The Effect of Magnetic Fields and Ambipolar Diffusion on Core Mass Functions

0.91 MB·English

by  Nicole D. Bailey

Draftversion January31,2013
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
THE EFFECT OF MAGNETIC FIELDS AND AMBIPOLAR DIFFUSION ON CORE MASS FUNCTIONS
Nicole D. Bailey and Shantanu Basu
Department ofPhysicsandAstronomy,UniversityofWesternOntario
1151RichmondStreet, London,Ontario,N6A3K7
Draft version January 31, 2013
ABSTRACT
Linear analysis of the formation of protostellar cores in planar magnetic interstellar clouds yields
3
information about length scales involved in star formation. Combining these length scales with var-
1
ious distributions of other environmental variables, (i.e., column density and mass-to-flux ratio) and
0
applyingMonteCarlomethodsallowustoproducesyntheticcoremassfunctions(CMFs)fordifferent
2
environmentalconditions. OuranalysisshowsthattheshapeoftheCMFisdirectlydependentonthe
n physical conditions of the cloud. Specifically, magnetic fields act to broaden the mass function and
a
develop a high-mass tail while ambipolar diffusion will truncate this high-mass tail. In addition, we
J
analyzethe effectofsmallnumberstatisticsonthe shapeandhigh-massslopeofthesynthetic CMFs.

0 We find that observed core mass functions are severely statistically limited, which has a profound
3 effect on the derived slope for the high-mass tail.

Subjectheadings: diffusion–ISM:clouds–stars: formation–stars: luminosityfunction,massfunction
]
R – ISM: magnetic fields – ISM: structure
S
h. 1. INTRODUCTION formingregionsarethe sameand undergothe same pro-
p cess to form stars, however recent observations and sim-
Observations of the stellar initial mass func-
- ulationshavestartedtorevealcracksinthisassumption.
tion (IMF) and the core mass function (CMF)
o
In a study of the effect of turbulence on the formation
r show similarities in the shape and high mass
t slope of these two functions (Motte et al. 1998; of the CMF, Hennebelle & Chabrier (2008, 2009) find
s that comparisons between their IMF and observations
a Testi & Sargent1998;Johnstone et al.2000;Alv´es et al.
fordifferentcloudconditionssuggestthatstarformation
[ 2007; Nutter & Ward-Thompson 2007; Simpson et al.
should predominantly occur in clouds five times denser
2008;Enoch et al.2008;Sadavoy et al.2010,amongoth-
1 than characterized by Larson (1981). This led them to
ers). As such, much theoretical effort has been in-
v question the universality of the IMF since, as they say,
vested in order to explain these similarities. Vari-
0 choosing different cloud parameters would lead to a dif-
ous different approaches to this problem have been
0 ferentCMF/IMF. Severalrecentstudies ofthe IMF also
explored, including analytic and numerical studies
3 tendtodisagreewiththeassumeduniversality. Observa-
which invoke gravitational fragmentation or accretion
7
tionsofdifferentstarclustersinboththeMilkyWayand
. (Silk 1995; Inutsuka 2001; Basu & Jones 2004), tur-
1 bulence (Padoan et al. 1997; Padoan & Nordlund 2002; the Large Magellanic Cloud (LMC) show a wide scatter
0 of slopes: α = 0.5 −2.0 (Elmegreen 1999). A survey
Ballesteros-Paredeset al. 2006; Hennebelle & Chabrier
3 of high mass slope values for different stars (i.e., clus-
2008, 2009), independent stochastic processes (Larson
1 terstarsversusassociationstarsversusfieldstars)yields
1973; Elmegreen 1997) and magnetic fields (Dib et al.
: a wide range of values; α = 2.0−4.0 for extreme field
v 2008), among others. Results of these studies vary from
starsto α=1.5−2.0forcluster stars(Elmegreen1997).
i thosewhichseemtoagreewiththefiducialSalpeterform,
X dN/dlogM ∝ M−α where α = 1.35 is the value of the Further to this, Elmegreen (1999) shows that through
stochastic fractal sampling of a cloud, the derived IMF
r Salpeter slope, to those that do not.
a slopes can vary from α as low as 1.0 to as high as 1.7.
The high mass slope of the IMF was initially derived
Clark et al. (2007) note that if the lifetime of a more
by Salpeter (1955) and later improved upon by Kroupa
massive core is longer than a less massive one, the slope
(2002) and Chabrier (2003a,b, 2005). Despite variations
of the CMF should be shallower in order to obtain the
in observed and theoretically derived IMF slope values,
IMF.Finally,Zaritsky et al.(2012)showthattheremay
it is often assumed that the shape and high mass slope
be evidence for twodistinct stellar IMFs thatdepend on
of the IMF and CMF are identical and universal. From
the age and metallicity of the cluster in question. Based
a theoreticalview, sucha one-to-onecorrespondencebe-
ontheaboveevidenceandarguments,itisnotclearwhy
tween these two functions implies that high-mass cores
oneshouldinsistonusingα=1.35astheuniversalslope
beget high-mass stars and likewise for low-mass cores.
for both the CMF and IMF.
