On the fundamental line of galactic and extragalactic globular clusters
M. Pasquato^{1}  G. Bertin^{2}
1  Dipartimento di Fisica, Università di Pisa, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy
2  Dipartimento di Fisica, Università degli Studi di Milano,
via Celoria 16, 20133 Milano, Italy
Received 21 July 2009 / Accepted 2 November 2009
Abstract
Context. In a previous paper, we found that globular clusters in our Galaxy lie close to a line in the (
,
,
)
parameter space, with a moderate degree of scatter and remarkable
axisymmetry. This implies that a purely photometric scaling law
exists, that can be obtained by projecting this line onto the (
,
)
plane. These photometric quantities are readily available for large
samples of clusters, as opposed to stellar velocity dispersion data.
Aims. We study a sample of 129 Galactic and extragalactic clusters on this photometric plane in the Vband. We search for a linear relation between
and
and study how the scatter around the bestfit relation is influenced by
both age and dynamical environment. We interpret our results in terms
of testing the evolutionary versus primordial origin of the fundamental
line.
Methods. We perform a detailed analysis of surface brightness
profiles, which allows us to present a catalogue of structural
properties without relying on a given dynamical model.
Results. We find a linear relation between
and
,
in the form
,
where
is measured in mag/arcsec^{2} and
in parsec. Both young and old clusters follow the scaling law, which has a scatter of approximately 1 mag in
.
However, young clusters display more of a scatter and a clear trend in
this with age, which old clusters do not. This trend becomes tighter if
cluster age is measured in units of the cluster halflight relaxation
time. Twobody relaxation therefore plays a major role, together with
passive stellar population evolution, in shaping the relation between
,
,
and cluster age. We argue that the

