OSSOS. IX. Two Objects in Neptune's 9:1 Resonance—Implications for Resonance Sticking in the Scattering Population

Table 1 lists the best-fit orbits for o5m72 and o4h39 based on all available OSSOS astrometry through 2017 October along with the libration center and amplitude for the resonant angle ϕ_9:1 = 9λ_TNO − λ_Neptune − 8ϖ_TNO (where λ is the mean longitude and ϖ is the longitude of perihelion); the libration amplitude is defined as A_ϕ = (ϕ_9:1,max − ϕ_9:1,min)/2. We note that, following typical dynamical conventions (such as those in, e.g., Murray & Dermott 1999), this resonant argument would be described as being 8th order in the TNO's eccentricity; in this theoretical context, one might expect the phase space volume of this resonance to be too small to host a significant number of resonant objects. Numerical simulations of transient resonance sticking show, however, that distant n:1 resonances in fact host the largest populations of transiently stuck objects (Lykawka & Mukai 2007b; Yu et al. 2018). This apparent disconnect is due to the fact that our typical interpretation of resonance order in the solar system is based upon the assumption of small eccentricities. In the small eccentricity limit, the order of the resonance describes the number of planet-TNO conjunctions that occur over the course of a single resonance cycle; high-order resonances lead to large numbers of conjunctions, weakening the resonant perturbation. However, in the case of distant, highly eccentric TNO orbits, the vast majority of conjunctions occur at such large physical separations between the planet and the TNO that they are dynamically unimportant; the most important interaction between the TNO and Neptune occurs when the TNO is at perihelion. TNOs in n:1 resonances have one perihelion passage per resonance cycle, n:2 resonant TNOs have two, and so on. This has led to the suggestion (e.g., Pan & Sari 2004) that distant n:1 resonances should be called "first" order resonances rather than (n − 1)-order resonances. Furthermore, among resonances at similar orbital separations, average transient sticking timescales are largest for n:1 resonances, followed by n:2, and so on. Yu et al. (2018) suggest that these longer stick times result from longer resonance libration periods. Because simulations predict that the instantaneous population of transiently resonant scattering objects is largest for n:1 resonances, the detection of objects in the 9:1 is not surprising. The objects described here are the most distant of these n:1 objects yet to be securely identified via an analysis of the orbital element uncertainty, but we expect future observations to reveal many more resonant objects in even more distant resonances.

The best-fit orbit and the uncertainties in the orbital parameters for o4h39 and o5m72 are taken from the Bernstein & Khushalani (2000) orbit fitting code. The libration characteristics are determined from a 10 Myr integration of the best-fit orbit and 250 clones whose orbits span the uncertainty range for the best-fit orbit; these additional clones are generated using the covariance matrix for the orbital parameter uncertainties generated by the Bernstein & Khushalani (2000) orbit fitting code. All integrations were performed using the rmvs3 subroutine in the SWIFT numerical integration package (Levison & Duncan 1994). The integrations included the four giant planets and the Sun (augmented by the mass of the terrestrial planets) as the only massive bodies in the system, and the planets initial positions and velocities were taken from JPL Horizons (Giorgini et al. 1996) for the epoch for the orbit fits. All clones of o5m72 and o4h39 were included as massless test particles in the simulations. We used a base integration step size of 0.5 years (with smaller, adaptive time-steps used if planets are approached), and test particles were discarded if they reached heliocentric distances larger than 1500 au or smaller than 5 au. Orbital elements were output every 1000 years for analysis of the resonant behavior.

