There are many really great things about astronomy. One is that one can get quite far just with elementary physics and basic math.

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Take asteroids, for example. "How fast can an asteroid rotate without being ripped apart?" - answering that needs a lot of high-powered math, right?

Wrong. It's really trivial.

OK, OK, I have to modify that statement. It's trivial if we can make some assumptions: (1) the asteroid must be of the rubble pile type, i.e., a pile of rock held together by mutual gravitational attraction, as opposed to a massive monolith and (2) the density distribution inside the asteroid should be more or less homogeneous. But those assumptions apply to the overwhelming majority of asteroids, so it's OK to make them. 

The underlying physics is quite straightforward: rotation creates a centrifugal force. The faster something rotates, the larger the centrifugal force. If we assume the rubble pile model, the asteroid is composed of a heap of rocks of various shapes and sizes. These have mass, so they exert gravity. Each rock attracts the others, so they end up nestling against one another, with some gaps in between. If the centrifugal force, which tends to pull the rocks outwards, away from the centre of mass, exceeds gravity, which acts towards the centre of mass and holds the rubble pile together, the asteroid will be ripped apart.  If the centrifugal force does not exceed gravity, the asteroid will not be ripped apart.

In between, there must be a rotation speed where the centrifugal force is exactly equal to gravity. This corresponds to the maximum rotation rate that we are looking for. It's that simple.

I'm not going to attempt to write the entire calculation in html. Was it Stephen Hawking who said that every equation reduces the potential reader base by half? I'm going to need seven equations, so I'd lose over 99% of my readers, and obviously, I don't want that. So I resorted to a trick. The equations are still there, but I put them in a separate document here.

The results were amazing, also to me. In the end, the limiting rotation period depends only on the density. I get a shortest possible period P of 375780 divided by the square root of the density in kilograms per cubic meter. If you hear a hissing sound now, don't worry, it's just physicists who are hyperventilating when they see my cavalier treatment of the units without which, let me hasten to add, no meaningful calculation are possible. Please relax, dear physicists. I'll be really rigorous next time. I promise.

Anyway. We get this very simple formula that shows that the minimum rotation period is a function of the asteroid's density. That's all there is to it, provided they're not monoliths. It doesn't matter whether the asteroid is large or small.

2000 kilograms per cubic meter is a good guess. Then the minimum rotation period is 8403 seconds or 2.33 hours. An asteroid that that takes a longer time to rotate will maintain gravitational cohesion.

None of this is new. If even I could compute this, many others could, and did. But I like it, because it's such a simple, elegant computation, and it turns out to be very consistent with observations. Asteroids are normally somewhat irregular bodies, so when you observe them, there will be a ripple in their apparent brightness. This is what is called the "light curve". The period of the light curve is the asteroid's rotation period.

Until lately hardly any asteroids with periods of less than around 2.2 hours were known. Most rotate a lot more slowly than that. Only recently have a few fast rotators been discovered; all of these are very small. This gives rise to the conjecture that the overwhelming majority of asteroids must be held together by gravitation. They can be rubble piles or ex-monoliths that have been fissured by an impact. 

No large, fast-rotating asteroids have been observed. This does not constitute proof that there are no large monoliths, but even so, it can be seen as a strong indication. All known monoliths (or metallic bodies) - fast rotators with periods as low as several minutes - are small. That's why they weren't detected earlier.

These fast rotators must be interesting bodies. No pebble, not even a speck of dust could remain on them, and of course, no spacecraft could land there, it would just be swept away by the massive centrifugal forces.

Such bodies likely constitute the remnants of impacts between asteroids that led to their mutual destruction. A fascinating idea ... and all of that follows from one simple mathematical equation. 

Not bad, huh?

Further Information 

Planetary Database - Small Bodies Node

Harris: The Rotation Rates of Very Small Asteroids: Evidence for Rubble Pile Structure (LPI, 1996): A statistical analysis of the rotation rates of small asteroids known until the, lending support to the conjecture of rubble pile composition due to the absence of rotation rates that would require tensile rather than gravitational cohesion (i.e., a monolithic asteroid). Although many more small asteroids have been detected since then, Harris' observations still hold. Fast rotators are few in numbers and all of these are small. 

Harris, Wisnewski: paper on asteroid rotation rates

Donnison, Wiper: Statistical analysis of asteroid rotation data (MNRAS 1999) 

Fujiwara, Kawaguchi, Yeomans et al.:  The Rubble-Pile Asteroid Itokawa as Observed by Hayabusa (Science, 2006):This paper summnarizes what was learned through the Japanese sample return mission Hayabusa to asteroid 25143/Itokawa, including conjecture that Itokawa was created by collisional break-up of a parent body and re-agglomeration of debris

Scheeres, Hartzell, Sánchez: Scaling Forces to Asteroid Surfaces: The Role of Cohesion (ArXiv, 2010): This paper looks specifically at very small, gfast rotating asteroids, suggesting that for small bodies, Van der Waals Forces maybe dominant and thus allow a degree of cohesion suprior to that explained by gravitational cohesion

Vernazza, Binzel et al.: Solar wind as the Origin of Rapid Redding of Asteroid Surfaces (Nature 2009):This paper analyzes the time scale for solar-wind induced weathering of silicate-rich asteroid regolith and the associated reddening, stating a figure of 1 million years for this process. From the observation that many near-Earth asteroids lack this reddening, suggesting recent action of a mechanism of surface re-shaping, it is conjectured that this must be due to gravitational perturbations during near-Earth flybys

Levasseur-Regourd, Hadamcik, Lasue: Interior Structure and Surface Properties of NEOs: What is Known and what Should be Unterstood to Mitigate Potential Impacts (Advances in Space Research, 2005): A comprehensive compilation of data on NEO properties and a summary of what can be derived regarding their interior composition, cohesion and porosity

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