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Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 30 October 2018 (MN LATEX style file v1.4)

The Effects of Photoionization on Galaxy Formation — II:

Satellite Galaxies in the Local Group

A. J. Benson1, C. S. Frenk2, C. G. Lacey3, C. M. Baugh2 & S. Cole2

1. California Institute of Technology, MC 105-24, Pasadena, CA 91125, U.S.A. (e-mail: abenson@astro.caltech.edu)2. Physics Department, University of Durham, Durham, DH1 3LE, England3. SISSA, Astrophysics Sector, via Beirut 2-4, 34014 Trieste, Italy

30 October 2018

ABSTRACT

We use a self-consistent model of galaxy formation and the evolution of the intergalac-tic medium to study the effects of the reionization of the universe at high redshift onthe properties of satellite galaxies like those seen around the Milky Way. Photoion-ization suppresses the formation of small galaxies, so that surviving satellites arepreferentially those that formed before the universe reionized. As a result, the numberof satellites expected today is about an order of magnitude smaller than the numberinferred by identifying satellites with subhalos in high-resolution simulations of thedark matter. The resulting satellite population has an abundance and a distributionof circular velocities similar to those observed in the Local Group. We explore manyother properties of satellite galaxies, including their gas content, metallicity and starformation rate, and find generally good agreement with available data. Our modelpredicts the existence of many as yet undetected satellites in the Local Group. Wequantify their observability in terms of their apparent magnitude and surface bright-ness and also in terms of their constituent stars. A near-complete census of the MilkyWay’s satellites would require imaging to V≈ 20 and to a surface brightness fainterthan 26 V-band magnitudes per square arcsecond. Satellites with integrated luminos-ity V = 15 should contain of order 100 stars brighter than B = 26, with central stellardensities of a few tens per square arcminute. Discovery of a large population of faintsatellites would provide a strong test of current models of galaxy formation.

Key words: cosmology: theory - galaxies: formation - Local Group - intergalacticmedium

1 INTRODUCTION

High-resolution N-body simulations of the formation of darkmatter halos in the cold dark matter (CDM) cosmogony re-veal a large number of embedded subhalos that survive thecollapse and virialization of the parent structure (Klypinet al. 1999a; Moore et al. 1999). Although their aggre-gate mass typically represents less than 10% of the totalhalo mass, these substructures are very numerous. In richclusters, the abundance of surviving subhalos is compara-ble to the abundance of bright galaxies. In galaxy halos, onthe other hand, the number of subhalos exceeds the num-ber of faint satellites observed in the Local Group by wellover an order of magnitude. This discrepancy has recentlybeen highlighted as a major flaw of the CDM cosmogonyand has prompted investigation of alternative cosmologicalmodels. These range from non-standard models of inflation(Kamionkowski & Liddle 2000) to models in which the uni-verse is dominated by warm, self-interacting or annihilatingdark matter which may not generate small-scale substruc-ture in galactic halos, but may do so on cluster scales (Hogan

1999; Spergel & Steinhardt 2000; Moore 2000; Yoshida etal. 2001; Craig & Davis 2001).

That hierarchical clustering theories predict many moresmall dark matter halos than there are faint galaxies in thelocal universe has been known for a long time, as has onepossible solution to this apparent conflict (White & Rees1978). Feedback generated by the energy injected into galac-tic gas in the course of stellar evolution can regulate starformation in small halos rendering their galaxies too faintto be detected in local surveys (White & Rees 1978; Dekel& Silk 1986; Cole 1991; White & Frenk 1991; Lacey &Silk 1991). The overabundance of dark subhalos around theMilky Way predicted by CDM models was also known be-fore the high resolution N-body simulations highlighted thediscrepancy (Kauffmann, White & Guiderdoni 1993). Us-ing a semi-analytic model of galaxy formation based on theextended Press-Schechter theory for the assembly historiesof dark halos (Bond et al. 1991; Bower 1991), Kauffmannet al. recognized the Milky Way “satellite” problem and ex-amined various possible solutions (see their Fig. 1). Theyconcluded that the best way to reconcile their models withthe luminosity function of the Milky Way’s satellites was to

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2 A. J. Benson, C. S. Frenk, C. G. Lacey, C. M. Baugh & S. Cole

assume that gas is unable to cool within dark matter halosof circular velocity less than 150 kms−1 at redshifts between5 and 1.5. They suggested that this effect might result froma photoionizing background at high redshift, but also notedthat the required suppression threshold of 150kms−1 wasmuch larger than was expected based on physical calcula-tions of the effects of photoionization. Moore (2001) hasexpanded on the reasons why standard supernova feedbackof the kind invoked to explain the relative paucity of faintfield galaxies in CDM models does not, on its own, solvethe Milky Way satellite problem. He points out that the ob-served satellites of the Milky Way of a given abundance havecircular velocities which are about three times smaller thanthe circular velocities of subhalos of the same abundance inthe N-body simulations.

The lack of a Gunn-Peterson effect in the spectra ofhigh redshift quasars indicates that the Universe was reion-ized at z >∼6 (Fan et al. 2000). The reionization of theuniverse raises the entropy of the gas that is required to fuelgalaxy formation, preventing it from accreting onto smalldark matter halos and lengthening the cooling time of thatgas which is accreted. The inhibiting effects of photoion-ization have been investigated in some detail (Rees 1986;Babul & Rees 1992; Efstathiou 1992; Shapiro, Giroux &Babul 1994; Katz, Weinberg & Hernquist 1996; Quinn,Katz & Efstathiou 1996; Thoul & Weinberg 1996; Abel &Mo 1997; Kepner, Babul & Spergel 1997; Weinberg, Hern-quist & Katz 1997; Navarro & Steinmetz 1997; Barkana &Loeb 1999), with the conclusion that galaxy formation isstrongly suppressed by reionization in halos of circular veloc-ity, VC <∼60 kms−1. Although this value is smaller than thevalue assumed by Kauffmann, White & Guiderdoni (1993),their general picture remains valid: the number of satellitesaround bright galaxies is much smaller than the number ofdark matter subhalos because only those subhalos that werealready present before reionization were able to acquire gasand host a visible galaxy. This idea has recently been inves-tigated further by Bullock, Kravtsov & Weinberg (2000).These authors followed the formation of dark halos using amerger tree formalism similar to that of Kauffmann, White& Guiderdoni (1993), but taking into account the tidal ef-fects experienced by substructures when they are accretedinto their parent halo. Combining their halo model with asimple argument based on the mass-to-light ratio of satel-lite galaxies, they calculated their observability, concludingthat, for a reasonable redshift of reionization, the number ofvisible satellites would indeed be close to that observed. Asimilar conclusion was reached by Somerville (2001) using asemi-analytic model similar to that of Cole et al. (2000) butwith more limited model of photoionization than used in thiswork (specifically, the model use by Somerville (2001) doesnot self-consistently evolve the properties of galaxies and theIGM and does not account for the effects of photoheating ofvirialized gas in halos or tidal disruption of satellites).

