Artist's illustration of disintegrating asteroid P/2013 R3
This illustration shows one possible explanation for the disintegration of asteroid P/2013 R3.
It is likely that over the past 4.5 billion years the asteroid was fractured by collisions with other asteroids. The effects of sunlight will have caused the asteroid to slowly increase its rotation rate until the loosely bound fragments drifted apart due to centrifugal forces. Dust drifting off the pieces makes the comet-looking tails. This process may be common for small bodies in the asteroid belt.