Characterizing the Brown Dwarf Formation Channels from the Initial Mass Function and Binary-star Dynamics

0.46 MB·English

by  Ingo Thies

DRAFTVERSIONFEBRUARY19,2015
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
CHARACTERIZINGTHEBROWNDWARFFORMATIONCHANNELSFROMTHEINITIALMASSFUNCTIONAND
BINARY-STARDYNAMICS
INGOTHIES1,JANPFLAMM-ALTENBURG1,PAVELKROUPA1,MICHAELMARKS1
DraftversionFebruary19,2015
ABSTRACT
Thestellarinitialmassfunction(IMF)isakeypropertyofstellarpopulations.Thereisgrowingevidencethat
theclassicalstar-formationmechanismbythedirectcloudfragmentationprocesshasdifficultiesreproducing
5 theobservedabundanceandbinarypropertiesofbrowndwarfsandvery-low-massstars. Inparticular,recent
1 analyticalderivationsofthestellarIMFexhibitadeficitofbrowndwarfscomparedtoobservationaldata. Here
0 wederivetheresidualmassfunctionofbrowndwarfsasanempiricalmeasureofthebrowndwarfdeficiency
2
inrecentstar-formationmodelswithrespecttoobservationsandshowthatitiscompatiblewiththesubstellar

b partoftheThies-KroupaIMFandthemassfunctionobtainedbynumericalsimulations. Weconcludethatthe
e existingmodelsmaybefurtherimprovedbyincludingasubstellarcorrectiontermthataccountsforadditional
F formationchannelslikediskorfilamentfragmentation.Theterm“peripheralfragmentation”isintroducedhere
forsuchadditionalformationchannels. Inaddition,wepresentanupdatedanalyticalmodelofstellarandsub-
8
stellarbinarity. Theresultingbinaryfractionandthedynamicallyevolvedcompanionmass-ratiodistribution
1
are in good agreement with observational data on stellar and very-low-mass binaries in the Galactic field, in

] clusters,andindynamicallyunprocessedgroupsofstarsifallstarsformasbinarieswithstellarcompanions.
A Cautionarynotesaregivenontheproperanalysisofmassfunctionsandthecompanionmass-ratiodistribution
and the interpretation of the results. The existence of accretion disks around young brown dwarfs does not
G
implythattheseformjustlikestarsindirectfragmentation.
.
h Subjectheadings:binaries: general—browndwarfs—methods: numerical—methods: statistical—stars:
p low-mass—stars: luminosityfunction,massfunction—
-
o
r
t 1. INTRODUCTION Analytical approaches by Padoan & Nordlund (2002,
s PN02, from here) and Hennebelle & Chabrier (2008, 2009,
a The stellar initial mass function (IMF) is a key tool for
HC08,andHC09,hereafter)deducestheIMFfromananaly-
[ star-formationresearchbecauseitmirrorstheprocessesofthe
tical description of the distribution of prestellar cloud cores.
formationofstellarpopulations(Bastianetal.2010,Kroupa
2 While the stellar IMF is reproduced by these approaches, a
etal.2013).Consequently,theIMFhasbeensubjecttoexten-
v significant deficit of BDs and very-low-mass stars (VLMSs,
siveresearchbothobservationallyandtheoretically. Inrecent
0 between about 0.08 and 0.2 M ) with respect to observa-
years the majority of the star-formation community has fa- (cid:12)
4 tionallyconstrainedIMFdescriptions(Kroupa2001,Chabrier
vored the assumption of continuous star formation from the
6 2005, Thies & Kroupa 2007, 2008, the latter three from
lowest-mass brown dwarfs (BDs) to the most massive stars
1
here on referred to as C05, TK07, and TK08, respectively,
0 (Padoan & Nordlund 2002, 2004, Hennebelle & Chabrier
and Kroupa et al. 2013) appears if realistic properties of the
. 2008). However, there is evidence for at least two separate
1 prestellar clouds are assumed. The fundamental reason why
formation channels for most BDs on the one hand and most
0 directfragmentationofaturbulentmolecularcloudrarelypro-
stars on the other. For instance, careful analysis of the ob-
5 duces BDs is that the formation of a BD requires a high-
servational data reveals a disagreement between the theoret-
1 density gravitationally self-bound but very low mass fluctu-
icalpredictionsofthebinaryseparationdistributionsofBDs
: ationthatcannotdrawsignificantamountsofadditionalmass
v andstarsandtheobservedones(Bouyetal.2003,Burgasser
fromanaccretionreservoir(seealsoAdams&Fatuzzo1996).
i etal.2003,Martínetal.2003andCloseetal.2003). Inthis
X However, HC08 and HC09 speculate that this deficit might
lightthekeyassumptionofauniformorcontinuousformation
r mechanism assumed in most star-formation models needs to be solved by a refinement of their models by including tur-
a bulent fragmentation descriptions. A different interpretation
bequestioned. BasedonobservationaldataandN-bodycom-
is that their model is overall correct but reveals that an ad-
putations, Parker & Goodwin (2011) conclude that the birth
ditional formation channel is required to match the observa-
populationofvery-low-massbinarieswithsystemmassesbe-
tions. TheIMFbyC05isanempiricalandalmostequivalent
low 0.2 M must be very different from that of M-dwarfs.
(cid:12) update of the IMF by Chabrier (2003). Fully hydrodynam-
Anotherimportantissueisthe“BDdesert”(McCarthy,Zuck-
icalcomputationsofwholestar-formingcloudsbye.g. Bate
erman,&Becklin2003),anobserveddearthofBDcompan-
etal.(2003)andBate(2009a)reproducetheformationofBDs
ions to stars. Unlike earlier studies like e.g. by Grether &
largely from fragmenting circumstellar disks, although these
Lineweaver (2006) who interpret this as being related to the
simulationstendtooverproduceBDsunlessradiativeheating
companionmassratioratherthantheabsolutemass,themore
isincluded(Bate2009b,2012).
recentsurveybyDieterichetal.(2012)deducealowerabso-
In this paper we introduce the residual mass function
lutemasslimitofcompanionstostarscloseto0.1M .