In this paper, we investigate the abundance and proper-ties of satellite galaxy populations using a model of galaxyformation which self-consistently calculates the physics ofreionization and the process of galaxy formation. We usethe semi-analytic techniques developed by Cole et al. (2000)which we have recently extended to enable calculation of thecoupled evolution of the intergalactic medium (IGM) andgalaxies (Benson et al. 2001; hereafter Paper I). The model

follows the formation and evolution of stars, the productionof ionizing photons from stars and quasars, the reheating ofthe IGM, and the associated suppression of galaxy formationin low mass halos. The model also incorporates a detailedtreatment of the dynamics of satellite halos under the in-fluence of dynamical friction and tidal forces. Like the halomodel of Bullock, Kravtsov & Weinberg (2000), our modelagrees with the results of the high-resolution N-body simula-tions. In Paper I, we demonstrated that reionization reducesthe number of faint field galaxies in the local Universe, flat-tening the faint end slope of the galaxy luminosity function,in good agreement with the most recent observational de-terminations. In this paper, we will explore the propertiespredicted by this very same model (i.e. with the same param-eter values) for the population of satellites around galaxieslike the Milky Way. We will carry out as detailed a com-parison as is possible with current observational data andpresent tests of our model which rely on the prediction of alarge and as yet undetected population of faint satellites inthe halo of the Milky Way.

The remainder of this paper is organised as follows. In§2, we briefly describe our model. In §3, we calculate the ex-pected luminosity function and circular velocity function ofsatellites around galaxies like the Milky Way. In this section,we also discuss observational strategies for discovering thelarge faint satellite population predicted by our model. In §4,we compare the gas content, star formation rate, metallic-ity and structure of our model satellites with observationaldata. Finally, in §5 we present our main conclusions.

2 MODEL

We begin by briefly describing our model, a full descriptionof which may be found in Paper I. We represent the IGMas a distribution of gas elements whose density evolves asa result of the expansion of the Universe and the formationof structure. Each element ‘sees’ a background of ionizingphotons emitted by stars (as given by the star formationhistory in our model of galaxy formation) and by quasars (asobtained from the observational parametrization of Madau,Haardt & Rees 1999), which ionize and heat the gas. Fromthis, we derive the thermal and ionization history of theIGM. The hot IGM acquires a significant pressure whichhampers the accretion of gas into low mass dark matterhalos. From the inferred thermal state of the IGM we derivethe filtering mass, defined as the mass of a dark matter halowhich accretes a gas mass equivalent to 50% of the universalbaryon fraction (Gnedin 2000), as a function of time.

Knowledge of the filtering mass allows us to determinehow much gas is available for galaxy formation in a givendark matter halo at any time. The increase in entropy pro-duced by reionization causes lower mass halos to contain asmaller fraction of their mass in the form of gas than largermass halos. The ionizing background which accumulates af-ter reionization also heats gas already present in dark matterhalos, preventing it from cooling into the star forming phase.We include this heating in our estimates of the mass of gaswhich can cool, resulting in a further suppression of galaxyformation. When a galaxy falls into a larger halo becominga satellite in it, we compute its orbit through the halo indetail, accounting for the effects of dynamical friction, tidal

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The Effects of Photoionization on Galaxy Formation — II: Satellite Galaxies in the Local Group 3

limitation and gravitational shocking (as described by Tay-lor & Babul (2000) and in Paper I). As was demonstratedin Paper I, our model of satellite dynamics reproduces theresults of high-resolution N-body simulations with reason-able accuracy in both of the cosmological models for whichnumerical results are available.

In Paper I, we performed calculations using our ex-tended galaxy formation model in a ΛCDM cosmology(mean mass density Ω0 = 0.3, cosmological constant termΛ/3H2

0 = 0.7, mean baryon density Ωb = 0.02, and Hub-ble parameter h = 0.7.) The values of the model parame-ters required to describe the relevant physical processes arestrongly constrained by a small subset of the local galaxydata; we showed that the model also reproduces many otherproperties of the galaxy population at z = 0 (see also Cole etal. 2000). We will retain exactly the same parameter valuesthroughout the present work. The result is a fully-specifiedmodel of galaxy formation that incorporates the effects ofreionization and tidal limitation and which we now use toexplore in detail the properties of satellite galaxies aroundthe Milky Way.

3 THE ABUNDANCE OF SATELLITES

We study the population of satellite galaxies that form in anensemble of halos harbouring galaxies similar to the MilkyWay. We class a galaxy as being “similar to the Milky Way”if the circular velocity of its disk (VC,d, measured at the diskhalf-mass radius) is between 210 and 230 kms−1, and if itsbulge-to-total mass ratio (including stars and cold gas) isin the range 0.05 to 0.20 (which is approximately the rangefound by Dehnen & Binney (1998) in mass models of theMilky Way).

Using our model of galaxy formation, including all theeffects of photoionization and tidal limitation described in§2, we construct 1800 realizations of dark matter halos withmass in the range 4.0× 1011 to 2.3× 1012h−1M⊙ (the rangein which we find galaxies similar to the Milky Way) at z = 0.We ensure that our calculation resolves all halos into whichgas is able to accrete and cool in the redshift interval 0 to25. From the set of simulated halos we select those whichcontain a central galaxy similar to the Milky Way. We findapproximately 70 such halos in our sample. These have “qui-eter” merger histories than is typical for halos of their mass,as was shown, for example, by Baugh, Cole & Frenk (1996).All other halos are discarded and, for the remainder of thispaper, we consider only the satellite populations of haloshosting Milky Way type galaxies. We refer to this sampleof satellites as our “standard model.” For comparison, wealso generated a sample of Milky Way satellites using ourmodel with no photoionization (but still including the ef-fects of tidal limitation of satellites), which we will refer toas the “no photoionization” model, and a sample of MilkyWay satellites with the original model of Cole et al. (2000),which includes neither photoionization nor tidal effects.

To test our model, we make use of observational dataon Local Group galaxies taken from the compilation by Ma-teo (1998). Mateo points out that the census of Local Groupdwarfs is almost certainly incomplete. Unfortunately, the ob-servational selection effects are not quantitatively well un-derstood, and so we do not attempt to correct for them here,

but note that they could possibly alter the observational re-sults significantly. Some galaxies classed as members of theLocal Group lie outside the virial radii of the dark matter ha-los thought to be associated with the Milky Way and M31.For the Milky Way, we can estimate the virial radius us-ing the three-component mass models of Dehnen & Binney(1998), some of which assume dark matter halos with theNFW (Navarro, Frenk & White 1997) profile appropriateto the CDM universe that we are considering here. Of these,their model 2d gives the best fit to the observational datathat they consider. Assuming a spherical top-hat collapsemodel for the Milky Way halo then implies a virial mass andradius of MMW = 1.11 × 1012M⊙ and RMW =272 kpc re-spectively (for the particular set of cosmological parametersconsidered in this paper). This is in good agreement with thevirial radii of the halos that end up hosting Milky Way typegalaxies in our model, which typically have Rvir ≈ 300 kpc.(The model predicts a distribution of Rvir because MilkyWay type galaxies can be found in halos with a range ofmasses). Lacking any better estimate, we also take RMW asthe virial radius of M31, since the measured circular velocityof M31 is quite close to that of the Milky Way. Throughoutthis section, we will distinguish between Local Group satel-lites and satellites lying within RMW of either the MilkyWay or M31.

