Caption: How the Earth's equatorial bulge and Earth's axial rotation leads to the axial precession.

Features:

1. The short explication for the Earth's axial precession is that the axial precession is caused by the Earth's axial rotation in combination with the tidal force (in order decreasing importance) the Moon, the Sun, and, to a much lesser degree, the other Solar-System bodies on the Earth's equatorial bulge (see Wikipedia: Axial precession: Cause) which are in turn caused by the Earth's axial rotation.

2. Now for the long explication.

3. The Earth is actually NOT exactly round: it is a slightly oblate spheroid: i.e., it is slightly oblate in that it has an equatorial bulge due to the centrifugal force caused by the Earth's rotation. The centrifugal force pulls the equatorial region outward causing aforesaid equatorial bulge.

   The oblateness is NOT very large: the Earth equatorial radius R_eq_⊕ = 6378.1 km and the Earth polar radius R_po_⊕ = 6356.8 km: the difference is only 21.3 km (see Wikipedia: Earth Table; Wikipedia: Earth: Shape; Wikipedia: Earth's radius). The flattening ratio is

       f = (a-b)/a = 0.0033528 = 1/298.257222101 ,

   where a is the semi-major axes and b is semi-minor axis of the oblate spheroid Earth: i.e., respectively, Earth equatorial radius and Earth polar radius (see Wikipedia: Earth Table).

4. The slightly different gravitational forces (i.e., the differential gravitational force) on the different parts of the equatorial bulge of the Earth (stronger on the near side to the source of external gravity, weaker on the far side to the source of external gravity) of the Moon, Sun, and and, to a much lesser degree, the other Solar-System bodies try to align approximately the equatorial bulge with the ecliptic plane, and thereby align approximately the Earth's axis with the ecliptic axis.

   The differential gravitational force, in fact, the tidal force.

5. The tidal force in this case causes a torque in the physics terminology. A torque is a force-like quantity used to describe the "twisting" effect of a force.

   For example, people sometimes talk of torquing a screw.

6. If the Earth were NOT rotating, the Earth's axis would tend to oscillate (at least for some initial conditions yours truly thinks) in the plane of the illustration if the outside gravitational force were in the plane of the illustration.

   The oscillation might be damped out in in time due to other perturbations leading to approximate alignment of the Earth's axis and the ecliptic axis.

7. But the Earth is rotating on its axis.

   The Earth's rotation is the cause of the equatorial bulge via the centrifugal force as aforesaid.

8. With rotation, one gets NOT oscillation, but a axial precession. See the figure below (local link / general link: axial_precession.html).

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9. There seems NO simple way to understand why there is axial precession or precession in general. Precession just defies any in-words explanation.

   However, the situation is somewhat similar to why orbiting objects do NOT fall inward under gravity.

   They keep falling, but keep missing because they have a sideways motion or, in physics terminology, angular momentum (which is a measure of rotation).

   Similarly, the axis of rotating Earth keeps trying to align perpendicular to the direction of gravity, but keeps missing because the Earth has angular momentum about its axis

   The "missing" is the axial precession.

10. Precession is just very tricky. But it is quite commonly seen.

    For example, precession is exhibited by toy tops. See the figure below (local link / general link: precession_toy_top.html).

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11. Another interesting example of precession is exhibited by gyroscopes. See the animation in the figure below (local link / general link: precession_gyroscope.html).

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12. Yet another interesting example of precession is provided by bicycles. When the wheels are vertical and rotating there is NO precession. But there is still the stabilizing effect that comes with having angular momentum which is a measure of rotation. Angular momentum is actually a pseudovector aligned with the axis of rotation. Trying to change the direction of angular momentum requires a torque which requires a twisting force which takes an effort and hence the stabilizing effect of angular momentum. In turning a bicycle, a cyclist torques the front wheel directly with the handlebar, but also by leaning the bicycle which causes gravity to provide a torque. The combined torques do give a tendency to precession which helps effect the turn. Cyclists usually know how do this without studying physics.

    See the demonstration of bicycle wheel precession in the video Bicycle Wheel with gyroscope precession | 0:26 shown below in Precession Videos.

13. Precession Videos:

    Axial precession videos (i.e., Axial precession videos):

    1. Milankovitch Cycles | 0:10: The axial precession is the motion of the Earth relative to the fixed stars. It has a period of approximately 26000 years, but there is NO fixed period due to gravitational perturbations and other astrophysical perturbations. Short enough for the classroom.
    2. Bicycle Wheel with gyroscope precession | 0:26: What happens to the Earth also happens to a bicycle wheel. They are somewhat different cases of precession, but there are basic similarities. Both are hard to explain though precession is very common. In the video, note that the precession turns the bicycle wheel in the direction of the leaning of the wheel. In cycling, this precession tendency helps turn the bicycle in the direction the cyclist leans. Cyclists usually know how do this without studying physics. Good and short enough for the classroom.