Constraints on Planet Nine's Orbit and Sky Position within a Framework of Mean-motion Resonances

A useful quantification of clustering stems from the degree to which the KBO longitudes of perihelion, ϖ_i, are spread about anti-alignment with Planet Nine. If

$$\begin{eqnarray}&&| {\varpi }_{i}-{\varpi }_{P9}-180^\circ | =| {\rm{\Delta }}{\varpi }_{i}-180^\circ | \end{eqnarray} \tag{ 2 }$$

is the absolute deviation of the ith KBO's longitude of perihelion from perfect anti-alignment with Planet Nine (in a range from 0° to 180°), then

$$\begin{eqnarray}&&m=\mathop{\mathrm{median}}\limits_{1\leqslant i\leqslant 7}| {\rm{\Delta }}{\varpi }_{i}-180^\circ | \end{eqnarray} \tag{ 3 }$$

is a measure of the typical deviation of the seven KBOs at one slice in time. The median over T time slices,

$$\begin{eqnarray}&&s=\mathop{\mathrm{median}}\limits_{1\leqslant t\leqslant T}\{m\},\end{eqnarray} \tag{ 4 }$$

is thus an average measure of the scatter across the duration of the integration. We wish to favor configurations that produce small values of s. A suitable merit function is

$$\begin{eqnarray}&&f(s)={e}^{-{s}^{2}/1000},\end{eqnarray} \tag{ 5 }$$

where the constant 1000 was chosen through experimentation.

A conservative merit function only demands KBO clustering rather than both clustering and orbital anti-alignment with Planet Nine. This may be parameterized by alignment of the KBO argument of perihelion vectors,

$$\begin{eqnarray}&&{{\boldsymbol{e}}}_{\omega }=(\cos \omega )\hat{i}+(\sin \omega )\hat{j}.\end{eqnarray} \tag{ 6 }$$

Provided a set of orbital evolutions for seven KBOs, we quantify the clustering as follows. We first calculate the seven bodies' unit argument of perihelion vectors ${{\boldsymbol{e}}}_{1}(t),...,{{\boldsymbol{e}}}_{7}(t)$ and their mean unit vector ${{\boldsymbol{e}}}_{m}(t)$ at all times t. We compute the sum of the time-varying dot products,

$$\begin{eqnarray}&&{S}_{\omega }(t)=\displaystyle \sum _{i=1}^{7}{{\boldsymbol{e}}}_{i}(t)\,\cdot \,{{\boldsymbol{e}}}_{m}(t),\end{eqnarray} \tag{ 7 }$$

which quantifies the degree to which the vectors are clustered about their mean. If a KBO gets ejected from the system before the integration is over, we set ${{\boldsymbol{e}}}_{i}(t)$ to zero, such that this KBO stops contributing to ${S}_{\omega }(t)$. We also evaluate the time-varying anti-alignment with respect to Planet Nine as

$$\begin{eqnarray}&&{A}_{\omega }(t)={{\boldsymbol{e}}}_{m}(t)\,\cdot \,{{\boldsymbol{e}}}_{P9}(t).\end{eqnarray} \tag{ 8 }$$

If the seven bodies are perfectly anti-aligned with respect to Planet Nine for all times, then S_ω(t) = 7 and A_ω(t) = −1. The present-day value of S_ω for the seven KBOs we are considering is S_ω(0) = 6.52. The time average $\langle {S}_{\omega }(t)\rangle$ is an effective measure of confinement. Assuming that the present-day observed configuration of the KBOs is not a unique moment in time, $\langle {S}_{\omega }(t)\rangle$ should be close to ${S}_{\omega }(0)$. Furthermore, an integration that maintains KBO anti-alignment with P9 will yield $\langle {A}_{\omega }(t)\rangle \approx -1$.

