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Mon.Not.R.Astron.Soc.000,1–12(2005) Printed2February2008 (MNLATEXstylefilev2.2) Thermal physics, cloud geometry, and the stellar IMF ⋆ Richard B. Larson YaleAstronomyDepartment,Box208101,NewHaven,CT06520-8101,USA Submitted2004December15;Accepted2005January31 5 ABSTRACT 0 0 Thethermalpropertiesofstar-formingcloudshavean importantinfluenceonhowthey 2 fragment into stars, and it is suggested in this paper that the low-mass stellar IMF, which n appearstobealmostuniversal,isdeterminedlargelybythethermalphysicsoftheseclouds. a Inparticular,itissuggestedthatthecharacteristicstellarmass,alittlebelowonesolarmass, J is determined by the transition from an initial cooling phase of collapse to a later phase of 1 slowly rising temperature that occurs when the gas becomes thermally coupled to the dust. 3 NumericalsimulationssupportthehypothesisthattheJeansmassatthistransitionpointplays animportantroleindeterminingthepeakmassoftheIMF.Afilamentarygeometrymayalso 2 playakeyroleinthefragmentationprocessbecausetheisothermalcaseisacriticaloneforthe v collapseofacylinder:thecollapseandfragmentationofacylindercancontinuefreelyaslong 7 asthetemperaturecontinuestodecrease,butnotifitbeginstoincrease.Thelimitedavailable 5 results on the dependenceof the thermal propertiesof cloudson metallicity do not suggest 3 a strong dependenceof the IMF on metallicity, but the far-infraredbackgroundradiation in 2 1 starburst regions and in the early universe may significantly shift the peak mass to higher 4 massesinthesesituations. 0 Keywords: stars:formation–stars:luminosityfunction,massfunction / h p - o r 1 INTRODUCTION have been found in many different star-forming environments in t s ourGalaxyandothers,andnocleardependencehasbeenfoundon a The problem of understanding the distribution of masses with anyplausiblyrelevantastrophysicalparametersuchasmetallicity. : which stars are formed, or the stellar initial mass function v Thus,notonlymustsomefeatureofthephysicsofstarformation (IMF),remainsunsolved.However,observationshavemademuch i lead to a characteristic mass scale, it must operate in a relatively X progressinrecentyearsinclarifyingthemainfeaturesoftheIMF universal way that depends only weakly on the environment and r thatneedtobeunderstood.Muchevidencenowsupportsageneral mostastrophysicalparameters.Thisfactposesafurtherchallenge a form of the IMF that is similar to the original Salpeter (1955) toanytheoreticalunderstandingofstarformation. powerlawatmassesaboveonesolarmassbutflattensbelow one One possibility for explaining a universal mass scale might solar mass and declines below 0.1 solar masses when expressed beintermsof someuniversality intheproperties ofstar-forming intermsofthenumberofstarsperunitlogarithmicmassinterval. cloudsortheinitialconditionsforstarformation.Forexample,if The IMF in logarithmic units is thus a peaked function, peaking prestellar cloud cores are created by turbulent compression (Lar- atafewtenthsofasolarmass(Miller&Scalo1979;Scalo1986, son 1981; Mac Low & Klessen 2004), a characteristic turbulent 1998;Kroupa2001,2002;Chabrier2003).Thismeansthatnature pressure might translate into a characteristic mass scale for star makes stars with a preferred mass of this order. The amount of formation;theinferredturbulentpressuresinnearbystar-forming massthatgoesintostarsineachlogarithmicmassintervalisalso cloudsareoftherightmagnitudeforsuchanexplanationtoseem apeakedfunctionthatpeaksatabout 0.5solarmasses,according feasible(Larson 1996, 2003). Alternatively, acharacteristic mag- totheapproximationsuggestedbyKroupa(2002).Thus,interms neticfieldstrengthmightalsoimplyamassscaleforstarformation, of where most of the mass goes, there is a characteristic stellar withmagneticpressuretakingtheplaceofturbulentpressure;our massoftheorderofhalfofasolarmass.Thisisperhapsthemost limitedknowledge ofmagneticfieldstrengthsinmolecular cloud fundamentalfactaboutstarformationthatneedstobeexplainedby cores seems consistent with this hypothesis as well (Shu, Li & anytheory:somefeatureofthephysicsofstarformationmustyield Allen2004). However, star-formingenvironments vary sowidely acharacteristicstellarmassthatisalittlelessthanonesolarmass. in their properties, and turbulence and magnetic fields are in any AfurtherremarkablefactisthattheIMFshowsaconsiderable case such variable and poorly defined phenomena, that explana- degree of universality; similar and often indistinguishable results tionsbasedonturbulenceormagneticfieldsdonotclearlyoffera compellingexplanationforauniversalmassscale. ⋆ E-mail:[email protected] Another possibility is that a characteristic stellar mass scale (cid:13)c 2005RAS 2 RichardB. Larson mightarisefromsomeuniversalityinthebasicphysicsofcollaps- Allstudiesagreethatthemassesofthecoresaresimilarinorderof ing clouds, or in the physics of forming stars, that yields a mass magnitudetothemassesoflow-massstars. scaledependingonlyonfundamentalphysics.Forexample,ithas If low-mass stars form by the direct collapse of prestellar beensuggestedthatprotostellaraccretionmightbeterminatedata cloudcoresand acquiremassessimilartothoseof thecores,this characteristicstellarmassbyfeedbackeffectssuchasoutflowsthat suggests that the characteristic stellar mass is determined largely dependonlyoninternalstellarprocessessuchasdeuteriumburning bytheprocessesofcloudfragmentation.Understandingtheorigin (Shu, Adams & Lizano 1987; Adams & Fatuzzo 1996; Meyer et of stellar masses then requires understanding how the material al. 2000). Whileoutflowsprobably do play somerole inlimiting in star-forming clouds becomes divided up into individual star- protostellaraccretion,theoriesoftheIMFbasedonthishypothesis formingunits.Theclearest evidence thatthecharacteristicstellar tend to contain many uncertain parameters, and this makes the massdependsmainlyonthescaleofcloudfragmentationmaybe importanceofsucheffectsveryuncertain. providedbythefactthatmoststarsforminsmallclusterscontain- In this paper, we address another aspect of the physics of ing a few hundred stars where the efficiency of star formation is star formation that has so far received relatively little attention moderately high, of the order of 25 to 30 percent (Lada & Lada inthiscontext, namelythedetailedthermal physicsofcollapsing 2003).Thismeansthattheaveragestellarmass,whichisjustthe andfragmentingclouds.High-resolutionmappingofstar-forming totalmassofthestarsineachregiondividedbythenumberofstars cloudsatmillimeterwavelengthshassuggestedthatlow-massstars present,isdeterminedtowithinafactorof3or4justbythenumber formfrom,andacquiretheirmassesfrom,observeddenseprestel- ofstarsformedineachregion.Theproblemofunderstanding the lar cloud cores that have a mass spectrum resembling the stellar characteristic stellar mass then becomes basically a problem of IMF (Andre´, Ward-Thompson & Barsony 2000; Motte & Andre´ understanding the number of prestellar cores that form in each 2001).Iflow-massstarsindeedderivetheirmassesfromthoseof region. theobservedprestellarcores,theprocessesofcloudfragmentation Whatcontrolsthescaleofcloudfragmentation,orthenumber must play an important role in determining the IMF. The scale ofobjectsformed?Classically,thishasbeenthoughttobethebal- of cloud fragmentation is expected to depend on the Jeans mass, ancebetweenthermalpressureandgravity,andaminimumlength which in turn depends strongly on the temperature, and recent scaleλ forgrowingdensityfluctuationsinamediumwithafinite J numerical simulationshave shown that theamount of fragmenta- pressurewasfirstderivedbyJeans(1902,1929).Acorresponding tionthat occurs isvery sensitive totheexact temperature-density minimum massor Jeans mass M can be derived fromthe Jeans J relationincollapsingclouds(Li,Klessen&MacLow2003). We lengthbyassumingageometryforthegrowingdensityfluctuations discussfurtherinSection2thelikelyimportanceoffragmentation (Spitzer1978;Larson1985,2003).Hereweadoptforconvenience scalessuchastheJeanslengthandmass,andinSection3wedis- theJeansmassintheusualformgivenbySpitzer(1978),whichis cussthekeyroleofthetemperature-densityrelationindetermining justthemassinacubewhosesideistheJeanslength.Althoughthe theformoftheIMFandthecharacteristicstellarmass. Thether- analysis of Jeans was inconsistent in that it neglected the overall malphysicsofstar-formingcloudsandthepredictedtemperature- collapse of the background medium, rigorous stability analyses densityrelationarereviewedinSection4onthebasisoftheexist- have been made for equilibrium configurations such as sheets, ing literature, and the role of the geometry