(cid:12) (RMF)asacorrectiontermforthisBDdeficiencyintheana-
lytical approaches by PN02 and HC08 with respect to the
1Helmholtz-InstitutfürStrahlen-undKernphysik(HISKP),Universität observation-based IMFs in C05 and TK07. In Section 2 the
Bonn,Nussallee14–16,D-53115,Germany
2
model is described and the RMF is defined. The results are assumingastar-formationefficiencyof30%–50%.Themod-
presented in Section 3, followed by an analysis of the com- elshavebeentestedintherange3 20. Inthispaper,
≤M≤
panion mass-ratio distribution in the overlap region of these werestrictthemodeltotheMachnumber =12inthecase
two populations and a discussion of the BD desert and its ofHC08(Figure5toptherein)and =1M0inPN02because
M
claimed consistency with empirically determined IMF mod- theseMachnumbersaresuggestedasthemostlikelyvaluesin
els (Reggiani & Meyer 2011), in Section 4. In addition, the thesepapers(seep. 873inPN02andp. 406inHC08). With
termperipheralfragmentationisintroducedasthemainfor- these values both models match each other well in the stel-
mationchannelofBDs. larregimedespitetheirdifferentformulations. HigherMach
numberswouldmainlyshiftthemassspectrumtowardlower
2. METHOD masses and decrease the MF toward the high-mass end. As
shown in Figure 1 the PN02 =10 mass function (long-
The basic idea is to quantify the deficit of analytically
M
dashed curve) and the = 12 HC08 mass function (solid
derived mass functions with respect to observationally con-
M
curve)closelymatchinthestellarregimebutareslightlydif-
strainedIMFs. TheresultingdeficitdefinestheRMF.
ferent at the low-mass end. The resulting mass function de-
2.1. ObservationalIMFmodels viatessignificantlyfromtheobservedstellar+substellarmass
function in the substellar mass regime, as shown in Figures
ThestellarIMFisbasedonanextensiveanalysisofobser-
2and3forboththeoreticalmodelscomparedtotheobserva-
vational data from young stellar clusters and in the Galactic
tionallyconstrainedIMFsbyC05andTK07. Toquantifythis
field(Kroupaetal.1993,Kroupa2001). Becausethemajor-
differencethemodelIMF,eitherHC08orPN02,issubtracted
ityofmultiplesystemsremainunresolvedsuchIMFsneedto
from an observational reference mass function that is taken
be interpreted as system mass functions unless a careful nu-
from C05 and TK07 in order to determine two estimates of
mericalbinarycorrectionisused.
theRMF:ξ ,
res
TheempiricalIMFbyC05usedhereisasystemIMFbased
on the earlier IMF by Chabrier (2003), assuming BDs and ξ (m)=ξ (m)−ξ (m), (1)
res obs theo
starsformingthesameway,therebysimplyoccupyingdiffer-
entregionsinthesamecontinuousmassfunction,anassump- where ξtheo is any of the theoretical IMFs and ξobs refers to
tionalsomadeinthepastbyKroupa(2001,2002)TK07,on any of the observationally constrained IMFs. In general, a
theotherhand,provideanindividual-bodyIMFthathasbeen mass function ξ is defined as the differential number N over
transformed into the corresponding system IMF by Monte thedifferentialobjectmassm:
Carlo random pairing among the star-like and the separate
dN
BD-like population introduced in TK07. This accounts for ξ(m)= , (2)
the observational evidence for two separate (albeit related) dm
formationchannels. Inparticular,thereisalackofobserved and,inthelogarithmicscale
BDcompanionstostars,especiallyforsmallseparations(Mc-
Carthyetal.2003,Grether&Lineweaver2006),whereasthe dN
ξ (m)= =(ln10)m/M ξ(m). (3)
statistical properties of binary BDs differ largely from stel- L dlog m/M (cid:12)
10 (cid:12)
larones(Bouyetal.2003,Burgasseretal.2003,Martínetal.
2003,Closeetal.2003).AsarguedinThies&Kroupa(2007), The TK07 IMF is the canonical IMF (Kroupa et al. 2013)
thedifferentbinarypropertiesofBDsandVLMSsontheone that takes into account that BDs and some VLMSs need to
hand, and stars on the other, consequently suggest the exis- beaddedasanadditionalpopulationcalledBD-like,whereas
tenceoftwoseparateformationchannels. Thisthusleadsto moststarsbelongtothestar-likepopulation:
therequirementfortwoseparateIMFs,eachcorrectedforun-
resolved binaries, to describe the real individual-body mass ξBD(m)=Rpopk (cid:0)0.m07(cid:1)−0.3, 0.01<m<0.15,
functionofastar-formingevent. (cid:40)(cid:0) m (cid:1)−1.3, 0.07<m ∼0.5, (4)
allTyhmesoetiIvMatFeddeasncdritphteiorenfso,rheodwoevneor,taerxeplpauinreltyheobusnedrevraltyioinng- ξstar(m)=k k0.0(cid:0)7m (cid:1)−2.3, 0.5<m ≤m ,
physicalprocessesleadingtotheactualmassspectrum. m 0.5 ≤ max
where is the BD-like to star-like population ratio
pop
R
2.2. AnalyticalIMFmodels ( 0.2, TK07) and k = (cid:0)0.5(cid:1)−1.3 ensures continuity
Inrecentyears,severalattemptshavebeenmadetounder- oRf pthope≈stellar IMF at 0.5 Mm. H0e.r0e7, m follows from the
(cid:12) max
stand the physics behind star formation and to reproduce its m -M relation(Weidneretal.2010,2013,Equation(10)
max ecl
outcome. This section deals with the analytical models by inPflamm-Altenburgetal.2007)andapproaches150M for
(cid:12)
PN02andHC08.InbothPN02andHC08theprestellarclump themostmassiveclusters. Notetheoverlappingmassranges
massfunctionisdescribedbyananalyticalfunctionwiththe ofthepopulationsindicatingthatbodiesbetweenabout0.07
Jeansmass,thelengthscaleoftheinitialinhomogeneitiesand and0.15M maybelongeithertothestar-likeortheBD-like
(cid:12)
theMachnumber, , asthemainparameters(seeEquation population.
M
(24) in PN02 and Equation (44) in HC08). Both studies de- Atthehigh-massendoftheBD-likepopulation, thesharp
rivetheIMFfromthemassdistributionofgravitationallyun- truncation used in TK07 has been replaced in this study by
stable clumps based on empirical power spectra of turbulent a steep power-law function to reduce numerical artifacts in
flows within the molecular clouds. While the particular for- thesumIMF.Wechoseapower-lawexponentof10tokeep
malism differs in PN02 and HC08, fragmentation due to the the effect on the BD-like to star-like ratio negligibly small.
supersonicinteractionofgassheetsistheengineforforming There is no such declining function applied to the star-like
Jeans-unstableclumpsinbothmodels. Theresultingprestel- populationthatisintrinsicallysmootherduetothedynamical
larclumpmassfunctionisthentransformedintoastellarIMF populationsynthesis(DPS)methoddescribedinSection2.3.