Analogous metrics may be constructed with the longitude of perihelion vectors,

$$\begin{eqnarray}&&{{\boldsymbol{e}}}_{\varpi }=(\cos \varpi )\hat{i}+(\sin \varpi )\hat{j},\end{eqnarray} \tag{ 9 }$$

for which the present-day value is ${S}_{\varpi }(0)=5.63$. A successful configuration must produce clustering in both ω and ϖ in order to match the observations. We note that it is insufficient to enforce clustering in ϖ or the Runge–Lenz eccentricity vectors alone because these metrics do not guarantee confinement in ω.

In the analysis that follows, the merit function f(s) will be used to guide a Markov Chain Monte-Carlo procedure; $\langle {S}_{\omega }(t)\rangle$, $\langle {A}_{\omega }(t)\rangle$, $\langle {S}_{\varpi }(t)\rangle$, and $\langle {A}_{\varpi }(t)\rangle$ will be used to further analyze the resulting samples.

Having defined merit functions by which we can judge the KBO orbital evolution associated with a given set of Planet Nine parameters, we next discuss how these parameters are sampled. We begin with initial parameter distributions that are motivated by the strong suggestion of commensurability in Section 2 and by the constraints provided by the simulations described by Brown & Batygin (2016). The distributions are listed in Table 3, where we define ${ \mathcal N }(\mu ,\sigma ,a,b)$ as a truncated Gaussian probability distribution with mean μ, standard deviation σ, and lower and upper bounds, a and b. ${ \mathcal N }(\mu ,\sigma )$ and ${ \mathcal U }[a,b]$ are Gaussian and uniform distributions, respectively.

Table 3.  Initial Parameter Distributions

Parameter	Distribution
Semimajor axis, a (au)	$a\sim { \mathcal N }(654,20,590,720)$
Mass, m $({M}_{\oplus })$	$m\sim { \mathcal N }[10,7.5,(a-600)/20,$ $(a-200)/30]$
Eccentricity, e	$e\sim { \mathcal N }[0.5-0.05(m-10),0.2,0.1,0.9]$
Inclination, i (°)	$i\sim { \mathcal N }(30,20,-10,70)$
Longitude of ascending node, Ω (°)	${\rm{\Omega }}\sim { \mathcal U }[0,360]$
Argument of perihelion, ω (°)	$\omega \sim { \mathcal N }(230-{\rm{\Omega }},25)$
Mean anomaly, M (°)	$M\sim { \mathcal U }[0,360]$

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The dependence of the m distribution on a corresponds to the constraints derived by Brown & Batygin (2016). This mass/semimajor axis relationship was also employed in Section 2 when we used mass constraints within the calculation of the MMR semimajor axis distribution. Similarly, the relationship of the eccentricity distribution to the mass has been adapted from Brown & Batygin's constraints. The main feature is the negative correlation between e and m. Finally, we note that the longitude of perihelion has been constrained to $\varpi \sim { \mathcal N }(230^\circ ,25^\circ )$ in accord with anti-alignment with the KBOs. ω and Ω are allowed to vary through a full 360°, subject to the constraint that $\varpi =\omega +{\rm{\Omega }}$.

Using these initial parameter distributions to seed a collection of Markov Chains, we carried out a parameter exploration using a DE-MCMC algorithm (Ter Braak 2006). The DE-MCMC technique is most often used to sample from a posterior parameter distribution, typically for Bayesian inference; it iterates many Markov Chains in parallel and uses the inter-chain differences to inform the parameter jumps from one iteration to the next. In our analysis, however, the merit function takes the place of a likelihood function. As a result, this technique will not produce true posterior parameter distributions because the merit function is not a true probability density function. Rather, the DE-MCMC is simply a tool that we use to efficiently sample the seven-dimensional parameter space, assuming large parameter covariance and non-smooth parameter space structure, both of which DE-MCMC is helpful in accounting for.

