I understand what you are saying, but where on the data sheet is the free air number?
Thermal resistance junction to ambient is what you are looking for, but I gotta tell you I don't see it on that data sheet. TO220 packages (like this one) are 62.5 C per watt.
I'm not trying to dissipate 50W, only 1/100 of that in conductive mode (0.5Watts conductive power dissipation). I would not expect to need a heatsink but what should I expect the temperature to be in the MOSFET without a heat sink.
Normally you would be correct, but in your case you must have like a 5 volt lamp if you are driving it from the high side with the micro? If thats the case your gate to source voltage is only 5 volts or so. The 25 Mohms is with 10 volts gate to source with your FET. -- See figure 3. If you want to drive it with 5 volts you need a
logic level FET. So you really have 2 problems: The gate to source voltage and the micro can't drive much current so it can't discharge the gate capacitance very fast so the FET won't switch fast. You can calculate switching time using Qg. This is the time it takes (in ns) for the FET to switch with 1 amp of gate drive. So in this case it's 197 divided by say 20 ma from the micro it would take 10 usec. to switch it.
But I also noticed the capacitor was getting hot to the touch, especially at the really low frequency (120Hz). Can somebody explain this and why the cap would get hot?
You will need to look at the ripple current in the cap. At 120 HZ it is really high.
I consider your suggestion to remove the inductor, cap, and diode...but would this have negative effects on the bulb lifespan? I'd image this would be harder on the bulb filament without the smoothing element of the buck converter arrangment.
It might make a slight difference in life but not much if you keep the frequency up to the point where the bulb doesn't have time to totally cool between cycles.