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Scott, As Paul indicated you can make it work if the az circle is stuck to the baseplate: here's how I used to use an ancient Celestron 8 in altaz mode many years ago:
1. Choose a bright star low near the east or west horizon, and calculate the azimuth of this.
2. Turn your scope so it reads this azimuth value on the base. Now turn the whole base so the scope is pointing at the star - you can move the tube up and down (altitude) as much as you like.
3. Repeat for a star near the other west/east horizon.
You will probably find that you won't be able to eliminate the errors because this is sensitive to any lateral offset of the circle with respect to the centre of rotation in azimuth, however you may be able to get it with 2 degrees or so.
If you're good at mental arithmetic, an alternative solution is:
1. Choose a star and calculate its azimuth, low in the east/west as per above,
2. Plonk the base down on the ground with the circle roughly aligned to north, within say 30 degrees;
3. Point the scope directly at the selected star and note the azimuth as indicated on the circle. Subtract this from the calculated azimuth to get the azimuth error.
4. You can write this down, or do as I did - using some electrical tape or similar, put a marker on the circle corresponding to this error - this will effectively point to true north.
5. From then on each time you compute a star position add this offset to the azimuth.
Notes:
a) if your azimuth circle could slip around the base, at step 3 above the thing to do is to rotate the circle so that it reads the calculated azimuth of the star. But you said you glued it in place... ah well, it's a modification to carry out on a rainy day to fix this.
b) All of the above only works reasonably well if the base of the scope is level within the accuracy of the altitude circle (0.5 degree). If its tilted in some direction this will affect your calculations of the altitude position and you'll have to allow for this tilt - or level the scope.
c) In addition if the altitude circle is not concentric with the axis of rotation in altitude this will also create another source of error.
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