The numb er luminosity and mass density of spiral

galaxies as a function of surface brightness

Stacy S McGaugh

Institute of Astronomy University of Cambridge Madingley Road Cambridge CB HA

ABSTRACT

I give analytic expressions for the relative numb er luminosity and mass density of

disc galaxies as a function of surface brightness These surface brightness distributions

are asymmetric with long tails to lower surface brightnesses This asymmetry induces

systematic errors in most determinations of the galaxy luminosity function Galaxies

of low surface brightness exist in large numb ers but the additional contribution to the

integrated luminosity density is mo dest probably

Key words galaxies formation galaxies fundamental parameters galaxies general

galaxies luminosity function mass function galaxies spiral galaxies structure

Accepted for publication in the Monthly Notices of the Royal Astronomical Society

Present address Department of Terrestrial Magnetism Carnegie Institution of Wash

ington Broad Branch Road NW Washington DC USA

INTRODUCTION

The space density of Low Surface Brightness LSB galaxies has long b een a contro

versial and confusing sub ject It is basic to our inventory of the contents of the universe

and is crucial to many asp ects of extragalactic astronomy For example the density of

LSB galaxies has an impact on the luminosity function faint galaxy numb er counts Ly

absorption systems and theories of galaxy formation In this pap er I derive analytic

expressions which quantify the density of discs of all surface brightnesses

In order to clarify some of the p oints of confusion it is imp ortant to quantify what

is meant by the term LSB Dierent p eople use it to mean dierent things leading to

contradictory interpretations of otherwise consistent data One frequent but misleading

use is treating LSB and dwarf as synonymous Dwarf galaxies may b e a subset of

galaxies which are low in surface brightness but there exist many LSB galaxies which can

not b e considered dwarfs either in terms of size or luminosity Romanishin Strom Strom

Phillipps Disney Bothun et al Davies et al Imp ey Bothun

Bothun et al Irwin et al Knezek Sprayb erry et al McGaugh

Bothun de Blok van der Hulst Bothun a Some seem to refer to LSB

galaxies as things which are totally invisible optically Others use the term to refer to any

galaxy which is fainter than some canonical value like that of Freeman or the night

sky

In order to provide a consistent denition it is necessary to quantify the luminosity

proles of galaxies Sersic found that the radial luminosity distribution could b e t

by

r h

r e

where is the central surface luminosity density in L p c h is a physical scale length

in kp c and is an index which varies the shap e of the prole This is a generalization

of the exp onential disc de Vaucoulers and r de Vaucoulers

proles

Since disc and bulge are b elieved to b e distinct physical comp onents and most debate

concerns the numb ers of LSB disc galaxies I will limit the discussion to proles with

ie spiral irregular and dE galaxies It is straightforward to generalise the results

presented b elow for dierent prole shap es but in principle a multiparamter approach

which adequately represents the full app earance of galaxies is required see McGaugh

Bothun Schomb ert a

In magnitude units the exp onential prole is

r

r

where is the central surface brightness and is the angular scale length pro jected on

the sky by a galaxy of physical scale length h at distance d The surface brightness and

size of a disc galaxy are characterised by and h For pure disc systems is simply

related to the eective surface brightness I will therefore use to characterise the global

surface brightness of disc galaxies

Using it is now p ossible to dene what is meant by low and high surface bright

ness HSB Freeman asserted that all spiral discs had the same central surface

brightness

B mag arcsec

This has b ecome widely known as Freemans law and is often used as a working denition

of HSB To b e slightly more sp ecic I will dene as high surface brightness any disc with

B mag arcsec which corresp onds to L p c Any disc which satises

this criterion is likely also to fall within the narrow range sp ecied by Freemans law and

the particular value is chosen to bracket it on the faint side On the bright side I will

describ e as very high surface brightness VHSB anything which fails to ob ey Freemans

law by b eing to o bright with mag arcsec or L p c

The most obvious dividing line for dening LSB is the brightness of the night sky one

magnitude fainter than the Freeman value in the b est of conditions To give a linear factor

of two b etween this and HSB and to b e clearly fainter than the night sky let us describ e

as LSB any disc galaxy with mag arcsec Fig shows examples of idealised

high and low surface brightness galaxy proles

Though I would prefer to avoid the p ollution of excessive nomenclature further quan

titative distinctions are useful The transition region b etween HSB and

LSB I will call intermediate surface brightness ISB since ob jects in this range do not

ob ey Freemans law but are not particularly dim either It is also useful to denote the

extremes of the LSB realm by very low surface brightness VLSB and

extremely low surface brightness ELSB mag arcsec Most LSB galaxies which

are known in the eld are to o bright for either of these categories though some examples

do exist Nonetheless it is p ossible to see VLSB galaxies on survey plates if one lo oks hard

enough It is not until the ELSB regime that galaxies b ecome eectively invisible to the

deep est existing photographic surveys only one galaxy this faint has so far b een rep orted

GP Davies Phillipps Disney

The nomenclature utilised here is summarized in Table While such quantitative

denitions are necessary to make sensible use of terms like LSB it should b e realised that

there is some distribution of central surface brightnesses Obtaining this should b e our

goal

THE RELATIVE NUMBERS OF LSB GALAXIES

To b egin in this direction it is p ossible to make a crude estimate of the relative numb er

of galaxies in two widely separated surface brightness bins Recent surveys Schomb ert

Bothun Schomb ert et al have identied large numb ers of LSB galaxies by

visual examination of the plates of the new Palomar Sky Survey The selection criterion of

the Schomb ert et al sample henceforth referred to as the LSB sample mimic that

0

of the diameter limited UGC Nilson The only dierence is the greater depth

and lower grain noise of the new plates Ob jects are not included if already catalogued in

the UGC

Extending the UGC in this way will incorp orate galaxies which just missed the original

diameter criterion This will include HSB galaxies which are on average slightly more

distant as well as lower surface brightness galaxies In order to reject the former and

isolate from the surface brightness distribution some p ortion which is truly LSB only

galaxies were included which were essentially undetected on the POSSI This guarantees

that only shallow prole LSB galaxies are selected Surface photometry of a subset of

galaxies from the LSB sample shows that this pro cedure selects a p opulation which has a

typical central surface brightness of B mag arcsec McGaugh Bothun

de Blok et al a The distribution is very sharply p eaked around this value with a

standard deviation of and a high kurtosis

Note that is fainter than the Freeman value If Freemans law were

correct and the surface brightness distribution really was a sharply p eaked function then

these LSB galaxies should not exist Yet they represent an enhancement over

the numb er of galaxies in the UGC Schomb ert Bothun and the UGC already

contains a fair numb er of LSB galaxies Romanishin et al McGaugh Bothun

de Blok et al a

For a direct comparison to b onade HSB galaxies a sample is required which has

surface photometry so that the adopted denition of HSB can b e quantitatively satised

Such a sample exists in the data of van der Kruit who claimed to conrm Freemans

law for large early typ e discs This HSB sample contains galaxies which have

B mag arcsec the dierence b etween this and the Freeman value

J

b eing exactly comp ensated by the dierence b etween the B and B bands There are

J

0

disc galaxies larger than over square degrees of sky of which were suciently face

on to p erform surface photometry Of these qualify as Freeman discs the remainder

b eing classied as LSB dwarfs Assuming that an equivalent fraction of the discs without

surface photometry are HSB gives a surface density of Freeman discs of N p er

HSB

square degree

0

In comparison Schomb ert et al found LSB galaxies larger than over

square degrees of sky Thirteen of these are classied as p eculiar ellipticals with

shells or some other LSB extension leaving b ona de LSB galaxies Only seven of

these are dEs the rest are spirals and irregulars Since dEs are low surface brightness and

generally have exp onential proles I retain them but obviously this makes no dierence

