Dark matter substructure and dwarf galactic satellites
DDark matter substructure and dwarf galactic satellites
Andrey Kravtsov Dept. of Astronomy & Astrophysics, Kavli Institute for Cosmological Physics,The University of Chicago, Chicago, IL 60637, USAreview paper submitted to the special issue “Dwarf Galaxy Cosmology” of Advances in Astronomy
Abstract.
A decade ago cosmological simulations of increasingly higher resolu-tion were used to demonstrate that virialized regions of Cold Dark Matter (CDM)halos are filled with a multitude of dense, gravitationally-bound clumps. Thesedark matter subhalos are central regions of halos that survived strong gravitationaltidal forces and dynamical friction during the hierarchical sequence of mergingand accretion via which the CDM halos form. Comparisons with observationsrevealed that there is a glaring discrepancy between abundance of subhalos andluminous satellites of the Milky Way and Andromeda as a function of their circu-lar velocity or bound mass within a fixed aperture. This large discrepancy, whichbecame known as the “substructure” or the “missing satellites” problem, begs foran explanation. In this paper I review the progress made during the last severalyears both in quantifying the problem and in exploring possible scenarios in whichit could be accommodated and explained in the context of galaxy formation in theframework of the CDM paradigm of structure formation. In particular, I show thatthe observed luminosity function, radial distribution, and the remarkable similar-ity of the inner density profiles of luminous satellites can be understood withinhierarchical CDM framework using a simple model in which e ffi ciency of star for-mation monotonically decreases with decreasing virial mass satellites had beforetheir accretion without any actual sharp galaxy formation threshold. [email protected] a r X i v : . [ a s t r o - ph . C O ] J un Introduction
In the hierarchical scenario of galaxy forma-tion [1], theoretically rooted in the Cold DarkMatter (CDM) structure formation model [2],galaxies form via cooling and condensation ofgas in dark matter halos, which grow via an hi-erarchical sequence of mergers and accretion.The density perturbations in these models haveamplitude that increases with decreasing scaledown to ∼ Λ CDM model is illustrated in Figure 1, whichshows collapse of a ≈ M (cid:12) object. The fig-ure shows that during the early stages of evo-lution the matter that is incorporated into thefinal halo collapses into a large number of rel-atively small clumps with a filamentary, web-like spatial distribution. Further evolution, me-diated by the competition between gravity andexpansion of space, is a sequence of accretionand mergers that builds objects of progressivelylarger mass until the single system is formedduring the last several billion years of evolu-tion. The figure also shows that cores of someof the small clumps that merge with and areincorporated into larger objects survive untillater epochs and are present in the form of halosubstructure or subhalos : small dense clumpswithin virialized regions of larger halos.The CDM model of structure formation is re- markably successful in explaining a wide rangeof observations from temperature fluctuationsof the cosmic microwave background [8] togalaxy clustering and its evolution [9] bothqualitatively and, in many cases, quantitatively.Nevertheless, many key details of the model arestill being developed [10, 11] and its testing isby no means complete.One area of active investigation is testingpredictions of the CDM models at scales froma few kpc to tens of pc (i.e., the smallest scalesprobed by observations of galaxies). In par-ticular, there is still tension between predic-tions of the central mass distribution in galax-ies [12, 13] and sizes and angular momentaof galactic disks and observational results [14].Notably, this tension has not gone away duringthe past 10-15 years, even though both theo-retical models and observations have improveddramatically.Another example of tension between CDMpredictions and observations that has been ac-tively explored during the last decade is the factthat satellite systems around galaxies of di ff er-ent luminosity are qualitatively di ff erent, eventhough their dark matter halos are expected tobe approximately scaled down versions of eachother [15], with their total mass as the scal-ing parameter. Faint dwarf galaxies usuallyhave no luminous satellites at all, Milky Wayand Andromeda have a few dozen, but clustersof galaxies often have thousands of satellitesaround the brightest cluster galaxy.The number of gravitationally-bound satel-lite subhalos at a fixed mass relative to themass of their host CDM halo, on the otherhand, is expected to be approximately the same[16, 17, 18]. This is illustrated in Figure 2,which shows distribution of dark matter outto approximately two virial radii around thecenters of two CDM halos of masses di ff erentby two orders of magnitude. It is clear thatit is not easy to tell the mass of the halo bysimply examining the overall mass distributionor by counting the number of subhalos. This1 ig. 1— Formation of a Milky Way-sized dark matter halo in a cosmological simulation of flat Λ CDM cosmology( Ω m = − Ω Λ = . h = . σ = . z =
0) in the form of ”substructure”. The size of the region shown is about 3 comoving Mpc at z = ≈ z = is a visual manifestation of approximate self-similarity of CDM halos of di ff erent mass. Ifwe would compare similar images of distribu-tion of luminous matter around galaxies andclusters, the di ff erence would be striking.The manifestly di ff erent observed satellite populations around galaxies of di ff erent lu-minosities and expected approximately self-similar populations of satellite subhalos aroundhalos of di ff erent mass is known as the sub-structure problem. [19, 20, 16]. In the case ofthe best studied satellite systems of the Milky2 ig. 2— Comparison of two z = × M (cid:12) and 3 × M (cid:12) formed in flat Λ CDM cosmology( Ω m = − Ω Λ = . h = . σ = . Way and Andromeda galaxies, the discrepancybetween the predicted abundance of small-massdark matter clumps and the number of observedluminous satellites as a function of circular ve-locity (see §
2) has been also referred to as the “missing satellites problem.” The main goalof this paper is to review theoretical and ob-servational progress in quantifying and under-standing the problem over the last decade.
In order to connect theoretical predictions andobservations on a quantitative level, we needdescriptive statistics to characterize populationof theoretical dark matter subhalos and ob-served luminous satellites. Ideally, one wouldlike theoretical models to be able to predictproperties of stellar populations hosted by dark matter halos and subhalos and make compar-isons using statistics involving directly observ-able quantities, such as galaxy luminosities. Inpractice, however, this is di ffi cult as such pre-dictions require modeling of still rather uncer-tain processes shaping properties of galaxiesduring their formation. In addition, the simu-lations can reach the highest resolution in theregime when complicated and computationallycostly galaxy formation processes are not in-cluded and all of the matter in the universe ismodeled as a uniform collisionless and dissi-pationless component. Such simulations thusgive the most accurate knowledge of the darkmatter subhalo populations, but can only pre-dict dynamical subhalo properties such as thedepth of their potential well or the total mass
3f gravitationally bound material. Therefore,in comparisons between theoretical predictionsand observations so far, the most common strat-egy was to find a compromise quantity that canbe estimated both in dissipationless simulationsand in observations.
