How small is your dog? Could have been a hawk or owl. If your dog is small like a squirrel or rabbit and out in yard by itself you might be concern since it could mistaken your dog for small wild animal. Anything larger I wouldn't be worried. Could have been a rodent of some kind in same area as you and that's what it was circling.
I think that big bird hovering around your dog is a vulture which feeds on carrion or dead meat. There are also the owls and hawks which are capable of preying on small dogs. So you should actually establish what type of bird is it so as to be able to offer maximum protection to your dog.
Where I live, large birds of prey are known to take small cats and very small dogs from a persons yard. It's mostly the Owls that concern my neighborhood. And It would have to be a small dog. ( a lap dog) .