3.1 Abundance as a function of luminosity and

stellar mass

We begin by comparing the luminosity function of the popu-lation of satellites in the model with the observed luminosityfunction of satellites of the Local Group. Fig. 1 shows theV-band luminosity function of satellites⋆ , normalised to thenumber of central galaxies (i.e. the Milky Way and M31),as filled circles. We include only satellites within RMW of ei-ther the Milky Way or M31. Throughout this paper, we usemagnitudes corrected for foreground extinction by the MilkyWay using the reddening values listed by Mateo (1998). Themedian luminosity function from all pairs of Milky Way typegalaxies in our standard model is shown by the solid line,with errorbars enclosing 10% and 90% of the distribution ofluminosity functions. (We calculate the median and intervalsfor pairs of halos since we are comparing to the combinedMilky Way and M31 datasets and the scatter in model pre-dictions for pairs of halos is, of course, smaller than for sin-gle halos.) The satellite galaxies in our model typically haveonly very small amounts of internal dust-extinction (all buta tiny fraction have less than 0.1 magnitudes of extinction inthe V-band). Nevertheless, we include the effects of internaldust-extinction in all model magnitudes.

It is immediately apparent in Fig. 1 that photoioniza-tion does help to reduce the number of satellite galaxies:our model predicts many fewer satellites than the model ofCole et al. (2000). Furthermore, the dashed line shows thattidal stripping also acts to reduce the number of satellitesof a given luminosity, but this is a much smaller effect. ForMV ≈ −10, the combined effects of tidal stripping and pho-toionization reduce the number of satellites by about a factorof 10 compared to the Cole et al. (2000) model. Note that

⋆ We include the SMC, LMC and M33 in this category.

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4 A. J. Benson, C. S. Frenk, C. G. Lacey, C. M. Baugh & S. Cole

Figure 1. The V-band luminosity function of satellite galaxies(per central galaxy). The Local Group data, taken from the com-pilation of Mateo (1998), are shown as filled circles, except forbins in which the luminosity function is zero which we indicateby an open circle at arbitrary position on the y-axis (faintwardsof MV = −9 there are no known satellites so we plot no sym-bols). Only satellites within RMW of the Milky Way or M31 are

included and no correction for any possible incompleteness hasbeen applied. The median luminosity function of satellites aroundpairs of Milky Way type galaxies in our model is shown by thesolid line, with errorbars indicating the 10% and 90% intervals ofthe distribution of luminosity functions found in approximately70 realizations of the satellite galaxy population. Where the 10%interval corresponds to zero galaxies, a single-headed downward-pointing arrow is plotted. Where there is no connecting line, themedian is zero, and the 90% interval is shown by an errorbar andarrow. Where the 90% interval corresponds to zero galaxies (i.e.less than one in ten of the simulated halos contained any galaxiesof this magnitude), we show a double-headed downward-pointingarrow (at an arbitrary position on the y-axis). The dashed linecorresponds to the model in which the effects of photoionizationare ignored (but tidal limitation of satellites is included), whilethe dotted line shows the prediction from the model of Cole et al.(2000) and the dot-dashed line shows results from our standardmodel with the reduced escape fraction of fesc = 10%. For claritywe have omitted error bars from these three models; these aretypically 10–20% smaller than for the standard model.

the model predicts significant variation in the satellite lumi-nosity function from halo to halo. Once this scatter is takeninto account, our standard model is in reasonable agreementwith the Local Group luminosity function, except perhapsat the brightest magnitudes, where the model underpredictsthe number of satellites. However, the statistics are poor atthese magnitudes, because there are so few galaxies. For ex-ample, the −17.5 ≥ MV > −18.5 bin in Fig. 1 contains asingle galaxy (the LMC) within RMW, implying a mean of0.5 such satellites per central galaxy. Our model predicts amean of approximately 0.08. Since no halo in the samplecontains more than one such galaxy, 8% of the model ha-los contain a satellite galaxy in this luminosity range. Thus,

such bright satellites are not impossible in our model, butthey are quite rare.

As was noted in Paper I, the disk velocity dispersionused in the Taylor & Babul (2000) model (σd = VC,d/

√2)

is rather high for the Milky Way. This quantity determinesthe magnitude of the dynamical friction force felt by satel-lites which pass close to the disk of the central galaxy, andso may affect the rate of merging for satellites on highly ec-centric orbits. An overly large σd would result in a reduceddynamical friction force and a lower rate of mergers, po-tentially leading us to overestimate the number of satellitesremaining. However, adopting the more appropriate value,σd = 0.2VC,d, makes virtually no difference to the predictednumber of satellites, since the the disk velocity dispersion isonly important for satellites very close to the central galaxy,and which therefore tend to merge shortly afterwards in anycase. On the other hand, the abundance of the faintest satel-lites is fairly sensitive to two uncertain model assumptions,the escape fraction of ionizing photons, fesc, and the lumi-nosity of the galaxies that we have classed as resembling theMilky Way.

In Paper I, we found that adopting an escape fraction,fesc = 100% (i.e. all ionizing photons produced by stars es-cape from their galaxy into the IGM), results in a reasonableredshift of reionization (z ≈ 8), but also in an ionizing back-ground which is significantly higher than current observa-tional estimates. On the other hand, a fraction, fesc = 10%,results in a more reasonable ionizing background and is inbetter agreement with observational determinations of fesc(Leitherer et al. 1995; Steidel, Pettini & Adelberger 2001),but implies a lower redshift of reionization (z ≈ 5.5), per-haps uncomfortably close to the lower limit imposed bythe lack of a Gunn-Peterson effect in the highest redshiftquasars. We find that the lower redshift of reionization asso-ciated with fesc = 10% leads to a larger number of satellitesof a given luminosity (as shown by the dot-dashed line inFig. 1), producing a luminosity function which, faintwardsof MV = −11, is a factor of about two too high comparedwith the data (even after allowing for halo-to-halo varia-tions). However, possible incompleteness at the faint end ofthe observed luminosity function makes this comparison in-conclusive. We will adopt fesc = 100% as our standard valuefor all results presented in this paper, unless stated other-wise.

A second source of uncertainty is the identification ofmodel galaxies with the Milky Way. As discussed in Pa-per 1 (see Fig. 9), our modelling of the structure of disksis crude and produces circular velocities for spiral galax-ies that are typically 30% larger for their luminosity thanis implied by the observed Tully-Fisher relation. As a re-sult, the model galaxies that we have identified with theMilky Way according to their circular velocity are some-what fainter than galaxies lying on the mean Tully-Fisherrelation at that circular velocity. We could alternatively se-lect Milky Way galaxies according to luminosity. Direct es-timates of the luminosity of the Milky Way are rather un-certain and so we instead estimate the luminosity from theobserved Tully-Fisher relation as advocated by Binney &Merrifield (1998, §10.1). The observed Tully-Fisher relation(Matthewson, Ford & Buchhorn 1992) implies an I-bandmagnitude in the region of −21.5 >∼MI − 5 log h >∼− 22.0 fora typical galaxy with the circular speed of the Milky Way.

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The Effects of Photoionization on Galaxy Formation — II: Satellite Galaxies in the Local Group 5

Model galaxies with I-band magnitude in this range (andbulge-to-total ratio in the range discussed above) typicallyhave higher circular velocities, and inhabit higher mass ha-los than the Milky Way type galaxies of our original sample.As a result, this new sample has a substantially larger num-ber of satellites. Its luminosity function is compatible withobservations for magnitudes brighter than MV ≈ −13, butfaintwards of this, it overpredicts the number of satellites.Again, possible incompleteness in the faint data makes it dif-ficult to exclude this model, but for it to be compatible withobservations, the data would have to be severely incompleteso that at least 70% of MV ≈ −10 satellites should havebeen missed.

