Astron. Astrophys. 347, 769–798 (1999) ASTRONOMY AND ...aa.springer.de/papers/9347003/2300769.pdf · the formation rates of cold clouds through thermal instability, as will be seen

Astron. Astrophys. 347, 769–798 (1999) ASTRONOMYAND

ASTROPHYSICS

Hierarchical, dissipative formation of elliptical galaxies:is thermal instability the key mechanism?

Hydrodynamical simulations including supernova feedback, multi-phase gasand metal enrichment in CDM: structure and dynamics of elliptical galaxies

John Hultman and Arnaud Pharasyn

Uppsala Astronomical Observatory, Box 515, S-751 20 Uppsala, Sweden (apharasy, [email protected])

Received 30 April 1998 / Accepted 8 February 1999

Abstract. We present numerical simulations of galaxy forma-tion, including star formation, in a standard, hierarchical cosmo-logical scenario (CDM). The gas dynamics is followed with theSmooth Particle Hydrodynamics (SPH) method, and the grav-itational interaction with a tree method. The scales involvedrange from 20 Mpc down to less than 1 kpc in massive objects.The gas component is treated as a two-phase medium, where thehot component cools radiatively and forms cold clouds throughthe thermal instability mechanism. Star formation subsequentlytakes place in the cold clouds. Supernovae heat the hot compo-nent and evaporate cold clouds. Metal enrichment is included,and its effect on radiative cooling rates is taken into account.

For a certain degree of supernova evaporation, several gen-eral properties of elliptical galaxies are reproduced, like shapes,r1/4-profiles, half-light radii, slow rotation and anisotropic ve-locity dispersions. A Faber-Jackson like relation is observed,being of the formM ∼ σ2.6. Objects with circular velocitiesdown to around 150 km s−1 can be dynamically and structurallyresolved. No disc-shaped objects form.

The elliptical objects form predominantly by early hierar-chical merging. The thermal instability is the main mechanismfor breaking an otherwise purely gaseous collapse. The SFRhas a rapidly rising and later exponentially decaying behaviourin individual objects. Cosmologically, the stellar contributionto the closure density is around one percent. The co-movingaverage SFR increases with redshift, in rough agreement withobservations. Low mass objects are strongly suppressed, havingsmall stellar content, and stop forming stars early. Generally, themore massive objects are in place atz ∼ 0.5–2. Although manyproblems and questions still remain, the overall impression isthat, surprisingly, many properties of ellipticals are reproduced.

These results are also largely in agreement with those pre-sented by Yepes et al. (1997), but the simulations presented hereare of higher spatial resolution, and can thereby resolve at leastthe more massive galactic objects (vcirc > 150 km s−1).

Send offprint requests to: A. Pharasyn

Key words: hydrodynamics – methods: numerical – galaxies:elliptical and lenticular, cD – galaxies: formation – galaxies:structure

1. Introduction

Hierarchical scenarios are still the most favoured for structureformation in the universe. The characteristic time of collapsefor a proto-galactic object is an increasing function of mass.More massive galactic halos thus form at later cosmologicalepochs. This general picture is consistent with observations,indicating that galaxy clusters are younger than the containedgalaxies, and that the Universe is homogeneous on very largescales. This scenario fits the standard picture of White & Rees(1978), where the baryonic gas component cools and falls intothe potential wells of the dark matter (DM) halos, where it canform stars. A major problem in this type of hierarchical scenarios(White & Frenk 1991; Cole et al. 1994) is that they tend to lockup most of the baryonic mass in the universe in small galacticobjects at early times. The cooling processes are already veryefficient at an early stage, when the density is high, because thecooling time scales roughly with inverse density, resulting in an“over-production” of dwarf galaxies which is in contradictionwith the observed galaxy luminosity function.

There have been various mechanisms envisaged to over-come this so called “over-cooling” or “faint galaxy excess”problem. One idea, proposed by Efstathiou (1992), argued thata strong background UV radiation field might suppress the for-mation of dwarf galaxies. A photo-ionizing background field,induced by quasars, would heat the gas and, by keeping thegas at a high degree of ionization, it would suppress collisionalline cooling by neutral atoms. Observational evidence for such afield is indeed suggested by the lack of a Gunn-Peterson effect athigh redshiftsz ≈ 5 (Webb et al. 1992), indicating that the Uni-verse was highly ionized at these redshifts. Another possibilityis that supernova driven winds (Dekel & Silk 1986; Tegmarket al. 1993) and photo-ionizing UV radiation from high massstars (Miralda-Escude & Ostriker 1990) are able to suppress the

770 J. Hultman & A. Pharasyn: Dissipative formation of ellipticals

formation of dwarf galaxies. Other ideas include other forms of“pre-heating” of the baryonic component, for example Comp-ton heating (Collin-Souffrin 1991) and the decay of an exoticdark matter field (Asselin et al. 1987).

The effects of an UV ionizing field have now been testedquite extensively in direct simulations (Vedel et al. 1994; Thoul& Weinberg 1996; Quinn et al. 1996; Weinberg et al. 1997;Navarro & Steinmetz 1997; Kallander & Hultman 1998). Theseauthors used different numerical approaches, initial conditionsand strengths of the photo-ionizing field. Vedel et al. used threedimensional SPH (Smooth Particle Hydrodynamics) simula-tions of a Milky Way sized object, but could therefore not drawconclusions on objects of other masses. Thoul & Weinberg usedone dimensional models, and had therefore much higher reso-lution. The remaining authors used 3D SPH simulations andemphasised numerical resolution effects, which are shown tohave critical effects to the results in these simulations. In spiteof the different implementations, the main conclusions are verysimilar: a background photo-ionizing field does have some ef-fect on lower mass objects, but cannot alone be responsible forsuppressing the formation of dwarf galaxies and make the hier-archical (CDM) models consistent with observations.

In hindsight, these results were perhaps not totally unex-pected. It has been argued that the photo-ionization from highredshift quasars is not strong enough to ionize the medium to theobserved level, (e.g., Miralda-Escude & Ostriker 1990). Also,the redshift distribution of quasars, assumed in the above men-tioned simulations, falls off at redshifts larger thanz ≈ 6. Ear-lier, there are simply no quasars, or they are not observed forsome reason (Miralda-Escude & Ostriker 1990). However, thereis plenty of time for the gas to cool and form stars beforez ≈ 6,thus the over-cooling problem still remains. This does not nec-essarily mean that the effects of an UV ionizing field have beenruled out all together, as will be further discussed in Sect. 7.3.

In this article we have chosen to focus on feedback fromstar formation (SF), mainly because it now seems to be the sim-plest solution to the over-cooling problem. Furthermore, anyapproach to galaxy formation must - sooner or later - take SFproperly into account in order to be able to predict luminos-ity functions, etc. The difficulty is that SF cannot be imple-mented from first principles, since it is not yet fully understoodas a physical process. Previous implementations of SF havetherefore been of highly phenomenological character, at least inthe context of galaxy formation. For example, pioneering SPHsimulations by Katz (1992) and Steinmetz & Muller (1995) pro-duced reasonable results in limited cosmological environments,i.e. formation of a single, isolated galactic object, but did notexplain the over-cooling problem. This was mainly because su-pernova energy feedback was implemented purely as depositionof thermal energy into the gas, immediately being radiated awayby the efficient cooling. In certain implementations momentumwas also injected (e.g., Navarro & White 1993; Mihos & Hern-quist 1994), showing that supernova feedback would affect SFrates etc., but this was highly dependent on the details of theimplementation in the SPH method and the resolution. Katz etal. (1996) performed cosmological star forming SPH simula-

tions and excluded the momentum injection. Their resolutionwas however too low to resolve low mass objects.

Recently, Yepes et al. (1997) (YKKK henceforth) proposeda SF model based on the theory for the multiphase interstellarmedium (ISM) (McKee & Ostriker 1977), including supernovaenergy feedback, cold cloud evaporation and thermal instabil-ity (e.g., Field 1965; Fall & Rees 1985; Nulsen 1986; Begel-man & McKee 1990; Mucket & Kates 1996). YKKK presented3D hydrodynamical simulations of standard CDM models us-ing a reliable Eulerian grid method, the Piecewise ParabolicMethod (PPM). Their main result was that supernova feedbackstrongly influences SF rates, and indeed – for certain parametervalues – produces reasonable SFRs and amounts of luminousmatter, compared to observed amounts. Dwarf galaxies werestrongly suppressed in that model, although low mass objectswere formed, and they had very low luminosities and smallfractions of stars.

The multiphase ISM approach to SF has previously beenused by Theis et al. (1992) and Samland et al. (1997) to studychemodynamical evolution of galaxies, using one, or more re-cently two-dimensional models. This approach has been capableof explaining several gross properties of the formation of bothdisc and spherical galaxies. Full chemodynamical evolution isstill somewhat too intricate to implement in three-dimensionalcosmological models, but the YKKK supernova evaporationmodel, here denoted the “SNE model”, significantly reducesthe gap between these models, by picking out the presumablydominant mechanisms of the ISM – star formation and feed-back loops. It should be noted that the formation mechanismsfor cold clouds are different, being thermal instability in theSNE model and condensation of hot gas in the chemodynami-cal evolution models. On the other hand, both these theoreticalapproaches are closely related (Begelman & McKee 1990), thethermal instability approach being a simplified version of thecondensation process.

Supernovae enrich the ISM with chemical elements. Ob-servations of intra-cluster gas indicate metallicities of [Fe/H]= −1/2. Damped Lymanα galaxies, at redshifts around 1–4,typically have around [Fe/H] =−2 to −1, increasing with de-creasing redshift (Lu et al. 1996). The cooling rate of a gaswith solar metallicity can be more than an order of magnitudelarger than for a gas of primordial composition (Sutherland &Dopita 1993), especially in the interval of temperatures of thehot phase ISM. Thus, metal enrichment can have a strong ef-fect on the amount of gas that is able to cool, and it will affectthe formation rates of cold clouds through thermal instability,as will be seen. Kallander (1996) performed SPH simulationsusing a simple model for the metal enrichment of halo gas. Hefound that the increase in cooling rate due to line cooling ofmetals was enough to produce a cooling flow in the hot halogas in roughly one Hubble time. Without the increased cool-ing rate, the cooling flow was significantly smaller and essen-tially no disc formed. Kallander & Hultman (1998) obtained aless pronounced effect. White & Frenk (1991) used an intricatesemi-analytic framework to compare different models of hier-archical galaxy formation with observations and found that the

J. Hultman & A. Pharasyn: Dissipative formation of ellipticals 771

results were sensitive to the assumptions made for the metalenrichment of the gas.

The purpose of the present article is to present an imple-mentation of the SNE model into the SPH + gravitational treemethod, and to use it for simulations of hierarchical galaxy for-mation (see Hultman & Kallander 1997 for details on the actualSPH tree code used). The SNE model is a more realistic descrip-tion of the SF process, and it will probably improve the solutionof several (numerical) resolution-related problems encounteredin previous SF models used in cosmological galaxy formationsimulations. Although including a more detailed description ofthe physics, the SNE model is still surprisingly simple in itsform and therefore very suitable for this type of approach. Fur-thermore, due to its adaptive nature, the SPH method has theadvantage of achieving higher “dynamical resolution”, i.e. con-stant mass resolution, as well as higher resolution in time due toindividual particle time-steps. Equivalent grid methods that usea constant cell size will therefore have a lower spatial resolu-tion in high density regions, for comparable computational time.This would then give a valuable comparison with the results ofYKKK, since different numerical methods are used. One of theother improvements made here is a specialized set-up of cosmo-logical initial conditions, in order to increase resolution. Also, amore consistent model for the metal enrichment of the gas (andthereby increased cooling rates) was adopted, as well as a morecorrect implementation of the formation of “star particles” inthe code.

We will focus mainly on the strength of the supernova evapo-ration and the strong effect it turns out to have on the formationof objects of elliptical type. This in turn calls for an analysisof these objects, in order to make a comparison with what istheoretically and observationally known.

Elliptical galaxies show some very distinct systematic prop-erties (Binney & Tremaine 1987; de Zeeuw & Franx 1991): thede Vaucouleursr1/4 or the Hubble-Reynolds law for the sur-face brightness profiles, Schechter’s law for the luminosities, in-cluding a range of luminosities of some107, the Faber-Jacksonrelation or the more recent “fundamental plane”. Also, ellipti-cals show essentially no present-day SF and contain little or nogas or dust, although the massive ones show extensive X-rayemitting halos of hot gas. Stellar populations in ellipticals aregenerally very old, although there are indications that bluer el-lipticals do exist, and that SF may have recently occurred in theircores (Silk & Wyse 1993). Also, elliptical galaxies rotate, butthey generally do so very slowly, even though faint ellipticalscan rotate faster (de Zeeuw & Franx 1991). The slow rotationimplies that they cannot be rotationally flattened, but are sup-ported by anisotropic velocity dispersions. It is yet not clearwhether – or to what extent – elliptical galaxies form throughmergers, but many models, both analytical and numerical, showthat merging is a viable mechanism (e.g., Barnes 1992; Mihos& Hernquist 1996). Exclusive work has been done on ellipticalgalaxies, and equilibrium models in particular. Dissipationlessmodels can reproduce several properties, like ther1/4-law, butwith “realistic” cosmological initial conditions it seems difficultto produce the extremely high phase space densities, implied by

the velocity dispersions, and half-light radii (White 1987). Dueto its complexity, relatively little work has been done on dis-sipational formation of ellipticals. However, it has been shown(Carlberg 1984; Carlberg 1987; Lake 1987; Lake & Carlberg1988a; Lake & Carlberg 1988b) that dissipation can solve manyof the difficulties with high phase space densities, in the contextof cosmological initial conditions.

The layout of this paper is as follows: in Sect. 2 an outlineof the SNE model, and a comparison with previous approachesto SF in cosmological galaxy formation models, is presented. InSect. 3, a description of the implementation of the SNE modelinto the SPH method is given. Sect. 4 presents simple tests ofthe SNE model, and in Sect. 5, the dependence of the dominantsupernova evaporation parameterA is investigated for a simplegalaxy formation model. Finally, in Sect. 6, a set of simulationsin a more realistic cosmological setting are performed. Sect. 7contains the conclusions and a discussion of the results are pre-sented. Sect. 8 contains a discussion on the effects of numericalresolution.