Start-ing with the first studies that made such com-parisons using results of numerical simulations[20, 16] the quantity of choice was the maxi-mum circular velocity, defined as V max = max (cid:32) Gm ( < r ) r (cid:33) / , (1)where m ( < r ) = π (cid:82) ρ ( r ) r dr is the spheri-cally averaged total mass profile about the cen-ter of the object. V max is a measure of the depthof the potential (the potential energy of a self-gravitating system is W ∝ V ) and can befairly easily computed in a cosmological sim-ulation once the center of a subhalo is deter-mined. The attractive feature of V max is thatit is well defined and does not require estimateof a physical boundary of subhalos, which isoften hard to determine. The price is that res-olution required to get the V max correctly for asubhalo is higher, compared for example to thetotal bound mass of subhalo, because V max isprobing the mass distribution in the inner re-gions of halos.The total gravitationally bound mass of asubhalo, m sub , is less sensitive to the resolu-tion, but requires careful separation betweenreal subhalo particles and unbound particles ff erent algo-rithms are used in the literature [21, 7, 22, 23, 24, 25, 26],all algorithms boil down to the automated search for den-sity peaks (most often in configuration space, but some-times in the phase space) field smoothed at a scale com-parable to or smaller than the size of the smallest sub-halos in the simulations. Once the peaks are identified,the gravitationally bound material around them is usuallyfound by iteratively removing the unbound particles. from the di ff use halo of the host dark matterhalo. This can be quite di ffi cult in the inner re-gions of the host system where density of thebackground di ff use halo is comparable to theinternal density of subhalo or when two subha-los overlap substantially.An alternative option is to define mass of asubhalo within a fixed physical radius. For suit-ably chosen radius value, the mass can be mea-sured unambiguously both in simulations and inobservations. We will discuss the measurementof the enclosed mass and comparisons betweensimulations and observations below in § § refsec:models (see Figs. 8 and 12).Figure 3 shows the cumulative circular ve-locity and mass functions (CVF and CMF) ofsubhalos within the virial radius of a simu-lated Milky Way sized halo, formation of whichwas illustrated in Figure 1. Both cumula-tive functions can be approximated by powerlaws over the ranges of circular velocity and inunits of V max and virial mass of the host: ν ≡ V max / V hostmax (cid:46) . µ ≡ m sub / M hostvir (cid:46) . − . ÷ − ≈ − . ÷ − . Λ CDM cosmology [28, 29,30] show that the power laws with the slopesin the range indicated above describe the CVFand CMF down to µ ≈ − and ν ≈ − .Note, however, that over a wider range of sub-halo masses the power law can be expectedto change slowly reflecting the changing slopeof the rms fluctuations as a function of scale,which controls the abundance of halos as afunction of mass [31, 32].The amplitude of the mass and velocity func- R ∆ = (3 M ∆ / π ∆ ¯ ρ ) / , where ¯ ρ is the meanmatter density in the universe and ∆ = z = Λ CDM cosmology [27]. ig. 3— Cumulative circular velocity and mass functionof subhalos within the virial radius R =
328 kpc of ahalo of virial mass M = × M (cid:12) at z =
0. Thedashed lines show power laws with the slopes indicatedin the legend of each panel. tions is sensitive to the normalization of thepower spectrum on small scales [33, 28] andis thus sensitive to the cosmological parametersthat control the normalization (such as tilt andnormalization σ ).For a given cosmology, the normalizationof the CVF and CMF scales approximatelylinearly with the host halo mass [17]: N ( >µ | M h ) ∝ M h . The halo-to-halo scatter in thenormalization of CVF and MCF for a fixed hosthalo virial mass is described by the Poisson dis-tribution [17]: σ N ( >µ ) = (cid:112) N ( > µ ). The frac-tional scatter is therefore quite small for small µ and ν (large N ).The mass and circular velocity functionswithin a given radius describe the overall abun-dance of subhalos of di ff erent mass, but nottheir radial distribution. The latter dependsrather sensitively on how the subhalo samples are selected [34]. This is because subhalos atdi ff erent distances from their host halo centeron average experience di ff erent tidal mass loss,which a ff ects di ff erent subhalo properties bydi ff erent amount. Subhalo mass is the most af-fected quantity as large fraction of halo masswhen it accretes is relatively loosely bound andis usually lost quickly. Although circular veloc-ity is determined by the inner mass distributionin the inner mass of subhalos, it is still a ff ectedby tidal stripping (albeit to a lesser degree andslower than the total mass [35]).The average mass loss experienced by sub-halos increases with decreasing distance to thecenter of the host halo [34]. Therefore, select-ing subhalos based on their current bound massor circular velocity biases the sample againstsubhalos at smaller radii and results in the ra-dial distribution much less concentrated thanthe overall mass distribution of the host halo[21, 36, 23, 37, 35, 34, 38]. Conversely, onecan expect that if the selection of subhalos ismade using a quantity not a ff ected by stripping,the bias should be smaller or even disappear al-together.The top panel of Figure 4 shows the radialdistribution of subhalos (the same population asin Figure 3) selected using their current boundmass or circular velocity and density profile ofdark matter within a MW-sized sized host halo.The figure shows that the subhalo distribution isless radially concentrated compared to the over-all density profile because selection using cur-rent subhalo properties a ff ected by tidal evolu-tion biases the sample against the inner regions.The bottom panel shows the radial distributionof subhalos in the same host halo but now se-lected using circular velocity and mass the sub-halos had before accretion (which are of coursenot a ff ected by the tides). In this case the ra-dial profile is very close to that of the dark mat-ter distribution. This dependence of the radialprofile on the property used for subhalo selec-tion should be kept in mind when the observedand predicted radial distributions are compared.5 ig. 4— Radial distribution of subhalos (solid lines)selected using their di ff erent properties ( V max and totalmass — solid red and blue lines, respectively) comparedto the matter density profile in their host halo (dashedlines). The upper panel shows the profiles for subhalosselected using circular velocity and bound mass a the cur-rent epoch, while the lower shows radial distribution ofsubhalos selected using the corresponding quantities be-fore subhalo was accreted una ff ected by the subsequenttidal mass loss. Note that minimum threshold values forsubhalo selection, µ min and ν min , are di ff erent in the lowerpanel because for a typical mass loss many subhalos withsmaller circular velocities and masses at the time of ac-cretion fall below the completeness limit of the simula-tion by z = The latter are selected based on their luminosity(i.e., the stellar mass), which may be a ff ected bytides much less than either the total bound massor circular velocity [34, 39].Finally, the spatial distribution of satellitesis not completely spherically symmetric, but istriaxial, which reflects their accretion along fil-aments and subsequent evolution in the triaxialpotential of their host halos [40, 41]. Although we currently know only a few dozen of nearby satellite galaxies aroundthe Milky Way and Andromeda, these galaxiesspan a tremendous range of the stellar densitiesand luminosities. The two brightest satellitesof the Milky Way, the Large and Small Magel-lanic Clouds (LMC and SMC), are easily visi-ble by the naked eye in the southern hemisphereand have therefore been known for many hun-dreds of years, while the faintest satellites havebeen discovered only very recently using so-phisticated search algorithms and the vast datasets of stellar photometry in the Sloan Digi-tal Sky Survey and contain only a few hundredstars [42, 43]. Up until late 1990s only a dozendwarf galaxies were known to exist within 300kpc of the Milky Way, with a similar num-ber around the Andromeda [44]. These galax-ies have luminosities L (cid:38) L (cid:12) and mor-phologies of the three types: 1) dwarf irreg-ular galaxies (dIrrs, e.g., LMC and SMC) —low surface brightness galaxies of irregular ap-pearance which have substantial amount of gasand thus exhibit continuing star formation, 2)dwarf spheroidal galaxies (dSphs, e.g., Dracoor Fornax) — low surface brightness galax-ies with spheroidal distribution of stars and no(or very little) ongoing star formation and 3)dwarf elliptical galaxies (dEs, e.g. M32) —high-surface brightness, low-luminosity ellip-ticals with no gas and no current star forma-tion. dSph and dE galaxies tend to be locatedwithin 200 kpc of their host galaxies, while dIrrgalaxies are distributed more uniformly. Thistendency is called the “morphological segrega-tion” ([44]) and appears to exist in other nearbygroups of galaxies [45]. The properties of these“classical” dwarf galaxies are reviewed exten-sively by [44] (see also recent study of scalingrelations of dwarf galaxies by [46].Despite the wide range of observed proper-ties, all of the nearby dwarfs share some com-mon features in their star formation histories(SFHs). The SFHs of all classical dwarfs ischaracterized by a rather chaotically varyingstar formation rates. Most bright dwarfs form6 − − − − − − − − M V R c o m p [ k p c ] SDSS SatellitesClassical SatsLG Galaxies L [L ⊙ ] Fig. 5— The distance to which the current samples ofdwarf satellite galaxies around the Milky Way are com-plete as a function of galaxy luminosity (absolute V-bandmagnitude on the bottom scale and physical luminosityin units of solar luminosity at the top scale). Note thatthe completeness distance of the faintest recently discov-ered satellites is (cid:46)
50 kpc. Adopted from [51]. stars throughout their evolution, although themajority of stars may be formed in several mainepisodes spread over ten or more billion years[47, 48, 49, 50], and have at least some frac-tion of stars that formed in the first two billionyears of the evolution of the universe. In termsof their SFHs, the main di ff erence between thedwarf irregular and dwarf spheroidal galaxies isin the presence or absence of star formation inthe last two billion years [49].The radial distribution of the classical satel-lites around the Milky Way is rather compact.For the dwarf galaxies within 250 kpc of theMilky Way, the median distance to the cen-ter of the Galaxy is ≈
70 kpc [35, 52], whilethe predicted median distance for subhalos is ≈ −
140 kpc [35] (see Figure 13 below).The distribution of the satellites around the − − − − − − − − M V N ( < M V ) Combined CorrectionFixed Coverage CorrectionUncorrected/Observed L [L ⊙ ] Fig. 6— The luminosity function of dwarf galaxiesaround the Milky Way. The function includes all ob-served galaxies within 417 kpc of the Milky Way. Thered circles connected by the dotted line show the lu-minosity function as observed, without any correctionsfor incompleteness. The green circles connected by thedashed line show the observed luminosity function cor-rected for limited coverage of surveys on the sky. Theblue circles connected by the solid line show luminos-ity function corrected for the radial bias using radial dis-tribution of subhalos from the Via Lactea I simulation.Adopted from [51].
