I was always under the assumption the alt runs the car while the engine is on.
If the alternator is a three-diode design then once spinning it is self sustaining. I have read of early alternator designs where the Field Current source is solely the battery. Remove it from the circuit and the alternator ceases to produce power. Same is true for a truly dead battery (versus just a drained one). Jump the vehicle - up and running. Remove the booster cables and the engine dies . . .
Here's a link to an article that describes a three diode alternator circuit including how to interpret the warning light (dim/bright) . . .
Understanding Alternators
