Knowing that our solar system does not just stay in one place as it revolves around the center of the galaxy, how long will it take for our solar system to move out of the spiral arm it is currently in (the Orion arm) and into a different one? Also, will it stay roughly the same distance away from the center of the galaxy or will it move back (towards) and forth (away) from the center?
The solar motion on top of its circular orbit about the centre of the Galaxy (which has a period of about 230 million years) can be described by how fast it is going in three different directions:
U = 10 km/s (radially inwards)
V = 5 km/s (in the direction of Galactic rotation)
W = 7 km/s (northwards out of the plane of the Galaxy)
Of course the Sun won't keep on going in this direction forever. In fact we approximate its motion by an 'epicycle' on top of the mean motion around the Galaxy. The period of oscillation in and out of the plane of the galaxy (up and down) is about 70 million years. This means that we pass through the Galactic midplane about every 35 million years which some people have compared with the period between mass extinctions on Earth to come up with yet another doomsday theory. In fact it is true that the number of cosmic rays which hit the Earth will increase during the (about a) hundred thousand years we are closer to the Galactic plane. There have also been some plausible theories about the overall temperature of the Earth increasing (with the relevent climatic changes that implies).
In the plane of the galaxy the Sun is located in the small spiral arm we call the Orion arm (or local arm) which is really just connection between the two nearest major spiral arms (the Sagitarius and Perseus arms). There is a neat page on these structures: SEDS Milky Way Spiral Structure page. We pass through a major spiral arm about every 100 million years, taking about 10 million years to go through. During the transit, there would be a higher rate of 'nearby' supernova and possibly other so called 'environmental stresses' which could alter the climate of the Earth.
There is an interesting review of this (and other external influences on the climate of the Earth with reference to possible causes of the extinction of the dinosaurs) from which I get most of my figures. It's Russell, 1978 in the Annual Review of Earth and Planetary Science (ADS link).
This page was last updated on April 18, 2016.