Andromeda galaxies is consistent with that ofthe Milky Way satellites, but is less accuratelydetermined due to larger errors in distances.The spatial distribution of satellites about theMilky Way and Andromeda is also manifestlynon-isotropic with the majority of the satellitesfound in a flattened structure nearly perpendic-ular to the disk [53, 54, 55, 56, 57].In 1994, a new faint galaxy was discov-ered in the direction toward the center of theGalaxy (in the Sagittarius constellation, [58]).The galaxy is similar to other nearby dwarfspheroidal galaxies in its properties but is re-markably close to the solar system (the distance7f only ≈
28 kpc) and is in the process of be-ing torn apart by the tidal interactions with theMilky Way. This interaction has produced aspectacular tidal tail, which has wrapped sev-eral times along the orbit of the Sagittariusdwarf [59].The discovery of this new satellite has alertedresearchers in the field to the possibility thatother satellites may be lurking undiscovered inour cosmic neighborhood. The advent of widefield photometric surveys, such as the SloanDigital Sky Survey and the targeted surveysof regions around the Andromeda galaxy, andnew search techniques has resulted in dozens ofnew satellite galaxies discovered during the lastdecade [60, 61, 42, 62, 63, 64, 65, 66, 43] withmany more discoveries expected in the near fu-ture [51, 67]. The majority of the newly dis-covered galaxies are fainter than the “classicaldwarfs” known prior to 1998. Due to their ex-tremely low luminosities (as low as ∼ L (cid:12) in the case of the Segue 1 [43]) they have col-lectively been referred to as the “ultra-faint”dwarfs. Such low luminosities (and impliedstellar masses) indicate an extreme mode ofgalaxy formation, in which the total popula-tion of stars produced during galaxy evolutionis smaller than a star cluster formed in a singlestar formation event in more luminous galaxies.More practically, the extreme faintness of themajority of dwarf satellites implies that we havea more or less complete census of them onlywithin the volume of ∼ −
50 kpc of theMilky Way [68, 51]. Figure 5 shows the dis-tance to which the dwarfs of a given luminos-ity are complete in the SDSS survey, in whichthe faintest new dwarfs have been discovered.The figure shows that we have a good censusof the volume of the Local Group only for therelatively bright luminosities of the “classical”satellites. At the fainter luminosities of the ul-tra faint dwarfs, on the other hand, we can ex-pect to find many more systems at larger radiiin the future deep wide area surveys. The ex-act number we can expect to be discovered de- pends on their uncertain radial distribution, butgiven the numbers of already discovered dwarfsand our current knowledge of the radial distri-bution of brighter satellites (and expected radialdistribution of subhalos) we can reasonably ex-pect that at least a hundred faint satellites existwithin 400 kpc of the Milky Way. This is illus-trated in Figure 6, which shows the luminosityfunction of the Milky Way satellites correctedfor the volume not yet surveyed under di ff er-ent assumptions about radial distribution of thesatellites [51].The basis for considering these extremelyfaint stellar systems as bona fide galaxies isthe fact that, unlike star clusters, they are darkmatter dominated: i.e., the total mass withintheir stellar extent is much larger than the stellarmass expected for old stellar populations [43].The total dynamical masses of these galaxiesare derived using kinematics of stars. Highresolution spectroscopy of the red giant stars inthe vicinity of each galaxy provides the radialvelocities of these stars. The radial velocitiescan then be modeled using using the Jeans equi-librium equations to derive the total mass pro-file [69, 70, 71, 72, 73, 74]. This modeling re-quires certain assumptions about the unknownshape of the stellar distribution and velocity dis-tribution of stars, as well as assumptions aboutthe shape and radial profile of the dark matterdistribution. The resulting mass profile there-fore has some uncertainty associated with theseassumptions [69, 72, 74].Additionally, the ultra-faint dwarfs followscaling relations of the brighter classical satel-lites such as the luminosity-metallicity relation[75] and therefore seem to be the low luminos-ity brethren within the family of dSph galaxies. Defining the substructure problem.
