Structural parameters for globular clusters in M31
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Preprint typeset using L A TEX style emulateapj v. 5 / / STRUCTURAL PARAMETERS FOR GLOBULAR CLUSTERS IN M31 S ong W ang , J un M a AJ, in press
ABSTRACTIn this paper, we present surface brightness profiles for 79 globular clusters in M31, using images observedwith
Hubble Space Telescope , some of which are from new observations. The structural and dynamical param-eters are derived from fitting the profiles to several di ff erent models for the first time. The results show that inthe majority of cases, King models fit the M31 clusters as well as Wilson models, and better than S´ersic models.However, there are 11 clusters best fitted by S´ersic models with the S´ersic index n >
2, meaning that they havecuspy central density profiles. These clusters may be the well-known core-collapsed candidates. There is abimodality in the size distribution of M31 clusters at large radii, which is di ff erent from their Galactic coun-terparts. In general, the properties of clusters in M31 and the Milky Way fall in the same regions of parameterspaces. The tight correlations of cluster properties indicate a “fundamental plane” for clusters, which reflectssome universal physical conditions and processes operating at the epoch of cluster formation. Subject headings: galaxies: individual (M31) – galaxies: star clusters general – galaxies: stellar content INTRODUCTION
It is well known that, studying the spatial structures anddynamics of globular clusters (GCs) is of great importancefor understanding both their formation condition and dy-namical evolution within the environment of their galaxies(Mclaughlin et al. 2008). For example, these clusters are ideallaboratories for detailed studies on two-body relaxation, masssegregation, stellar collisions and mergers, and core collapse(Meylan & Heggie 1997). The correlations of structures withgalactocentric distance can provide information on the role ofthe galaxy tides towards the clusters, while the distributionof ellipticity can shed light on the primary factor for clusterelongation. In addition, comparisons of structures of GCs lo-cated in di ff erent environment of galaxies o ff er clues to di ff er-ences in the early formation and evolution of the galaxies orin their subsequent accretion histories (Bellazzini et al. 2003;Mackey et al. 2007). The “fundamental plane” for clustersin parameter space reflects universal cluster formation con-ditions, regardless of their host environments (Barmby et al.2009).The structural and dynamical parameters of clusters are of-ten determined by fitting the surface brightness profiles tostructure models, combined with mass-to-light ratios esti-mated from velocity dispersions or population-synthesis mod-els. An accurate and well resolved density profile can beobtained by studying the distribution of integrated light cou-pling with star counts (Federici et al. 2007). Several modelsare often used in the fits: the empirical, single-mass, mod-ified isothermal spheres (King 1962, 1966; Wilson 1975);the isotropic multi-mass models (Da Costa & Freeman 1976);the anisotropic multi-mass models (Gunn & Gri ffi n 1979;Meylan 1988, 1989); the power-law surface-density profiles(S´ersic 1968; Elson, Fall & Freeman 1987).The nearest large GC system outside the Milky Way (MW)is that of M31, with a distance of ∼
784 kpc from us National Astronomical Observatories, Chinese Academy of Sciences,Beijing, 100012, China;[email protected] University of Chinese Academy of Sciences, Beijing, 100039, China Key Laboratory of Optical Astronomy, National Astronomical Obser-vatories, Chinese Academy of Sciences, Beijing, 100012, China (Stanek & Garnavich 1998). It is so close to us that mostGCs in it can be well resolved with
Hubble Space Tele-scope ( HST ). Battistini et al. (1982) first estimated core radiifor several clusters in M31, and subsequently, a numberof studies (Pritchet & van den Bergh 1984; Crampton et al.1985; Spassova et al. 1988; Bendinelli et al. 1990, 1993;Cohen & Freeman 1991; Fusi Pecci et al. 1994; Barmby et al.2002; Ma et al. 2006, 2007, 2012) focused on the internalstructures of M31 GCs, including the core radius, half-lightradius, tidal radius, and ellipticity, using the images from largeground-based telescopes and
HST . Barmby et al. (2007) de-rived structural and dynamical parameters for 34 M31 GCs,and construct a comprehensive catalog of these parametersfor 93 M31 GCs with corrected versions of those in a previousstudy (Barmby et al. 2002). Combined with the structures anddynamics for clusters from the MW, Magellanic Clouds, theFornax dwarf spheroidal, and NGC 5128, these authors foundthe GCs have near-universal structural properties, regardlessof their host environments. Barmby et al. (2009) found thatbright young clusters in M31 are larger and more concen-trated than old ones, and are expected to dissolve within afew Gyr and will not survive to become old GCs. With mea-surements of structural parameters for 13 extended clusters(ECs) in the halo regions of M31, Huxor et al. (2011) pre-sented that the faintest ECs have magnitudes and sizes similarto Palomar-type GCs in the MW halo. Wang & Ma (2012)measured structures and kinematics for 10 newly discoveredGCs in the outer halo of M31, and found that they have largerellipticities than most of GCs in M31 and the MW, which maybe due to galaxy tides from satellite galaxies of M31 or maybe related to the merger and accretion history that M31 hasexperienced. Using the same sample clusters in Wang & Ma(2012), Tanvir et al. (2012) found that some GCs in M31 ex-hibit cuspy cores which are well described by S´ersic (1968)models. These authors also confirmed the exist of luminousand compact globulars at large galactocentric radii of M31,with no counterparts found in the MW. The last three stud-ies extended the structural analysis of clusters in M31 out to R gp ∼
100 kpc, providing important information on the accre-tion history of M31 outer regions.In this paper, we determined structures and kinematics for Wang et al.79 clusters in M31 by fitting several structural models to theirsurface brightness profiles. This paper is organized as follows.In Section 2, we present the
HST observations for the sampleclusters, and the data-processing steps to derive the surfacebrightness profiles. In Section 3, we determine structures andkinematics of the clusters and make some comparisons withprevious studies. A discussion on the correlations of the de-rived parameters is given in Section 4. Finally, we summarizeour results in Section 5. DATA AND ANALYSIS PROCEDURES
Globular Cluster Sample
HST has imaged a large fraction of globular clusters (GCs)in M31. Barmby et al. (2002) used HST / Wide Field Plan-etary Camera 2 (WFPC2) images to measure ellipticities,position angles, and best-fit King (1966) model (hereafter“King model”) parameters for a large sample of M31 GCs.Barmby et al. (2007) determined structures and kinematicsfor M31 GCs by fitting surface brightness profiles to di ff er-ent models, however, for all the GCs but G001 studied inBarmby et al. (2002), only King model was used. In addi-tion, there were some clusters located at the edges of theimages or observed with only one filter. With updated ob-servations by HST, new data can be derived for them now.So, we decided to re-estimate the structure parameters forthese GCs in Barmby et al. (2002). However, Barmby et al.(2007) determined structures for G001 using three modelsand new observation, while Barmby et al. (2009) determinedstructure parameters for five clusters (B315, B318, B319,B368, and B374) using updated HST data. In addition, thereis no new observation for B077, which is at the edges ofthe HST images. We also noticed that BH20 and BH21 hadbeen classified as stars, while BH24 a galaxy (Caldwell et al.2009), which would not be discussed again in this paper.The remaining 69 clusters from Barmby et al. (2002) wouldbe included in the sample. Mackey et al. (2007) estimatedmetallicities, distance moduli and reddening values for 10newly-discovered halo GCs in M31 using the ACS / Wide FieldCamera (WFC) images. Although Wang & Ma (2012) andTanvir et al. (2012) determined structures for the 10 GCs,Wang & Ma (2012) only used King model, while Tanvir et al.(2012) presented few structure parameters. So, we re-estimated the structure parameters for these GCs. Finally,there are 79 clusters in our sample. We obtained the com-bined drizzled images from the Hubble Legacy Archive. Theimages in the bandpass close to V band (F555W or F606W)and I band (F814W) were preferred, otherwise the images ofF300W, F435W or F475W were selected. The images withhigh resolution were preferentially adopted (WFPC2 / PC orACS / WFC), followed by those obtained with WFPC2 / WFC.Figure 1 shows the spatial distribution for the sample GCs.