We ran in parallel a large collection of DE-MCMC instances of five chains. We used jumping parameters γ = 0.1 and γ_s = 0.001 (Ter Braak 2006), such that one chain's proposal jump during a given iteration was approximately one-tenth of the vector between two other randomly selected chains. As stated above, the merit function f(s) took the place of the likelihood. The sampling of e and ω used the transformations $e\cos (\omega )$ and $e\sin (\omega );$ the other orbital elements were unaltered.

For all of our N-body dynamical integrations, we used the hybrid symplectic-Bulirsch–Stoer algorithm from the Mercury N-body integration package (Chambers 1999). We modeled the gravitational potential of the inner three giant planets (Jupiter, Saturn, and Uranus) as an orbit-averaged quadrupole by giving the Sun a physical radius equal to Uranus's orbital radius and a J₂ moment equal to (Burns 1976)

$$\begin{eqnarray}&&{J}_{2}=\displaystyle \frac{{m}_{J}{{a}_{J}}^{2}+{m}_{S}{{a}_{S}}^{2}+{m}_{U}{{a}_{U}}^{2}}{2{M}_{\odot }{{a}_{U}}^{2}}.\end{eqnarray} \tag{ 10 }$$

Neptune was followed in direct N-body fashion, and the integration timestep was set to one-tenth of Neptune's orbital period. The integrations were carried out for one billion years. While this is too short to ensure apsidal confinement lasting over the age of the solar system, it is certainly long enough to observe KBO orbital instability and loss of apsidal confinement due to differential precession (that is, within a configuration in which Planet Nine's orbital parameters are unsuitable).

While monitoring the orbital clustering of the seven KBOs is straightforward, the KBOs' own orbital uncertainties must also be accounted for. Inspection of the bodies listed in Table 1 using the the JPL Small-Body Database Browser³ indicates that the semimajor axis uncertainties for the distant KBOs are of similar magnitude to the resonant libration widths. If MMRs are indeed the primary mechanism for the orbital clustering, the uncertainties must be considered when evaluating whether the bodies can coexist in resonance.

To accommodate the uncertainties, all of our integrations were performed not only on the KBOs themselves, but also on a set of "clones." For each integration, we randomly generated five clones per KBO with identical values for e_i, i_i, Ω_i, ω_i, and M_i but with a Gaussian perturbation to the semimajor axis, a_i, consistent with the currently estimated semimajor axis uncertainties. Ideally, one would draw perturbations on all the KBO orbital elements, but the semimajor axis is clearly the principal parameter in determining the proximity to MMR.⁴ We retained one copy of the original KBO with no perturbation, for a total of six clones per KBO, or 42 bodies total to be followed in each integration.

A benefit of representing seven KBOs as 42 independent bodies is a dramatic increase in sample size enabled through a multiplexing process. In keeping with the hypothesis of Malhotra et al. (2016), we expect that the clone with semimajor axis closest to exact resonance will generally perform the best at maintaining apsidal alignment in time. For a given integration, we seek the set of seven best-performing clones. There are 6⁷ = 279,936 such sets of possible combinations. In effect, each dynamical integration of 42 bodies that is performed can be recast as a set of 279,936 independent integrations of seven bodies, each of them with the same set of parameters for Planet Nine but with different parameters for the seven KBOs of interest.

To summarize, we a posteriori generated the 279,936 multiplexed samples from each billion-year integration that was performed using the procedure discussed in Section 3.2. In each multiplexed sample's combination of seven KBOs, we calculated the measures of argument and longitude of perihelion vector clustering and anti-alignment that were defined in Section 3.1.

We now present the results of ∼250,000 individual billion-year orbital integrations. Rather than considering all 6⁷ multiplexed samples for each integration, which would produce a prohibitively large set of 70 billion one-Gyr integrations to analyze, we first consider a subset of the best-performing samples. This subset consists of each integration's single multiplexed sample that exhibits the largest value of $\langle {S}_{\omega }(t)\rangle$, defined in Section 3.1 as the time average of the sum of the dot products of the seven KBO argument of perihelion vectors with their mean vector. In this procedure, we opted to maximize $\langle {S}_{\omega }(t)\rangle$ as opposed to $\langle {S}_{\varpi }(t)\rangle$ because, as we will show, obtaining clustering close to the present-day observations is much more challenging for ω than ϖ.

