The dynamics of the solar system


Image courtesy of NASA

Historically, the first description of the orbits of the Solar System's planets was empirical, given by the three Kepler's laws. These empirical laws were demonstrated by Newton, as a simple 2-body application of the theory of universal gravitation. But, in reality, the Solar System is a n-body system and not a 2-body one, since the orbit of each planet is not only determined by the Sun's attraction. Every planet is also infleuenced by the other planets. So, considering the Solar System a 2-body system is a very rough estimation of the reality.
This problem can be overridden in two ways:
- first it is possible to use computers to simulate to a certain extent a n-body problem;
- a second solution is to consider that the planets' orbits can be described by Kepler's laws, applying to the orbits some corrections due to the fact that planets interact gravitationally.


The solar system described by Kepler's Laws: a simple 2-body problem
Kepler's laws are three empirical laws that describe the orbits of the planets of the Solar System as simple ellipses, with a not constant speed and a period proportional to the characteristics of the orbit .
Newton was the first to demonstrate mathematically these laws, applying the law of universal gravitation to a very simple, approximated 2-body problem, where each planet is only attracted by the Sun, and where some approximations are made.
These approximations are quite simple: first of all, since the Sun is extremely more massive than the other objects of the solar system, in a first approximation the motion of the planets can be studied only considering the interaction between the Sun and every single planet. Furthermore, in a first approximation it is possible to ignore the interactions between planets .
To formulate this simple demonstration, Newton invented the infinitesimal calculus.
 In the reality, planets do interact between them, and these interactions cannot be ignored. The motion of the planets is therefore much more complicated than the Kepler's laws can describe.
 

 Planetary perturbations: correcting Kepler
A good description of the Solar System can be given considering the gravitational interactions between the planets as simple corrections of orbits determined applying Kepler's laws. In fact, even if the interactions between planets cannot be completely ignored, they are sufficiently light to be considered as mere perturbations.
These perturbations are essentially of two kinds:

Short period perturbations: these are small variations around the mean Keplerian orbit of the planet (called here the main body). These perturbations don't have effects on the orbit of the main body over long periods. In fact the perturbing planets move all around the main body, in different directions. Therefore,  these perturbations have characteristic periods comparable to the orbital periods of the planets: if the effect of these perturbations is summed over longer periods the overall effect is absent. The typical period of this kind of perturbation goes form months to years.

Secular (or long period) perturbations: due to the action of the surrounding planets, in addiction to its keplerian orbit, the main body will follow a second very slow motion that can be seen as the sum of two distinct motions of the Keplerian orbit:
- a rotation of the line where the major axis of the orbit lies on the orbital plane (animation on the left);
- a precession of the orbital plane around a fixed direction (animation on the right).

This secular perturbation can be seen as a very slow periodical variations of the values of  the characteristics of the Keplerian orbit: the values of eccentricity, inclination  and major semiaxes fluctuate periodically around their mean values.
The typical period of these perturbations is of 10.000 years.