Yes, using long long integers might work in some parts of the code, but there are operations that need to be carried out as floating point (taking exponents, trig functions, etc.) or would involve significant code-contortions. If it's possible to just load a separate library and get bigger floats, I would prefer that approach to writing obfuscated mathematical code (i.e., trying to take the cosine of an integer and square it, etc.).
It's true that this doesn't need to be fast -- anything in the sky is going to make only one full revolution in 86400 seconds (1 day). I'm also reasonably confident that this is well within the capabilities of the Arduino, since -- although the required precision is high -- there are relatively few variables at any given point and the speed requirement is minimal.
Regarding what will be done with the results: that depends on whether I end up even getting good results -- this is my first adventure with Arduino. Eventually I'd like to drive a heliostat (servo- or stepper-controlled tracking mount), but that hasn't been built yet nor have parts even been purchased.
-Jon