February 19, 2026
identifying trends between P9 orbital elements and the statistical properties of the syn- thetic KBOs. A key goal of this exercise is not only to delineate the dependence of fa, y, µ and σ (summarized in Figure 16) on a9, e9, i9, and m9, but also to identify P9 parameters that yield a distant population of small bodies that match the observations most closely.
We consider the total apsidal confinement fraction fa as the first constraint. As a starting point, however, we draw on above results to reduce the number of independent variables by one. Recall from section 5.1 that for a given P9 mass, a combined choice of a9 and e9 yields a value of a¯c, which is delineated in Figure 15. Employing this relationship, we can restrict our analysis to systems characterized by 200 AU . a¯c .
300 AU, and thus (approximately) eliminate a9 as a free parameter in favor of e9. The
degree of apsidal confinement – computed using the two-bin approach discussed above – is shown as function of P9 eccentricity, and color-coded by P9 inclination in the top left panels of Figures 20, 21, and 22, for the m9 = 5, 10, and 20 M⊕ simulation suites, respectively. For each value of the eccentricity, J2NP9 and JSUNP9 results are portrayed side by side, with the former plotted on the left.
Three trends immediately emerge upon examination of these plots. First, it is evi- dent that (perhaps counter-intuitively) simulations with m9 = 5 M⊕ generally produce better confinement of the longitude of perihelion than their higher-mass counterparts. In fact, values of fa close to unity are readily achieved for e9 & 0.15 in m9 = 5 M⊕ experiments, while simulations with a more massive Planet Nine exhibit a more non- uniform dependence of fa on P9 eccentricity, resulting in only select runs (e9 & 0.35 and e9 & 0.55 for m9 = 10 and 20 M⊕ respectively) that attain satisfactory results. A second trend, already pointed out in Brown and Batygin (2016), is that the degree of
apsidal confinement degrades with increasing P9 inclination. Particularly, on all three Figures, the color-gradient of the illustrated simulation points suggests that P9 orbital solutions with i9 & 30 deg are simply not viable. Finally, although not universally true, provided the same P9 parameters, J2NP9 simulations tend to exhibit marginally tighter apsidal confinement than those performed within the fully resolved JSUNP9 cal- culations. Recalling from section 3 that among dynamically stable long-period KBOs
( fa)data = 8/9 89% (shown on the top left panels with a horizontal dashed line), here we adopt a slightly less stringent value of fa ≥ 80% as a criterion for success.
In the vast majority of our fa ≥ 80% simulations, apsidal confinement ensues
sufficiently close to ∆a 180 deg that the compatibility of the mean longitude of
perihelion in numerical experiments with theory (section 4) does not entail a practically useful constraint. On the contrary, the forced equilibrium angle µ – which corresponds to the mean value of ∆Ω – varies significantly with e9. As already discussed above, simulations with µ 0 0 fail to generate an inclination dispersion of the distant belt that is in good agreement with the observations, and as a quantitative cut here we have chosen to disregard simulations with µ ≥ 10 deg. We emphasize that this cut is
not motivated by a specific observational constraint, and instead stems from empirical
examination of simulation results.
The distribution of µ as a function of e9 is shown on the right top panels of Fig- ures 20-22. Although the specifics of each figure differ considerably, a common thread emerges, wherein µ remains close to zero for nearly circular P9 orbits, but tends to- wards a strongly negative value at high Planet Nine eccentricities (e9 & 0.65 for
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Figure 20: A summary of the m9 = 5 M⊕ simulation ensemble. The top left plot shows the perihelion con- finement fraction, fa (equation 16), as a function of P9 eccentricity (recall from Figure 15 that e9 and a9 are linked together by the requirement that the critical semi-major axis lies in the range 200 AU . a¯c . 300 AU). While the current census of stable long-period KBOs is characterized by ( fa)data = 8/9, as a rudimentary cut on the results, we disregard any simulation that generates a distant Kuiper belt with fa < 80%. The top right panel depicts the forced equilibrium angle, µ (a measure of the vertical offset of the center of the yellow circle in the right panels of Figure 17 away from the positive p-axis) of the simulated distant Kuiper belt, as a function of e9. Reducing the aggregate of successful simulations further, we ignore any P9 parameter combi- nation that produces a Kuiper belt with µ ≥ 10 deg. The bottom panel shows a bubble chart where the y-axis corresponds to the magnitude of the forced equilibrium, y, and the size of the individual bubbles informs the ratio of the forced equilibrium amplitude to the rms dispersion (meaning that larger bubbles correspond to tighter clustering of the orbital poles). Only simulations that satisfy the aforementioned fa and µ criteria are shown with colored circles (those that do not are shown with transparent bubbles), demonstrating that there exists only a limited eccentricity range that produces distant Kuiper belt architecture that is compatible with observations. The parameters of the two best-fit simulations for this choice of m9 are labeled, although given
observational uncertainties on the values of fa, y, and σ, it is clear that these parameter combinations are not unique. Note further that for each value of e9, we plot the results from JSUNP9 and J2NP9 simulations side-to-side, and that semi-averaged simulations tend to systematically exhibit marginally better confinement in both fa, and y.1.0
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Figure 21: Same as Figure 20, but for m9 = 10 M⊕.
m9 = 5, 10 M⊕ and e9 & 0.75 for m9 = 20 M⊕). This limitation is particularly con- straining for m9 = 20 M⊕ simulations because in this case the apsidal confinement criterion restricts P9 eccentricity from below at e9 & 0.55, leaving only a limited parameter range where such a massive planet can even approximately reproduce the real data. Moreover, even at modest values of e9 and m9 ≤ 10 M⊕, simulations with i & 30 deg tend to produce µ significantly in excess of 10 deg. We note however, that this behavior clashes with the apsidal confinement criterion, since high-inclination P9
simulations also generate particle disks with fa considerably below 0.8, and are there- fore incompatible with observations anyway.