Surface Brightness Profiles
The data analysis procedures to measure surface bright-ness profiles of clusters have been described in Barmby et al.(2007). When the center positions of these clusters were de-termined using the imcentroid task in IRAF, the ellipse taskwas run in two passes to derive the surface brightness pro-files. The ellipse showed inability to converge for several in-dividual clusters. In these cases we first smoothed the im-ages with a boxcar filter (Larsen et al. 2002), and then ranthe ellipse to derive the density profiles. The overall ellip-ticity and position angle (PA) were determined by averaging F ig . 1.— Location of our sample GCs in relation to M31. The inner el-lipse delineates M31’s main stellar disk ( i = ◦ and R = ◦ ) while theouter ellipse has a radius of 55 kpc and is flattened to b / a = .
6, as given inRichardson et al. (2009). The filled circles and asterisks represent the sampleGCs from Barmby et al. (2002) and Mackey et al. (2007), respectively. Thetwo small ellipses near the M31 center represent M32 (bottom) and NGC 205(top-right). the ellipse output in the first pass, with the errors estimatedas the standard deviation of the mean. Several clusters (B330,B468, BH04, BH11, BH29, and NB39) showed errors of P.A.slarger than 15 degrees. We checked the images for these clus-ters, and found that the random fluctuations due to individ-ual stars (Larsen et al. 2002) may account for the high errors,leading to a more di ffi cult business for accurate measurementsof PAs Table 1 lists the average ellipticity, P.A. and some ad-ditional integrated data for the sample clusters. Consideringthat the metallicities for young ( < ff erent, first, we averaged the ages for the sam-ple 79 clusters from several previous studies (Caldwell et al.2009, 2011; Kang et al. 2012; Wang et al. 2010, 2012) to dis-tinguish young from old clusters. Of which, there are 13 clus-ters (BH23, BH29, NB39, and the 10 GCs from Mackey et al.2007) with no available age values in the literatures, and weassumed them as old ones. Finally, there are six young clus-ters (B097D, B324, BH05, BH12, DAO38, and M091) in oursample, all of which are from Barmby et al. (2002). For the 69GCs from Barmby et al. (2002), the metallicities with uncer-tainties from Kang et al. (2012) were preferentially adoptedas our reference, followed by those of Caldwell et al. (2009,2011) for old clusters, while the solar metallicity was as-sumed for young clusters (Barmby et al. 2009). The redden-ing values were from Kang et al. (2012), while the other in-tegrated data were from RBC V.5 (Galleti et al. 2004, 2006,2009). For the 10 GCs from Mackey et al. (2007), we usedthe integrated data presented in Wang & Ma (2012), includ-ing the VI magnitudes, galactocentric distances, reddeningvalues, and metallicities (see Wang & Ma 2012, and refer-ences therein). In addition, B magnitudes for six of the 10GCs are from RBC V.5. Old clusters with no metallicity mea-surements are assigned with a mean M31 GC metallicity of[Fe / H] = − . / H] are assumed to be 0.6 as for the standard devi-roperties of M31 globular clusters 3ation of the metallicity distribution of the M31 GC system(Barmby et al. 2000). Clusters with no reddening values areassigned with the Galactic reddening in the direction of M31of E ( B − V ) = .
08 (van den Bergh 1969).Raw output from package ellipse is in terms of countss − pixel − , which needs to multiply by a number (400 forACS / WFC, 100 for WFPC2 / WFC, and 494 for WFPC2 / PC)to convert to counts s − arcsec − . A formula was used to trans-form counts to surface brightness in magnitude calibrated onthe vegamag system, µ/ mag arcsec − = − . − arcsec − ) + Zeropoint . (1)As noted by Barmby et al. (2007), occasional oversubtrac-tion of background during the multidrizzling in the automaticreduction pipeline led to “negative” counts in some pixels,so we worked in terms of linear intensity instead of surfacebrightness in magnitude. With updated absolute magnitudesof the sun M ⊙ (C. Willmer, private communication) listedin Table 2, the equation for transforming counts to surfacebrightness in intensity was derived, I / L ⊙ pc − ≃ Conversion Factor × (counts s − arcsec − ) . (2)Converting from luminosity density in L ⊙ pc − to surfacebrightness in magnitude was done according to µ/ mag arcsec − = − . I / L ⊙ pc − ) + Coe ffi cient . (3)Table 2 presents the Zeropoints, Conversion Factors, andCoe ffi cients used in these transformations for each filter. Ta-ble 3 gives the final, calibrated intensity profiles for the 79clusters but with no extinction corrected. The reported in-tensities are calibrated on the vegamag scale. Column 7gives a flag for each point, which has the same meaningas Barmby et al. (2007) and Mclaughlin et al. (2008) defined.The points flagged with“OK” are used to constrain the modelfit, while the points flagged with “DEP” are those that maylead to excessive weighting of the central regions of clusters(see Barmby et al. 2007; Mclaughlin et al. 2008, for details).In addition, points marked with “BAD” are those individ-ual isophotes that deviated strongly from their neighbors orshowed irregular features, which were deleted by hand. Point-spread Function
The point-spread function (PSF) models are critical to accu-rately measure the shapes of objects in images taken with HST(Rhodes et al. 2006). Barmby et al. (2002) found that by fit-ting models without PSF convolution, the scale radii were sys-tematically larger, and the concentrations smaller than thoseestimated from the convolved models. Compared to ground-based telescopes, the PSF of HST is very stable, although it isalso known to vary with time (Krist et al. 2011). Tiny Tim hasbeen the standard modeling software for HST PSF simulationfor 20 years, with a variety of uses ranging from deconvolu-tion, model convolution, PSF fitting photometry and astrome-try, and PSF subtraction (Krist et al. 2011). In this paper, wederived the ACS / WFC and WFPC2 PSF models using TinyTim , and then the models were fitted using a function of theform I PSF / I = [1 + ( R / r ) α ] − β/α , (4) http: // tinytim.stsci.edu / cgi-bin / tinytimweb.cgi. where r , α , and β for each filter are given in Table 4. Itcan be seen that the parameters in Table 4 are slightly dif-ferent with those from Barmby et al. (2007) for ACS / WFC inthe F606W and F814W filters. Barmby et al. (2007) selecteda number of isolated stars on a number of images, and com-bined them to produce a single, average PSF for each filter.Here we derived a few model PSFs at di ff erent positions ofthe camera, and averaged them to produce the final PSF foreach filter. The discrepancies of these parameters from thetwo studies are due to the di ff erent methods, but are negligi-ble. The PSF variation over the cluster extent was ignoredsince the clusters are small compared to the camera field ofview (Barmby et al. 2009). MODEL FITTING
Structure Models
We used three structural models to fit star cluster surfaceprofiles, including King model, Wilson (1975), and S´ersic(1968) model (hereafter “Wilson model” and “S´ersic model”).Mclaughlin et al. (2008) have described the three structuralmodels in detail, here we briefly summarized some basic char-acteristics for them.King model is most commonly used in studies of star clus-ters, which is the simple model of single-mass, isotropic,modified isothermal sphere. Barmby et al. (2007, 2009)found that M31 clusters are better fitted by King models. Thephase-space distribution function for King model is defined as f ( E ) ∝ ( exp[ − E /σ ] − , E < , , E ≥ , (5)where E is the stellar energy, σ is a velocity scale.Wilson model is an alternate modified isothermal spherebased on the ad hoc stellar distribution function ofWilson (1975). These models have more extended enve-lope structures than the standard King isothermal spheres(Mclaughlin et al. 2008). Several studies presented that Wil-son models fit the majority of GCs in the Milky Way (MW)and some of its satellites and NGC 5128 as well as orbetter than King models (Mclaughlin & van der Marel 2005;Mclaughlin et al. 2008). The phase-space distribution func-tion for Wilson model is defined as f ( E ) ∝ ( exp[ − E /σ ] − + E /σ , E < , , E ≥ . (6)S´ersic model has a R / n surface-density profile, and hasbeen the standard model for parameterizing the surface bright-ness profiles of early-type galaxies and bulges of spiral galax-ies (Baes & Gentile 2011). Tanvir et al. (2012) found thatsome classical GCs in M31 which exhibit cuspy core pro-files are well described by S´ersic models of index n ∼ − I ( R ) = I exp[ − ln(2) × ( R / r ) (1 / n ) ] . (7) Fits
Before we fitted models to the brightness profiles of thesample clusters, the intensity profiles were corrected for ex-tinction. Table 2 lists the e ff ective wavelengths of the ACSand WFPC2 filters from the Instrument Handbook. Withthe extinction curve taken from Cardelli et al. (1989) with R V = .
