arX

iv:a

stro

-ph/

0001

308v

2 2

9 M

ay 2

000

The infrared side of galaxy formation.

I. The local universe in the semi-analytical framework.

G.L. Granato1,4,5, C.G. Lacey2,3,4, L. Silva4,1, A. Bressan1, C.M. Baugh3, S. Cole3,

C.S. Frenk3

ABSTRACT

We present a new evolutionary model for predicting the far-UV to sub-mm properties of thegalaxy population. This combines a semi-analytic galaxy formation model based on hierarchi-cal clustering (GALFORM, Cole et al. 2000) with a spectro-photometric code which includesdust reprocessing (GRASIL, Silva et al. 1998). The former provides the star formation andmetal enrichment histories, together with the gas mass and various geometrical parameters, fora representative sample of galaxies formed in different density environments. These quantities,together with a few other assumptions concerning the spatial distribution of dust and its opticalproperties, allow us to model the spectral energy distributions (SEDs) of galaxies, taking intoaccount stellar emission and also dust extinction (absorption plus scattering) and re-emission.In the spectro-photometric code dust is considered only in the disk, but the general radiationfield is contributed by both the disk and the bulge components with their own distinct age andmetallicity distributions. Two phases are considered for the dust: molecular cloud complexes,where stars are assumed to be born, and the diffuse interstellar medium. The model includesboth galaxies forming stars quiescently in disks, and starbursts triggered by galaxy mergers. Wetest our models against the observed spectro-photometric properties of galaxies in the local Uni-verse, assuming a CDM cosmology with Ω0 = 0.3 and Λ0 = 0.7. The models reproduce fairly wellthe SEDs of normal spirals and starbursts from the far-UV to the sub-mm, and their internalextinction properties. The starbursts follow the observed relationship between the FIR to UVluminosity ratio and the slope of the UV continuum. They also reproduce the observed starburstattenuation law (Calzetti et al. 1999). This result is remarkable, because we use a dust mixturewhich reproduces the Milky Way extinction law. It suggests that the observed attenuation lawis closely related to the geometry of the stars and dust. We compute galaxy luminosity functionsover a wide range of wavelengths, which turn out to be in good agreement with observationaldata in the UV (2000A), in the B and K bands, and in the IR (12 − 100µm). Finally, we inves-tigate the reliability of some star formation indicators which are based on the properties of thecontinuum SEDs of galaxies. The UV continuum turns out to be a poor star formation indicatorfor our models, whilst the infrared luminosity is much more reliable.

Subject headings: Galaxies: evolution; Galaxies: formation; Galaxies: fundamental parameters; Galaxies:

interactions; Galaxies: starburst; Infrared: galaxies; Ultraviolet: galaxies

1Osservatorio Astronomico di Padova, Vicolo

dell’Osservatorio, 5, I-35122 Padova, Italy2Theoretical Astrophysics Center, Juliane Maries Vej

30, 2100 Copenhagen Ø, Denmark3Physics Department, Durham University, South Road,

Durham DH1 3LE, UK4SISSA, Via Beirut 2-4, I-34014 Trieste, Italy5granato@pd.astro.it

1. Introduction

In recent years, our understanding of galaxyformation and evolution has advanced very rapidly,as a result of both observations and theory. Onthe observational side, new instruments have al-lowed the direct study of galaxy populations at

1

different wavelengths out to z . 5. By combiningobservations in the UV, optical, IR and sub-mm,we can now start to reconstruct the history ofstar formation in galaxies over the epochs whenthe bulk of the stars have formed (e.g. Madauet al. 1996; Steidel et al. 1999; Hughes et al.1998). On the theoretical side, models based onthe paradigm of structure formation through hi-erarchical clustering (which has successfully con-fronted a wide range of observations on large scalestructure and microwave background anisotropies)have now been developed to the point where theycan make definite predictions for the observableproperties of galaxies (luminosities, colours, sizes,morphologies etc) at all redshifts, starting from anassumed initial spectrum of density fluctuations.The key technique for making these predictionshas been that of semi-analytical modelling (White& Frenk 1991; Lacey & Silk 1991; Kauffmann etal. 1993; Cole et al. 1994; Somerville & Primack1999). In this technique, one applies simplified an-alytical descriptions of the main physical processesof gas cooling and collapse, star formation, feed-back effects from supernovae, galaxy merging etc,with the backbone being a Monte Carlo descrip-tion of the process of formation and merging ofdark matter halos through hierarchical clustering.The predicted star formation histories are thencombined with detailed stellar population modelsto calculate galaxy luminosities at different wave-lengths. Conversely, direct numerical simulationshave been enormously successful in studying theevolution of structure in the dark matter on a hugerange of scales (e.g. Jenkins et al. 1998), but cur-rently do not have sufficient spatial resolution tosimultaneously follow all the processes involved ingalaxy formation.

The semi-analytical models have been success-ful in predicting and/or explaining a large range ofgalaxy properties, both at low and high redshift,for instance, luminosity functions and colours indifferent optical and near-IR bands (Lacey et al.1993; Kauffmann et al. 1993; Cole et al. 1994),the mixture of galaxy morphologies and the evolu-tion of elliptical galaxies (Kauffmann et al. 1993;Baugh et al. 1996; Kauffmann 1996), the prop-erties of Lyman-break galaxies at high redshift(Baugh et al. 1998; Governato et al. 1998), thesizes and circular velocities of galaxies (Cole et al.2000), and galaxy clustering evolution and the na-

ture of the clustering bias (Kauffmann et al. 1997;Baugh et al. 1999; Diaferio et al. 1999; Benson etal. 2000). However, with very few exceptions,these semi-analytical models have ignored bothextinction and emission by interstellar dust, andcalculated only the direct stellar emission in theUV, optical and near-IR. This has been partly be-cause the importance of dust was generally under-appreciated, especially for high redshift galaxies,but also because of the lack of physically realisticmodels for predicting dust effects.

This situation has now begun to change. Onthe one hand, there have been several observa-tional discoveries demonstrating the importanceof dust effects for building a complete picture ofgalaxy formation. (1) The discovery of a cosmicfar-IR/sub-mm background by the COBE satel-lite (Puget et al. 1996; Guiderdoni, et al. 1997;Dwek, et al. 1998; Fixsen et al. 1998; Hauseret al. 1998; Schlegel et al. 1998), whose energydensity indicates that, as suggested already byWang (1991) and Franceschini et al. (1994), alarge fraction of the energy radiated by stars overthe history of the universe has been reprocessedby dust. (2) The discovery that the population ofstar forming galaxies at z ∼ 2 − 4 that have beendetected through their strong Lyman-break fea-tures are substantially extincted in the rest-frameUV (Pettini et al. 1998; Steidel et al. 1999). (3)The discovery of a population of sub-mm sourcesat high redshift (z & 1) using SCUBA, whoseluminosities, if they are powered by star forma-tion in dust-enshrouded galaxies, imply very largestar formation rates (∼ 102M⊙yr−1), and a to-tal star formation density comparable to what isinferred from the UV luminosities of the Lyman-break galaxies (Smail et al. 1997; Hughes et al.1998; Lilly et al. 1999). (4) The ISO detection of apopulation of strong IR sources; 15 µm ISOCAM(e.g. Oliver, et al. 1997; Elbaz, et al. 1999) and175 µm ISOPHOT surveys (e.g. Kawara, et al.1998; Puget, et al. 1999) indicate a populationof actively star forming galaxies at 0.4 . z . 1.3,which boosts the cosmic star formation density bya factor ∼ 3 with respect to that estimated in theoptical from the CFRS (Flores et al. 1999). For(1) and (3), there is the caveat that the contribu-tion from dust-enshrouded AGNs to the sub-mmcounts and background is currently uncertain, butprobably the AGNs do not dominate (Granato et

2

al. 1997; Almaini et al. 1999; Madau 1999).These discoveries demonstrate that in order to un-derstand the history of star formation in the uni-verse from observational data, one must have aunified picture that covers all wavelengths fromthe far-UV to the sub-mm. The UV and the far-IR are expecially important, since young stellarpopulations emit most of their radiation in therest-frame UV, but a significant fraction of this isdust reprocessed into the rest-frame far-IR.

On the theoretical side, it is now possible forthe first time to construct true ab initio models inwhich the galaxy formation itself and stellar emis-sion and dust absorption and emission are calcu-lated from first principles, based on physical mod-els, and avoiding observational parameterizationsfor various key ingredients (e.g. shape of the lumi-nosity function, dependence of dust temperatureon galaxy properties). These new models, whichprovide a unified treatment of emission from starsand dust, and predict the evolution of galaxy lu-minosities from the far-UV to the mm, are thesubject of this paper.

The effects of dust on galaxy luminosities atdifferent wavelengths have been included in someprevious galaxy evolution models, at various lev-els of sophistication, but mostly in the contextof backwards evolution models, where one tries toevolve back from observed galaxy properties atthe present day, in contrast to the semi-analyticalmodels, where one evolves forward from cosmo-logical initial conditions. In backwards evolutionmodels, one starts from the observed luminosityfunctions of different types of galaxy at the presentday, assumes a different star formation history foreach type, and calculates the luminosity evolutionfor each type, to predict what the galaxy popula-tion would have looked like in the past. Guider-doni & Rocca-Volmerange (1987) were the firstto include dust absorption in a model of this type,based on a 1D slab model for the star and dustdistribution, and calculating the dust content self-consistently from a chemical evolution model. Thesame treatment of dust was later used in the semi-analytical galaxy formation models of Lacey et al.(1993). In both cases, the models were used to cal-culate galaxy luminosities and number counts inthe UV and optical. Mazzei et al. (1992) were thefirst to try to model the evolution of stellar emis-sion and dust emission together in a consistent

framework based on stellar population synthesismodels and a physical calculation of dust absorp-tion. This model was then used by Franceschini etal. (1994) to calculate galaxy evolution and num-ber counts in bands from the optical through tothe far-IR, based on the backwards evolution ap-proach. However, these models still made a num-ber of simplifying assumptions (e.g. slab geometryfor disks), and set a number of present-day proper-ties of galaxies from observations (e.g. the opticaldepth of galactic disks, and the intensity of theradiation field heating the dust), rather than pre-dicting them. Simpler backwards evolution mod-els, where the luminosity evolution is parameter-ized as a simple function of redshift, have beenconsidered by, e.g. Pearson & Rowan-Robinson(1996).

Recently, dust absorption has been includedin several different semi-analytical models (Kauff-mann et al. 1999; Somerville & Primack 1999;Cole et al. 2000). The first two calculate dusteffects only for present-day galaxies, using a 1Dslab model, and taking the dust optical depthfrom observational measurements. On the otherhand, Cole et al. (2000) predict the dust opticaldepth and how it evolves, based on chemical evo-lution and a prediction of disk sizes, and use the3D disk+bulge radiative transfer models of Fer-rara et al. (1999) to calculation the dust atten-uation. The only previous semi-analytical modelto calculate dust emission as well as absorption isthat of Guiderdoni et al. (1998). However, thatmodel also has several limitations: the galaxy for-mation model does not include merging of eitherdark halos or visible galaxies, and the fraction ofstar formation occuring in bursts is simply an arbi-trary function; dust absorption is again modelledassuming a 1D slab geometry; and the dust tem-perature distribution is not predicted. Instead, thedust emission spectrum is modelled as the sum ofseveral components, whose temperatures and rel-ative strengths are chosen so as to reproduce theobserved correlations of IR colours with IR lumi-nosity found by IRAS.

The present paper represents a major advanceover this earlier work in terms of scope, physi-cal self-consistency and predictiveness. We com-bine the semi-analytical galaxy formation modelof Cole et al. (2000) with the stellar population+ dust model of Silva et al. (1998). The galaxy

3

formation model includes formation of dark halosthrough merging, cooling and collapse of gas inhalos to form disks, star formation in disks regu-lated by energy input from supernovae, merging ofdisk galaxies to form elliptical galaxies and bulges,bursts of star formation triggered by these merg-ers, predictions of the radii of disks and spheroids,and chemical enrichment of the stars and gas. Thestellar population + dust model includes a realis-tic 3D geometry, with a disk and bulge, two phasedust in clouds and in the diffuse ISM, star forma-tion in the clouds, radiative transfer of starlightthrough the dust distribution, a realistic dustgrain model including PAHs and quantum heat-ing of small grains, and a direct prediction of thedust temperature distribution at each point in thegalaxy based on a calculation of dust heating andcooling. The output is the luminosity and spec-trum of the stellar populations attenuated by dust,and of the dust emission from grains at a rangeof temperatures. From this, we can calculate thedistribution of galaxy properties at any redshift,including the complete spectrum of each galaxy inthe model from the far-UV to the sub-mm.

