The large offset of Neptune's magnetic field in an offset tilted dipole (OTD) approximation allows some fraction of magnetospheric field lines to have monotonically increasing field magnitude over their entire length in the magnetosphere. The usual magnetic trapping is impossible on such field lines, meaning that all charged particles on these field lines will precipitate after mirroring at most once. These field lines can be thought of as the part of the global magnetic anomaly where no particles on the corresponding drift shells remain trapped for an entire rotation of the planet. In this paper we examine both the morphology and size of the affected surface area in the OTD2 and O8 magnetic field models. For regions where there are indications that part of the aurora could be due to monotonic field line precipitation we analyze whether these field lines could be sufficiently populated to produce auroral signatures by considering photoelectron production rates and radial diffusion. We conclude that monotonic field line precipitation may contribute to the observed Neptune aurora but does not explain the global phenomenon. ¿ American Geophysical Union 1993