Gyroscope angular momentum: direction and curvature

[HansH]

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.

 

[HansH]

not sure if everybody understands the question and setup (we had some discussion about that on a Dutch forum) so here perhaps a better description plus picture.

I consider a thought experiment in which a rapidly spinning gyroscope is forced to follow the trajectory of a laser beam. The gyroscope is actively adjusted in position, under the following conditions:

1. Constant velocity: the gyroscope moves at a fixed velocity v relative to the sun.
2. Orientation remains free: the steering forces only correct the position, but leave the orientation of the gyroscope itself unaffected—similar to the operation of a gyroscope in an airplane, which maintains its own orientation.

Given this setup:
Does the angle α between the direction of the laser beam (r₂) and the direction in which the gyroscope is pointing (r₁) change throughout the entire journey?

 

[A.T.]

What is the ratio between the change of direction of the laserbeam compared to the change in direction of the gyroscope.

Intuitive guess: 2

and if possible of course some reasoning behind why.

The laser beam is bend by both:
a) local free fall (that you also have in a linearly accelerating rocket, relative to the rocket)
b) the overall spatial geometry around the Sun (Flamm's paraboloid)
https://www.mathpages.com/rr/s8-09/8-09.htm

The gyro orientation will only be affected by b), because a falling gyro doesn't change orientation in an linearly accelerating rocket relative to the rocket. But the gyro is affected by the "angular defect" (a).
See Figure 11-25 here:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n191/mode/2up

For objects traveling at c the two effects cause the same amount of deflection. But the deflection depends on speed only for a). So since the gyro is only affected by b), it will only have that half of the light's deflection, no matter how fast it moves along the light's spatial path.