On the structure of Small Magellanic Cloud star clusters
AAstronomy & Astrophysics manuscript no. paper © ESO 2021January 11, 2021
On the structure of Small Magellanic Cloud starclusters
Andrés E. Piatti , (cid:63) Instituto Interdisciplinario de Ciencias Básicas (ICB), CONICET-UNCUYO, Padre J. Contr-eras 1300, M5502JMA, Mendoza, Argentina; Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Godoy Cruz 2290,C1425FQB, Buenos Aires, ArgentinaReceived / Accepted
ABSTRACT
It has been recently shown from observational data sets the variation of structural parametersand internal dynamical evolution of star clusters in the Milky Way and in the Large MagellanicCloud (LMC), caused by the di ff erent gravitational field strengths that they experience. We reporthere some hints for such a di ff erential tidal e ff ects in structural parameters of star clusters in theSmall Magellanic Cloud (SMC), which is nearly 10 times less massive than the LMC. A keycontribution to this study is the consideration of the SMC as a triaxial spheroid, from which weestimate the deprojected distances to the SMC center of the statistically significant sample of starclusters analyzed. By adopting a 3D geometry of the SMC, we avoid the spurious e ff ects causedby considering that a star cluster observed along the line-of-sight is close to the galaxy center.When inspecting the relationships between the star cluster sizes (represented by the 90% lightradii), their eccentricities, masses and ages with the deprojected distances, we find: (i) the starcluster sizes are not visibly a ff ected by tidal e ff ects, because relatively small and large objects arespread through the SMC body. (ii) Star clusters with large eccentricities ( ≥ ∼ M / M odot ) ∼ Key words.
Methods: observational - Galaxies: Magellanic Clouds - Galaxies: star clusters:general (cid:63) e-mail: [email protected]
Article number, page 1 of 12 a r X i v : . [ a s t r o - ph . GA ] J a n ndrés E. Piatti: SMC star clusters
1. Introduction
The structure of star clusters evolves over their lifetime, mainly because of the stellar evolution,two-body relaxation and tidal e ff ects caused by the host galaxy’s gravitational field (e.g., Heggie &Hut 2003; Lamers et al. 2005; Kruijssen & Mieske 2009; Gieles et al. 2011; Webb et al. 2013, 2014;Shukirgaliyev et al. 2018). Although mass loss due to tidal heating has long been treated theoreti-cally and from numerical simulations (e.g., Gnedin et al. 1999; Baumgardt & Makino 2003; Gieleset al. 2006; Lamers & Gieles 2006; Gieles & Baumgardt 2008; Kruijssen et al. 2011; Gieles &Renaud 2016), the magnitude of this phenomenon on di ff erent star clusters has been more di ffi cultto measure. Indeed, the observed variation across the body of a galaxy of the core, half-mass, andJacobi radii, cluster eccentricity, half-mass relaxation time, cluster mass, among other star clusterparameters, has relatively recently been studied in some limited number of cases.Piatti et al. (2019) analyzed the extent in shaping the structural parameters and internal dynam-ics of the globular cluster population caused by the e ff ects of the Milky Way gravitational field.They employed a homogeneous, up-to-date data set with kinematics, structural properties, currentand initial masses of 156 globular clusters, and found that, in overall terms, cluster radii increaseas the Milky Way potential weakens. Core radii increase at the lowest rate, while Jacobi radii doat the fastest one, which implies that the innermost regions of globular clusters are less sensitiveto changes in the tidal forces with the Galactocentric distance. The Milky Way gravitational fieldalso di ff erentially accelerates the internal dynamical evolution of globular clusters, with those to-ward the bulge appearing dynamically older. Globular clusters with large orbital eccentricities andinclination angles experience a higher mass loss because of more tidal shocks at perigalacticon andduring disc crossings (Piatti 2019).Milky Way open clusters are also subject to tidal heating. Because they are younger than glob-ular clusters, mass loss due to stellar evolution can be more important, particularly if they areyounger than few hundred million years, while two-body relaxation becomes important as the massloss rate due to stellar evolution continues to decrease (Lamers et al. 2005). Nevertheless, shockswith giant molecular clouds are known to be the dominant source of mass-loss over the open clus-ter’s lifetime (Lamers & Gieles 2006). Joshi et al. (2016) studied a statistically complete sample ofopen clusters located within 1.8 kpc from the Sun and found that their present-day masses followa linear relationship with their respective ages. Assuming that the gravitational field does not varysignificantly within such a circle, stellar evolution could be responsible for such a trend.The Large Magellanic Cloud (LMC) is nearly 10 times less massive than the Milky Way (Dea-son et al. 2020) and di ff erential tidal e ff ects are also seen within its population of globular clusters.Piatti & Mackey (2018) built extended stellar density and / or surface brightness radial profiles foralmost all the known LMC globular clusters and found that those located closer than ∼ ∼ Article number, page 2 of 12ndrés E. Piatti: SMC star clusters - o - o . h . h r d e p r o j ( k p c ) Fig. 1.
Equal-area Hammer projection of the SMC in equatorial coordinates. Three ellipses with semi-majoraxes of 1 ◦ , 2 ◦ , and 3 ◦ are superimposed. Symbols are colored according to the star cluster distance to the SMCcenter, while their sizes are proportional to the star cluster 90% light radii. tions in the galaxy, in the sense that the closer the globular cluster to the LMC center, the smaller itssize. Although the masses of the LMC globular clusters are commensurate, the outermost regionsof globular clusters located closer than ∼ ff erent mean distances from itscenter along their lifetime, the distinction of their structural properties reflect the di ff erential tidale ff ects between them.We wonder whether tidal heating still has an important role in the structural parameters of starclusters in galaxies less massive than the LMC. We focus here on the Small Magellanic Cloud,which is nearly 10 times less massive than the LMC (van der Marel & Kallivayalil 2014; Sta-nimirovi´c et al. 2004), because it has a statistically complete sample of studied star clusters toexplore this issue. Gieles et al. (2007) analyzed a sample of 195 star clusters in the SMC and foundno evidence for cluster tidal dissolution in the first gigayear. They arrived to this conclusion bycomparing the observed star cluster frequency with that predicted by stellar evolutionary models,assuming no tidal dissolution.The paper is organized as follows. In Section 2 we present the data set used and di ff erent starcluster parameters obtained from it. Section 3 deals with the analysis of the variation of structuralparameters as a function the star cluster distance to the SMC center. Finally, Section 4 summarizesthe main results of this work. Article number, page 3 of 12ndrés E. Piatti: SMC star clusters
2. SMC star cluster properties
We gathered information from two main sources: the recent catalog of star clusters compiled byBica et al. (2020), from which we retrieved star cluster ages; and Table 2 of Hill & Zaritsky (2006),from which we used star cluster coordinates (RA and Dec), 90% light radii ( r ), integrated V mag-nitudes, and cluster eccentricities ( (cid:15) ). We would like to note that di ff erent SMC imaging surveyshave been carried out since the Magellanic Clouds Photometric Survey (Zaritsky et al. 2002) usedby Hill & Zaritsky (2006), e.g., VMC (Cioni et al. 2011), OGLE (Udalski et al. 2015), SMASH(Nidever et al. 2017), VISCACHA (Maia et al. 2019), among others. As far as we are aware, noneof these surveys have been exploited yet in order to update the parameters derived and analysisdone by Hill & Zaritsky (2006), which justifies our choice. We computed the cluster masses usingthe relationships obtained by Maia et al. (2014, equation 4) as follows:log( M / M (cid:12) ) = a + b × log(age / yr) - 0.4 × ( M V − M V (cid:12) )with a = ± b = ± Z = M V (cid:12) = σ (log(M / M (cid:12) )) ≈ ff ect their calculatedmasses, since metallicity di ff erences imply mass values that are within the uncertainties (see figures10 and 11 in Maia et al. 2014). We checked that our cluster masses are in very good agreementwith those calculated by Hill & Zaritsky (2006, see their figure 16). As for the completeness of thepresent star cluster sample, we refer the reader to the work by Gieles et al. (2007), which showsthat the sample is magnitude limited.As far as we are aware, the frequent geometry considered to analyze the spatial distributionsof SMC star clusters is the elliptical framework proposed by Piatti et al. (2007) as a simple rep-resentation of the orientation, dimension and shape of the SMC main body. This framework doesnot consider the SMC depth, which is much more extended than the size of the galaxy projected inthe sky (Ripepi et al. 2017; Muraveva et al. 2018; Graczyk et al. 2020). In an attempt to representthe SMC main body more realistically, we devised a 3D geometry, considering the SMC as anellipsoid, as follows: x a + y b + z c = , (1)where x and y directions are along the semi-minor and semi-major axes in the Piatti et al. (2007)’sframework, respectively, and the z axis is along the line-of-sight. The SMC center is adopted as theorigin of this framework, i.e., (RA S MC ,Dec
S MC ) = (13 ◦ . , − ◦ . S MC = ◦ and a a / b ratio of 1 / Article number, page 4 of 12ndrés E. Piatti: SMC star clusters
The PAs of the star clusters in this rotated coordinate system were calculated using the positionAngle routine from
PyAstronomy (PyA, Czesla et al. 2019), and the observed distances in the sky to theSMC center in R.A. ( x ) and Dec. ( y ), respectively, as follows: x = -(RA - RA S MC ) cos(Dec) cos(PA
S MC ) + (Dec - Dec S MC ) sin(PA
S MC ), y = (RA - RA S MC ) cos(Dec) sin(PA
S MC ) + (Dec - Dec S MC ) cos(PA
S MC ).We assumed that the spatial star cluster distribution is a function of their ages (see figure 8 inBica et al. 2020, and references therein), so that each ellipsoid corresponds to a fixed age. Usingthe age gradient of figure 8 in Bica et al. (2020), we entered the star clusters’ ages to estimate theircorresponding semi-major axis. We additionally used a mean SMC distance of 62.5 kpc (Graczyket al. 2020), and an average b / c ratio of 1:2.3 (Ripepi et al. 2017; Muraveva et al. 2018; Graczyket al. 2020, and references therein) to find the projected distance r = ( x + y ) / and z values forwhich:(1 + × sin ( PA )) × ( r / b ) + . × ( z / b ) − = , (2)where b (kpc) = × log(age / yr) -10.85 (age < ∼ x = r × sin ( PA ), y = r × cos ( PA ), a / b = / b / c = / = x = x and y = y . The r and z values that comply witheq. (2) for each star cluster were obtained by evaluating eq. (2) 17600 times, from a grid of valuesof r from 0.0 up to 11.0 kpc, in steps of 0.1 kpc, and z from 0.0 up to 16.0 kpc, in steps of 0.1kpc, and then looking for the r and z ones which correspond to the smallest of the 17600 absolutevalues of eq. (2), which were always smaller than 0.01. We note that, theoretically speaking, theresulting r and z value should lead eq. (2) to be equal to zero. Finally, the linear distance of a starcluster to the SMC center is calculated as r depro j = ( r + z ) / . We estimated the uncertainties in r depro j by performing the procedure described above for 1000 realizations with b values randomlychosen in the interval [ b - σ ( b ), b + σ ( b )]. Then, we adopted σ ( r depro j ) = / FWHM of the r depro j distributions, which resulted to be typically ≈ ff erent deprojected distances to the SMCcenter are revealed. Some star clusters projected close to the SMC center are relatively distanceobjects, while others apparently placed in the outer galaxy regions turned out to be closer to theSMC center.The analysis of the variation of star cluster structural parameters as a function of their depro-jected distances to the SMC center supersedes previous ones, which are based on the star cluster https: // github.com / sczesla / PyAstronomy Article number, page 5 of 12ndrés E. Piatti: SMC star clusters r deproj (kpc)0.00.10.20.30.40.50.60.70.8 2.02.53.03.54.04.55.0 l o g ( M / M ) Fig. 2.
