– alpha =0
In this case, the quadrupole contribution to the gravitational energy is always zero. Since the monopole energy is independent of the angle of the dumbbell with respect to the z-axis, all orientations of the dumbbell are equal in energy.
– alpha >0
Rewriting -alpha(2cos^2-sin^2) as -alpha(3cos^2-1) allows to predict the behaviour of this function in terms of theta. Take alpha=1 for simplicity. Then the quadrupole correction to the energy varies between -2 for theta=0° and 1 for theta=90°. The quadrupole correction is thus most negative for theta=0°. This means that the lowest-energy configuration is the one with the dumbbell parallel to the z-axis.
– alpha <0
The quadrupole correction now varies between 2 for theta=0° and -1 for theta=90° (for alpha=-1), making it the most negative for theta=90°. Thus, in this case the lowest-energy orientation for the dumbbell is within the xy-plane.