2. The supernova evaporation model comparedto previous SF models

The SNE model is a step on the way towards improving someof the previous phenomenology by modelling some key aspectsof the physical processes believed to be important in the ISM. Itmay not yet beproven beyond doubtthat these processes occur,or are really the dominant ones. The thermal instability is per-haps the least well founded of these, on observational grounds,but offers a very plausible mechanism for the formation of coldclouds. Cold clouds and the effects of massive stars are, on theother hand, well supported, on both observational and theoret-ical grounds. We shall not go through the full details of theSNE model, as it is excellently presented by YKKK, but weshall highlight some important improvements over the previousmodels. It has sometimes been argued that there have been awide variety of SF models used, but this is not the case for SPHbased simulations. With few exceptions, the recipe has been mi-nor variations of Katz (1992) and Navarro & White (1993). Wewill refer to these as the “Katz recipe”, henceforth.

2.1. Star formation rates

In the Katz recipe, thestellar birth function(SBF) is given by

dρ∗dt

=c∗ρg

t∗(1)

or by a Schmidt law

dρ∗dt

= c∗ρng , (2)

whereρ∗ is the density of stars,ρg is the gas density,c∗ is aconstant, andt∗ is the SF time-scale, here typically chosen as thelocal dynamical time,tdyn = (4πGρg)−1/2 or the maximumof tdyn and the cooling-time. The indexn is constrained byobservations to be typically in the range1–2. With t∗ = tdyn


Eq. (1) is thus equivalent to a Schmidt law of indexn = 3/2. Inthe SNE model the SBF is given by

dρ∗dt

=(1 − β)ρc

t∗(3)

whereρc is the density of cold clouds, which are here treatedas a separate gas density component, of which the evolution isgoverned by the thermal instability and supernova evaporationmechanism. The other gas component is the hot gas density,denotedρh, which is typically at temperatures∼ 106 K (McKee& Ostriker 1977).β is the “supernova fraction”, that is the massfraction of massive stars that are assumed to explode as typeII supernovae on a short time-scale, compared to the SF time-scale. It can thus be regarded as instantly returned to the hotgas phase, theinstantaneous recycling approximation(Tinsley1980), which is assumed in all SF models described here (notethat Tinsley has a more consistent definition ofβ, and denotesit R). The SF time-scalet∗ is chosen as a constant of 0.1 Gyr.Assuming a Salpeter initial mass function (IMF) givesβ = 0.12(YKKK).

The SNE SBF should be a more realistic description if weaccept the thermal instability as the formation mechanism forcold clouds. It may also be argued that the SNE SBF should use aSchmidt-like law, but SF is commonly believed to self-regulate(Koppen et al. 1995) due to supernova feedback, and therebybecomes largely independent of the actual parametrization ofthe SBF. As we shall see later, there is also some support forself-regulation, in Sects. 3.1, 4 and 5.

2.2. Supernova feedback

A more serious difference between the Katz recipe and the SNEmodel lies in the treatment of supernova feedback. Both recipesadd a total supernova energy release of〈ESN〉 = 1051 ergs persupernova explosion. In the Katz recipe, the supernova energyrelease is sometimes divided into a fraction of kinetic energy,fv,and thermal energy,1 − fv. As previously mentioned, if onlythermal energy is injected it is quickly radiated away by thecooling, and thus has little effect on the dynamics. The kineticenergy is injected through a “velocity kick”, i.e. to each SPHparticle’s neighbours, a velocity change is added correspondingto the energy divided equally between each neighbour, and di-rected radially away from the particle in question. (The morephysically correct method of changing the neighbours veloc-ities in such a way that the total energy is conserved has thedisadvantage ofalwaysreversing convergent flows.)

The SNE model adds all the supernova energy as thermalenergy, but adds another important effect in that cold clouds areevaporatedby supernovae. It is assumed that a supernova heatsand evaporates a mass of cold clouds back to the hot gas phasethat is a factorA larger than the mass of the supernova itself.We will refer toA as thesupernova evaporation parameter. As-suming dynamical equilibrium, the previously given supernovaenergy release and hot phase temperatures∼ 106 K, YKKKderive a constraint onA to be in the range of0–250.

The main problem with the Katz recipe is not that it fails toconserve energy, but that the supernova feedback is very depen-dent on the spatial resolution, i.e. the SPH smoothing lengthsh (being the radius that encloses each particle’s neighbours).Typically, h varies likeρ

−1/3g , for a constant number of neigh-

bours. This means that supernovae in low density regions affecta larger volume. This may perhaps be argued for on physicalgrounds, but it is to a large extent just a consequence of the na-ture of the SPH method. Simulations with finer resolution usedfv = 10−4 for a simulation of the encounter of a compact objectwith a galactic disc (Mihos & Hernquist 1994), the aim being tokeep a constant scale height in the disc. The typical smoothinglengths were around 0.4 kpc in the disc. In simulations of lowerresolution (Navarro & White 1993) with smoothing lengths ofaround2–5 kpc, values as low as offv = 10−4 would mostlikely have no effect at all. Typically,fv = 0.1–0.2 was neededin order to show significant effects of supernova feedback. Thisindicates thatfv is not a physical parameter, but just a “fiddleparameter”.

In this respect the SNE model is to be preferred, since thereis less a priori intrinsic resolution dependence in the imple-mentation. However, this does not mean that it would be savedfrom all sorts of resolution effects (some effects are discussedin Sect. 8).

2.3. Other conditions for star formation

All SF simulations that take place in a cosmological setting (e.g.,Katz et al. 1996; YKKK) include a gas density threshold forSF to occur,

ρg > D ρb Ωbar, (4)

whereρb is the critical density in the universe.Ωbar is the bary-onic gas fraction andD is chosen in the range 50–100, to preventSF from starting too early or too intensely, due to the high den-sity at high redshift. YKKK argued that UV ionization fromquasars, active galactic nuclei, and “non-linear structures in theprocess of formation”, suppresses the thermal instability. In lightof the recent results on QSO UV ionization, other mechanismslike UV ionization by early SF should also be considered. How-ever this is not included into the SF model yet. Regardless ofthe nature and effects of these processes we still need a densitythreshold (this is further discussed in the conclusions, Sect. 7.3).

In some of the Katz type recipes, further restrictions areimposed on SF regions to be allowed to form stars, like Jeansinstability, h/c > tdyn (in SPH notation,c being the speed ofsound). Another requirement is that the region is converging,i.e. ∇ · v < 0. Unfortunately, these are largely meaningless,because the requiredreal physical phenomena cannot (yet) beresolved (see also Sect. 8).

2.4. Main parameters

As previously mentioned, there is more realistic physics in-cluded in the SNE model as compared to the Katz recipe, butthere are essentially no more free parameters. The main new


parameter isA. YKKK also introduce a further parameter, thecooling enhancement factorC, which accounts for increasedcooling rate, due to unresolved density and temperature inhomo-geneities. Following McKee & Ostriker (1977), YKKK assumeC = 10, in regions where thermal instability is present (this isdiscussed further in Sect. 8). Strictly speaking, the cold cloudtemperatureTc oruc is also a new parameter, but of much lesserinfluence.

The other parameters belong to the “usual” set assumed forSF in this context:D (needed in cosmological context),〈ESN〉(here represented byuSN), β, t∗, and/or possiblyn andc∗, inorder of estimated decreasing significance (the parametrizationof the SBF, i.e.t∗, n andc∗, have been “rated low”, as they arebelieved to self-regulate away, as previously mentioned).

For the Katz recipe, the corresponding parameters are,fv,corresponding toA but distinctly different to its effect, and theSF parameters.

3. Implementation of star formationand supernova feedback into SPH

3.1. System equations

We have made a straightforward implementation of the SNEmodel equations into the SPH + tree-code of Hultman & Kall-ander (1997). The main difference in respect to “normal” SPHis the separation of the gas densityρg into the hot and coldcomponents,

ρg = ρc + ρh. (5)

For “traditional” reasons we have chosen to evolve the follow-ing variables: the gas densityρg, the hot gas phase densityρh,the velocity fieldv, and the gas thermal energyug (we havea slight notational difference with YKKK, in that we useufor internal energy per unit mass, the traditional SPH notation,whereas YKKK useε). When needed, the cold phase densityρcis then obtained through Eq. (5), and the “hot” thermal energy,uh, throughρgug = ρcuc +ρhuh. The cold gas phase is treatedas isothermal, with a constant thermal energyuc correspondingto a temperature of 9000 K.

As previously mentioned, thermal instability governs therate of formation of cold clouds. Here, it shows its presence in thecontinuity equation for the hot density component. It is derivedunder the assumptions that it is only the hot gas phase thatcools, and when it does so, it produces cold clouds. The entropyof the remaining hot gas phase does not change (there are somearguments against this last isentropic assumption, but it doesnot have a qualitatively crucial impact on the model. This is alsofurther discussed in Sect. 7.4). In principle, thermal instabilitycan be said to occur, for ionized gas, when the cooling time isincreasing with the temperature (Begelman & McKee 1990).For the cooling rates used here (Sutherland & Dopita 1993)this could be simply expressed asTh > 105 K, but then therewould still be regimes below105 K that would be potentiallyunstable (for low metallicities). We have therefore chosen touse no further restrictions on the thermal instability except forEq. (4). Some implications of this will be discussed in Sect. 7.3.

In addition to the equations given in Hultman & Kallander(1997), the following continuity equations for the total and hotgas densities are evolved:

dρg

dt+ ρg(∇ · v) = −

[(1 − β)ρc

t∗

]SF

(6)

and

dρh

dt+ ρh(∇ · v) =

[β(A + 1)ρc

t∗

]SF

−[CΛr(ρh, uh, Z)

γuh − uc

]TI

+[dρh

dt

]CC

. (7)

The factor(A+1) in Eq. (7) accounts for both the mass exchangedue to evaporation of cold gas and transfer of supernovae massto the hot gas component.Λr is the radiative cooling rate andZis the gas metallicity.γ is the ratio of specific heats,γ = 5/3 forthe hot gas component that is assumed to be always ionized. Forclarity, we have put subscripts on the source terms, indicatingtheir origin: “SF” for star formation and supernova feedback,“TI” for thermal instability, and “CC” as follows:[dρh

dt

]CC

= −ρh/tcool Th < 2 × 104K;

0 otherwise. (8)

This term handles the case when the hot phase starts to be-come as cool as the cold phase. This typically happens duringthe so calledcatastrophic cooling, i.e. tcool tdyn, wheretcool ≡ uh/[duh/dt]cool. All the bracketed source terms arealso subject to the gas density threshold condition, Eq. (4), andthey are otherwise zero. Also the cooling enhancement factorC is unity unless Eq. (4) is fulfilled. The equation for the gasthermal energy is:

ρgdug

dt+ P (∇ · v) = ΓSF − Λ, (9)

where the pressure is given by

P = (γ − 1) ρhuh, (10)

and

Λ = C Λr(ρh, uh, Z) + Λc(ρh, uh, z), (11)

being the sum of the radiative and Compton cooling rates (theradiative cooling rates will be described in Sect. 3.2). ForΓSF,the supernova and SF heating source term, we have

ΓSF =ρc

t∗[βuSN + (1 − β)(ug − uc)] , (12)

whereuSN = 〈ESN〉 / 〈MSN〉 is the supernova thermal en-ergy. Assuming a mean supernova mass of〈MSN〉 = 22M,this gives a corresponding supernova temperature of〈TSN〉 ≈108 K. Note that the supernova evaporation parameterA doesnot appear in Eq. (12), and thus nor in Eq. (9). It only affects theevaporation of cold gas to the hot component.

The previously given constraint of0 < A < 250 did not takecooling into account. It is easy to see, however, from Eq. (12)


that thermal equilibrium is not possible without cooling. As-suming that cooling balances the SN heating in Eq. (9), i.e. self-regulation (SR), we get:

ρc ≈ Λt∗β uSN

(13)

to a very good approximation, ifuSN uh uc. Thus, if self-regulation is in effect the amount of cool clouds is just directlyproportional to the cooling rates, andnot even dependentonA!As mentioned, the SBF is thereby roughly proportional toρ2

g, nomatter what parametrization we choose. SR also means that theevaporation in Eq. (7) cancels out the thermal instability, giving

uh ≈ uSN

γ(A + 1) − (1 − β)/β≈ uSN

γ(A + 1), (14)

where we have assumedug ≈ uh uc. So, in principle the hotISM temperature is fixed byA. We obviously needγ(A+1) >(1 − β)/β ≈ 7, for SR to be at all possible. A typical ISM (hotmedium) temperature ofTh = 5.5 × 105 K (McKee & Ostriker1977) givesA ≈ 120. Local conditions for SR can in principlebe derived by linear stability analysis, but global behaviour cangenerally only be investigated numerically (Koppen et al. 1995).Depending on the initial conditions it takes for example sometime before SR can set in.

Like YKKK, we havenotchosen to treat the cold and hot gasasseparatefluids, but assume that the thermal instability, ISMstructure, and SF are processes that occur below the currentlyavailable numerical resolution. Hence, the cold cloud densityρc is better thought of as representing a mass fraction of coldclouds. This is also reflected in the equation of state, Eq. (10),where only the hot component contributes to the pressure. Sincethe cold clouds and hot gas are expected to be roughly in pressureequilibrium (McKee & Ostriker 1977), this prescription shouldnot be too unrealistic. However, it should be noted that effectslike “drag” between hot gas and the cold clouds (Samland et al.1997), or effects on cloud velocity dispersions by supernovae,can then not be directly included in the present method.

Each SPH particle is assigned an additional value forρh, andthe separate gas phases thus share phase-space coordinates, etc.An alternative approach would be to use two separate SPH par-ticle species, for the cold and hot gas components. This would,however, introduce further complications, in terms of creationand destruction of particles, during the transfer between the hotand cold gas phases. Worse is that it would contradict part ofthe working hypothesis: that, although the cold clouds and SFoccur below resolution, we might still hope to resolve the hotgas phase, this being the dynamically important one.

