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

#### tony873004

Gold Member
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. :?

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#### 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.

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#### 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.

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#### 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.

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#### 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

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 ).

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