Sami DibNBI, STARPLAN

Unveiling the diversity of the MW stellar clusters

+

Sacha Hony (ITA/Heidelberg)

Stefan Schmeja (ARI/ Heidelberg)

Dimitrios Gouliermis (ITA/Heidelberg)

Åke Nordlund (NBI, STARPLAN)

The multi-exponent power law: Kroupa 2002

The lognormal distribution+power law: Chabrier 2003, 2005

The TPL IMF: de Marchi & Parecse 2002; Parravano et al. 2010

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ξ logM( ) = kM −Γ 1− exp −M

Mch

⎛

⎝ ⎜

⎞

⎠ ⎟

γ +Γ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪= kM −1.35 1 − exp −

M

0.42

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

0.57+1.35 ⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

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0.1≤ M /Msol ≤120

The Galactic field IMF

Maschberger (2013): L3 function (low mass + high mass power laws)

Basu et al. (2015): MLP (modified lognormal+ power law)

Luhman & Rieke 1999; Luhman 2000, ++

rho-Ophiucus

Taurus

The IMF of young clusters

comparing the slope at the intermediate-to-high mass end

Massey et al. 1995a,b Massey 2003

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−1.35,Salpeter

The IMF of Open Stellar Clusters

Sharma et al. 2008

Figer 1999; Stolte et al. 2005, Kim et al. 2006 Espinoza et al. 2010, Habibi et al. 2014

The IMF of Starburst clusters: Arches, NGC 3603: shallower than Salpeter

Stolte et al. 2006

Characterizing The IMF of stellar clusters with Bayesian statistics

Avoids:

The effects of the bin size

Making subjective choices about break points

Allows for: Including the effects of individual uncertainties

on masses

Effects of completness

Use of prior information

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P M i D( ) =P M i( )P DM i( )

P D( )

Test on synthetic data• Likelihood: TPL

• Flat prior functions on the parameters (in linear space)

• MCMC

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ξ logM( ) = kM −Γ 1− exp −M

Mch

⎛

⎝ ⎜

⎞

⎠ ⎟

γ +Γ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Application to young clusters: TPL Likelihood

Dib 2014

€

ξ logM( ) = kM −Γ 1− exp −M

Mch

⎛

⎝ ⎜

⎞

⎠ ⎟

γ +Γ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

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σIMF = σΓP,σγP

,σM P( ) = 0.60,0.25,0.27Msol( )

Application to young Galactic Stellar Clusters

Case of “Chabrier”-like likelihood function

Dib 2014

No overlap in the parameters at the 1σ confidence limit

Effect of completeness & mass uncertainties: ONC

Dib 2014

The statistics of massive stars in young clusters

The MWSC (Kharchenko et al. 2012,13 , Schmeja et al. 2014, Schmeja, Dib et al. 2015) Based on 2MASS + PPXML data; 3145 clusters within < 2 kpc; complete to ~ 1.8 kpc from the Sun

Extract statictics from the MWSC catalogue that can be contrasted with large sample of synthetic clusters

extract order zero statistics i.e, scalars

order 1: distributions of specific quantities

Order 2: relations between 2 quantities

Some statistics (among others)

The fraction of “lonely” O stars of masses

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fO =NO,lonely

No,tot

€

M ≥ MO,lim

The fraction of “lonely” O stars of masses

and no massive B stars (between 10 and 15 Msol)

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M ≥ MO,lim

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f2,O =NO,lonely,nomassiveB

No,tot

Construct a statistical sample of clustersModel chart

Define how much mass is present in clusters that contains stars with M > MO,lim

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Σcl,O = SFR × τO,lim

Define cluster mass function (slope, mininum & maximum clusterMasses) And distribute over this distribution

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dNcl

dMcl

∝ Mcl−β

€

Σcl,O

For each cluster of mass

Define the distributions of the IMF parameters

From which they are draw for each cluster.€

Mcl

€

Mcl,min

€

Mcl,max

Sample the masses of stars: i.e., the IMF with the chosen set of parameter for each cluster

assign an age to each cluster €

dN*

dM*

∝ M −ΓP 1 − exp −M

MP

⎛

⎝ ⎜

⎞

⎠ ⎟

γ P +ΓP ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

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0 ≥ τ cl ≥ τO,lim

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ΓP ,γP ,MP( )

For each O system of mass in each cluster: Define a binarity probability For binary system, define a mass

ratio

Correct for stellar evolution

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q =M2

M1€

M*,O

€

M1 + M2 = M*,O

Correct for CLMF completeness

Measure the fractions & other statistics

Sampling the IMF parameters

Which distributions of the IMF parameters allow us to match the statistics from the MWSC ?

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ΓP

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σIMF = σΓP,σγP

,σM P( )

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ΓP , field

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dNΓP

dΓP

How does the sample look like ?

ICLMF, CLMF, CLMF-lonO System IMFs

Comparing the statistical sample to the MWSC Sampling the IMF parameters

no Mmax-Mcl relation. IMF sampled between 0.02 and 150 Msol

Reproducing the observations is only compatible with the IMF parameters having

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σIMF = σΓP,σγP

,σM P( )obs

Dependence onthe sampling parameters

Effect of the CLMF exponent.

Best match for

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β =−2

Effect of the choice of Mcl,min

Best match for Mcl,min=50 Msol

with a M*,max-Mcl relation (Weidner& Kroupa 2013)

Summary(1) The IMF of Galactic clusters is not universal !..given the data

(2) Better not describe the IMF by a set of 3 (or more) parameters, but by distribution functions of the parameters that may vary from galaxy to galaxy

(3) The cluster mass function has a slope of -2 (4) Minimum mass cutoff of ~50 Msol in young cluster masses required (or a

turnover)

(5) Mmax-Mcl relation incompatible with the matching of statisticals models and the MWSC data

-------------------------Implications/suggestions: • In local simulations: aim should be to recover distributions

of IMF parameters rather than single values• In global simulations: implement distributions of the IMF

rather than single values

Variations of CMF and IMFEffect of accretion on cores

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dN(r,M, t)accdt

= −∂N

∂MM

.

−∂ M

.

∂MN

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟r,M, t( )

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dN(r,M, t)accdt

= −∂N

∂MM

.

−∂ M

.

∂MN

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟r,M, t( )

Dib et al. 2010

Data points: Orion A+B cloudJohnstone & Bally 2006

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dN(r,M, t)accdt

= −∂N

∂MM

.

−∂ M

.

∂MN

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟r,M, t( )

Dib et al. 2010

In low-mass clusters: no tail at high stellar masses

Application to the Orion Cloud and the ONC

Dib et al. (2010)

Dib et al. 2010

Variations with the cores properties: making a Taurus clusterEffect of varying the lifetimes of the cores+stopping effect of feedback

=0.4

=1.8

Slope= -3/(4-)-1=-2.33

calculate instanteneous cross section of collision between contracting objects of Masses Mi and Mj and integrate over the mass spectrum.

Initial conditions from turbulent fragmentation (Padoan & Nordlund 2002)

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++=))()((2

21)()(),,,(

2

2

tRtRv

MMGtRtRtrMM

ji

jijiji πσ

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dN(r,M, t)coaldt

=1

2η r( ) N r,m, t( )N r,M −m, t( )σ (m,M −m,r, t)v(r)dm

M min

M −M min

∫

( ) ( ) ( ) ( ) ( )∫ −−max

min

,,,,,,,M

M

dmrvtrmMmtmrNtMrNr ση

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→

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↓

Variation of the IMF

Effects of core coalescence

Dib et al. 2007

Core coalescence: application to Starburst Clusters

Dib et al. 2007

time