OBSERVATIONAL CONSTRAINTS ON THE ORBIT AND LOCATION OF PLANET NINE IN THE OUTER SOLAR SYSTEM

Since the time of the discovery of Sedna, it has been clear that a large perturbing mass either is or was present in the outer solar system at some time (Brown et al. 2004). With a perihelion distance of 76 au, Sedna is essentially immune to direct interactions with the known planets; thus, unlike all other Kuiper belt object (KBO) orbits, it cannot have been placed onto its orbit by perturbation from any of the known planets. Proposals for the perturber required to have created Sedna's orbit have included sibling stars in the Sun's birth cluster (Brown et al. 2004; Brasser et al. 2006; Dukes & Krumholz 2012), a single passing star (Kenyon & Bromley 2004; Morbidelli & Levison 2004; Rickman et al. 2004) as well as a small former or extant planet in the outer solar system (Brown et al. 2004; Gladman & Chan 2006; Gomes et al. 2006). Progress on understanding the cause of Sedna's perturbed orbit, however, was not possible because of a lack of additional high perihelion objects.

With the discovery of 2010 GB174 (Chen et al. 2013) and 2012 VP113 (Trujillo & Sheppard 2014)—the second and third high-perihelion Sedna-like objects—additional patterns began to emerge. Most importantly, Batygin & Brown (2016) point out that all well-determined orbits of KBOs beyond Neptune with semimajor axis, a, larger than 227 au approach perihelion within 94° of longitude of each other. Moreover, these objects also share very nearly the same orbital plane, which is tilted an average of 22° to the ecliptic. The combined probability of these two occurrences happening simply due to chance is less than 0.01%. Importantly, of all KBOs with $a\gt 100\;{\rm{au}}$, the five with the largest perihelion distances are likewise confined to the same perihelion region and orbital plane.

Batygin & Brown (2016) show that a distant massive eccentric planet will cause clustering of the perihelion and orbital planes of distant KBOs in the manner observed, and, additionally, will naturally lead to the creation of objects with high perihelion orbits like Sedna. Surprisingly, these clustered and high perihelion objects have orbits that are anti-aligned with the giant planet. That is, the clustered KBOs come to perihelion 180° away from the perihelion position of the planet. Despite chaotic evolution, the crossing orbits maintain long-term stability by residing on a interconnected web of phase-protected mean-motion resonances.

The distant eccentric perturber studied in Batygin & Brown (2016)—which we refer to as Planet Nine—modulates the perihelia of objects in the anti-aligned cluster and naturally creates objects like Sedna, in addition to the other high perihelion KBOs. Additionally, the existence of Planet Nine predicts a collection of high semimajor axis eccentric objects with inclinations essentially perpendicular to the rest of the solar system. Unexpectedly, this prediction is strongly supported by the collection of low perihelion Centaurs with perpendicular orbits whose origin had previously been mysterious (Gomes et al. 2015).

Here, we make detailed comparisons between dynamical simulations that include the effects of Planet Nine, and solar system observations, to place constraints on the orbit and mass of the distant planetary perturber. We then discuss observational constraints on the detection of this distant giant planet and future prospects for its discovery.

The inclined orbits of the aligned KBOs (and thus, presumably, of the distant planet) render ecliptic-referenced orbital angles awkward to work in (particularly when we consider the Centaurs with perpendicular orbits later). Accordingly, we re-cast the three ecliptic-referenced parameters—argument of perihelion, longitude of ascending node, and inclination—into simple descriptions of orbit in absolute position on the sky: the ecliptic longitude of the point in the sky where the object is at perihelion (which we call the "perihelion longitude," not to be confused with the standard orbital parameter called "longitude of perihelion" which, confusingly, does not actually measure the longitude of the perihelion except for zero inclination orbits), the latitude of the perihelion ("perihelion latitude"), and an angle that measures the projection of the orbit pole onto the plane of the sky ("pole angle" perhaps more easily pictured as the direction perpendicular to the motion of the object at perihelion).

