5 Mass function

The mass function (MF) may be defined as the number of stars observed in a certain mass bin. The mass function in stellar populations is most frequently fitted by a power law, whose slope depends on the mass range analysed (e.g., Kroupa 2001). In the logarithmic representation, the mass function is defined as

$${\rm d} (\log N)/{\rm d} (\log M) \sim M^\Gamma$$

where M is usually given in solar masses, and $\Gamma$ is the slope of the mass function. The exact shape of the mass function and, in particular, its slope, has been subject of intense discussion (see, e.g., Massey 1995a, 1995b; Scalo 1986; and Scalo 1998 for a review). Massey et al. (1995a, 1995b) report slopes between -0.7 and -1.7 with a weighted mean of $-1.1 \pm 0.1$ for $M > 7\ M_\odot$in several starforming regions in the Milky Way, and $-1.1 < \Gamma < -1.6$with a mean of $-1.3 \pm 0.1$ for $M > 25\ M_\odot$ in the LMC. From these studies in young clusters and associations, the mass function has been suspected to be rather universal, following a Salpeter (1955) power law with a slope of $\Gamma = -1.35$ for stars with masses $M \ge 1~M_\odot$ (Kroupa 2001). For the two starburst clusters studied in the Milky Way, Arches (FKM) and NGC 3603 (Eisenhauer et al. 1998), a flat slope of $\sim$-0.7 is observed for masses $M > 10~ M_\odot$ in Arches, and $M > 1~ M_\odot$ in NGC 3603. These values have to be kept in mind as a comparison for the results presented in the following sections. The mass regime which will be discussed here spans a mass range of $6\!<\!M\!<\!65\ M_\odot$ for Arches cluster stars.

5.1 Integrated mass function

The present-day mass function (Fig. 13) of the Arches cluster has been derived from the colour-magnitude diagram by transforming stellar luminosities into masses via a 2 Myr isochrone from the Geneva basic set of stellar evolution models (Lejeune & Schaerer 2001) using the method described in Grebel & Chu (2000). Enhanced mass loss models were also used, but did not alter the resultant mass function. Stellar evolution (mass loss, giant evolution) is not important on timescales of the Arches age of $\sim$2 Myr for stars with initial masses of $M < 50~M_\odot$ ( $\log (M/M_\odot) < 1.7$, i.e. stellar evolution affects the two upper mass bins of the MF at most). No attempt has been made to reconstruct the initial mass function (IMF) from the present-day MF for $M > 50~M_\odot$. A distance modulus of 14.5 mag and an extinction of A_V = 24.1 mag have been applied.

The slopes of all mass functions discussed have been derived by performing a weighted least-squares fit to the number of stars per mass bin. The size of the mass bins was chosen to be $\delta\log (M/M_\odot) = 0.1$ as the best compromise between mass function resolution and statistical relevance. This bin size is significantly larger than the photometric uncertainty in the considered mass and thus magnitude ranges. Only mass bins with a completeness factor of $\ge$75% have been included in the fit.

Note that we have not attempted to subtract the field star contribution. As can be seen in Fig. 1, the Gemini field is mostly restricted to the densest cluster region. When comparing to an arbitrary part of the GC field, we do not expect to observe the same distribution of stars as in the Arches field, as the faint, reddened background sources are negligible due to the high density of bright sources in the cluster area.

In addition, the stellar density in the GC is strongly variable, imposing additional uncertainties on the field contribution. Neither the Gemini nor the HST field covers enough area to estimate the field star population in the immediate vicinity of the cluster. A main sequence colour cut ( 1.15 < H-K < 1.90 mag) has been applied to the colour-corrected CMDs to select Arches members, excluding blue foreground and red background sources.

