tokens: π is readily apparent. A simple way to quantify this apsidal alignment is to compare with the number fraction f even of an even π distribution. If we have N Trojans in total, since the width of each π bin is adopted to be 30 ○ , a sample size of N bin = (30/360) N per bin is expected, i.e. the mean fraction . Considering the random variations about the value of f even due to counting statistics, the uncertainty from Poisson distribution can be calculated by . Accordingly, in the upper panels of Fig. 1 , the horizontal solid and dashed lines are plotted to indicate the mean fraction f even and its 5 σ upper uncertainty, respectively. As the peak of the histograms can achieve the value of ∼0.21 (L5)–0.23 (L4), much higher than the dashed line, the clustering of π shows strong significance over 5 σ confidence level. Another key point shown in these two figures is that the number peak corresponds to the bin of π = 30 ○ –90 ○ (300 ○ –330 ○ ) for the L4 (L5) Trojan swarm, giving a difference of around +60 ○ (−60 ○ ) with Jupiter’s longitude of perihelion π J (indicated by the red line). Figure 1. Distribution of the longitudes of perihelia π (upper panels) and the longitudes of ascending nodes Ω (lower panels) of the observed Jupiter Trojans, at the epoch of 2020 May 31. Left-hand column: There are 4625 objects in the L4 region, and the values of their π are clustered around π J + 60 ○ ( π J is the longitude of perihelion of Jupiter). Right-hand column: There are 2729 objects in the L5 region, and the π clustering is around π J − 60 ○ . For reference, the horizontal solid lines are plotted in the upper two panels for the uniform distribution in π , and the dashed lines indicate the 5 σ upper uncertainty from Poisson statistics. In order to further confirm the newly found characteristic of apsidal clustering, we need to consider the evolving orbits of Jupiter Trojans at different epoch. Since the periodicity in the variation of Jupiter’s longitude of perihelion π J is ∼0.17 Myr, we propose that the 1-Myr time-scale is long enough to check the persistence of the apsidal confinement of Jupiter Trojans due to different precession rates of their π . Then the π distribution of the Jupiter Trojans after 1-Myr evolution is shown in Fig. 2 , from the integration mentioned at the beginning of this section. One can immediately notice the similar patterns depicted in Fig. 1 : (1) the longitudes of perihelia π are clustered around π J + 60 ○ and π J − 60 ○ for the L4 and L5 swarms, respectively; and (2) the number fraction of the clustered samples within a π bin could be over 0.2, which is significantly larger than f even ∼ 0.083 (>5 σ confidence level) in the case of a uniform π distribution. Therefore, the apsidal asymmetric-alignment of Jupiter Trojans should be intrinsic but not due to the observational bias. We suppose that the gravitational perturbation of Jupiter with eccentricity e J ∼ 0.05 may induce and maintain this intriguing orbital
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