Annu. Rev. Astron. Astrophys. 1992. 30: 613-52
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2.1 Galaxy Counts

2.1.1 TECHNIQUES Charge-coupled devices (CCD's) have extended the practical range of galaxy photometry by several magnitudes, allowing galaxies of even ordinary luminosity to be detected at redshifts greater than unity. Measurements of color, surface brightness, and morphology, and indirect arguments based on the distributions of these quantities, can help constrain models at fluxes that are too faint for spectroscopic work.

Various techniques have been devised for automatically identifying and classifying faint images in digital data. Galaxy images are usually assumed to be contiguous and centrally concentrated, and a search for local maxima then provides a first-cut basis for image detection. To guard against noise peaks, one imposes a lower limit to the number of contiguous pixels that appear above a set threshold. To guard against incompleteness, one sets the threshold as low as possible. Thus, galaxy counts tend not to be strictly magnitude-limited, but rather are limited simultaneously in area and in surface brightness. If the detection area were, say, 2 arcsec2, and if the threshold in surface brightness were 3 x 10-3 of the B-band sky background, then galaxies with measured B = 28.5 could be detected. The magnitude is then measured in some way that maintains a particular signal-to-noise ratio, or allows a relatively convenient interpretation of the results, or has some other desirable attribute. Isophotal magnitudes are computed by integrating the flux above a chosen surface brightness threshold; aperture magnitudes are computed within an area of fixed size; quasi- total or asymptotic magnitudes attempt to collect close to all the light; and a number of other schemes have also been tried (see Kron 1980b).

Photometry of galaxies is not a well-defined procedure and requires attention to many technical details, since a number of systematic errors or biases emerge at very low signal-to-noise ratio: 1. Small errors in the subtracted sky can easily dominate the net photometric errors. 2. Crowding of images becomes a significant problem at faint limits (by B = 26, the average separation is only 12 arcsec, with many galaxies overlapping). Software must be able to distinguish cases of confused images and to account properly for the flux in the individual components. 3. Ultraviolet images of nearby spiral galaxies, which will approximate the appearance of faint galaxies in the U and B bands, show distributed patches of high surface brightness (Bohlin et al. 1991). Thus real images might not conform to the centrally-concentrated, contiguous idealization of image-detection algorithms. 4. Some effects are peculiar to isophotes in noisy data. Due to photon shot noise, even a centrally concentrated image may break up into small islands that become difficult to reassemble without bias in the derived fluxes. Moreover, objects will appear artificially bright because statistically low pixels will be excluded from the isophotal area. 5. There will be a statistical redistribution of objects resulting in steeper counts because there are more faint galaxies than brighter ones and the photometric errors increase with increasing magnitude. 6. Competing with this effect is increasing incompleteness with increasing magnitude. 7. A different problem is the photometric zero-point, which is based on bright stars that have spectral shapes quite different from those of high-redshift galaxies.

Accurate interpretations of faint galaxy counts should thus be based on detailed simulations of the selection function, random errors, and known systematic errors. We prefer to include these aspects in the model rather than applying correction factors to the data. In this review, we adopt this viewpoint by emphasizing the common elements of observations by independent groups and by plotting uncorrected data.

2.1.2 BRIGHTER GALAXY COUNTS Galaxy counts at B < 19 provide the normalization for the faint-galaxy expectations. However, in several respects bright-galaxy photometry is harder than faint-galaxy photometry because the range of surface brightnesses is greater. Also, with the lower surface density of bright galaxies, setting up a spatially uniform photometric system is more difficult, especially with variable Galactic extinction (100 deg2 must be sampled to get roughly 100 galaxies at B ~ 15). Moreover, one needs to correct for the vastly larger number of contaminating stars.

Essentially all modern bright galaxy counts are derived from Schmidt plates, notably the UK Schmidt (see Heydon-Dumbleton et al. 1989 and Maddox et al. 1990 for examples) and the Palomar/Oschin Schmidt. The counts of Sebok (1986) and Picard (1991) are illustrative of some of the differences between surveys. Both used IIIa-F plates with the Schmidt telescope at Palomar and surveyed contiguous northern regions at b ~ +50°. Sebok analyzed 4 plates, and Picard 7 plates. Both made direct calibrations of their photographic photometry with selected galaxies in their fields and reduced their results to the Gunn/Thuan r-band system. The star counts are similar in both studies, yet the galaxy counts, A(m), are apparently different: over the interval 16.5 < r < 19, Sebok's counts are steeper than Picard's. At r = 16.5, DeltalogA = 0.42, and at r = 19, DeltalogA = 0.08, in the sense: Picard (north) - Sebok. The difference between these two northern fields, which are separated by about 60 degrees, is in fact greater than the difference between Picard's northern and southern fields (DeltalogA ~ 0.15). The areas and depths are large enough that the effect of large-scale structure at the level of 1 s is expected to be less than DeltalogA = ± 0.04 (S. M. Kent, private communication).