Figures 4–7 show the performance of the KBOs in maintaining clustering in their arguments of perihelion as a function of Planet Nine's orbital elements. In each scatter plot, the x-axis is the argument of perihelion clustering measure $\langle {S}_{\omega }(t)\rangle$, with the dashed line marking the present-day value, S_ω(0) = 6.52. The colorbar variable $\langle {A}_{\omega }(t)\rangle$, as defined in Section 3.1, quantifies the degree to which the KBOs maintained anti-alignment with Planet Nine, with $\langle {A}_{\omega }(t)\rangle =-1$ being perfect anti-alignment. Figures 4 and 7 include an additional scatter plot using $\langle {S}_{\varpi }(t)\rangle$, the longitude of perihelion clustering measurement, as the colorbar.

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We first make some general observations. The figures indicate that an optimization scheme of this nature is a viable technique for constraining Planet Nine's orbital elements. Obtaining large values of $\langle {S}_{\omega }(t)\rangle$ is challenging, and the vast majority of orbital configurations do not work out, yet there are constrained regions of the parameter space where significant clustering can be maintained.

$\langle {S}_{\omega }(t)\rangle$ is generally anti-correlated with $\langle {A}_{\omega }(t)\rangle$, though the samples with the largest values of $\langle {S}_{\omega }(t)\rangle$ are not exclusively anti-aligned with Planet Nine. We note that anti-alignment with Planet Nine is not required by the KBO observations, which simply exhibit an observed clustering in ω and ϖ. It is difficult to conceive, however, of a configuration that is not anti-aligned and that can survive for billions of years. Either secular or resonant confinement mechanisms require ${\rm{\Delta }}\varpi \approx 180^\circ$, with ${\rm{\Delta }}\omega \ne 180^\circ$ implying a nodal asymmetry. When longer time integrations are performed, we expect that there will be a decrease in the number of samples that perform well without an anti-aligned configuration.

Figure 4 presents the results for the eccentricity. Samples with the largest values of $\langle {S}_{\omega }(t)\rangle$ have e ∼ 0.4–0.55, with the left panel showing that anti-aligned configurations have e ∼ 0.4–0.5. This is slightly lower but not in disagreement with the eccentricities favored by Brown & Batygin (2016) for a ∼ 600–700 au. The right panel of Figure 4 depicts similar results, except the colorbar now shows clustering in the KBO longitudes of perihelion. Given that ${S}_{\varpi }(0)=5.63$, clustering is evidently much easier for ϖ than for ω. Interestingly, a large degree of clustering in ϖ is most easily attainable for highly eccentric orbits.

Figure 5 presents the distributions of ω, Ω, and ϖ. As discussed in Section 3.2, the initial distribution restricted ϖ to initial anti-alignment with the KBOs but let ω and Ω vary through 0°–360°. It is not surprising to see strongest anti-alignment occurring for ω ∼ 150° because this corresponds to present-day anti-alignment with the KBOs. It is still significant, however, as in the absence of the perturbing planet, the KBOs randomize their alignment in tens of millions of years due to differential precession. Moreover, the region where ω ∼ 150° also produces the strongest clustering in the KBO arguments of perihelion. The longitude of the ascending node displays strongest clustering and anti-alignment for $\varpi \sim 205^\circ$. By extension, the node is best at Ω ∼ 50°.

