Solenoid, Starter, Alternator Oh my!

[aldedmon]

My 93 500E has a problem.
Whenever it sits for an extended period, the car turns over slowly like the lights were left on for an hour. No really, really slow, but just not as happy as it should. Now that the cold has sat in, If I park the car outside in 38 degreee weather for 12 hours, it starts as though the lights were left on for maybe a 4 hours or so hours.(the lights on analogy will hopefully make sense to those reading. Its really slow , but still a smooth spin and not the Whaou, Whaou, Whaou, Whaou..tick, tick, tick...of the proverbial dead battery.
So, Its a new battery. Even after putting it on, it still never really spun free, and fast like my 400E does. ...Well maybe when I first put it in, but not after the first week of storage. I only drive it on the weekend.
So how can I differentiate if its the Solenoid, Starter, or the Alternator