The bar pattern speed of the Large Magellanic Cloud

In the section Kinematic analysis of the Large Magellanic Cloud we presented the study of the internal kinematics of the different clean Large Magellanic Cloud (LMC) samples of Gaia DR3 data. As a follow-up work, in Jiménez-Arranz+24a we used the internal kinematics maps of the LMC complete sample to determine the LMC bar pattern speed.

In the context of galactic dynamics, a barred spiral galaxy is characterized by the presence of a linear, elongated structure known as a bar, which extends from the galactic center. The bar, composed of stars, rotates as part of the overall galactic rotation. The bar pattern speed refers to the angular rotation speed Ω𝑝 of this central bar around the galactic nucleus. It is a fundamental parameter in the dynamics of galaxies from which the principal bar-disc resonances can be identified, and the structure of stellar orbits can be studied in a given gravitational potential.

As observed in Fig. 1, it is trivial to determine the bar pattern speed of a simulated galaxy using time-centred finite-differences from consecutive snapshots, i.e. using the explicit temporal evolution of the system. However, since observations only provide a static picture of the entire process, extracting the bar pattern speed from observational data becomes a considerably more intricate task. It requires sophisticated techniques like kinematic analyses and dynamical modeling.

Historically, bar pattern speeds have been inferred exclusively by means of the Tremaine-Weinberg method (Tremaine & Weinberg 1984, hereafter TW) using line-of-sight velocities. Its direct application makes use of integrals of kinematics and positions of a tracer that should obey the continuity equation, like stars.

Alternatively, Dehnen et al. (2023) proposed a new method for determining bar pattern speeds in single time snapshot numerical simulations. This involves measuring the Fourier amplitudes of particle positions and velocities within the bar region. In another work, Gaia Collaboration et al. (2023) proposed an indirect measurement of the bar angular speed by fitting a bisymmetric model to the tangential velocities (hereafter BV method) to get the bar phase angle and corotation radius (distance from the galactic center where the rotational speed of the stars matches the bar pattern speed), and from here inffer the bar pattern speed. 

Unfortunately, these last two methods have limited applicability due to their reliance on the availability of objects with individually measured planar (tangential and radial) velocities, which are rare. Only the Milky Way (MW) and the LMC (see Jiménez-Arranz+23a) offer this possibility. Using this to our advantage, we will use the three methods (the TW method using line-of-sight velocities, the Dehnen method, and the BV model), along with a variation of the TW method, to use the in-plane velocities, to infer the LMC bar pattern speed.

Testing the methods with simulations

1. Simulation of a MW-mass galaxy (B5, from Roca-Fàbrega+13), with no external perturbations
2. Simulation of a LMC-like system interacting with a SMC-mass and MW-mass systems  (K6, from Jiménez-Arranz+, in prep.). 

The Dehnen and BV methods both successfully recover (✅) the bar pattern speed when validated for both simulations. Values are summarised in Table 1.

However, a problem (❌) is found when utilizing both line-of-sight (LTW) and in-plane velocities (IPTW) to analyze the TW method. First, let's focus on the in-plane version of the TW method, IPTW. We need to know the velocity field in the 𝑥 − 𝑦 plane, being the 𝑧−axis perpendicular to the disc. Since the continuity assumption is independent on the choice of the Cartesian frame, this implies that we can choose arbitrarily the orientation of the reference 𝑥 − 𝑦 plane by rotating it around the 𝑧−axis, and measure the TW integrals (see Eq. 4 of Jiménez-Arranz+24a) at various orientations. Only astrometric data can make such analysis possible, unlike spectroscopic data. This is thus a good opportunity for us to assess for the first time the effect of the viewing angle of the bar in the disc plane on the TW integrals, and on the LMC bar pattern speed.

One should expect that the recovered pattern speed should be insensitive to the arbitrary 𝑥 − 𝑦 reference frame. However, that is not the case when tested in the B5 and KRATOS simulations. Figure 3 shows the variation of the pattern speed Ω𝑝 as as function of the reference frame orientation Δ𝜙 for the B5 (left) and KRATOS (right) simulation.

