tokens: gradient. The negative radial gradient of is expected because is associated with the surface mass density (e.g. Freeman Bland-Hawthorn 2002 ; Binney Tremaine 2008 ), which decreases with increasing Galactocentric radius (e.g. Xiang et al. 2018 ). The positive vertical gradient mainly reflects the increase of surface mass density enclosed by increasing height above the disc. The dispersion of J Z , however, has less dependence with R . Again, this is likely to be because of the nature of J Z , as discussed above. The skewness of V Z is close to 0 and uniformly distributes across the R – Z plane, except for the outermost region ( R > 11 kpc), whereas the skewness in J Z shows strong patterns. These characteristics are similar to the case of skewness for V R and J R , and they share similar physical reasons (see Section 3.2.1 ). 4 ORBITAL PARAMETERS AND STELLAR MIGRATION In this section, we present the orbital parameters of the RGB stars across the age–metallicity plane and the disc R – Z plane. We discuss the impact of non-circular orbital motion on the stellar distribution by comparing the current position of the stars with the guiding-centre radii of their orbits. We further estimate the birth radii of the stars based on their age and metallicity in the same way as Chen et al. ( 2019b ), initially proposed by Minchev et al. ( 2018 ), and we study the effect of stellar migration (churning). Note that here we refer to the stellar migration as the churning process, in which stars change their orbital radii through angular momentum transfer via scattering or perturbation (e.g. Sellwood Binney 2002 ; Minchev et al. 2013 ). 4.1 Orbital parameters As mentioned in Section 2 , the orbital parameters of our sample stars are derived using positions and proper motions from Gaia DR2, distances from BJ18 and radial velocities from LAMOST DR4. The orbit calculation is implemented with galpy , assuming the MWpotential2014 (Bovy 2015 ). Fig. 10 plots the median, dispersion and skewness of the stellar eccentricity distributions in the age–[Fe/H] plane and the disc R – Z plane. The figure illustrates that the young or old but metal-rich thin disc stars have mean orbital eccentricities smaller than 0.2, and have small (0.1) dispersion among stars in each mono-age and mono-[Fe/H] bin, which is consistent with previous work (e.g. Kordopatis et al. 2015 ). The old, metal-poor thick disc stars have larger eccentricities and broader distribution, and their mean eccentricities vary from 0.2 to larger than 0.5. For the thin disc stars, the eccentricity steadily increases with stellar age and exhibits little change with [Fe/H] at a fixed age, possibly suggesting that the steady disc heating processes have played a decisive role in determining the orbital evolution of these stars (e.g. Ting Rix 2019 ). At a given age, the [Fe/H] of a star is an indicator of its Galactocentric radius because of the existence of radial [Fe/H] gradient for mono-age populations (e.g. Boeche
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