Figure 1(a) shows the perihelion longitude and latitude as well as the pole angle for all objects with $q\gt 30$ and $a\gt 60\;{\rm{au}}$ and well-determined orbits. The seven objects with $a\gt 227\;{\rm{au}}$ are highlighted in red. The clustering in perihelion location as well as pole angle is clearly visible. In Figure 1(b), we plot the perihelion longitude of all well-constrained orbits in the Kuiper belt that have perihelia beyond the orbit of Neptune as a function of semimajor axis. The seven objects with $a\gt 227\;{\rm{au}}$ cluster within 94° of perihelion longitude. Batygin & Brown (2016) showed that this clustering, when combined with the clustering in pole angle, was unexpected at the 99.993% confidence level. The clustering is consistent with that expected from a giant planet whose perihelion is located 180° in longitude away from the cluster, or an ecliptic longitude of 241° ± 15°. A closer examination of Figure 1(b) makes clear a possible additional unlikely phenomenon. While KBOs with semimajor axes out to 100 au appear randomly distributed in longitude, from 100 to 200 au, 13 objects are loosely clustered within 223° of each other. While this clustering is not as visually striking, the probability of such a loose clustering of 13 objects occurring in randomly distributed data is smaller than 5%. As will be seen, such a loose clustering of smaller semimajor axis objects can indeed be explained as a consequence of some orbital configurations of Planet Nine.

Zoom In Zoom Out Reset image size

To understand how these observations constrain the mass and orbit of Planet Nine, we performed a suite of evolutionary numerical integrations. Specifically, we initialized a planar, axisymmetric disk consisting of 400 eccentric planetesimals that uniformly spanned semimajor axis and perihelion, q, distance ranges of a = 150–550 au and q = 30–50 au, respectively. The planetesimal population (treated as test particles) was evolved for 4 Gyr under the gravitational influence of the known giant planets, as well as Planet Nine.

Perturbations due to Planet Nine and Neptune were accounted for in a direct N-body fashion, while the secular effects of the remaining giant planets were modeled as a suitably enhanced quadrupolar field of the Sun. As shown in Batygin & Brown (2016), such a numerical setup successfully captures the relevant dynamical phenomena, at a substantially reduced computational cost.

In these integrations we varied the semimajor axis and eccentricity of Planet Nine from ${a}_{9}$ = 200–2000 au and ${e}_{9}$ = 0.1–0.9 in increments of ${\rm{\Delta }}{a}_{9}=100\;{\rm{au}}$ and ${\rm{\Delta }}{e}_{9}=0.1$ (here, and subsequently, the nine subscript refers to the orbital parameters of Planet Nine, while unsubscripted orbital parameters refer to the test particles). The a₉–e₉ grids of synthetic scattered disks were constructed for Planet Nine masses of m₉ = 0.1, 1, 10, 20, and $30{M}_{e}$ (Earth masses), totaling a suite of 320 simulated systems. All calculations were performed using the mercury6 N-body integration software package (Chambers 1999), employing the hybrid symplectic-Bulirsch-Stoer algorithm with a time step equal to one-tenth of Neptune's orbital period.

We assess the success of each simulation with two simple metrics. First, we collect the orbital elements of all remnant objects at each 0.1 Myr output time step from 3 to 4 Gyr after the start of the simulation (in order to assure that the objects we are considering are stable over at least most of the age of the solar system), and we restrict ourselves to objects with instantaneous perihelion $q\lt 80\;{\rm{au}}$ (to restrict ourselves to objects that are most likely to be observable). In this fashion we are examining stream functions of orbital elements that fit into an observable range of parameter space, rather than examining individual objects at a single time step. This approach is used in all subsequent discussions of simulations below. We then select 13 objects at random in the $a=[100,200]$ au range and 7 objects at random in the $a=[227,600]$ au range and calculate the smallest angles that can be used to confine the two populations. We perform this random selection 1000 times and calculate the joint probability that, like the real data, the 13 objects in $a=[100,200]$ au range are confined within 223° and the 7 objects a = [227–600] au range are confined within 94°. Additionally, we examine whether any objects exist in the range $a=[200,300]$ au, as many simulations remove all objects in this range. We assign these simulations a probability of zero. This probability calculation has the advantage that it is agnostic as to whether or not our observations of clustering are significant or even physically relevant. It simply calculates the probability that a given simulation could reproduce some of the apparent features of the real data, even if by chance.