3
3 HPNC0028,, MMaacchh==1102 numberofsystems,Nsys. Here,thetermsystemincludesmul-
tiple systems and singles (their number being noted as N )
sng
aswell. Then
mg ()ξ10L 2 f = NNbsyins = NsnNg+binNbin. (6)
o
l 1 Forthestar-likepopulationweapplythebinaryDPSmethod
developed by Marks, Kroupa, & Oh (2011) and Marks &
BDs stars
Kroupa (2011), hereafter referred to as dynamical or DPS
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 pairing. In DPS the binary stars are formed in a popula-
log10 m / M tion of embedded clusters within which they are dynami-
⊙
FIG.1.—ComparisonoftheanalyticalsystemIMFbyPN02forM=10 cally processed to yield the Galactic disk stellar single-plus-
(their preferred value, dashed curve) with that by HC08 (M=12, solid binary population. An attractive feature of this theory is its
curve). Bothfunctionsarescaledinthisplotforequalpeakheightforcom- underlying assumption of the universality of binary proper-
parisononly.Althoughtheynearlymatchinthestellarregimethereareslight
tiesoflate-typestarsbeingconsistentwithobservationaldata
deviationsinthesubstellarregimeaswellasinthepositions(i.e.masses)of
thepeak. (Marks & Kroupa 2012, Leigh et al. 2014). For initial bi-
naries with intermediate to large separations the DPS pair-
ingmethodappliesrandompairing2 belowaprimarymassof
5 M and ordered pairing (such that q 0.9) above. Here,
Inadditiontothis, star-likeobjectsbelow0.06M(cid:12) andBD- q =(cid:12)m /m 1, where m is t≥he companion mass
likeonesabove0.3M(cid:12)arenotconsidered. Itshouldbenoted and mcomp isptrhime≤mass of the pcroimmpary star. This initial bi-
herethatthetheoreticalmassfunctionobtainedbyThiesetal. prim
narypopulationisthenalteredbydynamicalevolution. Close
(2010)fromsmoothedparticlehydrodynamics(SPH)simula-
binaries with orbital periods below about 10 days undergo
tionsalsoshowsasteepdeclineabove0.1M ratherthanan
(cid:12) eigenevolution (Kroupa 1995) and tend to equalize the com-
exacttruncation.
panion masses. Note that this eigenevolution term alters the
SincetheC05stellarIMFusedbyHC08isasystemmass
very-low-massendofthestarlikeIMF.Forthepurposeofthis
function rather than an individual-body-mass function, the
work,howevertheseeffectsonlyplayanegligiblerole. Here,
TK07canonicalIMFhasalsobeentransformedintothecorre-
the initial or primordial binary fraction is 100%, i.e. it is
spondingTK07systemmassfunctionusingtheMonte-Carlo
assumed that all stars form in binaries. The final (after dy-
modeldescribedinSection2.3. Thestellarcomponentofthe
namical processing in the embedded cluster) overall binary
canonicalTK07systemIMFisnormalizedtotheC05system
fraction is about 40% (i.e. f =0.4) but varies as a function
IMFinthewholemassregime(0.01–150M ),i.e.
(cid:12) oftheprimary-starmass. ForM-dwarfs,inparticular,itisas
15(cid:90)0M(cid:12) 15(cid:90)0M(cid:12) low as 25%while G-dwarfs show about 56% binarity. The
binary fraction approaches 90% for O stars. For the BD-
ξ (m)dm= ξ (m)dm. (5)
TK07,sys C05,sys like population we chose an overall binary fraction of 20%
0.01M(cid:12) 0.01M(cid:12) (i.e. f =0.20), inaccordancewithTK07, TK08. Abouthalf
of the members of observed average stellar populations are
Here it is assumed that in the C05 IMF stars and BDs share
binaries, most of them remaining unresolved in typical star-
thesameIMFwithoutadiscontinuityoroverlap,sothemass
clustersurveys. However,veryyoungandlikelydynamically
regime above 0.08 M corresponds to the stellar component
(cid:12) unevolvedpopulationsliketheTaurus-Aurigaassociationex-
intheTK07model.
hibit almost 100% binarity (Kroupa et al. 2013, Duchêne &
The BD multiplicity fraction is only about 10%-20%
Kraus 2013, Reipurth et al. 2014). The number of systems
(Bouyetal.2003,Closeetal.2003,Martínetal.2003;Kraus,
must not be confused with the number of individual bodies,
White, & Hillenbrand 2006 and Law et al. 2008) and is as-
N =N +2N .Sincehigher-ordermultiplesarerelatively
sumedtobeequalforbothC05andTK07IMFs. bod sng bin
rare (Goodwin & Kroupa 2005) they are summarized within
Further, it has to be noted here that the RMF is based on
thebinarypopulationinthiswork,sothetotalnumberofbod-
themassrangebetween0.01andabout0.3M (log m/M
(cid:12) 10 (cid:12) iesis
between−2and−0.5)only,becausetheresidualsinthestel-
N =2N +N . (7)
larregimeareduetoslightlydifferentfunctionaldescriptions bod bin sng
ratherthanbeingphysicallyconstrainedandarethereforene-
The CMRD describes the relative number of binaries as a
glectedhere. Theresidualsbetweentheobservationalstellar
function of the companion-to-primary mass ratio. Observa-
IMF fits of C05 and TK07 and the HC08 and PN02 models
tions reveal a continuous decline of f as a function of the
areofpotentialinteresttolaterwork,buttheyarenotfurther
primary-object mass which has been interpreted as a contin-
consideredinthispaper.
uous transition from the stellar to the substellar regime (Jo-
ergens 2008, Kraus & Hillenbrand 2012, but see Thies &
2.3. Monte-Carlomodelofthebinarityofstarsandbrown
Kroupa2008). Thereisalsoashifttowardsmoreequal-mass
dwarfs
binaries (q=1) for VLMS and BDs (Dieterich et al. 2012).
Besides the mass function itself, the binarity as a function Thesepropertiesofthestellarpopulationarewellreproduced
of the primary-star mass and the CMRD are also important byDPSsuchthattheoriginandpropertiesofbinarypopula-
characteristics of stellar populations. They are studied here tionsarewellunderstood.
usingaMonteCarloapproachwithstar-likeandBD-likeob-
jectsdrawnfromseparateIMFs. 2Marksetal. (2014,inprep.) showthatrandompairingwithsubsequent
Thebinarityorbinaryfraction, f,isdefinedastheratioof dynamicalprocessingdoesindeedreproduceatleasttheobservedlow-mass
thenumberofbinaryorhigher-ordersystems,N ,tothetotal stellarpopulation(seealsoKroupa1995).
