OBSERVATIONAL CONSTRAINTS ON THE ORBIT AND LOCATION OF PLANET NINE IN THE OUTER SOLAR SYSTEM
Michael E. Brown & Konstantin Batygin Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125
Draft version June 13, 2016
ABSTRACT We use an extensive suite of numerical simulations to constrain the mass and orbit of Planet Nine, the recently proposed perturber in a distant eccentric orbit in the outer solar system. We compare our simulations to the observed population of aligned eccentric high semimajor axis Kuiper belt objects and determine which simulation parameters are statistically compatible with the observations. We find that only a narrow range of orbital elements can reproduce the observations. In particular, the combination of semimajor axis, eccentricity, and mass of Planet Nine strongly dictates the semimajor axis range of the orbital confinement of the distant eccentric Kuiper belt objects. Allowed orbits, which confine Kuiper belt objects with semimajor axis beyond 380 AU, have perihelia roughly between 150 and 350 AU, semimajor axes between 380 and 980 AU, and masses between 5 and 20 Earth masses. Orbitally confined objects also generally have orbital planes similar to that of the planet, suggesting that the planet is inclined approximately 30 degrees to the ecliptic. We compare the allowed orbital positions and estimated brightness of Planet Nine to previous and ongoing surveys which would be sensitive to the planet’s detection and use these surveys to rule out approximately two-thirds of the planet’s orbit. Planet Nine is likely near aphelion with an approximate brightness of 22 < V < 25. At opposition, its motion, mainly due to parallax, can easily be detected within 24 hours. 1. INTRODUCTION 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 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 (Morbidelli & Levison 2004; Kenyon & Bromley 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 welldetermined orbits of Kuiper belt objects (KBOs) beyond Neptune with semimajor axis, a, larger than 227 AU approach perihelion within 94 degrees of longitude of each other. Moreover, these objects also share very nearly the same orbital plane, which is tilted an average of 22 degrees 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 > 100 AU, the five with the largest perihelion distances are likewise mbrown@caltech.edu, kbatygin@gps.caltech.edu 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 Kuiper belt objects 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 Kuiper belt objects come to perihelion 180 degrees 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. 2. CONSTRAINTS ON MASS, SEMIMAJOR AXIS, AND ECCENTRICITY 2 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 which 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 1a shows the perihelion longitude and latitude as well as the pole angle for all objects with q > 30 and a > 60 AU and well-determined orbits. The seven objects with a > 227 AU are highlighted in red. The clustering in perihelion location as well as pole angle is clearly visible. In Figure 1b we plot the perihelion longitude of all well constrained orbits in the Kuiper belt which have perihelia beyond the orbit of Neptune as a function of semimajor axis. The 7 objects with a > 227 AU cluster within 94 degrees 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 degrees in longitude away from the cluster, or an ecliptic longitude of 241 ± 15 degrees. A closer examination of Figure 1b 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 degrees 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 a Planet Nine. 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 a9 = 200 − 2000 AU 360 270 180 90 0 perihelion longitude -60 -40 -20 0 20 40 60 perihelion latitude 0 100 200 300 400 500 semimajor axis (AU) 0 90 180 270 360 perihelion longitude Fig. 1.— Orbital parameters of distant Kuiper belt objects. (a) The standard orbital parameters argument of perihelion, longitude of ascending node, and inclination can be transformed into nonstandard, but more readily interpretable ecliptic longitude and latitude of the point where the object comes to perihelion and an angle which is a projection of the orbits pole position on the sky. In this representation, the collection of all objects with q > 30 and a > 60 AU is shown. The six objects with the highest semimajor axis are highlighted in red. The KBO 2000 CR105, which has the seventh largest semimajor axis and has an elevated perihelion of 44 AU, is shown in green. The blue points are all of the object with a > 227 AU and i > 50 degrees. All of these objects are Centaurs with perihelia inside of 15 AU. (b) A plot of the semimajor axis versus the ecliptic longitude at which the object comes to perihelion for all KBOs with well determined orbits and with q > 30 AU shows that the seven KBOs with the largest semimajor axes are clustered within 94 degrees of each other. The green points additionally highlight all objects with a > 100 AU and q > 42 AU, showing that these objects, too, are similarly