Apsidal asymmetric-alignment of Jupiter Trojans

Presently there are 14 extreme Kuiper belt objects (KBOs) discovered with semimajor axes exceeding 250 AU (e.g., Brown et al., 2004; Chen et al., 2013; Trujillo & Sheppard, 2014; Sheppard & Trujillo, 2016). It was noted that all these objects are clustered both in the longitude of perihelion and in the longitude of ascending node, and the confidence level can be as high as $99.8\%$ (Brown & Batygin, 2019). This distinct orbital feature leads to the existence of an additional planet, referred to as Planet 9, which resides on a highly eccentric and inclined orbit (Batygin & Brown, 2016; Batygin et al., 2019). The secular perturbation of Planet 9 can be a dynamical mechanism which forces the orbital alignment of the known extreme KBOs, as well as the effect of its mean motion resonances (MMRs) (Beust, 2016).

We then speculate that the similar orbital alignment could possibly be observed elsewhere much closer to the Sun in the solar system. For instance, for some known planet moving on the eccentric orbit, it could also confine the apsidal configuration of the objects trapped in its MMRs. The first planet came into our thought is Jupiter, which has the largest mass and a relatively large eccentricity ($e_{\mbox{\scriptsize{J}}}=0.05$) among the four outer planets. Thus we intend to reinvestigate the orbital structure of objects in the low order MMRs with Jupiter.

The Jupiter Trojans share the semimajor axis of Jupiter, around the leading (L4) or trailing (L5) triangular Lagrangian point, and they are said to be settled in the 1:1 mean motion resonance (MMR) with Jupiter. This population comprising thousands of observed asteroids is currently the second largest group of minor planets in the solar system, only fewer than the main belt asteroids. As such a large group of small bodies, they may serve as a good field experiment for the dynamical theories developed for Planet 9. It is also well known that the 2:1 Jovian MMR contains a population of asteroids, i.e., the so-called Hecuba gap centered at $\approx 3.27$ AU (Schweizer, 1969). Although the number of this resonant population has increased to several hundreds, merely 124 objects (i.e., Zhongguos) were identified on stable orbits by Chrenko et al. (2015). So these stable Zhongguos are too few to be used for the statistical analysis at present, and their orbital distribution would not be discussed in this paper.

Over the past decade, many studies have been done for better understanding of the physical and orbital distributions of the Jupiter Trojans. The Wide-field Infrared Survey Explorer (WISE) project has provided the most complete measures of physical properties for approximately 1900 Jupiter Trojans down to $\sim 10$ km (Grav et al., 2011, 2012). According to the size and color observations, the Trojans are proposed to originate from the same primordial population as the Kuiper belt objects (Fraser et al., 2014; Wong & Brown, 2016). While as for the albedo distribution, this property shows no statistical difference between the L4 and L5 swarms, probably indicating the identical chemical and dynamical evolutions (Emery et al., 2011). A more exhaustive review can be found in Emery et al. (2015).

The known Jupiter Trojans have been better characterized in terms of their orbital features, and accordingly they were thought to be captured into the Lagrangian regions from the primordial planetesimal disk during the mass growth of Jupiter, by the dissipative forces such as gas drag or collisions (Shoemaker et al., 1989; Kary & Lissauer, 1995; Marzari & Scholl, 1998; Fleming & Hamilton, 2000; Kortenkamp & Hamilton, 2001). However, this process can not explain the broad inclination distribution of the Jupiter Trojans up to $40^{\circ}$, which was later reproduced in the framework of the Nice model (Tsiganis et al., 2005). Once Jupiter and Saturn crossed their mutual 1:2 MMR, there was a period of time that the planetesimals can be scattered and trapped around L4 and L5 through chaotic paths, and acquire large inclinations; as well, the observed orbital distributions of the eccentricity and libration amplitude can be successfully generated (Morbidelli et al., 2005).

