Are cosmic rays effective for ionization of the solar nebula?

In this paper, we argue that the effectiveness of cosmic rays to ionize the bulk of the nebular gas may be further impaired by the influence of the magnetic field on the propagation of cosmic rays. When cosmic rays enter the nebular disk they ionize the gas and make the dynamo generation of magnetic fields possible. However, once magnetic fields are embedded in the nebular gas, the upcoming cosmic rays can no longer penetrate directly into the nebular disk because they start to interact with the magnetic field and lose their energy before propagating significantly toward the midplane. That, in turn, undercuts the ionization source within the bulk of the gas stopping the dynamo action. Nebular dynamo models ignored this back reaction of magnetic fields on cosmic rays. We calculate this back reaction effect, but for the sake of mathematical simplicity, we ignore the effect of magnetic field weakening due to diminishing ionization by cosmic rays.