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Astron. Astrophys. 328, 682-688 (1997)

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4. Results

First we summarize some global characteristics of the umbral core under investigation: its size, the number of UDs, and filling factor of UDs (ratio of the total area of UDs to the area of the umbral core). The area A (in pixels) of the umbral core was defined using the iso-intensity level of [FORMULA], as described in Sect. 3. From this we calculated the effective diameter [FORMULA]. It decreased during the 4 hour 26 minute period from 8:006 to 8:000. In the linear approximation (least-squares fit) the umbral core area decreased at a rate of -3.1 % per hour.

To obtain the number and filling factor of UDs in the umbral core we excluded all bright objects formed by only 2 or 3 pixels, i.e. with [FORMULA] smaller than the resolution limit of the telescope (0:0025). The average number of UDs in the umbral core was [FORMULA]. It varied between 16 and 47 UDs in correlation with the image sharpness (correlation coefficient [FORMULA]), but did not show any regular trend in time. The observed filling factor (based on the "observed sizes" of UDs) was [FORMULA] ; its temporal variations were also correlated with the image sharpness ([FORMULA]), with no regular time-trend. We conclude that during the 4 1/2 hour period the number of UDs and the filling factor remain practically constant and their temporal variations are due mostly to changes in image quality.

4.1. Effective diameters of umbral dots

We remind the reader that the effective diameters [FORMULA] of UDs presented here correspond to the "observed sizes", which are influenced by image blurring (see Sect. 1). To obtain a histogram of [FORMULA] (Fig. 3) we utilized 11758 observations of instantaneous sizes of UDs disregarding their temporal evolution. This has the disadvantage of collecting statistically dependent elements due to the short time spacing between images, but it has the strong advantage of providing a large sample to achieve high statistical significance.

[FIGURE] Fig. 3. Normalized number of UDs vs. observed effective diameter [FORMULA] for 11758 instantaneous measurements.

The average value of [FORMULA] is [FORMULA], in good agreement with the mean observed diameter reported by Sobotka et al. (1993) ([FORMULA]). The histogram in Fig. 3 is asymmetric, with the number of UDs strongly increasing toward the resolution limit. The tail of the histogram for [FORMULA] is probably due to unresolved clusters of UDs rather than to individual ones. This size distribution of UDs is not sensitive to image quality or the level of segmentation: Increasing the image quality selection criterion from 6.6% to 8.5% reduced the number of frames from 360 to 52, but showed nearly no change in the shape of the histogram; increasing the segmentation level from 0.015 to 0.02 in the differential image induced only slight changes in the histogram. We conclude that UDs do not have a "typical" diameter; rather, the smaller the UDs, the more numerous they are.

To show the spatial distribution of UDs of different [FORMULA] (Fig. 4) we calculated the time-averaged effective diameters [FORMULA] of the 662 UDs. We see that large UDs ([FORMULA]) preferentially appear in the bright parts of the umbral core, while small UDs are distributed more or less randomly. The histogram of [FORMULA] has a shape similar to that of [FORMULA] (Fig. 3). From this we conclude that Fig. 3 shows a real distribution of observed sizes of UDs - and is not distorted by transient phenomena like speckles.

[FIGURE] Fig. 4. Spatial distribution of 662 UDs with different time-averaged effective diameters [FORMULA]: Symbol "+" represents UDs with [FORMULA], triangles correspond to [FORMULA], squares to [FORMULA], and bold squares to [FORMULA]. The underlying grey-scale image of the umbral core is the average of 360 frames of the series with contours corresponding to intensities 0.24, 0.26, 0.28, 0.30, and 0.45 [FORMULA]. The coordinates are in pixels (scale 0:00125 per pixel).

4.2. Lifetimes of umbral dots

The histogram of the lifetimes is shown in Fig. 5. The minimum lifetime, 89 s, was set by the tracking algorithm, which, for the sake of reliability, excluded all objects which appeared only in one or two frames. The number of UDs rapidly decreases with increasing lifetime. We find that 66% of UDs have lifetimes shorter than 10 minutes, 27% between 10 and 40 minutes, 6% between 40 and 120 minutes, while only 1% of UDs exist longer than 2 hours. We cannot see any peaks in the histogram. We conclude, therefore, that the 662 UDs in this sample do not have a "typical" lifetime. The average lifetime is 13.8 minutes, and the median 5.9 minutes.

[FIGURE] Fig. 5. Number of UDs vs. lifetime for 662 UDs.

We display in Fig. 6 the spatial distribution of UDs with different lifetimes. We see that long-lived UDs (lifetime [FORMULA] minutes) appear in bright parts of the umbral core at locations similar to those of UDs with large [FORMULA]. We show the relation between lifetimes and time-averaged sizes of UDs with a scatter diagram (Fig. 7). It indicates that the minimum size of UDs increases with lifetime.

[FIGURE] Fig. 6. Spatial distribution of UDs with different lifetimes t: Symbol "+" represents UDs with [FORMULA] minutes, triangles correspond to [FORMULA] minutes, squares to [FORMULA] minutes, and bold squares to [FORMULA] minutes. The underlying image and scale are as in Fig. 4.
[FIGURE] Fig. 7. Scatter diagram of effective diameter [FORMULA] vs. lifetime.
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© European Southern Observatory (ESO) 1997

Online publication: March 26, 1998