relation and hence the fundamental line scaling law does not have a
primordial origin at cluster formation, but is rather the result of a
combination of stellar evolution and collisional dynamical evolution.
Key words: globular clusters: general  Galaxy: structure
1 Introduction
In a previous paper (Pasquato & Bertin 2008, hereafter Paper I), we demonstrated that Galactic globular clusters (Galactic GCs) occupy a narrow region around a line in the ( , , ) parameter space. This fundamental line is located within the fundamental plane of Galactic GCs (see Djorgovski 1995) and its existence was noted by Bellazzini (1998) based on principal component analysis techniques.
Bellazzini (1998) divided the observed Galactic GCs into two sets of inner GCs and outer GCs with respect to the Sun's orbit, and found that outer GCs have a dimensionality of 1 in the ( , , ) space^{}, while the dimensionality rises to 2 for inner GCs. He argued that inner GCs, which are closer to the Galactic bulge and interact more frequently with the disk, are more dynamically disturbed than outer GCs. Therefore, he suggested that dynamical interactions with the environment are more likely to disrupt than to preserve a fundamental line, which was then speculated to be primordial in origin.
By projecting the fundamental line onto the ( , ) plane, in Paper I we predicted the existence of a purely photometric scaling law in the form of a linear relation between and . This relation can be practically studied for a much larger sample than the fundamental plane, because velocity dispersion data is not required. In particular, bright clusters (both young and old) in the LMC, SMC, and Fornax galaxies may be included in the study, together with Galactic GCs. This allows a comparison to be made between the behaviors in the ( , ) plane, of clusters that have significantly different ages and are associated with different environments.
The choice of using only photometric variables allows us to consider a relatively large sample of clusters with a substantial age spread. We are then able to attack the problem of the origin and evolution of the  scaling law by studying its dependence on cluster age and to look for evidence for or against the picture in which this scaling law originates primordially and diminishes with time because of dynamical evolution.
In the present paper, we follow Paper I in deriving cluster structural parameters in a modelindependent way, in particular by avoiding fitting a given dynamical model to observational data. Some minor changes in the modelindependent algorithm used are described in Sect. 2, where a detailed account of the data that we collect from the literature is also given. We compile a catalogue of fractionoflight radii, average surface brightness, and absolute magnitudes, as a machinereadable table. Central cusps in the surface brightness profile have been observed in both old, Galactic GCs (Noyola & Gebhardt 2006) and extragalactic clusters (Noyola & Gebhardt 2007) and are predicted to form after corecollapse by Nbody simulations (Trenti et al. 2010). Extended cluster envelopes have also been observed in young clusters in external galaxies (e.g., Larsen 2004; Schweizer 2004). The simple, singlemass, spherical isotropic King (1966) models that are usually taken to fit cluster surface brightness profiles have a flat core and sharp tidal cutoff and therefore have difficulties in modeling clusters with these features. The problem of extended envelopes prompted McLaughlin & van der Marel (2005) to refit a sample of Milky Way, LMC, SMC, and Fornax clusters with King models, powerlaw models, and Wilson (1975) models, finding that, for a sizeable fraction of the old GC population, King models do not perform more effectively than either Wilson (1975) or powerlaw models in fitting surface brightness profiles. More generally, the construction of physicallymotivated selfconsistent models continues to advance, by including features such as tidal flattening (e.g., see Bertin & Varri 2008), but modelindependent approaches remain a valid complement to modelbased analyses (see Kron & Mayall 1960). While a certain dependence on specific assumptions, such as the details of the extrapolation of the external part of surface brightness profiles, cannot be avoided, the catalogue of modelindependent GC structural parameters presented in this paper naturally emerges as a tool useful for avoiding the potentially limiting assumptions required by dynamical models. The fundamental line and its scatter are presented in Sect. 3. Discussion and conclusions are given in Sect. 4.
2 The data
2.1 Distance moduli
We combined the distance moduli by RecioBlanco et al. (2005) and Ferraro et al. (1999) for the Galactic GCs in our sample. RecioBlanco et al. (2005) and Ferraro et al. (1999) list 72 and 61 objects respectively, with an overlap of 34, which provides a sample of 99 Galactic GCs with measured distance moduli. Both the apparent distance moduli in the F555W band in RecioBlanco et al. (2005) and the true distance moduli in Ferraro et al. (1999) are calibrated on the horizontalbranch standard candle. The distance moduli from Ferraro et al. (1999) are taken from Col. 8 of their Table 2, i.e., are based on the metallicity scale. We converted the distance moduli by RecioBlanco et al. (2005) to true distance moduli using the adopted E(BV) reddenings (see following subsection) and the relation between reddening and extinction in the F555W band from Table 12 of Holtzman et al. (1995). For the data in the overlap of the Ferraro et al. (1999) and RecioBlanco et al. (2005) samples, we took the errorweighted average value of the true distance modulus. In Paper I, we adopted quite conservative uncertainties in distance moduli, with a rescaling in the error bars to include systematic errors and an estimate of the error caused by the limited knowledge of the amount of enhancement. This procedure allowed us to prove that the scatter about the fundamental plane is not caused by observational errors alone, even though conservatively estimated. For the purpose of compiling the catalogue described in Sect. 2.8, we consider the errorbars quoted by the authors of the respective measurements and combine them in the standard way. For the extragalactic GCs in our sample, we adopt the distance to the host galaxy as the cluster's distance, following McLaughlin & van der Marel (2005), with the exception of the two clusters NGC 1466 and NGC 1841 which, while being listed as part of the LMC GC system by McLaughlin & van der Marel (2005), have an independent distance modulus measurement by RecioBlanco et al. (2005). The extragalactic sample of McLaughlin & van der Marel (2005) comprises 68 GCs. Thus, the sample of both Galactic and extragalactic GCs with measured distance moduli comprises 165 objects.
2.2 Reddening
For Galactic GCs, we combined reddening data by Ferraro et al. (1999) and RecioBlanco et al. (2005). For reddenings present in both listings, we adopted the errorweighted average. Errors are assumed to be 10% of the adopted reddening value, as in Paper I. The clusters NGC 6304, NGC 6316, NGC 6342, NGC 6356, NGC 6388, NGC 6441, NGC 6539, NGC 6624, and NGC 6760 have their distance modulus measured by RecioBlanco et al. (2005), but reddening was listed for them by neither RecioBlanco et al. (2005) nor Ferraro et al. (1999). Therefore, for these clusters, we chose to use the reddening values provided by Harris (1996).
2.3 Surface brightness profiles: apparent magnitudes and effective radii
The modelindependent procedure based on splinesmoothing that we used for extracting integrated apparent magnitudes and fractionoflight radii from the surface brightness profiles of GCs does not differ substantially from the one described in Paper I. Three differences are worth mentioning:
 The smoothing parameter of the splinefitting algorithm is now adjusted interactively by directly examining how much the resulting spline matches the GC surface brightness data.
