Dissolution is the solution: on the reduced masstolight ratios of Galactic globular clusters
J. M. D. Kruijssen^{1,2}  S. Mieske^{3}
1  Astronomical Institute, Utrecht University, PO Box 80000, 3508TA Utrecht, The Netherlands
2  Sterrewacht Leiden, Leiden University, PO Box 9513, 2300RA Leiden, The Netherlands
3  European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile
Received 1 December 2008 / Accepted 27 March 2009
Abstract
Context. The observed dynamical masstolight (M/L) ratios of globular clusters (GCs) are systematically lower than the value expected from ``canonical'' simple stellar population models, which do not account for dynamical effects such as the preferential loss of lowmass stars due to energy equipartition. It has recently been shown that lowmass star depletion can qualitatively explain this discrepancy for globular clusters in several galaxies.
Aims. To verify whether lowmass star depletion is indeed the driving mechanism behind the M/L decrease, we aim to predict the M/L_{V} ratios of individual GCs for which orbital parameters and dynamical Vband masstolight ratios M/L_{V} are known. There is a sample of 24 Galactic GCs for which this is possible.
Methods. We used the SPACE cluster models, which include dynamical dissolution, lowmass star depletion, stellar evolution, stellar remnants, and various metallicities. We derived the dissolution timescales due to twobody relaxation and disc shocking from the orbital parameters of our GC sample and used these to predict the M/L_{V} ratios of the individual GCs. To verify our findings, we also predicted the slopes of their lowmass stellar mass functions.
Results. The computed dissolution timescales agree well with earlier empirical studies. The predicted M/L_{V} are in agreement with the observations for 12 out of 24 GCs. The discrepancy for the other GCs probably arises because our predictions give global M/L ratios, while the observations represent extrapolated central values that are different from global ones in the case of mass segregation and a long dissolution timescale. The GCs in our sample that likely have dissimilar global and central M/L ratios can be excluded by imposing limits on the dissolution timescale and King parameter. For the remaining GCs, the observed and predicted average M/L_{V} are 78
^{+9}_{11}% and 78
2% of the canonically expected values, while the values are 74
^{+6}_{7}% and 85
1% for the entire sample. The predicted correlation between the slope of the lowmass stellar mass function and M/L_{V} drop is found to be qualitatively consistent with observed mass function slopes.
Conclusions. The dissolution timescales of Galactic GCs are such that the % gap between canonically expected and observed M/L_{V} ratios is bridged by accounting for the preferential loss of lowmass stars, also when considering individual clusters. It is concluded that the variation in M/L ratio due to dissolution and lowmass star depletion is a plausible explanation for the discrepancy between the observed and canonically expected M/L ratios of GCs.
Key words: Galaxy: globular clusters: general  Galaxy: stellar content  galaxies: star clusters
1 Introduction
The topic of dynamical masstolight (M/L) ratios of old compact stellar systems has attracted increasing attention during recent years (Mieske et al. 2008; Evstigneeva et al. 2007; Forbes et al. 2008; Hilker et al. 2007; Hasegan et al. 2005; Rejkuba et al. 2007; Baumgardt & Mieske 2008; Kruijssen 2008; Dabringhausen et al. 2008; McLaughlin & van der Marel 2005; Chilingarian et al. 2008). The outcome of these studies can be summarised as follows.
 For the mass regime of ultracompact dwarf galaxies (UCDs,
), dynamical M/L ratios tend to be some 50% above predictions from stellar population models (Mieske et al. 2008; Forbes et al. 2008; Dabringhausen et al. 2008).
 For the mass regime of globular clusters (GCs,
), dynamical M/L ratios tend to be some 25% below predictions from simple stellar population (SSP) models that assume a canonical IMF (Kruijssen 2008; Mieske et al. 2008; Kruijssen & Lamers 2008; Rejkuba et al. 2007).
 As a consequence, the dynamical M/L ratios of UCDs are on average about twice as high as those of GCs, at comparable metallicities.
As noted already in Kruijssen & Lamers (2008, hereafter KL08), the next step is to apply these analytical cluster models including preferential loss of lowmass stars to individual clusters. This would then account for variations in metallicity and dissolution timescale. Such a study will naturally be restricted to GCs with measured M/L ratios for which realistic estimates of their individual dissolution timescale are available from information on their actual orbit within the Milky Way potential. With the database of individual dissolution time scales at hand, the loss of lowmass stars can be quantified according to the prescriptions of Kruijssen (2008) and KL08, leading to predictions for the drop of M/L for individual GCs. Those predictions are to be contrasted with the actual observed M/L ratios of these GCs. This will allow us to quantitatively test the hypothesis that the loss of lowmass stars is responsible for the too low M/L ratios of GCs, and hence also partially for the discrepancy of M/L between GCs and UCDs.
Previous studies assessing the preferential loss of lowmass stars in Galactic GCs focus both on observations (e.g., De Marchi et al. 2007) and theory (e.g., Baumgardt et al. 2008). In De Marchi et al. (2007), the slopes of the stellar mass functions in GCs are measured for stars between 0.3 and 0.8 , thereby directly reflecting possible lowmass star depletion. The study by Baumgardt et al. (2008) predicts the same slopes using Nbody models and different degrees of mass segregation, assuming dissolution by twobody relaxation. The aforementioned papers both do not consider the M/L ratios of the GCs in question.
Table 1: Observed properties for the cluster sample, together with their standard errors.
Table 2: Orbital parameters for the cluster sample, together with their standard errors.
In this study, the reference sample for dynamical M/L ratios of Galactic GCs is that of McLaughlin & van der Marel (2005), obtained by the fitting of singlemass King profiles. It contains data for 38 GCs. Only a subsample can be used for our analysis, namely those clusters for which accurate proper motions and radial velocities are measured and can be translated to orbital parameters. Next to the destruction rates due to twobody relaxation, also the influence of disc shocking on the cluster dissolution needs to be taken into account. Both have to be derived from the orbital parameters. Several studies in which orbital information is derived and used to compute destruction rates are available in the literature (Allen et al. 2008,2006; Dinescu et al. 1999; Gnedin & Ostriker 1997), all with certain benefits and tradeoffs. Specifically, Gnedin & Ostriker (1997) assign statistically sampled orbits conforming to the bulk motion of the GC system in an axisymmetric potential to derive destruction rates of 119 globular clusters. Dinescu et al. (1999) use proper motions and radial velocities to compute the orbits and destruction rates of 38 clusters in two axisymmetric potentials (Johnston et al. 1995; Paczynski 1990). Allen et al. (2008,2006) follow the same procedure, but consider both axisymmetric and barred potentials (Pichardo et al. 2004; Allen & Santillan 1991, respectively) for 54 globular clusters. They do not find a significant deviation between the results for both potentials. However, they do note that their calculated destruction rates are multiple orders of magnitude lower than others in literature and recommend combining their orbital information with the more rigorous FokkerPlanck approach used by Gnedin & Ostriker (1997) to derive the destruction rates.
We choose to adopt the orbital data and destruction rate due to disc shocking from Dinescu et al. (1999). Our study cannot be based on statistically assigned orbits but requires the actual orbits of individual clusters, thus excluding the estimated orbits from Gnedin & Ostriker (1997). In addition, the Dinescu et al. (1999) destruction rates seem to be in better agreement with the observations than those from Allen et al. (2008,2006).
In Table 1 the observed properties are listed of the 24 Galactic globular clusters for which the Vband masstolight ratios (M/L_{V}) and orbital parameters are available, i.e., the sample that is covered both by Dinescu et al. (1999) and McLaughlin & van der Marel (2005). The masses and observed M/L_{V} ratios represent dynamical values. For all clusters, the standard error in [Fe/H] is assumed to be 0.15 when computing the error propagation (see Sect. 3), which represents a conservative accuracy estimate (see, e.g., Carretta & Gratton 1997). These errors determine the uncertainty on in the last column, since the canonical M/L ratio only depends on metallicity. The galactocentric radii used to compute the orbits in Dinescu et al. (1999) are from Zinn, private communication. In extreme cases this may cause a small disagreement between the galactocentric radius quoted here and the apogalactic distance predicted by Dinescu et al. (1999, see Table 2).
A first inspection of the observed M/L_{V} ratios in Table 1 can be made by comparing them to the (``canonical'') M/L_{V} ratios from SSP models, which only depend on metallicity due to the invariance of the shape of the stellar mass function in these models. In Fig. 1 (left), the observed M/L_{V} ratios of our sample are plotted versus the canonically expected values that were computed by interpolating SSP models. These were emulated with the models from KL08 neglecting the preferential loss of lowmass stars. The discrepancy between observed and expected M/L_{V} ratio is evident, as the canonical M/L_{V} are constrained to a much narrower and generally higher range than the observed ones. The number histogram of the two M/L_{V} (Fig. 1, right) further substantiates this dissimilarity. The observed M/L_{V} ratios are on average 74 ^{+6}_{7}% of the canonically expected values.
Figure 1: Left: observed masstolight ratio versus the canonically expected masstolight ratio , together with their standard errors (see Table 1). The dotted line follows the 1:1 relation. Clusters for which the disagreement is larger than 1 are plotted as dots. Right: number histogram of (diamonds, shaded area) and (triangles, hashed area). For comparison, canonically expected M/L_{V} ratios from Bruzual & Charlot (2003) for a Chabrier (2003) IMF are overplotted (squares, dotted line). Again, the error bars denote deviations, which were determined from 30k random realisations of the underlying data. 

