Your most recent planets classify as borders, and they are denoted by a red "b" next to them.
The best way to insure you keep a good planet is to always leave yourself enough turns to plunder unwanted planets below the good one on your list, and to explore and/or capture junk planets to have above the good planet, as a buffer.
This process is a key part of the tension and fun of the game, as you are often left scrambling to protect a prize you just found.
Clustering is also a big part of this process. Each race only has so many "slots" for planets, and after a certain number of planets are aquired, it is impossible to aquire more until some of the current planets are "clustered". You can make additional room in your planet slots by clustering.
The ability to cluster is something you research.
Good luck and welcome to the game. I play on the Ultra server. Cheers.