Sources of chaos

What is causing the appearance of the chaotic layer?  In previous simulations (see Low-resoultion survey: results) we showed how perturbations from the outer solar system are responsible for the appearance of a layer of chaos at the separatrix between circulating and librating orbits.   Among the different terms included in our simulations, what is the main responsible for the appearance of this chaotic boundary?   There are essentially five possibilities:

  1. Saturn eccentricity
  2. Jupiter indirect perturbations
  3. Jupiter perturbations
  4. Jupiter + Uranus
  5. Jupiter + Uranus + Neptune
The first mechanism is different from the secular model because, contrary to that case, there is no averaging over the mean motions of both satellites and planets and there is no truncation in the series approximating the perturbing function to second order in \rho=a/asat.   According to the secular model, there should be no chaos in this case because we are dealing with a one degree of freedom model.   It can be interesting to check if the  direct numerical simulation with Saturn alone on an eccentric orbit can actually cause the appearance of the chaotic boundary.  Another possible mechanism could be due to the indirect perturbations of Jupiter on Saturn and its satellites.  By inderect perturbations we mean the following: Jupiter is gravitationally influencing both Saturn and its satellites, but there is a small difference between the two perturbations due to the different relative positions.  The main effect of jovian perturbations is to make the eccentricity of Saturn varies with a period of ~200,000 yrs.  If we substitute the gravitational influecne of Jupiter on the satellite with the equivalent perturbation on Saturn, we have what we define as indirect perturbations.   Direct perturbations are just the system of Sun + Saturn + Jupiter + satellites, and analogously for the other two remaining cases.

In the next figures we show the results of simulations for the first and third case.   For the first case, we have a system of just Sun + Saturn + satellite, with Saturn on a orbit in which e=<e> over the cycle of oscillations of 200,000 yrs.  The  initial conditions for the test particles are equal to the case seen in Low-resoultion survey: results.   This system is not completely equivalent to the case seen in  Low-resoultion survey: results
because the intial orbital elements of the satellites and planets should be those at the time in which, in the simulations with the OSS, the eccentricity of Saturn was equal to its average value, not the initial values.  Nevertheless, if a negative result should be obtained (no chaos) we claim that this should be extensible to the equivalent real Sun + Saturn + satellite system.
 

Integrations done for the third cases have trivial initial conditions: we just took the same initial conditions for the integrations described in  Low-resoultion survey: results, but limited the number of planets to three (Saturn, Jupiter, and the Sun).
 

Results seems to indicate that perturbations from Jupiter are enough to cause the appeareance of the layer of chaos.  The next step is to check for which terms in the perturbing potential of Jupiter are responsible for the presence of chaos in the system (see Bretagnon model).