It is imp ortant that the sample consists mostly of spiral galaxies as it is often stated that

Freemans law do es not apply to dwarfs This term is rarely quantied in this context

and can follow from any of morphology size or luminosity The LSB sample do es not

consist of dwarfs by any of these denitions The surface density of LSB spiral galaxies is

N p er square degree

LSB

An estimate of the relative space density can b e made by noting that the actual

numb er density n is related to the surface density by n N V where is the solid

angle and V is the volume sampled by a survey The ratio V can b e estimated for

these diameter limited surveys from equation given values of the characteristic of

each sample and the isophotal level at which the diameter r is

measured Since hd the maximum distance at which a galaxy can lie and meet the

survey requirement is

h

d

so the volume prob ed by the survey is

d V

This suces for the HSB survey but not for the LSB survey b ecause galaxies already

catalogued by the UGC are sp ecically excluded This places a minimum as well as max

imum limit on the volume prob ed to calculate this prop erly the volume already surveyed

by the UGC must b e subtracted o In eect the LSB survey only prob es a shell of width

h

UGC LSB

d d

LSB UGC

stemming from the dierence in the isophotal levels at which the diameters are measured

From CCD data the diameters of the LSB survey are determined to b e measured at

LSB

mag arcsec Schomb ert Bothun McGaugh and those

UGC

Cornell et al see also Paturel Fouque in the UGC at

Paturel The volume sampled by the LSB survey is thus

Z

d

R dR V d d

d

which reduces to the familiar approximation V d for d

The relative numb er density of LSB to HSB galaxies now follows from the observed

surface densities and the volumes sampled by each survey n n N N

LSB HSB LSB HSB

V V Substituting the ab ove expressions for V gives

HSB LSB

L

N n h

L LSB H

H

n N h

HSB H L

H H

h i

L L L U L L L U L U

where for brevity of sup erscripts the value sp ecic to each catalog is denoted by L for

LSB H for HSB and U for UGC All of the relevant values have b een discussed except

for the ratio of scale lengths h h The LSB catalog is dominated by ob jects with

HSB LSB

scale lengths similar to the HSB galaxies studied by van der Kruit so I assume

h h Schomb ert et al McGaugh Bothun de Blok et al a

HSB LSB

If instead HSB galaxies are on average larger than LSB galaxies h h this

HSB LSB

will increase the relative numb er of LSB galaxies since n n h h

LSB HSB HSB LSB

LSB HSB

Inserting the relevant values N N

LSB HSB

HSB LSB HSB LSB UGC

h h into

HSB LSB

equation yields

n

LSB

log

n

HSB

That is there are approximately the same numb er of galaxies with as with

This result is obviously very dierent from the notion that all spiral discs have the

same central surface brightness Integrating the sharply p eaked Gaussian distributions of

Freeman and van der Kruit to obtain the density exp ected for galaxies with

mag arcsec ie those deviant Freemans law predicts

n

LSB

log

n

HSB

There are several uncertainties in this calculation but none which come close to six and a

half orders of magnitude Aside from the ratio of scale lengths variation of the parameters

in equation over reasonable ranges do es not cause the answer to change by more than

a factor of a few see Table Even if several factors conspire together the result is still

not much aected

The variations considered in Table are rather extreme It is very unlikely that

U GC

has b een misestimated by a magnitude and it do es not really matter if it has

since the volume already surveyed for LSB galaxies by the UGC is essentially zero for

U GC

is even a little fainter than the numb er of If on the other hand

LS B U GC

It is not very sensible to consider LSB galaxies rises sharply since as

HSB

since all that matters to the calculation is the dierence b etween this variations in

HSB

This dierence is not sensitive to zero p oint errors only to improp erly corrected and

nonlinearities The selection isophote of the LSB sample is well determined and uniform

but it is p ossible that the subset of galaxies for which CCD photometry is available is not

LS B

statistically representative Even so varying do es not alter the answer much and if

anything it is more likely that the sample as a whole is fainter than those ob jects which

have b een studied in detail If so this again raises rather than lowers the LSB numb er

density

The ratio of typical scale lengths can make a large dierence to this calculation As

already noted it go es in the wrong sense required to save Freemans law There is no

evidence that the distribution of scale lengths in the LSB sample of Schomb ert et al

is distinguishable from that in the HSB sample of van der Kruit It is thus

imp ossible to escap e the conclusion that Freemans law is erroneous by arguing that it

do es not apply to dwarfs

An estimate of the error in the relative density of LSB galaxies can b e made from

counting statistics and Table The galaxies in the LSB sample contribute dex

to the error in the relative density while the b ona de Freeman discs in the HSB sample

contribute dex This results in a combined statistical error of From Table a

reasonable estimate of the errors due to uctuations in the various is dex and

an equal amount comes from p ossible small deviations of the scale length ratio from

unity Combining all these sources of error in quadrature gives a total error of dex

Using this numb er and equations and Freemans law is formally rejected with

a condence of Though this error includes some estimate of the systematic as well

as random uncertainties the systematic problems could of course b e larger However it

is imp ossible to consider a space density of LSB galaxies less than indicated by the raw

surface densities The volume correction is fairly mo dest a factor of and the LSB

catalog is if anything less complete than the HSB one Hence a very hard lower limit is

logn n This is still over ve orders of magnitude larger than predicted

LS B HSB

by Freemans law

AN ILLUSTRATIVE SURFACE BRIGHTNESS DISTRIBUTION

In order to understand why studies which app ear to conrm Freemans law are so

misleading let us imagine a particular form for the surface brightness distribution and ask

what we would exp ect to observe As we shall see Freemans law arises from the failure

to apply volume corrections to the observed apparent distribution in order to recover the

intrinsic distribution see Allen Shu as well as Disney Disney Phillipps

McGaugh et al a

In order to illustrate the eect surface brightness has on the volume sampled by

a complete survey let us hyp othesise a simple distribution of disc galaxy central surface

brightnesses I dene to b e the space density of galaxies at each central surface

 

brightness relative to that at some ducial value n n Motivated by

the results of x let us assume that galaxies exist in equal numb ers faintwards of the

ducial value Brightwards of this let us assume a sharp exp onential cut o Hence



for





0

0

e for

The exp onential cut o at the bright end is made to b e consistent with the lack of

galaxies observed with central surface brightness brighter than the Freeman value Allen



Shu The ducial value corresp onds to the Freeman value To b e consistent

with a narrow apparent distribution the exp onential cut o must b e sharp

In order to illustrate the eects of on diameter and magnitude selected samples

b elow I make two further simplifying assumptions The rst is that is not correlated

with h so that on average only the surface brightness matters to the selection of galaxies at

each The second is that complete catalogs can indeed b e constructed over suciently

large volumes of space that they provide a fair representation of the universe containing

the hyp othesised

Diameter Selection

For a complete diameter limited survey the volume correction asso ciated with any

given galaxy is obtained from V V Using equation can b e decomp osed

max

into h and giving the volume sampled as a function of the galaxy parameters h

and the survey parameters The volume over which a particular galaxy can exist

and satisfy the selection criteria follows from equation V h Larger

and higher surface brightness galaxies can b e detected over larger volumes and will b e

numerically overrepresented in catalogs relative to their space density The eect is strong

in b oth and h though it is stronger in in the sense that no galaxy with will b e

included in the catalog while it is in principle p ossible to detect galaxies with arbitrarily

small h Note also that V only increases as b ecomes brighter It never reaches a

maximum at a preferred value of as suggested by Disney see McGaugh et al

a

Sp ecic cases are illustrated by Fig At a particular surface brightness changing

the size h by a factor of two also changes the isophotal diameter by a factor of two So

the relative volume over which these galaxies could b e found diers by a factor of eight

regardless of the isophotal level at which the diameters are measured providing that this

is not so faint that it is b eyond the edge of the discs At a xed size changing the

surface brightness alters the isophotal diameter in a way that dep ends on the dierence

b etween and The brighter the more serious the eect For the illustrated case of

LSB HSB

and shifting by mag arcsec the ratio of isophotal diameters are

LSB HSB

Hence the relative volumes over which galaxies of the same size can

b e detected vary by a factor of and for and resp ectively This

illustrates the enormous sensitivity to and the imp ortance of measuring it consistently

and accurately when attempting to construct complete catalogs Note also that it is

essentially imp ossible to construct a truly complete catalog when the level is chosen as

is usually done to b e at the ultimate limit of the survey material since the signal to noise

for low surface brightness images will go to zero as

Rather than sp ecify a particular set of survey parameters let us fo cus on the relative

