Chin. Astron. Astrophys. (1995)19/4,405-413 A translation of Acta Astrophys. Sin. (1995) 15/3,197-204

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Faint blue galaxy count and the dwarf

population of starburst galaxiest

CHEN Shil MA Er’t2 YU Yun-qiang’v3 ‘Institute of Th eoretical-Physics, Chinese Academy of Sciences, Beijing 100080

2 Beijing Astronomical Observatory, Chinese Academy of Sciences, Beijing 100080

3Department of y Ph sits, Peking University

Abstract The dwarf population of starburst galaxies is analyzed by the method

of evolving population synthesis. The results show that the existence of an

additional population can give a good fit to the available number counts and

redshift surveys. These dwarf galaxies readily evolve into low surface brightness

objects and become undetectable in our local neighbourhood.

Key words: dwarf galaxies-galaxy counts-galaxy redshift

1. INTRODUCTION

In recent years, the problem of galaxy counts has aroused wide interest. The crux of the

matter can be summarized under three heads:

1) By B magnitude 26-27, the count of faint galaxies has already reached N 3x 105/sq deg.

Starting from the known local luminosity function within the standard cosmological frame,

this value far exceeds the expectations on the no-evolution model. If we consider only pure

luminosity evolution, then the situation is not much improved. For example, Cowiel’l using

the pure luminosity model of Yoshii et a1.121, found that merely decreasing the value of

~0 will not extend space sufficiently to accommodate so many galaxies, which can only be

achieved by greatly modifying the geometry, by taking qo = 0.1, A = 0.9, while Guiderdoni

et al.131 showed that, under their evolving model, one must, in addition to decreasing qo, make the time of formation of galaxies, tr,,r, much earlier: for qo 5 0.15, .zror has to be as

much as 10, even then the fit to the observations is still forced; for a best fit, we have to‘put

qrJ = 0.05, rr,, = 30.

2) At present, complete redshift samples14~5~61 are available down to B magnitudes 20-

22.5 and 23-24. Ref. [S] shows, where the count is already 3-5 times the no-evolution model

value, the majority of the galaxies are still quite local (z M 0.4). Ealeslfl, based on a

direct interpretation of available redshift data in terms of luminosity evolution (cf. Ref. [S]),

t Supported by National Natural Science Foundation Received 1994-03-17

406 CHEN Shi et al.

pointed out that the luminosity function in the redshift range 0.0-0.4 varies by a factor of 3,

while in the range 0.15-0.20, the increase in the amplitude is already obvious. These results

demonstrate forcefully that cosmological effects cannot be the main factor in the problem

of faint blue galaxy count.

3) The observations also showed that these faint galaxies are rather blue in colour. While

the count in the B band in the magnitude range 26-27 exceeds the no-evolution model by

a factor of 3-5, the count in the K band in the same magnitude range shows hardly any

excess. While this fact by itself already implies that these “excess” galaxies are bluer than

the normal galaxies, the small sample of Cowie et al. further details that, of the 11 dwarf

galaxies, five have an average B - K value near the normal (4.9), while six have a value as

small as 3.3.

Various solutions have been proposed for the problem of faint blue galaxy count. Besides

modifying the geometry of the universe, such as the introduction of a cosmological constant

of nearly 1 in Ref. [l], to enlarge the comoving volume, some believe that the galaxies in

recent times have had over-frequent merging, some believe that the luminosity of dwarf

galaxies in these times has undergone strong evolution, giving rise to bright objects at the

corresponding redshifts, while others believe that the recent epoch saw the formation of an

additional dwarf galaxy population, which has since become invisible or disappeared.

Simply modifying the geometry of the universe cannot explain the different behaviour of

the counts in the B and K bandsl’~sl. Although the introduction of a nonzero cosmological

constant has been one of the means adopted by many authors in recent years when tackling

the problem of the formation of cosmic structures, it also raises many theoretical problems.

Recently, it is also pointed outllOl that a cosmological constant near 1 may already conflict

with the constraints given by the statistics of gravitational lensing.

Because a considerable number of cases of galaxy merging have been observed in recent

years, there has been much discussion on merging modelsls~‘ll, some of which incorporate

also the advantages of strongly evolution models. The merging models are phenomenological

in character, and although they can solve the number count problem, some require rates of

merging at z = 0.1 so large as to be probably inconsistent with the observed rate, and the

amount of mass that the disk of a spiral galaxy can accrete is limited in any casel121. An

important difficulty faced by the merging models is, observations have shown that faint blue

galaxies are not located in dense environment of high merging ratel’3~‘41.

