The Dependence of the Galactic Star Formation Laws on Metallicity
aa r X i v : . [ a s t r o - ph . GA ] S e p SF2A 2011
G. Alecian, K. Belkacem, S. Collin, R. Samadi and D. Valls-Gabaud (eds)
THE DEPENDENCE OF THE GALACTIC STAR FORMATION LAWS ONMETALLICITY
S. Dib , L. Piau , S. Mohanty and J. Braine Abstract.
We describe results from semi-analytical modelling of star formation in protocluster clumps ofdifferent metallicities. In this model, gravitationally bound cores form uniformly in the clump following aprescribed core formation efficiency per unit time. After a contraction timescale which is equal to a few timestheir free-fall times, the cores collapse into stars and populate the IMF. Feedback from the newly formed OBstars is taken into account in the form of stellar winds. When the ratio of the effective energy of the windsto the gravitational energy of the system reaches unity, gas is removed from the clump and core and starformation are quenched. The power of the radiation driven winds has a strong dependence on metallicity andit increases with increasing metallicity. Thus, winds from stars in the high metallicity models lead to a rapidevacuation of the gas from the protocluster clump and to a reduced star formation efficiency, as comparedto their low metallicity counterparts. We derive the metallicity dependent star formation efficiency per unittime in this model as a function of the gas surface density Σ g . This is combined with the molecular gasfraction in order to derive the dependence of the surface density of star formation Σ SF R on Σ g . This feedbackregulated model of star formation reproduces very well the observed star formation laws in galaxies extendingfrom low gas surface densities up to the starburst regime. Furthermore, the results show a dependence ofΣ SF R on metallicity over the entire range of gas surface densities, and can also explain part of the scatterin the observations.Keywords: Stars: massive, winds; ISM: clouds, galaxies: star formation: star clusters
Over the last two decades, the dependence of the star formation rate surface density (Σ
SF R ) on the gas surfacedensity (Σ g ) and eventually on other physical quantities has been extensively investigated both observationally(e.g., Kennicutt 1998; Wong & Blitz 2002; Boissier et al. 2003; Bigiel et al. 2008; Blanc et al. 2009; Onodera etal. 2010; Tabatabaei & Berkhuijsen 2010; Heiner et al. 2010; Schruba et al. 2011; Bolatto et al. 2011) as wellas theoretically and numerically (e.g., Tutukov 2006; Krumholz & Thompson 2007; Fuchs et al. 2009; Silk &Norman 2009; Krumholz et al. 2009a; Papadopoulos & Pelupessy 2010; Gnedin & Kravtsov 2011; Narayananet al. 2011; Feldmann et al. 2011; Vollmer & Leroy 2011; Braun & Schmidt 2011; Monaco et al. 2011; Kimel al. 2011; Dib 2011a,b). Determining the rate of star formation in a given tracer of the gas surface densityrequires quantifying the fraction of that tracer as a function of the global gas surface density and a descriptionof the efficiency at which the star forming gas is converted into stars per unit time. The relationship betweenΣ SF R and the surface density of the molecular hydrogen gas Σ H is given by:Σ SF R = Σ g f H SF E τ τ , (1.1)where f H = Σ H / Σ g is the molecular hydrogen mass fraction is the atomic-molecular star forming complexes,and SF E τ is the star formation efficiency over the timescale τ . Krumholz & McKee (2005) proposed a theoryin which supersonic turbulence is the dominant agent that regulates star formation in giant molecular clouds(GMCs). They derived a core formation efficiency per unit free-fall time, CF E ff , which is given by CF E ff = Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ; [email protected] LATMOS, 11 Boulevard d’Alembert, 78280 Guyancourt, France Laboratoire d’Astrophysique de Bordeaux, Universit´e de Bordeaux, OASU CNRS/INSU, 33271 Floirac, Francec (cid:13)
Soci´et´e Francaise d’Astronomie et d’Astrophysique (SF2A) 2011
38 SF2A 20110 . α − . vir M − . , where α vir and M are the virial parameter and the rms sonic Mach number of the GMC,respectively ∗ . By assuming that only a fraction of the mass of the cores ends up in the stars, this CF E ff can be converted into a star formation efficiency SF E ff = η × CF E ff ( η ≤ The model follows the formation of dense gravitationally bound cores in a protocluster clump. Cores form inthe clump with a given core formation efficiency per unit time and follow a local mass distribution that is theresult of the gravo-turbulent fragmentation of the clump. In their series of models, Dib et al. (2011) varied thecore formation efficiency per unit free-fall time (
CF E ff ) between 0 . .