As I noted above, comparison of theory and ob-servations in terms of the directly observablequantities such as luminosities is possible onlyusing a galaxy formation model. These mod-els, although actively explored ([76, 77, 78, 35,79, 80, 81, 82], see also § ff ect on the distribution of the dynam-ically dominant dark matter. Fruitful compari-son between simulation predictions and obser-vations is therefore possible if a quantity relatedto the total mass profile can be measured in thelatter.The first attempts at such comparisons [7, 16]assumed isotropy of the stellar orbits and con-verted the line-of-sight velocity dispersion ofstars in dSph satellites, σ r , to estimate theirmaximum circular velocities as V max = √ σ r . The admittedly over-simplistic conversion wasadopted simply due to a lack of well measuredvelocity profiles and corresponding constraintson the mass distribution at the time. Figure 7shows such a comparison for the classical satel-lites of the Milky Way and subhalo popula- V max values are muchmore uncertain and because their total number withinthe virial radius requires uncertain corrections from thecurrently observed number that probes only the nearestfew dozen kpc. The velocity dispersions of the ultra-faint dwarfs are very similar to each other ( ∼ / s) andthey therefore formally have similar V max values accord-ing to this simple conversion method (hence, they wouldall be “bunched up” at about the same V max ∼ / svalue). The maximum circular velocity of the halos ofthese galaxies is expected to be reached at radii wellbeyond the stellar extent and its estimate from the ob-served velocity dispersions requires substantial extrapo-lation and assumptions about the density profile outsidethe radii probed by stars. The errors of the derived valuesof V max can therefore be quite substantial [69, 83]. I will Fig. 7— Comparison of the cumulative circular velocityfunctions, N ( > V max ), of subhalos and dwarf satellites ofthe Milky Way within the radius of 286 kpc (this radiusis chosen to match the maximum distance to observedsatellites in the sample and is smaller than the virial ra-dius of the simulated halo, R =
326 kpc). The subhaloVFs are plotted for the host halos with max. circular ve-locities of 160 km / s and 208 km / s that should bracket the V max of the actual Milky Way halo. The VF for the ob-served satellites was constructed using circular velocitiesestimated from the line-of-sight velocity dispersions as V max = √ σ r (see discussion in the text for the uncer-tainties of this conversion). tions in Milky Way-sized halos formed in theconcordance Λ CDM cosmology.The observed velocity function is comparedto the predicted VF of dark matter subhaloswithin a 286 kpc radius of Milky Way-sizedhost halos. In the literature, “Milky Way-sized”is often used to imply a total virial mass of M vir ≈ M (cid:12) and maximum circular veloc-ity of V max ≈
200 km / s. However, there issome uncertainty in these numbers. Thereforethe figure shows the VFs for the host halos with V max =
208 km / s and 160 km / s. The former ismeasured directly in a simulation of the haloof that circular velocity, while the latter VF compare the predicted luminosity function of the lumi-nous satellites using a simple galaxy formation model in § N ( > V max ) ∝ M vir , / M vir , = ( V max , / V max , ) . , using scaling measured statis-tically in the simulations [17].The simple conversion of σ r to V max hasjustly been criticized as too simplistic [84]. In-deed, the conversion factor η ≡ V max /σ r re-quires a good knowledge of mass profile fromsmall radii to the radius r max . The mass pro-file derived from the Jeans equation has errorsassociated with uncertainties in the anisotropyof stellar orbits, as well as with uncertainties ofspatial distribution of stellar system and / or itsdynamical state [85, 70]. Most importantly, themass profile is only directly constrained withinthe radius where stellar velocities are measured, r ∗ . If this radius is smaller than r max , conver-sion factor η depends on the form of the densityprofile assumed for extrapolation. The uncer-tainties of the derived mass profile within thestellar extent will of course also be magnifiedincreasingly with increasing r max / r ∗ ratio [85].Thus, for example, Stoehr et al. [84] haveargued that the conversion factor can be quitelarge ( η (cid:38) −
4) if the density profile is shal-low in the inner regions probed by the stars.Such large conversion factor would shift thelow V max points in Figure 7 to the right closerto the subhalo VF [84, 86, 72] and would implythat there is a sharp drop in the luminosity func-tion of satellites below a certain threshold circu-lar velocity ( V max ≈
30 km / s). High-resolutionsimulations of individual satellites, on the otherhand, have demonstrated that the CDM satel-lites retain their cuspy inner density profiles,even as they undergo significant tidal stripping[87]. This implies that η is likely not as largeas advocated by Stoehr et al. The main uncer-tainty in its actual value for a specific satellite isthen due to the uncertainty in the density profileand ratio r max / r ∗ .Recently, Pe˜narrubia et al. [72] combinedthe measurements of stellar surface density and σ r ( R ) profiles to estimate V max values for indi-vidual observed MW satellites under assump-tion that their stellar systems are embedded into NFW dark matter potentials. Such procedureby itself does not produce a reliable estimateof V max , because NFW potentials with a widerange of V max can fit the observed stellar pro-files. To break the degeneracy Pe˜narrubia etal. have used the relation between V max and r max expected in the concordance Λ CDM cos-mology. They showed that this results in es-timates of V max which imply η ≈ − ff ects of tidal stripping on the evolu-tion of the r max − V max relation. Typically, sub-halos located in the inner regions of the haloare expected to have lost ∼ −
90% of theirinitial mass by z = V max changesonly by ≈ −
30% [35, 72] but r max shouldchange by a factor of ≈ − ff ect their inferred conversion factor η (see their Fig. 9). This conclusion, however,was drawn based on the systems in which bothstellar system and DM halo were significantlystripped. In such system r max is close to the stel-lar radius and σ and V max evolve in sync. Forsystems with more realistic mass loss and withstars deeply embedded within r max , however,stellar system (and σ ) may not be a ff ected,while V max can evolve significantly. For suchsystems the method of [72] will lead to a sig-nificant overestimate of η . Indeed, the systemsin Fig. 9 of [39] for which the method overesti-mates η (by a factor of ≈ .
4) the most are thesystems with moderate total mass loss and leasta ff ected stellar systems. Note that even thesesystems have likely experienced more tidal lossthan most of the real dSph satellites.Another factor in estimates of V max isanisotropy of stellar velocities in dSph (e.g.,[87]). For example, recent analysis of observedvelocity dispersion profiles of “classical” dSphby [73], in which the anisotropy of stellar or-bits was treated as a free parameters, results in10 ig. 8— The mass function of dwarf satellites of theMilky Way, where masses of subhalos and observedsatellites are measured within a fixed physical radius of0 . estimates of V max of their host subhalos in therange ∼ −
25 km / s, smaller than would besuggested if correction factor was η ≈ − ff erence between the observed andpredicted VFs – the large di ff erence in theirslope – unless η strongly depends on V max (thereis no observational evidence for this so far).Another promising approach is to abandonattempts to derive V max altogether and to mea-sure instead the observed mass within the ra-dius where the uncertainty of the measuredmass profile is minimal. Such radius is close tothe stellar extent of observed galaxies [85, 83,72, 74]. Figure 8, adopted from [85], showscomparison of the mass functions of subha-los in the Via Lactea I simulation [89] andobserved satellites of the Milky Way, wheremasses are measured within a fixed physicalradius of 600 pc (see also discussion in § ff erent,the conclusion similar to that derived from the comparison of the circular velocity functions.Thus, the discrepancy that is clearly seenin the comparison of circular velocity func-tions, measured with more uncertainty in ob-servations, persists if the comparison is doneusing a much better measured quantity. Un-fortunately, the stellar distribution in most ofthe newly discovered ultra-faint dwarf galax-ies does not extend out to 600 pc radius and m ( r < . ≡ m . therefore cannot be mea-sured as reliably for these faint systems as forthe classical dwarfs. Similar comparisons haveto be carried out using masses within smallerradii. This puts stringent requirements on theresolution of the simulations, as they need toreliably predict mass distribution of subhaloswithin a few hundred parsec radius. Such high-resolution simulations are now available, how-ever [28, 29, 30].We can draw two main conclusions from thecomparisons of the circular velocity functionsand the more reliable m . mass functions ofsubhalos and observed satellites presented inthe previous sections, even taking into accountexisting uncertainties in deriving circular veloc-ities and the total dynamical masses for the ob-served satellites. First, the predicted abundanceof the most luminous satellites is in reasonableagreement with the data, even though the statis-tics are small. Most MW-sized halos simulatedin the concordance Λ CDM cosmology have 1-2 LMC sized ( V max ≈ −
70 km / s) subhaloswithin their virial radius.This is not a trivial fact because the abun-dance of the most massive satellites is deter-mined by a subtle interplay between the accre-tion rate of systems of corresponding circularvelocity and their disruption by the combinede ff ects of the dynamical friction and tidal strip-ping [90]. Dynamical friction causes satellitesto sink to the center at a rate which dependson the mass and orbital parameters of satelliteorbit. Orbital parameters, in turn, depend onthe cosmological environment of the accretinghost halo and are mediated by the tidal strip-11ing which reduces satellite mass as it sinks,thereby rendering dynamical friction less e ffi -cient [91, 92, 93, 94]. The fact that the concor-dance Λ CDM model makes an ab initio predic-tion that the number of massive satellites thatcan host luminous dwarfs is comparable to ob-servations can therefore be viewed as a successof the model.Second, the slopes of both the circular ve-locity function and the m . mass function aredi ff erent in simulations and observations. Thisimplies that we cannot simply match all ofthe luminous satellites to the subhalos with thelargest V max and m . , as was sometimes advo-cated [84, 86]. The m . mass function compar-ison, in particular, indicates that there should besome subhalos with the m . ∼ M (cid:12) that donot host the luminous galaxies, and some thatdo. As I discuss in § ff erence in terms of galaxy formation scenar-ios.In summary, the substructure problem canbe stated as the discrepancy in the slopes ofthe circular velocity and m . mass functions in-ferred for observed satellites of the Milky Wayand the slopes of these functions predicted fordark matter subhalos in the MW-sized host ha-los formed in the concordance Λ CDM cosmol-ogy.