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With the present paper we aim to quantify the contribution of dynamical effects such as the preferential loss of lowmass stars and the selective loss of stellar remnants (see KL08) to the discrepancy between the observed and canonically expected M/L ratios. In Sect. 2, we summarise the cluster models from KL08 and highlight the aspects that are particularly relevant to this study. The dissolution timescales for the cluster sample are computed in Sect. 3, whereas the predicted mass and M/L_{V} evolution are considered and compared to the observations in Sect. 4. We predict slopes of the lowmass stellar mass function and discuss observational tests to verify the preferential loss hypothesis for appropriate clusters in Sect. 5. In the final Sect. 6, we discuss the results and present our conclusions.
2 Cluster evolution models and M/L evolution
In order to study the evolution of clusters on specific orbits, we use analytical cluster models (SPACE, see KL08) that incorporate the effects of stellar evolution, stellar remnant production, cluster dissolution and energy equipartition. They are summarised here and are treated in more detail in KL08. In the second part of this section, the dependence of masstolight ratio evolution on initial mass, metallicity and dissolution timescale is assessed (for a more detailed description, see Kruijssen 2008).
2.1 Summary of the models
In the SPACE cluster models, clusters gradually lose mass due to stellar evolution and dissolution. The total cluster mass evolution is determined by
where the first term denotes mass loss due to stellar evolution and the second represents mass loss by dissolution. Additionally, the formation of stellar remnants and the massdependent loss of stars by dissolution are taken into account, thus providing a description of the changing mass function and cluster mass in remnants.
Stellar evolution is included by using the Padova 1999 isochrones^{}. These are available for metallicities Z = 0.0004, 0.004, 0.008, 0.02, 0.05 (or [Fe/H] = 1.7, 0.7, 0.4, 0.0, 0.4), which thus restricts our model computations to these values. Stellar evolution removes the most massive stars from the cluster and increases the nonluminous cluster mass by turning stars into remnants, which is included by adopting an initialremnant mass relation^{}. A Kroupa (2001) IMF is assumed.
Cluster dissolution represents the dynamical cluster mass loss due to stars passing the tidal radius, which acts on the timescale
:
where represents the present day cluster mass and the second equality follows from the relation derived by Lamers et al. (2005a):
The characteristic timescale t_{0} depends on the environment and determines the strength of dissolution. For example, in the case of dissolution by twobody relaxation t_{0} depends on tidal field strength and therefore on the angular velocity of the cluster orbit. Typical values are t_{0}=10^{5}10^{8} yr (e.g., Lamers et al. 2005b), translating into a total disruption time 10^{11} yr for a 10 cluster. The exponent is found to be , both from observations (Boutloukos & Lamers 2003; Gieles et al. 2005) and from the Baumgardt & Makino (2003) Nbody simulations of tidal dissolution for clusters with King parameter W_{0}=5 (Lamers et al. 2005b). However, it is recently derived by Lamers et al. (2009) that for King parameter W_{0}=7. Since this concentration more closely resembles the mean King parameter for Galactic GCs (see Table 1), we adopt throughout this study.
The effect of dissolution on the mass function depends on the dynamical state of the cluster. As it evolves towards energy equipartition, lowmass stars are preferentially lost from the cluster. This mass loss (in the ``preferential mode'', KL08) is approximated by increasing the minimum stellar mass (Lamers et al. 2006), while evaporation that is independent of stellar mass (mass loss in the ``canonical mode'', KL08) is accounted for by decreasing the normalisation of the mass function. In our models, both modes coexist to allow for intermediate modes of mass loss. Their relative contributions are fitted such that the M/L_{V} ratio evolution matches the Nbody simulations by Baumgardt & Makino (2003).
Cluster photometry is computed by integrating the stellar mass function over the stellar isochrones, yielding cluster magnitude evolution for a passband and a cluster with initial mass .
2.2 Dependence of masstolight ratio on model parameters
The models described in Sect. 2.1 yield a masstolight ratio evolution that depends on the dissolution timescale, metallicity and initial cluster mass. In canonical models, i.e., without the preferential loss of lowmass stars, M/L monotonously increases with time. For a given age and metallicity, these models provide M/L ratios that are independent of cluster mass. On the other hand, our models including dynamical effects predict a massdependent drop in masstolight ratio due to the ejection of lowmass, highM/L stars (Kruijssen 2008, KL08). In Fig. 2, the Vband masstolight ratio evolution M/L_{V} is shown for two metallicities and several initial cluster masses. In both panels, the upper curve marks the canonical masstolight ratio evolution, while the others correspond to cluster evolution including the preferential loss of lowmass stars for different initial masses. Since lowmass clusters evolve on shorter timescales than massive ones, the deviation of their masstolight ratio evolution with respect to canonical models arises at earlier times than for massive clusters.
Figure 2: Left: Vband masstolight ratio evolution M/L_{V} for t_{0}=1 Myr, Z=0.0004 and initial masses in the range 8 with 0.25dex intervals. Right: same graph, but for Z=0.004. From top to bottom, different curves represent the M/L_{V} evolution for decreasing initial cluster masses. 

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The masstolight ratio decrease can be quantified by considering the ratio of the observed or predicted M/L to its canonical value to divide out their metallicity dependence:
Figure 3 shows the predicted fraction of the canonical M/L ratio as a function of for clusters with initial masses in the range 10^{4} , dissolution timescales Myr and metallicities . It shows that is independent of metallicity, initial cluster mass and dissolution timescale when considered as a function of the elapsed fraction of the total disruption time . The threecomponent linear approximation illustrates the welldefined uniform correlation and is given by
which applies for all initial conditions, i.e., is independent of the cluster properties or environment^{}. Equation (5) does not include possible effects of primordial mass segregation on the change of the mass function. From model runs where we did assume the preferential depletion of lowmass stars from t=0 on we know that its effects become about 10% stronger with respect to purely dynamically induced lowmass star depletion (KL08). This number should be treated with some care, because our models are based on Nbody simulations of clusters that did not start out in a masssegregated state (Baumgardt & Makino 2003).
Figure 3: The ratio of predicted to canonical masstolight ratio as a function of the elapsed fraction of the total disruption time . Dotted curves denote model predictions for a broad range of initial conditions (varying initial masses, dissolution timescales and metallicities), while the solid line describes a threecomponent linear approximation to the models. 