The second metric we use to assess the success of the simulations relies on the observation of Batygin & Brown (2016) that a distant massive eccentric perturber will cause secular perturbations which lower the eccentricity and thus raise the perihelion of moderate semimajor axis objects at a wide range of perihelion latitudes. This effect increases strongly with increasing perturber eccentricity and with decreasing perturber semimajor axis. Of the 15 known KBOs with $100\lt a\lt 220\;{\rm{au}}$ and $q\gt 30$, zero have perihelion greater than 42 au (whereas 5 of the 7 objects with $a\gt 227\;{\rm{au}}$ have such elevated perihelia). These simulations were not designed in such a way that simple assessment of the probability of such a distribution is straightforward. We approximate this probability by collecting all objects from time steps between 2 and 4 Gyr with $30\lt q\lt 40\;{\rm{au}}$ and $100\lt a\lt 200\;{\rm{au}}$, randomly selecting 15 objects from this set, and calculating the product of the fraction of the time that each of these objects spends with $q\lt 42$ and $100\lt a\lt 200\;{\rm{au}}$. These probabilities range from unity, when Planet Nine is distant or only mildly eccentric, to 10⁻¹² for higher eccentricities and lower semimajor axes. Because these calculated probabilities are difficult to straightforwardly compare to the real data, we instead impose a simple threshold and assert that simulations with a 99% or higher probability of creating at least one high perihelion object in the $100\lt a\lt 200$ au range are effectively ruled out. We assign their overall probability to zero.

Figures 2 and 3 show grids of probabilities for the 10 and 20 M_e simulations (the 0.1, 1, and 30 M_e simulations have no acceptable solutions). The probabilities should be taken as more qualitative than quantitative, as these simulations are exploratory and attempt to cover large ranges of phase space by including a limited number of particles per simulation and excluding three-dimensional effects. Nonetheless, the overall trends are clear. The lack of high perihelion objects between $100\lt a\lt 200\;{\rm{au}}$ strongly rules out all of the low semimajor axes and nearly all of the highest eccentricities, while the need to confine objects in perihelion longitude requires moderately high eccentricities or moderately low semimajor axes. The combined effect of these two constraints makes for a rather narrow combination of a₉ versus e₉ with acceptable solutions. In fact, the range is sufficiently narrow that it is clear that the grid spacing of our simulations is often too large to capture acceptable solutions at all semimajor axes or eccentricities.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Nonetheless, from the 10 M_e simulations we can discern the narrow range of acceptable results (Figure 2). In general, all simulations that cannot be excluded at at the 99% confidence level fall with a range of ${a}_{9}=[500,800]$ au approximately along an empircally defined line of ${e}_{9}=0.75-{(450{\rm{au}}/{a}_{9})}^{8}$, shown as the dashed line in Figure 2. Such orbits have a perihelion, q₉, in the range $\sim [200,340]$ au. The acceptable q₉ range is greater at smaller semimajor axis; at 600 au and beyond, all solutions have ${q}_{9}\sim 200\;{\rm{au}}$.

For the 20 M_e simulations, the locus of acceptable a–e combinations shifts outward and can be fit with a similar empirical function ${e}_{9}=0.75-{(650{\rm{au}}/{a}_{9})}^{8}$, but only from ${a}_{9}=[800,1000]$ au. Lower semimajor axes perturb the low semimajor axis KBOs to high perihelia, while higher semimajor axes fail to cluster the high semimajor axis KBOs appropriately. Higher-mass simulations cannot match the observations at all.

In Figure 4, we show, as an example, the Planet Nine–centered perihelion longitudes as a function of a of all objects that have $q\lt 80\;{\rm{au}}$ and that have survived at least 3 Gyr, for the case of ${M}_{9}=10{M}_{e}$, ${a}_{9}=500\;{\rm{au}}$, and e₉ = 0.6, one of the simulations along the acceptable locus. Objects anti-aligned with the planet have a longitude of 180° in this simulation, while those aligned will be at 0° longitude. This simulation shows the major effects that we have previously identified in the real data. Inside of 100 au little perturbation is visible. From 200 to 600 au the longitudes are confined around 180°, that is, they are anti-aligned with Planet Nine. And from 100 to 200 au there is a slight tendency for a broad cluster centered on the longitude of the planet.

Zoom In Zoom Out Reset image size

As a counter example, Figure 5 shows a simulation with ${M}_{9}=10{M}_{e}$, ${a}_{9}=700\;{\rm{au}}$, and e₉ = 0.3, which depicts many of the same general phenomena, but these phenomena do not develop until larger semimajor axes. For example, the anti-alignment does not begin until 400 au, while a broad aligned cluster can be seen from about 300 to 400 au. Even with the crudeness of these simulations, these basic effects are clear. The semimajor axis at which anti-alignment begins and the range where broad confinement is evident are strong indicators of the combination of semimajor axis and the eccentricity of Planet Nine.

