Assuming there is an (infinitely) large sphere (the sky) that rotates counterclockwise as seen from above around its axis of rotation (which goes through its center point) at a fixed rate ω_sky, with a point (star) on its inner surface at 0° ≤ φ_star < 360°, -90° ≤ θ_star ≤ 90°.
In the center, a vector (a startracker) is aligned with the axis of rotation toward the north pole, but misaligned by a degree (say, θ_tracker = 90°- , =1°, φ_tracker=0°) and rotates at the same rate and in the same direction as the sky.
Finally, a vector (camera) is fixed to the rotation of that vector, but is initally pointed to the star.
Over a duration (full rotation of both sky + (tracker+camera)), from the perspective of the camera, the point/star describes a closed path.
Viewed from the outside of the sphere, fixed on the changing point on the sphere the camera points to (red dot):
From the inside, showing how the star/fixed point on the sphere changes position from the perspective of the camera:
Which when stacked in a long exposure can look like this:
Typical exposures are much shorter, so their resulting trail is just a very short segment of this closed path.
Is it possible to calculate the length of this path (star trail) in degrees over a certain range of (exposure) time analytically, for a given star position and misalignment ?