The need for extensive simulations of how a complex of
The majority of the work in this area has focused on
cores turns into a cluster of stars is simplified tremen-
the effects of turbulence within the molecular clouds on
dously if it is assumedthat eachcore will collapse into a
the formation and shape of the CMF. Research which
single star with some mass loss to account for the mass
considers the effect of magnetic fields and ambipolar
shift between the CMF and IMF.
diffusion on the CMF is sparse. Kunz & Mouschovias
The underlying tenet of universality is that all star-
(2009) used the results of a non-ideal MHD linear
analysis of a partially ionized sheet (Morton 1991;
[email protected](NDB);[email protected](SB)
2 Bailey & Basu
Ciolek & Basu 2006) to generate a broad CMF, assum- The key ingredient to this analysis is the assumed
ing ambipolar-diffusion initiated core formation. Their length scale for the core. This length scale for collapse
model assumed subcritical to critical initial conditions can be derived through linear analysis. The nonaxisym-
with a uniform distribution of mass-to-flux ratios be- metricequationsofCiolek & Basu(2006)andBasu et al.
tween 0.1 and 1.0 times the critical value for gravita- (2009a,b) include the effect of ambipolar diffusion. This
tional instability (see Section 5 for more discussion of is quantified by the timescale for collisions between ions
their model). bound to the magnetic field and free neutral particles.
Inthisstudy,weusetheresultsofthelinearanalysisof This timescale is
apartiallyionizedsheetalongwithalognormaldistribu-
m +m 1
tionofinitialcolumndensityandvariousdistributionsof τ =1.4 i H2 . (1)
ni
(cid:18) m (cid:19)n hσwi
mass-to-flux ratio. We explore both subcritical and su- i i iH2
percritical initial conditions. Mildly supercritical initial Here, m is the ion mass, n is the number density of
i i
conditionsarethemostlikelytoleadtomassivecorefor- ions and hσwi is the neutral-ion collision rate. The
mation,asseenine.g.,Figure2ofCiolek & Basu(2006). typical atomiciaHn2d molecular species within a molecular
Furthermore, we use a lognormaldistribution of column cloud are singly ionized Na, Mg and HCO which have
densities, as expected in molecular clouds on both the- a mass of 25 amu. Assuming collisions between H and
2
oretical grounds for a turbulent medium (Padoan et al. HCO+, the neutral-ion collision rate is 1.69×10−9 cm3
1997) and from observations (Kainulainen et al. 2009). s−1 (McDaniel & Mason 1973). Collisions between neu-
The aim of this paper is two fold. In the first part trals and ions transfer information about the magnetic
we show the effects of a magnetic field on the shape of field to the neutral particles. The threshold for whether
theCMF.Startingfromanassumptionoflognormalcol- aregionofamolecularcloudisstableorunstableto col-
umn density probability we show the broadening effect lapseisgivenbythemass-to-fluxratioofthebackground
of neutral-ion drift via ambipolar diffusion and differing reference state
mass-to-flux ratio distributions. In the second part, we µ ≡2πG1/2σn,0, (2)
address the inherent limitations of observed core mass 0 B
ref
functions, i.e. sample size and bin size. Specifically, we
where (2πG1/2)−1 is the critical mass-to-flux ratio for
aim to compare small sample synthetic CMFs to large
gravitational collapse in the adopted model and B is
sample synthetic CMFs to show effect of small number ref
the magnetic field strength of the reference state. Re-
statisticsontheobservedfeaturesoftheCMF.InSection
gionswithµ <1aredefinedassubcritical,regionswith
2weoutlineourmodelandmethodsforconstructingour 0
µ > 1 are defined to be supercritical and regions with
synthetic CMFs. Section 3 shows the results for the dif- 0
µ ∼1 are transcritical.
ferent distribution models considered. Section 4 shows 0
A dispersion relation for the governing magnetohy-
the effect of small number statistics and the variance in
drodynamic equations can be found via linear analysis
derivedanalyticslopes. FinallySections5and6giveour
(Ciolek & Basu 2006; Basu et al. 2009b; Bailey & Basu
discussion and conclusions.
2012) . Here we follow the analysis as described in
2. SYNTHETICCOREMASSFUNCTIONS Bailey & Basu (2012). For a model with ambipolar dif-
fusion, the resulting dispersion relation is
To better understand the effects of the environment
on the shape and peak of the core mass function, we
produce synthetic CMFs (synCMFs) based upon vary- (ω+iθ)(ω2−Ce2ff,0k2+2πGσn,0k)
ing physics and properties of molecular clouds. These =ω(2πGσ kµ−2+k2V2 ) (3)
include the column density (σ ), ionization fraction n,0 0 A,0
n,0
(χi = log[ne/nH]), mass-to-flux ratio (µ0), and neutral where
ion-collision time (τni). The synCMFs are produced by θ =τni,0(2πGσn,0kµ−02+k2VA2,0). (4)
randomlysamplingpredefinedcolumndensityandmass-
to-fluxratiodistributions(whereapplicable)andusinga Here, ω is the angular frequency of the perturbations,
preferredfragmentationlengthscaletocalculatethecore τni,0 is the initial neutral-ion collision time, k is the
mass. Wechoosetousesuchmethodsduetotherandom wavenumberinthe z-direction,VA,0 is the Alfv´enspeed,
nature of molecular cloud properties. This allows us to where
B2
statistically determine the shape of the CMF for a wide V2 ≡ ref =2πGσ µ−2Z , (5)
range of randomly chosen σ −µ pairs. A,0 4πρ n,0 0 0
n 0 n,0
Z is the initial half-thickness of the sheet, and C is
2.1. Physical Model 0 eff,0
the local effective sound speed, such that
We consider the formation of cores and the resulting
CMF within ionized, isothermal, interstellar molecular C2 = πGσ2 [3Pext+(π/2)Gσn2,0]c2. (6)
clouds. These clouds are modelled as planar sheets with eff,0 2 n,0[P +(π/2)Gσ2 ]2 s
ext n,0
infiniteextentinthex-andy-directionsandalocalver-
tical half thickness Z. The nonaxisymmetric equations Here, cs = (kBT/mn)1/2 is the isothermal sound speed,
and formulations of our assumed model have been de- kB is the Boltzmann constant, T is the temperature in
scribed in detail in several papers (Ciolek & Basu 2006; Kelvins and mn is the mean mass of a neutral particle
Basu et al.2009a,b; Bailey & Basu2012). For this work (mn = 2.33 amu). For this analysis, we assume a tem-
weconsiderthreemodels: nonmagnetic,flux-frozenmag- perature T = 10 K and a normalized external pressure
netic fieldandamagnetic fieldwith ambipolardiffusion. P˜ ≡2P /πGσ2 =0.1.
