(2) CONVERTING H LUMINOSITIES INTO STAR FORMATION RATES For a given global SFR, the total number of massive stars based on an IGIMF is less than in the standard procedure. Thus, the expected IGIMF H luminosity is less than the H luminosity calculated in the standard way, implying that SFRs of galaxies determined by the standard method are systematically underestimated. After a summary of the IGIMF basics, we develop a relation between the SFR and the produced H luminosity for different IGIMF models, and provide easy-to-use fitting functions (x 2). In x 3, we apply our LH -SFR relation to the dwarf irregular galaxies in the Sculptor cluster, and discuss the changes of some of their parameters. Finally, we explore the detection limits for star formation (x 4). 2. LH -SFR RELATION For a given SFR, we assume that star formation takes place in star clusters distributed according to the embedded cluster mass function (ECMF) of stellar masses (i.e., the function of the stellar mass content in clusters before they expel their residual gas), ecl (Mcl ). These star clusters are formed within a time span, t, which is called the ‘‘star formation epoch’’ of the galaxy (Weidner & Kroupa 2005). The total mass of all formed stars, MSFR , in all star clusters within the epoch t is then given by Z. Mecl; max. MSFR ¼ SFR t ¼. Mecl ecl ðMecl Þ dMecl ;. ð1Þ. Mecl; min. where Mecl;min and Mecl;max are, respectively, the least massive and most massive embedded clusters able to form. The total ensemble of all newly formed stars has a mass spectrum described by the IGIMF, IGIMF (m), constructed below. A star with mass m produces a total number Nion;t (m) of ionizing photons during the epoch t. The total number of ionizing photons emitted by all young stars within t is given by Z mmax IGIMF (m)Nion;t (m) dm; ð2Þ Nion;t ¼ mlow. where mlow and mmax are, respectively, the minimum and maximum stellar masses. The calculation of Nion;t will be described in x 2.2. Assuming that a fraction, , of all ionizing photons leads to H emission in the surrounding gas, the resulting H luminosity is then LH ¼ 3:0207 ; 1012 erg Nion =t;. ð3Þ. where the energy of one H photon is 3:0207 ; 1012 erg. Throughout this paper, we assume that ¼ 1. It can be argued that the time span t can be removed in this formulation by considering rates directly. But the total mass, MSFR , defined by equation (1) and depending explicitly on t, is an upper limit of the ECMF, as no cluster more massive than the available material can be formed. 2.1. IGIMF Larsen (2000, 2002b, 2002a) and Larsen & Richtler (2000) found that the V-band luminosity of the brightest young cluster correlates with the global SFR. Based on the conclusion by Larsen (2002b) that the so-called superclusters are just the young and massive upper end of a cluster mass function, Weidner et al. (2004). 1551. derived a relation between the maximum embedded cluster mass, Mecl;max , in a galaxy and the current global SFR, log Mecl;max ¼ log k ML þ 0:75 log SFR þ 6:77;. ð4Þ. where k ML is the mass-to-light ratio, typically 0.0144 for young (<6 Myr) stellar populations. They showed that this relation can be reproduced theoretically if the ECMF is a power law with a slope of 2.35, and the entire star cluster population ranging from Mecl;min 5 M to Mecl;max is born within a time span of t 10 Myr, independently of the SFR. The concept of a relation between the SFR of a galaxy and the mass of the most massive star cluster has been developed further to derive the star formation history of galaxies (Maschberger & Kroupa 2007). Each cluster with mass Mecl then forms N stars between the mass limits mlow and mmax , according to the canonical IMF, Z. mmax. Mecl ðmÞ dm:. N¼. ð5Þ. mlow. This canonical IMF, Mecl (m), is a two-part power law, and is here calculated using the algorithm from Pflamm-Altenburg & Kroupa (2006). The normalization constant of the IMF in an individual cluster, Mecl (m), with a mass Mecl and a maximum stellar mass, mmax , in this star cluster is determined by the two equations ( Weidner & Kroupa 2004) Z mmax mMecl (m) dm; ð6Þ Mecl ¼ mlow. Z. mmax. 1¼. Mecl (m) dm:. ð7Þ. mmax. The existence of an upper physical stellar mass, mmax , of about 150 M has been found in statistical examinations of several clusters (Weidner & Kroupa 2004; Figer 2005; Oey & Clarke 2005; Koen 2006; Maı́z Apellániz et al. 2007). The canonical IMF, Mecl (m) ¼ km i ; has the slopes 8 > < 1 ¼ þ1:30; 0:08 m=1 M 0:50; 2 ¼ þ2:35; 0:50 m=1 M 1:00; > : 3 ¼ þ2:35; 1:00 m=1 M mmax;. ð8Þ. ð9Þ. as used by Weidner & Kroupa (2005), who constructed different IGIMF models. The third slope, above 1 M, has been introduced for varying the IMF slope and constructing noncanonical models (as in Table 1). The pair of the two integral expressions (eqs.  and ) cannot be solved analytically to obtain the relation between the star cluster mass, Mecl , and its most massive star, m max , but the existence of a unique solution can be shown (Pflamm-Altenburg & Kroupa 2006). Using a simple bisection method, this relation can be calculated numerically (Fig. 1, crosses). The numerical solution can be very well fitted by the function y ¼ axðb n þ x n Þ1=n þ c;. ð10Þ. where y ¼ logðm max /1 M Þ, x ¼ logðMecl /1 M Þ, a ¼ 2:56, b ¼ 3:82, c ¼ 0:38, and n ¼ 9:17 (Fig. 1, solid line)..