of collapsing clouds filaments,andspheres,andhaveyieldeddimensionallyequivalent is emphasized in Section 5. Section 6 discusses the dependence results(Larson1985,2003).Forexample,ifsphericalsymmetryis of the results on parameters such as metallicity and ambient far- assumedanditissupposedthatcollapsebeginswithamarginally infraredradiationinstarburstregionsandtheearlyuniverse,with stable isothermal sphere or ‘Bonnor-Ebert sphere’, then the mass theimplicationthatatop-heavyIMFmaybeplausibleinthelatter ofsuchasphereisdimensionallythesameastheJeansmass,but situations. A summary of the main conclusions is presented in itisnumericallysmallerbyafactorof4.7becauseaBonnor-Ebert Section7. spherecontainsonlymatterwhosedensityisabovethebackground density,whilearegiononeJeanslengthacrossalsocontainsmatter of lower density that may or may not collapse with the denser gas (Larson 2003). In numerical simulations of cloud collapse 2 THESCALEOFCLOUDFRAGMENTATION andfragmentation,thenumberoffragmentsthatformisgenerally Much evidence now indicates that low-mass stars form in small comparabletothenumberofJeansmassespresentinitially(Larson dense prestellar ‘cores’ in star-forming molecular clouds; these 1978;Monaghan&Lattanzio1991;Klessen,Burkert&Bate1998; cloudcoreshaveapproximatelystellarmasses,andtheyareoften Klessen 2001; Bate, Bonnell & Bromm 2003); in fact, Bate & closely associated with newly formed stars (Myers 1985, 1999; Bonnell(2005)findthatthemedianfragmentmassscalesclosely Benson&Myers1989;Lada,Strom&Myers1993;Evans1999; with the initial Jeans mass. These results suggest that the Jeans Williams,Blitz&McKee2000;Ward-Thompson2002).Thisclose massmaybeofquitegeneralimportanceindeterminingthescale associationsuggeststhatthesecorescollapsedirectlytoformstars, of cloud fragmentation, at least whenever thermal pressure isthe perhaps typically in binary or small multiple systems. Several mostimportantforceresistinggravity. detailedmappingstudieshavefoundthattheprestellarcoreshave Observationally, ithasbeen arguedthat thesmallbut appar- a mass spectrum that resembles the stellar IMF, and it has been ently real differences in the mass spectra of the young stars in suggested on this basis that low-mass stars acquire their masses differentnearbyregionsofstarformationcanbeunderstoodifthe directly from those of the observed cores (Motte, Andre´ & Neri peakmassoftheIMFdependsontheJeansmassintheassociated 1998; Testi&Sargent 1998; Luhman&Rieke1999; Motteetal. clouds. In particular, denser clouds in which the Jeans mass is 2001; Motte & Andre´ 2001). Other similar studies have yielded smaller appear to produce stars with a mass spectrum that peaks coremassesthataresystematicallysomewhatlarger(Johnstoneet at a lower mass and contains more brown dwarfs, qualitatively al.2000,2001),buttheresultingmassspectrumagainhasashape as expected if the Jeans mass plays an important role in cloud similar tothe stellar IMF,and it isstill consistent withthedirect fragmentation(Bricenõet al.2002; Luhmanet al.2003; Luhman collapseofthesecoresintostarswithasomewhatlowerefficiency. 2004). (cid:13)c 2005RAS,MNRAS000,1–12 Thermalphysics,cloudgeometry, andthestellarIMF 3 The relevance of the Jeans mass has, however, been much in which thermal pressure provides the dominant support against debated because effects such as turbulence and magnetic fields gravity.Turbulenceeventuallybecomes unimportant because itis areexpectedtobemuchmoreimportantthanthermalpressurein dissipativeandispredictedtodecayrapidlyevenifitisdominated resisting gravity during the early stages of cloud evolution (Mac by MHD waves. Magnetic fields can be dissipated by ambipo- Low & Klessen 2004). The angular momentum of star-forming lar diffusion and field reconnection, and magnetic forces are in cores has also been thought to present a serious obstacle to star anycaseanisotropicandcannotpreventmatterfromaccumulating formation(Larson2002).However,turbulenceispredictedtodecay alongthefieldlines.Bycontrast,thermalpressurecannotbedissi- rapidly as a cloud evolves, even if it consists predominantly of patedorreducedbelowaminimumsetbythelowestattainabletem- MHD waves (Mac Low & Klessen 2004). Dissipation of turbu- perature;moreover,asanisotropicforce,itcannotbeovercomeby lence may even be required to allow the gas to condense into anyanisotropicmotions.Thermalpressureisthusafinalirreducible bound prestellar cores (Nakano 1998; Myers & Lazarian 1998); barrier to star formation that remains even after turbulence and theturbulenceinprestellarcoresisinfactobservedtobesubsonic magneticfieldshavebeendissipated,andthatcannotbeovercome and therefore less important than thermal pressure in supporting exceptbytheaccumulationofatleastaJeansmassofmaterial.The them against gravity (Myers 1983; Goodman et al. 1998). The Jeansscalemustthereforeplayakeyroleinatleastthefinalstages structuresofmanycoresappeartobeadequatelyapproximatedas ofthestarformationprocess,regardlessofwhathappensduringthe thermally supported Bonnor-Ebert spheres (Williams et al. 2000; earlierevolutionofstar-formingclouds. Ward-Thompson 2002.) Rotation is also too small an effect to influencesignificantlytheearlystagesofcorecollapse(Goodman etal.1993),anditsmaineffectonthelaterstagesisprobablyjust 3 IMPORTANCEOFTHETHERMALPHYSICS theformationofbinaryor smallmultiplesystems(Larson2002). Numericalsimulationsofturbulent collapsingcloudshaveshown Most numerical simulations of cloud collapse and fragmentation that the number of bound clumps that form is not very sensitive have assumed that the gas remains isothermal during the early tothe way inwhich theturbulence istreated, or even towhether stagesofcollapse;thisapproximationhasbeenbasedonearlystud- turbulence isinitiallypresent atall;thenumber of bound clumps iesofthethermalbehaviorofstar-formingcloudswhichhadfound formedisalwayscomparabletothenumberofJeansmassespresent thatthetemperaturedoesnotchangemuchovertherelevantrange initially,although fragment massesmaybesomewhat reduced by indensity(McNally1964;Hayashi&Nakano1965;Hayashi1966; compression during the collapse (Klessen 2001; Bonnell & Bate Hattori, Nakano & Hayashi 1969; Larson 1973). As was noted 2002;Bateetal.2003.) above,simulationsofisothermalcollapseandfragmentationhave The effect of a magnetic field on cloud fragmentation has shown that the number of bound fragments formed is generally beenlesswellexplored,butsimulationssuggestthatthenatureof similartothenumberofJeansmassespresentinitially(e.g.,Larson theturbulenceinstar-formingcloudsisnot greatlyalteredbythe 1978; Monaghan &Lattanzio1991; Klessenet al.1998; Klessen presenceofamagneticfield,sincesimilarfilamentaryandclumpy 2001; Bate et al. 2003; Bate & Bonnell 2005). The Jeans mass structures are still found (Ostriker, Gammie & Stone 1999; Mac dependsstronglyonthetemperature,varyingeitherasT3/2ρ−1/2 Low & Klessen 2004; Li et al. 2004). A few limited simulations orasT2P−1/2accordingtowhetherthedensityorthepressureis offragmentationthathaveincludedbothturbulenceandmagnetic specified,andthereforethethermalpropertiesofcollapsingclouds fieldssuggestthatthepresenceofamagneticfieldmayreducethe mustplayanimportantroleindeterminingthescaleoffragmenta- efficiencyofstarformation,buttheydonotshowacleareffectof tion.Theusual isothermalapproximation isactuallyonlyafairly themagneticfieldonthemassspectrumoftheobjectsformed(Liet crudeoneandmaynotbeadequate,sincetheabovestudiesfound al.2004;Va´zquez-Semadenietal.2005).Inanycase,star-forming thatthetemperatureincollapsingcloudscanvarybyasmuchasa cloudcoresareexpected tolosemuchof theirmagneticfluxata factorof2ormoreaboveandbelowtheusuallyassumedconstant relativelyearlystagebyambipolardiffusion,aprocessthatmaybe value of 10 K. If the temperature varies significantly during the acceleratedbyturbulence(Heitschetal.2004).Nakano(1998)has collapse, this might have an important effect on the amount of arguedthattheobservedprestellarcorescannotbepredominantly fragmentation that occurs; for example, Monaghan & Lattanzio magnetically supported because they would then not show their (1991), Turner et al. (1995), and Whitworth et al. (1995) found observed large enhancements in column density. Direct measure- thatfragmentationisgreatlyenhancedifsignificantcoolingoccurs ments of the magnetic field strengths in cloud cores are difficult duringthecollapse. andhavemostlyyieldedonlyupperlimits,buttheavailabledataare Larson (1985) suggested that the detailed thermal behavior consistentwithalevelofmagneticsupportthatmaybecomparable of a fragmenting cloud might play a key role in determining the