As can be clearly seen from Fig. 1, our model predictsthat the satellite galaxy luminosity function should continueto rise at magnitudes fainter than the faintest satellite yetobserved. The luminosity function peaks at MV ≈ −3 andthen falls off at fainter magnitudes. This cut-off is causedby the inability of gas below 104K to cool when no molec-ular hydrogen is present. The cut off occurs at significantlyfainter magnitudes in our standard model than in the modelwith no photoionization or the Cole et al. (2000) model sincephotoionization reduces the efficiency of galaxy formation inhalos of a given mass (or, equivalently, virial temperature)

making the galaxies in those halos fainter†. It is interestingtherefore to consider the observability of the galaxies thatmake up the faint end of the satellite luminosity function.In Fig. 2 we show the number of satellite galaxies per haloas a function of their apparent V-band magnitude. To inferthe distances to the satellites, we have made use of theirorbital positions, assuming that they are observed from alocation 8 kpc from the centre of the host halo (i.e. at aboutthe position of the Sun in the Milky Way). (The mean dis-tance for MV ≤ −10 satellites is 95 kpc and this changes

only slowly with MV)‡. The solid line shows all satellites,

while the dotted and dashed lines show only those whosecentral surface brightness is brighter than 26 and 22 mag-nitudes per square arcsecond respectively. Central surfacebrightnesses for model galaxies were computed using thepredicted luminosities and sizes, with disk components mod-elled as exponential disks, and spheroids as King profileswith rt/rc = 10. The faintest known Local Group galaxyis Tucana (mV = 15.15 with almost no foreground extinc-tion; note that this point is not seen in Fig. 2 since Tucanalies outside of the Milky Way’s virial radius), only slightlybrighter than the peak in the predicted apparent magni-tude distribution. Surface brightness adds a further limit tothe detectablility of satellites. Currently the lowest surface

† We are able to predict the properties of these very faint galaxiesby extrapolating our standard rules for star formation, feedbacketc. to these very small objects. However, it should be kept inmind that these very faint galaxies typically contain only a fewthousand Solar masses of stars. It is not clear how well our sim-plified rules for star formation describe reality in such systems,where the entire galaxy is less massive than a single giant molec-ular cloud.‡ The number density of satellites within the virial radius of aMilky Way type halo in our model scales roughly as r−2 forr >∼10 kpc with a rapid cut-off towards Rvir, but flattens sig-nificantly at smaller radii since these inner satellites are stronglyaffected by tidal forces and dynamical friction.

Figure 2. V-band number counts of satellites. The solid lineshows the results from our standard model with no cut on cen-tral surface brightness, while dotted and dashed lines show theeffects of applying a central surface brightness cut of 26 and 22magnitudes per square arcsecond respectively. Lines show the me-dian counts in the standard model, with errorbars indicating the10% and 90% intervals of the distribution for the results with

no surface brightness cut. (The errorbars for the 26 magnitudesper square arcsecond sample are comparable to these, exceptfor mV < 16 where they are about twice as large.) Where the10% interval corresponds to zero galaxies, a single-headed down-ward pointing arrow is shown. Where only an errorbar or arrowis shown (i.e. there is no connecting line), the median is zero,and where the 90% interval corresponds to zero galaxies we showa double-headed downward-pointing arrow (at arbitrary positionon the y-axis). Note that unlike the luminosity function (Fig. 1),the results here refer to a single halo (as opposed to pairs of ha-los). Apparent magnitudes for satellites are computed from theirposition in the host halo, assuming the observer is located 8 kpcfrom the centre. The filled circles show the observed counts fromMateo (1998), including only those galaxies within RMW of theMilky Way. Where a bin contains no observed satellites we showan open circle at arbitrary position on the y-axis (faintwards ofthe mV = 12 bin there are no observed satellites, so we do notplot any symbols).

brightness member of the Local Group that is known is Sex-tans, with µ0 = 26.2± 0.5mag arcsec−2 (Mateo 1998). Thisis close to the dotted line in Fig. 2. It is clear from Fig. 2that the faint satellites have very low surface brightness andthat finding them will require very deep photometry.

Several satellites have now been detected as an excessof stars against the background of the Milky Way (e.g. Irwin& Hatzidimitriou 1995). From the star formation historiesof our model satellites, we can calculate the number andsurface density of stars visible today. We use the stellar evo-lutionary tracks of Lejeune & Schaerer (2001) to obtain thenumber of stars brighter than a certain luminosity, Lc, in asingle stellar population of unit mass, as a function of ageand metallicity:

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6 A. J. Benson, C. S. Frenk, C. G. Lacey, C. M. Baugh & S. Cole

NLc(t, Z) =

∫ Mmax

Mmin

SLc(M, t, Z)

dn

dMdM, (1)

where dn/dM is the stellar Initial Mass Function (IMF; weuse the IMF of Kennicutt (1983), as assumed in our calcu-lations of galaxy luminosities), normalized to unit mass andwith minimum and maximum masses, Mmin and Mmax (0.1and 125M⊙ respectively), and

SLc(M, t, Z) =

0 if L(M, t, Z) < Lc

1 if L(M, t, Z) ≥ Lc,(2)

where L(M, t, Z) is the luminosity of a star of zero-age massM , age t and metallicity Z. The number of stars more lu-minous than Lc in a galaxy at the present day is then givenby

N =

∫ t0

0

ρ⋆(t)NLc(t0 − t, Z[t])dt, (3)

where ρ⋆(t) is the star formation rate in the galaxy at time tand Z[t] is the metallicity of the stars being formed at timet. For each satellite, we compute the luminosity correspond-ing to a given apparent magnitude limit at its position in thehalo and determine N . From the known size of the disk andspheroidal component of each model galaxy, we also derivethe central number density of these stars. Fig. 3 shows theresult of this calculation. (We do not include the effects ofdust extinction on the stellar luminosities since this variesacross the galaxy, but we do include them in the total galaxymagnitude. As noted above, these internal extinction cor-rections are typically small in any case.) The central stellardensities predicted by our model are in excellent agreementwith those measured by Irwin & Hatzidimitriou (1995) fromscanned photographic plates with a limiting magnitude ofB= 22: compare the open stars with the open triangles inFig. 3. Imaging two magnitudes deeper (circles) would allowthe detection of galaxies to V=17–18 at the same centralsurface density, although the higher density of backgroundobjects may make detection more difficult.

3.2 Abundance as a function of circular velocity

In this subsection we derive the abundance of model satel-lites as a function of their circular velocity and compare itto the Local Group data. This is the statistic originally em-ployed by Klypin et al. (1999a) and Moore et al. (1999) tohighlight the apparent discrepancy between the CDM cos-mogony and the properties of the satellites of the MilkyWay. Their motivation for using the distribution of circularvelocities of satellite halos it is that can be straightforwardlyobtained from N-body simulations including only dark mat-ter, whereas the luminosities of satellite galaxies, while fairlyeasy to measure observationally, cannot be so easily pre-dicted theoretically, because of the complicated dependenceon the processes of cooling, star formation, feedback andphotoionization. The main drawback of making the compar-ison in terms of circular velocities is that measuring circularvelocities is a difficult task for the majority of the LocalGroup satellites which do not have gas disks, and in anycase these measurements give the circular velocity withinthe visible galaxy, not the value at the peak of the rotationcurve of the dark halo, which is typically what is measuredin the simulations. In addition, the comparisons by Klypin

Figure 3. The number of stars (filled symbols) and the surfacedensity of stars (open symbols) brighter than B⋆=20, 22 and 24(squares, triangles and circles respectively) for satellites of MilkyWay type galaxies as a function of the apparent magnitude ofthe galaxy. The measured central stellar densities for a subset ofthe Local Group satellites measured by Irwin & Hatzidimitriou(1995), with a limiting magnitude B = 22, are shown as open

stars.

et al. (1999a) and Moore et al. (1999) implicitly assumeda tight relation between subhalo circular velocity and satel-lite luminosity, since otherwise the observational selectionon luminosity modifies the form of the circular velocity dis-tribution, as we will see below.

