# Other planets interaction on Earth-Moon System

1. Sep 15, 2009

### moon_sphinx

Hi again :)

I need some help in a not difficult question (I hope).

I would like to know how other planets can interfere in the system Earth-Moon - I suppose it is mainly gravitational force acting.
However, is this interaction from other planets big enough to cause changes in the route of the Moon, or distance from Moon to Earth, or even the speed of rotation or translation of the Moon?

How can I compute such interactions?

Do I have to calculate the F= G M1 M2 / (d^2)
for each planet on Moon?

Thank U

2. Sep 15, 2009

### fatra2

Hi there,

You're right on the dot. The main interaction of other planets is gravitational. This interaction is incredibly weak, due to the "small" mass of the planets and their distance to us. On the other hand, the Sun has a much bigger gravitational pull than planets in our system.

The system Moon-Earth has been form with these interactions (planets and Sun gravitational pull). Therefore, to think that they might suddenly change something drastically in this system is not realistic.

Cheers

3. Sep 15, 2009

### D H

Staff Emeritus
This is not an easy question. Yes, you can calculate the forces of the Sun, the planets, and the Moon on each other at some point in time using Newton's law -- if you know where everything is. Now, what does that tell you? The answer: Not a whole lot.

To see the effect of these forces are one needs to integrate these forces over time. This cannot be done analytically. This is a many body problem. Numerical integration techniques are needed. For high accuracy, some aspects of general relativity must be incorporated into the model, and the model needs to encompass small bodies as well as large. What will this tell you? You will get the positions of the solar system bodies as a function of time over the period during which you performed the integration. This is the approach taken by JPL and the Russian space agency in formulating their planetary ephemerides model.

This approach doesn't say much about orbits (orbital elements). If you want to know about orbits you will need to take a different approach. That approach is what people used before computers. The approach taken by Lagrange, Dalaunay, Hill, Newcomb, and Brown (these were the key developers) was to describe the behavior of the Moon in the form of Hamiltonian written in terms of the orbital elements. Lagrange started this by developing Lagrange's planetary equations. Dalaunay's planetary equations refined Lagrange's approach. Hill wrote his own planetary equations because Lagrange's and Dalauney's approaches converged very slowly. Newcomb added some refinements to Hill's work. Brown put the capstone on the whole thing.

Some of the seminal papers:

Brown, E.W., "On the theoretical values of the secular accelerations in the lunar theory," Monthly Notices of the Royal Astronomical Society 57:5 342-349 (1897)

Hill, G.W., "On the application of Delaunay transformations to the elaboration of the secular perturbations of the solar system," Astronomical Journal, 22:527 183-189 (1902).

Newcomb, S., "On the action of the planets on the moon," Astronomical Journal 25:592 129-132 (1907).

A couple more recent papers that use a planetary equations formulation:

Standaert, D., "Direct perturbations of the planets on the moon's motion," Celestial Mechanics, vol. 22:40 357-369 (1980)

Chapront, J. & Chapront-Touze, M., "Planetary Perturbations of the Moon in ELP 2000," Celestial Mechanics, 26:1, 83-94 (1982)

4. Sep 15, 2009

### moon_sphinx

So, as I understand, the forces from other planets and sun in the system Earth-Moon will not affect much, besides what it has been doing.

Is it correct to say that in the system Earth-Moon, this interactions are minimum?

I also know that the moon is "getting away" from Earth. Are the other planets or Sun responsible for this, or do they take a big part?

As far as I know the Earth as been loosing angular momentum to the moon. And the consequences are: the Earth is slowing down (because of the lost of momentum) and the moon is raising the radius of it's orbit (so moving away from Earth due to the gain of momentum). So, I don't have to worry about other planets in this matter, right?

Thank U
moon_sphinx

5. Sep 15, 2009

### D H

Staff Emeritus
Did you read my post, moon_sphinx?

What exactly are you trying to do? If you want to compute the Moon's orbit for a month or so you can ignore the Sun. Beyond that, you can't ignore the Sun. If you want to compute the Moon's orbit for a few months or so you can ignore the shape of the Earth. Beyond that, you can't ignore the shape of the Earth. The planets have lesser effects, but those effects do exist. The reason the Moon's orbit is eccentric rather than circular is because of the Jupiter and Venus. See http://adsabs.harvard.edu/abs/2007DDA....38.0303C.