  Silicate dust grains in the interstellar medium are known to be mostly
amorphous, yet crystalline silicate grains have been observed in many
long-period comets and in protoplanetary disks. Annealing of amorphous silicate
grains into crystalline grains requires temperatures > 1000 K, but exposure of
dust grains in comets to such high temperatures is incompatible with the
generally low temperatures experienced by comets. This has led to the proposal
of models in which dust grains were thermally processed near the protoSun, then
underwent considerable radial transport until they reached the gas giant planet
region where the long-period comets originated. We hypothesize instead that
silicate dust grains were annealed in situ, by shock waves triggered by
gravitational instabilities. We assume a shock speed of 5 km/s, a plausible
value for shocks driven by gravitational instabilities. We calculate the peak
temperatures of micron and submicron amorphous pyroxene grains of chondritic
composition under conditions typical in protoplanetary disks at 5 - 10 AU. Our
results also apply to chondritic amorphous olivine grains. We show that {\it in
situ} thermal annealing of submicron and micron-sized silicate dust grains can
occur, obviating the need for large-scale radial transport.