Figure 3. Variation of Ω𝑝 as as function of the reference frame orientation Δ𝜙 for the B5 (left) and KRATOS (right) simulations for the IPTW method. Top left: Surface density where the bar region (obtained using Dehnen method) is indicated by the green dashed circle. A scatter plot representing the value of ⟨𝑥⟩ for each slice in 𝑦 is overlapped. The scatter plot varies its colour as function of the distance to the centre in the 𝑦-axis, being the red points close to the centre and the green close to the external parts of the galaxy. Top right: Scatter plot of the TW integrals ⟨𝑥⟩ and ⟨𝑣𝑦⟩ for the different slices in the 𝑦-axis. The colour of the scatter plot is the same in both left and right panels. The points with a black circle are inside of the bar region and, therefore, the only considered for fitting Ω𝑝.

The same problem occurs (❌) for the line-of-sight version of the TW method. We simulate a galaxy observation by projecting the particles from each simulation onto a galactic plane with arbitrary inclination and position angle of the semi-major axis of the receding half. For the result described here, we adopted a disc projected with PA = 60º, and inclinations of 𝑖 = 25º, 45º and 75º. Assuming a mock disc distance of 10 Mpc, which is well suited to mock galaxies on which the LTW method can been applied, the maps of the projected density and line-of-sight velocity have 512×512 pixels sampled at 1′′ (∼ 50 pc/pixel). We chose the 𝑥−axis aligned with the line of nodes, so that the reference of the azimuthal angle 𝜙 = 0º is along the semi-major axis of receding disc half.

Again, we investigate the impact of the variation of the orientation of the Cartesian frame on the results, by rotating the reference 𝑥 and 𝑦 axes around the 𝑧−axis in the simulation. The rotation of the Cartesian frame before projection on the sky plane allows the TW integrals to view the bar and spiral perturbations through various angles. Then, new density maps and line-of-sight kinematics and Ω𝑝 can be inferred.

Figure 4 (upper panel) shows the resulting bar pattern speed as a function of the frame orientation Δ𝜙 for the case 𝑖 = 45º only (red open symbols) and the B5 simulation. The quoted uncertainties correspond to the 1𝜎 error of the covariance matrix of the fitting. We highlight the frame orientations parallel and perpendicular to the bar major axis as orange and navy vertical lines, respectively. Results for the KRATOS simulation are shown in the lower panel of Fig. 2 for the three assumed inclinations.

Qualitatively, the IPTW and LTW methods show similar trends: the agreement with the ground-truth bar pattern speed is rarely observed, no symmetry with respect to the bar major axis is found, and stronger discrepancies are near the positions of the major and minor axes of the bar. For the IPTW, the B5 simulation (upper panel) shows that Δ𝜙 ∼ 135º is also a location of stronger disagreement, which was not observed for the LTW method. Since the IPTW method works directly with coordinates and velocities in the Cartesian frame of the disc, no variation with inclination needs to be evaluated here. It is interesting to note that the LTW pattern speeds with better agreement with the IPTW method are for the intermediate inclination of 45º (lower panel for the KRATOS simulation). It is important to remind that the Dehnen method does not show such systematic variation with Δ𝜙, as its results are invariant with the frame orientation.

Finally, we can estimate the incidence of finding a bar pattern speed consistent with the ground-truth value for both the LTW and IPTW methods, with the two simulations. We define this likelihood as the number of frame orientations where the measured and real Ω𝑝 agree within the quoted (1𝜎) uncertainties on measured and ground truth values. For the B5 simulation, the IPTW and LTW methods give a correct Ω𝑝 in 5% and 8% of the cases only. For the KRATOS simulation, the incidence is 37% (IPTW case), 57% (LTW case at 𝑖 = 25 ◦ ), 48% (LTW case at 𝑖 = 45º) and 42% (LTW case at 𝑖 = 75º). Larger inclinations are thus less prone to the LTW method. More generally, our two sets of simulations show it is highly unlikely to find a consistent Ω𝑝 by means of the TW method. It is also hard to reconcile the strong variations with Δ𝜙 seen here, i.e the bar orientation with respect to the disc 𝑥−axis, with the wide range of “allowed” orientations quoted in other studies (e.g. Zou et al. 2019).

