THE C H A R A C T E R I S T I C SIZE AND B R I G H T N E S S OF F A C U L A R

P O I N T S IN T H E Q U I E T P H O T O S P H E R E

R I C H A R D M U L L E R * and S T E P H E N L. K E I L

Air Force Geophysics Laboratory, Sacramento Peak Observatory**, Sunspot, NM88349, U.S.A.

(Received 28 June, 1982; in revised form 20 January, 1983)

Abstract. Using two very high resolution, white-light plates of the solar granulation, we measure a characteristic size and intensity for facular points. The plates were obtained with the 50 cm refractor at Pic-du-Midi Observatory using a 60 A bandpass filter center at 5750 A. After adjustment for atmospheric and instrumental smearing, we find a characteristic size of 0.22 arc sec and a characteristic intensity of t.3 to 1.5 times the mean continuum intensity.

1. Introduction

Filtergrams taken in photospheric lines or in the wings of strong lines, like Ha or Ca n K, shown that the bright photospheric network is visible across the entire solar disk, in both active and quiet regions. The network has been identified with white-light faculae (Mehltretter, 1974) and with the so-called solar filigree (Dunn and Zirker, 1973). These authors describe the network as formed by 'crinkles', while Mehltretter (1974) finds that it is made up of facular points in addition to crinkles, especially far from sunspots. In our observed quiet region of the Sun, the network is formed by small points, of size generaly smaller than 0': 5, sitting in intergranular lanes. They appear as isolated points or in the form of associations of several adjacent points, which look like the 'crinkles' described by the authors quoted above. A careful study of the evolution of such associations in the quiet photosphere confirms that crinkles are really formed by individual adjacent points (Muller, 1983). Thus, at least in the quiet Sun, the basic constituent of the photospheric network is the facular point.

In this work we attempt to find the true characteristic size and brightness of facular points at the white-light level in the quiet photosphere. These parameters are of importance for modeling facular points, computing magnetic flux tube models, and interpreting magnetic field measurements. Previously, photometric measurements have been made only on a few peculiar points (Dunn and Zirker, 1973; Mehltretter, 1974; Koutchmy, 1977) and have tittle statistical significance. They mainly show that the size of facutar points is of the order or smaller than the instrumental resolution (0'.'25); their white-light brightness, i.e., near the level % = 1, has never been measured.

* NAS/NRC Resident Research Associate, on leave from Pie du Midi Observatory, France. Present address: Observatoire du Pic-du-Midi et de Toulouse, 65200 Bagneres de Bigorre, France. ** Operated by the Association of Universities for Research in Astronomy, Inc., under contract AST 78-17292 with the National Science Foundation.

Solar Physics 87 (1983) 243-250. 0038-0938/83/0872-0243501.20. �9 1983 by D. Reidel Publishing Co., Dordrecht and Boston

244 RICHARD MULLER AND STEPHEN L. KEIL

2. Observations and Reduction Procedure

The observations were obtained with the 50 cm refractor at Pic-du-Midi Observatory on July 9, 1978; they consist of nearly-simultaneous filtergrams taken through a 60 bandpass white-light filter at 5750 .~ and taken through a 15 A passband interference filter centered on the CaII K (3933 ~.) line, (hereafter referred to as 'calcium pictures'), at disk center. The seeing during the observations was excellent and the selected white-light pictures are of the best apparent quality currently obtainable with the instrument. The resolution of the calcium pictures is not as high, as it is somewhat degraded by the fairly strong chromatism of the refractor in the near UV. Nevertheless, facular points are visible as bright dots on the calcium pictures. We have analyzed two white-light frames taken two seconds apart. To insure proper identification of facular points, several calcium pictures taken before and after the white-light pictures were used. The identification is very easy and unambiguous, because the granulation pattern is visible in both calcium and white-light-pictures and serves as a reference frame (Figure 1).

We digitized an area 110" by 88" on each frame, using the PDP densitometer at Kitt Peak. The frames were sampled every 0.055" in both the horizontal and vertical directions, using a 0.055" x 0.055" square slit. The same area of the solar surface was digitized on both frames. A set of calibration wedges was photographed on the same film and also digitized on the PDP.

A computer routine, using local gradients in density to locate bad pixels due to film scratches or dust, was applied over 0':4 • 0':4 squares. The routine would have

3933 5700

Fig. 1. A subse t of the obse rved regions is shown in bo th ca lc ium and in whi te light. The locat ions of some facular points are ind ica ted wi th arrows.