bin
4
In this study we used a Monte Carlo approach with the bulentMachnumber. OtherparametersyielddifferentRMFs,
TK07IMF(Equation(4)),i.e.,withtwoseparatepopulations, so only the general functional shape for a likely parameter
BD-likeandstar-like,rangingfrom0.01to0.15M andfrom set is presented here. It is further shown that such an RMF
(cid:12)
0.06 to the maximum stellar mass of 150 M , respectively. providesacorrectiontermtothestellarIMFsinordertofita
(cid:12)
TheslightlylargermassrangescomparedtoEquation(4)are reasonableoverallIMFwithouttheBDdeficitobservedinthe
due to the steep power-law decline added to the mass bor- purely stellar ones. Figure 2 shows the results of our calcu-
derstoreducenumericalartifacts. Themassesaredrawnran- lationsfortheanalyticalmodelfromHC08withtheC05and
domly from each IMF where the relative normalization be- TK07 system IMF as the observational reference (upper and
tween both populations is simply obtained by the number of lower panels, respectively). In both cases the RMF is lim-
objectsdrawnfromeachIMF. ited to the mass range below 0.3 M (log m/M <−0.5)
(cid:12) 10 (cid:12)
BD-like objects are assumed to form as single substel- becausetheslightlydifferentshapesoftheIMFsinthestellar
lar cores some of which are subsequently paired to binaries region are beyond the scope of this paper. The local min-
within a dense dynamically preprocessed environment like ima near log m/M =−0.8 (upper panel) and −1.4 (lower
10 (cid:12)
a massive extended accretion disk (Stamatellos et al. 2007a, panel) occur because the difference between the HC08 mass
Thies et al. 2010, Basu & Vorobyov 2012). Similarly, stel- functionandtheempiricalmassfunctionalmostvanisheslo-
lar binaries are assumed to be assembled from individually cally. This behavior is highly sensitive to the normalization
formed and subsequently paired stars. After the individual- oftheIMFsandismostprominentinthecaseofthediscon-
bodypopulationshavebeendrawnfromtheIMF,twodiffer- tinuousTK07IMF.Peakingnearthehydrogen-burningmass
ent methods are used for assembling the binaries. For stars limittheRMFdeclinessteeplyandeffectivelyvanishesabove
weapplytheDPSpairingmethodmentionedabove. log m/M =−0.5. Thepeakmass,however,isbetterrepre-
10 (cid:12)
BD binaries, on the other hand, are created by first draw- sentedintheresidualtoTK07inFigure2whereastheRMF
ing two objects from their separate population characterized fromC05appearstobeshiftedtowardlowermassesbyabout
by ξ (Equation 4) with the pairing probability p being afactoroftwo(i.e. anoffsetofabout-0.3onthelogarithmic
BD pair
determinedbyapowerlaw, scale).
InthecaseofthePN02analyticalmodel,asshowninFig-
p =constqγ, (8)
pair ure 3, the result with respect to the C05 IMF covers a mass
following the power-law bias scheme used by Goodwin range similar to HC08 but is also continuous. This is ex-
(2013) for the mass ratio distribution of binaries from sec- pectedbecausebothanalyticalmodelshaveafunctionalform
ond (i.e. binary-forming) fragmentation. Whereas the case similartotheC05IMF,namelyanextendedlog-normal-type
γ=0correspondstorandompairingfromtheIMF,γ>0de- shape. In the case of the TK07 system IMF (Figure 3) there
scribesabiasedpairingrulewithanincreasingpairingprob- is a prominent “dip” in the RMF around log10m/M(cid:12) =−1.3
ability toward equal-mass components. The extreme case (i.e. m=0.05M(cid:12)). Thereasonisthenear-equalityofTK07
of perfect equal-mass pairing corresponds to γ = but is andPN02atthispoint,whichvariessensitivelywiththenor-
in disagreement with data from the Very-Low-Mas∞s Binary malizationoftheTK07systemIMFanditssubstellarcompo-
Archive (VLMBA; Burgasser et al. 2007) which also con- nent. AswiththeHC08model,theRMFofPN02versusC05
tains a few unequal substellar binaries. In practice, first an peaksatlowermassesbyafactorofabouttwo.
arrayofBD-likesisgeneratedbydrawingrandomlyfromξ
BD 3.2. ComparisonwithSimulationData
(Equation 4). All subsequent pairings to make binaries are
performedonthearrayonly.Randompairingisperformedby TheRMFobtainedintheprevioussectioniscomparedhere
randomly drawing a companion from the array for each ob- with the mass spectrum of substellar clumps formed in the
jectofthesamepopulation.Ifthecompanionismoremassive SPHmodelsofThiesetal.(2010). Thereintheinducedfrag-
thantheconsideredobjectitbecomestheprimary,andother- mentationofmassiveextendedcircumstellardisksduetoper-
wiseitisthesecondarycomponent.Inthecaseofbiasedpair- turbation by passing stars in an embedded cluster has been
inganadditionaldecisionismadewhetheracompanionran- studied. UsinganSPHcodewitharadiativecoolingapprox-
domlydrawnfromthepopulationisacceptedtobeapartner imation (Stamatellos et al. 2007b) gravitational instabilities
orrejected,dependingontheprobabilityp inEquation(8). have been demonstrated to form through tidal perturbations
pair
Arejectedpartnermaylaterbechosenasapartnertoanother and to form compact clumps with typical masses between
object. Objectsthat, ontheotherhand, arealreadyboundin 0.01 and 0.15 M(cid:12). The mass function of the thus-formed
abinaryareskippedinthepairingprocedurehenceforth. The clumps, the “SPH mass function” (SPH MF) hereafter, has
biased pairing procedure is iterated until the required binary been shown in Thies et al. (2010) to be in agreement with
fraction f isachieved. Becausetheobjectsarechoseninran- the substellar component of the observed system IMF from
domorder,thisdoesnotintroduceanyadditionalbiasbesides TK07. In the current study, the SPH mass function is based
p to the binary populations. The method ensures that the on80objectsformedin29computations,someofthemper-
pair
slopeofthecanonicalIMFisretained,incontrasttomethods formedincontinuationofTK07.