ext ext n,0
Effect of Magnetic Fields on the CMF 3
Figure 1. Wavelength withminimumgrowth timeas afunction
of initialmass-to-flux ratio. Displayedcurves are forτni,0/t0 =0 Figure 2. Modellognormalcolumndensitydistribution.
(solidcurve,fluxfreezing)andτni,0/t0=0.2(dotted curve).
The addition of ambipolar diffusion complicates the
In the limit of flux freezing, τ → 0, which gives the process somewhat. In these cases, the gravitationally
ni,0
reduced dispersion relation unstablemodecorrespondstooneofthe rootsofthefull
dispersion relation (Equation 3). However since it is a
ω2+2πGσ k(1−µ−2)−k2(C2 +V2 )=0. (7)
n,0 0 eff,0 A,0 cubic function, there is no simple expression to describe
theseroots. Therefore,eachlengthscaleiscomputednu-
Thegravitationallyunstablemode correspondstooneof
the roots of ω2 < 0 and occurs for µ > 1. The growth merically. The value of this length scaleis relatedto the
0 degree of ambipolar diffusion i.e., the degree of ioniza-
time for this mode can be written as
tion within the cloud, and the mass-to-flux ratio of the
λ
region. Previous studies show that the ionization frac-
τ = (8)
g 2π[Gσ (1−µ−2)(λ−λ )]1/2 tion within a molecular cloud resembles a step function
n,0 0 MS
(Ruffle et al. 1998; Bailey & Basu 2012) such that the
for λ≥λ , where
MS outer layers are highly ionized due to UV photoioniza-
C2 +V2 tion while ionization of denser inner regions is primarily
λ = eff,0 A,0 . (9) due to cosmic rays. For this study, we choose to fix
MS Gσn,0(1−µ−02) the neutral-ion collision time to the dimensionless value
τ /t = 2πGσ τ /c = 0.2 ; a value typical of the
The length scale corresponding to the minimum growth ni,0 0 n,0 ni,0 s
denser inner regions where most cores are likely to form
time is λ = 2λ . This is the length scale used to
g,m MS (Basu et al. 2009a, and references within). This corre-
produceoursynCMFsformodelswithfluxfreezing. The
spondstoanionizationfractionχ =5.2×10−8ataneu-
variationofthislengthscaleasafunctionofµ isshown i
0 tralcolumn density σ =0.023 g cm−2. Figure 1 (dot-
by the solid line in Figure 1. For the case with no mag- n,0
ted line) shows the relation between the collapse length
netic field, Equation 9 reduces down to the thin disk
scale and the mass-to-flux-ratio for this neutral-ion col-
equivalent of the Jeans length,
lision time. By fixing the neutral-ion collision time, our
C2 ambipolar diffusion models have only two free parame-
λ = eff . (10)
J Gσ ters, the column density and mass-to-flux ratio distribu-
n,0
tions. Ourchoicesforthesetwoparametersarediscussed
Again, the length scale corresponding to the minimum in the following sections.
growth time is λ = 2λ , which is the scale used in
g,m,J J
our nonmagnetic model.
4 Bailey & Basu
Figure 3. Modelmass-to-fluxdistributionsforfluxfreezingmod-
Figure 4. Model mass-to-flux distributions for ambipolar diffu-
els. Left: Broad Lognormal Distribution (FF2). Right: Narrow
sion models. Left: Broad Lognormal Distribution (AD4), Right:
Lognormaldistribution(FF3).
NarrowLognormal(AD5).
