tokens: has a small impact on the vertical and radial density distributions of the stellar discs. These two last parameters will be constrained with observations and not by the dynamical self-consistency. Thus, these two parameters and are currently fixed before they can be more tightly constrained by kinematical observations. Then we assume that and we fixed for each stellar disc the velocity dispersion ratio σ U / σ W at the solar position, respectively 2.1 for the thin discs ( Gomez et al. 1997 ; Robin et al. 2017 ), 1.47 for the young thick disc, and 1.38 for the old thick disc ( Soubiran et al. 2003 ; Robin et al. 2017 ). Figure 1 (left) shows the vertical density distribution of the old thin disc (scale height around 230 pc) at three Galactic radii 4, 8, and 12 kpc, and Fig. 2 (left) its radial distribution at four z = 0, 200, 400, and 600 pc. The adjustment is satisfying. This is expected at R = 8 kpc since the BGM2014 version is built to be dynamically consistent up to z = 1 kpc. The fact that the fit is also satisfying at other galactic radii confirms that the choice of Einasto profiles in the BGM2014 version for the disc density was valuable with respect to the dynamics. For the other discs, the agreement between the BGM2014 and the new version is correct and is better for the thinner disc components with smaller vertical velocity dispersions. For the old thick disc, the vertical and radial density distributions (corresponding to a 795 pc scale height in the case of a s e c h 2 density law) are shown in Figs. 1 and 2 (right). The agreement between both models is still correct. We note that at R = 8 kpc and below z = 1 kpc the fit is excellent, illustrating the exactness of the dynamical consistency of the BGM2014 version. However, at larger height above the Galactic plane, both models differ, and at two scale-heights, or 1.6 kpc height, the BGM2014 density for the old thick disc is 17 % too small. Concerning the density scale length, the agreement between the two models is nearly perfect for the old thin disc. For the old thick disc, the agreement is obtained only for z lower than 1 kpc. We note that the flare present in the outskirts of the Galaxy could be naturally modelled by adjusting the vertical velocity dispersions at large Galactic radii. Table 1 gives the relative difference in percentage between the new and the old models at different heights z and at R = 8 kpc, for the old thin and old thick discs. The differences increase with z ; they become significant at three scale heights for the old thin disc and beyond 1 kpc for the old thick disc. At high z , the differences become of the order of the expected density of the stellar halo, and this illustrates the necessity of using dynamically self consistent distribution functions to properly identify and separate the Galactic stellar components. However, only observations will tell us whether our choice of nearly isothermal modelling of stellar discs is the correct one since small
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O B-CelestialObject O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Model O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Model I-Model O O B-Model O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Model O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O