  Recent N-Body simulations are in favor of the presence of a co-rotating Dark
Disk that might contribute significantly (10%-50%) to the local Dark Matter
density. Such substructure could have dramatic effect on directional detection.
Indeed, in the case of a null lag velocity, one expects an isotropic WIMP
velocity distribution arising from the Dark Disk contribution, which might
weaken the strong angular signature expected in directional detection. For a
wide range of Dark Disk parameters, we evaluate in this Letter the effect of
such dark component on the discovery potential of upcoming directional
detectors. As a conclusion of our study, using only the angular distribution of
nuclear recoils, we show that Dark Disk models as suggested by recent N-Body
simulations will not affect significantly the Dark Matter reach of directional
detection, even in extreme configurations.
