What is it that makes police power legitimate?

Answer 1

The power of law. That is to say, that you can trace authority to a source of law. If you can't trace it back, it is not legitimate. I can only speak to the U.S.

The U.S. Constitution is the supreme law of the land. It provides the federal givernment with certain enumated powers. The remained are reserved by the states (Amendment 10). The remainder typically includes the traditional policing power. So, for most police entities (outside of federal law enforcement entities, which derive their power from federal law passed by congress), derive thir power from state law (passed by state legislatures). Some states delegate, by law, police functions to municipalities (i.e. cities) or counties. In any event, it can all be traced back to the same document the U.S. constitution.