(3) 1552. PFLAMM-ALTENBURG, WEIDNER, & KROUPA. Vol. 671. TABLE 1 Slopes and Mass Limits of the IMF and ECMF for Four IGIMF Models Parameter. Standard. Minimal-1. Minimal-2. Maximal. m1 /1 M .............. 1 ......................... m2 /1 M .............. 2 ......................... m3 /1 M .............. 3 ......................... M1 /1 M .............. 1 .......................... M2 /1 M .............. 2 ........................... 0.08 1.30 0.5 2.35 1.0 2.35 5 2.35 50 2.35. 0.08 1.30 0.5 2.35 1.0 2.35 5 1.0 50 2.0. 0.08 1.30 0.5 2.35 1.0 2.35 ... ... 50 2.0. 0.08 1.30 0.5 2.35 1.0 2.7 5 2.35 50 2.35. Notes.—The IMF slopes i refer to a power law, (m) / m i , on a mass interval between mi and mi þ1 , where mi þ1 mmax . The ECMF slopes i refer to a i , on a mass interval between Mi and Mi þ1 , where Miþ1 ¼ power law, ecl / Mecl Mecl;max (SFR) (eq.  ).. This function behaves linearly for xTb [ y ¼ (a /b)x þ c], and is constant for x 3b ( y ¼ a þ c). The transition occurs at x b, whereas a large n describes a rapid transition. Numerical simulations of star formation in clusters ( Bonnell et al. 2004) also indicate that the mass of the most massive star scales with the system mass. The IGIMF is then calculated by adding all stars in all clusters, as already noted by Vanbeveren (1982, 1983) Z. Mecl;max (SFR). IGIMF (m) ¼. Mecl (m)ecl ðMecl Þ dMecl :. ð11Þ. Mecl;min. 2.2. Number of Ionizing Photons The total number of ionizing photons, Nion;t (m), emitted by a star of mass m within a time t ¼ 10 Myr, is calculated by integrating the emission rate of ionizing photons, Nion (m; t), over t, Nion;t. Z ¼. Fig. 2.— Total number of ionizing photons, Nion;t , emitted by a star with a mass m in solar masses within the time t ¼ 10 Myr at the beginning of the life of the star.. In order to derive the emission rates of ionizing photons, a grid of 4000 stars linearly spaced between 0.01 and 150 M is evolved from 0 to 20 Myr in time steps of 0.1 Myr. For stars below 50 M , the stellar evolution package by Hurley et al. (2000) is used. Above that limit until 120 M , the stellar parameters are interpolated from models by Meynet & Maeder (2003) without rotation. For the most massive stars above 120 M , formulae fitted to the massive star models by Schaller et al. (1992) are used in extrapolation.1 The physical parameters provided by the models are then used to find appropriate stellar spectra for each star at every time step from the BaSeL stellar library (Lejeune et al. 1997, 1998; Westera et al. 2002). From these spectra, the part of the luminosity that can ionize hydrogen (all radiation with a wavelength k 912 8), i.e., the ionizing luminosity, Lion (m; t), is deduced for each star. Finally, the number of ionizing photons, Nion (m; t), can be calculated from Nion (m; t) ¼ Lion (m; t) ; 1010:5 ;. t. Nion (m; t) dt:. ð12Þ. 0. Stars with lifetimes shorter than t contribute all of their emitted ionizing photons.. ð13Þ. from Stahler & Palla (2005). The resulting Nion;t (eq.  ) is shown in Figure 2. In order to compare the results obtained with the set of stellar evolution models used here 2 with other models, the same procedure described above was applied to the Padova94 tracks for solar metallicity ( Bressan et al. 1993), which are preferred by Bruzual & Charlot (2003). As Bruzual & Charlot (2003) also use the BaSeL library of stellar spectra, it is possible to use our procedure to obtain Nion , but with the different stellar models. The dependence of Nion on stellar mass is plotted in Figure 3 using the Padova94 tracks (dashed line) and using our models (solid line). The two compare very well. 2.3. Models Although the IMF of young star clusters seems to be universal and widely independent of their environment (Kroupa 2001, 2002), the universality of the ECMF slope is not an established result, as summarized in Weidner & Kroupa (2005). To be consistent with our previous work, the four IGIMF models here are the same as those defined in Weidner & Kroupa (2005). Both. Fig. 1.— The mmax -Mecl relation defined by eqs. (6 ) and ( 7 ) with mmax ¼ 150 M and fitted by eq. (10) (solid curve). Only each 30th point of the numerical solution is marked by a cross.. 1 An extensive description of the fitting formulae can be found in Weidner & Kroupa (2006). 2 A more detailed description of the models will be given in C. Weidner & P. Kroupa (2008, in preparation)..
