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2.1 How Surface Brightness Is Measured

In simple terms, surface brightness is very similar to surface air pressure. The amount of air molecules in a three dimensional column of air in the atmosphere determines the total amount of pressure which is exerted at a point on the surface of the earth. If we imagine a disk galaxy as an optically thin cylinder, then the surface brightness is a measure of the space density of stars as projected through a cylindrical cross section. The mean luminosity density through this cylinder, which is determined both by the stellar luminosity function and the mean separation between stars, is what observers measure as a projected surface brightness. Since the number of stars per Mpc3 has a strong radial dependence, the projected surface brightness profile shows a fall off with radius. This fall off is generally exponential in character and can be expressed as

µ (r) = µ 0 + 1.086 r / alpha l ,

where 1.086 = 2.5 log e-1. In this formulation two parameters completely characterize the light distribution: µ 0 is the central surface light intensity and alpha l is the scale length of the exponential light fall off. In what follows µ 0 will refer to the central surface brightness in the blue. If Freeman's law is correct, the number of parameters relevant to galaxy selection reduces to one as variations in size modulate those in luminosity. Since µ 0 is a measure of the characteristic surface mass density of a disk, Freeman's Law requires that all the physical processes of disk galaxy formation and evolution conspire to result in this very specific value for all galaxies. Either the surface mass density must be the same for all galaxies (in itself a peculiar result) with little variation in the mass to light ratio, or variations in the star formation history, collapse epoch and initial angular momentum content must all conspire to balance at this arbitrary value.

The historical importance of Freeman's Law is that it was the first real attempt at quantifying the surface brightness distribution of spiral disks. As such Freeman (1970) has been cited over 700 times indicating that it has had an enormous impact on the field, particularly on studies of the galaxy luminosity function, which, until recently has assumed to be independent of surface brightness. As a consequence of its popularity and wide spread influence in extragalactic astronomy there were several attempts to show that Freeman's Law wasn't correct. While Disney (1976) dismissed Freeman's Law as an artifact of selection, others were not so sure. For instance, Kormendy (1977) asserted that Freeman's Law could be an artifact of improper subtraction of the bulge component in disk galaxies. Boroson (1981) suggested that Freeman's Law was a conspiracy of dust obscuration, since if galaxies had appreciable optical depth then we could only see their front surfaces. This could result in the appearance that disk surface brightnesses were fairly constant. Bothun (1981) discovered from a thorough survey of disk galaxies in the Pegasus I cluster that there were equal numbers of galaxies per magnitude bin with B band central surface brightness values in the range of 21-23.5 mag arcsec-2. The existence of any disk galaxy with µ0 fainter than 23.0 mag arcsec-2 would represent a 4sigma deviation from Freeman's Law and hence should not have been found so easily. Unfortunately, Bothun (1981) was not sufficiently astute to notice or appreciate the significance of this at that time so the effect went unnoticed, until recently (e.g., Figure 1).

In principle, the surface brightness profiles of galaxies can be traced to arbitrarily large radii. Recent data by Zaritsky (1997) show that the dark matter halos of typical spiral galaxies may be extremely large (up to 200 h100-1 kpc in radius) and hence disk galaxies could extend out to approx 50 scale lengths! Integration of Eq. 1, as a function of scale length, shows that 1alpha l contains 26% of the total luminosity, while 4alpha l and 5alphal contain 90 and 96%, respectively. In practical terms, a diameter defined by four scale lengths provides a good measure of the total luminosity of the system.

For a Freeman disk, this corresponds to an isophotal level of 26.05 mag arcsec-2. The darkest night skies that can be found for terrestrial observing have µ 0 ~ 23.0 mag arcsec-2. Thus, the 90% luminosity isophote is some 3 mags below even the darkest skies or 6% of the sky level. We have consistently defined a LSB disk as one which has µ 0 fainter than 23.0 mag arcsec-2 or a 90% luminosity isophote of 27.4 mag arcsec-2 which is 2% of the night sky brightness. The Freeman value for µ 0 is about 1 magnitude brighter than the surface brightness of the darkest night sky. That the number of galaxies with faint central surface brightnesses appears to decline rapidly as µ 0 -> µ sky is suspicious and if true of the real galaxy population implies that our observational viewpoint is privileged in that we are capable of detecting most of the galaxies that exist, at least when the moon is down. This is the essence of the argument voiced by Disney (1976) in characterizing the Freeman Law as a selection effect.

So if surface brightness selection effects are important, what is the best way to measure surface brightness? Consider the following two hypothetical galaxies which have the same total luminosity (MB = -20):

Galaxy A: µ 0 = 21.0 alpha l = 3 kpc.

Galaxy B: µ 0 = 24.0 alpha l = 19 kpc.

There are three conceivable ways of measuring the surface brightness in these disks: (1) central surface brightness (µ 0), (2) average surface brightness within a standard isophote (= 25.0 mag arcsec-2, µ iso), or (3) effective surface brightness (measured within the half light radius = 1.7 alpha l = µ eff instead of measured at reff). Note that methods 2 and 3 do not require that the light profile is well fit by an exponential function. The results of applying each of these three measures of surface brightness are the following:

Galaxy A: µ 0 = 21.0; µ iso = 23.2; µ eff = 22.3.

Galaxy B: µ 0 = 24.0; µ iso = 25.8; µ eff = 24.8.

