High accuracy proper motions in crowded fields

The image subtraction method is a powerful tool to analyze the light variations in crowded fields. This method is able to achieve a nearly optimal differential photometry, even in very dense regions. However, image subtraction is not limited to photometry, and it is shown in this paper that the method can be generalized to the measurement of the differential proper motions. It is important to emphasize that image subtraction can re-construct an un-biased frame to frame astrometric transform. A nearly optimal determination of this astrometric transform can be performed by expanding the spatial variations of the kernel using the derivatives of the constant kernel solution. It is demonstrated that an expansion using first and second derivatives is optimal. Differential refraction can be corrected easily using an artificial image and the first derivatives of the kernel. To illustrate the ability of the method to measure proper motions, a small sub-area near the center of a Galactic Bulge field covered 265 times by the OGLE II experiment has been selected. The resulting astrometric accuracy is very close to the photon noise for the faint objects, while for the brighter ones it approaches closely the limit set by the residual seeing fluctuations on the smaller scales. The accuracy obtained with this new method is compared to the accuracy achieved using classical methods. The improvement obtained varies from a factor of $\simeq$ 2 for the brighter objects, to more than a factor of 3 for the fainter ones. This improvement is typical of the improvement that was obtained for the measurement of the photometric variations.