Asteroid Nutation Angles

Approximate expressions are derived for (a) the characteristic time τ that it takes to align the rotation axis of an asteroid with its body axes, and (b) the average expected value of the nutation angle $$\overline\alpha$$. Alignment occurs because of internal energy dissipation: a portion of the stored strain energy, due to the bending stresses caused by the gyroscopic torque and due to the strains associated with the displacement of the centrifugal bulge during the wobble motion itself, is lost during each wobble period. For an asteroid of radius r, spinning with an angular velocity $$\omega, \tau = \mu Q/(\rho K_3^2r^2\omega^3)$$, where µ is the asteroid rigidity, Q is its quality factor and ρ its density. K₃² is a shape factor which is about 10⁻² for nearly spherical bodies and 10⁻¹H² for non-spherical bodies with oblateness H. For reasonable parameter choices $$\tau \approx 5 \times 10^4 T^3/(rK_3)^2\enspace \text {yr}$$ with T the rotation period in hours and r in kilometres. The asteroids, which are represented by a power law mass distribution, continually collide with one another, producing misalignment. A statistical treatment of the transferred angular momentum permits a rate of misalignment to be computed. From this and the alignment rate, $$\overline\alpha$$ is found to $$\sim r^{-2.6}\omega ^{-2.5}$$ and to be very small for almost all observed asteroids, thus explaining why asteroid lightcurves are observed singly periodic. Small irregular asteroids, such as Geographos and 1971 F, are suggested to be the most likely candidates to be seen precessing.