Zoom In Zoom Out Reset image size

The planar simulations provide no constraints on inclination, i₉, argument of perihelion ${\omega }_{9}$, or longitude of ascending node, ${{\rm{\Omega }}}_{9}$, of Planet Nine. To examine the effects of these orbital elements on the Kuiper belt, we perform a second, fully three-dimensional suite of simulations. In these simulations we fix the semimajor axis and eccentricity to be 700 au and 0.6, respectively, values which are within our acceptable range of parameter space. The inclination dynamics are unlikely to be unique to the specific values of a₉ and e₉, so we deem these simulations to be representative. We allow the inclination of Planet Nine to take values of ${i}_{9}=1$, 10°, 20°, 30°, 60°, 90°, 120°, and 150°.

Unlike the planar suite of calculations, here we start all planetesimals with their longitude of perihelia anti-aligned to that of Planet Nine. The starting values of planetesimals' longitudes of ascending node, on the other hand, are taken to be random. As demonstrated by Batygin & Brown (2016), dynamical sculpting of such a planetesimal population yields a configuration where long-term stable objects have longitudes of ascending node roughly equal to that of Planet Nine. In turn, this ties together ${\omega }_{9}$ and ${{\rm{\Omega }}}_{9}$ through a fixed longitude of perihelion (which is the sum of these parameters).

Upon examination of these simulations, we find that efficiency of confinement of the distant population drops dramatically with increased inclination of Planet Nine. To quantify this efficiency, we sample each simulation 1000 times, picking seven random objects from the sample of all objects in the range $a=[300,700]$ au (as previously shown, these ${a}_{9}=700\;{\rm{au}}$ simulations do not begin strong perihelion confinement until $a\;\sim$ 300 au; as we are more interested in understanding the cluster than in specifically simulating our data at this point, we increase our semimajor axis range of interest). As before we restrict ourselves to time steps after 3 Gyr in which an object's orbit elements have $q\lt 80\;{\rm{au}}$, and we add the constraint that $i\lt 50^\circ$, to again account for observability biases. We calculate the fraction of times that the seven randomly selected objects are clustered within 94° (Figure 6). The confinement efficiency drops smoothly until, at an inclination of 60° and higher, it scatters around 20%. These results suggest, but do not demand, that Planet Nine has only a modest inclination.

Zoom In Zoom Out Reset image size

One of the striking characteristics of the seven aligned distant KBOs is the large value of and tight confinement in the pole angle (22° ± 6°; Figure 1). In examining the simulations that exhibit good confinement in longitude, we note that the polar angles of the simulated orbits are approximately perpendicular to the plane of Planet Nine, particularly for objects that come to perihelion in the plane of the planet. The implication of this phenomenon is that a pole angle of 22° suggests a minimum planet inclination of approximately 22° and an orbital plane (which is controlled by ${{\rm{\Omega }}}_{9}$) similar to the plane of the observed objects. To quantify this observation, we plot the median pole angle of our simulation objects that meet the criteria described above and that have perihelion latitudes between −25° and 0° like the real distant objects. We restrict ourselves to objects with perihelia south of the ecliptic both because the observed objects all have perihelia south of the ecliptic and also because we want to avoid any bias that would occur by a loss of observed objects north of the ecliptic due to the proximity of the galactic plane close to the perihelion positions. It is currently unclear whether the lack of clustered objects with perihelia north of the ecliptic is a dynamical effect or an observational bias. Figure 7 shows this mean polar angle as a function of argument of perihelion of Planet Nine. The maximum median polar angle occurs for ${\omega }_{9}\sim 150^\circ ,$ a configuration where the plane of the planet passes through the perihelion positions of the objects, and that maximum is approximately equal to the inclination of the orbit, confirming our observation that the clustered objects are along the same orbit plane as the planet.

Zoom In Zoom Out Reset image size

Based on the confinement probability and the large average pole angle of the real objects, we can infer that the inclination of Planet Nine is greater than ∼22° and less than the inclination at which confinement becomes improbable, which, based on an interpolation of the data from Figure 5, occurs approximately around 40°. For inclinations of ∼22°, ω must be quite close to 150°. For inclinations of 30°, the allowable range for the argument of perihelion appears to be ${\omega }_{9}$ ∼ 120°–160°.

