When a spacecraft approaches a moon or planet at high speed, it makes a close approach, curves around, and returns nearly the way it came. If they approach head-on, the spacecraft can gain speed, like a ping-pong ball bouncing from an approaching paddle. This is termed a slingshot maneuver.
I am about to experience a slingshot maneuver with Moscow. My visa expires on Saturday, and the last flight I could get was Friday. 5 other visitors and I had originally gotten single-entry visas for Russia. Our friends here did a great deal of research, and were told by the authorities that, since we were visiting the special zone for rocket launches, we would not need visas for Kazakhstan; and indeed would not actually be leaving Russia. The first was true; the second was not, in practice. The official at the exit insisted on stamping our visas as we left, thus invalidating them. We could not legally re-enter Russia.
After a a great many phone calls, and 3 hours of working on forms and finding the correct officials, the Russian consul at Domodedovo Airport agreed to issue us 5-day visas. They can't be extended while I am in Russia. The trouble and expense of flying to another country to obtain a third visa didn't seem worth it. Consequently, I will leave about 9 days earlier than expected. This is far better than the alternatives, of being deported or issued visas for a few hours. It's a change in plans. However, the scientists here are extremely busy in the aftermath of the launch, and are getting busier as the spacecraft is readied for operation. So, it's probably just as well that I am leaving early.