0920009

We propose to observe Her X-1 during INTEGRAL s AO-9 for a total observing time of 600 ksec: covering one Main-On with two INTEGRAL orbits the first near the maximum of the Main-On the second during its decay plus one INTEGRAL orbit during the following Short-On . Our general objective is to continue the long-term covering of this enigmatic source by INTEGRAL in order to investigate two (intimately related) aspects: A. the physics of generating the cyclotron line and the reasons behind the variability of Ecyc as a function of time luminosity 35 day precession-phase and 1.24 s pulse-phase and B. the question whether the neutron star (NS) does really precess with a period of 35 days in strict synchronization with the accretion disk. If the observed systematic variations of the shape of the pulse profiles are indeed due to the NS s precession then we would have to conclude that the precessional period is changing by a few percent every few years which most likely requires some changes in the structure of matter and/or magnetic field of the NS s interior. This may open a new channel to the understanding of the properties of matter at supra-nuclear densities. On the other hand some observational results seriously challenge the idea of NS precession in Her X-1. [truncated! Please see actual data for full text]