The large lower panels on Figures 20-22 encode the characteristics of the KBO in- clination degree of freedom from the simulations, as a function of P9 eccentricity. The vertical coordinate of each simulation result in this plot denotes the value of the forced equilibrium, y. Meanwhile, the size of the individual bubbles represents the ratio of this magnitude to the rms dispersion of observable particles, y/σ. Consequently, smaller bubbles correspond to synthetic belts where the orbital poles of the KBOs are more1.0
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Figure 22: Same as Figure 20, but for m9 = 20 M⊕. Note that unlike the m9 = 5 M⊕ and m9 = 10 M⊕ cases, in this simulation suite, none of the successful simulations exhibit clustering of the orbital planes that is as good as the data, suggesting that the mass of Planet Nine is considerably smaller than 20 M⊕.
randomly distributed, while larger bubbles correspond to simulations where clustering of orbital planes (and by extension, clustering of the longitude of the ascending node) is strong. Simulations that satisfy the criteria fa ≥ 0.8 and µ ≤ 10 deg are shown with colored bubbles (where as before, the color-scale informs P9’s inclination), while those that do not conform to aforementioned benchmarks are depicted as nearly-transparent
circles. As with the top plots in these Figures, J2NP9 and JSUNP9 results are illustrated next to one another, with the fully resolved calculations shown on the right.
The magnitude of the mean ( p , q )data vector corresponding to the long-term sta- ble observed KBOs is shown with a dashed line, and the measure of rms dispersion inherent to the observational data (ydata/σdata) is shown by the yellow band that encom- passes this line. Accordingly, a numerical experiment that constitutes an ideal match to the data in the (p, g) plane would be represented with a bubble that fits perfectly inside this yellow band. Examination of simulation data depicted in this sequence of plots reveals some of the same trends that we already highlighted for fa. Across theboard, J2NP9 simulations tend to yield slightly higher values of y than their JSUNP9 counterparts. More importantly, there exists a significant, and non-trivial dependence of the degree of orbital clustering on Planet Nine’s mass and eccentricity.
Among acceptable m9 = 5 M⊕ calculations (Figure 20), y exhibits a nearly mono- tonic inverse dependence on e9, such that simulations in the e9 0.15 0.25, a9
400 500 AU and i9 15 25 deg range result in the closest agreement between nu- merical experiments and data. The characteristic range of best-fit P9 parameters shifts to higher eccentricities and somewhat lower inclinations for m9 = 10 M⊕ runs, with e9 0.35 0.45, a9 600 800 AU and i9 15 20 deg appearing most favorable. We note, however, that in this set of calculations, synthetic disks that exhibit the tight-
est clustering of the orbital planes are only as clustered as the real data. This suggests that at higher masses, the degree of angular momentum vector clustering displayed by the real objects simply cannot be reproduced, implying that ten Earth masses should be viewed as a working upper limit on Planet Nine’s mass. Accordingly, the m9 = 20 M⊕ simulation suite indeed exhibits rather poor agreement with the observations. Even
when the degree of apsidal confinement is satisfactory – which already constrains the eccentricity and semi-major axis to the e9 0.55 0.65 and a9 900 1200 AU range – inclination dynamics of observed long-period KBOs are not well reproduced by the numerical experiments since the µ 0 requirement restricts P9 inclination to i9 10 deg, preventing adequate excitation of y.
For each P9 mass, we emphasize two simulations in Figures 20-22 that generate synthetic KBOs that are in closest agreement with the real distant Kuiper belt. These orbital parameters are highlighted on a9 e9 diagrams in Figure 15 with circles, and are marked by their corresponding values of i9. We further note that because the (Γ, ç) and (Z, z) phase space structure of each pair of simulations (corresponding to a given value of m9) is rather similar, we only depict the larger semi-major axis calculations in Figures 17-19 to avoid unnecessary redundancy. Overall, the analysis carried out above points to an orbital solution where a m9 5 10 M⊕ Planet Nine resides on a
mildly eccentric (e9 0.1 0.5) and moderately inclined (i9 15 25 deg) trajectory.
This set of properties results in good agreement between theory, simulation, and data, in contrast to solutions with higher values of m9. Let us now examine one additional aspect of P9-induced evolution in these simulations – namely, the generation of highly inclined TNOs.
High-Inclination Dynamics
As a first step in examining the high-inclination component of the distant Kuiper belt generated within the simulations, we inspect the orbital footprints of the long-term
stable particles on the i – ∆Ω plane. This projection is shown on the right-hand-side of Figure 23 for both of the aforementioned m9 = 5 M⊕, a9 = 500 AU, e = 0.25, i = 20 deg and m9 = 5 M⊕, a9 = 400 AU, e = 0.15, i = 20 deg calculations, with the more eccentric simulation plotted on the top panel19. Unlike Figures 17-19, here we
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