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The presence of a stellar companion can place constraints on occurrence andorbital evolution of satellites orbiting exoplanets, i.e., exomoons. In thiswork we revise earlier orbital stability limits for retrograde orbits in thecase of a three body system consisting of star-planet-satellite. The latterreads $a_{\rm sat}^{\rm crit} \approx 0.668(1-1.236e_{\rm p})$ for $e_p \leq0.8$ in units of the Hill Radius and represents the lower critical orbit as afunction of the planetary eccentricity $e_{\rm p}$. A similar formula isdetermined for exomoons hosted by planets in binary star systems, where $e_{\rmp}$ is replaced with the components of free and forced eccentricity fromsecular orbit evolution theory. By exploring the dynamics of putative exomoonsin $\alpha$ Centauri AB we find that the outer stability limit can be much lessthan half the Hill Radius due to oscillations in the planetary orbitaleccentricity caused by the gravitational interaction with the binary star. Weshow, furthermore, how the resulting truncation of the outer stability limitcan affect the outward tidal migration and potential observability of exomoonsthrough transit timing variations (TTVs). Typical TTV (RMS) amplitudes inducedby exomoons in binary systems are $\lesssim$10 min and appear more likely forplanets orbiting the less massive stellar component. A GitHub repository(saturnaxis/exomoon-in-binaries) is available to reproduce figures.

Exomoons in Systems with a Strong Perturber: Applications to $\alpha$ Cen AB