# The rotating movement of an asteroid

1. ### zsero

9
I almost all movies where you could see an animation about an asteroid, they move in a very distinct way.

I don't know how to explain better, but I think what we can see in the movies is that the asteroid is rotating around not one, but around two or three axis at a given time.

What is this kind of motion? Is it correct?

According to the Euler's rotation theorem any 3D rotation that has a fixed point also has a fixed axis. So if they are rotating around more then one axis, does it mean that they don't have any fixed point? But then what is the motion what their center of mass is doing?

I would suppose, that the correct movement would be when asteroids are rotating like planets, around one fixed axis. But to me the rotation commonly seen is not like that. What is the correct way?

Sorry, but I am confused, I have to ask this question: is it possible for any solid object to rotate around more then one axis? Or every possible rotation for a solid object in 3D is actually around one axis?

2. ### DaveC426913

16,141
Yes. Solid objects, especially asteroids, frequently rotate about two axes simultaneously.

I'm pretty sure that it has to do with
a] the size of planets and
b] the relative infrequency of large disturbances
that causes them to eventually stabilize to single axial rotation.

Asteroids are small and disturbed quite frequently, so their rotations will be likewise chaotic (small c, not big C).

3. ### zsero

9
Sorry, I'm not a physics student, and thats why I asked this explicitly:
"is it possible for any solid object to rotate around more then one axis? Or every possible rotation for a solid object in 3D is actually around one axis?"

4. ### DaveC426913

16,141
Explicitly (again): yes.

Explicitly: no. One or two. Not three. Any apparent rotation around three axes can be represented as two.

5. ### zsero

9

I am glad I found this forum, because since I have started reading about it I got just more and more confused. I am quite sure I learned the opposite in school, but I never understood why would one axis be "special".

So is there a definite difference in the movement between smaller and larger objects in space? I mean an asteroid or a spacecraft vs. a planet or a star? Can I say that a totally random movement in 3D space is around two axis, while planets and stars usually have one axis?

But then I don't understand Euler's rotation theorem:
"In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about a fixed axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation."

Can you explain to me how can something rotate around two axis, while not going against Euler's rotation theorem?

Last edited: Jun 10, 2011
6. ### DaveC426913

16,141
'Usually' is the key. I cannot speak with authority but I believe two axial rotation is unstable and will eventually settle to single axis rotation if left undisturbed long enough.

I don't see how Euler's theorem forbids two axis rotation. It addresses single-axis rotation.

7. ### zsero

9
Thanks, the physics part is now clear!

With Euler's rotation theorem, this is what makes me confused:

"Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about a fixed axis that runs through the fixed point."

Do you mean that those asteroids don't have any fixed point?

8. ### DaveC426913

16,141
Correct.

If you throw a football properly, it will spin along its longitudinal axis; this is what gives it stability. The fixed point (actually, two) are on its tips, which do not change their orientation.

If that football taps a player and gets knocked end-over-end, it will now be spinning along two axes, perpendicular to each other. And no point on the football will be stationary wrt orientation.

9. ### vanhees71

3,503
The motion of a rigid body is described by the motion of one fixed point within the body (three position vectors of this point) and the rotation of an arbitrary coordinate system fixed in the body with respect to a coordinate system in the observer's rest frame (three angles, e.g., the Euler angles of the rotation).

### Staff: Mentor

One consequence of Euler's theorem is that at any point in time you can find a single axis about which the body is rotating at that instance. This is why we can treat angular velocity as a pseudovector.

That does not mean that some time later some object object will still be rotating about the same axis -- even if no torques act on the object. It is angular momentum, not angular velocity, that is conserved here. The angular velocity will remain constant only if the body in question is perfectly rigid and if the angular velocity vector is perfectly aligned with one of the eigenvectors of the object's inertia tensor. The odds of a perfectly rigid body, such a perfect alignment, and exactly zero torque are rather small. It is a space of measure zero.

Small asteroids aren't massive enough to have pulled themselves into a spherical shape. They are lumpy potatoes. That means that small asteroids will typically have three distinct principal moments of inertia. If asteroids were rigid bodies, they would be tumbling. The precession rate would typically be on the small side. The movie depiction is not close to accurate.

Rigid bodies have two stable axes of rotation: The eigenaxes corresponding to the largest and the smallest principal moment of inertia. Asteroids are not rigid bodies. Non-rigid bodies have but one stable rotation axis, the eigenaxis with the largest moment of inertia. Over time, an asteroid (or a spacecraft that has lost attitude control) will eventually settle into a state where it is rotating fairly close to that axis. The precession rate is going to be very small because this axis is a also stable axis for rigid bodies and because the instantaneous rotation is very close to this axis.

Back to the lumpy potato analogy. Imagine skewering the potato so you can barbecue it. Most people skewer a potato through its longest axis, but some do use the shortest axis. Now imagine rotating that lumpy potato about this axis. Voila! You have a good model of the rotational behavior of most asteroids. This shortest axis is the axis with the largest moment of inertia.