clustered. The weaker potential clustering of objects ∼180 degrees away is also evident between 100 and 200 AU. and e9 = 0.1 − 0.9 in increments of ∆a9 = 100 AU and ∆e9 = 0.1 (here, and subsequently, the 9 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 Me (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 symplecticBulirsch-Stoer algorithm with a timestep equal to a 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 3 we restrict ourselves to objects with instantaneous perihelion q < 80 AU (to restrict ourselves to objects which are most likely to be observable). In this fashion we are examining stream functions of orbital elements which 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 degrees and the 7 objects a = [227 − 600] AU range are confined within 94 degrees. 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 < a < 220 AU and q > 30, zero have perihelion greater than 42 AU (whereas 5 of the 7 objects with a > 227 AU have such elevated perihelia). These simulations were not designed in such a way that simple assessment of the probability of such a distrubtion is straightforward. We approximate this probability by collecting all objects from time steps between 2 and 4 Gyr with 30 < q < 40 AU and 100 < a < 200 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 < 42 and 100 < a < 200 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 straightwordly 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 < a < 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 Me simulations (the 0.1, 1, and 30 Me 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 < a < 200 AU strongly rules out all of the low semimajor axes and nearly all of the highest eccentricities, while the need 200 400 600 800 1000 semimajor axis (AU) 0.2 0.4 0.6 0.8 eccentricity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.11 2.33 4.75 3.17 0.00 3.32 0.00 0.00 0.00 0.13 0.17 0.09 0.00 5.14 0.00 0.00 0.00 0.00 0.03 0.02 0.00 0.05 0.00 0.00 0.94 0.00 0.00 0.01 0.05 0.01 0.04 0.04 0.00 1.88 0.00 0.00 0.02 0.01 0.00 0.02 0.15 0.17 0.00 0.00 0.00 0.00 0.00 0.02 0.02 0.16 0.02 0.00 0.00 0.00 Fig. 2.— Only a limited set of simulated 10 Me Planet Nine orbits provide an adequate fit to the known orbits of the distant KBOs. For each semimajor axis - eccentricity combination, we calculate the probability that 7 objects selected at random with 227 < a < 600, q < 80 AU and at times t > 3 Gyr are clustered within 94 degrees of each other combined with the probability that 13 objects randomly selected with 100 < a < 200 and q < 80 AU. In addition we discard simulations with an unacceptably large probability of creating high perihelia in low semimajor axis objects. Acceptable simulations (with a probability of greater than 1%) occupy a narrow range of a9-e9 space. 600 800 1000 1200 1400 semimajor axis (AU) 0.2 0.4 0.6 0.8 eccentricity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.01 0.00 0.12 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.01 0.05 0.00 1.03 0.00 0.00 0.00 0.00 0.00 0.02 0.01 0.01 0.00 0.00 0.00 0.00 0.01 0.01 0.00 0.03 0.01 0.28 2.78 0.00 0.00 0.02 0.02 0.01 0.05 0.00 0.02 0.12 0.00 0.00 0.04 0.01 0.04 0.04 0.03 0.31 0.00 0.00 0.00 0.00 0.00 0.02 0.01 0.03 0.02 0.00 0.00 0.00 0.02 0.03 0.02 0.01 0.00 0.01 0.04 0.12 0.00 Fig. 3.— For a 20 Me Planet Nine with a9 = [800, 1000] AU and e9 = [0.6, 0.7] a marginal fit appears. At higher masses no simulations can reproduce the observations. 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 vs. e9 with acceptable solutions. In fact, the range is sufficiently narrow that the it is clear that the grid spacing of our simulations is often too large to capture acceptable solutions at all semimajor axes or eccentricities. Nonetheless, from the 10 Me simulations we can discern the narrow range of acceptable results (Figure 2). In general, all simulations which cannot be excluded at at the 99% confidence level fall with a range of a9 = [500, 800] AU approximately along an empircally defined line of e9 = 0.75 − (450AU/a9) 8 , shown as the dashed line in Figure 2. Such orbits have a perihelion, q9, in the range ∼ [200, 340] AU. The acceptable q9 range is greater at smaller semimajor axis; at 600 AU and beyond, all solutions have q9 ∼ 200 AU. For the 20 Me simulations, the locus of acceptable a-e combinations shifts outward and can be fit with a similar empircal function e9 = 0.75 − (650AU/a9) 8 , but only from a9 = [800, 1000] AU. Lower semimajor axes perturb 4 0 200 400 600 800 1000 semimajor axis (AU) 0 90 180 270 360 perihleion longitude a=500 AU e=0.6 Fig. 4.— One of the successful simulations showing the orbital evolution of all objects with q < 80 AU at times t > 3 Gyr. A collection of resonant stable high semimajor axis objects anti-aligned with Planet Nine (centered at ∆ longitude of 180 degrees) appears beyond 200 AU, while from 100-200 AU there is a slight preference for aligned orbits. 