The Planet Nine Hypothesis (Part 12)

 February 21, 2026

employ a color-scale that is solely a measure of the perihelion distance, with blue points

¹⁹Equivalent i ∆Ω plots for the favorable m₉ = 10 M_⊕ solutions look quite similar to the top right panel of Figure 23, so we omit them to avoid redundancy.

^m9 ^{= 5M a}9 ^{= 500 AU e}9 ^{= 0.25 i}9 ^{= 20 deg} ₁₈₀ ^m9 ^{= 5M a}9 ^{= 500 AU e}9 ^{= 0.25 i}9 ^{= 20 deg}

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Figure 23: High-inclination dynamics induced by a m₉ = 5 M_⊕ Planet Nine. Akin to plots illustrated in Figure 11, the right panels show the orbital footprints of simulated KBOs on the (i, ∆Ω) plane for the two optimal P9 orbital solutions emphasized in Figure 20. Note that the a₉ = 500 AU, e₉ = 0.25 solution produces high-inclination objects more readily than its lower eccentricity counterpart. The panel on the left

shows a density histogram of particles footprints, projected into (θ, Θ) phase-space (see equation 11 and Figure 12). Unlike the color-scheme employed in Figures 14 and 17, here we adopt a blue-gray gradient to exclusively represent perihelion distance.

 corresponding to q = 30 AU, and gray points signifying q ≥ 100 AU. The full census of observed a ≥ 250 AU TNOs is also shown on the panels, with high-inclination (i > 40 deg) Centaurs plotted as large orange points, and the KBO 2015 BP₅₁₉ shown

with a pink dot, as in Figure 9.

Because we are showing only long-term stable particles in Figure 23, simulated orbits that achieve q < 30 AU are essentially absent from the plot. This renders the comparison between the numerical experiments and the present-day orbits of high- inclination Centaurs inexact, leaving 2015 BP₅₁₉ as the only high-inclination TNO which conforms strictly to the depicted numerical results (recall that severe observa- tional biases exist against detecting high-inclination, high-perihelion KBOs). Nev- ertheless, it is very likely that the observed Centaurs originate as strongly inclined q > 30 AU Kuiper belt objects that are scattered inwards by Neptune, meaning that the numerical data depicted in Figure 23 represents the simulated source population of high-i KBOs from which the observed Centaurs are derived.

As can be deduced from examination of Figure 23, the simulation with e = 0.25 produces retrograde TNOs in much greater proportions than its lower eccentricity counterpart. Qualitatively, this can be attributed to the fact that the harmonic term responsible for orbit-flipping behavior is octupolar in nature (section 4.1) and thus ne- cessitates a significant eccentricity to drive high-inclination dynamics. In light of the strong observational biases that act against the detection of high-inclination TNOs, the 

 true occurrence ratio of distant high-i objects to low-i objects is not known. Unfor- tunately, this means that at present, we cannot observationally favor either of the two shown simulations, since they both produce strongly inclined TNOs in some propor- tion. To complement the discussion presented in section 4.4, the panel on the left-hand- side of Figure 23 presents the simulation results as a density histogram in phase-space, where the transparency of the cells corresponds to a logarithmic measure of the num- ber of orbital footprints contained within each cell and the color corresponds to typical values q of the constituent points. The regions of phase-space occupied by real TNOs is also heavily populated by simulated particles, suggesting that the agreement between the observed and synthetic populations of distant TNOs is satisfactory.

Although highly inclined long-period Centaurs embody an unexpected consequence of P9-induced dynamics, TNOs residing on nearly orthogonal and retrograde orbits are not confined to the distant edges of the Kuiper belt, and are observed on shorter- period orbits as well (Figure 9). As with the large semi-major axis Centaurs them- selves, these objects are envisioned to have acquired large inclinations through in- teractions with Planet Nine at large semi-major axes, and to have subsequently been scattered inwards by Neptune. N-body simulations that elucidate this process for a

m₉ = 10 M_⊕, a₉ = 600 AU, e₉ = 0.5 Planet Nine were reported in Batygin and Brown (2016b). Accordingly, let us now examine the generation of high-inclination TNOs with a ≤ 100 AU by a lower mass, lower eccentricity Planet Nine derived from the preceding analysis.

