OASA Stargazer - July 2017

Front cover Image taken By Pete Williamson - A member of the Online Astronomy Society

Image Of The Month

Taken By Pete Williamson - A member of the Online Astronomy Society

Written By Russell Adam Webb

The Hubble Space Telescope has been getting a good look at the nearby Luhman 16 AB system for the past few years, providing further details its behavior and composition.

The Luhman 16 AB system wasn’t discovered until 2013, which is surprising given that the system is the third known closest system to our own star, the Sun. Since its discovery, Hubble has been keeping tabs on the system and recently revealed a stack of images that show the system comprising of 2 brown dwarfs, and not 3 as originally suspected. Sometimes in astronomy, it’s our closest neighbors that provide us with the biggest surprises.

Between August 2014 and October 2016, Luigi Bedin and his dedicated team spent time viewing and studying the behavior and composition of Luhman 16A and Luhman 16B. They managed to image the stars 12 times during this period and made several derminations using the data gathered. They managed to accurately confirm the distance from our own star, the orbital behavior of the stars and whether or not any exoplanets were likely to be lurking in the system. The results will be published by Royal Astronomical Society.

The astronomers used the images and described the stars as dancing around each other across the night sky. A third potential body was suspected within the system, but this has been ruled out completely. A large exoplanet was suspected at first, given data from initial observations made by European Southern Observatory’s Very Large Telescope. The data ruled out this prediction, stating that if a planetary body existed in the system, it would be smaller than Neptune and take more than a few years to orbit the brown dwarf stars.

Many refer to Brown Dwarf stars as failed stars, because they aren’t big enough to keep up the fusion process in their core.

We have suspected that they might be able to use a rare isotope known as deuterium, which is used in matter-antimatter reactions on Federation starships, but this reaction time doesn’t last very long in cosmic terms. The star will cool off, slow down and grow dark and cold over time.

We do think, however, that there are more brown dwarf-style objects in the universe than high-mass object like our Sun. It is therefore expected that brown dwarfs are a more fertile hunting ground in the search for exoplanets.

How close is Luhman 16 AB? The system is located around 6.5 lightyears from us, or 2 parsecs. Barnard’s Star and Alpha Centauri are the only known systems that are closer to us.

Every 20-40 years, Luhman A and B circle each other at a distance of about 3AU, with one AU (Astronomical Unit) being the distance from us to our Sun.

The Luhman 16 AB system is a great place for scientists to study because it is very close to us. Brown dwarf stars are often difficult to spot as they are small and dim, but the data they give us could prove useful in a variety of astronomical purposes. The team plan to continue studying this system and look for smaller exoplanets that could be lurking in the dark

Understanding Gravity Written By Andrew Richens It is over 100 years since the publication of Albert Einstein’s theory of general relativity. The implications of this theory rocked the world of physics both then and now.

But what exactly is it, and why is it so important?

Gravity affects us all; it keeps our feet firmly on the ground. If we let go of an object it will also fall down towards the ground at a very specific rate of acceleration. Rather strangely, this rate is not determined by how much the object weighs – all objects will fall to the Earth at the exact rate (if dropped in a vacuum). If you were able to hold a pea in one hand and an elephant in the other and then let them fall, they would both hit the ground at the exact same time. But why, and, more importantly, what does this actually tell us about the universe?

Einstein’s general theory of relativity explains that what we perceive as the force of gravity pulling down on us in fact arises from the curvature of space and time.

Gravity isn’t actually a force which acts on an object. Gravity is a change in the geometric shape of space-time. The more massive the object (such as a planet, star or galaxy), the greater space-time is curved. So, if the Earth was more massive, you would weigh more, and things would fall with a greater rate of acceleration. It can be hard to visualise, but it is a concept that is worth understanding.

Imagine two intrepid pilots. One pilot lives in England, and another lives in Canada – both at the same latitude. Since they are at the same latitude to each other, they each have the same distance to travel in order to reach the Pole. Since both pilots wish to reach the North Pole at exactly the same time, they point their aircraft towards the north and fly in a straight line at the same speed. Although the pilots begin their journey several thousand miles apart, their paths will eventually converge with each other at the pole- despite the fact that they were each flying in a straight line.

The aircraft were not being pulled together by a force, but by the curvature of the Earth. If the Earth was more curved, they would have converged sooner. If the Earth was infinitely flat, they would never have met. However, if they could see nothing but the other aircraft, they would have no point of reference to show that they were moving. All they would see is the other aircraft falling towards them!

Another concept that is often difficult to visualise is that not only do objects fall at the same rate regardless of how much they weigh, objects with no mass also fall at the same rate too. Hard as it may be to imagine, but light also falls to the ground under the influence of gravity in exactly the same way as the falling pea (or elephant).

As objects, such as massive stars, become increasingly heavier, they add a greater curvature to space-time. This in turn increased the effect of gravity upon objects near to the star and so the objects weight also increases. As the mass of the star increases, so too does the escape velocity (the speed an object must travel in order to get away). Some objects in the universe are so massive (several million times more massive than the sun), that their escape velocity is such that even light is unable to escape. These massive objects are known as black holes and their existence was predicted by Einstein’s theory. Remarkably these objects have been photographed tearing entire stars apart that stray too close.

Since the first publication of Einstein’s theory or General Relativity, the passing years have done little to dampen the awe of our universe.

Astronomy Things To See During July 2017 (For UK Observers) Earth reaches aphelion (furthest point from the Sun) on 3rd July

Moon: First Quarter: Full: Last Quarter: New: First Quarter:

1st July 1:51am 9th July 5:06am 16th July 8:26pm 23rd July 10:45am 30th July 4:23pm

The Lunar “X” and “V” are visible twice in July, but only one of these is observable from the UK. The first time they are visible is around 8:00am UT (9am BST) which is several hours before the Moon rises from the UK. However, we get a second chance on 30th July at around 8pm UT (9pm BST) which is 3 hours before the Moon sets. The X and V will be visible for about 2 hours

Lunar conjunctions & occultations: Note: When the Moon is waxing it is visible in the western sky after sunset. When near Full Moon it is visible most of the night.

When it is waning, it is visible in the eastern sky before sunrise 1st July First Quarter Moon lies near to Jupiter & Spica th 5 July Waxing Gibbous Moon lies near to Antares 6th / 7th July Waxing Gibbous Moon lies near to Saturn th 8 July Nearly Full Moon lies near to The Teaspoon asterism 14th July Waning Gibbous Moon lies near to Neptune & Lambda Aquarii th 17 July Waning Crescent Moon lies near to Uranus 18th July Waning Crescent Moon lies near to Mu Ceti 20th July Waning Crescent Moon lies near to Aldebaran & Venus 25th July Waxing Crescent Moon forms triangle with Mercury and Vesta, with daytime occultation of Mercury th 28 July Waxing Crescent Moon lies near to Jupiter & Spica 29th July Waxing Gibbous Moon lies near to Spica

Planetary Observations: Mercury – if you have a clear north western horizon, you may catch Mercury after sunset this month around an hour after sunset.

It reaches greatest wester elongation on 30th July. It begins the month at mag -1.0 but will fade to mag -0.5 by the end of July. On 9th & 10th July Mercury lies close to M44 the Beehive Cluster. On the evening of 25th May, look for Mercury, Vesta & the Waxing Crescent Moon forming a small triangle Venus – located in Taurus, it will be difficult to miss mag -4.0 Venus as it dominates the dawn sky this month, rising around 3 hours before the Sun. On the morning of 13th July, it passes extremely close to Aldebaran. On 20th July the Waning Crescent Moon lies close to Venus Mars – is not observable this month Jupiter – located in Virgo, Jupiter is visible after sunset, setting at around midnight. At mag -1.8 it should be easy to spot Saturn – located in Ophiuchus, mag +0.2 Saturn is visible after sunset, setting at around 3am. With binoculars or a small telescope, you should be able to observe Saturn’s rings and also its largest moon, Titan. On 7th July Saturn lies just 3 degrees from the Waxing Gibbous Moon Neptune – located in Aquarius, Neptune rises at around 11pm and remains visible until dawn. At mag +7.8 you will need binoculars or a small telescope to spot it. On 14th July the Waning Gibbous Moon lies close to Neptune Uranus – located in Pisces, Uranus rises at around midnight and remains visible until dawn. At mag +5.8 you will need binoculars or a small telescope to spot it. On 17th July the Waning Crescent Moon lies near to Uranus Pluto – located in Sagittarius, Pluto reaches opposition this month, so it is well placed for observation. It rises at around 11pm and remains visible until dawn. At mag +14.1 you need a moderate telescope to spot it. Overnight on 11th/12th July it lies close to the Waning Gibbous Moon

Ceres – located in Gemini, Ceres rises at around 3:30am & sits to the lower left of Venus, low in the north east. At mag +8.5 you will need binoculars or a telescope to spot it Vesta – located in Leo, Vesta lies not far from Mercury, very low in the west-north-west after sunset. It sets at around 10:30pm.

At mag +7.6, you will need binoculars or a small telescope to spot it. On the evening of 25th May, look for Mercury, Vesta & the Waxing Crescent Moon forming a small triangle Juno – located in Scutum, this minor planet reaches opposition this month so is well placed for observation. It is visible all night long. At mag +9.6 you will need binoculars or a small telescope to spot it

Other Observations:.

Noctilucent Cloud Season is Here! – June/July is the peak of northern hemisphere noctilucent cloud season. They sit at an altitude of around 8 times higher than other clouds, which puts them on the edge of space. They are the edge of polar stratospheric clouds which are believed to be seeded by meteor dust. They can sometimes be seen around 60 – 120 minutes after sunset in the north west or 60 – 120 minutes before sunrise in the north east, but only between the end of May and mid August. They appear to glow a gorgeous white/blue whilst all the other clouds are in shadow, giving them their name “night shining clouds”. They are unpredictable, but if you get a good display, you will agree that they are well worth staying up late or getting up early for!

Lunar Occultation of Mercury – this occultation occurs in daylight so it will be a challenge! The Waxing Crescent Moon and Mercury rise about 3 hours after the Sun. At 8:30am BST, Mercury disappears behind the shadow side of the Moon, and it reappears again on the western limb at 9:00am BST. Exact timings will vary depending on your location Southern Delta Aquarid Meteor Shower – if you have a flat southern horizon you may catch a few meteors from this shower overnight on 28th/29th July. The First Quarter Moon will set at around midnight giving us a better chance of seeing fainter meteors as well as brighter ones Perseid Meteor Shower – the peak of the Persieds isn’t until the middle of August, but the shower begins to become active at the end of July, so keep your eye out for early Perseids. Rates will be low at first but will begin to increase as we move into August Binocular Tour – This month’s Sky at Night Binocular Tour by Stephen Tonkin is focused on the sky around the southern Milky Way. There are 5 targets if you have 10 x 50 binoculars. First is M11 the Wild Duck Cluster, an open cluster which contains nearly 3,000 stars. Another open cluster is M25, which contains a Cepheid variable star U Sagittarii. If you have dark, transparent sky, see if you can spot M8 the Lagoon Nebula with its associated open cluster NGC 6530. Next is the globular cluster M4, which will show some lovely detail even in small binoculars. The final target is Rho Ophiuchi, which is part of a triple star system which gives it the characteristic Mikey Mouse appearance. If you have 15 x 70 binoculars, look for M17 the Swan Nebula. For full details on how to find these objects, look at this month’s edition of Sky at Night Magazine Deep Sky Tour – This month’s Sky at Night Deep Sky Tour is centred on the area around Cygnus. The first 4 targets are all within the veil nebula region. NGC 6960 is the Western Veil (also known as the Witches Broom). This object is just visible with a 6” telescope, but the use of an OIII or UHC filter and averted vision will help visually. NGC 6992 is the eastern side of the Veil and this is easier to see visually. To the south of NGC 6992 is NGC 6995. Both are observable with small telescopes, but larger apertures will reveal more detail. If you have a larger aperture telescope, see if you can spot NGC 6974, Pickering’s Triangle. Away from the Cygnus Loop, look for NGC 7013. It is a galaxy which has been classified as both a spiral with restricted arms or as a lenticular. At mag +12.1 you will need a large aperture to see it. Finally is NGC 6040, an open cluster which contains about 170 stars. As this is quite a large cluster you can observe it with a smaller aperture. For full details of where to find these objects and how best to see them, pick up the current issue of Sky at Night magazine Crescent Nebula – Astronomy Now’s object of the month is NGC 6888 the Crescent Nebula. It is a Worlf-Rayet nebula which more closely resembles a bubble than a crescent. Visually you will probably need at least an 8” telescope to see the crescent shape. Use of OIII or UHC filters will aid visual observations. Due to its size, it can be imaged with most telescopes. It responds well to imaging with DSLRs and CCD cameras, but best results come from imaging through narrowband filters, particularly OIII and H-alpha. For more information on how to observe, image or sketch this object, take a look at the current edition of Astronomy Now magazine

Sky Tour – Astronomy Now’s sky tour this month takes us on a tour of the summer nebulae. There are 10 nebulae in the region of sky between Sagittarius and Draco, including M8 the Lagoon Nebula, M20 the Trifid Nebula, M27 the Dumbell Nebula, M57 the Ring Nebula, as well as the North America, Pelican, Cat’s Eye and Veil Nebulae. For more information about these summer nebulae and where to find them, take a look at the current edition of Astronomy Now magazine Solar Observations – the long days this month give us plenty of hours for solar observing. A white light filter will show sunspots, faculae and maybe some granulation. A specialist hydrogen-alpha telescope will show filaments, prominences and if you are lucky you may catch a solar flare in action. Also, if there is a lot of high level cirrus cloud around, keep a look out for solar optical phenomena such as parhelia (sundogs), 22 degree haloes and the various arcs associated with ice haloes SAFETY WARNING: Never attempt to observe or photograph the Sun without the correct equipment. Failure to do so will result in permanent damage to your eyes or even blindness!

International Space Station –The ISS returns to our skies during the 2nd week of July for some early morning passes and by the last week of the month there will be some evening passes. For the exact timings of the passes from your location, visit www.heavens-above.com You can also check the Iridium flare times for your location at Heavens Above

Comets Visible This Month: Comet C/2015 V2 Johnson – located in Virgo, at the beginning of July this comet becomes visible at dusk about 20 degrees above the south western horizon, then sets at around 1:30am. However, it rapidly sinks lower as the month progresses & by mid-July it will become very difficult to observe. The last reported visual observation had this comet at mag +8.2 and fading.

Click here to view the finder chart: http://bit.ly/2kcgAN3 Comet C/2015 ER61 (PanSTARRS) – located in Aries, at the start of July this comet rises at around 1:30am in the north east and remains visible until dawn. As the month progresses it rises earlier and by the end of July it will be rising at around 12:30am. The last reported visual observation had this comet at mag +9 and fading.

Click here to view the finder chart: http://bit.ly/2kL122C There are several other comets in the mag +11 to +15 range. Details of these can be found in the links below.

For up to date information about the fainter comets which are visible, please visit: https://in-the-sky.org/data/comets.php, the BAA Comets Section: https://www.ast.cam.ac.uk/~jds/ or Seiichi Yoshida’s home page: http://www.aerith.net/index.html

NB: All of the information in this sky guide is taken from Night Scenes 2017 by Paul L Money, Philips Stargazing 2017 by Heather Couper and Nigel Henbest, 2017 Yearbook of Astronomy by Richard Pearson and Brian Jones, Astronomy Now Magazine, Sky at Night Magazine, Stellarium, the BAA Comets Section website https://www.ast.cam.ac.uk/~jds/, www.inthesky.org and www.heavens-above.com Information collated by Mary McIntyre. For regular updates about the events happening in the sky this month, follow the Nightscenes Monthly Night Sky Facebook page at www.facebook.com/AstrospacePublications

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OB SER V AT IO N AL B IAS AN D TH ECL UST ER IN G OF D IS T AN T ECCEN TR IC KU IPER B ELT OB JECTS Mich a e lE.Bro wn

arXiv:1706.04175v1 [astro-ph.EP] 13 Jun 2017

Div isionofGeologicalandPlanetarySciences,C aliforniaInstituteofT echnology,Pasadena,C A9112 5 Draftversio nJ un e14,2017

AB ST R ACT T h e h ypoth e sis th at a m assive Plane t N in e e xists in th e oute r solar syste m on a distant e c c e ntric orbit was in spire d by obse rvations sh ow in gth at th e obje c ts w ith th e m ost distant e c c e ntric orbits in th e Kuipe r be lt h ave orbits w h ic h are ph ysicaly aligne d, th at is, th e y arec lu ste re d in longitude of pe rih e lion and h ave sim ilar orbital plane s. Que stions h ave re m ain e d, h ow e ve r, about th e e ffe c ts of obse rvational bias on th e se obse rvations, partic u larly on th e longitude s of pe rih e lion. Spe c ifi c aly, distant e c c e ntric Kuipe r be lt obje c ts te nd to be fain t and only obse rvable near th e ir pe rih e lia, sugge stin g that th e longitudes of perih elion of th e known distant objects could be strongly biased by th e lim ite d num be r ofloc ations in th e sky w h e re de e p surve ys h ave be e n c arrie d out. W e h ave de ve lope d a m e th od to rigorously e stim ate th e longitude of pe rih e lion bias for Kuipe r be lt obse rvations. W e find th at th e probability th at th e 1 0 know n Kuipe r be lt obje c ts w ith se m im ajor axis be yond 230 AU are draw n from a population w ith uniform longitude of pe rih e lion is 1 .2%. Com bin e d w ith th eobse rvation that th e orbitalpole s of th e se obje c t are alsoc lu ste re d, th e ove ral probabilityof de te c tin g th e se tw oin de pe nde nt c lu ste rin gs in arandom ly distribute d sam ple is 0.025%. W h ile obse rvational bias is clearly present in th ese obse rvations, it is unlikely toexplain th e observed alignm ent of th e distant e c c e ntric Kuipe r be lt obje c ts. am ple , a m agnitude lim ite d surve y w il pre fe re ntialy find obje c ts w h ic h are ne ar th e ir pe rih e lion position w h e re th e y 1.I NTRODUCTION are brigh te st,and if suc h surve ys are biase d in th e l o ngi t ude s at w h ic h th e yare carrie d out, th at bias w ilbe re R e c e ntly, B atygin & B row n (2016, h e reafte r B B 1 6) de m onstrate d th at th e m ost distant know nobje fle c te d in th e longitude of pe rih e lion distribution found.

