1. There is a lunar tidal cycle that
synchronizes the
slow precession of the lunar line-of-apse with the
seasons and the Synodic cycle (i.e. the Moon's phases).

The tidal cycle is called the 31/62 year Perigee-Syzygy
Cycle.
This tidal cycle is the time required for a
full (or new moon) at Perigee to re-occur at or very near to
the same point in the seasonal calendar.

It is highly recommended that readers go to the following link to get a fuller understanding of the parameters that are used to define the lunar orbit as well a better understanding of the 31/62 year Perigee-Syzygy tidal cycle, before proceeding with this post.

II. Seasonal Peak Tides - The 31/62 year Perigee-Syzygy Tidal Cycle.

2. The current post needs to be read in the light of the fact that a previous post at:

**El Niño events in New Moon epochs preferentially occur near times when the lunar line-of-apse aligns with the Sun at the times of the Solstices..**

**El Niño events in the Full Moon epochs preferentially occur near times when the lunar line-of-apse aligns with the Sun at the times of the Equinoxes. **

**The Changing
Aspect of the Lunar Orbit & its Impact ****Upon the Earth's Length of
Day.**

a. Inter-Annual Changes in the
Earth's LOD

The blue curves in figures 1a, 1b, and 1c
(below) show
the Earth's LOD over a six year period from January 1966 through to December
1971. These plots use daily LOD values that are available online from the
International Rotation and Earth Reference System Service (IERS) covering the
period from January 1962 until the present.

1a LOD
- 1966 to 1968

1b LOD
– 1968 to 1970

1c LOD
– 1970 to 1972

It is evident from these three figures
that there are abrupt periodic slow downs in the
Earth's rotation rate (corresponding
to an increase in LOD
~ 1 msec) once every 13.66 days (blue curve)* that are accompanied by much smoother
longer-term changes in LOD that are associated with the
annual seasonal cycle (red curves).

The smoother
longer-term seasonal variations in LOD are primarily the result of changes in
the angular momentum of the Earth that are a response to the slow (north-south)
seasonal movement of the Earth's atmosphere and its wind patterns.

The green LOD curve is
a crude seasonally de-trended version of the blue LOD
curve.

[Note: Half of a lunar tropical month
= 27.32158 /2 = 13.66079 days.*]

###
b. Reason for the Peaks in the Earth's LOD

A
more detailed
investigation
show
that the spikes in LOD (in
the blue and green curves) occur

within a day or two of the time that the
Moon crosses the Earth's Equator.

This tells
you that the slow down in the rotation rate is a direct result of the lunar
tidal bulge in the

Earth's oceans (and atmosphere) passing across
the Earth's Equator.

The slow
down occurs for much the same reason that a twirling ice-skater slows down
their rate of

spin by extending their arms i.e. by the conservation of angular
momentum.

###
c. The Change in the Ratio of Consecutive Peaks in LOD

In
early
1966 (figure 1a) the
peaks in LOD associated with transits of the Moon across

the Equator from the northern to the southern hemisphere, are roughly twice as large

as the next peaks in LOD (13.66 days later) that are associated with transits of the

Moon across the Equator from the southern to the northern hemisphere.

By early 1969 (figure 1b), the consecutive peaks in LOD are almost equal in size.

By late 1971 (figure 1c), the peaks in LOD that are associated with transits of the

Moon across the Equator from the southern to the northern hemisphere, are

roughly twice as large as the next peaks in LOD (13.66 days later) that are associated

with transits of the Moon across the Equator from the northern to the southern

hemisphere.

2.

Figure 2 shows that the absolute
size of the slow down in the Earth’s rotation rate is

determined by the
proximity of the Moon as its crosses the Earth’s Equator

3.

Figure 3 shows the ratio of consecutive peaks in
LOD versus lunar distance (in kilometres) for the

numerator of the ratio, for
the years from 1966 to 1971.

Figures 2 and 3 indicate that:

Whenever the
ratio of consecutive peaks in LOD is close to 1.0, the distance of the

Moon
from the Earth at consecutive transit crossings of the Equator are close to the

Moon's average
distance from the Earth of approximately 380,000 km.

However,
whenever the ratio of consecutive peaks in LOD is far from 1.0 (i.e. either

2.0
or 0.5), the distance of the Moon from the Earth at one transit crossing is at
the

distance of closest approach (i.e. the distance of lunar perigee = 356,000
km), and the

distance of the Moon at the other transit crossing is at its
furthest from the Earth

(i.e. the
distance of lunar apogee = 407,000 km).

4.

Figure 4 shows the difference in lunar
distance for consecutive transits of the Earth’s

Equator from 1962 to
1976.

Figure
4 confirms that whenever the perigee of the lunar orbit is pointed at Sun at
the time of the

Summer or Winter solstices, the difference in lunar distance
for consecutive transits of the Earth’s

equator is near zero kilometres i.e.
the ratio of consecutive peaks in LOD are ~ 1.0 because the lunar

distances at consecutive crossings are both close to the average
lunar distance of 380,000 km.