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The relation between the predicted fraction of the canonical masstolight ratio and the elapsed fraction of the total disruption time is expected, since in our models the decrease of M/L is the result of dynamical evolution. It is in agreement with studies by Richer et al. (1991) and Baumgardt & Makino (2003), who find that the depletion of the lowmass stellar mass function in globular clusters is closely related to the elapsed fraction of the total disruption time. Considering the physical processes driving dissolution, the result is not surprising either. Twobody relaxation is known to preferentially eject lowmass stars and tidal shocks remove the outer parts of the cluster, which in the case of mass segregation are constituted by lowmass stars.
The evolution of cluster mass and M/L_{V} can both be considered in the plane. The resulting ``evolutionary tracks'' are shown in Fig. 4 for two different dissolution timescales and again for two metallicities and a range of initial cluster masses as in Fig. 2. Clusters start with their initial masses and with M/L_{V} ratios close to zero, corresponding to an initial position on the xaxis of Fig. 4. As time progresses, clusters initially evolve to lower masses and increasing M/L_{V} due to the death of massive stars, translating into up and leftward motion in the plane. When the preferential loss of lowmass stars becomes an important mechanism (the onset of which is marked by dots for each evolutionary track), the M/L_{V} increase is turned into a decrease instead, as also illustrated in Fig. 2. In Fig. 4, the thus attained maximum in the M/L_{V} evolution is best visible for low cluster masses and t_{0}=1 Myr.
Figure 4: Cluster evolution in the plane for Myr and . Solid curves represent cluster evolutionary tracks for initial cluster masses in the range 10^{8} with 0.5dex intervals. Cluster isochrones at t=12 Gyr are described by dashed lines, while these at t=10 and t=14 Gyr are denoted by dotted lines (bottom and top, respectively). Dots denote the onset of the preferential loss of lowmass stars for each evolutionary track. 

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Since Galactic globular clusters generally share the same ages (e.g., Vandenberg et al. 1990), the observed distribution of GCs in the plane would follow curves of equal age in Fig. 4 if there were no spreads in metallicity and dissolution timescale. These curves, or cluster isochrones, are shown for ages Gyr. Along the isochrones, M/L_{V} increases with cluster mass since massive clusters have spent a smaller fraction of their total disruption time than lowmass clusters and will therefore have experienced a smaller M/L_{V} decrease due to lowmass star depletion. The curves flatten at the highest masses, since these clusters have not yet exhibited significant preferential lowmass star ejection.
From Fig. 4 we infer the influences of dissolution timescale and metallicity on the masstolight ratio evolution. The dissolution timescale sets the cluster mass for which the downbend of the cluster evolutionary tracks can occur and therefore also determines the location of the ``knee'' in the cluster isochrones. The metallicity determines the vertical extent of the cluster evolutionary tracks and thus the M/L_{V}normalisation of the cluster isochrones. As set forth in Kruijssen (2008), the natural spread in dissolution timescale and metallicity thus explains the scatter around the relation between M/L and cluster mass observed by Mandushev et al. (1991).
3 Determining the dissolution timescale
To assess the influence of the preferential loss of lowmass stars on the low observed masstolight ratios, the orbital parameters of individual clusters are to be translated into the appropriate dissolution timescales t_{0} for use in our cluster models. The computation is treated in this section.
3.1 Cluster dissolution timescales from orbital parameters
For globular clusters, dissolution due to twobody relaxation in the Galactic tidal field^{} and disc shocking are the main dissolution mechanisms (e.g., Chernoff et al. 1986). The total dissolution timescale
can be written as
where denotes the dissolution timescale due to twobody relaxation or evaporation (carrying the subscript ``evap'') and the dissolution timescale due to disc shocking.
For the dissolution timescale due to twobody relaxation, we use the expression for the total disruption time from Baumgardt & Makino (2003, Eq. (10)) as approximated by Lamers et al. (2005b) to write
with the dissolution timescale due to twobody relaxation for a circular orbit at the solar galactocentric radius, the apogalactic radius of the cluster orbit, the circular velocity of the gravitational potential at the distance of apogalacticon and e the orbital eccentricity. Values for are taken from Dinescu et al. (1999), while the circular velocities are computed for the galactic potential from Paczynski (1990). This potential, as well as the one from Johnston et al. (1995), is used by Dinescu et al. (1999) in the determination of the cluster orbits. By comparing our models to the Nbody simulations by Baumgardt & Makino (2003) we find Myr for clusters with W_{0}=5 King profiles, in very close agreement with earlier reported values of 20.9 Myr (Lamers et al. 2005a) and 22.8 Myr (Lamers & Gieles 2006). Using the same method for , corresponding to W_{0}=7 King profiles (see Sect. 2.1), we obtain Myr. This is the adopted value in this paper.
The dissolution timescale due to disc shocking can be obtained from the globular cluster destruction rates due to disc shocking
from Dinescu et al. (1999). Following from Eq. (3), a present destruction rate
is related to a dissolution timescale t_{0} by
with in units of , t_{0} in years, and denoting the cluster mass at age t. The denominator represents an estimate for the total cluster lifetime. This expression can be inverted to obtain from . However, in Dinescu et al. (1999) constant is assumed to compute the cluster masses. Since their destruction rates are derived from a relation , these should be corrected for the actual masstolight ratios. We define the correction factor
with the numerator the constant masstolight ratio and the denominator the observed dynamical masstolight ratio from McLaughlin & van der Marel (2005) (see Table 1). This allows us to express the dissolution timescale due to disc shocking as
Substitution of Eqs. (7) and (10) into Eq. (6) then yields the total dissolution timescale .
Table 3: Computed dissolution timescales (for ) due to twobody relaxation ( ), disc shocking ( ) and both mechanisms ( ), together with their standard errors. All values are rounded to one decimal.
In Table 2, our cluster sample is listed with the orbital parameters from Dinescu et al. (1999) for the Paczynski (1990) potential and our computed circular velocities of the gravitational potential at the distance of apogalacticon. The circular velocities are computed in the galactic plane (z=0). Because the gravitational potentials of the disc and bulge decrease with z, this implies that for clusters with kpc the actual can be 515% lower.
The corresponding dissolution timescales can be found in Table 3. The values for the dissolution timescale range from 20 Myr, corresponding to total disruption times for a 10 cluster in the range 300 Gyr. This is is in good agreement with the range that is required for lowmass star depletion to explain the observed masstolight ratio drop (Kruijssen 2008). By comparing the dissolution timescales for twobody relaxation and disc shocking , we can see that the latter destruction mechanism is important (i.e., lowers the total dissolution timescale by more than 40% with respect to ) for the clusters NGC 288, 5139 (Cen), 6362, and 6779. These clusters all have perigalactic radii smaller than 3 kpc (see Table 2). For the error analysis of Tables 2 and 3 and of the rest of this paper we refer to Appendix A.
4 Predicted and observed masstolight ratios
In this section we combine our cluster models and the derived dissolution timescales to study the masstolight ratio evolution for our sample of 24 Galactic globular clusters. Presentday M/L_{V} ratios are predicted for the cluster sample and are compared to the observations. We also discuss the possible causes for the individual clusters that still lack convincing agreement.
4.1 Predicted masstolight ratios for the cluster sample
We employ the cluster models treated in Sects. 2 and 3 to predict M/L_{V} ratios for the cluster sample. The input parameters for the models are the dissolution timescale t_{0} and metallicity Z. The latter is derived from the iron abundance [Fe/H] (see Table 1) according to
with , while the dissolution timescale is taken from Table 3 ( ). Since all clusters in the sample have metallicities Z<0.004, for each cluster the models are computed with metallicities and the appropriate dissolution timescales. The evolution is computed for a grid of initial cluster masses, yielding cluster evolution tracks for the mass and Vband masstolight ratio M/L_{V}. For both metallicities, at t=12 Gyr the tracks are interpolated over the mass grid to match the observed cluster mass. This provides predictions for M/L_{V}, the initial cluster mass and the total disruption time for two metallicities. These are then interpolated over metallicity to obtain the model predictions for the appropriate metallicity.
Figure 5: Top left: timeevolution of M/L_{V} for NGC 5466. The solid line represents the Z=0.0004 model with the dissolution timescale of the cluster (t_{0}=10.2 Myr), while the dotted curve indicates the canonical M/L_{V} evolution, i.e., if the preferential loss of lowmass stars were omitted. The dashed line denotes constant age of t=12 Gyr. The predicted M/L_{V} of NGC 5466 is marked by a cross and the onset of the preferential loss of lowmass stars is specified with a dot. Top right: evolution in the plane for NGC 5466. Curves and symbols have the same meaning as in the topleft panel. Bottom left: same as top left, but for NGC 6779 (t_{0}=1.2 Myr). Bottom right: same as top right, but for NGC 6779. 

