  The High Mass X-ray Binary (HMXB) SMC X-1 demonstrates an orbital variation
of 3.89 days and a super-orbital variation with an average length of 55 days.
As we show here, however, the length of the super-orbital cycle varies by
almost a factor of two, even across adjacent cycles. To study both the orbital
and super-orbital variation we utilize lightcurves from the Rossi X-ray Timing
Explorer All Sky Monitor (RXTE-ASM). We employ the orbital ephemeris from
Wojdowski et al. (1998) to obtain the average orbital profile, and we show that
this profile exhibits complex modulation during non-eclipse phases.
Additionally, a very interesting ``bounceback'' in X-ray count rate is seen
during mid-orbital eclipse phases, with a softening of the emission during
these periods. This bounceback has not been previously identified in pointed
observations. We then define a super-orbital ephemeris (the phase of the
super-orbital cycle as a function of date) based on the ASM lightcurve and
analyze the trend and distribution of super-orbital cycle lengths. SMC X-1
exhibits a bimodal distribution of these lengths, similar to what has been
observed in other systems (e.g., Her X-1), but with more dramatic changes in
cycle length. There is some hint, but not conclusive evidence, for a dependence
of the super-orbital cycle length upon the underlying orbital period, as has
been observed previously for Her X-1 and Cyg X-2. Using our super-orbital
ephemeris we are also able to create an average super-orbital profile over the
71 observed cycles, for which we witness overall hardening of the spectrum
during low count rate times. We combine the orbital and super-orbital
ephemerides to study the correlation between the orbital and super-orbital
variations in the system.