2.2. Column Density Distribution
A survey of column density σ distributions within
n
various molecular clouds shows that they generally ex- within molecular clouds are difficult to obtain. Due
hibit log-normal distributions either with or without a to limitations in techniques and resolution, studies
high density tail (Kainulainen et al. 2009). Correlation of magnetic fields within clouds are generally on a
of these different shapes with the conditions within the more global scale (see Crutcher 1999; Heiles & Troland
cloudssuggestthatregionswitha purelognormaldistri- 2004; Troland & Crutcher 2008; Falgarone et al. 2008;
butiontendtobequiescentwhilethosewithhighdensity Crutcher et al. 2010; Chapman et al. 2011, among oth-
tails show signs of active star formation. ers) which does not give much insight into the exact na-
Since the aim of this paper is to investigate the shape ture of µ0 within denser small scale regions. Therefore,
of the core mass function as an initial condition for themass-to-fluxratioofspecificregionsarenotgenerally
star formation, we choose a simple lognormal distribu- known,let alone a distribution overan entire cloud. Re-
tion as shown in Figure 2. This plot shows the distri- cent simulations of cloud formation with magnetic fields
bution as a function of both the column density (σ , (Va´zquez-Semadeni et al. 2011) show that the mass-to-
n
lower axis) and the visual extinction (A , upper axis). flux ratio distribution seems to exhibit a lognormal
v
Following the prescription of Pineda et al. (2010), the shape. On the other hand, analysis of the likelihood
conversion from visual extinction to column density is of different magnetic field distributions (Crutcher et al.
achievedbycombiningthe ratioofH columndensityto 2010) show that the magnetic field strengths for various
2
color excess (Bohlin et al. 1978) with the total selective regions(HIdiffuseclouds,OHdarkclouds,etc)exhibita
extinction (Whittet 2003) to yield a conversion factor uniformdistributionrangingfromverysmallvaluesupto
N(H ) = 9.35 × 1020A cm−2 mag−1. Although this a maximum value. This seems to disagree with the sim-
2 v
conversion is specifically for H , the abundance ratio of ulations of Va´zquez-Semadeni et al. (2011). With these
2
CO to H is ∼ 10−4 and other molecular contributions results in mind, we choose to explore both options (i.e.,
2
are even smaller, so they do not add significantly to the uniform and lognormaldistributions).
numberdensityofH . Thereforeweassumethisnumber As shown by the linear analysis results presented in
2
densityisrepresentativeofallspecies. Assumingamean Bailey & Basu (2012) and Figure 1, the length scale for
molecularweightof2.33amu,thistranslatesintoamass collapse is dependent on the value of the mass-to-flux
column density conversionof the form ratio. The value of µ0 is selected from a predefined
distribution that is independent of the distribution of
σn =3.638×10−3Av g cm−2 mag−1. (11) σn. This implies that the magnetic field strength is not
constant and varies according to the choices of σ and
The variance and mean (σ2 and µ) of this distribu- n
µ . The independent sampling of values of σ and µ
0 n 0
tion were chosen based upon observational information.
does not then allow for any systematic dependence of
Previous studies of molecular clouds show visual ex-
one quantity on the other. We believe this is an accept-
tinction thresholds for core and star formation to be
able first approximation since the initial conditions of
on the order of A = 5 mag (Johnstone et al. 2004;
v the mass-to-flux ratio distribution in a molecular cloud
Kirk et al. 2006) and A = 8 mag (see Johnstone et al.
v are poorly constrained. We test several possible µ dis-
0
2004; Froebrich & Rowles 2010, among others) respec-
tributions in an attempt to determine if the shape of
tively. As such, we adopted a mean visual extinction
an observed CMF could reveal information about the
value of 8 magnitudes for our lognormal density distri-
underlying mass-to-flux ratio distribution. We consider
bution. The variance reflects the typical width of the
bothuniformandlognormaldistributions. Figures3&4
lognormal fits to cloud density functions presented by
show the adopted lognormal mass-to-flux ratio distribu-
Kainulainen et al. (2009).
tions for the flux freezing and ambipolar diffusion mod-
els respectively. Specifically, all distributions sample the
2.3. Mass-to-Flux Ratio Distributions
transcritical peak in fragmentation scale, λ (see Fig-
g,m
Although density/visual extinction maps are fairly ure 1). The properties of all µ distributions considered
0
commonplace, measurements of magnetic field strengths are given in Table 1.
Effect of Magnetic Fields on the CMF 5
Table 1
ModelParameters
ModelName µ0 Distribution Mean(µ) Variance(σ2) µ0 Range
FluxFrozenModels
FF1 Uniform - - 1.0-3.0
FF2 Broadlognormal 0.01 1.0 1.0-10
FF3 Narrowlognormal 0.01 0.01 1.0-1.5
AmbipolarDiffusionModels
AD1 SubcriticalUniform - - 0.1-1.0
AD2 SupercriticalUniform - - 1.0-3.0
AD3 Uniform - - 0.7-3.0
AD4 Broadlognormal 0.01 1.0 0.3-10
AD5 Narrowlognormal 0.01 0.01 0.6-1.5
Figure 5. Syntheticcoremassfunctionforanonmagneticcloud.
Left: Total core mass function. Right: Contributions to the core Figure 6. Syntheticcoremassfunctionsforafluxfrozenmagnetic
massfunctionfromcoreswithAv<8mag(dashedline)andcores cloud assuming a uniformly distributed mass-to-flux ratio (FF1).
withAv>8mag(dotted line). Left: Total core mass function. Right: Contributions to the core
massfunctionfromcoreswithAv<8mag(dashedline)andcores
withAv >8mag(dotted line).
