Lowest energy configuration of the dumbbell for different alpha’s

[Tibo]

In the video it is said that the lowest energy configuration for the situation alpha > 0 is achieved when the dumbbell is parallel to the z-axis as this gives the most negative value to the quadrupole correction term to the potential gravitational energy.
It would therefore follow that if alpha < 0, the lowest energy configuration would be achieved in the case where the dumbbell lies in the xy-plane, perpendicular to the z-axis.
In the case of alpha = 0, the quadrupole correction is zero for all configurations of the dumbbell with respect to the z-axis with neither minimum nor maximum.

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