Figure 1 shows the range of resonant behavior seen in the 10 Myr integrations of both o4h39 and o5m72. Like other n:1 resonances, the 9:1 has three possible centers of libration for the resonance angle (see, e.g., Beauge 1994). The symmetric librators have an averaged value of $\langle {\phi }_{9:1}\rangle =180^\circ$, which means that over a libration cycle, they explore a large range of perihelia offsets (relative to Neptune), completely covering the range 180° ± A_ϕ away from Neptune; the libration amplitude for the symmetric libration mode is typically large (A_ϕ ≳ 100°). The two asymmetric libration islands are centered near ϕ_9:1 ∼ 90° (leading asymmetric) and ϕ_9:1 ∼ 270° (trailing asymmetric). Asymmetric librators have much smaller libration amplitudes (typically ≲80°) than symmetric ones. The libration period for ϕ_9:1 ranges from ∼(3 × 10)⁴–10⁵ years for these two objects. We note that these two objects can experience libration for additional 9:1 resonant arguments beyond the eccentricity-type argument we call ϕ_9:1. For mixed eccentricity-inclination type resonant arguments (e.g., ϕ = 9λ_TNO − λ_Neptune − 6ϖ_TNO − 2Ω_TNO), we typically see libration similar to those of ϕ_9:1, but about a libration center that slowly circulates on 50–100 Myr timescales, corresponding to the secular regression of the objects' longitudes of ascending node. Because the eccentricity-type resonance is the strongest and most clearly librating for these highly eccentric TNOs, we only consider the libration amplitudes for ϕ_9:1.

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Over the 10 Myr simulations, all of the o4h39 clones are small-amplitude leading asymmetric librators with libration amplitude ${A}_{\phi }={35}_{-7}^{+6}$ degrees; the uncertainty in A_ϕ represents the 1σ range from the distribution in the lower right panel of Figure 1. The clones of o5m72 are more mobile and spend most of the 10 Myr simulation switching between trailing asymmetric and symmetric libration; some clones also spend time in the leading asymmetric island. Thus, o5m72 has a less well-defined libration amplitude, and we provide only the median amplitude for the symmetric and trailing asymmetric clones in Table 1 rather than an uncertainty. Neither object is currently experiencing so-called Kozai libration (libration of the argument of perihelion, ω) within the 9:1 resonance; because this libration occurs on longer timescales than mean motion resonant libration, a 100 Myr integration was used to check for ω libration.

To determine how long o4h39 and o5m72 are likely to remain in the 9:1 resonance, we integrated their orbits forward in time for 4 Gyr using 1000 clones that span the orbital uncertainties from the best-fit orbit's covariance matrix. We use the same integration method outlined in Section 2.1, but with an output frequency decreased to every 10⁴ years. Figure 2 shows some example semimajor axis histories for clones of o4h39. We tracked the semimajor axis history of each clone over a sliding 5 Myr time window for the duration of the simulation to determine if or when it left the resonance. We consider a test particle to have left the resonance if the average semimajor axis over the 5 Myr window deviated from the expected resonant value (130.06 au) by more than 0.3 au and the minimum or maximum semimajor axis value over the window deviated by more than 1 au; we take the midpoint of the time window to be the time at which the clone left the resonance. Some clones leave the resonance but then re-enter it after a sequence of gentle scattering event with Neptune (as shown in the bottom panel of Figure 2); in these instances we consider the clone to have left the resonance as of the first excursion out of resonance. We use the semimajor axes of the test particles to determine resonance membership rather than the evolution of ϕ_9:1 because individual libration cycles are not fully resolved at this reduced output frequency and high-amplitude libration can be difficult to distinguish from circulation. The semimajor axis limits described above were determined by visually examining the semimajor axis and resonant angle evolution of 100 clones of each object and adjusting the limits until they reproduced the visually determined departure times from the resonance.

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Figure 3 shows the distribution of resonance sticking timescales for the clones of o4h39 and o5m72. The clones of o4h39 have a median resonance sticking time of 910 Myr, with ∼16% of the clones remaining continuously resonant at 4 Gyr. An additional ∼8% of the clones have scattered back into the 9:1 resonance at 4 Gyr. The clones of o5m72 have a median sticking time of 1.37 Gyr, and ∼17% of the clones are still continuously resonant at 4 Gyr; an additional ∼6% of the clones have scattered back into the 9:1 at 4 Gyr.

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The fact that o4h39 has a slightly shorter sticking timescale despite currently being deeper in resonance than o5m72 is likely at least partly due to its lower perihelion distance (∼39 au) compared to o5m72 (∼44 au). However, we note that longer-term variations in eccentricity while o5m72 remains resonant can lower its perihelion to ∼38 au, allowing clones to be scattered out of resonance. Over the 4 Gyr simulations, ∼31% of the o4h39 clones and ∼44% of the o5m72 clones were removed from the simulation because they were scattered far enough outward to reach heliocentric distances larger than 1500 au.

