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We present the results of precision gamma-ray timing measurements of thebinary millisecond pulsar PSR J2339$-$0533, an irradiating system of "redback"type, using data from the Fermi Large Area Telescope. We describe an optimizedanalysis method to determine a long-term phase-coherent timing solutionspanning more than six years, including a measured eccentricity of the binaryorbit and constraints on the proper motion of the system. A major result ofthis timing analysis is the discovery of an extreme variation of the nominal4.6-hour orbital period $P_{\rm orb}$ over time, showing alternating epochs ofdecrease and increase. We inferred a cyclic modulation of $P_{\rm orb}$ with anapproximate cycle duration of 4.2 years and a modulation amplitude of $\DeltaP_{\rm orb}/ P_{\rm orb} = 2.3 \times 10^{-7}$. Considering different possiblephysical causes, the observed orbital-period modulation most likely resultsfrom a variable gravitational quadrupole moment of the companion star due tocyclic magnetic activity in its convective zone.

Gamma-ray Timing of Redback PSR J2339-0533: Hints for Gravitational Quadrupole Moment Changes