February 15, 2026
approximation, which maintains the dominance of the leading-order term, rendering equation (10) an adequate model for understanding the relevant process (i.e., bending of the Laplace plane) in the context of the Planet Nine hypothesis.
30
a = 300 AU
30
e9 = 0.25 i9 = 20 deg
20 20
10 10
0
0 90 180 270 360
∆⌦ (deg)
0
0 90 180 270 360
∆⌦ (deg)
30
20 20
10 10
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0 90 180 270 360
∆⌦ (deg)0 90 180 270 360
∆⌦ (deg)
Figure 11: Secular inclination dynamics induced by Planet Nine. Like Figure 10, these panels depict the contours of the secular Hamiltonian (10), which provides an analytic approximation to slow variations of KBO inclination and longitude of ascending node facilitated by Planet Nine. As can be discerned from the left-right, top-down progression of panels, prominent regions of nodal libration associated with clustering of the angular momentum vectors develop with increasing KBO semi-major axis. Unlike the case of apsidal confinement delineated in Figure 10, however, no new equilibrium points emerge with increasing a. Corre- spondingly, the clustering of the orbital poles ensues smoothly, and simply coincides with the bending of the distant solar system’s equilibrium angular momentum plane by Planet Nine.
Generation of Highly Inclined TNOs
Unlike the eccentricity-perihelion and inclination-node evolution described above, high-inclination dynamics induced by Planet Nine involves coupled modulation of two degrees of freedom (related to e and i), with no clear symmetry inherent to the motion. Moreover, even in the simplified simulations of Batygin and Morbidelli (2017); Li et al. (2018), no discernible separation of timescales materializes, rendering the system intrinsically non-adiabatic. As a result of these complications, it is very difficult (if not impossible) to construct an integrable model for P9-driven high-inclination dynamics that will have any reasonable degree of quantitative accuracy. Thus, in the follow- ing discussion, we will limit ourselves to an essentially qualitative description of the secular evolution, with the hope of highlighting its most basic attributes.
Despite being non-integrable in nature, high-inclination dynamics induced by P9 are primarily driven by a well-defined secular harmonic. This harmonic, θ, is identified as being the octupole-level secular angle (Batygin and Morbidelli, 2017)
θ = 2v – ∆a = a + a9 – 2Ω
Θ = ,1 –
2
where Θ is the action conjugate to θ, that emerges if we adopt ∆a as the secular angle for the other degree of freedom (see Appendix B). We note that the physical meaning of θ is nothing other than the difference between the longitudes of perihelion of Planet Nine and a counter-revolving KBO, for which the retrograde longitude of perihelion is defined as aj = Ω v = 2Ω a (Gayon et al., 2009). In reality, the secular evolution in (θ, Θ) is intimately coupled to motion in (e, ∆a), meaning that during large-scale
excursion of KBO inclination, the eccentricity changes in concert (while ∆a executes complex librations around 180 deg; see Li et al. 2014 for a related discussion). As a very crude illustration of the underlying dynamics, however, we can choose to ignore this reciprocity and freeze the evolution in (e, ∆a), which allows us to examine the level curves of the Hamiltonian as before.
For definitiveness, let us take α = a/a9 = 1, ∆a = 180 deg, and set e = 0.5 (note from the right-bottom panel of Figure 10 that secular eccentricity modulation in the plane can cover almost the entire e (0, 1) range). With these approximations in hand, we compute the function
8 m a2
7 = – 1 G Ms 1 X j j + a˙ ,G M a .1 – ,1 – e2 cos(i)
4u2 |r – r9| e=0.5,∆a=u
in closed form on the (θ, Θ) plane, and project its contours on Figure 12.
A key feature that is immediately evident in Figure 12 is the existence of trivial circulating trajectories at low values of Θ, and the emergence of a prominent island of θ-libration at higher values of Θ. Recalling the definition of Θ from equation (11), the picture outlined in Figure 12 implies that as long as the eccentricity is high and
0.8
0.6
0.4
0.2
0
0 90
180 270 360
2⌦ — $ — $9 (deg)
Figure 12: Like Figures 10 and 11, this diagram shows an analytic exemplification of large-inclination dynamics facilitated by Planet Nine. Unlike results depicted in Figures 10 and 11 however, this secular diagram only provides a very crude reproduction of the real high i behavior exhibited by TNOs within the framework of the Planet Nine hypothesis, and should be considered highly approximate (see text for details). Qualitatively, this figure illuminates the existence of two families of trajectories: at low values of the action Θ – which corresponds to high eccentricity and low inclination – the angle θ circulates, meaning that the (θ, Θ) resonance plays little role in the dynamical evolution. If the action Θ is increased, however, the KBO may land on a secular trajectory characterized by libration of θ, which in turn translates to large-scale variations of the orbital inclination.
inclination is low – which translates to low values of Θ – the (θ, Θ) resonance plays no role in the dynamical evolution. In other words, the low-Θ regime of secular motion is the one where the integrable models outlined in the previous two sub-sections apply.
Conversely, if Θ is allowed to reach a high enough value by some dynamical pro- cess, the system can transition into a regime where θ begins to undergo bounded oscil- lations, resulting in coupled variations in Θ14. In turn, these excursions in Θ correspond to large changes in the orbital inclination. Although rudimentary and unsystematic, this interpretation of the dynamics is consistent with numerical results, where the orbit- flipping behavior of distant objects induced by Planet Nine is almost always triggered at the minimum of the KBO eccentricity cycle (which translates to a maximum in Θ; Batygin and Morbidelli 2017).