Give n th atbiase sin surve y longitude sare know n to e xc ts in th e Kuiperbelth ave orbitsw h ic h are ph ysic alyaligned, ist(m ain ly, butnote xc lu sive ly, cause d by avoidanc e of that th e galac tic plane ), th e possibilty of abias in m e asure is, th e y are c lu ste re d in longitude of pe rih e lion. Suc han d alignm e nt is une xpe c te d, as diffe re ntial pre c e ssion w il longitude of pe rih e lion sh ould be c are fully conside re d. destroy any suc h alignm enton a 1 0-1 00 M yrtim e scale.

B B 1 6 m a de a si m pl e a rgum e nt th a t th e m o st di s ta nt B B 1 6 de m onstrate d th at a distantgiant plane tin an e c e c c e ntri c KB Os sh o ul d

no t be

s i g ni fi c a ntl y

m o re biase d c e ntric orbit – re fe rre d to h e re as Plane t N in e th an not-q uite -as-distante c c e ntric KB Os – w h ic h sh ow – c ould an e sse ntialy uniform distribution oflongitude of pe rih e m ain tain an alignm e nt for th e age ofth e solar syste m if lion – but it is c le ar th at th e bias tow ards findin g obje c ts longitude of pe rih e lion of th e e c c e ntric orbitof Plane t atpe rih e lion grow s w ith e c c e ntric ity, soit isnotobvious N ine is orie nte d 1 80 de gre e s aw ay from th ose of th e clush ow applicable th issim ple argum e ntis, partic ularly for te re d Kuipe r be lt obje c ts(KB Os). T o date , no oth e r th e m o st e c c e ntri c orbits. viable e xplanation fora ph ysicalalignm e ntof th e orbits B e cause of th is pote ntial unc e rtain ty about obse rof distante c c e ntric KB Os h as be e n propose d. vational bias, th e spe c ulation th at th e longitude of Pre viously, T rujilo& Sh e ppard (2014, h e re afte r pe rih e lion c lu ste rin g m igh t be pure ly an obse rvational T S1 4)had note d th at distant KB Os w e re c lu ste re d se le c tion e ffe c t h as be e n sugge ste d (Sh e ppard & T rujilo in ar201 6; Shankm an e tal.

201 7; L aw le r e t al. gumentof pe rih e lion, ω ,a param e te r w h ic hc 201 7; orre sponds B anniste re tal. 2017). Asse ssin g th e im pac t of obnot to ph y sic al alignm e n t but to a spe c ifi c in te rnal­ orie ntation­w ith­re se rvationalbias in longitude of pe rih e lion is c ritic al spe c t-to-itse lfof an orbit.W h ile T S1 4 to unde rstandin g w h e th e r th e obse rvations point to a speculated that adistant planet m igh t be responsible, no m e c h se lf-gravitatin g m assive oute r disk or to th e pre se nc e anism for c lu ste rin g ω of a population of KB Os of a giant ninth planet. H ere we develop a rigorous m e th od toe by aplanetary perturber w ith out alsoh avin g a ph ys-ic al longitudeof stim ate th e longitude of pe rih e lion bias for distant e c c e ntric pe rih e lion c lu ste rin g h as be e n found.

KB Os. W e apply th e m e th od to th M adigan & M cCourt (2016, h ereafter M M 1 6) in stead de m e onstrate d th ata m assive disk of e c c e ntric KB Os w il distant e c c e n tric KB O s origin aly ide n tifi e d by B B 1 6 generate an in clin ation in stability wh ic h w il naturaly a nd t o t h o se th a t h a ve be e

n

i d e n ti fi e d sin c e to asse ss lead to c lu ste rin gin ω . T o date , no oth e r viable e xplath e possibilty ofth e pre se nc e of Plane tN in e . nation for aclu sterin g of ω (that does notalsoin clu de c lu ste rin g of longitude of pe rih e lion)of distante c c e ntric KB Osh 2.SIGNIFICANCEOFωCLUSTE RING asbe e n propose d.

W h ile th e ω c lu ste rin g re porte d by T S1 4is Be fore disc ussin g biase s in th e longitude of pe rih e lion,w e robust and cannotbe caused by any observationalbias (see below ), th quic kly disc ussth e argum e nt of pe rih e lion, ω , and sh e longitude of pe rih e lion c lu ste rin g re porte d by B B ow h ow th e obse rve d c lu ste rin g around ω ~ 0 c annot 1 6is c e rtain ly subje c t toobse rvationalbias.

As a sim ple e xmbrown@caltech.edu

2 be cause d by obse rvational bias, e ve n th ough th e re are c lear obse rvational biase s in ω for e c c e ntric obje c ts.

In partic ular, e c c e ntric obje c ts w ith ω near 0 or 1 80 de gre e s com e toperih elion and are thus brightest around th e h e avily obse rve d e c liptic , soone would e xpe c te c c e ntric obje c tstobe found pre fe re ntialyaround ω = 0and 1 80de gre e se ve n forauniformly distribute d population.Atth e mom e ntof disc ove ry, h ow e ve r, anobje c t w ithω =ω andone w ith ω = 1 80−ω diffe r only in th e dire c tion of th e ecliptic latitudin alcom ponent of th eir velocity vec-tors. As poin ted out by TS1 4, th ere is nopossible way tode sign a surve y tobe biase d in favor of findin g obje c ts c lose to ω = 0 at th e e xpe nse of obje c ts w ith ω c lose to1 80 (or vic e ve rsa), ye t th e distant e c c e ntric KB Os sh ow th is e ffe c t strongly.

W hile calc ulatin g th e full obse rvationalse le c tion bias of ω isnotpossible , itistrivialtocalc ulate th e probability th atobje c ts w ould be e xc lu sive ly c lu ste re d around ω = 0 or around 1 80. In th e origin alanalysis, TS1 4 found th at th e 1 2 m ost distant e c c e ntric KB Os – th ose w ith se m im ajoraxis 1 50 AU and greate r – c lu ste r w ith in 43 de gre e s of ω = 0. T h e probability th at 1 2 suc h obje c ts would clu steraround eith er 0or1 80issim ply 2×2−12,or 0.04%. N ote that h ere and th roughout th is paper we re-fe r toKB Os as alm ulti-opposition solar syste m obje c ts w ith pe rih e lion distanc e be yond N e ptune ’s orbit.

Sin c e th e origin alw ork of T S1 4, 9 ne w KB Os w ith sem im ajoraxis1 50 AU orgre ate rh ave be e n found.

Of th e se , 7 are c lose rto ω = 0 th an to ω = 1 80.T h e probability th at 1 9or more of 21obje c ts would be soc lu ste re d is 2 ×2−21×[C(21 ,2)+ C(21 ,1 )+ 1 ] w h e re C(n ,m )is th e num be r of in de pe nde ntcom bin ationsof mobje c tsfrom a population of n .

T h e probability ofth isoc c urre nc e is th us just0.022%.

N osoph istic ate d de biasin g ne e ds tobe done tosh ow th at th e clu sterin g in ω is h igh ly signifi c ant. T h is strong signal – une xplain e d by th e m e c h anism propose d by TS1 4 – led M M 1 6toth e realization that th is clu ster-in g c ould bec ause d by am assive distant disk c ausin g an in clin ation in stability in th e outer solar system . A distant e c c e ntric Plane t N in e , in c ontrast, c lu ste rs longitude of pe rih e lion and pole position, rath e r than ω (B row n & B atygin 2017). A population oflongitude-ofpe rih e lion­aligne d orbits w ith pole s c lu ste re d around a position offse t from th e north e c liptic pole w il ge ne raly, butnotexclu sively,alsoh ave clu stered ω . U nfortunately nosim ple c alculation givespole position bias,sowe con-tinue touse th e c lu ste rin g in ω asan im pe rfe c t statisticalproxy for c lu ste rin g in pole position.

Clu ste rin g of distant e c c e ntric KB Osin ω (or,alte rna-tive ly, in pole position) is firm ly e stablish e d. T o date , th e only viable e xplanations for th is c lu ste rin g is e ith e r th e in c lin ation in stability proposalof M M 1 6or th e Plane t N ine proposalof B B 1 6. T h e se proposals diffe r m ost in th e ir pre dic te d distribution of longitude of pe rih elion. T h e in c lin ation in stability sh ow s no pre fe re nc e for c lu s-terin g in longitude of perih e lion, w h ile Plane t N in e c on­ fi ne s th e longitude s.

Ifin fac tth e re is nolongitude of pe rih e lion c lu ste rin g, th e robustc lu ste rin g in ω is c urre ntly only e xplain able by th e pre se nc e of am assive oute r disk of mate rial in ducin g ain clin ation in stability th rough self-gravity, as propose d by M M 1 6.If, on th e oth e r h and, th e c lu ste rin g in longitude of pe rih e lion is a true e ffe c t, rath e r th an an

appare ntone cause d by obse rvational bias, Plane t N in e re m ain s th e only c urre ntly propose d e xplanation.

W e now e xamin e obse rvationalbiase s in longitude of pe rih e lion tode te rmine w h ic h of th e se h ypoth e se sappe ars m ore likely.

3.OBSE R VATIONALBIASINLONGITUDE OFPE RIHE L ION

T h e be st m e th od for de te rm ining th e e ffe c ts of ob­ se rvational bias on th e know n KB Os would be toh ave com ple te in form ation of al of th esurve ys conduc te d to date , in c lu din g th e ir de pth , pre c ise c ove rage , and th e ir e ffi c ie nc y. Suc h in form ation is unknow n for th e m ajorityof th e surve ys th atle d toth e disc ove rie s of th e cataloge d KB Os. In m any case s, noth in g is publish e d about th e disc ove ry surve y; th e e xiste nc e of th e obje c t is sim ply cataloge d by th e IAU M inor Plane t Ce nte r (se e h ttp:/w w w .m in orplane tc e nte r.ne t/iau/U nusual.h tm l). T h e c ataloge d in form ation is suffi c ie nt to de te rm in e th e e c lipticlongitude , e c liptic latitude , he lioc e ntric distant,and brigh tne ss of e ve ry obje c t at th e tim e of its disc ove ry.

W e develop anovelm eth od touse th e discovery cir-c um stanc e s of th e e nse m ble of al KB Os in th e catalog torigorously e stimate th e statisticaldistribution oflongi-tude sof pe rih e lion e xpe c te d fordistante c c e ntric KB Os.Conc e ptualy, th e m e th od re lie s on th e ide ath at eac h KB O disc ove ry can be th ough tof as a surve y th atcould h ave disc ove re d adistante c c e ntric KB O h ad th at obje c tbe e n brigh te nough and in th e sam e plac e .W e proceed asfolow s: foreac h distant e c c e ntric KB O (a ”pare ntobje c t”)w e construc tasynth e tic population of ne w obje c ts assum in g an ide nticalabsolu tem agnitude and ide ntical orbital e le m e ntsforauniform ly se le c te d longitude of pe rih e lion and m ean anom aly (in prac tic e w e also assum e sym m etry acrossth e ecliptic plane, soour construc te d population alsoin c lu de s orbits w h e re w e re plac e ω w ith −ω ). W e taly th e ecliptic longitu de,ecliptic lat-itude , h e lioc e ntric distanc e , and e xpe c te d m agnitude of eac h obje c tin th e synth e tic population. W e calth is th e ”uniform population” of th e pare nt KB O. N e xt, for e ve ry KB O disc ove ry in th e catalog w e asse ss w h e th e r or not one of th e uniform population of th e parent object existsat th e ec liptic longitude and latitude ofth e obser-vation and if that m em ber of th e uniform population is brigh te r than th e ac tualde te c te d KB O. If so, w e know n th at a surve y was be in g unde rtake n at th at pointth at could h ave de te c te d one of th e uniform population. (In prac tic e , w e look for m e m be rs of th e uniform population w ith in 1 de gre e of th e disc ove ry loc ation of th e KB O disc ove ry.) F in aly, w e tabulate th e longitude s of pe rih e lion of th e m e m be rs of th e uniform population th at c ould h ave be e n de te c te d at th at disc ove ry location.

We now know th at th e surve y th at re sulte d in th at partic ular KB O disc ove ry w as se nsitive tom e m be rs of ouruniform population if th e y h ad h ad apartic ular longitude of pe r-ih e lion.

T h is proc e dure is re peate d for e ve ry cataloge d KB O disc ove ry tode te rm in e th e probability distribution func tion of th e longitude of pe rih e lion of th e pare nt KB O assum in g that th e population isuniform ly distributed in longitude of perih e lion and m ean anomaly.

A c onc re te e xam ple m ake sthisproc e dure m ore c le ar.

Conside r 2013 R F 98, th e m ost e c c e ntric of th e

obje c ts origin aly ide n tifi e d by B B 1 6, as th e pare nt obje c tof

3 a uniformpopulation. N e xt, conside r th e disc ove ry of arandom ly cataloged KB O,2015 GP50,w h ic h , at th e m om e nt of disc ove ry, h ad an e c liptic latitude of -1 1 .2 de grees and amagnitude of 24.8. Exam in in g th e orbit of 201 3 R F 98, w e fin d th at it c rosse s ­1 1 .2 de gre e s tw ic e, onc e 27 de gre e s from pe rih e lion, w h e n it h as am agnitude of 24.6, and once closer toaph elion, wh en it has a magnitude of28.7. N ear aph e lion, th e uniform population w ould not h ave be e n de te c table atth is latitude , but at its m agnitude c lose r tope rih e lion it could h ave be e n dete c te d by th e obse rvation th at disc ove re d 2015 GP50. T h e KB O 201 5 GP50 was disc ove re d at alongitude of 1 96 de gre e s, th usth e m e m be r of th e uniform population th at is de te c table h as a longitudeof pe rih e lion of 1 96−27= 1 68 de gre e s. T h is spe c ifi c obse rvation is th us biase d to fi ndin g th is spe c ifi c longitude of pe rih e lion for th is spe c ifi c pare nt obje c t. If w e now c onside r not a sin gle KB O disc ove ry,but allK B O disc ove rie s, w e find a statistic aldistribution of th e longitude s of pe rih e lion in w h ic h disc ove rie s of th e 2013 R F98 uniformpopulation c ould h ave be e n m ade . W e th usc reate ase parate statistic al distribution of e xpe c te d longitude s of pe rih e lion for e ac h distant e c c e ntric KB O. Th is conc eptualfram ework relies on th e assum ption th at KB O disc ove rie s rough ly re pre se ntth e c ove rage and de pth of th ecom bin e d surve ys. T h is assum ption is c learly false for th e latitude distribution, w h e re more KB Os are disc ove re d at low e c liptic latitude s sim ply be -c ause of th e ir greater num bers. W e correct th is bias by scalin g by e xpe c te d de nsity of KB Os at agive n latitude .

T oapproxim ate th is e xpe c te d de nsity w e use th e m e th od de ve lope d by B row n (2001 ) tode te rm in e th e in c lin ation distribution and convert it toalatitudin aldistribution assum

in g c irc ular orbits. T h e final re sults are not se nsi­ tive to th e pre c ise latitudin al distribution c h ose n.

A se c ond w ay in w h ic h th e assum ption th at KB O dis-c ove rie s are uniform w ith se arc h areais violate d is in th e know n longitudin albiasin th e disc ove ry of re sonant KB Os, w h ic h are ove rdisc ove re d ne ar th e ir pe rih e lion positions, w h ic h are re late d to th e position of N e ptune .

Th e easiestway toavoid thisproblem istodiscard aldiscoveriesof Plu tin os, w h ic h are ,by far,th e most nu-m e ricaly prom ine nt and m ost spatialy corre late d of th e re sonant obje c ts. In prac tic e w e sim ply discard al disc ove re d obje c ts w ith se m im ajor axe s be low40 AU . T h is constraint alsoforces ustoonly retain m ulti-opposition KB Os w ith orbits know n ac c urate ly e nough tocalc ulate th is param e te r. W e note , h ow e ve r, th at re laxing th is assum ption m ake s th e final re sults of th is analysis more signifi c ant.

N one th e le ss, w e c onse rvative ly re tain th is c onstraint.

One oth er im portant assum ption is that adistant ecc e ntric KB O c ould alw aysbe disc ove re d if a c lose r but fainte r KB O wasdisc ove re d atits pre dic te d loc ation. T h is assum ption can be violate d if th e disc ove ry surve y is not se nsitive todistantobje c ts.

Som e ofth e ne are st KB Osand m any Ce ntaurs, for e xam ple , h ave be e n disc ove re din surve ys searc hin g for near-earth obje c ts, w hic h do n ot h ave obse rvational base lin e s suffi c ie ntly long to be se nsitive to m ore slow ly m ovin g distantobje c ts. T o e x-c lu de th e se surve ysw e w ilonly conside r disc ove rie sof objectsatdistances greaterthan 30 AU .Even fornormalKB O surve ys, som e of th e distant e c c e ntric KB Os m igh tnotbe de te c table due to th e irlow rate of motion e ve n

th ough th e yare stilbrigh te rthan th e m agnitude lim it.

Se dna, fore xam ple , w ould be visible toa distanc e of 225 AU toa surve y th atw e nttoa de pth of 25th m agnitude , but fe w surve ys are se nsitive toth e slow m otions of suc h distant obje c ts.

W ew il th us plac e an uppe r bound of 90 AU on th e m ost distant object that any survey could se e .

