Proper Motions and Origins of AXP 1E 2259+586 and AXP 4U 0142+61

Using high-resolution NIR images supported by laser guide star adaptive optics from the Keck II telescope from 2005 to 2012, we have measured the proper motions of two anomalous X-ray pulsars, AXP 1E 2259+586 and AXP 4U 0142+61. The proper motion of AXP 1E 2259+586 in the sky frame is (mu(alpha), mu(delta)) = (-6.4 +/- 0.6, -2.3 +/- 0.6) mas yr(-1) and that of AXP 4U 0142+61 is (mu(alpha), mu(delta)) = (-4.1 +/- 1, 1.9 +/- 1) mas yr(-1). After correcting for the velocity of the progenitors, we calculate the tangential ejection velocities of the magnetars to be 157 +/- 17 km s(-1) and 102 +/- 26 km s(-1) respectively. The proper motion vector of AXP 1E 2259+586 is directed away from the putative center of the supernova remnant CTB 109 that has long been proposed to be associated with AXP 1E 2259+586. This is significant evidence for linking the pulsar with CTB 109. We comment on the possible movement of CTB 109 after the explosion. We narrow the search cone for the birthsite or remnant of AXP 4U 0142+61 to an opening angle of 24 degrees. However, we are unable to find any suitable association.