I would guess that the most expensive part of such an endeavor wouldn't be the launch; it would be developing and building a spacecraft capable of capturing it and bringing it back.
Even the Space Shuttle wasn't necessarily a perfect fit for the job as-is. Hubble was serviced many times, but it was specifically designed for on-orbit capture and servicing by the Shuttle. Before they decommissioned the shuttle they actually had to install an extra piece of hardware to make it feasible to capture and de-orbit using future non-crewed spacecraft. And even then that's just to make sure it crashes in a safe place, not to bring it home intact.
There was also a mission to service a satellite that wasn't designed for the purpose, and they had a really hard time capturing it and very nearly had to give up after days' worth of failed attempts. It finally took simultaneous EVA by three astronauts to coordinate a successful capture (one to grab it by hand, two to get it onto a specialized adapter rig built just for that satellite so that the Canadarm could hold it), which is quite a thing considering that the Shuttle's only designed to allow two people on EVA at a time.
This craft is likely tumbling, which I presume would make it unacceptably dangerous for a crewed mission (and certainly rules out anyone just going out there and grabbing it with their hands), in addition to making successful capture that much more difficult.
No, when it tumbles, it does it around its centre of mass. You'd have to get a craft that has empty area inside.
If you know Elite, it has space stations where you dock by going inside, while matching station's rotation. That's only in one axis (and note the hangar goes through the axis of rotation, i.e. centre of mass). To add rotation around another axis would make the task impossible.
Only if it is only spinning along one axis. Very likely to be spinning on multiple axis. That said, with enough money and effort I'm sure we could figure something out, like shooting it strategically with small projectiles to slow the spin, etc.
Anything "spinning among multiple axes" is just spinning around a single compound axis? There's no reason for a spacecraft to limit itself to any earthbound reference frame when it comes to matching frame with an incoming body from outer space.
In three dimensions, the rotation around one axis can affect the distribution of mass around other axes of rotation. That change in the moment of intertia causes acceleration, which can result in chaotic motion even without the addition of any outside forces.
Reminds me of the tumbling T-handle. A small tool is spun up in one axis, and due to some interesting physics, ends up flipping over on another axis every few seconds.
It's about angular momentum and happens whenever the axis of rotation differs - even slightly - from the semi-major axis. Interaction with a fluid is not necessary.
You can demonstrate it at home with your smartphone (or, more canonically, a tennis racket), and see for yourself that the tumbling happens much too quickly to be explained by whatever force the air is imparting.
What do you mean by "other axes of rotation"? As long as the object is rigid and not acted upon by external forces, its axis should never change, since both the direction and magnitude of angular momentum are conserved.
Wikipedia talks of "chaotic rotation" of astronomical objects, but only over long timescales due to gravitational interactions and thermal effects. On short timescales, its axis shouldn't change much at all, unless you bump into it and apply an off-axis torque.
Alright, that makes more sense, the trick is that the (conserved) angular momentum vector need not be parallel with the angular velocity vector, the simplest example being torque-free precession [0]. It doesn't help that most examples of non-constant angular velocity have external forces in the mix to confuse the reader.
Unless acted on by an external force, all rigid objects only rotate about a single axis, do they not? That axis just might not be aligned with any useful parts you'd want to grab on to.
Does it really work this way? If the craft is rotating along any other axis than its direction of travel, wouldn't the matching craft have to be revolving around it, not just rotating?
Even the Space Shuttle wasn't necessarily a perfect fit for the job as-is. Hubble was serviced many times, but it was specifically designed for on-orbit capture and servicing by the Shuttle. Before they decommissioned the shuttle they actually had to install an extra piece of hardware to make it feasible to capture and de-orbit using future non-crewed spacecraft. And even then that's just to make sure it crashes in a safe place, not to bring it home intact.
There was also a mission to service a satellite that wasn't designed for the purpose, and they had a really hard time capturing it and very nearly had to give up after days' worth of failed attempts. It finally took simultaneous EVA by three astronauts to coordinate a successful capture (one to grab it by hand, two to get it onto a specialized adapter rig built just for that satellite so that the Canadarm could hold it), which is quite a thing considering that the Shuttle's only designed to allow two people on EVA at a time.
This craft is likely tumbling, which I presume would make it unacceptably dangerous for a crewed mission (and certainly rules out anyone just going out there and grabbing it with their hands), in addition to making successful capture that much more difficult.