Observational bias and the clustering of distant eccentric Kuiper belt objects

TL;DR

This study assesses whether observational bias can explain the apparent clustering of distant eccentric Kuiper belt objects, finding that bias is unlikely to account for the observed orbital alignments, supporting the Planet Nine hypothesis.

Contribution

Developed a rigorous method to estimate observational bias in Kuiper belt object surveys, demonstrating that the observed clustering is statistically significant and not solely due to bias.

Findings

Observed clustering unlikely due to observational bias

Probability of uniform distribution is only 1.2% for the objects

Combined clustering probability is 0.025%

Abstract

The hypothesis that a massive Planet Nine exists in the outer solar system on a distant eccentric orbit was inspired by observations showing that the objects with the most distant eccentric orbits in the Kuiper belt have orbits which are physically aligned, that is, they are clustered in longitude of perihelion and have similar orbital planes. Questions have remained, however, about the effects of observational bias on these observations, particularly on the longitudes of perihelion. Specifically, distant eccentric Kuiper belt objects tend to be faint and only observable near their perihelia, suggesting that the longitudes of perihelion of the known distant objects could be strongly biased by the limited number of locations in the sky where deep surveys have been carried out. We have developed a method to rigorously estimate the longitude of perihelion bias for Kuiper belt observations. We find that the probability that the 10 known Kuiper belt objects with semimajor axis beyond 230 AU are drawn from a population with uniform longitude of perihelion is 1.2%. Combined with the observation that the orbital poles of these object are also clustered, the overall probability of detecting these two independent clusterings in a randomly distributed sample is 0.025%. While observational bias is clearly present in these observations, it is unlikely to explain the observed alignment of the distant eccentric Kuiper belt objects.

Full text

1. Introduction

Recently, Batygin & Brown (2016, hereafter BB16) demonstrated that the most distant known objects in the Kuiper belt have orbits which are physically aligned, that is, they are clustered in longitude of perihelion. Such an alignment is unexpected, as differential precession will destroy any such alignment on a 10-100 Myr time scale. BB16 demonstrated that a distant giant planet in an eccentric orbit – referred to here as Planet Nine – could maintain an alignment for the age of the solar system if longitude of perihelion of the eccentric orbit of Planet Nine is oriented 180 degrees away from those of the clustered Kuiper belt objects (KBOs). To date, no other viable explanation for a physical alignment of the orbits of distant eccentric KBOs has been proposed.

Previously, Trujillo & Sheppard (2014, hereafter TS14) had noted that distant KBOs were clustered in argument of perihelion, $\omega$, a parameter which corresponds not to physical alignment but to a specific internal-orientation-with-respect-to-itself of an orbit. While TS14 speculated that a distant planet might be responsible, no mechanism for clustering $\omega$ of a population of KBOs by a planetary perturber without also having a physical longitude of perihelion clustering has been found. Madigan & McCourt (2016, hereafter MM16) instead demonstrated that a massive disk of eccentric KBOs will generate an inclination instability which will naturally lead to clustering in $\omega$. To date, no other viable explanation for a clustering of $\omega$ (that does not also include clustering of longitude of perihelion) of distant eccentric KBOs has been proposed.

While the $\omega$ clustering reported by TS14 is robust and cannot be caused by any observational bias (see below), the longitude of perihelion clustering reported by BB16 is certainly subject to observational bias. As a simple example, a magnitude limited survey will preferentially find objects which are near their perihelion position where they are brightest, and if such surveys are biased in the longitudes at which they are carried out, that bias will be reflected in the longitude of perihelion distribution found. Given that biases in survey longitudes are known to exist (mainly, but not exclusively, caused by avoidance of the galactic plane), the possibility of a bias in measured longitude of perihelion should be carefully considered. BB16 made a simple argument that the most distant eccentric KBOs should not be significantly more biased than not-quite-as-distant eccentric KBOs – which show an essentially uniform distribution of longitude of perihelion – but it is clear that the bias towards finding objects at perihelion grows with eccentricity, so it is not obvious how applicable this simple argument is, particularly for the most eccentric orbits.

