I love my Death Star. There is nothing more thrilling than pulverizing a planet and feeling the inhabitants screaming out in terror.

However, my Death Star seems to have developed a case of Kessler Syndrome. Fragments of the planets, moons, and ships that we have destroyed get pulled into orbit around our space station. They crash into each other, breaking up into smaller pieces, and forming a debris field.

This has caused problems with our fully armed and operational battlestation. Wreckage has damaged portions of the base's superstructure. Several TIE fighters sent out on patrol have been lost by collisions with debris. And every time a dignitary visits on a shuttlecraft, I cross my bionic fingers that they arrive and depart safely. I am so frustrated that I feel like strangling an admiral.

What can be done to rid my moon-sized fortress of this pitiful problem?

This question differs from other (1,2) kessler-syndrome questions, as this question is specific to the Death Star and because answers do not need to be science-based.

Your Death Star has a hyperdrive, right? Well, the debris doesn't.

Even if it maneuvers just at sublight speed, debris won't settle into orbit. If there is a debris problem, it is being hit by fragments, not being orbited by fragments. You'd probably want to avoid that.

If we think back to TESB, the Star Destroyer Avenger dumped trash which was presumably not taken along into FTL. So debris in close proximity but not in contact with the Death Star will not be taken along, either.

Well assuming its just like in Star Wars, you have the technology to control gravity. Just turn it off for a few hours, maybe a day, and all the debris will naturally move away under its own inertia.

Capitalism

Sell salvage rights to the worlds you don't destroy. They'll pay you for the privilege.

Most of the exploded planet bits will not stick around in your orbit as you move from star system to star system. There will still be a fair amount of debris, so salvage won't get rid of all of it. But it will make a sizable dent. Enough to greatly reduce damage to your shuttles and other craft, as well as to your Death Star's systems.

Every year or so, make a stop at Al's Pancake House and Super Salvage World. Give your crew a much needed vacation and let Al's equipment attract all that debris away from you. Leave lighter and freer and ready to take on new challenges in planet busting.

Do you not have a giant death ray at your disposal?

Dial it down a notch and use it to gently push the debris away, or dial it up and turn the debris near you into an ionised has you can then siphon into your very scientifically rigorous plasma conduits.

Either way: bye bye debris!

Do a fly-by near a planet (preferably with atmosphere).

You might think that your Death Star is huge, but actually that is just because it is yours.

When compared to most planets, your Death Star is small and, more importantly, light¹.

Just go near enough of a big orbital body and let that body gravity's do the work. There may be some inconveniences by part of the debris landing on your Death Star (instead of on the cleaning body), but if that was much of an issue your Death Star would already suffered a lot from the impacts when the Kessler body was created.

Additionally, if your selected celestial body has an atmosphere, the upper layers of it, while thin, can still help to slow the Kessler objects and make them fall from orbit (in either direction) faster.