February 18, 2026
m9 = 5M a9 = 500 AU e9 = 0.25 i9 = 20 deg
0.2
0
-0.2
-0.4
-1.0 -0.5 p
0 0.5 1.0
2 Г cos ∆$
-0.4 -0.2
0 0.2 0.4
2 Z cos ∆⌦
1.0
0.5
0
-0.5
-1.0
0.4
0
-0.2
-0.4
-1.0 -0.5 p
0 0.5 1.0
2 Г cos ∆$
-0.4 -0.2
0 0.2 0.4
2 Z cos ∆⌦
Figure 17: Comparison between simulated and observed dynamical architecture of the distant Kuiper belt. Results are depicted in phase-space, defined by cartesian analogs of reduced Poincare´ action-angle coordi- nates (equation 15; Figure 16). The panels on the left-hand-side show the degree of freedom related to the
eccentricity, such t,hat a circular orbit corresponds to the origin of the plot, and an e = 1 orbit resides at a
radial distance of 2. Meanwhile, the polar angle denotes the KBO longitude of perihelion, relative to that
of Planet Nine. The panels show all stable t ≥ 2 Gyr simulation data with a ≥ ac, employing the same color scheme as that adopted in Figure 14. The strong concentration of observable particles (blue points) at ∆a 180 deg is clearly evident on the left panels, and provides a good match to the observational data. The corresponding first, second, and third quartiles of the ∆a distribution of simulated particles are marked
on the outer black circle that encloses the Figure. For comparison, the first three quartiles corresponding to the observational data are shown on the inner purple circle (akin to that depicted in Figure 6). The right- hand-side panels show the degree of freedom related to KBO inclination. Similarly to the panels on the LHS, i = 0 orbits lie at the origin and the polar angle denotes the longitude ofascending node relative to that of P9. The quantitative attributes of the dynamics depicted in these panels are well summarized by the location of the forced equilibrium action (shown with a yellow dot) and the rms spread of the observable points about this equilibrium (shown with a solid yellow circle). For comparison, the correspondent char- acteristics of the observational dataset are shown with a target-like dot and a dashed circle. The amplitude of the forced equilibrium and the rms spread inherent to the simulation results as well as the observations are further shown with lines on top and on the right of each plot. In addition to numerical and observational data, the LHS and RHS panels also show the contours of Hamiltonians (9) and (10) respectively, evaluated at α = 1. The conformation of simulation results to analytical expectations inform the agreement between numerical experiments and secular perturbation theory.
9 = 10M a9 = 800 AU e9 = 0.45 i9 = 15 deg
apsidal confinement fraction: 87%
1.0
0.5
0
-0.5
-1.0
0.4
0
-0.2
-0.4
-1.0 -0.5 p
0 0.5 1.0
2 Г cos ∆$
-0.4 -0.2
0 0.2 0.4
2 Z cos ∆⌦
1.0
0.5
0
-0.5
-1.0
0.4
0
-0.2
-0.4
-1.0 -0.5 p
0 0.5 1.0
2 Г cos ∆$
-0.4 -0.2
0 0.2 0.4
2 Z cos ∆⌦
Figure 18: Same as Figure 17, but for P9 parameters m9 = 10 M⊕, a9 = 800 AU, e9 = 0.45, and i9 = 15 deg.
projected into cartesian analogs of the Poincare´ action-angle variables are shown on the right-hand-side panels of Figures 17-19. As with the panels depicting the dynamics in (Γ, ç), we overlay the observational data points as well as analytical contours of Hamiltonian (10) evaluated at α = 1 and e = 0.85 on the plots.
An advantage of portraying the simulation results, theory, and observations in the cartesian canonical coordinates (p, g) is that the interpretation of the underlying dynam- ics becomes straightforward. In these variables, the analytic contours of Hamiltonian
(10) simply appear as a succession of circles, centered on a point that resides on the
p axis, displaced to the right from the origin. Crucially, the distance between this point and the origin is the analytical approximation to the forced equilibrium, y = ,2Ze – a
quantity that characterizes the mean tilt of the orbital plane in P9-controlled domain of
the solar system.