 The extrapolation that is taken past the outermost observed point in the surface brightness profile assumes the projected luminosity density decreases as a power law with projected distance from the GC center. A discussion of this powerlaw behavior of the outer part of the GC surface brightness profiles is given in Sect. 3. The power law is truncated at , where is the outermost observed radius. The contribution to the integrated luminosity from the extrapolated profile is usually not influenced significantly by changes in this cutoff. The powerlaw exponent is determined by fitting the outer points in the surface brightness profile. The exact number of points used for each cluster is again defined interactively. GCs with a contribution to the integrated apparent magnitude from the extrapolated profile that exceeds 0.5 mag are removed from the sample.
 Automated outlier rejection based on comparison with literature data does not take place. Instead, the cluster surface brightness profiles are inspected visually, and as a result those of NGC 2419, NGC 4372, NGC 6101, NGC 6541, and NGC 6535 are excluded from the sample due to their poor quality.
2.4 Shape of the surface brightness profile at the GC center
In Paper I, we found an unexpected correlation between residuals to the fundamental plane of Galactic GCs and the central logarithmic slope of the GC surface brightness profile defined and measured by Noyola & Gebhardt (2006). In the present paper, the slopes measured by Noyola & Gebhardt (2007) for extragalactic clusters are combined with those measured by Noyola & Gebhardt (2006), producing a sample of 68 GCs with measured slopes.
2.5 Extrapolation of the surface brightness profile
Most GC surface brightness profiles can be accurately fitted in their outer parts with a powerlaw luminosity density I per unit area given by
We determine the value of the exponent a for the sample of 145 GCs with quality surface brightness profiles for which we compute the modelindependent structural parameters. This bestfit relation is then used to extrapolate the surface brightness profile and calculate the cluster integrated apparent magnitude.
2.6 Distance from the center of the host galaxy and orbital parameters
To assess the role of environmentdriven dynamical evolution in establishing or eliminating the  relation, we use distances from the center of the host galaxy listed by McLaughlin & van der Marel (2005) as a quantitative measure of the interaction. This approach is similar to that of Bellazzini (1998), who divided Galactic GCs on the basis of Galactocentric distance to study the fundamental line of GCs.
In principle, the size of the Roche lobe is a more accurate
estimator of the amount of tidal disturbance that the host galaxy
exerts on a cluster. A cluster of mass
in a circular orbit with an angular velocity
at distance
from the center of its host galaxy (taken to be spherically symmetric) has a tidal radius (see Bertin & Varri 2008)
where is a coefficient that depends on the rotation curve associated with the galaxy. For a flat rotation curve, and
where the constant of proportionality varies from one galaxy to another. The sample of clusters with modelindependent structural parameters, hostgalaxy distances, and masses estimated by McLaughlin & van der Marel (2005) consists of 114 GCs. We use Eq. (3) to calculate the tidal radius for 61 clusters in the Milky Way and 40 in the LMC; we do not consider either the SMC or Fornax objects because their sample size would be too small. Of course, a thorough analysis would require a discussion of how the available projected data truly constrain the relevant intrinsic quantities which appear in Eqs. (2) and (3) (see also Heggie & Hut 2003; Spitzer 1987) but this is beyond the scope of the present paper. McLaughlin & van der Marel (2005) list only skyprojected distances for LMC, SMC, and Fornax clusters. Full threedimensional distances are available only for Galactic GCs.
Measuring the strength of the dynamical interaction with the host galaxy using detailed orbital data instead of instantaneous distances would be more accurate, but these data are available only for a much smaller subset of Galactic GCs. Allen et al. (2006) calculate the orbits of 48 GCs in a model Milky Way potential, yielding average pericenter distances and orbital eccentricities. The sample of clusters with such an orbit determination and modelindependent structural parameters consists of 36 GCs. Allen et al. (2006) calculate orbits based on GC radial velocity and proper motion data, based on either an axisymmetric or a barred model Galactic potential. In the following, we use orbital data from the axisymmetric model, but we checked that our results do not differ significantly if the barred model is assumed instead.
2.7 Ages and timescales
MarínFranch et al. (2009) obtain GC relative ages from HST/ACS data acquired by the ACS Survey of Galactic globular clusters, while De Angeli et al. (2005) list ages from either groundbased photometry or HST snapshot data. For both samples, we adopt the ages based on the Carretta & Gratton (1997) metallicity scale, normalize both sets of relative ages to NGC 104, and adopt the average value for the combined samples. NGC 6287 is the oldest cluster in the combined sample, so we set its age to be 13 Gyr to determine the absolute ages of the combined set of Galactic GCs. The young clusters of the LMC, SMC, and Fornax are assigned ages following McLaughlin & van der Marel (2005). The final sample of Galactic and extragalactic GCs with measured ages contains 131 objects.
Relaxation times at the halflight radius are taken from McLaughlin & van der Marel (2005). Since McLaughlin & van der Marel (2005) analyze the GC surface brightness profiles in a modeldependent way, we adopt the values based on King (1966) model fits. The processes driven by twobody interactions, such as mass loss by means of the evaporation of stars, mass segregation of stars of different masses within the GC, and core collapse, are expected to occur over a timescale of several relaxation times. The ratio of age to relaxation time is thus a measure of the ``dynamical age'', i.e., the degree of relaxation of a given cluster.
Allen et al. (2006) use the orbits that they calculate for their sample of 48 GCs to compute destruction rates for bulge and diskshocking^{}. These are the reciprocals of the natural timescales against which cluster age is compared to quantify the amount of dynamical interaction with the host Galactic environment a that given GC has suffered.
In the following, we denote cluster ages by , halflight relaxation times by , diskshocking destruction rates by , and bulgeshocking destruction rates by .
2.8 The catalogue
For the adopted sample of 129 GCs, Table 1 lists the integrated apparent Vband magnitude obtained by using our modelindependent profile smoothing method, the adopted reddening and true distance modulus, the log projected halflight radius (in arcsecs), and the value of the exponent a obtained by fitting a powerlaw to the outermost part of the GC surface brightness profile (see Eq. (1)). Table 2 lists the average Vband surface brightness within the halflight radius (in mag/arcsec^{2}), halflight radius (in parsecs), absolute magnitude M_{V}, and residuals in to the fitted  linear relation.
To assess the impact of modeldependence on the GC structural parameters, we compare the luminosities that we obtained here with the values listed in the catalogue by Harris (1996) for Galactic GCs. The sample with parameters available both in the Harris catalogue and in this paper contains 76 GCs. Figure 1 compares our Vband integrated apparent magnitudes with the Harris (1996) values. The magnitudes derived in the present paper are fainter than those of Harris (1996) by 0.12 mag in the median. The rms scatter of the differences is 0.2 mag. Nine clusters exhibit a discrepancy of 0.4 mag or more between the integrated apparent magnitude that we derive here and the Harris (1996) value. These clusters are Arp 2, NGC 4590, NGC 5466, NGC 5634, NGC 6218, NGC 6316, NGC 6366, NGC 6584, and NGC 6809. With the exception of NGC 6584 and NGC 6316, a visual inspection of the Trager et al. (1995) surface brightness profiles of these clusters detects either a very noisy profile or puzzling features such as a nonmonotonic behavior.
Figure 1: Comparison between the integrated apparent Vband magnitudes of GCs derived in the present paper with the values listed by Harris (1996). The superimposed line represents the identity. The magnitudes we derive are systematically fainter by approximately 0.12 mag in the median with respect to the Harris (1996) values. 