In this paper we compare the predicted proper-ties for local galaxies with a wide range of obser-vational data. A future paper will be devoted tohigh-z galaxies (Lacey et al. 2000). In Sections 2and 3 we describe, respectively, the galaxy for-mation model and the spectrophotometric model.Section 4 describes how we generate model galaxycatalogues for both normal and starburst galaxies.The comparison with observations (SEDs, extinc-tion properties, colors, etc.) is presented in Sec-tion 5 for spiral galaxies, and in Section 6 for star-bursts. The model luminosity functions at differ-ent wavelengths are compared with observationsin Section 7. Section 8 uses the models to predictthe relationship between the star formation rateand the luminosities in various UV and IR bands,and to assess the accuracies of these as star for-mation indicators. Section 9 presents a summaryand conclusions.

2. Semi-analytical galaxy formation model

We calculate the formation histories and globalproperties of galaxies using the semi-analyticalgalaxy formation model (GALFORM) of Cole etal. (2000), a development of that described in Cole

Table 1: Adopted values for GALFORM parame-ters.

CosmologyΩ0 0.3Λ0 0.7h 0.7Ωb 0.02Γ 0.19σ8 0.93

Star formation and feedbackǫ⋆disk 6.7 × 10−3

α⋆ -1.5αhot 2.0Vhot (km/s) 150.0

Stellar populationsIMF Kennicutt (1983)Υ 1.4p 0.02R 0.29

Mergers and burstsfellip 0.3ǫ⋆burst 0.5η 0.1

et al. (1994) and Baugh et al. (1998). The princi-ple of the model is to calculate the formation andevolution of dark matter halos starting from an as-sumed cosmology and initial spectrum of densityfluctuations, and then to calculate the evolution ofthe baryons (gas and stars) within these evolvinghalos using a set of simple, physically-motivatedrules to model gas cooling, star formation, super-nova feedback and galaxy mergers. We describehere only the main features of the model, and re-fer the reader to Cole et al. (2000) for more detailsand for a discussion of the effects of varying pa-rameters with respect to standard values given inTable 1.

(a) Cosmology: The cosmology is specified bythe present-day density parameter Ω0, cosmolog-ical constant Λ0, baryon fraction Ωb (all in unitsof the critical density) and the Hubble constantH0 = 100h km s−1Mpc−1. We assume a cold darkmatter (CDM) model, with the initial spectrumof density fluctuations having shape parameter Γand amplitude σ8 (the r.m.s. density fluctuationin a sphere of radius 8h−1Mpc).

4

(b) Halo evolution: Dark matter halos formthrough a process of hierarchical clustering, build-ing up through merging from smaller objects.At any cosmic epoch, we calculate the numberdensity of halos as a function of mass from thePress-Schechter (1974) formula. We then calcu-late halo merger histories, describing how a halohas formed, for a set of halos of different masses,using a Monte-Carlo algorithm based on the ex-tended Press-Schechter formalism. We generatemany different realizations of the merger historyfor each halo mass. We then follow the processof galaxy formation separately for each of theserealizations.

(c) Cooling and collapse of gas in halos:Diffuse gas is assumed to be shock-heated to thevirial temperature of the halo when it collapses,and to then cool radiatively out to a radius de-termined by the density profile of the gas and thehalo lifetime. The gas which cools collapses toform a rotationally supported disk, for which thehalf-mass radius rdisk is calculated assuming thatdark matter and associated gas are spun up bytidal torques, and that the angular momentum ofthe gas is conserved during the collapse. The gassupply by cooling is assumed to be continuous overthe lifetime of the halo.

(d) Star formation in disks: Stars form fromthe cold gas in the disk, at a rate

ψ = Mcold/τ⋆disk, (1)

where the star formation timescale is assumed tobe

τ⋆disk = ǫ−1⋆diskτdisk

(

Vdisk/200km s−1)α⋆

(2)

where Vdisk is the circular velocity at the half-massradius of the disk, and τdisk = rdisk/Vdisk is the dy-namical time. ǫ⋆disk is the fraction of gas convertedinto stars in one dynamical time, for a galaxy withcircular velocity Vdisk = 200km s−1. The scaling ofthe star formation timescale with dynamical timeis motivated by observations of star formation innearby galaxies (Kennicutt 1998), but modified toreproduce the observed dependence of gas fractionon luminosity.

(e) Supernova feedback in disks: The energyinput from supernovae is assumed to reheat gas inthe disk and eject it into the halo at a rate

Meject = βdiskψ, (3)

where for βdisk we assume

βdisk = (Vdisk/Vhot)−αhot (4)

Gas which has been ejected is assumed to be un-available for cooling until the halo has doubled inmass through merging. The motivation for thethis parameterization is that the rate of gas ejec-tion should be proportional to the rate of super-novae, and also depend on the escape velocity fromthe disk, which in turn is related to the circularvelocity. Our standard case αhot = 2 is equiva-lent to the assumption that a constant fraction ofthe Type II supernova energy goes into ejectinggas from the disk, if the escape velocity is propor-tional to Vdisk.

(f) Galaxy mergers and morphology: Thegalaxy morphology (i.e. whether it is a spiral orelliptical) is determined by merging. Followingthe merger of two halos, the largest pre-existinggalaxy is assumed to become the central galaxyin the new halo, while the other galaxies becomesatellite galaxies. The central galaxy can continueto grow a disk by cooling of gas from the halo. Thesatellite galaxies merge with the central galaxy ona timescale equal to that for dynamical friction tomake the orbits decay. The merger is classed asa major merger if the mass ratio of the satelliteto central galaxy exceeds a value fellip, and as aminor merger otherwise. In a major merger, anypre-existing stellar disks are destroyed, producinga stellar spheroid (elliptical galaxy or bulge), andany remaining cold gas is consumed in a burstof star formation. The star formation timescalein the burst is related to the dynamical time ofthe bulge as described below. The spheroid cangrow a new disk by cooling of halo gas. In a mi-nor merger, the stars from the satellite galaxy addto the bulge of the central galaxy, while the coldgas adds to the disk, but no burst is triggered.In either case, the half-mass radius rbulge of thespheroid produced in a merger is calculated usingan energy conservation argument. Galaxies areclassified into different morphological types basedon their bulge-to-disk luminosity ratios.

5

(g) Star formation and feedback duringbursts: As already mentioned, star formationbursts are assumed to be triggered by major merg-ers of galaxies. In Cole et al. (2000), these burstswere modelled in a very simple way, with theconversion of gas into stars being assumed to beinstantaneous, since the galaxy properties exam-ined there were not sensitive to the detailed timedependence. In this paper, we model the burstsin more detail. We assume that star formationduring bursts follows a law analogous to that forstar formation in disks:

ψ = Mcold/τ⋆burst, (5)

with star formation timescale

τ⋆burst = ǫ−1⋆burstτbulge (6)

where τbulge = rbulge/Vbulge is the dynamical timeof the spheroid formed in the merger, Vbulge be-ing the circular velocity at rbulge. As in Cole etal. (2000), feedback is modelled as in disks exceptwith Vbulge replacing Vdisk in eqn.(4), assuming thesame values for Vhot and αhot, giving a feedbackfactor βburst. Since we assume that no new gas issupplied by cooling during the burst, the star for-mation rate and cold gas mass decay during theburst as exp(−t/τe), where

τe = τ⋆burst/(1 −R+ βburst), (7)

and R is the recycled fraction, discussed below.The burst is assumed to occur in a region of half-mass radius rburst, where

rburst = η rbulge (8)

More details on the geometry assumed for star-bursts are given in § 3. For simplicity, the metallic-ity of the gas in the burst and of the stars formedduring the burst are taken to be constant, andequal to the mean metallicity of the stars formedduring the burst as calculated by the GALFORMmodel. The star formation in a burst is truncatedat a time 5τe after the burst began, i.e. after 99%of the gas in the burst has either been convertedinto stars or blown out of the galaxy by super-nova feedback. At this time, the remaining gasand dust in the burst region are assumed to bedispersed. Star formation then starts again in anormal galactic disk surrounding the bulge, if onehas formed by cooling of halo gas since the majormerger that triggered the burst.

(h) Chemical evolution: We assume that afraction 1/Υ of the mass formed into stars goesinto visible stars (0.1 < m < 125M⊙), while theremainder goes into brown dwarfs (m < 0.1M⊙).For visible stars we adopt a universal IMF, similarto that in the solar neighbourhood. In Cole et al.(2000) and in this paper, we use the form proposedby Kennicutt (1983), which is consistent with the“best estimate” of Scalo (1998):

dN/d lnm ∝ m−0.4 (m < 1M⊙)

∝ m−1.5 (m > 1M⊙) (9)

We use the instantaneous recycling approxima-tion to calculate the evolution of the abundanceof heavy elements of the cold gas (Zcold) and stars(Z⋆) in each galaxy, together with that of the hotgas in the halo (Zhot), including the effects of in-flows and outflows between the galaxy and halo.The chemical evolution depends on the recycledfraction R and the yield of heavy elements p.

(i) Stellar population synthesis and dust ex-tinction: In Cole et al. (2000), we calculatedthe luminosity evolution of each galaxy at differ-ent wavelengths using the stellar population syn-thesis models of Bruzual & Charlot (2000). Theeffects of dust extinction were calculated in a sim-ple way using the dust models of Ferrara et al.(1999), which assume a smooth (unclumped) dis-tribution for both the dust (in a disk) and stars(in a disk and a bulge). In the present paper, weuse instead the combined stellar population anddust model GRASIL (Silva et al. 1998) to calcu-late the galaxy luminosities and spectra includingboth extinction and emission by dust. The stel-lar population part of GRASIL is similar to theBruzual & Charlot model, as both are based onsimilar stellar evolution tracks and stellar spectra.The dust part of GRASIL is however considerablymore sophisticated than the Ferrara et al. mod-els, in that GRASIL allows for clumping of bothdust and stars, and calculates the grain heatingand emission as well as the extinction.

The parameters we have chosen for the GAL-FORM model are the same as those of the stan-dard ΛCDM model of Cole et al. (2000), apartfrom ǫ⋆burst and η describing the timescale and ra-dius of bursts, which were not considered in Cole

6

et al. . These parameters values are given in Ta-ble 1, and were obtained by comparing the modelto observations of nearby galaxies, without anyconsideration of the far-IR or UV properties. Werefer the reader to Cole et al. for a complete discus-sion of the effects of varying the ’old’ GALFORMparamaters, and for a systematic presentation ofthe influence of these parameters on the optical-NIR properties of galaxies (LFs, Tully-Fisher re-lation, disk sizes, morphology, gas content, metal-licity, M/L ratios and colours). We only recallhere the main observational constraints used tofix each of these parameters: ǫ⋆disk - gas fractionof L⋆ galaxies; α⋆ - variation of gas fraction withluminosity; αhot - faint end of LF and Tully-Fisherrelation ; Vhot - faint end of LF and sizes of low-L spirals; IMF - observations of solar neighbour-hood; Υ - L⋆ in LF; p - metallicity of L⋆ ellipticals;fellip - morphological mix of L⋆ galaxies.

Values for ǫ⋆burst and η are instead obtainedlater in this paper by detailed comparison of theresults of the combined GALFORM+GRASILmodels with observed properties of bursting galax-ies.

3. The Stellar Population and Dust Model

Far-UV to mm SEDs of model galaxies arecalculated using the GRASIL code (Silva et al.1998), which follows both the evolution of stel-lar populations and absorption and emission bydust. GRASIL calculates the following: (i) emis-sion from stellar populations; (ii) radiative trans-fer of starlight through the dust distribution; (iii)heating and thermal equilibrium of dust grains (orthermal fluctuations for small ones); and (iv) emis-sion by dust grains.

