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Analyses of absorption from disk winds and atmospheres in accreting compactobjects typically treat the central emitting regions in these systems as pointsources relative to the absorber. This assumption breaks down if the absorbinggas is located within $few \times 1000\cdot GM/{c}^{2}$, in which case a smallcomponent of the absorber's Keplerian motion contributes to the velocity-widthof absorption lines. Here, we demonstrate how this velocity-broadening effectcan be used to constrain the sizes of central engines in accreting compactobjects via a simple geometric relationship, and develop a method for modelingthis effect. We apply this method on the Chandra/HETG spectra of threeultra-compact and short period neutron star X-ray binaries in which evidence ofgravitationally redshifted absorption, owing to an inner-disk atmosphere, hasrecently been reported. The significance of the redshift is above $5\sigma$ forXTE J1710$-$281 (this work) and 4U 1916$-$053, and is inconsistent with variousestimates of the relative radial velocity of each binary. For our mostsensitive spectrum (XTE J1710$-$281), we obtain a 1$\sigma$ upper bound of 310$\text{km}$ $\text{s}^{-1}$ on the magnitude of this geometric effect and acentral engine of size ${R}_{CE} < 60 ~ GM/{c}^{2}$ (or, $< 90 ~ GM/{c}^{2}$ atthe $3\sigma$ level). These initial constraints compare favorably to thoseobtained via microlensing in quasars and approach the sensitivity ofconstraints via relativistic reflection in neutron stars. This sensitivity willincrease with further exposures, as well as the launch of futuremicrocalorimeter and grating missions.

A Spectroscopic Angle on Central Engine Size Scales in Accreting Neutron Stars