1220032

We propose to observe Her~X-1 during INTEGRALs AO-12 for a total observing time 400 ksec: covering a substantial part of one ``Main-On with two INTEGRAL revolutions the first near the maximum of the ``Main-On the second during its decay.Our general objective is to continue the long-term covering of this enigmatic source by INTEGRAL in order to investigate the physics of generating the cyclotron line and the reasons for thevariability of E_cyc in particular as a function of time. There is now clear evidencethat E_cyc decreased by >4 keV between 1996 and 2012. The physics behind this behavior is not understood so far but is most likely connected to fundamental aspects of highly magnetized neutron stars: its structure and the physics of accretion. We speculate that the decrease of E_cyc is due to a progressing burial or screening of the magnetic field at the polar cap by the continuing accretion. The observational result of the secular decay of a cyclotron line energy is unique and was only possible through an intense effort of monitoring Her~X-1 with all satellite instruments capable of covering the hard X-ray range. A further objective is to continue to investigate the variation of E_cyc as a function of luminosity and of phase of the 35 day variation. In case ofacceptance of this INTEGRAL proposal NuStar will make every effort to schedule acoordinated observation. [truncated! Please see actual data for full text]