YKKK chose to create star particles of varying mass, ac-cording to Eq. (3), at each time-step. It may seem plausible thata high SBF produces heavier particles, better mimicking “starbursts”. (Since star particles represent groups of many stars, sev-eral millions typically, it may be reasonable to group stars thathave formed during a shorter period of time, and likely similarconditions.) However, it is equally dependent on the time-step,which may vary, and what is worse: light particles are very sus-ceptible to two-body relaxation effects. It is also more practicalto be able to control the number of star particles formed.

For these reasons, we chose to proceed in a fashion similarto the partition into cold/hot gas components. Each gas particleis considered to have a fraction of its mass,f∗, in stars, which iskept track of by integrating Eq. (3). This star mass-fraction,f∗,is thereforenot included when gas dynamical SPH quantitiesare calculated, but only in the gravitational interaction. Whenf∗ reaches 90%, the particle is excluded from the hydrodynamicinteraction, and treated as a pure star particle.

The SPH implementation of equations (6-12) is otherwisestraightforward, i.e. they are “written as they stand”, since nospatial derivatives appear, except for∇ · v, which is calculatedin the standard SPH way. Eq. (9) is solved with an implicit in-tegration method as in Hultman & Kallander (1997). Eq. (7) isgiven its own Runge-Kutta tolerance criterion, on the individ-ual time-steps. See Hultman & Kallander (1997) and Navarro& White (1993) for details. The gas density,ρg, is updated bySPH summation at each individual time-step, conserving thehot and cold gas fractions. Only at time-steps “in between” is itpredicted using Eq. (6).

3.2. Metal enrichment and radiative cooling

Given a prescription for the SBF, it is fairly straightforward toderive an equation for the local metal enrichment, here describedby the single metal abundance parameter, themetallicityZ, thatwill be generally interpreted as [Fe/H]. It will be assumed thatheavy element abundance ratios are scaling relative to the So-lar, as given by Sutherland & Dopita (1993). Following Tinsley(1980) and assuming the instantaneous recycling approxima-tion, we derive a continuity-like equation for the metallicity:

dZ

dt+ (Zf − Z)∇ · v =

[y (1 − Z)

ρg

dρ∗dt

]SF

, (15)

wherey is the yield, i.e. the mass of metals ejected per unitmass locked into stars.Zf is the mean metallicity of the gassurrounding the volume element in question (read: the particle’sSPH neighbours).

Although the term that takes into account the transport ofmetals into and out of different gas regions looks reasonable,it turned out to cause problems during trial tests (Sect. 5). Themetallicity attained unphysical values (i.e.Z > 1 andZ < 0).The cautious reader might object that∇ · v is already presentin the continuity equations, Eqs. (6) and (7), and apparentlycause no problems there. A more plausible cause may be thatthe larger scale motions, resolved at the actual resolution (i.e.h ∼ 1–10 kpc), are much larger than the motions at the physi-cal scale, of perhaps1–10 pc (molecular clouds), and thereforeoverestimate∇·v (see also Sect. 8). Also, it could be preferableto solve the continuity equation for the density of metals insteadof advecting the metallicity. Possible solutions to this problemhad to be left for future work.

In any case, the transport term had to be dropped, leavingus with

dZ

dt=

[y (1 − Z)

ρg

ρc

t∗

]SF

. (16)


104

105

106

107

10−36

10−35

10−34

K

Jm^3

/s

Fig. 1. The normalized cooling function as function of temperaturegiven by Sutherland & Dopita. Different curves correspond to differ-ent gas metallicity, primordial composition [Fe/H] = - 3 (solid curve),[Fe/H] = - 2 (dashed curve), [Fe/H] = - 1 (dotted curve) and [Fe/H] =0, solar composition (dot-dashed curve).

Thus, an individual particle is treated as a closed system, in themetal enrichment sense. However, this is not an overly crudeapproximation since it is equivalent to assuming that the metalcontent traces the gas flow. In that approximation the transportterm vanishes, i.e.Zf = Z.

For the yield we use the approximation given by Tinsley(1980), of y ≈ 0.8Z, for the solar neighbourhood, whereZ = 0.02. In reality, the yield is also dependent on, for exam-ple, the IMF but, in context of the other approximations made,this should be a reasonable choice. Eq. (16) is integrated foreach particle, using its corresponding Runge-Kutta tolerancecriterion.

The effect of the metallicity is incorporated into the radia-tive cooling rates, Eq. (11). A look-up table inTh and [Fe/H] (=log(Z/Z)) is used, as given by Sutherland & Dopita (1993)(see also Fig. 1). Collisional ionization equilibrium is assumed.This may overestimate cooling rates by a factor of∼ 3, butin comparison to other uncertainties, for example limited res-olution and the (yet) uncertain feedback effects by the thermalinstability on the cooling rates (Sutherland & Dopita 1993), thisshould still be a reasonable approach. The effects of an ioniz-ing UV background field, as in Kallander & Hultman (1998),wasnot included. The reason is that UV radiation from stars isnot yet included, nor is the feedback effect of UV radiation onthe thermal instability. Pure line cooling effects of an ambientUV background field would therefore be less meaningful in thiscontext.

4. Simple tests of the SNE model

In order to check the SF implementation and the parameter val-ues, and to compare with the simple test made by YKKK, acouple of similar simulations were performed. The analogy toa “single cell” as used by YKKK was here represented byoneSPH particle, with gravitational and adiabatic flows disabled.The density was set to correspond to 76% hydrogen (by mass),

Fig. 2. The time evolution of SF in a homogeneous medium, of solarcomposition, [Fe/H] = 0. The density corresponds to 76% hydrogen,with nH = 6 × 10−4 cm−3. Gravity and mass-flows were switchedoff, and the initial temperature wasTh = 106 K. The differently styledcurves shows the effect of different values of the supernova evaporationparameter,A. The cooling enhancement factor was set toC = 10.

Fig. 3. The same as Fig. 2, but with [Fe/H] varying as given by Eq. (16)

with nH = 6 × 10−4 cm−3, and the initial temperature toTh = 106 K. Solar composition and enhanced cooling,C = 10,was assumed. A number of runs were made with different val-ues of the supernova evaporation parameter,A = 0, 20, 100,200 (note that the case forA = 200 is not presented in YKKK).

The results agree rather well with YKKK, as can be seen bycomparing Fig. 2 with Fig. 3 of YKKK. There are minor differ-ences in terms of, for example, slightly smaller stellar fractionsbut this is not too surprising since slightly different implementa-tions of the cooling rates are used. In fact, it is rather reassuringthat the differences are not larger, since the cooling rates havea strong influence, as we shall see.

The general trend here is that higher supernova evaporationgenerally increases the time averaged SFR, and thus the stellarfractions. This is far from intuitive, but it is just a consequenceof the strongly non-linear model. One important effect is thata high supernova evaporation actually cools the hot phase. The


Fig. 4. The metal enrichment corresponding to Fig. 3.

Fig. 5. The hot phase temperatures corresponding to Fig. 3. The finaltemperatures corresponding to Fig. 2, constant solar composition, arevery similar, although SR sets in earlier.

inverse relationship betweenA andTh, Eq. (14), can now beseen in Fig. 5. As expected, theA = 0 case never becomes self-regulated, but the other cases do, most notably forA = 100 and200, and rather quickly so, after some 0.05 Gyr.

Some insight is given by the behaviour of the cooling rates(Fig. 1) at the hot phase temperatures. The cooling rates increasedramatically as we descend from around107 K down towards105 K and this therefore leads to a higher SFR, through theincreased formation rate of cold clouds according to Eq. (13).The “wiggling” behaviour of theA = 200 curve in Fig. 2, justwhere hot and cold curves touch, does not reflect numericalinstability, or any related problem. It just reflects the effect ofone of the peaks of the cooling curve, before the solution attainsthe self-regulated behaviour. All these results were tested fornumerical accuracy by varying the time-step accuracy parameterby orders of magnitude. One should refer to Navarro & White(1993) and Hultman & Kallander (1997) for a description of theintegration scheme used.

Although it is not very interesting to plot, it is worth mention-ing that in runs with primordial composition ([Fe/H]< −3) and

A = 100, the cooling rate was not enough to form any signifi-cant amount of cold clouds, and the final stellar fraction was lessthan 3%. This again indicates the importance of increased cool-ing, through metal enrichment, as previously demonstrated byKallander (1996) and Kallander & Hultman (1998). ReducingC = 10 to C = 1, using solar composition, had less influencebut it reduced the final stellar fraction to around1/2.

Fig. 3 shows the results of the corresponding set of runswith the more realistic metal enrichment described in Sect. 3.2,Eq. (16). The differences from the constant solar compositionruns, Fig. 2, are actually smaller than they may seem at a firstglance. There is a slight delay (exaggerated by the logarithmicscale) before SF starts, and lower peak cold fractions, but themain difference is the slightly larger final stellar fractions. Thisis because the metallicity quickly rises above solar levels, as canbe seen from Fig. 4. Note, however, that the delay is significantlylonger before SR sets in, after around 0.3–1.0 Gyr.

A factor of 100 higher density was also tried (not plotted).Higher SFRs, and somewhat more rapid transitions into self-regulation, were observed.

From this we conclude that the metal enrichment modelworks rather well, or in reverse, that the approximation usedby YKKK, assuming solar composition cooling rates, is rathergood. One might argue against low values ofA from the re-sulting rather high self-regulation temperatures of the hot gas.A ∼ 100 seems adequate, giving typical ISM hot medium tem-peratures. However, it must be remembered that dynamical ef-fects have not yet been taken into account.

5. Simulations in limited cosmological environment

Cosmological simulations commonly use a large box, with asize ranging from tens to hundreds Mpc as simulation volume,usually with periodic boundary conditions. The larger the box,the better the statistics, as many galaxies and clusters form. Thecomputational cost is however high, and becomes a problemwhen many simulations need to be performed. There are someother purely technical factors, seldom mentioned. The large datastorage needed becomes a severe problem, in particular whendetailed information over the entire time evolution is desired.Also, the simulation data analysis becomes significantly morecomplicated, as forming objects have to be found and traced.

5.1. Initial conditions

Before performing larger scale simulations, we wanted to ex-amine the dependence of the SNE model for several values ofthe supernova evaporation parameterA. Therefore, we chosethe initial conditions of Kallander & Hultman (1998) and Hult-man & Kallander (1997), being a development of those of Katz& Gunn (1991). In fact, we have used exactly the same initialconditions as in Sect. 4.4 of Hultman & Kallander (1997), us-ing a standard “flat” CDM cosmological model specified byΩ = 1, Ωbar = 0.05, Λ = 0, σ8 = 0.66 and a Hubbleconstanth100 = 0.5. The total mass was1012 M inside asphere, with an overdensity that corresponded to a3σ peak,


Fig. 6. The first star particles and the gas velocity field during themain collapse of theA = 100 simulation, at three orthogonal projec-tions centred on the highest density. A spherical region is shown. Longarrows show angular momentum vectors.

when the density field was smoothed at that same mass-scale.The system was started in solid body rotation corresponding toλ = 0.05, approximating the otherwise neglected tidal interac-tions. 4000 of each gas and DM particles were used, thus havingmasses of1.3× 107 and2.6× 108 M, and given gravitationalsmoothing lengths were 2 and 5 kpc, respectively. The simula-tions were started at a redshift ofz = 30, before the resolvedscales have grown non-linear. The gas was initially at a tem-peratureT = 9000 K, but considered to be in the hot phase,i.e. no initial cold clouds. The SF density threshold parameterwas set toD = 100, and the SF time-scale tot∗ = 0.1 Gyr.The cooling enhancement was not used, i.e.C = 1 (see Sect. 8for a discussion). Artificial viscosity parameters,αvisc = 1 andβvisc = 2.5, were used, and the number of neighbours was 64.

5.2. Results

A number of runs with different values ofA were made, andsix of these were later selected for study. They were denoted“TS” (for “truncated sphere”). The general evolution starts in aHubble-flow at redshiftz = 30 that retards due to the overden-sity, roughly following a spherical top-hat solution. A numberof smaller objects merge hierarchically to form a proto-galacticobject. Before turnaround, SF starts in the most dense regions,as can be seen in Fig. 6, representing an instant during the “maincollapse” of theA = 100 run. Star formation then proceeds asthe dark matter halo virializes. It is worth recalling that “col-lapse” is a rather poorly defined concept in hierarchical scenar-ios. The “main collapse” can be sharply defined here because ofspecialized initial conditions, but in the general case rather in-dividual perturbations collapse and/or merge at different times.As expected, the gas is quickly blown out of the potential wellin theA = 0 simulation. The other simulations proceed in a less

Fig. 7. Similar to Fig. 6. The star particles and the gas velocity field atthe end of theA = 100 simulation. The star particles have an ellipticaltriaxial distribution with an ellipticity,ε = 0.25. The remaining gas,constituting less than 0.5%, rotates relatively slowly at∼ 70 km s−1.Long arrows show angular momentum vectors.

Fig. 8. The time evolution of the SFR for the selected simulations(samplings are at 0.3 Gyr intervals).

violent manner. Star formation rates peak at around15 M/yrin generally all the simulations, and then falls off roughly ex-ponentially, with characteristic times around 1–2 Gyrs, notablylonger than the SF time-scale,t∗ = 0.1 Gyr, as can be seen inFig. 8. The forming stellar distributions have an elliptic (oblate-to-triaxial) shape, and generally follow the de Vaucouleursr1/4

surface density profile. This can be seen from Fig. 7 and 9. Forthe runs withA = 50–150 the fit is very good. Ellipsoidalshapes were determined using an iterative procedure (Dubinski& Carlberg 1991; Katz 1991; Warren et al. 1992).

It should here be noted that in the corresponding simulationin Sect. 4.4 of Hultman & Kallander (1997) a disc shaped objectformed, though quite centrally concentrated. This was however


Fig. 9. The surface density of the stellar component, plotted againstelliptical radius, for theA = 100 simulation. The solid line is a closestfit of the de Vaucouleurs,r1/4, profile (theelliptical radiusbeing de-fined asr = (ab)1/3(x2 +(y/a)2 +(z/b)2)1/2, wherea ≤ b < 1 arethe axis ratios; the normalization with the geometric mean,(ab)1/3,gives a better comparison with purely spherical estimates).

