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Abstract

We present the results of precision gamma-ray timing measurements of the binary millisecond pulsar PSR J2339$-$0533, an irradiating system of "redback" type, using data from the Fermi Large Area Telescope. We describe an optimized analysis method to determine a long-term phase-coherent timing solution spanning more than six years, including a measured eccentricity of the binary orbit and constraints on the proper motion of the system. A major result of this timing analysis is the discovery of an extreme variation of the nominal 4.6-hour orbital period $P_{\rm orb}$ over time, showing alternating epochs of decrease and increase. We inferred a cyclic modulation of $P_{\rm orb}$ with an approximate cycle duration of 4.2 years and a modulation amplitude of $\Delta P_{\rm orb}/ P_{\rm orb} = 2.3 \times 10^{-7}$. Considering different possible physical causes, the observed orbital-period modulation most likely results from a variable gravitational quadrupole moment of the companion star due to cyclic magnetic activity in its convective zone.