Measuring the LMC bar pattern speed

The LMC stars we use in this study are those in the NN complete sample of Jiménez-Arranz+23a. For each star, we apply the coordinate transformation detailed in Jiménez-Arranz+23a to express the deprojected positions and velocities in the in-plane coordinate system of the LMC. 

Figure 5 shows the results of the application of the Dehnen method to the LMC. The left, middle and right panels show the surface density, the median Galactocentric radial velocity, and residual of the median tangential velocity, respectively. The method was not able to find the bar region [𝑅0, 𝑅1] on its own, probably due to the fact that the quadrupole is not perfectly symmetric, or that the contrast of the bar region with respect to the disc is not as clear as in simulations because of the presence of dust lanes and spiral arms at low radius. By looking at the amplitude Σ2 and angle 𝜙2 of the 𝑚 = 2 Fourier coefficient (see Figure 7 of the main reference), we establish that the bar region is [𝑅0, 𝑅1] = [0.75, 2.3] kpc. Within this region, the Dehnen method gives a value of Ω𝑝 = −1.0 ± 0.5 km s −1 kpc −1, thus corresponding to an almost non-rotating stellar bar, seemingly in counter-rotation.

In Fig. 8 (left) we show the stellar line-of-sight velocity field corrected for the systemic motion of the LMC NN complete 𝑉los sub-sample, which contains 30 749 stars. These are predominantly the AGB stars from Gaia Collaboration et al. (2021b). This is in good agreement with the line-of-sight velocity field traced by carbon stars (van der Marel et al. 2002). We apply the LTW method to this line-of-sight velocity map with pseudo-slits parallel to the line-of-nodes. In Fig. 8 (right) we show the linear fit to the LTW integrals, yielding a bar pattern speed of Ω𝑝 = 30.4 ± 1.3 km s −1 kpc −1, using an inclination of 𝑖 = 34º.

Figure 8. Left: Stellar line-of-sight velocity field of the LMC NN complete 𝑉𝑙𝑜𝑠 sub-sample, corrected from the systemic motions. Data are from Gaia RVS (Katz et al. 2022; Jiménez-Arranz et al. 2023). Right: Result of the TW method applied to the line-of-sight velocity field of the LMC from Fig. 8. Only points within the radius 𝑅1 defined by the Dehnen method are shown. The red dashed line shows the result of the linear fit to the points, Ω 𝑝 sin 𝑖, with Ω 𝑝 = 30.4 ± 1.3 km s −1 kpc −1 .

Figure 9 shows the results of the IPTW method applied to the LMC Cartesian velocity fields. We recover as many values as adopted orientations Δ𝜙 of the Cartesian frame in the LMC plane. Interestingly, a good agreement is seen between the LTW Ω𝑝 (open red dot at Δ𝜙 = 0º) and the IPTW Ω𝑝 inferred at this orientation. However, and unsurprisingly, the estimated values of the IPTW method display a strong variation with the frame orientation. A wide range of possibilities is found for the LMC Ω𝑝, from 0 to 55 km s −1 kpc −1. Note also the clear correlation between the bar major and minor axis with the orientations where the shape of the Ω 𝑝 curve vary significantly. The median of all IPTW values seen in this graph is 23 ± 12 km s −1 kpc −1, adopting here the mean absolute deviation as the uncertainty.

Finally, in Fig. 10 we show the results of the BV method applied to the LMC tangential velocity map (right panel of Fig. 5). The LMC rotation curve from the NN complete sample (upper panel, light grey) is very similar to the 0th order Fourier component of the BV model (black solid line). The amplitude of the LMC bar perturbation is stronger at 𝑅 = 0.75 kpc (blue line). The orange dotted line showing the scatter in the residual tangential velocity is often larger than the bisymmetric mode. It thus shows that the bar is not the only perturber in the LMC disc, but this does not prevent the bisymmetry from being detected efficiently by the method.

