  The dynamical interactions between young binaries can perturb the material
distribution of their circumstellar disks, and modify the planet formation
process. In order to constrain the impact and nature of the binary interaction
in the RW Aur system (bound or unbound), we analyzed the circumstellar material
at 1.3 mm wavelengths, as observed at multiple epochs by ALMA. We analyzed the
disk properties through parametric visibility modeling, and we used this
information to constrain the dust morphology and the binary orbital period. We
imaged the dust continuum emission of RW Aur with a resolution of 3 au, and we
find that the radius enclosing 90% of the flux (R90%) is 19 au and 14 au for RW
Aur A and B, respectively. By modeling the relative distance of the disks at
each epoch, we find a consistent trend of movement for the disk of RW Aur B
moving away from the disk of RW Aur A at an approximate rate of 3 mas/yr (about
0.5 au/yr in sky-projected distance). By combining ALMA astrometry, historical
astrometry, and the dynamical masses of each star, we constrain the RW Aur
binary stars to be most likely in a high-eccentricity elliptical orbit with a
clockwise prograde orientation relative to RW Aur A, although low-eccentricity
hyperbolic orbits are not ruled out by the astrometry. Our analysis does not
exclude the possibility of a disk collision during the last interaction, which
occurred $295_{-74}^{+21}$ yr ago relative to beginning of 2024. Evidence for
the close interaction is found in a tentative warp of 6 deg in the inner 3 au
of the disk of RW Aur A, in the brightness temperature of both disks, and in
the morphology of the gas emission. A narrow ring that peaks at 6 au around RW
Aur B is suggestive of captured material from the disk around RW Aur A.