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The inclination distribution in Figure 6 illustrates that low inclinations i ≲ 20° are consistently disfavored. Strongest anti-alignment occurs for 30° ≲ i ≲ 40°, in agreement with the Brown & Batygin (2016) estimate, whereas clustering is strongest for very high inclinations. To understand this apparent preference for high inclination configurations, we examined the KBO trajectories of representative samples of this population. The KBOs in these configurations typically undergo large eccentricity oscillations and a rapid increase to highly inclined or retrograde orbits. In general, the bodies are only able to maintain inclinations close to their present-day values for i ≲ 40°, though some configurations with i < 40° can still cause the KBOs to evolve to high inclinations. This evolution to high inclination and retrograde orbits was discussed in Shankman et al. (2016) and in Batygin & Brown (2016b), who showed that Planet Nine is capable of producing the observed population of highly inclined, a < 100 au trans-Neptunian objects through a process involving Kozai-Lidov cycles and close encounters with Neptune. We also found a moderate correlation between KBO evolution to highly inclined orbits and nodal clustering, possibly providing a connection to the observation of nodal clustering in the population of a < 100 au TNOs reported in Chen et al. (2016). An extended exploration of these inclination dynamics is saved for a future discussion.

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In contrast to some of the other orbital elements, the mass distribution in Figure 7 exhibits less constraint. The KBO clustering and anti-alignment is possible for masses throughout the range 6 M_⊕ ≲ m ≲ 12 M_⊕. In agreement with previous studies, masses less than ∼6 M_⊕ are strongly disfavored.

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The semimajor axis distribution does not display much structure and is not shown here. The KBO semimajor axes are constantly perturbed, however, through the cloning procedure, meaning that the semimajor axis ratio a_P9/a_KBO or period ratio ${P}_{P9}/{P}_{\mathrm{KBO}}$ are more physically informative than a_P9. We pursue an exploration of the period ratio in Section 4.2.

If MMRs are responsible for the observed apsidal clustering of the KBOs, then the configurations that exhibit better maintenance of clustering than others should preferentially place the bodies in period ratio commensurabilities. We investigated this hypothesis in the following manner. For each of the ∼250,000 integrations that was performed, we extracted the single multiplexed sample combination of seven KBOs with the largest value of $\langle {S}_{\omega }(t)\rangle$. We also extracted one random combination of KBOs from the multiplexed samples. In the left panels of Figures 8 and 9, we plot the clustering merit $\langle {S}_{\omega }(t)\rangle$ versus the period ratio ${P}_{P9}/{P}_{\mathrm{KBO}}$ at t = 0 for the best-performing multiplexed samples. The right panels show the analogous plots for the random multiplexed samples.

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The diagrams confirm that MMRs are in fact an influential mechanism in the observed KBO clustering. The vertical overdensities in the left panels appear at distinct period ratios, signaling that the configurations that perform the best are preferentially near these commensurabilties. We have labeled the commensurabilities that manifest most strongly with their integer ratios. To further guide the eye, differenced histograms of the period ratios are plotted as black lines in the left panels. The differenced histograms are made by generating histograms of the period ratios of the best-performing samples (those in the left panels) with $\langle {S}_{\omega }(t)\rangle \gt 4$ and subtracting from them Gaussian fits to the period ratio histograms of the random samples (right panels). The peaks in the resulting differenced histograms indicate the locations of the overdensities.

2012 VP₁₁₃ exhibits the strongest feature at 4:1, with significant features also in 2010 GB₁₇₄ at 9:4, 7:3, and 5:2, and in Sedna at 3:2. Interestingly, Sedna displays several other overdense striations at nearby, higher order resonances.

