What is the cause of lunar nodal and apsidal precession?

[fizzy]

Over the last year or so I've been doing quite a bit of reading trying to find out what is actually causing lunar nodal precession. There seems to be a lot of handwaving but no definitive answer, which I find rather odd.

Maybe I've just not found the right source. Google seems obsessed with returning links to WonkyPedia these days and most articles there, rather than enlighten me, make we want to scream.

Firstly let me try to fend off some of the more erroneous likely answers.

The precession of a significantly sized satellite like the Earth's moon is NOT the same as a very small moon ( satellite in astronomy terms ) and man-made low Earth orbit satellites. For the latter. the precession is apparently mainly caused by the Earth's oblateness. So please do not reply saying this is the cause of lunar nodal precession unless you have some solid maths and refs to back it up.

One of the more credible explanations seems to be the torque exerted on the Earth-Moon angular momentum by the sun. Now if that is the case, we have the numbers and should be able to get something very close to 18.61 tropical years. I have only found some rather handwaving and admittedly approximate maths. I suspect this is only part of the story but I'll rather surprised that with the precission of astronomic calculations made these days we don't have a better account of something we have been studying for several millennia !

The corollary question is what is the most accurate assessment we now have for this period? Of course none of this is constant with the number of bodies in the calculation.

The best I have managed to find was fourth order polynomial credited to Chapront and relative to Y2K reference period:

# Lunar nodal cycle comes from (derived by T. Peter from Chapront [2002],
T_1000 = time from J2000.0 [1000 Year]
(6793.476501 + T_1000 * ( 0.0124002 + T_1000 * ( 0.000022325 - T_1000 * 0.00000013985 ) ) ) / 365.25

# thus evaluating this for year 2000 it is a constant :
print pNodal= 6793.476501/365.25
18.5995249856263

This is in sidereal years.

I did at one stage find some maths that produced a sin().cos() type formula for the torque on the E-M couple, though sadly I've lost the source of that information.

AFAICR, this ( counter intuitively ) gives zero torque with the moon at 0 , 90,180, 270 degrees from the line from sun to EM barycentre and max amplitude at the four "45" degree points .

So, if I am recalling this correctly there will be a cyclic change in the torque exerted on the E-M couple with a period half that of the nodal precession period.

ie there will be cyclically varying acceleration / deceleration with a period of circa 9.3 years.

The time series (3rd order) for Lunar apse cycle comes from (Chapront [2002], page 704)
(3232.60542496 + T_1000 * ( 0.0168939 + T_1000 * ( 0.000029833 - T_1000 * 0.00000018809 ) )) / 365.25

## evaluate at y2k:
=3232.60542496/ 365.25
=8.85039130721424

Since this is from the same source, I assume that this is also sidereal years.

Can anyone see any faults in this or add to this level of understanding?

Thanks.

 

[tony873004]

...Google seems obsessed with returning links to WonkyPedia these days and most articles there, rather than enlighten me, make we want to scream...

Wikipedia is actually quite good with astro articles. A few of their biggest editors visit this and other astro forums to confirm facts.

...The precession of a significantly sized satellite like the Earth's moon is NOT the same as a very small moon...

Yes it is, at least to first order.

...and man-made low Earth orbit satellites. For the latter. the precession is apparently mainly caused by the Earth's oblateness...

That is correct. Earth acts less like a perfect sphere for stuff that orbits very close to its surface. Oblateness does make a difference at the distance of the Moon, but the effects are negligable over timespans of decades or centuries.

One of the more credible explanations seems to be the torque exerted on the Earth-Moon angular momentum by the sun.

This is correct. To a first order, this is the ONLY thing that causes the Moon's Longitude of ascending node and its argument of perihelion to change at all. The Moon's ascending node progresses with a period of about 18 years. Its argument of perhielion progresses with a period of about 9 years. The Sun is responsible for almost all of this. Perturbations by the planets may add up over millions of years, but not decades or centuries.
All of this is easily verified with numerical simulation. The book "Solar System Dynamics" by Murray and Dermott can probably give you a more analytical approach.

