Precession and Forced Nutation of the Earth

Let
be the Earth's angular velocity vector due to its
daily rotation. This vector makes an angle with the -axis,
where
is the mean inclination of the ecliptic to the
Earth's equatorial plane. Suppose that the projection of
onto the ecliptic plane subtends an angle with the -axis,
where is measured in a counter-clockwise (looking from the north) sense--see Figure 45.
The orientation of the Earth's axis of rotation (which is, of course, parallel
to
) is thus determined by the two angles and .
Note, however, that these two angles are also *Euler angles*, in
the sense given in Chapter 8. Let us examine the Earth-Sun
system at an instant in time, , when : *i.e.*, when
lies in the - plane. At this particular instant, the -axis points towards the so-called *vernal equinox*,
which is defined as the point in the sky where the ecliptic plane crosses the projection of the Earth's
equator (*i.e.*, the plane normal to
) from south to north. A counter-clockwise (looking from the north) angle in the
ecliptic plane that is zero at the vernal equinox is generally known as an *ecliptic longitude*. Thus, is the
Sun's ecliptic longitude.

According to Equation (926), the potential energy of
the Earth-Sun system is written

(957) |

It is easily demonstrated that (with )

(958) |

(959) |

(960) |

(961) |

(962) |

where is a constant, and

Here,

(965) |

(966) |

According to Section 8.9, the rotational kinetic
energy of the Earth can be written

is a constant of the motion. Here, is the third Euler angle. Hence, the Earth's Lagrangian takes the form

where any constant terms have been neglected. One equation of motion which can immediately be derived from this Lagrangian is

(970) |

(971) |

Consider *steady precession* of the Earth's rotational axis, which is characterized by
, with both and constant. It follows, from the above equation, that
such motion must satisfy the constraint

(972) |

(973) |

(974) |

(975) |

(976) |

(977) |

(978) |

Using analogous arguments to those given above, the potential energy of the Earth-Moon system can be
written

is the Earth's angular velocity vector, and is the position vector of the Moon relative to the Earth. Here, for the moment, we have retained the dependence in our expression for (since we shall presently differentiate by , before setting ). Now, the Moon's orbital plane is actually slightly inclined to the ecliptic plane, the angle of inclination being . Hence, we can write

to first order in , where is the Moon's ecliptic longitude, and is the ecliptic longitude of the lunar

(982) |

where . Now, from (980) and (981),

(984) |

(985) |

to first order in . Given that we are interested in the motion of the Earth's axis of rotation on time-scales that are much longer than a month, we can average the above expression over the Moon's orbit to give

[since the average of over a month is , whereas that of is ]. Here, is a constant,

(987) | |||

(988) |

and

(989) |

(990) |

(991) |

(992) |

Two equations of motion that can immediately be derived from the above Lagrangian are

(993) | |||

(994) |

(The third equation, involving , merely confirms that is a constant of the motion.) The above two equations yield

respectively. Let

where is the mean inclination of the ecliptic to the Earth's equatorial plane. To first order in , Equations (995) and (996) reduce to

(999) | |||

(1000) |

respectively, where use has been made of Equation (983). However, as can easily be verified after the fact, , so we obtain

(1001) | |||

(1002) |

The above equations can be integrated, and then combined with Equations (997) and (998), to give

where

(1005) | |||

(1006) | |||

(1007) |

Incidentally, in the above, we have assumed that the lunar ascending node coincides with the vernal equinox at time (

According to Equation (1003), the combined gravitational interaction of the Sun and the Moon with the
quadrupole field generated by the Earth's slight oblateness causes the Earth's axis of rotation to
precess steadily about the normal to the ecliptic plane at the rate .
As before, the negative sign indicates that the precession is in the opposite direction to the (apparent) orbital
motion of the sun and moon. The period of the precession in years is given by

