Understanding Compact Object Formation and Natal Kicks. I. Calculation Methods and the Case of GRO J1655-40

In recent years proper-motion measurements have been added to the set of observational constraints on the current properties of Galactic X-ray binaries. We develop an analysis that allows us to consider all this available information and reconstruct the full evolutionary history of X-ray binaries back to the time of core collapse and compact object formation. This analysis accounts for five evolutionary phases: mass transfer through the ongoing X-ray phase, tidal circularization before the onset of Roche lobe overflow, motion through the Galactic potential after the formation of the compact object, binary orbital dynamics at the time of core collapse, and hydrodynamic modeling of the core collapse that connects the compact object to its progenitor and any nucleosynthetic constraints available. In this first paper we present this analysis in a comprehensive manner and apply it to the soft X-ray transient GRO J1655-40. This is the first analysis that incorporates all observational constraints on the current system properties and uses the full three-dimensional peculiar velocity constraints right after core collapse instead of lower limits on the current space velocity given by the present-day radial velocity. We find that the system has remained within 200 pc from the Galactic plane throughout its entire lifetime and that the mass loss and a kick possibly associated with the black hole formation imparted a kick velocity of ~=45-115 km s^-1 to the binary's center of mass. Right after black hole formation, the system consists of a ~=3.5-6.3 M_solar black hole and a ~=2.3-4 M_solar main-sequence star. At the onset of the X-ray phase the donor is still on the main sequence. We find that a symmetric black hole formation event cannot be formally excluded but that the associated system parameters are only marginally consistent with the currently observed binary properties. Black hole formation mechanisms involving an asymmetric supernova explosion with associated black hole kick velocities of a few tens of km s^-1, on the other hand, satisfy the constraints much more comfortably. We also derive an upper limit on the black hole kick magnitude of ~=210 km s^-1.