The Wobbling of the Earth and Other Curiosities

December 28, 1999

That beautiful full moon on the 21st of December 1999 rekindled my naive excitement about celestial mechanics. I hope you all saw it! It was a very special night, since there was a full moon with the moon as close to the earth as possible, and it was also the longest night of the year. In other words: full moon, perigee and winter solstice, all at once! The moon looked very big and bright. The last time this happened was in 1866.

(According to Phillip Helbig, the moon is about 14% brighter at perigee than at its mean distance from earth. I don't know if this is really all that noticeable... but it was a clear night out here in Riverside California, so the moon looked mighty nice.)

Okay... I'll wrap up my series of posts about the wobbling of the earth by quoting some stuff about precession and nutation. It's pretty cool: the modelling of the earth's motion is so precise now that folks use more than hundred separate terms in the formula!

```Precession and Nutation

Right ascension and declination, (alpha and delta), are the names of the
longitude and latitude in a spherical polar coordinate system based on
the Earth's axis of rotation.  The zero point of alpha is the point of
intersection of the celestial equator and the ecliptic (the apparent
path of the Sun through the year) where the Sun moves into the
northern hemisphere.  This point is called the first point of Aries,
the vernal equinox (with apologies to southern-hemisphere readers) or
simply the equinox.

This simple picture is unfortunately complicated by the difficulty of
defining a suitable equator and equinox.  One problem is that the Sun's
apparent motion is not completely regular, due to the ellipticity of the
Earth's orbit and its continuous disturbance by the Moon and planets.  This
is dealt with by separating the motion into (i) a smooth and steady mean
Sun and (ii) a set of periodic corrections and perturbations; only the
former is involved in establishing reference frames and timescales.
A second, far larger problem, is that the celestial equator and the
ecliptic are both moving with respect to the stars. These motions arise
because of the gravitational interactions between the Earth and the other
solar-system bodies.

By far the largest effect is the so-called ``precession of the equinoxes'',
where the Earth's rotation axis sweeps out a cone centred on the ecliptic
pole, completing one revolution in about 26,000 years. The cause of the
motion is the torque exerted on the distorted and spinning Earth by the Sun
and the Moon. Consider the effect of the Sun alone, at or near the northern
summer solstice.  The Sun `sees' the top (north pole) of the Earth tilted
towards it (by about 23.5 degrees, the ``obliquity of the ecliptic''), and
sees the nearer part of the Earth's equatorial bulge below centre and the
further part above centre.  Although the Earth is in free fall, the
gravitational force on the nearer part of the equatorial bulge is greater
than that on the further part, and so there is a net torque acting as if
to eliminate the tilt.  Six months later the same thing is happening in
reverse, except that the torque is still trying to eliminate the tilt.
In between (at the equinoxes) the torque shrinks to zero.  A torque acting
on a spinning body is gyroscopically translated into a precessional
motion of the spin axis at right-angles to the torque, and this happens
to the Earth.  The motion varies during the year, going through two
maxima, but always acts in the same direction.  The Moon produces the same
effect, adding a contribution to the precession which peaks twice per
month.  The Moon's proximity to the Earth more than compensates for its
smaller mass and gravitational attraction, so that it in fact contributes
most of the precessional effect.

The complex interactions between the three bodies produce a precessional
motion that is wobbly rather than completely smooth. However, the main
26,000-year component is on such a grand scale that it dwarfs the remaining
terms, the biggest of which has an amplitude of only 17 seconds and a period of
about 18.6 years. This difference of scale makes it convenient to treat
these two components of the motion separately.  The main 26,000-year effect
is called luni-solar precession; the smaller, faster, periodic terms are
called the nutation.

Note that precession and nutation are simply different frequency components
of the same physical effect.  It is a common misconception that precession
is caused by the Sun and nutation is caused by the Moon. In fact the Moon
is responsible for two-thirds of the precession, and, while it is true that
much of the complex detail of the nutation is a reflection of the
intricacies of the lunar orbit, there are nonetheless important solar terms
in the nutation.

In addition to and quite separate from the precession/nutation effect, the
orbit of the Earth-Moon system is not fixed in orientation, a result of the
attractions of the planets. This slow (about .5 seconds per year) secular
rotation of the ecliptic about a slowly-moving diameter is called,
confusingly, planetary precession and, along with the luni-solar
precession is included in the general precession.  The equator and
ecliptic as affected by general precession are what define the various
``mean'' reference frames.

The models for precession and nutation come from a combination of
observation and theory, and are subject to continuous refinement. Nutation
models in particular have reached a high degree of sophistication, taking
into account such things as the non-rigidity of the Earth and the effects of
the planets; SLALIB's nutation model (IAU 1980) involves 106 terms in each
of psi (longitude) and epsilon (obliquity), some as small as .0001 seconds.

SLALIB --- Positional Astronomy Library