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Appendix 2


Analytic Model without the ``fast'' top Assumption

At the cost of some complication a model without the fast top assumption may be constructed.

Joos [12] considers a symmetrical top that has been set into rotation about its figure axis, which is designated as the tex2html_wrap_inline764 axis. After the initial spinup no moments are applied about the figure axis insuring that tex2html_wrap_inline766 is constant. He then writes the angular velocity vector as the sum of a component along the figure axis of the top tex2html_wrap_inline482tex2html_wrap_inline770, and a component tex2html_wrap_inline772tex2html_wrap_inline774 normal to the figure axis.
displaymath760

The angular momentum vector
displaymath761
tex2html_wrap_inline586 is the moment of inertia about an axis perpendicular to the figure axis. Differentiating
equation252
Noting that
equation262
we also have tex2html_wrap_inline786. Solving for the derivative of tex2html_wrap_inline772tex2html_wrap_inline774,
equation275

With the derivatives of tex2html_wrap_inline772tex2html_wrap_inline774 and tex2html_wrap_inline764 in hand (equations 19 and 20) one may now integrate forward the motion of the top by standard differential equation solvers. This has been done with the torque tex2html_wrap_inline800, tex2html_wrap_inline444 in the tex2html_wrap_inline804 direction, and the motion of the center of mass determined by equation (2), and the model successfully predicts the observed high and the low frequency spin speed stability limits to within 20%. It also illustrates the mechanism of loss. At the low frequency limit the top tips over enough so the magnetic field gradient no longer supports it. At the high frequency limit the trapping in the horizontal plane gets softer and softer, and the top eventually wanders away.