 

h functional eects of the galaxy prole by dening a ducial galaxy of parameters

 

The obvious choice for is the maximum in the distribution x and h can b e chosen



to make this an L galaxy as in Fig Normalised to this ducial galaxy the volume

sampled by a diameter limited survey is

V h h

 

 

h V h

Prop erly this should b e displayed in a three dimensional plot as a surface of V which

 

declines rapidly for and h h see McGaugh et al a Since the fo cus

here is on surface brightness I shall restrict the discussion to simple two dimensional plots

which are pro jections along the surface brightness axis

The apparent numb er of galaxies observed at each surface brightness N will

obs

just b e the true distribution convolved with the sampling function V h For the

intrinsic distribution describ ed in x



for N

obs



and





0

0

e for N

obs



The result is shown in Fig for several values of



Even though the intrinsic distribution is at faintwards of the apparent distribution



is very strongly p eaked at regardless of the value of All that is gained by pushing

fainter is to extend the tail of the apparent distribution cf Allen Shu The

relative volume sampled b ecomes very small for two magnitudes brighter than so

nding any galaxies at all with and such ob jects do exist in b oth the UGC

and the list of Schomb ert et al immediately indicates a large space density of such

LSB galaxies

Flux Selection

For the case of galaxy selection by apparent magnitude the survey parameters which

need to b e sp ecied are the ux limit and the way in which the ux is measured Usually

isophotal uxes are used By denition these are not total uxes nor are they simply

related to total uxes except in the sp ecial case that the distribution of is a function

This has an imp ortant consequence the ux by which a galaxy is selected is not directly

related to its luminosity as with p oint sources so inverting the selection function do es not

directly yield the luminosity function

Nevertheless isophotal magnitudes do have the advantage that the p ortion of the

total ux which they represent is rigorously dened ie that within the isophote It

is harder to determine what is actually b eing measured by other ux measurement schemes

In principle one needs prole information to derive the total luminosity of a galaxy and

isophotal magnitudes have the virtue of b eing formally related to this

The total luminosity of a galaxy can b e obtained from the prole by integrating

equation For disc galaxies with

Z Z

r

max

r h

L e r dr d

For r this is simply

max

L h

1

The luminosity contained within some nite numb er of scale lengths x

x r h

max

is

x

L L xe

1

This is the true total luminosity for discs which truncate after x scale lengths Discs are

observed to remain exp onential for at least three scale lengths and usually more This

encompasses most of the light L L for x so the integration to innity is a

1

reasonable approximation of the true total luminosity Equation can also b e used to

relate the ux measured within any given isophote to the total ux

Galaxies with brighter will emit a larger fraction of their total ux ab ove any give

isophote than LSB galaxies even of the same total luminosity For the cases illustrated

in Fig the fraction of the total ux f F F measured by isophotal magnitudes are

1

F F and F F for HSB galaxies and F F and F F

1 1 1 1

for LSB galaxies Hence LSB galaxies will have their total uxes underestimated

and appear to b e much fainter than HSB galaxies even if they have the same luminosity

For the present case f f at and f f at

LS B HSB LS B HSB

This has a serious eect on the volume prob ed by magnitude limited surveys Since

V L at V V and at V V Hence

LS B HSB LS B HSB

b oth the numb er density and the luminosity of LSB galaxies will b e underestimated by

magnitude limited surveys in a way which dep ends sensitively on the eective isophotal

level at which the magnitudes are measured

Though typical of large photographic galaxy surveys these are arbitrary So to o is

the choice of for LSB galaxies Galaxies with fainter are known to exist including



the extreme case of Malin with and L L The imp ortant p oint is that

the identication of any galaxy of faint in a complete magnitude limited sample im

mediately demands a large numb er density of such ob jects There is nothing mysterious

ab out this selection eect it stems simply from the fact that surveys sample a very much

smaller volume of space for galaxies of low surface brightness These galaxies will thus

app ear rare even if common just as intrinsically faint stars are common even though the

naked eye p erceives mostly stars which are intrinsically bright The dierence is that LSB

galaxies will app ear rare even if intrinsical ly luminous

To quantify these eects for arbitrary and let us assume the same intrinsic

density distribution as b efore In general the case of ux selection is not as clean as

diameter selection for while the diameter is dominated by the disc the ux will also

contain light from any bulge comp onent which is not included in a simple exp onential

prole However the bulge is a small fraction of the total light for most disc systems so

the exp onential prole remains a reasonable if imp erfect approximation

x

The volume sampled go es as V L f h xe g Normalising as



  

0

0

b efore to a ducial galaxy of parameters h and noting that

the relative volume sampled by a ux limited survey as a function of surface brightness

and size is

x



V h xe h

0

0





   x

V h h x e

For the assumed intrinsic distribution the observed distribution N V

obs

is shown in Fig

The apparent distribution for ux selected catalogs is even sharp er than for the diam

eter limited case This is b ecause of the exp onential rather than p ower law dep endence on

Unlike diameter limited catalogs where the dierence b etween and is what mat



ters it is the dierence b etween and that matters most to ux selection Galaxies

with low surface brightnesses will b e very strongly selected against b ecause of the faintness

of regard less of This only enters through x and matters little Thus diameter

limited surveys in principle provide a much b etter picture of the true galaxy p opulation

PREVIOUS RESULTS

With the results of x it is p ossible to understand why the issue of the numb er density

of LSB galaxies and the surface brightness distribution in general has remained confused

for so long This results largely from unquantied use of the term LSB and from the

misuse or total lack of volume corrections In this section I review earlier work and show

that essentially all data sets provide consistent results when prop erly analysed

Optical Studies

Discussion of the role of surface brightness in galaxy selection go es back at least to

Zwicky Arp and de Vaucoulers The rst quantitative statement

ab out the distribution of galaxy surface brightnesses was in eect made by Freeman

Complete samples did not exist at the time so no correction for volume sampling was

considered

Many p eople esp ecially Disney Disney Phillipps Phillipps et al

and Davies p ointed out that selection eects could cause Freemans law

However these arguments were based on the assumption of no correlation b etween lu

minosity and surface brightness While galaxies cover a wide range in the L plane

current data show that lower surface brightness galaxies do tend to b e less luminous just

not smaller In this case the qualitative eects are very dierent see McGaugh et al

a for a detailed comparison The same eect had already b een noted by Allen Shu

who p ointed out that Freemans law is an overinterpretation of the data all that

can really b e said is that galaxies brighter than the Freeman value are rare the numb er

fainter remained unconstrained

Freemans law was apparently conrmed by many workers Freeman Schweizer

Thuan Seitzer Boroson van der Kruit but only van der Kruit

claimed to have a complete sample with surface photometry and further claimed

that Freemans law only applied to large early typ e galaxies He found a lack of galaxies

with B mag arcsec and h kp c for H km s Mp c The galaxies

J

which were LSB were also small and late Sd or later typ e opp osite the results of Kent