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Table 4: Our model predictions with their standard errors.
Examples of the M/L_{V} evolution with time and mass are shown in Fig. 5 for NGC 5466 and 6779. The predicted M/L_{V} are slightly offset with respect to the model curves because the models here are computed at Z=0.0004 while the predictions are interpolated over metallicity. However, the variation with metallicity is small for the displayed clusters, since their metallicities are close to Z=0.0004. It is evident that lowmass star depletion has a much stronger effect in the case of NGC 6779 than for NGC 5466. Considering their dissolution timescales (t_{0}=1.2 Myr versus t_{0}=10.2 Myr, respectively) and the resulting mass evolution, this is not surprising since NGC 6779 has suffered much stronger mass loss than NGC 5466.
The predicted masstolight ratios M/L_{V}, initial masses and remaining lifetimes are listed for our entire GC sample in Table 4. In addition, the observed and predicted fractions of the canonical M/L_{V} ratios and are shown, as well as the agreement between our predicted M/L_{V} and the observed values. Combining Tables 3 and 4, we see that GCs with short dissolution timescales indeed have low predicted M/L ratios.
4.2 Comparison of predictions to observations
The fifth column in Table 4 indicates the ratio between observed and predicted masstolight ratio . Analogously, the sixth column gives the ratio between predicted and canonical masstolight ratio . On average, the former ratio is 0.74 ^{+0.06}_{0.07}, while the latter ratio is 0.85 0.01 for the 24 GCs investigated. There are factors that introduce biases when comparing the predictions to the observations. Specifically, the observations are likely biased to central M/L ratios for some GCs, while we predict global values. In Sect. 4.3 a more detailed consideration is provided in which the comparison of the predictions to the observations is refined.
The seventh column in Table 4 gives the level of agreement between the observed and predicted masstolight ratios, which is defined as n if for and . Within the uncertainty, the predicted M/L_{V} agree with the observed values for 12 clusters out of the 24cluster sample. A Gaussian distribution of errors would yield an expected 16 out of 24 clusters to be found within .
Figure 6: Observed (crosses) and predicted (triangles) distribution of GCs in the plane. The solid line represents our linear fit to the observations, while the dotted line denotes the relation found by Mandushev et al. (1991).The error bars in the top left corner denote the average uncertainty on the observations, while the error bars in the bottom right corner represent the average uncertainty on the predictions. 

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As a first comparison and analogously to the presentation in Mandushev et al. (1991) and Rejkuba et al. (2007), in Fig. 6 the distribution of GCs in the
plane is shown for the observed and predicted masstolight ratios. Both populations fall within the same range and follow comparable trends of increasing M/L_{V} with cluster mass. Mandushev et al. (1991) already provided an expression for the observed logarithm of the mass as a function of magnitude, which allows for a derivation of the expected trend in Fig. 6. They fit
where M_{V} represents the Vband absolute magnitude of the cluster. Adopting a solar value of , the relation between M/L_{V} and mass from Mandushev et al. (1991) can then be expressed as
A firstorder Taylor expansion of M/L_{V} around then gives
which has a linear slope of 0.38. The best fitting slope for our sample is 0.41 0.28, thus agreeing with the value from Mandushev et al. (1991). The large uncertainty arises from the scatter in Fig. 6.
The trend of increasing masstolight ratio with mass is expected from the models shown in Fig. 4 and the discussion in Sect. 2.2. However, there the slope is 1.0 for metallicities Z=0.00040.004 and increases with Z. For some metallicities, the model slope is thus more than steeper than the fitted slope. This is not surprising, because the models each have a single dissolution timescale and metallicity, while in reality both quantities have a spread that causes horizontal and vertical scatter, respectively. It turns out that the spread in dissolution timescale has a stronger effect on M/L than the spread in metallicity (Kruijssen 2008), implying that the scatter in the horizontal direction is largest and that the slope fitted to the entire sample is shallower than that of a single model.
Figure 7: Left: observed masstolight ratio versus the predicted masstolight ratio , together with their standard errors. The dotted line follows the 1:1 relation. Clusters for which the disagreement is larger than 1 are plotted as dots. Right: number histogram of (diamonds, shaded area) and (triangles, hashed area). Again, the error bars denote deviations, which were determined from 30k random realisations of the underlying data. 

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In Fig. 7, a more specific comparison is made between the observations and model predictions using the same framework as for the canonical expectations in Fig. 1. Again, the lefthand panel plots the observed versus the predicted masstolight ratios, while the righthand panel shows the number histograms of the two. In the lefthand panel it is shown that the predictions for half of the clusters are such that they reach down to the appropriate masstolight ratios. When comparing this panel to its analog in Fig. 1, the improved agreement with the observations is evident. Nonetheless, there is an aggregate of deviating GCs below the 1:1 relation at , representing the clusters for which no strong lowmass star depletion is expected from the models due to their long disruption times. Consequently, the predicted M/L_{V} for these clusters are similar or equal to the canonical values. Except for NGC 6809, there are no clusters above the 1:1 relation that are inconsistent with the observations. The number histogram of the observed and predicted masstolight ratios in the righthand panel of Fig. 7 confirms both the improved agreement between observed and predicted M/L_{V} with respect to Fig. 1 and the accumulation of a number of clusters near the canonical M/L_{V} in the model predictions.
4.3 Discussion of discrepant clusters
In total, there are twelve clusters with a worse than agreement between the model predictions and observations. Five of these have worse than agreement, while we would expect only one. Here, we discuss possible reasons behind the discrepancy.
The deviant clusters below the 1:1 relation in the lefthand panel of Fig. 7, being NGC 104, 1851, 1904, 4147, 4590, 5272, 5904, 6205, 6341, 6934 and 7089, generally share properties such as relatively wide orbits and long dissolution timescales. Due to their long dissolution timescales, they are all predicted to have nearcanonical M/L_{V}. This is illustrated in Fig. 8, where the fraction of and with respect to the canonical is shown in panels similar to Fig. 7. In both the left and righthand panels of Fig. 8, the accumulation of too high predicted masstolight ratios occurs near or at . Since per definition , no values are found. In that range, the apparent disagreement between the observed and predicted histograms is disputable since all but one cluster (NGC 6809) are in agreement with their canonical masstolight ratios.
Figure 8: Left: the ratio of the observed masstolight ratio to the canonically expected versus the ratio of the predicted masstolight ratio to , together with their standard errors. The dotted line follows the 1:1 relation. Clusters for which the disagreement is larger than 1 are plotted as dots. Right: number histogram of the ratio (diamonds, shaded area) and (triangles, hashed area). Again, the error bars denote deviations, which were determined from 30k random realisations of the underlying data. 