3.1. Stellar Population Model

The single stellar population (SSP) libraries in-cluded in GRASIL are based on the Padova stellarmodels and cover a large range in age and metal-licity. They include the effects of dusty envelopesaround AGB stars (Bressan et al. 1998). The ageand metallicity distribution of a composite stellarpopulation is specified by the birthrate functionΨ(t, Z), where Ψ(t, Z) dt dZ gives the mass of starsthat were formed in the time interval (t, t+dt) withmetallicities in the range (Z,Z + dZ). The SEDfor the composite stellar population at time t is

then obtained using

Lλ(t) =

∫ t

0

dt′∫ 1

0

dZ lλ(t− t′, Z)Ψ(t′, Z) (10)

where lλ(τ, Z) is the SED of a SSP of age τ andmetallicity Z for the assumed IMF.

For our semi-analytical galaxy formation model,Ψ(t, Z) is calculated for each galaxy by summingover all the progenitor galaxies which have mergedto produce that galaxy, separately for the disk andbulge components. The progenitor galaxies eachhad their own star formation and chemical history,so that the composite Ψ(t, Z) obtained in generalhas a broad distribution of metallicity at each eachage, i.e. there is no unique age-metallicity relationZ(t).

Fig. 1.— Sketch of geometry of stars and dust inthe GRASIL model.

3.2. Dust Model

GRASIL computes the radiative transfer ofstarlight, the heating of dust grains, and the emis-sion from these grains with a self-consistent calcu-lation of the distribution of grain temperatures, foran assumed geometrical distribution of the starsand dust, and a specific grain model. The dustis divided into two components, dense molecularclouds and diffuse cirrus in a disk. Stars form in-side clouds and progressively leak out.

7

Table 2: Adopted values for the adjustableGRASIL parameters. Note that the results de-pend on the ratio Mc/r

2c rather than on the two

quantities individually (see § 3.2).hz/hR(disk) 0.1hz/hR(burst) 0.5hz(dust)/hz(stars) 1fmc 0.25Mc/r

2c 106M⊙/(16pc)2

tesc(disk) 2Myrtesc(burst) 10Myr

The details are given in Silva et al. (1998),but for convenience we summarize the main fea-tures here, focusing on the modifications intro-duced for the purposes of this application. ThoseGRASIL parameters which are not provided byGALFORM, and are in this sense additionaladjustable parameters of the combined GAL-FORM+GRASIL semi-analytic modelling, arelisted in Table 2, together with the adopted valuesfor our standard case. See Fig. 1 for a sketch ofthe geometry of our model.

(a) Geometry of stars: The stars are in twocomponents (Silva et al. (1998) considered onlypure disk and pure bulge systems): (i) a spher-ical bulge with an analytic King model profile,ρ ∝ (r2 + r2c )−3/2 for r < rt, with concentra-tion parameter log(rt/rc) = 2.2; (ii) a disk witha radially and vertically exponential profile, scale-length hR and scaleheight hz. As described in § 2,the disk and bulge masses, Mdisk and Mbulge, andhalf-mass radii, rdisk and rbulge, for any galaxyare predicted by the galaxy formation model. Thebulge core radius is related to the bulge half-massradius by rc = rbulge/14.6, while the disk scale-length hR is related to the disk half-mass radius byhR = rdisk/1.68. The star formation histories arealso calculated separately for the disk and bulge byGALFORM. However, the disk axial ratio hz/hR

is a free parameter of the GRASIL model.

As partially anticipated in § 2, in galaxies un-dergoing bursts, the burst star formation, as wellas the gas and dust, are assumed to be in an ex-ponential disk, but with half-mass radius rburst =η rbulge rather than rdisk. The axial ratio hz/hR

of the burst region is allowed to be different from

that for disks in non-bursting galaxies. The starswhich were formerly in the disks of the galaxies be-fore the galaxy merger which triggered the burstare assumed to become part of the bulge followingthe merger.

(b) Geometry of gas and dust: The gas anddust are in an exponential disk, with the same ra-dial scalelength as the disk stars (either rdisk fornormal galaxies or rburst = η rbulge for starbursts),but in general with a different scaleheight, so thathz(dust)/hz(stars) is a free parameter. The gasand dust are in two components within the disk,molecular clouds and the diffuse ISM. The lattercorresponds to the cirrus dust. The total gas massMcold and its metallicity Zcold are calculated bythe galaxy formation model, but the fraction ofthe gas in clouds, fmc, and the cloud mass Mc andradius rc are free parameters of GRASIL, thoughthe results actually depend only on their combina-tion Mc/r

2c , which determines, together with the

dust/gas ratio (see point (d) below), the opticaldepth of the clouds (Silva et al. 1998).

(c) Young stars and molecular clouds: Starsare assumed to form inside the molecular clouds,and then to escape on a timescale tesc. Specifically,the fraction of stars still inside clouds at time tafter they formed is assumed to be given by

F (t) = 1 (t < tesc)

= 2 − t/tesc (tesc < t < 2tesc)

= 0 (t > 2tesc) (11)

We allow tesc to take different values in normaldisks and in bursts, in keeping with the resultsof Silva et al. (1998). Indeed, given the smallsize scale and the intensity of the star formationactivity in bursts, it is conceivable that the star-forming environment is quite different from thatin normal spiral galaxies (see also § 3.3).

(d) Dust grain model: The dust is assumed toconsist of a mixture of graphite and silicate grainsand polycyclic aromatic hydrocarbon molecules(PAHs), each with a distribution of grain sizes.Absorption and emission properties are calculatedfor each grain composition and size. The grainmix and size distribution were chosen by Silva

8

et al. (1998) to match the extinction and emis-sivity properties of the local ISM, and are notvaried here. The dust/gas ratio δ in the cloudsand diffuse ISM is assumed to be proportional tothe gas metallicity, with a value δ = 1/110 forZ = Z⊙ = 0.02. Thus, the total dust mass in agalaxy scales as Mdust ∝ ZcoldMcold.

(e) Radiative transfer, dust heating andre-emission: The luminosities of the differentstellar components (bulge stars, disk stars, andyoung stars still in clouds) are calculated usingthe population synthesis model described above.The GRASIL code then calculates the radiativetransfer of the starlight through the dust distri-bution. Whilst in molecular clouds a full radia-tive transfer calculation is performed, the effectsof scattering by diffuse dust are included only ap-proximately, by assuming that the effective opticaldepth for absorption is related to the true absorp-tion and scattering optical depths τabs and τscat

by τabs,eff =√

τabs(τabs + τscat). Thus the dust-attenuated stellar radiation field can be calculatedat any point inside or outside the galaxy. ThenGRASIL calculates for each point in the galaxythe absorption of radiation, thermal balance andre-emission for each grain composition and size.Thus, the distribution of grain temperatures is cal-culated self-consistently for the assumed geometryof the stars and dust, including the effects of tem-perature fluctuations for small grains. The finalgalaxy SED Lλ is obtained by adding the con-tributions from the starlight (attenuated by dust)and from the dust re-emission, and depends on theinclination angle at which the galaxy is viewed.Emission from dust in the envelopes of AGB starsis included in the SSPs.

Our computations allow us to calculate theamount of energy emitted in the PAH bands, buttheoretical predictions of the detailed shapes of theemission features are rather uncertain. Thereforewe use the Lorentzian analytical fits to the ob-served PAH profiles for the Ophiuchus molecularcloud from Boulanger et al. (1998).

3.3. Choice of GRASIL adjustable param-eters and new GALFORM parame-ters

The values of GRASIL parameters (Table 2)not provided by GALFORM have been based ona variety of observational data for galaxies in thelocal universe. For some of them, the choices weremade by trying to match model predictions to theobservational data, as is discussed in more detailin the relevant sections of this paper. We nowsummarize the reasons for these choices and forthose of the two GALFORM parameters (ǫ⋆burst

and η) not considered in Cole et al. (2000)

(a) ǫ⋆burst: this is chosen mainly so as to repro-duce the bright end of the IR luminosity function,which is dominated by bursts triggered by galaxymergers (§7.4). A secondary (weak) constraint isto reproduce the relation between LIR/LUV andtotal luminosity or UV slope β observed for star-burst nuclei (§6.2). The value controls both theluminosity and lifetime (and thus number density)of starbursts.

(b) η = rburst/rbulge: the choice of this is mainlybased on the observational fact that starburst ac-tivity is usually confined to a nuclear region witha size much smaller than the galaxy as a whole,by about one order of magnitude (e.g. Sanders& Mirabel 1996, and references therein). Forinstance, in Arp 220 most of the molecular gasis found in the central ∼ 300pc (Scoville et al.1997), and the mid–IR light is dominated by moreor less the same region (Keto et al. 1992), whilethe half-light radius for the old stellar populationis ∼ 3kpc (Wright et al. 1990). The value ofη controls the amount of extinction of starlightfrom bursts by the diffuse ISM, which however isusually overwhelmed (in bursts) by extinction inmolecular clouds (see §6.4). Therefore our resultsare not very sensitive to the precise choice of thisparameter, nor to the value of hz/hR in starbursts(discussed below).

(c) hz/hR: for normal disks, we choose a valueof 0.1, consistent with observations of the stel-lar light distributions in edge-on spiral galaxies(e.g. Xilouris et al. 1999). It is also the typ-ical value used by Silva et al. (1998) to fit the

9

SEDs of spiral galaxies. This value is also im-portant, and the adopted value turns out to besuitable, to match the observed difference in ex-tinction between spiral galaxies seen edge-on andface-on (§5.2). Apart from this test, most pre-dicted properties are not very sensitive to hz/hR.The choice of hz/hR = 0.5 for starbursts is basedon general observational indications that they areonly moderately flattened.

(d) hz(dust)/hz(stars): this parameter has asignificant effect on how much starlight is ab-sorbed in the diffuse medium. From observationsof our own galaxy it is known that the scale-height of stars increases with the age of the stellarpopulation, so that there is no unique value forhz(dust)/hz(stars). The scaleheight of the gas iscomparable to that of the youngest stars. Sincewe are particularly interested in having a real-istic estimate of the extinction in the UV, bothbecause it is strongest there and because this isan important source for dust heating, we choosehz(dust)/hz(stars) = 1 to match what is seen forthe young stars.

(e) fmc: this can be estimated observationallyfrom the ratio of molecular to atomic hydrogen ingalaxies, since in normal spiral galaxies most ofthe hydrogen in molecular clouds is in H2, whilemost of the intercloud medium is atomic HI. Ouradopted fmc implies a ratio H2/HI similar to thetypical one for L⋆ spirals found by Sage (1993).Larger values reduce the extinction in the diffuseISM and produce a somewhat colder molecularclouds emission, but our results are in general notsignificantly affected as long as we keep fmc in therange 0.2–0.8.

(f) Mc,rc : as already remarked (§ 3.2) the pre-dicted SEDs depend on the ratio Mc/r

2c , rather

than Mc and rc separately. Thus Mc has beenchosen to match typical giant molecular clouds inour own and nearby galaxies, while rc is chosenbased on the results of Silva et al. (1998), whotuned Mc/r

2c to fit the SEDs of starburst galaxies

in particular. The resulting value for rc is consis-tent with direct measurements of cloud radii.

(g) tesc: this is a very important parameter inthe model, since it is this that mainly controls howmuch of the radiation from young stellar popula-tions is absorbed by dust. Silva et al. (1998) foundfrom detailed fits to 3 nearby spirals values of 2.5,3 and 8 Myr. For normal spirals, we favor a valueof 2 Myr, close to the lower limit of this range,rather than the average 5 Myr. Although the lat-ter provides an equally good overall fit to the LFs(somewhat better for IRAS colors and LFs, § 7.4,but somewhat worse for the UV LF), the formeris more consistent with the massive star censusin our own and nearby galaxies, which suggeststhat the time for which the stars are obscured bydust is about the 20 % of the total lifetime forthe brightest stars, above say 30 M⊙, whose life-time is around 6 Myr. For starbursts, the valuewe choose is based mainly on the comparison withproperties of UV-bright starbursts in §6.2. Thisleads us to a value closer to that of normal spiralsthan the values tesc = 20-60 Myr found by Silvaet al. (1998) from fitting 3 nearby starbursts, andsuggests that the starburst galaxies used by Silvaet al. may be not representative of the whole popu-lation. The difference could also be due in part tothe more complex geometry adopted in this paperfor starburst galaxies.

