I Gyroscope angular momentum: direction and curvature

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TL;DR
deviation laserbeam from say 10 light minutes distance from the sun across the edge of the sun and then to 10 light minutes distance to the opposite site of the sun compared to the deviation of a gyroscope
sending a laserbeam from say 10 light minutes distance from the sun across the edge of the sun and then to 10 light minutes distance to the opposite site of the sun gives a change in direction of the laserbeam of approx 1.7 arc seconds.

But now imagine the situation where I take an ideal gyroscope in a rocket where the rocket follows the laser beam by a tracking system that keeps the speed of the rocket at a fixed speed v compared to the position of the sun, while it forces the gyroscope to exactly follow the path of the laserbeam.

now the question is :
What is the ratio between the change of direction of the laserbeam compared to the change in direction of the gyroscope.
and if possible of course some reasoning behind why.
 
Being material observers, we do not expand with the universe. Our ruler for measuring its increasing size does not expand either - its scale does not change. If I identify the ruler with a metric, then from my perspective it should be invariant both spatially and temporally. If it expanded with the universe, then its size measured with this ruler would be constant. Why then do we use a metric with the spatial scale expanding with the universe and constant temporal scale to measure the...

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