tokens: OLR, corotation (CR), as well as the 4:1 and 6:1 resonances from higher order Fourier modes of the bar potential. We define a quasi-isothermal ( Binney 2010 ) unperturbed distribution function in action space to represent a young and cold ( σ R = 30 km s −1 ), as well as a hotter and older ( σ R = 50 km s −1 ) disc stellar population. Then we use a first canonical transformation to move to the slow and fast action-angle variables, which allows us to determine the regions of phase-space where orbits are trapped at resonances. In the vicinity of these resonant regions, and after averaging over the fast angles, the dynamics of the slow variables is that of a pendulum, for which we make a new canonical transformation to pendulum action-angle variables. We then phase-mix the original unperturbed distribution function over the pendulum angles to obtain the perturbed distribution function in the different resonant zones. We can then compute the perturbed distribution function at different positions and velocities, making use of the epicyclic approximation. We refer the reader to Monari et al. (2017a , 2019) for more details of the method. We note that it gives a very good representation of orbits trapped at resonances, but is not the best possible description of the linear deformation zones. In Fig. 2 , we present the variations in density of the v φ − R plane resulting from the response of both the cold (young) and hot (old) stellar populations to the bar potential of Portail et al. (2017) . As expected, the features are marked much more clearly for the dynamically young population. Interestingly, this is particularly true for the 4:1 resonance, which corresponds to the Sirius ridge, which is also seen very prominently in the data as a young feature in a large range of radii. Moreover, the density contrast of the ridge associated with corotation is most prominent for young stars around R ∼ 7 kpc, which is also where the Hercules ridge appears most clearly both in global density and as a young feature in the data. The ‘hat’ feature, which corresponds to the bar’s OLR in this model, appears prominently for the young and cold population at R ∼ 9 kpc, where it also peaks in differential age in the data. This analytical model thus provides a remarkable qualitative match to the trend uncovered in the data. Fig. 2. Normalised perturbed distribution function in the presence of the large bar perturbation of Portail et al. (2017) . Top panel : for an unperturbed distribution function representing a young and cold disc stellar population with σ R = 30 km s −1 at the Galactocentric solar radius R 0 . Bottom panel : for an older and hotter disc population with σ R = 50 km s −1 at the Galactocentric solar radius R 0 . 4. Arcturus and moving groups of the thick disc We note in Sect. 2 that the ridges from Arcturus down to Ella all have stellar ages that appear slightly but systematically younger than the surrounding population, which, combined with the regularity
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