Because of this potential uncertainty about observational bias, the speculation that the longitude of perihelion clustering might be purely an observational selection effect has been suggested (Sheppard & Trujillo, 2016; Shankman et al., 2017; Lawler et al., 2017; Bannister et al., 2017). Assessing the impact of observational bias in longitude of perihelion is critical to understanding whether the observations point to a self-gravitating massive outer disk or to the presence of a giant ninth planet. Here we develop a rigorous method to estimate the longitude of perihelion bias for distant eccentric KBOs. We apply the method to the distant eccentric KBOs originally identified by BB16 and to those that have been identified since to assess the possibility of the presence of Planet Nine.

2. Significance of $\omega$ clustering

Before discussing biases in the longitude of perihelion, we quickly discuss the argument of perihelion, $\omega$, and show how the observed clustering around $\omega\sim 0$ cannot be caused by observational bias, even though there are clear observational biases in $\omega$ for eccentric objects. In particular, eccentric objects with $\omega$ near 0 or 180 degrees come to perihelion and are thus brightest around the heavily observed ecliptic, so one would expect eccentric objects to be found preferentially around $\omega=0$ and 180 degrees even for a uniformly distributed population. At the moment of discovery, however, an object with $\omega=\omega\prime$ and one with $\omega=180-\omega\prime$ differ only in the direction of the ecliptic latitudinal component of their velocity vectors. As pointed out by TS14, there is no possible way to design a survey to be biased in favor of finding objects close to $\omega=0$ at the expense of objects with $\omega$ close to 180 (or vice versa), yet the distant eccentric KBOs show this effect strongly.

While calculating the full observational selection bias of $\omega$ is not possible, it is trivial to calculate the probability that objects would be exclusively clustered around $\omega=0$ or around 180. In the original analysis, TS14 found that the 12 most distant eccentric KBOs – those with semimajor axis 150 AU and greater – cluster within 43 degrees of $\omega=0$. The probability that 12 such objects would cluster around either 0 or 180 is simply $2\times 2^{-12}$, or 0.04%. Note that here and throughout this paper we refer to KBOs as all multi-opposition solar system objects with perihelion distance beyond Neptune’s orbit.

Since the original work of TS14, 9 new KBOs with semimajor axis 150 AU or greater have been found. Of these, 7 are closer to $\omega=0$ than to $\omega=180.$ The probability that 19 or more of 21 objects would be so clustered is $2\times 2^{-21}\times[C(21,2)+C(21,1)+1]$ where C(n,m) is the number of independent combinations of $m$ objects from a population of $n$. The probability of this occurrence is thus just 0.022%.

No sophisticated debiasing needs to be done to show that the clustering in $\omega$ is highly significant. This strong signal – unexplained by the mechanism proposed by TS14 – led MM16 to the realization that this clustering could be caused by a massive distant disk causing an inclination instability in the outer solar system. A distant eccentric Planet Nine, in contrast, clusters longitude of perihelion and pole position, rather than $\omega$ (Brown & Batygin, 2017). A population of longitude-of-perihelion-aligned orbits with poles clustered around a position offset from the north ecliptic pole will generally, but not exclusively, also have clustered $\omega$. Unfortunately no simple calculation gives pole position bias, so we continue to use the clustering in $\omega$ as an imperfect statistical proxy for clustering in pole position.

Clustering of distant eccentric KBOs in $\omega$ (or, alternatively, in pole position) is firmly established. To date, the only viable explanations for this clustering is either the inclination instability proposal of MM16 or the Planet Nine proposal of BB16. These proposals differ most in their predicted distribution of longitude of perihelion. The inclination instability shows no preference for clustering in longitude of perihelion, while Planet Nine confines the longitudes.

