Thermonuclear supernovae (SNe), a subset of which are the highly important SNe Type,Ia, remain one of the more poorly understood phenomena known to modern astrophysics. In recent years, the single degenerate helium (He) donor channel, where a white dwarf star (WD) accretes He-rich matter from a hydrogen-depleted companion, has emerged as a promising candidate progenitor scenario for these events. An unresolved question in this scenario is the fate of the companion star, which would be evident as a runaway hot subdwarf (He sdO/B) in the aftermath of the SN event. Previous studies have shown that the kinematic properties of an ejected companion provide an opportunity to closer examination of the properties of an SN progenitor system. However, with the number of observed objects not matching predictions by theory, the viability of this mechanism is called into question. In this study, we first synthesize a population of companion stars ejected by the aforementioned mechanism, taking into account predicted ejection velocities, inferred population density in the Galactic (Gal.) mass distribution and subsequent kinematics in the Gal. potential. We then discuss the astrometric properties of this population. We present $10^{6}$ individual ejection trajectories, numerically computed with a newly developed, lightweight simulation framework. A peak in the density distribution for close objects is expected in the direction of the Gal. center. If the entire considered mass range is realized, the radial velocity distribution should show a peak at 500kms. If only close US,708 analogues are considered, there should be a peak at ($sim750-850$)kms. We show that the puzzling lack of confirmed surviving companion stars of thermonuclear SNe, though possibly an observation-related selection effect, may indicate a selection against high mass donors in the SD He donor channel. (-abridged-)
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