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While our predicted masstolight ratios are global (i.e., clusterwide) values, the observations from McLaughlin & van der Marel (2005) are derived from central velocity dispersion measurements from Pryor & Meylan (1993) and are extrapolated to global values using surface brightness profiles (McLaughlin & van der Marel 2005, and references therein). They fit isotropic singlemass King models and thus neglect any radial gradients of M/L ratio or mass function slope. Consequently, the values of do not contain any information about such gradients and for some clusters only accurately reflect the M/L_{V} ratio in their central parts. The global M/L_{V} ratios of clusters with strong radial M/L gradients are at best approximated (McLaughlin, private communication). For instance, the centre of a masssegregated cluster may be populated with massive, i.e., luminous stars, yielding a lower M/L_{V} ratio than its global value.
The disagreement between the global and central M/L is expected to be largest for clusters that have suffered relatively weak mass loss but are internally evolved. In that case, the lowmass stars are outside the core but still bound to the cluster and are included in the global M/L, while they do not play a role in the value derived by McLaughlin & van der Marel (2005). This indeed applies for the discrepant GCs in our sample, which not only have long dissolution timescales but also higher King parameters W_{0}, implying that mass segregation can be reached on relatively shorter timescales. For the GCs with worse than agreement below the 1:1 line in Figs. 7 and 8 we have average King parameter , while for the consistent GCs we find , both with standard errors <0.1. This further validates our explanation for the difference between the central and global M/L ratios of these GCs.
In a recent study, De Marchi et al. (2007) find that extended (lowconcentration and lowW_{0}) GCs are depleted in lowmass stars, which they confirm to be in accordance with predictions by theoretical studies (Chernoff & Weinberg 1990; Takahashi & Portegies Zwart 2000), while GCs with high values of W_{0} have close to canonical mass functions. This is in agreement with our predicted M/L_{V} for these clusters and suggests that the observed M/L_{V} are indeed underestimated. A more precise check can be made by comparing the lowmass star depletion from De Marchi et al. (2007) with the observed and predicted fractions of the canonical M/L_{V} ratios . This can be done for four GCs with worse than agreement, being NGC 104, 5272, 6341 and 6809. For NGC 104 and 5272 the observed depletion is not strong enough to draw any definitive conclusions, while for NGC 6341 and 6809 the results from De Marchi et al. (2007) are clearly more consistent with our predictions than with the observed M/L_{V} (see also Sect. 5 and Fig. 9). This substantiates the claim that some GCs have observed M/L ratios that are biased to lower numbers. To test this assertion, global observational measurements of the velocity dispersion would be needed to enhance the accuracy of the present observed M/L ratios.
We now revisit the mean M/L_{V} fractions of the canonical value presented in Sect. 4.2 by leaving out the GCs that may have strongly different central and global M/L ratios. It was shown by Baumgardt & Makino (2003) that for a cluster core collapse is reached within a Hubble time if . This timescale is increased by a factor three for a GC with typical initial mass of , but mass segregation manifests itself on a shorter timescale than the core collapse time. The relative mass loss due to dissolution of a GC is smaller than 10% after 12 Gyr for dissolution timescales Myr. These limits could separate GCs with similar global and central M/L ratios from those with pronounced differences between the two. We exclude GCs with both Myr and , as well as NGC 6809 (which has a M/L_{V} ratio that cannot be explained by any model as it is 1.6 times the canonical value). This yields an average observed fraction of the canonical M/L_{V} of and a predicted value of 0.02. For the excluded GCs, we have and 0.01, reflecting the fundamental difference between both values. Although the cuts we made represent only ``educated guesses'', it is evident that the agreement between theory and observations is much better for those GCs for which we can be more certain that the central M/L_{V} reflects the global value. For these GCs, our models confirm an average M/L ratio drop of about 20% due to lowmass star depletion, corresponding to about 1/4 of the observed difference in M/L_{V} between GCs and UCDs.
Figure 9: Correlation between the observed slope of the lowmass stellar mass function and the relative masstolight ratio with respect to the canonical value . The dotted curves indicate the theoretically predicted relation between and (not a fit), with the dot at the righthand tip representing the canonical values of and . Values of representing the global MF are marked with triangles, while those for clusters with worse than agreement between and are denoted by squares. Left: for the observed masstolight ratio fraction . Right: for the predicted masstolight ratio fraction . 

