tokens: at 200 pc. We retain the functional form of the Bovy model’s bulge, ρ( r ) = ρ s ( r / r 1 ) −α exp [−( r / r c ) 2 ], but with revised parameters α = 2.0, r c = 1.0 kpc, ρ s = 6.58 × 10 8 M ⊙ kpc −3 , and r 1 = 1 kpc. This yields a total bulge mass of 7.33 × 10 9 M ⊙ . The model roughly matches the observed circular velocity data collected in Sofue ( 2013 ), for both large and small radii. It does not match this data’s prominent peak at R ∼ 500 pc and subsequent decline out to 2.5 kpc, but Chemin, Renaud Soubiran ( 2015 ) argue that this feature is likely an artefact of perturbations from the Galactic bar. The axisymmetry of our potential is clearly an approximation, as orbits on scales of a few kpc are strongly affected by the bar. However, at smaller radii, the observed physical components are close to axisymmetric and the potential contours generated in physical models such as Bissantz, Englmaier Gerhard ( 2003 ) are more rounded. Thus, axisymmetric models continue to be used to study orbits near the Galactic centre (Molinari et al. 2011 ; Kruijssen et al. 2015 ; Perera et al. 2019 ), and we leave non-axisymmetric refinements to future work. We integrate orbits in this potential using the package galpy 3 (Bovy 2015 ). We assume Sgr A* lies fixed at the GC. We adopt a distance to Sgr A* of 8.175 kpc from Gravity Collaboration et al. ( 2019 ), and do not consider in the computation its small observed error. We first define a modified Galactic coordinate system ( l ′ , b ′ ), which uses a slight rotation around the Sun’s position to place Sgr A* exactly at l ′ = 0, b ′ = 0, then shift by the solar distance to obtain Galactocentric coordinates. We choose a solar motion in the Galactocentric frame of v X = −11.0 km s −1 , v Y = 248.5 km s −1 , v Z = 7.8 km s −1 that produces the observed PM of Sgr A*, μ α cos δ = −3.156 mas yr −1 and μ δ = −5.585 mas yr −1 (Reid Brunthaler 2020 ). We use a Galactocentric convention where X points from Sgr A* to the Sun. Fig. B1 shows orbits of the two clusters with a series of initial distances, using our best values of the PM. Orbits starting well on the near side of the GC are highly eccentric. The eccentricity first decreases with the starting distance, then increases again as the orbits become almost radial near the GC. About 20 pc further out, the orbits become retrograde. We note none of the orbits are circular. As noted in Stolte et al. ( 2008 ), circular orbits require , which implies X = ( v X / v Y ) Y , radius R = [1 + ( v X / v Y ) 2 ] 1/2 Y , and circular velocity . For Quintuplet, this implies a velocity V c = 177 ± 4 km s −1 at R = 48 ± 1 pc, where the uncertainties incorporate our estimated PM and RV errors. For Arches, it implies a velocity V c = 168 ± 7 km s −1 at R = 40 ± 2 pc. Both of these circular velocities are comfortably above those implied by our potential, which are 108 and 98 km s −1 , respectively, at these radii. They also lie well above the scatter of different observational
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