The slow coning pattern of the spin axis of the earth's precession will move the end of this axis in a complete circle in a period

26000 years

Explanation:

Precession describes the angular motion of the Earth's body. Since the attitude of telescopes relative to the Earth's body can be controlled with high accuracy, and telescopes can measure the direction of incoming light also with high accuracy, the motion of Earth is under permanent high precision monitoring. Thus the basic numerical descriptor of precission, an angular rate of 5029.0966 seconds of arc per Julian century, traditionally denoted p (for precession) is a measured value from observed coordinate changes of thousands of stars over, say, two centuries. The understanding of this value in terms of forces acting on an oblate Earth from the Moon is well understood so that an extrapolation back and forth over a few full cycles contains little uncertainties. Of course, you can find details on the coordinate transformations mentioned above (the direct observational effect of precession) on the net. I was surprised to see that the Wikipedia article on precession covers the astronomical aspect very poorly. You thus better look for other sources.

distance = 150m

displ = root (50 ^2 + 100^2)

root(2500+10000)

root 125000