Draft version August 30, 2021
Typeset using LATEX preprint2 style in AASTeX63
The Orbit of Planet Nine
E. Brown1 and Konstantin Batygin1
1Division of Geological and PMlanetary Sciences California Institute of Technology
Pasadena, CA 91125, USichaelA
(Accepted 22 Aug 2021)
Submitted to AJ
ABSTRACT
The existence of a giant planet beyond Neptune – referred to as Planet Nine (P9) – has been inferred from the clustering of longitude of perihelion and pole position of distant eccentric Kuiper belt objects (KBOs). After updating calculations of observa- tional biases, we find that the clustering remains significant at the 99.6% confidence level. We thus use these observations to determine orbital elements of P9. A suite of numerical simulations shows that the orbital distribution of the distant KBOs is strongly influenced by the mass and orbital elements of P9 and thus can be used to infer these parameters. Combining the biases with these numerical simulations, we cal- culate likelihood values for discrete set of P9 parameters, which we then use as input into a Gaussian Process emulator that allows a likelihood computation for arbitrary values of all parameters. We use this emulator in a Markov Chain Monte Carlo analysis to estimate parameters of P9. We find a P9 mass of 6.2+2.2 Earth masses, semimajor axis of 380+140 AU, inclination of 16 ± 5○ and perihelion of 300+85 AU. Using samples
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of the orbital elements and estimates of the radius and albedo of such a planet, we cal-
culate the probability distribution function of the on-sky position of Planet Nine and of its brightness. For many reasonable assumptions, Planet Nine is closer and brighter than initially expected, though the probability distribution includes a long tail to larger distances, and uncertainties in the radius and albedo of Planet Nine could yield fainter objects.
INTRODUCTION
Hints of the possibility of a massive planet well beyond the orbit of Neptune have been emerg- ing for nearly twenty years. The first clues came from the discovery of a population of distant ec- centric Kuiper belt objects (KBOs) decoupled
Corresponding author: Michael Brown mbrown@caltech.edu
from interactions with Neptune (Gladman et al. 2002; Emel’yanenko et al. 2003; Gomes et al. 2006), suggesting some sort of additional grav- itational perturbation. While the first such de- coupled objects were only marginally removed from Neptune’s influence and suggestions were later made that chaotic diffusion could create similar orbits (Bannister et al. 2017), the dis- covery of Sedna, with a perihelion far removed from Neptune, clearly required the presence of
a past or current external perturber (Brown et al. 2004). Though the orbit of Sedna was widely believed to be the product of pertur- bation by passing stars within the solar birth cluster (Morbidelli & Levison 2004; Schwamb et al. 2010; Brasser et al. 2012), the possibility of an external planetary perturber was also noted (Brown et al. 2004; Morbidelli & Levison 2004; Gomes et al. 2006). More recently, Gomes et al. (2015) examined the distribution of objects with very large semimajor axes but with perihelia in- side of the planetary regime and concluded that their overabundance can best be explained by the presence of an external planet of mass ∼10 Me (where Me is the mass of the Earth) at a distance of approximately 1000 AU. Simultane- ously, Trujillo & Sheppard (2014) noted that distant eccentric KBOs with semimajor axis a > 150 AU all appeared to come to perihelion approximately at the ecliptic and always travel- ling from north-to-south (that is, the argument of perihelion, ω, is clustered around zero), a sit- uation that they speculated could be caused by Kozai interactions with a giant planet, though detailed modeling found no planetary configu- ration that could explain the observations.