Before proceeding with the calculation of the satellitecircular velocity distribution, it is important to check thatour model produces realistic sizes for the satellites, as wellas the correct luminosity-circular velocity relation. The sizeof a galaxy influences the circular velocity because it de-termines the self-gravity of the visible component (and theassociated compression of the halo which we treat using adi-abatic invariants, as discussed in Cole et al. 2000). It alsodetermines the part of the rotation curve which is accessi-ble to observations. The luminosity-circular velocity relationenters because, in practice, the distribution of circular ve-locity is measured for a sample selected to be brighter thana given luminosity.

In Fig. 4 we compare the half-light radii of our modelsatellites with observations. We estimate the half-light radiiof real satellites using published fits to their surface bright-ness profiles (typically exponentials or King models) in the

manner described in detail in the figure caption§. For the

§ For reference, half of the Local Group satellites listed by Mateo(1998), as well as the LMC, SMC and M33, are irregulars and halfare spheroidals; for galaxies brighter than MV = −8, the modelpredicts 78% spiral/irregulars and 22% spheroidals. It is likelythat the morphological evolution of satellites has been influencedby dynamical processes not included in our model (e.g. Mayer etal. 2001).

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The Effects of Photoionization on Galaxy Formation — II: Satellite Galaxies in the Local Group 7

model satellites, we take the half-light radius to be the half-mass radius of the stellar system (including both disk andspheroid) that remains within the effective tidal radius. (InPaper I we defined the effective tidal radius as the radiusin the original satellite density profile beyond which mate-rial has been lost.) We show results for our standard, nophotoionization and Cole et al. (2000) models. The sizesof satellites in our standard model are significantly smallerthan in the Cole et al. (2000) model, particularly fainterthan MV = −15. As the figure shows, much of this reduc-tion is due to tidal limitation which strips away the outerlayers of the galaxies. Photoionization appears to have lit-tle effect on the sizes of satellites. In reality, photoionizationdoes reduce the size, but it also reduces the luminosity, es-sentially preserving the size-luminosity relation, so that thegalaxies typically move in a direction almost parallel to themedian relation in Fig. 4. The model predictions are in goodagreement with the data and, given the rather crude ob-servational determinations available, this level of agreementseems sufficient at present.

The ‘Tully-Fisher’ relation between the circular velocityand the absolute V-band magnitude of satellites is plottedin Fig. 5. Whenever possible, the circular velocity of realsatellites was estimated from the rotation velocity of gas inthe ISM (corrected for inclination). Where a measurementof this is not available, we used instead the measured stel-

lar velocity dispersion multiplied by a factor√3.¶ For the

model satellites, we plot the circular velocity at the stellarhalf-mass radius, which is typically ∼ 25% smaller than thepeak circular velocity in the tidally limited halo. For thefainter satellites (MV ≃ −10), the circular velocity at thehalf-mass radius is about half the value at the virial radius,whereas for the brighter satellites (MV ≃ −15), the two aresimilar. The theoretical relations provide a good descriptionof the data, except in the magnitude range −9 <∼MV <∼− 14for which the model velocities are about a factor of 2 toohigh. In this range, the no photoionization model performsslightly better than the standard model since tidal limitationresults in the rotation velocity being measured at a smallerradius, an effect which is offset by the fact that photoion-ization requires galaxies of fixed luminosity to form in moremassive halos when photoionization is switched on (comparethe solid and dashed lines in Fig. 5). While it is possible thatthis discrepancy may reflect a shortcoming of the model, theestimation of circular velocities from current observationaldata is highly uncertain; improved determinations are ex-tremely important. In addition, there is also some theoreti-cal uncertainty in the calculation of the circular velocity. Forexample, as noted in Paper I, our model does not account

¶ The factor√3 follows from the assumption of isotropic stellar

orbits in an singular isothermal potential, for a stellar distribu-tion with an r−3 density profile, assumptions which may not berelevant to real galaxies. Alternatively, the average line-of-sightvelocity dispersion of the satellite’s spheroid can be estimated bymodelling the spheroid as a King profile (which is often a good fitto observed data) and then solving the Jeans equation (see Pa-per I, eqn. 21). Assuming isotropic orbits, we find Vbulge ≈ 1.2σ∗

for model satellites, albeit with large scatter, where Vbulge is thecircular velocity of the bulge at the stellar half-mass radius and σ∗

is the stellar line-of-sight velocity dispersion. Adopting this lattervalue does not change our conclusions for the velocity function.

Figure 4. The half-light radii of satellite galaxies as a functionof absolute V-band magnitude. The points show the radii of thesatellites of the Local Group, estimated from the data compiled byMateo (1998). For spiral and irregular galaxies, we convert theirmeasured exponential scalelength, rexp, to a half-light radius byassuming an exponential disk density profile, r1/2 ≈ 1.68rexp; weshow these points as squares. For ellipticals and dwarf spheroidals,

we use the quoted parameters of the King (1966) model fits toreconstruct the density profile of the galaxy, and so infer the half-light radius. The core and tidal radii listed by Mateo (1998) aremeasured along the major axis of each galaxy. We replace thesewith the equivalent radii for spherically symmetric systems bycomputing the geometric mean of the radii along the semi-majorand semi-minor axes (using the measured ellipticities, which weassume to be independent of radius). These points are shown astriangles. (In the few cases where a galaxy is classed as Irr/dSph,we include it in the Irr class for the purposes of this plot.) Filledpoints indicate satellites within a distance RMW ≈ 270 kpc fromthe Milky Way or M31, and open symbols indicate more distantsatellites. The solid line corresponds to our standard model, thedashed line to the model that ignores photoionization but whichincludes tidal stripping of satellite halos, and the dotted line tothe Cole et al. (2000) model. The lines show the median rela-tion and the errorbars indicate the 10% and 90% intervals of thedistribution in the standard model. In this and subsequent fig-ures errorbars are only shown where the magnitude bin containsenough model galaxies to estimate the 10% and 90% intervalsaccurately, and so do not appear at the brightest magnitudes.The scatter in the no-photoionization model is similar to this,but in the Cole et al. (2000) model it is typically 60-70% smaller.The half-light radius of the model satellites is taken to be thehalf-mass radius of the galaxy (including both disk and spheroid)remaining within the effective tidal radius. The half-mass radii ofthe disk and spheroid prior to tidal limitation were determined inthe manner described by Cole et al. (2000).

for changes in the density profile of the satellite after tidallimitation, which may reduce the rotation speed somewhat(Mayer et al. 2001).

