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If asteroid's rotation period provides greater centrifugal force than gravity, having tangential surface velocity greater than escape velocity, and/or if due to an oddly enough shape, it has no fixed polar region, or no principal rotation's axis.

How would one probe manage to hook/land to one of them, in order to stay synchronous relative to its surface for long exposure imaging, or for returning samples, or any other mission where matching rotation period is needed?

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    $\begingroup$ i would assume any such asteroids would fall apart under their own spin. $\endgroup$ – Sdarb Jan 17 at 23:51
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    $\begingroup$ @Sdarb They are small enough, and despite the "fast rotator" classification their rotation is slow enough, that if they aren't fractured up their material strength can hold them together. It is thought these are individual fragments of once-larger bodies that were broken apart by collisions or the YORP effect. They're essentially like the largest boulders created by a landslide—big, single pieces of rock, maybe metal in some cases. $\endgroup$ – Tom Spilker Jan 18 at 7:08
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    $\begingroup$ Land on the poles. $\endgroup$ – SF. Jan 18 at 14:07
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    $\begingroup$ @SF. There should be an area around a pole where tangential velocity is lower than escape velocity. $\endgroup$ – Uwe Jan 18 at 15:50
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    $\begingroup$ Looking for a difficult landing? Try one on a very non-spherical asteroid that isn't in principal axis rotation, like Toutatis! $\endgroup$ – Tom Spilker Jan 24 at 23:03
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In this paper Ryan & Ryan talk about the multiple instances of asteroids rotating faster than their "critical limit", the limit at which parts are moving faster than orbital velocity. This earlier paper also discusses this phenomenon.

All the ones rotating this fast seem to be small, 200 m or less in diameter. I say "seem" because these objects are too small to have their diameters measured by direct observation from Earth or near-Earth space. Instead, the astronomers measure the amount of reflected sunlight received at the telescope and the object's position in the solar system, then use an assumed albedo (fraction of sunlight falling on it that is reflected) to infer a size. If the albedo assumption is off, so is the inferred diameter. A very dark object will have its diameter underestimated, and conversely for a very bright object.

If the object's thermal emissions can be measured at infrared wavelengths you can get a somewhat better size estimate because infrared emissivities vary less than visual albedos.

Anyway, it appears that a rotation period of ~2.2 hours is the breakpoint. Anything inferred as larger than ~200 m diameter has a rotation period of that or slower. Only smaller objects have rotation periods shorter than that. Astronomers infer that anything rotating faster than that is a monolithic (non-fractured) body with non-zero tensile strength. Some have said for certain sizes and orientations of fractures a 200-m (or a bit larger) fractured body could still stay together at that rotation rate, but that gets into a fuzzy area. Since the sizes are fuzzy anyway, it's not worth pursuing to a gnat's eyelash. It's somewhere around 200 m.

I calculated for a homogenous spherical object of a given density what rotation period would give an equatorial centrifugal acceleration equal to the gravitational acceleration and came up with $$\tau = \sqrt{\frac{3\pi}{G\rho}}$$ where $\tau$ is the rotation period, $\rho$ is the object's mass density, and $G$ is the gravitational constant. Notably, $\tau$ is independent of the object's radius.

If I invert that equation to give $\rho$ as a function of everything else and plug in the 2.2-hour period it yields a density of ~2,250 kg/m^3, a bit lower than typical asteroidal materials. If the object is ellipsoidal you'd get a somewhat larger density. Densities of 2,500 - 3,100 kg/m^3 are typical densities of (non-porous) mineral types seen in rocky asteroidal material, measured from the meteoric fragments we have on Earth; porosity can decrease that.

Given that such a fast-rotating object is small, I see four basic ways to match its rotation rate: 1) attach to it with fasteners that rely on the object's tensile strength (like screws, pitons, concrete nails, etc.); 2) attach to it with a tool that doesn't rely on tensile strength (like a pair [or three, or four, or whatever] cables that wrap around behind the object); 3) apply a centripetal force propulsively (horribly inefficient for long stays!); or 4) slow down the object's rotation to sub-critical.

Methods 3 and 4 take a lot of propellant and are probably impractical, at least with current propulsion systems.

Method 2 is stable once in place and rotating. The trick is getting the system in place to start with. I suppose you could park the main spacecraft some short distance away (i.e., not orbiting) and send a drone craft to deploy cables around the object, keeping them from touching the object. When the cables are all ready the main spacecraft could begin reeling them in. Once contact is made on the far side (as seen from the spacecraft) the spacecraft does a high-acceleration burn to synchronous speed. The cables will naturally tighten to provide the centripetal force needed; some additional reeling in would probably be needed.

Method 1 could be tricky because we're not at all sure of important characteristics of the surface, such as porosity, brittleness, etc. that can greatly influence the effectiveness of various types of fasteners. Is the surface like basalt, or glass, or drywall, or something else? A lander intended to attach to the surface might carry a couple of types, and use propulsion to land and then hold itself down on the surface (hopefully not for long!) while it tries setting those fasteners. If successful, once the fasteners are set the propulsion is no longer needed and can be shut down. If the surface is rough you might even use some form of fast-setting glue!

Currently there are no missions funded to go to any of these objects. The science you'd get from such a mission is outweighed by more pressing objectives. And the engineering objective of seeing if you can divert an asteroid's orbit to avoid an Earth impact is better done with rubble-pile objects, since that is a more difficult task.

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