2.4. Producing Synthetic Core Mass Functions
ToproduceasyntheticCMF,werandomlysamplethe tion to correspond to the apparent visual extinction
columndensitydistributionforthenonmagneticcaseand threshold for the creation of star forming cores; A ∼ 8
v
boththe columndensity andmass-to-fluxratiodistribu- magnitudes. The righthand panelofFigure 5shows the
tions for the magnetic cases. These values are then used contributions from high density gas (A > 8 mag, dot-
v
to find the preferred length scale for collapse from the ted line) and low density gas(A <8 mag, dashedline).
v
linear analysis. Finally, the mass is determined by mul- As expected from the Jeans theory, the core mass dis-
tiplying the column density by the square of the cor- tribution mimics the column density distribution, with
responding length scale. By randomly sampling each high mass cores formed from low density gas and low
model distribution 106 times, a synthetic CMF is pro- mass cores formed from high density gas. The distri-
duced. bution of masses for this model peaks at a value of
3. MODELSANDRESULTS lwoigt(hMo/bMse⊙rv)at=ion0s.4(NourtMter≃& W2.5aMrd⊙-Twhohmicphsoisnc2o0n0s7is)t.ent
Our analysis coversseveraldifferent mass-to-fluxratio
distributions andassumptionsaboutthe neutral-ioncol- 3.2. Flux Frozen Magnetic Model
lision time and column density distribution. As stated Amainaimofthispaperistoshowtheeffectofamag-
earlier,thecolumndensitydistributionisthesameforall neticfieldonthe CMF.Afluxfrozenfieldrepresentsthe
models (see Figure 2) and the neutral-ion collision time
simplestcase. Suchascenarioarisesinhighlyionizedre-
for the ambipolardiffusion models is setto a normalized gions where frequent collisions between ions and neutral
value, τni,0/t0 = 0.2. In addition to the models listed particles would ensure perfect coupling to the magnetic
in Table 1, we also present a nonmagnetic (NM) fidu- field. Figures 6-8 show the resulting synthetic core mass
cial case. The following subsections present the results functionforthethreemodelsFF1,FF2,andFF3respec-
foreachmodelindividually. Anindepthcomparisonbe- tively. Under the assumption of a uniform mass-to-flux
tweenallthemodelsandimplicationsregardingobserved ratio distribution (FF1), the resultant CMF (Figure 6,
CMFs will be discussed in Sections 3.4 & 4 respectively. left) exhibits a narrow peak with a distinct high mass
tail. The right hand panel of Figure 6 again shows the
3.1. Non-Magnetic Model
contributions to the CMF from the two column density
Thenonmagneticmodelservesasabaselineforourin- regimes(A <8mag(dashedline)andA >8mag(dot-
v v
vestigation. TheleftpanelofFigure5showstheresulting ted line)). This composite plot shows that like the NM
core mass function from this technique. As discussed in case, and in line with the Jeans theory, the low density
Section 2.2, we choose the peak of our density distribu- gas forms high mass cores, while high density gas forms
6 Bailey & Basu
Figure 7. Synthetic core mass functions for a flux frozen mag- Figure 9. Synthetic core mass functions for a magnetic cloud
neticcloudassumingabroad,lognormalmass-to-fluxratio(FF2). including the effects of ambipolar diffusion assuming a uniform
PanelsdepictthesamecurvesasFigure6. subcriticaldistributedmass-to-fluxratio(AD1). Panelsdepictthe
samecurvesasFigure6.
Figure 8. Synthetic core mass functions for a flux frozen mag-
neticcloudassuminganarrow,lognormalmass-to-fluxratio(FF3). Figure 10. Synthetic core mass functions for a magnetic cloud
PanelsdepictthesamecurvesasFigure6. including the effects of ambipolar diffusion assuming a uniform
supercritical distributed mass-to-flux ratio (AD2). Panels depict
thesamecurvesasFigure6.
low mass cores. However, unlike the Jeans theory and
nonmagnetic case, we see that with the addition of a
magneticfield,thehighdensitygasalsocontributestothe ThisdistributionresultsinaCMFthatissimilartothat
formation of high mass cores, albeit to a lesser extent. of model FF1 (Figure 6), with a few minor differences.
Compared to the NM case, the peak of this core mass First, the high mass tail exhibits a steeper slope that
function is shifted to M ≃ 100.7 M⊙ ≃ 5.0 M⊙. On the results in a more pronounced peak region. Second, the
right hand side of this peak, the trend can be described peakofthemassfunctionhasshiftedtoaslightlysmaller
by two distinct slopes. For 0.7 < log(M/M⊙) < 1.2, valueofM ≃100.5 M⊙ =3.16M⊙. As before, the trend
α = 0.8 while for log(M/M⊙) > 1.2 the slope becomes of the high mass side can be described by two distinct
shallower; α ∼ 0.6. Neither of these values correspond slopes. For 0.5 < log(M/M⊙) < 1.0, α = 1.31 while for
to the typical Salpeter and observational values. This log(M/M⊙)>1.0theslopebecomesshallower;α=0.63.