We shall attempt, within the standard cosmological model with R = 1, use an additional

dwarf galaxy population generated by starburst for a phenomenological discussion of the

faint galaxy count excess. For, once such a dwarf galaxy is formed, the method of synthesis

of evolving stellar populations li5*isl will provide rather sure knowledge of its luminosity evo-

lution. We can therefore see wether or not such an additional galaxy population introduced

in the z 5 1 stage can consistently explain the different behaviours of the number counts in

the optical and infrared bands and the redshift distribution of faint galaxies. If the answer

is positive, then we can reverse the argument and find the constraints the observations will

place on the mass, luminosity, starburst duration etc. of this kind of dwarf galaxies. In fact,

as we shall see below, a population of starburst dwarf galaxies, lit up through starburst and

with a mass of 106-107M0can successfully account for the observed number count, colours

and redshifts of the faint galaxies.

Faint Blue Galaxy Count 407

Recently, McGaughI’4 put forth a new, possible explanation of the faint galaxy count:

in the local luminosity function, because of selection effects, the number of galaxies with

low surface brightness is greatly underestimated, while the colour and luminosity of these

galaxies agree with those of the “excessive” faint galaxies, hence, what we are faced with

is not excess of faint galaxy count near z N 0.4, rather, it is the loss of a large quantity of

low surface brightness galaxies in the local luminosity function. If this surmise is confirmed

by observations, then our proposal above can precisely provide a possibility of explaining

how come we have so many low surface brightness galaxies: for a galaxy formed in the late

recent period the medium density must have been low, and for low-density dwarf galaxies,

the feedback by starburst-generated stellar populations must be all the greater, making

it easier to form low surface brightness galaxies. We shall be discussing these ideas in a

subsequent paper and further embarking on the origin of the additional galaxy population

within the larger framework of formation of cosmic structures.

The method of population synthesis has been greatly developed and widely applied in

recent years. For a dwarf galaxy very blue in colour and with a small total mass, the

characteristic features of its multi-wavelength luminosity are controlled by young stellar

populations generated by the starburst process. Because the duration of starburst is far

smaller than the life time of the stellar population under discussion, we can regard the star-

burst as instantaneous, and hence can take the approximation of simple stellar population

(SSP). As long as th e initial mass function (IMF) is not of an extreme form, the luminosities

of SSP at various wave bands are all mainly determined by a small number of stars located

at or near the main-sequence turn-off point, hence are sensitively dependent on the age of

the population. In order to discuss an additional population of small-mass galaxies, lit up

by the formation of short, eruptive stars, which would later disappear from the local lumi-

nosity function, we specially chose a very young stellar populations, a set of models with

luminosities determined mainly by blue giants.

2. MODEL AND METHOD

Assume in the late recent stage t 5 1, there is a kind of primitive dwarf galaxies which for

some reason became bright through starburst. Let the rate of formation of such starburst

dwarf galaxies per unit comoving volume at time t be 4(t), so that $(i)At is the number

of galaxies formed in the interval (t, t + At). We shall first consider the case where they all

have the same mass and later the case where there is mass distribution.

Let the total mass of the stars formed at a constant rate over a duration 70 by starburst

be MT. For a fixed IMF, such as the Salpeter spectrum, calculation by population synthesis

will give the spectrum of the dwarf galaxy so formed and its evolution in time:

M-M&r) (I)

where M is the absolute magnitude, r is the age of the stellar population and X is the

wavelength.

Let the luminosity function of the additional galaxy population-the ensemble of such

dwarf galaxies, be ~(MA, t), so the number of galaxies per unit comoving volume at time 1

in the magnitude interval (MA, MA + AMA) is

408 CHEN Shi et al.

4of,,0~M’ (2) According to our assumptions of instantaneous formation of the starburst population

and the total mass of SSP, the galaxies (2) must have ages in the interval (7, r + AT), i.e.,

they must have formed at time t - 7. So we have

dWrrtY& - (t(t - ZW (3)

Because galaxies that can effectively contributes to the luminosity function of the additional

galaxy population have ages less than 10” yr (see below), and within a range of t - 10’ yr

$(t) should be a slowly varying function, we have

Thus,

44 - 4 =s (t(t) we have, approximately,

(4)

where the first factor represents the cosmic evolution of the formation rate of starburst

galaxies and the second factor, the time of remaining at various luminosities after the for-

mation of the galaxy population, leading to variation in the observable number count. For

convenience in comparing with observations, the cosmic time t will be changed into the

redshift z, t = (2/3)&l (1 + z)-~/~.

