  As next-generation imaging instruments and interferometers search for planets
closer to their stars, they must contend with increasing orbital motion and
longer integration times. These compounding effects make it difficult to detect
faint planets but also present an opportunity. Increased orbital motion makes
it possible to move the search for planets into the orbital domain, where
direct images can be freely combined with the radial velocity and proper motion
anomaly, even without a confirmed detection in any single epoch. In this paper,
we present a fast and differentiable multimethod orbit-modeling and planet
detection code called Octofitter. This code is designed to be highly modular
and allows users to easily adjust priors, change parameterizations, and specify
arbitrary function relations between the parameters of one or more planets.
Octofitter further supplies tools for examining model outputs including prior
and posterior predictive checks and simulation-based calibration. We
demonstrate the capabilities of Octofitter on real and simulated data from
different instruments and methods, including HD 91312, simulated JWST/NIRISS
aperture masking interferometry observations, radial velocity curves, and grids
of images from the Gemini Planet Imager. We show that Octofitter can reliably
recover faint planets in long sequences of images with arbitrary orbital
motion. This publicly available tool will enable the broad application of
multiepoch and multimethod exoplanet detection, which could improve how future
targeted ground- and space-based surveys are performed. Finally, its rapid
convergence makes it a useful addition to the existing ecosystem of tools for
modeling the orbits of directly imaged planets.
