Undergrad Can a Gyroscope in a Satellite Detect Orbit?

[Sagittarius A-Star]

In this video at 06:30 the astronaut places the gyroscope on the string attached to the walls of the ISS cabin. I'm not sure whether the gyroscope was rotating about itself when placed on the string or was not.

It's the ISS, which rotated, not the gyroscope axis:

The ISS rotates about its center of mass at a rate of about 4 degrees per minute so that it will complete a full rotation once per orbit. This allows it to keep its belly towards the Earth.

Source:
https://www.forbes.com/sites/quora/2017/10/03/how-does-the-iss-travel-around-the-earth/

 

[cianfa72]

It's the ISS, which rotated, not the gyroscope axis:

Yes, definitely. What I can't see clearly from the video is: was the gyroscope spinning about its axis when the astronaut placed it on the string ?

 

[A.T.]

was the gyroscope spinning about its axis when the astronaut placed it on the string ?

What would be the point of a non-spinning gyroscope?

 

[cianfa72]

What would be the point of a non-spinning gyroscope?

Yes, indeed (maybe it wasn't that evident in the video, though).

As far as I understand, the ISS's center of mass (CoM) is free-falling, i.e. it travels along a geodesic in spacetime. ISS rotates about its CoM at a rate of 4 degree/minute. The spinning gyroscope's gyro-stabilized axis -- by definition -- is Fermi-Walker transported along the gyroscope's CoM worldline (since this is geodesic, then FW transport coincides in this case with parallel transport along it).

As you said, it is the ISS which rotated (since the gyroscope axis actually defines locally what "non rotating" means).

P.s. very basic notion: the gyroscope's axis is gyro-stabilized only when the gyroscope is spinning about it

 

[Sagittarius A-Star]

P.s. very basic notion: the gyroscope's axis is gyro-stabilized only when the gyroscope is spinning about it

The gyroscope in the video is spinning about it's axis (even while it's housing is not spinning), but it is difficult to see. How this specific gyroscope works, can be seen in the video from 01:20 on:

 

[pervect]

The video shows a gyroscope experiment in the ISS. The gyroscope axis rotates relative to the ISS with 4 degrees per minute.

Source (see at 05:38):

As far as I know, this is just due to the ISS being deliberately locked to a constant orientation relative to the Earth, for practical reasons such as keeping the ground communications antenna's pointed at the ground station. So in one 90 minute orbit, the station rotates through 360 degrees.

 

[cianfa72]

So in one 90 minute orbit, the station rotates through 360 degrees.

You mean in one revolution about the Earth (90 minute orbit) the ISS rotates through 360 degree about itself.

 

[PeterDonis]

You mean in one revolution about the Earth (90 minute orbit) the ISS rotates through 360 degree about itself.

He means that it rotates 360 degrees relative to the gyroscope's axis, which is Fermi-Walker transported along the ISS's worldline.

 

[pervect]

No. The spatial geometry has an angular defect, which will be detected by a sufficiently precise gyroscope.

See pages 177-179 here:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n189/mode/2up

See also my previous post in this thread:

I agree with your no answer to the OP's question, but you haven't mentioned Thomas precession at all.
The best reference I have at the moment is https://arxiv.org/abs/0708.2490v1, "Gyroscopic precession in special and general relativity from basic principles". Unfortunately, it's not as B-level friendly as I would like, I would put it at the upper I level.

Section VII of the paper talks a little bit about the full GR case, but it starts to use concepts like "parallel transport" that I put at the A-level. Also, I'd need to study it more myself, it's been a while. It looks like it does shows how to combine Thomas precession with your correct observations about the spatial curvature to get the total effect.

Because I think Thomas precession is important to the question, I'll try and describe what it means by a specific example.

If we replace the satellite by an accelerating elevator with a flat floor, Thomas precession means that a gyroscope sliding across said flat floor will precess relative to the "fixed stars", whereas any of the gyroscope at rest on said floor won't. We can say the floor is flat in the limit when the fixed star is infinitely far away, and all gyroscopes on said floor point at the same fixed star. (If the star is a finite distance away, the floor would be curved, but if it's infinitely far away, in the limit it's flat). ln this illustrative example, we also put the direction of the acceleration of said elevator towards said fixed star.

The "sliding" gyroscope has a constant linear momentum, and the relative velocity between the floor and the sliding gyroscope is always constant.

