  Accretion of matter onto the surface of a freely precessing NS with a complex
non-dipole magnetic field can explain the change of X-ray pulse profiles of Her
X-1 observed by RXTE with the phase of the 35-day cycle. We demonstrate this
using all available measurements of X-ray pulse profiles in the 9-13 keV energy
range obtained with the RXTE/PCA. The measured profiles guided the elaboration
of a geometrical model and the definition of locations of emitting poles, arcs,
and spots on the NS surface which satisfactorily reproduce the observed pulse
profiles and their dependence on free precession phase. We have found that the
observed trend of the times of the 35-day turn-ons on the O-C diagram, which
can be approximated by a collection of consecutive linear segments around the
mean value, can be described by our model by assuming a variable free
precession period, with a fractional period change of about a few percent. We
propose that the 2.5% changes in the free precession period that occur on time
scales of several to tens of 35-day cycles can be related to wandering of the
principal inertia axis of the NS body due to variations in the patterns of
accretion onto the NS surface. The closeness of periods of the disk precession
and the NS free precession can be explained by the presence of a
synchronization mechanism in the system, which modulates the dynamical
interaction of the gas streams and the accretion disk with the NS free
precession period.