CHARACTERISTIC SIZE AND BRIGHTNESS OF FACULAR POINTS 245

suppressed facular intensities, if the facular point occupied only a single pixel. However, seeing conditions normaly spread the facular point over at least four pixels. Film densities were converted to intensities using the calibration wedges. All of the observed densities fell on the linear portion of the calibration curve, which had a gamma of 2.35.

Facular points were located by displaying 11" x 11" subsets of the data on a TV monitor using a Grinnell Imaging system. With a joystick, we moved a set of cross-hairs to facular-point locations. The co-ordinates of each facular point were stored and later passed to another routine, which searched a 0.22" x 0.22" area around the given co-ordinate to insure we had found a local maximum. The program then plotted horizontal, vertical, and two diagonal cross sections through the facular point. Before plotting the cross section, each pixel was replaced by a weighted mean over a 3 x 3 pixel square, where the center pixel was weighted 2, the edge pixels weighted 1, and the comer pixels 1/xf2. This was done to suppress noise introduced by film granularity and microphotometer electronics. Typical facular-point cross sections are shown in Figure 2. We next found a size for the facular point along each of the four cross sections and averaged them to produce a mean size for each facular point. Sizes were found by locating the nearest local minimum in each direction along a cross section from the local maximum. The locations of the half intensity points between local maximum and minimum were found and the distance between these points is defined as the size for

S U B - - I M A G E ~ 7 8 I N T E N S I T Y = 1. 108 S I Z E = 0 . 2 9 2

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1.2 �9

0 . 8 I I I I I I I I I I I t I I I I - - t 6 5 0 . 0 0 1.65

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1.0

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Fig. 2. Four cross sections through a representative facular point are shown. With respect to Figure 1, the cross sections are: (a) vertical, (b) horizontal, (c )and (d) diagonal, at 45 ~ and 135 ~ respectively. Intensities are plotted relative to the mean intensity of the surrounding photosphere averaged over a

11" • 11" area.

246 RICHARD MULLER AND STEPHEN L. KEIL

that cross section. Because the above process was displayed graphically, we could intervene if an oddly-shaped facular point caused the program to go astray. Intervention was required in less than 2~o of the cases and usually occurred when a facular point was near a boundary of the digitized area. The maximum brightness (from the unsmoothed data) and mean diameter (from the smoothed data) were determined in this way for each of the 77 facular points visible in the 110" x 88" digitized area. The effect of the triangular filter used above is to increase the measured diameters by some amount less than the pixel separation. For the observed sizes (see Figure 2 and the discussion in Section 3) the filter would increase the mean diameters by less than about 20~o of the pixel separation or 0.01"

Unfortunately, the present observations provide no direct method for finding the true modulation transfer function for the combined smearing due to atmospheric and instrumental effects. Below we use the facular points themselves to estimate the restoration function. The root-mean-square amplitude of the granular intensity fluc- tuations (Alrms) provides a guide to the resolution of the observations and the required restoration function. The mean value ofAIrm s averaged over both frames is 8.3 ~/o. When A/rm s is computed for 11" • 11" subframes of the data, its value varies between 7.8 ~o and 8.8 ~o. This result indicates that seeing conditions are highly uniform over the two frames. After correction for film granularity, the mean value of A/rm s is 7.5~o. Comparison of this value with previously published measurements of A/rm s (see Wittman, 1979, for a complete summary) indicates the resolution of the present observations is extremely good. Comparison with other 'corrected' values (Wittman, 1979) indicates our value of AIrm ~ should be multiplied by a factor lying between 1.3 and 1.8. The digitizing process above gives a correction factor of approximately 1.35.

3. Observed Size

Figure 3 shows the histogram of mean sizes of the 77 facular points measured on the observed quiet region of the Sun. No point has a ~ize significantly larger than 0': 5; a few points were measured with a size smaller than the half-width of the Airy function of the 50 cm refractor at 5750 ~, (0':22). This is probably due to uncertainties in measuring sizes, inherent to irregularities of profile intensities and to uncertainties in locating the nearest minima. The average observed size is 0'.' 33 (because of the narrow distribution of sizes observed, it makes sense to define the maximum of the distribution as a characteristic size). We must determine whether this histogram represents only variations of seeing across the observed field and uncertainties in measurement, or whether it reflects the true size spectrum of facular points. This was done by measuring the size of each facular point separately on both white-fight pictures, which were taken only 2 s apart. If we reasonably assume that the true size remains constant during such a short time interval, any measured difference results from seeing variations and errors in our size measurement. In Figure 4, the size of each facular point measured on one frame is plotted against its size measured on the other frame. It is apparent that each point, large or small, has nearly the same size on both pictures, implying that the