(4) No. 2, 2007. CONVERTING H LUMINOSITIES INTO STAR FORMATION RATES. Fig. 3.— Total number of ionizing photons (Nion;<10Myr ) vs. stellar mass for the models used here (solid line) and the Padova94 tracks (dashed line).. the IMF and the ECMF are multipower laws, (m) / m i and i . All slopes and mass limits of all four IGIMF ecl (Mecl ) / Mecl scenarios are summarized in Table 1. 1. The Standard IGIMF is based on a canonical IMF (eq. ) and an ECMF with a power ¼ 2:35 between 5 M and Mecl, max(SFR). A slope of 2.35 provides the best agreement between the theoretical and the observed relations between the actual SFR and the mass of the heaviest young star cluster ( Weidner et al. 2004). Furthermore, the theoretical formation timescale of star clusters has a constant value of 10 Myr independent of the cluster mass if a slope of 2.35 is chosen. In order to explore the dependence of the IGIMF on the ECMF, we test three additional scenarios. 2. The Minimal-1 scenario has a slope of 1 ¼ 1 for small cluster masses (5 Mecl /1 M 50) and 2 ¼ 2 for higher masses (50 Mecl /1 M Mecl; max ) and a canonical IMF (eq. ). 3. The Minimal-2 scenario has a truncated ECMF with ¼ 2 (50 Mecl /1 M Mecl;max ), i.e., no star clusters below 50 M are formed, and a canonical IMF (eq. ). 4. The Maximal scenario consists of an IMF (eq.  ) with a steeper high-mass star slope of 3 ¼ 2:7 and an ECMF with ¼ 2:35 between 5 M and Mecl, max(SFR). Both minimal scenarios produce a larger number of massive star clusters, and therefore more massive stars than the Standard scenario. The Maximal scenario produces fewer massive stars than the Standard scenario. Thus, for a given SFR, both minimal scenarios produce a higher H luminosity, and the Maximal scenario a lower H luminosity, than the Standard scenario. In order to explore the change of the classical linear LH -SFR relation due to the effect of the IGIMF, we compare the IGIMF models with two scenarios in which the IGIMF is invariant, i.e., independent of the SFR. These invariant scenarios, as well as the linear scenarios by Kennicutt et al. (1994) provide theoretical H luminosities based on the assumption that the galaxy-wide distribution function of newly formed stars corresponds to the IMF observed in young clusters. Thus, the slope of the IMF is independent of the total SFR, and the IGIMF is identical to the IMF in young star clusters. In our first invariant model, the IMF is chosen to have the canonical form defined by equation (9), and uses the same stellar properties calculated in x 2.2, where m max ¼ m max ¼ 150 M. Our second invariant model consists of a Salpeter IMF with a slope of 2.35 between 0.1 and 100 M and the same stellar. 1553. Fig. 4.— Normalized IGIMF in the Standard scenario for different SFRs (105, 104, 103, 102, 1, and 100 M yr1 ), and the canonical IMF (dashed line). The arrow indicates the mass at which all IGIMFs start to deviate from the underlying canonical IMF [m max Mecl;min ¼ 1:25 M ]. Note that the IGIMFs are not exactly identical for m 1:25 M due to the normalization by the total number of stars in each case.. properties calculated in x 2.2. This IMF is the basis of the widely used linear model derived by Kennicutt et al. (1994), SFR(total) ¼. LH M yr1 : 1:26 ; 1041 erg s1. ð14Þ. 2.4. Results The linearity of the widely used relation, equation (14), implies that if the total SFR is reduced by a certain factor, then the produced H luminosity is reduced by the same factor, because the portion of ionizing massive stars is constant if the IGIMF is identical to a universal IMF. But our concept of the IGIMF is a combination of two effects: first, with decreasing SFR, the upper limit of the ECMF falls (eq. ), and second, the fraction of ionizing massive stars is higher in heavy star clusters than in light star clusters due to the relation between the mass of a star cluster and its most massive star (eq. ). Therefore, the galaxy-wide fraction of massive stars is expected to decrease with decreasing total SFR. The resulting normalized Standard IGIMF, i.e., the Standard IGIMF divided by the total number of stars, is plotted