Although these period ratio diagrams establish the importance of proximity to orbital period commensurabilities, we have not yet demonstrated true resonant libration. If the KBOs are in resonance with Planet Nine, their motions will display librations of a critical resonant angle of the form

$$\begin{eqnarray}&&{\phi }_{r}=(p+q){\lambda }_{P9}-p{\lambda }_{\mathrm{KBO}}-(q-r){\varpi }_{P9}-r{\varpi }_{\mathrm{KBO}}\end{eqnarray} \tag{ 11 }$$

where λ is the mean longitude and ϖ is the longitude of perihelion. p and q are integers that are fixed for a given $(p+q):p$ resonance, and r is an integer ranging from 0 to q. We monitored the resonant angles of the KBOs that appear most strongly in Figures 8 and 9: Sedna in 3:2, 2012 VP₁₁₃ in 4:1, and 2010 GB₁₇₄ in 9:4, 7:3, and 5:2. We also monitored 2004 VN₁₁₂ in 3:1, 2014 SR₃₄₉ in 7:2, 2007 TG₄₂₂ in 8:5, 2001 FP₁₈₅ in 5:1, and 2000 CR₁₀₅ in 5:1. (The last two are members of the 200 au < a_KBO < 250 au set of KBOs that we integrated but did not include in the $\langle {S}_{\omega }(t)\rangle$ clustering measurements.) Crucially, within KBO semimajor axis uncertainties, all of these resonances can coexist with a_P9 ∼ 654 au.

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In order to investigate both temporary and long-lasting resonant libration, we divided the billion-year integrations into 10 sections of 100 Myr and checked for libration in each slice of time. Remarkably, all KBOs we monitored were capable of experiencing resonant libration given relatively similar Planet Nine parameters. While the existence of resonant libration should not be surprising, we note that this is the first direct numerical confirmation that these observed high-eccentricity KBOs are capable of librating stably in an MMR with a distant perturber.

Figure 10 depicts the coexistence of resonant libration. We first restricted the sample space to the subset of Planet Nine configurations for which at least one of 2012 VP₁₁₃'s critical resonant angles, ϕ_r, was librating with z < 1.95 in the first 100 Myr. Here, $z\equiv \min \{{\rm{\Delta }}\cos {\phi }_{r},{\rm{\Delta }}\sin {\phi }_{r}\}$, where ${\rm{\Delta }}\cos {\phi }_{r}\equiv {(\cos {\phi }_{r})}_{\max }-{(\cos {\phi }_{r})}_{\min }$ and likewise for ${\rm{\Delta }}\sin {\phi }_{r}$. For each KBO and within each 100 Myr section of time, we then calculated the percentage of samples for which at least one clone was librating during that 100 Myr slice of time. The results are shown in Figure 10. The scales of the percentages for each KBO are relatively unimportant because they depend on the prior distributions of a_P9 and a_KBO, but the shapes of the curves are more informative. The curves for Sedna demonstrate that it is capable of libration from the beginning of the integration, starting at 0–100 Myr, in a configuration in which 2012 VP₁₁₃ is also librating from t = 0. The sharp increases in the percentages for most of the other resonances indicate that these bodies are difficult to maintain in resonance starting from t = 0, but they can be captured into libration as time advances.

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We also examined long-lived libration, that is, resonance starting from t = 0 and lasting for an arbitrary length of time, rather than libration within a 100 Myr section of time. Figure 11 shows the decay in the maintenance of resonant libration of Sedna and 2012 VP₁₁₃ as a function of time. The percentage of samples with the bodies remaining in resonance decays predictably as time advances, though there is a substantial fraction of the initial population that maintains libration for the full billion-year integration. The libration zones of Sedna and 2013 VP₁₁₃ are illustrated on an a_P9/e_P9 diagram in Figure 12. Here, the spread in a_P9 for the resonant configurations reflects both the intrinsic libration width and the uncertainties propagated through the KBO semimajor axes. The libration zones shrink in time because there is a decay in the number of bodies in resonance from t = 0. However, the overlap in the libration zones remains at values of a_P9/e_P9 that are consistent between the two resonances.

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Batygin & Brown (2016a) were the first to propose that the mean-motion resonant perturbation may be responsible for the perihelion clustering of the distant KBOs. Subsequently, Malhotra et al. (2016) observed that many KBOs would exhibit small integer commensurabilities if the semimajor axis of Planet Nine was at ∼654 au. Our work has provided evidence that supports both of these claims.

