February 17, 2026
Remarkably, our semi-averaged simulation suite reveals that ac exhibits only a weak dependence on Planet Nine’s inclination. The bottom right panel of Figure 15 depicts the fractional variation of the critical semi-major axis, ac/ ac as a function of i9 across the entire simulation suite, where ac represents a simple average of ac over i9 computed at constant values of a9 and e9. Magenta, brown, and dark yellow points
correspond to m9 = 5, 10, and 20 M⊕ simulations respectively. Put simply, the results illustrated in this panel show that at a given combination of m9, a9, and e9, changes in
ac due to variations in i9 are only on the order of 10%. While the dependence of ac upon i9 is so shallow that it can in principle be ignored, here we correct for it by fitting a linear regression to the numerical data shown in Figure 15, and computing a corresponding weighted average of the critical semi-major axis a¯c.
Figure 15 shows a¯c on a (a9, e9) simulation grid for our three choices of the mass of Planet Nine (m9), with the solid lines on each plot corresponding to a¯c = 200, 250, and 300 AU as labeled. The simulations show a clear relationship, which bounds the P9 semi-major axis from below at a9 & 300 AU, and requires progressively higher eccentricities to generate an observationally acceptable value of a¯c at larger semi-major axis. Moreover, in the e9 . 0.5 range, the solid curves shown on each panel are
essentially identical to one-another, meaning that for q9 & 300 AU, the critical semi-
major axis is approximately m9-independent. This finding is quantitatively consistent
with analytical estimates, wherein a¯c is interpreted as the minimum semi-major axis at which Hamiltonian (9) possess an elliptic equilibrium at ∆a = 180 deg (Batygin and Morbidelli, 2017).
The results summarized in Figure 15 elucidate the acceptable range of a9 and e9 combinations for a given value of m9 within the Planet Nine hypothesis. Simultane- ously, however, the derived loci of a¯c point to a fundamental degeneracy between these two orbital elements i.e., the same value of critical semi-major axis can be reproduced by a broad array of (a9, e9) pairs, meaning that additional information is needed to further constrain Planet Nine’s orbit. The following section considers such additional constraints from a targeted set of fully resolved simulations.
With the P9 parameter range corresponding to 200 AU . a¯c . 300 AU identified for the semi-averaged simulations, we have carried out a second suite of calculations that fully resolve the orbital evolution of Jupiter, Saturn, and Uranus, in addition to
Planet Nine and Neptune (we will refer to calculations employing this physical setup as the JSUNP9 suite of simulations). Given the higher computational cost associated with these numerical experiments, we limited our choice of P9 orbital elements to those generating values of a¯c that roughly fall into the desired range, as informed by J2NP9 results. For each value of the mass m9, this domain is shown with a yellow outline in Figure 15 (corresponding to 660 additionalsimulations). With the exception of a smaller timestep necessary to properly model the orbit of Jupiter (which we set to
∆ t = 1 year, modifying the absorbing radius to = 4.5 AU and setting J2 = 0), we adopted all other parameters of the JSUNP9 calculations to be the same as those of the corresponding J2NP9 runs.
JSUNP9 analogues of the J2NP9 calculations that are depicted on the RHS panels of Figure 14 are shown on the LHS of the same Figure. Note that although JSUNP9 and
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Figure 15: Critical KBO semi-major axis, ac, corresponding to a transition from perihelion-randomized to apsidally confined orbital distribution, as a function of Planet Nine parameters. The lower right panel demonstrates the weak dependence of ac on Planet Nine’s inclination. Because this dependence is shallow, we model it out with a simple linear regression, and define a weighted average of the critical semi-major axis a¯c. The remaining three panels portray this quantity on the (a9, e9) plane, and explicitly mark the contours corresponding to a¯c = 200 AU, 250 AU, and 300 AU. The yellow outlines on each graph delineate the domain over which full-fledged JSUNP9 simulations were carried out. The two circles on each panel mark the (a9, e9) combination for the specific value of m9 that yield optimal fits to the data, with the embedded numbers reporting the value of i9 in degrees.
