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 Jupiters 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.
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