in Figure 4 for different SFRs. As expected, all IGIMFs are steeper than the canonical IMF in the high-mass regime, and the mass of the most massive star of the entire galaxy decreases with decreasing SFR. All IGIMFs start to deviate (Fig. 4, arrow) from each other above a mass threshold that is the maximum stellar mass of the smallest star cluster [mmax (Mecl;min ¼ 5 M ) ¼ 1:25 M ]. Stars below this threshold are formed in all star clusters. Note also that the IGIMF has a lower upper mass limit than the canonical IMF. Because the functional form of the IGIMF is a function of the total SFR, the produced H luminosity is expected to not depend linearly on the total SFR. The resulting relation between the SFR and the produced H luminosity is plotted in Figure 5. This includes all four IGIMF models and the two invariant models described above. In addition, the widely used linear LH -SFR relation, equation (14) (Kennicutt et al. 1994; solid line), and the complete sample of linear models based on different stellar evolution models and different IMFs (Kennicutt et al. 1994; gray shaded area), are plotted as well. The IGIMF models are nearly linear above an H luminosity of about 1040 erg s1, which is the lower value of the normal disk galaxies explored by Kennicutt (1983), whereas the relations.
(5) 1554. PFLAMM-ALTENBURG, WEIDNER, & KROUPA. Vol. 671. For dwarf irregular galaxies having H luminosities between 1036 and 1039 erg s1, the underestimation of the SFRs varies by a factor of at least 20Y160 for the Minimal scenario. In the Maximal scenario, the underestimation of SFRs varies between a factor of 800 and 5000. The SFRs of normal disk galaxies with an H luminosity of about 1040Y1042 erg s1 may be underestimated by a factor of 4Y50. 2.5. Fitting Functions For convenient use of the obtained LH -SFR relation, we provide fitting functions for each IGIMF model. The general form of the fitting function is a fifth-order polynomial, y ¼ a5 x 5 þ a4 x 4 þ a3 x 3 þ a2 x 2 þ a1 x þ a0 :. Fig. 5.— Calculated relation between the SFR and the resulting H luminosity for four different IGIMF models and two invariant models in which the IGIMF is assumed to be the canonical IMF and the Salpeter IMF. The solid line represents the widely used linear relation, eq. (14), by Kennicutt et al. (1994). The gray shaded area marks the full set of linear LH -SFR relations (Kennicutt et al. 1994), combining different stellar evolution models and different IMFs (for details see text).. become much flatter below 1040 erg s1. As expected, the IGIMF models lie above the linear relations. The order of magnitude of the deviation of the IGIMF LH -SFR relation from the linear LH -SFR relation is independent of the explicit choice of the IGIMF model. For high H luminosities, our invariant models are strongly linear, like the Kennicutt relations. Below a SFR of about 105 M yr1, they become shallower. In our invariant model, the IGIMF is replaced by the canonical IMF. But the IMF is normalized such that the total mass contained in it is still given by Z mmax m(m) dm; ð15Þ SFR t ¼. This function is used to fit the data points of each IGIMF model (Fig. 5). The argument and the function value of the fitting function are x ¼ log LH 1041 erg s1 ; ð18Þ f (x) ¼ log SFR M yr1 : The obtained values of the fitting parameters are listed in Table 2 for each model and the fitted models are plotted in Figure 6. The linear parts of our invariant models can be described by LH M yr1 ; 1:89 ; 10 41 erg s1 for LH 2:8 ; 1036 erg s1 ;. SFR ¼. SFR ¼. mmax 150 M ¼ ¼ 104:82 M yr1 ; t 10 Myr. ð16Þ. our LH -SFR relations diverge from linearity. The simple physical boundary condition that stars more massive than SFRt cannot form leads to nonlinear behavior of the LH -SFR relation.. ð19Þ. in the case of the canonical IMF and LH M yr1 ; 3:3 ; 1041 erg s1 for LH 4:9 ; 1036 erg s1 ;. SFR ¼. mmin. where mmax is the minimum of the physical upper mass limit, mmax , and the total mass content, SFR t (a star cannot be more massive than the total mass out of which it is formed ). Note that (m) ¼ Mecl (m) (eq.  ) for mmax ¼ mmax. If the mass available for star formation within the epoch t is less than the physical upper mass limit mmax , then the upper limit of the IMF is given by the amount of available mass. Thus, at a SFR of. ð17Þ. ð20Þ. in the case of the Salpeter IMF. 3. SFRs OF DWARF IRREGULAR GALAXIES To study the effect of the IGIMF in real systems, we now calculate the SFRs of the Sculptur dwarf irregular galaxies. The H luminosities are taken from Skillman et al. (2003), and are already corrected for galactic extinction, but not for N ii contamination, as these dwarf galaxies have typically very low nitrogen abundances. The SFRs are obtained using the fitting function equation (17). The SFRs in Skillman et al. (2003) are based on the classical relation equation (14). Observed H luminosities, linear and IGIMF SFRs, and H i masses, MH i , for 11 Sculptor dwarf irregular galaxies are listed in Table 3. The relation between the total galaxy gas mass, Mgas , and. TABLE 2 List of the Fitting Parameters of the L H -SFR Relation. Model. a5. a4. a3. a2. a1. a0. Standard ................ Maximal ................ Minimal-1.............. Minimal-2............... 2.67E05 2.94E05 2.32E05 3.84E05. 9.30E04 9.97E04 7.51E04 2.43E04. 8.47E03 9.13E03 5.70E03 6.58E03. +3.21E02 +2.67E02 +4.23E02 +3.50E02. +9.64E01 +9.63E01 +8.85E01 +8.94E01. +4.38E01 +1.11E00 1.58E01 1.91E01. LH -range (erg s1) 1:7 ; 1025 7:4 ; 1025 3:1 ; 1025 8:6 ; 1032. Y Y Y Y. 4:1 ; 1043 1:2 ; 1043 2:2 ; 1044 2:3 ; 1044. Note.—Fit parameters of the fitting function (eq.  ) for the four different IGIMF models: Standard, Maximal, Minimal-1, and Minimal-2, and the luminosity range of data points used for the fit..
(6) No. 2, 2007. CONVERTING H LUMINOSITIES INTO STAR FORMATION RATES. 1555. Fig. 6.— Fitted plots of the four IGIMF LH -SFR relations (crosses) using the fit function eq. (17) and the parameters of Table 2 (solid lines).. the SFR can be seen in Figure 7. To account for helium, the total galaxy gas masses are calculated from Mgas ¼ 1:32 MH i ;. ð21Þ. following Skillman et al. (2003). The conventional data based on the Kennicutt relation (asterisks in Fig. 7) can be divided roughly into two parts. Above a. galaxy gas mass of about 2 ;107 M, the SFRs are uniformly distributed around a value of about 7 ; 103 M yr1, with a large scatter ranging from 2:8 ; 104 to 5:0 ; 102 M yr 1. Below a galaxy gas mass of about 2 ; 107 M, there is a steep decline of the SFR with decreasing galaxy gas mass. This relation is changed by the Standard IGIMF effect (Fig. 7, filled squares) in three ways. First, all SFRs are increased. This increase is larger for low SFRs, and therefore mainly affects small galaxy gas masses. Second, the. TABLE 3 SFRs of Sculptor Dwarf Irregular Galaxies. Galaxy. M Hi (106 M ). log LH (erg s1). SFRskill (M yr1). SFRstd (M yr1). SFRmin1 (M yr1). SFRmin2 (M yr1). SFRmax (M yr1). ESO 347-G17 .................. ESO 471-G06 .................. ESO 348-G09 .................. SC 18 ............................... NGC 59............................ ESO 473-G24 .................. SC 24 ............................... DDO 226.......................... DDO 6.............................. NGC 625.......................... ESO 245-G05 ................... 120 163 84.3 5.0 16.7 63.8 2.1 33.9 15.4 118 289. 38.9 38.4 37.5 36.5 38.6 38.2 35.4 38.2 37.6 39.8 39.1. 6:3 ; 103 2:0 ; 103 2:8 ; 104 2:4 ; 105 3:1 ; 103 1:3 ; 103 2:1 ; 106 1:3 ; 103 2:9 ; 104 5:0 ; 102 1:0 ; 102. 4:1 ; 102 1:8 ; 102 5:0 ; 103 1:5 ; 103 2:5 ; 102 1:3 ; 102 5:8 ; 104 1:3 ; 102 5:7 ; 103 2:2 ; 101 5:9 ; 102. 1:6 ; 102 7:9 ; 103 2:6 ; 103 9:3 ; 104 1:0 ; 102 6:0 ; 103 4:0 ; 104 6:0 ; 103 2:9 ; 103 7:1 ; 102 2:2 ; 102. 1:4 ; 102 6:6 ; 103 2:2 ; 103 8:4 ; 104 8:8 ; 103 5:0 ; 103 4:0 ; 104 5:0 ; 103 2:4 ; 103 6:3 ; 102 1:9 ; 102. 1:9 ; 101 8:0 ; 102 2:1 ; 102 6:2 ; 103 1:1 ; 101 5:8 ; 102 2:2 ; 103 5:8 ; 102 2:4 ; 102 1:0 ; 100 2:7 ; 101. Notes.—Integrated H luminosities and determined SFRs by Skillman et al. (2003), using the Kennicutt relation (eq. ). The SFRs of the dwarf irregular galaxies for the IGIMF models are calculated using our fitting functions..