thatselectcomponentsaccordingtoamass-ratiodistribution InFigure4the(scaled)SPHmassfunctioniscomparedto
only. thecorrespondingRMF.TheoverallshapeoftheRMFessen-
tially matches the SPH mass function which features a gen-
3. RESULTS eralincreasewithincreasingmass,apeakaround0.1M(cid:12)and
a rapid decay for higher masses, thus characterizing a pop-
3.1. ResidualMassfunctionforSemianalyticalModels
ulation that is restricted to the substellar and very-low-mass
In this section we present the RMFs obtained from the stellarregime. AsmentionedbyThiesetal.(2010),thesere-
BD/VLMSdeficitoftheanalyticalIMFmodelsbyPN02and sults are quite similar to those in Stamatellos & Whitworth
HC08. TheRMFisderivedforaparticularand,accordingto (2009) who computed BD formation in self-fragmenting
PN02 and HC08, typical parameter set, in particular the tur- disks.Therefore,theresultsfromtriggeredfragmentationcan
5
3 HC08 2 HC08 BD RMF Cha05
C05 system IMF BD RMF TK07
BD residual BD RMF mean
SPH MF
2
1
1
m) m)
(ξL 0 (ξL 0
log10 3 TK07 BsyDs treemsHi dCIMu0aF8l log10 2 PN02 BBBDDD R RRMMMFFF C mThKea0a07n5
SPH MF
2
1
1
0 0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0
log10 m/M log10 m/M
⊙ ⊙
FIG.2.—Upperpanel: theanalyticalIMFmodelforM=12byHC08 FIG.4.— ResidualmassfunctionsofHC08(upperpanel,fromFigure2)
(solidline)comparedtotheempiricalIMFbyChabrier(2005)(dashedline). andPN02(lowerpanel)withrespecttothesystemmassfunctionsbyC05
Thesefunctions,originallydefinedassystemmassfunctions,havebeennor- (dashedline,fromFigure3)andTK07(dottedline)aswellastheaverage
malizedtomatchinthestellarregime.Thedottedlinerepresentstheresidual ofboth(solidline)incomparisontotheSPHmassfunctionfromThiesetal.
massfunction,i.e. thedifferencebetweenbothmassfunctions. Thegapsin (2010,includingpost-populationSPHdata),shownasahistogram(normal-
theRMFsarecausedbytheintersectionoftheHC08modelIMFwiththe izedinlog10ξLtotheRMF).
empiricalIMFs. TheyarehighlysensitivetothenormalizationoftheIMFs.
Lowerpanel: sameasintheupperpanelbutwiththecombinedbimodal
IMFaccordingtoThies&Kroupa(2007)(dashedline). NotethattheRMF
centralstar. Theimpactofthiseffectonthemassdistribution
istruncatedforlog10m/M(cid:12)≥−0.5becausethedifferentfunctionalshapes
ofthestellarpartsarenotconsideredhere. willbesubjecttofutureresearch.
Given the uncertainties of the PN02 and HC08 MFs the
RMF based on them can be considered to be in agreement
3 PN02, Mach = 10 withthesubstellarIMFinbothTK07andThiesetal.(2010).
C05 system IMF Interestingly, the average of both RMFs (each weighted as
BD residual
50%)providesanevenbetterfittotheSPHdata(seethesolid
2 curvesinFigure4). Useofsuchanaveragemaybemotivated
bytheexpectationthatthetrueIMFbelow 0.2M islikely
(cid:12)
≈
betweentheTK07andC05quantifications.
1
m) 3.3. MonteCarloStudyAppliedtoObservations
(ξL 0 TheIMFfromtheMonteCarlostudydescribedinSection
og10 3 TPKN0072 ,s Mysatecmh =IM 1F0 2.3 is shown in Figure 5 in comparison to the system mass
l BD residual function(dash-dottedcurve),theprimary-bodymassfunction
(dashed curve) and the individual-body mass function (solid
2
curve). TheupperpaneldisplaysthesumIMFwithbothpop-
ulationscombined,asitwouldbeseenbyanobserver. Inthe
1 lowerpanel,bothpopulationsareplottedseparately.
We note that Figure 12 in Kirk & Myers (2012), which
shows the observed average mass functions of young stel-
0 lar clusters, groups and isolated objects across the stellar–
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
substellarboundary,revealsaprominentstepnearthestellar–
log m/M
10 substellar border that is very similar to the one that emerges
⊙
FIG.3.—Upperpanel: thesameasinFigure2butwiththeclumpmass from our analysis as plotted in the upper panel of Figure 5.
functionmodelbyPN02insteadofHC08. Lowerpanel:sameasFigure2,
The system MF (dash-dotted curve) is to be compared with
lowerpanel, butwiththeclumpmassfunctionmodelbyPN02insteadof
the mass function by Kirk & Myers (2012). Because Kirk
HC08.Here,theRMFdoesnotcontainanygaps.
& Myers (2012) only studied low-density, dynamically un-
evolvedpopulations,thestellarsystemIMFismodelledhere
withabinaryfractionof80%(seealsofig. 10inMarksetal.
be taken as representative for the general disk fragmentation 2011). FortheGalacticfield, forwhichtherestofourstudy
outcome. Both scenarios assume that the substellar clumps hasbeenperformed,thebinaryfractionisonlyabout40–50%
formedinafragmentingdiskareejectedbyeithermutualdy- forlow-massstars.
namical interactions of clumps in the disk or in subsequent Figure6depictsasthesolidlinethebinarityasafunction
stellar encounters. According to Basu & Vorobyov (2012) oftheprimary-bodymass(i.e. thesystemmassinthecaseof
someoftheseclumpsaredisruptedbythetidalforcesduring single objects) from the DPS pairing model for the Galactic
the ejection or are destroyed because of migration onto the fieldpopulationforembeddedclusterswithahalf-massradius
6
bodies 3.4. BinarityandCompanionMass-ratioDistribution
0 primaries
systems TheobservableBDCMRD,whichistheCMRDforallpri-
KM12
-0.5 maries (BD-like and star-like) below 0.08 M(cid:12), is shown in
Figure 7. As can be seen the case of strongly biased pair-
-1 ing with γ =2 (solid line) matches the measured data from
the VLMBA better (and also see Dieterich et al. 2012) than
-1.5
m) the random-pairing q (γ =0, dotted line) and weakly biased
(ξL -2 pairing models (γ = 1, dashed line). In addition, the case
og 10 0 primboadriieess γ =4 (Duchêne & Kraus 2013; dash-dotted line) is shown.
l systems ThebinarityofBDsnearthehydrogen-burningmasslimitof
-0.5 sum 0.08 M(cid:12) is quite low in the model, only about 11%–12%
whichislargelyduetothe“dip”inthebinarityintheVLMS
-1 region of the DPS pairing method (see Figure 6). This lo-
calminimumisadirectconsequenceofthetwooverlapping
-1.5
populations. Reidetal.(2008)indeedreportasimilarbinary
-2 fractionof12.5+5%forLdwarfs. Ifalocalminimumiscon-
−3
-2 -1.5 -1 -0.5 0 0.5 1 firmedbyfuturesurveyswithsufficientprecisioninmassthis
log10 m/M wouldfurthersupportanoverlapoftwoseparatepopulations
⊙
FIG.5.—Upperpanel:thecombinedIMFofBD-likeandstar-likeobjects inthismassregion. However,inthecurrentdatathisfeature
fromourMonteCarlocalculationsforindividual-bodymasses(solidcurve), iswithintheuncertaintiesoftheobservationaldata.
primary-bodymasses(dashedcurve),andsystemmasses(dash-dottedcurve).