1, we derived the A λ values for each filter. Wang et al. F ig . 2.— Surface brightness profiles and model fits to one sample clusterB006, with the data of F555W and F814W band from top to bottom. Thethree panels in each line are fits to, from left to right: King model, Wilsonmodel, and S´ersic model. We first convolved the three models with PSFs for the filtersused. Given a value for the scale radius r , a dimensionlessmodel profile e I mod ≡ I mod / I was computed, and then the con-volution was carried out, e I ∗ mod ( R | r ) = Z Z ∞−∞ e I mod ( R ′ / r ) e I PSF (cid:2) ( x − x ′ ) , ( y − y ′ ) (cid:3) dx ′ dy ′ , (8)where R = x + y , and R ′ = x ′ + y ′ . e I PSF was approxi-mated using the equation (4) (see Mclaughlin et al. 2008, fordetails). The best fitting model was derived by calculatingand minimizing χ as the sum of squared di ff erences betweenmodel and observed intensities, χ = X i [ I obs ( R i ) − I e I ∗ mod ( R i | r ) − I bkg ] σ i , (9)in which a background I bkg was also fitted. The uncertaintiesof observed intensities listed in Table 3 were used as weights.As an example, we plotted the fitting for one sample clus-ter in Figure 2. The observed intensity profile with extinc-tion corrected is plotted as a function of logarithmic projectedradius. The open squares are the data points included in themodel fitting, while the crosses are points flagged as “DEP” or“BAD”, which are not used to constrain the fit (Wang & Ma2012). The best-fitting models, including the King model,Wilson model, and S´ersic model are shown with a solid linefrom the left to the right panel, with a fitted I bkg added. Thedashed lines represent the shapes of the PSFs for the filtersused. There are some clusters showing individual isophoteswith ellipse intensities that showed irregular features or de-viated strongly from their neighbors. As Mclaughlin et al.(2008) reported, such bumps and dips may skew the follow-ing model fits. In these cases, we first derived the ellipse out-put through a boxcar filter to make a smoothed cluster profile(Mclaughlin et al. 2008), and then fitted these surface profilesusing structural models. If some individual isophotes still can-not be well fitted, these points were deleted by hand, whichwere masked with “BAD” in Table 3.Most profiles of the sample clusters were well fitted bythe models, except for several clusters with di ff erent rea-sons. There are one or several bright objects at the inter-mediate radii of B097D, GC9, and M091; the shape is veryloose for DAO38; the signal-to-noise ratio (SNR) is low forB205; there are one or several bright objects near the outerregion of B328, B331, and BH11; the images are not wellresolved for B092, B101, B145, BH05, and NB39; sev- F ig . 3.— Images (F555W or F606W) of eight “ring cluster” candidates.From the upper left, these are B324, B330, B333, B468, GC3, GC7, BH04,and BH29. eral clusters (B324, B330, B333, B468, GC3, GC7, BH04,and BH29) are lack of a central brightness concentration,which may be candidates of “ring clusters”. The “ring clus-ters” have been reported in the Magellanic Clouds (MCs)and M33 (Mackey & Gilmore 2003; Hill & Zaritsky 2006;Werchan & Zaritsky 2011; San Roman et al. 2012), with ir-regular profiles such as bumps and dips which may not beattributed to the random fluctuations due to a few luminousstars. The images of these “ring cluster” candidates in oursample are displayed in Figure 3. There are three clusters(B018, B114, and B268) showing bumps only in the luminos-ity profiles of the F814W, with the intrinsic color of ( V − I ) = I bkg . We checked the im-ages carefully, and found that several reasons may account forthose high errors: most of these clusters are not well resolvedin the images; there are one or several bright objects in theintermediate and outer part of clusters B331 and M091; clus-ters B064 and B205 only have images in one band (F300W),of which the SNRs are low. We should notice that if the fit-ted background is too high, the tidal radius may be estimatedsmaller artificially. Only one band of observations can bederived for six clusters (B009, B020D, B064, B092, B101,B205), and none of the bands is close to V band. So, we wouldnot include the six objects in the following discussions.We used the same method mentioned in Barmby et al.(2009) and Wang & Ma (2012) to transform the magni-tudes from ACS and WFPC2 to V on the vegamag scale(Holtzman et al. 1995; Sirianni et al. 2005). For clusters withavailable data of F555W or F606W band, we briefly usedthe extinction-corrected color ( V − I ) or ( B − V ) to do thetransformations, while for clusters with no data of F555W orF606W band, we first transformed the ACS or WFPC2 mag-nitudes to I magnitude using the color ( B − I ) , and then com-puted the V magnitude using the color ( V − I ) . The BVI magnitudes and the reddening values are listed in Table 1.We estimated a precision of ± .
