February 14, 2026
9Formally speaking, the secular normal form is attained by canonically averaging the Hamiltonian over the mean longitudes, which are the fast angles of the system (Morbidelli, 2002; Touma et al., 2009)
potential. In terms of Keplerian elements, the octupole-level expansion of this function has the form (Mardling, 2010):
¯ 1 GMs a 2 1 " 3 2 . 2 . 2
1 + e 3 cos (i) – 1 3 cos (i9) – 1
29 = – 16 a9 a9 , 3 2
1 – e9 x
Pre˛cezssion x
+ 15 e2 sin2(i) cos(2v) + 3 sin(2i) sin(2i9) cos(∆Ω) + sin2(i) sin2(i9) cos(2∆Ω)
9 x Eccentric˛ityzCoupling x
x High-Inclin˛atizon Dynamics x
where the subscript 9 refers to parameters of P9. Each harmonic in the above expansion governs a particular dynamical effect, and if dominant, entails a specific orbital archi- tecture of the distant Kuiper belt. Let us briefly consider each of these terms and their physical meanings in isolation, as this will help us make sense of simple perturbative models for P9-induced dynamics that will follow.
Precession. The first term in equation (8) governs apsidal precession and nodal re- gression of a KBO forced by P9. Akin to the discussion surrounding equations (4) and (5), on its own this effect only acts to randomize the physical orientation of the orbits, leading to an un-clustered orbital distribution. Thus, irrespective of its magnitude, this component of Planet Nine’s potential only accelerates the evolution already enabled by the quadrupolar fields of the known giant planets.
Kozai-Lidov Effect. The second term in the above expansion possesses (two times) the argument of perihelion as its critical angle, and governs the Kozai-Lidov mechanism, which can facilitate a periodic exchange between eccentricity and inclination of a test- particle while conserving the zˆ-component of the orbital angular momentum vector
hz = ,1 e2 cos(i) (Lidov, 1962; Kozai, 1962). If dominant, this resonance can lead
to clustering among the arguments of perihelion (as first discussed in the context of the distant Kuiper belt by Trujillo and Sheppard 2014), such that v librates about 90 and 270 deg for well-separated orbits, or alternatively about 0 and 180 deg for orbits close to the perturber (Thomas and Morbidelli, 1996). It is worth noting, however, that the Kozai-Lidov resonance is easily destroyed by external sources of perihelion precession (such as that arising from the canonical giant planets) and even if active, would yield an orbital distribution in the distant Kuiper belt that is not compatible with the data.
Interactions of the Planes. The third harmonic in equation (8) has the difference of the longitudes of ascending nodes of the TNO and P9 as the critical argument, ∆Ω = Ω Ω9, and governs the interactions between the orbital planes of Planet
Nine and the TNOs it shepherds. The relative importance of this effect compared to the nodal
regression forced by Jupiter, Saturn, Uranus and Neptune (equation 5) dictates the equi- librium (Laplace) plane around which the orbits of the KBOs precess. That is, in the region of parameter space where forcing due to P9 dominates, the angular momen- tum vectors of the KBOs precesses around Planet Nine’s orbit normal. Conversely, the angular momentum vectors of shorter period KBOs, whose evolution is primarily regulated by Neptune, will tend to precess around the total angular momentum of the canonical giant planets. This suggests that in order to perturb the planes of the distant KBOs, the orbit of P9 itself must be appreciably inclined with respect to the ecliptic10.
Eccentricity Coupling. The fifth term in the expansion contains the difference of lon- gitudes of perihelion, ∆a = a a9, as its driving angle, and describes the coupling between the eccentricity (Runge-Lenz) vectors of P9 and the KBO orbits. Accordingly, oscillations in the KBO’s eccentricity and orbital orientation relative to Planet Nine’s apsidal line are regulated by this term. Unlike the preceding quadrupolar harmonics, this effect is octupolar in nature, and therefore explicitly depends on Planet Nine’s eccentricity. This simple fact alone already implies that in order for P9 to facilitate any degree of apsidal confinement via bounded libration of ∆a, its orbit must have non-zero eccentricity.
