other outer planets is almost constant during these very long-term integration periods， which is demonstrated in the time–frequency maps on our webpage.
In these two integrations， the relative numerical error in the total energy was ∼10−6 and that of the total angular momentum was ∼10−10.
5.1 Resonances in the Neptune–Pluto system
Kinoshita & Nakai (1996) integrated the outer five planetary orbits over ± 5.5 × 109 yr . They found that four major resonances between Neptune and Pluto are maintained during the whole integration period， and that the resonances may be the main causes of the stability of the orbit of Pluto. The major four resonances found in previous research are as follows. In the following description，λ denotes the mean longitude，Ω is the longitude of the ascending node and ϖ is the longitude of perihelion. Subscripts P and N denote Pluto and Neptune.
Mean motion resonance between Neptune and Pluto (3:2). The critical argument θ1= 3 λP−
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