Lunar cycles, the Chandler Wobble and the 1:81 ratio

In the paper THE CHANDLER WOBBLE OF THE POLES AND ITS AMPLITUDE MODULATION (here) by N.Sidorenkov, the abstract says: ‘It is shown that the period of the Chandler wobble of the poles (CWP) is a combined oscillation caused by three periodic processes experienced by the Earth: (a) lunisolar tides, (b) the precession of the orbit of the Earth’s monthly revolution around the barycenter of the Earth – Moon system, and (c) the motion of the perigee of this orbit.’ The focus in this post will be on (b).

The introduction says: ‘It is well known that the Earth and the Moon revolve around their center of mass (barycenter) with a sidereal period of 27.3 days. The orbit of the Earth’s center of mass (geocenter) is geometrically similar
to the Moon’s orbit, but the orbit size is roughly 1/81 as large as that of the latter [bold added]. The geocenter is, on average, 4671 km away from the barycenter. In the Earth’s rotation around the barycenter, all its constituent particles trace the same nonconcentric orbits and undergo the same centrifugal accelerations as the orbit and acceleration of the geocenter.’ Another glimpse of the physics there.

Following on from the two earlier lunar cycles posts (here and here), we expand the lunar chart (see below) by a factor of 5, as discussed in the second of those posts. This allows the Chandler Wobble and its cycle period to be related to the other whole number ratios. It also gives the lunar nodal cycle (LNC) two components, one of 106 LNC as per the original lunar chart, and the other of 5 LNC. We already found in the ‘square year’ post that ‘In every 5 lunar nodal cycles a sequence of 13 evections in latitude will occur 81 times.’

From the ‘square year’ post we know it meant a period of 385 draconic years (DY). In the lunar chart here we have 2079 DY. Since 2079/385 = 5.4, we multiply the whole chart by 5 (5.4*5=27), giving a result for DY of 385*27 = 2079*5 = 10395 DY.

The new chart is shown on the right. Referring back to the 2015 chart (here):
RLA has become the de Rop cycle, same quasi-6-year cycle but different name. It’s the period when the number of lunar nodal and apsidal cycles sums to 1, as described and illustrated in a short paper by Belgian astronomer Willy de Rop, featured at the Talkshop here. Thus 530 LNC plus 1115 LAC now sum to 1645 de Rop cycles.

The graphic below gives us a visual aid. As the line of nodes precesses west, the line of apsides precesses east, the lines returning to the same point in just under 6 years.
On the chart it’s 9865/1645 = 5.9969604 tropical years (TY).

So that’s the background to the post. We think the model can be extended as shown below.

The Chandler Wobble (CW)

Wikipedia says: ‘The Chandler wobble or Chandler variation of latitude is a small deviation in the Earth’s axis of rotation relative to the solid earth, which was discovered by and named after American astronomer Seth Carlo Chandler in 1891. It amounts to a change of about 9 metres (30 ft) in the point at which the axis intersects the Earth’s surface and has a period of 433 days.’

The 433 days is the nearest whole number of days, but the CW period doesn’t have to be a whole number.

One way to obtain a period for it is by using the lunar tropical year (LY) of 13 tropical months (TM). Since 35.3 tropical years (TY) = 36.3 LY, that gives us the beat period (difference of 1) of that pair, and 2 CW = 1 quasi-biennial oscillation (QBO):
35.3 TY = 13*36.3 TM = 1.8965 lunar nodal cycles
QBO in 35.3 TY = 13+1.8965 = 14.8965
1 QBO = 35.3/14.8965 = 2.3696841 TY
1 CW = QBO/2 = 2.3696841/2 = 1.1848421 = 432.7543~ days.
That gives us 8326 CW per 9865 TY:
8326*1.1848421 TY = 9864.9953 TY

The remainder of 9865-8326 = 1539.
That’s the number of Chandler Wobble cycles (CWC), i.e. the beat period of the CW and the tropical year (1 here):
CWC = 1.1848421*1 / (1.1848421-1) = 6.4100229 TY.
1539*6.4100229 = 9865.025 TY.

That tells us the CW number is a shade too low.
9865/1539 = 6.4100065 TY
The number of CW in the CWC is one less, by definition:
6.4100065/5.4100065 = 1.18484266~ TY = 432.75453 days.
Then: 5.4100065*1539 = 8326 CW.

The 1539 CWC reduces to 19*81, meaning a set of 19 CWC occurs 81 times in the full period. A further point is that the full period is also 385*27 draconic years (DY), and 19*81 = 57*27. Therefore we have exactly 57 CWC in 385 DY, which is the ‘square year’ from the earlier post, with 19 CWC being 1/3rd of that. Since the square year has 4134 evections in latitude, 1/3rd of that number (1378) = 19 CWC. The 1378 evections = 13*106.

There’s more to all these numbers, which will be the focus of the next post in the series, shifting back to the much shorter 5 LNC period (just over 93 years). Fundamentally the moon is one body so its various cycles, periods etc. should integrate with each other, or at least tend to do so. The 1:81 ratio suggests one way it relates itself to the motions of the Earth.
* * *
Footnote:
For most purposes it seems accurate enough to say 5.41 CW = 6.41 tropical years (TY) = 1 CW cycle. Removing the decimal places leads to:
541*81*19 CW = 832599 CW
641*81*19 CWC = 986499 TY (= 641*81 Metonic cycles).
Difference = 153900 CWC.

Putting the decimals back: 8325.99 CW = 9864.99 TY.
Compare to our model: 8326 CW = 9865 TY.
Either way the number of CWC is the same: 1539 (81*19).

Source: https://tallbloke.wordpress.com/2025/11/05/lunar-cycles-the-chandler-wobble-and-the-181-ratio/