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 Ninecentered perihelion longitudes as a function of a of all objects that have q < 80 AU and which have survived at least 3 Gyr, for the case of M9 = 10Me, a9 = 500 AU, and e9 = 0.6, one of the simulations along the acceptable locus. Objects anti-aligned with the planet have a longitude of 180 degrees in this simulation, while those aligned will be at 0 degrees 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 degrees, 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. As a counter example, Figure 5 shows a simulation with M9 = 10Me, a9 = 700 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. 3. CONSTRAINTS ON INCLINATION AND ARGUMENT OF PERIHELION The planar simulations provide no constraints on inclination, i9, argument of perihelion ω9, or longitude of ascending node, Ω9, 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 i9 = 1, 10, 20, 30, 60, 90, 120, and 0 200 400 600 800 1000 semimajor axis (AU) 0 90 180 270 360 perihleion longitude a=700 AU e=0.3 Fig. 5.— A low probability simulation. Clear anti-alignment does not develop until beyond 400 AU, while aligned orbits appear only from approximately 300 to 400 AU. 150 degrees. Unlike the planar suite of calculations, here we start all planetesimals with their longitude of perihelia antialigned 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 ω9 and Ω9 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 7 random objects from the sample of all objects in the range a = [300, 700] AU (as previously shown, these a9 = 700 AU simulations do not begin strong perihelion confinement until a ∼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 < 80 AU, and we add the constraint that i < 50 degrees, to again account for observability biases. We calculate the fraction of times that the 7 randomly selected objects are clustered within 94 degrees (Figure 6). The confinement efficiency drops smoothly until, at an inclination of 60 degrees and higher, it scatters around 20%. These results suggest, but do not demand, that Planet Nine has only a modest inclination. One of the striking characteristics of the 7 aligned distant Kuiper belt objects is the large value of and tight confinement in the pole angle (22 ± 6 degrees; Figure 1). In examining the simulations that exhibit good con- finement in longitude, we note that the polar angles of the simulated orbits are approximately perpendicular to the plane of Planet Nine, particularly for objects which come to perihelion in the plane of the planet. The implication of this phenomenon is that a pole angle of 22 degrees suggests a minimum planet inclination of approximately 22 degrees, and an orbital plane (which is controlled by Ω9) similar to the plane of the observed objects. To quantify this observation, we plot the median pole angle of our simulation objects which meet the 5 0 50 100 150 inclination 0.0 0.2 0.4 0.6 0.8 1.0 probability of confinement Fig. 6.— For simulations with a9 = 700AU and e9 = .6, the probability of confinement of 7 randomly selected objects with 300 < a < 700 and q < 80 AU varies strongly with inclination of the planet. 0 90 180 270 360 argument of perihelion -40 -20 0 20 40 polar angle (deg) Fig. 7.— For the inclined simulations, the average polar angle of anti-aligned orbits varies systematically with ω9. The average polar angle of the seven most distant objects is 22 degrees, shown as the dashed line. For simulations with high probability of confinement (i9 < 60 degrees), the average polar angle only reaches values as high as 22 degrees for ı9 = 30 degrees and ω9=150 degrees. The black, red, green, and blue points correspond to the i9=1, 10, 20, and 30 degree simulations, respectively. criteria described above and which have perihelion latitudes between -25 and 0 degrees 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 ω9 ∼ 150 degrees, 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. 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 degrees 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 degrees. For inclinations of ∼22 degrees, ω must be quite close to 150 degrees. For inclinations of 30 degrees, the allowable range for the argument of perihelion appears to be ω9 ∼ 120 − 160 degrees. 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 6 randomly selected objects (300 AU < a < 700 AU, q < 80 AU, i < 50 degrees, survival time greater than 3 Gyr, and perihelion latitude between -25 and 0 degrees) having perihelion longitudes clustered with 94 degrees and having an average polar angle greater than 20 degrees with an RMS spread of less than 6.2 degrees. Almost all simulations can be ruled out at greater than the 99% confidence level. The only simulations which cannot are, unsurprisingly, those with inclination of 30 degrees and argument of perihelion of 150 – which is the single best fit – and 120 degrees and, additionally, a few other seemingly random combinations of [i9, ω9]: [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 which 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 degree inclination (again, restricting ourselves to 300> a >700 AU, q < 80 AU and t > 3 Gyr; note that 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 > 200 AU and i > 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 < q < 15 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 which allow us to study these high semimajor axis Centaurs are critical. The perihelion locations of the perpendicular high semimajor axis Centaurs effectively rule out the possible higher inclination orbits for Planet Nine. The i9 = 90 and i9 = 150 degrees 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 i9 = 30, ω9 = 150 degrees and i9 = 30, ω9 = 120 degrees are viable. 