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Another option could be that the dissolution timescales of GCs on wide orbits are overestimated (as suggested for different reasons by Kruijssen & Portegies Zwart 2009), possibly due to a dissolution mechanism that is not included in our analysis. White dwarf kicks (Fregeau et al. 2009) could be a candidate for such a mechanism. This would imply that some of our predicted dissolution timescales and M/L ratios are overestimated. Also, we do not assume clusters to be initially masssegregated. Some of the clusters under consideration here are likely not to have reached energy equipartition within a Hubble time, but still exhibit evidence of mass segregation (e.g., Anderson & King 1996). This points to primordial mass segregation in these cases, which is shown by Baumgardt et al. (2008) to effect additional lowmass star depletion that we did not account for (see also Sect. 2.2). The additional modeled M/L ratio decrease would be % (KL08). However, this is not sufficient to lift the discrepancy for any of the deviating GCs.
5 Observational verification
If the decrease of M/L ratio with respect to the canonical value is indeed due to lowmass star depletion, one would expect a correlation between the observed slope of the lowmass MF and the ratio of the predicted and canonical M/L_{V} ratios . Specifically, for a powerlaw MF with , a low value of would be signified by a reduced value of .
In a study by De Marchi et al. (2007), MF slopes are determined in the stellar mass range m=0.3
for several Galactic globular clusters, based on a compilation of results
from HST imaging of different sources. By reanalysing the Baumgardt & Makino (2003) Nbody data, Baumgardt et al. (2008) conclude that for a Kroupa (2001) IMF the canonical slope in that mass range is
,
which is thus expected to be measured for clusters with canonical M/L ratios or
.
In addition, they provide a fourthorder powerlaw fit to the Nbody
simulations from Baumgardt & Makino (2003) for
as a function of
,
the elapsed fraction of the total disruption time. By inverting our relation between
and
(Eq. (5)) and inserting the outcome into
from Baumgardt et al. (2008, Eq. (4)), we obtain an expression for the predicted MF slope
between 0.3 and 0.8
as a function of the fraction of the canonical M/L_{V} ratio
.
Inversion of Eq. (5) yields
where the uncertainty for arises due to the range of over which it is constant in our models. Combination of this expression and Eq. (4) from Baumgardt et al. (2008) then provides the relation between and :
with the coefficients listed in Table 5 and again the uncertainty emerging from the degeneracy of that was mentioned earlier^{}.
Table 5: Coefficients for the fourthorder powerlaw approximation of as a function of (see Eq. (16)).
In Fig. 9, the correlation between and the observed lowmass MF slope is assessed for the subsample of clusters from the present study that is also considered in De Marchi et al. (2007). For comparison, the relation for the predicted lowmass MF slope as a function of is included as well. Most of the observed data match the predicted relation between and within their error bars, albeit with substantial scatter. This is due to the large uncertainties of the observations and possibly also related to biases introduced by comparing central and global masstolight ratios (see Sect. 4.3). The poor quality of the observations is illustrated by this spread and by the large error bars. For the predicted masstolight ratios the trend is more welldefined, but for low values of it does not extend down to the masstolight ratios that are predicted by theory. This could imply that either or are biased. If the latter is true, it suggests that some GCs perhaps dissolve more rapidly than presently included in the models.
As shown in Sect. 4.3, comparison of with the observed and predicted fractions of the canonical M/L_{V} ratios for the GCs with agreement parameter (see Table 4) provides an independent check of our predicted M/L_{V} ratios. While for NGC 104 and 5272 this does not allow for any definitive conclusions, for NGC 6341 and 6809 the observed mass functions are clearly more consistent with our predicted M/L_{V} ratios than with the observed values.
Table 6: Observed and predicted stellar mass function (MF) slopes in the range m=0.3 .
With Eq. (16), we can also predict the slope of the lowmass MF for clusters that where not considered by De Marchi et al. (2007). The predicted slopes are listed in Table 6. For most clusters with observed values of , the agreement between observed and predicted is reasonable. Only for NGC 6218 and 6712 there is a strong discrepancy. For NGC 6218, we expect the deviation to arise from the observed value of , since the predicted and observed M/L_{V} are in excellent agreement (see Table 4). On the other hand, for NGC 6712 the incompatibility may be due to a slight overestimation of and thus of and .
In this context it must be noted that the compilation of values from De Marchi et al. (2007) is drawn from a sample of literature estimates, most based on HST data, observed in somewhat different radial regions of each cluster. Four of the eleven GCs that coincide with our sample of 24 GCs do have a direct estimate for their global mass function (see Table 6). For the remaining seven other GCs from De Marchi et al. (2007), that estimate is taken from measurements restricted to the region around the halfmass radius , of which it is known that the shape of the MF is comparable to the global (i.e., clusterwide) MF (De Marchi & Pulone 2007; Baumgardt & Makino 2003; Richer et al. 1991). However, the uncertainty of these slopes is larger, and they do not provide a selfconsistent way to derive the global MF. It is clear that a direct determination of the global stellar mass function for most of the 24 GCs investigated in this study would allow one to verify the predictions of the present study with much higher confidence.
In Fig. 10 we investigate how feasible it is to observationally verify the predicted drop of for our full sample of 24 GCs. We plot the apparent Vband magnitude of stars with 0.3 solar masses V_{0.3} for each cluster versus the predicted slope of the stellar mass function for . V_{0.3} is obtained from the distance modulus of each GC and from the assumption that M_{V0.3}=9.8 mag (Baraffe et al. 1997). We also plot the angular size of each cluster versus . As a consequence of their generally higher galactocentric distance, those GCs with the faintest V_{0.3}>25 mag would not be expected to exhibit a strong lowmass star depletion. The angular halfmass diameters of the GCs with V_{0.3}<25 range between 2 and 8 arcmin. To obtain a representative estimate of the global mass function, it is clear that widefield groundbased imaging is required for most GCs. For this widefield imaging, a completeness magnitude of V 26 mag is desirable, which will allow moderately precise photometry already for mag. For 8 m class telescopes and with optical seeing in the range 0.8 to 1.0'', this requires 12 h integration time per filter, or 24 h for a twoband exposure. With widefield imagers such as VIMOS@VLT, IMACS@Magellan, or SuprimeCam@SUBARU, singleshot images will be sufficient to cover at least 23 halflight radii for most clusters. From Fig. 10 we conclude that the best candidate that also complements the compilation by De Marchi et al. (2007) is NGC 6779, followed by NGC 6362 and possibly NGC 6171.
Figure 10: Feasibility of observational tests of lowmass star depletion. Left: for the 24 GCs investigated in this paper, the apparent Vband magnitude of stars with 0.3 solar masses V_{0.3} is plotted versus the predicted slope of the stellar mass function for . Data points with large circles are those with available observational data from HST imaging (compiled by De Marchi et al. 2007). Right: the halfmass radius in arcminutes (Harris 1996) is plotted versus . Data points with crosses indicate GCs with V_{0.3}>25 mag. Data points with large squares indicate GCs whose predicted M/L_{V} deviates by more than 1 from the observed value. These are all GCs with agreement parameter in Table 4. 

