3.1. Proper motions of penumbral grains
The trajectories of PGs were determined from the positions tracked in time and smoothed by cubic splines. Of the 469 PGs, 341 (almost 3/4) move inward toward the umbra. We label them INW. More than 1/4 (128 PGs) move outward toward the photosphere. We call them OUT. There appears to be a dividing line (DL) in the penumbra, approximately 0.7 of the distance from the umbra to the photosphere. Outside the DL most PGs (75 %) are of type OUT; inside most (89 %) are INW. This phenomenon is illustrated in Fig. 1, in which we show trajectories of the INW and OUT PGs as black and white lines, respectively. The lengths of trajectories are in the range 0:003-4:008 for INW and 0:003-3:005 for OUT PGs. The mean values are 1:004 and 1:001, respectively, in a penumbra of average width 8:007. We did not observe any PG crossing the whole penumbra. Some INW PGs moved into the umbra but tracking was terminated when their brightness decreased below the 0.8 limit - such features are not the subject of this study.
In addition to tracking individual PGs we applied an LCT algorithm with a tracking window of 1". The flow maps were then averaged over the whole time series. We obtain the same results as reported by Wang & Zirin (1992): Both bright and dark penumbral features move inward in the inner part of the penumbra and outward in the outer part. The boundary between inward and outward motions, defined as the line connecting the LCT zero-velocity points, is plotted in Fig. 1. It separates most trajectories of INW and OUT PGs and is located very close to the position of the DL that we would have chosen by eye (though such a hand-drawn line would not have had the convoluted shape of the contour line). This indicates a consistency between LCT flow maps and the motions of individual PGs as determined by feature tracking.
The histogram of time-averaged proper motion speeds is shown in Fig. 2 (left). Speeds for INW PGs range from 0.0 to 2.0 km s-1 with a maximum around 0.3-0.4 km s-1 and a median of 0.43 km s-1. Speeds of the OUT PGs are in the same range but the maximum is around 0.5-0.6 km s-1 and the median is 0.53 km s-1. These values do not differ substantially from those measured by Muller (1973a) and Tönjes & Wöhl (1982).
To study the dependence of time-averaged velocities on position, we divided the penumbra into 10 bins, each covering 0.1 of the relative distance from the P/U (position 0.0) to the P/Ph boundary (position 1.0). Velocities of all PGs whose time-averaged positions fell into a given bin were averaged. The results are plotted in Fig. 3 (top) together with error bars which characterize the scatter of individual values.
For the INW PGs speeds increase from about 0.4 km s-1 at the P/U boundary to a maximum of 0.75 km s-1 at the relative distance 0.7-0.8 (just outside the DL) and then drop to about 0.5 km s-1 near the P/Ph border. Speeds of the OUT PGs show a strong increase from 0.2 km s-1 in the middle penumbra (position 0.4-0.5) to a maximum of 0.7 km s-1 near the P/Ph boundary. Note, however, that the numbers of OUT PGs at positions less than 0.6, and of INW PGs at positions greater than 0.8, are very small, as can be seen in Fig. 3.
The observed dependence of INW speeds on relative position in the penumbra is qualitatively similar to the prediction given in the model by Schlichenmaier et al. (1998b, Fig. 7). To compare PG motions in detail with the results of the model, we calculated instantaneous velocities of each PG as derivatives of positions smoothed by cubic splines. Individual velocity curves, showing variations of speed with time, were divided into five types:
Numbers of INW and OUT PGs belonging to the different speed types are summarized in Table 1. For INW PGs, decelerating (35 %) and accelerating (30 %) types are most common, while 33 % have more complex behavior ("" and ""). Only decreasing of INW speed with time, i.e. type "", agrees with the model prediction. About 1/3 of OUT PGs increase their speed with time. The maximum instantaneous speed for both INW and OUT PGs is in the range 2-3 km s-1, matching the value predicted by the model.