T h is valu e is probably a c onse rvativee stim ate and w il h ave th e e ffe c t of m akin g longitude of pe rih e lion biases stronger than they mightbe in reallife. In total, w e use th e obse rvationsof 1 248obje c ts tode te rm in e our e xpe c te d distributions.

U sin g alof th e se c onstraints, w e calc ulate e xpe cte d statisticaldistributionsoflongitude of pe rih e lion for eac h of th e 1 0 know n KB Os w ith se m im ajor axis be yond 230 AU . T h e se in c lu de th e six origin aly ide ntifi e d by B B 16 and th e four th at h ave be e n disc ove re d sin c e that tim e (F igure 1 ). T h e e xpe c te d longitudeof pe rih e lion distributionsare h igh ly structuredandh igh ly in div idual. One tre nd is e asily se e n. T h e brigh te st obje c ts (Se dna, 2007 T G422) are am ong th e m ost uniform in th e ir e xpe c te d disc ove ry distributions. M any surve ys could h ave found th e ve ry brigh t Se dna e ve n quite far from its pe rih e lion, for e xam ple . T h e struc ture se e n in th e re m ain in g obje c ts is only understandable after analysis. The distributions of201 2V P1 1 3and201 3 RF 98,forexam ple,are the most non-u niform . T h e se tw oobje c ts both com e tope rih e lion at h igh e c liptic latitude , soth e ir populationsare prim arily obse rvable at h igh latitude , ye t fe w surve ys reac h th e re quire d de pth at th e se latitude s. T h e longitude s of pe rih e lion of th e se obje c ts are h igh ly biase d by th e lim ite d num be r of surve ys th at could h ave de te c te d suc ha population. T h e struc ture s in th e oth e r distributions are sim ilarly func tionsof pe rih e lion latitude , brigh tne ss,and th e distribution of surve ysin th e sky.

4.C OMPARISONTOOBSER VATIONS

W e now se e th atm e asure m e ntof th e longitude of pe r-ih e lion for a distant e c c e ntric population is h igh ly biase d by th e spe c ifi c are a of th e sky and de pth of in div idual sur­ ve ys, as e xpe c te d. W ith ourde te rm in ation of th isbias,w e can now examin e w h eth erth e discoveriesof distante c c e ntric KB Os are c onsiste ntw ith be in g se le c te d from adistribution w h ic h is uniform in longitude of pe rih e lion or if,inde e d, th e y are c luste re d.

W e first c onside r th e six distant KB Os w h ic h B B 1 6 re porte d as c lu ste re d in longitude of pe rih e lion: Se dna, 2004 V N 1 1 2, 2007 T G422, 201 0 GB 1 74, 201 2 V P1 1 3, and 201 3 R F 98. At th e tim e , th e se w e re alof th e know n KB Os w ith se m im ajor axis be yond 230 AU . T o unde rstand th e statisticaly e xpe c te d distribution oflongitude s of pe rih e lion for th e se bodie s assum in g auniform ly dis-tribute d population,w e pe rform 1 00,000 population sam -plin gs in w h ic h w e c reate a ne w se le c tion of 6 de te c te d KB Os by random ly c h oosin g alongitude of perih elion from th e e xpe c te d probability de nsity func tion for e ac h of th e 6obje c ts.W e th e n e xam in e th e statistic sof th e se 1 00,000 re alizations.

Th e longitudes of perih elion of th e 6realobjects are distribute d suc h th at th e m axim um angle be tw e e n any pair of angularly adjac e nt obje c ts is 260.9 de gre e s (F igure 2). For th e 1 00,000 realizations of th is population assumin g a uniform distribution in longitude of pe rih e lion, th e m axim um angle be tw e e n tw oangularly adjac e nt obje c ts is 260.9 de gre e s or h igh e r in only 1 437 c ase s.

If

4 360

2004 VN112

probability distribution function

2007 TG422 2010 GB174 2012 VP113 2013 FT28 2013 RF98 2013 SY99 2014 FE72 2014 SR349

360 270 180 90 0 longitude of perihelion

F i g .

1 .

— Calculatedprobabilitydistrib utio nfu nctio nsfo rtheexp ecteddistrib utio noflo ngitu d e o f p e r i h e l i o n a s s u m i n g a p o p ulationofob jectswithid enticalorbitalelementsbutuniformly distributedin longitude o f p e r i h e l i o n a n d m e a n a n o m a l y .T h e c o l oreddotshowstheactuallongitudeofperih eliono feachobject.Theblu edotsnot ethe6KBOsorigin allydiscussedby BB16,thereddotsshowthenewerdiscoverie sofSheppard& T rujilo(2 016),whilethegreendotshowsthediscoveryofBanniste retal(2 . 017).Whileobservationalb iasesarestrongin theexp ecteddistrib utionoflongit udeofp erih elion,itisclearthatfornearlyallob jectsthebiastowardsdiscoverin gtheob jectwithitsactuallongitudeofperih elionisnotsevere.

th e longitude sof pe rih e liaof distant e c c e ntric obje c tsare uniform ly distribute d, w e would e xpe c t a longitude c lu s-te rin g as tigh t as th e one obse rve donly 1 .4%of th e tim e . W e also com pute th e Rayleigh zstatistic ofth e dataand th e random sam ple and fi nd th at R ayle igh zvalu e of0.80 of th e datais e xc e e de d only 2.0%of th e tim e in th e ran-dom data. In contrast, th e sim ple estimate from B B 1 6(w h ic h also took in to ac c ount th e c lu ste rin g in pe rih e -lion latitude w h ic h is ignore d h e re ) sugge ste d th at th e c lu ste rin g shouldonly be obse rve d 0.7% of th e tim e . W e re gard th e rough agre e m e nt from in de pe nde nt w ays of e stim atin g th e signifi c anc e of th is re sult as e nc ouragin g.

W ith only th e six obje c ts de fine d h e re , th e probability of ω c lu ste rin g be com e s 2 ×2−6or 3.1 %. R e statin g th e se tw o fi ndin gs, w e se e th at th e re is only a 3.1 % c h anc e th at th e valu e s forω are e qualy distribute d about 0 and 1 80 de gre e s, sugge stin g th at som e m e c h anism is c lu ste rin g th e distant KB Os in ω .W e like w ise find th at th e re is only a 1 .4% c h anc e th at th e longitude s of perih eliaare distribute d uniform ly, sugge stin g that th e se valu e s are like w ise c luste re d. T h e c om bine d probability th at both of these clu sters would be found in random datais thus 0.043%.

Sin ce th e origin alanalysis of B B 1 6,B row n & B atygin (2016)dem onstrated thatth e PlanetN in e h ypoth esis,in

longitude of perihelion

Sedna

270

180

90 0 0

200

400 600 semimajor axis (AU)

2000

2200

F ig .2.— Lo n g itu d e o fp e rih e lio n o fK B O sa sa fu n c tio n o fse m im a jo ra x is(n o teth ea x isch a n g eto in c lu d e2 014 FE 72 w ith a se m im a jo ra x iso f2 055A U ).T h e b lu e o b j e ctsa re th e o rig in a lsix d iscussedbyBB16,whichin clu dedallKBOswithsemimajoraxesgreaterthan2 30A U k n o w n a t t h e t i m e .T h e r e d p o i n t s a r e s u b s e quentdiscoveriesfromSheppard&T rujilo(2016)whilethesin gle greenp ointisfromBannisteretal.(2 017).AngulardistancesbetweensubsetsofKBOsdiscussedin thetextarenoted.

addition to c lu ste rin g distant e c c e ntric KB Os in longi-tude s of pe rih e lion opposite toth at of Plane t N in e (th e ”anti-aligne d population”), asmale r population of ob-jects w ith longitudes of perih e lion alignedw ith Planet N in e shouldalso e xist (th e ”aligne d population”). Four ne w distant e c c e ntric KB Os (w ith se m im ajoraxis greate r than 230 AU )h ave be e n disc ove re d sin c e th e in itialanaly- sis (Sh e ppard & T rujilo 201 6; B anniste r e t al. 201 7). Of th e se , th re e fit w e ll w ith th e anti­aligne d population (201 3 F T 28, 201 4 F E72, and 201 3 SY 99), w h ile one is c onsiste nt w ith be in g th e first re c ognize d m e m be r of th e aligne d population (2014 SR 349). T h e realization that w e now e xpe c t tw ose parate opposite ly -orie nte d populations re quire s a diffe re nt m e tric for asse ssin g th e m atc h be tw e e n th e e xpe c te dandobse rve d population. In stead of e xam in in g th e large stangle be tw e e n th e longitude s of pe rih e lion ofany tw oangularly adjac e ntobje c ts, w e look at th e se c ond large stangle be tw e e n any tw oangularly ad-jac e nt obje c ts. In a population w ith tw o w e lse parate d opposite lyorie nte d groups, th is se cond large st angle w ilbe maxim ize d. In prac tic e , w e w ould h ave also c onsid-e re d th e obse rve d population ofobje c ts c lu ste re d if th e re had only be e n a sin gle c lu ste r rath e r than tw o. Suc h aclu ster would h ave a large largest angle but asmalse cond large st angle . T oove rcom e th is proble m w e take e ith e r th e se cond large st angle or half of th e large st an­ gle , e ffe c tive ly m im ic kin g th e e ffe c ts of tw o populationse ve n if only one is obse rve d. In th e re al population of 1 0 distant e c c e ntric KB Os, th e se paration in longitude of pe rih e lion be tw e e n th e anti-aligne d group and 2013 SR 349 is 1 39.5 de gre e sin one dire c tion and 1 1 8.1 in th e oth e r dire c tion. Our se cond large st angle is th us 1 1 8.1 de -gre e s. W e again pe rform 1 00,000 random ite rations and com pare th is e xpe c te d population to th e realobse rva-tions. In only 1 201 case sis th e se cond-large st se paration between longitudes of perih elion as large or larger than 1 1 8.1 de gre e s. T h e probability that th e distant e c c e ntric KB Os w ould be distributed in longitude of perih elion as e xtre m e ly as th e obse rvations are if th e unde rly in g dis-tribution w e re uniform is 1 .0%. T h e c alc ulate d R ayle igh zstatistic of th ispopulation (w h ic h isonly sensitive toaunim odaldistribution) w ith a valu e of 0.62is e xc e e de d in 2.2%of th e random sam ple .W h ile th e disc ove ryof four

5 ne w distant e c c e ntric KB Os m igh t h ave be e n e xpe c te d to in c re ase our c onfide nc e in th e se populations signifi c antly, th e realization that w e are obse rvin g tw oopposin g rath e rth an one sin gle c lu ste re d population ne c e ssarily dilu te sth e signal.

One of th e se obje c ts (201 4 F T 28) h as ω = 286 de gre e s, w h ile th e re st are c luste re d about ze ro.

Sh e ppard & T rujilo(2016) use th is fac ttoexc lu de 2014 FT 28 from conside ration as partof th e c lu ste re d population,howeverB row n & B atygin (2017)show thatsuc h obje c ts are in de e d e xpe c te d in th e Plane t N in e h ypoth e -sis. T h e yare sim ply pre c e ssin garoundadisplac e d pole but h ave te m porarily c irc ulate d past th e north e c liptic pole .

Ec liptic -base d Ke ple rian orbitale le m e nts are apoor de sc riptor of th e orbits in th is c ase . N one th e le ss, we c ontin ue touse the sim ple m easure of ω as aproxy for pole c lu ste rin g and find th at if th e true valu e s of ω are uniform about 0 and 1 80, th e probability that 9or more of 1 0 valu e s forω would be c lu ste re d about e ith e r is 2.1 % . T h e com bine d probability of both ofth e se pa-ram e te rsbe ing c luste re dis0.025%.

T h e origin al ω c lu ste rin g disc usse d by T S1 3 w hic h le d to th e M M 1 6 h ypoth e sis of in c lin ation in stability in -c lu de salKB Os w ith se m im ajoraxis 1 50 AU and large r.

T odate, 21 suc h KB Osare know n, and 1 9 h ave ω closer to 0th an to 1 80 de gre e s. T h e probability of th is c luste rin g oc c urrin g random ly is am e re 0.022%. Exam ination of th e longitude of pe rih e lion (Figure 2) sh ow s that th is param eter, too,h as som e structure down toasemim ajoraxis of 1 50 AU,but it is clear that th e clus-terin g in longitude of pe rihelion is be gin nin g tobreak dow n. T h is be h avior w as se e n in th e population sim u - lations of B row n & B atygin (2017)w h e re it was note d th at th e longitude of pe rih e lion m e rge d from be in g h igh ly c luste re d at large se m im ajoraxis, tom ode rate ly tonot-at-alc luste re d as se m im ajor axis de c re a s e d. W e th us donote xpe c t th e longitude of pe rih e lion c lu ste r of obje c ts be yond 1 50 AU to be as signifi c ant as th ose be yond 230 AU that w e in itialy c onside re d. W e none th e le ss asse ss th e obse rvationalbiase s. Onc e again pe rform in g 1 00,000 iterationsofapopulation uniform ly distributed in in lon-gitude of pe rih e lion w e find th at 43590 h ave a se c ond­ large st longitude of pe rih e lion angle be tw e e n tw o KB Os as large or large r th an 48.2 de gre e s, th e valu e se e n in th e realdata(w h ile 39753 h ave alarge st angle of 67.9 or greater like th e data). The Rayleigh ztest likew ise sh ow s no signifi c anc e to th is c lu ste rin g.

In sh ort, for obje c ts w ith se m im ajor axis 1 50 AU and large r th e re is no statistic aly signifi c ant clu sterin gof longitude of per-ih elion intoone or twogroups, yet th e clu ster of ω is

h igh ly signifi c ant. T h is disc re panc y sh ow s, w e be lie ve ,th e e xpe c te d ble ndin g of th e h igh se m im ajor axis longi-tudin aly c lu ste re d obje c ts in toth e low e r se m im ajor axisbac kground population and th e unc e rtain ty astow h e re pre c ise ly to draw th e line for distantobje c tsw h ic h are and w h ic h are not affe c te d by Plane t N in e . At gre ate r se m im ajor axe s, th e longitude of pe rih e lion c lu ste rin g isrobustase xpe c te d.

5.C ONCLUSIONS

W e h ave sh ow n th atm e asure m e ntof th e longitude of pe rih e lion of apopulation of distant e c c e ntric KB Osis subje c ttoconside rable obse rvationalbias, butthatth is biasis unlike ly tobe re sponsible for th e obse rve d c lu ste rin g in longitude of pe rih e lion. If distante c c e ntric KB Os w e re uniform ly distribute d in longitude of pe rih e lion, obse rvations of th e origin alsix obje c ts w ith se m im ajor axis be yond 230 AU w h ic h le d B B 1 6 tosugge stth e e xiste nc e of Plane t N in e w ould only find th e e xtre m e c lu ste rin g obse rve d 1 .4% of th e tim e .

In c lu din g th e four m ost re -c e ntly disc ove re d KB Os w ith se m im ajor axe s be yond 230 AU drops th at probability to1 .2%.

Clu ste rin g in ω of distant e c c e n tric KB O s is fi rm ly e stablish e d. T h usde te rmin ation of th e ve rac ity of th e c lu ste rin g in longitude of pe rih e lion isc ritic altounde r-standin g th e gravitationalforc e s sc ulptin g th e oute r solarsyste m . W ith nolongitude of pe rih e lion c lu ste rin g, th e only c urre ntly propose d viable m e c h anism for causin g ω c lu ste rin g is th e M M 1 6sugge stion of amassive oute r disk causin g an in clin ation in stability and ω clu sterin g.

If, on th e oth e rh and, longitude of pe rih e lion and pole position (w h ic h rough ly m anifests itself asω for an off­ se tpole ) are c lu ste re d, Plane tN in e is th e only c urre ntly propose d viable h ypoth e sis. B y rigorously e stim ating th e e ffe c ts of obse rvationalbias, w e h ave sh ow n h e re that th e Plane tN ine h ypoth e sisis by farth e more like ly of th e se sc e narios. T h e probability thatth e com bin ation of th e alignm e ntof th e longitude s of pe rih e lion w ith th e c lu ste rin g in pole position (usin g th e ω proxy) th atis se e n in th e KB Os w ith se m im ajor axe s be yond 230 AU w ould oc c urby c h anc e in a uniform ly distribute d population is only 0.025%.W h ile e xplanationsoth e r than Plane t N in e m igh tone day be found to e xplain th e se obse rvations, th e signifi c anc e of th e obse rvations th e m se lve s appe ars se c ure .

W e w ould like tothank Ann-M arie M adigan forth e dis-c ussion w h ic h in spire d th is analysisand Elizabe th B aile y, Ko nsta ntiAn.- B atygin , and Ian W ong for criticalreadin gs of th e m Madi gan, M. &McCourt,M.2 016, a nusc ripMt.NRAS,457,L89Shankman,C.,Kavelaars,J.J.,Lawler,S .M.,Gla

dman,B.J.,& RE FE RE NCEBann S ister,M.T.2 017,AJ,153,63 Sheppard,S .S.& T rujilo,C.2 016,AJ,152 ,2 2 1T rujilo,C.A.& Sheppard,S .S .

Bannister,M.T.,S hankman,C.,Volk ,K.,Chen,Y.2 014,Natu re,507,471 T.,Kaib ,N.,Gladman,B.J.,Jakubik ,M.,Kavelaars,J.J.,Fraser,W.C.,Schwamb,M.E .,Petit,J.M.,Wang,S .-Y.,Gwyn ,S .D.J.,Alexandersen,M.,& Pike,R.E .2 017,ArXiv e-prin ts Batygin,K.&Brown,M.E .2 016,AJ,151,2 2 Brown,M.E.2 001,AJ,1 2 1,2 804 Brown,M.E.& Batygin,K.2 016,ApJ,82 4,L2 3— .2 017,submittedtoAJ Lawler,S.M.,S hankman,C.,Kaib ,N.,Bannister,M.T., Gladman ,B.,&Kavelaars,J.J.2 017,AJ,153,33

A Review of the Next Step You’ve been at it a while and now you wish to upgrade? Again it is that long hard review of what is out there.