Nonetheless van der Kruit concluded that selection eects as formulated

by Disney Phillipps were not o ccurring Phillipps et al state that van der

Kruit misapplies their formalism a p oint pursued by Davies et al The latter

claim to b e able to repro duce the apparent distribution of the data of van der Kruit

with the Disney Phillipps visibility formalism if there is an implicit magnitude

limit in his diameter limited catalog b ecause of signal to noise constraints However if this

reasoning were correct van der Kruit would not have found the LSB dwarfs which

he did

There is no obvious reason why van der Kruit missed large LSB galaxies His

study is unique in this p oint Romanishin et al Davies et al Irwin et al

McGaugh Bothun de Blok et al a de Jong van der Kruit If anything

the largest scale length galaxies tend to b e low rather than high surface brightness Bothun

et al Knezek Sprayb erry et al a the only consensus p oint b eing that

the lack of large HSB disks is not a selection eect de Jong van der Kruit The

large diameter limit employed by van der Kruit can only survey a very limited

volume of space for galaxies dimmer than the Freeman value so p erhaps this region do es

not constitute an adequate sample of the universe

Though the statistics are p o or the apparent distribution of central surface bright

nesses in the data of van der Kruit is roughly what is exp ected from Fig There

is one large LSB galaxy in his sample NGC which when corrected for volume sam

pling implies a greater density than all the large Freeman discs combined see his Fig

Since there was only one such galaxy in his sample van der Kruit chose to ignore

it Obviously this invalidates his primary conclusion conrming Freemans law even when

restricted to large discs

Aside from the p ossibility of Freemans law b eing a real eect or due to selection

eects it has also b een suggested that it could b e caused by improp er discbulge deconvo

lution Kormendy Phillipps Disney Schomb ert Bothun Davies

Ronnback or inclination and internal extinction eg Peletier Willner see

also Giovanelli et al While imp ortant neither of these eects are really p ertinent

here They are usually small and almost never as large as the mag dierence b etween

Freemans law and the LSB galaxies of Schomb ert et al Moreover they do not

matter much to selection which is by far the dominant eect

More recently Bosma Freeman and Roukema Peterson have ad

dressed this problem directly with further surveys The results of Bosma Freeman

are examined in detail b elow Roukema Peterson claim to nd a small numb er

of LSB galaxies However they fail to quantitatively dene LSB or to isolate

a p opulation which could b e describ ed as such though their Fig do es illustrate the

absurdity of Freemans law Their selection criteria are sub jective as is their estimate

of the redshift of completeness This has a strong inuence on the results since V z

c

a no less arbitrary choice of z would give a numb er of In essence they make

c

b oth ma jor mistakes which have b efuddled this eld no quantitative denition of terms

and improp er correction for volume sampling When these problems are considered es

sentially al l published data are consistent despite the many contradictory and misleading

interpretations given them

The Diameter Distribution

Bosma Freeman approach the problem of the central surface brightness

distribution by examining the distribution of diameter ratios on plates of dierent depth

They measure only the apparent distribution without correction for the volume sampled

In this section I derive this correction and apply it to their results

The ratio of the diameter of the same galaxy measured on two plates of dierent

limiting isophotes and is simply related to the central surface brightness In

1 2

eect this is a very crude form a surface photometry which ts a straight line to two

p oints to estimate Bosma Freeman dene the parameter to b e the observed

diameter ratio which is related to by

1 1

2 2

The samples are selected by diameter on the deep er plates by choice

1 2

Bosma Freeman employ plates from the rst Palomar sky survey Pal the

ESOB survey ESO and SRCJ survey SRC They derive values for the limiting

S RC ESO P al

essentially and isophotes of these materials of

by demanding that Freemans law b e true This causes their derived values to dier

ESO P al

Cornell et al signicantly from the actual measured values

S RC

Corwin de Vaucoulers de Vaucoulers Laub erts Valentijn and

However the numerical value of is not as imp ortant as their failure to consider

what one should exp ect to see

The exp ected apparent distribution for the intrinsic mo del of x can b e derived by

substituting for in equations and From the denition of

1 2

For a at intrinsic surface brightness distribution the apparent distribution will b e

2 1

1

N

obs



2 1



1

 

where corresp onds to This is plotted in Fig together with the data of Bosma

Freeman

Even for a at intrinsic distribution the exp ected apparent distribution N is very

obs



strongly p eaked at Indeed the exp ected distribution is so sharp that it is necessary

to consider the spread intro duced by random errors in without which values of

are imp ossible This is complicated b ecause is not directly measured but rather is

the ratio of two diameter measurements This results in a dep endence of on and

a nonGaussian error distribution This combined with the small exp ected tail towards

large causes a fair numb er of p oints to b e scattered to high A reasonable t

to the data is obtained for which is slightly smaller than the random error

estimate of Bosma Freeman If the errors were this large a greater

numb er of galaxies would b e scattered to than actually are

From the apparent distribution Bosma Freeman conclude that of spirals

ob ey Freemans law They also claim that a quarter of galaxies p ossess discs which truncate

after only two scale lengths in order to reconcile the observed spike with a Freeman law

These are b oth erroneous In fact the observations have precisely the form which is

exp ected for an intrinsicly at distribution Their observation of some p oints with

suggest a high density of VLSB galaxies The identication of a galaxy similar to Malin

in their complete sample also implies a large density of such galaxies

However provides only a very crude measure of with errors dominating the

entire shap e of the distribution Indeed the sharp spike do es not even provide a go o d

  

since the error in this go es as Since is always near one small estimate of

 

Thus all that can really b e said uncertainties in the p osition of lead to big ones in

from these data is that they are consistent with an approximately at distribution Any

galaxies with rule out a Freeman law

Constraints from cm Surveys

An imp ortant constraint on the p opulation of galaxies which are dicult to detect

optically is provided by cm surveys These dep end on the H I rather than optical prop

erties of galaxies so in principle suer no bias against gas rich but low surface brightness

galaxies However sensitivity limits have restricted these surveys to very small volumes

The cm surveys fall into two categories The rst typ e are serendipitous detections

of signal in the o b eams of p ointed observations of optically known galaxies Fisher

Tully Briggs These provide the apparently most impressive constraints on gas

rich but optically faint galaxies The second typ e are blind surveys largely indep endent of

known optical galaxies Kerr Henning Weinb erg et al Szomoru et al

The latter are in principle more useful but are to date very limited in extent

Strictly sp eaking the rst typ e of survey only constrains the mass density of inter

galactic H I clouds Fisher Tully ie ob jects which are totally invisible optically

If a galaxy is visible at all at the serendipitous cm p osition regardless of its actual

surface brightness it do es not count towards this invisible p opulation The eye is very

go o d at detecting extended coherent structure if told where to lo ok so the requirement of

invisibility is quite restrictive probably

The diameters listed in the UGC are typically measured at the mag arcsec

level but are sometimes as faint as mag arcsec Cornell et al Hence quite LSB

galaxies are at least discernible on the POSSI survey prints even if their app earance

thereon may seriously underestimate their true size An ob ject must b e well into the

ELSB regime to guarantee evading optical detection entirely Thus the rst set of surveys

provide no real constraint at all on the density of LSB galaxies However data of this sort

would b e very useful if the asso ciated optical identication criteria were quantied and

the surface brightnesses of detected galaxies actually measured

The case of the large H I cloud in Virgo H I is a go o d example of the

problems involved in using these surveys to place limits on the LSB galaxy p opulation

Originally thought to b e the rst discovery of an HI cloud totally devoid of an optical

counterpart Giovanelli Haynes H I do es have an asso ciated optical

comp onent McMahon et al Imp ey et al Though the H I prop erties of this

galaxy are extreme the optical comp onent is actually quite prominent by the standards