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2.9 Fraction of light radii and nonparametric concentration measurements
Fraction of light radii for fractions other than 1/2 are also obtained for each GC and listed in Table 3, thus providing an update and extension of the study by Trager et al. (1995), who list the fraction of light radii (from 10% to the halflight radius) for their sample of Galactic GCs. We recall that the ratio of two different fraction of light radii defines dimensionless shape parameters for the surface brightness profile, which for Kingmodel clusters should be on a onetoone relation with the concentration parameter c. McLaughlin & van der Marel (2005) call for ``a more generally applicable concentration index able to represent the spatial extent or potential depth of any cluster in a more modelindependent way (so as to allow, e.g., for a combined analysis of clusters that may not all be described well by the same type of model)''.
Using our fraction of light radii, we obtain the coefficients , , and . These concentration indicators do not rely on modelbased assumptions on the shape of the surface brightness profile, and can easily be calculated for a given model surface brightness profile. We also obtain the socalled Third Galaxy Concentration (TGC) index introduced by Graham et al. (2001) to improve the stability of nonparametric indicators of galaxy concentration as a function of the radial extension of the observed luminosity profile. The TGC coefficient is defined to be the ratio of the light contained within to the light contained within the halflight radius ^{}. To the best of our knowledge, this is the first calculation of these indices for GCs, while the use of nonparametric concentration indicators for galaxies dates to the studies of Fraser (1972) and de Vaucouleurs (1977).
Figure 2 shows the TGC index as a function of for the adopted sample of 145 GCs with quality surface brightness profiles. We find a strong correlation with r = 0.87. This is an indirect check that GC surface brightness profiles are described well by a oneparameter family of models. The observed linear correlation still leaves room for ingredients such as mass segregation or IMBHs to introduce deviations from this simple oneparameter picture.
Figure 2: Relation between the C_{31} and the TGC nonparametric concentration indices for the adopted sample of 145 GCs with quality surface brightness profiles. 

