  Future missions like Roman, HabEx, and LUVOIR will directly image exoplanets
in reflected light. While current near infrared direct imaging searches are
only sensitive to young, self-luminous planets whose brightness is independent
of their orbital phase, reflected light imaging will reveal changes in planet
brightness over the course of an orbit due to phase variations. One of the
first objectives will be determining the planet's orbit via astrometry, the
projected position of the planet with respect to its host star in the sky
plane. We show that phase variations can significantly improve the accuracy and
precision of orbital retrieval with two or three direct images. This would
speed up the classification of exoplanets and improve the efficiency of
subsequent spectroscopic characterization. We develop a forward model to
generate synthetic observations of the two dimensional position of the planet
with respect to its host star on the sky plane, and the planet/star flux ratio.
Synthetic data are fitted with Keplerian orbits and Henyey-Greenstein phase
variations to retrieve orbital and phase parameters. For astrometric
uncertainties of 0.01 AU in projected separation and flux ratio uncertainties
of 10^-12, using photometry in orbit retrieval improves the accuracy of
semi-major axis by 47% for two epochs and 61% for three epochs if the phase
curves have a known shape, but unknown amplitude. In the more realistic
scenario where the shape and amplitude of the phase curve are a priori unknown,
photometry still improves accuracy by 16% for two epochs and 50% for three.