4. Generation of model galaxy catalogues

The GALFORM model is run for a set of darkmatter halos covering a large range in mass, andgenerates a catalogue of model galaxies, includ-ing information about the following properties foreach galaxy at the chosen epoch: stellar massesMdisk, Mbulge, and half mass radii rdisk and rbulge

of the disk and bulge, mass Mcold and metallicityZcold of gas in the disk, and the star formationhistories Ψ(t, Z) of the disk and bulge separately,including both star formation in disks and duringbursts, and specifying the metallicity distributionof the stars of each age. In addition, each galaxyhas a weight or number density n, such that thatgalaxy should appear N = nV times in an averagevolume of the universe V .

10

The GALFORM code outputs all the galaxiesfor each different halo that is calculated, down toa minimum mass controlled by the mass resolu-tion of the merger tree. In practice, this meansthat the model catalogue contains many more lowmass galaxies than high mass galaxies. Runningthe GRASIL code on every galaxy in the originalcatalogue is neither feasible (because of computertime) nor necessary. We therefore select a subsetof galaxies from the catalogue chosen to samplegalaxies more evenly in mass, and redistribute theweights to give the same total number density ineach mass range. The GRASIL code is then run oneach galaxy in this reduced catalogue to give theSED Lλ including both stellar emission and dustabsorption and emission, and statistical proper-ties (e.g. luminosity functions) are then calculatedmaking use of these weights. In fact, we calculate2 samples of galaxies, a “normal” sample and a“burst” sample, as follows:

(a) Normal galaxies: By “normal” galaxies, wehere simply mean galaxies not selected to have hada recent burst. From the parent GALFORM cat-alogue, we select a sample with equal numbers ofgalaxies in equal bins in logM∗, M∗ being the to-tal stellar mass of the galaxy. Within each massbin, galaxies are randomly selected (allowing formultiple selection of the same galaxy) with prob-ability proportional to their weight n. The se-lected galaxies are then assigned new weights ni,such that each galaxy within the same bin hasthe same weight (multiply selected galaxies beingcounted as separate objects), and that the sum ofthe weights (i.e. number densities) within a bin isthe same as in the parent catalogue. We have usedbins with ∆ logM∗ = 0.3 and about 40 galaxiesper bin.

(b) Burst galaxies: By “burst” galaxies we meangalaxies which have had a burst in the recentpast, at whatever redshift we are looking. Burstshave short durations compared to the age of theuniverse, so the fraction of galaxies undergoing aburst at any one time is very small, but they canbe very luminous, and so may dominate the galaxyluminosity function at the highest luminosities. Inpractice, our “normal galaxy” catalogue containstoo few galaxies in total to provide a represen-tative sample of galaxies seen during their burstphase. Rather than use a greatly enlarged “nor-mal galaxy” sample, it is more efficient to calcu-

late a separate sample of “burst” galaxies, as fol-lows: for a redshift z, we choose a subsample ofgalaxies which have had bursts during the time in-terval t(z) > t > t(z) − T , where t(z) is the ageof the universe at redshift z, with equal numbersof galaxies in equal bins in logMburst, Mburst be-ing the mass of stars formed in the most recentburst. The galaxies are assigned new weights ni

analogously to the case of normal galaxies, butnow conserving the total number density in binsof Mburst for the galaxies which have had burstsmore recently than T . For each burst galaxy, wethen run GRASIL to calculate the total galaxy lu-minosity at a set of times after the start of theburst, chosen to sample all phases of the burstevolution, including the highest luminosity phaseof short duration. If T << t(z), then the rate ofbursts per unit volume during the time interval Tcan be taken as constant. Then, for the ith galaxyin the jth phase in the burst evolution that lastsa time ∆tj , the number density of galaxies thatshould be found in this phase is

nij = ni

(

∆tjT

)

(12)

These weights can then be used to calculate sta-tistical properties such as luminosity functions.When combining the “normal” and “burst” galaxysamples, the normal galaxies with bursts morerecent than T are explicitly excluded, to avoidstatistical double-counting. In practice, we choseT = t(z)/20 at all z, with bins ∆ logMburst = 0.3,around 10 galaxies per bin, and around 10 out-put times per galaxy, for 0 < t − tburst . 100τe.For many calculations of statistical distributions,we then interpolate between these output times tohave more burst phases.

5. Properties of spiral galaxies

In this section, we test the model predictionsfor disk galaxies against observed emission and ab-sorption properties of nearby spirals.

5.1. SEDs of face-on spirals

We compared the predicted near-UV to far-IRSEDs of our model galaxies with the broad-bandSEDs of a complete sample of nearby spiral galax-ies (de Jong & van der Kruit 1994), consistingof a diameter-limited sample of 86 nearly face-on,

11

Fig. 2.— Upper panels: SEDs (normalized to the K-band) of a representative sample of spiral galaxies fromthe models. Middle panels: Comparison of the models with the observed SEDs of a sample of face-on spiralgalaxies obtained by de Jong & van der Kruit (1994). The median and the 10% and 90% percentiles of themodel SEDs are plotted as lines. Lower panel: comparisons between observed SEDs of individual galaxiesfrom the sample and selected model SEDs. The left-hand panels are for galaxies in the luminosity range−23 > MK − 5 log h > −24, and the right-hand panels for −22 > MK − 5 logh > −23.

12

disk-dominated galaxies. de Jong & van der Kruitmeasured fluxes of these galaxies in the BVRIHKbands. We have supplemented these with U-band magnitudes from the literature and IRAS12, 25, 60, 100µm fluxes from Saunders (1997).

We considered only those model galaxies withbulge to total light ratio B/T ≤ 0.5 in the B-band,corresponding to the range of types in the de Jongsample (e.g. Simien & de Vaucouleurs 1986).From Figure 2 it is apparent that the models re-produce the observed spectral trends reasonablywell. This is particularly impressive since the ra-tio between the infrared and the optical-UV spansmore than one order of magnitude, both in the ob-served and in the theoretical SEDs. The predictedinfrared emission peaks at wavelengths somewhatlarger than those sampled by IRAS, in agreementwith recent ISO observations (e.g. Alton et al.1998). The emission in the mid-infrared is domi-nated by PAH molecular bands.

Figure 3 shows the effects on typical SEDs offactor 2 variations in the molecular cloud fractionfmc, their mass Mc (i.e. their optical depth, hav-ing fixed the radius) and the escape timescale tesc.The effects are mostly confined to the mid-IR be-tween 8 and 40 µm and in the UV below 0.4 µm.In these spectral regions, the predicted flux maychange by up to a factor ∼ 2, while the effectsare almost negligible elsewhere. In the mid-IRthe most important parameter is the cloud opticaldepth, while in the UV the effects of tesc dominate.

Fig. 3.— Effects on typical SEDs of factor 2 vari-ations of fmc, Mc (effectively, the cloud opticaldepth) and tesc. In each case only one parameteris varied with respect to the standard model.

Fig. 4.— Predicted dependence of dust opti-cal depth on luminosity for spiral galaxies in themodel. τV 0 is the central face-on extinction opticaldepth in the V-band for the diffuse dust compo-nent in the disk. The line shows the median, andthe error bars show the 10% and 90% percentiles.Model galaxies are selected to have B/T < 0.4 inthe B-band.

13

5.2. The global extinction in spiral galax-ies

The models predict that the extinction ingalaxy disks should increase strongly with galaxyluminosity, as shown in Figure 4. Clearly, in com-paring predictions of dust extinction with observa-tions, one must be careful to specify the luminosityof the objects concerned.

There have been many attempts to measure orobservationally constrain the total dust extinctionin galaxy disks, using a variety of techniques: theinclination dependence of magnitudes or colours(e.g. de Vaucouleurs et al. 1991; Giovanelli et al.1995), surface brightness distributions in edge-ongalaxies (e.g. Kylafis & Bahcall 1987), colourgradients in face-on disks (e.g. de Jong 1996c),and the ratio of far-IR to UV luminosities (e.g. Xu& Buat 1995; Buat & Xu 1996; Wang & Heckman1996). In general, different techniques have givensomewhat different answers.

Xilouris et al. (1999) estimate dust extinctionsby fitting detailed models of the star and dust dis-tributions to the observed surface brightness dis-tributions of edge-on spiral galaxies. Their dustmodels include scattering. From their six Sb-Scspirals with luminosities in the range −17.5 >MB − 5 log h > −19.0, we obtain a median cen-tral face-on extinction optical depth τV 0 = 0.6. Incomparison, for edge-on galaxies in our model inthe same luminosity range (after extinction), andwith 0.1 < B/T < 0.3 in the B-band, we finda median value τV 0 = 2.2, which is significantlylarger. There could be several reasons for this dif-ference between the models and the observations:there may be a problem with the Xilouris et al.method for deriving τV 0 from the observations, orthe Xilouris et al. sample may not be representa-tive, or the problem might be with our assumptionthat the dust and stars have the same exponen-tial scalelength. The extinction-inclination obser-vational test discussed next implies extinctions foredge-on galaxies in this luminosity range which areat least as large as those predicted by our model.

We considered the dependence of the net ex-tinction on the inclination angle at which a galaxyis viewed. This has been studied in many pa-pers using different methods, most recently byTully et al. (1998), who also summarize the re-sults from the earlier studies. Tully et al. mea-

sure the dependence of B −K, R−K and I −Kcolours on galaxy inclination at a given K-bandluminosity, the K-band being chosen to minimizeextinction effects. They have a complete sam-ple of spirals covering a large range in luminos-ity, −18.5 & MK − 5 logh & −24.5. They finda strong luminosity dependence, with a differencein B-band extinction between edge-on and face-ongalaxies of about 2 mag for the brightest galaxies,and negligible for the faintest ones.

Tully et al. (1998) follow the usual practice andparameterize the extinction relative to that for thegalaxy seen face-on as

Ai−0λ ≡ mλ(i) −mλ(0) = γλ log(a/b) (13)

where γλ is a function of the passband. The axialratio a/b is assumed to be related to the inclinationangle i by

cos i =

√

(b/a)2 − q2

1 − q2(14)

where i = 0 for a face-on system, and q is the axialratio of a galaxy seen edge-on.

The models are compared with observations inFig. 5. We use equation (14) to convert from themodel inclination angle to the axial ratio, assum-ing q = 0.1, which is the ratio hz/hR adopted inour galaxy models. We considered model galaxiescorresponding to the morphological types Sa-Scd,and four ranges in K-band luminosity, correspond-ing to the ranges chosen by Tully et al. (1998),indicated by the different symbols in the figure.

The three lines in the figure correspond to dif-ferent values of the slope γB. The model galax-ies approximately follow the linear dependence onlog(a/b) (equation (13)), but with slopes γB thatare somewhat shallower, at any given luminosity,than those observationally inferred by Tully et al.(1998). For instance, for the luminosity range−23.0 < MK − 5 logh < −22.0, our models fol-low an average slope 〈γB〉 ≈ 0.9, while Tully et al.find γB = 1.1 ± 0.5, after allowing for the K-bandextinction. The slope predicted by our models de-pends on the value chosen for the parameter hz/hR

(see also § 3.3). We have checked that increasinghz/hR from our adopted value of 0.1 decreases theslope, while reducing it does not increase the slopesignificantly.

14

Fig. 5.— Dependence of extinction on inclinationangle predicted by our galaxy models. The quan-tity Ai−0

B = MB(i)−MB(0) is plotted against theapparent axial ratio a/b, for the Sa-Scd models(with 0.04 < B/T < 0.4 in the B-band). Thegalaxies are plotted with different symbols in fourK-band luminosity ranges, as detailed in the fig-ure. The same galaxies are plotted seen at differ-ent inclinations. For clarity, small horizontal off-sets have been applied to the galaxies in differentluminosity ranges. Three lines are plotted corre-sponding to equation (13) with the slopes γB =0.5, 1 and 1.5.

The agreement for the inclination test is any-way acceptable. Part of the discrepancy could bedue to our simplified treatment of scattering bythe diffuse dust (§ 3.2). In our models, the abso-lute extinction is often dominated by the molecu-lar clouds, but the difference between the face–onand the edge–on extinction is entirely due to thecirrus. Comparisons of our model with that of Fer-rara et al. (1999), where the treatment of scatter-ing is more accurate, show that for the brightestobjects this effect can account for about 0.1-0.2mag of the differential extinction in the B band.