(7) 1556. PFLAMM-ALTENBURG, WEIDNER, & KROUPA. Fig. 7.— Dependence of SFRs in Sculptor dwarf irregular galaxies on the total galaxy gas mass. The galaxy gas masses and the SFRs calculated using the Kennicutt relation are taken from Skillman et al. (2003, Table 3). These SFRs are compared with the ones obtained using the Standard IGIMF relation (‘‘Standard’’ values in Table 2) and with our invariant model in which the IGIMF is replaced by a canonical IMF. The solid line marks the bivariate regression to the data calculated with the Standard IGIMF described by eq. (22).. scatter of SFRs above a total galaxy gas mass of about 2 ; 107 M reduces by about 1 order of magnitude. Third, the decrease of the SFR with decreasing total galaxy gas mass becomes less steep. A bivariate linear regression between the SFRs derived in the Standard IGIMF model and the total galaxy gas mass (Fig. 7, solid line) leads to the relation log. Mgas SFR ¼ 1:05 log 10:05: 1 1 M yr 1 M. ð22Þ. This means that the SFRs of the Sculptor dwarf irregular galaxies depend linearly on the total gas mass: 1:05 SFR ¼ 8:9 ; 1011 M yr1 Mgas : ð23Þ The corresponding gas depletion timescales, gas ¼ Mgas =SFR;. ð24Þ. Vol. 671. Fig. 8.— Dependence of gas depletion times in Sculptor dwarf irregular galaxies on the total galaxy gas mass. The galaxy gas masses and the SFRs calculated using the LH -SFR relation by Kennicutt are taken from Skillman et al. (2003, Table 3). These SFRs are compared with the ones obtained using the Standard IGIMF LH -SFR relation (Table 2).. limit to be distinguishable from background radiation and noise, and to be detectable. Assuming that star formation takes place in star clusters distributed according to an ECMF with star cluster masses down to a few solar masses, then the least luminous observed H ii region is not the smallest star cluster, but the cluster producing the lowest observable H radiation. Thus, the lowest observed H flux indicates the detection limit. In the work by Skillman et al. (2003), this detection limit is about 4 ; 1016 erg cm2 s1 (H ii region ESO 348-G9 No. 2). In the work by Miller & Hodge (1994), the lowest measured H flux in M81 group dwarf galaxies is about 6:5 ; 1016 erg cm2 s1 (IC 2574 MH 135). Assuming that each star cluster forms its own H ii region created by all ionizing stars formed in this star cluster, a mass-luminosity relation (Fig. 9) between the embedded star cluster mass and the averaged produced H luminosity within the star formation epoch t can be constructed as described in x 2.2. This can be well approximated by y ¼ (x þ 34:55) 1 1=exp 0:27x2 þ 0:46x þ 0:32 ; ð27Þ. are shown in Figure 8. While the gas depletion times vary by a factor of 400 in the linear (Kennicutt) picture, the IGIMF leads to a shrinkage of the range down to only a factor of 20. In addition, the strong increase of the gas depletion time with lower galaxy gas mass disappears, and the gas depletion time becomes constant at a few Gyr. This is not surprising, as a strictly linear dependence of the SFR on the total gas mass, SFR ¼ AMgas ;. ð25Þ. directly implies a constant gas depletion timescale, gas ¼ Mgas =SFR ¼ A1 :. ð26Þ. 4. H -INVISIBLE SF The determination of the SFR of a galaxy is based on the measurement of the total H luminosity of the galaxy and a reliable theoretical relation between the current SFR and the produced H radiation. The H emission is confined within distinct H ii regions. These regions have to be identified manually after H imaging. The emission of these H regions have to exceed some detection. Fig. 9.— Mass-luminosity relation between the stellar mass of the cluster and the produced total H emission. The solid line shows the fit by eq. (27). The observed position of the Orion Nebula cluster and 30 Doradus are indicated for comparison. Note that the steep decline of LH for Mecl P1000 M is a result of the mmax-Mecl relation ( Fig. 1)..
