Applications of the IGIMF-theory to the astrophysics of galaxies
aa r X i v : . [ a s t r o - ph . C O ] N ov UP: Have Observations Revealed a Variable Upper End of the Initial Mass Function?ASP Conference Series, Vol. **Volume Number****Author** c (cid:13) Applications of the IGIMF-theory to the astrophysics of galaxies
Jan Pflamm-Altenburg , Carsten Weidner , and Pavel Kroupa Argelander-Institut f¨ur Astronomie (AIfA), Universit¨at Bonn, Auf dem H ¨ugel71, D-53121 Bonn, Germany Scottish Universities Physics Alliance (SUPA), School of Physics andAstronomy, University of St. Andrews, North Haugh, St. Andrews, Fife KY169SS, UK
Abstract.
The functional form of the galaxy-wide stellar initial mass function isof fundamental importance for understanding galaxies. So far this stellar initial massfunction has been assumed to be identical to the IMF observed directly in star clusters.But because stars form predominantly in embedded groups rather than uniformly dis-tributed over the whole galaxy, the galaxy-wide IMF needs to be calculated by addingall IMFs of all embedded groups. This integrated galactic stellar initial mass function(IGIMF) is steeper than the canonical IMF and steepens with decreasing SFR, leadingto fundamental new insights and understanding of star forming properties of galaxies.This contribution reviews the existing applications of the IGIMF theory to galactic as-trophysics, while the parallel contribution by Weidner, Pflamm-Altenburg & Kroupa(this volume) introduces the IGIMF theory.
1. Introduction
Since Salpeters fundamental study it is known that stars do not form with arbitrarymasses, but their masses follow a universal distribution function, which is called theinitial stellar mass function (IMF). The IMF, ξ ( m ), determines the number of newlyformed stars, dN , in the mass interval [ m , m + dm ], and is mathematically defined by ξ ( m ) = dN / dm .Following its definition, i.e. counting stars, the IMF has only been determineddirectly in individual star clusters and clusterings. But the interpretation of galaxy-wideobservable quantities, e.g. chemical abundances or luminosities in di ff erent pass bands,and the transformation of them into physical quantities such as total stellar masses, starformation rates, or chemical evolution histories require the knowledge of the galaxy-wide mass function of all newly formed stars.Given the lack of galaxy-wide star number counts it has been assumed that thegalaxy-wide IMF is identical to the IMF in individual star clusters. But the fact thatall stars form in a clustered mode requires the galaxy-wide IMF to be calculated byadding all IMFs of all embedded star clusters or groups. This leads to the concept ofthe integrated galactic stellar initial mass function (IGIMF). It is steeper above about Only about 10 per cent of the ”embedded clusters” survive to become bound long-lived star clusters. -10-8-6-4-2 0 2-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 l og ( ξ ( m ) / N t o t ) log (m / M ⊙ ) c anon i c a l I M F -5 -4 -3 -1 -2 SFR (M ⊙ / yr) -6-4-2 0 2 -10 -5 0 5 l og ( S F R / M ⊙ y r - ) log (L H α / 10 erg s -1 ) Milky Way-typeSMC-typedIrrs
IMFIGIMF
Figure 1. Left: The IGIMF for di ff erent SFRs in M ⊙ / yr. Each IGIMF is nor-malised such that R ξ IGIMF ( m ) dm =
1. Right: The recalibrated H α -SFR tracer. M ⊙ than the individual IMF in the embedded star clusters and steepens with de-creasing SFR. The functional form of the IMF in each star cluster is universal. Butthe combination of two facts, i) that only high-mass star clusters host high-mass stars,and ii) that only high-SFR galaxies host high-mass star clusters leads to a decrease ofthe high-mass star fraction and a steepening of the IGIMF with decreasing SFR (Fig 1,left) . This decrease of the OB-star fraction with decreasing SFR is called the IGIMF-e ff ect.
2. IGIMF - Applications
In the following we summarise all applications where the constant galaxy-wide IMFhas already been replaced by the IGIMF and list them in Tab. 1.
