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Timescales of Kozai-Lidov oscillations at quadrupole and octupole order in the test particle limit

Kozai-Lidov (KL) oscillations in hierarchical triple systems have found application to many astrophysical contexts, including planet formation, type Ia supernovae, and supermassive black hole dynamics. The period of these oscillations is known at the order-of-magnitude level, but dependences on the initial mutual inclination or inner eccentricity are not typically included. In this work I calculate the period of KL oscillations ($t_{\textrm{KL}}$) exactly in the test particle limit at quadrupole order (TPQ). I explore the parameter space of all hierarchical triples at TPQ and show that except for triples on the boundary between libration and rotation, the period of KL oscillations does not vary by more than a factor of a few. The exact period may be approximated to better than 2 per cent for triples with mutual inclinations between 60$^{\circ}$ and 120$^{\circ}$ and initial eccentricities less than $\sim$0.3. In addition, I derive an analytic expression for the period of octupole-order oscillations due to the `eccentric KL mechanism' (EKM). I show that the timescale for EKM oscillations is proportional to $ε_{\textrm{oct}}^{-1/2}$, where $ε_{\textrm{oct}}$ measures the strength of octupole perturbations relative to quadrupole perturbations.