his introduction into observational and computational aspects of the image reconstruction technique. We feel indebted to Thomas Kentischer, observations would not be possible without his manpower. We thank Michael Knölker for his contributions and for financial support (for AT) granted by the High Altitude Observatory. Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG.