Dependence of the old star clusters' dynamical clock on the host galaxy gravitational field
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Dependence of the old star clusters’ dynamical clock on the host galaxy gravitational field
Andr´es E. Piatti
1, 2 Instituto Interdisciplinario de Ciencias B´asicas (ICB), CONICET-UNCUYO, Padre J. Contreras 1300, M5502JMA, Mendoza,Argentina Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Godoy Cruz 2290, C1425FQB, Buenos Aires, Argentina
ABSTRACTI report outcomes of the analysis of the A + parameter, which measures the level of radial segregationof blue straggler stars in old star clusters, commonly known as the dynamical clock for the long-terminternal dynamical evolution. I used A + values available in the literature for 48 Milky Way globularclusters. I found that the relationship of A + and the number of central relaxation times which haveelapsed ( N relax ) shows a non negligible dependence on the strength of the host galaxy gravitationalpotential, in addition to depending on the two-body relaxation mechanism. Indeed, a measured A + value corresponds to relatively smaller or larger N relax values for star clusters located farther or closerto the galaxy center. From an observational point of view, this finding reveals the possibility ofdisentangling for the first time the dynamical evolutionary stage due to two-body relaxation and tidaleffect, that affect the whole star clusters’ body concurrently. Keywords:
Galaxy: globular clusters: general – Methods: observational. INTRODUCTIONThe internal dynamics of star clusters has long beenaddressed in the literature from numerical and obser-vational studies (Meylan & Heggie 1997; Heggie & Hut2003; Krause et al. 2020). Recently, Ferraro et al. (2018,see also references therein) found a strong correlationbetween A + , defined as the area enclosed between thecumulative radial distribution of blue straggler stars andthat of a reference population, and the number of cen-tral relaxation times ( N relax =age/ t rc ) of old star clus-ters. Because of the observed correlation between A + and the central relaxation time of old star clusters, A + has been used as a powerful dynamical clock. For thesake of the reader, I refer to a review by Ferraro et al.(2020).Because A + measures the overall internal dynamicalstage of an old star cluster, the evolutionary stage due totwo-body relaxation and tidal effects are both includedin the A + values. Particularly, tidal effects acceleratemass segregation and two-body relaxation by increasingthe mass loss rate. As far as I am aware, the impact oftidal effects on A + has not been explicitly mentioned.The novelty of this work consists in disentangling ob- Corresponding author: Andr´es E. Piattie-mail: [email protected] servationally, for the first time, the dependence of A + on two-body relaxation and tidal effects, so that thedynamical evolutionary stage of a star cluster due totwo-body relaxation can be estimated for star clustersin different orbits. In doing this, I evaluate the contri-bution of tidal effects to the measured values of A + . ANALYSIS AND DISCUSSIONFigure 1 (left panel) reproduces the relation obtainedby Ferraro et al. (2018). They have been colored ac-cording to the semi-major axis ( a ) of the cluster’s or-bits around the Galactic center. I use a because ismore representative of the mean orbital distance of theglobular clusters than the perigalactic and apogalac-tic distances (Piatti 2019). A not subtle dispersion inlog( N relax ) at a constant A + value is observed, whichwould seem to change with a , in the sense that the largerthe log( N relax ) value, the smaller the mean semi-majoraxis. Such a trend, observed in Figure 1, reveals thecorrelation with the semi-major axis or, in other words,that depending on the position in the Milky Way, A + corresponds to slightly different internal dynamic evolu-tionary stages.In order to show the effects of tides in the Ferraroet al. (2018)’s relation we evaluate the range of A + bycomputing the difference between the A + values andthose located on the solid line of Figure 1 for the same a r X i v : . [ a s t r o - ph . GA ] D ec Andr´es E. Piatti A + l o g ( N r e l a x ) l o g ( a / k p c ) a /kpc)0.30.20.10.00.10.20.3 A + M d i s / M i n i A + l o g ( N r e l a x ) F - E q . ( ) Figure 1.
Left:
Relation between A + and log( N relax ) for the 48 Galactic globular clusters analyzed by Ferraro et al. (2018),and their derived least square fit drawn with a solid black line. Middle: ∆ A + as a function of the semi-major axis. Error barsare included. The solid line represents eq.(1). Right:
Difference between N relax values calculated from Ferraro et al. (2018)’srelation and from eq. (2). N relax values, called ∆ A + . The result, depicted in Fig-ure 1 (middle panel), shows the correlation of ∆ A + withthe positions of the globular clusters in the Milky Way.Points have been colored according to the ratio betweenthe cluster mass lost by tidal disruption and the initialcluster mass ( M dis /M ini ), which we use as the indica-tor for tidal field strength, following Piatti et al. (2019)’sresults. M dis /M ini and a are interchangeable. Uncer-tainties in ∆ A + were computing by adding in quadra-ture those of A + values and the rms error of the solidline derived by Ferraro et al. (2018). Figure 1 (mid-dle panel) also shows that ∆ A + ≈ A + values at a fixed N relax - is not random, but caused by the dependence of A + on a . A + is larger than the mean A + value for thecorresponding N relax for globular clusters located to-ward the outer Milky Way regions, and is smaller thanthat mean A + value for globular clusters located towardthe inner Milky Way regions. This behavior of A + is ex-pected, because A + is a measure of the level of spatialsegregation of blue straggler stars. Similarly to otherglobular cluster stellar populations, they differentiallyexperience the effects of the Milky Way gravitationalfield, so that they can more easily reach larger distancefrom the globular cluster center when the Galactic po-tential well is weaker, and role reversal. Therefore, pa-rameters that measure the spatial distribution of stellarpopulations of globular clusters (e.g., core, tidal radii,relaxation time, etc) reflect both the internal dynami-cal evolutionary stage of a globular cluster and the tidal effects, concurrently. From a purely stellar dynamicalpoint of view, a globular cluster located in the innerMilky Way region would appear dynamically acceler-ated, or in a more advanced internal dynamical evo-lutionary stage compared to a globular cluster locatedin the outer Milky Way regions (see Piatti & Mackey2018; Piatti et al. 2019). For completeness purposes Iperformed a quadratic least square fit between ∆ A + andlog( a /kpc), as follows:∆ A + = C + C × log ( a/kpc ) + C × ( log ( a/kpc )) . (1)I obtained C = -0.17 ± C = 0.24 ± C = -0.04 ± χ = 0.65, and a correlationcoefficient = 0.60 (see solid line in Fig. 1, middle panel).I note that a linear least square fit would produce similarresults with a slightly smaller correlation coefficient.Eq. (1) can be useful to compute dynamical ages of astar clusters due to two-body relaxation, as it were notaffected by tidal effects. In doing this, the calculated∆ A + value must be subtracted from the observed A + value, and the resulting one to be entered in eq.(2) ofFerraro et al. (2018). Because ∆ A + ≈ A + , N relax and a , as follows: log ( N relax ) = C + C × A + + C × log ( a/kpc ) , (2)and obtained C = 1.52 ± C = 4.57 ± C = -0.77 ± χ = 0.16, and a coefficient of ld star cluster dynamical clock = 0.81. I obtained for eq.(2) a Spear-man rank correlation coefficient of 0.87 and a Pearsoncorrelation coefficient of 0.90, which represent an im-provements over the values of 0.82 and 0.85 obtained by Ferraro et al. (2018), respectively, for their eq. (2). Forthe sake fo the reader Figure 1 (right panel) depicts thedifference between N relax values calculated from Ferraroet al. (2018)’s relation and from eq.(2) as a function of A + .REFERENCES.REFERENCES