My old 1250 Hydrostatic Cub is having some electrical problems and I am
trying to figure out if it is the alternator or battery (or maybe
something else).

Observations:
- I can successfully jump start the tractor with my truck, and it seems
to run just fine, however, the Discharge/Charge meter is showing
slightly Discharge.

- When I turn on the PTO, it runs for a little while (few minutes),
then just dies. If I turn off the PTO, it typically comes back to
life, but turning on PTO just causes it to shutdown. PTO will run
again after a jump start.

- I took off the right (sitting in the seat) cover panel to reveal what
is labeled "Alternator Electrical System." I assume this is the
alternator. Anyway, when I jump started the tractor, this device
moved, but the entire time the tractor was running (attempting PTO
again and jogging throttle) the device did not appear to move. Of
course, all I can see of it is the rubber part that you can rotate with
your hand (maybe something internal is moving?).

- While the tractor is running, the voltage across the battery
terminals is around 8V.

- The tractor will not continue to run if the battery is disconnected.

- I put an ohmmeter on the battery and did not read any resistance
under 20 KOhms (I was possibly trying to see if the plates were
shorted).

- The battery voltage (no load) was reading about 11.8V, but as soon as
I turned the key, it dropped to 4V.

Questions:
- Is the alternator running and I can't tell?
- Is the alternator always supposed to run?
- Could a bad battery cause it to not run?
- Is the alternator bad?
- Is the battery bad?

Thanks in advance-

Chris