05 mag in the transforma-tion, which was propagated through the parameter estimates(Barmby et al. 2007; Wang & Ma 2012).The mass-to-light ratios ( M / L values), which were used toderive the dynamical parameters, were determined from thepopulation-synthesis models of Bruzual & Charlot (2003), as-suming a Chabrier (2003) initial mass function. The ages andmetallicities used to computed M / L values in V band werederived as follows. An age of 13 Gyr with an uncertainty ofroperties of M31 globular clusters 5 ± M / L of old clusters include the uncertainties in ageand metallicity, while for young clusters, we simply adopted10% in M / L as the errors as Barmby et al. (2009) did.The basic structures and various derived dynamical param-eters of the best-fitting models for each cluster are listedin Table 5 to Table 7, with a description of each parame-ter / column at the end of each table (see for details of theircalculation in Mclaughlin et al. 2008). The uncertainties ofthese parameters were estimated by calculating their varia-tions in each model that yields χ within 1 of the globalminimum for a cluster, while combined in quadrature withthe uncertainties in M / L for the parameters related to it (seeMclaughlin & van der Marel 2005, for details). Comparison with Previous Determinations
In this paper, we determined structural parameters for 79clusters in M31 by comparing their surface brightness pro-files with three structural models. In Figure 4, some estimatedstructural parameters for clusters in this paper were comparedwith those from previous studies (e.g., Barmby et al. 2002,2007; Wang & Ma 2012). Barmby et al. (2002, 2007) pre-sented structural and dynamical parameters in V band for 51clusters in our sample, while Wang & Ma (2012) determinedstructures for 10 GCs using King model. So we used the re-sults on the bandpass close to V band (e.g., F475W, F555W,and F606W) and fitted by King model for comparison. It isnot unexpected to see that most of our parameters are largerthan the results from Barmby et al. (2002, 2007), since theisophotes flagged as “DEP” in this paper may not be excludedfrom the fitting process in Barmby et al. (2002), resulting inexcessive weighting of the inner regions in the fits. In addi-tion, the ellipticities presented by Barmby et al. (2002) wereaveraged over di ff erent filters for each cluster, while our re-sults in Figure 4 were on the bandpass close to V band. Thecluster with the largest discrepancy of ellipticity is BH05,with an estimate of 0.19 by Barmby et al. (2002), and 0.55 inthis paper. As discussed above, the image of BH05 is not wellresolved, so it is di ffi cult to derive accurate ellipticities andstructural parameters. Barmby et al. (2007) also concludedthat the shapes of outer parts of faint clusters are strongly af-fected by the galaxy background or low SNRs, leading to adi ffi cult business to accurately measure the ellipticities. Thelargest scatter of the R h is B018, which is ∼ ∼
30 pc in this paper. Al-though the PSFs and some calibration factors adopted hereare slightly di ff erent from those of Wang & Ma (2012), theparameters derived here are in good agreement with their re-sults.Strader et al. (2009, 2011) presented observed velocity dis-persions for a number of GCs in M31 using new high-resolution spectra from MMT / Hectochelle. These authorsestimated the M / L values in V band for clusters using thevirial masses and luminosities, since the virial masses arethe nominal estimates of the “global” mass of the systemand are less sensitive to the accuracy of the King model fit(see Strader et al. 2009, 2011, for details). Here we estimatedthe M / L values in V band from population-synthesis modelsby giving their metallicities and various ages (Barmby et al.2007, 2009). In Figure 5, We presented the comparison ofsome parameters derived by Barmby et al. (2007, 2009) andthis paper with those from Strader et al. (2011). There is a F ig . 4.— Comparison of our newly obtained cluster structural parameterswith previous measurements by Barmby et al. (2002, 2007) (open circles)and Wang & Ma (2012) (filled circles).F ig . 5.— Comparison of some dynamical parameters from Barmby et al.(2007) (crosses), Barmby et al. (2009) (triangles), and this paper (filled cir-cles) with those from Strader et al. (2011). large discrepancy between the M / L values derived from thetwo methods, and most of the M / L values from population-synthesis models are larger than those from observed veloc-ity dispersions. The model masses from Barmby et al. (2007,2009) and this paper are slightly larger than the virial massesfrom Strader et al. (2011), which were estimated using thehalf-mass radius and global velocity dispersion σ ∞ . The σ p , obs values from Strader et al. (2011) were the central veloc-ity dispersions estimated using the observed velocity disper-sions by integrating a known King model over the fiber aper-ture, and are consistent with the predicted line-of-sight veloc-ity dispersions at the cluster center from Barmby et al. (2007,2009) and this paper. However, there are few young clusterswith velocity dispersion measured by Strader et al. (2011), somore observation data and analysis are needed to check theconclusion of the comparisons.Figure 6 plots the correlations of velocity dispersion andmass with M / L for M31 clusters. The left two panels showthese parameters of clusters from Strader et al. (2011), whilethe right two panels show those derived by King model fromBarmby et al. (2007, 2009) and this paper. We can see thatthe σ p , and M mod from King-model fits show large depen-dence on the M / L . The old and young clusters show a dis-tinct boundary of the M / L values. There is no clear correla-tion for M / L and σ p , obs from Strader et al. (2011), while thedecrease in M / L toward lower masses is expected from dy- Wang et al. F ig . 6.— Correlations of velocity dispersion and mass with M / L in V band. Left panels show clusters from Strader et al. (2011) (asterisks) whileright panels show clusters from Barmby et al. (2007) (crosses), Barmby et al.(2009) (triangles), and this paper (filled circles). namical evolution like evaporation, which is the steady lossof low-mass (high M / L ) stars from the cluster driven by re-laxation (Portegies et al. 2010). Considering the sensitive de-pendence on M / L of σ p , , in the following discussion of thecorrelations of velocity dispersion with ellipticity and [Fe / H],we decided to use the σ p , obs values from Strader et al. (2011)for clusters in Barmby et al. (2007, 2009) and this paper. Comparison of Three Model Fittings
In order to determine which model can describe the struc-ture of clusters best, Mclaughlin & van der Marel (2005) andMclaughlin et al. (2008) defined a relative χ index, whichcompares the χ of the best fit of an “alternate” model withthe χ of the best fit of King model, ∆ = ( χ − χ ) / ( χ + χ ) . (10)It is evident that the “alternate” model is a better fit thanKing model if ∆ is negative, while King model is a better fitif ∆ is positive.Figure 7 shows the relative quality of fit, ∆ for Wilson- andS´ersic-model fits versus King-model fits for the sample clus-ters in this paper. The ∆ values are plotted as a function of ageand some structures, including the half-light radius R h , the to-tal model luminosity L mod , and the surface brightness over thehalf-light radius in the V band < µ V > h . The circles refer toclusters with R last ≥ R h , while triangles present those with R last < R h , where R last is the most large radius for the avail-able observation data. It can be seen that most clusters with R last < R h are those having fainter luminosity L mod or sur-face brightness < µ V > h . Mclaughlin & van der Marel (2005)presented that the fitting data out to R last ≥ R h can e ff ectivelyconstrain the model fitting to the outer regions, which are es-sential to determine which structural model best describes theclusters. However, when R last is small ( < R h ), few data areavailable at large cluster radii, and the fitting are mostly de-pendent on the inner part. So, these clusters cannot be usedto determine a preference of one model or the other. Simi-larly, we cannot conclude that these clusters are well fitted byboth two models even ∆ for them are small. Mclaughlin et al. F ig . 7.