Last edited: Jun 11, 2011
11. ### Quinzio

558
Asteroids are non rigid bodies ?

### Staff: Mentor

A lot of asteroids are believed to be rubble piles. Itokawa, for example, is almost certainly a rubble pile. A rubble pile asteroid is exactly what the name suggests: A loose collection of small rocks weakly bound together by the self-gravitation of those rocks toward one another. A rubble pile is anything but a rigid body.

Other asteroids such as Eros are more or less monolithic. That does not mean they are rigid bodies. There is no such thing as a perfect rigid body in nature. The speed of sound in a perfect rigid body is infinite. Something made of matter has a finite speed of sound. It is not perfectly rigid.

For either a rubble pile or a monolith, the rigid body assumption can a very good assumption if the internal stresses are small and if the time span of interest is neither too short nor too long. So what do those qualitative words such as small, too short, too long mean?

Small internal stresses. The internal stresses can be quite large if the asteroid has a fairly high rotation rate. Small asteroids are subject to the YORP effect, an external torque due to the lag between heating by the Sun on the sunlit side of the asteroid and cooling by radiation on the dark side of the asteroid. Some binary asteroids are now believed to result from a single parent asteroid that split apart because that parent eventually built up a sufficiently large rotation rate.

Short time span. It takes a finite amount of time for a change in state at one point of the asteroid to propagate throughout the asteroid, and an even longer time for the transients to die out. The rigid body assumption is not valid if the time span of interest is not significantly larger than this relaxation time.

Long time span. This is where things get interesting. Rigid bodies have two stable rotation axes. You can demonstrate this to yourself by tossing a book a number of times, giving it different rotations about different axes with each toss. Non-rigid bodies have but one stable rotation axis. The rotational kinetic energy of a rigid body is given by

$$E_{\text{rot}} = \frac 1 2 \, \mathbf L^\top \boldsymbol I ^{-1} \mathbf L$$

The configuration with the angular momentum aligned with the largest eigenvector minimizes this rotational energy. This is the configuration to which a slightly non-rigid body will eventually relax. The mechanism that leads to this is that the propagation of changes across the body is not perfect. Think of breaking a wire coat hanger by repeatedly bending it. The hanger becomes quite hot at the point where it breaks. The fictitious torque $\boldsymbol{\omega}\times(\boldsymbol I \times \boldsymbol{\omega})$ manifests as real stresses in the asteroid. These are not propagated perfectly across the body; some of the mechanical energy is lost as heat. High energy states such as rotation about the smallest eigenvector become forbidden. Eventually, the only allowed state is rotation about the largest eigenvector.

13. ### Quinzio

558
Ok, I appreciate you enlightenment, but surely the OP wanted to talk about a solid rock which is free in the space.

I agree we could spend years by just understanding and studying asteroids, but it's not this 3d goal.

### Staff: Mentor

In general, a rock rotating in free space will not have a constant axis of rotation.

This youtube video shows Poinsot's construction:

Here's another demonstration, this time with a book.

You might also want to google the phrase "The polhode rolls without slipping on the herpolhode lying in the invariable plane."

Last edited by a moderator: Sep 25, 2014
15. ### zsero

9
Thanks for all the detailed explanations!

What I am most interested in is that what would be the proper way to simulate the flying asteroids in a computer generated movie, as I would like to create one myself.

The only example I found was here:
http://csep10.phys.utk.edu/astr161/lect/asteroids/features.html

http://csep10.phys.utk.edu/astr161/lect/asteroids/toutspin.mpg
http://csep10.phys.utk.edu/astr161/lect/asteroids/toutspin2.mpg

OK, tell me if I understand it right: Supposing that the object is a rigid-body, it will have two axes of rotation. If I want to animate the object using Euler angles (or rotational matrices, or quaternions if those are easier) there is a method for finding out how to get an Euler angle from two axes and two angles for a given moment.

Does this mean that a rigid body cannot rotate freely around any axis going through its center of mass? I mean I would think that you can take a potato, rotate it around a random axis going through its center of mass and let it go in space. Do you say that this rotation will eventually turn into a rotation around one or two different axes?

- Is "tumbling" defined somehow, or it is just a common way of describing the strange movement we see?

- When is this "wobbling" or "tumbling" effect the most obvious, depending on the angles and the angular velocities? I mean for example when they are perpendicular or when angular velocities are close to each other? Do you have any information or guess about the angles and angular velocities for Asteroid Toutatis, as seen in the video?

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And the case when they are non-rigid

Does it happen frequently (say in our solar system)? If yes, does it give even more complicated movements compared to the rigid version? I mean you wrote that they only have one stable axis, so in theory the movement would be more simple, because it is just around one axis. But I would guess that a body behaving like if it was filled with some kind of fluid or gel can produce extremely complicated movements. Is the movement of a semi-rigid-body in space very complicated or just around one axis?