Fig. 10. Theeffective radiusRe of the stellar component, plotted forthe simulations. The fit is sometimes too bad to be used (i.e. for theA = 0 case), and then the closely related half mass radiusrh is usedinstead.

very typical of these simulations, and an artifact caused by theover-cooling problem.

The most striking feature in the simulations here, is the im-pact ofA on the resulting effective radiiRe (Re is defined asthe iso-contour radius containing half the surface mass of ther1/4 fit, cf Binney & Tremaine 1987) of the stellar component,see Fig. 10. There is a distinctive non-monotonic behaviour ofthe scale-lengthRe. As mentioned, theA = 0 simulation istotally blown apart. This is also the case forA = 250, but to alesser extent, and it still follows ther1/4 law quite well, withRe ∼ 40 kpc. Moderately low values ofA = 50 and 75 re-sult in very compact and massive objects withRe = 0.6 and1.4 kpc, respectively. Intermediate-to-high values, ofA = 100and 150 give equally massive, but less compact objects with

Fig. 11. The average gas temperature.

Re = 5.9 and 8.0 kpc, respectively. This is close to 6 kpc asobserved for bright ellipticals (Binney & Tremaine 1987). Highvalues,A = 150 and 250, continue a trend of “blow up”. This isalso the case for low values,A = 0 and 25, with an even moredramatic blow up. In fact, theA = 0 and 25 runs never reachthe expected high temperatures, but blow apart as soon as SFstarts.

The physical reasons for this are not entirely trivial. Thereare actually very small spatial differences during the initial col-lapse, apart from theA = 0 and 25 cases. Our interpretationis that self-regulation is the main mechanism for understandingthe evolution of the system. There is little evidence that self-regulation is strictly maintained during the collapse. This may,to some extent, be caused by the fact that SR does not set infast enough. Cooling and heating rates were examined, and SFheating exceeds cooling by up to a factor of 5, peaking aroundthe collapse but otherwise fairly equal. This will therefore notcompletely ruin the self-regulation argument. A mere factor of 5will not make the temperatures, and in particular the cold phasedensities (Eqs. 14 and 13), change by more than that amount. Inthis way a type of quasi self-regulation is still maintained.

Low values ofA = 50 and 75 more efficiently halt thecollapse before the main bulk of SF occurs. The reason is that theself-regulation process drives the temperature, Eq. (14), closerto the local virial temperature. A simple estimate for the virialtemperature of forming CDM halos (e.g., Rees 1986; Evrardet al. 1994; Kallander & Hultman 1998) can be expressed as

Tvir = 9 × 105 (1 + z)(

Mhalo

1012 M

)2/3

K. (17)

TheA = 50 and 75 runs have, for example, the highest “collapsetemperatures” (Fig. 11) as well as pressures. It is evident fromFig. 12, that the runs withA = 50 and 75 form less stars earlyin the simulation.

The runs with intermediate and high values,A = 100–250,on the other hand, form more starsbeforethe collapse is halted.This has at least two important effects. First, star particles areformed at higher velocities (Fig. 13). Second, and more impor-


Fig. 12. The mass fraction of the stellar component, within a radius of40 kpc, centred on the most dense region.

Fig. 13. Velocity dispersion (line of sight) for the star particles. Notethat this is the velocity dispersion for all star particles and will thereforebe an underestimate of the more commonly used central values. Theseare here harder to obtain due to poor statistics.

tant, small perturbations that collapse early are sensitive andeasily disrupted by the supernova heating. The smallest struc-tures that we can hope to resolve consist of about 50 particles.Thus, according to Eq. (17), they will have “disruption temper-atures” at aboveT ≈ 4 × 105(1 + z) K.

A higher A causes lower temperatures, and this in turncauses a higher “pre-collapse” SFR, as we could see in Sect. 4.The valuesA = 100 and 150 apparently give just the right tim-ing, setting the phase space densities to form a stellar distributionthat is comparable to what is observed.

It is remarkable how sensitive the whole process is. Thedifference in SFR, just as it rises (Fig. 8), might seem totallyinsignificant (except forA ≤ 25) but the impact on, for example,the total baryonic binding energy is substantial, as can be seenfrom Fig. 14.

Fig. 14. Total binding energy for the gas and stars (the baryonic com-ponent).

Fig. 15. The mass fraction of gas and stars (baryonic component),within a radius of 40 kpc, centred on the most dense region. The darkmatter halo is fairly stationary when it has virialized after around3–4 Gyrs.

Fig. 15 shows the fraction of gas and stars within a 40 kpcradius, containing the galactic object. There is a substantial lateinfall (i.e. a cooling flow) of gas for theA = 50–150 sim-ulations. Late infall and non-negligible present-day SFRs aregenerally not connected with elliptical galaxies, but there areseveral indications of the contrary (Silk & Wyse 1993).

Fig. 16 shows the gas fraction within 40 kpc. At the end ofthe simulations most of the gas has either been converted intostars, or blown out. The remaining gas fraction is perhaps notas low as what is observed in elliptical galaxies, or rather therelatively high final SFRs in these simulations may be a problem(this will be discussed further in Sects. 6 and 7.4).

There is a strong impression that these objects form“outside-in”, in the sense that the outer parts (i.e. with widerstellar orbits) are formed first to a high degree, and that the cen-tral parts form by star-forming gas falling into the centre of the


Fig. 16. The mass fraction of gas, within a radius of 40 kpc, centredon the most dense region.

Fig. 17. The total angular momentum of the gas and stars (baryoniccomponent).

DM potential well, as the DM halo relaxes. This can to someextent be seen from Fig. 10, where at least forA < 200 theeffective radii quickly shrink to their final sizes.

The baryonic component loses some angular momentum tothe DM halo (cf Fig. 17). This is due to the sub-clumps of gas(and stars) being affected by dynamical friction effects fromthe DM halo during the collapse. The loss here is smaller whencompared to Kallander & Hultman (1998), because the clumpsare, in this case, less dense, due to the supernova heating thatwas not taken into account in previous simulations.

The average gas metallicity rises quickly to around [Fe/H]≈−0.5 (Fig. 18), then slowly decreases due to metals being lockedup in stars, but also due to infall of less metal rich gas. Fig. 19shows the age - metallicity relationship for the star particles.The main bulk of star particles have formed quickly, are olderthan 8 Gyrs and have high metallicities (a tiny fraction have lowmetallicities, thus there is no “G-dwarf problem”). There is alsosome scatter, at around 0.2 dex. The metallicity decreases for

Fig. 18. The average metal content of the gas, within a radius of 40 kpc,centred on the most dense region.

Fig. 19. Age - metallicity relationship for the stellar component, at theend of theA = 100 simulation. The “age” is here defined for the timewhen the star particle is formed, i.e. when 90% of the gas in an SPHparticle have been consumed. This also goes for the metallicity, whichis simply inherited from the parent gas particle. Thus, theaveragemetallicity of a star particle will be overestimated (perhaps by a factorof 2 or 0.3 dex, assuming linearly increasing metallicity). See text.

stellar ages younger than 10 Gyrs, due to the later infall of non-enriched gas. It is not fully clear whether this is reasonable ornot. Stellar ages generally can not be determined for ellipticalgalaxies (i.e. the “age - metallicity degeneracy”). The situationhere may well be a consequence of neglect of the transportterm in Eq. (16), and that infalling gas is not mixed with thepresumably metal rich gas already present.

It must be stressed that star particles can of course not beinterpreted as individual stars. Metallicities and ages for starparticles have been recorded when only 10% gas remains, andthus represent the stellar populations that form late in the SF his-tory of the co-moving fluid element (read: particle) in question.Keeping this in mind, we may still make a comparison with


Fig. 20. Metallicity plotted against radius for the gas and star particles,within a radius of 20 kpc, centred on the most dense region.

observations of bulge stars in the Galaxy (Sadler et al. 1996;Castro et al. 1996). Average [Fe/H] and the scatter is not all thatdifferent. At the same time the situation here is not totally un-like what has been observed by Edvardsson et al. (1993) for oldstars in the disc. A large proportion of the young disc stars havelow metallicities [Fe/H]≈ −0.3. From this we may concludethat the situation in ellipticals may not be all that different ascompared to discs, at least in this respect.

Fig. 20 shows the metallicity plotted against radius. Possiblythere is a gradient of around−0.05 dex in the gas, but the scatteris large. For the star particle distribution, there is a rather positivegradient. This is again due to infalling gas forming stars in thecentre of the potential.

We conclude that values aroundA ∼ 100–150 are plausible,giving reasonable ISM hot medium temperatures, and produc-ing elliptically shaped objects with scale-lengths similar to theobserved ones for bright elliptical galaxies.

6. Simulations in more extended cosmological environment

6.1. Initial conditions

In order to set up more realistic cosmological initial conditions,but still be able to use equivalent or higher resolution as in theTS simulations in Sect. 5, we have taken inspiration from themethod of Katz & White (1993) and Navarro et al. (1994), usinghigher resolution only in regions that are especially selected forcloser study. Using the same cosmological model as in Sect. 5.1,a coarse resolution “trace simulation” was first performed, inorder to identify regions where structure forms. Previous exper-imenting has shown that a sphere about 5–10 co-moving Mpc inradius is needed to form a group of bright (∼ 1012M) objects,since one of these objects occupies roughly a sphere of radius1.5 Mpc. For regions smaller than 5 Mpc radius, most masstends to end up in one object, due to the lack of surroundingobjects, that can keep them apart.

Around 20,000 collisionless (DM) particles were set up on acubic lattice inside a sphere of 10 Mpc in radius, and zero over-

density. The particle positions and velocities were perturbedaccording to the CDM spectrum, and evolved fromz = 30 toz = 0. A subset of these particles were then selected, providedthey belonged to a group, and the group itself was not too closeto the spherical boundary, thereby avoiding edge effects. Particlegroups were determined using a “friends of friends” (FOF) al-gorithm, on subsets of particles having overdensities exceeding100, and linking lengths equal to the particles’ SPH smoothinglengths. Groups were finally corrected to be spherical regions ofradii r200, having mean overdensities exceeding 200, similarlyto YKKK. These selected particles then define a suitable regionfor increased resolution, as they are relatively isolated from theenvironment.

Since the particles are initially placed on a cubic lattice,they define a volume of cells. The cells corresponding to the se-lected particles thus define a “higher resolution” (HR)-volume,when the particles were traced back to their original positions.It was made sure that the HR-volume was compact, i.e had no“holes” in it, and an extra padding layer of cells was added tothe boundary. This lessens the likelihood of contamination bylower resolution (LR) particles. The increased resolution allowshigher accelerations, as well as additional density fluctuations,and therefore there is always a tendency for LR particles to bepulled into the HR-volume. The padding decreases this problemsignificantly. Due to its simplicity, we use the technique of split-ting the LR particles into powers of 8 new HR particles, placedat cubic positions, and with a factor of two smaller gravitationalsmoothing lengths. The new configuration was then perturbedwith the original CDM spectrum, but now allowing for higherfrequency density fluctuations, due to the increased resolution.The procedure described above might seem somewhat compli-cated, but it is worthwhile to keep the HR-volume as tightlyfitting as possible, as the HR particles become very many andare of course very computationally expensive to evolve.

A different aspect is that it may seem dangerous to haveparticles, 64 or even 512 times more massive, relatively closeby. Other methods (e.g., Katz & White 1993) have used shellsof particles of radially increasing mass. We have not found thisnecessary with this method, since the likelihood of contamina-tion actually becomes very small. More intermediately massiveparticles in the immediate surroundings only increase the con-tamination likelihood, as they also tend to become very numer-ous, i.e. computationally too costly and not much use. Shouldcontamination still occur, it is not as catastrophic as it may seem,because the LR particles have correspondingly larger gravita-tional smoothing lengths, and will only gradually degrade massresolution, not cause “catastrophic two-body scattering”, as onemight perhaps expect. As we shall see, this scheme is reason-ably efficient, in terms of the HR particles ending up where theyare useful (Table 1).

In the first set of two simulations, denoted CR2, two suc-cessive splittings were used, resulting in approximately 120,000particles in the HR-volume, in addition to the around 20,000 sur-rounding LR particles that essentially only provide tidal forces.This corresponded to a total mass of2.7 × 1013 M in the HR-volume being somewhat irregular 7 co-moving Mpc across. An


equal number of gas particles was added, corresponding to abaryonic fraction of 5%, and DM particle masses were corre-spondingly adjusted. The gas and DM particles then had massesof 1.1 × 107 and2.1 × 108 M, and gravitational smoothinglengths of 1.9 and 5.0 kpc, respectively. Other parameters werethe same as in Sect. 5.1. The only difference between these twosimulations was the value ofA, which was 100 and 200 respec-tively.

For the second set of two simulations, denoted CR3, threesuccessive splittings were used, and a different realization ofinitial CDM fluctuations, since the resolution of the CR2 real-ization was too low. The purpose of these simulations was alsoto check for resolution effects, using further increased resolu-tion. The number of HR particles was similar (130,000), butthe HR region was here chosen much smaller,3.7 × 1012 M,selected to lie along a “filament arm” in the forming structure.It was roughly 3–4 co-moving Mpc across, the goal being toform one massive object. The padding was here more extended,to safely avoid contamination by the now 512 times more mas-sive LR particles (note that if the entire simulation volume hadbeen filled with HR particles, there would have been around2 × 107 of these). Here gas was only added in the HR region.The particle masses were1.4 × 106 and2.6 × 107 M for gasand DM particles, respectively. Gravitational smoothing lengthswere 0.92 and 2.5 kpc, respectively.A = 150 was tried in thefirst of these simulations, andA = 200 in the second.

6.2. Global results

Table 1 shows some global quantities for the different simula-tions. These results are roughly in agreement with those reportedby YKKK. The fractions of stars relative to the closure density,Ω∗, is around one percent, close to accordance with observations(Binney & Tremaine 1987). Still, they are only around one halfthe values of YKKK. This may be partially explained by thefact that YKKK used a higher baryon fraction ofΩbar = 0.08(which is on the higher end of what is allowed by nucleosyn-thesis constraints). Taking this into account, the CR2 runs agreerather well with the YKKK model # 3. The average stellar den-sities〈ρ∗〉 are roughly in agreement. Atz = 0, average SFRsare significantly lower than those of YKKK, and thus in betteragreement with the value of 0.013M kpc−3 Gyr−1 by Gal-lego et al. (1995). At higher redshifts, the〈SFR〉 is very highif compared to observations and hierarchical models (Madau1997; Pettini et al. 1997). The increase in〈SFR〉 with redshiftis reproduced, but the slope is rather high.