6. Properties of starburst galaxies

Starburst galaxies are broadly defined as galax-ies in which the current star formation rate is much

greater than its time-averaged value, and the starformation timescale correspondingly much shorterthan the age of the universe. This definition in-cludes objects with a wide range of properties,from bursting dwarf irregular galaxies (e.g. Thuan& Martin 1981) to the ultra-luminous IR galaxies(ULIGs) found by IRAS (e.g. Sanders & Mirabel1996). In practice, a large variety of observationalcriteria have been applied to select samples of star-burst galaxies, ranging from optical morphologiesand spectra (e.g. Balzano 1983) to IR colours andluminosities (e.g. Armus et al. 1990; Lehnert &Heckman 1995). In our galaxy formation model,bursts are assumed to occur following major merg-ers of galaxies, producing elliptical galaxies fromdisk galaxies. For the ultra-luminous IR galax-ies, the link between the starburst activity andgalaxy mergers is clearly established (e.g. Sanders& Mirabel 1996), while for low-luminosity star-bursts, additional triggering mechanisms probablyoperate, which are not included in our model. Inthis section we will compare the properties of star-bursts predicted by our model with those of vari-ous observational samples.

15

6.1. Properties of starbursts in the model

Figure 6 shows how various properties of thebursts in our model vary with the stellar mass ofthe galaxy, Mstar, after completion of the burst.The total mass of new stars formed in the burst,Mburst, is seen to increase with the galaxy mass,with the fraction of stars formed in the burst be-ing typically between ∼ 1% and ∼ 50%. An ex-ception to this trend is the group of points inthe lower right corner of Fig.6a corresponding tosmall bursts occuring in large galaxies. Thesesmall bursts are produced by mergers betweengas-poor elliptical galaxies. The main trend inpanel (a) is produced by mergers between diskgalaxies containing significant fractions of gas,and these dominate the statistics at all burstmasses. The star formation rate during the burstis (Mburst/τe) exp(−t/τe), with t measured fromthe start of the burst. The peak star formationrate is thus Mburst/τe, and occurs at the begin-ning of the burst. This peak SFR is seen alsoto increase with the host galaxy mass. The half-mass radius rburst and exponential decay time τeof the burst are assumed to scale with the half-mass radius and dynamical timescale of the hostgalaxy, and also increase with galaxy mass. Largebursts, with Mburst ∼ 1010h−1M⊙, are predictedto occur in galaxies with Mstar ∼ 1011M⊙, andto have radii rburst ∼ 0.5h−1kpc, star formationtimescales τe ∼ 5 × 107yr, and peak star forma-tion rates ∼ 200h−1M⊙yr−1. These are similarproperties to those inferred observationally for theULIGS (e.g. Sanders & Mirabel 1996).

6.2. Properties of UV-bright starbursts

A large amount of work has been done on sam-ples of UV-bright starbursts selected from the cat-alogue of UV spectra of star-forming galaxies ofKinney et al. (1993). Various correlations havebeen found, for instance between the bolometricluminosity, the UV/IR ratio, the slope of the UVcontinuum and the metallicity (e.g. Meurer et al.1995; Heckman et al. 1998). In this section, wecompare the properties of our model starburstswith some of this observational data.

The observational sample that we use for ourcomparison is that of Heckman et al. (1998),who selected 45 starburst and star-forming galax-ies from the original atlas of Kinney et al. . The

criteria for a starburst galaxy to appear in theKinney et al. catalogue are (a) that it has beenpreviously classified as a starburst based on op-tical data, usually meaning that it has a com-pact optical morphology and strong optical emis-sion lines (but no AGN activity) (e.g. Balzano1983); and (b) that it has been observed by IUEand has a high enough surface brightness withinthe IUE aperture to produce a reasonable qualityUV spectrum. The catalogue is not in any sensestatistically complete. The starburst activity inthese galaxies is generally confined to the centralregions. (The galaxies have mostly been selectedso that the starburst activity fits within the IUEaperture, 20” × 10”, while the optical diametersof the underlying galaxies are typically a few ar-cminutes.)

For the galaxies in their sample, Heckman etal. (1998) measured a UV luminosity LUV ≡λLλ(1900A) and mean continuum slope β between1250 and 1850 A (defined by Lλ ∝ λβ) from IUEspectra, and a far-IR luminosity LFIR from IRASmeasurements. Heckman et al. use the definitionof LFIR from Helou et al. (1988), which can beexpressed as in terms of the luminosities in the 60and 100 µm IRAS bands as

LFIR = 0.65νLν(60) + 0.42νLν(100) (15)

LFIR provides an estimate of the 40 − 120µmluminosity. The quantity LFIR + LUV is simi-lar to the bolometric luminosity in the case ofstarbursts, where most of the radiation is emit-ted in either the UV or the FIR. The galaxiesin the Heckman et al. sample cover the rangeLFIR + LUV ∼ 108 − 1011L⊙.

The evolutionary tracks of a selection ofmodel starbursts, with burst masses covering therange Mburst ∼ 107 − 1010M⊙, in LFIR + LUV ,LFIR/LUV and β are shown in Figure 7, togetherwith observational data for the Heckman et al.sample. We have calculated these quantities fromthe model SEDs to match the way they are cal-culated from the observational data. The bolo-metric luminosities of the model bursts, as mea-sured by LFIR + LUV , peak soon after the startof the burst, following which they evolve towardssmaller values. At the same time, the amount ofdust reprocessing of the radiation, as measured byLFIR/LUV , also decreases. This results from twoeffects: the escape of young stars from the dense

16

Fig. 6.— Properties of starbursts in the model at z = 0. (a) The mass of new stars formed in the burst,Mburst vs the stellar mass of the galaxy after the burst, Mstar. (b) The peak star formation rate in the burstMburst/τe vs Mstar. (c) The half-mass radius of the burst rburst vs Mstar. (d) The exponential decay time ofthe burst τe vs Mstar.

molecular clouds, and the decrease in the opticaldepth of the diffuse dust component as the gas inthe burst is consumed. The UV slope β initiallyevolves towards more negative values, i.e. bluer,as the net dust opacity falls. However, as the rateof formation of new stars drops and the dominantstellar population becomes older, the intrinsic un-absorbed stellar spectrum becomes redder, so the

evolution in β reverses, the models becoming red-der with time even though the dust attenuationis falling. This happens after 20 − 30Myr, con-trolled mainly by the stellar evolution timescale.As long as the evolution in β is dominated by thedeclining dust opacity, the models stay close tothe locus of observed points in the LFIR +LUV vsLFIR/LUV and LFIR/LUV vs β panels, but when

17

Fig. 7.— Upper panel: LFIR + LUV (roughlycorresponding to the bolometric luminosity) inh−2erg s−1 versus the ratio of the IR to the UVluminosity LFIR/LUV . Lower panel: the rela-tion between LFIR/LUV and the slope β of theUV continuum. In each panel, the filled sym-bols show observed local starbursts from Heck-man et al. (1998), while the lines show the evo-lutionary tracks of five different model starbursts.The bursts start towards the upper right part ofeach panel, and evolve towards the lower left (toppanel) or lower right (bottom panel). The crossesmark the age since the start of the burst in stepsof 5 Myr.

the intrinsic stellar spectrum starts to redden withage, the models move away from the observed lo-cus. This is not in itself in contradiction withobservations, since there are selection effects inthe observational sample, as we discuss below.

The burst evolution involves the interplay be-tween two timescales, the lifetime of massive stars,∼ 107yr, and the exponential decay time τe of thestar formation rate and gas mass in the burst. Thelatter varies with burst mass, being larger than thestellar evolution timescale for large bursts, andcomparable for small bursts, as shown in Fig 6.The model starbursts begin their evolution with alarge infrared excess and a flat UV slope (upperand lower panel of Fig. 7). Fainter bursts, which

Fig. 8.— Same as in Fig. 7 but for a random sam-ple of model bursts (crosses), as described in thetext. The filled pentagons are the same observa-tional data from Heckman et al. (1998).

have lower gas column densities and are on averagealso more metal poor, quickly exhaust their gascontent and evolve toward a low infrared excessand a negative UV slope, along the locus definedby observations (lower panel of Fig.7). Conversely,brighter bursts, having larger gas column densitiesand higher metallicities, remain highly enshroudedby dust until, after a few tens of Myr, the dom-inant stellar population has become intrinsicallyold. Their UV continuum slopes always remainflat, at the beginning because of reddening and atlater times because of age.

In summary, the model bursts lie close to theregion occupied by observed bursts in Fig. 7 aslong as the stellar population is young, in the senseof the UV light being dominated by very massivestars. The position of bursts along the observa-tional locus is then determined mostly by the netdust opacity in the UV, in agreement with theinterpretation of Meurer et al. (1995) and Heck-man et al. (1998). This in turn depends both onthe initial gas mass, radius and metallicity of theburst, and on its evolutionary stage.

18

A detailed comparison with the Heckman et al.observations would require us to construct a mockcatalogue of model starbursts obeying the sameselection criteria as the observed sample. Unfor-tunately, the observational selection criteria arerather ill-defined. In addition, one of the selectioncriteria is the presence of strong HII region emis-sion lines, and the GRASIL code at present doesnot calculate these emission line properties. In-stead, we simply select starbursts with ages sincethe start of the burst less than tmax, to accountroughly for the effect that as soon as most of themassive stars have evolved away, the galaxy willno longer produce strong emission lines, and so nolonger be classified as a burst in the observationalsample. Fig. 8 shows the resulting distribution ofpoints for the choice tmax = 50Myr. The modelstarbursts are seen to follow similar relations tothe observational sample. The results do not de-pend sensitively on the choice of tmax.

Several parameters may in principle affect thethe spectral properties of a model starburst galaxyand therefore the location of our models in theabove plots, but the most critical are the ratiobetween the star formation timescale and the dy-namical time (ǫ−1

⋆burst = τ⋆burst/τbulge), and the es-cape time (tesc) for newly born stars to escapefrom their parental clouds. The former affectsthe bolometric luminosity, which is almost di-rectly proportional to the star formation rate, andthus inversely proportional to the star formationtimescale. The latter affects the fraction of lightabsorbed inside clouds, and so may affect both theslope of the UV spectrum and the ratio betweenthe IR and UV luminosities. The distribution ofmodel points in Fig. 8 can therefore be used toconstrain the values of ǫ⋆burst and tesc. However,we found that changes in either of these parame-ters by a factor ∼ 2 either way would only slightlyworsen the match with observations.

6.3. Infrared colours

We now consider the infra-red and sub-mmcolours of starbursts and normal galaxies. Fig-ure 9 shows the dependence of the mean IRAScolours on infra-red luminosity. This plot includesall model galaxies, both normal and starbursts.Their IRAS band luminosities are calculated byconvolving the SEDs with the IRAS response func-tions. In calculating the mean colours, the models

Fig. 9.— Infra-red colours versus total infra-redluminosity. The three panels show the 12/25,12/60 and 60/100 µm colours respectively, definedas the log of the ratio of νLν in each band. Ineach panel, the solid line with error bars shows themean and dispersion from the models, while thedashed line with error bars shows the mean and er-ror on the mean measured by Soifer & Neugebauer(1991) from the IRAS Bright Galaxy Survey.

are weighted by their number density and by a fac-

tor L3/2ν (60µm) to account approximately for the

volume within which a galaxy would be visible ina 60µm flux-limited sample. LIR is the standardestimate of the total 8 − 1000µm IR luminosityfrom the luminosities in the four IRAS bandpasses

19

(Sanders & Mirabel 1996):

LIR = 0.97νLν(12) + 0.77νLν(25)

+ 0.93νLν(60) + 0.60νLν(100). (16)

The model predictions are compared with the ob-served mean colours calculated by Soifer & Neuge-bauer (1991) from the IRAS bright galaxy sample(IRAS BGS, Soifer et al. 1989), which is a com-plete sample flux-limited at 60µm. Models andobservations, in particular the 12/60 µm colour,show in general similar trends.

Fig. 10.— Average SED of model starbursts withLIR ≥ 1011h−2L⊙ (empty circles) compared tothe observational data (crosses) for 14 luminousstarbursts from Lisenfeld et al. (2000). (We ex-clude galaxies with only upper limits at 850µm.)All SEDs are normalized to the 60µm luminosity.The error bars show the dispersion in the models.