0

of Schomb ert et al having a diameter Salzer et al This is twice

the limit of Schomb ert et al and HI would have b een incorp orated

as one of the more conspicuous LSB galaxies had that survey extended far enough south

Clearly claims of strict limits on the LSB galaxy p opulation based on these sort of data are

meaningless without rigorous quantication of b oth terminology and optical search limits

Blind surveys are in principle much more useful but to date cover only very small vol

umes This causes large uncertainties in normalization which make interpretation of their

results dicult see Schade Ferguson Ferguson McGaugh In particular

these surveys fail to recover the HI mass function which is known to exist in optically

bright galaxies Hence their ability to constrain optically dim galaxies is susp ect at b est

Contrast the last two sentences of the abstract of Weinb erg et al

Szomoru et al present more complete results from the same survey as Wein

b erg et al including measurements of the optical surface brightnesses of of the

H I selected galaxies Of these three are LSB This is a fairly high fraction esp ecially

considering that the elds were not entirely randomly selected Those centered on known

bright galaxies are unlikely to detect LSB galaxies which tend to b e very isolated on the

relevant scales Bothun et al Mo et al Two of the three LSB galaxies were

discovered in blank elds

Obviously not a great deal can b e said with such limited statistics Nonetheless

the conclusion of Szomoru et al are consistent with those presented here Numb er

density and mass density are not the same thing LSB galaxies could b e quite numerous and

still contain only a mo dest amount of mass Moreover as noted ab ove the vast ma jority of

LSB galaxies should b e visible on sky survey prints Thus they do not represent some kind

of mysterious entirely new p opulation but rather a familiar if neglected one consisting

predominantly of late morphological typ es which span and extend a continuum of galaxy

prop erties

THE DISTRIBUTION FUNCTIONS

In this section I derive expressions for the relative numb er luminosity and mass

density of disc galaxies as a function of central surface brightness

The Numb er Density

The numb er density of galaxies can b e extracted from the apparent distribution of

central surface brightnesses in a complete survey by inverting the pro cedure describ ed

in x That is given N one can obtain the relative numb er density distribution

obs

from N V Unfortunately very little data exist which meet b oth

obs

requirements of completeness and surface photometry particularly for lo cal samples of

eld galaxies

One study that do es is Davies These data b ehave precisely as exp ected from

Figs and see McGaugh et al a The distribution obtained from these

data is shown in Fig

The distribution on either side of the p eak is well t by a straight line It is therefore

p ossible to write an analytic expression that gives the relative numb er density of disc

galaxies as a function of surface brightness



log m



The p eak is at B mag arcsec and by least squares t giving equal weight to

J

diameter and ux selected samples the slop e is



for

m



for

Formally the errors on the t are small for the slop e on the faint end and for

the bright end However systematic eects plus the relatively small numb ers of galaxies

in the fainter bins makes the actual uncertainty rather larger

The numb er density derived from the data of Schomb ert et al in x suggest a

faint end slop e closer to m There are nearly as many galaxies in this bin at

as in the entire sample of Davies so m is at least as signicant

Correlations Between Surface Brightness and Size

The systematic eect of greatest concern is that the volume sampling function de

p ends on size as well as surface brightness So far I have assumed that size and surface

brightness are uncorrelated so that on average variations in h cancel out and V h

can b e approximated by V This assumption is the natural one to make given the

functional separation of the volume sampling into size and surface brightness terms More

imp ortantly it is motivated by the data which show no indication of a correlation b etween

these two parameters Romanishin Strom Strom Davies et al Irwin et al

McGaugh Bothun de Blok et al a McGaugh Schomb ert Bothun

b However a weak trend one way or the other is not ruled out so it is imp ortant

to examine the consequences of such a relationship

The volume sampling dep ends on scale length as V h h for b oth diameter and

ux selection Small galaxies are hard to nd just as dim ones are Strictly sp eaking the

slop e m describ es the pro jection of the bivariate distribution h along the surface

brightness axis To account for p ossible correlations with size we should write

h



log m log



h

If size and surface brightness are correlated in the sense suggested by much unwritten

common lore namely that dwarfs are b oth smaller and lower in surface brightness than

brighter giant galaxies then b oth size and surface brightness discriminate against them

Correcting for this additional factor due to h would require even more low surface bright

ness galaxies raising the distribution in Fig and causing the t m to b e less negative

On the other hand if size and surface brightness are correlated in the sense that

lower surface brightness galaxies tend to b e larger then their increased size tends to oset

their faint surface brightness This would push the distribution in Fig downwards

towards more negative values of m However this would imply that the lowest surface

brightness galaxies are quite large and luminous Lo cal surveys with relatively bright

limiting isophotes would give a very misleading impression of this p opulation

There is some evidence p ointing in b oth directions The largest disc galaxies are

inevitably those lowest in surface brightness Imp ey Bothun Knezek Spray

b erry et al a so that disc galaxies exist all over the surface brightnessluminosity

diagram up to maximums in b oth McGaugh However there may b e a slight ten

dency for the scale length distribution to b ecome steep er with fainter surface brightness

This would b o ost b oth the numb er of low surface brightness galaxies in Fig and steep en

the faint end slop e of the luminosity function see Bothun et al McGaugh

There is not a suciently strong eect in either direction to alter the conclusions reached

here though obviously these eects will mo dulate the precise value of m somewhat

Cosmological Dimming

Another systematic eect of concern for the Davies sample is cosmological

dimming The magnitude selected sample is limited at B where the median

J

redshift is z At this redshift z dimming amounts to mag plus the

k correction appropriate to each galaxy Hence it is p ossible that the galaxies in the

faintest surface brightness bins are actually higher surface brightness galaxies which have

b een dimmed to app ear low surface brightness In this case it is obviously inappropriate

to apply a large volume correction to derive the numb er of galaxies in that low surface

brightness bin

In order to test the p otential eects on the Davies sample I have taken the

empirical redshift distribution for the observed magnitude range Ko o Kron and

attributed the maximum amount of dimming to each galaxy Starting in the faintest bins

and working brightwards each galaxy is corrected by the maximum amount allowable for

the observed redshift distribution This is not a statistical pro cedure it is the worst case

scenario for the analysis presented here The results of this exercise are shown in Fig

If maximum credit is given to cosmological dimming in this fashion something resem

bling a Freeman law plus a tail to lower surface brightness can b e regained However this

is a by choice a pathological case the most signicant asp ect of which is that it is still

imp ossible to recover a true Freeman law Indeed a universe full of Freeman discs would

not dim to have the observed apparent distributions McGaugh et al a Moreover

surface brightness and redshift are not observed to b e correlated in any published sample

and the distribution derived from this contrived exercise is inconsistent with the results of

x The sample of Schomb ert et al gives a high density of LSB galaxies which are

known to b e lo cal with negligible cosmological dimming

Indeed one might exp ect the opp osite eect to b e at least as imp ortant That is from

the nature of the volume sampling function the b onade LSB galaxies which are found in

any complete sample should on average b e of lower redshift than higher surface brightness

galaxies simply b ecause the latter can b e seen further away Rep eating the ab ove pro cedure

in reverse this time attributing the highest redshifts to the highest surface brightness

galaxies we nd that the distribution is not so strongly aected This is b ecause the bulk



of the observed galaxies are clustered around the same surface brightness and most

of the redshift distribution is lled b efore the low surface brightness bins are reached This

stretches the surface brightness distribution causing it to b ecome approximately at This

would bring it into agreement with the density derived from the data of Schomb ert et al

so if anything the latter case is likely to b e more imp ortant However neither of

these extreme cases are likely and mostly dimming eects probably just shue the high

surface brightness bins around a bit

The most signicant eect of dimming is not the redistribution of the shap e of the sur

face brightness distribution so much as the shift caused by the median amount of dimming