C H A R A C I ' E R I S T I C S I Z E A N D B R I G H T N E S S O F F A C U L A R P O I N T S 247

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i 0 -

1

m

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Fig. 3. The solid line is the combined histogram of the facular point sizes measured on each of the two white-light frames. The dashed line is the dispersion of sizes on each frame from the mean size between the two frames. The dispersions have been averaged over 0."05 bins between 0':2 and 0':4 and over 0'.' 1

bins between 0':4 and 0."6.

atmospheric smearing function is uniform over the observed fields, and its width is not

much larger than the true size of the facular points. Therefore, the observed histogram

of Figure 3 reflects a real spectrum of sizes of facular points, and it will be possible to

estimate their true characteristic size with the help of Figures 3 and 4. It is noteworthy

that we find the same average size on both pictures, (0'.'335), which, as discussed in

Section 2, shows they are of the same quality with respect to atmospheric and

instrumental smearing.

Fig. 4.

. 7 ' -- -

< i <3

,~ . 5 - co

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| | |

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d I (o f -c secor~d.s)

The size of faeular points measured on frame 2 (ordinate) is plotted against the size of the same facular point measured on frame 1 (abscissa).

248 RICHARD MULLER AND STEPHEN L. KEIL

4. Characteristic Size of Facular Points in the Quiet Sun

A. AVERAGE SMEARING FUNCTION

The dispersion of points on Figure 4 is due to uncertainties in measuring the size of the facular points and to seeing variation between the two pictures. It is possible to estimate the smearing function from this dispersion and from the size of the smallest measured points. Let d 1 be the size of a point measured on frame 1 and d2 the corresponding size of the same point measured on frame2, d = (d 1 + d2)/2 its average size, and 6 d = [d 1 - d] the deviation of the measurement d 1 from d. To determine whether the deviation ~Sddepends, statistically, on the average size d, we compute the mean deviation bd,, of the points whose size is in the interval n < d < n + 0"05, taking the successive values 0'.'20, 0'.' 25, 0': 30 . . . . ; the mean size of the points in each of these intervals is denoted by dn. We have lumped the points between 0 ': 4 and 0': 5 and the points between 0.5 and 0.6 into single intervals to increase their statistical significance, bd,, is plotted as a function ofdn in Figure 3. The dispersion of measured sizes remains fairly constant up to 0.'4; above 0'.'4 the dispersion is seen to increase. This increase results partially from the smaller number of observed points and partially from greater irregularities in

shape for the larger facular points. The mean size deviation of points smaller than 0'.' 4 is 0'.' 015; for a gaussian distribution of deviations, the maximum deviation is approxi- mately twice the mean deviation, i.e. 0':03. The halfwidth of the Airy profile of the instrument (0'.' 22) is the smallest size that can be measured, and it is actually reached and surpassed (measuring error) by a few measurements. If we assume that these smallest points lie at the extreme deviation of the smearing function, we then find 0'.'22 + 0"03 -- 0"25 as a typical value for the width of the smearing function.

B. CHARACTERISTIC SIZE

We may now estimate the t rue characteristic size D O , from the simple relation:

Do 2 = do 2 - a 2 '

which is valid for gaussian profiles, where d o is the observed characteristic size and a the half-width of the smearing profile. The use of a more sophisticated restoration procedure is worthwhile only when the smearing profile is accurately known; but even then, restoration procedures give rise to large uncertainties in the results, especially when the features to be corrected are of scales similar to or smaller than the halfwidth of the correction function. For these reasons we only correct the characteristic size, which in any case, is an important parameter for models of facular points.

Correcting the observed size of 0':33 with a gaussian smearing function, where a = 0'.'25, yields a corrected characteristic size of 0"22, only slightly smaller than the half-width of the estimated smearing function. This value should be representative of the true characteristic size of facular points in the quiet Sun, as demonstrated by the following discussion. The true characteristic size cannot be smaller than, say, 0':1, because then Figure 4 would not show a linear trend, but, rather, points randomly

CHARACTERISTIC SIZE AND BRIGHTNESS OF FACULAR POINTS 249

spread around the size 0':25, due to measurement uncertainties and local changes of the seeing between the two pictures. Furthermore, the true size cannot be larger than the observed value of 0'233. Thus, the true characteristic size is larger than 0': 1 and smaller than 0':3, probably close to 0':2.

