  The accreting X-ray pulsar Her X-1 shows two types of long-term variations,
both with periods of ~35 days: 1) Turn-on cycles, a modulation of the flux},
with a ten-day long Main-On and a five-day long Short-On, separated by two
Off-states, and 2) a systematic variation in the shape of the 1.24 s pulse
profile. While there is general consensus that the flux modulation is due to
variable shading of the X-ray emitting regions on the surface of the neutron
star by the precessing accretion disk, the physical reason for the variation in
the pulse profiles has remained controversial. Following the suggestion that
free precession of the neutron star may be responsible for the variation in the
pulse profiles, we developed a physical model of strong feedback interaction
between the neutron star and the accretion disk in order to explain the
seemingly identical values for the periods of the two types of variations,
which were found to be in basic synchronization. In a deep analysis of pulse
profiles observed by several different satellites over the last three decades
we now find that the clock behind the pulse profile variations shows exactly
the same erratic behavior as the turn-on clock, even on short time scales (a
few 35 d cycles), suggesting that there may in fact be only one 35 d clock in
the system. If this is true, it raises serious questions with respect to the
idea of free precession of the neutron star, namely how the neutron star can
change its precessional period every few years by up to 2.5% and how the
feedback can be so strong, such that these changes can be transmitted to the
accretion disk on rather short time scales.