Of what choices you can make. Do you need something for observing or for imaging? Or even for both!

So where to start?

Social media is very again helpful. There are always people that are further along the journey and are willing to part with their advice. Just maybe you took mine, and joined a club where the total beginner and the avid expert will mix so thoroughly.

You have saved your pennies.

And once again, I recommend you save as much as you can. Always better to have too much than too little and the ideal telescope being just out of your reach.

So what shall you get?

So I made a ‘what I want’ and ‘what I do not want’ list. And after all of that typed into a search engine, it came back with what it thought was my ideal telescope.

So this review is on the Meade LX90-AF 10” Cost: Circa £2700 for the 10” model, £3400 for the 12” model Specification: Model: 1010-90-03 Brand: Meade Design: SCT Size: 10" (254mm) Computerized: Computerized GOTO Level: Advanced Mount Type: Alt Az The Gumpf (from a website): The LX90ACF offers large aperture portability and superb coma free performance with significantly lower weights than the LX200 meaning one person can setup the telescope for field operation. Includes two year official UK warranty through us directly for peace of mind.

The NEW LX90-ACF represents the latest step in the evolution of the LX90, featuring Meade's f/10 ADVANCED COMA-FREE OPTICS: Building from a classic RC design, Meade has created a new optical design with the same coma-free pinpoint star images and flatter field that discerning astrophotographers and most professional observatories expect. The NEW LX90-ACF with Meade’s Advanced Coma-Free system also reduces the astigmatism and eliminates diffractions spikes found in classical RCs. No competing Schmidt-Cassegrain Telescope can make those claims. The LX90-ACF is the perfect platform for the demanding visual observer and imaging enthusiast with telescopes available in apertures of 8 inches, 10 inches, and 12 inches. When it comes to industry-leading optics, depth of features, and computerized operation, the LX90-ACF is the best all-around telescope value you can find.

The new LX90-ACF now has it all.

OVERSIZED PRIMARY MIRROR Only Meade manufactures their primary mirrors in diameters larger than their listed aperture (e.g. the diameter of the 8" LX90’s primary mirror is actually 8.25").This additional 1/4" yields a wider, fully illuminated field-of-view, and allows you to see the light other telescopes leave behind.

8 X 50 VIEWFINDER Quickly and easily locate and centre deep sky objects.

Features: Diffraction limited ACF optics Oversized primary mirror Rigid cast aluminium fork mount LX200 - series steel tripod Smart drive Autostar controller 30,000 objects 8 X 50 optical finder Includes 26mm super plossl & 1.25" diagonal

Assembly: As I purchased this second hand it was already assembled. However, you would still need to mount the main telescope (mount and finder) onto the field tripod or pillar and this is achieved very easily.

The mount: The mount is a GOTO altazimuth type – with the GPS module. Very easy to use. Just switch on, align north and let the GPS module do its thing.

The tube: Heavy. 20kg. Recommend that you get additional ‘accidental breakage’ coverage on the house insurance.

This will require to be a named item because of its value.

The finderscope: Easy to set up. **TOP TIP** Pick an electricity pylon over a mile away in daylight to help you align the finderscope to the main tube.

Lens: The 10mm & 25mm standard lens that come with the telescope and are great to start with. **TOP TIP** This telescope can be used with DSLR cameras (with the correct adaptors: T-Ring and T-Adaptor) and can capture some great single shot images of the moon (or the sun with a white light filter). If you can, invest in some better quality lens as the standard lens are always lacking.

Conclusion: Apart from the obvious disadvantage of the weight I find this telescope easy to use. The GPS module is certainly a marvellous stroke of luck, as it reduces the need to know the exact location, time and date. A white light filter enabled me to utilises this kit during the summer months when there was astronomical dark.

Connected to camera to obtain excellent images of targets including deepsky targets.

2 1. IN TR OD U C TION

Overthe pasttw o decades,discoveriesof m in orplan etsin the outersolarsystem have re vealed com plex dyn am ical feature s an d prom pted n ew theore tical m odels of the form ation an d evolution of the solar system . O n e of the m ost surp risin g fi n din gs is thatthe orb ital plan es of Kuiperb eltob je cts (K BOs) are w idely dispersed.W hile m an y inves- tigators have re m ark ed on the w ide dispersion of KBOs’orb ital in clin ation s,on ly a few have attem pted to accurately m easure theirm ean plan e.It m ight b e supposed apriorithat the m ean plan e of the Kuiperb eltshould b e close to the m ean plan e of the solarsystem itself.Thisplan e,also k n ow n asthe “in variab le plan e”,isn orm al to the total orb ital an gularm om en tum of the solarsystem ;ithas b een determ in ed to b etterthan on e m illiarcsecon d accuracy based on the eightplan ets,M ercury–N eptun e,plusthe dw arf plan ets C ere s an d Pluto an d the tw o largestasteroids,V esta an d Pallas (S ouam i & Souchay 2 012). U n seen large m asses in the outer solar system w ould affect the m ean plan e of the solar system , e specially if those m asses are on sign ifican tly in clin ed orb its (e .g.Goldre ich & W ard 1972 ).There fore ,accurately m easurin gthe m ean plan e of the ob served Kuiper b e lt ob je cts offers a poten tially sen sitive prob e of un seen plan etary m assob je ctsb eyon d N eptun e.

The earliest attem pt to determ ine the m ean plane of the observed Kuiperbelt appears to have been b y C ollan der-Brown et al.(2 003). These authors defin ed several sub sets of the k n ow n KBOs that m ight b est defin e the m ean plan e, focusin gon the so-called classical KBOs w ith sem i-m ajoraxes in the ran ge 40–47au;they con cluded that the average orb ital an gularm om en tum of the classical KBOs (a sam ple of 141 at the tim e) w as con sisten t w ith theirm ean plan e b ein g very close to the in variab le plan e,b ut they did n ot assess the errorin theirm easure m en t.M easuringthe m ean plan e of an en sem b le of m in or plan ets’orb its b y averagingtheir an gularm om en tum vectors (in practice,averagin gthe un it orb itn orm alvectors)is susceptib le to serioussystem atic errorsdue to ob servation al b iases.

Brown & Pan (2 004) u sed a m ore rob u st m ethod of fin din g the m ean plan e b y determ in in g the plan e of sym m etry of the KBOs’sk y-plan e m otion vectors.They applied this m ethod to all 72 8 KBOs then k n ow n (of m edian sem i-m ajoraxisa= 44 au) an d con cluded thatthe Kuiperb elt’s m ean plan e is n ot con sisten tw ith the in variab le plan e b utis in stead con sisten t w ith the local L aplacian plan e (i.e ., the plan e forced b y the secular e ffects of the plan e ts) at a= 44 au pre dicted b y lin earsecularperturbation theory.A sim ilardetailed study b y Elliot et al.(2 005) re ached a differe n t con clusion , fi n din g that the Kuiper b elt’s m ean plan e is m ore con sisten t w ith the in variab le plan e than w ith the local L aplacian plan e ata=44 au.A subsequen t study b y C hian g& C hoi (2 008)poin ted out that the local L aplacian plan e of the classical Kuiperbelt should be w arp ed b y several degre es n eara= 40.5±1 au;this w arp is ow ed to the ν18n odal secularre son an ce,w hich is driven m ain ly b y N eptun e.

Sin ce the tim e of the pre vious studies,the n um b erof k n ow n Kuiperb elt ob je cts has m ore than doub led,an d the n ew discoveries n ow en com passa largerran ge of sem i-m ajoraxes,m otivatin gare -exam in ation of the Kuiperb elt’splan e.W e are also m otivated b yre cen t in triguin gb utn otcon clusive eviden ce of a large un seen plan eton an in clin ed orb itin the distan t solarsystem T ( ru jillo & Sheppard 2 014;Batygin & Brow n 2 016;Brow n & Batygin 2 016;M alhotra et al.

2 016;H olm an & Payn e 2 016b ,a;S heppard & Tru jillo 2 016)as w ell as pre vious suggestion s of un seen plan etary m ass obje cts perturbin gthe orbits of Kuiperbelt obje cts (e .g.,Gladm an et al.2 002 ;L ykaw ka & M ukai 2 008).Un seen plan etary m ass ob je cts in the outersolarsystem could re sult in KBOs havin ga m ean plan e that deviates from that expected due to the k n ow n plan ets.

W e are addition ally m otivated b y the desire to have a m ore com plete an d accurate m odel forthe overall distrib ution of KBO orbital plan es.Thisdistribution hasim portan tim plication sforthe dyn am ical history of the outersolarsystem .The ob servation ally derived w idths of the KBOs’in clin ation distribution fun ction have been used as con strain ts on theore tical m odels of the dyn am ical excitation history of the outersolarsystem (e .g.,Gom es 2 003;N esvorn y´2 015).In the literature ,the m easure s of the effective w idth of the in clin ation distrib u tion fun ction are b ased on KBOs’ in clin ation s re lative to the ecliptic plan e (e .g.,Brow n 2 001), re lative to the in variable plan e (e .g.,Petit et al.2 011),orre lative to the m ean plan e of the en tire Kuiperbelt (u sually assum ed to be con sisten t w ith the in variable plan e,e.g.,E lliot et al.2 005;Gulbis et al.2 010).H ow ever,the m ost dyn am ically m ean in gful m easure of the effective w idth of the in clin ation distrib u tion w ill b e that m easure d for in clin ation s re lative to the local forced plan e. If there are population s w ithin the Kuiperb elt w here the local forced plan e,i.e .,the tru e m ean plan e, is sign ifican tly in clin ed (as w e show in Section 5),then the distribution of in clin ation s re lative to the ecliptic orin variable plan es could b e a m isleadin gm easure of the tru e distribution of KBO orbital plan es;it could lead to poten tially in corre ct con clusion s w hen usin g the effective w idth of the in clin ation distrib u tion as an ob servation al test of Kuiper b elt form ation m odels.A n accurate m easure m en t of the Kuiperb elt’s m ean plan e as a fun ction of sem i-m ajoraxis w ill allow fora m ore dyn am ically m ean in gful re pre sen tation of the distrib ution of KBO orb ital plan es.

3 In the pre sen tw ork ,w e an alyze the curre n tdata to m easure the Kuiperb eltplan e as a fun ction of sem i-m ajoraxis,an d w e com pare the re sults w ith theore tical expectation s. The paper is organ ized as follow s. W e b riefly describ e in Section 2 the expected m ean plan e forthe classical Kuiperb eltb ased on stan dard lin earseculartheory an d the expected m ean plan e forthe scattere d disk based on sym m etry argum en ts.The datasetof KBO orbitsisdescrib ed in Section 3,w ith gre aterdetail given in A ppen dix A .W e describ e in Section 4 the m ethodsw e use to m easure the m ean plan e of KBOsan d how w e estim ate the associated un certain ties.The re sultsof these calculation sare pre sen ted in Section 5.In Section 6, w e sum m arize our fi n din gs an d discuss their im plication s.

2. E X PE C TE D M E A N

PL A N E

The orb ital distrib u tion of m in or p lan ets in the solar system is stron gly in fluen ced b y lon g­term perturb ation s from the plan ets. The plan etary effects can b e com plicated in m an y w ays, b u t the m ean plan e en forced b y their lon g­term perturb ation s can b e iden tified w ith the so­called “forced in clin ation vector”, q ( 0 ,p0 ) = (sin i0sin Ω 0 ,sin i0cosΩ 0 ), w hich is the hom ogen eous solution to the lin earsecularperturb ation equation s fora m in orplan et.This is re latively sim ply calculated in the L aplace-Lagran ge lin earseculartheory M ( urray & Derm ott1999), (q0 ,p0 )=(sin i0sin Ω 0 ,sin i0cos Ω 0 ) 8

= i=1

(1)

µi (sin γi,cosγi), fi −f0

w here f0is the n odal pre cession rate in duced b y the orb it-averaged quadru polarpoten tial of the plan ets,fi an d γi are the fre quen cies an d phases of the secularm odes of the solarsystem ’s eight m ajorplan ets,an d µ i are w eightin gfactors foreach secularm ode; fi an d γi depen d on ly upon the plan etary param eters,w hile f0an d µ i depen d addition ally on the m in orplan et’ssem i-m ajoraxis. (Follow in g stan dard practice,w e use the heliocen tric coord in ate system of the J 2 000.0ecliptic-equin ox throughoutthis paper,un less in dicated otherw ise.) W e n ote thatthe forced in clin ation vector,(q0 ,p0),is re lated to the ecliptic proje ction of the un itvector,ˆ n 0 = (sin i0 sin Ω 0 ,−sin i0 cosΩ 0 ,cosi0 ),w hich isn orm alto the local L aplacian plan e.Overseculartim escales,asthe plan ets’orb itsevolve un dertheirow n m utual perturb ation s,the forced in clin ation vectorchan ges.Even so,fora population of m in orplan etsw ith som e dispersion in sem i-m ajoraxes an d orb ital plan es,the m ean plan e w ill coin cide w ith the forced plan e at that epoch C ( hian g& C hoi 2 008).

The forced in clin ation an d lon gitude of ascen din gn ode,i0an d Ω 0 ,re spectively,of a testparticle are a fun ction of itssem i-m ajoraxis. W e ploti0an d Ω 0forthe sem i-m ajoraxisran ge 30–150au in F igure 3,accordin gto the lin earsecularsolution in M urray & Derm ott(1999). This defin es the local L aplacian plan e as a fun ction of sem i­ m ajor axis, an d is also the theore tically expected m ean plan e of n on -re son an tKBOs,asdeterm in ed b y the k n ow n m ajorplan ets,M ercury –N eptun e.W e observe thatforlarge sem i-m ajoraxes,a 40au,the expected m ean plan e asym ptotically approachesthe in variab le plan e; the latterhasin clin ation an d lon gitude of ascen din gn ode of 1.58◦ an d 107.6◦ ,re spectively S ( ouam i & Souchay 2 012 ). W e also ob serve that the expected m ean plan e in the Kuiper b e lt is n ot flat: it has a prom in en t w arp (of several degre es) n eara= 40.5au,ow ed to the ν18n odal secularre son an ce.C hian g& C hoi (2 008) attem pted to m easure this w arp in the observed KBOsw ith partial success.In the pre sen tw ork ,w e re -visit thisprob lem w ith the largerob servation al sam ple n ow availab le.W e also exam in e w hetherthe m ean plan e of the m ore distan tKBOsiscon sisten tw ith the in variab le plan e.

The secularsolution forthe forced in clin ation vectorof a m in orplan et is form ally valid in the approxim ation of sm all in clin ation s.The m in orplan et’s sem i-m ajoraxis m ustalso b e w ell separated from b oth the plan ets’sem i-m ajor axes an d from stron gm ean m otion re son an ces w ith the plan ets.Because secularperturb ation theory con siders on ly orb itaveraged perturb ation s on the in clin ation vector,the m in orplan ets’orb its m ustalso n ot b e plan et-crossin g.These con dition sare m etfairly w ell in the classical Kuiperb eltw hich con sistsof the n on -re son an tKBOsw ith 42 a/au 48.

H ow ever,itis question able w hetherthe lin ear seculartheory is applicable to the m ore distant KBOs.The curre nt ob servation al sam ple of distan tKBOs,w ith a 50au,is dom in ated b y the “scatterin gdisk ” an d t h e “scattere d disk ” population s.M an y of these ob je cts have curre n tperihelion distan ces w ithin a few au of N eptun e’s orbital radius,an d are expected to gravitation ally scatterw ith N eptun e on ~ 10 M yr tim escales. This tim escale is n ot m u ch differe n t from the secular pre cession tim escale of their orb ital plan es (w hich ran ges from a few M yr fora~ 50au to several ten s of M yrfora~ 100au).In this circum stan ce,the L aplace-Lagrange seculartheory is n ota good description of the dynam ics of these obje cts,so w e m ust ask:w hat is the theore tically expected m ean plane of the scattering

4 an d scattere d disk ob je cts? Provided that the gravitation al scatterings w ith N eptun e are n ot corre lated overlongtim escales oram on gstthe KBOs,the plan e of sym m etry forthe population of scatterin gan d scattere d ob je ctsm ustb e close to the tim e-averaged orb ital plan e of N eptun e, i.e ., the in variab le plan e. There is n o other p re ferre d plan e for this population , b arin g the effects of un seen distan tm asses.W e exam in ed the n um erical sim ulation s of the scatterin gpopulation of KBOs thatw ere carried out b y V olk & M alhotra (2 008) an d V olk & M alhotra (2 013),an d foun d that the m ean plan e of the sim ulated scatterin gKBOs re m ain s close to the in variable plan e (w ithin ~ 1◦ for 2 0 d iffere n t sub sets of ~ 2 50sim ulated scatterin gKBOs).This supports our argum en tabove thatthe expected m ean plan e of the scatterin gan d scattere d population is the in variab le plan e.

3 . KU IPE R BE LT OBSE R V A TION A L DA TA

Ourstartin g poin t is the list of m in orplan ets in the outersolarsystem cataloged in the datab ase of the M in orPlan et C en ter1 , as of Octob e r 2 0, 2 016. W e gathere d all the availab le astrom etric ob servation s for these ob je cts an d com p uted the b e st­fit barycentric orbit foreach obje ct usingthe Bernstein & Khushalani (2 000) orb it fittin g cod e. F igure 1 show sthe sem i-m ajoraxes,eccen tricities,an d in clin ation s forthese ob je cts (black crosses).W e then selected those ob je cts w hose perihelion distan ces are b eyon d N eptun e an d w hose sem i-m ajoraxis un certain ty,δa/a,did n ot exceed 5%.W e n um erically in tegrated their b est­fit orb its forw ard for 107years un der the in fluen ce of the Sun an d the four gian t plan ets to check for orb ital re son an ces w ith N eptun e.(W e do this b y check in gforlibration of are son an t an gle forall m ean m otion re son an ces up to 30th ord er).W e excluded ob je cts in orb italre son an ces from furtheran alysis b ecause theirorb ital in clin ation s an d pre cession rates are affected b y re son an t perturb ation s that are n ot describ ed b y secular theory. The re m ain in g sam ple of ob je cts is used in our Kuiper b elt m ean plan e calculation s;these are show n asgre en dotsin F igure 1. The com plete listin g of this sam ple, in cludin g their b e st fi t orb ital param eters an d sk y location s,is provided in Table 1 in A ppen dix A .