Continuing with my detailed analogy, while all stationary gyroscopes on the floor point at the fixed star and don't precess as the elevator accelerates, the sliding gyroscope DOES precess, and the rate of precession is proportional to both the acceleration of the elevator and the velocity at which the gyroscope slides across the floor.

That is my best attempt to describe Thomas precession, and my TL/DR summary is that "gyroscopes are a bit more complicated in SR than they are in Newtonian mechanics". And it has nothing to do with the spatial curvature effects that also exist that you point out in General Relativity, this is purely an effect of special relativity.

For the possible benefit of the advanced reader, I will mention a few A-level buzzwords, these being the aforementioned "parallel transport", and a couple of generalizations of it, notably Fermi-walker transport (applicable to gyroscpes that are accelerating), and M-transport, a further generalization of Fermi-walker transport that I've heard mentioned but haven't studied myself.

The paper I cite above, https://arxiv.org/abs/0708.2490v1, "Gyroscopic precession in special and general relativity from basic principles", starts out at the I-level, but has to spend quite a bit of time to get anywhere and is not easily summarized in a short post :(.

It can be shortened at the A-level to note that Lorentz boosts just don't commute, the order of the boost matters, and that this ties in with rotation because rotation is the commutator of the non-commuting boosts, intuitively rotation is WHY they don't commute.

I'll note that I tried to look at your Epstein reference, but it was pretty hard to view. Do you have a recommendation for a good viewer program for it by any chance?

 

[PeterDonis]

I think Thomas precession is important to the question

Yes, it is, but note that all of these "precessions" are relative to an observer at rest at infinity. If all you have is local observations of the gyroscope, you can't detect any of them. There are in principle three of them: Thomas precession, de Sitter (geodetic) precession, and (if the central mass is rotating) Lense-Thirring precession.

See my posts #7 and #8 earlier in the thread.

 

[PeterDonis]

If we replace the satellite by an accelerating elevator with a flat floor

Note, though, that an orbiting satellite is in free fall, so locally it is not the same as the accelerating elevator. So you can't really understand the precession that is usually called "Thomas precession" for an orbiting satellite using your analogy. Note that in flat spacetime, which is the framework in which Thomas precession is usually derived (see, for example, the first two Insights articles referenced in my post #8), it is done for an object moving in a circle, which is not in free fall in flat spacetime. It takes more work to see how there is still a similar precession for an object moving in a circle in a curved spacetime where the circle is a free-fall orbit.

 

[pervect]

Section 7 of the reference I quoted explains this in more detail:

In a static spacetime such as that of a Schwarzschild black
hole, the global static reference frame locally corresponds
to the accelerated reference frames we have considered in
special relativity. If we integrate the infinitesimal rota-
tions from Ω given by either Eqs. (12) or (14), we can
find the net rotation of a gyroscope that is transported
along a given spatial path. Note, however, that Ω de-
scribes how the gyroscope grid rotates relative to a frame
that is parallel transported with respect to the local spa-
tial geometry associated with the reference frame. Thus
directly integrating the effects of rotation from Ω gives
the rotation relative to a frame that is parallel trans-
ported with respect to the global spatial geometry.

So yes I agree that Thomas precession is just one component of the total precession it's not sufficient in and of itself. We need to add to this the effect of parallel transporting a curve through the _spatial_ (emphasis mine) geometry to the Thomas precession to get the total precession.

There are more remarks in the paper in subsequent sections that I haven't processed fully at this point. I think it'd be helpful to put some actual numbers into these formula, but I have only barely started, one of the obstacles is untangling the sign conventions and coordinate conventions the paper uses :(.

What I have so far is that for GPB, a precession of 6606.6 millarcseconds/year was predicted (source- the Nasa website and google). I work this out as a precession of 360 degrees in about 200,000 years.

Meanwhile, Thomas precession alone, we can set gamma $\approx$ 1. The precession omega due to thomas precession alone would be 1/2 a v / c^2, giving a period of $4 \pi c^2 / a v$. For GPB, I find on the www that a is about 8.1 m/s^2, and v is about 7500 m/s. This gives a period of 589 000 years. Thus the Thomas precession alone significantly underestimates the actual precession. Since I haven't paid close attention to the signs, I suppose things could be even worse, I've been assuming the two effects are in the same direction (as opposed to opposing each other).

 

[A.T.]

I'll note that I tried to look at your Epstein reference, but it was pretty hard to view. Do you have a recommendation for a good viewer program for it by any chance?

When you scroll down on this page:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/mode/2up
You will find DOWNLOAD OPTIONS for the entire book on the right.