discrepancy will be discussed further in Section 4. Figure8showstheresultingsynCMFforanarrowlog-
The formation of the high mass tail is due to the re- normal µ distribution (FF3). Unlike the previous two
0
lationship between µ and λ as defined by Equation 9. models, this one does not exhibit a narrow log-normal
0
Forµ −σ pairswhichhavemass-to-fluxratioscloserto type peak, but rather shows a broad peak that leads
0 n
the critical value (µ = 1), the corresponding length is directly into a high mass tail. As a result, the post peak
0
up to 23 times larger than the thermal Jeans length for trend for this model can be described by a single slope,
the samecolumndensity(seeFigure1). Thisincreasein α = 0.44. Also, note that the function itself has been
length scale has a direct effect on the mass of the core shifted toward higher masses as compared to the other
that is formed. Conversely, the low mass distribution is two flux frozen models. As such, this CMF peaks at
formed by µ0 −σn pairs that have low column density M ≃101.3 M⊙ ∼20 M⊙. This shift in the mass rangeis
andmass-to-fluxvalues thatarecloserto the otherlimit dueentirelytothenarrowpeakdistributionofthemass-
(µ =3),whereλisonlyabout1.5timeslargerthanthe to-flux ratio; all of the chosen mass-to-flux ratios result
0
thermal length scale. in length scales that are ∼ 6−23 times larger than the
Figure 7 shows the resulting synCMF for a broad log- thermallengthscale(seeFigure1)andtherefore,thelow
normal µ distribution (FF2). The two panels again mass cores that are formed in the other two models are
0
show the total and composite CMFs as describedabove. absent in this model. Overall, as shown by all three
Effect of Magnetic Fields on the CMF 7
Figure 11. Syntheticcoremassfunctionsforamagneticcloudincludingtheeffectsofambipolardiffusionassumingauniformlydistributed
mass-to-flux ratio(AD3). Left: Total core mass function. Middle: Contributions to the core mass function from cores with Av <8 mag
(dashed line) and cores with Av >8 mag (dotted line). Right: Contributions to the core mass function from cores with µ0 <1 (dashed
line)andcoreswithµ0>1(dotted line).
above. Specifically,thisCMFshowsthesamepeakedna-
turewithhighmasstailasthefluxfrozenmodel,however
models, the effect of adding a flux-frozen field is the ap-
pearance of a broadshallow tail at the high mass end of this tail abruptly declines at about 100 M⊙. This trun-
cationmakestheoverallshapeoftheCMFresemblethe
the core mass function.
nonmagneticcase,albeitbroader,withthebeginningsof
a “shoulder” feature between 10 and 100 M⊙. Looking
atthecompositecolumndensityCMF(Figure10,right),
weseethatthelowestandhighestmasscoresareformed
by the highest and lowest density gas respectively, while
3.3. Ambipolar Diffusion Magnetic Model
the middle has contributions from both density regimes.
In the previous section we looked at the effect of a The peak of the mass function for this model occurs at
simple flux-frozen field on the shapes of the resulting about log(M/M⊙)=0.7.
CMF(s). Here we look at how the addition of neutral- Model AD3 assumes a uniform mass-to-flux ratio dis-
ion slip via ambipolar diffusion affects the shape of the tributionthatsamplesthepeakoftheλversusµ graph
0
CMF. As discussed above, we have fixed the normalized (see Figure 1). The resulting CMF (Figure 11, left) is
neutral-ion collision time to τni,0/t0 = 0.2. This implies very similar to the one produced by AD2. Looking at
a high degree of ambipolar diffusion and therefore less the contributions from the low and high column density
frequentcollisionsbetweentheneutralsandions. Sucha gas,we againsee that the lowestand highest mass cores
situationwouldoccurinthe inner regionsofamolecular areformedby the highestand lowestdensity gas respec-
cloud where the main ionization mechanism is cosmic tivelywhilethemiddlerangehascontributionsfromboth
rays. density regimes.
Figures 9-13 show the resulting synCMFs for all five The right panel of Figure 11 shows the contributions
mass-to-flux ratio distributions respectively. To estab- from the subcritical (µ ≤ 1, dashed line) and super-
0
lish how the sub- and supercritical regions of the mass- critical (µ > 1, dotted line) gas. We see that the total
0
to-flux ratio affect the shape of the CMF, we start our synCMF for AD3 (Figure 11, left) is a combination of
analysisbypresentingtwocasesthatisolateeachregime. models AD1 and AD2. Specifically, we see that the ma-
Figures 9 & 10 show the resulting synCMFs for the sub- jorityofthecoresareformedfromsupercriticalgas,while
critical and supercritical uniform mass-to-flux ratio dis- thesubcriticalgasyieldsaminorcontributiontothepop-
tributions (AD1 and AD2) respectively. The two panels ulation of low mass cores. By mentally combining the
show the total and constituent core mass functions as middle andrighthand plots inFigure 11, one candeter-
described in the previous section. minethatthehighestmasscoresareformedbysupercrit-
FocusingonmodelAD1,Figure9,theleftpanelshows ical gas and fall into the non-star-forming regime while
that the core mass function is very similar to the non- low-masscoresareformedbybothsupercriticalandsub-
magnetic model (see Figure 5, left). This is due to the criticalgas,and fall into both the star-formingand non-
factthatthecurveonthesubcriticalsideofFigure1con- star-forming regimes. The peak of the mass function for
vergesto the nonmagnetic limit faster thanin the trans- this model occurs at about log(M/M⊙) = 0.7 and the
and supercritical regions. Upon closer comparison, AD1 averageslope of the high mass ‘tail’ is α=1.42.
peaks at the approximatelythe same value as NM, how- Finally, Figures 12 & 13 show the resulting synCMFs
everthedensitycompositeCMF(Figure9,right)reveals for the two lognormal µ distributions, AD4 and AD5,
0
differences between these two models. Unlike the non- respectively. The broad lognormaldistribution (AD4) is
magnetic model, AD1 shows evidence that a portion of similar to models AD2 and AD3, however this model
the high column density gas goes toward forming high shows a more distinct ‘peak’ and ‘shoulder’ region as
mass cores (Figure 9, right). compared to the other two. Looking at the composite
Figure 10 shows the resulting synCMF under the as- mass-to-flux ratio plot (Figure 12, right) we see that
sumption of a uniform supercritical distribution (AD2). the peak region is mainly formed by subcritical gas
The left panel shows that the total CMF is a hybrid be- while the shoulder region is formed mainly by contri-
tweenthenonmagneticandflux-frozenmodelspresented
8 Bailey & Basu
Figure 12. Synthetic coremassfunctions foramagnetic cloudincludingthe effects ofambipolar diffusionassumingabroad, lognormal
mass-to-fluxratiodistribution(AD4). PanelsdepictthesamecurvesasFigure11.