I believe that stated this way the problem iswell-defined. Defining the problem in terms ofthe di ff erence in the actual number of satellitesand subhalos is confusing at best, as both num-bers are fairly strong functions of subhalo massor stellar luminosity. Thus, for example, eventhough the discovery of the ultra-faint dwarfsimplies the possible existence of hundreds ofthem in the halo of the Milky Way [49,75] (thisfact has been used to argue that the substructureproblem has been “alleviated”), the most recentsimulations show that more than 100,000 sub-halos of mass m sub > M (cid:12) should exist in the Milky Way [28, 29]. The substructure prob-lem stated in the actual numbers of satellites istherefore alive and well and has not been alle-viated in the least.I would like to close this section by a briefdiscussion of the comparison of spatial distri-bution of observed satellites and subhalos. AsI noted above, the radial distribution of the ob-served satellites of the Milky Way is more com-pact than the radial distribution of subhalos se-lected using their present day mass or circularvelocity [35, 52, 97]. In addition, the observedsatellites are distributed in a quite flattenedstructure with its plane almost perpendicular tothe disk of the Milky Way [53, 54, 55, 56, 57].Although the spatial distribution of all subha-los is expected to be anisotropic, reflecting theanisotropy of their accretion directions alongfilaments [40, 41] and, possibly, the fact thatsome of the satellites could have been accretedas part of the same group of galaxies [98], it isnot as strong as the anisotropy of the observedMilky Way satellites.Thus, both the radial distribution andanisotropy of the observed satellites do notmatch the overall distribution of subhalos inCDM halos. This is likely another side of thesame substructure problem coin and the overallspatial distribution of observed satellites needsto be explained together with the di ff erences inthe circular velocity function. I will review afew possible explanations for the substructureproblem and di ff erences in the spatial distribu-tion in the next section. One pos-sible way to account for the di ff erences of the a priori that subhalos of mass10 − M (cid:12) are too small to host luminous stellar sys-tems of stellar mass M ∗ ∼ − M (cid:12) [95] — thestellar masses corresponding to the luminosities of thefaintest recently discovered dwarfs. After all, the halosof this mass are expected to be hosting formation of thevery first stars [96]. m . mass functions is to assume that Λ CDMmodel is incorrect on the small scales probedby the dwarf galactic satellites. Indeed, theabundance of satellites is sensitive to the ampli-tude of the power spectrum on the scales corre-sponding to the total mass of their host halos.For a halo of mass M the comoving scale offluctuations that seed their formation is d = R = (cid:32) M π Ω m0 ρ crit0 (cid:33) / (2) = . (cid:32) M M (cid:12) . Ω m (cid:33) / (cid:18) H (cid:19) − / , where Ω m is the present-day total matter den-sity in units of the present-day critical density, ρ crit0 ≡ H / π G and H is the current Hubbleconstant in units of km / s / Mpc.If the amplitude of density fluctuations onsuch scales is considerably suppressed com-pared to the concordance Λ CDM model usedin most simulations, the abundance of subha-los can then also be suppressed. Such suppres-sion can be achieved either by suitably vary-ing parameters controlling the amplitude of thesmall-scale power spectrum within the Λ CDMmodel itself [33], such as the overall normal-ization of the power spectrum or its large-scaletilt, or by switching to models in which theamplitude at small scales is suppressed, suchas the warm dark matter (WDM) structure for-mation scenarios [99, 100, 101, 33, 102]. Inthese models the abundance of satellites is sup-pressed both because fewer halos of dwarf massform in the first place (due to smaller initial am-plitude of fluctuations) and because halos thatdo form have a less concentrated internal massdistribution, which makes them more suscep-tible to tidal disruption after they accrete ontotheir host halo. Models in which dark matterwas assumed to be self-interacting, a propertythat can lead to DM evaporation, have also beenproposed and discussed [103, 104], but thesemodels both run into contradiction with other observational properties of galaxies and clus-ters [105, 106, 107, 108] and are now stronglydisfavored by observational evidence indicatingthat dark matter self-interaction is weak [109].The problem, however, is more subtle thansimply suppressing the number of satellites. Asdiscussed above, di ff erences exist between ob-served and predicted slopes of the circular ve-locity and mass. The slope is controlled bythe slope of the primordial fluctuation spectrumaround the scale corresponding to the massesof satellite halos and structural properties of theforming halos. It has not yet been demonstratedconvincingly whether both the circular veloc-ity and the m . mass functions can be repro-duced in any of the models alternative to CDM.In fact, recent measurements of mass distribu-tion in the central regions of observed satel-lites put stringent constraints on the phase spacedensity and “warmness” of dark matter [69].At the same time, measurements of the small-scale density power spectrum of the Lyman α forest indicate that fluctuations with the ampli-tude expected in the Λ CDM model at the scalesthat control the abundance of dwarf mass halosare indeed present in the primordial spectrum[110, 111].While the inner density distribution in ob-served satellites may still be a ff ected by darkmatter warmness in the allowed range of pa-rameter space [71] (see, however, [73]), themodels with such parameters would not sup-press the overall abundance of satellites con-siderably. In fact, observations of flux ratios inthe multiple image radio lenses appear to re-quire the amount of substructure which is evenlarger than what is typically found in the CDMhalos [112, 113, 114], disfavoring models withstrongly suppressed abundance of small masssubhalos.There is thus no compelling reason yet tothink that the observed properties of galacticsatellite populations are more naturally repro-duced in these models. In the subsequent dis-cussion, I will therefore use Occam’s razor and13ocus on the possible explanations of the di ff er-ences between observed satellites and subhalosin simulations within the Λ CDM model. Theprime suspect in producing the discrepancy isthe still quite uncertain physics of galaxy for-mation. After all a similar problem exists forobjects of larger masses and luminosities if wecompare the slope of the luminosity functionand the halo mass function [115, 116] or thepredicted and observed abundance of galaxiesin the nearby low density “field” regions [117].
