Robust ellipticity measurements of 29 Galactic globular clusters

Abstract

Globular clusters (GCs) exhibit varying degrees of flattening (ellipticity), which may provide insight into their internal dynamics and evolution histories. Commonly used methods to measure ellipticity, such as ellipse fitting of density contours and principal component analysis, often produce biased results, especially for clusters that are nearly round or have few observable stars. Using a combination of ground-based and space-based photometry, we investigate the shapes of 29 Galactic GCs. To that end, we test two commonly used methods: an ellipse fit to a kernel density profile and a principal component analysis. We find that both methods suffer from bias arising when the number of stars is small or the cluster is close to round. To solve this issue, we develop a robust method to measure the ellipticity of GCs, test it extensively on mock data, and apply it to the 29 Milky Way GCs in our sample. Using the $V/σ$ diagram used in the isotropic oblate rotator framework, we examine potential causes for the flattening, including rotation and velocity anisotropy. For ten clusters: NGC~104, NGC~1261, NGC~2808, NGC 3201, NGC 5286, NGC 5904, NGC 5986, NGC 6205, NGC 6341, and NGC 7078 we identify a very good agreement between the rotation angle and semi-minor axis of the ellipse, further corroborating the findings that rotation is the main driver of the ellipticity. The $V/σ$ diagram reveals that velocity anisotropy or tides could also be important in shaping the GCs. The robust method developed provides reliable measurements of the ellipticity of GCs, emphasising the importance of taking into account the flattening in theoretical models and simulations. It also offers a promising way to investigate the shapes of multiple stellar populations within GCs, where only small samples are usually available.

Figures (9)

• Figure 1: Recovered ellipticity as a function of the true ellipticity using the KDE method (upper-left panel), the PCA method (upper-right panel), the robust PCA method (lower-left panel) and the robust PCA method afer bias correction (lower-right panel). The points are colored according to the number of stars in the mock GCs.
• Figure 2: Impact of outliers on the recovered ellipticity and position angle for the KDE (circles), PCA (squares), and robust PCA (diamonds) methods for two sets of true parameters. The colour of the points indicates the location of the outliers, in pink with positive x and negative y, corresponding approximately to the direction of the semi-minor axis and in orange with positive x and positive y, corresponding approximately to the direction of the semi-major axis of the ellipse. Outliers were added within fixed quadrants relative to the cluster centre, not strictly along the ellipse axes, which leads to small asymmetries in recovered angles when the ellipse position angle changes. The green dashed line represents the true value of the ellipticity and angle, taken to 0.15 and 45° (left panel) and to 0.25 and 60° (right panel). We note that, for the Robust PCA method, only one point is visible, thus highlighting the robustness of the method. Regardless of the direction where outliers are added, the ellipticity and position angle are accurately recovered.
• Figure 3: The ellipticities of the 29 Galactic GCs in our sample, measured by using the robust PCA method, are here compared with the estimates from the literature, provided by 1987White (left panel), Chen_2010 (middle panel), and 2024Cruz_reyes (right panel). The solid lines indicate equal ellipticities; each GC is represented by a coloured point (see legend on top), and error bars are indicated.
• Figure 4: The ellipticity $e$ of 29 Galactic GCs, obtained using the robust PCA method, is shown here against the relaxation time at the half-mass radius from 1996Harris. The points are colored according to the last crossing time through the Galactic disk, taken from 2024Pancino. The name of each cluster is indicated close to the corresponding point. Both times are given in years. Dynamically young clusters tend to be more elliptical.
• Figure 5: This plot shows the rotation strength, taken from 2024Leitinger, as a function of the flattening we measured. The names of GCs are indicated close to the corresponding points, colored according to the inclination angle of the cluster, also taken from 2024Leitinger. Low inclinations are plotted in blue while high inclinations are colored in yellow. The uncertainties associated with the inclination angles are not shown in this figure. Instead, we report them in the last column of Table \ref{['tab:cluster_results']}. The dashed lines indicate the relation for isotropic oblate rotators viewed at different inclination angles.