For the CR3 runs, the differences with YKKK are larger.Ω∗is here between YKKK model # 7 and 9, atz ≈ 1, but lower thanmodel 6, atz = 0 (note, however, that YKKK were generallynot able to continue their corresponding runs toz = 0, due tothe small box size). There is a trend where the higher resolutiongives less SF. The volume averaged stellar densities aremuchlower. The〈SFR〉 is here rather too low, but the slope is inbetter agreement. On the other hand, since only ellipticals areformed in these simulations, we should expect lower SFRs atlow redshifts, in order to be observationally consistent. Note,

Table 1. Some global quantities for the cosmological simulations (val-ues are atz = 0 if not otherwise indicated).

Quantity CR2 CR2 CR3 CR3

Evaporation parameterA 100 200 150 200DM particlem (106M) 209 209 26 26Gas particlem (106M) 11 11 1.4 1.4DM smoothingε (kpc) 5.0 5.0 2.5 2.5Gas smoothingε (kpc) 1.9 1.9 0.92 0.92HR mass(1012M) 27 27 3.7 3.7HR size (Mpc) 7 7 3 3fDM group (%) 36 37 41 41fHR−DM group (%) 43 42 26 26fgas group (%) 15 14 29 18Ω∗ (z = 0) (%) 1.2 1.3 0.89 0.60Ω∗ (z = 0.83) (%) 0.78 0.89 0.40 0.36Ω∗ (z = 1.88) (%) 0.37 0.47 0.18 0.17〈SFR〉1 (z = 0) -1.6 -1.4 -2.8 -3.8〈SFR〉 (z = 0.83) -0.39 -0.96 -1.7 -2.5〈SFR〉 (z = 1.88) -0.071 -0.44 -1.5 -1.9〈ρ∗〉 (M kpc−3) 1.9 1.4 0.038 0. 1

1) Co-moving volume-averaged SFR, inlog10 units of M kpc−3

Gyr−1.HR mass is the total mass in the HR regions, and HR size the approx-imate size across.fDM group is the mass fraction of DM ending up ingroups, i.e. the “collapsed” fraction with mean overdensity above 200.fHR−DM group is the corresponding fraction of the higher resolutionDM, ending up in HR-groups, i.e. the “useful” fraction of particles thatcan be used for analysis.fgas group is the mass fraction of gas endingup in groups.Ω∗ is estimated as the mass fraction of stellar materialdivided by the total HR mass. Volume averaged quantities〈...〉 arecalculated for the SPH volume the particles occupy.

however, that for the CR3 runs, the HR region is a selected“filament region”, only∼ 3–4 Mpc across, and may thereforenot be large enough to make reliable volume averages. This isto some extent also valid for the CR2 simulation, representing agroup of galaxies, and is clearly not representative as a cluster.

For the CR3 runs, there is a reverse trend in that a highersupernova evaporationA gives less SF. This was, in principle,seen already in the TS runs in Sect. 5. Higher resolution may ofcourse account for this effect. YKKK do not observe this trend,but it must be emphasised that the simulations presented hereare of significantly higher spatial resolution, in dense regions,and may thus probe into a regime not explained by the YKKKsimulations.

The comparison of resolution is, however, not all straight-forward, mainly because different numerical methods are beingused. The PPM/PM code of YKKK had a higher mass resolutionfor the DM (105 M), but had lower spatial resolution, 5.8 kpcas the highest, atz = 1. If continued toz = 0 these valueswould have been twice as large because of the fixed co-movingcell size. The spatial resolution for SPH varies with the localdensity, but was here typically less than 1 kpc for the stellarcomponent in massive objects, which should be compared tothe 6–40 kpc of YKKK.


Fig. 21. A projection of the HR region in the CR2,A = 100simulation.The upper left frame shows the sizes (r200, being the radius where themean overdensity exceeds 200) of the objects (the numbers are groupidentification numbers of some of the more massive objects). The upperright frame shows the DM distribution. The lower right and left framesshow the distribution of star and gas particles, respectively. Close tothe edges, some large and massive LR-objects can be seen. For clarity,a sampled subset of only 5000 particles of each species is plotted.

6.3. Results for objects formed in the CR2 simulations

Fig. 21 shows the structure that has formed in HR region of theCR2,A = 100 simulation. Stars have preferentially formed inthe more massive DM halos, that tend to lie along filaments. Thegas is much less clustered. This is also confirmed by the massfraction of gasfgas that has fallen into the groups (Table 1).Hot (∼ 106 K) diffuse gas surrounds the DM halos. At an ear-lier redshiftz = 1.88 (Fig. 22), the structure is less clustered,and the objects are not yet as large or massive. The growingobjects obviously move around considerably, and cannot easilybe directly recognised in the final state. Gas was also presentin the surrounding LR region, but essentially no stars formedin this region, due to the much lower resolution. This was nottotally unexpected, as some resolution effects should be present(see discussion in Sect. 8). The CR2,A = 200 simulation hasa very similar appearance to theA = 100 simulation. The onlydifference is that the hot diffuse gas is slightly less clustered.

Resolved HR objects were identified as groups for whichthe mass fraction of HR particles was exceeding95 %. The HRmass fraction was typically either100 % or less than10 %. Therelative efficiency of the padding procedure can be seen from

Fig. 22. Same as Fig. 21, but at an earlier redshift ofz = 1.88. Theco-moving frame size is the same. Note that object numbers have notbeen matched, and that since the LR particles are relatively few incomparison, they hardly show when only 5000 particles are sampled.

fHR−DM (Table 1), being around40 % (note for reference, thata100 % efficiency would mean thatall HR particles ended up inexactly spherical, non-contaminated regions, above 200 meanoverdensity, and would thus require a highly intricate form ofset-up wizardry). Out of around 200 objects, approximately 40were HR objects for the CR2 runs. Generally, only low massobjects at the edges of the HR region were contaminated byLR particles. The originally selected objects were otherwisecompletely free of contamination, except for a single case wherea stray LR particle had sneaked in.

Fig. 23 shows the stellar masses in HR objects atz = 0, andFig. 24 atz = 1.88. The massive objects have obviously formedmore stars, as well as become more massive overall, sincez =1.88. The low mass objects, on the other hand, seem to haveceased forming stars sincez = 1.88. This is also confirmed tosome degree by Fig. 26, which shows the average ages when thethe gas particles stopped forming stars. Some low mass objectsare old, and the higher mass objects are younger than those. Astriking difference is that Fig. 24 shows many more low massobjects. It is of course tempting to identify this as the observedabundant population of dwarf galaxies, at high redshifts, thatlater seems to disappear (Silk & Wyse 1993). Unfortunately,however, it is just a selection effect. At low redshifts, manylow mass objects become contaminated by LR particles, andtherefore cannot safely be used for analysis (this is essentially


Fig. 23. The mass of the stellar component plotted as function of totalobject mass for the HR objects, at the end of the CR2,A = 100simulation. Masses are defined asM = M(r200)

Fig. 24. Same as Fig. 23, but at redshiftz = 1.88.

the only undesirable aspect that we have found of the resolutionenhancement technique). However, this does not contradict theobservation that thereis an abundant population of low massobjects at high redshifts.

Fig. 25 shows the mass fraction of the stars. Many of thelow mass objects have very low stellar fractions (some none atall). This is roughly in agreement with YKKK (note though, thatYKKK have better statistics, i.e. more objects, due to better massresolution). The low stellar fractions are rather due to the DMhalos growing in mass. The suppression of low mass objects isnot only due to lower stellar masses, but also because low massobjects have stopped forming stars early (Fig. 26), and thereforethese populations will tend to be old, and presumably very faint.

Fig. 25. The mass fraction of the stellar component vs. the total objectmass, for the HR objects, at the end of the CR2,A = 100 simulation.

Fig. 26. The average age of the stellar component, for the HR objects inthe CR2,A = 100simulation. Note again that “ages” have been definedwhen star particles are formed, i.e. SF stops, and are therefore not areliable average stellar age estimate, but appropriate for the analysishere.

Fig. 29 shows the ellipticityε of the formed HR objects.There is a clear trend of decreasing ellipticity with increasingmass, as expected for a merger rate that increases with mass.

As noted in Sect. 4 and 5.2, gas metallicities rise ratherquickly. This can be seen in Figs. 27 and 28, showing that al-ready at redshiftz = 1.88, there is a population of relativelymetal rich objects. This can be compared to the observations ofLu et al. (1996). The gas metallicity later decreases for the moremassive objects, due to metals being locked up in stars as thegas is consumed, as well as infall of metal poor gas. This is nottotally inconsistent with what is reported by Matsushita et al.(1997).


Fig. 27. The average metal content of the gas component (hot + cold)in the HR objects.

Fig. 28. Same as Fig. 27, but at redshiftz = 1.88.

Fig. 29. Ellipticity as function of mass. The highest mass point is a“freak point”, because that particular object actually consists of twoobjects, in the beginning of a merging process (the lowest mass pointshould not be taken too seriously either).

Fig. 30. Stellar velocity dispersions (line of sight) as function of objectmass.

Fig. 31. Stellar velocity dispersions, atz = 1.88

Many observational properties of elliptical galaxies are re-lated to velocity information. Fig. 30 shows line of sight velocitydispersions, plotted as a function of mass. The “slope” in Fig. 30is around 2.6. Assuming the luminosity - mass relation givenby YKKK, L ∼ M8/5, it is within the limits of the slope of theFaber-Jackson relation (de Zeeuw & Franx 1991), which canthen be expressed asM ∼ σα, where2 < α < 3. The scatter israther small. At the earlier redshiftz = 1.88, the slope is closerto 2 (Fig. 31). The reason for this is probably that higher massobjects are not yet as massive, and later grow.

It would of course be very interesting to also examine thefundamental plane, but the statistics of the simulated objectsare not very impressive, and there are some other difficultieswith the effective radiiRe, as we shall see. This will also bethe case for theDn − σ relation (Dressler et al. 1987). Notealso that velocity dispersions are not central butaverageforall the star particles in the object in question. Therefore theywill systematically underestimate central velocity dispersions,though giving better statistics.

The objects formed have marked anisotropic velocity dis-persions, as can be seen from Fig. 32. The low mass objects con-


Fig. 32. Velocity dispersion anisotropies vs. velocity dispersion, forthe star particles (δ = 1 − σ2

min/σ2max, σ2

i , being the eigenvalues ofthe velocity dispersion tensor).

Fig. 33. The rotation of the star particles plotted as a function of thedimensionless spin parameterλ.

tain very few particles, and are therefore not reliable in termsof structural or dynamical information. However, for the highermass objects, the anisotropy parameters have reasonable values,and there might be a real trend of decreasing anisotropy withincreasing velocity dispersion.

The stellar objects formed rotate slowly, as is observed forellipticals,〈v/σ〉 ≈ 0.2 (de Zeeuw & Franx 1991). The rotationwas estimated by calculating the “ordered” rotational kineticenergy as2Krot = L · I−1 · L, whereL is the angular momen-tum, andI−1 is the inverse of the inertia tensor. The rotationvelocity can then be obtained throughv2 = 2Krot/M (Binney& Tremaine 1987). There is a clear trend that the less massiveobjects rotate faster, but there is quite a lot of scatter. This is alsothe case for the rotation vs. ellipticity, but this is not unexpectedsince the anisotropies vary quite a lot (Binney & Tremaine1987). A stronger dependence of the rotation measure is onthe dimensionless spin parameter,λ = (L E1/2)/(GM5/2), Ebeing the binding energy. This is shown in Fig. 33. In spite of

Fig. 34. The mass fraction of the gas component (cold + hot gas) vs.r200, for the HR objects at the end of the CR2,A = 100 simulation.

some scatter there is a reasonable agreement with a linear rela-tionshipλ ∼ 0.3 v/σ, expected for ar1/4-profile (Silk & Wyse1993).

The above analysis has been illustrated for the CR2,A =100 simulation. TheA = 200 simulation is similar but hasslightly higher velocity dispersions and anisotropies.

6.4. Detailed results for individual objectsin the CR2 simulations

There are some undesirable aspects of the procedure used tofind groups (and thereby define galactic objects), according tothe overdensity criterion. Since it is mainly based on the DMdistribution, it has rather little to do with observable quantities.Typically, r200 (depending roughly on a power of total mass)becomes rather large compared to the stellar distribution, andsometimes DM halos even overlap. A better galaxy identifica-tion system, based on the stellar distribution, is clearly desirable,but this would make the comparison with previous work (e.g.YKKK) more complicated. Most objects show a rather high gascontent when measured within ther200 radii, as can be seen fromFig. 34. At smaller radii the gas fraction decreases significantly.Gas is certainly blown out by supernovae, but not much beyondther200 radius. Therefore, we have used a fixed radius of 50 kpcwhen taking a closer look at some of the more massive objectslisted in Table 2. At still smaller radii∼ Re, where observationsare typically able to probe, stellar to DM fractions become aslow as50 %, and gas to stellar mass fractions become lower thanone percent. The most massive object in the CR2 simulationsactually consisted of two objects (see Fig. 35), but has been splitup in Table 2 for easier study. In this table, two-body relaxationtimes have been estimated for the objects. It is evident that lowermass objects (i.e.vcirc < 150 km s−1, M < 1011M) have re-laxation times that are shorter than orders of Friedmann times,and should therefore not be taken too seriously in the analysis,at least not in terms of structural and dynamical information.Mass fractions are probably less affected.


Table 2. Some quantities for some more massive objects, formed in the cosmological simulations atz = 0 (unless otherwise indicated).