Similarly,
figure 4 confirms that whenever the perigee of the lunar orbit is pointed
at Sun at the time

of the Vernal or Autumnal equinoxes, the
difference in lunar distance for consecutive transits of the

Earth’s equator
approaches the maximum 50,000 km i.e.
the ratio of consecutive peaks in LOD are

far from 1.0
because the Moon is near perigee and apogee at consecutive crossings.

**d. Reason for the Change in the Ratio of Consecutive Peaks.**

5. Lunar Perigee pointing at the Sun
near Summer or Winter Solstices

The
yellow tilted elliptical orbits in figure 5 above, represent the apparent movement of the
Sun about

the sky as seen from the Earth. The Sun takes a full tropical year (365.242189
days) to
move once

about the yellow orbit, crossing the Earth’s equatorial plane
(the
grey plane) once every

six months at
the Spring and Autumnal (Fall) equinox, respectively.

The
red tilted elliptical orbits above, represent the apparent movement of the Moon
about the sky

as seen from the Earth. The Moon takes a full tropical month (27.32158
days) to
move once about

the red orbit, crossing the Earth’s equatorial plane roughly
once every 13.66 days where
the red

orbit crosses the black line [N.B. the ~ 5 degree tilt of the lunar orbit with respect to the ecliptic is

ignored as a second order effect at this stage of the argument.]

Clearly,
if the perigee of the lunar
orbit points at the Sun at either the Winter (i.e. the top diagram)

or Summer
Solstice (i.e. the bottom diagram), the Moon will be roughly at or near its
average

distance from the Earth (i.e. ~ 380,000 km). This means that when it
crosses the Earth’s equatorial

plane at consecutive transits of the Equator
(i.e. the intersection of the red orbit and the black line),

the difference in
lunar distance for consecutive transits will approach zero (i.e. the ratio of

consecutive peaks in LOD will be close to 1.0)

6. Lunar Perigee pointing at the Sun
near Vernal or Autumnal Equinox

The
yellow tilted elliptical orbits in figure 6 above represent the apparent movement of the
Sun about

the sky as seen from the Earth. The Sun takes a full tropical year (365.242189
days) to
move once

about the yellow orbit, crossing the Earth’s equatorial plane (the
grey plane) once every **six**

months at
the Spring and Autumnal (Fall) equinox, respectively.

The
red tilted elliptical orbits above, represent the apparent movement of the Moon
about the sky

as seen from the Earth. The Moon takes a full tropical month (27.32158
days) to
move once about

the red orbit, crossing the Earth’s equatorial plane roughly
once every **13.66 days** where
the red

orbit crosses the black line.

Clearly, if the perigee of the lunar
orbit points at the Sun at either the Vernal (i.e. the bottom

diagram) or Autumnal
Equinox (i.e.
the top diagram), the Moon will be
at or near perigee and apogee

at consecutive transits of the equatorial plane
(i.e. where the red orbit crosses the black line).

This means
that when it crosses the Earth’s equatorial plane at consecutive transits,
the difference
in

lunar distance for consecutive transits will approach its
maximum value of 50,000 km (i.e. the ratio

of consecutive peaks in
LOD will be as
far from 1.0 as possible).

*Hence, figure 4 tells us that the perigee moves from pointing at **the Sun at a **Solstice(/Equinox)*

*to **pointing at the **Sun ***at ****the ****following ****Equinox(/Solstice) in the seasonal calendar once every **

**2.0 ****Full ****Moon ****Cycles ***(FMC) = 2.0 x **411.78444836 **days = 2.2547695 sidereal** years = 2.2548570*

*tropical years*

**e. The 9.019 Tropical Year Cycle**

7.

Figure 7 shows the ratio of the consecutive 13.66 day peaks in LOD from 1962 to 1988.

It is evident from this plot that the long-term variation in the ratio of consecutive peaks is dominated

by a periodicity of ~ 9.0 years. In addition, there is an approximate 18.0 year periodicity that

modulates the ~ 9 year periodicity.

The ~ 9.0 periodicity results from the fact that **4.0 x 2 FMC ****= 9.019428 tropical years ** is the

time required for slowly drifting perigee of the lunar orbit to return to pointing at the Sun at the same

Solstice or Equinox (since it takes **2.2548570 topical years for **the alignment of the perigee with the

Sun at a Solstice/Equinox to move to the following Equinox/Solstice in the seasonal calendar**.**

*(N.B. The *~ *9.0 period is close to but not necessarily the same as the 8.85 years that it takes the*

*lunar **line-of-apse to precess once around the Earth with respect to the star. In addition, the*

~ *18.0 **periodic modulation of the 9.0 year period is close to but not necessarily the same as the 18.6*

*year period that it **takes the **lunar line-of-nodes to precess once around the Earth with respect to the*

*stars. The 18.6 **year **period is caused by the slowly changing tilt of the Moon’s **orbit with respect to*

*the Earth’s **equator **caused by precession of the lunar-line-of-nodes - Please see the Addendum at the*

*end of this post about **the possible long-term interaction of these two lunar cycles).*

###
**f. A Gleissberg Cycle in the Ratio of Consecutive P****eaks in ****LOD.**

The slow pro-grade precession of the perigee of the lunar orbit through the seasonal calendar leads to

the perigee
moving from pointing at the Sun at one Solstice(/Equinox) to pointing at the Sun
at the

following Equinox(/Solstice) once every 2.0 Full Moon Cycles (FMC) = 2.0 x 411.78445750

days = **2.2547695 sidereal years** = 2.2548570 topical years.

