What if we just built a really long, straight particle accelerator in space, and we just shot all our highly accelerated particles outside of the solar system? Aside from really ruining someone aliens day in a billion years, what’s the disadvantage of doing this?
The nice thing about rings or other methods like racetracks etc is that the particles come around and go through the same accelerator structure multiple times. Secondly, if you want to collide two beams, the advantage of a ring is that each particle has multiple chances of scattering of the other beam, as it "comes around" to the intersection region millions of times.
That being said, linear accelerators are a thing, as they have other advantages.