Run/ M(50 kpc) vcirc trelax,∗ fgas fstars 〈T 〉 [Fe/H] SFR Re δ v/σ ShapeObject (1011M) (km s−1) (Gyrs) (%) (%) (106 K) (M/yr) (kpc)

CR2, A = 100:97 (a) 5.6 226 89 0.85 9.1 1.9 +0.01 0.34 2.9 0.14 0.08 E1-P97 (b)1 4.1 201 55 0.83 9.9 1.6 +0.01 0.22 2.7 0.20 0.02 E1-O115 (c) 1.4 115 8.8 0.29 1.0 0.78 −0.30 0.0055 16 0.35 0.21 E3-O116 (d) 6.6 240 114 0.29 8.4 2.0 +0.03 0.19 3.5 0.27 0.17 E2-T132 1.7 123 17 0.37 1.0 0.68 −0.37 0.018 17 0.48 0.13 E1-O185 2.8 154 41 0.56 3.1 0.92 −0.22 0.12 6.7 0.42 0.07 E3-Pz = 1.88:69 (b) 4.1 190 30 3.3 2.9 0.92 −0.23 13 2.6 0.42 0.25 E4-O71 (d) 4.7 210 33 3.3 1.9 1.0 −0.47 8.8 23 0.71 0.46 E4-O75 (a) 4.3 190 29 3.2 1.8 0.98 −0.45 6.6 27 0.79 0.21 E3-O

CR2, A = 200:85 (a) 6.1 228 102 0.10 3.3 3.9 −0.13 0.0021 19 0.26 0.22 E2-T85 (b) 4.3 192 109 0.23 2.9 2.4 −0.17 0.14 18 0.39 0.10 E3-O98 0.34 61 3.6 0.26 1.8 0.79 −0.99 0.0 15 0.49 0.19 E6-O101 (d) 6.1 230 85 0.35 3.9 3.4 −0.17 0.0 17 0.43 0.11 E3-T123 (c) 1.3 105 24 0.15 1.5 1.5 −0.25 0.025 20 0.46 0.19 E4-T139 0.19 51 1.4 1.7 2.6 0.48 −0.75 0.0068 5.7 0.53 0.13 E3-T

CR3, A = 150:39 2.7 161 324 0.95 6.9 0.73 +0.05 0.88 3.0 0.28 0.07 E2-T

CR3, A = 200:42 2.5 146 212 0.15 1.8 0.77 −0.02 0.053 16 0.38 0.11 E2-T

1) Object 97 (b) was contaminated by one (annoying) LR particle.All values (including mass) are calculated for the objects within a radius of 50 kpc, centred on the highest density in the object. The most massivegroup in both CR2 runs turned out to actually contain two objects, denoted (a) and (b). Other objects that could be matched have been denoted (c)and (d). In the CR3 runs, there is only one massive object. Circular velocity is calculated as the maximum ofv2

circ = GM(r)/r which is “flat”,and thereby a more useful “mass indicator”, than the mass itself, which increases roughly linearly with the radius.trelax,∗ is the (numerical)two-body relaxation time for the star particles, estimated with the Fokker-Plank approximation (Binney & Tremaine 1987), taking into accountthe different gravitational smoothing lengths, and masses of the particle species. [Fe/H] is calculated as the mean metal content for gas and starparticles, correcting the stellar contribution with a factor of 0.5, as not to overestimate the total metal content.δ measures the anisotropy of thevelocity dispersion tensor,δ = 1−σ2

min/σ2max. Shapes are estimated according to the ellipticity, and the triaxiality parameter, e.g., accordingly

(Warren et al. 1992). P, O and T, denote nearly prolate, oblate and triaxial.

Differences between theA = 100 and 200 simulations canbe seen in Table 2 (results for the CR3 simulations are also listed,and these will be discussed later). There are both more stars andgas in theA = 100 case. The overall SF seems to be more ef-fective in theA = 100 case, as can be seen from metallicitiesand SFRs. The effect on the half-light or effective radiiRe isobvious. Here,A = 100 has produced more compact objects,with Re ≈ 3 kpc for an object of around 200 km s−1 circular ve-locity, rather than the previous 6 kpc. Since the resolution andmass, (i.e circular velocity) is similar for the TS simulations(Sect. 5.2), this may be due to the different, and more realistic,initial conditions. The lower mass objects show larger effectiveradii and this may of course cause some worry but, as men-tioned, these have likely been affected by two-body relaxation.In the A = 200 case,Re is correspondingly larger at around20 kpc, indicating that a more reasonable value forA may liesomewhere in between 100 and 200. This was indeed the reasonfor choosingA = 150 in one of the CR3 runs. The present-daySFR is rather high in the massive objects for theA = 100 caseand this may contradict what is typically observed for ellipti-

cal galaxies. Again, however, a somewhat higherA may curethis, as theA = 200 case has generally much lower present-daySFRs.

Figs. 36, 38 and 40 show some details of the “97 (a)” object,one of the three more massive objects in the CR2,A = 100simulation, and rather typical for the objects formed here. Thesurface density follows ther1/4-law strikingly well. There isa departure in the centre, but this is below the gravitationalresolution. The departure is however not seen in Fig. 9, Sect. 5.2,and this may be a real effect due to the different initial conditions.The corresponding object “85 (a)” in theA = 200 case does notshow as ordered gas-flow patterns, but more random motions.The r1/4 behaviour is very similar, but with a larger effectiveradius, as mentioned. Fig. 37 and 39 show object 69 (b) atz =1.88, that is corresponding to object 97 (b), when it has juststarted to form. The stellar fraction is around3 %. The objecthas already atz = 1.88 attained its final effective radius. Otherobjects, e.g. (a) and (d), grow in the “outside-in” sense, as canbe seen from the large effective radii atz = 1.88, that latershrink (cf Table 2).


Fig. 35. Object nr. 97 in the CR2,A = 100 simulation, actually con-sists of two similar objects, denoted (a) and (b) in Table 2. Likely, theywill later merge but atz = 0 they are still well separated.

Fig. 36. A closeup of object 97 (a) atz = 0. The gas flow shows asimilar pattern as in Fig. 7, rotating at around∼ 80 km s−1. See Table 2for other object quantities.

The gas surrounding objects 97 (a) and (b) are hot and dif-fuse, as can be seen from Fig. 41. It contains almost no or littlecold clouds, which is in principle predicted by Eq. (13). It isnot as hot as in an X-ray cluster (Katz & White 1993), butit is clear that this is not a “rich cluster”. Circular velocitiesare never above 250 km s−1. It is also evident from Table 2that temperatures decrease with object mass. This indicates that“self-regulation temperatures” are not maintained when virialtemperatures are higher. However, recall that this is still withina relatively large radius (50 kpc). Fig. 42 shows an age - metal-licity relationship that is similar to that in Sect. 5.2, with similardecreasing metallicities for younger star particles. The scatter

Fig. 37. A closeup of object 97 (b) atz = 1.88 (where it was giventhe number 69 (b), by the FOF algorithm). The stellar distribution havejust started to form.

Fig. 38. The surface density of the stellar component, for object 97(a), plotted against elliptical radius, as in Fig. 9.

is around 0.3 dex. Fig. 43 shows a radial metallicity distributionthat is similar to Fig. 20 of Sect. 5.2.

6.5. Detailed results for the CR3 simulations

As mentioned, the CR3 simulations were performed in orderto investigate the effect of higher resolution on a single object.Fig. 44 shows the structure in theA = 150 simulation. It is avery distinctive filament. Again stars have only formed in themore massive halos. At the earlier redshift ofz = 1.88, Fig. 45,things look very different as the filament has not yet collapsed.Some star particles have already formed, and they seem fairlyevenly distributed. This is partly due to the plot sub-sampling,and they are not formed at random. Note the very large amount


Fig. 39. The surface density of the stellar component of object 69 (b)atz = 1.88, corresponding to Fig. 37.

Fig. 40. The rotation curve, for object 97 (a). Different lines show eachcomponents contribution to the circular velocity, which is calculatedasv2

circ = GM(r)/r.

of low mass objects. Obviously, many of these merge later. Notethat there is no selection effect here. Fig. 44 and 45 show all ob-jects, and atz = 1.88 contamination is still small. Atz = 0 theobjects – contaminated or not – are much fewer. At the end ofthe simulation only one massive resolved HR object has formed(object 39), together with a handful of very light HR objects, ascan be seen from Fig. 46. These “dwarfs” have very low stel-lar fractions, and there are about an equal number (not seen inthe plot) that have no stars whatsoever. So, “dwarf suppression”seems to increase in efficiency with increased resolution, butthis is of course in a special environment, being selected on afilament arm, and may not be conclusive in general. (As pre-viously explained the “dwarfs” are also likely to be “two-bodyrelaxed”, and therefore dynamical information is not useful. Al-though the mass resolution was about 8 times higher, compared

Fig. 41. The temperature of the hot gas phase surrounding object 97,within a radius of 500 kpc, plotted against density (in units of the closuredensity).

Fig. 42. Age - metallicity relationship for object 85. See caption ofFig. 19.

Fig. 43. Metallicity plotted against radius for the gas and star particlesin object 97 (a), within a radius of 20 kpc.


Fig. 44. The final appearance of the HR region in the CR3,A = 150simulation. Layout is corresponding to Fig. 21. The HR region wasselected on a collapsing filament. At the top, in the upper left frame,the massive (vcirc ≈ 600 km s−1) “central cluster” LR object can beeseen. The regions outside the filament seem completely void of matter,but this is just because of the very coarse sampling of 5000 particles.Few of the LR particles show at all.

to the CR2 and TS simulations, it was possible to resolve objectsdown to aroundvcirc ≈ 50 km s−1 andM ≈ 1010 M.)

Object 39 shows a similar gas flow pattern to the CR2 ob-jects, Fig. 47, but the rotation speed increases towards the centre.There is a quite high present-day SFR, as can be seen in Table 2.As previously mentioned this may be a problem, but there is asimilar behaviour to the CR2 simulations. A higherA seemsto both reduce high SFRs, and increase low effective radii.Awas increased to 150 in one of the CR3 runs, and the effec-tive radius ended up at around 3 kpc. That may actually be areasonable value, because object 39 is about half as massive asobject 97 (a) of the CR2,A = 100 run. If we assume the previ-ous mass-version of the Faber-Jackson relation,M ∼ σ2.6 andL ∼ M8/5, the fundamental plane (de Zeeuw & Franx 1991)givesM ∼ Re, suggesting that 3 kpc might be about right (thepoint here is not toexactlypin downA to giveRe = 6 kpc forbright galaxies, but to investigate the gross effect).

Otherwise, object 39 shows similar properties as the pre-vious objects. Ther1/4-law is shown in Fig. 48, and the ro-tation curve in Fig. 49. The temperatures are somewhat lower(Fig. 50). However, object 39 is formed on a filament arm, beingfar from the cluster centre, and thus has lower virial temperature.

Fig. 45. The collapsing filament of Fig. 44 atz = 1.88. Note theamount of small (low mass) objects.

Fig. 46. The mass fraction of the stellar component vs. the total objectmass.

It should be mentioned that although the “central cluster object”seen in Fig. 44 mainly consists of LR particles, the temperaturesof the HR gas that has fallen in is on the order of107 K (vcirc ≈600 km s−1).

Fig. 51 shows the age–metallicity relationship for object 39.The previous tendency of falling metallicities for younger starparticles is less notable here. Although most star particles areold, there is a substantial amount of younger star particles, con-firming the previously observed high SFR at late times. YKKK


Fig. 47. A closeup of object 39, in the CR3,A = 150 simulation. Thegas flow is rotating at around∼ 80 km s−1, faster towards the centre(long arrows show direction of the angular momentum). See Table 2 forother object quantities. It may seem like the gas forms a disc, but thisis exaggerated by the velocity arrows. However, the gas has a higherellipticity of around 0.3, compared to 0.20 for the stellar component.

Fig. 48. The surface density of the stellar component, for object 39,plotted against elliptical radius, as in Fig. 9.

also observed this effect in terms of “bluer” populations in mas-sive objects.

The peculiar positive metallicity gradient is now gone (cfFig. 52). The gradient in the star particle distribution is around−0.02 dex, and−0.07 dex in the gas distribution. This givessome indications that increased resolution removes some of thepotentially unusual features seen in the TS and CR2 simulations.

Fig. 53 shows that the SF is occurring closer to the centre,confirming that the hot halo mainly consist of hot gas, and almostno cold clouds.

Fig. 49. The rotation curve for object 39. Different lines show eachcomponents contribution to the circular velocity, which is calculatedasv2

circ = GM(r)/r.

Fig. 50. The temperature of the hot gas component, surrounding object39. Note the hot “cavern” in the centre. There is still cold gas a largerradius, that has not yet exceeded the density threshold for SF to start.

7. Conclusions and discussion

Several intricate feedback processes are believed to govern theevolution of the ISM. The most important of these, the supernovafeedback loop, is incorporated in the SNE (supernova evapora-tion) model in a very simple fashion. The SNE model can stillwith some confidence be applied to the evolution of the ISM, ina hierarchical cosmological scenario. We made an implementa-tion of the SNE model into the combined SPH and gravitationaltree method, and performed a set of simulations in a realisticenvironment. This turned out to have some striking implica-tions, since the objects that form have several properties thatare similar to what is observed for elliptical galaxies.

It should be clear that the simulations presented here arenot to be strictly considered as large scale cosmological sim-


Fig. 51. The metallicity–age relationship for object 39. See caption ofFig. 19.

Fig. 52. Metallicity plotted against radius for the gas and star particlesin object 39, within a radius of 20 kpc.

Fig. 53.The SFR plotted for each gas particle, in object 39, as a functionof radius.

ulations. We have not covered topics that are typical to largescale scenarios (i.e. correlation functions, dark halo shapes, andbiasing, etc.), because the statistics, in terms of the number ofobjects formed, is too poor, anyway. The present approach ismainly to study a sample of individual objects, though formedin a realistic environment. Some gross cosmological propertiescan still be reproduced, or possibly predicted.

7.1. Comparison with observations and theory

While “playing god” and creating one’s own little universe, oneencounters not only the problem of designing the creation itself,but also the problem of observing one’s creation. Fortunately,there are not billions of potential galaxies created here, but thetask of comparing with every relevant observation made is, ofcourse, still impossible. We have tried to look at a few key obser-vational properties. First are some properties of cosmologicalnature.