Table 1. IGIMF - applicationsApplication Reference L H α –SFR relation Pflamm-Altenburg, Weidner, & Kroupa (2007)SFR– M gas relation Pflamm-Altenburg & Kroupa (2009)Gas depletion time scales Pflamm-Altenburg & Kroupa (2009) H α / FUV–SFR relation Pflamm-Altenburg, Weidner, & Kroupa (2009)Mass–metallicity relation K¨oppen, Weidner, & Kroupa (2007)[ α / Fe]–velocity dispersion relation Recchi et al. (2009)Colours of star-forming galaxies Haas & Anders (2010)Solar neighbourhood properties Calura et al. (2010)Radial H α cuto ff in disk galaxies Pflamm-Altenburg & Kroupa (2008) A tool to calculate an IGIMF and to fit it with a multi-part power law as required as the inputIMF for P egase has been presented in Pflamm-Altenburg et al. (2009) and can be downloaded from pplications ofthe IGIMF-theory tothe astrophysics ofgalaxies 3 -6 -5 -4 -3 -2 -1 S F R ( M ⊙ y r - ) M gas (M ⊙ ) M gas1.26 M gas1.87 IMF -6 -5 -4 -3 -2 -1 S F R ( M ⊙ y r - ) M gas (M ⊙ ) M gas0.99 IGIMF
Figure 2. The SFR of galaxies based on H α luminosities as a function of theirtotal gas mass (from Pflamm-Altenburg & Kroupa 2009). Left: For the case of aconstant galaxy-wide IMF. Right: For the IGIMF-theory. L H α –SFR relation Because the H α luminosity, L H α , of star forming galaxies depends on the productionrate of ionising photons by high-mass stars, it is expected that the usage of L H α as astar formation tracer needs recalibration in the IGIMF context. This recalibration ispresented in Pflamm-Altenburg et al. (2007) and can be seen in Fig. 1 (right). MilkyWay-type galaxies have the same H α luminosity for the same SFR in both contexts,constant IMF or IGIMF (grey shaded rectangle). The di ff erences start to appear forSMC-type galaxies (grey shaded circle). The SFRs of dIrrs with the faintest H α lumi-nosities are underestimated by up to two orders of magnitude (gray shaded triangle). M gas relation of star forming galaxies In the classical constant galaxy-wide IMF picture, total H α luminosities transform lin-early into SFRs. It is found that in this case SFRs of galaxies decrease faster than theirtotal gas mass. Furthermore the SFR– M gas relation steepens for galaxies less massivethan SMC-type galaxies (Fig. 2, left). In the IGIMF-theory the SFR– L H α relation be-comes non-linear for galaxies which are fainter in H α than SMC-type galaxies. Whenusing the recalibrated H α star formation tracer in order to correct for the IGIMF-e ff ectthe IGIMF-theory reveals a linear SFR– M gas relation over four orders of magnitude ofthe gas mass (Fig. 2, right). As a consequence of the revised H α star formation rates in the IGIMF theory, the cor-responding gas depletion time scales of star forming galaxies change, too. For an as-sumed constant galaxy-wide IMF the gas depletion time scale, τ gas = M gas / SFR, in-creases with decreasing galaxy gas mass as a result of a steeper than linear SFR– M gas relation. This has lead to the interpretation that dwarf irregular galaxies are ine ffi cientin forming stars compared to large disk galaxies. The IGIMF based SFR– M gas rela-tion is linear and therefore the gas depletion time scales are constant at about 3 Gyr(Pflamm-Altenburg & Kroupa 2009). In the IGIMF-context all star forming galaxieshave the same star formation e ffi ciency. Jan Pflamm-Altenburg, Carsten Weidner, and PavelKroupa -4 -3 -2 -1 -22-20-18-16-14-12-10-8-6 S F R [ M ⊙ y r - ] M FUV [AB mag]
IGIMF IMF 22 23 24 25 26 27 28 29 35 36 37 38 39 40 41 42-5-4-3-2-1 0 1-6 -5 -4 -3 -2 -1 0 L F U V / e r g s - H z - l og ( S F R F U V , K / M ⊙ y r - ) L H α / erg s -1 log (SFR H α ,K94 / M ⊙ yr -1 )IMFIGIMF Figure 3. Left: The SFR–FUV-magnitude relation for the constant IMF case andpredicted by the IGIMF-theory for solar metallicity (Pflamm-Altenburg et al. 2009).Right: H α and FUV luminosities expected in the case of a constant IMF and pre-dicted by the IGIMF-theory compared with observation by Lee et al. (2009) of localvolume star-forming galaxies. The stellar-mass buildup time, τ ∗ = M ∗ / S FR , is a measure for the ratio of the averagepast and the current SFR of a galaxy. Using the canonical SFRs calculated with aninvariant IMF it follows that τ ∗ becomes longer than a Hubble time for dwarf galaxies.This is unphysical and is also in contradiction to the observationally found downsizingresult. Using instead the IGIMF-based SFRs, τ ∗ becomes shorter than a Hubble timefor virtually all galaxies and decreases towards less-massive galaxies being consistentwith downsizing (Pflamm-Altenburg & Kroupa 2009). H α / FUV–SFR relation
