I was contemplating days and years. A day is one rotation of a celestial object. A year is one orbit around the center(ish) celestial object. So...
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Individual asteroids in the belt orbit the sun, each on a slightly different path. The positions of the asteroids relative to one another is continuously shifting, because the further out an orbit is, the slower it is.
The main population of the asteroid belt orbits between about 2.1 and 3.3 AU from the sun. The inner asteroids of that region orbit the sun in about 3.3 Earth years, and the outer ones in about 6 years.
Yes, the asteroid belt does indeed orbit the sun.
Not quite sure how the asteroids would stay out there if they were not in orbit. Some individual asteroids are not in the belt however.
Most asteroids in the Asteroid Belt have an orbital period of about 3-6 Earth years. This means that it takes these asteroids 3-6 times longer than Earth to make a trip around the sun.
Not all asteroids are in the Asteroid Belt. Sometimes collisions between asteroids, combined with the gravitational effects of Jupiter, can cause asteroids to leave the belt.