To allow for a direct comparison with the results obtained in FKM, we have used isochrones calculated for a metallicity of Z = 0.04 for all MF derivations. The derived MFs are displayed in Fig. 13. The overall mass function derived from the Gemini data displays the same slope as derived from NICMOS within the uncertainties, namely $\Gamma_{\rm Gemini} = -0.77 \pm 0.16$and $\Gamma_{\rm HST} = -0.82 \pm 0.14$ fitted for $10\!<\!M\!<\!65\ M_\odot$ (Fig. 13), which may be extrapolated down to $6\ M_\odot$ when taking into account the incompleteness correction. The present-day upper mass of $65~M_\odot$ corresponds to an initial mass of about $100~M_{\odot}$ according to the Geneva models. FKM derive an overall slope of $\Gamma = -0.7 \pm 0.1$ in the inital mass range $6\!<\!M\!<\!120\ M_\odot$, in good agreement with our results. The remaining difference in the maximum mass is due to the different extinction and the extinction corrections applied, which represent the largest uncertainties in the mass function derivation. As in particular the correction of the K magnitude for the varying extinction is uncertain due to the unknown extinction law of the NICMOS filters, we have also derived the mass function for uncorrected K magnitudes, with only the colour correction applied, which is independent of the extinction law assumed. In this case, the MF appears flatter with a slope of $\Gamma \sim -0.5 \pm 0.2$ (Fig. 13, lower panel). The discrepancy in the derived slopes clearly shows that the effects of differential extinction are not negligible, especially when deriving mass functions for very young regions, where the reddening varies significantly.

Furthermore, we have checked the effect of the binning on the MF by shifting the starting point of each bin by one tenth of the bin-width, $\delta\log (M/M_\odot) = 0.01$. The resultant slopes range from $-0.69 \pm 0.13 < \Gamma_{\rm Gemini} < -0.90 \pm 0.15$ and $-0.79\pm 0.12 < \Gamma_{\rm HST} < -0.98 \pm 0.13$. The average slopes for Gemini and HST, $\Gamma_{\rm Gemini} = -0.77 \pm 0.15$ and $\Gamma_{\rm HST} = -0.86 \pm 0.13$, respectively, agree well within the errors. The slightly flatter slope observed in the Gemini data may reflect the more severe incompleteness due to crowding. Although all slopes are consistent within the errors, the range in slopes derived by scanning the bin-step shows that statistical effects due to the binning may not be entirely neglected in the MF derivation.

The metallicity within the immediate Galactic Center region has been a matter of discussion during the past decade. Several authors report supersolar metallicities derived from CO index strength and TiO bands in bulge stars (Frogel & Whitford 1987; Rich 1988; Terndrup et al. 1990, 1991). Carr et al. (2000) measure [Fe/H] $= -0.02 \pm 0.13$ dex for the 7 Myr old supergiant IRS 7, and Ramirez et al. (2000) derive an average of [Fe/H] $= +0.12 \pm 0.22$ dex for 10 young to intermediate age supergiants, both very close to the solar value.

Using a 2 Myr isochrone with solar metallicity Z=0.02, the average slope of all bin steps is $\Gamma_{\rm Gemini} = -0.84 \pm 0.13$ and $\Gamma_{\rm HST} = -0.91 \pm 0.12$. The mass function is thus not significantly altered when using solar instead of enhanced GC metallicity. We note, however, that a lower metallicity (i.e., in this case solar) steepens the MF slightly, thus working into the same direction as the incompleteness correction.

FKM report a flat portion of the MF in the range $15\!<\!M\!<\!50\ M_\odot$, which is not seen in the Gemini MF. This plateau can, however, be recovered, when we create a MF from K-band magnitudes uncorrected for differential extinction, and use a lowest mass of $\log (M_{\rm low}/M_\odot)= 0.25$. For $\log (M_{\rm low}/M_\odot) = 0.20$ the plateau is seen neither with nor without extinction correction. This, again, shows the dependence of the shape of the MF on the extinction corrections applied, as well as on the chosen binning.

