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Kozai-Lidov (KL) oscillations in hierarchical triple systems have foundapplication to many astrophysical contexts, including planet formation, type Iasupernovae, and supermassive black hole dynamics. The period of theseoscillations is known at the order-of-magnitude level, but dependences on theinitial mutual inclination or inner eccentricity are not typically included. Inthis work I calculate the period of KL oscillations ($t_{\textrm{KL}}$) exactlyin the test particle limit at quadrupole order (TPQ). I explore the parameterspace of all hierarchical triples at TPQ and show that except for triples onthe boundary between libration and rotation, the period of KL oscillations doesnot vary by more than a factor of a few. The exact period may be approximatedto better than 2 per cent for triples with mutual inclinations between60$^{\circ}$ and 120$^{\circ}$ and initial eccentricities less than $\sim$0.3.In addition, I derive an analytic expression for the period of octupole-orderoscillations due to the `eccentric KL mechanism' (EKM). I show that thetimescale for EKM oscillations is proportional to$\epsilon_{\textrm{oct}}^{-1/2}$, where $\epsilon_{\textrm{oct}}$ measures thestrength of octupole perturbations relative to quadrupole perturbations.

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