While this analysis yields useful constraints, we quantify these results further by again sampling each of the simulations 1000 times and determining the probability of six randomly selected objects (300 au < a < 700 au, $q\lt 80\;{\rm{au}}$, $i\lt 50$°, survival time greater than 3 Gyr, and perihelion latitude between −25° and 0°) having perihelion longitudes clustered with 94° and having an average polar angle greater than 20° with an rms spread of less than 62. Almost all simulations can be ruled out at greater than the 99% confidence level. The only simulations that cannot are, unsurprisingly, those with an inclination of 30° and an argument of perihelion of 150—which is the single best fit—and 120° and, additionally, a few other seemingly random combinations of [i₉, ${\omega }_{9}$]: [90, 60], [150, 0], [150, 210], and [150, 330], all in units of degrees. We examine all of these cases in detail below.

One strong prediction of the existence of a giant planet in the outer solar system is that it will cause Kozai-Lidov oscillations that will drive modest inclination objects onto high inclination perpendicular and even retrograde orbits and then back again. This effect can be seen, for example, in Figure 8, where we plot the evolution of perihelion longitude versus inclination for the simulations with 30° inclination (again, restricting ourselves to 300 > a > 700 au, $q\lt 80\;{\rm{au}}$ and $t\gt 3\;{\rm{Gyr}};$ note that the argument of perihelion of Planet Nine has no substantive effect on this plot, so we plot all arguments together). The five known objects in the outer solar system with $a\gt 200\;{\rm{au}}$ and $i\gt 50$ deg are also shown. The simulations reproduce their perihelion longitudes and inclinations well, although they are all on the outer edge of the predicted clustering regions. An important caveat to note, however, is that the five high inclination objects are all Centaurs with $8\lt q\lt 15\;{\rm{au}}$. The high inclinations of these objects mean that they penetrate the giant-planet region much more easily and so can maintain their alignments much more easily than lower inclination Centaurs. Our simulations remove all objects inside 20 au, so we have not explored the dynamics interior to Uranus, but we note a systematic trend where objects with smaller perihelion distances move to the outer edge of the clustering regions, just like the real low perihelion objects appear to be. Clearly, simulations including all of the giant planets that allow us to study these high semimajor axis Centaurs are critical.

Zoom In Zoom Out Reset image size

The perihelion locations of the perpendicular high semimajor axis Centaurs effectively rule out the possible higher inclination orbits for Planet Nine. The ${i}_{9}=90^\circ$ and ${i}_{9}=150^\circ$ cases do create high inclination objects, but their perihelia are sporadically distributed across the sky (Figure 9). We conclude that, of our simulated parameters, only the ${i}_{9}=30^\circ ,{\omega }_{9}=150^\circ$ and ${i}_{9}=30^\circ ,{\omega }_{9}=120^\circ$ are viable.

Zoom In Zoom Out Reset image size

In order to complete our analysis in a tractable amount of time, each of the simulations used to explore parameter space above was limited in either dimensionality, number of particles, or in the range of starting parameter of the particles. In order to check that these limitations did not influence the overall results, we perform a final fully three-dimensional simulation with a large number of particles with randomly chosen starting angles. We choose to simulate Planet Nine with a mass of 10 M_e, ${a}_{9}=700^\circ$, e₉ = 06, ${i}_{9}=30^\circ$, ${\omega }_{9}=0^\circ$, and ${{\rm{\Omega }}}_{9}=0^\circ$ (note that the planet precesses over the 4 billion years of the integrations, but at this large semimajor axis the precession in ${\omega }_{9}$ is only about 30° during the entire period, so we ignore this effect). These full simulations reproduce all of the relevant effects of the more limited simulations, giving confidence to our simulation results.

Based on comparison to our suite of simulations, we estimate that the orbital elements of Planet Nine are as follows. For 10 and 20 M_e planets, a₉ and e₉ are empirically defined in Section 2. We roughly bound these empirical functions with simple linear fits as a function of mass on the minimum and maximum semimajor axes of Planet Nine for the two masses and a simple linear fit to the empirical fitting function. We thus estimate that a₉ is in the range [200 au + 30${M}_{9}/{M}_{e}$, 600 au + 20${M}_{9}/{M}_{e}$], where M₉ is approximately in the range [5 M_e, 20 M_e], and that e₉ = 0.75 − $[(250\;{\rm{au}}+20{M}_{9}/{M}_{e}$)/${a}_{9}{]}^{8}$.

