  We employ differential astrometric methods to establish a small field
reference frame stable at the micro-arcsecond ($\mu$as) level on short
timescales using high-cadence simulated observations taken by Gaia in February
2017 of a bright star close to the limb of Jupiter, as part of the relativistic
experiment on Jupiter's quadrupole. We achieve sub$\mu$as-level precision along
scan through a suitable transformation of the field angles into a small-field
tangent plane and a least-squares fit over several overlapping frames for
estimating the plate and geometric calibration parameters with tens of
reference stars that lie within $\sim$0.5 degs from the target star, assuming
perfect knowledge of stellar proper motions and parallaxes. Furthermore, we
study the effects of unmodeled astrometric parameters on the residuals and find
that proper motions have a stronger effect than unmodeled parallaxes. For e.g.,
unmodeled Gaia DR2 proper motions introduce extra residuals of $\sim$23$\mu$as
(AL) and 69$\mu$as (AC) versus the $\sim$5$\mu$as (AL) and 17$\mu$as (AC) due
to unmodeled parallaxes. On the other hand, assuming catalog errors in the
proper motions such as those from Gaia DR2 has a minimal impact on the
stability introducing sub$\mu$as and $\mu$as level residuals in the along and
across scanning direction, respectively. Finally, the effect of a coarse
knowledge in the satellite velocity components (with time dependent errors of
10$\mu$as) is capable of enlarging the size of the residuals to roughly 0.2
mas.
