Single wire alternators need excitation and get none of it if that wire is disconnected and can't get it from the battery. If your system was converted or had the alternator replaced only recently be sure an automotive alternator wasn't installed. There are different number sequences to tractor alts, partly to protect OEM suppliers from lo-buck automotive competition, but also for their unique differences. Car stuff operates at higher rpm and doesn't always get up to speed or generate full amperage on tractor installations.
My Deere has a car alternator because the PO saw a big saving when recognizing it's drop-fit compatibility and omitting a pricey regulator, a poor way to cut costs. As a result I get little to no charging within the rpm ranges I usually operate at. You'd have to check a single wire type with the battery connected. There are cheap LED testers that show what's going on, and you could hook up a cig lighter socket with alligator clips if you don't plan to install one permanently.
If you park your Ford outside, corrosion can cause brushes to seize and lose contact after wearing down a bit, or they could be worn short enough to lose spring tension. Boggers sometimes splash muddy water onto/into theirs, and I've seen guys rap the housings, as has long been done with starters, to break 'em loose & get 'em home. It's not too difficult to replace brushes, even if left with an auto repair manual that at least instructs on your particular type.
I suggest you connect a voltmeter to your battery, run up to 1500 rpm or more, and see if the voltage increases when you connect the alternator's wire. It should jump to ~13.5 to ~14.5v if everything's ok. If the battery isn't fully charged & there's no increase you have a problem for sure. Starter/generator repair shops usually understand tractor-specific alts, so be sure to ask if you call around. They'd be better able than auto part suppliers to test yours properly, and may be able to replace diodes or other internals. btw: you might have them look up the number stamped into the housing to verify whether you have a car alt. If yours fails a voltage test you may have to spring for the real deal.
Don't go on my word alone, I'm sure others will jump in. If there's a repair shop nearby, call & get it in for a checkup. They'd probably appreciate the business, and might save 1/2 the cost of new by rebuilding what you have. If it's 'car stuff' that's on your 851 you might have to start over.