Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://zebu.uoregon.edu/1997/ph410/preprint.ps Äàòà èçìåíåíèÿ: Wed Mar 12 19:20:40 1997 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 04:17:26 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 101

To appear in The Astrophysical Journal, Vol. 481, June 1, 1997
Low Surface Brightness Galaxies in the Local
Universe. III. Implications for the Field Galaxy
Luminosity Function
D. Sprayberry
Kapteyn Laboratorium, University of Groningen, Postbus 800,
9700 AV Groningen, The Netherlands
Email: dspray@astro.rug.nl
C. D. Impey
Steward Observatory, University of Arizona, Tucson, AZ 85721
Email: cimpey@as.arizona.edu
M. J. Irwin
Royal Greenwich Observatory, Madingley Road, Cambridge, UK CB3 0EZ
Email: mike@mail.ast.cam.ac.uk
and
G. D. Bothun
Department of Physics, University of Oregon, Eugene, OR 97403
Email: nuts@moo.uoregon.edu
ABSTRACT
We present a luminosity function for low surface brightness (LSB)
galaxies identified in the APM survey of Impey et al. (1996). These
galaxies have central surface brightnesses (¯(0)) in B in the range
22:0 Ÿ mu(0) Ÿ 25:0. Using standard maximum­likelihood estimators,
we determine that the best­fit Schechter function parameters for this
luminosity function (LF) are ff = \Gamma1:42, M \Lambda = \Gamma18:34, and OE \Lambda = 0:0036,
assuming H 0 = 100 h 100 km s \Gamma1 Mpc \Gamma1 . We compare the luminosity and
number densities derived from this luminosity function to those obtained
from other recent field galaxy studies and find that surveys which do
not take account of the observation selection bias imposed by surface
brightness are missing a substantial fraction of the galaxies in the local
universe. Under our most conservative estimates, our derivation of the
LF for LSB galaxies suggests that the CfA redshift survey has missed at
least one third of the local galaxy population. This overlooked fraction
is not enough by itself to explain the large number of faint blue galaxies
observed at moderate redshift under no­evolution models, but it does help
close the gap between local and moderate­redshift galaxy counts.

LSB Galaxies And The Field Luminosity Function 2
1 Introduction
The optical luminosity function (LF) of galaxies is one of the fundamental building
blocks of cosmology. Accurate knowledge of the luminosity function is necessary for,
among other things, estimating the mean luminosity density of the universe, and
predicting the redshift distribution of objects in various magnitude intervals (see
e.g., the review by Binggeli et al. 1988). The shape of the luminosity function also
provides an important test for theories of galaxy formation (e.g., Press & Schechter
1974). Further, considerable attention has been focussed of late on the large numbers
of blue galaxies found in deep surveys, first described by Kron (1980) and Hall
& Mackay (1984). The degree to which number counts of these galaxies exceed
those predicted from local observations (e.g., Bruzual & Kron 1980 and Guiderdoni
& Rocca­Volmerange 1990), and indeed whether an excess exists at all (compare
Koo et al. 1993 and McGaugh 1994), depend on the shape, normalization and color
dependence of the luminosity function.
One of the problems with building a galaxy luminosity function is that surveys
are limited in the detection of diffuse galaxies by the brightness of the night sky, and in
the detection of compact galaxies by the difficulty in distinguishing stars and galaxies.
As Disney (1976) and Disney & Phillipps (1983) have demonstrated, at a given
luminosity a survey will identify preferentially those galaxies that have the maximum
possible angular size above the limiting isophote. At a constant luminosity, galaxies
of high surface brightness (HSB) become indistinguishable from stars, and galaxies of
low surface brightness (LSB) fall below the limiting isophote over most of their extent.
Although they purport to be magnitude limited, galaxy surveys which do not take
account of surface brightness effects are missing an unknown but potentially large
number of galaxies in each magnitude bin. Recent surveys of the Virgo cluster by
Impey et al. (1988) and of the Fornax cluster by Irwin et al. (1990) and Bothun et al.
(1991) have taken account of this potential source of bias by deliberately searching for
LSB galaxies. They have found that previous surveys missed a significant fraction
of the cluster populations, particularly at fainter luminosities (MB ¸ ? \Gamma16), and
Impey et al. (1988) determined that inclusion of LSB galaxies in Virgo steepened
the low­luminosity tail of that cluster's luminosity function considerably. To date,
however, no estimates of the field galaxy luminosity function have addressed the
effects of surface brightness bias. However, McGaugh et al. (1995a) found that the
space density of galaxies as a function of central surface brightness appears to be flat
below ¯B (0) = 22:0. Also, Sprayberry et al. (1996) found a space density of galaxies
as a function of central surface brightness that appeared flat below ¯B (0) = 23:0
after descending from a peak around ¯B (0) = 21:75. Although many of these LSB
galaxies are not necessarily faint, the forms of these distribution functions strongly
suggest that the normalization of the galaxy space density at z = 0 has been strongly
influenced by surface brightness selection effects.
We have recently completed a survey for LSB galaxies in the region defined

