Title: 2604.08348 ArXiv: 2604.08348 3D kinematics of SMC star clusters: residual velocities disentangle kinematically perturbed clusters Title: Content selection saved. Describe the issue below: Description: License: CC BY 4.0 arXiv:2604.08348v1 [astro-ph.GA] 09 Apr 2026 1 1 institutetext: Instituto Interdisciplinario de Ciencias Básicas (ICB), CONICET-UNCuyo, Padre J. Contreras 1300, M5502JMA, Mendoza, Argentina; 2 2 institutetext: Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Godoy Cruz 2290, C1425FQB, Buenos Aires, Argentina 3D kinematics of SMC star clusters: residual velocities disentangle kinematically perturbed clusters Denis M.F. Illesca Andrés E. Piatti Matías Chiarpotti Roberto Butrón (Received / Accepted) Understanding the kinematic behaviour of the Small Magellanic Cloud (SMC) remains a challenge addressed by many authors using diverse approaches. Over time, increasing observational evidence has accumulated for tidal perturbations induced by the Large Magellanic Cloud (LMC) on the SMC, especially in its outer regions. In this study, we adopt star clusters as kinematic tracers of the SMC. We analyse 36 clusters distributed across the galaxy’s structural regions (Northern Bridge, Southern Bridge, Wing/Bridge, West Halo, Main Body and Counter-Bridge). From each cluster’s proper motions, radial velocity and heliocentric distance we estimate Cartesian velocities ( V x , V y , V z ) (V_{x},\,V_{y},\,V_{z}) in the SMC reference frame. We also compute the same velocity components under the assumption that the SMC behaves as a rotating disc. We then define the residual velocity Δ ​ V \Delta V for each cluster as the difference between the two velocities derived. Additionally, we perform a kinematic anisotropy analysis to characterise the distribution of kinetic energy across the SMC. We find that increasing values of Δ ​ V \Delta V correlate with increasing cluster distance from the SMC center, and that Δ ​ V ≈ 60 ​ km ​ s − 1 \Delta V\approx 60\ \mathrm{km\,s^{-1}} it appears to be a lower limit that separates, in kinematic terms, the areas of tidal origin from those with the best behavior. Key Words.: (Galaxies:) Magellanic Clouds – Galaxies: kinematics and dynamics - Galaxies: star clusters: general 1 Introduction The kinematic behavior of the Small Magellanic Cloud (SMC) has been investigated in recent years by several authors (e.g., Kallivayalil et al. , 2013 ; Zivick et al. , 2018 ; De Leo et al. , 2020 ; Niederhofer et al. , 2021 ; Dhanush et al. , 2025 ) .Understanding the internal kinematics of the SMC is essential for reconstructing its interaction with the Large Magellanic Cloud and the Milky Way. To trace the kinematic signatures of the SMC, different galactic constituents have been used, namely: HI gas, young stars, red giant stars, and massive stars, among others. The gas in the SMC exhibits considerable internal rotation (Stanimirović et al. , 2004 ) , while young stars show an orderly motion towards the Magellanic Bridge, with proper motions greater than that of the SMC main body (Oey et al. , 2018 ) . Indeed, Nakano et al. ( 2025 ) investigated the motions of massive stars ( > > 8 M ⊙ ) with ages ¡ 50 Myr, and found trajectories oriented towards the LMC and away from the SMC main body. In contrast, the oldest stellar population apparently shows little rotation (Harris and Zaritsky, 2006 ; Zivick et al. , 2021 ) , which make the whole SMC kinematics - to some extent - a still living conundrum. We note, however, that some of these results are based solely on radial velocity or proper motion measurements. The SMC is under tidal effects due to its interaction with the LMC (Mackey et al. , 2018 ; Zivick et al. , 2018 ; De Leo et al. , 2020 ; Niederhofer et al. , 2021 ; Omkumar et al. , 2021 ) . The magnitude and strength of tidal forces on the morphology and internal kinematics of the SMC were estimated from dynamic simulations by Besla et al. ( 2012 ) . They concluded that the Magellanic Clouds are in their first fall towards the Milky Way. In this context, Piatti ( 2021b ) used star clusters as tracers of the internal kinematics of the SMC and constructed a 3D image of the clusters’ motions from Gaia data (Gaia Collaboration et al. , 2016 ) and radial velocities obtained from the literature. The cluster motions derived by Piatti ( 2021b ) show some notable dispersion around the resulting rotating disk. This finding reveals that the kinematics of the SMC clusters is complex and cannot be fully captured by a representation of a rotating disk alone. In this work, we analyze 36 SMC star clusters with the aim of obtaining a comprehensive representation of the SMC’s internal kinematics, based on heliocentric distances obtained by Illesca et al. ( 2025 ) , proper motions retrieved from Gaia Data Release 3 (Gaia Collaboration et al. , 2016 ; Luri et al. , 2021 ) , and radial velocities available in the literature. Incorporating individual cluster heliocentric distances, rather than adopting a single SMC mean distance for all the clusters, makes the derived kinematic behaviors more robust. From this data set, we construct a three-dimensional velocity map of the SMC, following the formalism of van der Marel et al. ( 2002 ) . We then analyze the residual velocities and explore the resulting kinematic signatures across known tidally perturbed SMC structures. In Section 2, we describe the data collected and employed in the present analysis. In Section 3, we describe the results obtained, while in Section 4 we discuss the residual velocities of star clusters as indicators of kinematic perturbations caused by tidal forces. Section 5 summarizes the main conclusions of this work. 2 Data collection and processing Illesca et al. ( 2025 ) studied 40 SMC star clusters, mainly distributed across the outer SMC regions with the aim of investigating the connection between their ages, heliocentric