If in fact there is no longitude of perihelion clustering, the robust clustering in $\omega$ is currently only explainable by the presence of a massive outer disk of material inducing a inclination instability through self-gravity, as proposed by MM16. If, on the other hand, the clustering in longitude of perihelion is a true effect, rather than an apparent one caused by observational bias, Planet Nine remains the only currently proposed explanation. We now examine observational biases in longitude of perihelion to determine which of these hypotheses appears more likely.

3. Observational bias in longitude of perihelion

The best method for determining the effects of observational bias on the known KBOs would be to have complete information of all of the surveys conducted to date, including their depth, precise coverage, and their efficiency. Such information is unknown for the majority of the surveys that led to the discoveries of the cataloged KBOs. In many cases, nothing is published about the discovery survey; the existence of the object is simply cataloged by the IAU Minor Planet Center (see http://www.minorplanetcenter.net/iau/Unusual.html). The cataloged information is sufficient to determine the ecliptic longitude, ecliptic latitude, heliocentric distant, and brightness of every object at the time of its discovery.

We develop a novel method to use the discovery circumstances of the ensemble of all KBOs in the catalog to rigorously estimate the statistical distribution of longitudes of perihelion expected for distant eccentric KBOs. Conceptually, the method relies on the idea that each KBO discovery can be thought of as a survey that could have discovered a distant eccentric KBO had that object been bright enough and in the same place. We proceed as follows: for each distant eccentric KBO (a ”parent object”) we construct a synthetic population of new objects assuming an identical absolute magnitude and identical orbital elements for a uniformly selected longitude of perihelion and mean anomaly (in practice we also assume symmetry across the ecliptic plane, so our constructed population also includes orbits where we replace $\omega$ with $-\omega$). We tally the ecliptic longitude, ecliptic latitude, heliocentric distance, and expected magnitude of each object in the synthetic population. We call this the ”uniform population” of the parent KBO. Next, for every KBO discovery in the catalog we assess whether or not one of the uniform population of the parent object exists at the ecliptic longitude and latitude of the observation and if that member of the uniform population is brighter than the actual detected KBO. If so, we known that a survey was being undertaken at that point that could have detected one of the uniform population. (In practice, we look for members of the uniform population within 1 degree of the discovery location of the KBO discovery.) Finally, we tabulate the longitudes of perihelion of the members of the uniform population that could have been detected at that discovery location. We now know that the survey that resulted in that particular KBO discovery was sensitive to members of our uniform population if they had had a particular longitude of perihelion. This procedure is repeated for every cataloged KBO discovery to determine the probability distribution function of the longitude of perihelion of the parent KBO assuming that the population is uniformly distributed in longitude of perihelion and mean anomaly.

A concrete example makes this procedure more clear. Consider 2013 RF98, the most eccentric of the objects originally identified by BB16, as the parent object of a uniform population. Next, consider the discovery of a randomly cataloged KBO, 2015 GP50, which, at the moment of discovery, had an ecliptic latitude of -11.2 degrees and a magnitude of 24.8. Examining the orbit of 2013 RF98, we find that it crosses -11.2 degrees twice, once 27 degrees from perihelion, when it has a magnitude of 24.6, and once closer to aphelion, when it has a magnitude of 28.7. Near aphelion, the uniform population would not have been detectable at this latitude, but at its magnitude closer to perihelion it could have been detected by the observation that discovered 2015 GP50. The KBO 2015 GP50 was discovered at a longitude of 196 degrees, thus the member of the uniform population that is detectable has a longitude of perihelion of $196-27$=168 degrees. This specific observation is thus biased to finding this specific longitude of perihelion for this specific parent object. If we now consider not a single KBO discovery, but all KBO discoveries, we find a statistical distribution of the longitudes of perihelion in which discoveries of the 2013 RF98 uniform population could have been made. We thus create a separate statistical distribution of expected longitudes of perihelion for each distant eccentric KBO.