1 0.8

e

0.6 0.4 0.2 0 34

36

38

40

42

44

46

48

50

100

150

48

50

100

150

inclination (deg)

a (au) 50 40 30 20 10 0 34

36

38

40

42

44

46

a (au) Figu re1.Eccentricity (top panel)and inclination (low er panel)vs. sem i-m ajor axis for the d istant solar system ob jects listed in the M inor Planet Center (black crosses). The ob jects with q > 3 0 au and da/a < 0 .0 5 that appear not to be in resonance w ith Neptune(our criteriafor inclu sion in the m ean plane calc ulations)are show n as green dots. See Appendix A for additionalinform ation abou t these ob jects.

1http://www.minorplanetcenter.net/iau/lists/t_centaurs.htmland http://www.minorplanetcenter.net/iau/lists/t_tnos.html

5 4. M E TH OD S

4.1.Measuringthemeanplane A n um b er of differe n t m ethods have b een used to m easure the m ean plan e of the Kuiper b elt.C ollan der-Brown etal.

(2 003) used w hatm ightb e con sidere d the m ostin tuitive m ethod,n am ely com putin gthe average of the un itvectors n orm al to the orb ital plan esof the ob served sam ple of KBOs.The un itvector,ˆ h ,n orm al to a KBO’s orb it plan e is expre ssed hˆ =(sin isin Ω ,−sin icosΩ ,cosi).(2 ) The in clin ation ian d lon gitude of ascen din g n ode Ω of a KBO’s orb it can b e accurately determ in ed for e ven very short ob servation al arcs.H ow ever,m easurin gthe m ean plan e asthe average of ˆ h is susceptib le to serious system atic errorw hen applied to an observation ally biased sam ple.C on sider,forexam ple,a population of KBOsw ith a tru e m ean plan e in clin ed to the ecliptic an d havin gan in trin sically large dispersion of in clin ation s to its m ean plan e.If the ob served sub set of this population w ere discovere d on ly n earthe ecliptic (b ecause that is w here ob servation al surveys w ere perform ed),then the observed sam ple w ould b e system atically biased tow ard ob je cts w ith orbitplan es close to the ecliptic.The average of the un it orb it n orm al vectors forsuch a sub set of ob je cts w ould iden tify a m ean plan e w ith an in clin ation low erthan the tru e m ean plan e.This an d othersystem atic errors that can arise from averagin gthe un it orb it n orm alvectors are describ ed in m ore detail in A ppen dix B. Brown & Pan (2 004) defin ed the m ean plan e as the plan e of sym m etry of the sk y­plan e m otion vectors of KBOs, n otin gthaton average the KBOs’sk y-plan e velocity vectors should b e parallel to theirplan e of sym m etry n o m atterw here in the sk y v t, the KBOs are discovere d.The un itvectorin the dire ction of the sk y-plan e velocity of a KBO,ˆ can b e determ in ed from k n ow ledge of its un it orb it n orm al an d its sk y-plan e position .The sk y-plan e position of a KBO is given b y its ecliptic latitude an d lon gitude,βan d λ.The un it vectordire cted alon gthe heliocen tric position of the KBO is then given b y ˆ r=(cosβcosλ ,cosβsin λ,sin β ),(3) an d the sk y-plan e velocity vectorhas a dire ction given b y ˆ given b y

v t = ˆ h ׈ r. The ecliptic latitude an d lon gitude of ˆ v t are

βv t=arcsin (ˆ v t ·ˆ z) =arcsin [sin icosβ cos(λ −Ω )],λv t=arctan (ˆ v t ·ˆ y /ˆv t ·ˆ x ) sin βsin isin Ω −cosβcosλ cosi =arctan

.(4) sin βsin icosΩ + cosβsin λcosi

W e den ote w ith im an d Ω m the in clin ation an d lon gitude of ascen din gn ode of the m ean plan e of the KBOs (w here the subscriptmis to differe n tiate b e tw een the ob servation ally derived m ean plan e an d the theore tically expected m ean plan e,w hich has the subscript 0).The ecliptic latitude,βm ,at w hich this plan e intersects the sky as a fun ction of ecliptic lon gitude,λ,is given b y βm (λ;im ,Ω m )=arcsin [sin im sin(λ−Ω m )].(5) Brow n & Pan (2 004) m easure d the m ean plan e b y m in im izin gthe sum of the absolute value of the deviation s of the KBOs’ (βv t,λ vt) from the curve defin ed b y βm (λ )(E q.5).E lliot et al.(2 005)describ e several varian ts of this m ethod,in cludin gleast square s m in im ization an d m axim um lik e lihood fits b ased on m odeled distrib u tion fun ction s of the ecliptic latitudes βv t .In theirim plem en tation ,they used the low in clin ation approxim ation of E q.5, βm (λ;im ,Ω m )≈im sin(λ−Ω m ).(6) In gen eral,com putin gthe m ean plan e as the plan e of sym m etry of the sk y-plan e m otion vectors should b e m ore robust again st system atic errors(see section 2 .1 in Brown & Pan 2 004).W e show in A ppen dix B that this is in deed a m uch m ore rob ust m ethod than averagin gthe orb it n orm al un itvectors,b utitis n ot en tire ly fre e of system atic errors.

H ere w e im plem en t this approach in a com putation ally sim pler an d m ore effi cien t w ay than in pre vious studies.

L et ˆn den ote the un itvectorw hich is n orm al to the m ean plan e of the Kuiperb elt.Then a KBO w hose orb itplan e coin cides w ith the m ean plan e w ould have ˆ n ·ˆt v= 0.A n on -van ishin gvalue of ˆ n ·ˆ v t w ould m easure the deviation of the KBO’s orb itplan e from the m ean plan e.There fore ,to iden tify the m ean plan e of an ob servation al sam ple of

6 KBOs,w e m in im ize the sum of the absolute valuesof ˆ n ·ˆ v t of all KBOs, |ˆ n ·ˆ v t|,overa grid of all possible un it vectors,ˆ n .

A s a check , w e con firm ed that our m ethod y ields the sam e m ean plan e asBrow n & Pan (2 004)’sm ethod of m in im izin g |β v t(λ v t)−β m (λ v t)|w hen applied to the ob servation al data of the classical KBOs (n on -re son an t ob je cts in the sem i- m ajoraxis ran ge 42 –48 au).

4.2 .Uncertain tyofthemeasuredmeanplane The m easure m en t e ror of the com puted m ean plan e an d the sign ifican ce of its d eviation from the theore tically expected m ean plan e are n ot straightforw ard to determ in e dire ctly from the ob servation al sam ple of KBOs. The m easure m en t error depen ds upon the n um berof observation s available,the quality of those observation s,an d the distribution of those observation s on the sk y.It also depen ds upon the in trin sic dispersion of the orbital plan es about the m ean plan e;fora given n um berof ob served obje cts,apopulation w ithare lativelysm allinclination dispersion (such as the classical KBOs) w ill have a sm allerun certain ty in the m easure d m ean plan e than a population w ith a w iderin clin ation dispersion . H ow everthere isn o straightforw ard m ethod to estim ate the in trin sic in clin ation dispersion ab out the m easure d m ean plan e dire ctly from our d ata set b e cause the ob servation al selection effects for the m ajority of KBOs in the M in orPlan et C en terdatab ase are poorly k n ow n . L im itin g ourselves to ju st the sub set of KBOs from w ell-characterized surveysw ould dram atically re duce the n um berof available observation s,especially atlarge sem i-m ajoraxes.

W e there fore adopt a m odel distribution fun ction forthe in trin sic in clin ation dispersion ,as describ ed b elow .

Follow in g(Brown & Pan 2 004),w e estim ate the m easure m en tun certain ty of the m ean plan e due to the n um b erof observed ob je cts,theirobserved sk y position s,an d a m odel of the in trin sic in clin ation dispersion b y m ean sof M on te- C arlo sim ulation s. W e first gen erate a population of KBO orb its distrib uted ab out the m easure d m ean p lan e w ith a pre scrib ed in clin ation distrib ution .W e then gen erate a syn thetic ob servation al sam ple from thispopulation ,m easure the m ean plan e of this sam ple,an d re peat fora large n um b erof syn thetic ob servation al sam ples.The distrib ution of the m easure d m ean plan es of these syn thetic sam ples yieldsthe un certain ty associated w ith the com puted m ean plan e of the tru e ob servation al sam ple.

Sim ilarly, to assess the sign ifican ce of deviation s from the theore tically expected m ean plan e, w e first gen erate a population of KBOs distrib uted ab outthe expected m ean plan e w ith a pre scrib ed in clin ation distrib ution .W e gen erate m an y syn thetic sam ples from this population (again m atchin gthe sk y location distrib ution of the tru e ob servation al sam ple) an d m easure their m ean plan es; w e then quan tify how often w e w ould expect to fin d a deviation as large as that of the tru e ob servation al sam ple if the expected m ean plan e isthe tru e m ean plan e.

The m ostim portan tassum ption in these sim ulation s isthe pre scrib ed distrib ution of the sim ulated ob je cts’orb ital plan esab outtheirm ean plan e.The distribution of KBO orbitalplan eshasoften b een m odeled asa Gaussian m ultiplied b y the sin e of the in clin ation (e .g.,Brow n 2 001), f(i)=Csin(i)ex p(−i2 /2 σ 2 ),(7) w here Cis a n orm alization con stan t,iis the in clin ation re lative to a chosen re fere n ce plan e (u sually the ecliptic orin variab le plan e), an d the lon gitudes of ascen din g n ode, Ω , re lative to that plan e are assum ed to b e in trin sically ran d om ly d istrib u ted.

In pre viousstudies(e .g.,Brow n 2 001;Petitetal.2 011;Gladm an etal.2 012 ;Gulbis etal.2 010),this fun ction al form forthe in clin ation distrib ution has b een used to m odel the in trin sic in clin ation distrib u tion of the Kuiper b elt, an d differe n t values of the Gaussian stan dard deviation,σ, have b e en derived for d iffere n t dyn am ical classes of KBOs. W e n ote, how ever, that the in trin sic distrib ution of Ω w ill n ot actually b e un iform ran dom if the population ’s m ean plan e deviates from the chosen re fere n ce plan e plan e.In these pre vious studies,on ly the distrib ution of the in clin ation ,i,isdiscussed w hen m odelin gthe distrib ution of the KBOs’orb ital plan esw ith E q.7; the distrib u tion of Ω has gen erally n otb een discussed in detail,a poten tially im portan toversightform odelin gthe distrib ution of orb ital plan es.

In the pre sen t w ork , w e pre scrib e the distrib u tion of the sim ulated ob je cts’ orb ital plan e s as follow s. W e b e gin w ith the usual defin ition of the in clin ation vector, (q,p)=(sin icosΩ,sin isin Ω ),(8) w hich is the ecliptic proje ction of the un it vectorn orm al to the orb ital plan e (E q.2 ).Forsm all ecliptic in clin ation s,sin i≈i,the in clin ation distribution described b y f(i) (Eq.7) isthe w ell-k n ow n R ayleigh distribution .A R ayleigh

7 distrib u tion of in clin ation s, together w ith a un iform ran dom distrib u tion of Ω , is equivalen t to the in clin ation vector com pon en ts, qan d p,each havin ga Gaussian distrib ution of zero m ean .In ourM on te-C arlo sim ulation s,w e adopta Gaussian distrib ution of qan d p, b u t w ith an im portan t m odification : that the in clin ation distrib u tion is d efin ed ab out a plan e other than the ecliptic.

There fore ,to gen erate the pre scrib ed distrib ution of KBO orb ital plan es foroursim ulation s,w e gen erate the follow in gdistrib ution of in clin ation vectors: q=q0+ q1 p =p0+ p1,(9) w here (q0 ,p0 )=(sin i0 cosΩ 0 ,sin i0 sin Ω 0 )isthe pre scrib ed m ean plan e of the syn thetic population ,an d q1 an d p1 are ran dom n um b ers draw n from a Gaussian distrib ution of zero m ean an d a pre scrib ed stan dard deviation . (In k eepin g w ith stan dard term in ology,(q0 ,p0 ) can b e called the forced in clin ation vectoran d q ( 1 ,p1 ) can b e called the fre e in clin ation vector.) The choice of the stan dard deviation is based on pre vious studies as w ell as on re quirin gan acceptab le m atch b etw een the ecliptic in clin ation s of oursyn thetic KBO sam ples an d those of the re al ob served KBOs.(These choices,d escrib ed in Section s5.1 an d 5.2 , are differe n t for the classical K BOs an d for the m ore distan t KBOs.) The error b ars that w e re port for the m easure d m ean plan e’s in clin ation an d lon gitude of ascen din gn ode are derived from M on te-C arlo sim ulation s in w hich (q0,p0) is chosen to b e the m easure d m ean plan e of the ob served KBOs. To com pute the sign ifican ce level of the deviation of the m easure d m ean plan e from the theore tically expected m ean plan e (S ection 5.2 ),w e set (q0 ,p0)equal to the theore tically expected plan e.In Section 5.2 ,w e also con sidera n on -Gaussian distrib ution of q1an d p1to assessthe sen sitivity of the re sultsto the choice of pre scrib ed in clin ation distrib ution ;the n on -Gaussian distrib ution w e adoptis ob tain ed from an em pirical fit to approxim ately deb iased ob servation al data.

W e con stru ct syn thetic data sets b y selectin gob je cts from oursim ulated distribution of KBOs that are in n early the sam e sk y location s as the re al observed ob je cts.Determ in in gthe sk y position s of oursim ulated KBOs re quire s assign m en t of sem i-m ajoraxes,eccen tricities,lon gitudes of perihelion,an d m ean an om alies to the sim ulated ob je cts,in addition to Ω an d ias assign ed ab ove.W e assign to oursim ulated KBOs sem i-m ajoraxis an d eccen tricity distrib ution s sim ilarto those of the re al observed ob je cts;this en sure s that oursyn thetic sam ple w ill have sim ilarbiases in these elem en ts as the re al ob served population .W e then assum e that the distrib ution of perihelion lon gitudes an d m ean an om alies are ran dom in the ran ge 0–2 π .These assign m en ts fully determ in e the sk y position of each sim ulated ob je ct.

W e build up m an y sets of syn thetic data sam ples b y m atchin gtheirsk y position s to those of the re al ob je cts.By m atchin gthe synthetic sam ples’distribution of sky locations to that of the re al observed sam ple w e also naturally approxim ately accoun t forof the in clin ation biases in the re al observed sam ple.

Foreach pre scrib ed distrib ution of orb ital plan es,w e gen erate 40,000 syn thetic data sets,each havin g the sam e sam ple size as the on e re al data set.The distribution of m ean plan es m easure d in these synthetic data sets is then the distribution from w hich w e com pute the 1–σ,2 –σ,an d 3–σun certain ties of the m easure d values of im an d Ω m as w ell as the sign ifican ce levels for the m easure d deviation from the expected m ean plan e. C om plete details of our sim ulation s an d ourcalculation s of the m ean plan e m easure m en t un certain ties are given in A ppen dix C .

5. R E SU LTS

5.1.Themeanplan eoftheclassicalKuiperbelt A sof Octob er2 016,there w ere 62 1 k n ow n appare n tly n on -re son an tm ain classical Kuiperb eltob je cts,thatis,all KBOsw ith w ell­determ in ed sem i­m ajor axes in the ran ge 42 –48 au. W ith the m ethods describ e d in the pre vious section , w e fin d that ◦ the m ean plan e of thispopulation hasJ 2 000ecliptic-equin ox in clin ation an d lon gitude of as- cen din gn od e im = 1.8◦+0 .7 ◦ ◦ +18 −0.4 ◦an d Ω m = 77 −14 ◦(w ith 1–σun certain ty estim ates). W e can com pare thisw ith the forced plan e calculated w ith lin earseculartheory tak in gaccoun t of the k n ow n gian t plan ets on theircurre n t orb its M ( urray & Derm ott1999).A ta=45au (n earthe cen terof oursam ple ran ge),thatforced plan e has i0=1.7◦ an d Ω 0=92 ◦;thisisw ithin the 1−σun certain tiesof the m easure d m ean plan e.