The satellite cumulative velocity function is shown inFig. 6. In our model of photoionization, every dark matterhalo whose virial temperature is larger than the minimumtemperature for cooling (see Fig. 1 of Paper I) forms a galaxy

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8 A. J. Benson, C. S. Frenk, C. G. Lacey, C. M. Baugh & S. Cole

Figure 5. The relation between circular velocity and V-band ab-solute magnitude for satellite galaxies. For the real satellites of theLocal Group (Mateo 1998), circles indicate circular velocities in-ferred from the rotation speed of the ISM, while triangles indicatecircular velocities estimated from the stellar velocity dispersion asdiscussed in the text. Filled symbols denote galaxies within RMW

of the Milky Way or M31, while open symbols denote more dis-

tant galaxies. The median relation from our standard model isshown by the solid line with errorbars indicating the 10% and90% intervals of the distribution. For the model, we plot the cir-cular velocity at the half-mass radius of the satellite (includingboth disk and spheroid and taking only the mass remaining withinthe effective tidal radius). The dashed line shows the results fromthe model in which the effects of photoionization are ignored, buttidal limitation of halos is included, while the dotted line showsthe results from the model of Cole et al. (2000). The scatter inthe former is similar to that in the standard model, but in thelatter it is about 70% of that in the standard model.

(although in low mass halos the galaxies have extremelylow mass), and tidal stripping never completely destroys asatellite halo (although halos may be stripped to very smallradii). As a consequence, if we naively constructed the fullcumulative velocity function of satellites in our model, itwould be identical to that of Cole et al. (2000). In reality,many of the satellites in our model are extremely faint ei-ther because photoionization severely restricted their supplyof star-forming gas, or because tidal stripping removed mostof their stars, and so would be unobservable. We thereforeapply an absolute magnitude limit to our sample of satel-lites and construct the cumulative velocity function only forgalaxies brighter than this limit. The sample of known Lo-cal Group satellites does not have a well-defined absolutemagnitude limit (nor an apparent magnitude limit for thatmatter), but the faintest galaxy listed by Mateo (1998),Draco, has MV = −8.8. We therefore choose MV = −9 as asuitable cut for our models and apply the same cut to thedata.

Results for our standard, no photoionization and Coleet al. (2000) models are shown with solid, dashed and dottedlines respectively in Fig. 6. It is immediately apparent that

Figure 6.

while the model of Cole et al. (2000) substantially overpre-dicts the number of satellites, our photoionization model isin very good agreement with the data. As the figure clearlyshows, the main reduction comes from the effects of pho-toionization; tidal limitation alone has only a minor effecton the velocity function. The scatter from realization to real-ization is large: the central 80% of the distribution of velocityfunctions spans a factor of 3. Adopting a magnitude cut atMV = −10 produces equally good agreement (except, per-haps, at the lowest circular velocities, <∼13km/s), but takingMV = −8 leads to roughly twice as many satellites in the

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The Effects of Photoionization on Galaxy Formation — II: Satellite Galaxies in the Local Group 9

Figure 6. (cont.) The cumulative velocity function of satellitegalaxies. Data for the Local Group per central galaxy (i.e. satel-lites within RMW of the Milky Way or M31), taken from Mateo(1998), are shown by heavy lines without errorbars (in the toppanel we also plot squares to indicate the contribution from each

individual satellite). Top panel: The mean cumulative velocityfunction of satellites brighter than MV = −9 in our standardmodel is shown by the solid line, with errorbars enclosing 10%and 90% of the distribution of velocity functions (from a sampleof approximately 70 Milky Way type halos). The circular veloci-ties plotted are measured at the half-mass radius of the combineddisk and spheroid system after accounting for tidal limitation.The dot-dashed line shows the effect of reducing fesc to 10% (from100% in the standard model). The dashed line shows the modelin which the effects of photoionization are neglected but satellitelimitation by tidal stripping is included, while the dotted line cor-responds to the model of Cole et al. (2000). The scatter in theformer is similar to, and in the latter it is typically 70% of, thescatter in the standard model. All models have R0

c/Rvir = 0.5.Middle panel: Dependence on the absolute magnitude cut. Thethin lines show results for the standard model for the differentmagnitude cuts: solid for MV = −9 (as in the top panels), dot-ted for MV = −8, and dashed for MV = −10. The thick linesshow the Local Group data for the same magnitude cuts, withthe same line types (the MV = −10 line lies almost entirely un-der the MV = −9 line). Lower panel: As middle panel for thestandard model but with different values of R0

c/Rvir (as given inthe legend); R0

c/Rvir is the model parameter that controls theinitial orbital radii of satellites as defined in Paper I.

range 16km/s <∼Vc <∼40km/s as is observed. In all cases, thephotoionization model predicts far fewer satellites than theCole et al. (2000) model. Our results are weakly dependenton the assumed photon escape fraction. The result of as-suming fesc = 10% (instead of fesc = 100% in the standardmodel) is shown by the dot-dashed line in the upper panel.The number of satellites with Vc <∼30kms−1 is increased byabout 25% in this latter case.

Finally we show in the lower panel of Fig. 6 the effectof varying the ratio R0

c/Rvir, which parameterizes the initialorbital energy of a satellite in terms of the energy of a cir-cular orbit of radius R0

c . In Paper I, we demonstrated that avalue of R0

c/Rvir = 0.5 provides a good match to the num-ber of subhalos seen in high-resolution N-body simulations,and is comparable to the mean value measured for satellitesin those simulations at z = 0. Increasing R0

c/Rvir to 0.75produces somewhat too many satellites, while reducing itto 0.25 seriously underpredicts the number of satellites. In-creasing R0

c/Rvir makes satellite orbits begin at larger radiiwhere tidal forces are weaker and so allows more satellitesto survive (decreasing R0

c/Rvir has the opposite effect). Themagnitude of the changes induced by adopting these alterna-tive values for R0

c/Rvir is comparable to that seen for calcu-lations of satellite halo abundances in the pure dark mattercalculations reported in Paper I, and so can be seen to bedue almost entirely to the enhanced(reduced) tidal limita-tion and dynamical friction resulting from a smaller(larger)value of R0

c/Rvir, supplemented slightly by changes in theluminosities of satellites which alter whether they meet ourselection criteria for this figure.

The exact distribution of R0c/Rvir is therefore very im-

portant, and would be worth determining accurately fromnumerical simulations. We stress that the value R0

c/Rvir =

0.5 was chosen to match the results of N-body simulationsof dark matter, and also results in good agreement with theabundances of satellite galaxies.

4 OTHER PROPERTIES OF SATELLITES

In addition to the abundance, our model of galaxy formationpredicts many other properties of the satellite galaxy popu-lation and their associated dark halos. We now explore someof them, particularly those for which observational data areavailable.

4.1 Gas content, star formation rates,

metallicities and colours

We begin by considering the gaseous content and star forma-tion rates of satellites. The mass of hydrogen (atomic plusmolecular), normalized to the V-band luminosity, is shown,as a function of V-band magnitude, in Fig. 7. A rapid declinein the gas content towards faint magnitudes is predicted bythe models. This is a direct consequence of the strong effectsof supernova feedback in faint, low mass galaxies which ef-ficiently eject much of their cold gas and rapidly consumeany remaining fuel (these galaxies, being satellites, are un-able to accrete fresh gas in our model). The observationaldata show a band of almost constant gas-to-luminosity ratiowhich is occupied exclusively by galaxies beyond RMW ofthe Milky Way or M31 at faint magnitudes. Galaxies withinRMW frequently have only upper limits to their gas mass.Our model predictions are consistent with the observationsbut, since the data often consist only of upper limits, thiscomparison is not particularly conclusive.

