Constraints on models of fallback discs around millisecond magnetars

Some models of gamma-ray burst (GRB) afterglows invoke fallback discs interacting with the magnetospheres of nascent millisecond magnetars and spinning down with magnetic dipole radiation. Initially, the accretion rate in the fallback disc is very high, well above the rate required for the Eddington limit to match the observed luminosity, even if there is strong beaming. The inner parts of such a disc, even if it is cooling by the neutrino emission, get spherical due to the radiation pressure, which regulates the mass accretion rate within the spherization radius. Such a disc can not penetrate the light cylinder radius for the typical magnetic fields, and the initial spin frequencies invoked for the magnetars. As a result of the auto-regulation of the accretion flow, the fallback disc can not interact directly with the neutron star's magnetosphere within the first few days the GRB afterglows are observed. When the interaction of the disc with the magnetosphere starts, the system's luminosity is too low to address the X-ray afterglows. We conclude that a fallback disc model, which must include the effects of the radiation pressure, can only address GRB afterglows with unusually low luminosities, i.e., $\lesssim 10^{45}\,\mathrm{erg\,s^{-1}}$.