We prove that the mapping torus group of any automorphism of a free group of finite rank n => 2 is weakly hyperbolic relative to the canonical (up to conjugation) family of subgroups of the free group which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of the automorphism. Furthermore, we show that the mapping-torus group is strongly hyperbolic relative to the mapping torus of this canonical family.