This is mag dep ending on the colours of the galaxies This is the amount needed



to bring the value of determined from the Davies observations into agreement

with lo cal samples van der Kruit

Hence the surface brightness distribution is well describ ed by two straight lines



which meet at B mag arcsec The slop e of the faint end is probably in the

J

range m with the latter value p erhaps somewhat more likely The slop e is much

steep er on the bright end with m Note that this corresp onds to sharp er

even than assumed in x Signicant uncertainties remain in all of these parameters and



is not tremendously well determined even

Recent Data

Since the initial submission and presentation of these results McGaugh much

work has b een done which strongly conrms them Schwartzenb erg et al nd a

distribution which is at until a sudden dramatic rise in the numb ers of VLSB galaxies

Their data are very deep images obtained with the AAT f camera and the result is

sensitive to dimming eects Dalcanton identied a large numb er of VLSB galaxies

in deep drift scans consistent with a at distribution but is obliged to make the same

assumption ab out the distribution of scale lengths as I have done Though not quite as

deep there are two studies Sprayb erry de Jong which do have redshifts and

hence need make no assumptions Both these data sets are consistent with the distribution

derived here though p erceptible dierences exist The data of de Jong include

extensive CCD imaging of a sample selected from the UGC and indicate a slop e very

similar to that of the Davies data The data of Sprayb erry are selected

from plate scans by the APM and indicate something closer to m Part of the

dierence may b e attributed to morphological selection de Jong concentrates on

spiral galaxies as classied by Nilson It is my qualitative impression McGaugh

et al b that spiral structure tends to drop in amplitude or at least visibility with

surface brightness so that Sprayb erry may simply b e including more smo oth LSB

disks The data of b oth Sprayb erry and de Jong indicate a somewhat brighter



and steep er cut o than found here but these parameters are correlated and dep end

on the binning and photometric system Given the current state of the data all results

app ear to b e consistent

The Luminosity Density

Once the numb er density of disc galaxies as a function of surface brightness is known

it is straightforward to obtain the luminosity density contributed by galaxies of each central

surface brightness Let us dene the relative luminosity density J in analogy with the

surface brightness distribution so that it is normalised to the absolute luminosity

  

density j pro duced by galaxies That is J j j If the numb er density

of galaxies at each is known J is simply the pro duct of the numb er times the relative

luminosity of each galaxy L h ie J L h The relative luminosity

follows from equation giving

h

L h





h

and

h



logL h log



h

Combining equations and the relative luminosity density is

h



logJ m log



h



This is plotted in Fig for hh

Basically this just says that even if the number density of LSB galaxies is large

m the luminosity density pro duced by progressively dimmer discs declines with

their surface brightness Note that J do es not dep end strongly on the assumption

ab out h since the terms that account for the numb er h and luminosity L h

nearly oset The numerical eect is stronger so if lower surface brightness galaxies are

on average smaller they contribute even more to the luminosity density

Once J is known it is simply a matter of the denition of terms to determine

how much luminosity density is pro duced by low surface brightness galaxies If LSB is

taken to mean anything fainter than the Freeman value then well over half the luminosity

comes from low surface brightness galaxies For the opp osite extreme denition of essential

invisibility ELSB almost no light is pro duced by extrap olation of the trend in

Fig

Obviously very dierent interpretations will follow from the same data if the termi

nology is not quantitatively dened By the denitions adopted in Table most of the

luminosity density is pro duced by HSB and ISB discs while very little is pro duced by

either VLSB or VHSB discs A small but signicant amount is pro duced by

truly LSB galaxies

The Mass Density

The relative mass density follows from the pro duct of the luminosity density

and the mass to light ratio Typically this is assumed to b e the same for all galax

ies However one can do b etter with information ab out the systematics of the rotation

prop erties of galaxies with surface brightness Sprayb erry et al b Zwaan et al

de Blok McGaugh van der Hulst b McGaugh et al c Salp eter Homan

The observations require



q

q

0

0

where q is a bandpass dep endent parameter of order unity Given this relation b etween

the mass to light ratio and the surface brightness J is simply



log log J q

Substituting the ab ove results as b efore yields

h



log log m q



h



which is plotted in Fig for q and hh

The mass density contained in LSB galaxies is an even greater prop ortion of the total

than is the luminosity density Note however that the observed relation applies to

the mass enclosed within the edges of the discs Extrap olating this to the total mass of the

halos asso ciated with the galaxies would require knowledge of the total extent of the halos

relative to the discs The obvious assumption is that there is no systematic dep endence of

the halo extent on surface brightness ie R R

halo disc

The relatively large mass fraction in the halos of LSB galaxies suggested by Fig

is consistent with the predictions of the structure formation mo del develop ed by Mo et

al to explain the spatial distribution of LSB galaxies see their Fig and with

the exp ectations of galaxy formation theory generally Frenk et al McGaugh

Bothun et al AntonuccioDelogu Dalcanton et al

The Gas Density

In addition to the relation of enclosed dynamical mass to surface brightness there

exists a very similar relation for neutral gas mass fraction de Blok et al b Lower

surface brightness galaxies are progressively more gas rich This may indicate an evolu

tionary sequence McGaugh with galaxies increasing in surface brightness as they

convert gas into stars The scatter in the M L relation is rather larger than that in

HI

and it is p ossible that LSB galaxies with low gas content have b een selected against

Nevertheless the M L relation represents at least the upp er envelop e of the space

HI

o ccupied by galaxies In analogy with equation

h



log m q log

g as g as



h

If there remain many undetected gas p o or LSB galaxies q q and q could conceiv

g as g as

ably b e negative However from present data they are only marginally distinguishable de

Blok et al b with q q so Fig suces to display as well as

g as g as

DISCUSSION

The relationships set out in x quantify an imp ortant previously unappreciated asp ect

of the general eld galaxy p opulation These relations are as basic to understanding

galaxies as the luminosity and correlation functions As such they provide an imp ortant

new constraint on theories of galaxy formation and evolution They are also relevant to

various topical problems discussed b elow

The Luminosity Function

It has long b een recognized Zwicky de Vaucoulers Disney that

LSB galaxies can have an impact on determinations of the galaxy luminosity function

Generally these have implicitly treated the surface brightness distribution as a function

see McGaugh Ferguson McGaugh This is only adequate if the surface

brightness distribution is symmetric so it is interesting to see what eects the actual

may have

Galaxies of the same luminosity but dierent surface brightness are sampled over

dierent volumes recall Fig This means that the luminosity function can not b e ob

tained directly from the inversion of the selection function b ecause the selection magnitude

is not simply related to the total magnitude Since all surveys have an eective limiting

isophote direct construction of M from them always b egs the question of how much

light is pro duced by galaxies with However missing LSB galaxies entirely turns

out not to b e the most serious eect Most surveys are able to at least detect LSB and

even VLSB galaxies and there is little luminosity density b eyond that Fig The more

imp ortant eect comes from the systematic underestimation of the uxes of LSB galaxies

this happ ens by denition with isophotal magnitudes see equation This eect

can b e large mag McGaugh Bothun dep ending on and McGaugh

One exp ects LSB galaxies to comp ose a very small fraction of the total sample

Figs and so this eect can easily go unnoticed It is quite serious though b ecause

the few detected LSB galaxies carry a volume normalised weight comparable to the entire

rest of the survey Small errors in their luminosity are thus magnied enormously when

estimating M This wreaks havo c on the derived Schechter parameters Mc

Gaugh Ferguson McGaugh The real situation may b e even worse than

describ ed in these pap ers b ecause we assumed that everything had b een done correctly to

a uniform isophotal selection level In practice this is not the case as uctuates from

plate to plate over a range that causes signicant dierences in the fraction of the total

ux actually measured The eects of varying isophotal selection levels can clearly b e seen

in the combined data of Ellis et al Surveys with brighter limiting isophotes give



atter slop es and lower normalizations than do deep er surveys How much of this

is caused by surface brightness dep endent measurement eects large scale structure and

evolution is dicult to say probably all contribute

It is thus dicult to estimate the degree to which the integrated luminosity density is

underestimated by current surveys The required information is not usually published nor

even extracted and retained in the original work Fairly complete information exists for

the APM survey Maddox et al a for which For these measurement

isophotes the ux will b egin to b e signicantly underestimated in the ISB regime From