The H α luminosity is produced by recombining hydrogen ionised by the Lyman α con-tinuum photons of high-mass stars and shows therefore a strong IGIMF-e ff ect. Con-trary, the far ultraviolet (FUV) luminosity of galaxies is produced mainly by long-livedB-stars. Thus, the FUV-flux of galaxies also shows an IGIMF-e ff ect (Fig. 3, left) butless so than the H α luminosity. E.g. for a default SFR of 10 − M ⊙ yr − (10 − M ⊙ yr − )the IGIMF-e ff ect for H α is 2.0 dex (5.0 dex), whereas the IGIMF-e ff ect for FUV is1.22 mag (1.91 mag) or 0.48 dex (0.77 dex) (compare Fig. 1 right and Fig. 3 left). Inthis context the IGIMF-e ff ect means the di ff erence of the logarithm of the H α luminos-ity and FUV-luminosity between the IGIMF model and the constant-IMF case.It should be noted that due to the IGIMF-e ff ect in the FUV-luminosity the SFRsof galaxies which are calculated from FUV-luminosities and a classical linear FUV-luminosity–SFR relation resulting from an assumed constant IMF are closer to the trueunderlying SFR than H α based SFRs, but they are still lower.Combining the L FUV –SFR and the L H α –SFR relations predicted by the IGIMF-theory a relation between the L H α / L FUV -ratio and the H α luminosity has been predictedin Pflamm-Altenburg et al. (2009). This prediction is in quantitative agreement withthe observations by Lee et al. (2009) (Fig. 3, right).pplications ofthe IGIMF-theory tothe astrophysics ofgalaxies 5 The IGIMF-e ff ect describes the decreasing high-mass star fraction of all newly formedstars with decreasing SFR. Thus the amount of freshly produced chemical elements,which are synthesised by high-mass stars (e.g. oxygen in SNII), per total newly formedstellar mass must decrease with decreasing SFR. It is thus expected within the IGIMFtheory that the metallicity as well as the detectable, i.e. e ff ective, yield of oxygen ofgalaxies increase with increasing galactic SFR and therefore stellar mass. K ¨oppen et al.(2007) have shown that both relations are a natural consequence of the IGIMF-theoryand agree with the observations (Fig 4, left). In order to explain the observed mass-metallicity relation of galaxies with a constant galaxy-wide IMF the occurrence ofmetal-enriched outflows has to be speculated. This requires that a large fraction offreshly produced metals must escape from the galaxy and is in principle equivalent toreducing the number fraction of high-mass stars as pointed out by K ¨oppen et al. (2007).On first sight this might be plausible as lower-mass galaxies have shallower grav-itational potentials than large disk galaxies, and expanding supernova shells containingfreshly produced metals might escape easier from dwarf galaxies. In order to breakthe degeneracy between the IGIMF-model, on the one-hand side, and the constant IMFmodel plus metal enriched outflows, on the other hand side, one has to concentrateon galaxies with di ff erent SFRs but equal gravitational potentials. This can be doneby comparing low-surface brightness galaxies (LSBs) with normal disk galaxies hav-ing the same rotational velocity which is a proxy for the deepness of the gravitationalpotential.For the same potential, the constant-IMF model combined with metal enrichedoutflows would predict that the e ff ective yields are higher for LSBs than for normal diskgalaxies, because normal disk galaxies have higher SFRs and larger feedback by super-novae and metals should escape easier reducing the e ff ective, i.e. detectable, yields.The IGIMF-model, on the other hand, predicts that the e ff ective yields are higher fornormal disk galaxies than for LSBs because normal disk galaxies have higher SFRs andtherefore flatter IGIMFs and a larger fraction of high-mass stars.Fig. 4 (right) shows the e ff ective oxygen yields of normal disk galaxies and LSBs.At a rotational velocity of ≈