 

[fizzy]

Thanks for the reply Tony.

"Yes it is, at least to first order."

These course notes explain how the maths vary for small satellites ( natural and man-made ) and a large satellite like the moon.
astro.cornell.edu/academics/courses/astro6570/Precession_Free_and_Forced.pdf

You seem to agree that low Earth orbit satellites are dominated by oblateness and the moon is dominated by solar gravitation. That is what I meant by saying the two cases are NOT the same.

It is worth being more precise than "about 9 years" and "about 18 years" since there are several different but similar cycles at play. 18.61y nodal precession, 18.0y saros ( eclipse ) cycle ; 8.85y precession of the lunar apsides and 9.3 = 18.61/2 which is the periodic angular accel/decel variation in the nodal precession produced by the sun.

Do you have any information about cause of precession of lunar apsides? I have reason to think it is planetary but have not been able to find anything reliable about the cause.

 

[tony873004]

9.3 = 18.61/2

I'm not sure if this relevant or just a coincidence. Most of what I know is from numerical rather than analytic solutions. i.e. I care more about "what" rather than "why". I'll give you a more detailed response over the weekend. Welcome to PF!

 

[tfr000]

Mean annual motion of the: perigee node
caused by:
Principal solar action +146426.92 -69672.04
Figure of the Earth +6.41 -6.00
Direct planetary action +2.69 -1.42

from Roy, Orbital Motion, 3rd edition, 1988

Values are in arcseconds.
So as you can see... the vast majority of both is from perturbation by the Sun.

 

[tfr000]

Well... that didn't quite format right, but you get the idea. The first number is for lunar perigee, the second for lunar ascending node.

 

[fizzy]

Thanks for the input. That confirms that figure of the Earth is not significant, as I said at the top.
I think "Direct planetary action" is probably the key point, indicating how this is being analysed.

The solar effect on EM angular momentum is the key factor the direct contribution of planetary gravity is minimal. The effect of the planets on the Earth-Moon orbit and the position of the sun will cause periodic changes in the solar term.

What I was wanting to find was at least a first order mathematical explanation for the nodal and apsidal period.s. Does Roy show the derivation of those figures?

 

[tony873004]

You can try it here in a simulation:
http://orbitsimulator.com/gravitySimulatorCloud/simulations/1452313177243_lunarNodePeri.html

Press the Play button [>] on the "Time Step" interface to begin the simulation. Then you can watch the Moon orbit the Earth under the influences of the Sun and planets.

In the box labeled "Collision Log" it is recording the Moon's Longitude of ascending node and argument of perigee once per lunar orbit. Let it run for 19 years, pause the Sim, and copy this data and paste it into a spreadsheet. Then you can graph it, and you can clearly see the 18-year period of the LAN and the 9-year period of the peri.

Afterwards, refresh the Sim. Press E on your keyboard to edit objects. Set the mass of all the planets (except Earth and Moon) to 0 and repeat. You will likely see that the graphs are virtually identical. Hence the planets don't contribute much.

Finally, refresh the Sim, Press E to edit, and set the mass of the planets (except Earth and Moon) AND the Sun to 0. You will likely see no change at all in the Moon's elements.

You can also try changing the mass of the Moon to see what effect, if any, that has.

The objects are all treated as point masses, so you can't test for the effects due to Earth's shape.

 

[fizzy]

Many thanks Tony. I had thought that this must be possible with your sim but I've only briefly tried it and have no idea how to set it up and drive it. This should be a great help, I'll give it a try.