In order to populate the inner trans-Neptunian region with a sufficient number of synthetic KBOs, we re-ran the m₉ = 5 M_⊕, a₉ = 500 AU, e₉ = 0.25, i₉ = 20 deg numer- ical experiment, increasing the particle count to N = 20, 000. The simulated orbital dis- tribution of particles that attain a ≤ 100 AU throughout any point in the simulations is shown in Figure 24 as a density histogram, where once again transparency represents a

logarithmic measure of the density of points. Recalling that all initial conditions of our numerical experiments (including this one) are drawn from the a (100 AU, 800 AU) range, every particle that comprises the density histogram shown in Figure 24 has been emplaced into the a ≤ 100 AU region from a more distant orbital domain. The current

a ≤ 100 AU observational sample of TNOs is over-plotted on the Figure, and objects

with i ≥ 60 deg are emphasized.

Upon examination of Figure 24, two qualitative features are immediately evident.

First, P9-induced dynamics significantly boosts the inclination dispersion of compar- atively short-period TNOs, providing a natural explanation for the existence of nearly orthogonal orbits, such as those of Drac (Gladman et al., 2009) and Niku (Chen et al., 2016). Second, a distinct population of strongly retrograde particles with perihelion

distances in the 6 AU . q . 12 AU range is also produced. These synthetic objects seamlessly explain the almost planar, but backward orbits of the TNOs 2016 NM₅₆ and 2017 KZ₃₁. Comprehensively, the union of the two panels shown on Figure 24 demon- strates that the observed data is in excellent agreement with synthetic population of TNOs produced within the simulation.

More generally, the inability of standard Nice model simulations to produce KBOs with inclinations in excess of i & 40 deg has long been recognized as a shortcoming of the instability-driven scenario of outer solar system evolution (Levison et al., 2008; Nesvorny´, 2015b). Although the conflict between the observed aggregate of KBOs and

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Figure 24: Generation of a ≤ 100 AU high-inclination Centaurs by inward scattering of P9-perturbed distant objects. The current census of TNOs is shown with black points, and the i ≥ 60 deg objects that cannot be explained within the standard model of solar system evolution are emphasized. The underlying density

histogram delineates the regions of parameter space onto which long-period, P9-influenced particles are emplaced when they are scattered inwards by Neptune. The specific domains of a q i space most heavily populated by Planet Nine signal a strong consistency with the observational dataset.

the Nice model can in principle be resolved either by P9-facilitated pollution of the Kuiper belt with high-inclination objects that originate further out, or by inward deliv- ery of Oort cloud objects, the two hypotheses make distinct predictions regarding the orbital structure of the resulting high-inclination Kuiper belt. That is, while dynamical emplacement driven by Planet Nine generates a rather specific orbital architecture (as detailed in Figures 23 and 24), high-inclinations objects sourced from the Oort cloud should stem from a more uniform distribution of orbital elements. Continued map- ping of the high-i component of the Kuiper belt thus offers a direct avenue towards observational differentiation between these two hypotheses.

Solar Obliquity

A final piece of information that is readily informed by our aggregate of JSUNP9 simulations concerns the interactions between Planet Nine and the mean plane of the known eight-planet solar system (often referred to as the “invariable plane”; Souami & Souchay 2012). Shortly after the initial formulation of the Planet Nine hypothesis, it was pointed out by Bailey et al. (2016); Gomes et al. (2016); Lai (2016) that the sec- ular gravitational torque exerted by P9 upon the canonical giant planets would slowly perturb the orbital plane of the planets away from its initial state, thereby exciting a spin-orbit misalignment between the total angular momentum vector of the canonical giant planets and the spin-axis of the sun. Moreover, these authors found that given plausible P9 parameters (e.g., m₉ = 15 M_⊕, a₉ = 500 AU, e₉ = 0.5, i = 20 deg; Bai- ley et al. 2016), the entire 6-degree obliquity of the sun could be accounted for by P9 perturbations alone. 

 This effect is self-consistently captured in our simulations, and it is worthwhile to examine if the revised orbital properties of Planet Nine derived above remain consistent with a scenario where Planet Nine plays a dominant role in the excitation of solar obliquity. For the two best-fit P9 parameters combinations shown in Figures 17 and 18, the answer is a resounding ’no.’ Specifically, for m₉ = 5 M_⊕, a₉ = 500 AU, e₉ = 0.25,

i₉ = 20 deg, P9’s secular torque only leads to a ψ = 1.1 deg change in the inclination

of the solar system’s invariable plane over 4 Gyr. For a m₉ = 10 M_⊕ perturber on a a₉ = 800 AU, e₉ = 0.45, i₉ = 15 deg orbit, the induced solar obliquity is similarly small, evaluating to only ψ = 0.7 deg. This implies that some other process, unrelated

to the existence of Planet Nine, must be responsible the present-day obliquity of the sun.