Here a direct link to the PDF one:
https://dn710609.ca.archive.org/0/i...ity Visualized [elem txt] (1994, Insight).pdf

However, I don't think the book features explanations of Thomas precession in the free falling GR case.

Thus the Thomas precession alone significantly underestimates the actual precession.

From the Stanford website (emphasis mine):

In the framework of the gravito-electromagnetic analogy, the geodetic effect arises partly as a spin-orbit interaction between the spin of the test body (the gyroscope in the case of GP-B) and the "mass current" of the central body (the earth). This is the exact analog of Thomas precession in electromagnetism, where the electron experiences an induced magnetic field (in its rest frame) due to the apparent motion of the nucleus. In the gravitomagnetic case, the orbiting gyroscope feels the massive earth whizzing around it (in its rest frame) and experiences an induced gravitomagnetic torque, causing its spin vector to precess. This spin-orbit interaction accounts for one third of the total geodetic precession; the other two thirds arise due to space curvature alone and cannot be interpreted gravito-electromagnetically. They can, however, be understood geometrically. Model flat space as a 2-dimensional sheet, as shown in the diagram below (left).

 

[cianfa72]

He means that it rotates 360 degrees relative to the gyroscope's axis, which is Fermi-Walker transported along the ISS's worldline.

To take it simple, one could reasonably model the walls of the ISS as the rim of a 2D disk in which the center of mass (CoM) is free-falling. Points on the disk travel along non-inertial (proper accelerating) timelike worldlines and define a congruence in a region of spacetime.

The gyroscope's gyro-stabilized axis defines a direction within the spacelike subspace of the tangent space orthogonal to the gyroscope CoM worldline's tangent vector (4-velocity) at any point along it. This spacelike direction is, by definition, FW transported along the gyroscope CoM's worldline.

The fact that ISS rotates relative to the gyroscope's axis does mean that the above congruence has nonzero vorticity (let me say this should be similar to what happens for the Langevin congruence in flat spacetime).

 

[PeterDonis]

Points on the disk travel along non-inertial (proper accelerating) timelike worldlines

Strictly speaking, I suppose this is true, but the proper acceleration will be extremely small, because the only reasons it would be there would be (1) the nonzero vorticity (see below), which is extremely small because of the small angular velocity (only one rotation every 90 minutes) and (2) tidal gravity, which is extremely small within the confines of the ISS. The ISS as a whole is in a free-fall orbit.

The gyroscope's gyro-stabilized axis defines a direction within the spacelike subspace of the tangent space orthogonal to the gyroscope CoM worldline's tangent vector (4-velocity) at any point along it. This spacelike direction is, by definition, FW transported along the gyroscope CoM's worldline.

Yes.

The fact that ISS rotates relative to the gyroscope's axis does mean that the above congruence has nonzero vorticity

Yes.

(let me say this should be similar to what happens for the Langevin congruence in flat spacetime).

Sort of, but, as noted above, the angular velocity is small, so the vorticity will be small as well, and so will its effects.

 

[cianfa72]

Sort of, but, as noted above, the angular velocity is small, so the vorticity will be small as well, and so will its effects.

Yes the vorticity is small, however it suffices to provide the rotation of the ISS w.r.t. the gyroscope's axis.

 

[PeterDonis]

Yes the vorticity is small, however it suffices to provide the rotation of the ISS w.r.t. the gyroscope's axis.

The vorticity isn't a cause, it's a description. As I understand it, the rotation of the ISS wrt the gyroscope's axis is maintained by periodically firing thrusters.

 

[Sagittarius A-Star]

As I understand it, the rotation of the ISS wrt the gyroscope's axis is maintained by periodically firing thrusters.

They try to avoid the use of thrusters and use control moment gyroscopes most of the time, to not disturbe micro-gravity experiments.

Nominally, attitude control is provided by four control moment gyroscopes (CMGs). Each CMG contains a wheel that is 220 lbs (100 kg). That wheel spins at 6600 rpm, resulting in an angular momentum of 3500 ft-lb-s (4742.5 N-m-s). The basic idea is that if a torque induces a rotation on the ISS, those wheels can rotate about their gimbals to change the angular momentum of the ISS, creating a counter torque. Using CMGs is much more subtle than using thrusters, so microgravity experiments are not impacted. CMGs do have limits, though, so thrusters can assist, if needed. That assistance is needed whenever the torques are large.