Is that available as gsim file, it's rather slow in the browser: about 1h per year :(

 

[fizzy]

This seems to be similar to what I'd previously read. It deals with the precession of an oblate planet but the same principal can be applied to the E-M system as influenced by the sun ( at least to a first approx .)

http://astro.cornell.edu/academics/courses/astro6570/Precession_Free_and_Forced.pdf

the cartesian components of the torque are:
T_x = - T₀ i sin² u
T_y = T₀ i sin u cos u

Both of these terms have a frequency twice that of the orbit. The second one averages to zero over the cycle , which in this case is a year (not the precession period as I had incorrectly recalled). Now both LAN and the peri are showing a roughly 6mo variability but of course it is slightly longer as we know. This suggests that some other bodies are involved.

So far the test with gravity sim with J,S and V at zero mass is producing numbers identical to last digit. Which in view of relative magnitudes given by trf000 above seems a little odd.

 

[tony873004]

Is that available as gsim file, it's rather slow in the browser: about 1h per year :(

To do it in the Windows version, make sure you have the latest beta version available on the message board at gravitysimulator.com

Start with any sim that contains the planets and the moon.

Under the File menu, choose File > Output File...

Choose the Moon, and the desired orbital elements. Check "Create Data File" and "sample every" 2360584.6848 seconds.

Under the Integrator menu change from Euler to RK-4. Set the time step no higher than 1024.

When you run it, it will create a file with the same name as your .gsim file but with an extension of .dat or .txt.

What kind of computer and browser are you using?
The browser version should be must faster than 1h per year. I get about 15 seconds per year on my both my Windows 10 computer, and my Mac. But both these computers are very new. My 12-year old Windows XP computer does a year in about 3 minutes.

You can speed it up. I set the time step to 256, but you can increase it to 1024 and still get good results.
Also, under the "Preferences" menu, change "Do Events" to a higher number.
Both of these will cause fewer graphic updates, and your orbit might start looking like a stop sign instead of a smooth ellipse, but the data will still be good.

So far the test with gravity sim with J,S and V at zero mass is producing numbers identical to last digit. Which in view of relative magnitudes given by trf000 above seems a little odd.

I'm getting differences 4 places right of the decimal.
The data for August 30, 2015 gives me Longitude of ascending nodes of:
180.86465069850223 with Jupiter, Saturn and Venus at their correct masses
180.86450990382707 with Jupiter, Saturn and Venus masses set to 0

Make sure after you set them to 0 that you also press "Apply".
Re-choose them in the dropdown list to verify that their masses are 0.

Also, make sure your browser is the active window. It doesn't give very many CPU cycles to a buried tab.

 

[fizzy]

OK browser is firefox 43 ( latest update ) , OS is Linux, single core Athlon 64.
"windows version" , is there another version? I tried the Win version on XP , it ran the demo on installation but hung there after :(
I haven't used windoze in years and have no interest is messing around with it.

I'll have to check that I did the apply thing. I may have missed that. Ah, I've found I need to apply every change before moving to adjust another planet !

Thanks for the tuning tips. I'll have another try.

Is there a way to clear the collision log? I can "select all" but can't delete. File | Collision log seems to do nothing. I don't think this is working too well on firefox.

 

[tony873004]

The other version of Gravity Simulator, the one that makes .gsim files, only runs on Windows. It is written in Visual Basic 6.0.
That's why I made the browser version. It runs on anything.

 

[fizzy]

OK , I'm starting to get a first indication of this now. It seems the nodal data remains very close to the same without the planets. The circa 6mo cycle it half the draconic year ie 173.3d : the time it takes for the lunar node to coincide with the sun.
That fits the freq doubling I mentioned.
1/(1/drac-1/1) = 18.6 y

It looks like the perigee is more notably affected, though it does not look like the period has changed yet.

Thanks for the help, I think this sim is going to answer my questions.

 

[tony873004]

Is there a way to clear the collision log? I can "select all" but can't delete. File | Collision log seems to do nothing. I don't think this is working too well on firefox.