Because we infer the sticking timescale from an ensemble of clones, we must consider how the observational uncertainties in each objects' current semimajor axis affect the distribution of likely sticking timescales. (While the eccentricity will also affect stability in the resonance, the uncertainties in e and a are strongly correlated, so we need only examine the variation in sticking timescale across one parameter.) We examined the resonance sticking time as a function of initial semimajor axis for the clones of each object. For the o4h39 clones, this distribution is very similar across the entire range of a sampled in our simulations. In contrast, the observational uncertainty in o5m72's orbit spans a portion of the 9:1 resonance where the stability timescale is changing; the most stable o5m72 clones are not evenly distributed across o5m72's uncertainty range. Thus, we expect future observations, which will decrease the orbit-fit uncertainties, to have a much more significant effect on the inferred sticking timescale for o5m72 than for o4h39. We can confirm this expectation for o4h39, because (as noted in Table 1) additional observations were linked to o4h39 after the OSSOS observations were sent to the Minor Planet Center after the submission of this paper. These additional observations occurred in 2004 and 2007, and their inclusion in o4h39's best-fit orbit approximately halves the semimajor axis uncertainty. The new orbit has short-term evolution in the 9:1 very similar to the orbit fit we used throughout this work, with a libration amplitude well-within our uncertainty range. We performed a 1 Gyr integration of 100 clones of the new orbit and confirm that the fraction of clones remaining in the 9:1 at 1 Gyr is statistically indistinguishable from the fraction shown in Figure 3 for that time.

Lykawka & Mukai (2006, 2007b) investigated resonance sticking in simulations of actively scattering test particles. They identified several captures in the 9:1 with timescales ranging from a few to a hundred Myr, showing both symmetric and asymmetric libration behavior. These simulations demonstrate that the 9:1 is capable of temporarily capturing scattering objects, but they do not provide sufficient statistics to estimate the current expected population of transient 9:1 objects.

To better estimate the expected transiently stuck 9:1 population, we instead extrapolate the results of Yu et al. (2018) who investigate resonance sticking in the region a = 30–100 au. The orbital distribution of the initial population in this simulation is based on the Kaib et al. (2011) model of the scattering population, which has been shown to be consistent with observations of the current scattering population (Shankman et al. 2016). This same model is used to generate the population estimate for the scattering population (Lawler et al. 2018b) necessary to ultimately compare the predicted transient 9:1 population to the observationally derived 9:1 population estimate from Section 3. Yu et al. (2018) find that starting from this population of actively scattering test particles and looking at the time-averaged population on a timescale of 1 Gyr, ∼40% of 30 < a < 100 au TNOs should be transiently stuck in a resonance at any given time; this implies that, at any given time, there are about 1.3 times as many actively scattering (non-resonant) objects as there are objects transiently stuck to resonances in this range. These simulations indicate that about a quarter of the the transiently stuck particles are in n:1 resonances, and the transient populations of the n:1 resonances increase with semimajor axis.

Figure 4 shows the number of transiently stuck particles for the n:1 resonances with a < 100 au in the Yu et al. (2018) simulations (black dots) in units of the total actively scattering population from a = 30–100 au. We extrapolate these simulation results slightly beyond 100 au to estimate the expected population of transiently stuck 9:1 objects. We fit three different functional forms to the distribution of total sticking populations for the n:1 resonances, P_n:1: a linear fit (P_n:1 =a + bn), a second-degree polynomial fit (${P}_{{\text{}}n:1}=a+{bn}\,+{{cn}}^{2}$), and an exponential fit (P_n:1 = a exp bn). These fits are shown, extrapolated to P_9:1, in Figure 4. The linear fit is poor quality, but we use it as a lower-bound estimate; the polynomial and exponential fits are of similar quality. The exponential fit is intended to provide an upper bound, and we do not expect any of these extrapolations to hold out to arbitrarily large semimajor axes. The ability of Neptune's resonances to efficiently capture scattering objects should drop off for extremely distant resonances. However, the Lykawka & Mukai (2007b) simulations show temporary sticking out to ∼250 au, including sticks in the 9:1 resonance. Thus, we expect our conservatively wide range of extrapolations out to the 9:1 at a ∼ 130 au to encompass the true 9:1 sticking population. The extrapolated Yu et al. (2018) results imply that the population of transiently stuck 9:1 objects is ∼0.13–0.36 times as large as the population of actively scattering objects from a = 30–100 au.

