Staff Emeritus

Science Advisor

Homework Helper

HansH said:

I am not sure if I can interpret what you say.
you say: The spin axis will shift so as not to point towards the distant "fixed" star. so does it then drift away during 1 orbit? are you then referring to the frame dragging effect giving a very tiny drift?

But the original question was if you can detect the orbit of the satellite. So what is now the conclusion ? does the spin axis of the gyroscope follow the momentary direction of the orbit (then the conclusion is that you cannot detect the orbit with a gyroscope) or does it approximately keep pointing to the star, so then the conclusion is that you can detect the orbit with a gyroscope or at least almost completely (except for the frame dragging effect, but that is very small and costs 75.000000 dollar to detect it).

There's a couple of different questions here. Basically, using devices that spin (such as the Forward Mass detector), can detect nearby massive bodies. The forward mass detector works by detecting tidal forces. There's a short reference to the forward mass detector in the wiki entry on Forward, https://en.wikipedia.org/wiki/Robert_L._Forward. The method of operation is probably not really clearly described in the wiki reference, sadly, but the most intelligible thing I can say is to take a closer look at tidal forces and how they would affect the described device.

These devices are a bit more complicated than a gyrosope, though. If you only have a gyroscope, you need some external reference frame to be able to detect said precession, such as the "guide star" used in GPB.

The part about detecting nearby masses is relevant to your question because if one can detect that a mass is nearby with such a detector, one knows they are not just floating in space, one know they are either orbiting or falling. Falling can be tested for experimentally, the crude and easy to describe way is to say that one will notice when they hit the ground :).

The method of gyroscopic precession I described does, as I mentioned, require that one is able to see the fixed stars to establish a frame of reference. Note that it's possible to do physics in other frames of reference, the "fixed stars" method is just the one most commonly used to define what is mean by a frame of reference.

Methods such as the Forwards mass detector or similar methods based on tidal forces don't need the aid of these "fixed stars", however. We have other gravity gradiometers nowadays than Forwards design - they are commonly used by geologists to help detect oil fields. See for instance https://www.bellgeo.com/ desciption of their"Full Tensor Gravity Gradiometer". Look for the "technology" tab to get information on h ow it operates if you're interested. There are some quantum designs nowadays, too.

On a related point, there is an effect in special relativity known as "Thomas precession". This special relativity effect causes a gyroscope to precess in any accelerating frame of reference. There's a wiki article on it at https://en.wikipedia.org/wiki/Thomas_precession, and I've posted about it on PF in the past. At low non-relativistic speeds its' proportional to ##\frac{\vec {a} \times \vec {v}}{c^2}##, where ##\vec{a}## is the acceleration vector, ##\vec{v}## is the velocity vector, and the "times" signal is the vector cross product, https://en.wikipedia.org/wiki/Cross_product.

Doing some of my own calculations, I think I concluded that while the Thomas precession would have effect on the inertial guidance system of ballistic missiles, I concluded it would be smaller than the errors due to imperfections and "drift" of said inertial guidance systems with current technology. But I don't have a textbook reference on this, and my number crunching isn't as accurate as it used to be. And of course I only have publically available information on how good state of the art guidance systems are. It's an interesting topic, though - the V2 rockets used something called PIGA, precission gyrosopic integrating accelerometers, while more modern systems use different technology which you can look up the way I did if you are interested.

GR is a bit more complicated than SR. You will typically find this information in articles on Gravity probe B, the experiment that measured both of them. One of these effects is called geodetic precession, ans is similar to (but different in magnitude than) the Thomas precession of special relativity, the difference being due to subtle effects of GR. I'm afraid I know longer recall the details about the numerical relationship between Thomas precession and the geodetic effect. The other effect, frame dragging, is only present when the body being orbited is rotating. The frame dragging effect is MUCH smaller than the geodetic effect.

https://einstein.stanford.edu/MISSION/mission1.html is an article that talks about the geodetic and frame dragging effects in the context of Gravity probe B, the current (and only) experiment we've done to detect frame dragging (it also detected the geodetic effect). The orbit of the GP-B satellite was chosen to make the frame dragging precession orthogonal to the much larger geodetic precession.

I would say that the geodetic effect would be the answer to your question, the "frame dragging" part is a smaller additional effect due to rotation of the body around which one is orbiting.

Thomas precession shows that gyroscopes in SR just don't act the same way as they do in Newtonian physics, without bringing in the complications due to gravity and GR that the full treatment of the geodetic effect involves. It's hard to guess exactly what your interest is, the simplest in theory might be to tackle Thomas precession, which involves an advanced knowledge of SR. Understanding the intricacies of GR will be sigificantly harder than understanding Thomas precession. Depending on your background, understanding Thomas precession might not be a piece of cake either.

But if you're interested in actual experiments done with gyroscopes around orbiting bodies, Gravity Probe B is the obvious resource. However, the theoretical details to treat this case require more than special relativity, it requires General relativity, which is significantly more complicated, most especially if you want enough knowledge to be able to calculate things numerically.