— Relative quality of fit for Wilson and S´ersic models (grey and opencircles refer to the clusters with R last ≥ R h , while grey and open trianglesthe clusters with R last < R h ) versus King models for the sample clusters inthis paper. (2008) determined a catalogue of structural and dynamical pa-rameters for GCs in NGC 5128 using the three models, andshowed that the bright clusters ( L mod > L ⊙ ) are better fit-ted by Wilson model and S´ersic model, indicating that thehalos of clusters in NGC 5128 are more extended than whatKing model describes. There is no evident correlation of ∆ with L mod and < µ V > h for clusters in this paper. Severalstudies (Elson, Fall & Freeman 1987; San Roman et al. 2012)showed that young clusters in the Large Magellanic Cloud(LMC) and M33 do not appear to be tidally truncated andseem to be better fitted by power-law profiles than King mod-els, while old clusters show no clear di ff erences between thequalities of the fittings. However, there is no correlation of ∆ with age in Figure 7. We should notice that there are only sixyoung clusters in our sample, and 13 clusters with no ages es-timated in previous studies are assumed to be 13 Gyr. A largesample of young star clusters with precise age estimates areneeded for the study on correlation of ∆ with age. However,we do see that the King- and Wilson-model fits are better thanthe S´ersic-model fits. We concluded that clusters in M31 canbe well fitted by both King model and Wilson model (alsoreported in Barmby et al. 2007, 2009).Figure 8 compares the relative quality of fit, ∆ values witha number of structure parameters ( R c , R h , µ V , , L mod , σ p , ,and E b ) for the sample clusters in this paper. The grey andopen circles show the physical properties of clusters with R last ≥ R h derived from Wilson- and S´ersic-model fits com-paring to King-model fits, respectively. The triangles referto clusters with R last < R h . There are some clusters withcomparable χ , but large discrepancy of R h and L mod val-ues for King- and Wilson-model fits. We can see that mostof these clusters have R last < R h . As discussed above,the few constrain of the fitting to outer regions results inmuch di ff erent extrapolations of models, and it is hard to de-roperties of M31 globular clusters 7 F ig . 8.— Comparison of some parameters for Wilson and S´ersic modelsversus King models for the sample clusters in this paper, including the pro-jected core radius R c , the projected half-light radius R h , the central surfacebrightness in the V band µ V , , the total model luminosity L mod , the predictedcentral line-of-sight velocity dispersion σ p , , and the global binding energy E b . Symbols are as in Fig. 7. termine which model does the correct fitting in those cases(Mclaughlin & van der Marel 2005). Most parameters de-rived from Wilson model are slightly larger than those fromKing model, while parameters derived by S´ersic model aresmaller than those from King model, especially for R c , σ p , ,and E b . Barmby et al. (2007) concluded that rather than an in-trinsic di ff erence between clusters in M31 and other galaxies,the preference for King models over Wilson models for M31clusters is due to some subtle features of the observations. DISCUSSION
We combined the newly derived parameters here with thosederived by King-model fits for M31 young massive clusters(YMCs) (Barmby et al. 2009), M31 globulars (Barmby et al.2007), M31 extended clusters (ECs) (Huxor et al. 2011),and MW globulars (Mclaughlin & van der Marel 2005) toconstruct a large sample to look into the correlations be-tween the parameters. The ellipticities and galactocentricdistances for the MW GCs are from Harris (1996) (2010edition). The parameters used in the following discussionfor M31 GCs (Barmby et al. 2007) are those derived onthe bandpass close to V band (e.g., F555W and F606W),while the data from bluer filters are preferred for the YMCsin Barmby et al. (2009). The metallicities for most youngclusters in Barmby et al. (2009) were assumed to be solarmetallicity, while metallicities from Perina et al. (2010) wereadopted for some older clusters (B083, B222, B347, B374,and NB16). Huxor et al. (2011) derived structure parame-ters of 13 ECs, including the core radius, half-light radius,tidal radius, and the central surface brightness in magnitude. The metallicities of 4 ECs (HEC4, HEC5, HEC7 and HEC12)were determined by Mackey et al. (2006), and the integratedcluster masses for them were derived here using the abso-lute magnitudes and M / L values in V band from population-synthesis models (see Wang & Ma 2012, for the details). Galactocentric Distance
Figure 9 shows structural parameters as a function of galac-tocentric distance R gc for the new large sample clusters inM31 and the MW. Some global trends can be seen. Both R h and r t increase with the R gc as expected, although the trend islargely driven by the ECs from Huxor et al. (2011). The starclusters with large R gc can keep more stars with weaker tidesrelatively to those located at bulge and disk. Georgiev et al.(2009) explained that the R h and r t of the clusters in the haloof galaxies, which may be accreted into the galaxies fromdwarf galaxies, can expand due to the change from strong toa weaker tidal field. Barmby et al. (2002) concluded that thecorrelation of R h with R gc reflects physical formation condi-tions as suggested by van den Bergh (1991) for GCs in theMW. However, Strader et al. (2012) found that no clear cor-relation between R h and R gc exists beyond R gc ∼
15 kpc forGCs in NGC 4649, and they suggested that the sizes of GCsare not generically set by tidal limitation. It can be seen thatthere are no compact clusters at large radii ( R gc >
40 kpc) inthe MW, while in M31, there is a bimodality in the size dis-tribution of GCs at large radii (also reported in Huxor et al.2011; Wang & Ma 2012). It is interesting that there are fewGCs having R h in the range from 8 to 15 pc at large radii( R gc >
40 kpc) in M31. There are three clusters (B124,B127, and NB39) with large R h and r t at small R gc , leadingto more di ff use trends. However, the total model masses forthem are 10 . , 10 . , and 10 . M ⊙ , respectively, meaningthat they are massive clusters. It is not unexpected that thesemassive clusters can contain more stars than other clusters, al-though they are at small R gc . The luminosities of clusters de-crease with increasing R gc , implying that either strong dynam-ical friction drives predominantly more massive GCs inwards,or massive GCs may favor to form in the nuclear regions ofgalaxies with the higher pressure and density (Georgiev et al.2009). However, the lack of faint clusters with small galac-tocentric distance may be due to selection e ff ects, since theseclusters are di ffi cult to detect against the bright backgroundnear M31 center, which is also reported by Barmby et al.(2007) using the correlation of central surface brightness withgalactocentric distance. The metallicities of clusters decreasewith increasing R gc , indicating that metal-rich clusters are typ-ically located at smaller galactocentric radii than metal-poorones, although with large scatter. van den Bergh (1991) foundthat metal-rich clusters ([Fe / H] ≥ − .
8) in M31 seem to formin a rotating disk extending to R gc ≃ R gc , which are at the bottom-left in the [Fe / H]- R gc panel. The di ff use distribution of these parameters vs. R gc may be caused by the di ff erent galactocentric distance used,which is true three-dimensional distances for Galactic GCswhile projected radii for M31 clusters. Ellipticity
Figure 10 shows the distribution of ellipticity with galacto-centric distance, metallicity, and some structure parameters Wang et al. F ig . 9.— Structural parameters vs. galactocentric distance R gc . Thefilled circles are the sample clusters in M31 (this paper), the open circlesare Galactic GCs (Mclaughlin & van der Marel 2005), the crosses are M31GCs (Barmby et al. 2007), the open triangles are M31 YMCs (Barmby et al.2009), and the open stars are M31 ECs (Huxor et al. 2011).F ig . 10.— Ellipticity as a function of galactocentric distance, metallicity,the observed velocity dispersion, and structure parameters. Symbols are as inFig. 9. for clusters in the MW and M31, which may show cluesto the primary factor for the elongation of clusters: rota-tion and velocity anisotropy, cluster mergers, “remnant elon-gation”, and galaxy tides (Larsen et al. 2001; Barmby et al.2007). Geyer et al. (2009) presented that the tidal forces canonly distort a cluster’s outer region, while the elongation ofthe inner parts are due to its own gravitational potential andthe total orbital angular momentum.1) Rotation and velocity anisotropy. Cluster rotationis the generally accepted explanation for cluster flattening(Davoust & Prugniel 1990). Dynamical models show that in-ternal relaxation coupled to the external tides will drive acluster toward a rounder shape over several relaxation times(see Harris et al. 2002, and references therein). Barmby et al.