A Two Micron All Sky Survey View of the Sagittarius Dwarf Galaxy. III. Constraints on the Flattening of the Galactic Halo

M giants selected from the Two Micron All Sky Survey (2MASS) have been usedto trace streams of tidal debris apparently associated with the Sagittariusdwarf spheroidal galaxy (Sgr) that entirely encircle the Galaxy. While the SgrM giants are generally aligned with a single great circle on the sky, wemeasure a difference of 10.4 +- 2.6 degrees between the mean orbital poles ofthe great circles that best fit debris leading and trailing Sgr, which can beattributed to the precession of Sgr's orbit over the range of phases exploredby the data set. Simulations of the destruction of Sgr in potentials containingbulge, disk and halo components best reproduce this level of precession alongthe same range of orbital phases if the potential contours of the halo are onlyslightly flattened, with the ratio between the axis length perpendicular to andin the disk in the range q = 0.90-0.95 (corresponding to isodensity contourswith q_\rho ~ 0.83 - 0.92). Oblate halos are strongly preferred over prolate(q_\rho > 1) halos, and flattenings in the potential of q <= 0.85 (q_\rho <=0.75) and q >= 1.05 (q_\rho >= 1.1) are ruled out at the 3-sigma level. Moreextreme values of q <= 0.80 (q_\rho <= 0.6) and q >= 1.25 (q_\rho >= 1.6) areruled out at the 7-sigma and 5-sigma levels respectively. These constraintswill improve as debris with larger separation in orbital phase can be found.