I ran into the same thing a few weeks back.
http://forums.arcade-museum.com/showthread.php?t=232980&highlight=asteroids+ring
Everything pointed to a new tube. Ultimately I feel that is the best solution. I did, however, come up with something that worked in my situation.
Started with a dead monitor. After a rebuild, and eventually a diode cleaning, I had a beautiful picture but a ring and dot after a bit of run time. During the rebuild I did rejuvenate the tube on my Sencore. I noticed it would not pass a lot of current. Barely a blip on the meter. Having seen this on B/W's in the past I did not think much of it.
After seeing the ring and dot, and after the research, I was at the point of a tube swap. I decided, since the tube was on the ropes as it was, to beat the bat snot out of it with the rejuninator. My thinking was the beam crosses through the center of the tube some untold number of times a second as it draws play field elements. As it leaps element to element, passing through 0,0 (center), the Z Amp is off...but what if the gun had so much crud on it that it retained some heat and could not react fast enough to totally kill the beam? Not enough to be visible like retrace, but enough that it would sluff off just enough to tickle the small, dime sized, bit of phosphor that it was consistently sweeping across. If you have ever worked on an electric hot water heater in an area with hard water such a phenomenon occurs. Enough build up on the elements traps heat in the element so that it can not dissipate it in the water. The build up acts and an insulator and slows down the heat transfer. Eventually, with enough build up, the element overheats and it pops the thermal protection breaker. Could be way off base, and obviously the comparison is somewhat odd, but it seemed plausible.
So, I pulled the monitor, put it face down, and hit it with the max rejuvenation setting on my Sencore, let it cool about a minute, and zapped it again. I probably hit it 4 to 6 times. Eventually, I started to get reasonable readings on the meter representing current passing through the gun. I put it back together and was happy with the result. It was still there, but very, vary faint even after hours of run time. I had to point it out to the guy that bought the game. He is a huge Asteroids fan, a very good player, and he did not notice it on his own even after I had described it as being there via a phone call.
One other thing I did was pay attention to brightness and contrast settings. Double check the manual but I want to say you should let it warm up, dial both all the way down, then tick up brightness until an image is just barley visible. At that point you use contrast to bring the image the rest of the way up. I did not go whole hog with contrast to make the image Sun bright. It was not dim either. The intent was to make it look great but not get it so bright I was causing my problem to worsen. I did notice that the brighter I left the final image the faster, and the brighter, the ring and dot appeared. In my game room, with my personal Asteroids, I don't keep it super bright either as, with the room lights down, the game elements leave trails as they move. It's less noticeable if the room is nicely lit.
