[TML] TML Digest, Vol 2007, Issue 254

[Jerry W Barrington]

On 10/5/07 5:45 AM, "tml-request at travellercentral.com"
<tml-request at travellercentral.com> wrote:

> On Thu, Oct 04, 2007 at 11:42:11PM -0400, Jerry W Barrington wrote:
>> Nothing in space "tumbles" like that.  In freefall, you have a fixed
>> angular momentum about a fixed spin axis thru the center of gravity.
>> (For the purists yes, there are gravity effects that can cause
>> wobble, but not on this sort of time/size scale.)  The spin axis
>> *IS* the angular momentum axis, by definition!
> 
> Not true.  The angular momentum axis is fixed (on these scales), but
> is only in very special cases the same as the spin axis.
> 
> The relationship between the two is determined by the moments of
> inertia, which are in general represented by a symmetric rank-2
> tensor, not a scalar.  *If* (and only if) the spin axis is aligned
> with one of the three orthogonal principal axes of the body, then the
> angular momentum will be along the same axis.
> 
> There is a rather simplified account on Wikipedia
> (http://en.wikipedia.org/wiki/Moment_of_inertia), but a decent
> dynamics textbook would give a fuller explanation.
> 
> Regarding the chaotic tumble, I cannot quickly find an authoritative
> online reference.  Most rotational dynamics references are either
> research papers (which take undergraduate and even most postgraduate
> knowledge for granted) or simplified high-school physics resources.
> However, it is derivable (with some nontrivial math) from
> http://en.wikipedia.org/wiki/Euler%27s_equations
> and you might derive some physical intuition into the motion from
> http://en.wikipedia.org/wiki/Poinsot%27s_construction
> 
> My primary information source is lecture notes for the University of
> Tasmania course "Dynamical Systems and Chaos", prepared and delivered
> by Prof.  Robert Delbourgo.

OK, I'll agree I overstated a bit.  It's possible for a body to start out in
such a tumble, where the rotation is not thru the principal rotation axis.
On the other hand, there are forces that shift the spin axis to the
principal axis, and as the reference below states "the vast majority of
asteroids are in uniform rotation about their largest moment of inertia".
Given that, it would take an immensely more valuable rock for the miners to
bother with the rare tumblers.  And you're just not going to find a 1km gold
nugget.  :)

http://www-personal.umich.edu/~scheeres/conferences/AIAA-2004_1446.pdf