General model of vertical distribution of stars in the Milky Way using complete Jeans equations

The self-consistent vertical density distribution in a thin, isothermal disc is typically given by a sech^2 law, as shown in the classic work by Spitzer (1942). This is obtained assuming that the radial and vertical motions are decoupled and only the vertical term is used in the Poisson equation. We argue that in the region of low density as in the outer disc this treatment is no longer valid. We develop a general, complete model that includes both radial and vertical terms in the Poisson equation and write these in terms of the full radial and vertical Jeans equations which take account of the non-flat observed rotation curve, the random motions, and the cross term that indicates the tilted stellar velocity ellipsoid. We apply it to the Milky Way and show that these additional effects change the resulting density distribution significantly, such that the mid-plane density is higher and the disc thickness (HWHM) is lower by 30-40% in the outer Galaxy. Further, the vertical distribution is no longer given as a sech^2 function even for an isothermal case. These predicted differences are now within the verification limit of new, high-resolution data for example from GAIA and hence could be confirmed.