3. This is consistent with the rangeof
CF E ff measured in numerical simulations which describe the gravo-turbulent fragmentation of magnetised,turbulent, and self-gravitating molecular clouds (e.g., Dib et al. 2008; Dib et al. 2010a). The gravitationallybound cores that are generated at every epoch have a mass distribution that is given by the gravo-turbulentfragmentation model of Padoan & Nordlund (2002). In this work, we leave out, for simplicity, the role playedby gas accretion and coalescence in modifying the mass distribution of the cores. The interested reader isreferred to Dib et al. (2007) and Dib et al. (2010b) for such models. Cores contract over a lifetime which isa few times their free-fall time before collapsing to form stars. Feedback from the most massive stars ( M ⋆ ≥ ⊙ ) is taken into account in the form of stellar winds. The formation of cores in the protocluster clump, andconsequently star formation, are terminated whenever the fraction of the wind energy stored into motions thatoppose gravity exceeds the gravitational energy of the clump. In order to calculate reliable estimates of thefeedback generated by metallicity dependent stellar winds, we proceed in two steps. In the first step, we usea modified version of the stellar evolution code CESAM (see appendix 1 in Piau et al. 2011) to calculate agrid of main sequence stellar models for stars in the mass range [5-80] M ⊙ (with steps of 5 M ⊙ ) at variousmetallicities Z/Z ⊙ = [1 / , / , / , / , ,
2] ( Z ⊙ = 0 . ∼ T eff , the luminosity L ⋆ , and the stellar radius R ⋆ are then used in the stellar atmospheremodel of Vink et al. (2001) in order to calculate the stellar mass loss rate ˙ M ⋆ . Vink et al. (2001) did not derivethe values of the terminal velocities of the winds ( v ∞ ), therefore, we use instead the derivations of v ∞ obtainedby Leitherer et al. (1992).The power of the stellar winds is given by ˙ M ⋆ v ∞ . This quantity is displayed in Fig. 1 for the models withdifferent metallicities. The values of ˙ M ⋆ v ∞ are fitted with fourth order polynomials (overplotted to the data)and whose coefficients are provided in Dib et al. (2011). The ˙ M ⋆ v ∞ − M ⋆ relations displayed in Fig. 1 allow forthe calculation of the total wind energy deposited by stellar winds. The total energy from the winds is givenby: E wind = Z t ′ = tt ′ =0 Z M ⋆ =120 M ⊙ M ⋆ =5 M ⊙ N ( M ⋆ ) ˙ M ⋆ ( M ⋆ ) v ∞ dM ⋆ ! dt ′ . (2.1)We assume that only a fraction of E wind will be transformed into systemic motions that will oppose gravityand participate in the evacuation of the bulk of the gas from the proto-cluster clump. The effective kinetic windenergy is thus given by: E k,wind = κ E wind , (2.2)where κ is a quantity ≤ κ = 0 . E k,wind is compared at every timestepto the absolute value of the gravitational energy, E grav , which is calculated as being: E grav = − π G Z R c ρ c ( r ) r dr. (2.3) ∗ Padoan & Nordlund (2011) found a different dependence of the SFR on α vir and M . The results of their numerical simulationssuggest that the SFR decreases with increasing α vir but also that it increases with increasing M . he dependence of star formation rates in galaxies on metallicity 239 Fig. 1.