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 6 360 270 180 90 0 perihelion longitude 0 30 60 90 120 150 180 inclination Fig. 8.— For simulations with a9 = 700 AU, e9 = 0.6, and i9 = 30 deg, examination of all objects with 300 < a < 700 AU and q < 80 AU shows a low inclination population anti-aligned with Planet Nine and a high inclination population on either side of the antialigned population. The colored points are the same as in Figure 1a, with the blue points showing that the highly inclined large semimajor axis Centaurs are closely aligned with the predicted high inclination locations. 360 270 180 90 0 perihelion longitude 0 30 60 90 120 150 180 inclination Fig. 9.— The simulations with i9 = 150 degrees produce little confinement of either the low or high inclination population. 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 Me, a9 = 700, e9 = 0.6, i9 = 30 degrees, ω9 = 0 degrees, and Ω9 = 0 degrees (note that the planet precesses over the 4 billion years of the integrations, but at this large semimajor axis the precession in ω9 is only about 30 degrees 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. 4. SKY POSITION Based on comparison to our suite of simulations, we estimate that the orbital elements of Planet Nine are as follows. For 10 and 20 Me 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 [200AU+30M9/Me,600AU+20M9/Me], where M9 is approximately in the range [5 Me, 20 Me], and that e9=0.75-[(250AU + 20M9/Me)/a9] 8 The inclination is between approximately 22 < i9 < 40 degrees, and the argument of perihelion is between 120 < ω9 < 160 degrees. We fix the perihelion longitude at 241±15 degrees. 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 to 20 Me. 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 Me with 10% atmospheric mass fraction, the radius is approximately fit as R9 = [2.4 + 0.1(M9/Me)] Re, which we will use as our nominal relationship through this range and extending to 20 Me. 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. 5. CURRENT OBSERVATIONAL CONSTRAINTS While most wide field surveys of the Kuiper belt have not been sensitive to sky motions smaller than about 1 arcsecond per hour (Millis et al. 2002; Brown 2008; Petit et al. 2011, i.e.), 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 The WISE spacecraft surveyed the entire sky twice in its 3.4 and 4.6 µm (W1 and W2) bands, allowing Luhman (2014) to rule out Saturn-sized planets – which have substantial enhanced short wavelength emission owing to emission from internal heat – out to a distance of ∼30,000 AU. A 10 Me planet, however, would not be expected to have this enhanced short wavelength emission. For example, ISO detected nothing but reflected sunlight from Neptune from 2.5 to 5 µm (Burgdorf et al. 2003), with an average flux of about 5 mJy. The Luhman (2014) W1 limit corresponds to approximately 0.2 mJy, which suggests that Neptune itself could only be detected to ∼70 AU. As a confirmation, we examined the catalog of WISE single-source detections of Neptune itself. Neptune is detected 16 times with a signal-to-noise (S/N) of approximately 50 in the W1 band. Assuming that all 7 -40 -20 0 20 40 declination 0 500 1000 1500 2000 distance (AU) 14 16 18 20 22 24 26 V magnitude 360 270 180 90 0 RA 0.0 0.2 0.4 0.6 0.8 speed ("/hr) Fig. 10.— Using all constraints on the orbital and physical parameters of Planet Nine, we can predict the location, distance, brightness, and speed of the planet throughout its orbit. Regions within 10 degrees of the galactic plane are outlined in red, and the ecliptic plane is shown in blue. The colored portions show regions where Planet Nine would have been or should be detected by previous or ongoing surveys. Light blue shows limits from the CRTS reanalysis, yellow shows Dark Energy Survey limits and coverage, dark blue shows Pan-STARRS transient analysis limits, green shows Pan-STARRS moving object analysis current limits, and red shows eventual Pan-STARRS expected limits. Orange shows the region exclusively ruled out by lack of observed perturbation to Saturn (Fienga et al. 2016; Holman & Payne 2016). The black regions show regions of phase space where Planet Nine could not have been or will not be detected in previous or currently planned surveys. of those images are coadded, Neptune could be detected with a S/N of 10 only to ∼63 AU, consistent with our estimate above. Fortney et al. (2016) suggest that the W1 brightness of Planet Nine could be enhanced above its blackbody level and