Collision Log was designed to only let the program write to it.
But you can use Quick Help:
Close Collision Log
Use the Help menu to open "Quick Help"
This text area is user-editable.
Under the Autopilot menu, choose "per graphic update" and change the line:

txtAreaCollisionLog.value = txtAreaCollisionLog.value + UTC(elapsedTime * 1000 + startDate) + "," + L + ',' + P + String.fromCharCode(10);

to

txtAreaQuickHelp.value = txtAreaQuickHelp.value + UTC(elapsedTime * 1000 + startDate) + "," + L + ',' + P + String.fromCharCode(10);

Then it will store the data in Quick Help instead of Collision Log.

 

[fizzy]

Thanks, what is the brouwser version written in , java? I would have thought that text window object would have its own built-in functionality, it has a menu with cut , copt etc but they don't seem to work. Oh well.

I let it run all night and I got about 100y which is enough to prove the point. : still seeing 8.85 and 18.61y periods with other planets at zero mass. However, if I zoom out and watch the planets, I see they are still orbiting ! This implies positional information is coming from something other than a gravity calculation and this will also apply to E-M position and motion.

Now I would expect a planet with zero mass to carry on moving in a straight line with the position and velocity of the initial conditions or not move all since it has no inertia. If the planets are still orbiting then I am not surprised that lunar cycle has not changed. It seems that setting mass to zero is only removing the direct planetary effects. Which as tfr000 showed are minimal.

Is there an explanation somewhere of the basis of what the sim is calculating? Can you explain why zero mass planet is still orbiting ?!

 

[tony873004]

Now I would expect a planet with zero mass to carry on moving in a straight line with the position and velocity of the initial conditions or not move all since it has no inertia.

You've got a Physics misconception here.

Positional info is simply from a gravity calculation.

zero mass objects can still orbit. Acceleration on an object by the Sun is
a = GM/ d^2
Where M is the mas of the Sun and d is the distance between an orbiting object and the Sun. The orbiting object's mass is not in the formula.
If the mass of the orbiting object made a difference in how hard gravity accelerated it, then if you dropped a big rock and a small rock, they would hit the ground at different times.

The only difference is how much a massless particle can accelerate the Sun:
a = GM/d^2
Now M = mass of the massless particle. i.e. the sun won't accelerate.

If you want to cause the orbiting objects to carry on in a straight line, you need to set the mass of the Sun to 0.

 

[fizzy]

Physics misconception: Gravitational attraction is a mutual attraction which depends upon the mass of both objects.

How much is a zero mass accelerated by a zero gravitational attraction , and in what direction ? Does a zero force vector have a direction? Interesting concepts.

The correct formula is a=GmM/(d²m) = 0/0 , the answer is undefined.

For the sake of argument, I suppose we could set the planetary masses to one gramme and they would orbit as shown. It raised some questions when I saw gravitational attraction acting on zero mass objects but if that is the way it's coded, that makes sense.

Is it possible to get the collision log to show perihelion distance ?

Thanks.

 

[fizzy]

One of the reasons I posted this question is that there are some strange relationships between these periods that I do not find any explanation for is the simplistic 'it's the sun' answers.

Taking the most accurate figures I have found for J2000 epoch, given in the original post here:
pNodal= 6793.476501
pApsides=3232.60542496

Calculating the difference of the two frequencies:
1 / (1/pApsides - 2/pNoda) / 365.25 = 183.159356370069 julian years.

2 / (1/pApsides - 2/pNoda) / 365.25 = 366.318712740139 julian years.

Now that means that the two cycles are drifting by about 1 day per year. More importantly it is not a solar day but a Earth rotation period.

If this was simply an E-M-S interaction the relationships should remain in that frame. Where does the linkage to the sidereal Earth rotation come from ? Does this imply some fixed frame influence ?

Why half the Nodal period, is this to do with sin²u frequency doubling I mentioned?

Maybe there is a banal explanation to this which I am missing.

 

[tfr000]

What I was wanting to find was at least a first order mathematical explanation for the nodal and apsidal period.s. Does Roy show the derivation of those figures?

