  In [5], Holroyd, Levine, M\'esz\'aros, Peres, Propp and Wilson characterize
recurrent chip-and-rotor configurations for strongly connected digraphs.
However, the number of steps needed to recur, and the number of orbits is left
open for general digraphs. Recently, these questions were answered by Pham [6],
using linear algebraic methods. We give new, purely combinatorial proofs for
these formulas. We also relate rotor-router orbits to the chip-firing game: The
number of recurrent rotor-router unicycle-orbits equals the order of the Picard
group of the graph, defined in the sense of [1], and during a period, the same
chip-moves happen, as during firing the period vector in the chip-firing game.
