In conventional theodolites (which is basically what you have) 0,0 is true north and pointed at the horizon. Therefore there will be many trignometric algorithms available for conversion to equatorial coordinates. Your Library will no doubt have a book of algorithms for the real world. The Two star alignment algorithm will extract your 0,0 coordinates which can be refined through further observation and correction. I have seen that published here on the web a few times. Of course hard stop resets for 0,0 will speed setup in future.

Bob is right though. At a one second interval most satellites will be long gone. Perhaps you mean that they apply a correction to the existing tracking every second.

Checking for drift will prove your system but as typically applied, no it will not be useful for polar alignment because you do not have a polar axis

When you get this working, send us a spec list.

Cheers,
-Rick