These disparate observations were finally uni- fied with the realization by Batygin & Brown (2016) that distant eccentric KBOs which are not under the gravitational influence of Nep- tune are largely clustered in longitude of peri- helion, meaning that their orbital axes are ap- proximately aligned, and simultaneously clus- tered in the orbital plane, meaning that their angular momentum vectors are approximately aligned (that is, they share similar values of in- clination, i, and longitude of ascending node, Ω). Such a clustering is most simply explained by a giant planet on an inclined eccentric or- bit with its perihelion location approximately
180 degrees removed from those of the clus- tered KBOs. Such a giant planet would not only explain the alignment of the axes and or-
bital planes of the distant KBOs, but it would also naturally explain the large perihelion dis- tances of objects like Sedna, the overabundance of large semimajor axis low perihelion objects, the existence of a population of objects with orbits perpendicular to the ecliptic, and the ap- parent trend for distant KBOs to cluster about ω = 0 (the clustering near ω = 0 is a coin- cidental consequence of the fact that objects sharing the same orbital alignment and orbital plane will naturally come to perihelion at ap- proximately the same place in their orbit and, in the current configuration of the outer solar system, this location is approximately centered at ω ∼ −40○). The hypothesis that a giant planet on an inclined eccentric orbit keeps the axes and planes of distant KBOs aligned was called the Planet Nine hypothesis.
With one of the key lines of evidence for Planet Nine being the orbital clustering, much emphasis has been placed on trying to assess whether or not such clustering is statistically significant or could be a product of observa- tional bias. In analyses of all available contem- porary data and their biases, Brown (2017) and Brown & Batygin (2019, hereafter BB19) find only a 0.2% chance that the orbits of the dis- tant Kuiper belt objects (KBOs) are consistent with a uniform distribution of objects. Thus the initial indications of clustering from the origi- nal analysis appear robust when an expanded data set that includes observations taken over widely dispersed areas of the sky are considered. In contrast, Shankman et al. (2017), Bernar- dinelli et al. (2020), and Napier et al. (2021) using more limited – and much more biased – data sets, were unable to distinguish between clustering and a uniform population. Such dis- crepant results are not surprising: BB19 showed that the data from the highly biased OSSOS survey, which only examined the sky in two dis- tinct directions, do not have the sensitivity to detect the clustering already measured for the full data set. Bernardinelli et al. (2020) recog- nize that the sensitivity limitations of the even- more-biased DES survey, which only examined the sky in a single direction, precluded them from being able to constrain clustering. It ap- pears that Napier et al., whose data set is dom- inated by the combination of the highly-biased OSSOS and DES surveys, suffers from similar lack of sensitivity, though Napier et al. do not provide sensitivity calculations that would allow this conclusion to be confirmed. Below, we up- date the calculations of BB19 and demonstrate that the additional data now available contin- ues to support the statistical significance of the clustering. We thus continue to suggest that the Planet Nine hypothesis remains the most viable explanation for the variety of anomolous behaviour seen in the outer solar system, and we work towards determining orbital parameters of Planet Nine.
Shortly after the introduction of the Planet Nine hypothesis, attempts were made to con- strain various of the orbital elements of the planet. Brown & Batygin (2016) compared the observations to some early simulations of the ef- fects of Planet Nine on the outer solar system and showed that the data were consistent with a Planet Nine with a mass between 5 and 20 Earth masses, a semimajor axis between 380 and 980 AU, and a perihelion distance between 150 and 350 AU. Others sought to use the possibility that the observed objects were in resonances to determine parameters (Malhotra et al. 2016), though Bailey et al. (2018) eventually showed that this route is not feasible. Millholland & Laughlin (2017) invoked simple metrics to com- pare simulations and observations, and Batygin et al. (2019) developed a series of heuristic met- rics to compare to a large suite of simulations and provided the best constraints on the orbital elements of Planet Nine to date.
Two problems plague all of these attempts at deriving parameters of Planet Nine. First, the
metrics used to compare models and observa- tions, while potentially useful in a general sense, are ad hoc and difficult to justify statistically. As importantly, none of these previous meth- ods has attempted to take into account the ob- servational biases of the data. While we will demonstrate here that the clustering of orbital parameters in the distant Kuiper belt is un- likely a product of observational bias, observa- tional bias does affect the orbital distribution of distant KBOs which have been discovered. Ignoring these effects can potentially bias any attempts to discern orbital properties of Planet Nine.