Figure 10 shows the quite good agreement be-tween the average IR and sub-mm SED of modelstarburst galaxies, and the observed SEDs of lumi-nous infra-red galaxies from the sample of Lisen-feld et al. (2000). The limit LIR ≥ 1011h−2L⊙ forthe models has been chosen to approximately re-produce the selection for the Lisenfeld et al. sam-ple. Note that the dust opacity in our models de-creases as λ−b with b ≈ 2 for 100 . λ . 1000µm,while Lisenfeld et al. , by fitting optically thin sin-gle temperature models to the data at λ ≥ 60µm,

derived b values in the range 1.5–2. Our modelsdemonstrate that the shallower slopes can insteadbe explained by the distribution of dust tempera-tures within each galaxy.

6.4. Extinction in starburst galaxies

An important problem in the study of star-forming galaxies is to determine the amount ofattenuation of starlight by dust, especially in theUV. This bears directly on the determination ofstar formation rates in galaxies from their UV lu-minosities. For our own and a few nearby galax-ies, the extinction law of the dust can be mea-sured directly from observations of backgroundstars, where the dust acts as a foreground screen.The differences found between the shapes of theextinction curves of the Galaxy, the Large Magel-lanic Cloud and the Small Magellanic Cloud be-low λ ≤2600A (e.g. Fitzpatrick 1989) are oftenascribed to the different metallicities in these sys-tems, covering the range Z ∼ 0.1 − 1 Z⊙. Re-cently, Calzetti et al. (1994) (see also Calzetti1997, 1999) have analyzed the dust extinction instarburst galaxies. In this case, the measurementof the extinction is more complicated, since onemeasures the integrated light of the whole system,where stars and dust are mixed in a complex way.From the optical and UV spectra of a sample ofUV-bright starbursts, Calzetti et al. derive an av-erage attenuation law characterized by a shallowerfar-UV slope than that of the Milky Way extinc-tion law, and by the absence of the 2175 A feature.This is at first sight quite surprising, because themetallicities of these galaxies are mostly similar tothat of the Milky Way, and so they might be ex-pected to have similar dust properties. The ques-tion is then to what degree the differences betweenthe starburst attenuation law and the Milky Wayextinction law are due to the geometry of the starsand dust, and to what degree they can only be ex-plained by differences in dust properties.

Figure 11 compares the average attenuationcurves for galaxies from our model with the em-pirical “attenuation law” obtained for starburstsby Calzetti et al. (1999). The attenuation Aλ forthe models is defined as the difference in magni-tudes of the stellar luminosity Lλ of a galaxy withand without dust, and is normalized to the colourexcess E(B − V ) = AB − AV of the stars to givean attenuation “law” k(λ) = Aλ/E(B−V ), equiv-

20

Fig. 11.— The average dust attenuation curvesfor starlight in different classes of galaxies (normaland starburst, SB) in the model compared withthe average Milky Way extinction law (solid line)and with the Calzetti “attenuation law” (filled cir-cles, Calzetti et al. (1999), with R′

V = 4.05). Theattenuation curves are all normalized to the netreddening of the stellar population, E(B − V ).The error bars show the dispersion of the mod-els around the mean attenuation curve.

alent to the definition of Calzetti et al. . As de-scribed in §3.2, the dust properties we adopt implyan extinction law characterized by a distinct 2175A feature produced by graphite grains, and wellmatching the average Milky Way extinction curve.The model extinction law (solid line in Fig. 11) isthe attenuation law that would be measured if allthe dust were in a foreground screen in front of thestars and no scattered light reached the observer.This geometry is clearly not realistic as applied tothe integrated light from galaxies. In our models,we have instead a complex and wavelength depen-dent geometry, where the UV emitting stars areheavily embedded inside molecular clouds, whilethe older stars, mainly emitting in the optical andnear infrared, are well mixed with the diffuse in-terstellar medium.

Figure 11 shows average attenuation curves for3 classes of model galaxies: (a) normal galax-ies with E(B − V ) > 0.05; (b) starbursts with

5 × 108 < LIR < 5 × 1010h−2L⊙; and (c) star-bursts with LIR > 5 × 1010h−2L⊙. The star-burst models are all chosen to have ages < 50Myrsince the start of the burst, as discussed in §6.2.Sample (b) corresponds roughly to the galaxies forwhich Calzetti et al. measured their attenuationlaw. The model attenuation law depends signifi-cantly on the sample, but all 3 classes show a weakor completely absent 2175A feature. In particular,the predicted attenuation curve for the lower lu-minosity starbursts is remarkably close to the em-pirical “Calzetti law”. This result is an entirelygeometrical effect, and did not require us to as-sume for starbursts dust properties different fromthose of the Galaxy. This conclusion is contrary tothat of Gordon et al. (1997), who argued that theobserved shape is only produced with dust thatlacks the 2175A feature in its extinction curve.The reason is presumably that Gordon et al. onlyconsidered clumping of dust, not of stars, and as-sumed a spatial distribution for stars independentof stellar age. Our results follow naturally fromthe assumption that stars are born inside densedust clouds and gradually escape.

Fig. 12.— Predicted attenuation curves for twonormal galaxies (left panels) and two starbursts(right panels) from the models.

To further illustrate the importance of geomet-rical effects in determining the attenuation law,

21

we show in Fig. 12 the attenuation laws of 2 nor-mal and 2 starburst model galaxies. The globalattenuation law, and the separate contributionsfrom the molecular clouds and diffuse dust, areshown in each case, normalized to the colour ex-cess E(B−V ) produced by that dust component.The global (g), molecular cloud (MC) and diffusedust (d) contributions are related by

(Aλ/E)g =(Aλ/E)MCEMC + (Aλ/E)dEd

Eg(17)

In the far-UV, including the spectral regionaround the 2175A feature, the global attenua-tion in the models is strongly contributed, or evendominated, by the MCs. The shape of the attenu-ation curve there has little to do with the opticalproperties of grains, because our MCs usually havesuch large optical depths that the UV light fromstars inside the clouds is completely absorbed.The wavelength dependence of the attenuationlaw of the MC component instead arises from thefact that the fraction of the light produced by veryyoung stars increases with decreasing wavelength,and at the same time, the fraction of stars whichare inside clouds increases with decreasing age,as given by eqn. (11). The additional attenuationarising in the cirrus component can sometimes im-print a weak 2175A feature, but this is not the casefor the starbursts, where the primary UV stellarlight is dominated by very young populations.

7. Galaxy Luminosity Function

7.1. Method

The luminosity function of galaxies at differentwavelengths is a basic property of the galaxy pop-ulation which a galaxy formation model should ex-plain. We calculate the galaxy luminosity functionat different wavelengths by combining the modelSEDs with the weights for the individual galax-ies (Section 4). For the normal galaxy sample wehave, for the number density of galaxies per lnLat some wavelength λ

dn

d lnLλ=

1

∆ lnL

∑

| ln Li−lnL|< 1

2∆(lnL)

ni (18)

where ni is the number density for the ith galaxy,Li is its luminosity at wavelength λ, the centreof the bin is at L and its width is ∆(lnL). For

the burst galaxy sample, we have to sum over theburst phase j also, giving

dn

d lnLλ=

1

∆ lnL

∑

| ln Lij−lnL|< 1

2∆(lnL)

nij (19)

where nij is the number density for galaxy i atevolutionary phase j, and Lij its the luminosityat that phase.

Galaxy luminosity functions are measured inspecific bands defined by a filter+instrument re-sponse function, e.g. the standard B or K bands,or the IRAS bands. Thus we convolve the modelSEDs with the appropriate response functionto calculate the luminosity Lν in that band.We use absolute magnitudes on the AB system,MAB = −2.5 log10(Lν/4.345 × 1020erg s−1Hz−1).The model luminosity functions have statisticaluncertainties due to the finite size of the modelgalaxy catalogue. We estimate these statisticalerrors by bootstrap resampling of the catalogue.

7.2. Optical and Near Infra-Red

At optical and near-IR wavelengths the emis-sion is mostly from older stars and the effects ofdust obscuration are generally modest. Figure 13shows the local luminosity function in the B-band(λ = 0.44µm), compared to the observed lumi-nosity function measured from the ESP redshiftsurvey by Zucca et al. (1997). The predicted lu-minosity function agrees well with the observedone, except at the highest luminosities. Extinctionby dust makes galaxies around 0.6mag fainter onaverage, for bright (L & L⋆) galaxies. Galaxieswhich have had recent bursts (i.e. in the last 1/20of the age of the universe, 0.7 Gyr) do not dom-inate the luminosity function at any luminosity,when the effects of dust are included.

As described in Cole et al. (2000), the B-band luminosity function is used as one of theprimary observational constraints for setting theparameters in the GALFORM model, in particu-lar, the parameters αhot and Vhot controlling feed-back, and the parameter Υ which sets the fractionof brown dwarfs in the IMF. The good agreementwith the observed B-band luminosity function istherefore not a surprise, but it was not guaran-teed, since the stellar population and dust mod-els used in Cole et al. (2000) are not identicalto those used here. Cole et al. used the stellar

22

Fig. 13.— Galaxy luminosity functions in the optical and near-IR. The left panel shows the luminosityfunction in the B-band (0.44µm), compared to observational data from Zucca et al. (1997). The rightpanel shows the luminosity function in the K-band (2.2µm), compared to observational data from Gardneret al. (1997). In both panels, the solid line shows the total luminosity function including the effects ofdust, while the long-dashed and short-dashed lines show the contributions to this from galaxies with andwithout recent bursts, and the dotted line shows the luminosity function without dust. The error bars on themodel luminosity functions are the statistical errors resulting from the finite sample size, calculated usingbootstrap resampling. The observational data have been converted to AB magnitudes assuming B = BJ +0.2(to convert from measured to standard magnitudes) and BAB = B−0.12, and KAB = K+1.87 respectively.

population models of Bruzual & Charlot (2000),and calculated the effects of dust using the mod-els of Ferrara et al. (1999). The stellar popula-tion model in GRASIL is based on similar stellarevolution tracks and spectra, but the treatmentof dust extinction is significantly different. TheFerrara et al. models assume that stars and dustare smoothly distributed, while in GRASIL a frac-tion of the dust is in clouds, and young stars areconfined to these clouds. The B-band luminos-ity functions, both with and without dust, calcu-lated by GALFORM using the Bruzual & Char-lot (2000) and Ferrara et al. (1999) models agreevery well with those computed using the GRASILstellar population+dust model, demonstrating theconsistency of the procedure of using the galaxyformation parameters derived in Cole et al. in com-bination with the GRASIL model. The effects ofdust computed using the two models are quite sim-ilar in the B-band, in spite of the differences in thestar and dust geometry. This is because most ofthe B-band light is produced by stars which are oldenough to have escaped from the clouds in which

they formed, so in GRASIL the attenuation is duemostly to the diffuse component of the dust, whichis modelled in a similar way to that in the Ferraraet al. models.

Figure 13 also shows the model and observed lu-minosity functions in the K-band. In this case, theeffects of dust are very small, so the comparisonis essentially independent of assumptions aboutdust. Again, the model agrees well with obser-vations over most of the luminosity range, as wasalso found by Cole et al. (2000). The contributionof galaxies with recent bursts is very small at allluminosities.