This difference may be partly due to the details of the photometry and its calibration. The measuring machines, the software, the detectors used for calibration, and the technique of calibration, were all quite different. Considering the fundamental limitation of photographic plates to record faithfully intensity levels over a wide dynamic range; the possibility of the data being affected by scattered light in the respective microdensitometers; and the wide range of galaxy profile shapes, even direct calibration of integrated magnitudes on photographic plates may not always be reliable.

Counts of brighter galaxies in the blue band have been studied by many more investigators. The work of Maddox et al. (1990) is important because the area covered is 4300 deg2, far larger than any other photometric survey. Since other surveys lie within the Maddox et al. (1990) region, there is the opportunity to compare photometry on a galaxy- by-galaxy basis: differences in the zero points for each common field are of order 0.15 mag (Maddox et al. 1990).

Maddox et al. (1990) called attention to the steeper slope of their counts, DeltalogA / Deltam, in the general range 17 < B < 19, than predicted by Ellis's (1987) no-evolution model. In fact, the steep slope in this region is characteristic of a number of such studies, e.g. Butcher and Oemler (1985), Stevenson, Shanks, and Fong (1986), and Heydon-Dumbleton et al. (1989). Butcher and Oemler (1985) presented a summary of the Kirshner, Oemler, and Schechter (1978) and Kirshner et al. (1983) field-galaxy counts and color distributions. Stevenson, Shanks, and Fong (1986) presented data from 17 Schmidt fields covering 355 deg2. Heydon-Dumbleton et al. (1989) surveyed 100 deg2, and there are over 3000 galaxies in their survey in the interval 17.5 < B < 18.5. All of these surveys are characterized by a slope that is as steep as that of Maddox et al. for B < 19. Even more extreme are Sebok's counts, which reach deeper than the blue counts discussed so far, and yet are almost as steep as the Euclidean case DeltalogA/ Deltam = 0.6 up to r = 18, where confusion with stars begins to become important. We will return to this result in Section 3.3.

2.1.3 FAINTER GALAXY COUNTS Counts in the U-band have been presented by Koo (1986), Majewski (1988, 1989), Guhathakurta, Tyson, and Majewski (1990b), Songaila, Cowie, and Lilly (1990), and Jones et al. (1991). Majewski's (1989) counts suggest a possible flattening fainter than U ~ 24 - a potentially important result that needs confirmation. Counts in blue and red bands have received the most observational attention, and the current status has been reviewed by Metcalfe et al. (1991). I-band counts have been made by Hall and Mackay (1984), Koo (1986), Tyson (1988), Hintzen, Romanishin, and Valdes (1991), and Lilly, Cowie, and Gardner (1991). K-band counts have been summarized by Broadhurst, Ellis, and Glazebrook (1992) and by Cowie et al. (1992). The limit of extreme crowding and extremely low signal-to-noise ratio, B > 26, has been studied so far only by Tyson (1988) and by Lilly, Cowie, and Gardner (1991).

Figure 1. Differential galaxy counts A(m) in number per square degree per magnitude in the blue (bJ), red (Gunn r), and near-infrared (K) bands. The magnitude scale for the bJ counts is at the top. To cover the greatest range in magnitude, the plot shows bright photographic surveys and faint CCD surveys. Sources for the blue counts: Butcher and Oemler (1985); Stevenson et al. (1986); Ciardullo (1987); Heydon-Dumbleton et al. (1989); Maddox et al. (1990); Jones et al. (1991); Tyson (1988); Metcalfe et al. (1991); Neuschaefer et al. 1991; Lilly, Cowie, and Gardner (1991). Sources for the red counts: Stevenson et al. (1986); Sebok (1986); Picard (1991); Hall and Mackay (1984); Yee, Green, and Stockman (1986); Tyson (1988); Metcalfe et al. (1991). Sources for the K-band counts: Mobasher et al. (1986); Broadhurst, Ellis, and Glazebrook (1992); Cowie et al. (1992). The curves give the predicted counts for the no-evolution model discussed in the text, all sharing a common normalization.

Figure 1 shows the differential counts log A(m) (number per deg2 per mag) for surveys in the blue, red, and near- infrared bands, transformed where necessary to the adopted standards of photographic bJ, Gunn r, and K, according to information presented in the original papers. Some differences do appear between different investigators, but the overall agreement is obvious, and we do not distinguish between the separate surveys in Figure 1; sources are given in the figure caption. To prevent excessive crowding of the points, in general we have omitted photographic photometry for bJ and r in the intermediate range of magnitudes. We have plotted data only brighter than conservative limits where the statistical noise in the sky background is comparable to the flux itself (and beyond which the corrections for incompleteness become large).

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