The largest difference in surface brightness occurs when µ 0 is used as the measure. Hence we have adopted this to define the disk galaxy surface brightness. While we realize that this definition requires that the galaxy be adequately fit by an exponential profile, most LSB galaxies at all luminosities meet this criterion (see McGaugh and Bothun 1994; O'Neil et al. 1997a; Sprayberry et al. 1995a). This exercise also makes the trivial point that galaxies with low µ 0 require surface photometry out to very faint isophotal radii in order to determine a total luminosity.

2.2 A Censored View of the Galaxy Population

Much of our knowledge of galaxies has stemmed from detailed studies of objects that populate the Hubble Sequence. Over 70 years ago, Hubble (1922) warned of relying too much on this venture:

``Subdivision of non-galactic nebulae is a much more difficult problem. At present and for many years to come, their classification must rest solely upon the simple inspection of photographic images, and will be confused, by the use of telescopes of widely differing scales and resolving powers. Whatever selection of types is made, longer exposures and higher resolving powers will surely cause a reclassification of many individual nebulae ...''

In this quote Hubble establishes that galaxy classification, and therefore implicitly galaxy detection, is highly dependent upon observing equipment and resolution. The essential issues are: (1) how severe is the bias in terms of the potential component of the galaxy population that has been missed to date, and (2) how would this affect our current understanding of galaxy formation and evolution? In hindsight it is somewhat mysterious why this issue of galaxy detection wasn't considered more seriously 25 years ago. Tinsley's (e.g. Tinsley 1975) elegant and accurate modeling of the stellar populations of galaxies in the late 60's and early 70's certainly indicated that galaxies could undergo significant luminosity evolution, thereby producing faded and diffuse galaxies at z = 0. Alternatively, there might be a population of intrinsically low surface mass density systems whose evolution is quite different from ``normal'' galaxies, but which nevertheless are important repositories of baryonic matter. Disney was the most resonant voice to suggest that such diffuse systems could exist, and therefore that we could be missing an important constituent of the general galaxy population.

A very simple way of describing the effect of surface brightness selection is offered below. While Disney and Phillips (1983) and McGaugh et al. (1995) have quantified these effects to produce Figure 1, the essential point is that LSB disks, at any luminosity/circular velocity, are detectable out to a significantly smaller distance than HSB disks. Consider the five hypothetical galaxies listed below. The first four have pure exponential light distributions and similar total luminosity (MB approx -21.1); the fifth galaxy has the same scale length has Galaxy B but a factor of 10 lower total luminosity. This adheres to the McGaugh et al. (1995) assumption that scale length and total luminosity are uncorrelated in a representative sample of disk galaxies.

Galaxy A: alpha l = 0.5 kpc, µ 0 = 16.0 mag arcsec-2,

Galaxy B: alpha l = 5.0 kpc, µ 0 = 21.0 mag arcsec-2,

Galaxy C: alpha l = 25.0 kpc, µ 0 = 24.5 mag arcsec-2,

Galaxy D: alpha l = 50.0 kpc, µ 0 = 26.0 mag arcsec-2,

Galaxy E: alpha l = 5.0 kpc, µ 0 = 23.5 mag arcsec-2.

Assume that the intrinsic space density of these five galaxies are equal, and suppose that we conduct a survey to catalog galaxies which have diameters measured at the µ 0 = 25.0 mag arcsec-2 level (D25) of greater than one arcminute. Under these conditions, we are interested in determining the maximum distance that each galaxy can be detected:

Galaxy A: This galaxy is quite compact (ratio of 1/2 light diameter to D25 = 0.33) and would fall below the catalog limit beyond a distance of 60 Mpc.

Galaxy B: This is a typical large spiral (like M31); D25 corresponds to 3.74 alphal which is 18.7 kpc or a diameter of 37 kpc. This projects to an angular size of 1 arcminute at a distance of 125 Mpc.

Galaxy C: D25 corresponds to 0.45 alphal or 11.5 kpc. This projects to an angular diameter of 1 arcminute at a distance of 76 Mpc.

Galaxy D: D25 doesn't exist and this galaxy would never be discovered in such a survey.

Galaxy E: D25 corresponds to 0.92 alphal or 4.6 kpc. This projects to an angular diameter of 1 arcminute at a distance of 30 Mpc.

The total survey volume is defined by Galaxy B, as they can be seen to the largest distance. The ratio of sampled volumes for each of the other Galaxy types is considerably smaller. For instance, the volume ratio of Galaxy B to Galaxy E is a factor of 70! Hence, a survey like this would take the real space density distribution (which is equal) and, through the survey selection effect, produce a catalog which would contain 72% type B galaxies, 18% type C galaxies, 9% type A galaxies and 1% type E galaxies. Type D galaxies would not be represented at all. This is a severe bias which would lead us to erroneously conclude that there is predominately one type of disk galaxy in the universe, and that this naturally leads to Freeman's Law. Now in the real zoology of LSB galaxies, type C and D galaxies are quite rare but type E galaxies are common. Hence, their detection locally automatically means the space density is relatively large because they are so heavily selected against and the volume correction factors are significant. This was the essence of Disney's original argument but there was no real data to support it at that time. Twenty years of progress enables us to plot Figure 1, which has the remarkable property that there is no indication of a rapid fall of in µ0.

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