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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