thus potentially observable to a greater distance, but the sensitivity of the WISE data to Neptune-sized planets except at the closest possible distances remains low. The Luhman (2014) results thus provide no constraints on the position or existence of Planet Nine. 5.2. Catalina real time transient survey While near earth object searches are performed at cadences poorly matched to the detection of objects in the outer solar system, they cover the sky multiple times in a year, allowing the possibility of detecting objects by their weekly or monthly motion. In Brown et al. (2015), we performed such an analysis from the Catalina real time transient survey (Drake et al. 2009, CRTS), which itself repurposed the Catalina Sky Survey near earth asteroid search into a transient survey. We collected all one-time transients over an 8 year period, that is, all instances in which an object was detected at a spot in the sky only once, and attempted to fit all combinations of 4 or more detections to Keplerian orbits. The Keplerian filter is strong. From ∼ 1019 potential combinations, we narrowed the detections down to eight known Kuiper belt objects and zero false positives. Every bright Kuiper belt object in the survey fields was detected, often dozens of times. The survey was determined to be essentially 100% complete to V ∼ 19.1 in the north and V ∼ 18.6 in the south. For some of the smaller potential values for semimajor axis and larger values of planetary radius, for example, Planet Nine would have been visible to this survey over a substantial portion of its orbit, though it was not detected. 5.3. The Pan-STARRS 1 Survey The Pan-STARRS 1 telescope has surveyed large amounts of sky multiple times to moderate depths at declinations greater than -30 degrees. We consider the analysis of the data in two stages. The Pan-STARRS Survey for Transients. Like the earlier, CRTS, the Pan-STARRS Survey for Transients (PST) (Smartt et al. 2014) quickly disseminates detections of transients sources detected in the sky. We have performed a similar analysis on the reported PST data, searching for viable Keplerian orbits. As of 15 May 2016, no series of transients can be found which fit an outer solar system body on any bound Keplerian orbit. Typical transient depths reached are g = 21.0, and, based on the collection of reported transients, the survey appears to efficiently cover the sky north of -30 declination and at galactic latitudes greater than about 10 degrees. This survey rules out much of the sky within about 45 degrees of the predicted perihelion point, with the exception of the region near the galactic plane. The Pan-STARRS Outer Solar System Key Project. A survey for objects in the outer solar system was one of the initial goals of the Pan-STARRS survey. Holman et al. (2015) have now completed a preliminary analysis of the survey data and report no detections out to 600 AU. While detailed sensitivity studies have yet to be completed, it is estimated that the survey is complete to approximately r ∼ 22.5, though limits in the galactic plane are worse. An extended analysis is currently underway which will have the same brightness limits, but will remove the artificial restriction to objects closer than 600 AU (Holman, private communication). If these sensitivity estimates are correct, the Pan-STARRS 1 moving object survey has or will rule out a substantial fraction of the non-aphelion sky. 5.4. The Dark Energy Survey The Dark Energy Survey (DES) is performing the largest deep southern hemisphere survey to date. Some of the DES region covers the orbital path of Planet Nine (indeed one of the 7 cluster objects, 2013 RF98, was detected in the DES). While cadences are not designed for ease of outer solar system detection, it is clear that the data will be sensitive to Planet Nine if it is in the survey area. The DES team estimates a Planet Nine detection limit of r ∼ 23.8. (Gerdes, private communication). The survey should be completed in 2018. 5.5. Additional surveys Additional surveys covering wide areas of the sky have been performed, but in all cases they are insensitive to 8 the slow expected speeds of Planet Nine, they have too low of a survey efficiency to consider the region effectively surveyed, or they cover little or none of the region of the predicted orbital path. The large community surveys which are concentrating on specific areas of the sky, such as the VISTA surveys in the southern hemisphere and the Subaru Hyper-SuprimeCam survey along the celestial equator unfortunately do not overlap with the required search region. In the future, the Large-Scale Synoptic Telescope is expected to survey much of the sky observable from its Chilean site to a single-visit depth of approximately r ∼ 24.5 magnitude. The current survey strategy does not include visits to fields as far north as those at the extremes of the predicted Planet Nine orbital path, but if Planet Nine has not yet been found by the expected start of the LSST survey operations in 2023, a simple extension could quickly rule out nearly