LSB Galaxies And The Field Luminosity Function 3
by \Gamma3 ffi Ÿ ffi Ÿ 3 ffi and jbj ? 30 ffi , surveying about 786 square degrees of sky with
the Automated Plate Measuring (APM) system at Cambridge. 1 We have identified
693 galaxies, most previously uncataloged and most with central surface brightness
¯B (0) ? 22 mag arcsec \Gamma2 . The complete catalog of this survey appears in Impey
et al. (1996) (Paper I). The selection effects and completeness corrections for the
survey are analyzed in detail in Sprayberry et al. (1996) (Paper II).
In this paper, we present the luminosity function for LSB galaxies from the APM
survey and compare that luminosity function to those obtained from the CfA redshift
survey. We also review suggestions by Phillipps et al. (1990), McGaugh (1994),
McLeod (1994), and Ferguson & McGaugh (1995) that LSB galaxies might account
at least partially for the large numbers of faint blue galaxies seen in deep surveys.
Section 2 describes the survey data and presents the samples used for determining the
luminosity function and the corrections applied to those samples. Section 3 covers the
methods used to develop the luminosity functions. Section 4 presents the luminosity
functions and compares the results to those obtained from the CfA redshift survey.
Section 5 reviews the consequences of this LSB luminosity function for the general
field luminosity function and for the question of local counterparts to the faint blue
galaxies. Finally, Section 6 summarizes our conclusions. Throughout this paper, we
assume H 0 = 100 h 100 km s \Gamma1 Mpc \Gamma1 . Also, all magnitudes and surface brightnesses
used here are in the Johnson B band.
2 Samples Used
The APM survey for LSB galaxies is presented in Paper I, and Paper II describes the
details of how LSB galaxies were identified and calibrated. Paper II also presents a
selection function that gives the completeness of the survey as a function of galaxy
central surface brightness and scale length (hereafter, ``the APM selection function'').
We conducted followup optical spectroscopy at the Multiple Mirror Telescope 2
and 21 cm H I spectroscopy at Arecibo Observatory 3 to obtain radial velocities for
as many of the galaxies as possible. To date we have measured recessional velocities
for 332 of the 693 galaxies on the list, of which 190 come from H I spectroscopy
and 142 from optical spectroscopy. These heliocentric velocities are presented in
Paper I. For developing the luminosity function, we have further corrected these
heliocentric velocities to the rest frame of the Local Group, using the standard
1 The APM is a National Astronomy Facility, at the Institute of Astronomy, operated by the
Royal Greenwich Observatory. A general description of the APM facility is given by Kibblewhite
et al. (1984).
2 The Multiple Mirror Telescope is a facility jointly operated by the Smithsonian Institution and
the University of Arizona.
3 The Arecibo Observatory is part of the National Astronomy and Ionosphere Center. The
NAIC is operated by Cornell University under a cooperative agreement with the National Science
Foundation.

LSB Galaxies And The Field Luminosity Function 4
Figure 1: Structural properties of the complete LSB sample and the subset with radial
velocities. (a) shows the distribution as a function of B central surface brightness, and
(b) shows the distribution as a function of half­light radius. In the upper panels, the
dotted histogram is the distribution of the complete sample, and the solid histogram
is the distribution of the subset with velocities. In the lower panels, the filled circles
show the fraction of galaxies with velocities for each bin, with error bars from counting
statistics. The solid lines show the parametrizations described in the text.
correction v corr = v hel + 300 sin l cos b. No correction was applied for Virgocentric
infall since the median velocity of the sample places most of the galaxies well beyond
the Local Supercluster. These corrected velocities were then used to estimate distance
moduli using the relation:
m \Gamma M = 5 [log v corr \Gamma log H 0 + 5] (1)
assuming as noted above that H 0 = 100 h 100 km s \Gamma1 Mpc \Gamma1 .
The galaxies with velocities do not form a random subset of the overall survey.
For reasons of observational efficiency, like all other galaxy surveyers we favored
galaxies of higher central surface brightness and larger angular size. Figure 1 shows
the distributions of central surface brightness and half­light radius for the complete
sample and for the subset with velocities, along with the ratios of the two sets by bin.
We assume that the galaxies for which we have measured redshifts are representative
of all galaxies in a given bin of surface brightness and angular size. This additional
source of bias must be taken into account in preparing a luminosity function. We
have parameterized this bias in the simple forms depicted in Figure 1: three separate

LSB Galaxies And The Field Luminosity Function 5
linear fits in the different regions of the ¯(0) distribution
p ¯ =
8 ? !
? :
1:000; ¯(0) ! 20:25
4:950 \Gamma 0:194 ¯(0); 20:25 Ÿ ¯(0) Ÿ 25:0
0:111; ¯(0) ? 25:0
(2)
where ¯(0) is in mag arcsec \Gamma2 , and in the different regions of the half­light radius
distribution
p re =
8 ? !
? :
0:667; r eff ! 3
\Gamma0:130 + 0:076 r eff ; 3 Ÿ r eff Ÿ 13
0:773; r eff ? 13
(3)
where r eff is in arcseconds. The final probability that an LSB galaxy will be detected
by the APM and included in the subset with velocities is given by
p tot = pAPM \Theta p ¯ \Theta p re (4)
where pAPM is the probability derived from the APM selection function of Paper II.
Equation 4 assumes that the corrections in ¯(0) and r eff are separable. This
assumption is reasonable for our sample, because ¯(0) and r eff are uncorrelated:
Pearson's r = 0:075 and the Spearman rank correlation coefficient s = \Gamma0:121, and
neither coefficient is significantly different from zero.
We note that Figure 1 shows that the surface brightness range 23:5 Ÿ ¯(0) Ÿ 24:5
includes a large number of identified galaxies, but that a very small fraction of those
galaxies were observed spectroscopically. Also, the observed fraction as a function
of angular size declines sharply at small sizes. These features are artifacts of the
two stages in which the APM survey was performed. The first stage identified LSB
galaxies of large angular size, and all the followup spectroscopy was performed on
galaxies in this first list. The second stage identified small angular size galaxies, which
also tended to be predominantly in the surface brightness range 23:5 Ÿ ¯(0) Ÿ 24:5.
The interested reader is referred to Paper II for a more complete discussion of the
survey mechanics. Here we note only that the actual observed fraction in the range
23:5 Ÿ ¯(0) Ÿ 24:5 lies below the parametrization of Equation 2, which implies
that the parametrized correction is too small for those two surface brightness bins.
Any bias introduced by this effect is ``conservative'', in that it will result in an
underestimation of the total number of LSB galaxies.
We can estimate the completeness of our sample of galaxies using the hV=V max i
test of Schmidt (1968). For the complete set of 693 galaxies identified by the APM,
the test yields hV=Vmax i = 0:15 \Sigma 0:04 with no corrections for incompleteness,
and hV=V max i = 0:44 \Sigma 0:06 after correcting for incompleteness using the APM
selection function described in Paper II. For the subset of 332 galaxies with velocities,
the test gives hV=V max i = 0:04 \Sigma 0:05 with no corrections for incompleteness,
hV=V max i = 0:34 \Sigma 0:07 after applying just the APM selection function, and
hV=V max i = 0:50 \Sigma 0:07 after applying the APM selection function and the further
correction for incompleteness in the velocity observations from Equations 2, 3,