5. Characteristic Brightness

The 77 facular points measured on our white-light pictures have intensities, measured at the maximum of the intensity profile, ranging from 0.97 to 1.18, with a characteristic value of 1.08 (these values are relative to the intensity of the mean surrounding photosphere). In order to apply corrections for smearing, we must define a background intensity above which both observed and corrected facular point intensities lie. Because the facular points occur in intergranular lanes, we need to estimate the characteristic intensity of these lanes. The intensity in the largest intergranular spaces is affected less by parasitic light than is the intensity in smaller lanes, and can be taken as the background in a first approximation. This brackground has an average intensity of 0.87; the characteristic intensity of the observed facular points is, thus, in excess o f A i = 0.21 over the background. Assuming a gaussian intensity profile, the corrected intensity excess over the background, AI, is expressed by

A I = Ai 1 + c~.

Using a Gaussian smearing function with a = 0':25, a tree facular point size of D o = 0': 22, and taking into account smearing by the microphotometry and the reduction procedure (see Section 2) through the factor c~ (e ~ 1.35), one finds A I = 0.65 and a corrected characteristic intensity Ioorr = I b + A I = 1.5. This correction procedure sup- poses that a point is completely isolated, which is not true in the case of facular points imbedded in intergranular lanes, but closely surrounded by granules. The restflting corrected values are in that case overestimated, and we have to consider the value of Ioorr = 1.5 as an upper limit for the true characteristic intensity of facular points. On the other hand, the intensity in the intergranular spaces directly adjacent to the facular points is, on the average, 0.97. A significant amount of this intensity results from stray light, making it well above the true background. If we take 0.97 as the background intensity against which facular points are measured, the correction procedure described above will underestimate the corrected values. Corrected characteristic intensities obtained in this way have to be considered as a lower limit. With Ai = 0.11, one finds A I = 0.34 and Ioo~r = 1.3, which is a lower limit on the true characteristic intensity of facular points. Thus the true characteristic intensity of facular points should lie between 1.3 and 1.5.

6. Discussion

From our observations we find that in the quiet photosphere a typical facular point has a size of 0':2 = 150 km and a white-light (5700 A_) brightness between 1.3 and 1.5 (the

250 RICHARD MULLER AND STEPHEN L. KEIL

brightness is given relative to that of the average photosphere). The size distribution is fairly narrow, as only a few points are larger than 0"5, and no point has a mean size (dl + d2)/2 larger than 0'.'6. In contrast, Spruit and Zwaan (1981) find, on filtergrams taken in the Mg b line, a much wider size distribution in an active region, including facular elements of diameter as large as 1 "5; they call the elements smaller than 0" 5 facular points and the elements in the range 0'.'5-1'.'5 facular knots; the latter are probably the facular granules described by Mehltretter (1974) or by Muller (1977). The wider size distribution in active regions is confirmed by high resolution pictures obtained at Pic-du-Midi Observatory; moreover, it seems that the size distribution is dependent on the local degree of activity- narrower in the quiet photosphere, wider in young active regions, intermediate in old active regions. As an example, Figure 2 of Muller (1981) shows an old active region with facular elements lying in the range 0"2-1 '.'0. This figure also clearly shows that, even near sunspots," 'crinkles' are resolved into facular points. A similar dependence of size distribution vs local degree of activity probably exists for the associated small-scale magnetic fields.

Foukal et al. (1981) find that faculae appear darker than the average photosphere when observed at disk center in true continuum windows; however, facular fine structure was not resolved. Compared to their continuum observations, our white-light observations refer to a level only a few kilometers higher in the photosphere, which can hardly explain the large discrepancy in brightness obtained in our two respective works. It would be of great interest to measure the intensity of resolved facular elements in the true continuum. Koutchmy (1977) finds an intensity of at least twice the continuum intensity for one particular point observed at Ha + 2 A, i.e., at a photospheric level slightly higher than that we have observed. A few of our facular points reach this level of intensity; however, they are not representative of a characteristic facular point.

References

Dunn, R. B. and Zirker, J. B.: 1973, Solar Phys. 33, 281. Foukal, P., Duvall, T., and Gillespie, D.: 1981, Astrophys. J. 249, 394. Koutchmy, S.: 1977, Astron. Astrophys. 61, 397. Mehltretter, J. P.: 1974, Solar Phys. 38, 43. Muller, R.: 1977, Solar Phys. 52, 249, Muller, R.: 1981, in F. Monyama and J. C. Henoux (eds.), Proceedings of the Japan-France Seminar on Solar

Physics, p. 142. Muller, R.: 1983, Solar Phys. 85, 113, Spruit, H. C. and Zwaan, C.: 1981, Solar Phys. 70, 207. Wittman, A.: 1979, in Small-Scale Motions on the Sun, Mitteilungen aus dem Kiepenheuer-Institut Nr. 179,

p. 29.