Nevertheless, an outstanding issue remains, i.e., the number difference between the two Trojan swarms. With the samples detected by the WISE, Grav et al. (2011) estimated a number ratio of $N(\mbox{L}4)/N(\mbox{L}5)\sim 1.4\pm 0.2$. As a matter of fact, even the possible selection effects are involved, there are still significantly more Trojans in the L4 swarm (Szabó et al., 2007). Recently, Di Sisto et al. (2019) calculated the difference in the escape rate between the L4 and L5 Trojans, which is only as small as $\sim 2\%$ over the age of the solar system. In additional, Hellmich et al. (2019) considered the Yarkovsky force on the Jupiter Trojans, since only objects with radii $\leqslant 1$ km are significantly influenced, the number asymmetry existing at large size can not be explained. Consequently, the source of the asymmetry problem should be due to the initial capture implantation, but not the later long-term dynamical evolution. For instance, the Jumping Jupiter and capture mechanism started at $\sim 3-10$ Myr after the birth of the Sun, could potentially create a number ratio of $N(\mbox{L}4)/N(\mbox{L}5)=1.3-1.8$ (Nesvorný et al., 2013).

Since the small Trojans with absolute magnitudes $H>11$ are far from observationally complete, their intrinsic distributions need to be further improved by increasing the number of the observed samples. According to our previous work (Li & Sun, 2018), the total number of Jupiter Trojans with diameters $>1$ km (i.e., $H\lesssim 19$) is estimated to be approximately half a million. This suggests that there is a huge Trojan population yet to be discovered in future surveys. Hence it is of special interest to investigate that whether the orbital orientation of the Jupiter Trojans has some particular characteristic. As we know, in the papers published so far for the various distributions of the Jupiter Trojans, the longitude of perihelion and ascending node have not been explicitly considered. If the clustering of either of these two orbital orientations can be detected, it can directly place constrains on preparing the future observation plans.

The rest of this paper is organized as follows. In Section 2, we reveal the apsidal asymmetric-alignment of the observed Jupiter Trojans. In Section 3, we develop the theoretical approach to understand the dynamical mechanism that accounts for this anomalous orbital structure of the Jupiter Trojans, and correspondingly a detailed explanation is provided. In Section 4, we perform numerical simulations to reproduce the apsidally aligned Trojans, sculpted by the eccentric orbit of Jupiter, from an initially uniform distribution of the longitudes of perihelia. The conclusions and discussion are given in Section 5.

As of 2020 May 27, there are more than 8100 Jupiter Trojans have been registered in the Minor Planet Center¹¹1https://minorplanetcenter.net/iau/lists/JupiterTrojans.html (MPC). This population includes many objects that only have short-arc observations which may induce inaccurate orbit determinations for them. Thus we only select the Trojans that have been observed over multiple oppositions, resulting 7354 in total. To represent the true members of the Jupiter Trojans, we then numerically evolve the orbits of these samples under the perturbations of four planets in the outer solar system. The planets’ masses, initial positions, and velocities are adopted from DE405, in the heliocentric frame referred to the J2000.0 ecliptic plane at epoch 1969 June 28 (Standish, 1998). Then the planets are integrated to the specified epoch of the Trojans. Finally, we compute the orbital evolution of the considered Trojans²²2Here and later in Sections. 3.2 and 4, the SWIFT_RMVS3 symplectic integrator (Levison & Duncan, 1994) is used to numerically compute the orbital evolution of the Trojans. We adopt a time step of 0.5 year, which is about 1/24 of the shortest orbital period (i.e., Jupiter’s period) in our model.. At the end of the 1 Myr integration, all of them can survive around the individual Lagrangian points. Then these 7354 multioppositional and stable Jupiter Trojans could be used for our analysis. In this way, we can detect and exclude the possible transient Jupiter Trojans such as P/2019 LD2 which only can stay on Trojan-like orbit for about ten years (Hsieh et al., 2021).