The a ∼ 654 au resonant hypothesis will be rapidly strengthened or undermined by future KBO observations. Participation in resonant libration generates predicted bounds on the KBO semimajor axes. These bounds are substantially smaller than current measurement uncertainties such as those for 2012 VP₁₁₃, which we have shown can exhibit long-lasting libration in a 4:1 resonance. Repeated observations of these KBOs to refine their period ratios can therefore permit rapid testing of the predictions, to either refute or substantially strengthen the validity of our assumed scenario. Moreover, the semimajor axes of as-yet-undiscovered, distant KBOs and their proximity to commensurabilities with a ∼ 654 au will continue to test the claim.

The MMRs impose restrictions not only on the orbit of the perturbing planet, but also on its position within the orbit. We can therefore evaluate the constraints our results yield on the sky position of Planet Nine, and compare the results with those of other studies.

In Figure 13, we display with large black points the sky positions of configurations where the KBOs maintained strong anti-alignment with Planet Nine, $\langle {A}_{\omega }(t)\rangle \lt -0.97$, and strong clustering in the arguments of perihelion, $\langle {S}_{\omega }(t)\rangle \gt 5.2$. The large gray points correspond to resonant configurations with the following constraints: Sedna and 2012 VP₁₁₃ librating from t = 0 for at least 100 Myr, 2010 GB₁₇₄ librating in one of its three dominant resonances, and 0.4 < e_P9 < 0.5. The faint gray points are the sky locations of all samples, which is not a uniform distribution in R.A. due to the orientation of the orbit in space from the restrictions on ϖ. Finally, the orbit of our fiducial model is shown with coloration according to distance.⁵ Though the plot is sparse, it is clear that the region with the highest density of favored points occurs at R.A. ∼ 30° and decl. ∼ −10°, or near the aphelion of our fiducial orbit.

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Furthermore, analytic considerations of the KBO resonances also point toward aphelion as a likely location. In our integrations, we found that resonant configurations were the most stable and long-lasting when the critical resonant angle ${\phi }_{q}=(p+q){\lambda }_{P9}-p{\lambda }_{\mathrm{KBO}}-q{\varpi }_{\mathrm{KBO}}$ exhibited libration about centers at ϕ_q = 0° or ϕ_q = 180°. We calculated histograms of ϕ_q libration amplitudes exhibited by the KBOs when they were in resonance. Then, under the assumption of libration about ϕ_q = 0° or ϕ_q = 180°, we transformed these observed distributions of KBO libration amplitudes into a joint probability distribution of the present-day mean anomaly for Planet Nine. The resulting most likely locations are perihelion or aphelion, which may be understood as a symmetry resulting from the fact that the KBOs are all currently near their perihelion passages. Moreover, observational constraints significantly disfavor perihelion for Planet Nine, making aphelion the most probable location.

If the planet has the size and the reflectivity of Neptune, our fiducial model suggests that it will have V ∼ 23 at aphelion, with a parallactic motion of $0\mathop{.}\limits^{\prime\prime }5\,{\mathrm{hr}}^{-1}$. To visualize the favored location more concretely, a 3D plot of the KBO orbits and our fiducial Planet Nine model is shown in Figure 14. A manipulable version of the 3D figure is available at https://smillholland.github.io/P9_Orbit/.

