  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.

High Energy Astrophysical Phenomena