all but the faintest and most distant Planet Nine predictions. At its most distant predicted locations, Planet Nine is faint and in the northern hemisphere. Subaru HyperSuprime Cam will be the instrument of choice for detecting the planet at these locations. We began a survey of these regions in the fall of 2015 and will attempt to cover all of this part of the predicted orbital path. 5.6. Additional constraints: Fienga et al. (2016) perform full fits to the locations of all planets and nearly 300 asteroids observed from ancient times to the present with and without a 10 Me Planet Nine at various positions along an orbit with a9 = 700 AU and e9=0.6, consistent with the nominal orbit suggested in Batygin & Brown (2016). They find that the strongest constraint on the existence of Planet Nine comes from the very precise measurements of the distance to Saturn as measured by the Cassini spacecraft over the past decade. With no Planet Nine, their best fit to the position of Saturn has Earth-Saturn distance residuals which roughly follow a sinusoid with a 12-year period and a full amplitude of ∼70 m. Previous fits to the ephemeris of Saturn using the identical data, however, found smaller residuals and no evidence of systematic variation (Hees et al. 2014), while even more recent fits, leading to the creation of the DE435 JPL planetary ephemeris, put even tighter constraints on any remaining residuals. It thus remains unclear whether any residual is present in the Earth-Saturn distance. We remain agnostic about the existence of residuals in the distance to Saturn, but instead assume that the signal apparently detected by Fienga et al. (2016) is at or near the level of the systematic errors in this type of analysis and that larger signals from Plant Nine could be detected. At its the nearest possible locations, in particular, Planet Nine would cause such a large effect on the Earth-Saturn distance that the Fienga et al. (2016) analysis and an extension by Holman & Payne (2016) (which assumes the same residuals) can rule out any of the Planet Nine solutions near perihelion over the range of Right Ascension of RA9 > 255, RA9 < 2 deg. A recent paper (Malhotra et al. 2016) makes no attempt to explain spatial alignments, but instead attempts to simplistically look for mean-motion commensurabilities in the distant KBOs, in hopes of being able to constrain both a9 and the location of Planet Nine within its orbit. Specifically they assume that the four most distant KBOs are in N:1 and N:2 resonances, and examine the implications for Planet Nine. Such an approach could, in principle, work in the circular restricted three-body problem, but, as shown in (Batygin & Brown 2016), highly elliptical orbits are required to explain the spatial confinement of the orbits, and no specific resonances dominate the disturbing function in this elliptical problem. Indeed, no particular preference for type of critical angle or even resonance order can be identified in the dynamical simulations shown here. Rather, the crossing orbits evolve chaotically but maintain long term stability by residing on a interconnected web of phaseprotected mean motion resonances. The assumption of simple low order resonance is thus unlikely to be justi- fied. Not surprisingly, the Planet Nine orbits produced by these assumptions do not produce the spatial confinements of the KBOs that are observed. Thus, it appears that no useful constraint on the orbit or position can be drawn from this method. 5.7. Joint constraints In Figure 10 we show the regions of the potential orbits of Planet Nine that have been ruled out by the above constraints (or, in the case of the ongoing DES and final PanSTARRS analysis, where the planet might still be found by these surveys). The existence of Planet Nine can be ruled out over about two thirds of its orbit. The vast majority of the orbital region in which Planet Nine could be located is beyond about 700 AU and within about 60 degrees of its aphelion position. For the eccentric orbits considered here, Planet Nine spends greater than half of its time at these distances, so finding it currently at these locations near aphelion would be expected. At its most distant allowed location and with a Neptune-like albedo, a 20 Re Planet Nine is approximately V = 25. While faint, such an object would be well within the limits of 10-m class telescopes. 6. CONCLUSIONS 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 Kuiper belt objects, 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 substantial 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 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 9 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 re- fining the orbital parameters of Planet Nine. Each addition Kuiper belt object (or Centaur) with a > 100 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 Kuiper belt objects) 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 ω9 and i9 and so can be used to better refine these estimates. 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