to but not greater than thermal pressure in importance (Crutcher extent to which fragmentation can continue. As was shown in 1999; Bourke et al. 2001). Furthermore, models of magnetized that paper, the temperature is predicted to decrease significantly cloud cores fit the observations best if they have magnetic fields with increasing density at low densities, causing the Jeans mass that arecomparable tobutnot dominant over thermal pressurein to decrease strongly as the density rises; this suggests that frag- importance (Ciolek & Basu 2000, 2001). If the magnetic field is mentationmightbestronglyfavoredattheselowdensities.Athigh notstrongenoughtocompletelysuppressgravitationalinstability, densities,however,thetemperatureispredictedtoincreaseslowly the minimum scale for fragmentation is not very different from withincreasingdensity,causingtheJeansmasstodecreasemuch the Jeans scale (Larson 1985), and the collapse of a magnetized moreslowly,andthissuggeststhatcontinuingfragmentationmight cloud core proceeds in a way that is qualitatively similar to the bemuchlesslikelyatthesehigherdensities.Thedensityatwhich non-magnetic case but issomewhat slowed bythe magnetic field thetemperaturereachesitsminimumvalueandbeginstoincrease (Basu1997,1998;Larson2003). may therefore be the highest density at which fragmentation is In summary, both simulations and observations suggest that probable, anditmayimplyapreferredmassscaleforfragmenta- nature eventually disposes of most of the initial turbulence and tion.Larson(1985,1986)suggestedthatthispreferredmassscale magnetic fields in star-forming clouds to form prestellar cores mightleadtoapeakintheIMFataboutthismass,andnotedthat (cid:13)c 2005RAS,MNRAS000,1–12 4 RichardB. Larson theJeansmassatthetemperatureminimumispredictedtobeabout atarateperunitmassthatincreaseswithdensitycausesthetemper- 0.3 solar masses, similar to the mass at which the observed IMF aturetodecreasewithincreasingdensity,whileathighdensitiesthe peaks. The temperature minimum occurs when the gas becomes coolingrateperunitmassnolongerincreaseswithdensityandthe thermally coupled to the dust, and the preferred mass scale may temperaturebeginstoriseslowly(seeSection4.)Again,itmight therefore depend on the conditions under which the gas and dust beexpectedthattheJeansmassatthetemperatureminimumwould becomethermallycoupled.Thepossibilitythattheonsetofthermal beapreferredscaleoffragmentation.Ifthepredictedvariationof couplingbetweengasanddustmightresultinapreferredscalefor temperaturewithdensityincollapsingcloudsisindeedimportant fragmentation was previously suggested by Whitworth, Boffin & in determining the stellar IMF, it is then necessary tounderstand Francis(1998). as well as possible the detailed thermal physics of star-forming The workof Liet al. (2003) shows that theamount of frag- clouds;thissubjectwillbediscussedfurtherinthefollowingsec- mentation that occurs is indeed very sensitive to the assumed tionwheretherelevantliteraturewillbereviewed. temperature-densityrelation.Theseauthorssimulatedthecollapse Another mass scale in star formation that depends only on andfragmentationofturbulentcloudswithsimplepolytropicequa- fundamental physics is the ‘opacity-limit’ mass, a lower limit on tionsofstateoftheformP ∝ργ,correspondingtoatemperature- the mass scale for fragmentation that is determined by the onset densityrelationoftheformT ∝ρ(γ−1).Avalueofγ <1thuscor- at high density of a high opacity to the thermal emission from respondstocoolingduringthecollapse,whileγ > 1corresponds dust (Low & Lynden-Bell 1976). Low & Lynden-Bell estimated toheating.Lietal.(2003)foundthatthenumberoffragmentsthat that the opacity-limit mass is about 0.007 solar masses, almost formsdependsstronglyonγ,especiallyforvaluesnearunity;for twoordersofmagnitudesmallerthanthecharacteristicstellarmass example, a simulation with γ = 0.7 produced about 380 bound mentionedaboveinSection1.Thus,theopacitylimitisevidently clumps,whileanotherwiseidenticalonewithγ = 1.1produced notwhatdeterminesthecharacteristicstellarmassoraccountsfor onlyabout18boundclumps,afactorof20fewer.Thesevaluesof the peak inthe IMF, but it may nevertheless set a lower limit on γarerelevanttostarformationbecausetheyareapproximatelythe stellarmasses,ormorepreciselyonthemassesofobjectsthatform valuesthatareexpectedtoapplyatlowandhighdensitiesinstar- like stars by independent fragmentation, as distinct from objects formingclouds:atlowdensitiesγ ≃0.73(Larson1985),whileat thatformlikeplanetsinprotostellardisks(althoughthisdistinction highdensitiesγ ≃1.07(Section4.)Itmightthenbeexpectedthat may not be entirely sharp.) Thus there may be two fundamental muchmorefragmentationwouldoccuratthelowdensitieswhere massscalesinstarformation,acharacteristicmassandaminimum γ ≃0.73thanatthehighdensitieswhereγ ≃1.07,andthisshould mass, both of which are determined by the thermal physics. The haveimportantconsequences forthestellarIMF,favoringmasses characteristicmassmayreflecttheJeansmassat thepoint where similartotheJeansmassatthetransitionpointwhereγ increases thecoolingrateperunitmassstopsincreasingandthetemperature fromγ <1toγ >1.Thecharacteristicstellarmassmightthenbe reaches a minimum, while the minimum mass reflects the Jeans determinedlargelybythedetailedthermalphysicsofstar-forming massatthepointwherecoolingiscompletelysuppressedbyopac- clouds. ityandthecollapsebecomesadiabatic. An exampleof acasewhere thethermal physicshas aclear impactonthescaleoffragmentationisprovidedbystudiesofthe formationofthefirststarsintheuniverse.Intheabsenceofheavy 4 THERMALPROPERTIESOFSTAR-FORMING elements, the thermal physics of the first star-forming clouds is CLOUDS relativelysimpleandiscontrolledbytraceamountsof molecular hydrogen.AsreviewedbyBromm&Larson(2004),severalgroups Observations and theory both indicate that there are two basic have made detailed calculations of the collapse of the first star- thermal regimes in star-forming clouds (Larson 1973, 1985): (1) forming clouds and have arrived at a consistent picture of their Atlow densities, i.e.below about 10−19gcm−3, thetemperature thermalbehavior(Abel,Bryan&Norman2002;Bromm,Coppi& decreases strongly with increasing density, controlled by a bal- Larson2002;Nakamura&Umemura2002).Afteraninitialphase ance between radiative heating and cooling processes; heating is of rapid collapse and heating, cooling by line emission from H2 due mainly to photoelectric emission from dust grains, possibly moleculesreducesthetemperaturetoaminimumvalueofaround with a contribution from cosmic rays, while cooling is due to 200 K set by the level spacing of the H2 molecules. When the collisionally excited line emission from common atoms and ions densityrisesaboveabout104cm−3,coolingbecomeslessefficient such as C+ and O. Because the associated cooling rate per unit because the upper levels of the molecules become thermalized; mass increases with density, the predicted temperature decreases thetemperaturethenbeginstoriseslowlywithincreasingdensity, withincreasingdensity,eventuallyfallingbelow10Katdensities causing the contraction of the star-forming clumps to decelerate. above10−19gcm−3.(2)Athigherdensities,cloudcoresbecome The Jeans mass at this point is of the order of 103M⊙, and the opticallythicktotheheatingradiationandatomiclineemissionthat numericalsimulationsallyieldclumpmassesoftheorderofseveral areimportantatlowdensities,andtheseprocessesbecomeunim- hundredtoathousandsolarmasses.AparameterstudybyBromm portant.Coolingbymoleculesmayplaysomeroleatintermediate et al. (2002) showed that this mass scale is relatively robust and densities, but this role is limited because most of the molecules dependsmainlyonthephysicsoftheH2 molecule,i.e.onfunda- soonfreezeoutontothedustgrains(Goldsmith2001).Atdensities mentalatomicandmolecularphysics,withonlyweakdependences above about 10−18gcm−3 the gaseventually becomes thermally on other parameters such as the cosmological initial conditions. coupled tothedust, which subsequently controls thetemperature Althoughthefinalevolutionofthesefirststar-formingclumpshas by its far-infrared thermal emission (see also Whitworth et al. notyetbeendetermined,itseemslikelythattheresultwillbethe 1998).Atthesehighdensities,heatingisexpectedtobeduemainly formationof averymassive starwhosemassmaybe100M⊙ or tocompressionatnearlythefree-fallrate;sincetheassociatedheat- more(Bromm&Larson2004). ingrateincreaseswithincreasingdensity,thetemperaturebeginsto In present-day clouds, a qualitatively similar situation is increaseslowlywithdensity.Figure2ofLarson(1985)summarizes