Navigator wrote:

Perhaps you might tell Doug, Jeff and other people interested in tides
what Nutation has to do with an annalemma first?

I'll give it a go...

An analemma is a bendy curve thing which plots the declination of a body
against time. The analemma for the sun is a figure-8 shape, the crossover of
the figure-8 indicating when the sun crosses the equator twice a year. An
analemma for a lunar month would have a similar shape. However, because the
moon's orbit nutates, the curve is rather more complex and covers a longer
period. The complexity comes about because the point at which the moon
crosses the equator constatly changes - the path of the orbit itself slowly
rotates around the earth. This rotation of the orbit is called nutation. The
nutation period of the moon's orbit is generally given as 18.61 years.

The tides are nominally a function of the gravitational pulls of the moon
and sun working together (modifed by things like local geography). Given
that there are springs and neaps, we can see that the relative positions of
the two bodies in the sky have an effect on how much 'pull' there is at a
given time. Note that, if there was no moon, the gravitational factor in the
tidal cycle would be solar - much more constant and easy to predict. It
strikes me that neaps don't occur simply as a result of 'less pull' on the
tidal bulge, but that there is some pull from one body, and some from
another which is at a different point in the sky - the main bulge mght be
following the moon, but there is a secondary bulge following the sun. Were
it otherwise, there would be no springs and neaps.

As noted in another post, the sun and moon don't simply make their way
around the equator, but travel through paths that are each at an angle to
the equator. Logically, this means that the direction of their gravitational
pulls is not perpindicular to the equator, but is nominally in the direction
of each body.

Although the primary effect, springs and neaps, is seen in terms of the
position of each body around the sky - the Right Ascension (RA) of each -
there must be a similar, though less pronounced, effect as each body follows
its path above and below the equator. For instance, there are springs at
many times, but springs when there is a solar eclipse wll be different from
springs when the sun is, say at it's highest declination (summer in the
northern hemisphere) and the moon is at its lowest. In other words, the
bodies might be at the same RA, but their disparate declinations will
'spread' the gravitational pull, some above the equator, some below. When
they're perfectly aligned during a solar eclipse, the pulls are concentrated
into a single direction.

While the sun's path is straightforward, the combined effect of the sun and
moon goes through a huge range of alignments - the sun might pass over a
given point every year, but, due to nutation, the moon will be in a
different relative position compared to last year. Also, because the moon's
nutation period is 18.61 years, when the moon's cycle starts to repeat, it's
out of sync with that of the sun.

Since this stuff, with its talk of periods, is all very cyclical, one
naturally wonders if there is maybe a 'grand cycle' that can be derived,
even if it's only an approximation. Note that the sun has a yearly period,
and that the moon's nutation period is measured in years - 18.61. To derive
an approximate period for a 'grand cycle', one simply has to multiply 18.61
by a number that will give a result that is close to a whole number of
years. The first resonable approximation is arrived at when we multply 18.61
by 5, giving 93.05 years.

Rounding for convenience, this means that, for a given start point, the
combined solar and lunar cycle starts to repeat after 93 years, or 5
nutations - the two bodies will have the same delination and RA that they
had at our start point.


--
Wally
www.forthsailing.com
www.wally.myby.co.uk