This conceptual framework relies on the assumption that KBO discoveries roughly represent the coverage and depth of the combined surveys. This assumption is clearly false for the latitude distribution, where more KBOs are discovered at low ecliptic latitudes simply because of their greater numbers. We correct this bias by scaling by expected density of KBOs at a given latitude. To approximate this expected density we use the method developed by Brown (2001) to determine the inclination distribution and convert it to a latitudinal distribution assuming circular orbits. The final results are not sensitive to the precise latitudinal distribution chosen.

A second way in which the assumption that KBO discoveries are uniform with search area is violated is in the known longitudinal bias in the discovery of resonant KBOs, which are over-discovered near their perihelion positions, which are related to the position of Neptune. The easiest way to avoid this problem is to discard all discoveries of Plutinos, which are, by far, the most numerically prominent and most spatially correlated of the resonant objects. In practice we simply discard all discovered objects with semimajor axes below 40 AU. This constraint also forces us to only retain multi-opposition KBOs with orbits known accurately enough to calculate this parameter. We note, however, that relaxing this assumption makes the final results of this analysis more significant. Nonetheless, we conservatively retain this constraint.

One other important assumption is that a distant eccentric KBO could always be discovered if a closer but fainter KBO was discovered at its predicted location. This assumption can be violated if the discovery survey is not sensitive to distant objects. Some of the nearest KBOs and many Centaurs, for example, have been discovered in surveys searching for near-earth objects, which do not have observational baselines sufficiently long to be sensitive to more slowly moving distant objects. To exclude these surveys we will only consider discoveries of objects at distances greater than 30 AU. Even for normal KBO surveys, some of the distant eccentric KBOs might not be detectable due to their low rate of motion even though they are still brighter than the magnitude limit. Sedna, for example, would be visible to a distance of 225 AU to a survey that went to a depth of 25th magnitude, but few surveys are sensitive to the slow motions of such distant objects. We will thus place an upper bound of 90 AU on the most distant object that any survey could see. This value is probably a conservative estimate and will have the effect of making longitude of perihelion biases stronger than they might be in real life. In total, we use the observations of 1248 objects to determine our expected distributions.

Using all of these constraints, we calculate expected statistical distributions of longitude of perihelion for each of the 10 known KBOs with semimajor axis beyond 230 AU. These include the six originally identified by BB16 and the four that have been discovered since that time (Figure 1). The expected longitude of perihelion distributions are highly structured and highly individual. One trend is easily seen. The brightest objects (Sedna, 2007 TG422) are among the most uniform in their expected discovery distributions. Many surveys could have found the very bright Sedna even quite far from its perihelion, for example. The structure seen in the remaining objects is only understandable after analysis. The distributions of 2012 VP113 and 2013 RF98, for example, are the most non-uniform. These two objects both come to perihelion at high ecliptic latitude, so their populations are primarily observable at high latitude, yet few surveys reach the required depth at these latitudes. The longitudes of perihelion of these objects are highly biased by the limited number of surveys that could have detected such a population. The structures in the other distributions are similarly functions of perihelion latitude, brightness, and the distribution of surveys in the sky.

4. Comparison to observations

We now see that measurement of the longitude of perihelion for a distant eccentric population is highly biased by the specific area of the sky and depth of individual surveys, as expected. With our determination of this bias, we can now examine whether the discoveries of distant eccentric KBOs are consistent with being selected from a distribution which is uniform in longitude of perihelion or if, indeed, they are clustered.

We first consider the six distant KBOs which BB16 reported as clustered in longitude of perihelion: Sedna, 2004 VN112, 2007 TG422, 2010 GB174, 2012 VP113, and 2013 RF98. At the time, these were all of the known KBOs with semimajor axis beyond 230 AU. To understand the statistically expected distribution of longitudes of perihelion for these bodies assuming a uniformly distributed population, we perform 100,000 population samplings in which we create a new selection of 6 detected KBOs by randomly choosing a longitude of perihelion from the expected probability density function for each of the 6 objects. We then examine the statistics of these 100,000 realizations.

