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[en] We evaluated the prospects of quantifying the parameterized post-Newtonian parameter β and solar quadrupole moment with observations of near-Earth asteroids with large orbital precession rates (9 to 27 arcsec century−1). We considered existing optical and radar astrometry, as well as radar astrometry that can realistically be obtained with the Arecibo planetary radar in the next five years. Our sensitivity calculations relied on a traditional covariance analysis and Monte Carlo simulations. We found that independent estimates of β and can be obtained with precisions of 6 × 10−4 and 3 × 10−8, respectively. Because we assumed rather conservative observational uncertainties, as is the usual practice when reporting radar astrometry, it is likely that the actual precision will be closer to 2 × 10−4 and 10−8, respectively. A purely dynamical determination of solar oblateness with asteroid radar astronomy may therefore rival the helioseismology determination.