distances and metallicities. We used their cluster collection as a starting point to build a sample of SMC star clusters with the three mentioned fundamental parameters, in addition to radial velocities (RVs) and proper motions. Unfortunately, as far as we are aware, 12 star clusters do not have RVs available in the literature (B88, B139, BS116, HW64, HW67, HW73, HW77, IC1655, L2, L3, L73, and L95). In contrast, we added other 8 star clusters with the required information (L1, L8, L12, L68, L113, NGC 339, NGC 361, and NGC 419), with their fundamental parameters taken from Piatti ( 2023 ) . For the final sample of 36 star clusters (accurate individual cluster heliocentric distance was required), we extracted the clusters’ right ascension (RA), declination (Dec.), and radii from Bica et al. ( 2020 ) . As for the astrometric information, we retrieved from Gaia DR3 proper motions in right ascension (pmra), proper motions in declination (pmdec), parallaxes ϖ \varpi , excess noise (epsi) , significance of excess noise (sepsi) , and G G , B ​ P BP , and R ​ P RP magnitudes for every star located within three times the respective cluster’s radius. We applied a filter to the proper motion errors to retain those stars with σ \sigma ≤ 0.1 ​ mas ​ yr − 1 \leq 0.1\penalty 10000\ \mathrm{mas}\penalty 10000\ \mathrm{yr}^{-1} , following the procedure described in Piatti et al. ( 2019 ) . We favored the selection of extragalactic stars by applying the condition | ϖ | / σ ​ ( ϖ ) < 3 |\varpi|/\sigma(\varpi)<3 . Furthermore, in order to improve our data quality, we limited sepsi < < 2, epsi < < 1, RUWE ≤ \leq 1.4, and G G ≤ \leq 18 mag, respectively (see, e.g. Ripepi et al. , 2019 ) We then used the procedure devised by Piatti and Bica ( 2012 ) , originally designed to clean star cluster color-magnitude diagrams from field star contamination, to statistical remove SMC field stars from the vector point diagrams (VPDs) of the star clusters. The statistical cleaning method makes use of comparison field regions surrounding each cluster. Figure 1 illustrates the locus of the cluster circle with respect to 8 different circular comparison fields of the same area as the cluster’s circle. The method superimposes the cluster and one comparison field VPDs, and for each star in the latter it subtracts the closest one in the cluster VPD. The proper motion errors of the stars were also taken into account when searching for a star to subtract from the cluster’s VPD. To do this, we allowed the proper motions of the stars in the cluster’s VPD to vary within a range of ± 1 ​ σ \pm 1\sigma . We repeated the procedure described above for one thousand comparison fields placed around the cluster’s circle at randomly chosen position angles. Finally, we assigned to the stars in the cluster’s VPD probabilities of being cluster members as P ( % ) = 100 × S / 1000 P\,(\%)=100\times S/1000 , where S represents the number of times the star has not been subtracted after a thousand different runs. An example of the results obtained is illustrated in Figure 2 . Stars with different P P values were plotted with different colors. In the subsequent analysis, we retained only stars with P > 50 % P>50\% . Figure 1: Stars selected from Gaia DR3 distributed in the field of NGC 416. The red circle corresponds to the cluster field, while the black circles correspond to 8 different comparison fields placed adjacent to the cluster. The size of the symbols is proportional to the brightness of the stars in the G G filter. For the number N N of stars that satisfy the above restriction in each cluster, we applied the concept of effective sample size introduced by Kish ( 1987 ) . We defined an effective N N ( N eff N^{\mathrm{eff}} ) as a representative and comparable measure of the star-by-star kinematics. We computed N eff N^{\mathrm{eff}} using the expression: N eff = N ~ max ⁡ ( N ~ ) , N^{\mathrm{eff}}=\frac{\widetilde{N}}{\max(\widetilde{N})}, (1) where: N ~ = P ¯ ​ N ​ ( ∑ k = 1 N p k ) 2 ∑ k = 1 N p k 2 . \widetilde{N}=\bar{P}\,\sqrt{N}\,\frac{\left(\sum_{k=1}^{N}p_{k}\right)^{2}}{\sum_{k=1}^{N}p_{k}^{2}}\qquad\qquad. (2) In Eq. (2), N ~ \widetilde{N} combines the individual star membership probabilities p k p_{k} , k = 1 , … , N k=1,...,N , with an average value P ¯ \bar{P} . The first factor P ¯ \bar{P} in Eq. (2) penalizes star clusters with averaged low membership probabilities, while the second factor N \sqrt{N} penalizes clusters with smaller numbers of stars; the ratio corresponds to the Kish ( 1987 ) ’s effective sample size. N eff N^{\mathrm{eff}} is the normalized version of N ~ \widetilde{N} and we used it in the subsequent analysis as a relative quality weight in our star cluster kinematic results. We applied a maximum likelihood statistical method (Meylan and Pryor, 1993 ; Walker et al. , 2006 ) to estimate the mean proper motions and dispersion of the studied clusters. In practice, we optimized the probability ℒ \mathcal{L} such that a given set of stars with proper motions ( p ​ m i pm_{i} ) and errors σ i \sigma_{i} is extracted from a population with mean proper motion ⟨ p ​ m ⟩ \langle pm\rangle and dispersion W, as follows: ℒ = ∏ i = 1 N ( 2 ​ π ​ ( σ i 2 + W 2 ) ) − 1 2 ​ exp ⁡ ( − ( p ​ m i − ⟨ p ​ m ⟩ ) 2 2 ​ ( σ i 2 + W 2 ) ) , \mathcal{L}=\prod_{i=1}^{N}\left(2\pi\left(\sigma_{i}^{2}+W^{2}\right)\right)^{-\frac{1}{2}}\exp\left(-\frac{\left(pm_{i}-\langle pm\rangle\right)^{2}}{2\left(\sigma_{i}^{2}+W^{2}\right)}\right), where the mean and dispersion errors were calculated from the respective covariance matrices. The resulting mean cluster proper motions are shown in Table 1 , alongside the number of stars ( N N ) used to compute them. Figure 2: VPD for selected Gaia DR3 stars distributed in the field of NGC 416. Color symbols vary according to the assigned membership probability. 