expectedtooccurbecauseatlowdensities,efficientatomiccooling theobservationalandtheoreticalresultsthenavailable,anditshows (cid:13)c 2005RAS,MNRAS000,1–12 Thermalphysics,cloudgeometry, andthestellarIMF 5 thatinthelow-densityregime,theoryandobservationsareingood turesthataresystematicallyabout 25percentlower.Thesecalcu- agreement and indicate a similar trend of decreasing temperature lationsdidnotincluderadiativeheatingofthedustbyambientfar- with increasing density. At higher densities, the gas temperature infraredradiation,whichislikelytoraisetheminimumtemperature becomesmoredifficulttomeasurebecausemostofthemolecules toatleast6to7K,aswasnotedabove.Radiativeheatingofthedust used for this purpose freeze out onto the grains, and it becomes becomes even more important in regions of active star formation necessarytorelymainlyontheoryfortemperatureestimates.The suchasstarburstregions,aswillbediscussedfurtherinSection6.2. theoretical prediction shown in Figure 2 of Larson (1985), taken Mostoftheabovecalculationsdidnotincorporateadistribu- from Larson (1973), shows the temperature reaching a minimum tion of grain sizes, but assumed grains of a single size. At high valueof5Katadensityof2×10−18gcm−3,afterwhichitbegins densitieswherethegasisthermallywellcoupledtothedust,only toriseslowlywithincreasingdensity. the total mass in dust is relevant for the thermal balance and the Koyama & Inutsuka (2000) have studied again the thermal size distribution is unimportant, but at intermediate densities the propertiesofstar-formingcloudsinthelow-densityregime,assum- size distribution can be important because it affects the collision ing that the heating is due primarily to the photoelectric effect ratebetweengasanddust andhencethedensityatwhichthegas rather than to cosmic rays, as had been assumed in most pre- anddustbecomethermallycoupled.Amongthestudiesmentioned vious work. Their predicted temperature-density relation is sim- above,onlyOmukai(2000)incorporatedarealisticdistributionof ilar to those shown by Larson (1985) for densities up to about grainsizes,andaswasnotedabovehefoundlowertemperatures, 2×10−20gcm−3,whileyieldingsomewhat higher temperatures including an unrealistically low minimum temperature of about at higher densities. At these higher densities, predictions become 2.6Katadensityof2×10−19gcm−3.Asanotherillustrationof more uncertain for several reasons: optical depth and radiative thepossibleeffectofsmallergrainsizes,thetemperaturespredicted transfer effects become important, and cooling by molecules and byLarson(1973)wererecalculatedassumingthesamedustmass smalldustgrainsmayalsobecomeimportant.Observationsshow but grain sizes 10 times smaller; the resulting temperatures are that low-mass prestellar cores with such densities generally have unchanged at the lowest and highest densities, but reach a lower temperaturesaround 10K,withsomevariationbetweendifferent minimumof4Katadensityof2×10−19gcm−3.Again,however, regions of star formation (Myers 1985, 1999; Benson & Myers such low temperatures seem unlikely to be realistic because of 1989; Evans 1999). As was reviewed by Evans (1999), the typ- radiativeheatingofthedustbyambientfar-infraredradiation.Since ical temperature of a dense low-mass core with a density of theuncertaintiesareparticularlylargeattheseintermediatedensi- 10−19gcm−3 isabout8.5K,andthisisconsistentwithasmooth ties,itmaybebetterforapproximatepurposestouseinterpolations extension of the temperature-density relation seen at the lower basedonthesimplepower-lawfitsthatworkwellatthelowerand densities,andwiththecontinuingvalidityofthepolytropicapprox- higherdensities. imationwithγ ≃0.73suggestedbyLarson(1985)uptoadensity Within the uncertainties, the observational and theoretical ofatleast10−19gcm−3. resultsdiscussedaboveforthelow-andhigh-densityregimescan Itisunclear observationally whether thetemperaturecontin- be represented by the following approximate temperature-density ues to decrease to even lower values at higher densities, as is relationconsistingoftwopowerlaws: predictedbymostofthetheoreticalcalculations.Larson(1985)and T =4.4ρ−0.27K, ρ<10−18gcm−3 Masunaga&Inutsuka(2000)bothpredictedaminimumtempera- 18 ture of 5 K at a density of 2 to 3×10−18gcm−3, while cloud T =4.4ρ+0.07K, ρ>10−18gcm−3 18 models that include radiative heating of the dust by ambient far- infraredradiationpredictasomewhathigherminimumtemperature whereρ isthedensityinunitsof10−18gcm−3.Thisapproxima- 18 ofabout6to7Katasimilardensity(Zucconi,Walmsley&Galli tiontotheequationofstate,inwhichγchangesfrom0.73to1.07 2001; Evans et al. 2001; Galli, Walmsley & Gonçalves 2002). atadensityof10−18gcm−3,isvalidtowithinabout±0.1dexfor Observationshavegenerallyindicatedstillhighertemperaturesof most densities between 10−22gcm−3 and 10−13gcm−3, except at least 8 K and typically about 10 K (Benson & Myers 1989; thatatdensitiesbetween3×10−20gcm−3and3×10−17gcm−3 Ward-Thompson, Andre´ &Kirk2002), eveninthedensest cloud the uncertainty is larger, perhaps ±0.2 dex. The actual tempera- cores(Tafallaetal.2004).Thusthetemperaturemaynotinreality tureminimumwillbesomewhatsmoothedoutcomparedwiththis becomequiteaslowasispredictedtheoretically,althoughthepos- simple two-part approximation, especially if radiative heating of sibilityhasnotbeenexcludedthatthedensestcloudcorescontain thegrainsisimportantandraisestheminimumtemperaturesignif- verycoldgasthathasnotyetbeendetected. icantly.Moredetailedtreatmentsofthethermalphysicsandbetter Atthehighestdensities,whereitisnecessarytorelyentirely observational constraintswillbeneeded toobtainmoredefinitive ontheoryfortemperatureestimates,anumberofcalculationshave results. predictedverysimilartemperature-densityrelations(Larson1973; The details may, in any case, not be as important as the Low & Lynden-Bell 1976; Masunaga & Inutsuka 2000; Omukai basic fact that thetemperature instar-formingclouds isexpected 2000). In all cases it has been assumed that heating iscaused by to decrease strongly with increasing density in the low-density dynamicalcompressionatnearlythefree-fallrate,andthatcooling regime, i.e. at densities below about 10−19gcm−3, while it is isduetofar-infraredthermalemissionfromthedust.Becauseofthe expected to vary much less in the high-density regime, at least strongtemperaturedependenceofthedustemission,whichvaries until the opacity limit is reached at a density of 10−13gcm−3. asthe6thor7thpowerofthetemperature,theresultingtemperature According to Li et al. (2003), the largest drop in the efficiency depends only weakly on the density or on the grain properties. of fragmentation occurs whenγ isincreased from0.7 to1.0; the For densities between 10−18gcm−3 and 10−13gcm−3, the first number of fragments decreases by a factor of 7 over this range threeoftheabovestudies(Larson1973;Low&Lynden-Bell1976; of γ, but only by another factor of 3when γ isincreased further Masunaga&Inutsuka2000)agreetowithin0.1dexinpredicting to1.1.Thus,themostimportantfeatureofthethermalbehaviorof temperaturesthatincreaseslowlyfromabout5Kto13Koverthis star-formingcloudsmayjustbethechangefromaninitialcooling density range, while the fourth (Omukai 2000) predicts tempera- phase of collapse to a later nearly isothermal phase. Even if the (cid:13)c 2005RAS,MNRAS000,1–12 6 RichardB. Larson temperatureneverdropsanylowerthan,say,8K,whichisapprox- a cylinder can collapse indefinitely toward its axis and fragment imatelythelowesttemperaturedirectlyinferredfromobservations, indefinitely into very small objects, while if γ > 1, this is not andifitstaysapproximatelyconstantathigherdensities,theJeans possiblebecausepressurethenincreasesfasterthangravityandthe mass at the transition point between the cooling phase and the collapseishaltedatafinitemaximumdensitywheretheJeansmass nearlyisothermalphaseshouldstillbeanimportantmassscalefor hasafiniteminimumvalue (Mestel 1965). Fragmentation should fragmentation. According to the approximation given above, the thereforebefavoredwhenγ < 1andsuppressedwhenγ > 1,as temperatureinthelow-densityregimefallsto8Katadensityof wasindeedfoundbyLietal.