No, he doesn't, but some other authors do. I don't think their simplified derivations would give you any kind of reasonable approximation of reality.
The definitive semi-analytical work on lunar theory is: https://books.google.com/books?id=CKvQAAAAMAAJ
Good luck with it. He worked on it all of his life, as far as I know. It would probably take years just to understand it.

 

[tony873004]

A force may be a mutual attraction that depends on the mass of the 2 interacting objects, but acceleration is not. If I hold a 7-kg bowling ball 1 meter above the ground, the Earth is pulling the bowling ball with a force of 68.6 Newtons down. The bowling ball ispulling the Earth with a force of 68.6 Newtons up. F=GMm/r^2 = 6.67e-11(5.97e24)(7)/6371000^2 = 68.6 Newtons. When I let go, the ball accelerates down at a = GM/r^2 = 6.67e-11(5.97e24)/(6371000^2) = 9.8 m/s^2. The Earth also accelerates up at a rate of Gm / r^2 = (6.67e-11)(7)/6371000^2 = 1.1502934948353E-23 m/s^2. That is unnoticable.

The correct formula is a=GmM/(d2m) = 0/0

From this point you treat it like a Limits problem from Calculus class. The m's cancel, just like your goal in any "definition of the first derivative" problem is to get an h in the numerator to cancel with the h in the denominator. Then you are allowed to cancel the h's and continue.

$$f'\left( x \right) = \lim _{h \to 0} \frac{{f\left( {x - h} \right) - f\left( x \right)}}{h}$$

As m approaches 0, acceleration of m approaches GM/d^2
As m approaches 0, acceleration of the Sun approaches 0.

Light is massless. Eddington demonstrated that its trajectory gets bent by gravity the same as it would if it had mass by measuring the position of a star that appeared very close to the Sun during a solar eclipse.

 

[tony873004]

Is it possible to get the collision log to show perihelion distance ?

Yes. q = perihelion, a = semi-major axis, e = eccentricity.
q = a (1 - e)
Under the Autopilot menu, choose "per graphics update"
Now you can modify the code to output q.

if (elapsedTime > APnv1) {
ComputeOrbitalElements(11, 4);
var L = (oeLAN * 180 / Math.PI).toString();
var P = (oePeri * 180 / Math.PI).toString();

var a = oeSMA;
var e = oeEcc;
var q = a * (1 - e);

txtAreaCollisionLog.value = txtAreaCollisionLog.value + UTC(elapsedTime * 1000 + startDate) + "," + L + ',' + P + String.fromCharCode(10);
APnv1 = APnv1 + 27.321582 * 86400;
}

Then in the line below var q = a * (1 - e);
change either L or P to replace either LAN (L) or argument of perihelion (P) with perihelion distance (q),
or add

+ ',' + q

after P to output all 3 values.

Also, like I mentioned a few posts ago, you can output this stuff to the Quick Help interface instead of Collision Log. That way the data is also user-editable, so you can clear it if you like.

 

[fizzy]

Thanks for the link trf000. Apart from the horrible scan quality it looks fairly well explained and easy to follow. However, google books keeps chopping out pages so it's not really much use. I wasn't able to find anything that looked particularly relevant to what I'm seeking.

I'm sure there must be a lot of more detailed stuff on this, maybe I need to find a more specific astronomy forum or get in direct contact with academics but I want to get a better understanding to avoid wasting anyone's time.

 

[fizzy]

"Light is massless. Eddington demonstrated that its trajectory gets bent by gravity the same as it would if it had mass by measuring the position of a star that appeared very close to the Sun during a solar eclipse."

A photon has zero rest mass. The ones Edison was watching were not at rest ! Photons have energy and thus an equivalent mass. That is what Edison demonstrated.

The idea of a zero force accelerating a zero mass is in an undefinable direction is a pretty meaningless situation to discuss. The usual thing is to talk of test particle of small but finite mass. That would orbit as in your model. Like I said, the model seems to function correctly, I was just surprised to see zero masses displaying gravitationally driven behaviour.

