  In 2015, K2 observations of the bright (V = 8.9, K = 7.7) star HIP 41378
revealed a rich system of at least five transiting exoplanets, ranging in size
from super-Earths to gas giants. The 2015 K2 observations only spanned 74.8
days, and the outer three long-period planets in the system were only detected
with a single transit, so their orbital periods and transit ephemerides could
not be determined at that time. Here, we report on 50.8 days of new K2
observations of HIP 41378 from summer 2018. These data reveal additional
transits of the long-period planets HIP 41378 d and HIP 41378 f, yielding a set
of discrete possible orbital periods for these two planets. We identify the
most probable orbital periods for these two planets using our knowledge of the
planets' transit durations, the host star's properties, the system's dynamics,
and data from the ground-based HATNet, KELT, and WASP transit surveys. Targeted
photometric follow-up during the most probable future transit times will be
able to determine the planets' orbital periods, and will enable future
observations with facilities like the James Webb Space Telescope. The methods
developed herein to determine the most probable orbital periods will be
important for long-period planets detected by the Transiting Exoplanet Survey
Satellite, where similar period ambiguities will frequently arise due to the
telescope's survey strategy.