(2002) presented that more compact clusters which expe-rience more relaxation processes, and clusters with larger velocity dispersions which rotate more slowly, should berounder. Barmby et al. (2007) presented that dynamical evo-lution could reduce the initial flattening caused by rotationor velocity anisotropy, indicating that more evolved clus-ters would be rounder. In order to check these predic-tions, we show the correlations of r , c , the observed ve-locity dispersion σ p , obs , and t r , h with the ellipticity. The ob-served velocity dispersion σ p , obs for M31 clusters are fromStrader et al. (2011), while σ p , obs for clusters in the MW arefrom Mclaughlin & van der Marel (2005). All the σ p , obs havebeen extrapolated to their central values with an aperture of 0.There is no clear correlation of ellipticity with these parame-ters. However, we do see that more compact clusters–smaller r and larger c –are more rounder, which is consistent withprevious studies (Barmby et al. 2002).2) Cluster mergers and “remnant elongation”.van den Bergh & Morbey (1984) presented that elliptic-ity correlates strongly with luminosity for clusters inLMC: more luminous clusters are more flattened, whilevan den Bergh (1996) presented that the most flattened GCsin both the MW and M31 are also brightest. This may beexplained by the cluster mergers and “remnant elongation”,which is from some clusters’ former lives as dwarf galaxynuclei. So, the larger ellipticities for clusters located at largegalactocentric radii of M31 may be related to the merger oraccretion history that M31 has experienced (e.g. Bekki 2010;Mackey et al. 2010; Huxor et al. 2011). However, no clearcorrelation of ellipticity with luminosity can be seen.3) Galaxy tides. A strong tidal field might rapidly de-stroy the velocity anisotropies, and force an initially triax-ial, rapidly rotating elliptical GC to a more isotropic distri-bution and spherical shape, while weak tidal fields are un-able to change the initial shapes of GCs (Goodwin 1997). Itseems plausible that clusters located at di ff erent galactocen-tric radii or di ff erent galaxies have various distributions of el-lipticities, due to the diverse galaxy tides. Harris et al. (2002)and Barmby et al. (2007) found the distributions of elliptici-ties for M31 and NGC 5128 are very similar, but di ff er fromthe MW distribution, which has few very round clusters. Noclear correlation of ellipticity with R gc can be seen for clustersin these galaxies (also see Barmby et al. 2007). However, theinnermost clusters are slightly more spherical (Barmby et al.2002), which may due to the strong tidal field near galaxycenter that reduces ellipticities of them. Some clusters lo-cated at large projected radii of M31 do have larger elliptici-ties than most GCs in M31 and the MW, which may be causedby galaxy tides coming from satellite galaxies (Wang & Ma2012).San Roman et al. (2012) summarized the orientation of alarge sample of clusters in M33 and found that the distribu-tion of PAs shows a strong peak at − ◦ , which is close tothe direction towards M31. These authors suggested that theelongation of clusters in M33 may be attributed to the tidalforces of M31, considering that a recent encounter betweenM33 and M31 (McConnachie et al. 2009; Putman et al. 2009;San Roman et al. 2010; Bernard et al. 2012) may have led tosignificant e ff ects on the properties of M33 disk. Figure 11depicts the distribution of PAs of star clusters in this paperand 34 clusters from Barmby et al. (2007), which were notincluded in Barmby et al. (2002). No clear trend is present inthe orientation vectors towards M33, although a small peakat − ◦ does exist in the distribution of the PAs. Here weconcluded that the elongation of clusters seems to be due toroperties of M31 globular clusters 9 F ig . 11.— ( Le f t panel ) Cluster elongations and orientations shown withrespect to M31. The vector sizes are correlated with the ellipticities. Theellipse has a radius of 10 kpc and is flattened to b / a = .
6. (
Right panel )Distribution of the P.A.s of star clusters in M31. various factors (Barmby et al. 2007).
Metallicity
Figure 12 plots structural parameters as a function of [Fe / H]for the new large sample clusters in M31 and the MW. Thetrends of the parameters with [Fe / H] nearly disappear whenwe add the data for the M31 YMCs, which were fitted withsolar metallicities (Barmby et al. 2009). However, we noticedthat the metallicities for most of these YMCs obtained byKang et al. (2012) are poorer than [Fe / H] = − .
1. If we do notconsider the YMCs in Barmby et al. (2009), it is evident thatthe metal-rich clusters have smaller average values of R h thanthose of metal-poor ones (Larsen et al. 2001; Barmby et al.2002). Strader et al. (2012) presented that the sizes of metal-rich GCs are smaller than the metal-poor ones in NGC 4649,which is a massive elliptical galaxy in the Virgo galaxy clus-ter. These authors explained that as an intrinsic size di ff erencerather than projection e ff ects. Sippel et al. (2012) carried outN-body simulations of metallicity e ff ects on cluster evolution,and found that there is no evident di ff erence for the half-massradii of metal-rich and metal-poor cluster models. So, theyexplained that metal-rich and metal-poor clusters have similarstructures, while metallicity e ff ects combined with dynamicale ff ects such as mass segregation produce a larger di ff erenceof the half-light radii. Barmby et al. (2007) also found that R h decreases with increased metallicity for GCs in the MW, theMCs, Fornax dwarf spheroidal, and M31, except for GCs inNGC 5128. An evident feature is that these four ECs are allmetal-poor and have large R h . No clear correlation of modelluminosity L mod with [Fe / H] is present, but we do see that themetal-rich clusters tend to be more luminous. Both the ob-served velocity dispersion σ p , obs and central “escape” veloc-ity ν esc , increase with metallicity. Georgiev et al. (2009) pre-sented that the higher ν esc of more metal-rich clusters in theobservation may reflect the metallicity dependence of the ter-minal velocities of the stellar winds, since the ν esc of a metal-rich cluster should be higher to retain the fast stellar winds. Core-collapsed Clusters
Core-collapsed clusters in general show a power-law slopein the central surface brightness profiles, which can be betterfitted by a power-law model than King model (Barmby et al.2002). Noyola & Gebhardt (2006) presented that the processof core collapse can be divided into two stages. In the firststage, stars are driven to the halo of the cluster due to closeencounters. Stellar evaporation occurs and the core contractsdue to energy conservation. In the second stage, the low-mass stars are scattered to high velocities and escape to the F ig . 12.— Structural parameters and the observed velocity dispersion as afunction of [Fe / H]. Symbols are as in Fig. 9. halo, while the high-mass stars sink to the core due to energyequipartition. The increasing core density also increases theinteraction rate of the binaries and single stars, which can gen-erate energy in the core, reverse the contraction process, andproduce an expansion. After a long time, the core shrinksagain, and the process repeats. M15, which has a centralcusp, may be at an intermediate state between the extremesof collapse and expansion (Dull et al. 1997). Chatterjee et al.(2012) presented that the core-collapsed clusters are those thathave reached or are about to reach the “binary burning” stage,while the non core-collapsed clusters are still contracting un-der two-body relaxation. Trager et al. (1995) presented a cat-alogue of surface brightness profiles of 125 Galactic GCs, andclassified 16% of their sample as core-collapsed clusters and6% as core-collapsed candidates. Mclaughlin et al. (2008) no-ticed that a number of clusters (more than half of their sample)in NGC 5128 have the index n > χ < χ and χ < χ ), and 11 of them have n > n > R c andlarger R h / R c than their counterparts with n <
2. Barmby et al.(2002) presented several clusters (B011, B064, B092, B123,B145, B231, B268, B343, and BH18) as core-collapsed can-didates in M31. However, most of these clusters are best fittedby King model or Wilson model in this paper except B145 andB343. We checked the images of these clusters and found that0 Wang et al. F ig . 13.— The distributions of the galactocentric distance and several struc-ture parameters derived from S´ersic-model fitting for clusters in this paperand M31 GCs in Barmby et al. (2007). the images of B123, B231, and B268 are not fully resolved,while the SNRs of images for B064 and B092 are low. Ac-tually, the discrimination between the core-collapsed clustersand the “King model clusters” (Meylan & Heggie 1997) mayoften become unclear for several reasons: (1) statistical noisedue to some unresolved bright stars in the cores of clusters;(2) the similarity between the high-concentration King-modeland power-law profiles (see Meylan & Heggie 1997, for de-tails). B145 is best fitted by S´ersic model in this paper, but n = .
5. The cluster B343 shows a cuspy density profile inthe center, and has been classified as a core-collapsed cluster(Bendinelli et al. 1993; Grillmair et al. 1996). We also find abetter fitting by S´ersic model for it with n = .
35. The distri-bution of core-collapsed cluster candidates is not constrainedto the center of M31. There are only two out of the 11 clusterswith R gc < R gc >
15 kpc.