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3 Results
3.1 The SB_{ e}log R_{ e} relation
As anticipated, based on the projected fundamental line argument of Paper I, we do find a linear relation between Vband
and
for the adopted sample of 129 GCs, of the form
where the coefficients are obtained by biweight (e.g., see Mosteller & Tukey 1977) fitting as a function of , and errors are derived by bootstrap resampling. The scatter about the relation is 1.04 mag in , as quantified by the standard deviation of its residuals. Figure 3 shows the results for our catalogue of 129 GCs in the ( , ) plane, different symbols representing the various host galaxies.
Figure 3: Relation between Vband and for our catalogue of 129 GCs. Open circles are Galactic GCs, upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters, and diamonds are Fornax clusters. The solid line is the linear biweight fit of versus over the entire sample (see Eq. (4)). The thinner dashed line corresponds to a slope of exactly 5 (see Eq. (6)). 

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Equation (4) results from neglecting the errors in the individual datapoints, an approach which is usually taken in the literature (e.g., Djorgovski 1995). If, on the other hand, these errors are included in an ordinary leastsquares regression fit, we obtain
The slopes of Eqs. (4) and (5) are consistent with each other within 3.
3.2 Interpretation
Equation (4) is compatible to within 1 with
which, for a given k, describes the locus of constant absolute magnitude within the ( , ) plane. Therefore, the clusters of our sample show no systematic trend of absolute magnitude with halflight radius, even though the spanned magnitude range is large.
This behavior is evident from Fig. 4, where we plot Vband integrated absolute magnitude M_{V} as a function of for GCs (identified by parent galaxy symbols as in Fig. 3) and for nearby dwarf spheroidal galaxies (the data for the latter are taken from van den Bergh 2008). While dwarf spheroidals show a clear trend of absolute magnitude with halflight radius, GCs form an almost perfect scatter plot, independent of their parent galaxy.
Quantitatively, by fitting a linear relation to M_{V} as a function of
for the full sample of 129 GCs, we obtain
where the coefficient does not differ significantly from 0.
Figure 4: Absolute integrated Vband magnitude M_{V} as a function of (in parsec) for our sample of 129 GCs (open circles are Galactic GCs, upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters, and diamonds are Fornax clusters) and the sample of 21 nearby dwarf spheroidal galaxies by van den Bergh (2008) (crosses). 

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Therefore, the existence of a linear  relation appears to reflect the statement that GCs have a luminosity independent of size. For a constant masstolight ratio M/L, this requirement dictates that GCs have a mass distribution that does not depend on their physical size. For Galactic GCs, this is a wellknown fact (e.g., see Djorgovski & Meylan 1994), which has also been recently confirmed for extragalactic clusters (see Barmby et al. 2007) and is at the basis of the explanation proposed by Bellazzini (1998) for the existence of a fundamental line.
Even if Galactic, LMC, SMC, and Fornax clusters are considered separately, no significant trend of M_{V} with emerges, i.e., the independence of total luminosity on cluster size is apparently universal.
3.3 A trend of residuals with age for young clusters
Figure 5: Trend of residuals to the  relation with cluster age. Open circles are Galactic GCs, upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters, and diamonds are Fornax clusters. The solid line is the linear least squares fit of residuals versus log age for clusters younger than 4 Gyr. The old GCs are not located along the extrapolation of the correlation line for the younger clusters, which indicates that the young, massive clusters that are in the SMC, LMC, and Fornax now, are not likely to be similar to the YMCs that were the progenitors of the old GCs (see text for a discussion). 

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The residuals to the fitted  relation are plotted against cluster age in Fig. 5. The residuals are defined as the difference between the true surface brightness and that predicted by the fitted linear relationship, so that a positive residual means a fainter surface brightness. Based on the assumption of a constant masstolight ratio, this implies a lower surface density. The plot shows a rather different behavior for young and old clusters. Old clusters, which we define to have age in excess of 4 Gyr, are mainly Galactic GCs and their residuals show no significant trend with age (the correlation coefficient is r = 0.08, compatible with 0 according to a ttest). The residuals of clusters younger than 4 Gyr, instead show a clear trend with age, with r = 0.79 and a highly significant ttest pvalue of . Moreover, Fig. 6 shows that these young clusters exhibit a larger scatter about the fitted line, as quantitatively confirmed by the standard deviation of their residuals with respect to it, which is 1.37 mag in as opposed to 1.04 mag for the scatter about the relation given by Eq. (4) for the whole sample. The presence of a greater scatter and the indication that it is driven by a third parameter clearly infers that the behavior of young clusters and old clusters in the ( , ) plane is different. Moreover, Fig. 5 shows that old GCs are not located along the extrapolation of the correlation line for the younger clusters. While a firm conclusion regarding this matter would require additional studies, this occurrence may be an indication that massive clusters that are in the SMC, LMC, and Fornax now, are either not similar to the young massive clusters that were the progenitors of the old GCs, or that evolutionary phenomena capable of explaining the nonlinearity observed in Fig. 5 are at work.
Figure 6: Relation between Vband and for the 91 GCs in our sample with a reliable age determination. The solid line is the biweight linear fit to this restricted sample, with a slope of 5.09. Open circles are clusters younger than 4 Gyr, solid circles are GCs older than 4 Gyr. 