LSB Galaxies And The Field Luminosity Function 6
and 4 (as depicted in Figure 1). The corrections thus substantially remove the
incompleteness in both the complete set and in the subset chosen for spectroscopy.
There is yet another source of bias to be found in the magnitudes measured for
LSB galaxies. The magnitudes measured in our survey are isophotal magnitudes,
not extrapolated or asymptotic. The median limiting isophote is ¯ lim ú 27:4
mag arcsec \Gamma2 . As authors from Disney (1976) to McGaugh (1994) have pointed out,
use of isophotal magnitudes will cause galaxy luminosities to be underestimated, and
the underestimation becomes more severe with decreasing central surface brightness.
Most LSB galaxies are well­described by exponential surface brightness profiles
(Impey et al. 1988, Bothun et al. 1991, and McGaugh & Bothun 1994) of the form
¯(r) = ¯(0) + 1:086 r
l
(5)
where ¯(0) is the central surface brightness in mag arcsec \Gamma2 and l is the exponential
scale length in arcseconds. This simple analytical form allows a direct calculation of
the ratio of the total galaxy flux to that observed within the limiting isophote, as
F obs
F tot
= 1 \Gamma (1 + n l )e \Gamman l (6)
where n l is the number of scale lengths l observed within the limiting isophote.
This simple approximation will clearly understate the ratio for galaxies with central
condensations, such as spirals with bulges. The isophotal aperture in units of the
galaxy scale length is then given by
n l = ¯ lim \Gamma ¯(0) \Gamma 10 log(1 + z) \Gamma k(z)
1:086 (7)
where ¯ lim is the surface brightness of the limiting isophote. The first term involving
z accounts for the (1 + z) 4 cosmological dimming in surface brightness, and the
second corrects for the redshifting of the galaxy's spectral energy distribution (the
k correction). The k correction of course depends on galaxy type as well as
redshift. The magnitudes and surface brightness for the LSB galaxies with velocities
have been corrected as described in Paper II using the tabulated k corrections of
Coleman et al. (1980). The B \Gamma V and V \Gamma R colors for galaxy types Sbc, Scd,
and Irr closely match the range of colors observed among the galaxies for which we
obtained CCD photometry. The absolute magnitudes have been corrected according
to Equations 7 and 6, so as to avoid skewing the luminosity function by this tendency
to underestimate galaxy luminosities.
Of course, our set of LSB galaxies is not itself a fair sample of the local galaxy
population, precisely because it excludes most galaxies with ¯(0) ¸ ! 22 mag arcsec \Gamma2 .
However, it is still useful to derive a luminosity function for this set, so that this
LF can be compared to one derived from higher surface brightness galaxies. In this
way, it is possible to obtain some idea of how surface brightness selection effects have
influenced estimates of the density of local galaxies (see also McGaugh et al. 1995

LSB Galaxies And The Field Luminosity Function 7
and Paper II). To validate such a comparison, it is necessary first to compare the
range of surface brightnesses covered by the present set of LSB galaxies with the range
covered by other surveys. Unfortunately, no other recent galaxy redshift surveys have
published surface brightness data for their galaxies. Thanks to the recent release of
a digitized version of the original Palomar Observatory Sky Survey (the Digitized
Sky Survey 4 or DSS), it is now possible to make independent measurements of the
basic photometric parameters of any object visible on the original survey, when the
celestial coordinates of the surveyed galaxies are known. The CfA Redshift Survey
described by e.g., Marzke et al. (1994b) is based on Zwicky's Catalog of Galaxies and
Clusters of Galaxies, which was in turn created by visual examination of the Palomar
Observatory Sky Survey plates, so every object included in that survey should be
visible on the DSS. Most importantly, the coordinates of galaxies surveyed by the
CfA are publicly available, so that it is possible to retrieve images of the surveyed
galaxies from the DSS. Thus it should be possible to measure the surface brightness
range covered by the CfA Redshift survey. The lack of publicly available coordinates
prevents us from making a similar analysis of other recent redshift surveys.
We recovered from the Astrophysics Data System listing of the CfA Redshift
Survey the coordinates of every galaxy listed in the regions of sky used by Marzke
et al. (1994b). We subdivided that list according to the morphological categories used
by Marzke et al. (1994a), and we randomly selected 10% of the galaxies within each
morphological class to keep the number of galaxies manageable. This selection yielded
a list of 579 galaxies. We then retrieved images from the DSS of this randomly chosen
subset and analyzed the images using the same algorithms used in our APM LSB
galaxy survey. In this way, we obtained extrapolated central surface brightnesses for
the CfA galaxies that are directly comparable to those obtained in the course of the
APM survey. Paper II contains a complete description of the process of estimating
the extrapolated central surface brightness. As a check on the calibrations, we also
retrieved from the DSS images of a randomly chosen subset of the APM LSB galaxies
and analyzed them. After cross­calibration, the results for the APM LSB galaxies
were consistent with those obtained from the deeper UKST plate materials used in
the APM LSB survey, with the exception that the lowest surface brightness objects
were not visible on the DSS.
The surface brightness distribution for the CfA Redshift survey is shown in the
upper panel of Figure 2. The solidly drawn smooth curve represents the best Gaussian
fit to the CfA distribution. The lower panel shows the complete SB distribution
obtained by the APM for one UKST field. Also drawn for illustration in each panel
is a dashed curve representing the canonical ``Freeman Law'', a Gaussian centered at
4 Based on photographic data of the National Geographic Society -- Palomar Observatory Sky
Survey (NGS­POSS) obtained using the Oschin Telescope on Palomar Mountain. The NGS­
POSS was funded by a grant from the National Geographic Society to the California Institute
of Technology. The plates were processed into the present compressed digital form with their
permission. The Digitized Sky Survey was produced at the Space Telescope Science Institute under
US Government grant NAG W­2166.

