Relations between image centroid motion and large-telescope long-exposure image size are given for the cases of absolute motion measured with a single aperture and differential motion measured with two apertures. The effect of a finite exposure time in the measurement of image positions is included in the analysis, and is shown to be of crucial importance, especially in the measurement of differential image motion. For example, the contribution of the free atmosphere to the mean square differential motion can be underestimated by more than a factor of ten in realistic circumstances if an exposure time of 1/30 s is used.

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We present the first scientific images obtained with a deformable secondary mirror adaptive optics system. We utilized the 6.5m MMT AO system to produce high-resolution (FWHM=0.07 ′′ ) near infrared (1.6µm) images of the young (∼ 1 Myr) Orion Trapezium θ 1 Ori cluster members. A combination of high spatial resolution and high signal to noise allowed the positions of these stars to be measured to within ∼ 0.003 ′′ accuracies. We also present slightly lower resolution (FWHM∼0.085 ′′ ) images from Gemini with the Hokupa'a AO system as well. Including previous speckle data (Weigelt et al. 1999), we analyze a six year baseline of high-resolution observations of this cluster. Over this baseline we are sensitive to relative proper motions of only ∼ 0.002 ′′ /yr (4.2 km/s at 450 pc). At such sensitivities we detect orbital motion in the very tight θ 1 Ori B 2 B 3 (52 AU separation) and θ 1 Ori A 1 A 2 (94 AU separation) systems. The relative velocity in the θ 1 Ori B 2 B 3 system is 4.2 ± 2.1 km/s. We observe 16.5 ± 5.7 km/s of relative motion in the θ 1 Ori A 1 A 2 system. These velocities are consistent with those independently observed by Schertl et al. (2003) with speckle interferometry, giving us confidence that these very small (∼ 0.002 ′′ /yr) orbital motions are real. All five members of the θ 1 Ori B system appear likely gravitationally bound (B 2 B 3 is moving at ∼ 1.4 km/s in the plane of the sky w.r.t. B 1 where V esc ∼ 6 km/s for the B group). The very lowest mass member of the θ 1 Ori B system (B 4 ) has K ′ ∼ 11.66 and an estimated mass of ∼ 0.2M ⊙ . There was very little motion (4 ± 15 km/s) detected of B 4 w.r.t B 1 or B 2 , hence B 4 is possibly part of the θ 1 Ori B group. We suspect that if this very low mass member is physically associated it most likely is in an unstable (non-hierarchical) orbital position and will soon be ejected from the group. The θ 1 Ori B system appears to be a good example of a star formation "mini-cluster" which may eject the lowest mass members of the cluster in the near future. This "ejection" process could play a major role in the formation of low mass stars and brown dwarfs.