Is shooting down a drone a civil offense?

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You don't own the airspace over your house, think about planes that fly over. I'm not certain where in the air your property line starts though.

I just read an article that stated you own somewhere between 80-500ft of airspace over your home. The Supreme Court has refused to explicitly state the number, though.

Before air travel you owned an infinite column of airspace over your home.

LINK

There was a drone thread a while back posted by a guy who suspected his neighbor was taking pics of his wife while she was sunbathing. He wanted to break out his goose gun and start shooting. I suggested letting his kids throw wet nerf footballs at it. If hit, it would come down and he wouldn't get busted for discharging a firearm in the neighborhood.

Prolly.

La. C.C. art. 2315. Liability for acts causing damages
A. Every act whatever of man that causes damage to another obliges him by whose fault it happened to repair it.

If someone pulls into your driveway past a "No Trespassing" sign can you shoot then? Or their car?

For the drone, criminal violation, probably felony, and likely responsibility for paying for the damage you cause, i.e., buying them another drone and whatever it might have been carrying.

Are they breaking the law, probably. Call the police. Unless they try to fly it into your window, in which case, fire away.

All of my guns shoot themselves. Often at drones for no reason. I tell them to stop but they don't listen.