Fig. 1 shows the orbital element distributions of the L4 (left column, 4625 in total) and L5 (right column, 2729 in total) populations. It can be seen in the upper two panels that, for either the L4 or L5 swarm of Jupiter Trojans, the clustering in the longitude of perihelion $\varpi$ is readily apparent. A simple way to quantify this apsidal alignment is to compare with the number fraction $f_{even}$ of an even $\varpi$ distribution. If we have $N$ Trojans in total, since the width of each $\varpi$ bin is adopted to be $30^{\circ}$, a sample size of $N_{bin}=(30/360)N$ per bin is expected, i.e., the mean fraction $f_{even}=N_{bin}/N\approx 0.083\%$. Considering the random variations about the value of $f_{even}$ due to counting statistics, the uncertainty from Poisson distribution can be calculated by $\sigma=\pm f_{even}\cdot\sqrt{N_{bin}}/N_{bin}$. 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$\sigma$ upper uncertainty, respectively. As the peak of the histograms can achieve the value of $\sim 0.21$ (L5) $-$ 0.23 (L4), much higher than the dashed line, the clustering of $\varpi$ shows strong significance over $5\sigma$ confidence level. Another key point shown in these two figures is that, the number peak corresponds to the bin of $\varpi=30^{\circ}-90^{\circ}$ ($300^{\circ}-330^{\circ}$) for the L4 (L5) Trojan swarm, giving a difference of around $+60^{\circ}$ ($-60^{\circ}$) with Jupiter’s longitude of perihelion $\varpi_{\mbox{\scriptsize{J}}}$ (indicated by the red line).

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 $\varpi_{\mbox{\scriptsize{J}}}$ is $\sim 0.17$ Myr, we propose that the 1 Myr timescale is long enough to check the persistence of the apsidal confinement of Jupiter Trojans due to different precession rates of their $\varpi$. Then the $\varpi$ 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 $\varpi$ are clustered around $\varpi_{\mbox{\scriptsize{J}}}+60^{\circ}$ and $\varpi_{\mbox{\scriptsize{J}}}-60^{\circ}$ for the L4 and L5 swarms, respectively; (2) the number fraction of the clustered samples within a $\varpi$ bin could be over 0.2, which is significantly larger than $f_{even}\sim 0.083$ ($>5\sigma$ confidence level) in the case of a uniform $\varpi$ 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 $e_{\mbox{\scriptsize{J}}}\sim 0.05$ may induce and maintain this intriguing orbital distribution, and an explanation will be offered in the next section.

However, the longitudes of ascending nodes ($\Omega$) of Jupiter Trojans are not visually clustered in any $\Omega$ bin, as shown in the lower panels of Fig. 1. The element $\Omega$ measures the azimuthal direction of an object’s orbital plane. Since Jupiter has a very low inclination, which is only about $0.3^{\circ}$ relative to the invariable plane of the solar system, this planet could not induce a significant clustering of the orbital planes of its Trojan population. Then it is quite understandable that there appears such a nearly uniform $\Omega$ distribution of Jupiter Trojans.

The planetesimals could have been chaotically captured into Jupiter’s co-orbital regions during the early evolution of the solar system within the Nice model (Tsiganis et al., 2005; Morbidelli et al., 2005). At the end of this process, their longitudes of perihelia should cover all values of $\varpi=0-360^{\circ}$. Therefore, we would like to investigate that whether the apsidal asymmetric-alignment of the currently observed Jupiter Trojans, could be generated from an even $\varpi$ distribution in the later long-term evolution. As the study in the previous sections is indicative of this peculiar orbital structure, the numerical experimentation is to be carried out to evaluate the effect of the secular perturbation of the eccentric Jupiter.

In this section, we still adopt the present outer solar system model, consisting of the Sun and four giant planets (i.e., Jupiter, Saturn, Uranus and Neptune). As in recent dynamical explorations of the Jupiter Trojans (e.g., Hellmich et al. (2019), Holt et al. (2020)), to construct robust simulations, we follow the long-term evolution of 30,000 test Trojans around the L4 point. Initially, all particles have the same semimajor axis ($a\approx 5.2$ AU) with Jupiter. And similar to the observed Trojan population, they have $e$ ranging from $0-0.3$ and a broad inclination distribution of $i=0-40^{\circ}$. The initial orbital angles are chosen such that particles start from $\Omega=0$ and $\psi=\lambda-\lambda_{\mbox{\scriptsize{J}}}=30^{\circ}-90^{\circ}$, while the values of their $\varpi$ are randomly selected between 0 and $360^{\circ}$. We then monitor the orbital evolution of these test Trojans in the long-term numerical integration.