(8) No. 2, 2007. CONVERTING H LUMINOSITIES INTO STAR FORMATION RATES. 1557. If the Sculptor dwarf galaxy SC 24 were located at a distance greater than 2.3 Mpc, then star formation in SC 24 would not have been detected. Therefore, very moderate star formation can be overlooked. The effect of H -dark star formation on the cosmological SFR due to a possible large number of dwarf galaxies with low SFRs needs further study. 5. CONCLUSION. Fig. 10.— Relation between the stellar mass of embedded star clusters and the maximum distance of the star cluster within which the star cluster can be detected with H observations, for three different detection limits: 1015, 1016, and 1017 erg s1 cm2.. where y ¼ log (LH /1 erg s1 ) and x ¼ logðMecl /M Þ. For comparison, the Orion Nebula cluster, with a total mass of about 1800 M ( Hillenbrand & Hartmann 1998) and an observed H luminosity of 1037 erg s1 (Kennicutt 1984), and 30 Doradus, with a total mass of about 272,000 M (Selman et al. 1999 ) and an observed H luminosity of 1:5 ; 1040 erg s1 (Kennicutt 1984), are marked, lying well on the mass-H luminosity relation. The Sculptor dwarf irregular galaxy SC 24, for example, has only one observable H ii region. Its flux is 8 ; 1015 erg cm2 s1 (Skillman et al. 2003), being just above the detection limit. Given a distance of SC 24 of about 2.14 Mpc, the H luminosity of the most luminous H ii region in SC 24 is 2:5 ; 1035 erg s1. Given our mass-luminosity relation (Fig. 9), the corresponding star cluster stellar mass is about 280 M. Equation (4) then determines the current SFR to be about 5:0 ; 104 M yr 1, in good agreement with the Standard IGIMF scenario, rather than the classical one (Table 3). The classical SFR of 2:1 ; 106 M yr1 implies that only 1 M of stellar material will have formed within 1 Myr, which is inconsistent by 2 orders of magnitude with the expected cluster mass (280 M). However, our SFR of 5:8 ; 104 means that 580 M must have formed within 1 Myr, showing an internal consistency of our picture. An H ii region with the luminosity LH in a galaxy at a distance D produces an H flux of j¼. LH : 4D2. ð28Þ. Using the mass-H luminosity relation (eq. ) and a detection limit, jlimit , a relation between the star cluster mass and the maximum distance, Dmax , within which the star cluster can be observed, can be constructed (Fig. 10).. Using the integrated galactic initial mass function (IGIMF) instead of an invariant IMF for calculating the produced H luminosity for a given galaxy-wide SFR, we revise the linear LH SFR relation by Kennicutt et al. (1994). The order of magnitude of the deviation of the IGIMF LH -SFR relation from the linear LH -SFR relation is independent of the explicit choice of the IGIMF model. Because the fraction of massive stars determined by the IGIMF is always less than that given by the underlying invariant IMF of individual star clusters, SFRs are always underestimated when using linear LH -SFR relations. The linearity between the H luminosity and the SFR is broken, too. Therefore, the SFRs of normal disk galaxies with an H luminosity of about 1040Y1042 erg s1 may be underestimated by a factor of 4Y50. For dwarf irregular galaxies having H luminosities between 10 36 and 10 39 erg s1, the underestimation of the SFRs varies by a factor of at least 20Y160 for the Minimal scenario. In the Maximal scenario, the underestimation of SFRs of dwarf galaxies varies between a factor of 800 and 5000. Evolutionary parameters of dwarf galaxies such as the gas depletion timescale are affected by the same order of magnitude. In fact, we find that the IGIMF formulation implies a linear dependence