Thanks for the tips on following perihelion. I had noted your comments about using the help window.

 

[fizzy]

BTW the 100y run I did with zeroed out planets produced a kind of torus. I was expecting the lunar orbit to cycle right around with precession. This is presumably viewed in a rotating frame, what is the horizontal direction that plot?
Thanks.

 

[tony873004]

That is what Edison demonstrated.

That is what Einstein predicted and Eddington demonstrated. Edison was the light bulb dude, born much earlier. Too many E's :)
Regardless, if you're not comfortable with zero mass, do what you you suggested earlier... don't set the mass to 0. Set it to 1 gram... actually, why stop at 1 gram? Just set it it to 9e-99 micronanomilligrams. Then you have a non-zero number to cancel the numerator with the denominator. It will just take the program longer to get you results since it knows to ingore 0.

No, it is not a rotating frame. The program allows rotating frame, but it is not set in your starting simulation. With what period do you want it to rotate with? I can help you set this.

Fascinating conversation. But the weekend is over, so I may be slower to respond in the next few days.

 

[fizzy]

I've been thinking about how well zero mass planets indicates their effect on lunar orbits. As I said initially it only really tells about the direct gravitational effects. So where are the indirect ones?

Well they are in the initial conditions of the run which are provided by JPL ephemeris. The position, velocity etc of the planets are the result of billions of years of mutual interactions. Just switching off their mass will not instantly provide what would happen without them. That would take billions of years of sim time, or at least some massively long run to find new equilibrium of the solar system.

For example an orbital resonance will probably take a very long time to break up once it is programmed in by the initial conditions. As we've seen the "zero mass" planets carry on orbiting much the same as their massive counterparts.

Similarly, all the 'elements' of the E-M orbit have been honed by billions of years of mutual gravitational interaction of all the planets in the solar system as well as the sun. The period, distance and eccentricity of the EM orbit which affect the sun driven torque : the primary cause of nodal precession. are themselves determined by the planets.

So the period of the EM orbit which determines the period of the nodal precession is itself a result of the system as a whole.

 

[fizzy]

"No, it is not a rotating frame. The program allows rotating frame, but it is not set in your starting simulation. With what period do you want it to rotate with? I can help you set this."

I'm having trouble seeing the nodal precession. The plane of the lunar orbit rotates 360 deg in 18.61y, in the plot it just looks like a wobble not a full turn. :?

 

[fizzy]

OK, I think I get it. It's looking down on the plane of the ecliptic, it's just not very clear with a 2D image with just two grey tones. Once there's five nodal periods on top of each other its not at all clear as an image. If we could rotate the view point it would probably be a lot clearer but that looses all the plotted data.

Maybe means to stock all the data points and then play with the sliders would be good for visualising this sort of thing.

I'm still trying to understand the 366.33 year repetition. pNodal is essentially E-M-S so this suggests that precession of the aspides is timed to the stella inertial frame. And why is this reflecting Earth rotations. Surely someone must have spotted this and explained it.

 

[tfr000]

Thanks for the link trf000. Apart from the horrible scan quality it looks fairly well explained and easy to follow. However, google books keeps chopping out pages so it's not really much use. I wasn't able to find anything that looked particularly relevant to what I'm seeking.

I'm sure there must be a lot of more detailed stuff on this, maybe I need to find a more specific astronomy forum or get in direct contact with academics but I want to get a better understanding to avoid wasting anyone's time.

Well if you're comfortable with some math,
http://www.willbell.com/math/mc7.htm
has a (brief) chapter on the Moon's motion. They didn't include the CDs back when I bought my copy. Danby can be a little terse - by which I mean, I sometimes feel like I would have liked a little more explanation before he goes off into a set of problems for you to solve. Most of the other books I have around are either old and out of print, or new and don't include lunar theory...