Mean-Motion Resonances
The entirety of the theoretical discussion presented above is framed within the con- text of orbit-averaged perturbation theory. This approach to understanding dynamical
he fact that Θ cannot exceed ,1 e2 in Figure 12 is an unphysical consequence of the constant eccen- tricity assumption inherent to our computation of equation (12).
evolution is sensible when the Keplerian motion of the interacting bodies is in essence, uncorrelated. Notably, however, this assumption is invalidated if the orbital periods of P9 and the KBO become commensurate, such that their ratio can be expressed as a fraction of two nearby integers. In this case, harmonics of the Hamiltonian involv- ing the mean longitudes of P9 and the TNO can execute bounded oscillations, causing P9 perturbations to become resonant (as opposed to secular) in nature, and facilitate a coherent exchange of orbital energy and angular momentum.
Are mean-motion resonances (MMRs) with Planet Nine relevant to the dynamical evolution of the distant Kuiper belt? Strictly speaking, the answer is yes (Millholland and Laughlin, 2017; Becker et al., 2017; Hadden et al., 2018; Bailey et al., 2018), al- though their observational consequences are thus far relatively insignificant. That is, even though resonant dynamics are implicated in ensuring long-term stability for KBOs that share Planet Nine’s orbital plane, it is secular interactions that are dominantly re- sponsible for sculpting the actual observed distribution of long-period KBOs (Batygin and Morbidelli, 2017). Let us expand upon this point further.
Examination of particle evolution in the simulation suite of Batygin and Brown (2016a) revealed that over the course of typical 4 Gyr integrations, KBO orbits can ex- hibit temporary capture into P9 MMRs, usually lasting 10 100 Myr. This resonance- hopping behavior was more thoroughly investigated by Becker et al. (2017), who fur- ther considered the long-term stability of observed KBOs as a constraint on P9’s in- ferred orbit. Beyond purely theoretical considerations, the potential prevalence of P9 resonances in the distant belt inspired Malhotra et al. (2016) to examine the numero- logical relationships between the orbital periods of observed KBOs, and note that 4 out of 6 objects known at the time lie close to N:1 and N:2 orbital period ratios with a putative perturber. Based upon these relationships, Malhotra et al. (2016) suggested that P9 may reside on an orbit with a 665 AU.
This line of reasoning was examined in a more quantitatively rigorous fashion by Millholland and Laughlin (2017), who carried out a large-scale Monte-Carlo explo- ration of P9-forced dynamics of known KBOs, and derived a probability density func- tion of P9 parameters that approximately maintains the KBOs in a clustered distri- bution, while temporarily preserving commensurabilities. However, Millholland and Laughlin (2017) simultaneously noted that resonant configurations generally have dy- namical lifetimes that are considerably shorter than the age of the solar system, im- plying that even long-term stable KBOs do not remain bound to a single mean mo- tion commensurability on Gyr timescales (which is also seen in Becker et al. 2017; Hadden et al. 2018). Practically, this means that even though some fraction of the observed KBOs can reasonably be expected to be locked resonance with Planet Nine, it is likely impossible to confidently identify P9 resonant vs non-resonant KBOs, or to determine the value of the specific resonant angle. Unfortunately, this tendency of long-period KBOs to stochastically skip between mean-motion commensurabilities es- sentially eliminates the promise of deriving P9’s semi-major axis from the present-day orbital period distribution of observed KBOs (Bailey et al., 2018).
If resonant effects can play a pronounced role in the dynamics of long-periods KBOs, then why is it sensible to adopt a purely secular Hamiltonian as aperturbative model for P9-induced evolution? Put simply, this is because the anomalous structure of the distant Kuiper belt is predominantly shaped by orbit-averaged interactions. More
a = 500 AU
1.0
m9 = 5M a9 = 500 AU e9 = 0.25
a = 500 AU
1.0
ϕres = λ — λ9 → 0
0.75 0.75
0.5
0.25
0.5
0.25
0 0
0 90 180 270 360 0
∆$ (deg)
90 180 270 360
∆$ (deg)
Figure 13: A comparison between purely secular and resonant-secular dynamics induced by Planet Nine. The left panel shows the a = 500 AU (co-orbital) phase-space portrait, arising from equation (9) (also shown as the right bottom plot of Figure 10). The right panel depicts an equivalent e ∆a phase-space portrait corresponding to an object also with a = 500 AU, but residing in a 1:1 mean-motion resonance with Planet Nine, where the orbit-averaging procedure was carried out under the condition дres = h h9 0 (see Batygin and Morbidelli 2017 for details). The similarity between the two panels highlights the inherent separation between the secular and resonant degrees of freedom.
specifically, while MMRs may modulate the long-term stability of KBOs through the phase-locking mechanism (Morbidelli, 2002), the clustering of orbits in physical space, as well as excitation of extreme inclinations of long-period TNOs is largely a secular effect. To demonstrate this separation of degrees of freedom, Batygin and Morbidelli (2017) considered a simplified 2D model of the solar system, where the gravitational fields of the canonical giant planets were replaced by an effective quadrupole (J2) mo- ment of the sun, leaving Planet Nine as the only direct perturber in the system. By restricting all orbital motion to a common plane, this physical setup renders all KBOs that are not phase-protected dynamically unstable, resulting in a fully resonant orbital distribution of distant KBOs.
Examining the dynamics of the simulated apsidally clustered objects, Batygin and Morbidelli (2017) pointed out that the resonant multiplets responsible for driving the KBO evolution typically have the form
дres = ph – qh9 – (p – q)a9. (13)
Resonances of this type (sometimes referred to as “corotation” resonances) modulate the KBO’s semi-major axis but not eccentricities (because de/dt b cos д/ba=0), leaving the e ∆a degree of freedom to be controlled by secular interactions. Ac- cordingly, by carrying out the orbit-averaging procedure under the resonant condition, Batygin and Morbidelli (2017) demonstrated that secular dynamics embedded within P9 MMRs have a very similar phase-space topology to