Source:
https://www.quora.com/How-does-the-...&share=6bbd3c52&srid=5Gqkk&target_type=answer

 

[cianfa72]

The vorticity isn't a cause, it's a description. As I understand it, the rotation of the ISS wrt the gyroscope's axis is maintained by periodically firing thrusters.

Ok, this is similar to the Langevin congruence: take the worldline at the center of the rotating disk as "reference/fiducial" worldline. The neighboring worldlines of the Langevin observers in the congruence are described by a nonzero vorticity, yet they rotate around thanks to a suitable force acting on them.

 

[A.T.]

They try to avoid the use of thrusters and use control moment gyroscopes most of the time, to not disturbe micro-gravity experiments.

Another potential reason: Gyroscopes use only electric energy, which can be obtained via solar panels. For thrusters you would need to spend fuel, as Ion thrusters would be probably too weak.

 

[pervect]

When you scroll down on this page:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/mode/2up
You will find DOWNLOAD OPTIONS for the entire book on the right.

Here a direct link to the PDF one:
https://dn710609.ca.archive.org/0/i...ity Visualized [elem txt] (1994, Insight).pdf

However, I don't think the book features explanations of Thomas precession in the free falling GR case.



From the Stanford website (emphasis mine):
.....

Thanks - I can see why if the majority of the effect is due to "the missing inch" aka "space curvature", one might neglect Thomas precession in an introductory treatment, as it's apparently only half as large. I was trying (and failing) to recall what I had heard of the relative magnitudes. Thus, while the "missing inch" treatment is incomplete, it gets the dominant effect. One of the downsides I see, though, is that this effect and explanation involves bringing GR into the anlaysis. The special relativity analysis is simpler, and illustrates that the behavior of gyroscopes in SR is a bit different than it is in Newtonian physiucs (though numerically the effects are usually small).

I find the "mass current" idea a bit strange, as for instance in my sliding block example of Thomas precession, there are no real masses to cause a "mass current" anyhwhere in the problem description. But the actual masses aren't needed, the pseudo-gravito-electric field on the accelerating "Einstein's elevator" when observed from the sliding block generates the same pseudo-gravito-magnetic field, which by the same argument causes a magnetic-like torque on the gyroscope.

One additional subtle point here is what happens in the limit as the gyroscope rotates slower and slower and eventually stops rotating. As far as I know, the change in axis remains the same regardless of how much the gyroscope spins, though I don't think I've seen an explicit reference on the topic. One can certainly observe that in the references cited in this thread (for instance eq 12 from https://arxiv.org/pdf/0708.2490v1 the amount of precession of the gyroscope does NOT depend on the speed of angular rotation of the gyroscope in question , but this is not emphasized. Because the precession doesn't depend on how fast it's spinning, , we expect the same effects even as the spin goes to zero.

The gravitomagnetic analogy is still somewhat helpful, but I think it's inferior to the idea that a pair of boosts in different directions generate a rotation, as in my arxiv reference tries to explain. But while I think it's truly the most fundamental approach, it's sheer abstractness elevates it a bit in difficulty in my opinion.

 

[A.T.]

One of the downsides I see, though, is that this effect and explanation involves bringing GR into the anlaysis.

Just part of GR, which can be understood qualitatively based on a simple paper-cone model, that anyone can build. That simplicity is likely the reason why Thorne ans Epstein focus on it when addressing the layman audience.

The special relativity analysis is simpler,

Yes, but as @PeterDonis noted above, while the 'simple' SR explanation of Thomas precession is based on subsequent boosts, the orbiting gyroscope is in free fall all the time. So how is the SR analysis applicable/transferable to the orbiting gyroscope?

(for instance eq 12 from https://arxiv.org/pdf/0708.2490v1 the amount of precession of the gyroscope does NOT depend on the speed of angular rotation of the gyroscope in question

For an idealized gyroscope that is somewhat unsurprising, because in the idealized case a gyroscope doesn't need to be rotating at all. It can just be pointer (or 3 orthogonal ones rigidly connected), that maintain their orientation, simply because they are initially non-spinning and torque free. Of course, in the real world "initially non-spinning" and "torque free" cannot be achieved. Thus we use spinning things, with large initial angular momentum, which doesn't change much in relative terms, under unavoidable small external torques.

But there are other types of gyroscopes:
https://en.wikipedia.org/wiki/Vibrating_structure_gyroscope
https://en.wikipedia.org/wiki/Ring_laser_gyroscope

That makes me wonder: Could the Sagnac effect (used by the ring laser gyroscope) be used to explain the Thomas precession part of the total effect for the orbiting gyroscope?