In previous work (Gates and Delgado-Aparicio 2012 Phys. Rev. Lett.108 165004), the onset criterion for radiation-driven islands (Rebut et al 1985 Proc. 10th Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research 1984 (London, UK, 1984) vol 2 (Vienna: IAEA) p 197) in combination with a simple cylindrical model of tokamak current channel behaviour was shown to be consistent with the empirical scaling of the tokamak density limit (Greenwald et al 1988 Nucl. Fusion28 2199). A number of the unexplained phenomena at the density limit are consistent with this novel physics mechanism. In this work, a more formal theoretical underpinning, consistent with cylindrical tearing mode theory, is developed for the onset criteria of these modes. The appropriate derivation of the radiation-driven addition to the modified Rutherford equation (MRE) is discussed. Additionally, the ordering of the terms in the MRE is examined in a regime near the density limit. It is hoped that, given the apparent success of this simple model in explaining the observed global scalings, it will lead to a more comprehensive analysis of the possibility that radiation-driven islands are the physics mechanism responsible for the density limit. In particular, with modern diagnostic capabilities detailed measurements of current densities, electron densities and impurity concentrations at rational surfaces should be possible, enabling verification of the concepts described above.