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Indeed, Jupiter is bright, but there are systematic ways of handling this in Gaia photometry. Gaia will observe over 350,000 objects in our solar system. The vast majority are main-belt asteroids, but this includes moons of other planets, implying to accomplish this they would have to account for the brightness of the planets themselves.

In the specific case of microlensing with Jupiter and Gaia, the seminal paper also cited above explains quite nicely: starting from the geodesic equation, they arrive at equations 10-12 which give the deflection vector containing the usual monopole term and the relevant quadrupole terms. These are simplified in the case of "near-grazing rays as, unlike with the solar deflection, the effect is too small to be observed at large angle from the planet" into equations 13 and 14, which depend on the orientation of the spin axis of the planet (i.e., the inner products with the spin vector in the $z$ direction). In equation 13, the monopole term is the first term that is independent of $J_2$ (the dimensionless coefficient of the second zonal harmonic), and the terms that subtract from it are quadrupole terms. Equation 14 is due to quadrupole. Clearly the quadrupole terms depend on $z$, implying that this is directly due to the oblateness of Jupiter.

EDIT: Short answers: Yes, it is possible (and increasingly more plausible!); it has not been tried with full rigor to the best of my knowledge; it is strongly expected to work, since microlensing with Jupiter has been successful for discovering, for example, exoplanets.

Indeed, Jupiter is bright, but there are systematic ways of handling this in Gaia photometry. Gaia will observe over 350,000 objects in our solar system. The vast majority are main-belt asteroids, but this includes moons of other planets, implying to accomplish this they would have to account for the brightness of the planets themselves.

EDIT: Short answer: No, not yet :)

In the specific case of microlensing with Jupiter and Gaia, the seminal paper also cited above explains quite nicely: starting from the geodesic equation, they arrive at equations 10-12 which give the deflection vector containing the usual monopole term and the relevant quadrupole terms. These are simplified in the case of "near-grazing rays as, unlike with the solar deflection, the effect is too small to be observed at large angle from the planet" into equations 13 and 14, which depend on the orientation of the spin axis of the planet (i.e., the inner products with the spin vector in the $z$ direction). In equation 13, the monopole term is the first term that is independent of $J_2$ (the dimensionless coefficient of the second zonal harmonic), and the terms that subtract from it are quadrupole terms. Equation 14 is due to quadrupole. Clearly the quadrupole terms depend on $z$, implying that this is directly due to the oblateness of Jupiter...

EDIT: .... which the oblateness ITSELF is due to the planet's rotation. In principle, a non-rotating planet would still cause light bending (b/c the Schwarzschild solution does), and a non-spherical mass distribution of the planet could cause extra light bending due to a mass-quadrupole moment of the non-sphericity, but this would be more similar to an effect of an extended body in general relativity, rather than spin, which is what is modeled in the case of Jupiter. So, specifically in the case of Jupiter, one is not wrong in saying that the oblateness directly causes more bending, but it comes with the nuance that the oblateness itself is due to the rotation: this is shown in the equations 13 and 14 above, by the quantity $J_2$ appearing everywhere there is spin. This paper explains this more technically.

This is an active area of research. For example, this paper presents recent developments in an estimation and calibration framework, where they demonstrate the sub$\mu$as-level stability "of a local reference frame composed of a few tens of comparison stars surrounding the bright target star that is expected to show a large value for the relativistic light deflection due to its proximity to Jupiter’s limb." The authors conclude by stating they plan next to apply the methodology "to actual observations of events in the GAREQ experiment," which is an acronym for "Gaia Relativistic Experiment on Jupiter's Quadrupole," the device aboard Gaia making these measurements.

In the specific case of microlensing with Jupiter and Gaia, the seminal paper also cited above explains quite nicely: starting from the geodesic equation, they arrive at equations 10-12 which give the deflection vector containing the usual monopole term and the relevant quadrupole terms, which can be simplified in the case of "near-grazing rays as, unlike with the solar deflection, the effect is too small to be observed at large angle from the planet." These are equations 13 and 14 (cleaned into equations 15 and 16, respectively), which depend on the orientation of the spin axis of the planet (i.e., the inner products with the spin vector in the $z$ direction), implying that this is directly due to the oblateness of Jupiter.

The Gaia SOC calibration team developed calibration tools with GEREQ (which is an acronym for "Gaia Relativistic Experiment on Jupiter's Quadrupole," the device aboard Gaia making these measurements) being the prototypical case.

This is an active area of research. For example, this paper presents recent developments in an estimation and calibration framework, where they demonstrate the sub$\mu$as-level stability "of a local reference frame composed of a few tens of comparison stars surrounding the bright target star that is expected to show a large value for the relativistic light deflection due to its proximity to Jupiter’s limb." The authors conclude by stating they plan next to apply the methodology "to actual observations of events in the GAREQ experiment."

In the specific case of microlensing with Jupiter and Gaia, the seminal paper also cited above explains quite nicely: starting from the geodesic equation, they arrive at equations 10-12 which give the deflection vector containing the usual monopole term and the relevant quadrupole terms. These are simplified in the case of "near-grazing rays as, unlike with the solar deflection, the effect is too small to be observed at large angle from the planet" into equations 13 and 14, which depend on the orientation of the spin axis of the planet (i.e., the inner products with the spin vector in the $z$ direction). In equation 13, the monopole term is the first term that is independent of $J_2$ (the dimensionless coefficient of the second zonal harmonic), and the terms that subtract from it are quadrupole terms. Equation 14 is due to quadrupole. Clearly the quadrupole terms depend on $z$, implying that this is directly due to the oblateness of Jupiter.

Indeed, Jupiter is bright, but there are systematic ways of handling this in Gaia photometry. Gaia will observe over 350,000 objects in our solar system. The vast majority are main-belt asteroids, but this includes moons of other planets, implying to accomplish this they would have to account for the brightness of the planets themselves.  This site describes the detectability of stars near Jupiter (relative to Gaia's view), where "the detectability is a function of the star angular position with respect to Jupiter." The image at this site is helpful: [![Detectability of stars close to Jupiter]9]9

Indeed, Jupiter is bright, but there are systematic ways of handling this in Gaia photometry. Gaia will observe over 350,000 objects in our solar system. The vast majority are main-belt asteroids, but this includes moons of other planets, implying to accomplish this they would have to account for the brightness of the planets themselves. 

This site describes the detectability of stars near Jupiter (relative to Gaia's view), where "the detectability is a function of the star angular position with respect to Jupiter." The image at this site is helpful, which shows the detection limits "for a star of magnitude G = 13.5 as it gets close to Jupiter in a wide range of relative orientations... Jupiter has an elongated shape because the Gaia pixels are rectangular, with a size three times smaller in the x-axis direction. The detection limits found for a G=13.5 star close to Jupiter are between 2" to 4"."