A tale of two periods: determination of the orbital ephemeris of the super-Eddington pulsar NGC 7793 P13.

We present a timing analysis of multiple XMM-Newton and NuSTAR observations of the ultra-luminous pulsar NGC 7793 P13 spread over its 65d variability period. We use the measured pulse periods to determine the orbital ephemeris, confirm a long orbital period with P_orb = 63.9 (+0.5,-0.6) d, and find an eccentricity of e <= 0.15. The orbital signature is imprinted on top of a secular spin-up, which seems to get faster as the source becomes brighter. We also analyse data from dense monitoring of the source with Swift and find an optical photometric period of 63.9 +/- 0.5 d and an X-ray flux period of 66.8 +/- 0.4 d. The optical period is consistent with the orbital period, while the X-ray flux period is significantly longer. We discuss possible reasons for this discrepancy, which could be due to a super-orbital period caused by a precessing accretion disk or an orbital resonance. We put the orbital period of P13 into context with the orbital periods implied for two other ultra-luminous pulsars, M82 X-2 and NGC 5907 ULX and discuss possible implications for the system parameters.