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3.4 Passive stellar evolution or dynamical aging?
The trend of the residuals to the fitted  relation with cluster age may be produced by the evolution of the cluster stellar population with age, whose characteristic timescales are influenced by cluster metallicity but do not display large variations from one cluster to another, or from dynamical effects, e.g., mass loss due to evaporation of light stars. To decouple the dynamical effects from stellar evolution, we studied the dependence of the residuals from the  relation on the ratio of cluster age to the halflight relaxation time . The timescale for cluster dissolution by the evaporation of stars is a multiple of , as well as the timescale for the onset of corecollapse and mass segregation. The ratio is therefore a measure of dynamical age, i.e., of how much a cluster has evolved by twobody relaxation processes.
Figure 7 shows a scatter plot of the residuals against . Only clusters younger than 4 Gyr are included in the plot. The trend shown here is stronger than the correlation of residuals with cluster age alone (not shown). The correlation coefficient for clusters younger than 4 Gyr is r = 0.90, which is quite tight, even though it remains consistent within its 95% confidence interval with the value of r = 0.79 found for the correlation with age alone. This finding suggests that, for young clusters, internal twobody relaxation phenomena play an important role in driving the evolution of the  relation.
Figure 7: Trend of residuals to the  relation with cluster age in units of the cluster halflight relaxation time. Only clusters younger than 4 Gyr are shown in the plot. Upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters. The superimposed line is obtained by performing a least squares linear regression. 

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3.5 Residuals and the central slope of the surface brightness profile
Figure 8 shows that there is no apparent correlation between residuals of the  relation and central surface brightness profile slopes measured by Noyola & Gebhardt (2007,2006). The correlation coefficient that we find is r = 0.19, which is not significant (with a pvalue of 0.16), i.e., r is compatible with 0 and no correlation is present. This result is somewhat unexpected, given the correlation found in Paper I between central surface brightness profile slope and residuals to the fundamental plane of GCs. A possible interpretation is that this correlation is caused by phenomena pertaining to the velocity dispersion . This would be the case if the presence of a cusp in the cluster luminosity density profile makes higher than expected from the fundamental plane relation.
Figure 8: Scatter plot of the residuals to the  relation versus Noyola & Gebhardt (2007,2006) central surface brightness profile slope s. Open circles are Galactic GCs, upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters, and diamonds are Fornax clusters. 

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3.6 Dynamical influence of the galactic environment
To measure environmental effects on the  relation, we study how the residuals vary with different indicators. To quantify the amount of influence that the galactic environment exerts on a sample of 114 GCs, we use distances from the center of each cluster's host galaxy, listed by McLaughlin & van der Marel (2005).
We find no significant trend of residuals to the  relation with log distance from the host galaxy center. A positive correlation coefficient of r = 0.22 is found, which corresponds to a ttest pvalue of 0.02.
For the sample of 61 Galactic GCs for which we calculated the radii of the relevant Roche lobes, we find a negative correlation coefficient with the residuals of the  relation, which is marginally significant with 0.01 pvalue. In the case of LMC clusters, no significant trend with tidal radii is found.
For the 48 Galactic GCs whose orbits were computed by Allen et al. (2006), the interaction with Galactic environment can be quantified in a more refined way, by using the computed orbital parameters. Allen et al. (2006) calculated their orbits over a period of several Gyr, and list average values of perigalactic distance and orbital eccentricity during this time. These two orbital parameters are understood to set the cluster tidal cutoff radii (e.g., see King 1962), thereby influencing the whole cluster density and surface brightness profile. Surprisingly, we find no correlation between the residuals and either perigalactic distance or orbital eccentricity. The correlation coefficients are consistent with 0 having pvalues of 0.19 and 0.30, respectively.
Allen et al. (2006) also computed GC destruction rates relative to the tidal interaction with either the Galactic bulge or the disk. Destruction rates depend on the computed GC orbit as well as the assumed Galactic potential model. Our results are similar for the two potential models considered by the authors of the above paper. We consider the ratio of GC age to the related timescales to measure the amount of environmentdriven dynamical evolution. A scatter plot of the residuals^{} of the  relation versus the log ratio of cluster age to Allen et al. (2006) bulge destruction time and disk destruction time shows only a mild trend, with correlation coefficients of 0.45 and 0.49, respectively.
3.7 Powerlaw behavior of surface brightness profiles in the outermost regions
Figure 9 shows the histogram of the bestfit coefficient a for the relation defined by Eq. (1), for the sample of 145 GCs with sufficientquality surface brightness profiles. The mean and median values of a for the sample are 3.5 and 3.2 respectively. A value of 3 would be expected for partially relaxed models produced by collisionless collapse (Bertin & Stiavelli (1993), see also Trenti & Bertin (2005)). Figure 10 shows a scatter plot of a versus the concentration parameter c; the values of c are taken from McLaughlin & van der Marel (2005) and obtained by fitting King models to the surface brightness profiles. No significant trend emerges from the sample of 121 GCs for which a and c are available.
Figure 9: Histogram of bestfit power law coefficients a for the outer part of the GC surface brightness profiles. 

