VW Jetta - New starter whine?

I recently had the starter motor on my 2002 VW Jetta die. When it started grinding, I opened it up and lubed the gears, which seemed to solve the problem for a month or so, but it started grinding again once it got cold, and the just wouldn’t start.

So, I put in a rebuilt motor that I got from Autozone, and the car starts fine. However, the new motor is /much/ louder than the old one, sounding something like a high-pitched whine (not grinding at all). The sound starts when I turn the key, and stops when the motor goes off, so I’m sure that it’s with the new motor, but since it isn’t grinding… what’s up?

Do I need to take it in to a mechanic and have them take a listen, or is this probably just a function of the new motor/solenoid being louder than the old one?