SFEs in clusters. Final value of the SFE. For an isolated clump SFE exp. (t exp. = SFE(t exp. M ( cluster. t ) exp M clump. (t) M gas,i.

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2 SFEs in clusters SFE(t) Final value of the SFE M cluster (t) M gas,i + M gas,acc (t) SFE exp = SFE(t exp ) M cluster (t exp ) M gas,i + M gas,acc ( t ) exp For an isolated clump SFE exp M ( cluster t ) exp M clump

3 Derived SFEs in clusters SFE cluster,obs ~ SFE cluster,obs M cluster M cluster + M gas e.g. Lada & Lada (2003) and references therein No established dependences on mass, metallicity, environment.

4 Derived SFEs in Galaxies SFE gal SFR gal M H 2 Dib et al. (2011) Data compiled from: average spirals (Murgia et al. 2002) M33 and NGC6822 (Gratier et al. 2010a,b) SMC (Leroy et al. 2006)

5 the pure Gravo-Turbulent Star Formation Model: Prediction for the SFE ff (Krumholz & McKee 2005) CFE ff 0.15εα 0.68 vir Ma 0.32 For Ma=10 and akpha_vir=2 = SFE ff =0.014 SFE ff 0.05α 0.68 vir Ma 0.32

6 Age spread: recent observational constraints Preibisch 2012 Kudryavtseva, Brandner, et al. 2012

7 Star formation in a protocluster clump with feedback regulation General properties A) Fraction of the mass of the clump converted into dense cores per free fall time CFE ff Clump properties C) Protocluster cloud mass-radius relation, M_cl-R_cl (constrained by the observations) D) Clump core Radius, R_c0 (0.02 pc) E) exponent of density profile in outer regions (-2) F) Larson relation exponent α, or exponent of turbulent vel. field power spectra β (constrained by the observations) Core properties G) Contration timescale t_cont = ν t_ff; ν =1-10 (constrained by the observations and theory) H) Cores peak density with respect to clump background density as a function ρ p0 (constrained by the observations and theory) I) Core to star efficiency 1/3 Feedback from massive stars (M > 5 M sol ) I) Mass loss rate of massive stars and the terminal velocity of the wind J) Fraction of wind energy that disperses the gas from the clump (unknown?)

8 Variation of the CFE: The input of numerical simulations Simulations of Turbulent, magnetized, and self-gravitating clouds

9 Influence of the magnetic field µ c = 8.8 CFE ff = 0.33 µ c = 2.8 CFE ff = 0.06 Dib et al. in 2010

10 Feedback: Stellar Winds Stellar mass loss rate Terminal wind velocity dm dt v Energy cumulated in winds E wind = t " =t t " =0 2 N(m)(dM /dt) (m)v dm dt " 2 m =80M sol m =5M sol Fraction of wind energy that counters gravity E k,wind = κe wind k <1

11 Power of stellar wind - Calculate main sequence models of OB stars ( 5 M ) (using CESAM, now MESA) - (T eff, L *, R * ) Stellar atmosphere model (Vink et al.) Ṁ Ṁ v 2

12 initial core population: gravo-turbulent fragmentation of the clump (Padoan & Nordlund 2002) α=0.4 β=1.8 Slope= -3/(4-β)-1=-2.33

13 The SFE is strongly regulated by feedback cores stars time

14 SFE exp on metallicity Dib et al. (2011)

15 Galactic SFEs Dib et al. (2011)

16 Stellar clusters mass function vs. Protocluster clumps mass function dn dm clusters η M emb,clusters dn dm protocluster,clump δ M protocluster,clumps γ M cluster M protocluster,clumps δ = γ(η +1) 1 We find Then, if γ =1 η 2 δ 2

17 The Star Formation Laws in Galaxies in the pure Gravo-Turbulent Star Formation Model t ff Σ SFR = Σ g f H 2 SFE ff * Star Formation occurs in GMCs * The characteritic mass is M char = M GMC M char is set by the gravitational instability in the disk *A description of f H 2 (Z ' ) (Krumholz, McKee, Tumlinson 2009; Gnedin & Kravtsov 2010,2011 and others) * A description of (Krumholz & McKee 2005) SFE ff 0.15εα 0.68 vir Ma 0.32

18 The Star Formation Laws in Galaxies in The Feedback Regulated Star Formation Model Star formation occurs in protostellar clumps (embedded in GMCs). 2 N(M clump ) M clump The characteristic mass is M char = Max(M cl,max,m GMC ) ( ) MN M dm A description of f H 2 (Σ g,z ' ) M cl,min

19 The Star Formation Laws in Galaxies in The Feedback Regulated Star Formation Model Σ SFR = Σ g f H 2 ( Σ g,z ' ) SFE exp t exp Σ SFR = Σ g f H 2 ( Σ g,z ' ) SFE exp n exp t ff f *, ff Σ SFR = Σ g f H 2 t ff

20 The Star Formation Efficiency per unit time in The Feedback Regulated Star Formation Model f *, ff max(m ( Z ' ) cl,max,m GMC ) = f *, ff (M,Z ' )N M M cl,min ( ) dm

21 The Star Formation Efficiency per unit time in The Feedback Regulated Star Formation Model f *, ff max(m ( Z ' ) cl,max,m GMC ) = f *, ff (M,Z ' )N M M cl,min ( ) dm

22 The Star Formation Laws in Galaxies: Feedback regulated vs. Turbulence regulated Dib (2011). Observational data: Kennicutt (1998), Bigiel et al. (2008, 2010)

23 Conclusions Turbulence, magnetic fields, geometry regulate the rate of core formation in a clump/cloud Feedback is a important regulator of the SFE in a protocluster clump. Stong metallicity dependence of the SFE. SFE decreases with increasing metallicity Implications/Predictions of the feedback regulated, metallicity dependent star formation Mass functions of protocluster clumps, and clusters slopes of 2 Can explain typical observed age spreads and SFE. Metallicity dependent star formation laws in galaxies over the entire surface density regime. Use observations are to: - Test the models (SFE, age spreads) - Test if star formation is uniform or accelerated - Test the dependence on metallicity