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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.

Under the assumption of a simple and time-invariant gravitational potential, many Galactic dynamics techniques infer the milky Way’s mass and dark matter distributions from stellar kinematic observations. These methods typically rely on parameterized potential models of the Galaxy and must take into account nontrivial survey selection effects, because they make use of the density of stars in phase space. Large-scale spectroscopic surveys now supply information beyond kinematics in the form of precise stellar label measurements (especially element abundances). These element abundances are known to correlate with orbital actions or other dynamical invariants. Here, we use the Orbital Torus Imaging framework that uses abundance gradients in phase space to map orbits. In many cases these gradients can be measured without detailed knowledge of the selection function. We use stellar surface abundances from the Apache Point Observatory Galactic Evolution Experiment survey combined with kinematic data from the Gaia mission. Our method reveals the vertical ( z -direction) orbital structure in the Galaxy and enables empirical measurements of the vertical acceleration field and orbital frequencies in the disk. From these measurements, we infer the total surface mass density, Σ, and midplane volume density, ρ _0 , as a function of Galactocentric radius and height. Around the Sun, we find ${{\rm{\Sigma }}}_{\odot }(z=1.1\,\mathrm{kpc})={72}_{-9}^{+6}\,{M}_{\odot }\,{\mathrm{pc}}^{-2}$ and ${\rho }_{\odot }(z=0)={0.081}_{-0.009}^{+0.015}\,{M}_{\odot }\,{\mathrm{pc}}^{-3}$ using the most constraining abundance ratio, [Mg/Fe]. This corresponds to a dark matter contribution in surface density of Σ _⊙,DM ( z = 1.1 kpc) = 24 ± 4 M _⊙ pc ^−2 , and in total volume mass density of ρ _⊙,DM ( z = 0) = 0.011 ± 0.002 M _⊙ pc ^−3 . Moreover, using these mass density values we estimate the scale length of the low- α disk to be h _R = 2.24 ± 0.06 kpc.

The universal rotation curve (URC) of disk galaxies was originally proposed to predict the shape and amplitude of any rotation curve (RC) based solely on photometric data. Here, the URC is investigated with an extensive set of spatially resolved RCs drawn from the PROBES-I, PROBES-II, and MaNGA databases with matching multiband surface brightness profiles from the DESI-LIS and Wide-Field Infrared Survey Explorer surveys for 3846 disk galaxies. Common URC formulations fail to achieve an adequate level of accuracy to qualify as truly universal over fully sampled RCs. We develop neural network (NN) equivalents for the proposed URCs that predict RCs with higher accuracy, showing that URC inaccuracies are not due to insufficient data but rather nonoptimal formulations or sampling effects. This conclusion remains even if the total RC sample is pruned for symmetry. The latest URC prescriptions and their NN equivalents trained on our subsample of 579 disk galaxies with symmetric RCs perform similarly to the URC/NN trained on the complete data sample. We conclude that a URC with an acceptable level of accuracy (Δ V _circ ≲ 15%) at all radii would require a detailed modeling of a galaxy’s central regions and outskirts (e.g., for baryonic effects leading to contraction or expansion of any dark-matter-only halo).