The longitudes of perihelion of the 6 real objects are distributed such that the maximum angle between any pair of angularly adjacent objects is 260.9 degrees (Figure 2). For the 100,000 realizations of this population assuming a uniform distribution in longitude of perihelion, the maximum angle between two angularly adjacent objects is 260.9 degrees or higher in only 1437 cases. If the longitudes of perihelia of distant eccentric objects are uniformly distributed, we would expect a longitude clustering as tight as the one observed only 1.4% of the time. We also compute the Rayleigh $z$ statistic of the data and the random sample and find that Rayleigh $z$ value of 0.80 of the data is exceeded only 2.0% of the time in the random data. In contrast, the simple estimate from BB16 (which also took into account the clustering in perihelion latitude which is ignored here) suggested that the clustering should only be observed 0.7% of the time. We regard the rough agreement from independent ways of estimating the significance of this result as encouraging.

With only the six objects defined here, the probability of $\omega$ clustering becomes $2\times 2^{-6}$ or 3.1%. Restating these two findings, we see that there is only a 3.1% chance that the values for $\omega$ are equally distributed about 0 and 180 degrees, suggesting that some mechanism is clustering the distant KBOs in $\omega.$ We likewise find that there is only a 1.4% chance that the longitudes of perihelia are distributed uniformly, suggesting that these values are likewise clustered. The combined probability that both of these clusters would be found in random data is thus 0.043%.

Since the original analysis of BB16, Brown & Batygin (2016) demonstrated that the Planet Nine hypothesis, in addition to clustering distant eccentric KBOs in longitudes of perihelion opposite to that of Planet Nine (the ”anti-aligned population”), a smaller population of objects with longitudes of perihelion aligned with Planet Nine should also exist (the ”aligned population”). Four new distant eccentric KBOs (with semimajor axis greater than 230 AU) have been discovered since the initial analysis (Sheppard & Trujillo, 2016; Bannister et al., 2017). Of these, three fit well with the anti-aligned population (2013 FT28, 2014 FE72, and 2013 SY99), while one is consistent with being the first recognized member of the aligned population (2014 SR349). The realization that we now expect two separate oppositely-oriented populations requires a different metric for assessing the match between the expected and observed population. Instead of examining the largest angle between the longitudes of perihelion of any two angularly adjacent objects, we look at the second largest angle between any two angularly adjacent objects. In a population with two well separated oppositely oriented groups, this second largest angle will be maximized. In practice, we would have also considered the observed population of objects clustered if there had only been a single cluster rather than two. Such a cluster would have a large largest angle but a small second largest angle. To overcome this problem we take either the second largest angle or half of the largest angle, effectively mimicking the effects of two populations even if only one is observed. In the real population of 10 distant eccentric KBOs, the separation in longitude of perihelion between the anti-aligned group and 2013 SR349 is 139.5 degrees in one direction and 118.1 in the other direction. Our second largest angle is thus 118.1 degrees. We again perform 100,000 random iterations and compare this expected population to the real observations. In only 1201 cases is the second-largest separation between longitudes of perihelion as large or larger than 118.1 degrees. The probability that the distant eccentric KBOs would be distributed in longitude of perihelion as extremely as the observations are if the underlying distribution were uniform is 1.0%. The calculated Rayleigh $z$ statistic of this population (which is only sensitive to a unimodal distribution) with a value of 0.62 is exceeded in 2.2% of the random sample. While the discovery of four new distant eccentric KBOs might have been expected to increase our confidence in these populations significantly, the realization that we are observing two opposing rather than one single clustered population necessarily dilutes the signal.