The number of the additional galaxies in the redshift interval dz and magnitude interval

~MA is

d’A(M&,x) - +(M&(l+a)’ zdM&dx

Corresponding to present-day (z = 0) observation in the wavelength interval AX is the

luminosity function of the starburst galaxy population at redshift t, wavelength X’ = X/( 1 +

z) and wavelength interval Al’ = A1x/(l + 2). Th is is expressed by the so-called E and K corrections. the first in included in our evaluating the luminosity function at d(M~j, z), and

the second, in taking rn~ to be

ml- Mg+Slog +l i- x - Jr+-L)M;f ] - 25 - 2.5log(i + t), (7) 8

For a fixed z, we have dmA = dMA#, and so

+(m~,=)dm~ - cb(Ml’,%)dMrr & &)ds, (3)

At this point, we have

d2A(m,,z) - +b(M *t,z)( l+~)~ * dMlfdz dz

, (9)

hence, we can calculate the contribution by the additional galaxy population to the galaxy

count 1

Faint Blue Galaxy Count 409

N(md - ,+(M,y)( 1 + 2)’ 5 dz z Jo (%)-I( l+z)j $ dz I (10)

and the redshift distribution of the additional galaxy population in a fixed magnitude inter-

val,

N(s) - J $(M lr,z)( 1 + z)‘$ dMp = cl(a)( 1 + z)’ 5 At (11)

where AT is the age spread of the starburst galaxies of a given apparent magnitude interval.

As stated in the Introduction, cosmological effects cannot be the main factor in the

problem of faint blue galaxy count. In this paper, we take 52 = 1, h = 0.5, and so the

comoving volume element is

4c3 <1/l + z- 1)’ (l+o$-~ (1+z)5/2 - 8.64 x 10” x

(Jl + z - 1)’ M3

(1 -I- z)5’2 pc (12)

As in the case of star luminosity evolution models, the key calculation here is to find

the relation between the galaxy spectrum and the age. For starburst dwarf galaxies, the

main point is to understand the properties of the young stellar population determined by

large-mass stars, and the difficulty is that the spectrum of large-mass stars is determined not

only by its evolving internal structure, the effect of the atmosphere must also be considered.

For the latter, we may not even be able to invoke local thermodynamic equilibrium.

Mas-Hesse et al.l’sl using a new model of evolutionary paths of stars, taking into account

the effects of the atmosphere, for various metal abundances including those appropriate to

young galaxies undergoing starburst for the first time and the case of low-mass, metal-poor

dwarf galaxies, and combining a large volume of observational data from ultraviolet through

infrared of large-mass stars, gave a rather reliable relation between the broad band (1285A-

36OOOA) spectrum of the stellar population and its age. We shall use the results of Table

6c of that paper, which assumes 7s = 0 and a metal abundance z = 0.1.~. Interpolations

were made for the wavelengths required in our calculation. The age range is from 2~10~

to 2x107yr, which we extrapolated to 4x107yr. What Ref. [18] gives is luminosity of the

population, normalized to 1 Ma, at different age and different wavelengths. Using this data,

we calculated, for a set of total mass of the population, the number of magnitudes at various

ages and wavelengths.

3. CALCULATIONS AND RESULTS

Since our aim here is to investigate the feasibility of explaining the faint blue galaxy excess in

terms of an additional population of starburst dwarf galaxies formed at z < 1, our emphasis

is on qualitative features, rather than specific numerical values. So, any choice or assumption

that had to be made in the course of the calculation were done in this spirit.

For t/~(z), we did not make any assumption regarding the mechanism of formation of

the additional galaxy population, and the choice of the function had only two requirements

derived from the observations: first, it must be sufficiently small for z < 0.1; second, it must

be sufficiently large for 0.1 < z < 1.0. We took

410 CHEN Shi et al.

Fig. 1 B-band number counts, observed and

calculated for 4 values of the total galactic

mass MT

Fig. 2 Redshift distributions, observed and

calculated for 4 values of the total galactic

mass MT

1 : 0.7 -’

z ) h’M-,3, 0 < z < 1; (13)

2 7 1.

where the nonzero part was taken from Coles et al.flg], $0 being an adjustable coefficient.