Several plau-sible physical processes can suppress gas ac-cretion and star formation in dwarf dark mat-ter halos. The cosmological UV background,which reionized the Universe at z (cid:38)
6, heatsthe intergalactic gas and establishes a character-istic time-dependent minimum mass for halosthat can accrete gas [118, 119, 120, 121, 122,123, 124, 125]. The gas in the low-mass ha-los may be photoevaporated after reionization[126, 127, 128] or blown away by the first gen-eration of supernovae [129, 130, 131, 132] (see,however, [133]). At the same time, the ion-izing radiation may quickly dissociate molecu-lar hydrogen, the only e ffi cient coolant for low-metallicity gas in such halos, and prevent starformation even before the gas is completely re-moved [134]. Even if the molecular hydrogenis not dissociated, cooling rate in halos withvirial temperature T vir (cid:46) K is consider-ably lower than in more massive halos [135]and we can therefore expect the formation ofdense gaseous disks and star formation sup-pressed in such halos. Another potential galaxyformation suppression mechanism is the gasstripping e ff ect of shocks from galactic out-flows and cosmic accretion [136, 137]. Finally, T vir is related to the virialmass by kT vir = µ m p GM vir / R vir , where isothermal tem-perature profile is assumed for simplicity. The virialmass and radius are related by definition as M vir = π/ × ∆ vir ¯ ρ R . Assuming ∆ vir =
178 appropriatefor z (cid:38) Ω , this gives M vir ≈ . × h − M (cid:12) ( Ω / . − / ((1 + z ) / − / ( T vir / K) / . even if the gas is accreted and cools in small-mass halos, it is not guaranteed that it will formstars if gas does not reach metallicities and sur-face densities su ffi cient for e ffi cient formationof molecular gas and subsequent star formation[35, 138, 139, 140, 141].The combined e ff ect of these processes islikely to leave most of dark matter halos withmasses (cid:46) few × M (cid:12) dark, and could haveimprinted a distinct signature on the propertiesof the dwarf galaxies that did manage to formstars before reionization. In fact, if all thesesuppressing e ff ects are as e ffi cient as is usu-ally thought, it is quite remarkable that galax-ies such as the recently discovered ultra-faintdwarfs exist at all. One possibility extensivelydiscussed in the literature is that they man-aged to accrete a certain amount of gas andform stars before the universe was reionized[76, 77, 78, 142, 143, 144, 79, 82]. Directcosmological simulations do show that dwarfgalaxies forming at z > ff ectsof the UV background.Cosmological simulations also clearly showthat the subhalos found within virial radii oflarger halos at z = Given the galaxy formation suppression mech-anisms and evolutionary scenarios listed above,the models aiming to explain the substructureproblem can be split into the following broadclasses: 1) the “threshold galaxy formationmodels” in which luminous satellites are em-bedded in the most massive subhalos of CDMhalos and their relatively small number indi-cates the suppression of galaxy formation insubhalos of circular velocity smaller than somethreshold value [84, 148, 72] and 2) “selectivegalaxy formation models” in which only a frac-tion of small subhalos of a given current V max and mass host luminous satellites while the restremain dark.In the second class of models the processesdetermining whether a subhalo hosts a lumi-nous galaxy can be the reionization epoch [76,145, 143, 149, 144, 82]: subhalos that assem-ble before the intergalactic medium was heatedby ionized radiation become luminous. The ob-served faint dwarfs can then be the “fossils” ofthe pre-reionization epoch [145, 146]. Subhalomay also form a stellar system if its mass as-sembly history was favorable for galaxy forma-tion [35]: namely, luminous subhalos are thosethat have had su ffi ciently large mass during aperiod of their evolution to allow them to over-come the star formation suppression processes.Several models using a combination of theprocesses and scenarios outlined above havebeen shown to reproduce the gross propertiesof observed population of satellites reasonablywell [35, 80, 82, 79]. How can we test di ff er-ent classes of models and di ff erentiate betweenspecific ones? First, I think the fact the m . mass func-tion for observed satellites has a di ff erent slopecompared to simulation predictions (Fig. 8) fa-vors the second class of the selective galaxy for-mation models, at least for the brighter “clas-sical” satellites. Indeed, given what we knowabout the average mass loss of subhalos, itis more natural to associate the observed sys-tems with the halos of the largest mass priorto accretion [35] rather than with the subha-los with the largest current masses. Second,the predicted number of the weakly evolvingpre-reionization objects [142, 144, 79], the ex-tended star formation histories of most of theobserved dwarf satellites [150, 48, 49, 50], andthe significant spread in metallicities and cer-tain isotope ratios [151] indicate that majorityof “classical” dwarfs have not formed most oftheir stars before reionization but have formedtheir stars over rather extended period of time( ∼
10 Gyr). It is still possible, however, that asizable fraction of the ultra-faint dwarfs are the“pre-reionization fossils” [142, 152, 146, 153],if star formation e ffi ciency in these objects isgreatly suppressed [79] compared to that ofbrighter dwarfs.Interesting additional clues and constraintson the galaxy formation models available forthe dwarf satellites of the Milky Way arethe measurements of the total dynamical masswithin their stellar extent. Observations showthat the total masses within a fixed apertureof the observed satellites are remarkably sim-ilar despite a several order of magnitude spanin dwarf satellite luminosities [154, 44, 155,72, 83]. For example, the range of masseswithin 0 . M ( r half ), and ff erencebetween the dIrr and dSph galaxies appears to be pres-ence or lack of star formation in the last 2 billion yearsbefore z = ig. 9— The mass within central 300 pc vs the totalvirial mass of an NFW halo predicted using the con-centration mass relation c ( M vir ) at di ff erent redshifts inthe concordance WMAP 5-year best fit cosmology. Thehorizontal dotted lines indicate the range of m . massesmeasured for the Milky Way dwarf satellites. the corresponding radius r half [73] for brightdSph galaxies implies a very similar inner darkmatter density profile of their host halos. Re-cently, L. Strigari and collaborators [83] haveshown that the mass estimated within the cen-tral 300 pc, m ( < . ≡ m . , for all of thedwarfs with kinematic data varies by at most afactor of four, while the luminosity of the galax-ies varies by more than four orders of magni-tude.These observational measurements put con-straints on the range of masses of CDM sub-halos that can host observed satellites. To esti-mate this range, we should first note that for theCDM halos described by the NFW profile [15]with concentration c ≡ R vir / r s (where r s is thescale radius – the radius at which the densityprofile has logarithmic slope of −
2) the depen-dence of the mass within a fixed small radius x ≡ r / R vir on the total virial mass is m ( < x ) = M vir f ( cx ) f ( c ) , where (3) f ( x ) ≡ ln(1 + x ) − x + x (4) and is quite weak for r =
300 pc: m . ∝ M . − . , as shown in Figure 9. Indeed, evenfor a halo with the Milky Way mass at z = m . ≈ × M (cid:12) , a value not toodi ff erent from those measured for the nearbydwarf spheroidals. Physically, the weak depen-dence of the central mass on the total mass ofthe halo reflects the fact that central regions ofhalos form very early by mergers of small-masshalos. Given that the rms amplitude of densityperturbations on small scales is a weak functionof scale, the central regions of halos of di ff erentmass form at a similar range of redshifts andthus have similar central densities reflecting thedensity of the universe when the inner regionwas assembled. At earlier epochs the depen-dence is stronger because 300 pc represents alarger fraction of the virial radius of halos.Note that the relation plotted in Figure 9 isfor isolated halos una ff ected by tidal stripping.Taking into account e ff ects of tidal stripping re-sults in even flatter relation [81]: m . ∝ M . ,which also has a lower normalization (smaller m . for a given M vir . This is likely due toa combination of two e ff ects: 1) the halos oflarger mass have lower concentrations and thuscan be stripped more e ffi ciently and 2) the ha-los of larger mass can sink to smaller radii af-ter they accrete and experience relatively moretidal stripping. Overall, the e ff ect of strippingon m . appears to be substantial and cannot beneglected.Finally, figure 9 shows that the virial massrange corresponding to a given range of m . isquite di ff erent for halos that form at z > m . can therefore be only interpreted inthe context of a model for subhalo evolutionaryhistories.Several recent studies have used such mod-els to show that the nearly constant centralmass of the satellite halos is their natural out-come [80, 81, 79]. This outcome can be un-16erstood as a combination of the weakness ofthe m . − M vir correlation and the fact that inthe galaxy formation models galaxy luminosity L must be a nonlinear function of M vir in orderto produce a faint-end slope of the galaxy lumi-nosity function much steeper than the slope ofthe small-mass tail of the halo mass function.For example, if the faint-end slope of the lu-minosity function is ξ (i.e., dn ( L ) / dL ∝ L ξ )and the slope of the halo mass function at smallmass end is ζ ( dn ( M ) / dM ∝ M ζ ) and we as-sume for simplicity a one-to-one monotonicmatching between galaxies and halos n ( > L ) = n ( > M ) (see [157, 158, 159] for the detailedjustification for such assumption), the impliedslope of the L − M vir relation is β = (1 + ζ ) / (1 + ξ ),which for the fiducial values of ζ ≈ − ξ ≈ − . β ≈
5. In semi-analytic mod-els, such a steep nonlinear L − M vir relation isusually assumed to be set by either suppressionof gas accretion due to UV heating or by gasblowout due to SN feedback, e.g., [78]). If I in-stead assume the faint-end slope of ξ ≈ . − . β ≈ L ∝ M β vir we have L ∝ m γ . , where γ ≈ β/ .