(2003). about10−19gcm−3,atwhichpointtheJeansmassisabout2M⊙ Insightintothedynamicsofcollapsingconfigurationscanbe whiletheBonnor-Ebertmass,anothercommonlyusedmassscale, obtained from similarity solutions, which can often approximate isabout0.5M⊙.Thesenumbersarestilloftherightmagnitudeto thebehavior of systemswithsimplepolytropicequations ofstate berelevanttothecharacteristicstellarmass. (Larson 2003). Similarity solutions for collapsing cylinders with To explore numerically the effect of an equation of state in polytropic equations of state have been derived by Kawachi & whichγchangesfrom0.7atlowdensitiesto1.1athighdensities, Hanawa(1998),andtheseauthorsfoundthattheexistenceofsuch Jappsen et al. (2005; see also Klessen et al. 2005) have made solutionsdepends onthevalueof γ:similaritysolutionsexistfor detailed simulations of cloud collapse and fragmentation that are γ < 1, but not for γ > 1. For the solutions with γ < 1, the similar to those of Li et al. (2003) except that γ is assumed to collapse becomes slower and slower as γ approaches unity from changefrom0.7to1.1atsomecriticaldensityρcrit.Thevalueof below, asymptotically coming to a halt when γ = 1. This result ρcrit isthenvariedtotesttheeffectofthisparameteronthemass showsinaparticularlyclearwaythatγ = 1isacriticalcasefor spectrum of the resulting bound clumps. The results of Li et al. the collapse of filaments. Kawachi & Hanawa (1998) suggested (2003)andthediscussioninSection3suggestthattheJeansmass that the deceleration of the collapse that is predicted to occur as at the point of minimum temperature should be a preferred scale γ approaches unitywillcauseafilamenttofragmentintoclumps forfragmentation,andthatitmaydeterminethepeakmassofthe because the timescale for collapse toward the axis then becomes IMF;if thisiscorrect, thetypical massof thebound clumpsthat longer than the timescale for fragmentation. If γ increases with formshouldscaleapproximatelywiththeJeansmassMJ,critatthe increasingdensity,asisexpectedfromthediscussioninSections3 densityρcritwherethetemperaturereachesitsminimumvalueand and4,thecollapseofafilamenttowarditsaxiswouldbeexpected thegasbecomesthermallycoupledtothedust(seealsoWhitworth to decelerate as γ approaches unity, favoring the breakup of the et al. 1998). The results of Jappsen et al. (2005) show a clear filamentintoclumpsatthispoint.Thesensitivityofcylinderfrag- dependence of the clump massspectrum on ρcrit in theexpected mentationtotheequationofstate,andtheterminationoffragmen- sense that as ρcrit increases and MJ,crit decreases, the number tationwhenγrisesaboveunity,werealsonotedbyBate&Burkert of bound clumpsincreasesandtheirmedianmassdecreases. The (1997) on the basis of numerical simulations. The Jeans mass at median mass of the clumps is also typically similar to the Jeans the point where γ rises above unity and the temperature reaches massMJ,crit atthecriticaldensity.Theseresultsconfirmthatthe aminimumispredictedtobeabout 0.3solarmasses(Section3), thermalpropertiesofstar-formingcloudsplayanimportantrolein similartothemassatwhichtheIMFpeaks.Thissimilaritysuggests determiningthestellarIMF,andtheyaddsupporttothehypothesis thatthecollapseandfragmentationoffilamentswithanincreasing thattheJeansmassatthepointofminimumtemperatureisimpor- valueofγmayplayacentralroleintheoriginofthestellarIMF. tantindeterminingthecharacteristicstellarmass. The isothermal equation of state is a special case also for anotherwell-studiedcollapseproblem,thecollapseofauniformly rotatingcloud toacentrifugally supported disk. If a uniformand uniformlyrotatingcloudcollapsesisothermallywithconservation 5 ROLEOFTHEGEOMETRYOFFRAGMENTING ofangular momentumtoformacentrifugallysupported disk,the CLOUDS resultisthesingularisothermaldiskstudiedbyToomre(1982)and A striking feature of the simulations of Jappsen et al. (2005) is Hayashi, Narita & Miyama (1982), which can be regarded as a the prominence of filaments and the fact that nearly all of the generalizationtofinitetemperatureandthicknessoftheinfinitely bound objects form in the filaments. Filamentary structure is a thin disk first studied by Mestel (1963). The singular isothermal verycommonfeatureofsimulationsofcollapseandfragmentation, disk, which has been generalized further tothe magnetic case by appearinginmanysimulationsregardlessofwhetherturbulenceor Li & Shu (1996), is a self-similar structure in which the rotation magnetic fields are included (e.g., Monaghan & Lattanzio 1991; speed is constant everywhere and the Toomre stability parameter Klessen2001;Klessen&Burkert2001;Bonnell&Bate2002;Bate Q is also constant everywhere. For a realisticinitial rotation rate etal.2003;Klessenetal.2004;Lietal.2003,2004;Padoanetal. thediskhasQ < 1whenfullyformed,anditshouldthereforebe 2005). Filamentary structure is also seen in many observed star- unstabletofragmentation,oratbestonlymarginallystable(Larson forming clouds, and much of the star formation in these clouds 1984). Detailed calculations of isothermal collapse with rotation appearstooccurinfilaments(Schneider&Elmegreen1979;Lar- haveshownthatasingulardiskisindeedformed,andthatitisbuilt son1985;Curry2002;Hartmann2002.)Manystar-formingcloud up from the inside out (Matsumoto, Hanawa & Nakamura 1997; coresareelongatedandprobablyprolate,sometimesclearlyform- Saigo&Hanawa1998). Instabilityshouldthereforeoccur firstin ing parts of larger elongated or filamentary structures (Myers et theinnerpartofthedisk,whereitmayresultintheformationofa al.1991).Thus,theformationandfragmentationoffilamentsmay binaryormultiplesystem. be an important aspect of star formation quite generally (Larson Aself-similardiskispossibleonlyfor γ = 1;ifγ 6= 1the 1992). If most of the fragmentation that forms stars occurs in self-similarity is lost and the value of Q becomes a function of filaments,thismayhelptoexplaintheresultofLietal.(2003)that radius.Ifγ < 1,theinnermostpartofthediskbecomesverythin the amount of fragmentation depends strongly on the value of γ, and unstable to fragmentation (Q ≪ 1), whereas if γ > 1 the especiallyforvaluesofγnearunity.Thisisbecausetheisothermal inner disk is strongly stabilized (Q > 1). Fragmentation of the caseγ =1isacriticaloneforthecollapseofacylinder:ifγ <1, inner disk is therefore unavoidable if γ < 1, but is prevented if (cid:13)c 2005RAS,MNRAS000,1–12 Thermalphysics,cloudgeometry, andthestellarIMF 7 γ >1.Calculationsofthecollapseofrotatingcloudswithvarious howeverinhibitedbytheincreaseofpressureforces,byanamount values of γ confirm that when γ < 1 the inner part of the disk thatdependsontheequationofstateandthegeometry.Forexam- becomesverythinandunstable,typicallyhavingQ∼0.35;infact ple,intheisothermalcasetheratioofpressuretogravitydecreases atorusthenalwaysformsnearthecenter,andsuchaconfiguration duringsphericalcollapse,remainsconstantduringcylindricalcol- ishighlyunstabletofragmentation(Saigo,Matsumoto&Hanawa lapse, and increases during planar collapse. Indefinite collapse to 2000).Whenγ > 1,thesecalculationsshowthattheinnerpartof a infinitely thin sheet is therefore not possible in the isothermal thediskisthickandstableagainstfragmentation. Theisothermal case.Infact,collapsetoathinsheetisnotpossibleforanyvalueof case γ = 1isthus aspecial marginal case between stabilityand γ > 0.Collapsetoathinfilamentisalsopreventedforγ > 1.In instability,andthismayexplainwhytheisothermalcollapseofa thelow-densityregimewhereγ ∼ 0.7,collapsetoathinsheetis rotating cloud has been such a challenging problem numerically, thereforenotpossiblebutcollapsetoathinfilamentremainspossi- withsomecalculationsyieldingringsandothersnot,dependingon ble,leavingtheformationofthinfilamentsasthemostpromising thenumericalresolutionused(Larson2003).Calculationsofrotat- allowed path to fragmentation. In the high-density regime where ingcollapseandfragmentationinwhichγ isassumedtoincrease γ >1,evencollapsetoathinfilamentisnotpossible,andtheonly from1to7/5whentheopacitylimitisreachedshow thatcentral remainingallowedmodeofcollapseisthenearlysphericalcollapse collapse is halted at this point, and fragmentation into a binary ofclumpsformedbythefragmentationoffilamentsatlowerdensi- or small multiple system is then likely to occur (Bonnell 1994; ties.Theseconsiderationsmayhelptoexplaintheapparentlyubiq- Bonnell & Bate 1994a,b; Whitworth et al. 1995; Matsumoto & uitousroleoffilamentsinthefragmentationofstar-formingclouds. Hanawa2003). Inthisview,fragmentationmaybeseenasinvolvinganinterplay Theisothermalcasethereforeplaysaspecialroleinrotating betweenthethermalphysicsandthegeometryofcollapsingclouds. cloudsaswellasinfilamentaryones.Inbothcases,axialcollapse Although indefinite collapse to a thin sheet cannot occur if is halted or slowed when γ rises above unity, and the scale of γ >0,thegasinacloudcanstillbecompressedone-dimensionally fragmentationislikelytobelargelydefinedatthispoint.Rotation, toathinsheetbyexternalpressuresproducedbyeventssuchasa however,isgenerallyobservedtobeasmalleffectinstar-forming cloudcollisionor supernova shock. Shockcompression hasoften clouds,anditbecomesimportantonlyathighdensitieswhereitis