Star cluster eccentricity versus deprojected distance from the SMC center, color-coded according tothe star cluster mass. positions projected on the sky. As far as we are aware, there are very few SMC star clusters withindependent distance estimates (see, e.g. Glatt et al. 2008; Dias et al. 2016). In general, a meanSMC distance modulus is adopted when fitting theoretical isochrones to the CMD of a star cluster,since changes in the distance modulus by an amount equivalent to the average SMC depth leads to asmaller age di ff erence than that resulting from the isochrones (characterized by the same metallic-ity) bracketing the observed star cluster features in the CMD. Nevertheless, there is still di ff erencesbetween individual star cluster estimates. Glatt et al. (2008) estimated distances for NGC 121, Lind-say 1 and Kron 3 of 64.9 ± ± ± ± ± ±
3. Analysis and discussion
The di ff erent gravitational field strengths experienced by star clusters a ff ect their structural parame-ters, and ultimately their internal dynamical evolutionary stages. For example, the increase of core,half-mass, and Jacobi radii as a function of the star cluster distance from the Milky Way centerwas predicted theoretically by Hurley & Mackey (2010) and Bianchini et al. (2015), among others.Star clusters in weaker tidal fields, like those located in the outermost regions of the Milky Waycan expand naturally, while those immersed in stronger tidal fields (e.g. the Milky Way bulge) donot. We here use the calculated deprojected distances as a proxy of the SMC gravitational field, toinvestigate whether some star cluster properties show any trend with it. Article number, page 6 of 12ndrés E. Piatti: SMC star clusters
Figure 2 shows the eccentricity versus deprojected distance plane for the studied star clustersample, from which some obvious features arise at a glance. The eccentricities span a wide rangeof values (0.0 < (cid:15) < < ∼ < (cid:15) < ∼ ff erent mechanisms, such as dynamical relaxation and decay of initial velocityanisotropy, cluster rotation, external perturbations, di ff erential interstellar extinction, etc (see Chen& Chen 2010, for a review). Milky Way globular clusters have a median eccentricity of ∼ M / M (cid:12) ) ∼ . ff erent eccentricity regimes mentioned above (for r depro j smaller or larger than ∼ z = ∼ ∼ ∼ ∼ Article number, page 7 of 12ndrés E. Piatti: SMC star clusters r deproj (kpc)510152025 r ( p c ) l o g ( a g e / y r ) Fig. 3.
Star cluster size ( r ) versus deprojected distance from the SMC center, color-coded according to theirages. Star clusters with (cid:15) > l o g ( M / M ) r d e p r o j ( k p c ) Fig. 4.
Star cluster mass versus deprojected distance from the SMC center, color-coded according to theirdeprojected distances from the SMC center. Star clusters with (cid:15) >
Figure 3 also tells us that the star cluster sizes do not show any correlation with the deprojecteddistances, i.e., they would not be a ff ected by the SMC gravitation field, as it is the case of MilkyWay and LMC globular clusters (Piatti & Mackey 2018; Piatti et al. 2019), which are bigger asthey are located further from the galaxy center. This finding puts a limit to the galaxy mass, a valuein between the LMC and the SMC mass, in order to the galaxy gravitational field can drive thesize of its star clusters. We point out that old globular clusters in the Milky Way and the LMCare on average one order of magnitude more massive than massive SMC star clusters (Piatti &Mackey 2018), so that the comparison between them could favor a minimum galaxy mass moresimilar to that of the LMC. This also could have its own impact in the computation of the clustermass lost by tidal disruption along the entire lifetime of star clusters stripped o ff the SMC by theLMC (Carpintero et al. 2013). In the standard cosmological scenario (Moore et al. 1999; D’Onghia& Lake 2008), accreted globular clusters are formed in small dwarf galaxies. Hence, most of thecluster mass lost by tidal disruption should have disrupted once the star cluster is under the e ff ectsof the Milky Way gravitational field, because low mass galaxies would not seem to a ff ect seriouslythe mass budget of its massive globular clusters. Nevertheless, the large eccentricity values foundonly in SMC star clusters located inside a volume of radius ∼ r , although a robust estimate of the star cluster size, does not represent thecluster Jacobi radius, which should strictly speaking be considered for monitoring any change inthe star cluster dimension with the deprojected distance. Typical errors in r are ∼ ∼ M / M (cid:12) ) ∼ .