LSB Galaxies And The Field Luminosity Function 8
Figure 2: Distributions of central surface brightness for (a) a randomly chosen sample
of galaxies from the CfA redshift survey, and (b) the complete list of galaxies identified
by machine scan of one UKST survey field. In (a) the solid curve represents the best
fit of a Gaussian to the CfA survey surface brightness distribution. In both panels
the dashed Gaussians illustrate the canonical ``Freeman Law'' of ¯(0) = 21:65 \Sigma 0:35.
The distribution in (b) is corrected for incompleteness of the detection algorithm for
¯B (0) ¸ ! 25 as described in Paper II.
¯(0) = 21:65 with oe = 0:35 (Freeman 1970). It is clear from Figure 2 that the range
of surface brightnesses covered by the CfA Redshift Survey is very narrow, narrower
even than the ``Freeman Law.'' The best­fit Gaussian to the CfA distribution has
a center at ¯(0) = 21:44 and oe = 0:19. This is completely consistent with the
investigation of the Zwicky magnitude scale by Bothun & Cornell (1990) who find
that this magnitude is not a sky­limited magnitude. In this case, one expects surface
brightness effects to completely dominate the magnitude estimates. In essence, the
Zwicky magnitude is very much a ``bulge'' or high surface brightness magnitude and
is insensitive to extended, low surface brightness light. In contrast, the APM LSB
survey has identified galaxies over a much broader range, as described in Paper II.
Clearly, the identification of galaxies for the CfA Redshift survey suffered from a

LSB Galaxies And The Field Luminosity Function 9
Figure 3: Distribution of absolute magnitudes for the LSB galaxies (¯(0) ? 22:0
mag arcsec \Gamma2 ) used to develop the LF. This distribution includes the effects of the
correction from isophotal to total magnitudes described in Equations 7 and 6.
substantial bias against LSB galaxies. In all the following analysis, we use only those
galaxies from the APM LSB survey with ¯(0) ? 22:0 mag arcsec \Gamma2 , or 3oe fainter than
the typical value found in the CfA Survey. This limitation assures that the resulting
LF covers a different regime of surface brightness parameter space from that covered
by the LFs of Marzke et al. (1994b) and Marzke et al. (1994a). We note that there
is a weak LSB tail in the CfA distribution: the overall ü 2
š of the Gaussian fit is 1.37,
virtually all of which is due to this tail. However, the very weakness of this tail, when
compared to the APM distribution in the lower panel, underscores the severity of the
SB selection bias inherent in the CfA survey. We note also that the CfA survey does
not identify nearly as many high surface brightness galaxies as does the APM. This
lack is most likely due to the general absence of galaxies smaller than 1 arcminute
from the Zwicky catalog; many of the high surface brightness galaxies identified by
the APM are smaller than 1 arcminute. Figure 3 shows the distribution of absolute
magnitudes for the LSB survey galaxies with ¯(0) ? 22:0 mag arcsec \Gamma2 .
3 Methods
The differential luminosity function of field galaxies OE(M)dM is defined as the
function giving, at each absolute magnitude M , the number of galaxies per Mpc \Gamma3
in the luminosity interval M +dM=2 Ÿ M Ÿ M \Gamma dM=2. Because the area surveyed
by the APM LSB survey covers a wide area of sky and cuts across several large scale
structures, we adopt two density­independent techniques for estimating the LF. The
first is the parametric maximum likelihood technique developed by Sandage et al.
(1979) (hereafter STY). The second is the stepwise maximum likelihood method

LSB Galaxies And The Field Luminosity Function 10
(hereafter SWML) developed by Efstathiou et al. (1988). Both methods assume that
the LF has a universal form, independent of position, allowing the probability of
a galaxy's inclusion in a complete catalog to be written simply in terms of the LF
itself. The STY method is continuous and uses all the galaxy data, but it requires
the assumption of a parametrized form for the LF. It therefore gives no information
as to the suitability of the parametrized form chosen to represent the LF. The SWML
method requires binning the data, but it requires no assumptions about the shape of
the LF. It can therefore be used in combination with the STY method to provide an
independent check on the goodness­of­fit of the chosen parametrization, as described
by Efstathiou et al. (1988). Like Marzke et al. (1994a) and virtually all others who
have used this combination of methods, we assume in the STY method a luminosity
function parameterization in the form first proposed by Schechter (1976), which is
written in absolute magnitudes as
OE(M)dM = 0:4 ln 10OE \Lambda
h
(10 0:4(M \Lambda \GammaM ) ) 1+ff e \Gamma(10 0:4(M \Lambda \GammaM ) )
i
dM (8)
Using the two methods together thus gives best­fit values for the Schechter function
parameters ff (the faint­end slope) and M \Lambda (the characteristic absolute magnitude
of the ``knee''), as well as a probability that the underlying galaxy population is
well­described by the best­fit Schechter function.
There is one major difficulty with applying these methods to the APM LSB
galaxy survey data. Both the STY method and the SWML method assume that
the galaxy catalog in use is magnitude limited, or that all galaxies with m ! m lim
have the same probability (p ! 1) of being included in the catalog, as in the case
of a redshift survey that uniformly samples a magnitude­limited catalog with 1=n
sampling. In our case, however, each galaxy has a unique probability of inclusion
that is determined from Equation 4, so the given forms of the STY and SWML
methods require modification. Zucca et al. (1994) recently addressed this problem.
They derived a simple modification to the STY estimator that accounts for the unique
observation probability assigned to each galaxy:
L =
N
Y
i=1
p w i
i (9)
where L is the likelihood to be maximized, the weight w i is defined as the inverse
of the probability that the ith galaxy will be included in the sample (i.e., for our
situation w i = 1=p tot;i , with p tot;i from Equation 4), and p i is as defined by STY:
p i = OE(M i )
, Z \Gamma1
M max(z i )
OE(M)dM (10)
The corresponding change to the SWML estimator of Efstathiou et al. (1988)
immediately yields:
ln L =
N
X
i=1
W (M i \Gamma M k )w i ln OE k \Gamma
N
X
i=1
w i ln
8 !
:
Np
X
j=1
OE j \DeltaM H
i
M max(z i ) \Gamma M j
j 9 =
; + const
(11)