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Figure 10: Scatter plot of power law coefficients a as a function of cluster concentration. 

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With the full sample of 145 GCs, we do not identify any trend in the slopes represented by a with either the halflight relaxation time or the ratio of GC age to relaxation time. This is unsurprising, since the surface brightness profiles of most clusters in this sample are not sufficiently extended in terms of the truncation radius (as obtained by fitting King (1966) models to the profile). We then restrict our attention to a sample of 24 GCs with a measured halflight relaxation time and an extended observed surface brightness profile (McLaughlin & van der Marel 2005). Figure 11 shows a trend in powerlaw exponent a with log halflight relaxation time, which is quantitatively confirmed by a correlation coefficient r = 0.58 that is significant to . The trend is mild but points to the existence of a relation between twobody relaxation phenomena and the shape of the outer envelope of GCs.
Figure 11: Correlation of bestfit power law coefficients a for the outer part of GC extended surface brightness profiles with halflight relaxation time by McLaughlin & van der Marel (2005). 

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4 Discussion and conclusions
By considering a sample of 129 Galactic and extragalactic GCs, we have found a linear relation between and , given by Eq. (4). The relevant coefficient is compatible with 5, i.e., with cluster absolute magnitude not showing a systematic linear trend with . Indeed, when examined in the ( , M_{V}) plane, the cluster absolute magnitude shows no trend with size (in contrast to the trend exhibited by dwarf spheroidal galaxies; see van den Bergh 2008).
The typical scatter about the relation is 1 mag in . Young clusters, of age below 4 Gyr, exhibit a larger scatter of about 1.4 mag. GC age seems to be an important factor in driving this scatter, because young clusters are found to display a clear trend of residuals to the relation with age, with a correlation coefficient of r = 0.79, while old GCs do not exhibit signs of such a correlation. If we assume that all the massive young clusters included in our sample are genuine GC progenitors, our finding suggests that the scaling law being considered shows signs of evolution with age, i.e., that GCs of different ages differ with respect to this scaling law. Moreover, the larger scatter displayed by young clusters points to an evolutionary origin of the relation between and , as opposed to a primordial one. This result is in contrast to the suggestion of Bellazzini (1998) that ``at earlier times, globular clusters populated a line in the threedimensional Sspace, i.e., their original dynamical structure was fully determined by a single physical parameter''.
In principle, modeling the processes that generate the relation between and would require taking into account passive stellar evolution, internal dynamical phenomena (evaporation and corecollapse driven by twobody relaxation), and dynamical interaction with the environment. We find that the trend with age becomes stronger for young clusters if age is measured in units of the halflight relaxation time of each GC, i.e., if dynamical age with respect to twobody relaxation is considered, the correlation coefficient rises to 0.90. Given this result, we argue that twobody relaxation is the key ingredient in shaping the evolution of the  scaling law, at least for young clusters. This may not be the case for older clusters, which do not show such a trend. The evolution of these clusters in the ( , ) plane is likely to be influenced by their host environment, but the available data is of limited use in resolving this issue. The distance to the host galaxy is only a simple proxy for the degree of dynamical disturbance that the GC suffers from it, while more detailed orbital data is available only for a sample of 48 old, Galactic GCs (Allen et al. 2006). In any case, we find no trend between the residuals to the relation and either distance from the host galaxy or pericenter distance and orbital eccentricity, when these data are available. When GC age is measured in units of the bulgeinteraction and diskinteraction destruction times based on the orbital data provided by Allen et al. (2006), weak ( ) trends are found with  relation residuals. While this suggests that dynamical interaction with the galactic environment may exert an influence on GCs withobservable effects in the ( , ) plane, to draw conclusive evidence, the correlation with age alone should be removed from the trend and a larger sample of clusters with accurately measured orbits is necessary.
In contrast to what we noted for fundamental plane residuals in Paper I, we find here that central surface brightness profile slopes measured by Noyola & Gebhardt (2006) and Noyola & Gebhardt (2007) do not correlate with residuals from the relation. This result suggests that velocity dispersion information is necessary to understand the mechanism producing cuspy GC surface brightness profiles.
AcknowledgementsWe wish to thank S. Degl'Innocenti, W. Harris, M. Lombardi, M. Trenti, E. Vesperini, and D. Kruijssen for their comments and helpful suggestions. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. Part of this work was carried out at the Kavli Institute for Theoretical Physics, Santa Barbara (CA), while we participated in the Program Formation and Evolution of Globular Clusters.
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Table 1: Adopted photometric quantities.
Table 2: Halflight photometric quantities (a and b), integrated absolute magnitudes (c) and residuals (d) to the  relation.
Table 3: Catalogue of projected fractionoflight radii (in arcsec) obtained by using the modelindependent method presented in Paper I.
Footnotes
 ... space^{}
 The photometric quantities considered by Bellazzini (1998) are core/central quantities (i.e., core radius and central surface brightness ), as opposed to the halflight ones adopted in Paper I and in this paper (i.e., halflight radius and average surface brightness within it ).
 ... diskshocking^{}
 The Allen et al. (2006) destruction rates differ significantly from other literature values (e.g., see Kruijssen & Mieske 2009). This should be kept in mind when considering our results in Sect. 3.6.
 ...^{}
 This is the choice that we adopt in the present paper, but in principle fractions other than 1/3 could be chosen, leading to slightly different definitions of TGC.
 ... residuals^{}
 both computed for an axisymmetric Galactic potential model
All Tables
Table 1: Adopted photometric quantities.
Table 2: Halflight photometric quantities (a and b), integrated absolute magnitudes (c) and residuals (d) to the  relation.
Table 3: Catalogue of projected fractionoflight radii (in arcsec) obtained by using the modelindependent method presented in Paper I.
All Figures
Figure 1: Comparison between the integrated apparent Vband magnitudes of GCs derived in the present paper with the values listed by Harris (1996). The superimposed line represents the identity. The magnitudes we derive are systematically fainter by approximately 0.12 mag in the median with respect to the Harris (1996) values. 