We now consider the star formation rates in satellitesas estimated from their Hα luminosities. In Fig. 8, we plotthe Hα luminosity per unit hydrogen mass for satellites withmeasurements or limits on Hα (all such galaxies also havemeasured gas masses). With so few data points (note thatmany of the galaxies in the plot lie further than RMW fromthe Milky Way or M31), it is difficult to quantify the levelof agreement between model and data. Of the six satelliteswithin RMW with measured Hα luminosity, four lie close tothe model prediction, while one fainter satellite has an Hαto hydrogen ratio an order of magnitude above the modelrelation, and another has an upper limit lying three ordersof magnitude below the model expectation.

In Fig. 9, we plot the metallicity of the ISM gas (upperpanel) and of the stars (lower panel). In both cases, the Lo-cal Group satellites exhibit a trend of increasing metallicitywith luminosity. Our models show a trend in the same sense,although for the ISM it appears somewhat weaker than forthe data. It is interesting that the satellites within RMW liemuch closer to the model predictions (which are only forsatellites within Milky Way-like halos) than those outsideRMW, but with the limited data available it is impossible tosay if this is a significant effect. The stellar metallicities inthe model also agree well with the data, except perhaps atthe faintest magnitudes where the model is slightly too high.While photoionization makes little difference to the metal-licities of galaxies brighter than MV ≈ −15, at fainter mag-nitudes the standard model predicts slightly higher metal-licities than the Cole et al. (2000) model. Photoionization

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10 A. J. Benson, C. S. Frenk, C. G. Lacey, C. M. Baugh & S. Cole

Figure 7. The mass of hydrogen per unit V-band luminosityin satellite galaxies as a function of V-band absolute magnitude.Data for the Local Group satellites are taken from Mateo (1998),except for the LMC and SMC for which Hi masses are taken fromWesterlund (1997), M33 for which the Hi mass comes from Cor-belli & Salucci (2000) and for NGC3109 and Antlia for which Hi

masses are taken from Barnes & de Blok (2001). Hi detections are

shown as squares, and upper limits as triangles. Galaxies withinRMW of the Milky Way or M31 are represented by filled sym-bols and more distant satellites by open symbols. Solid, dashedand dotted lines show results from our standard, no photoioniza-tion and Cole et al. (2000) models respectively. The lines showthe median model relations and the errorbars indicate the 10%and 90% intervals of the distribution in the standard model. Thescatter in the other models is comparable to this.

causes galaxies of a given absolute magnitude to form inhigher circular velocity halos. These have deeper potentialwells which reduce the effectiveness with which supernovaefeedback expels gas, thus increasing the effective yield (seeCole et al. 2000) and resulting in a higher metallicity. In Pa-per I, we considered the metallicity of the ISM for galaxies ingeneral and found that photoionization actually reduced thegas metallicity in faint galaxies by preventing pre-processingof that gas in smaller halos. In the case of satellites, pho-toionization has a much smaller effect because, unlike cen-tral galaxies, the satellites are not accreting gas at present,and so their metallicity is much less sensitive to any pre-enrichment.

Finally, we consider the B-V colours of satellite galaxies.These are plotted in Fig 10 for the Local Group, as a functionof satellite V-band absolute magnitude and compared withthe model predictions. The model predictions are rather in-sensitive to whether or not photoionization is included. Withor without photoionization, satellites are typically very oldsystems with little recent star formation and so their colourscorrespond to those of an old ( >∼10 Gyr), low metallicitystellar population (i.e. roughly B-V=0.5–0.7 for the IMFconsidered here). Filled squares in the figure indicate Lo-cal Group satellites within RMW of the Milky Way or M31.The colours of these galaxies are in good agreement with

Figure 8. The Hα luminosity per unit mass of hydrogen for satel-lite galaxies, as function of absolute V-band magnitude. Symbolsshow values for Local Group satellites taken from Mateo (1998)and from Westerlund (1997) for the Magellanic Clouds and Ken-nicutt (1998) for M33. Satellites within RMW of the Milky Wayor M31 are shown as filled symbols, with more distant satellitesshown as open symbols. Hα detections are denoted by squares,

and upper limits by triangles. Solid, dashed and dotted lines in-dicate results from our standard, no photoionization and Coleet al. (2000) models respectively. The lines indicate the medianrelation and the errorbars the 10% and 90% intervals of the dis-tribution in the standard model. The scatter in the other modelsis comparable to this.

the model predictions, except perhaps for two intrinsicallyfaint galaxies for which the observed colours are very uncer-tain. Interestingly, Local Group galaxies which lie beyondRMW of either the Milky Way or M31 have systematicallybluer colours, suggesting that they have experienced recentstar formation. This is generically expected in our modelbecause these objects are still the central galaxies in theirhost halos and, unlike genuine satellites, they are still ableto accrete gas to fuel star formation even at the present day.This interpreation is consistent with the higher overall gascontent measured in the most distant galaxies as seen inFig. 7

4.2 Structure

Dynamical quantities describing the structure of satellitegalaxies and their dark matter halos are readily available inour model of galaxy formation. In Fig. 11, we show some ofthese quantities as a function of the absolute V-band mag-nitude of the satellite. We remind the reader that in ourcalculations, the dynamical and structural properties of thesatellite halos within the effective tidal radius are unaffectedby the mass loss beyond that radius. In the upper panels,we plot the circular velocity at the virial radius (left-handpanel) and at the NFW scale radius (right-hand panel). Thevirial radius and corresponding circular velocity here are thevalues the satellite halo last had when it was still a separate

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The Effects of Photoionization on Galaxy Formation — II: Satellite Galaxies in the Local Group 11

Figure 9. The metallicity of satellite galaxies as a function ofabsolute V-band magnitude. Metallicities for Local Group satel-lites are taken from Mateo (1998), except for the MagellanicClouds (Skillman, Kennicutt & Hodge 1989) and M33 (Kobul-nicky, Kennicutt & Pizagno 1999). The upper panel shows themetallicity of gas in the ISM (relative to Solar, with Z⊙ = 0.02),while the lower panel shows the metallicity of the stars. Filled

circles indicate satellites within RMW ≈ 270 kpc of the MilkyWay or M31, with more distant satellites shown as open circles.Solid lines give the median relation in our standard model, witherror bars indicating the 10% and 90% intervals. Dashed linescorrespond to our no photoionization model (which includes tidalstripping of satellites) and dotted lines to the Cole et al. (2000)model. The scatter in these models is comparable to that in thestandard model.

halo, before it merged with the Milky Way halo. In evaluat-ing the circular velocity, we take into account contributionsfrom both dark and baryonic matter in the galaxy, includingthe contraction of the dark halo caused by the condensationof the galaxy (which is assumed to proceed adiabatically).For fainter satellites, the photoionization model predicts sig-nificantly higher circular velocities than the other models.The reason for this is one we have encountered before: theinhibiting effect of photoionization leads to satellites of agiven luminosity forming in a more massive halo than wouldbe the case in the absence of photoionization. Note that suchan effect is not seen in Fig. 5 where we plotted the satellitegalaxy “Tully-Fisher” relation. There, the increase in circu-lar velocity is offset by the reduction in the sizes of galaxies,which causes the visible matter to sample the rotation curveat smaller radii where the circular velocity is less.