3 Star cluster kinematic properties We choose star clusters as kinematic tracers because they provide with a robust methodology that distinguishes it from other approaches. Unlike the selection of field star populations Dhanush et al. ( 2025 ) , clusters are discrete, gravitationally bound objects. This allows estimating their ages, distances and velocities with a greater accuracy than for field stars. Moreover, our star cluster sample includes individual heliocentric distances, which constitute a valuable feature compared to kinematic models based on field stars that employ mean distances, thereby underestimating the role of distances. Stellar clusters, as kinematic tracers, provide an insightful view of the SMC kinematic, without the biases that different tracers might introduce because of lack of distance estimates. Although Dhanush et al. ( 2025 ) perform a differential analysis by populations to account for changes in geometry and, consequently, in the kinematic model, we here exploit the SMC kinematic model obtained by Piatti ( 2021b ) which is based on star clusters. He found that the SMC rotation disk is characterized by the right ascension and declination of its center (RA = 13.30 ∘ ± 10 13.30^{\circ}\pm 10 , Dec = − 72.85 ∘ ± 10 -72.85^{\circ}\pm 10 ), its distance to the center ( 59 ± 1.5 ​ kpc 59\pm 1.5\ \mathrm{kpc} ), radial velocity ( 150 ± 2 ​ km ​ s − 1 150\pm 2\ \mathrm{km\ s^{-1}} ), central proper motion in RA ( pmra center = 0.75 ± 10 ​ mas ​ yr − 1 \mathrm{pmra_{center}}=0.75\pm 10\ \mathrm{mas\ yr^{-1}} ), central proper motion in Dec ( pmdec center = − 1.26 ± 0.05 ​ mas ​ yr − 1 \mathrm{pmdec_{center}}=-1.26\pm 0.05\ \mathrm{mas\ yr^{-1}} ), disk inclination ( 70 ∘ ± 10 70^{\circ}\pm 10 ), position angle of the line of nodes ( 200 ± 30 200\pm 30 ), and rotation velocity ( 25 ± 5.0 ​ km ​ s − 1 25\pm 5.0\ \mathrm{km\ s^{-1}} ), respectively. We firstly subtracted the mean proper motion and radial velocity of the SMC center of mass (Piatti, 2021b ) from the resulting clusters’ mean proper motions and radial velocities, and calculated the residual linear velocities V RV , V RA and V Dec , the latter in units of [ km ​ s − 1 \mathrm{km\penalty 10000\ s^{-1}} ] through the expression 4.7403885 × \times D D [ mas ​ yr − 1 \mathrm{mas\penalty 10000\ yr^{-1}} ], where D D is the cluster heliocentric distance. To convert the vector (V RV , V RA , V Dec ) into one with components V ​ x Vx and V ​ y Vy in the plane of the SMC and V ​ z Vz perpendicular to it, we used the reference system defined by van der Marel et al. ( 2002 ) , and followed the procedure described in Piatti et al. ( 2019 ) . This comprised inverting the matrix A = B × \boldsymbol{\times} C , where B is the matrix: ( 1 0 0 0 b 1 b 2 0 b 3 b 4 ) \begin{pmatrix}1&0&0\\ 0&b_{1}&b_{2}\\ 0&b_{3}&b_{4}\end{pmatrix} with b 1 b_{1} , b 2 b_{2} , b 3 b_{3} , and b 4 b_{4} being the coefficients of the transformation Eq. (9), and C is the matrix defined in Eq. (5) of van der Marel et al. ( 2002 ) , respectively, so that : ( V x V y V z ) = 𝐀 − 1 ​ ( V RV V RA V Dec ) \begin{pmatrix}V_{x}\\ V_{y}\\ V_{z}\end{pmatrix}=\mathbf{A}^{-1}\begin{pmatrix}\mathrm{V_{\rm RV}}\\ \mathrm{V_{\rm RA}}\\ \mathrm{V_{\rm Dec}}\end{pmatrix} (3) The errors σ ​ ( V x ) \sigma(V_{x}) , σ ​ ( V y ) \sigma(V_{y}) and σ ​ ( V z ) \sigma(V_{z}) were estimated by performing Monte Carlos experiments using the uncertainties in V RV , V RA and V Dec . From Eq. (3) we calculated V rot = ( V x 2 + V y 2 ) 1 / 2 \mathrm{V_{rot}}=(V_{x}^{2}+V_{y}^{2})^{1/2} and V rot ​ .3 ​ D = ( V x 2 + V y 2 + V z 2 ) 1 / 2 \mathrm{V_{rot.3D}}=(V_{x}^{2}+V_{y}^{2}+V_{z}^{2})^{1/2} , and the resulting values are listed in Table 2 . On the other hand, we computed the velocity components ( V x ′ V_{x^{\prime}} , V y ′ V_{y^{\prime}} , V z ′ V_{z^{\prime}} ) with respect to the SMC center that the star clusters would have, if they rotated at their present positions in the SMC disk according to the rotation disk fitted by Piatti ( 2021b ) . The difference between ( V x V_{x} , V y V_{y} , V z V_{z} ) and ( V x ′ V_{x^{\prime}} , V y ′ V_{y^{\prime}} , V z ′ V_{z^{\prime}} ) is the so-called residual velocity vector ( Δ ​ V x \Delta V_{x} , Δ ​ V y \Delta V_{y} , Δ ​ V z \Delta V_{z} ), where Δ ​ V x \Delta V_{x} = V x − V x ′ V_{x}-V_{x^{\prime}} , Δ ​ V y \Delta V_{y} = V y − V y ′ V_{y}-V_{y^{\prime}} , and Δ ​ V z \Delta V_{z} = V z − V z ′ V_{z}-V_{z^{\prime}} , respectively. The resulting values are listed in Table 3 . The module of the residual velocity vector ( Δ ​ V = ( Δ ​ V x 2 + Δ ​ V y 2 + Δ ​ V z 2 ) 1 / 2 \Delta V=(\Delta V_{x}^{2}+\Delta V_{y}^{2}+\Delta V_{z}^{2})^{1/2} ) was introduced by Piatti ( 2021a ) as a measure of the kinematic perturbation experienced by a star cluster, i.e, how much the cluster’s motion departs from an ordered rotation. Finally, following van der Marel and Cioni ( 2001 ) , we computed the Cartesian coordinates ( x , y , z x,y,z ) of the star clusters with respect to the SMC’s center: x \displaystyle x = D ​ sin ⁡ ρ ​ cos ⁡ ( ϕ − θ ) , \displaystyle=D\sin\rho\cos(\phi-\theta), (4) y \displaystyle y = D ​ [ sin ⁡ ρ ​ cos ⁡ i ​ sin ⁡ ( ϕ − θ ) + cos ⁡ ρ ​ sin ⁡ i ] − D 0 ​ sin ⁡ i , \displaystyle=D\left[\sin\rho\cos i\sin(\phi-\theta)+\cos\rho\sin i\right]-D_{0}\sin i, z \displaystyle z = D ​ [ sin ⁡ ρ ​ sin ⁡ i ​ sin ⁡ ( ϕ − θ ) − cos ⁡ ρ ​ cos ⁡ i ] + D 0 ​ cos ⁡ i . \displaystyle=D\left[\sin\rho\sin i\sin(\phi-\theta)-\cos\rho\cos i\right]+D_{0}\cos i. where D D , ρ \rho and ϕ \phi are the cluster heliocentric distances, the cluster projected distances from the SMC’s center and their position angles, respectively, the latter computed from the cluster celestial coordinates (RA, Dec.). D O D_{O} represents the mean heliocentric distance of the SMC’s center (62.44 kpc, Graczyk et al. , 2020 ) , while θ \theta and i i are the position of the line of nodes and the