beenpostulatedasamechanismtotriggerstarformation.However, likelytoresultonlyintheformationofbinaryormultiplesystems aslong asγ > 0, adense layer created byexternal compression (Larson2002,2003).Thismeansthatthefragmentationofrotating will not be self-gravitating and will not fragment gravitationally configurations is probably only of secondary importance for the unlessitismaintainedinitscompressedstateforasufficientlylong IMF,influencingstellarmassesbyfactorsof2or3butnotinorder timeforgravitationalinstabilitytodevelop;thistimecanbemuch of magnitude. Filamentary structure may therefore still play the longer than the sound crossing time or the free-expansion time. mostimportantroleinthefragmentationofstar-formingcloudsand This implies, for example, that a cloud collision can trigger star theoriginofstellarmasses. formation only if it is relatively slow and has a sufficiently long What might account for the prevalence of filamentarystruc- timescale(Smith1980). Butfor aslow collisionthecompression ture in star-forming clouds? One way to produce filaments is by factorisnotverylarge,andthislimitstheeffectivenessofcollisions purelydynamicalphenomenasuchasshearandvorticity,andsuch inpromoting fragmentation. Thus, even withthehelp of external effects might account for some of the wispy structure seen in compression, sheet formation does not appear a very promising simulationsofturbulentstar-formingclouds(MacLow&Klessen pathtowardfragmentationandstarformation. 2004) and perhaps for much of the filamentary structure seen in Aspecialcaseofsheetfragmentationthatmightnevertheless turbulentflowsgenerally.Purelygravitationaleffectscanalsocre- berelevantfortheformationofsomelow-massobjectsisthefrag- ate filamentary structure in self-gravitating clouds by amplifying mentationofcircumstellardisksaroundformingstars.Bate,Bon- departuresfromsphericalsymmetryduringcollapse;forexample, nell&Bromm(2002)foundthatmostofthe‘proto-brown-dwarfs’ a pressure-free oblate spheroid collapses to a disk and a prolate formedintheircollapsesimulationswerecreatedincircumstellar spheroid to a line (Lin, Mestel & Shu 1965). Even with a finite disksby thefragmentation of thin spiral filamentsthat formedin pressure,aprolateconfigurationcancollapseisothermallytoathin thesedisks.Acircumstellardiskcanavoidthetime-scalelimitation filamentifitslinedensityexceedsacriticalvalue(McCrea1957; mentioned above because it does not have to be self-gravitating Mestel1965;Larson1972).Arelatedpossibilityisthataflattened to survive, being confined by the gravity of the central star. This configurationwithanelongatedorirregularboundarycancollapse allowsthedisktogrowstablyinmassbyaccretingmaterialuntilit toformanirregularfilamentaryshape(Burkert&Hartmann2004). becomesunstabletofragmentation,atwhichpointitmayformone Thusgravityalonecancreatefilamentarystructures.Thismaybe or more small companion objects that could be either small stars especiallythecaseincosmology,wheresimulationsofthedynam- or large planets. A companion object formed in this way might icsofthecolddarkmatterhaveshowntheubiquitousappearance sometimes grow into abinary stellar companion of similar mass, offilamentarynetworks. helping to solve the angular momentum problem (Larson 2002). Anotherpossibilityisthattheequationofstatemayitselfplay Such mechanisms may help to populate the low-mass end of the aroleindeterminingthegeometryofcollapseandfragmentation. stellarIMF,buttheydonotseemlikelytochangemuchtheoverall Sphericalcollapsecanoccurforanyvalueofγ lessthan4/3,but formormassscaleoftheIMF. spherical collapse is not likely to result in much fragmentation Ageneralizationofthesimplecylindricalgeometrydiscussed because it quickly becomes strongly centrally concentrated and above would be amore complex fractal-likefilamentary network tends to produce a single central density peak. Even with the withahigherdimension;manynumericalsimulationshaveindeed additionof arealisticamount ofrotation, fragmentationprobably shown the formation of filamentary networks (e.g., Monaghan & stilldoesn’tproceedbeyondtheformationofabinaryormultiple Lattanzio 1991; Klessen et al. 1998; Klessen & Burkert 2001; system.Extensivefragmentationcanonlyoccurifmanycentersof Bonnell&Bate2002;Jappsenetal.2005).Acloudmodelbased collapsedevelop,andthisrequireslargedeparturesfromspherical onafractalfilamentarynetworkwassuggestedbyLarson(1992). geometry. The growth of departures from spherical symmetry is If a fractal filamentary network forms in a collapsing cloud, its (cid:13)c 2005RAS,MNRAS000,1–12 8 RichardB. Larson dimensionmaydependonthevalueofγ:smallervaluesofγmay solar-metallicity case, it occurs when the gas becomes thermally leadtomorecomplexspace-fillingnetworks ofhigher dimension coupledtothedust.Becauseofthereduceddustabundance,ther- because the stronger associated cooling may allow more small- malcouplingrequiresamuchhigherdensityinthelow-metallicity scalestructureandmorebranchingofthefilamentstodevelop.To case, about 10−15gcm−3. The temperature at this point is pre- illustratethepossiblerelationbetweenγ andtheattainablefractal dicted to be about 15 to 20 K, i.e. 3 to 4 times higher than the dimensionDoftheresultingconfiguration,considerthefollowing minimumtemperatureinthesolar-metallicitycase.Asimilarresult simple(non-fractal)specialcases:(1)sphericalcollapsetoapoint isobtainedwhentheresultsofLarson(1973)arerecalculatedwith mass of dimension D = 0 is possible only if γ < 4/3; (2) a metallicity reduced by two orders of magnitude; this yields a cylindrical collapse to a line with a higher dimension D = 1 minimumtemperatureof20Katadensityof3×10−15gcm−3. is possible only if γ < 1; and (3) collapse to a sheet with a Becauseboththetemperatureandthedensityatthisnewminimum still higher dimension D = 2 is possible only if γ < 0. In pointareconsiderablyhigherthaninthesolar-metallicitycase,the these special cases, the maximum attainable dimension D of the associated Jeans mass is not greatly different, but it is somewhat resultingconfigurationisrelatedtoγ byD = (4−3γ)/(2−γ). reducedtoabout0.05to0.1solarmasses.Ifthismassscaleiswhat Thisrelationcanbederivedinamoregeneralwaybynotingthat primarilydeterminesthepeakmassoftheIMF,thepeakmassmay theJeanslengthλ variesasρ(γ−2)/2,whiletheJeansmassM actually decrease somewhat with decreasing metallicity, contrary J J varies as ρ(3γ−4)/2; this implies that the number of Jeans-scale toprevious simplearguments that themassscaleshould increase substructures thatcanforminasystemof fixedtotalmassvaries withdecreasing metallicity(e.g.Larson1998). Thepossibilityof asρ(4−3γ)/2.ThenumberNofboundsubstructuresisthenrelated formingverylow-massobjectsremainsessentiallythesameasin to their size L by N ∝ L−D, where D = (4− 3γ)/(2− γ) the solar-metallicity case because the opacity-limit mass remains andtheexponentDcanbedefinedasthefractaldimensionofthe almostthesame,i.e.about0.01M⊙orless,evenwiththereduced system. If γ ≃ 0.73, as is expected for the low-density regime metallicity. However, since there isnow asecondary temperature instar-formingclouds,thedimensionpredictedbythisrelationis minimumatamuchlowerdensitywheretheJeansmassisseveral D≃1.43,whichcorrespondstoamoderatelycomplexfractalnet- tensofsolarmasses,theneteffectofalowmetallicityontheIMF work.Perhapsfortuitously,thisdimensionissimilartothefractal isnotobvious,anditcanonlybedeterminedbydetailednumerical dimension D ∼ 1.4 of the spatial distribution of newly formed simulationslikethoseofJappsenetal.