However, since 2.0 FMC is longer than 2.00 sidereal years (by 0.2547695 sidereal years), the perigee

of the lunar orbit will slowly drift from pointing at the Sun at a given point in the seasonal calendar,

only re-synchronizing with the seasonal calendar after:

(2.0000000 x 2.2547695) / (2.2547695 - 2.0000000) ≈ **17.70046823 sidereal** years

This comes about because there are 8.850234 lots of 2.00 sidereal year cycles in 17.70046823
sidereal years and 7.850234 lots of 2.2547695 sidereal year cycles in 17.70046823 sidereal years.
In effect, what this means is that there will be a precise realignment between when the lunar perigee

points at the Sun at a given Solstice/Equinox and when it does so again at the same point in the

seasonal calendar (i.e. the same Solstice/Equinox), once every 354.000 years:

157 x 2.0 FMC = 353.9988071

≈ **354.00 ****sidereal years**

Of course, it would take half of this time i.e. 177.0 sidereal years, if we broaden our criterion to
include the cases where the lunar Perigee points directly away from the Sun:
157 x 1.0 FMC = 176.9999404 ≈ **177.000 ****sidereal years**

And it would take half of this time again i.e. 88.5 years, if we only make the constraint that the

Perigee of the lunar orbit move from one Solstice/Equinox to the next, rather than go through the full

seasonal calendar:

**157 x 0.5 FMC = 88.499702 ≈ 88.500 sidereal years **

Hence,
the long-term variations of the ratio of the strength of the consecutive (13.66 day) slow

downs in
the Earth’s LOD that are caused by the Moon transiting across the Earth’s
Equator,
have a natural long term repetition cycle with respect to the seasons that matches that of the:

88 year Gleissberg Cycle

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Addendum 1 - ****A Possible 2340 Year Hallstatt-like Cycle **

In this blog post we have established that there is a quasi 9.0 year periodicity in the ratio of
consecutive increases in the Earth’s LOD that result from transits of the Moon across the Earth’s
Equatorial plane every 13.66 days.

There is a possibility that the quasi 9.0 periodicity could be the harmonic mean of the two factors in

figure 7 that are influencing the long-term variations in the ratio of consecutive increases in the

Earth’s LOD i.e. the 8.8505 tropical year **precession of the lunar line-of-apse (i.e. the Lunar**

**Anomalistic Cycle - LAC) **and the 9.3001 (= 18.6002/2.0) tropical year half cycle for the precession

of the lunar-line of nodes (i.e. the half Lunar Nodical Cycle - LNC) such that:

2.0 x (9.3001 x 8.8505) / (9.3001 + 8.8505) = 9.0697(3) tropical years

Since 9.0697(3) tropical years is slightly longer than 9.00 tropical years, it slowly drifts by 0.0697(3)

tropical years once every 9.06973 years, resulting in a forward shift by

one full tropical year once every:

9.06973 / (9.06973 - 9.00000) = 9.06973 / 0.06973 ≈ 130.069267 tropical years

In addition, the beat period between 9.0697(3) tropical year and 9.00000 tropical year gives:

9.00 x 9.06973 / (9.06973 – 9.00) = **1170.623 tropical year** ≈ **9.0 x 130.069267 tropical years**

Which is **half of a ****Hallstatt-like cycle of 2341.247 tropical years. **

Hence, the interaction between the LAC with LNC naturally produces a Hallstatt-like cycle.

__Addendum 2 - Some More Information on the 31 Year Perigee-Syzygy Lunar Tidal Cycle__
A1

Figure A1 shows the offset in days from a New/Full Moon at each instant where the Perigee of the

lunar

orbit points either towards or away from the Sun - up to 27.5 FMC = 31.00308 sidereal years.
This figure shows us that phase of the Moon goes from:
New --- Full --- New --- Full

every

0.0 --- 9.00 --- 18.00 --- 27.00 FMC's

0.0 --- 10.146463 --- 20.292925 --- 30.439388 sidereal years

with the Moon being less than one day past Full Moon at 27.5 FMC = 31.00308 sidereal years.
A2

Figure A2 shows the offset of the Moon from the lunar line-of-apse at each instant where the Perigee

of the lunar orbit points either towards or away from the the Sun - up to 27.5 FMC = 31.00308

sidereal years (N.B. the Perigee alternates between point directly at the Sun to pointing directly away

from the Sun once every 0.5 FMC).

Hence

New Perigee --- Full Apogee --- New Perigee --- Full Apogee every

0.0 --- 9.00 --- 18.00 --- 27.00 FMC's

0.0 --- 10.146463 --- 20.292925 --- 30.439388 sidereal years

with the Moon returning to perigee, less than one day past Full Moon, at 27.5 FMC = 31.00308

sidereal years.