7.3. Far Ultra-Violet

In Figure 14 we compare the predicted lumi-nosity function in the far-UV (λ = 0.2µm) withthat measured by Sullivan et al. (2000) from aUV-selected redshift survey, based on FOCA in-strument fluxes. This comparison has not previ-ously been made for any semi-analytical galaxyformation models. The effect of dust are much

23

Fig. 14.— Far-UV (0.2µm) luminosity function. (a) The left panel shows the results for our standardparameters. The solid line shows the model prediction for the LF at z = 0 including dust, while thelong-dashed and short-dashed lines show the contributions to this from galaxies with and without recentbursts. The dotted line shows the model luminosity function obtained if absorption by dust is neglected.The symbols with error bars show the observed 0.2µm luminosity function (without any dust correction)measured from a UV-selected redshift survey by Sullivan et al. (2000). FOCA magnitudes are convertedto the AB system, assuming mAB = m2000 + 2.26. (b) The right panel shows the effects of varying tescand rburst. The solid line is for our standard model (tesc = 2 and 10Myr in normal galaxies and burstsrespectively, and rburst/rbulge = 0.1), including dust. The short-dashed line shows the effect of increasingtesc(burst) to 30Myr, and the short-dash-long-dash line the effect of increasing tesc(normal) to 5Myr. Thelong-dashed line shows the effect of increasing rburst/rbulge to 0.5. The dotted line shows the LF withoutdust, which is the same in each of these cases. The observational data are as in the left panel.

larger than in the optical, as one would expect. Inthis case, the effects of the more realistic geometryfor the stars and dust assumed by GRASIL com-pared to the Ferrara et al. models (clumpy ratherthan smooth distributions for the stars and dust)are significant. The stars that produce most ofthe UV light spend a large fraction of their life-times in the molecular clouds where they form, sothe mean extinction is larger than in the case ofa smoothly distributed dust component with thesame total dust mass. Bursting and non-burstinggalaxies contribute roughly equally at the highestluminosities. This result is however sensitive tothe details of how bursts are modelled, since thisdetermines what small fraction of the UV light es-capes from currently or recently bursting galaxies.When we compare our model LF including extinc-tion with the directly observed LF, uncorrected forextinction, we find reasonable agreement at lowerluminosities, but at high luminosities, the model

LF is somewhat lower than the observed one. Thismight be partly an effect of evolution in the obser-vational sample, which covers a significant redshiftrange (z . 0.5), but it might also be that the UVextinction is over-estimated in the model.

Figure 14 shows also the effect of changing theburst radius rburst and the timescale tesc for starsto escape from clouds. Increasing rburst/rbulge

from 0.1 to 0.5 reduces the optical depth in thediffuse component during bursts, allowing more ofthe UV light from bursts to escape, and increas-ing the LF at the highest luminosities. Increasingtesc in bursts from 10Myr to 30Myr has negligi-ble effect on the total UV LF. Increasing tesc innormal galaxies from 2Myr to 5Myr slightly low-ers the amplitude of the luminosity function at thebright end.

24

Fig. 15.— The 60µm far-IR luminosity function. (a) The left panel shows the results for our standardparameters. The solid line shows the total model luminosity function, while the long-dashed and short-dashed lines show the contributions to this from galaxies with and without recent bursts. The points witherror bars show the observed 60µm luminosity functions measured from galaxies observed by IRAS, bySaunders et al. (1990) (filled circles) and Soifer & Neugebauer (1991) (open circles). (b) The right panelshows the effects of varying the burst or escape timescales: ǫ−1

⋆burst ≡ τ⋆burst/τbulge = 1 (dotted), 2 (solid), 4(dashed). The long dashed curve has ǫ−1

⋆burst = 2 (our standard value) but tesc = 2 Myr instead of 5 Myr.

7.4. Mid and Far Infra-Red

In the mid- and far-infrared, the luminosity ofgalaxies is dominated by re-emission from dust.Using the GRASIL code, we can now directly pre-dict the far-IR luminosities of galaxies from ourgalaxy formation model, and compare with obser-vations. The luminosity functions of galaxies at12, 25, 60 and 100µm have been measured usingIRAS data. The best determination is at 60µm,where IRAS was most sensitive. Figure 15 showsthat the predicted luminosity function agrees ex-tremely well with that observed by Saunders et al.(1990) and Soifer & Neugebauer (1991), except atvery low luminosities, where the measured LF isfairly uncertain. Above νLν(60) & 3×1010h−2L⊙,the model LF is dominated by galaxies undergo-ing bursts triggered by mergers. This is in agree-ment with observations of ultra-luminous IRASgalaxies, which are all identified as recent mergersbased on their optical morphology (e.g. Sanders& Mirabel 1996).

The right panel of Figure 15 shows the ef-fect on the 60µm LF of varying the parameterǫ⋆burst, which relates the star formation timescale

in bursts to the dynamical time of the bulge (equa-tion 6). Unlike the other parameters in the GAL-FORM model, Cole et al. (2000) did not tryto choose a best-fit value, because the observa-tional data in the optical and near-IR that theycompared with were not sensitive to its value.(The Cole et al. results were calculated assumingτ⋆burst = 0.) However, the far-IR LF is sensitiveto this and thus constrains the burst timescale forthe most luminous galaxies. Figure 15 shows pre-dictions for ǫ⋆burst = 1, 0.5, 0.25, correspondingto τ⋆burst/τbulge = 1, 2, 4 respectively. Increasingǫ⋆burst means bursts are more luminous, but lastfor a shorter time, and so have a lower numberdensity. This trend is seen at the high-luminosityend of the 60µm LF, which is dominated by burst-ing galaxies. A value ǫ⋆burst = 2 seems to fit some-what better than higher or lower values, so weadopt this as our standard value. Also shown inthe same panel is the somewhat better fit obtainedsetting tesc = 5 Myr. However, as explained in§ 3.3, our adopted standard value 2 Myr is favoredby stellar evolution timescale argument and by theUV LF. Increasing tesc in bursts from 10Myr to

25

30Myr has negligible effect on the LF.

Fig. 16.— The luminosity functions in the 12,25 and 100µm IRAS bands, compared to obser-vational data from Soifer & Neugebauer (1991).The line types and symbols are as in Fig.15.

The luminosity functions at 12, 25 and 100 µm,are compared with the observational data fromSoifer & Neugebauer (1991) in Figure 16. Thepredicted luminosity function agrees well with themeasured one in each case.

8. Star formation rate indicators

Here we examine the accuracy of several SFRindicators based on continuum UV or IR luminosi-ties (reviewed by e.g. Kennicutt 1999).

The luminosity Lν(2800) at 2800A has beenextensively used to estimate SFRs of high-redshiftgalaxies and to investigate the evolution of the cos-mic SFR density (e.g. Lilly et al. 1996; Conollyet al. 1997). In the top panel of Fig. 17, we plotthe SFR against Lν(2800) for the model galaxies,

Fig. 17.— Star formation rate versus UV and IRluminosities for galaxies from our model. Onlymodels corresponding to spirals and luminousstarburst galaxies are plotted. See text for details.

including the effects of extinction. Only modelscorresponding to spiral galaxies (B/T ≤ 0.5) andluminous starburst galaxies (LIR ≥ 1010h−2L⊙)are shown. At higher luminosities and SFRs,Lν(2800) & 3 × 1026erg s−1Hz−1, the modelsin the absence of dust follow a linear relationbetween SFR and Lν , with a rather small dis-persion, as would be expected if the UV lu-minosity is dominated by young stars and therecent SFR has been approximately constant.This linear relation, SFR/(M⊙yr−1) = 8.5 ×10−29Lν(2800A)/(erg s−1Hz−1), is indicated bythe solid line, and its extrapolation to lower lumi-nosity is shown by the dotted line. Dust extinction

26

shifts points to the left of this line. At lower lu-minosities, the effects of dust extinction are verysmall, because of the low gas contents of the galax-ies. On the other hand, because the SFRs are sosmall, the 2800A light has a significant contri-bution from post-AGB stars and old metal poorpopulations, and this causes the locus of pointsto bend to the right of the linear relation. Thegalaxies with very high metallicities (Z > 0.1)have very low gas fractions.

The figure also shows as a dashed line the linearrelation between Lν(2800) and SFR obtained byKennicutt (1999), using stellar population mod-els for a Salpeter IMF, and assuming a constantSFR for the last 108yr. The SFR/Lν ratio inour models without dust (solid line) is about 40%lower than Kennicutt’s value, but this difference isentirely due to the different IMF we adopt (equa-tion (9)).

Perhaps the most striking feature of this plotis that the starburst models are offset by morethan an order of magnitude from the average rela-tion holding for the normal spirals, because of thelarge UV extinctions in the starbursts. Further-more, their dispersion in SFR at a given luminos-ity is also quite large. Thus, the 2800A luminositywith no dust correction performs rather poorly asa quantitative SFR indicator, both for very highSFRs (because of extinction) and for very low ones(because of the light from older stars).

The middle panel of Fig.17 depicts the relationbetween the star formation rate and LIR (eq. 16),the estimated 8−1000µm luminosity based on theIRAS fluxes. The solid line represents the rela-tion derived by Kennicutt (1998) for starbursts,SFR/(M⊙yr−1) = 4.5 × 10−44LIR/(erg s−1), byassuming that the bolometric output in a continu-ous burst of age between 10–100 Myr is completelyreprocessed by dust, again for a Salpeter IMF. TheKennicutt relation is seen to fit our model galax-ies quite well at all luminosities (normal spiralsas well as starbursts), even though we assume adifferent IMF from Kennicutt.

The luminosity in the ISO 15µm band has alsobeen proposed as an approximate SFR indica-tor. The bottom panel of Fig. 17 shows the SFRvs Lν(15µm) for our galaxies. The line plot-ted is our best linear fit to the model points,SFR/(M⊙yr−1) = 5.6×10−30Lν(15µm)/(erg s−1Hz−1).Despite the important contribution from PAH

bands, the correlation of the SFR with Lν(15µm)in our models is still fairly good.

9. Summary and Conclusions

We have combined an ab initio model of galaxyformation (GALFORM, §2, Cole et al. 2000), withan ab initio model for stellar emission and dustemission and absorption in galaxies (GRASIL, §3,Silva et al. 1998). Both models are state-of-theart. We are able to predict, in the context of thecold dark matter cosmology, the luminosities andspectral energy distributions from the UV to thesub-mm for the whole galaxy population, and howthese change with cosmic epoch. Here we focusedon a wide range of spectrophotometric propertiesof present-day galaxies, from the UV to the sub-mm, for a flat low-density cosmology (Ω0 = 0.3,Λ0 = 0.7) with a CDM spectrum of density fluctu-ations. The model is remarkably successful in ex-plaining the UV, optical and IR spectrophotomet-ric and extinction properties of galaxies in the lo-cal universe. Future papers will investigate galaxyevolution in the UV, optical, IR and sub-mm outto high redshift.

Dust plays a dominant role in starburst galax-ies, where star formation proceeds in the centralregions of a galaxy on a short timescale. We didnot previously make any detailed comparison ofthe properties of the starbursts predicted by semi-analytical galaxy formation models with observa-tional data. Here we have shown that these prop-erties are nicely reproduced.

Our model predicts an average dust attenuationlaw for starburst galaxies that agrees remarkablywell with the empirical law found by Calzetti etal. (1999), although with a significant dispersionaround the mean. In particular, the 2175A bumpis absent when the net attenuation of the galaxylight is considered. This is entirely an effect of thegeometry of stars and dust in our model, and hasnothing to do with the optical properties of dustgrains. Indeed, our dust mixture would reproducethe average Milky Way extinction curve (with astrong 2175A feature), if it were arranged in aforeground screen geometry. The absence of thisfeature in the attenuation curves of model star-bursts is because in that case the dust attenua-tion is dominated by molecular clouds, with theshape controlled by the gradual escape of young

27

stars from the clouds.

The starburst galaxies are predicted to dom-inate the bright end of the luminosity functionin terms of bolometric luminosity, but because ofthe large extinctions in these objects, they do notmake a dominant contribution to the bright endof the luminosity function in either the UV, op-tical or near-IR once dust effects are included.At these wavelengths, the luminosity function isdominated by normal spiral and elliptical galax-ies. However, the starbursts completely dominatethe bright end of the luminosity function in themid- and far-IR (10–100µm), at total luminosi-ties LIR & 1011h−2L⊙. Overall, the luminosityfunction predictions from the far-UV to the far-IR are a remarkable success for the model, sincethe dust contents and galaxy radii are predicted apriori, and the only significant adjustable param-eter in the comparison was the ratio of the bursttimescale to the bulge dynamical time, which waschosen to fit the bright end of the luminosity func-tion at 60µm.

As expected, our models show that the UV con-tinuum is in general a poor star formation indica-tor, both because of the large variations in theamount of extinction, and also because of the con-tribution from old stellar populations in the mid-UV (∼ 3000A) in more quiescent galaxies. Theinfrared luminosity is a much more reliable SFRindicator.