(2005). stars that was found by Larson (1995) from an analysis of the Low&Lynden-Bell(1976)alsoconsideredtheeffectofhav- clustering of young stars in Taurus. Whatever may be the utility ingareducedfractionoftheheavyelementscondensedintodust, of such fractal descriptions, some collapse simulations do show andasalimitingcasetheyconsideredtheeffectofhavingnodust theformationofatleastmoderatelycomplexfilamentarynetworks at all, with a gas metallicitythat is still two orders of magnitude (e.g.Bonnell&Bate2002;Bateetal.2003;Jappsenetal.2005),so below solar. The primary temperature minimum associated with furtherstudyofthegeometryofcloudcollapseandfragmentation thethermalcouplingofgasanddustthendisappearsandthetem- maybefruitful. peraturebecomesmuchhigheratthesedensities,leavingonlythe secondaryminimumassociatedwithatomiccooling,whichoccurs at a density of 10−19gcm−3 where the temperature is predicted 6 DEPENDENCEONOTHERPARAMETERS to be about 40 K. The Jeans mass at this point is about 40 M⊙, andtheJeansmassdoesnotfallbelow15M⊙inthedust-freecase 6.1 Metallicity untilmuchhigherdensitiesandtemperaturesarereachedwhereH 2 If fragmentation and the IMF depend importantly on the thermal molecules control the thermal physics, as in the metal-free case propertiesofstar-formingclouds,theyshoulddependontheabun- (see below). This suggests that the formation of low-mass stars dancesoftheatoms,molecules,anddustgrainsthatareresponsible is suppressed, or becomes very unlikely, in the absence of dust. forthecooling,andhenceontheoverallmetallicity.Relativelyfew Thedustplaysacrucialrolebecauseonlythethermalcontinuous theoreticalstudieshaveconsideredtheeffectofmetallicityonthe emissionfromdustcanprovideenoughcoolingtoreducetheJeans thermalpropertiesofcollapsingclouds,buttwothathavedoneso mass below one solar mass at the intermediate densities where arethoseofLow&Lynden-Bell(1976)andOmukai(2000),with continuum optical depths are still low and fragmentation is still generally similar results. Both studies considered the effect of a probable; line cooling by atoms and molecules is unimportant at metallicitytwoorders of magnitude below solar, andboth found, these densitiesbecause the lineoptical depths become very large asmight beexpected, that thislower metallicityresults inhigher (Low&Lynden-Bell1976;Whitworthetal.1998).Animportant temperatures at all densities; the temperature is predicted to be implicationforcosmologymaybethat,whatevertheoverallabun- higherbyanorderofmagnitudeormoreatthelowerdensities,and dance of heavy elements, the formation of low-mass stars does byabout0.4dexatthehigherdensities.However,thetemperature not become probable until significant amounts of dust have been is also predicted to vary more strongly with density in the low- createdanddispersed. metallicitycase,andtherearenowtwotemperatureminimarather The completely metal-free case has been studied by many than one as in the solar-metallicity case. These two minima, one authorsinordertounderstandtheformationofthefirststarsinthe associated with atomic cooling and the other with dust cooling, universe(forareviewseeBromm&Larson2004).Inthisextreme occuratlowerandhigherdensitiesthanthesingleminimuminthe case, the only available coolant is molecular hydrogen, and the solar-metallicitycase,andtheirassociatedJeansmassesarehigher predictedminimumtemperatureofabout200Koccursatadensity and lower, respectively. Because there are now two temperature of the order of 10−20gcm−3. The Jeans mass at this point is of minimaandtwoassociatedmassscales,itisnotpossiblewithout theorderof103M⊙,andtheargumentsgivenearliersuggestthat furtherdetailedcollapsecalculationstomakesimplepredictionsof thisshouldleadtolargetypicalmassesforthefirststars.Although theeffectofmetallicityontheoverallmassscaleofthestellarIMF. feedbackeffectsaremoreimportantformassivestarsthanforlow The lowest temperature is actually attained in the higher- massstarsandmaysignificantlyreducetheefficiencyofstarforma- densityminimumthatisassociatedwithdustcooling,andasinthe tion,knownfeedbackeffectscannotpreventametal-freeaccreting (cid:13)c 2005RAS,MNRAS000,1–12 Thermalphysics,cloudgeometry, andthestellarIMF 9 protostarfromgrowingtoamassofatleast30M⊙(Tan&McKee andto1.2M⊙ataredshiftof10.Thustheformationofsolar-mass 2004);afinalmassoftheorderof100M⊙ormoreispossible,and starsmaybecompletelysuppressedatredshiftsabove10. a massas largeas 500 M⊙ cannot be excluded (Bromm &Loeb Anothersourceofbackgroundradiationthatcanbeimportant 2004).Theformationofanysignificantnumberoflow-massmetal- in star-forming clouds is the ambient far-infrared radiation from freestarsseemsunlikelybecauseveryhighcompressionwouldbe heated dust in regions of very active star formation. Strong far- required,forexampleinverythindensefilaments,butitcannotbe infrared emission is an observed property of such regions, and completely excluded because the opacity-limit massisstill much because optical depths are small at far-infrared wavelengths, this lowerthanasolarmasseveninthiscase(Omukai2000; Omukai radiation can penetrate the dense cloud cores and heat the dust &Yoshii2003;Bromm&Larson2004). inthemtotemperaturessignificantlyhigherthanwouldotherwise Theeffectofthefirstsmallamountsofheavyelementsinpro- be predicted. The effect of this ambient radiation on the thermal vidingenhancedcoolingandthuspossiblyallowingtheformation structureofdensecloudcoreswascalculatedbyFalgarone&Puget ofthefirstsignificantnumberoflow-massstarshasbeenstudiedby (1985), and itspossible implications for cloud fragmentation and Brommetal.(2001)andBromm&Loeb(2003),andtheyfindthat thestellarIMFwerediscussedbyLarson(1986),whospeculated forthemostimportantheavy-elementcoolant,carbon,aminimum thattheresultmightbeahigh-massmodeofstarformation.Falgar- abundanceof3.5dexbelowsolarisrequiredinorderforC+cool- one&Puget(1985)showedthattheobservedleveloffar-infrared ingtobecomemoreimportantthanH2 cooling.OnceC+ cooling radiationinstar-forminggiantmolecularcloudcomplexesissuffi- takesoverfromH2cooling,thegasrapidlycoolstoatemperature cient to heat the dust throughout these complexes to a minimum nearthecosmicbackgroundtemperature.Theresultinglargedrop temperature of 15 to 20 K. As a result, the gas temperature is in temperature, which may be by almost an order of magnitude, raisedtoatleast15Katthehighdensitieswherethegasbecomes maythenpermittheformationofsignificantnumbersoflow-mass thermally coupled to the dust, making the temperature at these stars with a more normal IMF, provided that some dust has also densitiesafactor of 2or 3higher thanwould bepredicted inthe beenproducedanddispersed,butdetailedcalculationsareneeded absence of dust heating (see Figure 1 of Larson 1986). The gas toexplorethispossibilityfurther. temperature is predicted to have a dip at an intermediate density of3×10−20gcm−3,whereitreachesaminimumvalueofabout 11K.AtthispointtheJeansmassisabout7M⊙ andtheBonnor- 6.2 Backgroundradiation Ebert mass is about 1.5 M⊙; these numbers are about 20 times Atanystageintheexpansionoftheuniverse,aminimumtemper- largerthanthosepredictedinSection4foraminimumtemperature ature is set by the cosmic background temperature 2.73(1 +z). of5Katadensityof2×10−18gcm−3,and3timeslargereven Atpresent,thisbackgroundtemperatureisbelowanypredictedor thanthosepredictedforaminimumtemperatureof8Kreachedat observed cloudtemperature, but not byalargefactor;thismeans a density of 10−19gcm−3. This difference suggests that the far- that the cosmic background radiation will become important for infrared ambient radiation inregions of active star formation can thethermalpropertiesofstar-formingcloudsatnotverylargered- have important effects on the thermal properties of star forming shifts.Forexample,ataredshiftofz = 5thecosmicbackground cloudsandhenceonthestellarIMF,possiblyfavoringatop-heavy temperatureis16K,highenoughtomakeaconsiderabledifference IMF in such regions. Such effects should be even larger in the tothethermalpropertiesandfragmentationofstar-formingclouds more extreme circumstances existing in the vicinity of starbursts asdiscussedinSections3and4.Ifthecosmicbackgroundradiation andAGNs. setsaminimumcloudtemperatureof16Kbutthephysicsofstar- Theabove discussion assumes that thethermal propertiesof forming clouds is otherwise unchanged, the transition from the thegasatdensitiesbelowthatatwhichthegasbecomesthermally initialcoolingphaseofcollapsetothelaterisothermalphaseoccurs coupledtothedustarenotalteredbytheenhancedambientradia- atadensityofonly10−20gcm−3,atwhichpointtheJeansmass tioninstar-formingregions,butthismightnotbethecaseifthere isabout20M⊙ andtheBonnor-Ebertmassisabout5M⊙.These is also enhanced gas heating by effects such as the photoelectric masses are an order of magnitude higher than those predicted in effect.Falgarone&Puget(1985)includedenhancedphotoelectric Section4evenforaminimumtemperatureashighas8K,which heating in their cloud models, but they found that its effect is is approximately the lowest temperature observed in present-day limitedbyhighopticaldepthsattherelevantwavelengths,andtheir clouds.Accordingtotheargumentsgivenearlier,thisshouldhave predicted temperature-density relation in the low-density regime an important effect on the stellar IMF, shifting the mass scale is similar again to that found in the various studies mentioned upward by at least an order of magnitude. This upward shift in in Section 4. The temperature-density relation in the low-density massscaleisevenlargerthantheshiftsuggestedtobepossibleby regimemaythereforenotdiffergreatlyindifferentregions,andthe Larson(1998)onthebasisofthesimpleassumptionthatpressures most important variableaffecting thestellarIMF may just bethe inpresent-dayandhigh-redshiftcloudsaresimilar;thisisbecause