LSB Galaxies And The Field Luminosity Function 11
where the OE k are the luminosity function values within each bin, N is the total number
of galaxies in the sample, N p is the number of steps, M max(z i ) is the maximum (i.e.,
the faintest) absolute magnitude visible at z i , \DeltaM is the bin width in magnitudes,
and the window functions are
W (x) =
(
1; jxj Ÿ \DeltaM=2
0; otherwise (12)
and
H(x) =
8 ? !
? :
0; x ! \Gamma\DeltaM=2
(x=\DeltaM + 1=2); jxj Ÿ \DeltaM=2
1; x ? \DeltaM=2
(13)
There is an implied sum over the doubled index k in the first term of Equation 11.
Finally, the survey biases must also be incorporated into the normalization. Both
the STY and SWML estimators are normalized in the manner described by Efstathiou
et al. (1988) using the unbiased minimum variance estimate of the mean density as
developed by Davis & Huchra (1982), but with a modification to the estimator to
incorporate the corrections for survey incompleteness. This normalization proceeds
in three steps. First, a selection function is defined as
S(x) =
Z M 1
max[M max(x) ;M 2]
OE(M)dM
, Z M 1
M 2
OE(M)dM (14)
for galaxies in the range M 1 ! M ! M 2 , where M max(x) is the maximum
(i.e., the faintest) absolute magnitude visible at distance x according to the
catalog limits. Second, this selection function is then corrected to incorporate the
incompleteness correction, so that it includes the combined probability of detecting
and spectroscopically observing an LSB galaxy in our survey:
S tot (x i ) = S(x i ) \Theta p tot;i (15)
where p tot;i is obtained from 4. Finally, the mean density of galaxies is obtained from
the corrected selection function as described by Efstathiou et al. (1988):
hni = 1
V
N
X
i=1
1
S tot (x i ) (16)
where the sum extends over all the galaxies in volume V . The mean density is
converted to a Schechter function normalization as:
OE \Lambda = hni
\Gamma (ff + 1; 10 0:4(M \Lambda \GammaM 2 ) ) \Gamma \Gamma (ff + 1; 10 0:4(M \Lambda \GammaM 1 ) ) (17)
where \Gamma is the Euler incomplete gamma function.
Zucca et al. (1994) also estimated the effects of failing to consider the individual
galaxy weights. Their simulations revealed that use of Equation 10 to determine the

LSB Galaxies And The Field Luminosity Function 12
Figure 4: Luminosity function for LSB galaxies from the APM survey. The solid line
represents the maximum likelihood Schechter function, and the points with error
bars represent the model­independent step­wise maximum likelihood function. Note
that for the LSB galaxies, the model­independent binned LF shows a significant
excess of low luminosity galaxies, beyond the level of the maximum likelihood
Schechter function. The dashed line shows the maximum likelihood Schechter
function estimated by Marzke et al. (1994a) for all morphological types in the CfA
redshift survey, and the dotted line shows the maximum likelihood Schechter function
estimated by Marzke et al. for irregulars in the CfA redshift survey
Schechter function parameters for a galaxy sample with significant incompleteness
(hV=V max i ¸ ! 0:3) would bias the results towards flatter faint­end slopes (i.e., lower
absolute values of ff) and brighter values of M \Lambda . We can objectively determine
individual galaxy weights from parameters of our survey technique (the APM
selection function) and from the internal statistics of our followup observations
(Figure 1 and Equations 2 and 3), so Equations 9 and 11 are the clear techniques of
choice for our data.
4 Results
Figure 4 shows the luminosity function for the LSB galaxies (¯(0) – 22:0) from
the APM survey. The solid line represents the maximum likelihood Schechter
function from the STY method, and the points with error bars represent the
model­independent SWML method. As is obvious from Figure 4, the maximum
likelihood Schechter function is a very poor representation of the ``true'' distribution

LSB Galaxies And The Field Luminosity Function 13
Figure 5: Histogram of T ­types for LSB galaxies from the APM survey. Galaxies
that appeared to be spirals but whose images on the APM scans were too small to
permit reliable further classification were assigned T = 5, so the number in that bin
is somewhat inflated. Interacting galaxies were assigned T = 11.
as determined by the SWML method: the reduced ü 2
š from the likelihood ratio test
of Efstathiou et al. (1988) is 14.04, which implies that the probability of exceeding
this ü 2
š by chance is ¸ 2:5 \Theta 10 \Gamma18 . The Schechter function is particularly poor at
the low­luminosity end. There, the model­independent SWML method finds two
to three times the galaxy density predicted by the maximum likelihood Schechter
function. The SWML bins at M = \Gamma16, \Gamma15, \Gamma14, and \Gamma13 contain 31, 9, 15, and
12 galaxies respectively. Across those four bins, the median correction due to the
APM selection function (Paper II) is 0.759, and the median correction due to the
incomplete spectroscopic observations (Equations 3 and 2) is 0.259; the median total
incompleteness correction (Equation 4) is therefore 0.197. Our sampling of galaxies
in these low­luminosity bins is quite sparse, and hence the uncertainties at this end
are large. The formal uncertainties shown in Figure 4 may well understate the true
range.
The dashed line in Figure 4 represents the Schechter function estimated by
Marzke et al. (1994a) for all galaxy morphologies in the CfA Redshift Survey.
At the faintest luminosities, LSB galaxies in the range 22:0 Ÿ ¯(0) Ÿ 25:0
are more numerous than the HSB galaxies sampled by the CfA Survey if the
comparison is based on the model­independent SWML points for the LSB galaxies,
or approximately as numerous if the comparison is based on the maximum likelihood
Schechter function. For all galaxies brighter than MB ! \Gamma15 the HSB galaxies are
significantly more numerous. Table 1 lists the maximum likelihood Schechter function
parameters for the LSB galaxies in the present study, along with similar parameters
for all morphological types and for irregular galaxies from the CfA Survey.
The LSB sample from the APM survey is not restricted as to morphological type.

