I am not a mechanical engineer, so just trying to figure this out. I want to attach a servo to my telescope focus, but the servo only has about 180 degrees motion, whereas you need to turn the knob about 540 degrees for the full range of focus. It has a rack and pinion gear, so my plan was to find a pinion gear with more teeth (maybe 48?) to get the full range in 180 degrees.
The pinion gear has 16 teeth and an outer diameter of about 8mm. I measured 10 teeth on the rack which was about 15mm. The diameter of the shaft is about 5mm. My question is, what size gear would I need, how many teeth, what pitch? Does it exists? My best guess is 48 teeth would be needed, but I am not sure about my pitch. Also, if the gear exists, do they make them to attach directly to a servo? Or will I need to get a shaft and a shaft to servo coupler?