minimum dust temperature set by thebackground radiation field. thetransitionfromthecoolingphasetotheisothermalphaseoccurs Larson(1985)notedthatifonlythisminimumtemperaturevaries, at a lower pressure when the background temperature is higher. thepredicted massscaleforfragmentationisquitesensitivetoit, An increase in the mass scale of the IMF at high redshifts can varyingasT3.35.Thisrelationpredicts,forexample,thatthemass min havemany important consequences for galaxyevolution, aswere scaleforfragmentationincreasesbyafactorof10foreachfactor discussedbyLarson(1998). of2increaseinTmin.ThenumericalexperimentsofJappsenetal. Anothereffectofthecosmicbackgroundradiationthatcould (2005)suggestaqualitativelysimilarbutquantitativelysomewhat beimportant isalargeincreaseintheopacity-limit mass,i.e.the weakerdependenceofthetypicalfragmentmassonminimumgas minimum fragment mass set bytheonset of high opacity at high temperature,findingthatthemedianfragmentmassvariesapprox- densities.Low&Lynden-Bell(1976) notedthattheopacity-limit imatelyasthe1.7power oftheminimumtemperatureorthe-0.5 mass increases strongly withthe cosmic background temperature poweroftheassociateddensity;thisisstillaveryimportanteffect, because of the strong increase of dust opacity with temperature; anditcanhavemajorconsequencesforthestellarIMF. thisminimumstellarmassincreasesto0.07M⊙ ataredshiftof5 Althoughtheabovediscussionsuggeststhattheambientradi- (cid:13)c 2005RAS,MNRAS000,1–12 10 RichardB. Larson ationinstar-formingregionscanbeimportantforthemassscaleof ‘bottom-light’IMFdeficientinlow-massstars,hasbeenfoundby fragmentation,itsneteffectontheIMFofallthestarsformedina Riekeetal.(1993)inthesmallstarburstgalaxyM82,anddetailed regionmaybereducedbecausethefar-infraredradiationthatheats studiesofthe‘superstarcluster’M82-Fhavealsosuggestedsuch the dust is produced only after a significant number of luminous abottom-light IMF in some clusters inM82 (Smith& Gallagher starshavebeen formed, whileitispossible thatmost of thelow- 2001; McCrady, Graham & Vacca 2005), although this evidence massstarshavealreadyformedbythistime,especiallyifthelow- tooisindirectandnotconclusive.IftheinferenceofanIMFshifted massstarstendtoformbeforethemoremassiveones.Theeffect tohighermassesinM82isindeedcorrect,andifstarburstactivity ofgrainheatingontheoverallIMFmaythenbereduced,andthe was more prevalent at earlier cosmological times, then the IMF resultingIMFmaynotbeveryanomalous,oritmaybeanomalous maygenerallyhavebeenmoretop-heavyatearliertimes,atleast onlyinlocalregionssuchasthecoresofdenseclusters.Theeffect inthedensestregionssuchasgalacticnuclei.Manyimportantcon- ofheatingbyfar-infraredradiationmightbegreaterifahighlevel sequenceswouldfollowfromanIMFthatwassystematicallymore ofambientradiationweresustainedforalongenoughtimeforlarge top-heavyatearliertimes,aswerereviewedbyLarson(1998). numbersofstarsofallmassestobeformed;suchasituationmight occur,forexample,inastarburstgalacticnucleuswheredensegas isretainedandcontinuestoformstarsforatimemuchlongerthan 7 SUMMARYANDCONCLUSIONS thelocaldynamicaltime. Observationally,itremainsunclear,evenafterextensivestudy, Many observations have shown that the stellar IMF has a nearly whether starburst regions have a systematically top-heavy IMF universal standard form that is similar to the original Salpeter (Brandl & Andersen 2005; Elmegreen 2005). Most well-studied functionatmassesaboveasolarmassbutflattensatlowermasses very luminous young clusters do not clearly show an anomalous and peaks (in logarithmic units) at a few tenths of a solar mass, IMF.Thestrongestevidencefortheexistenceofatop-heavyIMF decliningagaintowardthelowestmasses.Similarandoftenindis- maybeprovidedbytheArchescluster,thebest-studiedluminous tinguishableresultshavebeenfoundinmanydiversestar-forming youngclusterintheGalacticcenterregion,aregionofexception- environments, and this suggests that the basic characteristics of allyintensemassivestarformation(Figer2003,2005;Stolte2003; the IMF, including its peak mass, are determined by relatively Stolte et al. 2002, 2005). These authors find that the IMF of the universalfeaturesofthephysicsofstarformation.Numericalsim- Arches cluster is somewhat less steep than the Salpeter function ulations of cloud collapse and fragmentation show that the mass at masses above 10 M⊙, but a more important result may be the spectrum of the bound clumps that form is very sensitive to the tentative finding of Stolte et al. (2005) that the IMF flattens and assumed temperature-density relation, and this suggests that the possibly begins to decline at masses below about 6 M⊙. This IMFdepends, atleast for low-massstars,on thedetailedthermal change in the slope of the IMF at 6M⊙, if confirmed by further physicsofstar-formingclouds. studies over a larger area, would imply that the mass scale for Observations and theory show that there are two important starformationinthisregionisanorderofmagnitudehigher than thermalregimesinstar-formingclouds:alow-densityregimecon- itiselsewhere intheGalaxy. Other well-studiedluminous young trolledbyatomiccoolinginwhichthetemperaturedecreaseswith clusters that are outside the Galactic center region, such as NGC increasing density, and a high density regime controlled by dust 3603,donotshowanysimilarflatteningoftheIMFatintermediate coolinginwhichthetemperatureincreasesslowlywithincreasing masses(Stolte2003).TheapparentlyanomalousIMFintheGalac- density. Collapse simulations show that fragmentation isstrongly ticcenterregionmightresultfromthefactthatthisregioncontains favoredinthelow-densityregimewheretheJeansmassdecreases the largest concentration of massive young stars and clusters in strongly with increasing density, while much less fragmentation the Local Group (Figer 2003); the dust heating effect discussed occursinthehigh-densityregimewheretheJeansmassdecreases above should therefore be much more important in this region muchmoreslowlywithincreasingdensity. Thissuggeststhatthe thanelsewhere.ThemolecularcloudsneartheGalacticcenterare Jeansmassatthetransitionpoint between thesetworegimesisa observed to be much warmer than molecular clouds elsewhere, preferredmassscaleforfragmentation,andthatitmaydetermine typically about 70 K, and the dust temperature in this region is thepeakmassoftheIMF.TheJeansmassatthepredictedtemper- alsoquitehighbythestandardsdiscussedabove,about40Katthe atureminimumbetweenthetworegimesisabout0.3solarmasses, distanceoftheArchesclusterfromthecenter(Morris&Serabyn which is approximately the mass at which the IMF peaks. This 1996). Since both the gas temperature and the dust temperature coincidence may be partly fortuitous because observations have are considerably increased by local heating effects, the situation not confirmed the very low predicted minimum temperature, but is more complicated than the simple case discussed above where even if radiative heating effects raise the minimum temperature only the dust temperature is increased, but if the overall form of to8 K,which isapproximately thelowest value directly inferred the temperature-density relation is similar to the form discussed fromobservations,themassscaleforfragmentationatthetransition earlierexceptforupwardscaling(whichwillrequiremoredatato betweenthecoolingphaseofcollapseandthelaternearlyisother- confirm),argumentslikethosegivenearliersuggestthataverytop- malphaseisstilloftheorderofonesolarmass,andthereforeitis heavy IMF will still result, with a mass scale that is higher than stillofrelevancetothecharacteristicstellarmass. normalbyatleastanorderofmagnitude.TheIMFoftheArches Thecharacteristicstellarmassmaythusbedeterminedbyfun- clusterappearstoshowsuchanupwardshiftinmassscalebyabout damentalphysics,andmorespecificallybytheconditionsrequired anorderofmagnitude(Stolteetal.2005),althoughasthoseauthors forthethermalcouplingofgasanddust.Thetemperaturewhenthis note,furtherworkisrequiredbeforethisresultcanbeconsidered occursispredictedinmostcircumstancestobeverylow,lessthan definitive. 10K,becausethedustcoolseffectivelybyradiatinginacontinuum Evenmoreextremeconditionsmayexistinstarburstgalactic and not just in lines. The density at which the thermal coupling nuclei, but no examples of such regions are presently accessible occurs depends on the abundance of dust grains but is relatively todetailedstarcountstudiescapableofyieldingadirectdetermi- high, of the order of 10−18gcm−3 or more, because of the low nation of IMF.Indirect evidence for atop-heavy IMF,or reallya numberdensityofthegrains.Thelowtemperatureandhighdensity (cid:13)c 2005RAS,MNRAS000,1–12