J2NP9 results are qualitatively identical, panels corresponding to JSUNP9 simulations appear more “fuzzy” than their J2NP9 counterparts. This difference dominantly stems from the fact that (to maintain consistency with the observations) we have chosen to plot the dynamical evolution in osculating heliocentric, rather than barycentric orbital elements. Because our semi-averaged calculations do not resolve the orbital motion of Jupiter, Saturn, and Uranus around the sun, much of the short-periodic jitter that is present in the JSUNP9 results is filtered out in the J2NP9 simulation suite. In other words, the underlying dynamical evolution detailed in the two sets of simulations is even more similar than it appears.
As the next step in comparing the fully resolved and semi-averaged simulations, we re-computed the critical semi-major axis in the JSUNP9 runs, and found that the resulting contours of a¯c on the (a9, ee) plane are in close agreement with those obtained within the J2NP9 set of numerical experiments. In fact, a hint of this agreement was already evident in Figure 14, where JSUNP9 and J2NP9 experiments employing the same parameters are illustrated side-by-side. Consequently, we conclude that the black curves delineated on Figure 15 can be considered to be an accurate representation of the J2NP9 and JSUNP9 results alike.
Despite the considerable degeneracy that exists in P9 orbital elements that can gen- erate a value of a¯c close to 250 AU, the actual characteristics of the distant belt produced in the simulations exhibit a sensitive dependence on Planet Nine’s orbital properties. Therefore, a key objective of our analysis is to quantify the orbital architecture of the simulated a > a¯c Kuiper belt in each numerical experiment, and compare its statis- tical properties with the observations. For consistency, we will follow the procedure outlined in our discussion of the data (section 3), and characterize the attributes of the synthetic Kuiper belt in terms of apsidal confinement, mean inclination, and the rms dispersion of the orbital poles.
To ensure the orthonormality of our coordinate system, it is useful to introduce scaled Poincare´ action-angle variables (Murray and Dermott, 1999):
Γ = 1 – ,1 – e2 ç = –∆a
Z = ,1 – e2 1 – cos(i) z = –∆Ω, (15)
and carry out this analysis in phase-space, rather than in terms of Keplerian orbital elements. Note that in the above expressions, the conjugate angles are measured with respect to the corresponding parameters of Planet Nine’s orbit (as mentioned already in section 4, this choice leads to a trivial modification of the Hamiltonian, wherein additional terms are introduced to account for the slow apsidal precession and nodal regression of Planet Nine’s orbit; see Appendix B). Although the actions Γ and Z have well-defined physical meanings and naturally connect to the definition of the angular momentum deficit (Laskar, 1997), one limitation of these variables is that at i = 0,
the angle ∆Ω becomes ill-defined17. A convenient way to sideste,p this coordinate
singularity is to transform to the cartesian canonical variables h = 2Γ cos(∆a), k =
17The same issue technically arises for ∆a in relation to circular orbits, but this poses no practical limita- tions for the representation of the distant Kuiper belt.
h, k) ⌘ (p, g)
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Figure 16: Phase-space coordinates and statistical measures employed in our analysis of the distant (a > ac) Kuiper belt. The panel on the left-hand-side depicts the relationship between canonical cartesian (h, k) variables and the orbital eccentricity. Analogous relationship between (p, g) variables and inclination is shown on the right panel for e = 0.85 and e = 0.95 orbits (see equation 15 for definition). Domains of phase-space occupied by observable low-q, low-i KBOs that share the same orbital plane and are apsidally anti-aligned with respect to Planet Nine are marked on both panels by blue outlines. Additionally, the geometrical meanings of the statistical characteristics of the distant Kuiper belt fa, y, µ, σ are labeled on the panels, and their corresponding success criteria are summarized in the table below the plots.,2Γ sin(∆a) and p = ,2Z cos(∆Ω), g = ,2Z sin(∆Ω), such that radial distances from the origin on (h, k) and (p, g) plots become measures of eccentricity and inclination, while the polar angles translate to ∆a and ∆Ω, respectively. The relationship between these variables and standard orbital elements is illustrated graphically in Figure 16. The dynamical footprints of long-term stable particles in the a > a¯c domain for the same three combinations of P9 parameters as in Figure 14 are shown in Figures 17-19, where JSUNP9 and J2NP9 simulations are plotted on the top and bottom pairs of panels respectively. For consistency, here we continue to employ the same color scheme as that used in Figure 14.