• Mean luminous density fraction (Ω∗) is within observationallimits at around one percent.

• The mean SFR is roughly in agreement with observationsand hierarchical models. The absolute value varies for dif-ferent resolution and cosmological environment, but the in-creasing behaviour with redshifts in the rangez ∼ 0–1.5 isroughly reproduced.

• Low mass galaxies are strongly suppressed, in that they havelow stellar content. They have stopped forming stars quiteearly (z ∼ 1–2), and are presumably very faint. The statisticsis, however, too poor to make any meaningful quantitativeconclusions on the impact on the luminosity function.

• If these objects encountered here correspond to ellipticalgalaxies, they may form too late. Several objects are nowobserved at redshiftsz > 3 (Giavalisco et al. 1996; Steidelet al. 1996; Pettini et al. 1997). These objects are possiblyellipticals or spheroids of discs in the process of formation.The most massive objects (vcirc ≈ 200 km s−1) formed hereare in place no earlier thanz ∼ 0.5–2. Part of this may berelated to problems in the simulations with the relativelylow early SFRs and the high present-day SFRs, but this isalso highly related to the cosmological environment thatthese objects form in. It is still possible that these objectsrepresent a selected class of elliptical galaxies that form late.This is further discussed in Sects. 7.2 and 7.4.

• Ellipticalscanform independently. Late merging is not ruledout,nor needed. These objects forms bynature, butnurtureis possible.

• Dark matter fractions are typically 90% or above within50 kpc in individual objects, but is less dominant in centralregions at around 50% within one effective radius.

Some gross properties more specific for elliptical galaxies arereproduced:

• The formation mechanism for a massive (vcirc ≈200 km s−1) elliptical object is here realized as a dissipa-tional, star-forming collapse, in an “outside-in” sense. Apart


from the gravitational collapse (violent relaxation) of thedark component, and dissipational infall of the baryoniccomponent, the main mechanism is the thermally unstableformation of cold clouds, and subsequent formation of stars,balanced by energy feedback from supernovae.

• r1/4 profiles. The agreement is striking, and the transitioninto a r1/4-shaped structure is very rapid. Still, it is notso conclusive. As pointed out by White (1987): “However,there is nothing magic about ther1/4-law.” It can be seenas resulting from a semi-thermalized distribution function.Ther1/4-law could arise from a variety of different causes.

• The scale-lengths or effective radii have a distinct non-monotonic dependence on the degreeA of supernova evap-oration assumed. For a certain degree (A ≈ 150) the result-ing scale-lengths are close to observations for massive ob-jects. There might well be a problem in that the less massive(vcirc < 150 km s−1) objects in the simulations presentedhere have too large effective radii. However, these are notproperly resolved, and we might hope that two-body relax-ation effects, or poor resolution in general, is the cause ofthis.

• Ellipticities show rather typical observed values, and a de-creasing trend with increasing mass.

• Shapes of all the possible forms seem to occur, i.e. prolate,oblate and triaxial. There are considerable variations, how-ever, due to the discreteness of the particle representation,and the detailed results should therefore not be taken tooseriously.

• Velocity dispersions seems to show roughly correct values.There are again some minor method-related problems, inthat central values cannot be directly obtained with goodstatistics. Average velocity dispersions have been used in-stead.

• Velocity dispersions are markedly anisotropic, possibly in-creasing for less massive objects. The anisotropy parametergets reasonable values compared with what has been tried inanalytical models and observed (Binney & Tremaine 1987).

• Rotation is generally slow, which is consistent with disper-sion anisotropies. Low mass objects rotate faster. Strongestdependency is on the spin parameterλ.

• The Faber-Jackson relation. Although rough, there is someagreement, and the slope seems to fall between the limits.The fundamental plane or theDn − σ relation have notbeen explored, mainly due to the aforementioned problemswith low mass objects not being resolved, as well as poorstatistics.

• A significant amount of hot diffuse gas surrounds the ellip-tical objects, out to large distances on the order of 100 kpc.Since the SNE model tends to drive the temperature to∼ 106 K, this implies that X-ray clusters need massive darkhalos, or some other mechanism, in order to produce the ob-served temperatures. Observations are beginning to probe at∼ 106 K (Matsushita et al. 1997), and there is a lot of scatter.This can be explained by effects due to the hierarchical en-vironment and different halo sizes (this is further discussedin Sect. 7.2).

• At smaller scales, on the order of effective radii, thesurrounding gas is still hot and rotating, at around 40–100 km s−1. This gas is perhaps not distributed in a disc,but is more flattened than the stellar distribution. The frac-tion of gas is much smaller, at around one percent.

• Star formation rates. The SFR has a rapidly rising and laterexponentially decaying behaviour, as is expected in ellipti-cals. The total amount of stars formed seems reasonable, butthere is a potential problem with the SFR being too high atlate times. At present-day,SFR ∼ 0.2–1 M/yr in massiveobjects. This is further discussed in Sect. 7.2 and 7.4.

• Gas metallicity. The metal content rises quickly to〈[Fe/H]〉∼ −1 atz ∼ 2, somewhat increasing with mass. The scatteris around 0.5 dex, decreasing with mass. Atz = 0, 〈[Fe/H]〉is somewhat less than−1, increasing with mass to beginwith, then decreasing for the most massive objects. Thisis quite in accordance with observations. There are radialgradients in the remaining gas distributions atZ = 0, ataround−0.05 dex, but there is considerable scatter.

• Stellar metallicities and ages. Most star particles formearly (> 8 Gyrs old), and have generally high metallici-ties 〈[Fe/H]〉 ∼ 0.2–0.3. There is scatter on the order of0.2 − 0.3 dex. Such metallicities are fairly close to whathas been observed for the bulge of the galaxy, and the age –metallicity relation is not totally unlike that observed for olddisc stars. Almost no old star particles have low metallicities,indicating that there is apparently no “G-dwarf problem”.There is a considerable amount of younger stars, and thesehave lower metallicities, because they are formed by infallof non-enriched gas. The radial gradients are increasing butbecomes small when resolution is increased. It should berecalled, however, that the implementation of the metal en-richment is done in a very simplified fashion, and there areseveral severe purely method-related problems.

7.2. Influence of the cosmological environment

A potential problem is that the more massive objects presentedhere form relatively late, if they are to represent elliptical galax-ies. This may partly be due to the SFRs, discussed further inSect. 7.4, but the cosmological environment affects this to avery high degree. Elliptical galaxies, massive ones in particu-lar, are predominantly common in clusters but the largest regionstudied here can perhaps be said to represent a group of galax-ies. A standard CDM power spectrum peaks at wavelengths ofaround100h Mpc, perhaps even300h Mpc for models that areconsistent with COBE (Silk & Wyse 1993). This means thatwhen selecting regions of only 20 Mpc, filtering away longerwavelengths, the resulting structure will form later, due to theconsequent relatively low mean overdensity in the region. Aparticular example is the CR3 runs where, as late asz ≈ 2,there is no “massive” structure whatsoever. If applied in a richerenvironment, it may well be possible to reproduce also typicalmassive elliptical objects, forming at the right epoch. In general,anything that can increase the baryonic density early on wouldbe of help here. A higher amplitude power-spectrum would also


give more early structure and, in fact, more structure in general(Weinberg et al. 1997). This would also affect the formationtimes. A higher baryonic fraction also increases densities.

A non-zero cosmological constant could help this scenarioin at least two ways. First, it provides more time. Second, itwould stretch the comparatively late times of formation to higherredshifts.

A common idea is that proto-ellipticals may have formeddue to Compton cooling, which is extremely efficient at highredshiftsz > 5 (Silk & Wyse 1993). Although Compton coolingwas included here, its effect on the thermal instability was nottaken into account (YKKK). However, Compton cooling wouldhave had no effect because the objects simulated here starts toform later thanz = 5.

7.3. Ellipticals... Aren’t there supposed to be discs, also?

One may suspect that the tools, and not the physics, have de-signed the outcome of the experiments. Particularly in the caseof numerical simulations, there are always purely technical con-siderations. It might for example be a problem that stellar discsare more difficult to correctly represent in N-body like ap-proaches. Previous star forming simulations of the formationof disc galaxies (Katz 1992; Steinmetz & Muller 1995) haveused up to three times as many star particles, as those ending upin the objects formed here. Still, some faith has been needed toidentify the resulting stellar distributions with thin discs, mainlydue to resolution and discreteness effects. There are, however,so many more typical dynamical and structural properties of el-liptical nature than disc nature of the objects formed here, thatit seems highly unlikely they have just been artificially relaxedinto mimicking elliptical galaxies.

We think one can also rule out “bad luck”. One could ar-gue that, since a random realization is used for setting up ini-tial conditions, and the studied regions are still comparativelysmall, possibly no discs formed. However, previous simulationsincluding only gas cooling, i.e. no star formation, have formeddisc-like objects, for thevery sameinitial conditions (Hultman& K allander 1997). There is more reason to believe that theSNE model, in its present configuration, has captured some ofthe essential physics in the formation of elliptical galaxies, butnot for disc galaxies.

This implies that some crucial part of the physics might stillbe missing, and one can only speculate what that might be. Oneneeds a gaseous collapse in order to form a thin rotationallysupported disc. If too many stars are formed too early duringthe collapse there will simply be too little gas left to halt thecollapse through shocks and adiabatic compression. In the SNErepresentation, the SF process is mainly governed by the thermalinstability, and the supernova evaporation through the parameterA.

Before entering the following discussion of possible effectssuppressing the early formation of stars, we must keep in mindthat the underlying assumption of the SNE approach is that SFis assumed to take place inside cold clouds, at a fixed time-scale,and regardless of all the other intricate processes that affect the

formation of stars. The following discussion will first focus onwhat we may improve in the SNE model.

Let us first discuss the influence ofA. As pointed out byYKKK, it cannot be ruled out thatA could depend on the en-vironment, e.g. the supernova energy release or the cold clouddensity, a dependence that we cannot currently resolve. Thismust also be taken under consideration if it turns out that theincrease in effective radii, found for low mass objects, is notcaused by discreteness effects. Nor should we rule out a vary-ing supernova evaporation as a mechanism for the formation ofcompact, or low surface brightness objects. It is worth notingthatif self-regulation is active,A has only the effect of directingthe hot phase temperature, Eq. (14). Thus, the only essential wayto change the SFR, i.e. the amount of cold clouds (Eq. (13)), isby changing the cooling rates (the IMF, i.e.β, and the supernovatemperature would have less influence, as explained below).

Intuitively, a very effective supernova evaporation (A 200) could potentially suppress the early formation of coldclouds but, as demonstrated here, the effect is rather the op-posite. It would also give inconsistent temperatures of the hotphase ISM. There are more reasons to suspect that the clue to theabsence of discs in these simulations lies in the thermal instabil-ity itself. Apart from the density threshold, Eq. (4), there is nofurther restrictions on the thermal instability. A single sentenceconclusion to this article may be: “Anunrestrictedthermal in-stability is indeed a key mechanism to the formation of ellipticalgalaxies.” So, the answer to the question posed in the title wouldbe a yes, or at least a maybe. In this context it is also interestingto note that thermal instability has already been proposed as apossible mechanism behind the formation of globular clusters,and possibly for the formation of all spheroids (Fall & Rees1985; Kang et al. 1990).

As mentioned by YKKK, and shown by Mucket & Kates(1996), UV ionizing radiation can have a suppressive effect onthe thermal instability. This is related to, but not to be confusedwith, the effect of UV ionizing radiation on line-cooling. Thereare at least two possible sources of UV radiation, present at earlytimes: high mass stars and quasars. Presently, this effect is onlyincluded in an artificial way, by means of the density threshold,Eq. (4). As pointed out by Anatoly Klypin (private communi-cation), the density threshold could be raised to a much higherlevel, so that the initial collapse phase remains gaseous, and al-lows the formation of a disc. This might certainly be interestingto explore further, but in realityD would not be a constant, justa rather meaningless control parameter for explaining the splitin the Hubble sequence.

The density threshold is a weak part of the SNE model,and it would be better to investigate the effect of the UV ion-izing radiation on the thermal instability, in a fashion simi-lar to Mucket & Kates (1996). However, it might not be justone process affecting the other, but a whole system that mustbe treated self-consistently, especially since metallicity effectshave been shown to be important,if not crucial. Unless it isshown that some (non obvious) simplifying approximations canbe made, one must cope withthe many complications of fullnon-equilibrium evolution of hundreds of ionization states of


some 10–15 different atoms, with the additional complicationof forming sub-structure due to the thermal instability.

Before this programme is undertaken, we might of courselike to ask ourselves if this would really have the potential ofexplaining the morphological differences between discs and el-lipticals. One reason that we might believe so is that the UV ion-izing radiation is not homogeneously distributed. For example,the quasars are point-like sources on these scales. We have, forexample, the “proximity effect” (see Miralda-Escude & Ostriker1990 for references) where the quasar baths its environment inharsh ionizing radiation at close range. However, if we assumethat quasars cluster like the rest of the matter, and that the ion-izing radiation has the effect of imposing a density thresholdon the thermal instability, one comes to the unpleasant conclu-sion that discs would outnumber ellipticals in clusters, in directcontradiction with observations. So, it may appear that UV ra-diation from active sources have then been ruled out, but it maynot be that simple. Although our arguments may be convincing,it still seems worthwhile to keep the UV radiation mechanismalive a while longer.

An extreme form of the “proximity effect” is when thequasar isinside the core of the forming galaxy (Silk & Rees1998). It need not be a blazing quasar, but just an active nu-cleus that is intense enough (has the timing to “light up” at theright moment) to suppress the thermal instability, and allow fora smooth gaseous collapse into a thin disc. We know for ex-ample that ellipticals are associated with giant radio sourcesand Seyfert galaxies in connection with discs (Silk & Wyse1993), and that there might be a more or less active radiationsource in every galaxy. The cause and the sensitivity of the tim-ing is, of course, even more a subject of speculation, but givenenough range, it could explain even the observed range in sizesof spheroids, as well as morphological types. In support of this,there is an attractive delay in that quasars peak betweenz ≈ 2and 3, while the SF history of the Universe peaks betweenz ≈ 1and 2 (Silk & Rees 1998).