A roughly constant value of 𝜙2,kin ≃ 15 − 20º is seen out to 𝑅 = 2 kpc, followed with a smooth decrease out to 𝑅 = 3.95 kpc, as evidence of the impact of arms in the kinematics  even in the bar region. This radius is the location from where the amplitude 𝑉2 starts to increase. At this radius, 𝜙2,kin changes by ∼ 100º to recover a constant value comparable to the bar phase angle at low radius.

Following prescriptions from the numerical modelling, we adopt the radius just after the sharp transition  of phase angle as the bar corotation radius, placing the LMC bar corotation at 𝑅𝑐 = 4.20 ± 0.25 kpc. Relative to the angular velocity curve Ω (solid line in the bottom panel of Fig. 10), it corresponds to a LMC bar pattern speed of Ω𝑝 = 18.5 +1.2 −1.1 km s−1 kpc−1.

Summary & Conclusions

Table 2 is a summary of the LMC bar properties obtained with the different methods in the previous section. At first glance, it is difficult to conclude the pattern speed of the stellar bar of the LMC given the large range of values.

A second important result is that the TW method is extremely sensitive to the orientation of the x-y frame, and therefore to the way the integrals view the bar perturbation in the disc. The origin of the strong variations with the bar angle, and of the large discrepancy with true values, may be the impact of patterns other than the bar in the TW integrals, like spiral arms in the N-body simulation and the LMC data. 

Thus, without agreement among the trends found with the various simulations used in the previous works and our present study, and without an identifiable region of bar orientation where ground-truth and measured pattern speeds agree within simulations, the individual pattern speeds found by the TW method in Fig. 9 cannot be representative of the real LMC bar pattern speed. The agreement of Ω𝑝 found by the LTW method with the value found by the IPTW method for the LMC also indicates that the pattern speed of bars measured by means of the LTW method may likely be only representative of any value stemming from random frame orientations fixed by the position angle of the major axis of discs on the sky plane, but not of a global bar angular frequency.

The pattern speed obtained with the Dehnen method, when applied to the LMC data, it surprisingly results in a bar with null rotation, perhaps slightly counter-rotating with respect to the disc. Peculiar bars with such property exist in numerical simulations (e.g. Collier & Madigan 2023). However, this result does not come without issues. An almost non-rotating LMC bar would indeed not show any corotation within the disc since such Ω𝑝 should never cross the Ω curve. It is not an easy task to imagine how the orbits and the disc structure would respond to this peculiar circumstance. An absence of corotation could allow the bar to increase its length and strength out to the disc outskirts, that is, make the orbits of stars and the LMC stellar gravitational potential very elongated throughout the whole LMC disc. Indeed, nothing could prevent it here from growing significantly owing to the absence of corotation and the expected destructive orbits perpendicular to the bar beyond corotation. We think that the method may be sensitive to dust extinction and completeness effects in the inner LMC region, perhaps more strongly than the other methods.

Assuming that the corotation radius 𝑅𝑐 = 4.20 ± 0.25 kpc measured by the BV model is more representative of the bar properties, it corresponds to a pattern speed of 18.5 +1.2 −1.1 km s−1 kpc-1 . The LTW pattern speed of 30.4 ± 1.3 km s-1 kpc−1 would thus be discrepant by 64% from the one inferred here. When compared to its radius of 2.3 kpc, the LMC stellar bar has 𝑅𝑐 /𝑅1 = 1.8±0.1, thus corresponding to a slow bar, according to numerical methods (Athanassoula 1992). Finally, if we assume that the pattern speed has to be estimated using a velocity curve tracing more closely the circular velocity (the rotation curve of the younger stellar populations in Jiménez-Arranz+23a) than the tangential velocity of the whole sample dominated by older stars (upper panel of Fig. 11), then 𝑅 𝑐 = 4.2 kpc would translate into Ω𝑝 = 20.9 ± 1.1 km s −1 kpc −1, which still compares well with the value found for the whole sample.

For a more extensive discussion of the results obtained using the TW we refer the reader to the full paper.