Here, we perform the first rigorous statisti- cal assessment of the orbital elements of Planet Nine. We use a large suite of Planet Nine sim- ulations, the observed orbital elements of the distant Kuiper belt, as well as the observa- tional biases in their discoveries, to develop a detailed likelihood model to compare the obser- vations and simulations. Combining the likeli- hood models from all of the simulations, we cal- culate probability density functions for all or- bital parameters as well as their correlations, providing a map to aid in the search for Planet Nine.
The existence of a massive, inclined, and ec- centric planet beyond ∼250 AU has been shown to be able to cause multiple dynamical effects, notably including a clustering of longitude of perihelion, $, and of pole position (a combi- nation of longitude of ascending node, Ω, and inclination, i) for distant eccentric KBOs. Criti- cally, this clustering is only strong in sufficiently distant objects whose orbits are not strongly af- fected by interactions with Neptune (Batygin & Brown 2016; Batygin et al. 2019). Objects with perihelia closer to the semimajor axis of Nep- tune, in what is sometimes referred to as the “scattering disk,” for example, have the strong clustering effects of Planet Nine disrupted and are more uniformly situated (i.e. Lawler et al. 2017). In order to not dilute the effects of Planet Nine with random scattering caused by Neptune, we thus follow the original formula- tion of the Planet Nine hypothesis and restrict our analysis to only the population not inter- acting with Neptune. In Batygin et al. (2019) we use numerical integration to examine the or- bital history of each known distant object and classify them as stable, meta-stable, or unsta- ble, based on the speed of their semimajor axis diffusion. In that analysis, all objects with q < 42 AU are unstable with respect to peri- helion diffusion, while all objects with q > 42 AU are stable or meta-stable. Interestingly, 11 of the 12 known KBOs with a > 150 AU and q > 42 AU have longitude of perihelion clus- tered between 7 < $ < 118○, while only 8 of 21 with 30 < q < 42 AU are clustered in this re- gion, consistent with the expectations from the Planet Nine hypothesis. We thus settle on se- lecting all objects at a > 150 AU with perihe- lion distance, q > 42 AU for analysis for both the data and for the simulations below.
A second phenomenon could also dilute the clustering caused by Planet Nine. Objects which are scattered inward from the inner Oort cloud also appear less clustered than the longer- term stable objects (Batygin & Brown 2021). These objects are more difficult to exclude with a simple metric than the Neptune-scattered ob- jects, though excluding objects with extreme semimajor axes could be a profitable approach. Adopting our philosophy from the previous sec- tion, we exclude the one known object in the sample with a > 1000 AU as possible contam- ination from the inner Oort cloud. While we again cannot know for sure if this object is in- deed from the inner Oort cloud, removing the object can only have the effect of decreasing our sample size and thus increasing the uncertain- ties in our final orbit determination, for the po-
Table 1. Orbital elements of all reported ob- jects with 150 < a < 1000 and q > 42 AU.
name | a AU | e | i deg. | Ω deg. | a deg. |
2000CR105 | 218 | 0.80 | 22.8 | 128.3 | 85.0 |
2003VB12 | 479 | 0.84 | 11.9 | 144.3 | 95.8 |
2004VN112 | 319 | 0.85 | 25.6 | 66.0 | 32.8 |
2010GB174 | 351 | 0.86 | 21.6 | 130.8 | 118.2 |
2012VP113 | 258 | 0.69 | 24.1 | 90.7 | 24.2 |
2013FT28 | 312 | 0.86 | 17.3 | 217.8 | 258.3 |
2013RA109 | 458 | 0.90 | 12.4 | 104.7 | 7.5 |
2013SY99 | 694 | 0.93 | 4.2 | 29.5 | 61.6 |
2013UT15 | 197 | 0.78 | 10.7 | 192.0 | 84.1 |
2014SR349 | 302 | 0.84 | 18.0 | 34.8 | 15.7 |
2015RX245 | 412 | 0.89 | 12.1 | 8.6 | 73.7 |
Note—As of 20 August 2021.
tential gain of decreasing any biases in our final results.
