The Fast Radio Burst-emitting magnetar SGR 1935+2154 -- proper motion and variability from long-term Hubble Space Telescope monitoring

We present deep Hubble Space Telescope near-infrared (NIR) observations of the magnetar SGR 1935+2154 from June 2021, approximately 6 years after the first HST observations, a year after the discovery of fast radio burst like emission from the source, and in a period of exceptional high frequency activity. Although not directly taken during a bursting period the counterpart is a factor of ~1.5 to 2.5 brighter than seen at previous epochs with F140W(AB) = $24.65\pm0.02$ mag. We do not detect significant variations of the NIR counterpart within the course of any one orbit (i.e. on minutes-hour timescales), and contemporaneous X-ray observations show SGR 1935+2154 to be at the quiescent level. With a time baseline of 6 years from the first identification of the counter-part we place stringent limits on the proper motion of the source, with a measured proper motion of ${\mu} = 3.1\pm1.5$ mas/yr. The direction of proper motion indicates an origin of SGR 1935+2154 very close to the geometric centre of SNR G57.2+08, further strengthening their association. At an adopted distance of $6.6\pm0.7$ kpc, the corresponding tangential space velocity is ${\nu_T} = 97\pm48$ km/s (corrected for differential Galactic rotation and peculiar Solar motion), although its formal statistical determination may be compromised owing to few epochs of observation. The current velocity estimate places it at the low end of the kick distribution for pulsars, and makes it among the lowest known magnetar kicks. When collating the few-magnetar kick constraints available, we find full consistency between the magnetar kick distribution and the much larger pulsar kick sample