TheBD-likebinarityis20%here,andthestar-likeoneishereabout80%on
3.5. ContributionofPeripheralFragmentationtoBDs
average,tomatchthecaseofdynamicallyunevolvedpopulations(seetext).
For comparison, the histogram for grouped stars in Kirk & Myers (2012, One important implication of the model discussed in this
“KM12”),whichalsoshowsadiscontinuitynearthestellar–substellarborder,
paper is the hybrid nature of BDs and VLMSs. Although
isincludedasthedottedhistogram.Lowerpanel:theIMFsfortheBD-like
andstar-likepopulationsplottedseparately.Thelinepatternsarethesameas thereisaminorcontributionbythestar-likepopulationfrom
intheupperpanelforeachseparateIMF.Forcomparison,theindividual-body directfragmentation, themajorityofBDsarecontributedby
sumIMFissuperimposed(thindottedcurve,thesameasthesolidcurvein theBD-likepopulationthroughdynamicallypreprocessedgas
thetoppanel).Notethatthevery-low-massendofthestar-likeIMFisslightly
(e.g. fragmenting circumstellar disks (Stamatellos & Whit-
depletedduetoeigenevolution(Marksetal.2011).
worth 2009, Thies et al. 2010). Because this preprocessed
materialoftenoccursintheperipheralregionsofstarforma-
tion(e.g. intheouterpartsofdisks)weproposethetermpe-
of 0.1 pc for the underlying cluster population. The long- ripheralfragmentationfortheBD-likeformationchannel. Of
dashedlineindicatesthecaseofahalf-massradiusof0.2pc all BDs between 0.01 and 0.08 M 64% are contributed by
(cid:12)
whichyieldsanupto20%highermultiplicity. Thesemodel theBD-likepopulation,and19%ofMdwarfsbetween0.08
Galacticfieldpopulations,computedwithDPS(Section2.3), and0.45M areBD-like. Thisresult,however,ishighlysen-
(cid:12)
assume that all stars form in binary systems in a population sitivetothechosenlowermassborderofthestar-likeregime.
ofembeddedstarclustersthatevolveanddispersetheirstars Here, we assumed it to be 0.06 M . If, on the other hand,
(cid:12)
intothefield. a sharp truncation of the star-like regime at 0.08 M is cho-
(cid:12)
These two cases are not to be taken as statistically strict sen,theBD-likefractionofBDsis100%butstill18%ofM
confidencelimitsbutratherastwolikelycasesthathavebeen dwarfsareBD-like. Becausestar-likeBDsandVLMSsmay
studied in Marks & Kroupa (2011), and the initial stellar bi- be detectable by larger circumsubstellar disks or, if applica-
narypopulationisindicatedbytheshadedregionatthetop. ble, wide binary separations, future high-resolution observa-
The stellar binary fraction resulting from DPS is about tions will help to further constrain the BD-like and star-like
f =0.4 on average in the low-mass region below 1 M for massborders.
(cid:12)
the case of 0.1 pc. The single-hatched region indicates the
largeuncertaintyoftheobservedbinarityinthesubstellarre- 4. DISCUSSION
gion. For comparison, Galactic-field observational data are In this paper we have introduced the RMF as a correc-
overplottedforAstars(VASTsurvey,DeRosaetal.2014,as- tion term for the analytical star-formation models by PN02
terisks),Gandlate-Fstars(Duquennoy&Mayor1991,filled and HC08 to match the observationally constrained IMF by
squares), M to G stars (Kroupa et al. 2003; open squares), Chabrier (2005) and Thies & Kroupa (2007). The effec-
M dwarfs (Fischer & Marcy 1992; upright triangle), and L tive deficit of BDs and VLMSs in these theoretical models
dwarfs(Reidetal.2008;upside-downtriangle). with respect to the observed BD statistics suggests the re-
Despite consisting of two separate populations (BD-like quirement for additional formation channels in these analy-
and star-like) the binary fraction declines continuously from tical IMF models. HC08 speculate that this deficit, being at
the stellar toward the substellar regime. As discussed in least at the edge of significance (see uncertainties shown in
Marks & Kroupa (2011) and Marks & Kroupa (2012), this Figure 4–23 in Kroupa et al. 2013), may be solvable by an
is primarily due to the dynamical processing of the star-like improvedalgorithmthataccountsfortheeffectsofturbulence
population. In part, the mass overlap of the star-like and the andotherdynamicalprocessingoftheprestellargas. Theex-
BD-like population also contributes to this transition. It re- pectedcontributionoftheseadditionaleffectstotheIMFmay
mainstobeshowninfutureworkthattheapparentobserved be understood as a separate population or, at least, as an ad-
trend toward narrow binary separation distributions for late- ditional formation channel of BDs and VLMSs and can for-
type M dwarfs reported by Janson et al. (2014) can also be mallybedescribedbyanadditionalcorrectionterm. Itisalso
reproducedwithDPSbythisoverlap. applicableasatestforfuturestar-formationmodels.
7
1
initial stellar binary population
0.9
0.8
0.7 dynamical evolution
0.6
stars (r =0.1pc)
f 0.5 h
stars (r =0.2pc)
h
0.4 BDs
De Rosa (2014)
0.3 BD binaries
D+M (1991)
0.2 F+M (1992)
Reid+al (2008)
0.1
Kroupa+al (2003)
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5
log m/M
10
⊙
FIG.6.— BinaryfractionforBDsandstarsasafunctionoftheprimary-objectmass. ThesolidlineshowstheresultofourMonteCarlocomputationfor
starsviaDPSpairingforthefieldpopulationbasedontheintegrationofthestar-formationoutcomeoverallclustermasses. Ahalf-massradiusrh=0.1pcis
assumedforallembeddedhostclusters.Theaveragefield-starbinaryfractionis f≈0.5forlow-masstointermediate-massstars,inaccordancewithTK07.The
long-dashedlinereferstotheDPSmodelwithrh=0.2pc,correspondingtoahigherbinaryfraction.Theshort-dashedcurverepresentstheBD-likepopulation
withbiasedpairing. Thesingle-hatchedregioninthelowerleftcornerindicatestheuncertaintyofthesubstellarbinaryfractionintheobservationaldata. The
smoothlyshadedareaatthetopreferstotheinitialstellarbinarypopulationwithabinaryfractionofnear100%(Marks&Kroupa2011).Observationalfield-star
dataareoverplottedforAstars(VASTsurvey,DeRosaetal.2014,asterisks),Gandlate-Fstars(Duquennoy&Mayor1991,filledsquares),MtoGstars(Kroupa
etal.2003;opensquares),Mdwarfs(Fischer&Marcy1992;uprighttriangle),andLdwarfs(Reidetal.2008;upside-downtriangle).