The parameters values adopted here for theGALFORM model are those chosen previously byCole et al. (2000) to fit the properties of the localgalaxy population in the optical and near-infrared,apart the new ones specifying the timescales andradii of bursts. Cole et al. also discuss the effectsof varying the ’old’ GALFORM paramaters. Thepurpose of this paper was to present the effects ofincluding dust in a fixed galaxy formation model.The treatment of dust reprocessing with GRASILrequires some additional parameters to be set, butopens the possibility to test semi-analytical mod-els against the wealth of IR and sub-mm obser-vations already available or planned for the nearfuture. The adopted values for these parametershave also been guided by the results of Silva etal. (1998), who used GRASIL to reproduce de-tailed SEDs of several local normal and starburstgalaxies. Some of the effects of variations of theseparameters on SEDs, LFs and starburst properties

have been discussed in the relevant sections of thispaper. A more systematic parameter study willbe included in future investigations. Among thenewly introduced parameters, probably the mostimportant ones are ǫ⋆burst and tesc, describing thetimescales of bursts and the time for young stars toescape from their parent molecular clouds. Theyhave significant effects on (and are constrained by)the IR and UV luminosity functions respectively.

In conclusion, this paper is a stepping stonefor future work, which will apply the same mod-els to galaxies at high redshift. Now that semi-analytic models can be effectively compared withinfrared and sub-millimetre observations, as wellas UV and optical data, they can be used to worktowards an observationally and theoretically con-sistent picture for the history of galaxy formationand star formation in the universe.

We thank the anonymous referee for a con-structive report. We acknowledge the supportby the European Community under TMR grantERBFMRX-CT96-0086. SMC acknowledges thesupport of a PPARC Advanced Fellowship, andCSF a PPARC Senior Fellowship and a Lever-hulme Research Fellowship. CGL acknowledgesthe support of the Danish National ResearchFoundation through its establishment of the TAC,and a PPARC Visiting Fellowship at Durham.This work was partially supported by the PPARCrolling grant for extragalactic astronomy and cos-mology at Durham.

REFERENCES

Almaini, O., Lawrence, A., Boyle, B.J., 1999, MN-RAS, 305, L59

Alton, P. B., et al. 1998, A&A, 335, 807

Armus, L, Heckman, T., Miley, G, 1990, ApJ, 364,471

Balzano, V.A., 1983, ApJ, 268, 602

Baugh, C.M., Cole, S., Frenk, C.S., 1996b, MN-RAS, 283, 1361

Baugh, C. M., Cole, S., Frenk, C. S., Lacey, C.G.,1998, ApJ, 498, 504

Baugh, C.M., Benson, A.J., Cole, S., Frenk, C.S.,Lacey, C.G., 1999, MNRAS, 305, L21

28

Benson, A.J., Cole, S., Frenk, C.S., Baugh, C.M.,Lacey, C.G., 2000, MNRAS, 311, 793

Boulanger, F., Boissel, P., Cesarsky, D., Ryter, C.,1998, A&A, 339, 194

Bressan, A., Chiosi, C., Fagotto, F. 1994, ApJS,94, 63.

Bressan, A., Granato, G.L., Silva, L. 1998, A&A,332, 135.

Bruzual, G., Charlot, S., 2000, in preparation

Buat, V., & Xu, C., 1996, A&A, 306,61

Calzetti, D., 1994, AJ, 113, 162

Calzetti, D., 1999, UV Emission and Dust Prop-

erties of High Redshift Galaxies, Workshop onUltraluminous Galaxies: Monsters or Babies,Ringberg Castle, Germany, Kluwer, in press(astro-ph/9902107)

Calzetti, D., Armus, L., Bohlin, R.C., Kinney,A.L., Koornneef, J., Storchi-Bergmann, T.,1999, accepted for publication in ApJ(astro-ph/9911459)

Calzetti, D., Kinney A., L., Storchi-Bergmann, T.,1994, ApJ, 429, 582

Cole, S., Aragon-Salamanca, A., Frenk, C.S.,Navarro, J.F., Zepf, S.E., 1994, MNRAS,271,781

Cole S., Lacey C. G., Baugh C. M., Frenk C. S.,2000, submitted to MNRAS

Conolly, A.J., Szalay, A.S., Dickinson, M., Sub-baRao, M.U., Brunner, R.J., 1997, ApJ, 486,L11

Diaferio, A., Kauffmann, G., Colberg, J.M.,White, S.D.M., 1999, MNRAS, 307, 537.

Dwek, E., et al. 1998, ApJ, 508, 106

Elbaz, D., et al. 1999, A&A, 351, L37

Ferrara, A., Bianchi, S., Cimatti, A., Giovanardi,C., 1999, ApJS, 123, 437

Fitzpatrick, E. 1989, IAU Symp. 135: InterstellarDust, 135, 37

Fixsen, D.J., Dwek, E., Mather, J.C., Bennett,C.L., Shafer, R.A., 1998, ApJ, 508, 123

Flores, H, et al. 1999, ApJ517, 148

Franceschini, A., Mazzei, P., de Zotti, G., Danese,L., 1994, ApJ, 427, 140

Gardner, J.P., Sharples, R.M., Frenk, C.S., Car-rasco, B.E., ApJ, 480, L99

Giovanelli, R., Haynes, M.P., Salzer, J.J., Wegner,G., da Costa, L.N., & Freudling, W., 1995, AJ,110, 1059

Gordon, K.D., Calzetti, D., Witt, A.N., 1997,ApJ, 487, 625

Governato, F., Baugh C. M., Frenk C. S., Cole S.,Lacey C. G.,Quinn, T. Stadel, J., 1998, Nature,392, 359

Granato, G.L., Danese, L. & Franceschini, A.,ApJ, 486, 147

Guiderdoni, B., Rocca-Volmerange, B., 1987,A&A, 186, 1

Guiderdoni, B. , Bouchet, F. R., Puget, J. -L. ,Lagache, G. & Hivon, E. 1997, Nature, 390,257

Guiderdoni, B., Hivon, E., Bouchet, F.R., Maffei,B., 1998, MNRAS, 295, 877

Hauser, M.G., et al. , 1998, ApJ, 508, 25

Heckman T. M., Robert C., Leitherer C., GarnettD.R. & ven der Rydt F. 1998, ApJ, 503, 646

Helou, G., Khan, I.R., Malek, L., Boehmer, L.,1988, ApJS, 68, 151

Hughes, D.H., et al. , 1998, Nature, 394, 241

Jenkins, A., Frenk, C.S., Pearce, F.R., Thomas,P.A., Colberg, J.M., White, S.D.M., Couch-man, H.M.P., Peacock, J.A., Efstathiou, G.,Nelson, A.H., (the VIRGO consortium), 1998,ApJ, 499, 20

de Jong, R.S., 1996c, A&A, 313, 377

de Jong, R.S. van der Kruit 1994, A&ASupp 106,451

29

Kauffmann, G., White, S.D.M., Guiderdoni, B.,1993, MNRAS, 264, 201

Kauffmann, G., 1996, MNRAS, 281, 487

Kauffmann, G., Nusser, A., Steinmetz, M., 1996,MNRAS, 281, 487

Kauffmann, G., Colberg, J.M., Diaferio, A.,White, S.D.M., 1999, MNRAS, 303, 188

Kawara, K., et al. 1998, A&A, 336, L9

Kennicutt, R.C., 1983, ApJ, 272, 54

Kennicutt, R.C., 1998, ApJ, 498, 541

Kennicutt, R.C., 1999, ARA&A, 36, 189

Keto, E., Ball, R., & Arens, J., Jernigan, G.,Meixner, M., 1992, ApJ, 387, L17

Kinney, A.L., Bohlin, R.C., Calzetti, D., Panagia,N., Wyse, R.F.G., 1993, ApJS, 86, 5

Kylafis, N.D., & Bahcall, J.N., 1987, ApJ, 317,637

Lacey, C.G., Granato, G., Silva, L., Bressan, A.,Baugh, C., Cole, S., Frenk, C.S., 2000, in prepa-ration

Lacey, C.G., Guiderdoni, B., Rocca-Volmerange,B., Silk, J., 1993, ApJ, 402, 15

Lacey, C.G., Silk, J., 1991, ApJ, 381, 14

Lehnert, M., & Heckman, T., 1995, ApJS, 97, 89

Lilly, S.J., LeFevre, O., Hammer, F., Crampton,D., ApJ, 460, L1

Lilly, S.J., Eales, S.A., Gear, W.K., Hammer, F.,LeFevre, O., Crampton, D., Bond, J.R., Dunne,L., 1999, ApJ, 518, 641

Lisenfeld, U., Isaak K., G., Hills, R. 2000, MN-RAS, 312, 433

Madau, P., 1999, in “VLT Opening Symposium”(astro-ph/9907268)

Madau, P., Ferguson, H.C., Dickinson, M, Gi-avalisco, M., Steidel, C.C., Fruchter, A., 1996,MNRAS, 283, 1388

Madau P., Pozzetti L., Dickinson M., 1998, ApJ,498, 106

Mazzei, P., Xu, C., de Zotti, G., 1992, A&A, 256,45

Meurer, G.R., Heckman, T.M., Leitherer, C., Kin-ney, A., Robert, C., Garnett, D.R. 1995, AJ110, 2665

Oliver, S. J., et al. 1997, MNRAS, 289, 471

Pearson, C., Rowan-Robinson, M., 1996, MNRAS,238, 523

Pettini, M., Kellogg, M., Steidel, C., Dickinson,M., Adelberger, K.L., Giavalisco, M., 1998,ApJ, 508, 539

Press W. H., Schechter P., 1974, ApJ, 187, 42

Puget, J.L, et al. 1996, A&A, 308, L5

Puget, J. L., et al. 1999, A&A, 345, 29

Sage, L.J., 1993, A&A, 272, 123

Sanders, D.B., Mirabel, I.F. , 1996, Ann Rev 34,749

Scalo, J., ’The Stellar Initial Mass Function’, ASPconference series Vol. 142, eds. G. Gilmore &D. Howell, p.201 (ASP: San Francisco)

Saunders, W., 1997, private communication

Saunders, W., Rowan-Robinson, M., Lawrence,A., Efstathiou, G, Kaiser, N., Ellis, R.S., Frenk,C.S., 1990, MNRAS, 242, 318

Schlegel, D.J., Finkbeiner, D.P., Davis, M., 1998,ApJ, 500, 525

Scoville, N.Z., Yun, M.S., Bryant, P.M., 1997,ApJ, 484, 702

Silva, L., Granato, G.L., Bressan, A., Danese, L.,1998, ApJ, 509, 103

Simien, F., de Vaucouleurs, G., 1986, ApJ, 302,564

Smail, I., Ivison, R.J., Blain, A.W., 1997, ApJ,490, L5

Soifer, B.T., Boehmer, L., Neugebauer, G.,Sanders, D.B., 1989, AJ, 98, 766

Soifer, B.T., Neugebauer, G., 1991, AJ, 101, 354

30

Somerville, R.S., Primack, J.R., 1999, MNRAS,310, 1087

Steidel, C.C., Adelberger, K.L., Giavalisco, M.,Dickinson, M., Pettini, M., 1999, ApJ, 519, 1

Sullivan, M., Treyer, M.A., Ellis, R.S., Bridges,T.J., Milliard, B., Donas, J., 2000, MNRAS,312, 412

Thuan, T.X., Martin, G.E., 1981, ApJ, 247, 823

Tully, R.B., Pierce, M.J., Huang, J. Saunders, W.,Verheijen, M.A.W., Witchalls, P.L., 1998, AJ,115, 2264

de Vaucouleurs, G., de Vaucouleurs, A., Corwin,H.G., Buta, R.J., Paturel, G., Fouque, P., 1991,Third Reference Catalogue of Bright Galaxies(New York: Springer)

Wang, B., & Heckman, T., 1996, ApJ, 457, 645

White, S.D.M., Frenk, C.S., 1991, ApJ, 379, 25

Wright, G.S., James, P.A., Joseph, R.D., McLean,I.S., 1990, Nature, 344, 417

Xilouris, E.M., Byun, Y.I., Kylafis, N.D., Paleolo-gou, E.V., Papamastorakis, J., A&A, 344, 868

Xu, C., & Buat, V., 1995, A&A, 293, L65

Wang, B. 1991, ApJ, 374, 465

Zucca, E., et al. , 1997, A&A, 326, 477

This 2-column preprint was prepared with the AAS LATEXmacros v5.0.

31