 

[fizzy]

Thanks, that looks like a good reference ( Wow Borland Pacal code too, excellent ) though this sort of stuff must be available on line.
Seems like the reverent sections are :
13.3. The Couples Exerted on the Earth by the Sun and Moon 392
13.4. The Lunisolar Precession 395

I have plotted the numbers from Tony's gravsim which is quite enlightening.

The nodal precession is a steady drift with a circa 6mo oscillation. Since the precession is retrograde the period is found by summing the frequencies:
pNodal= 6793.476501 days

0.5/(1/pNodal+1/365.25) = 173.307172137645
= eclipse period = eclipse year / 2
This is time for sun to coincide with the lunar nodes, one of the conditions for an eclipse. Since the torque is the same whether the plane is inclined towards or away from the sun, it happens twice as fast.
This is also the sin² term I mentioned above that doubles the period. sin² looks like a sine the goes from zero to 1 with a period of π instead of 2π . The average drives the steady precession.

The apside is much more interesting. There is a similar 6mo oscillation but with a notable amplitude modulation. The drift is in the opposite sense. The 'beat' period of this modulation is the 8.85y apsidal period but this means that the modulation is twice that. This is standard interference patterns like tuning a guitar by harmonics.

This is interesting since it goes some way to explaining my question of why I had to halve on the frequencies.

Manually analysing the graph I count 167 little bumps in 10 of the modulation beats:
167 bumps in 10 cycles.
pApsides=3232.60542496
pAps_6mo =10*pApsides/167. = 193.569187123353

1/(2/365.25 - 1/pApsides) = 193.560116288636

Note here it is the difference of the frequencies because the precession is in the other direction.
A similar alignment is seen with the same doubling but in addition the modulation of the amplitude and a much stronger c. 6mo component.

For now I note the two are slightly different , this may be significant or it may be an error from reading the graph. I will need to do a fit or spectral analysis to check that more accurately.

When the eccentricity is greatest the 6mo oscillation in the apsidal precession is greatest.

 

[fizzy]

I've been giving some more time to this question.

The frequency doubling of torque equation for the nodal precession explains why that period gets halved. The eccentricity variation is one cycle between successive alignments of line of apsides , not two, so it stays whole. Now these half-the-sum and half-the-difference equations are what would apply to side-band frequencies in amplitude modulation to find the 'carrier' and modulation frequencies which cause them:

2 / (1/pApsides - 2/pNoda) / 365.25 = 366.318712740139 julian years.
2 / (1/pApsides + 2/pNoda) / 365.25 = 9.07julian years.

So these two periods could be seen as a 366y amplitude modulation of the 9.07 year cycle. I'm wondering whether there is an exchange of energy between these two oscillations.
So what could be happening physically with a period of 366.33 years? A figure which seems to point to the number of Earth rotations in a sidereal year.

 

[Quern]

# Lunar nodal cycle comes from (derived by T. Peter from Chapront [2002],
T_1000 = time from J2000.0 [1000 Year]
(6793.476501 + T_1000 * ( 0.0124002 + T_1000 * ( 0.000022325 - T_1000 * 0.00000013985 ) ) ) / 365.25

Chapront [2002], I think, you mean: J. Chapront, M. Chapront - Touze, G. Francou:
A new determination of lunar orbital parameters,..., A&A 387, 700 - 709, 2002

I assume you take that from Table 4 on p. 704, the polynomial for $\varpi$'

Who is T. Peter (source?) and how do you derive this equation? Is that similar to the equation given in this
article?

Quern

 

[fizzy]

Yes, that the A&A paper by Chapront et al. I can't find in my notes where I got the T. Peter name from. Seems I only noted the primary reference.

The WonkyPedia article will be slightly different since it is an older polynomial fit and based on j1900.5 not J2000 per Chapront.

since I did my calculations for a fixed date, I just used J2000.0 and it reduced to the first term 6793.476501 JD. I don't know how Peter got to that form and don't have time to dig the issue right now.

 

[fizzy]

I think this may be where I got the formula, credited to Tom Peter:

http://www.archaeocosmology.org/eng/moonfluct.htm