5 1
BDs, biased (γ=4)
Kroupa/TK07
BDs, biased (γ=2)
4 BDs, biased (γ=1) Chabrier
BDs, random pairing 0 Bochanski
3 VLMB Archive m)
fq() 2 (ξ10L -1
g
o
l
1 -2 valid range
of Bochanski
0
0 0.2 0.4 0.6 0.8 1 -3
q = m /m -2 -1 0 1
comp prim
log m/M
FIG.7.— Number distribution f(q) for binary BDs between 0.03 and 10 ⊙
0.08M(cid:12) asafunctionofthecompanion-to-primarymassratio. Thesolid FIG.8.— Comparison of the Thies & Kroupa (2007), Chabrier (2005),
lineshowstheresultofourcomputationforγ=2,andthedashedanddotted andtheextrapolatedBochanskietal.(2010)IMF,thelatterbeingusedby
linesrefertothecaseγ=1andrandompairing,respectively.Forcomplete- Reggiani&Meyer(2011). TheoriginalBochanskiIMFhasbeenderived
ness,alsothemoreextremecaseofγ=4(Duchêne&Kraus2013)isalso fromdatabetween0.1and0.8M(cid:12).NotethesteeperdeclineintheBochanski
shownasthedash-dottedline. Thehistogramshowstheobservationaldata IMFbelowitsvalidmassrangerelativetotheChabrierIMF.
fromtheVery-Low-MassBinaryArchive(Burgasseretal.2007). Thepeak
atq=1isprimarilyduetothecontributionofbinariesfromthelow-massend
ofthestar-likepopulationbecauseanybinarywithbothcomponentsnearthe
lowermassborderofapopulationcanonlyhavenearlyequalcomponent
tationprocessinmolecularclouds. Thesemodelscannot,by
masses.Theareabeloweachcurveandbelowthehistogramisnormalizedto
1. theirnature,accommodatetheperipheralfragmentation,e.g.,
inaccretiondisksaroundprotostars,whichyieldstheBD-like
population(seeSection3.5). Themostnaturalwaytoexplain
Thiscorrectiontermhasbeenidentifiedandquantifiedhere these results is, to the best of our knowledge, a composite
astheRMFwhichhasasimilarshapeforbothanalyzedobser- populationconsistingofastar-likeandaBD-likecomponent
vationalreferenceIMFsbyC05andTK07. Themoststriking suchthatasignificantfractionoftheBDpartcomesfroman
outcomeisageneralagreementbetweentheRMF(Equation additional process acting during star formation (Reipurth &
(1)) as obtained from the theoretical IMF and the observa- Clarke2001,Stamatellosetal.2007a,Thiesetal.2010,Basu
tional IMFs, and the mass function of fragments from SPH & Vorobyov 2012). Essentially this result follows for BDs
simulations (Thies et al. 2010). In other words, both PN02 beinglesslikelytoformdirectlyinastar-formingmolecular
andHC08IMFmodelsappeartodescribeessentiallythepop- cloud than from dynamically preprocessed material. This is
ulationofstarswithoutasignificantfractionofBDs.Theycan becauseinordertoformonlyaBDacloudcoreneedstohave
beconsideredtobesuccessfulmodelsofthedirectfragmen- ahighpre-collapsedensity(forittobegravitationallyunsta-
8
surveybyDieterichetal.(2012)furthersupportstheexistence
ofatrueBDdesert,i.e. adearthofsubstellarcompanionsto
primary stars ranging from M to G dwarfs. They found an
effectivelowermasslimitofstellarcompanionsnear0.1M
(cid:12)
which cannot be related to incompleteness but rather points
toward a discontinuity in the pairing statistics close to the
hydrogen-burningmasslimit. Theyparticularlypointoutthat
this discontinuity is essentially independent of the primary
starmass. Thisclearlysupportsaseparatesubstellarpopula-
tioncontributingthemajorityofBDsandisalsoinagreement
withSPHstudiesofwholestar-formingcloudsby,e.g.,Bate
et al. (2003) and Bate (2009a) who reproduce the formation
FIG.9.—SnapshotfromSPHmodelX002byThiesetal.(2010)showing of BDs largely from fragmenting circumstellar disks. They,
substellar clumps (with sink particles marked with white points) around a however,pointoutthatthesesimulationstendtooverproduce
fragmented accretion disk, all of them surrounded by their own accretion
BDsandattributethistothelackofradiativeheatingintheir
envelopes. Notethattheescapingobject(0.013M(cid:12),i.e. alow-massBD)
totherighthasretaineditsaccretiondisk(visibleasdiffusestructurearound model(Bate2009b,2012). ItshouldalsobenotedthatBate
thesinkparticle)evenafterdynamicalejection. Thehalf-massradiusofthe (2012) could reproduce a mass function in good agreement
envelopeisabout10AU,anditsmassis2·10−4M(cid:12),i.e. almost2%ofthe with the C05 IMF and that the previous overproduction of
masoftheescapingobject. BDswasavoidedtherebecausetheradiativeheatingreduced
thefrequencyofdiskfragmentation.
In a related Monte Carlo study performed here, we also
ble) while having no further supply of gas in order to limit
found a good agreement of the two-population composite
itsfurthergrowthinmass. Thishasalreadybeenemphasized
model with observational data on field very-low-mass stel-
by Adams & Fatuzzo (1996). While there has been a recent
lar and BD binaries (Figure 6). The binarity is well repro-
discoveryofapossibleproto-BDinOphB-11withamassof
duced down to the stellar–substellar border as a continuous
0.03M (Andréetal.2012)itsfurtherevolutionandpossible
(cid:12) function of the primary-star mass in agreement with the ob-
additional accretion from the surrounding molecular gas re-
servations. More importantly, the mass-ratio distribution of
mainsunclear. And, theexistenceoftheBD-likepopulation
very-low-mass binaries deduced from observational data is
formedthroughperipheralfragmentationdoesnotexcludethe
wellreproducedinourmodel(Figure7),oncemoresupport-
formationofBDsviathedirectfragmentationchannel,i.e. as
ingabimodalstarandBDformationscenario. Whereasstar-
star-like objects as is evident from the analytical PN02 and
likebinariesarewellrepresentedbyDPS(Marksetal.2015),
HC08IMFs.
