tokens: the eccentric disc at the centre of M31. In order to test whether the finding that σ z /σ R > 1 depends on our assumption that the velocity ellipsoid is aligned on cylindrical coordinates, we have repeated our analysis assuming that the velocity ellipsoid is aligned on spherical coordinates (see section 2.4 of Cappellari 2020 ). We found that models with σ θ /σ r > 1 are clearly favoured, which on the plane z = 0 corresponds to σ z /σ R > 1. Thus, the alignment of the velocity ellipsoid does not affect the results discussed here. There are two questions in relation to our finding that σ z /σ R > 1. The first is why are such values favoured by our models? Comparison of Fig. 9 with Fig. 13 shows that the reason is that a small value of b (i.e. a large σ z /σ R ) is needed to reproduce the drop in the observed μ obs near the centre (| l | ≲ ), which is present both in the APOGEE data and the SiO maser data (see middle row in Fig. 9 ). In Fig. 9 , which shows model 3 with b = 0.45, the drop is well reproduced, while in Fig. 13 , which shows the same model but for b = 1, the drop is not well reproduced. The ‘NSD only’ models favour a slightly larger value of b compared to the ‘NSD+NSC’ models because the absence of the NSC component in the middle lowers the velocity dispersion in the central regions compared to the outer parts. The ring models also favour a larger value of b (consistent with b ∼ 1, see Fig. A2 ) because the density is essentially zero for , and therefore those regions do not contribute to the integrals in equations ( 11 ) and ( 12 ). Assuming that the drop in the data is real (which ought to be confirmed with better data), this suggests that indeed b 1, or that our assumed tracer density population is not representative of the population from which the kinematics are drawn. We note that the observed kinematics directly constrain only the v R and v φ components of velocity: our constraints on the v z component come from the integral ( 11 ) of ρ and Φ along z . If the potential or density were much flatter than we have assumed, then the σ z given by ( 11 ) would increase and we could fit the observed kinematics with larger values of b . It is currently unclear whether such a strong flattening would be detectable given the extreme and strong differential extinction. Figure 13. Same as Fig. 9 , but for a model that is identical to model 3 except that b = 1. Note that the drop in second moment velocities at | l | ≲ is reproduced less well than in the model shown in Fig. 9 . Assuming that σ z /σ R > 1 then the second question is how would stars get such large vertical oscillations? In the solar neighbourhood stars are formed from gas clouds that move on almost closed orbits, beginning their lives with random velocities of the order of a few . There are a number of dynamical processes that inevitably cause these random velocities to increase over time (see Sellwood 2014 for a recent review). Each of these heating mechanisms has a different
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