From the considerations above, we conclude that the overall mass function of the Arches cluster has a slope of $\Gamma = -0.8$ to -0.9 in the range $6\!<\!M\!<\!65\ M_\odot$. Although the uncertainty of missing lower mass stars in the immediate cluster center remains, the incompleteness correction strongly supports the derived shape of the MF. If the flat slope would be solely due to a low recovery rate of low-mass stars in the cluster center, this should be visible in a much steeper rise of the incompleteness corrected MF in contrast to the observed MF. We thus conclude that the slope of the MF observed in Arches is flatter than the Salpeter slope of $\Gamma = -1.35$, assumed to be a standard mass distribution in young star clusters. Such a flat mass function is a strong indication of the efficient production of high-mass stars in the Arches cluster and the GC environment.

5.2 Effects of the chosen isochrone, bin size, and metallicity

Blum et al. (2001) estimate a cluster age of 2-4.5 Myr for Arches assuming that the observed high-mass stars are of type WN7. If we compare the Geneva basic grid of isochrones with fundamental parameters obtained for WN7 stars (Crowther et al. 1995), a reasonable upper age limit for this set of isochrones is $\sim$3.5 Myr. Crowther compares the parameters derived for WN stars with evolutionary models at solar metallicity from Schaller et al. (1992) and with the mass-luminosity relation for O supergiants from Howarth & Prinja (1989), yielding a mass range of $20{-}55~M_\odot$, but with high uncertainties at the low-mass end. The more reliable mass estimates for the colour and magnitude range observed for WN stars in Arches are restricted to $35\!<\!M\!<\!55\ M_\odot$. From spectroscopic binaries, the masses of two WN7 stars are determined to be $\sim$30 $M_\odot$ and >48 $M_\odot$(Smith & Maeder 1989). The theoretical lower limit to form Wolf-Rayet stars is $25\ M_\odot$ for the Geneva models (Schaerer et al. 1993).

In the Geneva basic grid of models with Z = 0.04, the 3.5 Myr isochrone is limited by a turnoff mass of $32~M_\odot$. We have thus calculated mass functions for isochrones with ages 2.5, 3.2, and 3.5 Myr in addition to the 2 Myr case discussed above. Though the derived slopes scatter widely, irrespective of the isochrone used, the slope of the MF tends to be even flatter for any of the older population models. We thus conclude that, regardless of the choice of model and parameters, the Arches mass function displays a flat slope.

   
5.3 Radial variation in the mass function

The radial variation (Fig. 14) of the mass function is particularly interesting with respect to YC evolution. We have analysed the stellar population in Arches within three different radial bins, $0\hbox{$^{\prime\prime}$ }< R < 5\hbox{$^{\prime\prime}$ }$, $5\hbox{$^{\prime\prime}$ }< R < 10\hbox{$^{\prime\prime}$ }$ and $10\hbox{$^{\prime\prime}$ }< R < 20\hbox{$^{\prime\prime}$ }$. The resulting mass functions for the Gemini and HST datasets, along with the radius dependent incompleteness correction for the Gemini MFs, are displayed in Fig. 14.

We confirm the flat mass function observed by FKM in the innermost regions of the cluster, which steepens rapidly beyond the innermost few arcseconds. Most of the bright, massive stars are found in the dense cluster center, where the mass function slope is very flat. FKM derive a slope of $\Gamma = -0.1~ \pm~ 0.2$ from the HST data in this region, which is consistent with Fig. 14. It is obvious from the lowest panel in Fig. 14 that we are crowding limited in the innermost region. We have thus not tried to fit a slope for $R < 5\hbox{$^{\prime\prime}$ }$.

In the next bin, $5\hbox{$^{\prime\prime}$ }< R < 10\hbox{$^{\prime\prime}$ }$, the mass function obtained from the weighted least-squares fit has already steepened to a slope of $\Gamma = -1.0 \pm 0.3$. Again, the MF in this radial bin remains significantly flatter ( $\Gamma = -0.5 \pm 0.4$) when no A_K-correction is applied. Beyond 10 $^{\prime \prime }$ (0.4 pc, upper panel), a power law can only be defined in the range $10\!<\!M\!<\!30\ M_\odot$ ( $1.0 < \log (M/M_\odot) < 1.5$), where a slope of $\Gamma = -1.69 \pm 0.66$ is found, consistent with a Salpeter ( $\Gamma = -1.35$) law. The large error obviously reflects the small number of mass bins used in the fit. Nevertheless, Fig. 14 clearly reveals the steepening of the MF very soon beyond the cluster center.