The Fundamental Plane
It has been widely noticed that GCs do not occupy the fullfour-dimensional parameter space (concentration c , scale ra-dius r , central surface brightness µ V , , and central M / L orvelocity dispersion σ ) instead locate in a remarkably nar-row “fundamental plane” (FP). It is interesting to learn whichof the structural and dynamical properties are either univer-sal or dependent on characteristics of their parent galaxies(Meylan & Heggie 1997). Djorgovski & Meylan (1994) ex-plored some tight correlations between various properties us-ing a large sample of Galactic clusters, and Djorgovski (1995)defined an FP for GCs in terms of velocity dispersion, ra-dius, and surface brightness. Saito (1979) investigated themass-binding energy relationship for a few GCs and ellipti-cals, while McLaughlin (2000) defined an FP using the bind-ing energy and luminosity, which is formally di ff erent fromthat of Djorgovski (1995). Harris et al. (2002) found that theNGC 5128 GCs describe a relation between binding energyand luminosity tighter than in the MW. Here we show the twoforms of the FP for the new large sample clusters in the MWand M31. F ig . 14.— Evidence of an FP of the cluster parameters, which is defined interms of central velocity dispersion σ p , , radius R c or R h , and surface massdensity < Σ > or < Σ > h . Symbols are as in Fig. 9. Figure 14 shows the mass-density-based FP relations. Theleft two panels show the correlations of the properties de-rived with the M / L values from the velocity dispersions(Strader et al. 2011), using King model and the same fittingprocess in this paper (Section 3), while the right two pan-els show the correlations of the properties derived with the M / L values from population-synthesis models. Barmby et al.(2009) presented the surface-brightness-based FP relationsand found a large o ff set between the young M31 clusters andold clusters. They explained that as a result from lower M / L values for the young clusters in M31. It is obvious that thevelocity dispersion, characteristic radii, and surface densityfor these clusters show tight relations, both on the core andhalf-light scales. The exist of FP for clusters strongly reflectssome universal physical conditions and processes of clusterformation.Figure 15 shows the correlation of binding energy with thetotal model mass. The left panel shows the M mod and E b de-rived with the M / L from the directly observed velocity dis-persions (Strader et al. 2011), using King model and the samefitting process in this paper (Section 3), while the right panelshows those properties derived with the M / L from population-synthesis models. All the clusters locate in a remarkably tightregion although in the widely di ff erent galaxy environments,which is consistent with the previous studies (Barmby et al.2007, 2009). Barmby et al. (2007) concluded that the scat-ter around this relation is so small that the structures of starclusters may be far simpler than those scenarios derived fromtheoretical arguments. SUMMARY
High-resolution imaging can be derived from
HST observa-tions for M31 star clusters. In this paper, we presented surfacebrightness profiles for 79 clusters, which were selected fromBarmby et al. (2002) and Mackey et al. (2007). Structural androperties of M31 globular clusters 11 F ig . 15.— Evidence of an FP of the cluster parameters, which is defined interms of binding energy E b and the total model mass M mod . Symbols are asin Fig. 9. dynamical parameters were derived by fitting the profiles tothree di ff erent models, including King model, Wilson model,and S´ersic model. We found that in the majority of cases,King models fit the M31 clusters as well as Wilson models,and better than S´ersic models.We discussed the properties of the sample GCs here com-bined with GCs in the MW (Mclaughlin & van der Marel2005) and clusters in M31 (Barmby et al. 2007, 2009;Huxor et al. 2011). In general, the properties of the M31 and the Galactic clusters fall in the same regions of param-eter spaces. There is a bimodality in the size distributionof M31 clusters at large radii, which is di ff erent from theirGalactic counterparts. There are 11 clusters in M31 best fittedby S´ersic models with index n >
2, meaning that they havecuspy central density profiles, which are classified as core-collapsed cluster candidates. We investigated two forms ofthe FP, including the correlation of velocity dispersion, radius,and surface density, and the correlation of binding energywith the total model mass. The tight correlations of clusterproperties indicate a tight FP for clusters, regardless of theirhost environments, which is consistent with previous stud-ies (Barmby et al. 2007, 2009). In addition, the tightness ofthe relations for the internal properties indicates some physi-cal conditions and processes of cluster formation in di ff erentgalaxies.We thank the anonymous referee for providing a rapid andthoughtful report that helped improve the original manuscriptgreatly. We would like to thank Dr. McLaughlin for hishelp in deriving the parameters of the three structure mod-els, and Richard Hook in understanding the TinyTim pack-age, and Pauline Barmby in understanding the PSF of HST ,and Christopher Willmer in providing the updated table ofabsolute solar magnitudes. This work was supported bythe Chinese National Natural Science Foundation grands No.10873016, and 10633020, and by National Basic ResearchProgram of China (973 Program), No. 2007CB815403.
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Name ǫ a θ a ǫ b θ b B V I R gc E ( B − V ) [Fe / H] Age(deg E of N) (deg E of N) (Vegamag) (Vegamag) (Vegamag) (kpc) (Gyr)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)B006 0 . ± . − ± . ± . − ± − . ± .
41 13.25B011 0 . ± .
01 21 ± . ± .
01 32 ± − . ± .
24 15.85B012 0 . ± .
02 24 ± . ± .
01 18 ± − . ± .
21 8.00B018 0 . ± . − ± . ± . − ± − . ± .
39 1.20B027 0 . ± .
02 11 ± . ± . − ± − . ± .
16 15.75B030 0 . ± . − ±
12 0 . ± . − ± − . ± .
26 9.62B045 0 . ± . − ± . ± . − ± − . ± .
50 11.40B058 0 . ± . − ± . ± . − ± − . ± .
21 8.01B068 0 . ± .
02 31 ± . ± .
02 37 ± − . ± .
17 7.90B070 0 . ± .
01 1 ±
10 0 . ± . − ±
11 17.61 16.76 15.72 2.46 0.12 − . ± .
43 8.75 a ǫ and θ of bluer filters. b ǫ and θ of redder filters. TABLE 2C alibration D ata for HST images .Filter Pivot λ R a λ Zeropoint b M c ⊙ Conversion Factor d Coe ffi cient e (Å)(1) (2) (3) (4) (5) (6) (7)Calibration Data for ACS / WFC imagesF435W 4318.9 4.20 25.779 5.459 3.1693 27.031F475W 4746.9 3.72 26.168 5.167 1.6926 26.739F555W 5361.0 3.19 25.724 4.820 1.8508 26.392F606W 5921.1 2.85 26.398 4.611 0.8207 26.183F814W 8057.0 1.83 25.501 4.066 1.1349 25.638Calibration Data for WFPC2 imagesF300W 2986.8 5.66 21.448 6.061 297.9614 27.633F450W 4555.4 3.93 24.046 5.263 13.0545 26.835F555W 5439.0 3.14 24.596 4.804 5.1542 26.376F606W 5996.8 2.81 24.957 4.581 3.0100 26.153F814W 8012.2 1.85 23.677 4.074 6.1342 25.646 a A λ = R λ × E ( B − V ). b Additive conversion between surface brightness in counts s − arcsec − and magnitude in mag arcsec − . c Updated absolute magnitude of the sun (C. Willmer, private communication). d Multiplicative conversion between surface brightness in counts s − arcsec − and intensity in L ⊙ pc − . e Additive conversion between surface brightness in magnitude in mag arcsec − and intensity in L ⊙ pc − .TABLE 3T he intensity profiles for sample clusters in M31.Name Detector Filter
R I
Uncertainty Flag(arcsec) ( L ⊙ pc − ) ( L ⊙ pc − )(1) (2) (3) (4) (5) (6) (7)B006 WFPC2 / PC F555W 0.0234 44001.773 155.603 OKWFPC2 / PC F555W 0.0258 43776.562 171.971 DEPWFPC2 / PC F555W 0.0284 43528.629 188.743 DEPWFPC2 / PC F555W 0.0312 43258.270 206.935 DEPWFPC2 / PC F555W 0.0343 42964.875 227.137 DEPWFPC2 / PC F555W 0.0378 42647.070 249.659 DEPWFPC2 / PC F555W 0.0415 42303.609 274.847 DEPWFPC2 / PC F555W 0.0457 41919.195 299.011 DEPWFPC2 / PC F555W 0.0503 41381.914 308.833 OKWFPC2 / PC F555W 0.0553 40666.176 309.009 DEPWFPC2 / PC F555W 0.0608 39884.902 322.641 DEPWFPC2 / PC F555W 0.0669 38966.234 328.825 DEPWFPC2 / PC F555W 0.0736 37925.949 316.808 OKTABLE 4C oefficients for the
PSF models .Detector Filter r α β (arcsec)(1) (2) (3) (4) (5)ACS / WFC F435W 0.068 3 3.80F475W 0.064 3 3.60F555W 0.057 3 3.39F606W 0.053 3 3.14F814W 0.056 3 3.05WFPC2 / WFC F300W 0.076 2 5.05F450W 0.073 2 4.89F555W 0.064 2 4.35F606W 0.059 2 4.11F814W 0.051 2 3.71WFPC2 / PC F300W 0.051 2 3.76F555W 0.045 2 3.10F606W 0.045 2 2.96F814W 0.059 2 3.18 r op e r ti e s o f M l obu l a r c l u s t e r s TABLE 5B asic parameters of sample clusters in M31.