Figure 13. Syntheticcoremassfunctionsforamagneticcloudincludingtheeffectsofambipolardiffusionassuminganarrow,lognormal
mass-to-fluxratiodistribution(AD5). PanelsdepictthesamecurvesasFigure11.
butions from supercritical gas. This model peaks at the lognormal peak with high mass tail as represented
M = 100.7 M/M⊙ ≃ 5.0 M⊙, and the average slope by FF1, FF2, and FF3. The appearance of these shapes
of the high mass tail is α = 1.18. Switching to the nar- are directly connected to the state of the magnetic field
row lognormal distribution (Figure 13), we see that this in the region. In the absence of a magnetic field, the
model results in a double peakedfunction. Examination CMF is a pure lognormal function. This shape is also
of the composite plots show that the low mass peak is observed in model AD1. As mentioned earlier, the rea-
formedbythe subcriticalmaterialwhilethesecondpeak son that this AD model shows such a shape while the
is formed by supercritical material. These peaks occur otheronesdo notisdue tothe shapeofthe λ−µ curve
0
atlog(M/M⊙)∼0.7andlog(M/M⊙)∼1.5respectively. onthe subcritical side ofFigure 1; the curve asymptotes
The formation of the high mass peak is due to the ex- to the nonmagnetic limit faster on that side than on the
tremely narrow mass-to-flux ratio distribution. It picks supercritical side. Therefore one would expect a model
out only large length scales from the peak of the λ−µ with only subcritical mass-to-flux values to look similar
0
curve (with τ /t =0.2) in Figure 1. to the nonmagnetic model, but with a slight broaden-
ni,0 0
ing due to a narrow region of mass-to-flux ratios with λ
larger than the non-magnetic limit.
Formodelswithanincreasingsupercriticalregime,the
3.4. Assessment of Synthetic Core Mass Functions
broadening becomes more pronounced as a shoulder de-
The previous subsections presented the overall results velops. This shoulder is due to an increase in higher
and features of each of the models. Within these results masscoresthataretheproductofthelargerlengthscales
we found three main features that changed between the picked out by the supercritical mass-to-flux ratios. The
different models. These are the overallshape of the core extent of the shoulder depends on the mass-to-flux ra-
mass function, the location of the peak(s) and the slope tio distributions. For uniform distributions, the CMF is
of the high mass tail (if it exists). Here we discuss these narrower with a less defined shoulder region, while for
three features across all models. a broad lognormal distribution, the shoulder region is
much broader and distinct. Finally, the appearance of
the double peaked CMF in AD5 is an example of an ex-
treme shoulder. This second peak is due solely to the
3.4.1. Shape
extremely narrow mass-to-flux ratio range used in this
Withintheninemodelspresented,therewerethreedis- model. This preferentially picks out only mass-to-flux
tinct recurringshapes; pure lognormalasrepresentedby ratios with length scales much larger than the nonmag-
the NM and AD1 models, lognormal peak with a shoul- netic model.
der as represented by AD2, AD3, AD4 and AD5, and The appearance of the pure high mass tail is entirely
Effect of Magnetic Fields on the CMF 9
Figure 14. Small sample core mass functions for three models: NM (top row), FF1 (middle row), and AD1 (bottom row). Number of
coresforeachpanel indicatedinthetoplefthandcornerofeachplot.
Figure 15. Bin size comparison for small sample core mass functions. Panels show the effect of the bin size on the resulting curve for
∆log(M/M⊙)=0.25bins(toprow)and∆log(M/M⊙)=0.1bins(bottom row). ModelusedinallpanelsisAD3.
a product offlux-freezing. This is due to the asymptotic magnetic models, the location of the peak was gener-
nature of the flux-frozen curve as it nears the critical ally larger than this value as long as the mass-to-flux
mass-to-flux ratio (see Figure 1). This allows for trans- ratio distribution was uniform with some contribution
critical mass-to-flux ratio values to produce much larger fromthesupercriticalregime(seemodelsFF1,AD2,and
masses for the same column density. AD3). Model AD1, although also assuming a uniform
mass-to-flux ratio distribution, exhibits a similar peak
3.4.2. Peak Location value to NM due to the exclusion of supercritical mass-
to-flux ratio values. When considering the lognormal
ThelocationoftheCMFpeakdependsonthedistribu-
mass-to-flux ratio distributions, we find that the peak
tionofthemass-to-fluxratio. Thelocationofthepeakin
location is dependent on the width of the distribution.