inclination of the SMC disk derived by Piatti ( 2021b ) . From Eq. (4), we computed the projected distance on the SMC plane R plane = x 2 + y 2 R_{\mathrm{plane}}=\sqrt{x^{2}+y^{2}} , and the space distance R 3 ​ D = x 2 + y 2 + z 2 R_{\mathrm{3D}}=\sqrt{x^{2}+y^{2}+z^{2}} , and listed the resulting values in Table 3 . At first glance, we found that most of the selected star clusters are distributed within R 3 ​ D ∼ R_{\mathrm{3D}}\sim 14 kpc, some few ones reaching R 3 ​ D ∼ R_{\mathrm{3D}}\sim 25 kpc (see Figure 3 ). Figure 3: Rotational velocity values for the studied star clusters as a function of their distances to the SMC’s center. The colors represent different structures of the SMC (Dias et al. , 2016 ) as indicated in the top-right panel (NB=Northern Bridge, MB= Main Body, SB=Southern Bridge, W/B=Wing/Bridge, WH=West Halo). Symbol size is proportional to N eff N^{\mathrm{eff}} . 4 Analysis and discussion Besla et al. ( 2012 ) showed that the irregular morphology and internal kinematics of the Magellanic System can more robustly explained by considering gravitational interactions between the LMC and the SMC. This outcome leads to question about the kinematic signatures witnessing the tidally disturbed structures of the SMC. We here addressed this issue by using star clusters as kinematic tracers, and their residual velocities as a measure of the perturbed kinematic signatures. In this context, star clusters located in tidally perturbed SMC regions are expected to have larger residual velocities. For instance, Piatti ( 2021b , see his Figure 3) found that star clusters pertaining to outer SMC regions (some of them with a known tidal origin) have Δ ​ V > \Delta V> 50 km ​ s − 1 \mathrm{km\,s^{-1}} . We built a similar figure (see Figure 5 ) using our sample of 36 star clusters. As can be seen, star clusters located outside the SMC main body tend to have Δ ​ V > \Delta V> 60 km ​ s − 1 \mathrm{km\,s^{-1}} , while smaller Δ ​ V \Delta V values are mostly seen for star clusters in the SMC main body. Moreover, the closer star clusters to the Sun, the larger their residual velocities, which could be a direct measure of the strength of the tidal interaction with the LMC (mean heliocentric distance ∼ \sim 49.9 kpc, de Grijs et al. , 2014 ) . Figure 6 shows the sky distribution of the studied star clusters with the different outer SMC regions separated by dashed lines, namely: Northern Bridge (NB), Wing/Bridge (W/B), Southern Bridge (SB), West Halo (WH), and Counter Bridge (CB), respectively (Dias et al. , 2016 ) . Star clusters have been colored according to their dispersion velocities, those with larger Δ ​ V \Delta V values being mainly distributed in the outer SMC regions. These regions are known to have been affected by LMC tides (e.g., Zivick et al. , 2018 ; Schmidt et al. , 2020 ; Dias et al. , 2022 ; Parisi et al. , 2024 ; Mackey et al. , 2018 ) , so that the derived larger Δ ​ V \Delta V values could represent a measure of the strength of the LMC tidal effects. For instance, L116, located in the Southern Bridge region, has a residual velocity of 225.73 km ​ s − 1 \mathrm{km\,s^{-1}} and is moving towards the LMC. In the West Halo, L4, 11, and 13 exhibit residual velocities greater than 110 km ​ s − 1 \mathrm{km\,s^{-1}} , with velocity vectors oriented in the opposite direction to the LMC. Both the Wing/Bridge and the Northern Bridge have also star clusters with relative high residual velocities pointing towards the LMC (see Table 2 ). Star clusters located in the SMC main body or surrounding it generally have residual velocities Δ ​ V \Delta V < < 60 km ​ s − 1 \mathrm{km\,s^{-1}} . A 3D space view of the residual velocities is depicted in Figure 4 . As can be seen, the SMC is more elongated al ong the x x axis (approximately parallel to the SMC line-of-sight), with increasing residual velocities from its center out to its outskirts. To characterize the kinematics of clusters in different substructures with a possible tidal origin, we analyze the dispersion of the 3D components of their residual velocities and compare them to the total dispersion. Following the work of Watkins et al. ( 2024 ) , we introduce the kinematic anisotropy in the SMC framework as follows: A i = σ 2 ​ ( Δ ​ V i ) σ 2 ​ ( Δ ​ V x ) + σ 2 ​ ( Δ ​ V y ) + σ 2 ​ ( Δ ​ V z ) A_{i}=\frac{\sigma^{2}(\Delta V_{i})}{\sigma^{2}(\Delta V_{x})+\sigma^{2}(\Delta V_{y})+\sigma^{2}(\Delta V_{z})} (5) for i i = x , y , z x,y,z Figure 7 shows the values of A x A_{x} , A y A_{y} , and A z A_{z} as a function of the galactocentric distance, for each of the SMC disk models proposed in Piatti ( 2026 ) . At first glance, star clusters pertaining to the outer regions of the SMC tend to show a larger anisotropy along the x x and z z axes, which suggests an overall agitated kinematics approximately parallel to the SMC line-of-sight and perpendicular to its plane. 4.1 West Halo The West Halo was proposed by Dias et al. ( 2016 ) as a substructure distant from the SMC main body, and confirmed by proper motion studies (Niederhofer et al. , 2018 ; Piatti, 2021b ) . Moreover, Tatton et al. ( 2020 ) suggested that the West Halo could be the tidal counterpart of the SMC Bridge (see also Zivick et al. , 2018 ) . We obtained A x A_{x} = 0.72, A y A_{y} = 0.17, and A z A_{z} = 0.11, and a depth in the spatial distribution of star clusters of ∼ \sim 17 kpc, which point to a clear elongation and predominant dispersion of motions along the x x axis (see Figure 4 ). These outcomes reinforce the hypothesis that the West Halo is a dispersed and disturbed substructure, possibly originated from a detachment of the SMC main body (Dias et al. , 2022 ) . Figure 4: 3D distribution of the studied star clusters. Star clusters projected onto different SMC substructures are represented with different symbols, while their colors correlate with their residual velocities. Symbol sizes are proportional to N eff N^{\mathrm{eff}} . Figure 5: Residual velocities as a function of the heliocentric distances of the studied star clusters. The vertical dashed lines represent the boundaries of the SMC main body (Piatti, 2021b ) , while the horizontal red line represents the lower residual velocity limit adopted in this work for star clusters located outside the SMC main body. Star clusters pertaining to different substructures (Dias et al. , 2016 ) are drawn with different colors as indicated in the top-right panel (NB=Northern Bridge, MB= Main Body, SB=Southern Bridge, W/B=Wing/Bridge, WH=West Halo). Symbol sizes are proportional to N eff N^{\mathrm{eff}} . 