One of these objects (2014 FT28) has $\omega=286$ degrees, while the rest are clustered about zero. Sheppard & Trujillo (2016) use this fact to exclude 2014 FT28 from consideration as part of the clustered population, however Brown & Batygin (2017) show that such objects are indeed expected in the Planet Nine hypothesis. They are simply precessing around a displaced pole but have temporarily circulated past the north ecliptic pole. Ecliptic-based Keplerian orbital elements are a poor descriptor of the orbits in this case. Nonetheless, we continue to use the simple measure of $\omega$ as a proxy for pole clustering and find that if the true values of $\omega$ are uniform about 0 and 180, the probability that 9 or more of 10 values for $\omega$ would be clustered about either is 2.1%. The combined probability of both of these parameters being clustered is 0.025%.

The original $\omega$ clustering discussed by TS13 which led to the MM16 hypothesis of inclination instability includes all KBOs with semimajor axis 150 AU and larger. To date, 21 such KBOs are known, and 19 have $\omega$ closer to 0 than to 180 degrees. The probability of this clustering occurring randomly is a mere 0.022%. Examination of the longitude of perihelion (Figure 2) shows that this parameter, too, has some structure down to a semimajor axis of 150 AU, but it is clear that the clustering in longitude of perihelion is beginning to break down. This behavior was seen in the population simulations of Brown & Batygin (2017) where it was noted that the longitude of perihelion merged from being highly clustered at large semimajor axis, to moderately to not-at-all clustered as semimajor axis decreased. We thus do not expect the longitude of perihelion cluster of objects beyond 150 AU to be as significant as those beyond 230 AU that we initially considered. We nonetheless assess the observational biases. Once again performing 100,000 iterations of a population uniformly distributed in in longitude of perihelion we find that 43590 have a second-largest longitude of perihelion angle between two KBOs as large or larger than 48.2 degrees, the value seen in the real data (while 39753 have a largest angle of 67.9 or greater like the data). The Rayleigh $z$ test likewise shows no significance to this clustering. In short, for objects with semimajor axis 150 AU and larger there is no statistically significant clustering of longitude of perihelion into one or two groups, yet the cluster of $\omega$ is highly significant. This discrepancy shows, we believe, the expected blending of the high semimajor axis longitudinally clustered objects into the lower semimajor axis background population and the uncertainty as to where precisely to draw the line for distant objects which are and which are not affected by Planet Nine. At greater semimajor axes, the longitude of perihelion clustering is robust as expected.

5. Conclusions

We have shown that measurement of the longitude of perihelion of a population of distant eccentric KBOs is subject to considerable observational bias, but that this bias is unlikely to be responsible for the observed clustering in longitude of perihelion. If distant eccentric KBOs were uniformly distributed in longitude of perihelion, observations of the original six objects with semimajor axis beyond 230 AU which led BB16 to suggest the existence of Planet Nine would only find the extreme clustering observed 1.4% of the time. Including the four most recently discovered KBOs with semimajor axes beyond 230 AU drops that probability to 1.2%.

Clustering in $\omega$ of distant eccentric KBOs is firmly established. Thus determination of the veracity of the clustering in longitude of perihelion is critical to understanding the gravitational forces sculpting the outer solar system. With no longitude of perihelion clustering, the only currently proposed viable mechanism for causing $\omega$ clustering is the MM16 suggestion of a massive outer disk causing an inclination instability and $\omega$ clustering. If, on the other hand, longitude of perihelion and pole position (which roughly manifests itself as $\omega$ for an offset pole) are clustered, Planet Nine is the only currently proposed viable hypothesis. By rigorously estimating the effects of observational bias, we have shown here that the Planet Nine hypothesis is by far the more likely of these scenarios. The probability that the combination of the alignment of the longitudes of perihelion with the clustering in pole position (using the $\omega$ proxy) that is seen in the KBOs with semimajor axes beyond 230 AU would occur by chance in a uniformly distributed population is only 0.025%. While explanations other than Planet Nine might one day be found to explain these observations, the significance of the observations themselves appears secure.

We would like to thank Ann-Marie Madigan for the discussion which inspired this analysis and Elizabeth Bailey, Konstantin Batygin, and Ian Wong for critical readings of the manuscript.