Primordial Orbital Alignment of Sednoids

The orbital evolutions of sednoids over the past ≈4 Gyr are primarily driven by secular precessions induced by gravitational effects of the four giant planets, assuming that no planetary mass is still present in the outer solar system. The analytically approximate apsidal precession rate as a function of the TNO's (a, e, i) orbital elements is given by Batygin et al. (2019):

$$\begin{eqnarray}&&\dot{\varpi }=\displaystyle \frac{3}{8}\ n\ \displaystyle \frac{3{\cos }^{2}i-1}{{\left(1-{e}^{2}\right)}^{2}}\displaystyle \sum _{j=5}^{8}\displaystyle \frac{{m}_{j}{a}_{j}^{2}}{{M}_{\odot }{a}^{2}},\end{eqnarray} \tag{ 1 }$$

where n denotes the TNO's mean motion, M_⊙ is the solar mass, and the index j denotes the jth planet. This equation estimates the apsidal precession periods of 2012 VP₁₁₃ (1.2 Gyr), Sedna (2.8 Gyr), and Leleakuhonua (7.1 Gyr). Assuming that their orbital a, q, and i have not changed significantly post formation, one can rewind their longitudes of perihelion back in time by applying the linear precession rate (Equation (1)). Surprisingly, we found that the only time their ϖ were all tightly clustered was 4–4.5 Gyr ago, at around ϖ ≈ 200°.

Intrigued by this approximate analytical result, we carried out backward numerical integration to validate the primordial alignment of the sednoids. For each sednoid, we generated 11 initial conditions, including one nominal orbit and 10 cloned orbits distributed within the orbital uncertainty, using SBDynT, ³ with the clones serving to diagnose plausible uncertainties in the apsidal angles. These initial test-particle orbits along with the four giant planets were integrated backward in time using the Mercurius integrator in Rebound (Rein & Liu 2012; Rein et al. 2019).

Figure 1 shows the past evolution histories computed of ϖ. Similarly to the analytical approximation, their apsidal lines were aligned at 200° 4.5 Gyr ago, with a circular standard deviation of only 8° (black curve in the lower panel). For reference, three randomly generated angles would have an average circular standard deviation of ≈60°.

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To test the statistical significance of the clustering, we applied a Rayleigh test of uniformity at all the output times. The p-value of the test (red shaded curve and right-hand scale in the lower panel of Figure 1) represents the probability that three random angles are more clustered than the three sednoids at that time. The Rayleigh test shows that there is a <5% chance that the primordial clustering is just a statistical coincidence. In particular, it would seem in principle that the alignment, if it were by chance, could occur at any time in the past; the fact that it occurs at the "special" time that corresponds to the planet formation epoch of solar system history is a "difficult to quantify" additional low-probability event.

The galactic disk creates additional dynamics for large-a bodies via tidal forces, inducing q and i oscillations and an apsidal precession around the galactic pole (Heisler & Tremaine 1986). The galactic tidal effect becomes strong beyond ∼2000 au, where the tidal torquing timescale is comparable to the planetary scattering timescale (Duncan et al. 1987), and thus should be tiny to small for the three sednoids. Sheppard et al. (2019) integrated forward the three sednoids considering the galactic tide and the four giant planets, and found that the galactic tide has almost no effect on Sedna and 2012 VP₁₁₃, but does create a small ±6 au oscillation of Leleakuhonua's perihelion (their Figure 7); the authors concluded that Leleakuhonua is stable to the galactic tide, but do not show its ϖ evolution with the tide.

To verify whether adding the tide would alter the −4.5 Gyr alignment, we back-integrated the same nominal and clone orbits, considering both giant planets and the galactic tide. Due to its dominance, we modeled only the vertical component of the tide (Heisler & Tremaine 1986), using the standard local density of the galactic disk of 0.1 M_⊙ pc⁻³ (Levison et al. 2006). We confirm that the inclusion of the tide barely affects Sedna and 2012 VP₁₁₃, but find it does lower Leleakuhonua's primordial ϖ by ≈20° (black triangle in Figure 1), while producing a slightly larger uncertainty range (marked by gray triangles in Figure 1) 4.5 Gyr ago.

We observe that the galactic tide does not visibly modify ϖ by imposing a different precession rate directly. The deviation between two nominal ϖ evolutions occurred when Leleakuhonua's q dropped to ≈60 au around 1 Gyr ago (consistent with Sheppard et al.'s 2019 Figure 7). A lower q allows stronger diffusion in a (Hadden & Tremaine 2023), which further disperses the ϖ precession induced by the four giant planets (Equation (1)).

In the presence of the tide, the p-value of chance primordial clustering is still <5%. We find that the ϖ uncertainty at −4.5 Gyr is currently dominated by the large a uncertainty in Leleakuhonua's orbit (rather than whether or not the tide is included). We thus conclude that the primordial alignment in ϖ remains an extremely interesting possibility, with (1) more sednoid orbits required to reach 3σ confidence, and (2) high-precision orbits being needed for these large-a objects.