UV radiation from massive stars in the proto-spheroid couldhave a similar effect to that of an active nucleus, and suppressthe thermal instability.

One should remember that the previous discussion has as-sumed that the thermal instability is the direct link to the for-mation of stars. However, the amount of clouds at temperaturesaround104 K is hardly all we need to know about star formation.Below temperatures of104 K, molecular cooling processes be-come important for the subsequent fragmentation of molecularclouds (Silk & Wyse 1993). Still, the simple argument for thesuppression of SF by UV radiation may be valid, because ion-izing UV radiation will also strongly suppress molecular cool-ing. Photo-dissociation ofH− will suppress the formation ofH2 molecules (Silk & Rees 1998), the dominant coolant be-low 104 K. Even if the thermal instability is not suppressed wemight still have a suppression of SF by UV radiation inside theclouds, and thus the subsequent collapse could still be gaseous.This suggests that molecular radiation processes should also beadded to the recipe.

7.4. Some other potential problems with the SNE model

If we still accept the SNE prescription as a mechanism for theformation of elliptical galaxies, there are still some problems.We consider the high SFR at late times to be the next largestproblem (Fig. 8) since this also strongly couples to the formationtimes. Possibly, this problem would also be cured by a richerenvironment but if not, we have to look for other means. It couldbe desirable to have an initially much more peaked SFR, with afaster decline. As previously indicated, we might achieve this,if we could increase the amount of cold clouds. The coolingenhancement parameterC has not yet been utilised. Its valueis not yet certain, and we cannot exclude the fact that it mayeven exceedC = 10 (see also discussion in Sect. 8). We mightalso consider a different, or possibly evolving, IMF but note,however, that the IMF only makes its presence through the frac-tion of massive starsβ, which would not likely fall outside therange of0.1–0.3 (Tinsley 1980), andβ = 0.12 is already closeto the lower limit. The supernova energy release seems fairlywell determined, but the supernova temperature, i.e.uSN, couldpossibly be modified, together with a proportional change inA,so as to not change the hot phase ISM temperature.

As mentioned, there might be a problem with larger effectiveradii for low mass objects. This is likely to be a discretenesseffect due to two-body relaxation, and has therefore to awaitsimulations with better resolution.

Some concern might be devoted to the assumptions made forthe effect of the thermal instability, specifically the isentropiccondition for the hot gas phase. This might not be a completelycorrect physical description, because at least some entropy mustbe taken away with the hot gas that cools, but the qualitativeeffect still may not be very large. An interesting comparison isgiven by changing the isentropic condition for an isothermal,which has the effect of removing the factorγ from Eq. (7), i.e.simply puttingγ = 1, as one might expect on physical grounds(in the equation of state, Eq. (10), there should of course stillbe the value for ionized gas,γ = 5/3). Klypin et al. (1997)propose another variant to this approach.

The problem with metallicity and the mass flow has beenmentioned. Little can be said about this at the present stage, andwill have to await future work. The prescription developed byMaeder (1987) might turn out to be a better approach.

7.5. For the future

Is Ω = 1? This is not probable. Observations point more in thedirection ofΩ ≈ 0.3. Ω = 1 was mainly chosen here because itis a standard template scenario. We might worry, however, thatthese results are strongly dependent on large amounts of darkmatter. Different cosmological models have to be examined infuture work.

A better treatment of the present density threshold for SF andthe thermal instability, in terms of a self consistent evolution ofthe radiative processes, has been mentioned. In principle, thiswould have to be evolved along with the hydrodynamics andgravity, and can well prove to be computationally prohibitive,


due to the seriously different time-scales and the mentioned res-olution dependence (see Sect. 8). But, as pointed out by Mucket& Kates (1996), a separate evolution for the ISM–radiation–thermal instability processes could be performed, possibly in-cluding the whole SF process. Reference tables could be pro-duced for later use in hydrodynamics – gravity simulations (suchreference tables would likely becomehugein size, due to manyvariables needed, but may well be possible to handle on existingparallel computers).

Another important aspect is the cosmological environmenteffects. They are clearly present, both reported by YKKK andseen here. A larger, richer environment is obviously needed inthe simulations. There is also a need for better statistics, as wellas higher mass resolution. Needless to say, at least the first twoof these imply a significant increase in computational volume,but the approaches will likely be different. More powerful com-putersareavailable (and perhaps relaxing the condition that cre-ation takes six days, being what your average galaxy designercan tolerate) an increase of the order of 50 should be possible toachieve. This, however, requires the present code to be adaptedto massively parallel computers.

Higher mass resolution can be investigated in so-called lim-ited cosmology (cf Sect. 5). This particular form of initial con-ditions have often been criticized, but keeping their limitationsin mind (e.g., unrealistic tidal interactions), they can still beextremely valuable for achieving higher resolution in a limitedregion (Kallander & Hultman 1998). A richer environment canbe incorporated by extending the resolution enhancement pro-cedure of Sect. 6.1, by adding larger surrounding regions of out-wards decreasing resolution. An alternative is to simply selectan overdense region to begin with (Evrard et al. 1994; Kallander& Hultman 1998).

A stellar population synthesis model, as used by YKKK,would greatly add to the predictive power of these simulations.Better treatment of stellar evolution parameters for stellar par-ticles would then have to be included, in order to utilise stellarpopulation synthesis to full potential.

8. On numerical resolution

A complete discussion of numerical resolution is way beyondthe scope of the article, and we will only cover a few topics ofimportance here (with “numerical resolution”, we will only re-fer to the typical code resolution, like cell sizes, SPH smoothinglengths or time-steps, and not numerical accuracy or propaga-tion of integration errors).

The problem with astrophysical numerical simulations isthat one often has physical processes included that occurfarbelow the available resolution. One could take the spatial res-olution of the density fieldρ(x) as an example, since it is afrequent basic ingredient for several processes.ρ(x) is the den-sity that is needed to physically describe a certain process. Thedensity field that we have available in our computer code can bedenoted〈ρ(x)〉s, since it is a “smoothed” estimate of theρ(x)density. If a process depends linearly onρ we can hope that oursmoothed estimate will at least be a reasonable volume average

of ρ. The SBF used here, Eq. (3), is a good example. It dependslinearly on the density of cool cloudsρc, of which the code cer-tainly doesnotknow its intricate (ISM), “sub-resolution” varia-tion, but knowing〈ρc〉s will still hopefully give us a reasonableaverage SBF.

If the process depends non-linearly onρ, things becomeworse. In principle, we can have any functional dependence,but let us take a square dependence as example, since it is whattypically occurs here. We can then calculate〈ρ〉2s , but ratherwhat we need is

⟨ρ2

⟩s. An example is the radiative cooling rate

Λ, which scales roughly with the square of the density (let us forthe moment forget about the temperature dependence, in orderto keep things simple). Simulations including only cooling ofthe gas worked fairly well when they were examined for reso-lution effects, even when there was substantial variations of thedensity, when the resolution was varied (see Hultman & Kall-ander (1997) for an example). The reason is, that in the equationfor the thermal energy, Eq. (9), the cooling functionΛ is dividedby the gas density, and the temperature cooling rate still scalesroughly linearly with the density. When a heating UV field wasincluded, the situation became very different, as mentioned inthe introduction. Weinberg et al. (1997) in particular, demon-strated strong resolution effects. The UV-heating scales linearlywith ρ, and there is a resolution dependent interplay betweenρandρ2 scaling processes.

Now, the situation might seem essentially similar here. Thesupernovae evaporation scales as the cold phase density, throughthe SBF∼ ρc, which we may assume is proportional to∼ρg, and we would have the sameρ vs. ρ2 interplay as before.But, if self-regulation is in effect,ρc will rather be proportionalto the cooling rate, according to Eq. (13), and, as previouslymentioned, this makes the functional form of the SBF largelyirrelevant. So, self-regulation – to the extent it is in effect –inhibits an otherwise resolution dependent behaviour, and infact seems to make the resolution problem here much smaller,as compared to the UV-heating case.

YKKK took sub-resolution effects on the cooling rates intoaccount by introducing the cooling enhancement parameterC = 10, i.e.Λ(ρh, Th) = C 〈Λ(〈ρh〉s , 〈Th〉s)〉s. However, thecooling enhancement turned out to fortunately have a surpris-ingly small effect, presumably due to self-regulation (a stifffactor of 10 on the cooling rate, weknowwould otherwise havehad a strong effect in the case when no heating was included,e.g., Kallander 1996).

The YKKK cooling enhancement approach is a simpleway to include a coupling between code resolution and a sub-resolution process, by just a single constant. One can envisagemore intricate schemes when the sub-resolution behaviour isknown to some extent. An example might be the behaviourof the velocity dispersion of cold clouds in the ISM, that isbecoming increasingly more investigated (e.g., Padoan et al.1997) and it might well be possible to utilise this knowledgeof sub-resolution mass flows, e.g. for a solution to the potentialmetallicity – mass flow problem of Sect. 3.2. Needless to say,such approaches have the unpleasant capability of producing


vastnumbers of new free parameters, and should probably beavoided for that reason alone.

Generally, one might expect that underestimates (e.g., of thecooling rates) might be a more common consequence of resolu-tion problems, but one might also encounter the opposite. Thepreviously mentioned gas mass flowρgv might be an exam-ple, but this is still just a speculation, and remains to be furtherinvestigated.

The cooling enhancement have not been utilised here, ex-cept for the tests (Sect. 4), to be able to compare the results withYKKK. The value ofC = 10 does not seem very precisely de-termined as it is just an estimate proposed by McKee & Ostriker(1977). In principle, it could also depend on other quantities, oron the numerical resolution itself. With the varying resolution ofSPH, this might become complicated. Generally, due to higherresolution, SPH achieves much higher densities anyway, whicheasily compensates for a mere factor of 10. Still, it is valuableto have the extra freedom of increased cooling, should it reallyprove to be needed.

An aspect related to the resolution is that all sub-resolutionprocesses have here been implemented on “isolated” SPH par-ticles, in the sense that the equations (e.g., Eqs. (6), (7) and (9))are solved without the influence of the neighbouring particles,except for mass flows through the∇ · v terms. There mightbe arguments against this approach, since it may induce un-necessary scatter between neighbouring particles. Other SPHapproaches utilise equations with more coupling in the equa-tions (e.g., Hernquist & Katz 1989) or by directly smoothingquantities over the SPH particles neighbours, e.g., the metallic-ity treatment of Steinmetz & Muller (1994). That is, in principle,good SPH formalism forresolvedquantities, but we find littlereason that this should be done also for sub-resolution processes.

Fig. 54 gives some impression of the spatial resolution usedfor these simulations. The plot shows a region within three effec-tive radii. Since the particles have around 64 neighbours within2h, this means that most particles, especially in the centre, havemore than 64 neighbours within the gravitational smoothinglengths (1 kpc for gas and star particles, 2.5 kpc for DM) andwill therefore produce a rather smooth potential, and thus lesstwo-body relaxation. For low mass objects, that contains as fewas 100 particles, this will not be as satisfactory, and these objectswill be artificially relaxed.

Before worrying too much about the scatter commonly seenin plotted SPH data, one should note, as pointed out by Kanget al. (1994), that if calculated on a grid, the SPH representationis as smooth as the results of grid methods.

Another aspect of the code used here is its behaviour onstrongly clustered particle distributions. In other codes (e.g.,Katz et al. 1996) the SPH smoothing lengthsh are limitedto never becoming smaller than the gravitational smoothinglengths or fractions of them. Though this is certainly plausi-ble from some physical points of view, it is also introduced fornumerical considerations. If the Courant-condition is used todetermine the time-steps, a very smallh would be computa-tionally prohibitive, since the Courant-condition scales likeh/c(c being the speed of sound). A downward smoothing floor has

Fig. 54. SPH smoothing lengths for, from top to bottom, the gas, DMand star particles (the gas and DM is almost coinciding), within a radiusof 10 kpc, in “object 39”, the most massive object in the CR3,A = 150simulation.

Fig. 55. The total CPU time plotted against physical time, for the morecomputationally demanding CR3,A = 150 simulation. The relatively“slow start”, is due to the rapid initial expansion. (The observant readermight notice that creationdoestake longer than 6 days. This is, how-ever, total CPU time, and since the Cray C90 have up to 7 processorsavailable, there are still several days to rest and party.)

another numerical disadvantage. The number of neighbours thenbecomes computationally prohibitive, in high density regions,unless this is avoided by sub-sampling, when it happens. It isoften reported that cosmological SPH codes cannot be contin-ued afterz = 1 or evenz = 2 (Weinberg et al. 1997), due to thesevere increase in computational time, as clustering increases.

We have put no downward smoothing floor on the SPHsmoothing lengths, and still there is virtually no increase in com-putational time, as clustering increases. This can be seen fromFig. 55. The reason is due to the time-step scheme (Navarro &


White 1993; Hultman & Kallander 1997), which is not particu-larly limited by small SPH smoothing lengths, nor extreme clus-tering. Another advantage is that densities are actually allowedto become as high as possible, which is of course important, asresolution effects are present.

In previous applications of the present code, a method ofmerging unnecessary gas particles in unresolved regions wasused (Hultman & Kallander 1997, Hultman 1998). This tech-nique was also utilised here but, due to the supernova heating,sub-resolution clustering is less of a problem, and in fact, nomerging events occurred.

Acknowledgements.We thank Nils Bergvall and GoranOstlin, of theExtragalactic research group, for several discussions on this work, theobservational implications in particular. Also Bengt Gustafsson, forgiving detailed comments on the manuscript. John Hultman is gratefulto the Theoretical astrophysics group, at Uppsala Astronomical Obser-vatory, in particular for providing the computational resources needed.Arnaud Pharasyn would like to thank Julie Ferguson, Heather Findlayand Sergio Gelato for a careful reading of the manuscript. The compu-tationally demanding cosmological simulations were performed on theCray C90 atNationellt Superdatorcentrum(NSC), Linkoping, Sweden(Project 96013).

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Astron. Astrophys. 347, 769–798 (1999) ASTRONOMY AND ...aa.springer.de/papers/9347003/2300769.pdf · the formation rates of cold clouds through thermal instability, as will be seen