H ow ever, lin ear secular theory pre dicts that the forced plan e varies sign ifican tly w ith sem i­m ajor axis in the 35–50 au ran ge thaten com passesthe classical b elt.To b etterassessw hetherthe m easure d m ean plan e,i(m ,Ω m ),of the classical beltiscon sisten tw ith lin earseculartheory,w e carried outM on te-C arlo sim ulation sof the population of observed 769 n on -re son an tclassical KBOs as follow s.W e m odeled the in trin sic in clin ation distrib ution of the classical Kuiperb elt

8 ab outitstheore tically expected m ean plan e asa sum of tw o R ayleigh com pon en ts, f1(i)

sin 2 i sin i ), exp ( − 2 σ12 σ12

f2(i)

sin 2 i sin i exp ( − 2 σ22 σ22

).(10)

W e chose param eters σ1 = sin 3◦ an d σ2 = sin 13◦ ,an d w e forced the syn thetic KBO sam plesto be approxim ately even ly split b etw een f1an d f2. These choices provide an acceptab le m atch (i.e . n ot re je ctab le at 95% con fiden ce w hen an A n derson -Darlin gtest is applied)b etw een these syn thetic sam ples’ecliptic in clin ation s an d the re al ob served ecliptic in clin ation s an d are roughly con sisten tw ith ob servation al con strain ts on σ1 an d σ2 forthe classical b eltin clin ation distribution (e .g., Brow n 2 001;Gulbis etal.2 010;Petitetal.2 011,2 017).W e gen erated 40,000sim ulated sets of769 ob je cts w ith the sam e sem i-m ajor axes asthe re al classical KBOs,each on e w as assign ed a forced in clin ation q ( 0 ,p0) equal to the theore tically expected forced in clin ation vectorfortheirgiven sem i-m ajoraxis,an d w e assign ed each on e a ran dom value of the fre e in clin ation ,q ( 1 ,p1) tak en from this m odel in clin ation distrib ution (see A ppen dix C forfurtherdetails).W e then com puted the m ean plan e of each syn thetic datasetb y m in im izin g |ˆ v t|.The distribution of n ·ˆ com puted m ean plan esforthe syn thetic population isshow n in the leftpan el of F igure 2 .The colore d m ap show sthe n orm alized den sity distrib ution of m ean plan es re covere d from the sim ulated data sets;also in dicated w ithin the den sity m ap are the 1–,2 –,an d 3–σellipses thaten com pass 68.2 %,95.4%,an d 99.7% of the sim ulated m easure m en ts,re spectively.W e con clude thatthe m easure d m ean plan e of the re al setof ob served ob je cts(in dicated w ith the gre en dot) is w ithin 1–σof the seculartheory pre d iction .

Forre fere n ce (an d forcom parison w ith pre vious studies),w e also carried out sim ilarM on te-C arlo sim ulation s in w hich the m ean plan e w as pre scrib e d to b e the ecliptic. W e fin d that the ob servation al sam ple is in con sisten t w ith the ecliptic as its m ean plan e at gre aterthan 3–σsign ifican ce (right pan e l of F igure 2 ).Sim ulation s w ith the in variable plan e as the pre scrib ed m ean plan e of the syn thetic datasets show that the observation al sam ple is in con sisten t w ith the in variab le plan e as its m ean plan e at gre aterthan 2 –σsign ifican ce.

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Figu re2.T h e m ean plane of the classical K B O s (non­resonant K B O s w ith sem i­m ajor axis in the rang e 3 5–50 au), as m easured by m inim izing Σ |ˆ vt·ˆ n p|. (Left panel)T he colored m ap show s the distribution of the m ean plane of synthetic random sam ples draw n from adistribution w hose forced inclination is prescribed by linear secular theory. (Right panel)T he colored m ap show s the d istribution of the m ean plane of synthetic rand om sam ples d raw n from a d istribution w hose forced inclination is d efined by the ecliptic. T h e m ean plane of the observed sam ple is show n as a g reen d ot. T h e colored m ap is the d ensity of recovered planes in each pixelnorm alized to the m aximum density point. The black ellipses represent the 1–, 2–, and 3 –σ (68.2%,95.4%,and 99.7%)limits of the expected m ean plane distribution.

W e observe in the right pan el of Figure 2 that the distribution of m easure d m ean plan es in ourM on te-C arlo sim ulation s is n ot pre cisely cen tere d on the pre scrib ed m ean plan e in the sim ulation s p ( = q= 0 in this case),an d the distrib ution also deviates slightly from circularsym m etry in (q,p).W e show in A ppen dix B that these deviation s are ow ed alm ost en tire ly to the n on ­un iform distrib u tion in ecliptic lon gitude of the ob served KBOs. For the classical Kuiper b e lt, this effect is re latively sm all because m ost of the population has been detected nearthe ecliptic and

9

inclination (deg)

the longitude coverage alon gthe ecliptic has on ly sm all gaps.H ow ever,as show n in Section 5.2 ,this bias is m ore pron oun ced in the m ore distan t ob servation al sam ple of KBOs. So farw e have tre ated the classical Kuiperbelt as on e population . W hile the sim ulation s above show that the en tire sam ple is con sisten t w ith lin earseculartheory,the sam ple is dom in ated b y KBOs in the sem i-m ajoraxis ran ge 42 –45au w here the pre dicted plan e chan ges very little as a fun ction of sem im ajoraxis.This m ean s that the m easure d m ean plan e of oura<50au sam ple can also b e acceptab ly m odeled as havin ga m ean plan e equal to the lin earseculartheory pre diction fora= 45 au. H ow ever, the ob servation al sam ple of KBOs in terior to 50 au is suffi cien tly large that w e can sub d ivide it in to sem i-m ajoraxis bin s to b ettertest w hetherthe observed m ean plan e as a fun ction of a con firm s the expectation from lin ear secular theory, in cludin g the expected w arp n ear 40–42 au. W e divided the sam ple of n on -re son an t classical KBOs in to 7sem i-m ajoraxis b in s:35–40.3 au (43 ob je cts),40.3–42 au (82 ob je cts),42 –43 au (100ob je cts), 43–44 au (186 ob e j cts), 44–45 au (141 ob je cts), 45–48 au (194 ob je cts), an d 45–50 au (2 17 ob je cts). The b oun dary b e tw een the first tw o b in s is chosen to b e the cen terof the ν18secularre son an ce;the outerm ost tw o b in s overlap b ecause there are too few ob je cts in 48–50au fora separate b in .Foreach of these sem i-m ajoraxis b in s,w e calculated the m ean plan e,i(m ,Ω m ),alon g w ith the 1-σun certain ties from M on te-C arlo sim ulation s w ith syn thetic data sets.The re sults are show n in F igure 3,togetherw ith the lin earseculartheory forthe expected m ean plan e as a fun ction of sem i-m ajoraxis.F igure 4 show s the sam e re sults in q,pspace.

18 16 14 12 10 8 6 4 2 0 34

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a (au) Fig u re3.T he plane of the K uiper belt predicted by linear secular theory (i0 and Ω 0, show n in black)com pared to the plane of sym m etry of the velocity vectors of the observed non-resonant ob jects (show n in purple);the left panels show the classicalK uiper belt region while the zoomed out right panels show the m ore distant K uiper belt (discussed in Section 5.2).T he horizontalgrayarrow s indicate the sem i-m ajor axis bin for each m ean plane m easurem ent. T he verticalerror bars are the 1-σ unc ertainties, obtained from M onte-Carlo simu lations.

E xam in in gF igure s 3 an d 4,the m easure d m ean plan e as a fun ction of aappears to follow lin earseculartheory fairly w ell in the classical beltre gion ;though som e deviation is appare n tforthe outerm ostbin s an d the bin ju stoutside the ν18 secularre son an ce.The w arp n eara ~ 40au at the location of the ν18 n odal secularre son an ce is clearly eviden t:w e see thatthe m easure d m ean plan e’sn ode un dergoesa dram atic tran sition acrossourin n erm osttw o b in san d thatthe 40.3–42 .0 au b in ’s m ean plan e is sign ifican tly in clin ed to the ecliptic. W hen w e com pare the m easure d m ean plan es for the in n erm ost tw o b in s, they differ from each other at>3-σcon fiden ce. H ow ever, the lon gitude of ascen din g n ode for the b in ju st outside the ν18do es n ot agre e w ith the value pre dicted b y the lin earseculartheory,an d the m easure d m ean plan e is in clin ed b y ~ 13◦ to the pre dicted plan e forthatb in ; thisisa n early 3-σdiscre pan cy (see the leftpan el in F igure 4).The cause of this discre pan cy is n otob vious,butw e n ote thatthere is an eccen tricity secular

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Fig u re4.The plane of the K uiper belt predicted by linear secular theory, (q0,p0), is indicated w ith the nearly straight but d iscontinuous coloredline). T he m easured m ean plane is show n as colored d ots,and the 1-σ uncertainty is ind icated w ith colored ellipses, for the sem i-m ajor axis binsin the range 3 5–50 au K B O s. B oth thesecular theory line and the m easured planes are color-c oded according to sem i-m ajor axis (color bar). (The color-c oded sem i-m ajor axis of the m easured m ean plane is set to the valu e in the center of the corresponding sem i-m ajor axis bin in F igure3 .)The left panelshow s the bins from 3 5–50 au,while the right panelis zoom ed in to show the classicalK uiper belt from 42–50 au. For reference, the ecliptic and invariable planes are ind icated by g ray d ots labeled “E” and “I”, respectively.

re son an ce, ν8, in the sam e vicin ity, an d lin earseculartheory does n ot accoun t forthe couplin gb etw een eccen tricity an d in clin ation .Such couplin gcould lead to addition al secularre son an ces in this re gion ,such as those in dicated in Figure 5of Kn ezevic et al.(1991).It is possible that higher-ord erseculartheory is n ecessary to accurately m odel the forced in clin ation s of Kuiperbelt obje cts in the vicin ity of the ν18 .A ltern atively,it is possible that this discre pan cy in dicates a sm all shift of the exact location of the ν18d ue to un seen m asses.

The m ean plan es m easure d forthe thre e sem i-m ajoraxis b in s in the m ain classical b elt (42 –45au) agre e very w ell w ith the secularpre diction ,but the outerre gion of the classical Kuiperbelt show s som e discre pan cies betw een theirm easure d m ean plan es an d the lin earseculartheory.The m easure d m ean plan e of the 45–48 au b in deviates from the seculartheory pre diction b y slightly m ore than 1-σ,w hile the slightly exten ded 45–50au b in deviates b y ~ 2 -σ(this is m ost visible in the right pan el of Figure 4). These deviation s are less statistically sign ifican t than those pre sen ted for the m ore distan t,a>50au,population in the n ext section,but perhaps are in dicative of the perturbation aw ay from the exp ected m ean plan e.

5.2 .ThemeanplaneofthemoredistantKuiperbelt The observation al sam ple of KBOsatlargersem i m ajoraxes,a 50au,is too sparse in sem i-m ajoraxis values to allow for the sm all b in s in athatw e exam in ed forthe classical KBOs.To m easure theirm ean plan e,w e choose tw o overlappin gsem i-m ajoraxisran ges:50 ≤a/au≤80an d 50 ≤a/au≤150.These ran gesare chosen b ased on the sem i-m ajoraxisdistribution of the observed KBOs,w hich isheavily w eighted tow ard low ersem i-m ajoraxesdue to observation al biases.(W e do n otcon siderKBOsw ith a>150au because they are few in n um beran d populate the large sem i-m ajoraxis range too sparsely.) There are 12 5KBOs w ith 50≤a/au ≤80 that m eet our orb it­fittin g accuracy re quire m en t an d are appare n tly n on -re son an t;exten din gthisran ge to 50≤a/au≤150in cre ases this n um b erto 162 .These ob je cts are all in cluded in Table 1 in A ppen dix A .These tw o bin s each have a large en ough n um b erof ob je cts to m easure theirm ean plan e an d to determ in e the statistical sign ifican ce of its deviation from the theore tically exp ected plan e.

Forthe sem i-m ajoraxis b in of 50–150au,the m easure d m ean plan e has J 2 000ecliptic-equin ox in clin ation an d lon gitude ◦ of ascen din g n ode im = 6.8◦ +5.6 ◦

= 2 2 2 ◦+41 −32 ◦;the sem i-m ajoraxisbin of 50–80au hasa ◦ ◦ +6.6 ◦+18 m easure d m ean plan e w ith im = 9.1 −3.8 ◦an d Ω m = 2 2 7 −44 ◦. These can b e com pare d to the expected plan e forKBOs ata=75 au from lin earseculartheory,w hich has i0= 1.6◦ an d Ω 0= 107◦. (This pre dicted plan e is very close to the in variab le plan e,an d it chan ges very little overthe 50–150au sem i-m ajoraxis ran ge.) The expected an d m easure d m ean plan es are show n in the right pan els of Figure 3;also show n are the 1–σun certainties quoted above,w hich are com puted from the M on te-C arlo sim ulation s.A s n oted in Section 4 an d A ppen dix B,usin gthe sk y ­plan e velocity vectors to m easure the m ean plan e does n ot en tire ly re m ove the effects of un even sk y coverage in KBO surveys.The M on te-C arlo distribution s of the m ean plan es forthese tw o sem i-m ajoraxis bin s are n otsym m etric ab outthe pre scrib ed m ean plan es;this is w hy the un certain ty estim ates show n in F igure 3 are n otsym m etric ab out ◦

−3.2 ◦an d Ω m

11 the m easure d values.Forthe M on te-C arlo sim ulation s,w e assum ed thatthe in trin sic in clin ation s (re lative to each pre scribed m ean plan e) are draw n from a R ayleigh distribution of sin iw ith param eterσ= sin 18◦ .This choice is con sisten tw ith observation al con strain ts of this population ’s in clin ation dispersion (e .g.Petit et al.2 011;Gulbis et al.2 010),an d it yields syn thetic datasets w ith ecliptic in clin ation an d lon gitude of ascen din g n ode distrib u tion s that are n ot re je cted at a 95% con fiden ce level w hen an A n derson -Darlin gtest is used to com pare them to the ob served population ’s in clin ation an d n ode distrib ution s.W e discuss otherpossib le choices forthe in clin ation distrib ution laterin this section .

The m ean plan e thatw e m easure forthe tw o sam ples(sem i m ajoraxisran ges50–80au an d 50–150au)are surp risin gly in clin ed re lative to the expected m ean plan e. To assess the sign ifican ce of this deviation , w e re peat the M on te­C arlo sim ulation s usin gthe secularpre diction i(0=1.6◦ , Ω 0=107◦)as the sim ulated population ’s pre scrib ed m ean plan e.W e gen erate sim ulated data setsforthe tw o sem i-m ajoraxisbin s.The re sultin gdistribution sforthe m easure d m ean plan esof the tw o syn thetic datasetsare show n in Figure 5. W e fin d that for b oth sem i­m ajor axis b in s, the m easure d m ean plan e of the ob servation al sam ple is differe n t from the theore tical pre diction at the ~ 99% con fiden ce level.

50 < a/au < 150

50 < a/au < 80 1

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99.3% confidence

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Fig u re5.The expected distribution of recovered m ean planes (colored m aps)for the distant K uiper belt:(left panel)50 < a/au < 80 population, and (right panel)the 50 < a/au < 150 population;the simulations assum e atrue m ean plane of i0 = 1.6◦and Ω 0 = 10 7◦ (black asterisks, thesecular theory prediction for 75 au),and an inclination distribution about them ean planed escribed by aRayleig h d istribution w ith param eter σ = sin 18◦. T hecolored m ap is thed ensity of recovered planes in each p,q bin norm alized to them aximu m density. T heblack ellipses enclose68.2%,95.4%,and 99.7% of thesimulated m ean planem easurem ents, representing the 1–, 2–, and 3 –σ confid ence lim its of the d istribution. T h e observed m ean plane for each popu lation is ind icated by the g reen d ots. T he confid ence ellipse for each observed m ean plane is show n in orang e.

It is im portan t to n ote that the calculation of the sign ifican ce level of the m ean plan e’s deviation from the theore tically pre dicted m ean plan e is b ased on w hat w e pre scrib ed forthe in trin sic distrib ution of in clin ation s ab out that plan e.A s n oted in Section 4.2 ,ourm easure m en t of the m ean plan e of the classical Kuiperb elt has re latively sm all un certain ties because that population ’s observed in clin ation dispersion is fairly sm all. In con trast,the m ore distan t observed KBOs w ith a>50 au have a m uch largerin clin ation dispersion ,an d theirsam ple size is sm aller. C on sequen tly,the un certain ty ellipses in F igure 5are quite large.The sm allersam ple size of this population also m ean s thatits intrinsic ecliptic inclination distribution islessw ell con strained than that of the classical Kuiperbelt.In pre vious studies,the in clin ation distribution fun ction has b een m odeled b y E quation 7,w ith in clin ation s m easure d from eitherthe in variable plan e orthe classical Kuiperb elt plan e;the re sultin gestim ates forthe m odel param eterσare poorly con strain ed an d ran ge from ~ 10◦ to ~ 2 5◦ (Gulb is et al. 2 010;Petit et al. 2 011).W hen approxim ate ob servation al biases (based on m atchin gthe observed ecliptic latitude distribution ) are applied to ourn om in al m odel in w hich the fre e in clin ation vectorcom pon en ts q ( 1 ,p1) have a Gaussian distrib ution of stan dard deviation σ= sin 18◦ ab out the in variab le plan e (e quivalen t to a R ayleigh distrib ution of in clin ation s),the re sultin g distrib ution is n ot statistically re je ctab le at 95% con fiden ce w hen com pare d to the re al ob servation s usin g an A n derson ­Darlin g test. H ow ever, it over­pre dicts the n um b erof ob served low in clin ation ob je cts an d produces a n arrow erin clin ation distrib ution than the observed on e;this is illustrated in the left pan el of Figure 6,w hich show s this m odelled distribution (in re d)com pare d w ith the ob served in clin ation distrib ution .A w iderm odel distrib ution (larger σ) can provide a b etterm atch to the

1 w idth of the observed distribution ,butitthen over-pre dictsthe n um berof ob je cts in the high in clin ation tail of the distrib ution .Our n om in al m ean plan e un certain tiesassum e σ=sin 18◦ forthe in trin sic distrib ution of in clin ation sof the population ’sm ean plan e b ecause this provides statistically n on -re je ctab le syn thetic ecliptic in clin ation an d n ode distrib ution s forb oth the expected, low er in clin ation m ean plan e an d for the ob served m ean plan e. Pre scrib in g a differe n t w idth param eter for the in clin ation distrib u tion , or a differe n t fun ction al form altogether (as Petit et al.2 017have re cen tly suggested m ight b e n ecessary for the dyn am ically hot population s in the Kuiper b e lt), w ould re sult in a diff ere n t estim ate of the un certain ties.

