The Origin of the IMF from Core Mass Functions
a r X i v :a s t r o -p h /0407106v 1 6 J u l 2004The Origin of the IMF from Core Mass Functions
S.P.Goodwin,A.P.Whitworth &D.Ward-Thompson Department of Physics &Astronomy,Cardi?University 5The Parade,Cardi?,CF243YB,UK We examine the initial mass functions (IMFs)of stars produced by di?erent molecular core mass functions.Simulations suggest that more massive cores produce more stars,so we propose a model in which the average number of stars formed in a core is equal to the initial number of Jeans masses in that core.Small-N systems decay through dynamical interactions,ejecting low-mass stars and brown dwarfs which populate the low-mass tail of the IMF.Stars which remain in cores are able to competitively accrete more gas and become more massive.We deduce the forms of the core mass functions required to explain the IMFs of Taurus,Orion,IC 348and NGC 2547.These core mass functions fall into two categories -one which peaks at a few M ⊙to explain Taurus and NGC 2547,and one that peaks at around 0.2M ⊙to explain
Orion and IC 348.
Keywords :Stars -formation,Stars -mass function
1Introduction
All stars form in dense molecular cores (eg.Andr′e et al.2000).Observations of the densest cores,known as prestellar cores (Ward-Thompson et al.1994),show that their mass functions are remarkably similar to the IMF of ?eld stars (Motte et al.1998;Testi &Sargent 1998;Motte et al.2001).This suggests that the form of the IMF may be directly related to the form of the core mass function (CMF).
Most stars >1M ⊙exist in binary or multiple systems (eg.Duquennoy &Mayor 1991).Most of these multiple systems must form as such,since it has been shown that dynamical evolution is unable to signi?cantly alter the initial binary properties or population (Kroupa 1995).Therefore many cores must produce multiple objects and so the IMF cannot be a simple mapping of the CMF.
In a previous paper (Goodwin et al.2004c)we showed that the IMF of Taurus could be explained if all of the stars in Taurus formed from cores of a few solar masses,a CMF similar to
Figure1:The average number of objects
initial number of Jeans masses in the core.
that observed by Onishi et al.(2002).In this contribution we investigate the e?ect of changing the CMF on the IMF and multiplicity of star forming regions.
2Multiple star formation in cores
Recent studies have shown that within massive(>5M⊙)turbulent cores,multiple star formation is the norm(Bate et al.2002,2003;Delgado Donate et al.2004;Goodwin et al.2004a,b).A signi?cant population of low-mass stars and brown dwarfs is formed by ejections from unstable multiple systems in these cores.
Delgado Donate et al.(2003)modelled the origin of the IMF by assuming that fragmentation in cores is scale-free:ie.that all cores produce the same number of objects(stars and brown dwarfs)and that the masses of these objects scale with the mass of the core.By convolving the outcome of star formation in a1M⊙core with a core mass function(CMF)they obtained an IMF.However,Goodwin et al.(in preparation)?nd that the number of objects that form depends strongly upon the mass of a core,with low-mass cores being far less able to form multiple objects than more massive cores.Fig.1shows the average number of objects that form in cores of di?erent masses but with the same initial thermal and turbulent virial ratios(the ratio of the initial thermal or turbulent energy to the initial potential energy).The average number of objects that form is approximately one per initial Jeans mass(～1M⊙).
Given that most stars form in multiple systems and the number of stars forming in a core might be expected to increase with the mass of the core,we propose a simple model for the formation of stars within cores and the relationship of stellar masses and multiplicities to the core mass(the core-to-star relationship):
?Cores form an average number of objects(stars and/or brown dwarfs)approximately equal to the initial number of Jeans masses in the core(eg.Goodwin et al.in prep).
?Multiple systems with≥3members are initial unstable and will decay to a stable sys-tem within a few×104yrs through the ejection of low-mass stars and brown dwarfs(cf.
Reipurth&Clarke2001;Bate et al.2002;Sterzik&Durisen2003;Goodwin et al.2004a).
The initial mass function(IMF)is then due to the convolution of the core mass function (CMF)and the core-to-star relationships in these di?erent cores.
Figure2:The IMFs of Taurus(histogram,from Luhman et al.2003a)and NGC2547(points,from Je?ries et al. 2004).NGC2547has been normalised to contain the same total number of stars as Taurus for ease of comparison.
We assume a form for the CMF-in this case a log-normal which may have di?erent variances above and below the mean-and randomly sample cores from that CMF.A core then produces N?objects where N?is drawn from a gaussian of mean M core(where M core is the core mass in solar masses)withσ=2.N?is then rounded to the nearest integer≥1.
If N?≤3then N?stars are formed of mean mass?M core/N?(we assume that the core-to-star e?ciency?=0.75in all cases).