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To translate this theoretical relative 9:1 population estimate to an absolute number (to compare with the observationally derived population estimate in Section 3), we need a population estimate for the actively scattering population. An analysis of the combined OSSOS and CFEPS actively scattering population detections implies that there are ∼(1.1 ± 0.2) × 10⁴ scattering objects with a < 100 au and H_r < 8.66 (Lawler et al. 2018b). This population estimate is model dependent but uses the same Kaib et al. (2011) orbital distribution for the scattering population as the Yu et al. (2018) simulations. Based on the extrapolation of the Yu et al. (2018) results, this translates to ∼(1.1–4.5) × 10³ objects with H_r < 8.66 transiently stuck in the 9:1 resonance. This should be compared to the ∼(0.4–3.3) × 10⁴ (95% confidence range) population estimate derived from the two OSSOS 9:1 objects in Section 3. We note that there are a few potential differences between the resonance sticking population in the scattering simulations and the nominal 9:1 model used to produce our observational population estimate. The first is that the inclinations in the Yu et al. (2018) simulations are generally lower than the i = 24°–40° range used to produce our nominal 9:1 population estimate. In the simulations, resonance stickiness does not depend strongly on inclination, so the number of transiently stuck 9:1 objects is unlikely to change with a different inclination distribution; however, the observational population estimate does depend on the modeled inclination distribution. The comparison between the expected 9:1 sticking population and the observed population improves slightly if we model the 9:1 with an off-center Gaussian inclination distribution, which slightly decreases the observed 9:1 population estimate (see Section 3). The second possible difference between the simulations and our nominal 9:1 model is the assumed eccentricity/perihelion distance distribution. The two observed 9:1 objects currently have perihelion distances that are larger than typical for the actively scattering population that served as the initial conditions for the Yu et al. (2018) simulations. However, as noted in Section 2.2, eccentricity cycling within the 9:1 causes longer-term variations in q for both objects; during the long-term simulations, clones of both objects do achieve perihelion distances consistent with the scattering population. Direct simulations of the transient 9:1 population would allow the calculation of a time-averaged perihelion distribution (and thus eccentricity distribution) that could be used in place of our uniform e = 0.6–0.7 distribution to generate an improved observational population estimate. However, with only two detections, the uncertainty in the population estimate for any 9:1 model is very large, so we leave this for future investigations.

With these caveats in mind, we note that the upper range of the theoretically estimated 9:1 transient population (∼(1.1–4.5) × 10³ objects with H_r < 8.66) is consistent with the lower range of the observationally derived population estimate (∼(0.4–3.3) × 10⁴ objects with H_r < 8.66). We thus conclude that the two OSSOS 9:1 detections are marginally consistent with the 9:1 population expected to be transiently stuck from the current scattering population.

Given the only marginal agreement between the OSSOS 9:1 population estimate and the expected number of 9:1 objects transiently stuck from of the current scattering population, as well as the fact that ∼15%–20% of the o5m72 and o4h39 clones remain resonant on 4 Gyr timescales, it is possible that these objects are the remnant of a larger primordial resonant population. The observed 9:1 objects could either have been captured during the era of planet migration or could represent extremely long-timescale "transient" sticks from earlier in the solar system's history when the scattering population was much more numerous. The primordial scattering population is expected to be one to two orders of magnitude larger than the current scattering population (see, e.g., Kaib et al. 2011; Brasser & Morbidelli 2013). The Yu et al. (2018) simulations only considered 1 Gyr timescales for both population decay and resonance sticking, but we can use these results to estimate whether longer-timescale sticks from the larger primordial population significantly enhance the expected 9:1 population.