0. More massive star clustershave eccentricities smaller than ∼ M / M (cid:12) ) ∼ .
0. Likewise, we wonder on thepresence of many star clusters less massive than log( M / M (cid:12) ) ∼ . ∼ ff erent rotationvelocities, or a di ff erential perturbation by the LMC during the last close passage to the SMC (Patelet al. 2020).Figure 4 also shows that the cluster mass distribution as a function of age is quite di ff erent fromthat of Milky Way open clusters located in a circle of radius 1.8 kpc from the Sun (Joshi et al. 2016,solid line). In the case of these open clusters, we can assume that the mass variation as a functionof their ages is mainly caused by evolutionary e ff ects, if the Milky Way gravitation field does nota ff ect di ff erently them in that relatively small circle. Furthermore, we can imagine straight linesparallel to that for Joshi et al. (2016)’s open clusters that correspond to star clusters under di ff erenttidal disruption regimes (Piatti et al. 2019), with those for weaker tidal fields located upward.Figure 4 shows a large number of SMC clusters that would seem to follow a similar trend, shifted by Article number, page 9 of 12ndrés E. Piatti: SMC star clusters ∆ (log( M / M (cid:12) )) ∼ ff erencecould reflect the much stronger tidal field of the Milky Way at the solar circle in comparison withthat of the SMC, assumed that the SMC star clusters are a ff ected by the same SMC tidal fieldstrength. We note that such a trend is followed by star clusters with some hundred Myr, for whichmass loss is mainly driven by stellar evolution, and also for some older star clusters, where two-body relaxation can have a more important role. Star clusters older than ∼ / yr) > ∼ ff ected by weaker gravitational field strengths. Wenote that most of them have eccentricities < ∼ > ∼ ∆ (log( M / M (cid:12) )) ∼
4. Concluding remarks
We made use of available data sets of structural properties for a statistically significant sample ofSMC star clusters with the aim of studying at what extend the SMC gravitational field are responsi-ble of the star cluster shapes and sizes. Recently, it was shown the observed dependence of the core,half-mass, and Jacobi radii, alongside relaxation time, cluster mass loss by tidal disruption, amongothers, with the position in the galaxy of old globular Milky Way and LMC clusters. Although theSMC does not harbor star clusters as old as the ancient globular clusters, the spatial coverage of starclusters spanning the whole age range allows us to probe for tidal e ff ects. Hill & Zaritsky (2006)performed an analysis of some structural properties of SMC star clusters. As far as we are aware,this is the first time that star cluster properties are analyzed in the context of the 3D geometry ofthe SMC.We adopted an ellipsoid as a representation of the SMC with the three axes having di ff erentextensions. They have been adjusted combining the known star cluster age gradient and the recentlySMC depth estimated from Classical Cepheids, RR Lyrae stars, and late-type eclipsing binaries. Inthis framework, each age is assigned to a unique ellipsoid. Therefore, by using the age of thestar clusters and their projected positions in the sky with respect to the SMC center, we estimatedtheir deprojected distances, which we used as a proxy of the SMC gravitational field. The use ofdeprojected distances solved the spurious e ff ect of considering a star cluster to be located close tothe SMC center, from its projected position in the sky.We sought any trend between the star cluster size (represented by the 90% light radius), theeccentricity, the mass and age with the deprojected distance. We did find that the size of the starclusters would not seem to be sensitive to changes in their positions in the galaxy, because starclusters spanning the entire observed range are found everywhere. We point out, however, thatJacobi radii would be appropriate for a more definitive answer. The star cluster eccentricities revealthat those more elongated objects ( (cid:15) > ∼ < ∼ Article number, page 10 of 12ndrés E. Piatti: SMC star clusters kpc. This finding could be a hint for di ff erential tidal e ff ects between star clusters located closer andfarther from the SMC center. However, we found a numerous population of stars clusters distributedinside the same volume that look like less elongated ( (cid:15) < ∼ M / M (cid:12) ) ∼ (cid:15) < ∼ / yr) ∼ ff erential tidal e ff ects. Likewise, there is a large number of star clusterslocated at deprojected distances < ∼ ff set of 0.7 toward more massive star clusters. We interpret this shift as originating from di ff erentgravitational field strengths. Acknowledgements.
I thank the referee for the thorough reading of the manuscript and timely suggestions to improve it.
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