Why would a broken alternator belt cause overheating?

As I was getting off the highway last night I noticed the battery light came on. I drove the a nearby parts store (~1 mile) and when I parked I was puking coolant from the overflow bottle. Had them test the battery, 97% charge. Had them test the alternator . . . nothing. The alternator belt was split and no longer on the pulley.

After I let the car cool off I made it the 2 miles home just fine, coasting with the car off on downhills to keep it cool. I don't think replacing the alternator belt will be a big problem.

Anyways, WHY would a broken alternator belt lead to running so hot? I was under the impression the water pump was driven off the timing belt, not the alternator belt. Is it an issue of insufficient power to run the fans without the alternator? Something else obvious I am overlooking?
 
the water pump is BEHIND the timing belt, which is why you would be best to replace it while you are in there for a timing belt. but, i "Think" the alternator belt spins the water pump pully.
 

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