35

1

number per bin

30

cumulative fraction of population

observed distribution simulated observed Rayleigh distribution simulated observed empirical distribution

25 20 15 10 5

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0 0

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ecliptic inclination (deg)

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ecliptic inclination (deg)

Fig u re6.Left panel:theobserved ecliptic inclination d istribution for the 50 < a/ au < 150 population (black histog ram , poisson errorbars)com pared to simulated observed ecliptic inclination distributions for apopulation w ith aRayleigh distribution in the inclination relative to the the invariable plane w ith σ = sin 18◦ (red histog ram ) and from an em piricaly fit inclination d istribution about the invariable plane (blu e histog ram ). Rig h t panel: cumulativ e intrinsic inclination distribution relative to the m ean plane for the Rayleigh distribution (σ = sin 18◦,red line)and for the em piricaldistribution (blu e line).

To illustrate how sen sitive the sign ifican ce level of the m ean plan e deviation is to the assum ed in trin sic in clin ation distrib u tion , w e re peated the M on te­C arlo sim ulation s for the 50–150 au population usin g an em pirically fit in trin sic in clin ation distrib ution ab outthe expected m ean plan e.This em pirical distrib ution w as con stru cted b y applyin gapproxim ate deb iasin g factorsto the ob served in clin ation distrib ution ,then sm oothin gthatdistrib ution b y averagin g it w ith a R ayleigh distribution tru n cated atthe in clin ation of the highestin clin ation ob je ctin the re al ob served population ;this em pirical in clin ation distribution fun ction is show n in blue in the rightpan el of F igure 6. (The differe n ces w ith the sim ple R ayleigh m odel,show n in re d,are ob viousto the eye.) U sin gthe em pirical distribution fun ction to assign in clin ation s (re lative to the theore tically pre dicted m ean plan e) re sults in syn thetic ecliptic in clin ation distrib u tion s that m atch fairly w ell the effective w idth of the re al ob served population ’s distrib u tion w ithout pro ducin ga high-in clin ation tail n ot seen in the re al population (F igure 6).M on te-C arlo sim ulation s w ith this em pirical in clin ation distrib ution yield a slightly w iderdistribution of m easure d m ean plan es of the syn thetic datasets,an d from these w e fin d the sign ifican ce level of the m easure d plan e’s deviation to b e ~ 97%. This dem on strates that the statistical sign ifican ce of the re sult depen ds on the assum ed in trin sic in clin ation distrib u tion .

The M on te­C arlo sim u lation s show that tw o diff ere n t b u t statistically acceptab le assum ption s ab out the in trin sic in clin ation distrib u tion of the m ore distan t KBOs fin d that the large deviation of the m easure d m ean plan e from the expected m ean plan e is statistically sign ifican t at~ 97% an d ~ 99% con fiden ce. W e caution that the exact con fiden ce level of the deviation is m odel depen den t.A ddition al observation s an d bettercon strain ts on the in trin sic dispersion of the orb ital plan es of the m ore distan t K uiper b e lt w ill b e n eeded to con firm this d eviation w ith higher con fiden ce.

6. SU M M A R Y A N D DISC U SSION

Our an alysis of the ob served classical Kuiper b elt ob je cts fin ds that, tak en as a sin gle sam ple, their m ean plan e is con sisten t w ith that pre dicted b y the secularperturb ation s of the k n ow n gian t plan ets. The n on -re son an t classical KBOs w ith sem i-m ajoraxesin the range 42 –48 au have a m ean plan e of im =1.8◦ an d Ω m =77◦ (m easure dre lative to the J 2 000ecliptic-equin ox an d w ith 1–σlim its of 1.2 –2 .2 ◦ an d 63–95◦ ,re spectively).This plan e is w ithin 1–σof the seculartheory pre diction forsem i m ajoraxisa= 45 au, an d it differs from the in variab le plan e at the 2 –σlevel.The ecliptic plan e is ru led outas the classical Kuiper b elt’s m ean plan e w ith gre aterthan 3–σsign ifican ce. These re sults are con sisten t w ith Brow n & Pan (2 004),w ho also foun d the m ean plan e to be closerto the seculartheory

13 pre diction than the in variab le plan e.H ow ever,w e do n otru le outthe in variab le plan e atgre aterthan 3-σsign ifican ce as Brown & Pan (2 004) did, a differe n ce lik ely due to differe n t criteria for in cludin g ob je cts in the calculation ; our sam ple is re stricted on ly to ob je cts w ithre ason ab ly w ell-determ in ed,n on -re son an t orb its w ith sem i-m ajoraxes in the classical belt sem i-m ajor axis ran ge,w hile theirs in cluded all KBOs discovere d at heliocen tric distan ces gre aterthan 30 au.Ourdeterm in ation of this population ’s plan e also overlapsw ith the re sultsof E lliotetal.(2 005)atthe 1–σlevel, though ourm easure d m ean plan e iscloserto the seculartheory pre diction than theirs.E lliotetal.(2 005) con cluded thatthe classical Kuiper b eltplan e agre ed b etter w ith the in variab le plan e than the seculartheory pre diction ,w hile w e com e to the opposite con clusion .Ourclassical belt sam ple size is n early fourtim es largerthan that available to E lliot et al.(2 005),w hich lik ely accoun ts for this diff ere n ce.

W hen w e divide the a<50 K BOs in to sm aller sem i­m ajor axis b in s, w e fin d further support for the secular theory pre diction s (F igure 3).W e clearly see a w arp of several degre es in the m easure d m ean plan e n eara ~ 40au,pre viously discussed b y C hian g& C hoi (2 008).The m easure d m ean plan e in teriorto the w arp associated w ith the ν18 secularre son an ce agre es w ith the lin earseculartheory,b utoutside the w arp ,the m easure d m ean plan e deviates from the secularpre diction b y n early 3-σ. This discre pan cy m ay b e due to higher­ord er secular e ffects, such as possib le couplin g b e tw een in clin ation an d eccen tricity n earthis re son an ce,orperhaps due to a sm all shiftin the exactlocation of the ν18due to un seen m asses. Tow ard the outerpartof the classical b elt re gion ,w e also see n oticeable discre pan cies b etw een the seculartheory pre diction s an d the m easure d m ean plan es forthe 45−48 au sem i-m ajoraxis bin ( 1-σ discre pan cy) an d the 45−50au sem i-m ajoraxisbin ( 2 -σdiscre pan cy).These discre pan ciespoten tially sign al the b egin n in g of the m ore sign ifican t deviation w e fin d in the higher sem i­m ajor axis population .

The m easure d m ean plan es of the m ore distan t KBOs,in the sem i-m ajoraxis ran ges of 50–80au an d 50–150au,are in clin ed to the expected m ean plan e b y ~ 7◦ (w ith a 1–σlow er-lim it of ~ 4◦).U n certain ties in the m odel forthe intrin sic in clin ation distrib u tion of this population ab out its m e an plan e lead to un certain ties in the statistical sign ifican ce of this surp risin gly large m easure d deviation .H ow ever,it isfairto say that the m easure d deviation is large;forre ason ab le assum ption s ab out the in trin sic in clin ation distrib u tion the deviation is sign ifican t at the ~ 97−99% level.

The sam ple of KBOs in the sem i­m ajor axis ran ges of 50–150 au are thought to b e affected b y gravitation al scatterin g w ith N eptun e. Gravitation al scatterin g b y N eptun e is n ot expected to re sult in a m ean plan e differe n t than the in variab le plan e.

These ob je cts are also too close to the Sun to have their an gular m om en tum sign ifican tly affected b y Galactic tides or stellar flyb ys (see, e .g.

C ollin s& Sari 2 010).It is there fore in tere stin gto con siderpossib le explan ation sforthe m easure d deviation from the expected m ean plan e of the m ore distan tKBOs. In the seculartheory discussed in this paper,w e tre at the KBOs as m assless test particles.So on e possib le explan ation for the discre pan cy could b e the effects of self­gravity am on gst the KBOs them selves.M adigan & M cC ourt(2 016)describ e a spon tan eouscollective tiltin gof a circum stellardisk of m assive b odiesfrom a state of in itially n early co-plan arbuthighly eccen tric orbits.

Thism echan ism re quire sa m assive disk of KBOs, 1M ,m uch largerthan estim atesof O(0.01M ) of the curre n t m ass of the Kuiperb elt (see,e.g.,Fraseret al.2 014).It also re quire s a speculative in itial state an d it is un clear that a tilted m ean plan e of KBOs w ould persist un der d iffere n tial orb ital pre cession in duced b y the gian t plan ets. Thus, the self-gravity of KBOs seem s an un lik ely explan ation .

A n otherpossib le explan ation is thatan im pulse-lik e ortran sien tperturb ation cohere n tly altere d the orb ital plan es of the distan t KBOs in the sem i m ajoraxis ran ge of 50–150au.H ow ever,such a perturbation w ould have to have occurre dre cen tly en ough that sub sequen t secular p re cession of these n ew orb it plan es ab out the in variab le plan e has n ot had suffi cien t tim e to re lax the KBOs’m ean plan e.The pre cession tim escale ata= 50au is~ 5M yran d it is ~ 15M yrata=80au.This m ean s that any tran sient im pulse perturbation of the m ean plan e of obje cts in this sem i-m ajoraxis ran ge w ill ten d to b e erased b y differe n tial secular p re cession ab out the in variab le plan e on tim escales of ~ 10M yr.This tim escale im plies a perturb ation m uch too re cen t to b e a re sult of stellar fl yb ys (e .g.,L evison et al.2 004) orrogue plan ets (e .g.,Glad m an & C han 2 006).Thus an im pulse perturbation seem s un lik ely.

A n otherpossib ility isthe pre sen ce of an in clin ed un seen plan etin the outersolarsystem ,w hich w ould chan ge the secularly forced plan e.R ecen t suggestion s T ( ru jillo & Sheppard 2 014;Batygin & Brow n 2 016;Brow n & Batygin 2 016;M alhotra et al.2 016;H olm an & Payn e 2 016b ,a;Sheppard & Tru jillo 2 016) of a ~ 10M plan etin the very distan tsolarsystem (b eyon d several hun dre d au)w ould n ot explain the ob served deviation of the m ean plan e in the 50–150au sem i-m ajoraxis range.

A n addition al 10M plan etata≈600 au has a n egligib le effect on the forced plan e of the Kuiper Belt at sem i­m ajor axes in teriorto ~ 100au,even if the addition al plan etis highly in clin ed.

14 A chievin g a sign ifican t chan ge of the m ean plan e in the 50–80 au sem i­m ajor axis ran ge re quire s a m uch closer p erturb er, b ecause, ab sen t a secular re son an ce, the forced plan e of a KBO is n early un affected b y an addition al plan et un less the KBO’s sem i­m ajor axis is suffi cien tly n ear that of the plan et. To exam in e the possib ility that the ob served chan ge in the Kuiper b elt’sm ean plan e isdue to a sm aller,n earb y perturb eron an in clin ed orb it,w e develop an an alytical estim ate b ased on lin earseculartheory forthe forced in clin ation of a testparticle asa fun ction of the m ass, sem i-m ajoraxis,an d in clin ation of such a perturber.In this calculation ,w e assum e (i) thatin the absen ce of the un seen plan et, the forced plan e of distan t KBOs is given b y the in variab le plan e (that is, w e n eglect the sm all differe n ce b etw een the forced plan e at fin ite sem i m ajor axes an d the in variab le plan e as defin ed b y the k n ow n plan ets), (ii) that the un seen plan et’s n odal pre cession rate is con trolled b y the k n ow n plan ets b u t that the un seen plan et is suffi cien tly low m ass an d distan t that it does n ot sign ifican tly affect the in clin ation secular m odes of the k n ow n plan ets, an d (iii) that the distan t KBOs’sem i-m ajoraxes are in close proxim ity to the un seen plan et’s sem i-m ajoraxis.In otherw ords,w e calculate how the in variab le plan e is perturb ed b y an in clin ed low m ass plan etin the sem i-m ajoraxis ran ge w ell b eyon d the k n ow n plan ets. W e then graphically solve the in verse prob lem of fin din g the com b in ation s of un seen plan et param eters (m ass, sem i m ajoran d in clin ation ) fora desire d forced in clin ation of KBOs of sem i-m ajoraxis in the ran ge of 50–100au.Details of this calculation are given in A ppen dix D. Figure 7illustrates som e com b in ation s of the m ass m ( 9 ),sem i-m ajoraxis a ( 9 ),an d in clin ation (i9 )of an un seen plan et that could pro duce a forced in clin ation (ata=65au)of the m agn itude w e observe forthe m ore distan t KBOs in oursam ple.(N ote that in this figure , all in clin ation s are re fere n ced to the local L aplacian plan e determ in ed b y the k n ow n plan ets; for the sem i-m ajor axis ran ge of in tere st here ,this re fere n ce plan e is very close to the solarsystem ’s in variable plan e.) Peru sin g this figure , w e see, for e xam ple, that a M ars m ass ob je ct (~ 0.1M ) on a m oderately in clin ed orb it at a sem im ajoraxis in the ran ge 65–80 au is suffi cien t to force a sub stan tial in clin ation re lative to the in variab le plan e. W e show in A ppen dix D that such a perturb er w ould produce sign ifican t forced in clin ation s over a zon e of ab out 10–2 0 au w idth aroun d its orb it (see F igure 11 in A ppen dix D).W e n ote that it is possib le form ore than on e perturb erto b e re spon sible forthe ob served deviation . H ow ever, the effects of a very large n um b e r of such perturb ers w ould ten d to average out an d lead to a m ean plan e close to the in variab le plan e,un less theirorb ital plan es w ere coin ciden tally align ed;thus,it is un lik ely that the observed deviation can b e accoun ted forb y m ore than a sm all n um b erof perturb ers.The sem i-m ajoraxis distribution of the KBOs observed ata>50au is con cen trated in the low era range;nearly 80% of the obje cts in our50–150au sam ple are alsoin the 50–80au sam ple (S ection 5.2 ).Thus,it is possible that the m easure d deviation of the m ean plan e could be the re sult of a localized perturbation due to a low er-m ass plan etary ob je ct curre n tly re siden t am on gst the scattere d an d scatterin gKBOs.Such a low er-m ass perturb erw ould b e m ore ak in to the extan t plan ets that have b een suggested to explain the population of detached KBOs w ithin the scattere d disk (e .g.,Gladm an et al. 2 002 ;L yk aw k a & M uk ai 2 008).

Figure 7also show s approxim ate visual m agn itude con tours as a fun ction of the perturber’s m ass an d distan ce.Forthis calculation ,w e assum e thatthe plan ethasalbedo 0.5(sim ilarto the albedosof the icy surfacesof dw arf plan etsin the Kuiperb elt), an d den sity 2 –5.5g cm −3(ran gin gfrom the den sity of dw arf plan ets up to the den sity of terre strial plan ets).

W e see thata perturberof M ars’m ass an d size in the distan ce ran ge of 65–80au w ould be brighterthan visual m agn itude ~ 17. C an pre vious ob servation al surveys of the outer solar system ru le out the existen ce of such an ob je ct? It is diffi cult to ascertain from the literature the prob ab ility thatsuch a perturb er w ould have re m ain ed un detected in pre vious ob servation al surveys.W e foun d on ly a brief re m ark in Brown etal.(2 015) statin ga ~ 30% chan ce thatthere is on e re m ain in gKBO brighterthan visual m agn itude 19 w ithin ~ 30◦ of the ecliptic thathas yetto be detected (m ostlik ely in the un -surveyed re gion s n earthe galactic plan e).There are also un -surveyed re gion s athigherecliptic latitudes w here re latively brightob je cts m ightre m ain un detected.Itappears n otim possible thata perturb eron the ord erof M ars’size an d m ass,at such close distan ces ~ ( 65–80au,as re quire d to perturb the Kuiperb elt’s m ean plan e)re m ain s to b e discovere d.

It isalso pertin en t to n ote thatH olm an & Payn e (2 016a) re cen tly iden tified ran ges of m ass an d distan ce com b in ation s for an un seen plan et w hose perturb ation s could im prove the orb it­fit re siduals for Pluto an d other KBOs. H ow ever, their con strain ts do n ot overlap w ith our iden tified ran ges of m ass an d distan ce of a perturb e r in terior to 100 au that could pro duce the re quire d in clin ation forcin gof the distan tKBOs. The observed population of KBOsin the 50–100au sem i-m ajoraxisran ge doesplace an upperlim iton the m assof such a plan etre sidin gthere : the plan etcan n otb e so m assive thatitw ould have dyn am ically com pletely cleare d thisre gion of sm all bodiesoverthe age of the solarsystem .Fora M ars-m assperturber,w e can scale the dyn am ical lifetim esof M ars-crossin gasteroids in the in n ersolarsystem to the lon gerorb ital periodsin the distan tKuiperb elt.

15

10

1

0.1

0.01 60

80

100

120

Fig u re7.The colored curves indicate the m ass and sem i-m ajor axis combinations for arelatively close-in planet-9 w ith the ind icated o r b i t a l inclination, i9, w hich w ould prod ucethe ind icated forced inclination, ifo rced, of test particles at a = 65 au (m agentacorresponds to i9 = 8◦ and ifo r ced = 7◦, green to i9 = 15◦ and ifo r ced = 7◦, cyan to i9 = 20◦ and ifo r ced = 4◦,and blu e to i9 = 3 0 ◦ and ifo r ce d = 2◦). The shaded zones indicate param eter regions w here the planet would be brighter than the indicated visualm agnitudes, assum ing an albedo of 0 .5 and the indicated bulk density (ρ in g ram s per cubic centim eter). Inc linations quoted in this fi g u re are relative to the invariable plane of the solar system .