For a more quantitative confirmation of the mass segregation present in the Arches cluster, we created cumulative functions for the mass distributions in the three radial bins (Fig. 15). We have applied a Kolmogorov-Smirnov test to quantify the observed differences of these functions. When the central, intermediate, and outer radial bin are compared pairwise, we obtain in each case a confidence level of more than 99% that the mass distributions do not originate from the same distribution.

Thus, the inner regions of the cluster are indeed skewed towards higher masses either by sinking of the high mass stars towards the cluster center due to dynamical processes or by primordial mass segregation, or both. The same effect is observed in the similarly young cluster NGC 3603 (Grebel et al. 2002, in prep.).

   
5.4 Formation locus of massive stars

We find several (5) high mass stars in the cluster vicinity. These stars fall onto the Arches main sequence after applying the reddening correction (Sect. 3.1). When the entire HST field is separated into two equalsize areas, the first area being a circle of radius 16 $^{\prime \prime }$ around the cluster center, and the second the surrounding field, no additional comparably high-mass stars are found with Arches main sequence colours except for the two bright stars found at the edges of the Gemini field. In the dynamical models of Bonnell & Davies (1998) there is a low, but non-zero probability that a massive star originating outside the cluster's half-mass radius might remain in the cluster vicinity. High-mass stars formed near the center show, however, a tendency to migrate closer to the cluster center. The disruptive GC potential might enhance the ejection process of low-mass stars, but according to equipartition, it is unlikely that the most massive stars gain energy due to dynamical interaction with lower mass objects. We have to bear in mind, however, that interactions between massive stars in the dense cluster core could cause the ejection of high-mass stars.

N-body simulations performed by Portegies Zwart et al. (1999) show that the inclusion of dynamical mass segregation in cluster evolution models enhances the collision rate by about a factor of 10 as compared to theoretical cross section considerations. For a cluster with 12 000 stars initially distributed according to a Scalo (1986) mass function, a relaxation time of 10 Myr, and a central density and half-mass radius comparable to the values found in Arches, about 15 merging collisions occur within the first 10 Myr ( $1~t_{\rm relax}$) of the simulated cluster. Shortly after the start of the simulations, frequent binary and multiple systems form from dynamical interactions leading to the ejection of several contributing massive stars. The flat MF in the Arches center as compared to a Scalo MF, containing a larger fraction of massive stars to interact, may even increase the collision rate.

Though it is likely that the high-mass stars seen in the immediate vicinity of the Arches cluster have formed from the same molecular cloud at the same time as the cluster, a final conclusion on the possible ejection of these stars from the cluster core due to dynamical processes can only be drawn when velocities for these cluster member candidates are available.

Figure 13: Arches mass function derived from the Gemini/Hokupa'a colour-magnitude diagram shown in Fig. 9. A 2 Myr main sequence isochrone from the Geneva set of models (Lejeune & Schaerer 2001) was used to transform magnitudes into stellar masses. The mass function has been derived for bins of $\delta\log (M/M_\odot) = 0.1$with a lowest mass bin $\log (M/M_\odot) = 0.20$. Upper panel: with AK correction. Lower panel: without AK correction.

Figure 14: Radial change of the mass function as observed in the Gemini/Hokupa'a data. A very flat mass function is seen in the inner cluster regions, where predominantly massive stars are found. The slope of the mass function increases towards the Salpeter value ( $\Gamma = -1.35$) already at a radial distance from the cluster center of >10 $^{\prime \prime }$ (0.4 pc at a distance of 8 kpc).