The sample with which we will compare our observations thus includes all known multi- opposition KBOs with 150 < a < 1000 AU and q > 42 AU reported as of 20 August 2021. Even after half of a decade of intensive search for dis- tant objects in the Kuiper belt, only 11 fit this stringent criteria for comparison with models. The observed orbital elements of these 11 are shown in Table 1. These objects are strongly clustered in $ and pole position, though obser- vational biases certainly can affect this observed clustering.
All telescopic surveys contain observational biases. Correctly understanding and imple- menting these biases into our modeling is crit- ical to correctly using the observations to ex- tract orbital parameters of Planet Nine. BB19 developed a method to use the ensemble of all known KBO detections to estimate a full geo- metric observational bias for individual distant
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a (AU)
Figure 1. Semimajor axes versus a for the 11 KBOs of our sample (green points). The points are plotted as ∆a, defined as a − a9, where here we plot the points for an assumed value of a9 = 254○. For each known distant KBO we show a one-dimensional projection of the bias with respect to a (blue). While consistent bias exists, the a cluster is approximately 90○ removed from the direction of bias. We also show the probability density of a versus semimajor axis in the maximum likelihood model with m9 = 5 Me, a9 = 300 AU, e9 = 0.15 and i9 = 16○ (red). The density plot is normalized at every semimajor axis to better show the longitudinal structure. Note that this comparison is simply for visualization; the full maximum-likelihood model compares the full set of orbit elements of each object to the simulations and also incorporates the observational biases on each observed objects.
KBOs. For each of the distant KBOs, they cre- ate the function
B(a,e,H)j [(i, $, Ω)|U ], (1)
where, for our case, j represents one of the 11 distant KBOs of the sample and B(a,e,H)j is the probability that distant KBO j, with semima- jor axis, eccentricity, and absolute magnitude (a, e, H)j would be detected with orbital angles i, $, and Ω, if the population were uniformly distributed in the sky, given U , the ensemble of all known KBO detections. The details of the method are explained in BB19, but, in short, it relies on the insight that every detection of ev-
ery KBO can be thought of (with appropriate caveats) as an observation at that position in the sky that could have detected an equivalent object j with (a, e, H)j if, given the required or- bital angles (i, $, Ω) to put object j at that po- sition in the sky, the object would be predicted to be as bright as or brighter at that sky position than the detected KBO. For each sample object j with (a, e, H)j, the ensemble of all KBO de- tections can thus be used to estimate all of the orbital angles at which the object could have been detected. This collection of orbital angles at which an object with (a, e, H)j could have been detected represents the bias in (i, $, Ω)
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relative probability
Figure 2. A comparison of the projection of the pole position of the distant detached KBOs (the green points show (sin i cos ∆Ω, sin i sin ∆Ω), where ∆Ω is the difference between the longitude of ascending node of the observed object and of the modeled Planet Nine, assumed to be 108○ here) and a density plot of their expected values in the maximum likelihood model (red background). In blue we show an average of the two-dimensional projection of the pole position bias of all of the objects. While strong bias in pole position exists, no preferential direction is apparent. White circles indicate 30 and 60 degree inclinations.
for object j. While biases calculated with this method are strictly discrete, we smooth to one degree resolution in all parameters for later ap- plication to our dynamical simulations.
Note that this method differs from bias calcu- lations using full survey simulators. It does not rely on knowledge of the survey details of the detections, but rather just the fact of the de- tection itself. Comparison of these bias calcula- tions with the bias calculated from a full survey
similar for the OSSOS survey shows comparable results (BB19).
Of the objects in our sample, all were included in the BB19 calculations with the exception of 2013 RA109, which had not been announced at the time of the original publication. We repro- duce the algorithm of BB19 to calculate a bias probability function for this object.
While the bias is a separate 3-dimensional function for each object, we attempt to give an approximate visual representation of these
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