One hypothesis to explain the alignment revolves around a temporarily present rogue planet born in the solar system. Gladman & Chan (2006) showed that scattering rogue objects could have raised TNO perihelia and created Sedna-like orbits; the secular q-lifting effect is dominated by the single most massive rogue. Huang et al. (2022) recently demonstrated that the rogue can also help populate distant TNOs below a < 100 au by collaborating with Neptunian mean-motion resonances. Previous studies have shown that the detachment dynamics of a highly eccentric (rogue) planet is correlated with its relative apsidal orientation Δϖ (Batygin & Morbidelli 2017; Huang 2023). It is thus worth exploring a proof of concept that a rogue can produce a primordial alignment.

Figure 2 displays the simulation result of a rogue planet that lifts objects from a massive early scattering disk (which has q < 40 au). The 2 M_⊕ rogue has an initial orbit a_r,0 = 400 au, q_r,0 = 32 au, i_r,0 = 15°, and was propagated along with the four giant planets for 185 Myr. To generate the initial disk of scattering particles, Neptune was forced to migrate from 24 to 30 au through an outer planetesimal disk of 500,000 test particles spanning 24.5 to 33 au; the migration timescale and disk parameters are similar to those of grainy migration simulations (Nesvorný & Vokrouhlický 2016; Nesvorný et al. 2016), but we find that these parameters play little role in the emplacement of detached TNOs beyond a > 200 au. As a result of Neptune scattering, TNOs were constantly fed to the large-a region, where the rogue's gravity can raise the TNO perihelia (Gladman & Chan 2006). The planetary simulation (rogue scattering and Neptune migrating) was performed using Rebound (Rein & Liu 2012), while the test-particle simulation was carried out by Glisser, a GPU N-body integrator based on Zhang & Gladman (2022).

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During the ∼100 Myr temporary presence, the rogue was able to reproduce sednoids in a wide range of semimajor axes (a = 200 to∼ 1000 au, color-coded in Figure 2). The detachment was strongest along the rogue's apsidal line (Huang 2023), which precesses ≈100° (blue lines) until ejection by Neptune. This 2M_⊕ rogue creates a strong clustering in the ϖ − q space, with a circular standard deviation of ≈25° for q > 50 au particles. While this is more dispersed than the primordial alignment observed in the three Sednoids, more Sednoid discoveries will be needed to measure the underlying spread of their primordial apsidal lines. A tightly dispersed primordial ϖ distribution can occur for a more massive rogue planet with a shorter lifetime before ejection.

After the rogue's removal, the now-detached objects/sednoids precess at different rates (determined by their a, q, and i as per Equation (1)). This eventually results in a nearly uniform ϖ distribution in today's surviving population. For icy bodies with q > 120 au, however, there is not enough time to homogenize their orbital orientations because their precession periods are comparable to the age of the solar system. If this general picture is true, then if any q > 120 au TNOs were to be discovered, they would probably spread from ϖ ≈ −120°(240°) to ≈60° (Figure 2's right panel), assuming a primordial ϖ clustering near 200°.

The close primordial sednoid alignment is only created in ϖ, not in Ω or ω; both angles have a larger circular standard deviation of ≈80° at 4.5 Gyr ago (but have an anticorrelation to produce the ϖ clustering). This is also the case in the rogue planet simulation, where the detached TNOs possessed various Ω and ω, instead of showing a strong clustered peak as in Figure 2's left panel.

Another scenario worth exploring is that the primordial alignment might correlate with a passing star's longitude of perihelion. Batygin et al. (2020) analytically showed that the particle's eccentricity evolution under a passing star is related to their ϖ difference. However, numerical studies of stellar encounters mainly focused on TNO a, q, and i distributions (e.g., Ida et al. 2000; Brasser et al. 2012; Nesvorný et al. 2023) instead of their apsidal lines. Therefore, we conducted a simulation with a solar-mass star passing through a primordial scattering disk (5 < q < 25 au), with the closest stellar approach at q_* = 300 au (Adams 2010; Batygin et al. 2020). Its hyperbolic trajectory has i_* = 15° relative to the solar system and v_∞ = 1 km s⁻¹, which is the typical velocity dispersion for young embedded clusters (Brasser et al. 2006).

Figure 3 shows the result of this simulation. Although such a passing star produced sednoids across a wide semimajor axis range, no strong clustering was formed in the Δϖ–q panel. One observes a weak preference for particles with Δϖ = 90° and 270° to be lifted from the scattering disk as the star passes (Figure 3's histogram) but this is insufficient to produce the correlation shown in Figure 1. While this preliminary simulation does not favor the stellar encounter for primordially clustered sednoids, further exploration of the star's parameters with a focus on the ϖ distribution it creates is warranted.

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