Since it is a National Historic Site, you should check with the rangers at the Visitors Center before using any metal detectors. Anything you find may be considered US government property.
People have been bringing metal detectors for years. If you find anything but jewelry or money that may have historical value, it belongs to the government. No you cannot use it on the fort itself. Only the outside grassy areas.
I don't think so, since its U.S. property!
Yes, it's not really fenced so you can probably bring it. I don't think you'll be able to perform any sort of digging in the historical area, though
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