# Lunar perturbations

1. Jun 10, 2007

### lewis198

Hi everyone. I was wondering if you know any web sites that describe perturbations on the moon's motion [due to differential gravitational forces] in detail.

Last edited: Jun 10, 2007
2. Jun 10, 2007

### lewis198

Also, if the sun-moon-earth is effectively 3 body, nay, the whole solar system is n-body, then how can any accurate predictions of planetary coordinates be made?

3. Jun 10, 2007

### lewis198

Sorry about the language, I got a bit carried away there.

The universe is just amazing.

4. Jun 10, 2007

### tony873004

They are pretty accurate considering that we can sent probes to other worlds.

The Sun is by far the biggest perturber on the Moon's orbit, causing its Longitude of Ascending node to rotate in 18 years, and its Longitude of Perihelion to rotate in 9 years.

5. Jun 10, 2007

### D H

Staff Emeritus
You are referring to the fact that the n-body problem does not yield a general solution in terms of elementary functions. This does not mean that the n-body problem is insoluble. Most real-world problems do not have simple solutions; the n-body problem is one of them. The differential equations that govern the behavior of the planets are well known. A wide variety of techniques exist to numerically solve differential equations to any desired degree of accuracy.

By way of analogy, even the cumulative normal distribution function $\Phi(x) = 1/\sqrt{2\pi} \int_{-\infty}^x \exp(-u^2/2) du$ cannot be expressed in terms of elementary functions. This does not mean that $\Phi(x)$ is insoluble.

In answer to your first question, I suggest you look at the JPL Solar System Dynamics web site for starters.

6. Jun 11, 2007

### lewis198

Thank you for your replies. What are those differential equations? Also, can you please give me a list of those techniques?

7. Jun 11, 2007

### lewis198

Since the recognition that the n body problem is not insoluble, has it been solved? If not, then what advances have been made to obtain a clearer picture?

The last milestone is probably to obtain an equation that predicts the coordinates of an n body system with the current coordinates as the integral constants. This surely would give us an accurate picture of planetary motion, and would revolutionize time-keeping. Using the coordinates of the solar system we could calculate the value of t, the only limitation to this method being the number of observations you would make (other stellar bodies would affect the value of t).

8. Jun 11, 2007

### D H

Staff Emeritus
Ignoring relativistic effects, the differential equations that govern a system of N point masses are Newton's Universal Law of Gravitation:

$$\frac{d^2x_i}{dt^2} = -\sum_{j\ne i} \frac{G m_j} { ||\vec x_i - \vec x_j||^3 } \, ( \vec x_i - \vec x_j )$$

Things get more complicated with relativistic effects, but its still a set of differential equations.

There is no known equation that solves the N-body problem. So how to solve it? Numerically. A couple of links:http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations and http://www.ma.utexas.edu/documentation/nr/bookcpdf/c16-0.pdf

9. Jun 11, 2007

### lewis198

Thank you for that Sir. I will apply myself to studying that.

10. Jun 12, 2007

### lewis198

Hey wait a minute, are numerical methods useful for predicting body coordinates way out in time? Are numerical methods only useful for approximations for small values of t?