In the lower left-hand panel of Fig. 11, we plot the ef-fective tidal radii, as defined above and in Paper I, of the

satellites‖ . For comparison, we plot the virial radius of the

‖ In our calculations, galaxies are modelled with surface densityprofiles which initially extend to the virial radius of the halo inwhich they formed. We can therefore define a tidal radius forsatellies even if this turns out to be much larger than the visibleextent of the galaxy. In practice, when this occurs, the tidal radius

Figure 10. The B-V colour of satellite galaxies as a function ofabsolute V-band magnitude. Colours for Local Group satellitesare taken from Mateo (1998), except for the Magellanic Clouds(Skillman, Kennicutt & Hodge 1989) and M33 (Kobulnicky, Ken-nicutt & Pizagno 1999). Filled squares indicate satellites withinRMW ≈ 270 kpc of the Milky Way or M31, with more distantsatellites shown as open squares. The solid line gives the median

relation in our standard model, with error bars indicating the 10%and 90% intervals. The dashed line corresponds to our no pho-toionization model (which includes tidal stripping of satellites)and the dotted line to the Cole et al. (2000) model. The scatterin these models is comparable to that in the standard model.

halo in the Cole et al. (2000) model, which does not includetidal limitation (and assumes, for calculations of dynami-cal friction, that the satellite retains all of its halo after itmerges with a large halo). Tidal forces strip the halos of thefaintest galaxies to less than 10% of their initial virial radius,but this effect becomes much less dramatic for the brightergalaxies. (The virial radii of the halos harbouring satellitesof a given MV are actually slightly larger than the valuesgiven by the Cole et al. (2000) model since photoionizationcauses galaxies of fixed MV to form in larger halos). In thelower right-hand panel, we show the NFW scale radius of thehalos (lower set of lines). For comparison, we also show themedian virial radius (i.e. the virial radius of the satellite’shalo before it merged with the Milky Way’s halo; upper setof lines). Comparison of the two shows that satellites typi-cally formed in halos with concentration parameters in therange c =5–6.

5 DISCUSSION

Small satellite galaxies like those that surround the MilkyWay are the descendents of some of the oldest objects in theUniverse. Thus, to understand their properties, it is nec-essary to take into account processes that occured at very

plotted should be considered to be the tidal radius of the satellite’shalo rather than that of the satellite itself.

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12 A. J. Benson, C. S. Frenk, C. G. Lacey, C. M. Baugh & S. Cole

Figure 11. Predicted dynamical and structural properties of satellite galaxies and their dark matter halos, as a function of the absoluteV-band magnitude of the satellite. In all the panels, the lines show the median model relation: solid lines for the standard model, dashedlines for no-photoionization model (which, however, includes tidal stripping of the satellites) and dotted lines for the Cole et al. (2000)model. As before, the errorbars indicate the 10% and 90% intervals of the distribution in the standard model; the scatter in the othertwo models is comparable. Upper left: The circular velocity at the virial radius. Upper right: The circular velocity at the NFW scaleradius, rs. This takes into account contributions from both dark and baryonic matter in the galaxy, including the contraction of the darkhalo caused by the condensation of the galaxy (assumed to proceed adiabatically). Lower left: The effective tidal radii of the satellites.For the Cole et al. (2000) model, which does not include tidal limitation, we plot instead the virial radius of the halo for comparison.Lower right: The NFW scale radius of the satellite halos (lower set of lines) and the virial radius of the halos (upper set of lines).

early times, like the reionization of the Universe, which areoften neglected when studying the properties of larger galax-ies. Furthermore, since these satellites orbit in the halo ofthe parent galaxy, it is also important to take into accountdynamical processes such as tidal effects and dynamical fric-tion. We have used an extension of the Cole et al. (2000)semi-analytic model of galaxy formation which includes allthese processes to study the expected abundance and prop-erties of satellite galaxies in the Local Group. Our work im-

proves upon earlier studies by Kauffmann, White & Guider-doni (1993) and Bullock, Kravtsov & Weinberg (2000)because it self-consistently calculates the physics of reion-ization and galaxy formation and includes a treatment ofthe main dynamical effects experienced by satellites orbit-ing in a dark matter halo.

We generated samples of halos containing galaxies sim-ilar to the Milky Way. Photoionization, which occurs atz ≈ 8, inhibits the formation of small galaxies and so the

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The Effects of Photoionization on Galaxy Formation — II: Satellite Galaxies in the Local Group 13

satellites that survive to the present tend to be those thatformed while the universe was still neutral. This has theconsequence of greatly suppressing the number of satellitesof a given luminosity relative to the number that would beexpected if photoionization were neglected. Furthermore, ata given satellite luminosity, we find that it is those satelliteswith the lowest circular velocity that are preferentially de-pleted by the effects of photoionization. The measured dis-tribution function of satellite circular velocity depends onthis differential destruction but also on the internal struc-ture of the satellite which determines the shape of its ro-tation curve. We take the ‘measured’ circular velocity of asatellite to be the value at the half-light radius. We find thatfor the fainter satellites (MV ≃ −10), the measured circularvelocity is about half the value at the virial radius becausethe half-light radius is within the rising part of the rotationcurve, whereas for the brighter satellites (MV ≃ −15), themeasured circular velocity is very similar to the value at thevirial radius. These effects work together to produce both aluminosity function and a circular velocity function whichare in good agreement with the available data. This result isremarkable because we have not had to adjust a single pa-rameter value in the Cole et al. (2000) model nor fine-tuneour treatment of photoionization.

Our model also predicts the sizes and metallicities ofsatellites and these also turn out to be in good agreementwith the limited amount of data currently available. Themodel predicts that photoionization has negligible effectsin galaxies with circular velocity above about 60km s−1, inagreement with earlier analytical (Efstathiou 1992; Thoul& Weinberg 1996) and numerical treatments (Quinn, Katz& Efstathiou 1996; Navarro & Steinmetz 1997).

Our model can, in principle, be tested on the basis of thepredictions it makes for as yet unobserved properties of thesatellite population in the Local Group. In particular, ourmodel suggests that there should be a large population offaint satellites around the Milky Way awaiting discovery. Anear complete census would require deep imaging (at least to26 V-band magnitudes per square arcsecond). Alternatively,satellites may be recognized as an excess of stars against thebackground. We have presented detailed predictions for theexpected number of stars and their surface density as a func-tion of the luminosity of the satellite to which they belong.These calculations may be useful in designing observationalstrategies aimed at discovering new satellites. A further pos-sibility is to try and detect these satellites by searching fortheir Hi content (Putman et al. 2001). Our model predictsthat MV = −10 satellites should typically contain 105M⊙ ofHi, with a rapid decline in Hi mass at fainter magnitudes.Such observations are difficult, but perhaps not impossible.

We conclude that speculative mechanisms such as non-standard inflation (Kamionkowski & Liddle 2000) or newcomponents of the Universe such as warm, self-interactingor annihilating dark matter (Hogan 1999; Spergel & Stein-hardt 2000; Yoshida et al. 2001; Craig & Davis 2001)are not required to explain the observed abundance of LocalGroup satellites. Instead, the low abundance of satellites isa natural consequence of galaxy formation in a CDM uni-verse when the physical effects of photoionization and tidalinteractions, two processes which are known to occur, aretaken into account. Continuing improvement in the obser-vational data for Local Group galaxies, particularly better

measurements of their structure and dynamical state and anassessment of the completeness of existing samples, togetherwith searches for new satellites, will provide a strong test ofcurrent models of galaxy formation.

ACKNOWLEDGMENTS

CSF acknowledges a Leverhulme Research Fellowship. CGLacknowledges support at SISSA from COFIN funds fromMURST and funds from ASI. CMB acknowledges a RoyalSociety University Research Fellowship. SMC acknowledgesa PPARC Advanced Fellowship. We also thank Ben Moorefor drawing our attention to the Milky Way satellite prob-lem.

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