Understanding Compact Object Formation and Natal Kicks: II. The case of   XTE J1118+480
  In recent years, an increasing number of proper motions have been measured for Galactic X-ray binaries. When supplemented with accurate determinations of the component masses, orbital period, and donor effective temperature, these kinematical constraints harbor a wealth of information on the system's past evolution. Here, we consider all this available information to reconstruct the full evolutionary history of the black hole X-ray binary XTE J1118+480, assuming that the system originated in the Galactic disk and the donor has solar metallicity. This analysis accounts for four evolutionary phases: mass transfer through the ongoing X-ray phase, tidal evolution before the onset of Roche-lobe overflow, motion through the Galactic potential after the formation of the black hole, and binary orbital dynamics due to explosive mass loss and possibly a black hole natal kick at the time of core collapse. We find that right after black hole formation, the system consists of a ~6.0-10.0 solar masses black hole and a ~1.0-1.6 solar masses main-sequence star. We also find that that an asymmetric natal kick is not only plausible but required for the formation of this system, and derive a lower and upper limit on the black hole natal kick velocity magnitude of 80 km/s and 310 km/s, respectively. 
T. Fragos, B. Willems, V. Kalogera, N. Ivanova, G. Rockefeller, C. L.   Fryer, and P. A. Young