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The self gravity effect on the orbital stability of Twotinos

We investigate how the self gravity changes the orbital stability of Twotinos using N-body simulations in which the gravitational interaction between planetesimals are fully calculated. We show the timescale in which the half of Twotinos becomes unstable, t_{half}, obey the formula, t_{half} = 4 \times 106(Mtot0.1M\oplus)-1(mp7.6 \times 1023g)-1(hi2i1/20.002)(years), if we assume the primordial planetesimals disk have the power low surface mass density, Σ= Σ_0 \times r-2/3. Where Mtot, mp, hi2i1/2 are the total mass of large bodies of Twotinos, the maximum mass of planetesimals, and the inclination dispersion. With this formulae, we conclude the total mass of Twotinos is reduced to the order of 0.01 M\oplus by the self gravity and secular perturbation of the Planets even if there was huge mass such as several order of the earth mass in 1:2 MMR of Neptune at the early age of the solar system. These results will invoke reexamination to many previous works which tried to explain the dynamical evolution of TNOs.