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6 Discussion and conclusions
In this section, we provide a summary and a discussion of our results. We consider the effects of the assumptions that were made and reflect on the implications of the results.
6.1 Summary
In this study, we have investigated the dynamical masstolight ratios of 24 Galactic globular clusters. We have tested the hypothesis of the preferential loss of lowmass stars as the main explanation for the fact that the average observed masstolight ratios of the Galactic GCs in our entire sample are only 74 ^{+6}_{7}% of the expectations from stellar population models. Accounting for the orbital parameters we derived dissolution timescales due to twobody relaxation and disc shocking for our globular cluster sample and calculated the evolution of their masses and photometry using the SPACE analytical cluster models from Kruijssen & Lamers (2008, throughout this paper KL08). These models account for the preferential loss of lowmass stars which is fitted to the Nbody simulations by Baumgardt & Makino (2003) and therefore provide noncanonical M/L_{V} predictions. We find the derived dissolution timescales to be in good agreement with the range required for lowmass star depletion to explain the observed M/L ratio decrease from Kruijssen (2008).
The presentday (t=12 Gyr) M/L_{V} ratios have been compared to the observed values from McLaughlin & van der Marel (2005), yielding agreement for 12 out of 24 GCs. We considered possible causes for the remaining discrepancies that occur for the other GCs. It is found that 11 of these clusters have predicted M/L_{V} very close to the canonically expected M/L_{V} ratios due to their long dissolution timescales and the correspondingly modest lowmass star depletion, while their observed M/L_{V} are lower. This is probably due to the method by which the observed M/L_{V} are derived, which is biased towards the central M/L_{V} while our models predict global M/L_{V}. For masssegregated GCs with long dissolution timescales, both values can be substantially different. The discrepant GCs have higher than average King parameters W_{0}, which should indeed reach mass segregation on shorter timescales (see e.g., Baumgardt & Makino 2003). This explanation for the discrepancy between some of the observed and predicted M/L ratios is confirmed by a study of lowmass star depletion in GCs by De Marchi et al. (2007), whose observed mass functions are in good agreement with our predictions. The average observed M/L_{V} ratio of 74 ^{+6}_{7}% of the canonical expectations would therefore be underestimated. Excluding GCs which likely have dissimilar global and central M/L ratios by making cuts in dissolution timescale and King parameter, we find that the observed and predicted M/L_{V} ratios are consistent at 78 ^{+9}_{11}% and 78 2% of the canonical values, respectively. For the entire sample, the average predicted fraction of the canonical M/L_{V} ratio is 85 1%.
To assess the imprint of lowmass star depletion on the slope of the lowmass stellar mass function, we compared the observed mass function slopes from De Marchi et al. (2007) for 11 GCs contained in our study to the values predicted by our models as well as to the observed and predicted masstolight ratio fractions of the canonical values and . Most of the measured slopes agree with the predictions, but exhibit considerable scatter. Since most of them are values derived at around the halfmass radius and are extrapolated to global values, we also discuss the feasibility of observations for directly measuring global mass functions of most of the GCs investigated. We show that deep (groundbased) widefield imaging would be necessary, with point source detection limits mag. The most suitable candidate for such a campaign would be NGC 6779.
6.2 Propagation of assumptions
In the course of the study presented in this paper, several assumptions were made that affect the results to different extents. Their implications are as follows.
 (1)
 We have adopted the SPACE cluster models (KL08), of which the stellar evolution and photometry are based on the Padova 1999 isochrones (see Sect. 2). Consequently, the predicted cluster photometry and corresponding masstolight ratios are affected by that choice. To indicate the level of the deviation with the cluster models from Bruzual & Charlot (2003), in Fig. 1 we compared the canonically expected M/L_{V} from SPACE to the Bruzual & Charlot (2003) values for our cluster sample. The difference between both is inadequate to explain any systematic tendency of low masstolight ratio with respect to the SPACE models. Therefore, we conclude that the adopted cluster models do not effect substantial implications for the predicted M/L ratios^{}.
 (2)
 Due to the treatment of the preferential loss of lowmass stars in the SPACE cluster models, there are indications that the predicted masstolight ratios could be underestimated during the final
of the total cluster lifetime (KL08). Table 4 shows that none of the GCs in our sample reside in this regime. On the other hand, we did not include primordial mass segregation, which could decrease the predicted M/L ratios by % (KL08).
 (3)
 By adopting the average cluster orbits from Dinescu et al. (1999), we assume constant orbital parameters over the total cluster lifetimes. Considering the ballistic nature of the orbits, such an assumption is legitimate as long as the external conditions do not strongly differ. The Galactic potential was only substantially different from its present state during the formation of the Milky Way. A more extended distribution of mass during these early epoch would obviously increase the dissolution timescale due to disc shocking, and would affect the dissolution timescale due to twobody relaxation in a similar way because of the reduced tidal field. Consequently, this would imply that the mass loss during the first Gyr of our models is overestimated, causing our initial masses to be overestimated as well. However, the extended nature of the Milky Way would cause dissolution due to giant molecular cloud encounters to become an important mechanism (e.g., Gieles et al. 2006), thereby counteracting the previous effect. Although we cannot rule out any consequences, a residual influence would only be relevant for a small fraction (the first %) of the total cluster lifetime, where mass loss by dissolution is much less effective than later on during cluster evolution. Therefore, this likely only affects our analysis within the error margins.
 (4)
 We have compared our predictions to the observed masstolight ratios from McLaughlin & van der Marel (2005), which are biased towards central M/L values. As treated more extensively in Sect. 4.3, this yields underestimated observed M/L ratios for masssegregated clusters with long dissolution timescales. Therefore, based on the earlier discussion and the parameter range in which discrepancies arise, we consider the dynamical from McLaughlin & van der Marel (2005) to be subject to improvement for GCs with both dissolution timescales Myr and King parameters .
6.3 Consequences and conclusions
The consequences of our findings are not only relevant to studies of the M/L ratios of compact stellar systems, but also to other properties of these structures. Here we list them together with the conclusions of this work.
 (1)
 When constraining our sample to the subset for which the observed M/L_{V} likely reflect the global values, we find that the preferential loss of lowmass stars can account for the 20% discrepancy between observed dynamical masstolight ratios of Galactic GCs and those expected from stellar population models that assume a canonical present day mass function (Kroupa 2001). This alleviates the factor of two offset in M/L between GCs and UCDs by about 25%. Still, some additional dark mass with respect to a canonical IMF is required to explain the M/L of most UCDs.
 (2)
 Accounting for the orbital parameters, presentday masses and chemical compositions of individual clusters, we find that there is good agreement between our model predictions and observations of the M/L_{V} ratios of these clusters. For the GCs with worse than agreement there are strong indications that the discrepancy is due to an underestimation of the observed M/L ratio. In masssegregated clusters with long dissolution timescales, the observed M/L_{V} ratios represent central values that do not reflect the global M/L_{V} ratio.
 (3)
 The ideal way to confirm the validity of our explanation for the reduced M/L ratios of GCs for individual clusters will be to obtain a homegeneous set of deep widefield imaging for most GCs. This would expand and complement the currently available heterogeneous data sets of space based GC imaging, which is restricted to small fields in each GC, at different radial ranges. By this, the global mass and luminosity functions could be measured directly for individual GCs and be compared quantitatively to the predictions of this paper regarding the lowmass star depletion due to dynamical evolution. In addition, velocity dispersion measurements would allow for the determination of global M/L ratios, thus providing an update to those from McLaughlin & van der Marel (2005).
 (4)
 The topic of globular cluster selfenrichment and multiple stellar populations can also be considered within the framework of this paper. In a recent study by Marino et al. (2008) it is shown that NGC 6121 contains two stellar populations that are probably due to primordial variations in their respective chemical compositions. It is mentioned that the presentday mass of NGC 6121 is an order of magnitude smaller than that of known multiplepopulation GCs such as NGC 1851, 2808 and 5139. Consequently, Marino et al. (2008) pose the question how the enriched material could have remained in such a shallow potential and argue that multiple populations are unlikely to be strictly internal to GCs, unless they are the remnant of much larger structures. In the case of NGC 6121, our calculations seem to explain the issue, as it is the initially fifth most massive GC of our sample ( ). As a result, mass could have been retained much more easily, implying that the multiple populations of NGC 6121 are no reason to invoke external processes for enrichment and to abandon the selfenrichment scenario.
Acknowledgements
We thank the anonymous referee for valuable comments that improved the manuscript. We are grateful to Dana CasettiDinescu for interesting discussions and for providing the disc shocking destruction rates from Dinescu et al. (1999). Christine Allen is acknowledged for providing the data from Allen et al. (2008,2006) in electronic form. We thank Guido De Marchi for sharing the uncertainties on the mass function slopes from De Marchi et al. (2007). We particularly enjoyed stimulating discussions with Holger Baumgardt, Mark Gieles, Dean McLaughlin, Simon Portegies Zwart, Frank Verbunt and Reinier Zeldenrust. J.M.D.K. is grateful to Henny Lamers for support, advice and constructive comments on the manuscript, and is supported by a TopTalent fellowship from the Netherlands Organisation for Scientific Research (NWO), grant number 021.001.038.
Appendix A: Error analysis
In this Appendix, the error propagation through our computations is discussed. The errors in Tables 2 and 3 are standard errors, most of them determined by computing the formal error propagation. For a function
f(x_{1},x_{2},...,x_{i}) this implies
with the error in the parameter i. Asymmetric errors on each parameter are both separately propagated by employing the same recipe, while inverse relations are accounted for by swapping the positive and negative errors. However, Eq. (A.1) assumes an approximately constant derivative over the standard error interval. For very large errors on nonlinear relations this assumption does not hold. The first of two parameters where we have to correct for this effect is . It is inversely related to the destruction rate from Dinescu et al. (1999), which is a parameter with very large relative errors, even to the extend that after computing the error propagation one can have . Because dissolution timescales below zero are not physical, instead the negative error on is determined by computing
where indicates the negative error in a parameter i. In the context of Eq. (A.1), this approach is equivalent to assuming the derivative equals the mean slope of f(x) over the interval . On two occasions (NGC 6093 and 6712), a strongly asymmetric error in propagates into such that . However, since a very large positive error in or would make the term vanish in the inverse addition of Eq. (6), it should not propagate into a similarly large error in . This brings up the second parameter we have to correct for the propagation of large errors through nonlinear relations. We define the error in for NGC 6093 and 6712 such that .
The error margins on our predictions in Table 4 are determined by numerically evaluating Eq. (A.1) for the desired quantities. Our predictions depend on the observed mass, metallicity and dissolution timescale. The derivatives of M/L_{V} with respect to the former two are trivial since M/L_{V} is determined by interpolating over these parameters. For the dissolution timescale, we compute additional models at
to obtain the numerical derivative of M/L_{V} with respect to
.
In fact, this is the differential rather than the derivative, because for long dissolution timescales M/L_{V} can be locally constant, while it varies over a larger range. The only case were a nonlinearity forces us to derive alternative errors is for the positive standard error on M/L_{V}. Although the uncertainty in metallicity could increase the predicted masstolight above its canonical value, the uncertainty in mass and dissolution timescale cannot due to the flattening of the cluster isochrones in the
plane (see Fig. 4). Therefore, the combined positive standard error of the masstolight ratio due to the uncertainty in mass and dissolution timescale
is defined as
with the standard error according to Eq. (A.1), the canonically expected masstolight ratio and the predicted value. This definition ensures that the positive standard error is never larger than the difference between the canonical and predicted masstolight ratios.
Except for the alternative error in Eq. (A.3) that is specific to M/L_{V}, the standard errors on the predicted initial masses are determined analogously to the above. For the remaining lifetimes, numerical derivatives with respect to mass, metallicity and dissolution timescale are simply obtained by reintegrating Eq. (1) for slightly different initial conditions.
Finally, for the predicted slopes in Table 6, the errors are computed using Eq. (A.1) and restricted such that (analogous to Eq. (A.3)).
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Footnotes
 ... isochrones^{}
 These isochrones are based on Bertelli et al. (1994), but use the AGB treatment as in Girardi et al. (2000).
 ... relation^{}
 For white dwarfs, this relation is taken from Kalirai et al. (2008), while for neutron stars the relation from Nomoto et al. (1988) is used. Black hole masses are assumed to be constant at , in agreement with observations (Casares 2007). For more details, see KL08.
 ... environment^{}
 Please note that a Kroupa IMF was assumed here. For substantially different IMFs the relation will vary.
 ... field^{}
 This includes the effect of bulge shocks, which occur in clusters on eccentric orbits.
 ... earlier^{}
 Consequently, it represents the same uncertainty as , with corresponding to and to . Since for most GCs under consideration the elapsed fractions of the total disruption time are closer to than to (see Table 4), we adopt if .
 ... ratios^{}
 Of course, if the preferential loss of lowmass stars is accounted for, i.e., noncanonical models are considered, the SPACE cluster models predict very different photometric evolution than canonical cluster models such as Bruzual & Charlot (2003).
All Tables
Table 1: Observed properties for the cluster sample, together with their standard errors.
Table 2: Orbital parameters for the cluster sample, together with their standard errors.
Table 3: Computed dissolution timescales (for ) due to twobody relaxation ( ), disc shocking ( ) and both mechanisms ( ), together with their standard errors. All values are rounded to one decimal.
Table 4: Our model predictions with their standard errors.
Table 5: Coefficients for the fourthorder powerlaw approximation of as a function of (see Eq. (16)).
Table 6: Observed and predicted stellar mass function (MF) slopes in the range m=0.3 .
All Figures
Figure 1: Left: observed masstolight ratio versus the canonically expected masstolight ratio , together with their standard errors (see Table 1). The dotted line follows the 1:1 relation. Clusters for which the disagreement is larger than 1 are plotted as dots. Right: number histogram of (diamonds, shaded area) and (triangles, hashed area). For comparison, canonically expected M/L_{V} ratios from Bruzual & Charlot (2003) for a Chabrier (2003) IMF are overplotted (squares, dotted line). Again, the error bars denote deviations, which were determined from 30k random realisations of the underlying data. 