theBD-likebinariesapparentlydofitabiasedpairingwhere
theprobabilityofpairingrisestowardmoreequalcomponent
4.1. IsThereaSeparateSubstellarPopulation?
masses following Equation (8). Here, the cases γ =0 (sim-
There is an ongoing discussion whether a separate popu- plerandompairing)and1 γ 2(biasedtowardequal-mass
lation is really needed to explain the observed properties. A ≤ ≤
binaries) were compared. The biased pairing case is in bet-
similar discontinuity between planets and BDs is widely ac-
ter agreement with the observational data. It remains to be
cepted. Chabrier et al. (2014) even report a possible mass
studiedwhetherpostformationdynamicalprocessingthrough
overlap of the BD and the giant planet regime as well and
stellar-dynamicalencountersintheirbirthembeddedclusters
suggest a distinction between BDs and planets by their for-
isresponsibleforthisapparentpreferenceofmoreequal-mass
mation history rather than by deuterium burning. This can
binariesovermoreunequalonesintheBDmassrange. How-
beseenasananalogtotheseparatesubstellarpopulationad-
ever,bothsimpleandbiasedpairingareinreasonableagree-
dressedbyourwork.Jumper&Fisher(2013)claimtobeable
mentwithintheuncertaintiesofthedataifalowbinaryfrac-
tomodelthetrendtowardmoretightlyboundbinariesinthe
tion of 10% within the BD-like population is assumed. A
BDregime. However,becausethisstudyalreadyassumesthe
binaryfractionoflessthan10%inthetheoreticaldiskfrag-
massestobedrawnfromacontinuousIMFitaprioriexcludes
mentation outcome and the very narrow range of semimajor
theoptionofatwo-componentIMF.TheAstraLuxsurveyre-
axes have also been found by Thies et al. (2010), as well as
ports a narrower binary separation distribution for M-dwarf
the survival of circumsubstellar accretion disks, as shown in
binaries compared to binaries with FGK primaries (Janson
Figure9. Theoverallbinaryfractionforsystemswithin0.03
etal.2012andJansonetal.2014,respectively). Suchanar- and 0.08 M , irrespective of the population to which they
(cid:12)
rowpeakintheseparationdistributionmayberelatedtosen-
belong, is around 20 % for both simple and biased pairing
sitivityartifactsaswellas,atleastforlate-typeMdwarfs,due
(i.e. thevaluechosenfortheBD-likepopulation),ifthelocal
tothemassoverlapwiththeBD-likepopulationthatextends
minimum in the binarity function near the stellar–substellar
uptoabout0.2M .
(cid:12) boundary(Figure6)isassumedtobeanartifact,whereasitis
Incontrast,inastudyofdynamicalbinaryevolutionParker
aslowas11%otherwise. However,observationsofVLMSs
& Goodwin (2011) found that the field populations of M
indicateabinaryfractionaslowas12.5%(Reidetal.2008),
dwarfsandvery-low-massbinariesmustreflectverydifferent
so this “dip” may indeed be real and thus be a result of the
birth populations, since their dynamical processing is essen-
overlapoftwoseparatepopulations.
tiallythesame.
4.3. CompanionMassRatioDistribution
4.2. TheBrownDwarfDesert
There is an ongoing discussion whether random pairing is
Similarly,theBDdesertitselfhasbeendisputedbyseveral applicable for stars. Reggiani & Meyer (2011) claim to rule
groups who suggest that it is more related to the companion outthismodelinfavorofauniversalCMRD.However,their
massratioratherthantotheabsolutemass. However,arecent resultsonlycoveranarrowrangeofvalidityoftheunderlying
9
model IMF (as specified by Bochanski et al. 2010, see Fig- sary to treat BDs separately which implies a separate albeit
ure 8) that only covers M, K and late-G stars, but no BDs. related formation channel for the majority of BDs. This al-
Consequently,theCMRDderivedfromitisquitelimited,es- ready follows from the work of Adams & Fatuzzo (1996).
pecially for M dwarfs, for which no hypothesis test on the ThetheoreticalevidencebyStamatellos&Whitworth(2009),
pairing rule can be made on this basis. Reggiani & Meyer Thies et al. (2010), and Basu & Vorobyov (2012) supports a
(2011) perform pairing experiments that actually exceed the separatepopulationbyfragmentationofextendedyoungcir-
rangeofvalidityoftheunderlyingmassfunctionbyBochan- cumstellar disks. If O stars are close by, the accretion enve-
skietal.(2010),andthereforetheirresultsarenotapplicable. lopeofVLMprotostarsmaybephotoevaporated, leavingan
Goodwin(2013)suggeststhatarandompartitionofprotostel- unfinished substellar embryo (Kroupa & Bouvier 2003). In
lar cloud fragments is in better agreement with observations addition, the embryo-ejection model by Reipurth & Clarke
ofstellarbinaritythananinitialbinarypopulationdrawnfrom (2001)givesanexampleofBDformationbyejectionofun-
theIMFwithsubsequentdynamicalandeigenevolution. Be- finished stellar embryos out of multiple protostar systems.
cause that study was based on the IMF by Chabrier (2003) Because these mechanisms are not covered by the analytical
without a separate BD treatment there cannot by any satis- cloudfragmentationmodelsbyPN02and HC08theydonot
factoryagreementwiththeobservedsubstellarCMRD.Even contribute to the resulting theoretical clump mass function.
more importantly, the initial binary population needs to be ThesameistrueforBDformationindensegaseousfilaments
modeledandtestedagainstdynamicallybarelyevolvedstellar that form due to the gravitational pull of surrounding stars
populationslikeTaurus-AurigaratherthanagainsttheGalac- (Bonnell et al. 2008) rather than being described by random
ticfield. Thefieldpopulationthenresultsfromanintegration density fluctuations assumed by PN02 and C05. Therefore,
of all dynamically evolved clustered populations following ananalyticalmodelalsocoveringBDsandVLMSsmustin-
theembeddedcluster-massfunction(Marks&Kroupa2011, cludesuchseparateformationchannelsviadynamicallypre-
Marksetal.2015). processed material (disks, unstable multiple stellar embryo
As a final remark, it is being argued that the existence of systems, photoevaporation, tidally shaped gas filaments and
disksaroundyoungBDsimpliesthattheseformlikestarssup- soon). Thenormalizationmaybeadjustedempiricallyorde-
posedlysupportingthecontinuousIMFscenario. Figure9is ducedfromphysicalrelationsbasedontypicaldisk-to-stellar
one of many examples where a young BD with an accretion mass relations and by the application of turbulence and the
diskformedfromperipheralfragmentationinacircumstellar Jeanscriterionwithinthispreprocessedmaterial. Thedevel-
diskisnudgedawaytobecomeafree-floatingBDwithadisk. opmentofsuchananalyticalorsemianalyticalmodelshould
be the subject of future studies. The observational data thus
5. CONCLUSION verystronglysuggestthatthedominantfraction(Section3.5)
ofBDsandVLMSsresultsfromperipheralfragmentationthat
Asthemainconclusionfromthefirstpartofthisstudywe
contributesaseparateIMFfromthatofstars.
emphasize the difficulties of theoretical star-formation mod-
ThisprojecthasbeenfundedbyDFGprojectKR1635/27.
els to describe both stars and BDs by a single mechanism,
namely from direct cloud fragmentation. It remains neces-
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