LSB Galaxies And The Field Luminosity Function 14
It includes a few dwarf ellipticals and early spiral types. However, it is dominated by
very late­type spirals and irregulars. As Figure 5 shows, over half the LSB sample
have de Vaucouleurs T ­types of 9 or 10. For that reason, we have also shown the
Schechter function derived by Marzke et al. (1994a) for irregulars (which they define
as 8 Ÿ T Ÿ 10) in Figure 4 and Table 1. The Schechter function for the CfA
irregulars bears a striking resemblance to that derived here for the LSB galaxies.
The steep low luminosity tail of the function for CfA irregulars seems to match the
model independent SWML points for the LSB galaxies quite well. This similarity in
LF slopes could be used as an argument that the high space density of star forming
irregulars are the parent population of the fainter LSBs. Unfortunately, photometric
surveys of LSBs continue to find no relation between SB and color which is required
to support such a fading model.
Table 1: Comparison of Luminosity Function Model Parameters
Model/Survey ff M \Lambda OE \Lambda
(1) (2) (3) (4)
Maximum­Likelihood Schechter Functions:
LSB \Gamma1:46 \Gamma18:66 0:0036
CfA (all types) \Gamma1:02 \Gamma18:90 0:0201
CfA (Sm ­ Im) \Gamma1:87 \Gamma18:79 0:0006
Schechter Function + Power Law:
LSB (giants) \Gamma0:92 \Gamma18:19 0:0060
LSB (dwarfs) \Gamma2:20 \Gamma16:00 0:0041
Notes: M \Lambda in B mag, OE \Lambda in h 3
100 Mpc \Gamma3 mag \Gamma1 .
Several authors have suggested that the LF for faint galaxies may exhibit an
upturn from the pure Schechter form at faint luminosities. The LF of Impey et al.
(1988) for LSB dE's in the Virgo cluster turns up at an apparent magnitude mB = 17;
the increase is so steep that they were unable to rule out a divergent faint­end slope
(i.e., ff = \Gamma2:0). Upturns from the Schechter form have been observed in Coma by
Thompson & Gregory (1993); in nearby local groups by Ferguson & Sandage (1991);
in four local Abell clusters by De Propris et al. (1995); and in Coma, Abell 2554
and Abell 963 by Driver & Phillipps (1996). These deviations from the Schechter
form are generally seen to begin in the range \Gamma17 Ÿ MB Ÿ \Gamma15, after adjustment to
the distance scale used here (H 0 = 100 h 100 km s \Gamma1 Mpc \Gamma1 ). In contrast, Ferguson &
Sandage (1988) found an LF for the Fornax cluster that was consistent with a single
Schechter function having a faint­end slope of ff = \Gamma1:34. We note that the form
of the SWML data points in Figure 4 is consistent with earlier findings of a sharp

LSB Galaxies And The Field Luminosity Function 15
Figure 6: Luminosity function for LSB galaxies from the APM survey, determined
using the two­component model described in the text (dashed curve), and maximum
likelihood Schechter function (solid curve). The points with error bars represent
the model­independent step­wise maximum likelihood LF. For galaxies brighter than
M = \Gamma16 the two­component model is a standard Schechter function with all three
parameters allowed to vary. For galaxies fainter than M = \Gamma16, the model is a
power law with the normalization at M = \Gamma16 constrained to match the value of the
Schechter component there. Note that this model was fit to the SWML binned LF,
and not to the individual galaxy data points. The fiducial line labeled ``Divergent''
illustrates the faint end slope ff = \Gamma2 where the integral of the LF becomes divergent.
change in slope at faint luminosities: the binned model­independent data points
clearly break up from a smooth Schechter form at M = \Gamma16.
To investigate this break in more detail, we also fit a two­component model to
the SWML LF representation. For absolute magnitudes M Ÿ \Gamma16 (``giants''), the
model followed the usual Schechter function form, with all three parameters (ff, M \Lambda ,
and OE \Lambda ) allowed to vary. For absolute magnitudes M – \Gamma16 (``dwarfs''), the model
followed a simple power law, with the normalization at M = \Gamma16 constrained to
match the Schechter model value there (i.e., only the slope ff was allowed to vary in
the fit). Results of this fit are depicted in Figure 6, and the fitted model parameters
are listed in Table 1. Two aspects of this fit deserve special comment. First, it is not
possible to compare directly the goodness­of­fit of this two­component model to that
of the STY maximum­likelihood Schechter function. The ü 2
š quoted above for the
STY result is obtained by a likelihood­ratio test which, like the STY model itself,
is computed from the individual galaxy data points. The two­component model is

LSB Galaxies And The Field Luminosity Function 16
Figure 7: Cumulative luminosity functions for LSB galaxies and for galaxies from the
CfA redshift survey, plotted linearly as a function of Log L. The top axis represents
the corresponding B magnitude. The cumulation runs from high to low luminosities
(i.e., from right to left). The points represent the binned model independent LF for
the LSB galaxies, the solid line represents the maximum likelihood Schechter function
for the LSB galaxies, and the long­dashed line represents the two­component model
LF for the LSB galaxies. The short­dashed line is the Schechter function determined
by Marzke et al. (1994) for all morphological types in the CfA redshift survey, and
the dotted line is the Schechter function determined by Marzke et al. (1994) for
irregulars.
a fit to the binned SWML LF, not to the individual galaxy data points, and thus
its much larger ü 2
š (ü 2
š = 294:8, due to the much smaller number of degrees of
freedom) is computed in a very different manner. Second, the two­component model
yields results which are not physically plausible. The resulting slope of the dwarf
galaxy power law is ff = \Gamma2:20, which implies an infinite total luminosity if the
two­component LF is integrated from zero to infinity. This faint­end slope is best
interpreted as a finding that the model­independent SWML indicates a very steeply
increasing density of faint galaxies, so steep in fact that a divergent LF cannot be
ruled out (cf. Impey, Bothun & Malin 1988).
Because the low luminosity tail of the LSB luminosity function rises so steeply,
the contribution of LSB galaxies to the overall number density of field galaxies locally
is quite large. Figure 7 shows the same luminosity functions as Figure 4 along with the