The inclination is between approximately $22^\circ \lt {i}_{9}\lt 40^\circ$, and the argument of perihelion is between $120^\circ \lt {\omega }_{9}\lt 160^\circ$. We fix the perihelion longitude at 241° ± 15°. While these choices of parameter ranges have been justified in the analysis of the simulations above, they cannot be considered a statistically rigorous exploration of parameter space. Indeed, any attempt at such statistical rigor is not yet warranted: substantial uncertainty comes not just from the statistics of the objects themselves, but from the currently small number of simulations in the best-fit region of parameter space. Clearly, significantly more simulation is critical for a better assessment of the path of Planet Nine across the sky.

The last parameter we consider is the mass of Planet Nine, which we assume is in the range of 5–20 M_e. To transform this mass into an expected brightness requires assumptions of both radius (and thus composition) and albedo (and thus surface composition), neither of which is constrained by any of our observations. Fortney et al. (2016) consider a range of Planet Nine masses and core fractions. For masses between 5 and 15 M_e with 10% atmospheric mass fraction, the radius is approximately fit as ${R}_{9}=[2.4+0.1({M}_{9}/{M}_{e})]{R}_{e}$, which we will use as our nominal relationship through this range and extending to 20 M_e. In addition Fortney et al. (2016) find a quite high albedo value of 0.75 primarily due to Rayleigh scattering in the atmosphere. We assume, to be conservative, a range between 0.3, the approximate albedo of Neptune, and the modeled value of 0.75.

We now use our estimated orbital parameters to predict the orbital path of Planet Nine across the sky. We carry out a simple Monte Carlo analysis selecting uniformly across all of the parameter ranges. Figure 10 shows the sky location, heliocentric distance, magnitude, and sky motion at opposition for our suite of predicted orbits.

Zoom In Zoom Out Reset image size

While most wide-field surveys of the Kuiper belt have not been sensitive to sky motions smaller than about 1 arcsecond per hour (i.e., Millis et al. 2002; Brown 2008; Petit et al. 2011), a few surveys have had the sensitivity and cadence to have potentially detected Planet Nine at some point in its orbit. We discuss all such surveys below.

5.1. WISE

5.2. Catalina Real-time Transient Survey

5.3. The Pan-STARRS 1 Survey

5.4. The Dark Energy Survey (DES)

5.5. Additional Surveys

5.6. Additional Constraints

5.7. Joint Constraints

The existence of a distant massive perturber in the outer solar system—Planet Nine—explains several hitherto unconnected observations about the outer solar system, including the orbital alignment of the most distant KBOs, the existence and alignment of high perihelion objects like Sedna, and the presence of perpendicular high semimajor axis Centaurs. These specific observations have been compared to suites of numerical integrations in order to constrain possible parameters of Planet Nine. The current constraints must be considered preliminary: our orbital simulations needed to cover substantial regions of potential phase space, and so were, of necessity, sparsely populated. At present, the statistical reliability of our constraints are limited as much by the limited survey nature of the simulations as by the small number of observed objects themselves. Continued simulation could substantially narrow the potential search area required. In addition, continued simulation is required in order to understand one effect not captured in the current models: the apparent alignment of the argument of perihelion of the 16 KBOs with the largest semimajor axes (Trujillo & Sheppard 2014). Some of this apparent alignment may come from yet unmodeled observational biases related to the close proximity of the perihelion positions of the most distant objects to the galactic plane, while some may be a true as-yet-unmodeled dynamical effect.

As important as continued simulation, continued detection of distant solar system objects is the key to refining the orbital parameters of Planet Nine. Each addition KBO (or Centaur) with $a\gt 100\;{\rm{au}}$ tightens the observational constraints on the location of Planet Nine (or, alternatively, if significant numbers of objects are found outside of the expected cluster location, the objects can refute the presence of a Planet Nine).

Interestingly, the detection of more high semimajor axis perpendicular objects (whether Centaurs or KBOs) has the possibility of placing the strongest constraints in the near term. While we have currently only used the existence of these objects as a constraint, their perihelion locations and values of ω change strongly with ${\omega }_{9}$ and i₉ and so can be used to better refine these estimates. Though there are only currently five known of these objects, they are being discovered at a faster rate than the distant KBOs, so we have hope of more discoveries soon. As with the distant KBOs, of course, detection of these objects also has the strong possibility of entirely ruling out the existence of Planet Nine if they are not found with perihelia in the locations predicted by the hypothesis.