4.2 Bridges and Wing For the Wing/Bridge region we obtained A x A_{x} = 0.68, A y A_{y} = 0.09, and A z A_{z} = 0.23, suggesting that the star clusters are moving towards the LMC, as is also the case of star clusters in the Southern Bridge ( A x A_{x} = 0.51, A y A_{y} = 0.06, and A z A_{z} = 0.44). Four out of the six star clusters analyzed in this latter region have residual velocities larger than the threshold value found in Figure 5 (62 km ​ s − 1 \mathrm{km\,s^{-1}} ) and heliocentric distances smaller than 53 kpc, which could be indicating escaping motions. On the other hand, star clusters in the Northern Bridge show a predominant motion dispersion perpendicular to the SMC plane ( A x A_{x} = 0.16, A y A_{y} = 0.29, and A z A_{z} = 0.55). Three of them are located close to the boundary of the SMC main body, while the other four are placed at heliocentric distances smaller than 51 ​ kpc 51\,\mathrm{kpc} . One again, the correlation between the amplitude of the residual velocities and the heliocentric distances reinforces their tidal origin (Piatti, 2022 ; Sakowska et al. , 2024 ) . Figure 6: Sky distribution of the studied star clusters, colored according to their residual velocities. The dashed lines delimit the different outer SMC regions (Illesca et al. , 2025 ) . Symbols sizes are proportional to N eff N^{\mathrm{eff}} . 4.3 Main Body The studied star clusters projected on to the SMC main body span ∼ \sim 23.5 kpc of heliocentric distance, B99 and H86-97 being the closer star clusters to the Sun ( D D < < 39 kpc). These two star clusters have Δ ​ V \Delta V > > 170 km ​ s − 1 \mathrm{km\,s^{-1}} , which highlight from those physically occupying the SMC main body ( Δ ​ V \Delta V < < 60 km ​ s − 1 \mathrm{km\,s^{-1}} ). 4.4 Kinematics under different SMC disk models As previously noted by Piatti ( 2026 ) , the estimation of Δ ​ V \Delta V depends on the adopted SMC rotation disk. Therefore, a comprehensive analysis of the kinematics of the studied star clusters requires considering different rotation disk models. Dhanush et al. ( 2025 ) used Gaia DR3 data to derive kinematic parameters for different SMC star populations. From young to old star populations, they found a change in the SMC disk inclination from ∼ 82 ∘ \sim 82^{\circ} to ∼ 58 ∘ \sim 58^{\circ} , and in the position angle of the line of nodes (LON) from ∼ 180 ∘ \sim 180^{\circ} to ∼ 240 ∘ \sim 240^{\circ} . Following the three SMC rotation disk models analyzed in Piatti ( 2026 ) (see Table 4 ), we computed, for each kinematic scenario, the corresponding Δ ​ V \Delta V and the anisotropy along each SMC axis using a Monte Carlo approach. The relations between anisotropy and the distance of each cluster from the SMC center for the three disk models are shown in Fig. 5. From the estimated global anisotropy, we obtain for the old disk model A x = 0.61 A_{x}=0.61 , A y = 0.15 A_{y}=0.15 , and A z = 0.23 A_{z}=0.23 . For the model of Piatti ( 2021b ) we find A x = 0.63 A_{x}=0.63 , A y = 0.13 A_{y}=0.13 , and A z = 0.24 A_{z}=0.24 , while for the young disk model we obtain A x = 0.72 A_{x}=0.72 , A y = 0.04 A_{y}=0.04 , and A z = 0.24 A_{z}=0.24 . Figure 7: Distribution of A x A_{x} , A y A_{y} , and A z A_{z} as a function of distance from SMC center ( R 3 ​ D {R_{\mathrm{3D}}} ).Anisotropy was estimated for each of the models studied in (Piatti, 2026 ) . As can be seen, the upper panel refers to the cluster disk model (Piatti, 2021b ) , the middle panel to the old disk model (age > 2 >2 Gyr), and the lower panel to the young disk model (age < 50 <50 Myr). Star clusters pertaining to different substructures (Dias et al. , 2016 ) are drawn with different colors as indicated in the top-right panel (NB=Northern Bridge, MB= Main Body, SB=Southern Bridge, W/B=Wing/Bridge, WH=West Halo). The red line ( A i A_{i} = 0.33, i = x , y , z i=x,y,z ) represents the expected value for an isotropic motion. Symbol size is proportional to N eff N^{\mathrm{eff}} . These values indicate a more dispersed and dynamically perturbed kinematics along the line of sight (the x x -axis) in the young disk scenario. In contrast, the old disk and Piatti ( 2021b ) models show a kinematic behavior with lower dispersion, although the dominant agitation still occurs along the line of sight of the galaxy. The results under the young disk scenario are fully consistent with the findings reported by Piatti ( 2026 ) and Dhanush et al. ( 2025 ) , where star clusters exhibit a gradient in kinematic agitation as their distances from the SMC center increase. On the other hand, when analyzing the results obtained for the Piatti ( 2021b ) and old disk models, we do not find a well-behaved kinematic distribution along the three axes of the galaxy, as might be expected considering that our cluster sample has a mean age of ∼ 3 \sim 3 Gyr. In this context, it is important to examine several key aspects. First, the clusters selected for this study are mostly located in external SMC substructures. Therefore, although the aforementioned models may capture the average agitation of older clusters, the presence of clusters in regions such as the West Halo or the Southern Bridge introduces a level