Open with DEXTER  
In the text 
Figure 2: Relation between the C_{31} and the TGC nonparametric concentration indices for the adopted sample of 145 GCs with quality surface brightness profiles. 

Open with DEXTER  
In the text 
Figure 3: Relation between Vband and for our catalogue of 129 GCs. Open circles are Galactic GCs, upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters, and diamonds are Fornax clusters. The solid line is the linear biweight fit of versus over the entire sample (see Eq. (4)). The thinner dashed line corresponds to a slope of exactly 5 (see Eq. (6)). 

Open with DEXTER  
In the text 
Figure 4: Absolute integrated Vband magnitude M_{V} as a function of (in parsec) for our sample of 129 GCs (open circles are Galactic GCs, upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters, and diamonds are Fornax clusters) and the sample of 21 nearby dwarf spheroidal galaxies by van den Bergh (2008) (crosses). 

Open with DEXTER  
In the text 
Figure 5: Trend of residuals to the  relation with cluster age. Open circles are Galactic GCs, upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters, and diamonds are Fornax clusters. The solid line is the linear least squares fit of residuals versus log age for clusters younger than 4 Gyr. The old GCs are not located along the extrapolation of the correlation line for the younger clusters, which indicates that the young, massive clusters that are in the SMC, LMC, and Fornax now, are not likely to be similar to the YMCs that were the progenitors of the old GCs (see text for a discussion). 

Open with DEXTER  
In the text 
Figure 6: Relation between Vband and for the 91 GCs in our sample with a reliable age determination. The solid line is the biweight linear fit to this restricted sample, with a slope of 5.09. Open circles are clusters younger than 4 Gyr, solid circles are GCs older than 4 Gyr. 

Open with DEXTER  
In the text 
Figure 7: Trend of residuals to the  relation with cluster age in units of the cluster halflight relaxation time. Only clusters younger than 4 Gyr are shown in the plot. Upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters. The superimposed line is obtained by performing a least squares linear regression. 

Open with DEXTER  
In the text 
Figure 8: Scatter plot of the residuals to the  relation versus Noyola & Gebhardt (2007,2006) central surface brightness profile slope s. Open circles are Galactic GCs, upwards facing triangles are LMC clusters, downwards facing triangles are SMC clusters, and diamonds are Fornax clusters. 

Open with DEXTER  
In the text 
Figure 9: Histogram of bestfit power law coefficients a for the outer part of the GC surface brightness profiles. 

Open with DEXTER  
In the text 
Figure 10: Scatter plot of power law coefficients a as a function of cluster concentration. 

Open with DEXTER  
In the text 
Figure 11: Correlation of bestfit power law coefficients a for the outer part of GC extended surface brightness profiles with halflight relaxation time by McLaughlin & van der Marel (2005). 

Open with DEXTER  
In the text 
Copyright ESO 2010