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J. Laskar (ASD/IMC-CNRS)
I present here a simplified model of planetary accretion, based on the conservation of mass, conservation of momentum, and AMD-stability. I show how, within the limitations of this model, the organization of generic planetary systems can be derived from the knowledge of their initial mass distribution. The angular Momentum Deficit (AMD) is the difference between the angular momentum and the circular angular momentum of the orbits : C = \sumk=1n_p \Lambdak (1-\sqrt{1-ek2}\cos ik) . This quantity is conserved in the secular system at all order, and bounds the possible excursion of the eccentricities resulting from planetary perturbations, even when the secular motion is fully chaotic.
Let us assume that we have a distribution of planetesimals of linear mass density \rho(a) =\zeta ap. We assume that the secular motion of these planetesimals is fully chaotic and that they are only bounded by their AMD. Many collisions wil occur, which we assume to be perfectly inelastic, with conservation of the mass and of the momentum. As the mass of the planetesimals increases, their excursion will be more limited by the AMD constraint, until they cannot encounter any longer any other particle.
It was possible to derive analytically the expression of the final distributions resulting from this simplified accretion model, for any initial mass distribution \rho(a) =\zeta ap. This derivation aggrees very well with the averaged result of a numerical simulation of the same system.
For an initial constant linear mass distribution (p=0), \sqrt{a} is proportional to the index of the planet n, which is in very good agreement with our innner and outer planetary system, and with the \upsilon-Andromedae planetary systems.
Laskar, J: 2000, `On the Spacing of Planetary Systems', Phys. Rev. Letters, 8415, pp 3240--3243
The author(s) of this abstract have provided an email address for comments about the abstract: laskar@bdl.fr