The power of the stellar winds, or wind luminosities, for stars in the mass range 5-80 M ⊙ on the main sequence,and for various metallicities. The stellar mass loss rates have been calculated using the stellar characteristics (effectivetemperature, stellar luminosity and radius) computed using the stellar evolution code CESAM coupled to the stellaratmosphere model of Vink et al. (2001). The values of v ∞ have been calculated using the derivation by Leitherer et al.(1992). Over-plotted to the data are fourth order polynomials. The parameters of the fit functions can be found in Dibet al. (2011). Adapted from Dib et al. (2011). Since higher metallicity stellar winds deposit larger amounts of energy in the clump than their lower metallic-ity counterparts, this leads them to evacuate the gas from the clump on shorter timescales. This in turn quenchesthe process of core and star formation earlier and sets a smaller final star formation efficiency,
SF E exp . Fig. 2displays the dependence of
SF E exp and of the expulsion time, t exp (expressed in units of the free-fall time t ff ),as a function of metallicity for clumps of various masses. Using the above described model, it is possible to derive the dependence of Σ
SF R on Σ g . The star formationrate surface density in the feedback regulated mode of star formation is given by:Σ SF R = Σ g f H h SF E exp ih t exp i , (3.1)where h SF E exp i and h t exp i are, respectively, the characteristic SF E exp and the epoch at which gas is expelledfrom the protocluster region for the clump mass distribution associated with a given Σ g . Writing h t exp i in termsof the clump free-fall time h t ff i , Eq. 3.1 becomes:40 SF2A 2011 Fig. 2.
Dependence of the quantities
SF E exp (final star formation efficiency) and n exp = t exp/t ff (ratio of the expulsiontime to the free-fall time) for selected values of the protocluster forming clump masses and metallicities. These resultsare based on the models of Dib et al. (2011). Σ SF R = Σ g f H h SF E exp ih n exp i h t ff i = Σ g f H h f ⋆,ff ih t ff i . (3.2)where f ⋆,ff is the dimensionless star formation efficiency and which corresponds to the mass fraction of themolecular gas that is converted into stars per free-fall time t ff of the clumps. h f ⋆,ff i and h t ff i representcharacteristic values of f ⋆,ff and t ff for the spectrum of clump masses found in the GMC for a given value of Σ g .The quantity f H is the mass fraction of the total gas that is in molecular form. In this work, we use the functionalform of f H obtained by Krumholz et al. (2009b) who derived f H as a function of the gas surface density andmetallicity (see their paper or Dib 2011 for the detailed formula). h t ff i can be approximated by the free-falltime of the clump with the characteristic mass t ff ( M char ) = 8Σ ′ − / cl M / char, Myr where M char, = M char / M ⊙ . The characteristic mass M char is given by :he dependence of star formation rates in galaxies on metallicity 241 Fig. 3.
Star formation efficiency per unit free-fall time in the protocluster clump in the metallicity-dependent feedbackmodel. The star formation efficiencies per free-fell use a core-to-star efficiency conversion factor of 1/3. The left paneldisplays f ⋆,ff as a function of both M cl and Z ′ = Z/Z ⊙ in the original data, while the right panel displays the analyticalfit function to this data set given in Eq. 3.5. Adapted from Dib (2011a). M char = Z max ( M cl,max ,M GMC ) M cl,min M cl N ( M cl ) dM cl , (3.3)where N ( M cl ) is the mass function of protocluster forming clumps which we take to be N ( M cl ) = A cl M − cl ,and A cl is a normalisation coefficient given by A cl R max ( M cl,max ,M GMC ) M cl,min N ( M cl ) dM cl = ǫ , where 0 < ǫ < ǫ = 0 . M cl,min is taken to be 2 . × M ⊙ (this guarantees, for final SFEs in the range of0.05-0.3 a minimum mass for the stellar cluster of ∼
50 M ⊙ ) and the maximum clump mass is 10 M ⊙ . Thecharacteristic GMC mass is determined by the local Jeans mass and is given by: M GMC = 37 × (cid:18) Σ g
85 M ⊙ pc − (cid:19) M ⊙ . (3.4)Fig. 4 (top) displays M char as a function of Σ g . The quantity f ⋆,ff = SF E exp /n exp is displayed in Fig. 3(left panel) as a function of mass and metallicity ( Z ′ = Z/Z ⊙ ). These models use a value of CF E ff = 0 . M cl , Z ′ ) data points with a 2-variables second order polynomial yields the following relation shown in Fig. 3,right panel: f ⋆,ff ( M cl , Z ′ ) = 11 . − . cl ) + 0 . cl )] − . Z ′ + 3 . Z ′ log(M cl ) − . Z ′ [log(M cl )] +2 . Z ′ − . Z ′ log(M cl ) + 0 . Z ′ [log(M cl )] . (3.5)Using Eq. 3.5, it is then possible to calculate h f ⋆,ff i : h f ⋆,ff i ( Z ′ , Σ g ) = Z max ( M cl,max ,M GMC ) M cl,min f ⋆,ff ( M cl , Z ′ ) N ( M cl ) dM cl . (3.6)42 SF2A 2011 Fig. 4.