There are two parameters in our model, the total mass of the stellar population MT

and $0. For given MT, the relative number densities in apparent magnitude intervals are

determined without reference to the value of $0. We found that, in order that the galaxy

count in B = 26-27 may reach 2-3x 10g/sqdeg, while not conflicting with the observed

densities in other,intervals, MT had to take a small value. Fig. 1 shows the expected number-

magnitude relations for different values of MT and the observed relation. The curves have

been normalized to the value of 2.5x105/sqdeg at B = 26-27. N(m) is the number of

galaxies per unit apparent magnitude interval per square degree and MT, = A&/lOsMo.

Fig. 1 shows that, when MT 2 1 x 107Ma, the counts at the brighter end will greatly

exceed the observed values. This is because when the total mass is large, the number of the

additional galaxies belonging to the same apparent magnitude interval can be distributed

over a large redshift range; the larger the total mass and the fainter the apparent magnitude,

the greater will be the spread. But we have assumed that the additional population was

formed in z 5 1, so the contribution from the higher redshift end has been excluded, and

the depletion at the fainter magnitude is the greater, the larger the mass. Then, when we

have normalized the counts at the faintest magnitude to the same value, the counts at the

brighter magnitudes of the large-mass galaxies will be much higher.

The observed and calculated redshift distributions are shown in Fig.2. N(r) is the

number of galaxies per unit redshift interval per square minute, normalized ss in Fig. 1.

We read: for MTs = 2.5,5,10,25, the redshift of the maximum count, zmax, and the mean

Faint Blue Galaxy Count 411

redshift, (z), are, z,, = 0.20,0.30,0.40,0.60 and (2) = 0.16,0.26,0.36,0.53. According to the small sample of Cowie et al., zmax is between 0.3 and 0.4, and (2) 21 0.26. Hence, if

the Cowie sample has a general significance, the redshift distribution will favour neither too

large nor to0 small Values Of MT.

We now extend the case of a single mass to a mass distribution. Following usual assump-

tion, we put

CL(Z, M,)a=M;’ (14

Figs. 3 and 4 give the number count and redshift distributions calculated for two mass

ranges of kfTs, 2.5-25 and 2.5-10. In either case, .z,,,,x is 0.30, while (z) is 0.35 and 0.28,

respectively. It can be seen the 2.5 < M Ts 5 10 case fits the observations better. When

the mass range is extended in the large mass end, the counts at the brighter end will be

too large and the average t will be too high. On the other hand, if we extend the mass

range to still smaller values, e.g., to l-25, then there will be too many low-redshift galaxies,

specifically, at B = 23-24, the .z < 0.2 count will be 82% the 0.2 < z 5 0.4 count, which

clearly inconsistent with the observations.

23.5 -74.5 25.5 ‘6.5

m2.S25mZ.5-11le Obs.

B Magnitutes

Fig. 3 B-band number counts, observed and

calculated for two mass ranges MT

is

Fig. 4 Redshift distributions, observed and

calculated for two mass ranges MT

Thus we see that it is feasible to use a population of starburst dwarf galaxies formed

in z 5 1 to solve the problem of the excess count of faint blue galaxies: both the observed

number counts in the B and K bands and the observed redshift distribution for B = 23-24

can be explained. Our calculation further shows that the mass of such starburst galaxies

must be small and the mass distribution must be narrow. Too small a mass will spoil the

redshift distribution, too large a mass, the number count distribution.

We also considered the case of $(z, MT) o( A4G1 ,2.5 5 MT~ 5 10 and we found: 1) that

the count in B = 27-28 will be 3.5x105/sqdeg, that is, the count continues to grow, but

at a smaller rate, 2) that the count in I< = 24-25 will be 1 x 105/sq deg, comparable to the

observed value in K = 22-23, thus again the excess, and 3) that the redshift distribution

for B = 24-25 will be as shown in Fig.5, with z,,,,~ = 0.40 and (z) = 0.40.

412 CHEN Shi et al.

26

2 It4 I I ! 1 9 I

0.1 0.2. 0.3 0.4 0.5 0.6 0.7 0.8

2

Fig. 5 Predicted redshift distribution of the additional population at

B = 24-25 for MT~ = 2.5-10.

4. DISCUSSION

1) Because different types of galaxies each have their own individual properties and make

different contributions, the calculation of number, colour and redshift counts of galaxies

is usually a multi-model, many-parameter affair. However, since the number count after

m > 23 is already several times greater than what the no-evolution model predicts, and the

change in the luminosity function takes place at very low redshifts, the additional galaxy

population must have made a dominant contribution. We have therefore concentrated on the

additional population and left aside the contribution by the normal population, requiring

only that the contribution by the additional population should leave some room for the

contribution by the normal population.