If the dynamics executed by stable KBOs in the numerical experiment conforms
m9 = 20M a9 = 1100 AU e9 = 0.65 i9 = 10
1.0
0.5
0
-0.5
-1.0
0.4
0
-0.2
-0.4
-1.0 -0.5 p
0 0.5 1.0
2 Г cos ∆$
-0.4 -0.2
0 0.2 0.4
2 Z cos ∆⌦
sidal confinement fraction: 72%
1.0
0.5
0
-0.5
-1.0
0.4
0.2
0
-0.2
-0.4
-1.0 -0.5 p
0 0.5 1.0
2 Г cos ∆$
-0.4 -0.2
0 0.2 0.4
2 Z cos ∆⌦
Figure 19: Same as Figure 17, but for P9 parameters m9 = 20 M⊕, a9 = 1100 AU, e9 = 0.65, and i9 = 10 deg.
to the expectations from analytic theory, then orbital footprints of simulated particles should encircle this equilibrium point as well. With this notion in mind, we compute the location of the equilibrium point in each simulation as the average of the vector
((p), (g)) = 1 XW (p , g )
, (17)
j j
W j 30AU≤q≤100AU, i<40 deg, a>a¯c
where the sum runs over all observable (blue) points depicted in Figures 17-19. Quan- titatively, the magnitude of the forced equilibrium is simply the norm of the two quanti- ties y = p 2 + g 2. In Figures 17-19, the coordinates of the average vector ( p , g ) are marked with a yellow dot, and the magnitude of y is shown on top of each right- hand-side panel with a black line.
We caution that the norm of the average vector ((p), (g)) alone does not fully characterize the clustering of orbital planes, because it does not specify the direction (i.e., longitude) of the mean angular momentum vector of the distant KBOs. This comple- mentary quantity is defined by the forced equilibrium angle µ = arctan( g / p ). The analytic picture described in section 4 suggests that within the framework of the P9 hypothesis, µ should generally be close to zero. While this expectation is well-satisfied in simulations presented in Figures 17-19, numerical experiments where µ is far away from zero are plentiful in our aggregate of simulations. Examining such large-µ sim- ulations on an individual basis, we have determined that they routinely provide a poor match to the observational data. Typically, this is because they represent calculations where y is so small that µ is essentially ill-defined, or because the low-inclination com- ponent of the distant belt is so depleted that the statistical measures y and µ become unintelligible. As a result, simulations with µ 0 0 are disfavored in our analysis. For this reason, in Figure 16, we simply denote y as being the horizontal displacement of the average ( p , g ) vector along the p axis.
A second quantity that characterizes the extent to which simulated objects follow theoretically expected dynamics is the rms dispersion of the (p, g) vector around the equilibrium point ((p), (g)):
σ = 1 XW
,.pj
– (p) 2 + .gj
– (g) 2
. (18)
W j 30AU≤q≤100AU, i<40 deg, a>a¯c
Graphically, the rms dispersion of numerical data is shown in Figures 17-19 with a yellow circle centered on the ( p , g ) equilibrium, and its magnitude is depicted on the right of each panel with a black line. It is worth noting that a less precise way to characterize the orbital state of the distant Kuiper belt that has been employed in the literature draws attention to the clustering of the longitudes of ascending node (shown in Figure 8). In terms of the above quantities, this clustering can be understood as a direct consequence of the equilibrium inclination forced by Planet Nine, and only arises when the rms dispersion of particle trajectories, σ, is comparable to, or smaller than y itself.
To complement the simulation quantities y and σ, on each right-hand-side panel of Figures 17-19, we illustrate analogous quantities corresponding to the observed long- term stable objects as follows. The forced equilibrium point is marked with a large yellow target-shaped point that resides on the p axis, and the extent of rms disper- sion is shown with a dashed yellow circle. The two quantities are further depicted with purple lines on the top and the right of each panel, respectively. As is evident from examination of Figures 17-19, among the depicted numerical experiments, the m9 = 5 M⊕ and m9 = 10 M⊕ simulations provide a notably better match to the current
observational census of distant Kuiper belt objects than the m9 = 20 M⊕ simulation. As
we will discuss further below, the incompatibility of m9 = 20 M⊕ simulations with the observations is a general result of our suite of numerical experiments.
Having outlined a sequence of criteria by which a given simulation can be quan- tified, let us now evaluate our entire ensemble of calculations with an eye towards