From the Yu et al. (2018) simulations, we find that the probability of a scattering object (with semimajor axis in the range 30–100 au) getting transiently stuck in a resonance for timescale t_stick goes approximately as $P({t}_{\mathrm{stick}}\gt t)\propto {t}^{-1}$ for sticks up to a few hundred Myr in length. The stick probability appears to fall faster for longer sticks ($P({t}_{\mathrm{stick}}\gt t)\propto {t}^{-2}$), although the statistics are poorer. The decay of the Kuiper Belt populations over time is often modeled as N(t) ∝ t^−b; for various models of the scattering population, b values in the range ∼0.7–1.3 appear to approximate the population reduction reasonably well (see, e.g., Kaib et al. 2011; Brasser & Morbidelli 2013; Greenstreet et al. 2016). Considering a 4 Gyr history for the scattering population, a transient stick from time t must stick to a resonance for a timescale longer than (4 Gyr–t) to still be seen as resonant today. The similarity in the exponents for the population decay and the sticking probability means that the much larger scattering population at early times is essentially balanced by the decreased probability of the very long sticking timescales required for those objects to remain resonant until today. The result is that the distribution of currently observed transiently stuck resonant objects from a decaying scattering population is expected to have roughly flat distribution of log(t_stick) (Yu et al. 2018).

The expected transiently stuck resonant populations in Figure 4 are based on a 1 Gyr simulation and resonance sticking timescales from 10⁵–10⁹ years. If we assume a relatively flat distribution of sticking timescales in log(t) from the above considerations, extending the population estimates to include transient sticks from as long as 4 Gyr ago will increase the resonant population estimates by ∼15%. This does bring the theoretically expected stuck 9:1 population into slightly better agreement with the observationally derived population estimate. However, we do not expect "transiently" stuck resonant objects from early in the solar system's history to be a large fraction of the current resonant population unless one or more of the following is shown to be true: the primordial scattering population was more than the expected one to two orders of magnitude larger than the current scattering population, the primordial scattering orbital distribution was such that resonance sticking was significantly more likely than in the current population, or the resonances themselves were sticker in the past (perhaps due to a larger orbital eccentricity for Neptune as in, e.g., Levison et al. 2008).

It is also possible that the observed 9:1 objects were emplaced onto stable resonant orbits during the era of giant planet migration. Given the previous lack of observed 9:1 objects, this has not been examined in the literature for this resonance. Pike et al. (2017) did find test particles in many high-a Neptune resonances (including the 9:1) in an analysis of the end state of a Nice model scenario for giant planet migration; however, the analysis did not differentiate between transiently stuck and primordially captured resonant objects and did not provide a 9:1 population estimate. It is possible that some scattering objects that stick to the 9:1 while Neptune is still migrating will evolve into more stable phase space within the resonances. However, Nesvorný et al. (2016) find that slow migration of Neptune results in objects with larger perihelia (similar to our 9:1 objects) being preferentially dropped out of distant resonances as Neptune migrates; thus it is unclear whether sticking during migration could increase the 9:1 population. If additional observations of the 9:1 population do not improve the currently marginal agreement between its total population and the resonance sticking hypothesis, the possibility of primordial capture should be examined in more detail.

Beyond 50 au, the n:1 populations with the best constrained population estimates are the 3:1 and 5:1 resonances. Alexandersen et al. (2016) estimates that the 3:1 contains at least ∼1200 objects with H_r < 8.66, using a similar approach to the survey simulation in Section 3 where the orbital model is based on the population having eccentricities and inclinations similar to the objects detected in the survey. Pike et al. (2015) derive a 5:1 population estimate of ∼2000 objects with H_g < 8 from three observed 5:1 objects. Assuming a typical g − r color (g − r = 0.5; see, e.g., Alexandersen et al. 2016) and α = 0.8, this corresponds to ∼4 × 10⁴ 5:1 objects with H_r < 8.66 (the 95% confidence lower bound on this estimate is ∼10⁴ objects); this would be the largest resonant population known (as suggested by Gladman et al. 2012).

While a 3:1 population ≃9 times smaller than the 9:1 is consistent with the transient resonant population ratios in Figure 4, the observations indicate that the 5:1 population is nominally equal to or larger than the 9:1 population. Thus, the observed 5:1 population appears to be inconsistent with transient sticking, despite the fact that the observed 5:1 resonators appear to be only transiently present in the resonance, escaping on 10⁸–10⁹ year timescales (Pike et al. 2015). It is possible that the 5:1 population estimate (based on detections in the CFEPS survey, Petit et al. 2011, 2017) is only large by chance and thus a significant overestimate; this is supported by the fact that the much larger OSSOS project discovered only one additional 5:1 resonator (Bannister et al. 2018c). Future constraints on these resonant populations will shed more light on whether they are consistent with the resonance sticking hypothesis.