Name Detector Band N pts a Model χ b I bkg c W d c / n e µ f log r g log r h ( L ⊙ pc − ) (mag arcsec − ) (arcsec) (pc)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)B006 WFPC2 / PC 555 49 K66 35.44 5 . ± .
54 7 . + . − . . + . − . . + . − . − . + . − . − . + . − .
49 W 5.53 − . ± .
92 7 . + . − . . + . − . . + . − . − . + . − . − . + . − .
49 S 130.08 3 . ± . . . . . + . − . . + . − . − . + . − . − . + . − . B006 WFPC2 / PC 814 49 K66 14.97 6 . ± .
36 8 . + . − . . + . − . . + . − . − . + . − . − . + . − .
49 W 1.88 − . ± .
25 7 . + . − . . + . − . . + . − . − . + . − . − . + . − .
49 S 32.24 4 . ± . . . . . + . − . . + . − . − . + . − . − . + . − . The number of points in the intensity profile that were used for constraining the model fits. b The minimum χ obtained in the fits. c The best-fitted background intensity. d The dimensionless central potential of the best-fitting model, defined as W ≡ − φ (0) /σ . e The concentration c ≡ log( r t / r ). f The best-fit central surface brightness in the native bandpass of the data. g The best model-fit scale radius r in arcseconds. h The best model-fit scale radius r in parsecs. W a ng e t a l . TABLE 6D erived structural and photometric parameters of sample clusters in M31.
Name Detector Band Model log r tid a log R cb log R hc log R h / R cd log I e log j f log L Vg V tot h log I hi < µ V > h j (pc) (pc) (pc) ( L ⊙ , V pc − ) ( L ⊙ , V pc − ) ( L ⊙ , V ) (mag) ( L ⊙ , V pc − ) (mag arcsec − )(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)B006 WFPC2 / PC 555 K66 1 . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . W 2 . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . S ∞ − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . B006 WFPC2 / PC 814 K66 1 . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . W 2 . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . S ∞ − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . The model tidal radius r t in parsecs. b The projected core radius of the model fitting a cluster, defined as I ( R c ) = I / c The projected half-light, or e ff ective, radius of a model, containing half the total luminosity in projection. d A measure of cluster concentration and relatively more model-independent than W or c . e The best-fit central ( R =
0) luminosity surface density in the V band, defined as log I = . . − µ V , ), where 26.358 is the “Coe ffi cient” corresponding to a solar absolute magnitude M V , ⊙ = + .
786 (C.Willmer, private communication). f The central ( r =
0) luminosity volume density in the V band. g The V -band total integrated model luminosity. h The total V -band magnitude of a model cluster, defined as V tot = . − . L V / L ⊙ ) + D /
10 pc). i The luminosity surface density averaged over the half-light / e ff ective radius in the V band, defined as log I h ≡ log( L V / π R h ). j The surface brightness in magnitude over the half-light / e ff ective radius in the V band, defined as < µ V > h = . − . I h . r op e r ti e s o f M l obu l a r c l u s t e r s TABLE 7D erived dynamical parameters of sample clusters in M31.
Name Detector Band Υ pop V a
Model log M tot b log E bc log Σ d log ρ e log Σ h f log σ p , g log ν esc , h log t r , hi log f j ( M ⊙ L − ⊙ , V ) ( M ⊙ ) (erg) ( M ⊙ pc − ) ( M ⊙ pc − ) ( M ⊙ pc − ) (km s − ) (km s − ) (yr) ( M ⊙ (pc km s − ) − )(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)B006 WFPC2 / PC 555 2 . + . − . K66 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . W 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . S 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . B006 WFPC2 / PC 814 2 . + . − . K66 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . W 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . S 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . The V -band mass-to-light ratio. b The integrated cluster mass, estimated as log M tot = log Υ pop V + log L V . c The integrated binding energy in ergs, defined as E b ≡ − (1 / R r t π r ρφ d r . d The central surface mass density, estimated as log Σ = log Υ pop V + log I . e The central volume density, estimated as log ρ = log Υ pop V + log j . f The surface mass density averaged over the half-light / e ff ective radius R h , estimated as log Σ h = log Υ pop V + log I h . g The predicted line-of-sight velocity dispersion at the cluster center. h The predicted central “escape” velocity with which a star can move out from the center of a cluster, defined as ν , /σ = W + GM tot / r t σ ]. i The two-body relaxation time at the model-projected half-mass radius, estimated as t r , h = . × yr ln(0 . M tot / m ⋆ ) M / R / h m ⋆ . m ⋆ is the average stellar mass in a cluster, assumed to be 0.5 M ⊙ . j The model’s central phase-space density, defined as log f ≡ log[ ρ / (2 πσ2
Model log M tot b log E bc log Σ d log ρ e log Σ h f log σ p , g log ν esc , h log t r , hi log f j ( M ⊙ L − ⊙ , V ) ( M ⊙ ) (erg) ( M ⊙ pc − ) ( M ⊙ pc − ) ( M ⊙ pc − ) (km s − ) (km s − ) (yr) ( M ⊙ (pc km s − ) − )(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)B006 WFPC2 / PC 555 2 . + . − . K66 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . W 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . S 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . B006 WFPC2 / PC 814 2 . + . − . K66 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . W 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . S 6 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . The V -band mass-to-light ratio. b The integrated cluster mass, estimated as log M tot = log Υ pop V + log L V . c The integrated binding energy in ergs, defined as E b ≡ − (1 / R r t π r ρφ d r . d The central surface mass density, estimated as log Σ = log Υ pop V + log I . e The central volume density, estimated as log ρ = log Υ pop V + log j . f The surface mass density averaged over the half-light / e ff ective radius R h , estimated as log Σ h = log Υ pop V + log I h . g The predicted line-of-sight velocity dispersion at the cluster center. h The predicted central “escape” velocity with which a star can move out from the center of a cluster, defined as ν , /σ = W + GM tot / r t σ ]. i The two-body relaxation time at the model-projected half-mass radius, estimated as t r , h = . × yr ln(0 . M tot / m ⋆ ) M / R / h m ⋆ . m ⋆ is the average stellar mass in a cluster, assumed to be 0.5 M ⊙ . j The model’s central phase-space density, defined as log f ≡ log[ ρ / (2 πσ2 c ) /2