  Given a smooth curve with weighted marked points, the Abel-Jacboi map
produces a line bundle on the curve. This map fails to extend to the full
boundary of the moduli space of stable pointed curves. Using logarithmic and
tropical geometry, we describe a modular modification of the moduli space of
curves over which the Abel-Jacobi map extends. We also describe the attendant
deformation theory and virtual fundamental class of this moduli space. This
recovers the double ramification cycle, as well as variants associated to
differentials.