of perturbation so high that their velocities exceed any average rotational behavior. For instance, the cluster L116 in the Southern Bridge exhibits a Δ ​ V \Delta V of 225.73 km s -1 , moving toward the LMC. These parameters likely place it outside any disk orbit, even a perturbed one. Another important aspect is that, although the Piatti ( 2021b ) and old disk models adopt different geometries compared to the young-cluster model, their geometries are still inferred from present-day observations. In other words, they do not fully represent the original disk geometry at the epoch when these clusters formed or when they were affected by the tidal forces of the LMC. Furthermore, we used individual heliocentric distances in the equations used to derive the velocities of each cluster. This provides additional robustness to the determination of Δ ​ V \Delta V and the corresponding anisotropy. It is therefore likely that the parameters of the Piatti ( 2021b ) and old disk models do not accurately reflect the magnitude of the kinematic agitation affecting old clusters located in the outer regions of the SMC. Our work does not aim to settle this debate, but rather to highlight the complexity involved in addressing the kinematics of the SMC. 5 Conclusions The SMC is currently understood to be gravitationally bound to the LMC. Their interaction has left imprints on the SMC’s formation and evolution. Star clusters are fundamental building blocks of any galaxy, so it is reasonable to expect that they may contain valuable information about the SMC dynamical history. In this work, we analyzed 36 star clusters in the SMC to derive their 3D velocities, with the aim of exploring the relationship between the star cluster kinematics and the tidal forces affecting the SMC, particularlly in the SMC’s outer regions. We used proper motions from Gaia DR3, radial velocities taken from the literature, and our derived heliocentric distances. From these data, we derived 3D velocities and their residual velocities. Our main findings can be summarized as follows: • The lower threshold for the residual velocities of star clusters located in outer SMC regions is Δ ​ V \Delta V ≈ \approx 60 km ​ s − 1 \mathrm{km\ s^{-1}} , in very good agreement with the value derived by Piatti ( 2021a ) . Star clusters belonging to the SMC main body mostly show lower Δ ​ V \Delta V values, thus confirming a more tightly disk-like kinematics. • We performed an anisotropy analysis for different SMC disk models (Piatti, 2026 ) , based on recent findings by Dhanush et al. ( 2025 ) linking the kinematics of the SMC with the age of the analyzed stellar sample. Although we found kinematic differences for each disk model, we also found certain regularities in relation to the kinematics and external substructures of the SMC: the West Halo, the Wing/Bridge,the Northern and the Southern Bridges show a preference for larger kinematic dispersion along the x x axis (approximately parallel to the SMC line-of-sight) and perpendicular to the disk, while star clusters in the SMC main body retains some amount of coherent rotation. • Our heliocentric distances (Piatti, 2023 ; Illesca et al. , 2025 ) allowed us to construct a more realistic internal reference frame for the SMC. We thus report a line-of-sight depth for the studied star cluster sample of ∼ \sim 25 kpc. • Building a 3D map of the SMC from the derived positions of each star cluster, combined with residual velocities and membership of star clusters to different SMC’s substructures, enabled us to identify spatial–velocity dispersion correlations. • The subregion-by-subregion analysis leads to an overall kinematic picture of the SMC with kinematically hot outer regions, a pattern consistent with tidal models and recent close-encounter scenarios between both Magellanic Clouds (Rathore et al. , 2024 ) . Acknowledgements. We thank the referee for the thorough reading of the manuscript and timely suggestions to improve it. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). 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Appendix A Collected and derived kinematic parameters of star clusters Table 1: Proper motions and radial velocities of the studied star clusters. Star cluster pmra pmdec N N D D RV Ref. (mas yr -1 ) (mas yr -1 ) (kpc) (km s -1 ) B99 0.78 ± 0.05 0.78\pm 0.05 − 1.21 ± 0.04 -1.21\pm 0.04 1 38.02 159.20 ± 2.60 159.20\pm 2.60 1 B168 0.94 ± 0.09 0.94\pm 0.09 − 1.15 ± 0.09 -1.15\pm 0.09 3 52.72 141.70 ± 4.60 141.70\pm 4.60 2 BS121 0.82 ± 0.06 0.82\pm 0.06 − 1.23 ± 0.04 -1.23\pm 0.04 11 60.26 164.10 ± 4.20 164.10\pm 4.20 3 BS188 1.25 ± 0.08 1.25\pm 0.08 − 1.35 ± 0.07 -1.35\pm 0.07 2 50.35 120.30 ± 3.50 120.30\pm 3.50 2 H86-97 0.80 ± 0.16 0.80\pm 0.16 − 1.26 ± 0.03 -1.26\pm 0.03 2 36.81 120.90 ± 2.80 120.90\pm 2.80 3 HW31 0.57 ± 0.06 0.57\pm 0.06 − 1.23 ± 0.05 -1.23\pm 0.05 2 47.86 125.50 ± 3.40 125.50\pm 3.40 4 HW41 0.79 ± 0.05 0.79\pm 0.05 − 1.35 ± 0.05 -1.35\pm 0.05 1 57.54 143.60 ± 1.60 143.60\pm 1.60 4 HW47 0.56 ± 0.12 0.56\pm 0.12 − 1.18 ± 0.06 -1.18\pm 0.06 5 52.48 122.90 ± 2.40 122.90\pm 2.40 3 HW56 0.99 ± 0.11 0.99\pm 0.11 − 1.27 ± 0.10 -1.27\pm 0.10 2 58.61 157.70 ± 5.40 157.70\pm 5.40 2 HW84 1.22 ± 0.03 1.22\pm 0.03 − 1.23 ± 0.05 -1.23\pm 0.05 3 49.43 135.60 ± 1.50 135.60\pm 1.50 5 HW86 1.19 ± 0.11 1.19\pm 0.11 − 1.28 ± 0.16 -1.28\pm 0.16 2 51.29 143.80 ± 1.60 143.80\pm 1.60 5 L1 0.58 ± 0.01 0.58\pm 0.01 − 1.53 ± 0.01 -1.53\pm 0.01 38 56.90 145.30 ± 1.60 145.30\pm 1.60 6 L4 0.38 ± 0.05 0.38\pm 0.05 − 1.30 ± 0.03 -1.30\pm 0.03 9 56.49 140.20 ± 1.60 140.20\pm 1.60 5 L6 0.50 ± 0.05 0.50\pm 0.05 − 1.33 ± 0.03 -1.33\pm 0.03 5 56.75 142.30 ± 2.80 142.30\pm 2.80 5 L7 0.50 ± 0.03 0.50\pm 0.03 − 1.13 ± 0.03 -1.13\pm 0.03 6 56.49 131.40 ± 2.60 131.40\pm 2.60 5 L8 0.67 ± 0.13 0.67\pm 0.13 − 1.32 ± 0.04 -1.32\pm 0.04 37 60.60 135.10 ± 0.70 135.10\pm 0.70 6 L9 0.43 ± 0.06 0.43\pm 0.06 − 1.12 ± 0.03 -1.12\pm 0.03 4 56.23 157.40 ± 2.10 157.40\pm 2.10 7 L11 0.42 ± 0.03 0.42\pm 0.03 − 1.28 ± 0.04 -1.28\pm 0.04 7 56.49 126.28 ± 1.60 126.28\pm 1.60 8 L12 0.58 ± 0.03 0.58\pm 0.03 − 1.28 ± 0.03 -1.28\pm 0.03 4 69.80 208.00 ± 1.30 208.00\pm 1.30 7 L13 0.46 ± 0.05 0.46\pm 0.05 − 1.14 ± 0.02 -1.14\pm 0.02 2 52.24 109.50 ± 3.10 109.50\pm 3.10 3 L17 0.62 ± 0.03 0.62\pm 0.03 − 1.12 ± 0.03 -1.12\pm 0.03 12 52.24 106.00 ± 1.60 106.00\pm 1.60 5 L19 0.54 ± 0.04 0.54\pm 0.04 − 1.29 ± 0.03 -1.29\pm 0.03 11 57.28 152.70 ± 2.10 152.70\pm 2.10 5 L27 0.76 ± 0.04 0.76\pm 0.04 − 1.46 ± 0.04 -1.46\pm 0.04 14 49.89 175.00 ± 2.60 175.00\pm 2.60 5 L58 0.47 ± 0.08 0.47\pm 0.08 − 1.32 ± 0.07 -1.32\pm 0.07 1 52.48 121.00 ± 9.30 121.00\pm 9.30 6 L68 0.71 ± 0.06 0.71\pm 0.06 − 1.24 ± 0.04 -1.24\pm 0.04 8 62.20 143.70 ± 0.83 143.70\pm 0.83 8 L100 0.81 ± 0.05 0.81\pm 0.05 − 1.17 ± 0.05 -1.17\pm 0.05 2 55.72 145.80 ± 1.40 145.80\pm 1.40 2 L108 1.09 ± 0.03 1.09\pm 0.03 − 1.37 ± 0.03 -1.37\pm 0.03 7 54.20 95.00 ± 4.00 95.00\pm 4.00 5 L110 0.79 ± 0.02 0.79\pm 0.02 − 1.18 ± 0.02 -1.18\pm 0.02 7 54.70 178.80 ± 3.00 178.80\pm 3.00 5 L113 1.33 ± 0.02 1.33\pm 0.02 − 1.22 ± 0.02 -1.22\pm 0.02 17 50.50 171.80 ± 4.50 171.80\pm 4.50 5 L116 1.63 ± 0.09 1.63\pm 0.09 − 1.10 ± 0.07 -1.10\pm 0.07 1 47.86 153.44 ± 2.55 153.44\pm 2.55 8 NGC 339 0.65 ± 0.03 0.65\pm 0.03 − 1.21 ± 0.03 -1.21\pm 0.03 20 57.60 103.30 ± 2.35 103.30\pm 2.35 8 NGC 361 0.83 ± 0.03 0.83\pm 0.03 − 1.28 ± 0.02 -1.28\pm 0.02 28 55.80 161.18 ± 1.24 161.18\pm 1.24 8 NGC 416 0.90 ± 0.03 0.90\pm 0.03 − 1.19 ± 0.04 -1.19\pm 0.04 7 50.35 155.00 ± 0.75 155.00\pm 0.75 9 NGC 419 0.87 ± 0.04 0.87\pm 0.04 − 1.22 ± 0.02 -1.22\pm 0.02 27 56.20 171.48 ± 2.53 171.48\pm 2.53 8 NGC 458 0.89 ± 0.01 0.89\pm 0.01 − 1.23 ± 0.02 -1.23\pm 0.02 24 54.20 149.00 ± 0.85 149.00\pm 0.85 9 OGLE 133 0.67 ± 0.07 0.67\pm 0.07 − 1.25 ± 0.03 -1.25\pm 0.03 3 54.95 145.40 ± 3.20 145.40\pm 3.20 7 Ref.: (1) Parisi et al. ( 2015 ) ; (2) Dias et al. ( 2021 ) ; (3) Dias et al. ( 2022 ) ; (4) De Bortoli et al. ( 2022 ) ; (5) Parisi et al. ( 2009 ) ; (6) Piatti ( 2021b ) ; (7) Parisi et al. ( 2015 ) ; (8) Parisi et al. ( 2022 ) ; (9) Song et al. ( 2021 ) . Table 2: Space velocity components of the star clusters. Star cluster V x V_{x} V y V_{y} V z V_{z} V rot V_{\mathrm{rot}} V rot , 3 ​ D V_{\mathrm{rot,3D}} (km s -1 ) (km s -1 ) (km s -1 ) (km s -1 ) (km s -1 ) B99 − 125.42 ± 11.17 -125.42\pm 11.17 − 31.88 ± 4.12 -31.88\pm 4.12 − 118.54 ± 9.28 -118.54\pm 9.28 129.48 ± 11.16 129.48\pm 11.16 175.72 ± 12.30 175.72\pm 12.30 B168 − 5.55 ± 23.75 -5.55\pm 23.75 − 37.01 ± 9.62 -37.01\pm 9.62 − 67.50 ± 21.07 -67.50\pm 21.07 43.78 ± 11.85 43.78\pm 11.85 81.06 ± 22.03 81.06\pm 22.03 BS121 4.90 ± 17.51 4.90\pm 17.51 5.22 ± 5.73 5.22\pm 5.73 − 26.49 ± 11.25 -26.49\pm 11.25 17.01 ± 10.08 17.01\pm 10.08 33.22 ± 10.74 33.22\pm 10.74 BS188 59.35 ± 48.49 59.35\pm 48.49 − 52.00 ± 12.72 -52.00\pm 12.72 − 41.72 ± 35.67 -41.72\pm 35.67 89.04 ± 28.49 89.04\pm 28.49 105.40 ± 25.36 105.40\pm 25.36 H86-97 − 136.05 ± 32.90 -136.05\pm 32.90 − 62.39 ± 6.75 -62.39\pm 6.75 − 99.12 ± 17.51 -99.12\pm 17.51 150.52 ± 29.56 150.52\pm 29.56 181.43 ± 27.31 181.43\pm 27.31 HW31 − 125.82 ± 19.05 -125.82\pm 19.05 − 42.04 ± 6.53 -42.04\pm 6.53 − 50.92 ± 16.59 -50.92\pm 16.59 132.87 ± 18.69 132.87\pm 18.69 142.99 ± 20.63 142.99\pm 20.63 HW41 − 7.90 ± 14.67 -7.90\pm 14.67 − 6.37 ± 5.04 -6.37\pm 5.04 1.31 ± 12.27 1.31\pm 12.27 16.20 ± 9.01 16.20\pm 9.01 20.34 ± 9.07 20.34\pm 9.07 HW47 − 109.96 ± 31.90 -109.96\pm 31.90 − 40.74 ± 6.79 -40.74\pm 6.79 − 43.91 ± 18.44 -43.91\pm 18.44 118.34 ± 28.45 118.34\pm 28.45 128.06 ± 26.12 128.06\pm 26.12 HW56 40.49 ± 33.42 40.49\pm 33.42 − 4.82 ± 11.78 -4.82\pm 11.78 − 31.37 ± 26.66 -31.37\pm 26.66 46.54 ± 27.44 46.54\pm 27.44 62.86 ± 25.72 62.86\pm 25.72 HW84 43.23 ± 11.23 43.23\pm 11.23 − 47.49 ± 4.55 -47.49\pm 4.55 − 70.53 ± 11.36 -70.53\pm 11.36 65.02 ± 6.58 65.02\pm 6.58 96.45 ± 8.55 96.45\pm 8.55 HW86 58.75 ± 28.59 58.75\pm 28.59 − 33.70 ± 13.15 -33.70\pm 13.15 − 64.93 ± 37.13 -64.93\pm 37.13 70.66 ± 24.19 70.66\pm 24.19 100.82 ± 31.74 100.82\pm 31.74 L1 − 39.17 ± 14.59 -39.17\pm 14.59 17.39 ± 4.46 17.39\pm 4.46 75.78 ± 10.67 75.78\pm 10.67 44.15 ± 10.97 44.15\pm 10.97 88.88 ± 5.07 88.88\pm 5.07 L4 − 127.63 ± 13.73 -127.63\pm 13.73 − 1.58 ± 3.58 -1.58\pm 3.58 38.86 ± 10.19 38.86\pm 10.19 127.70 ± 13.71 127.70\pm 13.71 133.79 ± 14.41 133.79\pm 14.41 L6 − 90.34 ± 19.52 -90.34\pm 19.52 − 0.41 ± 5.94 -0.41\pm 5.94 31.45 ± 15.44 31.45\pm 15.44 90.55 ± 19.46 90.55\pm 19.46 97.46 ± 17.53 97.46\pm 17.53 L7 − 120.02 ± 13.70 -120.02\pm 13.70 − 28.50 ± 4.68 -28.50\pm 4.68 − 16.75 ± 11.30 -16.75\pm 11.30 123.41 ± 13.98 123.41\pm 13.98 124.96 ± 14.82 124.96\pm 14.82 L8 − 22.55 ± 36.77 -22.55\pm 36.77 − 8.21 ± 6.83 -8.21\pm 6.83 24.10 ± 21.25 24.10\pm 21.25 37.46 ± 23.89 37.46\pm 23.89 48.77 ± 25.06 48.77\pm 25.06 L9 − 141.07 ± 17.54 -141.07\pm 17.54 − 2.09 ± 3.98 -2.09\pm 3.98 − 19.77 ± 11.07 -19.77\pm 11.07 141.14 ± 17.51 141.14\pm 17.51 143.06 ± 16.58 143.06\pm 16.58 L11 − 114.83 ± 12.48 -114.83\pm 12.48 − 18.75 ± 4.96 -18.75\pm 4.96 28.90 ± 12.73 28.90\pm 12.73 116.42 ± 12.79 116.42\pm 12.79 120.82 ± 10.81 120.82\pm 10.81 L12 − 5.71 ± 11.97 -5.71\pm 11.97 74.86 ± 3.82 74.86\pm 3.82 40.08 ± 10.40 40.08\pm 10.40 76.02 ± 3.88 76.02\pm 3.88 86.29 ± 7.94 86.29\pm 7.94 L13 − 144.55 ± 13.09 -144.55\pm 13.09 − 53.64 ± 3.73 -53.64\pm 3.73 − 21.08 ± 7.52 -21.08\pm 7.52 154.32 ± 11.92 154.32\pm 11.92 155.98 ± 11.30 155.98\pm 11.30 L17 − 106.79 ± 27.42 -106.79\pm 27.42 − 63.28 ± 7.31 -63.28\pm 7.31 − 44.07 ± 19.80 -44.07\pm 19.80 124.52 ± 26.66 124.52\pm 26.66 132.63 ± 30.96 132.63\pm 30.96 L19 − 82.68 ± 14.56 -82.68\pm 14.56 4.75 ± 4.88 4.75\pm 4.88 8.38 ± 13.14 8.38\pm 13.14 82.99 ± 14.36 82.99\pm 14.36 84.57 ± 13.60 84.57\pm 13.60 L27 − 51.56 ± 14.45 -51.56\pm 14.45 19.87 ± 5.29 19.87\pm 5.29 − 16.74 ± 12.83 -16.74\pm 