25, using the m . − M vir re-lation above taking into account e ff ects of tidalstripping. To account for a smaller than a fac-tor of four spread in central masses for approxi-mately four orders of magnitude spread in lumi-nosity one needs γ ≈ − β ≈ −
4, the val-ues not too di ff erent from the estimate above.Thus, the weak correlation of the m . and lu-minosity will be the natural outcome of anyCDM-based galaxy formation model which re-produces the slope of the faint end of the galaxyluminosity function. Within the framework I just described, theslope of the m . − L correlation depends on theslope of the L − M vir , acc correlation β . The mod-els published so far [80, 81, 79], as well as the Fig. 10— The mass within the central 300 pc vs luminos-ity for the dwarf satellites of the Milky Way (stars witherror bars, see [83]). The open symbols of di ff erent typesshow the expected relation for subhalos in three di ff erentMilky Way-sized halos formed in the simulations of theconcordance Λ CDM cosmology if the luminosity of thesubhalos is related to their virial mass at accretion epochas L = × L (cid:12) ( M vir , acc / M (cid:12) ) . (see text for discus-sion). simple model above with the slope β ≈ − L − m . relation is shallow butis nevertheless not zero. Constraining this slopewith future observations will tighten constraintson the galaxy formation models and will tell usmore about the L − M vir correlation if such ex-ists.To illustrate the points just made, Figure 10shows the m . − L relation for the observednearby dwarfs [83] and subhalos found within300 h − kpc around three di ff erent MW-sizedhalos formed in the concordance cosmologi-cal model (see [35, 142] for simulation details).To assign luminosity to a given subhalo I fol-low the logic of the model presented in [35],which posits that the brightest observed satel-lites should correspond to the subhalos which m . was thencomputed from the evolved density profile. before it was accreted onto theMW progenitor, M vir , acc : L V = × L (cid:12) (cid:32) M vir , acc M (cid:12) (cid:33) . . (5)The power law form of the relation is motivatedby the approximately power law form of thegalaxy luminosity and halo mass functions atfaint luminosities and small masses. The actualparameters were chosen such that luminositiesof the most massive subhalos roughly match theluminosities of the most massive satellites, suchas the SMC and LMC ( L ∼ − L (cid:12) ). Af-ter all, the first order of business for all the mod-els of satellite population is to reproduce theabundance and luminosities of the most mas-sive ( V max (cid:38)
40 km / s) satellites. The slope ofthe relation in eq. 5 was set to reproduce therange of observed satellite luminosities and flat-ness of the m . − L relation. Note that thismodel does not assume any threshold for for-mation of galaxies. It simply implies that the ef-ficiency with which baryons are converted intostars, f ∗ = M ∗ / M vir , steadily decreases with de-creasing M vir at the rate given by eq. 5.Figure 10 shows that the model with param-eters of eq. 5 is in agreement with observedmeasurements of the m . − L V relation. The re-sults will not change drastically if a somewhatsteeper ( ≈ − .
5) slope is assumed.
21 The figure does not show the most massive subhaloswhich would correspond to the systems such as the LargeMagellanic Clouds, which have luminosities L > L (cid:12) outside the range shown in the figure. This is justifiedbecause observational points shown in the figure includeonly the fainter dwarf spheroidal galaxies.2 The model of equation 5 is similar to the model 1B inthe recent study by Koposov et al. [79], which assumesthat stellar mass scales as M ∗ = f ∗ M ( M vir , acc / M ) + α ,These authors find that the model reproduces the lu-minosity function of satellites for f ∗ ≈ . × − , M = M (cid:12) , and α =
2, which gives M ∗ = . × Fig. 11— The cumulative luminosity function of subha-los in the three MW-sized halos shown in Fig. 10 withthe simple luminosity assignment ansatz of eq. 5. Theluminosity function includes all subhalos within 417 kpcfrom the center of each halo, the same radius as was usedto construct the luminosity function of the observed satel-lites shown in Fig. 6.
Having fixed the parameters of the L − M vir , acc relation, we can then ask the questionof whether the luminosity function of satelliteswould be reproduced self-consistently by sucha model. We can use the observed luminos-ity functions corrected for completeness for thefaintest dwarfs from [51] (shown in Fig. 6) totest this. Figure 11 shows the subhalo luminos-ity functions constructed using subhalos identi-fied within 417 kpc (the same outer radius usedin the construction of the observed luminos-ity function by [51]) in the simulations and thesimple luminosity assignment scheme of equa-tion 5. The luminosity functions in Fig. 11are in reasonable agreement with observations ( M sat / M (cid:12) ) , quite similar to the relation given byeq. 5. Koposov et al. adjust parameters to match the ob-served luminosity function and then show that L − m . relation is reproduced, while I adopted the opposite routehere. The key di ff erence between the models is thattheir model assumes a ceiling on the value of M ∗ / M vir , acc ,while I assume no such ceiling. Absence of the ceiling onstar formation e ffi ciency is actually important for brightsatellites (see Fig. 14). ig. 12— The cumulative m . function of subhalos inthe Λ CDM simulations of three MW halos (dotted lines)and for the observed dSph Milky Way satellites (points,connected by solid line) Sagittarius was excluded due toits very large mass errors), as measured by [85]. Thedashed lines show the mass function for subhalos with L ≥ . × L (cid:12) and d <
270 kpc (the same range ofluminosities and distances as for the observed satellites)with luminosities assigned using equation 5. I have ex-cluded the two most luminous objects from the modelluminous satellites to account for the fact that SMC andLMC are not included in the observational sample andone additional object to account for non-inclusion of theSagittarius dwarf in the comparison. within current uncertainties both in their ampli-tude and slope ( ≈ . m . massfunctions for the observed “classical” dwarfspheroidal satellites of the Milky Way [85]and predicted mass function for the entire sub-halo population of the MW-sized halos within270 kpc (the largest distance of the observedsatellite included in the comparison) and thesubhalos with L > . × L (cid:12) (the smallestdSph luminosity included in the observed sam-ple) with luminosities assigned using eq. 5. Ihave excluded the Sagittarius dwarf from thiscomparison as its m . mass has very large er-rors [85]. The number of the predicted lumi-nous satellites was reduced by three to accountfor exclusion of the Sagittarius, SMC, and LMC from the comparison. The figure showsthat the model predicts the range of m . quitesimilar to that measured for the observed lumi-nous dSphs. The shape of the mass functionis also in reasonably good agreement with thedata. Although there are somewhat more pre-dicted satellites at small masses, this is likelydue to the somewhat larger virial mass of thesimulated halos ( ≈ − × M (cid:12) ) comparedto the mass of the Milky Way ( ≈ M (cid:12) ). Weexpect the number of subhalos to scale approx-imately linearly with host mass and so the dif-ference in the virial mass of the Milky Way andsimulated halos can account for the di ff erencewith observations in Figure 12. There is somediscrepancy at the largest msixs values, but itis not clear just how significant the discrepancyis given that the typical errors on the m . mea-surements for these galaxies are ≈ − V max . The predicted distribution of brightluminous satellites is somewhat more radiallyconcentrated than the distribution of the V max -selected subhalos and is in reasonable agree-ment with the observed distribution both in itsmedian and in the overall shape.Thus, the observed m . − L relation, the lu-minosity function, the m . mass function, andthe radial distribution of the observed satel-lites can all be reproduced simultaneously withsuch a simple dwarf galaxy formation scenario.The observational uncertainties in these statis-tics are still quite