  The vertical kinematics of stars near the Sun can be used to measure the
total mass distribution near the Galactic disk and to study out-of-equilibrium
dynamics. With contemporary stellar surveys, the tracers of vertical dynamics
are so numerous and so well measured that the shapes of underlying orbits are
almost directly visible in the data through element abundances or even stellar
density. These orbits can be used to infer a mass model for the Milky Way,
enabling constraints on the dark matter distribution in the inner galaxy. Here
we present a flexible model for foliating the vertical position-velocity phase
space with orbits, for use in data-driven studies of dynamics. The vertical
acceleration profile in the vicinity of the disk, along with the orbital
actions, angles, and frequencies for individual stars, can all be derived from
that orbit foliation. We show that this framework - "Orbital Torus Imaging"
(OTI) - is rigorously justified in the context of dynamical theory, and does a
good job of fitting orbits to simulated stellar abundance data with varying
degrees of realism. OTI (1) does not require a global model for the Milky Way
mass distribution, and (2) does not require detailed modeling of the selection
function of the input survey data. We discuss the approximations and
limitations of the OTI framework, which currently trades dynamical
interpretability for flexibility in representing the data in some regimes, and
which also presently separates the vertical and radial dynamics. We release an
open-source tool, torusimaging, to accompany this article.
