A What is the cause of lunar nodal and apsidal precession?

[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[/color]?

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

 

[fizzy]

OK, I have finally got the bottom of this. The polynomials given on that link are obtained by differentiating Chapront's polynomial to get rotational speed and then using the first order term of the Taylor expansion : 1/(1-x) = 1+x to get a polynomial for period.

In the case of lunar apsidal period the result is accurate to 8s.f. back to 1000BC.

I have suggested the author include the second Taylor term and state the approximations being made and the expected accuracy ( giving 12s.f. in the coeffs is misleading ).

 

[Ant]

Did you ever figure out where the 366 came from?