large, which leaves significantfreedom in the parameters of eq. 5 and in itsfunctional form. It is also possible that all of L − M vir , acc relation can evolve with ig. 13— The cumulative radial distribution of the ob-served “classical” Milky Way satellites (solid points con-nected by the solid line) within 280 kpc and satelliteswith similar luminosities and within the same distancefrom their host halo in the model of eq. 5 (dashed lines).The figure also shows the cumulative distribution of allsubhalos selected using their current V max (dotted lines). these statistics may be reproduced in a drasti-cally di ff erent scenario. Nevertheless, the suc-cess of such simple model is encouraging andit is interesting to discuss its potential implica-tions.First of all, the equation 5 implies that allof the observed Milky Way dSph satellites hadvirial masses M vir , acc (cid:38) × M (cid:12) when theywere accreted and these masses may span therange up to ∼ × M (cid:12) (the actual range de-pends sensitively on the slope of the L − M vir , acc relation). This shows that progenitors of the ob-served satellites could have had a wide rangeof virial masses, even though the range of their m . and m . masses is narrow.An interesting implication of the value oflowest mass of the range of masses above is redshift.because luminosity may be determined both bythe mass of the halo at the accretion epoch and by the pe-riod of time before its accretion during which it was suf-ficiently massive to withstand star formation suppressingprocesses. Such redshift dependence would be an extraparameter which would generate scatter in the L − M vir , acc relation. that whatever gas the small-mass halos ( M vir (cid:46) × M (cid:12) ) are able to accrete, it should re-main largely unused for star formation, and ofcourse would not be blown away by supernovae(given that the model implies that such objectsshould have no stars or supernovae). If someof this gas is neutral, it can contribute to HIabsorption lines in the spectra of quasars anddistant galaxies. If this gas is enriched, it canalso produce absorption lines of heavier ele-ments. At lower redshifts, the neutral gas inthe otherwise starless or very faint halos couldmanifest itself in the form of the High VelocityClouds (HVCs) abundant in the Local Group[160, 161, 162, 163] and around other galaxies.Second, as I noted above the slope of the L − M vir relation required to explain the weakdependence of m . on luminosity is not surpris-ing, given what we know about the faint-endslope of galaxy luminosity function and whatwe expect about the slope of the mass functionof their host halos in CDM scenario [164]. Theimplied normalization of the L − M vir relation,however, is quite interesting. For example, itindicates that halo of M vir , acc = M (cid:12) shouldhave luminosity of L V = . × L (cid:12) . Convert-ing it to stellar mass assuming M ∗ / L V = M ∗ = . × M (cid:12) . Results of cosmologicalsimulations with UV heating of gas show thathalos of M ∼ M (cid:12) should have been ableto accrete almost all of their universal share ofbaryons, M b = ( Ω b / Ω m ) M vir , acc ≈ . × M (cid:12) (assuming Ω b / Ω m ≈ .
17 suggested by the
WMAP measurements [8]), even in the pres-ence of realistic UV heating [124, 125]. Thederived stellar mass thus implies that only F ∗ ≡ M ∗ / M vir , acc × ( Ω m / Ω b ) ≈ .
001 (i.e., 0 . ffi ciency F ∗ for systems accretion onto which is not sup-pressed by the UV heating implies that starformation is dramatically suppressed by someother mechanism. In fact, the implied e ffi ciency of baryon con-20 ig. 14— The e ffi ciency of gas conversion into stars, de-fined as the fraction of baryon mass, expected if the haloaccreted its universal fraction of baryons, converted intostars: F ∗ = ( Ω m / Ω b )( M ∗ / M vir , acc ). The points (as be-fore di ff erent symbols correspond to subhalos in threedi ff erent simulated host halos) show the dependence of F ∗ on the maximum circular velocity of each subhalo ataccretion according to luminosity assignment model ofeq. 5. The short-dashed line shows the functional formexpected if F ∗ was controlled by the suppression of gasaccretion due to UV heating of intergalactic gas (model3B of [79]), while long-dashed line shows scaling ex-pected in e ffi ciency was set by supernova feedback (e.g.,[131]). version into the stars, F ∗ , in this model is asteep, power law function of mass and circu-lar velocity ( F ∗ ≈ V ÷ , acc ), as shown in fig-ure 14. The figure also shows the functionalform one would have expected if the depen-dence of the e ffi ciency F ∗ on circular velocitywas determined by the fraction of gas halos areable to accrete in the presence of the UV radi-ation. This functional form is almost identicalto the fiducial model 3B of Koposov et al. [79](I used slightly larger value for critical velocitybecause I use V max rather than the virial circularvelocity used by these authors). The two mod-els have similar behavior at V max , acc (cid:46)
30 km / s,but the UV heating model asymptotes to a fixedvalue of F ∗ = − for more massive systems.This model would therefore underpredict lumi-nosities of the most massive satellites in MW- sized halos, which was noticed by Koposov etal. in its failure to reproduce the bright end ofthe satellite luminosity function. Moreover, theestimates of the e ffi ciency F ∗ for more lumi-nous galaxies, such as the Milky Way, are inthe range F ∗ ∼ . − . .These considerations indicate a very interest-ing possibility that while the UV heating canmediate accretion of gas into very small masshalos, the e ffi ciency with which the accretedgas is converted into stars in the known lu-minous galaxies is dramatically suppressed bysome other mass-dependent mechanism. Thisstrong suppression operates not only for halosin which gas accretion is suppressed but for ha-los of larger masses as well. Note that thissuppression mechanism is unlikely to be dueto blowout of gas by supernovae, which is ex-pected to give F ∗ ∝ V (e.g., [131]), a muchshallower relation than the scaling in Fig. 14.While discussion of the nature of this sup-pression mechanism is outside the scope of thispaper, the rapidly improving observational dataon the satellite population should shed light onthe possible mechanisms.The exercise presented in this section illus-trates just how powerful the combination ofluminosity function and radial distribution ofsatellites, high quality resolved kinematics dataand inferred dynamical constraints on the totalmass profile, measurements of star formationhistories and enrichment histories can be in un-derstanding formation of dwarf galaxies. Therapidly improving constraints on the mass pro-files of the dSph galaxies down to the small-est luminosities [83, 73] should further con-strain the range of subhalo masses hosting theobserved satellites and, by inference, the e ffi - F ∗ − V max relation could plausibly flatten at larger V max , as expected from the halo modeling of the galaxypopulation (e.g., [164]) ffi ciency is ac-tually a monotonic function of halo mass fromgalaxies such as the Milky Way to the faintestknown galaxies, such as Segue 1, indicates thatthe physics learned from the “near-field cos-mology” studies of the nearest dwarfs can po-tentially give us important insights into forma-tion of more massive galaxies as well. Acknowledgements.
I would like to thankBrant Robertson, Anatoly Klypin, NickGnedin, Jorge Pe ˜narrubia, Erik Tollerud,James Bullock for useful comments on themanuscript and stimulating discussions onthe topics related to the subject of this paper.This work was partially funded by the NSFgrants AST-0507666, and AST-0708154. Theresearch was also partially supported by theKavli Institute for Cosmological Physics atthe University of Chicago through grant NSFPHY-0551142 and an endowment from theKavli Foundation. I would like to thank KavliInstitute for Theoretical Physics (KITP) inSanta Barbara and organizers of the 2008 KITPworkshop “Back to the Galaxy II,” where someof the work presented here was carried out, forhospitality and wonderful atmosphere. I havemade extensive use of the NASA AstrophysicsData System and arXiv.org preprint serverduring writing of this paper.
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