Apsidal Confinement. The panels depicted on the left-hand-side of Figures 17-19 il- lustrate apsidal confinement of synthetic long-period KBOs. The vast majority of simulated objects that satisfy our detectability criteria (blue points) cluster around
∆a 180 deg. Note that the degree of confinement is never perfect – even at a > a¯c, contamination from particles that circulate in ∆a always exists. As with the obser- vational data itself, we characterize the degree of a–clustering by the total apsidalconfinement fraction, fa, (distinct from the previously mentioned running apsidal con- finement fraction, ζ) by splitting the observable output of the simulation into two bins, and computing the ratio (see Figure 16):
f = W .30AU ≤ q ≤ 100AU, i < 40 deg, ∆a e [90, 270) deg . (16)
While the apsidal confinement fraction of all but one simulation shown in Figures 14 and 17-19 is close to 90%, in general fa exhibits a rather sensitive dependence on P9 parameters, as will be discussed in greater detail below.
To complement fa, in Figures 17-19, we also mark the first, second, and third quar- tiles of the simulated data on an exterior outer circle that encompasses the graph, and report the corresponding quartiles of the long-term stable real objects (also shown in Figure 6) on the inner circle. Importantly, the characteristic spread of of the primary
∆a ~ 180 deg cluster exhibits considerable mass-dependence. That is, the m9 = 10 and 20 M⊕ anti-aligned clusters (Figures 18 and 19) have a noticeably smaller disper- sion than the real data, implying that the m9 = 5 M⊕ simulation appears to be favored by the observations18. Note further that at higher masses, a prominent collection of ob- servable orbital footprints with ∆a . 90 deg also emerges, implying a comparatively diminished fa.
More generally, we acknowledge that in addition to fa – which is a relatively crude measure of the degree of apsidal confinement – it is also possible to consider the an- gular width of the ∆a cluster as a meaningful constraint on the simulation results. Paired with more precise modeling of the observational biases, such an analysis may yield added insight into the most optimal fits of Planet Nine’s orbital parameters. For completeness, we overlay the simulation results on Figures 17-19 with observed data points, as well as the analytical contours of the Hamiltonian (9) (evaluated at α = a/a9 of unity, shown with black lines), which further elucidate the qualitative nature of sec- ular trajectories executed by the particles in the simulations.
The left top panels of Figures 17-19 provide a natural point of comparison of ap- sidal confinement entailed by the observational data, analytical theory, and fully re- solved numerical simulations. In section 4, we interpreted the observed grouping in longitude of perihelion as a consequence of secular libration of particle trajectories around ∆a = 180 deg, along contours of Hamiltonian (9). The prominence of this dynamical behavior is indeed clearly exemplified by the evolutionary tracks of the sim- ulated KBOs. The majority of the stable observational data points, in turn, appear to be fully consistent with simulated objects that are randomly drawn from the vicinity of the ∆a = 180 deg island of libration.
Clustering of the Orbital Planes. In terms of canonical variables (15), the dynamical state of the orbital planes of KBOs is described by the (Z, z) degree of freedom, where physically, the action Z represents a measure of an orbit’s angular momentum deficit in the direction normal to the ecliptic. The dynamical footprints of stable particles,
18The tendency towards a diminished range of ∆a among the apsidally confined subset of points is also evident in Figure 14.
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