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<H2><A NAME="SECTION00054000000000000000">
Precession and Nutation</A>
</H2>
<I>Right ascension and declination</I>, (<IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$">), 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 <IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img24.gif"
ALT="$\alpha$"> is the point of intersection of
the <I>celestial
equator</I> and the <I>ecliptic</I> (the apparent path of the Sun
through the year) where the Sun moves into the northern
hemisphere. This point is called the
<I>first point of Aries</I>,
the <I>vernal equinox</I> (with apologies to
southern-hemisphere readers) or simply the <I>equinox</I>.<A NAME="tex2html4" HREF="footnode.html#27833"><SUP><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="foot_motif.gif"></SUP></A>
<P>
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 <I>mean Sun</I>
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.
<P>
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 <IMG WIDTH="33" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img256.gif"
ALT="$23^{\circ}
\hspace{-0.37em}.\hspace{0.02em}5$">, the <I>obliquity of the
ecliptic</I>),
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.
<P>
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 <IMG WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img133.gif"
ALT="$17\hspace{-0.05em}^{'\hspace{-0.1em}'}$"> 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 <I>luni-solar precession</I>; the smaller,
faster, periodic terms are called the <I>nutation</I>.
<P>
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.
<P>
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
<IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img83.gif"
ALT="$0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.5$"> per year)
secular rotation of the ecliptic about a slowly-moving diameter is called,
confusingly, <I>planetary
precession</I> and, along with the luni-solar precession is
included in the <I>general precession</I>. The equator and
ecliptic as affected by general precession
are what define the various ``mean'' <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> reference frames.
<P>
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 <IMG WIDTH="14" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img105.gif"
ALT="$\psi$"> (longitude)
and <IMG WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img257.gif"
ALT="$\epsilon$"> (obliquity), some as small as
<IMG WIDTH="47" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img258.gif"
ALT="$0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.0001$"> .
<P>
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<ADDRESS>
<I>SLALIB --- Positional Astronomy Library<BR>Starlink User Note 67<BR>P. T. Wallace<BR>12 October 1999<BR>E-mail:ptw@star.rl.ac.uk</I>
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