  Ultra-short period (USP) planets are an enigmatic subset of exoplanets
defined by having orbital periods $<$ 1 day. It is still not understood how USP
planets form, or to what degree they differ from planets with longer orbital
periods. Most USP planets have radii $<$ 2 $R_{\oplus}$, while planets that
orbit further from their star extend to Jupiter size ($>$ 10 $R_{\oplus}$).
Several theories attempt to explain the formation and composition of USP
planets: they could be remnant cores of larger gas giants that lost their
atmospheres due to photo-evaporation or Roche lobe overflow, or they could have
developed through mass accretion in the innermost part of the protoplanetary
disk. The radius and mass distribution of USP planets could provide important
clues to distinguish between potential formation mechanisms. In this study, we
first verify and update the Kepler catalog of USP planet host star properties,
incorporating new data collected by the Gaia mission where applicable. We then
use the transit depths measured by Kepler to derive a radius distribution and
present occurrence rates for USP planets. Using spherical and tidally distorted
planet models, we then derive a mass distribution for USP planets. Comparisons
between the updated USP planet mass distribution and simulated planetary
systems offer further insights into the formation and evolutionary processes
shaping USP planet populations.
