  The CoGeNT and CRESST WIMP direct detection experiments have recently
observed excesses of nuclear recoil events, while the DAMA/LIBRA experiment has
a long standing annual modulation signal. It has been suggested that these
excesses may be due to light mass, m_chi ~ 5-10 GeV, WIMPs. The Earth's motion
with respect to the Galactic rest frame leads to a directional dependence in
the WIMP scattering rate, providing a powerful signal of the Galactic origin of
any recoil excess. We investigate whether direct detection experiments with
directional sensitivity have the potential to observe this anisotropic
scattering rate with the elastically scattering light WIMPs proposed to explain
the observed excesses. We find that the number of recoils required to detect an
anisotropic signal from light WIMPs at 5 sigma significance varies from 7 to
more than 190 over the set of target nuclei and energy thresholds expected for
directional detectors. Smaller numbers arise from configurations where the
detector is only sensitive to recoils from the highest speed, and hence most
anisotropic, WIMPs. However, the event rate above threshold is very small in
these cases, leading to the need for large experimental exposures to accumulate
even a small number of events. To account for this sensitivity to the tail of
the WIMP velocity distribution, whose shape is not well known, we consider two
exemplar halo models spanning the range of possibilities. We also note that for
an accurate calculation the Earth's orbital speed must be averaged over. We
find that the exposures required to detect 10 GeV WIMPs at a WIMP-proton
cross-section of 10^-4 pb are of order 10^3 kg day for a 20 keV energy
threshold, within reach of planned directional detectors. Lower WIMP masses
require higher exposures and/or lower energy thresholds for detection.