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Our current estimate of Planet Nine's sky location is consistent with previously derived constraints. Brown & Batygin (2016) used the observational limits of previous and present sky surveys to delineate regions of phase-space where Planet Nine should have been or will be detected. Figure 10 of their paper shows that the zone of R.A. ∼ 30°–90° corresponds to a largely unsurveyed region on the sky. Fienga et al. (2016) considered the nominal Batygin & Brown (2016a) Planet Nine with a range of orbital positions. They performed fits of the decade-long Cassini measurements of the Earth–Saturn distance to constrain the current location of Planet Nine. They rule out Planet Nine positions near perihelion, given by R.A. < 2° and R.A. > 255°, that worsen the fit to the Cassini residuals. The orbital position that leads to a maximum improvement of the residuals is located at R.A. ∼ 30° and decl. ∼ −20°, which is near our favored region. Holman & Payne (2016b) used a dynamical model of the tidal influence of Planet Nine to fit the Cassini ranging data. They significantly rule out large areas of the sky and find a most favored location of R.A. ∼ 40° and decl. ∼ −15°, extending ∼20° in all directions. Although these results depend sensitively on the assumed solar system model (Folkner et al. 2016), it is perhaps worth noting that the region correlates well with the highest density of favored points in Figure 13.

The sparseness of the resonant configurations in Figure 13 is a reflection of the computational infeasibility of using N-body integrations alone to pinpoint the perturber parameters that allow multiple KBOs to coexist in resonance. If the resonant hypothesis is correct, an optimization scheme that more heavily exploits the MMRs will effectively constrain Planet Nine's orbital elements and sky position. Specifically, we are interested in converging upon the parameters that allow all KBOs to exhibit long-lasting resonant libration simultaneously. This will require not just orbital integrations but also a better dynamical understanding of how these high-eccentricity resonances are arranged in phase-space.

The orbital element constraints in Section 4.1 were presented as constraints on Planet Nine conditional upon its existence. We now release this assumption and comment on the implications of the results for the evidence of the planet's existence. In this paradigm, there is opportunity for a much less optimistic interpretation.

As shown in Figure 4, for example, the time-averaged measure of clustering in the KBO arguments of perihelion, $\langle {S}_{\omega }(t)\rangle$, is almost always smaller than the present-day value, ${S}_{\omega }(0)=6.52$. In other words, sustained clustering that matches the present-day value is exceptionally difficult to obtain. The fact that it is so challenging could be viewed as evidence against the planet's existence.

One possible explanation of the discrepancy is that $\langle {S}_{\omega }(t)\rangle$ might not be a suitable comparison measurement to S_ω(0). This time-averaged quantity favors clustering throughout the entire integration, even though the bodies experience perturbations in their perihelion distances that would render them unobservable to present-day surveys. Moreover, it might be the case that these high-a, high-e KBOs are more clustered than as-yet unobserved counterparts with similarly large a but smaller e. If that is the case, a variant of $\langle {S}_{\omega }(t)\rangle$ that is restricted to bodies with perihelion distances below some limit of observability would be a more suitable comparison metric to ${S}_{\omega }(0)$. However, we checked the integrations for a connection between observability (i.e., small perihelion distances) and clustering in ω and ϖ and found little evidence for such a connection. There is thus no reason to expect that $\langle {S}_{\omega }(t)\rangle$ is a biased merit function, so the tension between $\langle {S}_{\omega }(t)\rangle$ and S_ω(0) remains.

One way to reconcile the difficulty the KBOs have in maintaining clustering is to take it as evidence against the existence of Planet Nine. An alternate, more optimistic interpretation involves two ideas: (1) we might be observing the KBOs at a time period in which they are more clustered than average, such that $\langle {S}_{\omega }(t)\rangle$ need not match S_ω(0); and (2) it is possible that only a small but nonzero domain of Planet Nine parameters are capable of producing long-lasting clustering in the observed KBOs.

The prospect that our solar system harbors an as-yet undetected super-Earth mass planet has generated an extraordinary wave of interest and attention. Two facets of the hypothesis seem particularly compelling. If the planet does exist, and especially if its mass is less than 10 M_⊕, it can provide a detailed ground-truth verification that theoretical models of the atmospheres and structures of planets in unexplored mass and temperature regimes are on the right track. Furthermore, there is a satisfyingly dramatic prospect of resolution. If a trans-Neptunian planet with the proposed properties is actually out there, it will be detected very soon.