Bearing in mind that, for Jupiter, the periodicity in the variation of its longitude of perihelion $\varpi_{\mbox{\scriptsize{J}}}$ is about $1.7\times 10^{5}$ yr. So given a relatively shorter timescale of $10^{5}$ yr, in the top panel of Fig. 7, we provide the first glance at the evolving distribution of $\Delta\varpi=\varpi-\varpi_{\mbox{\scriptsize{J}}}$. At this moment, there are a total of 26,117 particles (i.e., $87.1\%$) found on Trojan orbits, characterized by the librating critical arguments $\psi$. We notice that, although the distribution of $\Delta\varpi$ is seemingly accumulated around $60^{\circ}$, it is in fact very close to the uniform distribution indicated by the solid line.

As the time passes, due to the secular perturbation of the eccentric Jupiter, the simulated Trojans can reach the orbits which are more and more apsidally aligned. At the epoch of $10^{8}$ yr, as shown in the middle panel of Fig. 7, the clustering in $\Delta\varpi$ is quite obvious. And when the integration ends at $10^{9}$ yr, on the order of the age of the solar system, there remain 10,417 objects (i.e., $\sim 34.7\%$ of initial samples) librating around the L4 point. For these simulated Trojans, as shown in the bottom panel of Fig. 7, the largest fraction of objects clustered adjacent to the location of $\Delta\varpi=60^{\circ}$ is as high as $\sim 0.2$. We recall that the distribution of the longitudes of perihelia of the observed L4 Trojans, as displayed in the left panels of Figs. 1 and 2, also has a comparable peak next to the same location.

At the beginning of this section we assumed that the planetesimals deposited in Jupiter’s Trojan clouds in the framework of the Nice model have uniformly-distributed $\varpi$. Although it is a reasonable assumption since the evolution of the planetesimal is chaotic, the capture into libration around the L4 or L5 point could be dependent on the pre-capture orbit of the planetesimal. For instance, such a dependence may be that the perihelion distance of the planetesimal is smaller than that of Jupiter. Besides, there also may be other constraints imposed on the planetesimal’s orbit including the angle $\varpi$ describing its orientation. Then some additional simulations have been carried out, by assigning initial $\Delta\varpi$ values randomly from a Gaussian distribution

where $\mu$ is the mean, and the standard deviation $\hat{\sigma}$ is taken to be 0.3. The initial $\varpi$ of test Trojans can simply determined as $\varpi_{\mbox{\scriptsize{J}}}+\Delta\varpi$.

For each set of 3000 test Trojans, we choose the mean $\mu$ to be one of four representative values of $0,90^{\circ},-90^{\circ},180^{\circ}$, and the associated $\Delta\varpi$ distribution is plotted in the left column in Fig. 8 (from top to down). We can see that the initial number fraction of test Trojans within the bin next to $\Delta\varpi=\mu$ could peak over 0.2. Keeping the other initial conditions of the system, we repeat the $10^{9}$ yr numerical integrations for the simulated Jupiter Trojans, and the final distributions of $\Delta\varpi$ are summarized in the right column in Fig. 8. The main feature is that, starting from any $\Delta\varpi$ distribution, the simulated Trojans would eventually reach the state of apsidal alignment around $\Delta\varpi=60^{\circ}$. Through a closer look at the cases of $\mu=-90^{\circ},180^{\circ}$, one notices that although the region near $\Delta\varpi=60^{\circ}$ was nearly empty (two lower left panels), the expected apsidal alignment of the Trojans within this region can still be seen at the end of the $10^{9}$ yr evolution (two lower right panels). This provides support for the results obtained from our previous run with an initially uniform $\Delta\varpi$ distribution. We end by noting that the previous run using 30000 test Trojans is quite “expensive”, and more than a month of CPU time is needed. As a matter of fact, we found that an order of magnitude fewer objects could be a sufficient sample size to perform the statistical analysis. Therefore, in order to save computational time, we use a set of 3000 test Trojans (for each $\mu$) for the simulations with different set-up of $\Delta\varpi$ distribution, and the results are considered robust.