Fig I estimate that this is likely to lead to an integrated luminosity density which

is underestimated by approximately This pro cedure is highly uncertain and the

underestimate could b e as much as a factor of two Sprayb erry Dalcanton or

as little as It is certainly not an order of magnitude or zero

In principle one should construct the bivariate distribution h and integrate

over this more general quantity see So dre Lahav de Jong The slop e of

the faint end of the luminosity function is particularly sensitive to this pro cedure For a

at a at luminosity function eg Loveday et al requires a size

n

distribution h h with n see Bothun et al Most estimates of the size

distribution van der Kruit Hudson LyndenBell de Jong are closer

to n which corresp onds to but considerable uncertainty exists Moreover

there is little reason to exp ect plausible forms of the bivariate distribution to integrate to

precisely the Schechter form and while this might b e a tolerable approximation

more complicated shap es seem likely Phillipps Driver Ellis et al argue

that the slop e of the lo cal luminosity function remains at down to at least M and

suggest that surface brightness dep endent eects must act to preserve the at shap e of

M This o ccurs in the case of no correlation b etween surface brightness and luminosity

mo del A of Ferguson McGaugh but I nd this much less likely than the lack

of correlation b etween surface brightness and size indicated directly by the data This

must have an eect on the faint end slop e though the upturn need not o ccur immediately



faintwards of L

Faint Blue Galaxies

The large excess of blue galaxies in faint numb er counts eg Tyson led to many

extreme mo dels for galaxy evolution eg Broadhurst et al I argued McGaugh

that the corresp ondence of physical prop erties b etween p opulations of faint and low

surface brightness galaxies suggested a common nature removing the need for enormous

amounts of evolution at low redshift However the implication that there is a one to one

corresp ondence b etween lo cal LSB galaxies and distant faint blue galaxies is incorrect

numb er counts do not work this way in the cosmological context One can only discuss

p opulations statistically with the imp ortant eects b eing that of dierential luminosity

density measurements b etween surveys and p otential underestimation of the slop e of the

luminosity function Ferguson McGaugh

There is now unambiguous evidence of evolution in the galaxy luminosity function

Ellis et al Lilly et al a CFRS VI The CFRS is particularly p ersuasive

since it is deep and uniform and Lilly et al b CFRS I have thoroughly considered

the relevant eects including those discussed here Note that the CFRS do es not imply

a low density of LSB galaxies as might b e inferred from Fig of CFRS I Since it is

a ux selected sample one exp ects LSB galaxies to constitute a small fraction of the

total sample even though easily detected recall Fig With a measurement isophote of

I mag arcsec the CFRS detects nearly all the light for galaxies well into the

AB

VLSB regime suciently far that very little residual luminosity is missed Fig That

the CFRS indicates a steep faint end slop e for blue ob jects is thus no surprise However it

primarily constrains the p opulation at z where the observed amount of evolution is

reasonable if still large The situation lo cally continues to p ose the most serious problems



An upward revision of the lo cal as suggested by several lines of evidence see Ellis et

al cures many ills but still provides no explanation for the steep galaxy counts at

bright magnitudes Maddox et al b which imply a large amount of evolution very

recently z a signicantly bluer galaxy p opulation than usually assumed Ko o

Gronwall Bruzual see also Ferguson McGaugh Babul Ferguson

or systematic errors Metcalfe et al

The eects discussed in this pap er contribute to the solution of these problems with



an upward revision of b oth and seeming likely However they are not of sucient



magnitude to b e the sole solution A factor of in is required at the upp er limit of

what can b e attributed to surface brightness measurement eects It is unclear what the

faint end slop e actually is It seems plausible that red early typ e galaxies have a at

luminosity function while a steep er one would b e appropriate to the late typ es which make

up the excess in the counts Marzke et al Glazebro ok et al Driver et al

esp ecially as late typ es tend to b e ISB and LSB galaxies McGaugh et al b If so

the typ e dep endent redshift distribution will p eak at lower redshifts for later typ es or

p ossibly even have a bimo dal distribution if the classication scheme also includes strange

things at high redshift in the late typ e bin Since morphology is not quantitative one

really needs detailed knowledge of the trivariate distribution h col our in order to

interpret evolutionary or cosmological eects If the slop e of this distribution b ecomes

steep for blue low surface brightness galaxies it would cause an excess which is not well

constrained by lo cal surveys cf Phillipps Driver

Ly Absorption Systems

Another quantity which requires the bivariate distribution is the cross section of galax

ies as absorb ers along the sight of QSOs Indeed one really needs to know the bivariate

distribution of the gas discs However Fig provides a more direct way of estimating how

much absorption is caused by discs of various surface brightnesses as it gives an indication

of how much gas mass is contained in each p opulation Though LSB galaxies are gas rich

as individuals the total HI density do es decline with surface brightness

Damp ed Ly and Mg I I absorption systems contain most of the mass seen in absorp

tion From Fig it is clear that most of these would b e HSB and ISB systems readily

visible optically A signicant but small fraction should b e LSB galaxies but these to o

should b e fairly easily visible optically and so one do es not exp ect the absorb ers to remain

unidentied when closely examined Indeed one example of an LSB galaxy b eing the

absorb er has recently b een noted Steidel et al

There are low column density Ly absorb ers without optical counterparts Morris

et al These systems do not contain much mass but do exist in profusion By

extrap olation of Figures and into the ELSB regime one do es exp ect there to exist

some such invisible galaxies though they should contain very little mass One might exp ect

these ELSB galaxies to b e weakly clustered Mo et al consistent with observations

of the Ly clouds Mo Morris However the extrap olations involved are huge so

the uncertainties b ecome enormous At most I would say that it plausible that the low

column density absorption systems are related to a p opulation of ELSB galaxies It is just

as plausible that they are bubblelike structures unrelated to individual galaxies

Passband Biases A Cautionary Tale

The data discussed here are all selected from blue sensitive photographic plates The

known examples of very large massive VLSB galaxies similar to Malin tend to b e rather

more red than the normal sized LSB p opulation Sprayb erry et al a This raises the

concern that substantial numb ers of massive red LSB galaxies may still b e missed

Indeed despite the large amounts of eort that have gone into galaxy surveys the

situation even in the blue remains fairly abysmal In the red the appropriate data do not

exist at all In order to examine p ossible passband biases consider the case of selection on

blue sensitive plates for galaxies of the same b olometric surface brightness but a range of

colours In analogy with Fig imagine that equal numb ers of disc galaxies exist at each

bol

h app ear smaller b olometric surface brightness The redder discs will for the same

on blue sensitive plates than blue discs Hence fewer red discs will b e selected at any given