If N?>3then N??3stars are ejected with masses drawn uniformly from a logarithmic distribution between0.02M⊙and0.1?M core.The remaining three stars then distribute the rest of the mass in the core between themselves such that their individual masses are?(M core?M ej)/3 (where M ej is the mass of ejected stars).
3Results
3.1The IMFs of Taurus and NGC2547
Both Taurus(Luhman et al.2003a)and NGC2547(Je?ries et al.2004)have similar MFs. Both of these MFs show a signi?cant peak at～1M⊙,with a rapid drop above this peak,and a rather?atter decline into the brown dwarf regime,as illustrated in Fig.2(where the MF of NGC2547has been normalised to have the same total number of stars as Taurus for ease of comparison).The similarity between the two MFs is clear.
log M core=0.5 Fig3shows the results of applying our model to a log-normal CMF of meanˉ
andσlog M
=0.1(illustrated by the dashed-line in Fig3)which is a reasonable approximation core
to the CMF of Taurus as observed by Onishi et al.(2001).The hashed histogram is the observed IMF of Taurus(Luhman et al.2003a)and it compares well to the open histogram given by our model.The open circles show the contribution to the IMF from ejected stars and brown dwarfs.The binary fraction is very high in our model as the vast majority of stars have formed in multiple systems,only the ejected component has a low multiplicity.This again compares well with the high observed multiplicity in Taurus(Duch?e ne1999).
This agrees well with the results of Goodwin et al.(2004c),the IMF is a combination of a peak of bound systems with average stellar mass≈1M⊙which remain bound in the cores,and a?at low-mass tail of ejected brown dwarfs and low-mass stars.
Figure3:The open histogram shows the IMF resulting from the CMF shown by the dashed line.Open circles show the contribution to each bin of ejected stars and brown dwarfs.The hashed histogram shows the Luhman et al.(2003a)Taurus IMF.Both histograms are normalised to contain the same number of stars.
3.2The IMFs of Orion and IC348
Orion has an IMF that is very similar to the?eld(Muench et al.2002).Fig.4shows the?t to
=0.3(lower)and=0.7
log M core=?0.8andσlog M
the Orion IMF given by a CMF of meanˉ
core
(upper).
Figure4reproduces the IMF of Orion well with a wide,?at peak between0.1and0.6M⊙, falling at both ends,with an approximately Salpeter slope at high-masses.Fig.5shows the binary fraction as a function of primary mass.This model?ts the observed?eld binary fractions quite well,except at lower masses where it is assumed that all ejected stars and brown dwarfs are single(which is not always the case,a low fraction of ejected stars are multiples,see Goodwin et al.2004b).
IC348has an IMF that is very similar to Orion except that it is relatively de?cient in brown dwarfs(Luhman et al.2003b).Fig.6shows the?t to the IMF of IC348using a CMF of mean ˉ
=0.1(lower)and=0.7(upper).This is almost identical to the log M core=?0.8andσlog M
core
CMF used to model Orion,but the lower extent of the CMF is far smaller(0.1compared to 0.3in the Orion CMF).Almost no brown dwarfs are formed in cores in IC348,they are all the result of ejections from higher-mass cores.
4Conclusions
Using a simple model of fragmentation in cores we are able to match the IMFs of Taurus,NGC 2547,Orion and IC348with di?erent core mass functions.
The IMFs of Taurus and NGC2547are well-?tted with a CMF that peaks at a few solar masses,which matches the observed CMF of Taurus(Onishi et al.2002).This CMF reproduces the peaks in these IMFs at～1M⊙.Most solar-type stars are formed in multiple systems, explaining the very high observed binary fraction in Taurus(Duch?e ne1999).
To?t the IMFs of Orion and IC348requires CMFs that peak at only a few tenths of a solar mass.The lack of brown dwarfs in IC348as compared to Orion can be explained by a CMF that does not extend as far into the brown dwarf regime in IC348.The binary fractions in Orion and IC348are close to those observed in the?eld.
Figure4:The IMF of Orion(solid line,from Muench et al.2002)is well-?tted by the open histogram produced by the CMF shown by the dashed-line.As in?g.3,the circles show the contribution to the IMF from ejected
stars.
Figure5:The binary fraction of stars in the model of Orion as a function of primary mass(Open circles).The error bars show the observations of the?eld binary fraction adapted from Sterzik&Durisen(2003).
Figure6:The IMF of IC348(hashed histogram,from Luhman et al.2003b)is well-?tted by the open histogram produced by the CMF shown by the dashed-line.As in?g.3,the circles show the contribution to the IMF from
ejected stars.
Acknowledgements
SPG is a UK Astrophysical Fluids Facility(UKAFF)Fellow.
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