In summary, we conclude that the apsidal asymmetric-alignment of Jupiter Trojans should be a natural consequence of Jupiter’s secular perturbation. The orbital orientations of the Jupiter Trojans at the moment of capture could affect the extent of the clustering in their $\varpi$, but only slightly and would not alter the overall $\varpi$ distribution. The origin of the Jupiter Trojans still remains uncertain, and a thorough exploration goes beyond the scope of this paper.

This work is inspired by the hypothesis that the eccentric and inclined Planet 9 could cause the orbital clustering of the extreme KBOs beyond 250 AU (Batygin & Brown, 2016). As Jupiter has a relatively large eccentricity ($e_{\mbox{\scriptsize{J}}}\sim 0.05$) among the planets in the outer solar system, for the observed Jupiter Trojans with the number in excess of 7000, we statistically analyzed the distribution of their orbital orientations.

It is interesting to find that, the L4 and L5 Trojan swarms have longitudes of perihelia $\varpi$ gathered around $\varpi_{\mbox{\scriptsize{J}}}+60^{\circ}$ and $\varpi_{\mbox{\scriptsize{J}}}-60^{\circ}$ ($\varpi_{\mbox{\scriptsize{J}}}$ is the longitude of perihelion of Jupiter), respectively. And for the $\varpi$ bins with the width of $30^{\circ}$, either of these two swarms has a number fraction peaked over 0.2, which is much larger than the fraction of $\sim 0.083$ resulted from the uniform distribution, at the $5\sigma$ confidence level from Poisson statistics. This suggests that the apsidal asymmetric-alignment of Jupiter Trojans is noticeable. By integrating the orbits of Jupiter Trojans up to 1 Myr, the clustering in $\varpi$ can persist over time, thus this orbital characteristic should be robust but not due to the observational bias. The apsidal clustering of the Trojans is supposed to be caused by Jupiter which possesses a relatively large eccentricity of $e_{\mbox{\scriptsize{J}}}\sim 0.05$. However, the distribution of the longitudes of ascending nodes for the observed Trojans is rather uniform, since the inclination of Jupiter is very low.

Next, in the framework of the spatial elliptical restricted three-body problem, we developed a theoretical approach to understand the clustering in $\varpi$ for the Jupiter Trojans. The averaged Hamiltonian with three degrees of freedom is derived to describe the dynamics of the 1:1 mean motion resonance. By taking the resonant angle to be $\psi=60^{\circ}$ ($-60^{\circ}$) for the L4 (L5) Trojan swarm, the Hamiltonian can be reduced to a two degrees of freedom system, in which the momenta are in terms of the Trojan’s eccentricity $e$ and inclination $i$. Then we provide a global view of the dynamical aspects for the co-orbital motion in the phase space ($e$, $\Delta\varpi$), where $\Delta\varpi=\varpi-\varpi_{\mbox{\scriptsize{J}}}$. The portraits show that there are equilibrium points at ($e\sim e_{\mbox{\scriptsize{J}}},\Delta\varpi\sim\pm 60^{\circ}$), either for the coplanar or inclined cases with $i\leq 40^{\circ}$. This can nicely account for the distinctive apsidal asymmetric-alignment of Jupiter Trojans as revealed above. We further note that the apsidally aligned islands of libration exist only for the Trojan-like orbits with $e<e_{c}\sim 0.1$, and this is essentially determined by Jupiter’s eccentricity $e_{\mbox{\scriptsize{J}}}\sim 0.05$. However, only a small fraction of the known Jupiter Trojans satisfy this eccentricity condition.