High-Inclination Dynamics. Much like the ∆a harmonic, the final term of equation
(8) is also octupolar in nature, but its dynamical consequences are considerably more subtle. Like the Kozai-Lidov resonance, this term mixes longitudes of perihelion and node, facilitating a complex dynamical evolution that simultaneously modulates the de- grees of freedom related to the TNO’s eccentricity and inclination (Batygin and Mor- bidelli 2017; Li et al. 2018). As we will argue below, it is this harmonic that drives high-inclination dynamics and orbit-flipping behavior in the trans-Neptunian region of the solar system.
With a rudimentary description of secular forcing in place, we are now in a position to construct a sequence of simplified analytical models for P9-induced evolution, and examine how they connect to the three primary features of the anomalous structure of the distant Kuiper belt. Particular emphasis will be placed on inspecting the dynamics qualitatively, with the aid of integrable Hamiltonians. As in the previous section, we begin by considering apsidal confinement of long-period KBOs.
Our examination of the functional form of ¯ 9 implies that the apsidal confinement is dominantly driven by the secular angle containing the difference between the lon- gitudes of perihelion. Let us now consider a perturbative model, characterized by this critical argument. In order to tease out the pertinent dynamics, it is advantageous to restrict the evolution to the plane, by assuming that sin(i9), sin(i) → 0. From equation
(8), it is then evident that all but a single harmonic term – namely cos(a– a9) – vanish
10Notably, the harmonic term that follows is simply double the ∆Ω term, and qualitatively achieves a similar effect. As suggested by its steeper dependence on the KBO’s inclination, however, this cos(2 ∆Ω) correction only matters for highly inclined objects.
from the disturbing function. Indeed, this simplification is sufficient for us to write down an integrable model for the secular motion of KBOs, perturbed by the canonical giant planets and Planet Nine.
The relevant Hamiltonian has the form (e.g., Beust 2016; Batygin and Morbidelli 2017):
8 m a2
7 = – 1 G Ms 1 X j j + a˙ ,G M a .1 – ,1 – e2
4u2 |r – r9| i,i9 =0
where r is the position vector, and h is the mean longitude (i.e., a fast angle that varies on the orbital timescale and informs the object’s location on the orbit).
The physical meaning of the three terms that comprise the above Hamiltonian can be understood as follows: the first term governs the slow precession of the KBO’s longitude of perihelion due to the phase-averaged quadrupole fields of the canonical giant planets. The second term accounts for the fact that the reference frame as taken to be co-linear with the major axis of Planet Nine’s orbit, and is therefore, also slowly precessing at the rate a˙ 9, given by equation (4) (the generating function corresponding to the associated canonical contact transformation is spelled out in Appendix B). Most importantly, the third term governs the gravitational coupling between Planet Nine and the Kuiper belt object11.
Because the action conjugate to the fast angle h is a sole function of a, the process of averaging the Hamiltonian over the mean longitudes renders the semi-major axis of the KBO constant (e.g., Touma et al. 2009). This means that within the framework of a purely secular model, a acts as a parameter of the problem, rather than a variable. Consequently, the contours of equation (9) projected onto the e ∆a plane at a given semi-major axis fully encompass the evolution entailed by the Hamiltonian.
Figure 10 depicts a series of e ∆a phase-space portraits at KBO semi-major axes of a = 200, 300, 400, and 500 AU, where we have adopted the Planet Nine parameters a9 = 500 AU, e9 = 0.25 and m9 = 5 M⊕ (this particular choice of parameters is in- formed by the numerical simulations presented in section 5). A key qualitative attribute of KBO dynamics that is immediately evident from Figure 10 is that at large values of the semi-major axis (i.e., a & 250 AU), Hamiltonian (9) is characterized by a pair of stable (elliptic12) equilibrium points: one at ∆a = 0 and another at ∆a = 180 deg. Meanwhile, only a single equilibrium point (at ∆a = 0) exists for smaller semi-major axes. Note further that wherever present, the ∆a = 180 deg equilibrium resides at a higher eccentricity than its ∆a = 0 counterpart. Importantly, the appearance of the stable ∆a = 180 deg fixed point beyond a critical value of a is a robust feature of the
11Because in the assumed configuration the orbits can intersect, a series expansion of the form (8) does not always provide a good representation of the dynamics. This complication is, however, easily circumvented by computing the averaged potential in closed form on the e ∆a plane (Gronchi, 2005; Beust, 2016).