This is illustrated in Fig which shows the relative numb er of galaxies one would

exp ected to detect as a function of B I colour for representative combinations of b olo

metric central surface brightness and limiting B isophote The results dep end sensitively

bol

on the passband and the exact shap e of the sp ectral energy distribution esp ecially as

approaches There is no way to know what sp ectral energy distributions are really

appropriate for the putative p opulation of red LSB galaxies These could b e faded star

bursts and indeed one would exp ect signicant numb ers of such remnants from the high

numb er of known short duty cycle star burst dwarfs or something entirely dierent For

the sake of illustration the sp ectral energy distributions from the mo dels of Buzzoni

are utilised

Even for HSB galaxies selected at a fairly deep the apparent numb er of galaxies

declines steadily as their colours b ecome redder The eect b ecomes more severe as either

bol

b ecomes fainter or brighter Galaxies b egin to disapp ear entirely for only mo destly

bol

red colours B I for galaxies with esp ecially at the brighter limiting

isophotes typical of large surveys This is several magnitudes brighter than Malin so it

is worth emphasising that we would still b e unaware of this enormous galaxy had it lacked

a prominent bulge comp onent All of the known examples of galaxies similar to Malin

also have prominent bulges even though most LSB galaxies have little or no bulge

The distribution of colours for LSB galaxies in the Schomb ert et al catalog

roughly follows the appropriate line in Fig This implies a density of red LSB galaxies

as large as that of blue LSB galaxies ie a roughly at distribution with colour The

data are to o limited to say more b ecause the volume prob ed for red LSB galaxies by B

band surveys is extremely small Considering the brightness of the night sky at redder

wavelengths and that contrast for blue LSB galaxies is already a problem in B it could

well b e that only the most prominent observationally accessible comp onent of the galaxy

p opulation has b een identied An entire universe full of dim galaxies may remain to b e

discovered

CONCLUSIONS

I have quantied the relative numb er luminosity and mass density of disc galaxies

as a function of central surface brightness These relations can b e conveniently expressed

analytically as



log m

for the numb er density



logJ m

for the luminosity density and



log m

for the mass density This last refers to b oth the dynamical mass enclosed within the

optical radius and to the H I gas mass See equations and for further



details The values of the parameters and m obtained from extant data are



B mag arcsec

J



m for and



m for

These relations are consistent with all relevant data though substantial uncertainties

remain They represent a ma jor shift from the paradigm of the Freeman law which suggests



that spiral galaxies all have nearly the same central surface brightness The value of

corresp onds to Freemans value but a faint end slop e of m indicates roughly equal



numb ers of galaxies at each central surface brightness fainter than This conrms the

intuition of Disney that Freemans law is a selection eect but one which works

more in the fashion describ ed by Allen Shu

It is very striking that the surface brightness distribution is nearly at It is unclear

what signicance if any should b e attached to the particular value m The sharp cuto



brightwards of seems analogous to the turndown in the luminosity function brightwards

 

of L but the signicance of the particular value of is also unclear What is clear is

that these numb ers are fundamental to understanding spiral galaxies

ACKNOWLEDGEMENTS

This pap er was written over the course of two years while I was supp orted by a PPARC

PDRA for which I am grateful During that time I have aspired to keep the manuscript

current with the many relevant developments and ap ologize for any I may have inadver

tantly overlo oked I would like to thank the many p eople to o numerous to list who have

tolerated my p estering them ab out these issues and who resp onded with many lively and

often incredulous conversations Deserving of sp ecial mention are Greg Aldering Harry

Ferguson Ro elof de Jong David Schade Bob Abraham and Joseph Lehar

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FIGURE CAPTIONS

Figure Idealised galaxy luminosity proles for four exp onential discs The galaxies are char

acterised by their size h and central surface brightness The HSB galaxies are

Freeman discs with B mag arcsec and the LSB disks have

The scale lengths are chosen to illustrate the variation of isophotal quantities at xed

size and luminosity for dierent surface brightnesses The HSB disc with h kp c



has an integrated luminosity of L roughly L for H km s Mp c

The LSB disc with h kp c has this same luminosity while the LSB disc with

h kp c has the same physical size but only one quarter the luminosity The small



h kp c HSB disc also has L L L Note that when measured at a

particular isophotal level eg the horizontal lines at and an LSB

galaxy will always app ear smaller than an HSB galaxy of the same size and fainter

than one of the same luminosity

Figure The relative numb er of galaxies observed as a function of central surface brightness

for diameter limited catalogs The solid line shows the assumed intrinsic distribution



B mag arcsec with equal numb ers at every faintwards of the value

J



the intrinsic distribution is assumed to have a sharp exp onential cut Brighter than

o to b e consistent with observations The constant density assumed on the faint

side illustrates the eects of volume sampling for diameter limited catalogs selected at

dierent isophotal levels The resultant apparent distributions dashed lines are



for several representative lab eled These shown normalised to N at

obs

are typical of shallow plate material deep plate material and

very deep CCD images The apparent distribution is very sharply p eaked at

 

for all values of giving the app earance that all galaxies have the value



even if equal numb ers of galaxies exist at every

Figure As p er Fig but for catalogs limited by isophotal magnitude The apparent dis

tribution of is even sharp er than in the diameter limited case b ecause N

obs



0

0

rather than

Figure The distribution of diameter ratios which is related to the surface brightness

The histogram is the SRCESO data of Bosma Freeman The heavy solid

line shows the exp ected apparent distribution for a at intrinsic distribution This

is very sharply p eaked b ecause of the way is dened The dotted line gives the

exp ected distribution after convolution with errors indicating that these data suggest

an intrinsically at distribution

Figure The surface brightness distribution derived from the data of Davies Circles

are data selected by diameter while triangles are magnitude limited data Error bars

are from counting statistics The lines are least squares ts to the data giving equal

weight to p oints from diameter and magnitude limited samples The distribution

declines slowly faintwards of the Freeman value indicating a large space density of



low surface brightness galaxies It cuts o sharply brightwards of in analogy with



the turndown of the luminosity function at L

Figure The surface brightness distribution after maximally extreme corrections for the eects

of cosmological dimming Triangles the ux selected data shifted by the mean cor

rection Dashed line redistribution assuming all the dimmest galaxies are the highest

redshift ones Dot dashed line redistribution given the opp osite assumption see

text These lines give the maximum envelop e to which cosmological dimming can

mo dulate the surface brightness distribution derived from the data of Davies

This can not b e an imp ortant eect as it would cause inconsistency with the data of

Schomb ert et al

Figure The luminosity density pro duced by disc galaxies as a function of surface brightness

The luminosity density p eaks at the Freeman value and declines steadily towards

fainter surface brightnesses Substantially more luminosity is pro duced by discs which



are fainter than than is pro duced by those which are brighter In addition to the

data of Davies the data of Schomb ert et al are plotted as the solid

square For this the solid line shows the error bar due to counting statistics alone

while the dashed line shows the additional estimate of systematic error

Figure The mass density contained in disc galaxies as a function of surface brightness Points

as p er Fig A signicant amount of mass is contained in the low surface brightness

tail of the distribution in excess of that in Fig b ecause the mass to light ratio is

observed to increase as the surface brightness decreases Note that LSB discs with



contain the amount of mass that is in galaxies with

Figure The apparent numb er of galaxies of the same b olometric surface brightness that will

b e observed as a function of color Galaxies are assumed to exist with an intrinsically

bol

at distribution equal numb er density at each and color and are selected by

B

The plot is normalised diameter on blue sensitive plates at various isophotal levels

to N for B I the typical color of a B selected LSB galaxy Though

obs

B band surveys provide the b est contrast for LSB galaxies since the sky is reasonably

dark there they are quite insensitive to red LSB galaxies The exact sensitivity

dep ends on the limiting isophote and the shap e of the sp ectral energy distribution in

a complicated way

Table Nomenclature

Name

0 0

2 2

B mag arcsec L p c

VHSB

a

HSB

ISB

b

LSB

VLSB

c

ELSB

a

Satisfy Freemans Law

b

Brightness of darkest night sky

c

Practically invisible

Table Variations

L H U

h h l og n n

H L L H

0