For better understanding the influence of the Trojan’s eccentricity, the proper value $e_{p}$ and the maximum value $e_{\mbox{\scriptsize{Max}}}$ are computed in their 1 Myr evolution. Previous studies show that the two Lagrangian points have almost the identical dynamical features concerning the orbits around them, thus we then consider only the L4 Trojans. We introduced a time index $T_{frac}$ to measure the fraction of time that the L4 swarm has $\varpi$ placed in the range of $|\Delta\varpi-60^{\circ}|<\delta=60^{\circ}$. The results show that, for the population with $e_{p}<0.1$, the derived time fraction $T_{frac}$ is much larger than the value corresponding to a uniform variation of $\varpi$ in time. It suggests that although some objects evolve to the orbits with osculating eccentricities exceeding 0.1 (i.e., having $e_{\mbox{\scriptsize{Max}}}>0.1$), and accordingly could temporarily escape the libration islands of $\Delta\varpi$, they actually spend most of their time in the said $\Delta\varpi$ range. Because the more eccentric Trojans with $e_{p}>0.1$ are very few in number, the entire Trojan population would display the obvious apsidal alignment.

Finally, we proceed to investigate whether Jupiter Trojans could reach this peculiar orbital structure in the long-term evolution. Initially, tens of thousands of test Trojans around the L4 point are uniformly distributed between $\varpi=0-360^{\circ}$, given $e=0-0.3$ and $i=0-40^{\circ}$. Then by numerical simulations, we compute the evolution of these samples, sculpted by the eccentric Jupiter and the other three planets in the outer solar system. The orbital distributions obtained at different epochs show that, as time passes, the clustering of the simulated Trojans around $\Delta\varpi=60^{\circ}$ becomes more and more prominent. And at the end of the $10^{9}$ yr integration, the number fraction of the apsidally aligned objects is peaked about 0.2, for the $\Delta\varpi$ bins with the width of $30^{\circ}$, similar to what is observed in the real Trojan population. A series of additional runs, for which the test Trojans have initial $\varpi$ obeying Gaussian distributions with different means, have also been carried out. We find that the similar apsidal alignment around $\Delta\varpi=60^{\circ}$ can always be reproduced.

In this work, our theoretical approach built on the restricted three-body model suffers from a minor problem: for the 1:1 resonant angle $\psi$ of the co-orbital motion, the libration center $\psi_{0}$ does not fix at the classical Lagrangian point of $60^{\circ}$ (L4) or $-60^{\circ}$ (L5), but it actually varies with the Trojans’ eccentricities $e$ and inclinations $i$. By expanding the disturbing function to the fourth order in $e$ and $i$, Namouni & Murray (2000) studied the orbits librating around L4 or L5 with small amplitudes, and they showed that the displacements of equilibrium points from the equilateral configuration increase as a function of increasing $e$. The displaced equilibrium points were also calculated for the coplanar configuration of two massive planets in the co-orbital motion (Giuppone et al., 2010). As shown in Fig. 7 of their work, given a ratio $m_{2}/m_{1}\geq 1$ of planetary masses, the equilibrium value of $\psi$ (i.e., $\psi_{0}$) for the L4 solutions increases with the eccentricity $e_{1}$ of the less massive planet. Considering the limit of the mass ratio $m_{2}/m_{1}\rightarrow\infty$, i.e., in the restricted three-body problem, this trend is the same as that discovered by Namouni & Murray (2000). By using the semi-analytical method developed in our previous works for the resonant Kuiper belt objects (Li et al., 2014a, b, 2020), the libration center $\psi_{0}$ for the L4 co-orbital motion is determined as a function of the massless Trojan’s $e$ for different $i$. Considering the observed range of $e\sim 0-0.3$ and $i\sim 0-40^{\circ}$, Fig. 9 shows that $\psi_{0}$ could change between $57^{\circ}$ and $70^{\circ}$, indicating that the equilibrium point within the apsidal alignment islands (see Fig. 3a) may shift no further than $10^{\circ}$ from $\Delta\varpi=60^{\circ}$. Since the L4 Trojans are mainly clustered in a much wider range of $\Delta\varpi=0-120^{\circ}$, the $\lesssim 10^{\circ}$ displacement of $\Delta\varpi$ would not affect our conclusions. This argument can be supported by the apsidal alignment of Jupiter Trojans from both the observation and numerical reproduction.

This work was supported by the National Natural Science Foundation of China (Nos. 11973027, 11933001, 12073011 and 11603011), and National Key R&D Program of China (2019YFA0706601). We would also like to express our sincere thanks to the anonymous referee for the valuable comments.

The data underlying this article are available in the article and in its online supplementary material.