(1008) |

(1009) |

The point in the sky
toward which the Earth's axis of rotation points is known as the *north celestial pole*. Currently,
this point lies within about a degree of the fairly bright star *Polaris*, which is consequently sometimes known as the *north star*
or the *pole star*. It follows that Polaris appears to be almost stationary in the sky, always lying *due north*, and can thus
be used for navigational purposes. Indeed, mariners have relied on the north star for many hundreds
of years to determine direction at sea. Unfortunately, because of the precession of the
Earth's axis of rotation, the north celestial pole is not a fixed point in the sky, but instead traces out a circle,
of angular radius , about the north ecliptic pole, with a period of 25,800 years.
Hence, a few thousand years from now, the north celestial pole will no longer coincide with Polaris, and
there will be no convenient way of telling direction from the stars.

The projection of the ecliptic plane onto the sky is called the *ecliptic*, and coincides with the
apparent path of the Sun against the backdrop of the stars. Furthermore, the projection of the Earth's equator
onto the sky is known as the *celestial equator*. As has been previously mentioned, the ecliptic is inclined at to the
celestial equator. The two points in the sky at which the ecliptic crosses the celestial equator are
called the *equinoxes*, since night and day are equally
long when the Sun lies at these points. Thus, the Sun reaches the *vernal equinox* on about
March 21st, and this traditionally marks the beginning of spring. Likewise, the Sun reaches the
*autumn equinox* on about September 22nd, and this traditionally marks the beginning of autumn.
However, the precession of the Earth's axis of rotation causes the
celestial equator (which is always normal to this axis) to precess in the sky, and thus also causes the equinoxes to precess along the ecliptic. This
effect is known as the *precession of the equinoxes*. The precession is in the opposite direction to the Sun's apparent motion around the ecliptic, and is of magnitude per century. Amazingly, this miniscule
effect was discovered by the Ancient Greeks (with the help of ancient Babylonian observations). In about 2000 BC, when the science of astronomy originated in ancient Egypt and Babylonia, the vernal equinox lay in the constellation Aries. Indeed, the
vernal equinox is still sometimes called the *first point of Aries* in astronomical texts. About 90 BC,
the vernal equinox moved into the constellation Pisces, where it still remains. The equinox will move
into the constellation Aquarius (marking the beginning of the much heralded ``Age of Aquarius'') in about 2600 AD. Incidentally, the position of the vernal equinox in the
sky is of great significance in astronomy, since it is used as the zero of celestial longitude (much as
Greenwich is used as the zero of terrestrial longitude).

Equations (1003) and (1004) indicate that the small inclination of the lunar orbit to the ecliptic
plane, combined with the precession of the lunar ascending node, causes the Earth's axis of rotation to wobble
sightly. This wobble is known as *nutation*, and is superimposed on the aforementioned precession. In the absence of
precession, nutation would
cause the north celestial pole to periodically trace out a small ellipse on the sky, the sense of rotation being
*counter-clockwise*. The
nutation period is 18.6 years: *i.e.*, the same as the precession period of the lunar ascending node.
The nutation amplitudes in the polar and azimuthal angles and are

(1010) | |||

(1011) |

respectively, where . Given that , , , , , and , we obtain

(1012) | |||

(1013) |

The observed nutation amplitudes are and , respectively. Hence, our estimates are quite close to the mark. Any inaccuracy is mainly due to the fact that we have neglected to take into account the small eccentricities of the Earth's orbit around the Sun, and the Moon's orbit around the Earth. The nutation of the Earth was discovered in 1728 by the English astronomer James Bradley, and was explained theoretically about 20 years later by d'Alembert and L. Euler. Nutation is important because the corresponding gyration of the Earth's rotation axis appears to be transferred to celestial objects when they are viewed using terrestrial telescopes. This effect causes the celestial longitudes and latitudes of heavenly objects to oscillate sinusoidally by up to (

Note, finally, that the type of forced nutation discussed above, which is driven by an external torque, is quite distinct from the free nutation described in Section 8.9.