LSB Galaxies And The Field Luminosity Function 17
Figure 8: Cumulative luminosity densities for LSB galaxies and for galaxies from the
CfA redshift survey, plotted linearly as a function of Log L. The top axis indicates the
corresponding B magnitude. The points represent the binned model independent LF
for the LSB galaxies, the solid represents the maximum likelihood Schechter function
for the LSB galaxies, and the long­dashed line represents the two­component model
LF for the LSB galaxies. The short­dashed line is the Schechter function for all
morphological types in the CfA redshift survey determined by Marzke et al. (1994)
and the dotted line is the Schechter function determined by Marzke et al. (1994) for
irregulars. The cumulation runs from high to low luminosities (i.e., from right to
left).
two­component LF described above, but cumulated to show total number densities.
The cumulation runs from high to low luminosities (i.e., from right to left), and
the vertical axis scaling is linear. Across the whole range of luminosities, the LSB
galaxies are almost twice as numerous as all the HSB galaxies in the CfA survey if the
comparison is based on the SWML points for the LSB galaxies, or half as numerous
if the comparison is based on the maximum likelihood Schechter function. Thus,
even under the most conservative estimate, surveys like the CfA redshift survey have
missed at least one­third of the local galaxy population due to surface brightness
selection biases. The true missed fraction is almost certainly higher, even by the
most conservative estimator. The LSB LF presented here covers only the range
22:0 ! ¯(0) ¸ ! 25:0, but as McGaugh et al. (1995a) and Paper II showed, the
distribution appears flat for SB levels ¯(0) – 25:0. Thus, surveys sensitive to fainter
SB levels should find even higher number densities of galaxies.

LSB Galaxies And The Field Luminosity Function 18
Despite the significance of their total numbers, LSB galaxies contribute little
to the total luminosity density of the local universe, because the highest number
densities of LSB galaxies occur at the lowest galaxy luminosities. Figure 8 shows
the cumulative luminosity densities for LSB galaxies from the APM survey and HSB
galaxies from the CfA redshift survey. Again, the cumulation runs from high to low
luminosities (right to left) and the vertical scaling is linear. By any estimator, LSB
galaxies represent a small fraction of the total luminosity emitted by HSB galaxies
from the CfA survey.
5 Implications
Much attention has been devoted over the past 15 years to the population of faint
blue galaxies revealed in surveys sensitive to extended objects as faint as mB J ¸ 27.
These galaxies become bluer at fainter apparent magnitudes (Lilly et al. 1995). They
generally are not at extreme redshifts: Lilly et al. (1995) found a median redshift
of z med ú 0:56 for a sample of galaxies in the magnitude range 17:5 ! I AB ! 22:5.
These galaxies are clustered more weakly than are most local bright galaxies, though
their clustering strength is roughly comparable to that of local galaxies undergoing
rapid star formation, per Bernstein et al. (1994). Their numbers are significantly in
excess of expectations based on local galaxy populations in the absence of evolution
(Tyson 1988; Lilly et al. 1991; McLeod & Rieke 1995). This excess has led some
authors to suggest non­standard cosmologies as a possible explanation (Yoshii 1993),
and others to propose strong evolution in galaxy luminosities, perhaps with the rate
of evolution itself a function of luminosity (Broadhurst et al. 1988; Babul & Rees
1992; Babul & Ferguson 1996). Still another approach, taken by Gronwall & Koo
(1995), is to derive local luminosity functions by finding functions that can explain
as well as possible the faint galaxy number counts without invoking strong evolution.
The luminosity functions they derive predict more local low­luminosity galaxies than
are observed in existing surveys. At the very least, recent surveys for LSB galaxies
indicate that the galaxy density at z = 0 is higher than previously assumed which
means, at some level, the apparent excess of faint galaxies at high redshifts is at least
in part an artifact of improper normalization at z = 0. The issue is how large this
effect really is.
McGaugh (1994) suggested that LSB galaxies such as those in the present sample
could help reconcile the differences between observed local populations and this
population of faint blue galaxies (hereafter, FBGs). He noted that, like the FBGs,
LSB galaxies are generally blue (McGaugh & Bothun 1994) and weakly clustered
(Mo et al. 1994). Furthermore, if current models of slow, continuous star formation
LSB galaxies are correct (McGaugh & Bothun 1994), McGaugh (1994) argued that
LSB galaxies should become only slightly redder over the timescales of interest,
0 ! z ¸ ! 0:5. He also demonstrated through a simple analytic calculation that

LSB Galaxies And The Field Luminosity Function 19
the deep CCD surveys would be more sensitive to LSB galaxies at z ¸ 0:4 than
wide­field photographic surveys are to local (z ¸ ! 0:1) LSB galaxies. He argued that
including nearby LSB galaxies in the local luminosity function could reconcile the
number of low­luminosity galaxies in the local population with the FBG population.
More recently still, Driver et al. (1995b) and Driver et al. (1995a) have examined
the morphological mix of the faint field galaxies using data from the Hubble Space
Telescope, down to a flux limit of m I = 24:25. They compared the observed
differential number counts (number per square degree as a function of apparent
magnitude) for three different morphological groupings to the predictions of various
models. They concluded that differential number counts of ellipticals and early­type
spirals are consistent with the predictions of a no­evolution model based on standard
local LFs (after renormalization) for these types taken from Marzke et al. (1994a)
and Loveday et al. (1992). They also found that the observed number counts of
late­type spirals and irregulars were substantially in excess of similar no­evolution
predictions for these classes. They could reconcile prediction with observation for this
morphological class only by including a substantial amount of luminosity evolution
(1.3 magnitudes of brightening by z ¸ 0:5 for an Irr LF from Marzke et al. (1994))
or by using a sharply increased normalization (OE \Lambda = 3:5 \Theta 10 \Gamma2 h 3
100 Mpc \Gamma3 ; compare
with Table 1) for the Irr LF.
Our results underscore the uncertainty in the faint end slope of the field galaxy
luminosity function. Driver & Phillipps (1996) have shown that existing wide field
redshift surveys place few constraints on the shape of the luminosity function below
MB = \Gamma16:5 assuming H 0 = 100 h 100 km s \Gamma1 Mpc \Gamma1 . By reaching lower in surface
brightness, we have isolated a population of blue, LSB dwarfs that is absent from
published luminosity functions, and which contributes strongly below MB = \Gamma16.
Marzke et al. (1994a) also saw evidence for a sharp upturn in the dwarf and irregular
population at about the same luminosity. More recently, Zucca & et al. (1996)
have shown that the luminosity function of Loveday et al. (1992) calculated from
the Stromlo­APM survey is significantly incomplete. The new ESO Slice luminosity
function shows an upturn below MB = \Gamma16, due to blue, star­forming galaxies, made
up of a mixture of compact dwarfs and LSB galaxies. In all these studies, as in the
deeper HST surveys of Driver et al. (1995a), the data are well described by a hybrid
luminosity function consisting of a bright end Schechter function with ff = \Gamma1, and a
faint end (MB ¸ ? \Gamma16 assuming H 0 = 100 h 100 km s \Gamma1 Mpc \Gamma1 ) power law with a slope
\Gamma1:4 ! ff ! \Gamma1:8. These faint galaxies are not major contributors to the luminosity
density of the universe, but because the trend of M=L with luminosity is not well
understood their contribution to the mass density is an open issue.
There is also now evidence that some evolution may have occurred in the late­
type galaxy population over the range 0 Ÿ z Ÿ 1. Lilly et al. (1995) have studied
the evolution of the LFs over this range using date from the recent Canada­France
Redshift Survey (CFRS). They found that the LF for red galaxies shows little change
in number density or luminosity over this range in z, but that the LF for blue galaxies