100 km / s the regimes of normal disk galaxies and LSBsoverlap. In the region of the same rotational velocity normal disk galaxies have highere ff ective yields than LSBs in agreement with the IGIMF-theory (Pflamm-Altenburg &Kroupa – in prep.). α / Fe]–velocity-dispersion relation
If di ff erent chemical elements are produced in stars with di ff erent masses then the cor-responding e ff ective yields must have di ff erent IGIMF-e ff ects. E.g. oxygen (or any α -element as e.g Mg) is produced in SNII which have high-mass stars as progenitors,whereas iron is mainly produced in SNIa which have white dwarfs as their progenitors.Thus, the oxygen production in galaxies should decrease faster with decreasing SFRthan the iron production. In other words the IGIMF-e ff ect for oxygen is stronger thanthe IGIMF-e ff ect for iron.In early-type galaxies the observed velocity dispersion scales with the total stellarmass of the galaxies. As high-mass galaxies must have had a higher SFR than low-massgalaxies it is expected in the IGIMF-theory that the [ α / Fe] abundance ratio decreaseswith decreasing velocity dispersion of the galaxies. Recchi et al. (2009) have shownthat the chemical evolution of early type galaxies in the context of the IGIMF-theory is Jan Pflamm-Altenburg, Carsten Weidner, and PavelKroupa
STARS /M ) + l og ( O / H ) IMF IGIMFObservations -3.2-3-2.8-2.6-2.4-2.2-2-1.8-1.6 0 50 100 150 200 l og ( y i e l d e ff, O ) v rot / km s -1 Figure 4. Left: The mass–metallicity relation of galaxies in the IGIMF-theory(symbols) as calculated in K¨oppen et al. (2007) compared with the observations byTremonti et al. (2004) (dashed lines). Right: E ff ective oxygen yields of LSBs (opensquares) and normal spiral galaxies (black squares) based on a data compilation fromGarnett (2002). in quantitative agreement with the observations. Furthermore, the IGIMF-models pro-duce a steeper [ α / Fe]– σ relation in low-mass galaxies, as is observed, whereas standardconstant IMF models only provide theoretical [ α / Fe] ratios in agreement with observedratios in high-mass early-type galaxies, only reproduce the steepening of the [ α / Fe]– σ relation for low-mass early-type galaxies if element-selective outflows are speculatedto occur. The solar neighbourhood is the ideal test place to compare IGIMF-predictions withobservations as it is i) a result of the composition of numerous dissolved star clustersand thus represents a composite stellar population in terms of the IGIMF-theory, andii) due to its proximity star counts can be performed and metallicity determinations aremost accurate. Calura et al. (2010) vary systematically the slope of the embedded clus-ter mass function, β , and calculate theoretically the present-day stellar mass function(PDMF) and chemical enrichment in the IGIMF-context. A choice of β = The IGIMF-e ff ect of a galaxy-wide integrated property becomes smaller if the frac-tion of contributing long-lived low-mass stars increases. Therefore, optical bass bandsshould exhibit an IGIMF-e ff ect, too, but less strongly than the IGIMF-e ff ect for FUV.First studies to explore the IGIMF-e ff ect on widely-used pass bands as e.g. U, B, Vhave been presented in Haas & Anders (2010). Their main result is that optical coloursvary only slightly within di ff erent IGIMF-models and that this variation is smaller thanthe intrinsic scatter within a morphological class of galaxies. Therefore, optical coloursof galaxies can not be used to constrain details of the IGIMF theory.Long-lived low-mass stars are the main contributor to optical luminosities andtherefore the star-formation history (SFH) will have an important influence on theIGIMF-e ff ect integrated over time. Further studies are required in order to explorethe SFH dependence on galactic colours in the IGIMF context.pplications ofthe IGIMF-theory tothe astrophysics ofgalaxies 7 α cuto ff in star-forming disk galaxies The IGIMF-e ff ect in galaxies is a result of a decreasing mass of the heaviest star clus-ter with decreasing SFR. This can be physically understood as high-SFR galaxies havehigher gas densities and more material is locally available to form high-mass star clus-ters. Within galaxies the gas density decreases in general with increasing galactocentricradius. Therefore we would expect a local IGIMF-e ff ect, i.e. a composite