Measuring the Milky Way Vertical Potential with the Phase Snail in a Model-independent Way

The vertical phase-space spiral (snail) is a direct sign of disequilibrium of the Milky Way's disk. Nevertheless, the wrapping of the phase snail contains information on the vertical potential. We propose a novel method to measure the vertical potential utilizing the intersections between the snail and z/V _z axes, for which we know the maximum vertical heights (Z _max) or velocities ( ${V}_{z,\max }$ ). Using a refined linear interpolation method, we directly obtain $({Z}_{\max },\tfrac{1}{2}{V}_{z,\max }^{2})$ for these snail intersections to constrain the vertical potential profile empirically. Our method is model-independent, since no assumptions about the snail shape or the vertical potential have been made. Although the snail binned by the guiding center radius (R _g ) is more prominent, it traces a vertical potential shallower than that of the snail binned by the same galactocentric radius (R). We apply an empirical method to correct this difference. We measure the snail intersections in several R _g bins within 7.5 < R _g < 11.0 kpc for Gaia DR3 and apply the interpolation method to deduce the potential values at several vertical heights. The potential at the snail intersections, as well as the following mass modeling, is consistent with the popular Milky Way potentials in the literature. For the R _g -binned phase snail in the solar neighborhood, the mass modeling indicates a local dark matter density of ρ _dm = 0.0150 ± 0.0031 M _⊙ pc^-3, consistent with previous works. Our method could be applied to larger radial ranges in future works to provide independent and stronger constraints on the Milky Way's potential.