of the total SFR on the total galaxy gas mass, and that therefore the gas depletion timescale is constant, a few Gyr, and independent of the total galaxy gas mass. If this finding is correct, it imposes a problem for the theoretical understanding of galactic star formation in explaining the apparent low star formation efficiency of small galaxies (e.g., Kaufmann et al. 2007). Observational evidence confirming the IGIMF scenario is becoming available (Hoversten & Glazebrook, 2008), but further observational work is needed. In addition, more consistency checks with different methods of deriving SFRs should be done in the future. Further work is required to construct reliable conversions of IGIMF fluxes into other pass bands. Kennicutt (1998) mentioned that H measurements become unreliable for determining SFRs in regions with gas densities higher than 100 M pc2, as present in starburst galaxies. An extension of the IGIMF model to construct a relation between SFRs and far-IR luminosities is therefore clearly necessary.. We thank Ivo Saviane for stimulating discussions which encouraged us to write this paper.. REFERENCES Bonnell, I. A., Vine, S. G., & Bate, M. R. 2004, MNRAS, 349, 735 Kennicutt, R. C., Jr.———. 1998, ApJ, 498, 541 Bressan, A., Fagotto, F., Bertelli, G., & Chiosi, C. 1993, A&AS, 100, 647 Kennicutt, R. C., Jr., Tamblyn, P., & Congdon, C. E. 1994, ApJ, 435, 22 Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 Koen, C. 2006, MNRAS, 365, 590 Diehl, R., et al. 2006, Nature, 439, 45 Köppen, J., Weidner, C., & Kroupa, P. 2007, MNRAS, 375, 673 Figer, D. F. 2005, Nature, 434, 192 Kroupa, P. 2001, MNRAS, 322, 231 Hillenbrand, L. A., & Hartmann, L. W. 1998, ApJ, 492, 540 ———. 2002, Science, 295, 82 Hoversten, E., & Glazebrook, K. 2008, ApJ, in press Kroupa, P., Tout, C. A., & Gilmore, G. 1993, MNRAS, 262, 545 Hurley, J. R., Pols, O. R., & Tout, C. A. 2000, MNRAS, 315, 543 Kroupa, P., & Weidner, C. 2003, ApJ, 598, 1076 Kaufmann, T., Wheeler, C., & Bullock, J. S. 2007, MNRAS, submitted Larsen, S. S. 2000, MNRAS, 319, 893 (arXiv:0706.0210) ———. 2002a, in IAU Symp. 207, Extragalactic Star Clusters, ed. D. Geisler, Kennicutt, R. C., Jr. 1983, ApJ, 272, 54 E.K. Grebel, & D. Minniti (San Francisco: ASP), 421 ———. 1984, ApJ, 287, 116 ———. 2002b, AJ, 124, 1393.
(9) 1558. PFLAMM-ALTENBURG, WEIDNER, & KROUPA. Larsen, S. S., & Richtler, T. 2000, A&A, 354, 836 Lee, H.-c., Gibson, B. K., Flynn, C., Kawata, D., & Beasley, M. A. 2004, MNRAS, 353, 113 Lejeune, T., Cuisinier, F., & Buser, R. 1997, A&AS, 125, 229 ———. 1998, A&AS, 130, 65 Maı́z Apellániz, J., Walborn, N. R., Morrell, N. I., Niemela, V. S., & Nelan, E. P. 2007, ApJ, 660, 1480 Maschberger, T., & Kroupa, P. 2007, MNRAS, 379, 34 McKee, C. F., & Williams, J. P. 1997, ApJ, 476, 144 Meynet, G., & Maeder, A. 2003, A&A, 404, 975 Miller, B. W., & Hodge, P. 1994, ApJ, 427, 656 Oey, M. S., & Clarke, C. J. 2005, ApJ, 620, L43 Pflamm-Altenburg, J., & Kroupa, P. 2006, MNRAS, 373, 295 Scalo, J. M. 1986, Fundam. Cosm. Phys., 11, 1. Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS, 96, 269 Selman, F., & Melnick, J. 2005, A&A, 443, 851 Selman, F., Melnick, J., Bosch, G., & Terlevich, R. 1999, A&A, 347, 532 Skillman, E. D., Côté S., & Miller, B. W. 2003, AJ, 125, 593 Stahler, S. W., & Palla, F. 2005, The Formation of Stars ( Weinheim: WileyVCH ) Vanbeveren, D. 1982, A&A, 115, 65 ———. 1983, A&A, 124, 71 Weidner, C., & Kroupa, P. 2004, MNRAS, 348, 187 ———. 2005, ApJ, 625, 754 ———. 2006, MNRAS, 365, 1333 Weidner, C., Kroupa, P., & Larsen, S. S. 2004, MNRAS, 350, 1503 Westera, P., Lejeune, T., Buser, R., Cuisinier, F., & Bruzual, G. 2002, A&A, 381, 524.