12Note that at a & 350 AU, an unstable (hyperbolic) ∆a = 0 fixed point also emerges at large values of the eccentricity.
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)
a = 500 AU
0.75
0.75
apsidal libration
0.5
0.25
0.5
0.25
0 0
0 90 180 270 360 0
∆$ (deg)
90 180 270 360
∆$ (deg)
Figure 10: Secular eccentricity dynamics induced by Planet Nine. Level curves of Hamiltonian (9) are projected onto the (e, ∆a) plane, at various choices of KBO semi-major axes. All cases use canonical Planet Nine parameters of m9 = 5 M⊕, a9 = 500 AU, e9 = 0.25, and i9 = 20 deg. These equal- contours represent analytic approximations to slow variations experienced by KBO orbits within the framework of the Planet Nine hypothesis. That is, over timescales much longer than an orbital period, the KBO eccentricity and
longitude of perihelion gradually evolve along the depicted curves. Note that for a comparatively small KBO semi-major axis of a = 200 AU (upper left panel), the apsidal angle ∆a circulates, implying a nearly uniform distribution of longitudes of perihelion. At a = 300 AU (upper right panel), however, a stable apsidally anti- aligned equilibrium point appears at high eccentricity. The prevalence of secular trajectories that encircle this ∆a = 180 deg fixed point signifies the emergence of perihelion confinement in the distant Kuiper belt. Note further that at even higher semi-major axes of a = 400 AU (lower left panel) and a = 500 AU (lower right panel), an additional hyperbolic equilibrium at ∆a = 0 also emerges at high eccentricity.
model, and qualitatively explains the origin of the sharp changeover from randomized and apsidally confined sub-populations of the data.
The phase-plane portraits depicted in Figure 10 demonstrate that if P9 exists, long- period orbits with perihelion distances close to Neptune will predominantly populate the apsidally anti-aligned island of libration. Moreover, once entrained in a stable mode of ∆a 180 deg libration, orbits with initial q a8 will experience secular oscillations in eccentricity, periodically detaching from (and subsequently re-attaching to) Neptune. This effect provides a natural mechanism for generating the peculiar high-q orbits of Sedna, 2012 VP113, and 2015 TG387 from typical scattered disk objects. Conversely, long-period orbits that are not locked into a stable ∆a oscillation get driven to very high eccentricities, eventually crossing Neptune’s orbit and leaving the system (Khain et al., 2018a). Figure 10 further illustrates that comparatively short-period orbits oc- cupy secular trajectories that simply circulate in longitude of perihelion. Thus, put
simply, Figure 10 shows that while the phase-averaged potential of Planet Nine is of little consequence for orbits with a . 250 AU, under its influence, scattered disk ob- jects with a & 250 AU will become dynamically organized into an apsidally confined configuration with a broad distribution of perihelion distances.
Recall that in order to analytically describe apsidal confinement in the proceeding sub-section, we assumed exact coplanarity between the KBOs, Planet Nine, and the remainder of the solar system, which rendered inclination dynamics trivial. Let us now examine the consequences of abandoning this simplification, and introduce a small, but nevertheless finite tilt of P9’s orbit with respect to the ecliptic. In particular, let us presume that the qualitative nature for e ∆a dynamics is not strongly affected by this development, and focus our attention on characterizing slow changes in i and Ω in the distant Kuiper belt.