Table 1. Numbers of PGs by speed type
3.2. Photometric characteristics of penumbral grains
As mentioned earlier, we obtained the brightness maximum of each PG at each time step as one of the outputs of the feature-tracking routine. The temporal intensity fluctuations, characterized by , are small. The time-averaged intensities are in the range 0.84-1.10 for both INW and OUT PGs, but OUT PGs are brighter on average (0.96 ) than the INW ones (0.94 ). The intensities do not depend on relative distance from the umbra. They are uncorrelated with the speeds or lifetimes of PGs.
We used the best frame of the series to do complementary visual measurements of lengths, widths, and contrasts of 56 PGs. The lengths lie in a broad range from 0:006 to 3:007; the average value is 1:007. The mean and standard deviation of widths measured across the "heads" of PGs are . The intensity of the "heads" is typically 0.94 , 1.4 times brighter on average than adjacent dark fibrils. Since the intensities are influenced by scattered light, the real contrast is probably higher.
Balthasar et al. (1996) reported that the mean white-light image of a penumbra averaged over nearly 2 hours still showed radial structures. We confirm this result. Averaging frames over our 4.5 hour series we find a filamentary structure in the penumbra (of course without PGs, which have disappeared due to time-smoothing) with rms contrast of 7 %. For comparison, the penumbral rms contrast in our best frames is about 12 %. This remarkable persistence of high contrast over many hours must be due to the stability of the magnetic field configuration. This stability causes the paths of many individual PGs to follow nearly identical trajectories.
3.3. Lifetimes of penumbral grains
Since lifetimes are derived from the number of frames in which individual PGs are observed, histories of some PGs will be cut off at the beginning or end of the time series. However, this effect is small, because most of the lifetimes are substantially shorter than the duration of the series. We show in Fig. 2 (right) a histogram of lifetimes for the 341 INW and 128 OUT PGs that passed the visual consistency check. For the INW PGS the maximum lifetime is almost four hours (231 minutes) but only 17 % live longer than 1 hour; the median and mean lifetimes are 29 and 39 minutes, respectively. The OUT PGs live shorter than the INW ones: the maximum, median, and mean lifetimes are 59, 22, and 25 minutes, respectively.
The number of PGs increases with decreasing lifetime t to the limit of 12 minutes. (Recall that in Sect. 2 we had set the minimum lifetime of visually checked PGs to this value.) Of course, we expect a large population of PGs to have minutes, and try to estimate this number as follows: Of the total sample of 1027 PGs, 378 lived less than 12 minutes. Assuming the same fraction of PGs that have to be discarded (0.28) and relative numbers of INW and OUT PGs (0.73 and 0.27) as in the case of the minute population, we get about 198 (74) "good" short-lived INW (OUT) PGs. Adopting 7 minutes for the average lifetime of PGs with minutes, as derived from the statistics of the unchecked sample, we extrapolate the median and mean lifetimes for all INW (OUT) PGs to about 17 and 27 (15 and 19) minutes, respectively.
The average lifetime of PGs with minutes as function of position in the penumbra is shown in Fig. 3 (bottom). The mean lifetime of INW PGs rises from 35 minutes for those at the P/U boundary to a maximum of 47 minutes at the relative position 0.2-0.3, and then decreases to 22 minutes at positions 0.7-1.0 between the DL and the P/Ph border. The lifetime of OUT PGs exhibits little trend as function of position, and has a mean value about 25 minutes. The average lifetime of INW PGs that show deceleration at the beginning of their existence (speed types "" and "") is longer by a factor of 1.2 than that of the other INW PGs. For OUT PGs we observe no relation between lifetime and speed type.
In summary, while we agree with Muller (1973a) that the average lifetime of INW PGs in the inner part of the penumbra is larger than for those near the P/Ph boundary, we do not see the pronounced dependence of lifetime on penumbral position reported by Muller (1973a) and Tönjes & Wöhl (1982). More significantly, the lifetimes we measure are only about one-fourth those of the earlier measurements.
© European Southern Observatory (ESO) 1999
Online publication: July 26, 1999