LSB Galaxies And The Field Luminosity Function 20
appears to have brightened uniformly by about 1 magnitude by z ¸ 0:75.
Thus, the ``excess'' of FBGs consists of late­type spirals and irregulars, the same
types which dominate the population of LSB galaxies found by the APM survey
(Fig 5). These LSB galaxies have a Schechter function normalization approximately
6 times as large as that found by Marzke et al. (1994a) for HSB irregulars, so they
expand the known local population well beyond that used by Driver et al. (1995b) in
their modeling. The LSB normalization still is not as large as that found necessary
by Driver et al. (1995b) to account for the FBG counts with no evolution, but the
heretofore­uncounted LSB galaxies do help considerably to close the gap between
local population estimates and the FBG counts. Taking this increase together with
the modest evolution observed in blue galaxy LFs by Lilly et al. (1995), it may now
be possible to make an essentially complete reconciliation between local populations
and the FBGs. However, any such reconciliation will also require a model for the
evolution of the FBGs which accounts for the very blue colors of both the FBGs and
the local LSBs (see McGaugh & Bothun 1994, McGaugh 1994, and McGaugh et al.
1995a).
We can demonstrate the rough equivalence between the local LSB dwarf
population and the MDS population of Driver et al. (1995a), which we presume
to be at typical redshifts 0:3 ¸ ! z ¸ ! 0:6 based on Lilly et al. (1995). The LSB
dwarfs with MB ? \Gamma16 are at typical distances of 10 ¸ ! d ¸ ! 40 h \Gamma1
100 Mpc. They
have central surface brightness in the range 22:0 ! ¯B (0) ¸ ! 25 mag arcsec \Gamma2 , and
effective angular radii of 6 ¸ ! r eff ¸ ! 20 arcseconds. If they are related to the LSB
dwarfs in clusters, we expect them to have B \Gamma V ¸ 0:5 (Impey, Bothun, & Malin
1988). The late­type and irregular (Sdm/Irr) MDS galaxies have effective radii with
a median value of 0.4 arcseconds (Im et al. 1995), which would scale to 20 ¸ ! r eff ¸ ! 40
arcseconds for a local population. The central surface brightnesses of disk­dominated
categories in the MDS sample are 20 ¸ ! ¯ I (0) ¸ ! 22 mag arcsec \Gamma2 per Mutz et al.
(1994), which is equivalent to 22 ¸ ! ¯B (0) ¸ ! 24 mag arcsec \Gamma2 locally, assuming no
evolution. The Sdm/Irr galaxies have observed colors of V \Gamma I ¸ 1 per Casertano
et al. (1995), consistent with a local star­forming dwarf color of B \Gamma V ¸ 0:5, once
again assuming no evolution. There could well be a subset of the MDS population
which fade in surface brightness and redden to below the detection threshold for our
blue photographic survey. Such galaxies would be absent from all local catalogs.
6 Conclusions
We have estimated a luminosity function for galaxies with surface brightnesses fainter
than ¯(0) = 22:0 mag arcsec \Gamma2 , which is the approximate faint limit of ¯(0) for
galaxies covered by the CfA Redshift Survey. We find that this LSB LF has a steeply
rising tail at low luminosities(ff = 1:42), comparable to that found by Marzke et al.
(1994a) for galaxy types 8 Ÿ T Ÿ 10. The LSB LF has a normalization lower than

LSB Galaxies And The Field Luminosity Function 21
that found for the overall CfA survey, but much higher than that found for types
8 Ÿ T Ÿ 10. Thus estimates of the total population of local galaxies based on
the CfA survey are missing at least one­third of the total number of galaxies due
to surface brightness selection bias. These previously unaccounted­for LSB galaxies
can help considerably to resolve the apparent difference between estimates of the
local population and the large numbers of faint blue galaxies observed at moderate
redshift.
We are grateful to a number of our colleagues for many stimulating and
helpful discussions. We thank in particular: Frank Briggs, Erwin de Blok, Simon
Driver, Marijn Franx, Ron Marzke, Stacy McGaugh, Renzo Sancisi and Martin
Zwaan. We also thank the staffs of the Multiple Mirror Telescope Observatory,
the Steward Observatory Kitt Peak Station, and the Arecibo Observatory for their
expert assistance during the many observing runs carried out in connection with our
survey. This project made extensive use of the NASA Astrophysics Data System.
This work was supported in part by the National Science Foundation under Grant
AST­9003158.

LSB Galaxies And The Field Luminosity Function 22
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