Positions of the secular resonances in the primordial Kuiper Belt disk

The dynamically hot and cold populations of the Kuiper Belt probably formed from two distinct regions of the Solar System. The former originated from a massive planetesimal disk extending from the primordial position of Neptune to similar to 30 AU and the latter from a light extension of the planetesimal disk, prolonging beyond 30 AU. Previous studies on the dynamical evolution of the primordial cold population only accounted for the giant planets and did not consider its evolution under the influence of the massive part of the planetesimal disk. The latter affects it only indirectly through its interactions with the giant planets. Our goal is to introduce the perturbation of the massive part of the planetesimal disk on the apsidal and nodal precessions of both the giant planets and planetesimals, using the linear secular theory. We want to see how it affects the positions of the secular resonances. In the first place, we look at the positions of the secular resonances after the disappearance of the solar nebula, when the giant planets were locked in a multiresonant configuration. Because of this multi-resonant configuration, the linear secular theory allows us to compute only the nodal part. The existence of a massive disk of planetesimals makes the f(5) frequency non-zero. We show that the associated secular resonance is located in the region corresponding to the current cold Kuiper Belt in several multiresonant configurations of the giant planets. The efficiency of this secular resonance in rising the inclinations of the objects depends on the misalignment between the total angular momentum of the giant planets and the direction orthogonal to the massive planetesimal disk. If both are aligned, the amplitude associated to the f(5) frequency is null and the resonance has no effect. We illustrate this with simple numerical integrations, where the nodal precessions exerted by the massive disk is mimicked using fictitious forces. Then, we look at the positions of the apsidal and nodal secular resonances just before the instability between the giant planets. We find that taking into account the massive part of the disk only shifts the secular resonances by less than 1% compared with the case where we do not account for it and the f(5) secular resonance is located beyond 150 AU.