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

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## Main Question or Discussion Point

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.

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tony873004
Gold Member
...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.

"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.

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
Gold Member
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!

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.

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.

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
Gold Member
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.

jim mcnamara
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 :(

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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 .)

the cartesian components of the torque are:
Tx = - T0 i sin2 u
Ty = 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
Gold Member
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

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.

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.

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tony873004
Gold Member
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.

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

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
Gold Member
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.

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.

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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.

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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
Gold Member
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 approachs 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
Gold Member
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.
Code:
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),
Code:
+ ',' + 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.

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.

"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.