Rational Orbits and Gravitational-Wave Signatures around a Horndeski Black Hole

We study equatorial timelike geodesics and associated gravitational-wave (GW) emission for a static Horndeski black hole with secondary scalar hair, characterized by an effective pressure parameter peff. Using the rational-orbit taxonomy $(z,w,v)$, we analyze the innermost stable circular orbit (ISCO), marginally bound orbit (MBO), and periodic bound orbits. We find that increasing peff systematically shifts orbital radii outward and modifies the rational parameter q. Within the test-particle and quadrupole approximations, we estimate GW signals from an extreme mass-ratio inspiral (EMRI) system at a luminosity distance of DL=8000\,pc. The characteristic strains reach ~10⁻¹⁸ in the mHz band, partially overlapping the nominal LISA sensitivity range. These results demonstrate that Horndeski corrections systematically alter strong-field orbital dynamics and GW spectra, offering a phenomenological probe of deviations from general relativity.