2) When investigating the luminosity of galaxies, it is necessary to use the method of

synthesis of evolving stellar populations to discuss the combined spectrum and its evolution.

Usually, because the different types of galaxies have rather complicated histories of star

formation, and because we only have the colours of nearby galaxies as observational basis

while the behaviour of the older stellar populations provides no adequate guide to their

earlier stages, there is much uncertainty. In this paper, under the hypothesis of an additional

population of starburst dwarf galaxies, we considered only the properties of a young stellar

population, and there were only two free parameters, the total mass of the stellar population

and its age. Hence, our results are comparatively sure.

3) The results of our calculation show that, in the case where there are no observational

constraints other than the number count and redshift distribution, we still have rather

stringent constraints on the properties of the population of starburst galaxies. First, this

population is confined to a finite redshift range, otherwise the average redshift will grow

too fast. Next, there are also restrictions on the mass of the galaxies, too small a mass will

give too many low-redshift objects, while too large a mass will give too many objects at the

bright end of the magnitude distribution as well as a rather too high average redshift.

4) A population of starburst dwarf galaxies, formed by instantaneous starburst, with

Faint Blue Galaxy Count 413

an age between 2 x lo6 and 4 x lo7 yr and a mass range (2.5 - 10) x 106Mo, provided its

rate of formation is of the form (13), can reproduce the present observed data on the faint

galaxy count and redshift distribution. If the calculated B count is pushed to magnitude

27-28, it will continue to rise, though at a smaller amplitude. If the K count is pushed

two more magnitudes to 24-25, then the additional population will make a non-negligible

contribution. If the redshift sample is as faint as B magnitude 24-25, the redshift distribution

of the additional population will remain local. These predictions can all be checked with

observations in the near future.

5) When the normal bright galaxies are morphologically segregated into giant ellipticals

and large spirals, the luminosity function of each subtype is a narrow gaussian distribution,

thus, the requirement of a limited mass and luminosity range for the additional population

is nothing remarkable. In the standard cold dark matter model of galaxy formation, there is

sufficient amount of late-collapse, dwarf galaxy-sized perturbations that can act as the seeds

of the starburst dwarf galaxies. Because the collapse is late, their surface density is low,

and they are therefore more susceptible to feedback from the stellar population, resulting

in systems of very low surface brightness through mass loss and expansion. Thus, a natural

extension of this paper is the investigation of the origin, development and final outcome of

starburst galaxies in terms of usual theory of galaxy formation.

[ll [21 [31

141

151

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(131

1141

P51

Cowie L. L., Physica Scripta, 1991, T36, 102

Yoshii Y., Takahara F., ApJ, 1988,326, 1

Guiderdoni B, Rocca-Volmerange B., A&A, ISSO, 227, 362

Broadhurst T. J., Ellis R. S., Shanks T., MNRAS, 1988, 235, 827

CoIIess M., Ellis R. S., Taylor K., MNRAS, 1990, 244, 408

Cowie L. L., Sougaila A., Hu E. M., Nature, 1991, 354,460

EaIes S., ApJ, 1993,404, 51

Yoshii Y., Fukugita M., In: Shanks T. et al., eds., Observational Tests of Cosmological Inflation, 1991, 267

CoIin P., Schramm D. N., Peimbert M., Preprint FERMILAB-Pub- 93/17&A

Moaz D., Rix H. -W., ApJ, 1993,416,425

Broadhurst T. J., EIIis R. S., Glazebrook K., Nature, 1992, 355, 55

Toth G., Ostriker J. P., ApJ, 1992,389, 5

Efstathiou G., Bernstein G., Katz N. et al., ApJ, 1991, 380, L47

Roche N., Shanks T., MetcaIfe N. et d., MNRAS, 1993, 263, 360

Renzini A., In: Silk J. et al., eds., Galaxy Formation, Stellar Population Tools: Clues to Galaxy Formation

b31 BruzuaI G. A., Charlot S. et al., ApJ, 1993,405, 538

1171 McGaugh S. S., Nature, 1994, 367, 538

[181 Mas-Hesse J. M., Kunth D., A&AS, 1991,88,399

WI Coles S., Treyer M.-A., Silk J., ApJ, 1992, 385, 9

References