12.83 56.04 ± 12.20 56.04\pm 12.20 59.60 ± 13.50 59.60\pm 13.50 L58 − 121.29 ± 22.03 -121.29\pm 22.03 − 28.47 ± 11.21 -28.47\pm 11.21 1.15 ± 19.87 1.15\pm 19.87 125.21 ± 21.38 125.21\pm 21.38 126.81 ± 21.21 126.81\pm 21.21 L68 − 14.89 ± 19.37 -14.89\pm 19.37 − 7.21 ± 4.99 -7.21\pm 4.99 − 2.26 ± 14.21 -2.26\pm 14.21 22.41 ± 13.11 22.41\pm 13.11 26.70 ± 12.97 26.70\pm 12.97 L100 − 20.69 ± 12.90 -20.69\pm 12.90 − 23.89 ± 4.88 -23.89\pm 4.88 − 51.16 ± 12.05 -51.16\pm 12.05 33.38 ± 8.68 33.38\pm 8.68 61.51 ± 12.96 61.51\pm 12.96 L108 45.88 ± 17.29 45.88\pm 17.29 − 63.48 ± 6.20 -63.48\pm 6.20 − 5.23 ± 14.39 -5.23\pm 14.39 80.04 ± 8.16 80.04\pm 8.16 81.49 ± 8.13 81.49\pm 8.13 L110 − 26.78 ± 8.89 -26.78\pm 8.89 6.66 ± 4.08 6.66\pm 4.08 − 67.68 ± 8.52 -67.68\pm 8.52 28.16 ± 8.00 28.16\pm 8.00 73.58 ± 9.85 73.58\pm 9.85 L113 87.17 ± 13.24 87.17\pm 13.24 − 16.73 ± 4.99 -16.73\pm 4.99 − 91.18 ± 9.71 -91.18\pm 9.71 89.01 ± 12.55 89.01\pm 12.55 128.32 ± 4.82 128.32\pm 4.82 L116 169.15 ± 41.16 169.15\pm 41.16 − 57.11 ± 7.06 -57.11\pm 7.06 − 157.03 ± 27.13 -157.03\pm 27.13 179.25 ± 38.53 179.25\pm 38.53 241.56 ± 25.70 241.56\pm 25.70 NGC 339 − 59.48 ± 14.72 -59.48\pm 14.72 − 52.02 ± 4.88 -52.02\pm 4.88 − 10.85 ± 13.48 -10.85\pm 13.48 79.44 ± 13.16 79.44\pm 13.16 81.13 ± 14.21 81.13\pm 14.21 NGC 361 − 11.44 ± 11.35 -11.44\pm 11.35 − 0.18 ± 3.56 -0.18\pm 3.56 − 31.16 ± 8.67 -31.16\pm 8.67 14.04 ± 8.67 14.04\pm 8.67 34.89 ± 10.09 34.89\pm 10.09 NGC 416 − 31.07 ± 8.23 -31.07\pm 8.23 − 23.59 ± 3.46 -23.59\pm 3.46 − 77.48 ± 8.94 -77.48\pm 8.94 39.41 ± 6.98 39.41\pm 6.98 87.12 ± 9.75 87.12\pm 9.75 NGC 419 − 4.12 ± 10.27 -4.12\pm 10.27 3.62 ± 3.57 3.62\pm 3.57 − 52.34 ± 7.18 -52.34\pm 7.18 10.54 ± 6.10 10.54\pm 6.10 53.75 ± 7.05 53.75\pm 7.05 NGC 458 − 8.68 ± 11.80 -8.68\pm 11.80 − 20.57 ± 3.60 -20.57\pm 3.60 − 49.36 ± 9.41 -49.36\pm 9.41 24.55 ± 6.93 24.55\pm 6.93 55.23 ± 11.19 55.23\pm 11.19 OGLE 133 − 58.82 ± 17.77 -58.82\pm 17.77 − 14.26 ± 4.68 -14.26\pm 4.68 − 25.96 ± 9.26 -25.96\pm 9.26 61.01 ± 16.70 61.01\pm 16.70 67.43 ± 14.64 67.43\pm 14.64 Table 3: Residual velocity components of star cluster. Star cluster Δ ​ V x \Delta V_{x} Δ ​ V y \Delta V_{y} Δ ​ V z \Delta V_{z} Δ ​ V \Delta V R plane R_{\mathrm{plane}} R 3 ​ D R_{\mathrm{3D}} Projected (km s -1 ) (km s -1 ) (km s -1 ) (km s -1 ) (kpc) (kpc) region a B99 126.96 28.99 115.20 174.01 ± 10.17 174.01\pm 10.17 22.85 24.42 MB B168 15.23 42.29 62.64 82.42 ± 17.82 82.42\pm 17.82 9.84 10.19 NB BS121 1.03 9.20 23.43 31.59 ± 10.58 31.59\pm 10.58 2.13 2.38 MB BS188 49.76 48.39 37.79 95.44 ± 31.67 95.44\pm 31.67 11.62 12.44 NB H86-97 134.72 64.34 97.35 180.74 ± 26.56 180.74\pm 26.56 24.03 25.63 MB HW31 125.34 43.46 40.36 140.34 ± 17.94 140.34\pm 17.94 13.35 14.61 SB HW41 11.68 13.11 9.69 26.80 ± 9.70 26.80\pm 9.70 5.09 5.14 MB HW47 110.25 39.83 29.78 123.27 ± 28.54 123.27\pm 28.54 8.74 10.11 SB HW56 34.88 13.06 22.94 57.13 ± 24.18 57.13\pm 24.18 4.48 4.48 NB HW84 32.24 45.49 66.44 87.70 ± 9.72 87.70\pm 9.72 12.67 13.47 NB HW86 54.62 23.93 52.51 89.02 ± 29.65 89.02\pm 29.65 9.90 11.63 B L1 28.27 18.87 79.87 88.01 ± 10.73 88.01\pm 10.73 6.33 6.55 WH L4 119.26 4.32 42.21 126.86 ± 13.48 126.86\pm 13.48 5.97 6.39 WH L6 82.13 2.69 34.65 91.35 ± 18.97 91.35\pm 18.97 5.72 6.10 WH L7 112.23 31.82 13.42 117.89 ± 13.28 117.89\pm 13.28 5.88 6.31 WH L8 15.40 2.45 28.75 49.36 ± 22.73 49.36\pm 22.73 2.81 2.81 WH L9 133.92 7.99 15.36 135.59 ± 17.39 135.59\pm 17.39 6.01 6.59 WH L11 108.53 13.38 33.21 115.06 ± 12.43 115.06\pm 12.43 6.02 6.20 WH L12 2.64 75.63 43.16 88.43 ± 6.18 88.43\pm 6.18 7.22 7.68 WH L13 138.07 54.11 18.71 150.06 ± 12.31 150.06\pm 12.31 9.69 10.31 WH L17 101.65 66.98 41.19 131.37 ± 22.46 131.37\pm 22.46 9.55 10.28 MB L19 78.05 1.48 13.55 80.54 ± 14.49 80.54\pm 14.49 4.79 5.36 MB L27 48.36 22.18 14.88 57.27 ± 13.16 57.27\pm 13.16 11.87 12.57 MB L58 120.95 29.49 13.71 127.65 ± 21.50 127.65\pm 21.50 8.88 10.05 SB L68 15.73 4.80 9.96 27.95 ± 13.25 27.95\pm 13.25 0.39 1.43 SB L100 28.79 23.59 48.21 62.77 ± 11.37 62.77\pm 11.37 6.65 7.03 NB L108 36.52 59.31 1.20 72.61 ± 10.35 72.61\pm 10.35 8.14 8.74 NB L110 33.63 15.73 60.58 71.57 ± 8.53 71.57\pm 8.53 7.33 8.28 W/B L113 82.00 6.08 80.17 115.71 ± 11.72 115.71\pm 11.72 10.79 12.47 W/B L116 168.16 53.20 137.76 225.73 ± 34.71 225.73\pm 34.71 11.94 15.40 SB NGC 339 59.01 53.42 2.81 81.51 ± 11.40 81.51\pm 11.40 3.98 5.12 SB NGC 361 15.57 6.61 24.47 32.09 ± 8.92 32.09\pm 8.92 6.63 6.79 MB NGC 416 36.30 23.62 75.58 87.45 ± 8.67 87.45\pm 8.67 11.42 12.14 MB NGC 419 8.42 8.55 48.48 51.30 ± 7.17 51.30\pm 7.17 5.80 6.35 MB NGC 458 16.12 25.08 45.45 55.98 ± 8.64 55.98\pm 8.64 8.16 8.49 NB OGLE 133 63.00 16.71 23.85 70.39 ± 16.24 70.39\pm 16.24 7.16 7.54 MB a NB = Northern Bridge, W/B = Wing/bridge, SB = Southern Bridge, MB = Main Body, and WH = West Halo (Dias et al. 2016 ) Table 4: SMC rotation disk models. Parameter Cluster disk Young disk (age < < 50 Myr) Old disk (age > > 2 Gyr) SMC center RA ( ∘ ) 13.30 ± 0.10 13.30\pm 0.10 13.05 13.05 13.05 13.05 SMC center Dec ( ∘ ) − 72.85 ± 0.10 -72.85\pm 0.10 − 72.83 -72.83 − 72.83 -72.83 SMC center distance (kpc) 59.0 ± 1.5 59.0\pm 1.5 62.44 ± 0.47 62.44\pm 0.47 62.44 ± 0.47 62.44\pm 0.47 SMC center pmra \mathrm{pmra} (mas yr -1 ) 0.75 ± 0.10 0.75\pm 0.10 − 0.743 ± 0.027 -0.743\pm 0.027 − 0.743 ± 0.027 -0.743\pm 0.027 SMC center pmdec \mathrm{pmdec} (mas yr -1 ) − 1.26 ± 0.05 -1.26\pm 0.05 − 1.233 ± 0.012 -1.233\pm 0.012 − 1.233 ± 0.012 -1.233\pm 0.012 SMC center systemic velocity (km s -1 ) 150.0 ± 2.0 150.0\pm 2.0 145.6 ± 0.1 145.6\pm 0.1 145.6 ± 0.1 145.6\pm 0.1 SMC disk inclination ( ∘ ) 70.0 ± 10.0 70.0\pm 10.0 81.9 ± 0.7 81.9\pm 0.7 58.4 ± 1.4 58.4\pm 1.4 SMC disk position angle LON ( ∘ ) 200.0 ± 30.0 200.0\pm 30.0 185.7 ± 3.7 185.7\pm 3.7 207.6 ± 2.3 207.6\pm 2.3 SMC disk rotation velocity (km s -1 ) 25.0 ± 5.0 25.0\pm 5.0 10.0 ± 5.0 10.0\pm 5.0 10.0 ± 5.0 10.0\pm 5.0 BETA