Magnetically linked star-disk systems.: I.: Force-free magnetospheres and effects of disk resistivity

We consider the interaction between a magnetic star and its circumstellar disk under the assumption that the stellar magnetic field permeates the disk and that the system's magnetosphere is force-free. Using simplified axisymmetric models (both semianalytic and numerical), we study the time evolution of the magnetic field configuration induced by the relative rotation between the disk and the star. We show that if both the star and the magnetosphere are perfectly conducting, then there is a maximum disk surface conductivity Sigma(max) for which a steady state field configuration can be established. For larger values of conductivity, no steady state is possible, and the field lines inflate and effectively open up when a critical twist angle (which for an initially dipolar field is on the order of a few radian) is attained. We argue that for thin astrophysical disks, surface conductivities are likely to exceed the local Sigma(max) except in the immediate vicinity of the corotation radius in a Keplerian disk. If the disk conductivity is high enough, then the radial magnetic field at the disk surface will become large and induce radial migration of the field lines across the disk. We find, however, that the radial diffusion in the disk is generally much slower than the field-line expansion in the magnetosphere, which suggests that the opening of the magnetosphere is achieved before the diffusive outward expulsion of the field lines from the disk can occur. The effects of magnetospheric inertial effects and of field-line reconnection are considered in the companion paper.