IMF of manystar forming regions should be steeper in the outer disk of star forming galaxies than intheir inner regions.A local integrated galactic stellar initial mass function (LIGIMF) can be con-structed straightforwardly be replacing all relevant quantities in the IGIMF formula-tion by their corresponding surface densities (Pflamm-Altenburg & Kroupa 2008): TheLIGIMF then defines the number of newly formed stars, dN , in the mass interval from m to m + dm per unit area at the location (x, y) in a star-forming disk galaxy and iscalculated by adding all IMFs of all locally newly formed star clusters ξ LIGIMF ( m , x , y ) = Z M ecl , max , loc ( x , y ) M ecl , min ξ M ecl ( m ) ξ LECMF ( M ecl , x , y ) d M ecl , (1)where ξ LECMF ( M ecl , x , y ) is the local embedded cluster mass function surface density(LECMF), which defines the number of newly formed star clusters, dNecl , in the massinterval from M ecl to M ecl + dM ecl per unit area at the location (x, y) in a star-formingdisk galaxy, and ξ M ecl ( m ) is the IMF of an embedded star cluster with total stellar M ecl .In order to express the dependence of the local upper limit of the LECMF on thelocal gas surface density, the ansatz, M ecl , max , loc ( x , y ) = M ecl , max Σ gas ( x , y ) Σ gas , ! γ , (2)is made, where Σ gas ( x , y ) and Σ gas , are the gas surface densities at the location (x, y)and at the center of the galaxy and M ecl , max is the most massive star cluster in the galaxydetermined by the total SFR.The LECMF is locally normalised such that δ t Σ SFR ( x , y ) = Z M ecl , max , loc ( x , y ) M ecl , min ξ LECMF ( M ecl ) M ecl d M ecl , (3)where Σ SFR is the star-formation surface density, δ t ≈
10 Myr is the time span requiredto populate the cluster mass function completely (Weidner et al. 2004).For an exponential gas disk and a Kennicutt-Schmidt law with general exponent N the average radial star-formation rate surface density can be calculated (see supplementof Pflamm-Altenburg & Kroupa 2008 for details). As the the gas surface density de-creases with increasing galactocentric radius the local upper mass limit of the LECMFdecreases and the local fraction of high-mass stars decreases, too. Consequently, theLIGIMF steepens with increasing galactocentric radius and the radial H α surface den-sity decreases faster than the radial FUV surface density.For a choice of N = γ = of star-forming disk galaxies an agreementbetween i) the observed radial H α and FUV-luminosity profiles (Fig. 5 left), and ii) amatch of an apparent star formation cuto ff in low gas density environments (Fig. 5 left)can be achieved. Jan Pflamm-Altenburg, Carsten Weidner, and PavelKroupa -2-1 0 1 2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 l og ( Σ S F R / M ⊙ p c - G y r - ) r / r cutoff UV-IGIMFH α -IGIMF H α - ob s e r v a t i on s U V - ob s e r v a t i on s l og ( Σ H α / e r g s - p c - ) log ( Σ gas / M ⊙ pc -2 ) IMF (L)IGIMF
Observations
Figure 5. Left: Observed classical (constant IMF) FUV (thin solid lines) andH α SFRs (thin dotted lines) (Boissier et al. 2007) and expected classical SFRs inthe IGIMF-context (thick solid and dotted lines) (from Pflamm-Altenburg & Kroupa2008). Right: Local observed H α surface densities as function of the local gassurface density (black squares) compared with the expectations from an assumedconstant IMF and the IGIMF for a linear Kennicutt-Schmidt law ( N =
1) and nostar-formation cuto ff (from Pflamm-Altenburg & Kroupa 2008).
3. Conclusion
We have presented a summary of astrophysical topics where the constant galaxy-wideIMF used so far has been replaced by the IGIMF. The common result is that the usageof a constant galaxy-wide IMF requires including additional assumptions in order to ex-plain or understand physical properties of galaxies. The IGIMF-theory instead providesnatural explanations of galactic properties with no need of further ad-hoc assumptions.It should be noted that how the IGIMF varies with SFR is not arbitrary but rests entirelyon the clustered nature of star formation deduced from nearby star-forming events andis quantitatively determined by the galaxy-wide SFR or the local SFR density within agalaxy.
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