Towards Proper Motions in the Local Group
  Key and still largely missing parameters for measuring the mass content and distribution of the Local Group are the proper motion vectors of its member galaxies. The problem when trying to derive the gravitational potential of the Local Group is that usually only radial velocities are known, and hence statistical approaches have to be used. The expected proper motions for galaxies within the Local Group, ranging from 20 to 100 $\mu$as/yr, are detectable with VLBI using the phase-referencing technique. We present phase-referencing observations of bright masers in IC~10 and M33 with respect to background quasars. We observed the H$_2$O masers in IC10 three times over a period of two months to check the accuracy of the relative positions. The relative positions were obtained by modeling the interferometer phase data for the maser sources referenced to the background quasars. The model allowed for a relative position shift for the source and a single vertical atmospheric delay error in the correlator model for each antenna. The rms of the relative positions for the three observations is only 0.01 mas, which is approximately the expected position error due to thermal noise. Also, we present a method to measure the geometric distance to M33. This will allow re-calibration of the extragalactic distance scale based on Cepheids. The method is to measure the relative proper motions of two H$_2$O maser sources on opposite sides of M33. The measured angular rotation rate, coupled with other measurements of the inclination and rotation speed of the galaxy, yields a direct distance measurement. 
A. Brunthaler (MPIfR), M. Reid (CfA), H. Falcke (MPIfR), L.J.   Greenhill (CfA), C. Henkel (MPIfR)