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

Predicted spatial and velocity distributions of ejected companion stars of helium accretion-induced thermonuclear supernovae