To elaborate on this result further, it is useful to contextualize the sun’s spin-orbit misalignment within its broader, galactic context. Over the last decade, observations of the Rossiter-McLaughlin effect (see for example Winn et al. 2010; Triaud 2017) and doppler tomography (e.g., Marsh 2001; Johnson et al. 2017) have revealed that exo- planetary systems around sun-like stars generically exhibit a very broad range of spin- orbit misalignments, with projected stellar obliquities ranging from 0 to 180 deg. Importantly, this finding applies both to singly-transiting planets as well as to multi- transiting systems wherein the planets themselves are almost exactly coplanar, but are cumulatively inclined with respect to the spin-axis of their host star (e.g., the inner planets of Kepler-56, Li et al. 2014; Li & Winn 2016). Although the exact mechanism through which extrasolar spin-orbit misalignments are excited remains an active area of research, viable theories for the generation of planet-star misalignments include tur- bulence within the protostellar core (Bate et al., 2010; Spalding et al., 2014; Fielding et al., 2015), gravitational torques arising from primordial stellar companions (Batygin et al., 2012; Batygin & Adams, 2013; Lai, 2014; Spalding & Batygin, 2014, 2015), as well as magnetohydrodynamic interactions between the stellar magnetospheres and the inner edges of their circumstellar disks (Lai et al., 2011). In light of these results, we can comfortably attribute the 6-degree obliquity of the sun to the same primordial pro- cess that shapes the broad distribution of spin-orbit misalignments of generic sun-like stars throughout the Galaxy, decoupling it from the Planet Nine hypothesis.

Prospects for Detection 

Optical Surveys

An important aspect of the Planet Nine hypothesis is that all of its theoretical at- tributes are directly testable, through the astronomical detection, and characterization of P9 itself. Arguably, the most straightforward approach towards direct detection of Planet Nine is via conventional observations in reflected visible light. The reference Planet Nine envisioned by Batygin and Brown (2016a,b) and (Brown and Batygin, 2016) had a moderately high semi-major axis, eccentricity, and mass, making Planet Nine potentially as faint as 25th magnitude. The more detailed model comparisons shown here suggest a Planet Nine that is lower in all of these parameters. Let us exam- ine the effect of this refinement on the expected brightness.

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Figure 25: On-sky properties of a typical P9 orbital fit. The top panel depicts an example path of Planet Nine, along with the RA-DEC projection of the ecliptic and galactic planes. The bottom panel reports the expected range of the visual magnitude of Planet Nine as a function of the right ascension. While ongoing observational surveys such as Pan-STARRS yield important limits on the location of P9 along its orbit, such constraints are absent from the galactic plane, which remains scarcely explored by solar system surveys.

The brightness of Planet Nine depends on its size, albedo, and distance. For a mass range between 5 and 10 M_⊕, exoplanets follow an approximate power law where R R_⊕(M/M_⊕)^0.55, suggesting radii of 2.4 and 3.5R_⊕, respectively (Weiss and Marcy, 2014). Such estimates are further consistent with the interior modeling efforts of Linder and Mordasini (2016) who compute physical radii of ~ 1.9 – 3.7R_⊕ for an isolated 10 M_⊕ object (see also Ginzburg et al. 2016). The albedo of Planet Nine is unknown, but modeling by Fortney et al. (2016) suggests that at the inferred heliocentric distance,

the planetary H/He envelope will be free of all potential condensibles, rendering the atmosphere an essentially pure Rayleigh scatterer with a V-band albedo of almost unity. Neptune, in contrast, has an albedo of approximately 40%, which we take as a plausible lower limit.

As discussed above, for 5 M_⊕, the best-fit orbital solution is characterized by a₉ = 500 AU and e₉ = 0.25 which corresponds to an aphelion distance of 625 AU, where it would be between magnitude V ≈ 21.2 and 22.2 (it would be between magnitude 19.0 and 20.0 at perihelion), depending on the albedo. For 10 M_⊕, which we Part  

The Planet Nine Hypothesis (Part 11)

 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 f_a, 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 f_a 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 f_a 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 f_a 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

( f_a)_data = 8/9 89% (shown on the top left panels with a horizontal dashed line), here we adopt a slightly less stringent value of f_a ≥ 80% as a criterion for success.

In the vast majority of our f_a ≥ 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, f_a (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 ( f_a)_data = 8/9, as a rudimentary cut on the results, we disregard any simulation that generates a distant Kuiper belt with f_a < 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 f_a 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 f_a, 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 f_a, 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 f_a 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 f_a ≥ 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 (y_data/σ_data) 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 f_a. 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