thenonmagneticcase,whichoccursatlog(M/M⊙)=0.4 Specifically, broader distributions exhibit values closer
(M ∼ 2.5 M⊙) serves as the comparison point. For
10 Bailey & Basu
to the NM peak value, while narrower distributions ex-
-2
hibit peak values that are higher than the nonmagnetic
AD, 0.1
case. Model AD5 is an anomaly and does not fit within AD, 0.25
these trends given that it exhibits two peaks. -1.8 FFFF,, 00..125
)
3.4.3. High Mass Slope α
-1.6
(
As alluded to earlier,the shape and extent of the high e
mass slope was found to be variable and connected to p
o-1.4
the influence of the magnetic field. Specifically, the ap- Sl
pearance of the ‘shoulder’ feature is directly connected e
g
tothepresenceofambipolardiffusion. Thedegreeofthe a-1.2
r
shoulder in the ambipolar diffusion models was found to e
v
be dependent on the range of allowed mass-to-flux ra- A
-1
tio values. Overall, these differences in shapes result in
a wide range of slopes. For the flux-frozen models, the
slopes were as steep as α = 1.31 in the case of FF2, -0.8
and as shallow as α = 0.44 in the case of FF3. For the
102 103 104 105 106
ambipolar diffusion models, the averagehigh mass slope
Sample Size
ranges between α = 1.18 and α = 1.42. Although some
of these slopes are consistent with the Salpeter value, Figure 16. Average slope as a function of sample size. Sym-
α = 1.35 (Salpeter 1955), others are significantly differ- bols represent the derived slopes for the two models and two bin
ent. Further analysis of this discrepancy is given in the sizes: AD3,∆log(M/M⊙)=0.1(squares),AD3,∆log(M/M⊙)=
following section. 0.25 (circles), FF1, ∆log(M/M⊙) = 0.1 (diamonds), and FF1,
∆log(M/M⊙) = 0.25 (triangles). Average slopes computed over
aminimumof 2000 samples. Open symbolsindicate the slopes of
4. SCALINGTOOBSERVATIONS thefullsamplesize.
UnlikeoursynCMFs,typicalobservationalCMFsusu-
thebroadstrokes. Todeterminetheeffectofthebinsize
allycontainontheorderof200cores,not106. Therefore,
on the resulting CMF, we re-binned the histograms for
tomakeouranalysisrelevantfortypicalobservedCMFs,
AD3 in Figure 14 (bottom row) using ∆log(M/M⊙) =
wemustscalebackoursamplesizestothosetypicallyob-
0.25bins. Figure15showsthecomparisonoftheoriginal
served. The following two sections explore the effect of
bin size (∆log(M/M⊙) = 0.1) to the new bin size (top
two observational constraints, sample size and bin size,
row). As expected, with the larger bin size, the detail
on the shape and slope of observed CMFs.
becomes smeared out, resulting in an average curve.
4.1. Effect of Sample Size
To test the effect of the sample size on the resultant
CMF, we scaled three synCMFs (NM, FF1, and AD3) 4.3. Effect of sample size and bin size on CMF slopes
down to plausible observational sample sizes (100, 200,
The main piece of data generally extracted from a
300, 400, and 500 cores). Figure 14 shows the result-
CMFistheslopeofthehighmasstail. Thisinformation
ing synCMFs for each of the fifteen cases. In addition
is then used to compare different regions to each other,
to scaling the sample size, we have also truncated the
and to the initial mass function (IMF) in an attempt
massrangeconsideredtoonemoretypicallyfoundinob-
to determine the true nature of star formation and the
served CMFs (−1.0 < log(M/M⊙) < 1.3). Under these possible relation between the CMF and IMF. However,
scaled conditions, we see that the nonmagnetic CMFs
as discussed above, the sample size and bin size have a
stillmaintaintheoverallshapeexhibitedbythefullsam-
profound effect on the shape of the curve. This effect
plecurve(Figure5),howeverthetwomagneticcasesare
translates over to the derived slopes. To determine the
fairly different. The high mass tail and truncated shoul-
extent of this effect, we generate multiple small sample
der features present in the full sample curves for FF1
CMFs (2000+) for each sample size and compute the
and AD3 respectively are no longer quite as distinct at
average slope. Figure 16 shows the results of this anal-
these sample sizes. For a definitive difference between
ysis for models FF1 and AD3 for both mass bin sizes.
the ambipolar diffusion and flux-frozen cases, observa-
Thefilledsymbolsshowtheaverageslopeforeachofthe
tions would have to extend up to objects with masses
models while the open symbols depict the slope of the
between 102 and 103 solar masses. Therefore, on typical full sample (106). Tests with larger numbers of samples
observationalscales,conclusions aboutthe nature of the
for each sample size showed differences in the average
magnetic field from the shape of the CMF are possible,
slope of up to 0.01, which is encompassed in the size of
but highly uncertain.
the symbols.
AsshowninFigure16,thesizeofthebinclearlyaffects
4.2. Effect of Bin Size
the average slope. The larger bin size yields slopes that
Constructing histograms for the purposes of deter- are steeper than the Salpeter slope, while the smaller
mining a CMF requires binning data into predeter- bin size shows an overall shallower average slope. The
mined mass bins. For the above synCMFs, we used sizeofthe sample alsoeffects the slope. Smaller samples
∆log(M/M⊙) = 0.1 size bins. Variations in the bin generallyresultinsteeperslopesthanthosederivedusing
size acts to change the resolution of the resulting curve; the full sample. Furthering this analysiswe lookatboth
smallerbinsyieldmoredetailwhilelargerbinsshowonly the minimum and maximum slopes calculated for