The Stellar Distribution Function and Local Vertical Potential from Gaia DR2

We develop a novel method to simultaneously determine the vertical potential, force and stellar $z-v_z$ phase space distribution function (DF) in our local patch of the Galaxy. We assume that the Solar Neighborhood can be treated as a one-dimensional system in dynamical equilibrium and directly fit the number density in the $z-v_z$ plane to what we call the Rational Linear DF (RLDF) model. This model can be regarded as a continuous sum of isothermal DFs though it has only one more parameter than the isothermal model. We apply our method to a sample of giant stars from Gaia Data Release 2 and show that the RLDF provides an excellent fit to the data. The well-known phase space spiral emerges in the residual map of the $z-v_z$ plane. We use the best-fit potential to plot the residuals in terms of the frequency and angle of vertical oscillations and show that the spiral maps into a straight line. From its slope, we estimate that the phase spirals were generated by a perturbation $\sim540$ Myr years ago. We also determine the differential surface density as a function of vertical velocity dispersion, a.k.a. the vertical temperature distribution. The result is qualitatively similar to what was previously found for SDSS/SEGUE G dwarfs. Finally, we address parameter degeneracies and the validity of the 1D approximation. Particularly, the mid-plane density derived from a cold subsample, where the 1D approximation is more secure, is closer to literature values than that derived from the sample as a whole.