To approximately describe the secular evolution of the orbital planes of long-period KBOs, it suffices to consider the quadrupole component of the disturbing function (8) under the assumption that inclinations remain small i.e., sin(i) 1. Suitably, neglect- ing all terms with amplitudes that scale as sin2(i), we are left with an integrable Hamiltonian, containing only a single secular harmonic – the difference between the longitudes of the KBO’s and P9’s ascending nodes (see Appendix B):
4 2
7 = – 3 GM cos(i) X miai + Ω˙ ,G M a ,1 – e2 (1 – cos(i))
1 G m9 a !2 . 2 –3/2 1 3 2 2 2
+ 3 sin(2i9) sin(2i) cos(∆Ω) . (10)
In direct parallel with equation (9), the three terms comprising the Hamiltonian have well-defined physical meanings. Specifically, the first and second terms describe the regression of the KBO’s longitude of ascending node (equation 5), and the slow rotation of the reference frame (such that Ω9 is always zero) respectively. Meanwhile,the third term governs the angular momentum exchange between P9 and the KBO. Moreover, because Hamiltonian (10) describes a system with only a single degree of freedom, we may examine the corresponding secular evolution simply by projecting level curves of onto the (i, ∆Ω) plane, as before.
Figure 11 shows the i ∆Ω analog of Figure 10, and exemplifies the dynamics of KBO orbital planes at different values of a. An important characteristic of P9’s secular forcing that is clearly evident in the panels of Figure 11 is the bending of the Laplace plane i.e., the gradual change of the equilibrium inclination as a function of semi-major axis, and the appearance of a prominent island of ∆Ω libration that surrounds this equi- librium. Outside of this island, relative longitude of ascending node circulates and KBOs experience considerable modulation of orbital inclination. Numerical simula- tions (section 5) reveal that due to non-linear coupling between eccentricity and incli- nation dynamics, orbits residing far away from the i ∆Ω equilibrium either develop low perihelia and get ejected from the system, or experience large-scale e i excursions that remove them from the observable domain of the Kuiper belt (section 4.4). As a re- sult, apsidally-confined orbits can only maintain long-term stability if they lie close to the i ∆Ω fixed point in Figure 11, which approximately coincides with Planet Nine’s orbital plane for large KBO semi-major axes. The emergence of the stable island of
nodal libration at a & 250 AU thus qualitatively explains the observed clustering of the orbital planes of long-period KBOs, as orbits residing on secular trajectories that encircle the i ∆Ω fixed point remain confined to this region of phase-space, causing the orbit poles to group together in physical space13.
While the transition from a randomized to clustered distribution of KBO orbital planes is keenly reminiscent of the onset of apsidal confinement discussed in the pre- vious sub-section, it is worth noting that the two processes are theoretically distinct. In particular, the appearance of stable apsidally anti-aligned trajectories in Figure 10 stems from the emergence of a new equilibrium point in phase space, and therefore occurs via a sharp transition. Moreover, the appearance of homoclinic curves in phase-
space at a & 350 AU renders libration of longitude of perihelion around ∆a 180 deg a true secular resonance. Conversely, the inclination-node fixed point shown in Figure
11 is a forced equilibrium, meaning that clustering of the orbital planes arises smoothly, as a function of a. Moreover, it is noteworthy that within the framework of the envi- sioned secular model, the existence of the apsidally confined population of KBOs re- quires Planet Nine to be eccentric but not necessarily inclined, whereas the clustering of the planes would ensue even if P9’s orbit were circular. Therefore, to the extent that the perturbative Hamiltonians (9) and (10) approximate real motion, these two dynam- ical processes independently inform the necessary orbital eccentricity and inclination of Planet Nine, and only when considered simultaneously lead to an orbital distribution of long-period KBOs that exhibit large-scale clustering in physical space (Batygin and Brown, 2016a).
13Strictly speaking, Hamiltonian (10) is a leading order approximation to the full Hamiltonian, expanded as a power-series in α = a/a9, and is therefore not guaranteed to provide a good representation of the dynamics in the α 1 limit. Here, however, we are saved by the sin(i) 1 Part 6
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