Open with DEXTER  
In the text 
Figure 2: Left: Vband masstolight ratio evolution M/L_{V} for t_{0}=1 Myr, Z=0.0004 and initial masses in the range 8 with 0.25dex intervals. Right: same graph, but for Z=0.004. From top to bottom, different curves represent the M/L_{V} evolution for decreasing initial cluster masses. 

Open with DEXTER  
In the text 
Figure 3: The ratio of predicted to canonical masstolight ratio as a function of the elapsed fraction of the total disruption time . Dotted curves denote model predictions for a broad range of initial conditions (varying initial masses, dissolution timescales and metallicities), while the solid line describes a threecomponent linear approximation to the models. 

Open with DEXTER  
In the text 
Figure 4: Cluster evolution in the plane for Myr and . Solid curves represent cluster evolutionary tracks for initial cluster masses in the range 10^{8} with 0.5dex intervals. Cluster isochrones at t=12 Gyr are described by dashed lines, while these at t=10 and t=14 Gyr are denoted by dotted lines (bottom and top, respectively). Dots denote the onset of the preferential loss of lowmass stars for each evolutionary track. 

Open with DEXTER  
In the text 
Figure 5: Top left: timeevolution of M/L_{V} for NGC 5466. The solid line represents the Z=0.0004 model with the dissolution timescale of the cluster (t_{0}=10.2 Myr), while the dotted curve indicates the canonical M/L_{V} evolution, i.e., if the preferential loss of lowmass stars were omitted. The dashed line denotes constant age of t=12 Gyr. The predicted M/L_{V} of NGC 5466 is marked by a cross and the onset of the preferential loss of lowmass stars is specified with a dot. Top right: evolution in the plane for NGC 5466. Curves and symbols have the same meaning as in the topleft panel. Bottom left: same as top left, but for NGC 6779 (t_{0}=1.2 Myr). Bottom right: same as top right, but for NGC 6779. 

Open with DEXTER  
In the text 
Figure 6: Observed (crosses) and predicted (triangles) distribution of GCs in the plane. The solid line represents our linear fit to the observations, while the dotted line denotes the relation found by Mandushev et al. (1991).The error bars in the top left corner denote the average uncertainty on the observations, while the error bars in the bottom right corner represent the average uncertainty on the predictions. 

Open with DEXTER  
In the text 
Figure 7: Left: observed masstolight ratio versus the predicted masstolight ratio , together with their standard errors. The dotted line follows the 1:1 relation. Clusters for which the disagreement is larger than 1 are plotted as dots. Right: number histogram of (diamonds, shaded area) and (triangles, hashed area). Again, the error bars denote deviations, which were determined from 30k random realisations of the underlying data. 

Open with DEXTER  
In the text 
Figure 8: Left: the ratio of the observed masstolight ratio to the canonically expected versus the ratio of the predicted masstolight ratio to , together with their standard errors. The dotted line follows the 1:1 relation. Clusters for which the disagreement is larger than 1 are plotted as dots. Right: number histogram of the ratio (diamonds, shaded area) and (triangles, hashed area). Again, the error bars denote deviations, which were determined from 30k random realisations of the underlying data. 

Open with DEXTER  
In the text 
Figure 9: Correlation between the observed slope of the lowmass stellar mass function and the relative masstolight ratio with respect to the canonical value . The dotted curves indicate the theoretically predicted relation between and (not a fit), with the dot at the righthand tip representing the canonical values of and . Values of representing the global MF are marked with triangles, while those for clusters with worse than agreement between and are denoted by squares. Left: for the observed masstolight ratio fraction . Right: for the predicted masstolight ratio fraction . 

Open with DEXTER  
In the text 
Figure 10: Feasibility of observational tests of lowmass star depletion. Left: for the 24 GCs investigated in this paper, the apparent Vband magnitude of stars with 0.3 solar masses V_{0.3} is plotted versus the predicted slope of the stellar mass function for . Data points with large circles are those with available observational data from HST imaging (compiled by De Marchi et al. 2007). Right: the halfmass radius in arcminutes (Harris 1996) is plotted versus . Data points with crosses indicate GCs with V_{0.3}>25 mag. Data points with large squares indicate GCs whose predicted M/L_{V} deviates by more than 1 from the observed value. These are all GCs with agreement parameter in Table 4. 

Open with DEXTER  
In the text 
Copyright ESO 2009