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Fig. 2.
Case for 0.4 AU
≤ a< 1 . 0AU for 5Myr. The e
of the orbits with 0.70 AU
<a< 0 . 78 AU can be excited and in the 2:9 MMR at
0 . 76 AU, e can reach
0 . 90.
pumped up and may reach
0 . 76 AU, indicat-
ing that there may exist a gap near this resonance. Most of the Earth-like
planets about 1:4 MMR at
0 . 90 in the 2:9 MMR at
0 . 82 AU move stably in bounded motions with
low-eccentricity trajectories, except for two cases where the eccentricities
eventually grow to high values. Paper II pointed out that the secular reso-
nance b ν 2 arising from the outer companion (similar to ν 6 for Saturn) can
remove the test bodies. Would the ν 2 also influence the Earth-like planets
in this system? Nevertheless, we did not find this mechanism at work at
about
0 . 85 AU (see Paper II) when we examined the results, because the
terrestrial planets under study that all bear finite masses that may change
the strength of this resonance; on the other hand, the location of the secular
resonance is changed due to the orbital variation of the outer companion.
For a terrestrial planet with a mass of 10 M , the region for ν 2 secular
resonance is now shifted to
0 . 70 AU, where two eigenfrequencies for the
terrestrial body and outer giant planet given by the Laplace-Lagrange secu-
lar theory are, respectively, 211 . 37/yr and 225 . 48/yr. This indicates that
both planets almost have the same secular apsidal precession rates in their
motion. At the new location, the ν 2 resonance, together with the mean
motion resonance, can work at clearing up the planetesimals in the disk
b This means two bodies may bear almost the same precession frequencies in secular
orbital motion, Bois et al. 15 suggested to use the item “apsidal synchronous precession”.
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