Characteristic clump mass as a function of the gas surface density (Eq. 3.3, top panel) and the star formationefficiency per unit free-fall time in this feedback regulated model of star formation (lower panel). Adapted from Dib(2011a).
Fig. 4 (bottom) displays h f ⋆,ff i ( Z ′ , Σ g ) for values of Z ′ in the range [0 . − g = 85 M ⊙ pc − below which clumps are pressurised by their internal stellar feedback, suchthat Σ cl = Σ g,crit where Σ g < Σ g,crit and Σ cl = Σ GMC = Σ g when Σ g ≥ Σ g,crit . With the above elements, thestar formation law can be re-written as:Σ SF R = 810 f H (Σ g , c, Z ′ )Σ g × h f ⋆,ff i ( Z ′ ) M / char, ; Σ g
85 M ⊙ pc − < h f ⋆,ff i ( Z ′ ) M / char, (cid:16) Σ g
85 M ⊙ pc − (cid:17) / ; Σ g
85 M ⊙ pc − ≥ , (3.7)where Σ SF R is in M ⊙ yr − kpc − , M char is given by Eq. 3.3, and h f ⋆,ff i by Eqs. 3.5 and 3.6. Fig. 5 displays theresults obtained using Eq. 3.7 for Σ g values starting from low gas surface densities up to the starburst regime.The results are calculated for the metallicity values of Z ′ = [0 . , . , . , , ∼ . Fig. 5.
Star formation laws in the feedback-regulated star formation model. Overplotted to the models are the normaland starburst galaxies data of Kennicutt (1998) and the combined sub-kpc data (4478 subregions) for 11 nearby galaxiesfrom Bigiel et al. (2008,2010). The Bigiel et al. data is shown in the form of a 2D histogram with the colour codingcorresponding, from the lighter to the darker colours to the 1,5,10,20, and 30 contour levels. The displayed theoreticalmodels cover the metallicity range Z ′ = Z/Z ⊙ = [0 . , We have presented a model for star formation in protocluster clumps of difference metallicities. The modeldescribes the co-evolution of the dense core mass function and of the IMF in the clumps. Cores form uniformlyover time in the clumps following a prescribed core formation efficiency per unit time. Cores contract overtimescales which are a few times their free fall time before they collapse to form stars. Feedback from thenewly formed OB stars ( > ⊙ ) is taken into account and when the ratio of the cumulated effective kineticenergy of the winds to the gravitational energy of the system (left over gas+stars) reaches unity, gas is expelledfrom the clump and further core and star formation are quenched. The radiation driven winds of OB starsare metallicity dependent. Metal rich OB stars inject larger amount of energy into the clump than their lowmetallictiy counterparts and thus help expel the gas on shorter timescales. This results in reduced final starformation efficiencies in metal rich clumps in comparison to their low metallicity counterparts. Both the finalstar formation efficiency and the gas expulsion timescales are combined for a grid of clump models with differentmasses and metallicities in order to calculate the star formation efficiency per unit time ( f ⋆,ff ) in this feedbackregulated model of star formation. We calculate the characteristic value of f ⋆,ff for a clump mass distributionassociated with a gas surface density, Σ g . This is combined with a description of the molecular mass fraction asa function of Σ g and the assumption that there is a critical surface gas density (Σ g = 85 M ⊙ pc − ) above whichthe protocluster clumps and their parent giant molecular clouds switch from being pressurised from within bystellar feedback to being confined by the external interstellar medium pressure. The combination of these threeelements allows us to construct the star formation laws in galaxies going from low gas surface densities up tothe starburst regime. Our models exhibit a dependence on metallicity over the entire range of considered gassurface densities and fits remarkably well the observational data of Bigiel et al. (2008,2010) and Kennicutt(1998). This dependence on metallicity of the KS relation may well explain the scatter (or part of it) that isseen in the observationally derived relations.44 SF2A 2011 I would like to thank the organizers of the workshop
Stellar and Interstellar physics for the modelling of the Galaxy and itscomponents for the opportunity to speak, and would like to acknowledge the generous financial support from the SF2A. S.D. andS.M. acknowledge the support provided by STFC grant ST/H00307X/1.