M ichel et al.(2 000) determ in ed the dyn am ical half­lifetim e of various sub sets of M ars­crossin g asteroids, fi n din g half­ lifetim esran gin gfrom 45M yrto >100M yr.Scalin gthese re sultsto a M ars-m assperturberat~ 70au im pliesa factorof ~ 300lon gerhalf-lifetim es,or>13 Gyr,forthe distan tKBOs;this exceeds the 4.5Gyrage of the solarsystem .A ssum in gthatthe clearin gtim e w ould scale approxim ately inversely w ith the H ill radius of the plan et,w e can furtherestim ate that a perturb erof m ass exceedin g~ 2 .4 M w ould m ak e the half-lifetim es of the distan tKBOs to b e shorterthan the age of the solarsystem .Thus a dyn am ical upper lim itforthe perturb er’s m ass is~ 2 .4M .

Possib ly a stron gercon strain ton the m assan d orb itof a plan etin the ~ 50−100 au ran ge isprovided b y the existen ce of stab le population s of ob je cts libratin gin N eptun e’s m ean m otion re son an ces.In particular,there exists a large population of obje cts in the 5:2 re son an ce atsem i-m ajoraxisa= 55.4 au (Glad m an etal.2 012 ;V olk etal.2 016).The stab ility of these 5:2 re son an tob je cts has b een investigated b y L yk aw k a & M uk ai (2 008) w ho perform ed n um erical sim ulation s w ith hypothetical distan t M ars­to­E arth m ass plan ets on eccen tric orb its in specific re son an t con figuration s w ith N eptun e. They foun d, for e xam ple, that a M ars­m ass ob je ct on an eccen tric orb it in the 3:1 re son an ce w ith N eptun e a ( = 62 .5au) w ould destabilize the 5:2 re son an tKBOs on 4 Gyrtim escales,butthatm ore distan t an d m ore m assive re son an t plan ets could allow fora survivin g5:2re son an t population if the plan et’s perihelion distan ce exceeds the aphelion distan ces of the 5:2 re son an t KBOs. These authors did n ot re port on the effects of distan t p erturb erin m ore gen eral (n on -re son an t) orb its,an d theirm ass/ sem i-m ajoraxis com b in ation s do n ot fully overlap w ith the ranges w e con sider; thus it is n ot possib le to easily extrapolate their fi n din gs for the ran ge of param eters that w e have iden tified for the putative perturberto accoun tforthe in clin ed m ean plan e of the distan tKBOs.W e leave this to a future study.

Fin ally,w e n ote the possibility thatthe large m easure d deviation of the m ean plan e of the distan tKBOs is sim ply an un lik ely ob servation ,i.e .,the tru e m ean plan e re ally isthe invariab le plan e an d thatthe ob servation s thusfarhave

16 un luck ily sam pled a setof KBOs that in dicate otherw ise.Forre ason ab le assum ption s ab out the distrib ution of orb ital plan es, the prob ab ility of such a statistical fluk e is 1–3%.

W e con clude b y em phasizin gthat an accurate m easure m en t of the distan t Kuiperb elt’s m ean plan e provides a sen sitive prob e forthe existen ce of un seen plan etary -m ass ob je cts in the outersolarsystem .To im prove the pre cision of the m ean plan e m easure m en t pre sen ted here re quire s bettercon strain ts on the in trin sic distribution of the orbit plan es of the distan t KBOs,w hich re quire s an in cre ase in the sam ple size of w ell-characterized detection s as w ell as care ful m odelingof the population’s in clin ation distribution .This m odelingshould tre at the in clin ation as a vectorratherthan sim ply m easurin gthe am plitude of the in clin ation s re lative to the ecliptic orin variable plan es.C orre ct m odelin gw ill help re veal deviation s of the population ’s m ean plan e from the expected forced plan e,w ill provide a dyn am ically m ean in gfulre pre sen tation of the distrib ution of KBO orb ital plan es,an d w ill provide b etterob servation al con strain ts form odels of the dyn am ical history of the solarsystem .

W e than k ScottTre m ain e fordiscussion san d com m en tson an early draftof thispaperan d Dan iel Fabry ck y fora helpful re view .Thisre search hasm ade use of data provided b y the In tern ation al A stron om ical U n ion ’sM in orPlan etC en teran d b y N A SA ’sA strophysicsData System .W e ack n ow ledge fun din gfrom N A SA (gran tN N X 14A G93G),an d R.M .addition ally ack n ow ledges fun din gfrom N SF (gran t A ST-1312 498).

A PPE N DIX A. KU IPE R BE LT DA TA U SE D

W e fit orb its for all of the distan t ob je cts listed in the M in or Plan e t C en ter as of Octob e r 2 0, 2 016 usin g the Bernstein & Khushalani (2 000) orb it fittin g p roced ure . (For ob je cts w ith ob servation al arcs lon ger than ~ 10years,w e had to shorten the arc b y discard in gsom e of the olderastrom etry so the assum ption s un derlyin gthe Bern stein & Khushalan i (2 000) orb it fittin g routin e w ere n ot violated (see discussion in H olm an & Payn e 2 016a).Ob - je cts w ith such lon garcs ten d to have orb its that are so w ell determ in ed that the discard ed astrom etry has n o effect on w hether or n ot an ob je ct w ould b e in cluded in our sam ple.) These orb its are all show n in Figure 1 as b lack crosses;w e n ote that the excess of ob je cts n eare= 0 in that figure is an artifact of the assum ption s the Bern stein & Khushalan i (2 000) orb it fittin g code m ak es for e xtre m ely short ob servation al arcs.Ob je cts w ith sm all sem i-m ajoraxis un certain ty d ( a/a<0.05),sem i-m ajoraxisa>30au,an d perihelion distan ce q>30au w ere in tegrated forw ard un d er the in fluen ce of the Sun an d the four gian t plan ets for 107years usin g SW IF T L ( evison & Dun can 1994) to check fororb ital re son an ces w ith N eptun e.The re sultin gset of n om in ally n on -re son an t Kuiperb elt ob je cts is given in Tab le 1,w hich in cludes the follow in gin form ation foreach obje ct: pack ed M PC design ation ,sem i-m ajoraxis (a),sem i-m ajoraxis un certain ty d ( a),eccen tricity (e),in clin ation (i), lon gitude of ascen din g n ode (Ω ), argum en t of perihelion (ω ),ecliptic lon gitude (λ ),ecliptic latitude (β ),heliocen tric distan ce r(h ), an d the epoch of the orb it fit an d sk y position (in J D).A ll elem en ts are bary cen tric an d re fere n ced to the J 2 000ecliptic coord in ate system .N ote that the epoch foreach ob je ct’s orb it fit is slightly differe n t. The m ean plan e calculation w ould ideally b e don e w ith all ob je cts re fere n ced to a com m on epoch, b u t the orb it fits use the tim e of the ob servation s as the epoch. W here possib le w e have used the epoch of the m ost re cen t observation s forthe sk y position s an d orbits,so the m ajority of the epochs are w ithin a ~ 5yearspan ,but the full ran ge of epochs is~ 2 0 years. H ow ever, typical KBOs have such slow sk y m otion s that this differe n ce in epochs has a m in im al affect on the m ean plan e calculation . M ost of the KBOs in our sam ple m ove on ly ~ 1◦ / yearin lon gitude;theirm ovem en t in latitude depen ds on in clin ation ,b ut is typically less than a few degre es even on tim escalesof ~ 10 years. O ur e stim ate of the m ean plan e’s un certain ty in corp orates differe n ces in sk y position of this m agn itude (see A ppen dix C ).

B . E F FE C T OF OBSE R VA TIO N A L BIA SE S ON

TH E M E A N

PL A N E C A L C U L A

TIO N C om putin gthe m ean plan e b y sim ply averagin gthe un it vectors n orm al to the orb ital plan es does n ot w ork w ell for ob servation ally b iased population s. A s discussed in Section 4.1,the fact that m an y ob servation al surveys are perform ed n earthe ecliptic can b ias the in clin ation of a m ean plan e determ in ed in this m an n er.Sim ilarly,surveyin govera lim ited ecliptic longitude range w ould yield an averaged plan e w ith a b iased longitude of ascen din gn ode,b ecause the ob servation al b iases w ould favor p articular values of Ω in the ob served population . A sam ple of ob je cts could yield b oth a b iased n o de an d a biased in clin ation if they w ere discovere d in a survey w ith lim ited ecliptic latitude an d lon gitude cove rage.

17 Tab le1.Listob jects used to calc ulatem ean plan es M PC D es.

15760

a (au)

da (au)

e

Ω

4.39316E+ 01 2.012E-03 6.93630E-02 3.81529E-02 6.27292E+ 00 4.97942E-02

5.36863E-02 2.8 9374E+ 00 5.46686E+ 00 -0.655 0.022 -3.068 0.009 43.617 2457072.0 K 11O 60B 14J8 0M

i

ω

0.526 0.020

λ

β

rhepoch

41.185 2456571.9a3330 4.38022E+ 014.304E-034.76380E-02

45.718 2456450.9J94E02S 4.58115E+ 01 2.933E-03 1.15028E-01 1.8 5878E-02 2.70119E+ 00 1.74919E+ 00

1.00912E+ 02 7.941E-02 6.36351E-01 3.39501E-01 2.49069E+ 00 3.94581E+ 00 -0.391 -0.097 38.636 2456571.8

6.23061E+ 01 1.901E-02 2.62680E-01 3.57566E-01 3.18451E+ 00 1.69196E+ 00 -2.049 0.320

K

47.768 2457162.0

No te— Table 1ispublished in itsentirety in the m achin e-readable form at.A po rtion isshown here for guid ance regardin g its form and content.T he M PC designations are given in their packed form at (see http://www.minorplanetcenter.net/iau/info/PackedDes.html). T he orbital elem ents are the best­fit barycentric elem ents from a B ernstein & K hushalani(2000) orbit fit to the astrom etry available for each object from the M PC;da is the 1-σ sem i­m ajor axis uncertain ty (taken from the orbit fit covariance m atrix ). A l angles are given in radians, and the epoch (JD ) is for both the orbit fit and sky po sition.

U sin g the velocity vectors to determ in e the m ean plan e does n ot en tire ly elim in ate the effects of ob servation al b iases, b u t it doesre duce system atic errorsin the calculated m ean plan e.To illustrate this,w e con sidertw o hypothetical surveysof a classical Kuiperb eltpopulation w ith a tru e m ean plan e of i0= 1.8◦ an d Ω 0= 90◦ .Both surveyscover400deg2of sk y cen tere d 5◦ off ecliptic, w ith on e coverin gtw o patches in lon gitude n orth of the ecliptic an d the othercoverin gon e patch n orth an d on e patch south of the ecliptic at differe n t lon gitudes. F igure 8 show s w hatm ean plan es w ould b e re covere d from these tw o sim ulated surveys usin geitherthe velocity vectors (p urp le ellipses)orthe averaged orb itn orm als (b lue ellipses) of sim ulated ob served ob je cts. Itis clear that the m ean plan es foun d b y m in im izin g Σ |ˆ vt ·ˆ n |are m uch m ore con sisten tw ith the population ’stru e m ean plan e (gre en dots) than those foun d b y averagingtheir orb it n orm als. Ob servation al b iases can chan ge the shape of the expected distrib ution of the velocity vectorderived m ean plan es (d iscussed in m ore detail b elow ),b utthe velocity vectorm ethod isn otsub je ctto the sam e huge system atic errorseviden tin the averaged orbitn orm als.

0.1

200 deg2 centered at =5°, =20°

200 deg2 centered at =5°, =20°

and 200 deg2 centered at =5°, =140°

and 200 deg2 centered at = 5°, =120° 0.1 velocity vectors avg. orbit normals

0.05

0.05

0

0

p

p

avg. orbit normals velocity vectors

-0.05

-0.05

-0.1 -0.1

-0.05

0 q

0.05

0.1

-0.1 -0.1

-0.05

0 q

0.05

0.1

Fig u re8.M ean planes d eterm ined from tw o hypotheticalsurveys ofaclassicalK uiper belt population w ith atrue m ean plane of i0 = 1.8◦ and Ω 0 = 90 ◦(g reen dot). The purple ellipses show the 1–σ lim its on a m ean plane d eterm ined by m inim izing Σ |ˆ vt·ˆ n |forsets of simu a l ted observations whiletheblueellipses show the1–σ lim its of theaveraged orbitnorm als forthosesam esimu lated observations.

It is clear that averaging the orbit norm als ofabiased observationalsam ple is not areliable way to recover the m ean plane.

A setof observed ob je ctsdiscovere d in a biasfre e w ay w ould have a sym m etrical distribution of m easure d m ean plan es(i.e . the 1-,2 -,an d 3-σellipsesin p,qspace w ould b e circles).Through sim ulation sof ob servation s,w e foun d thatb iasesin ecliptic latitude atdiscovery do n otchan ge the shape of the m easure d m ean plan e distrib ution ,b ut

18 in tro ducin gecliptic lon gitude re striction s does.Figure 9 show s thre e 1-σun certain ty ellipses forthe expected m ean plan e distribution of a high-apopulation w ith 162 observed ob je cts.If n o re striction s are placed on w here ob je cts are observed in the sk y (i.e .,a bias fre e survey),the un certain ty in the m ean plan e is sym m etric about the tru e m ean plan e (the black circle). W hen the sim ulated observation s are re quire d to m atch the re al observation s in ecliptic latitude, the un certain ty is still sym m etrical, b ut very slightly offset re lative to the b ias­fre e survey (gre en dashed circle).H ow ever,w hen w e re quire that the sim ulated observation s m atch the ecliptic lon gitude distribution of the re al observation s,the un certain ty ellipse b ecom es n oticeab ly elon gated (p urp le ellipse).The re al observed population of KBOs has sign ifican t gaps in the ecliptic lon gitude distrib u tion due to the galactic plan e , as show n in F igure 10.A lm ost all of the asym m etry in the m ean plan e un certain ties w e calculate re sults from this ecliptic lon gitude bias.

The asym m etry is less pron oun ced in the largerobserved population of classical KBOs b ecause m ost of these ob je cts are observed n earthe ecliptic an d the longitude coverage has on ly the tw ogaps due to the galactic plan e.The sm allern um b erof observed high-aKBOs m ean s theirlongitude coverage is less un iform ;thus the asym m etry is larger.

0.15

0.1

p

0.05

0

-0.05

-0.1

-0.15 -0.15

bias-free latitude bias only longitude bias only -0.1

-0.05

0

0.05

0.1

0.15

q

Fig u re9.T he1-σ uncertainty ellipses for the expected d istribution of m ean planes fora50 ≤ a/ au ≤ 150 population of K B O s. The black circle represents theuncertainty from 162 bias-free observed ob jects. The green dashed circle is for 162 ob jects observed at thesam eecliptic latitudesas the realobserved ob jects, w ith no restriction on ecliptic longitude. T he purple ellipse is for 162 ob jects observed at the sam e ecliptic longitude as the real observed ob jects, w ith no restriction on ecliptic latitude.

50

ecliptic latitude (deg)

40 30 20 10 0 -10 -20 -30 -40 -50 -180

-120

-60

0

60

120

180

ecliptic longitude (deg) Figu re10.S ky positions for althe ob jects used in this work (listed in T able 1).

19 C. DETERM IN IN G M EA N PL A N E UN C ER TA IN TIES USIN G M ON TE-C A RL O SIM UL ATIO N S

A s discussed in Section 4.1,w e use the dire ction s of the velocity vectors proje cted on the sk y to determ in e the m ean plan e of a given set of KBOs. This approach defin es the m ean plan e as the plan e of sym m etry of the sk y­plan e proje ction of the velocity vectors B ( rown & Pan 2 004).This m ethodre duces the system atic un certain ties in the m ean plan e arisin g from ob servation al b iases; how ever the effects of b iases can n ot b e en tire ly re m oved (see A ppen dix B),an d there are still poten tially large un certain ties in the m ean plan e foun d b y m in im izin g Σ |ˆ vt ·ˆ n |due to the lim ited n um berof observation sthe sk y location s of those observation s,an d the in trin sic dispersion of the population ’s plan es ab outtheir m ean .W e estim ate these un certain ties usin gM on te-C arlo sim ulation s.

Foreach ob served ob je ct,the follow in gob servation al param eters are dire ctlyre quire d forthe m ean plan e calculation :in clin ation (i), lon gitude of ascen din g n ode (Ω ), ecliptic latitude (β ),an d ecliptic longitude λ( ). To gen erate the expected distribution of observed m ean plan es foran y given tru e m ean plan e,w e sim ulate an in trin sic population of ob je cts distributed in in clin ation about the assum ed tru e m ean plan e an d then select subsets of the sim ulated ob je cts that are foun d n earthe sk y location s β ( ,λ ) of the re al ob je cts.Forthis set of sim ulated “ob served” ob je cts,w e calculate the appare n t m ean plan e b y m in im izin g Σ |ˆ vt ·ˆ n |.This process is re peated 40,000tim es to b uild a distrib ution of sim ulated “ob served” m ean plan es; this is a large en ough sam ple to defin e a 3­σ, 99.7% con fiden ce b oun dary for the exp ected m ean plan e.

Foroursim ulation sto determ in e the un certain ty in the classical Kuiperb elt’sm ean plan e,w e use the follow in gprocedure to assign orbital elem en tsto oursim ulated in trin sic population : •sem i-m ajoraxisaisassign ed ran dom ly from the ran ge a obs,i(0.99,1.01),w here a obs,i isthe observed sem im ajoraxisof aran dom ly selected re al ob served ob je ct •m ean an om aly an d argum en tof pericen terare selected ran dom ly from the ran ge (0,2 π) •eccen tricity eisassign ed ran dom ly from the ran ge eobs,i(0.95,1.05),w here eobs,i isthe ob served eccen tricity of a ran dom ly selected re al ob served ob je ct •in the sim ulation s allow in ga sem i-m ajoraxis depen den t m ean plan e,the values of q0,p0forthe sem i-m ajoraxis of the sim ulated obje ct are determ in ed from the lin earseculartheory of the k n ow n solarsystem •in the sim ulation s w ith a sin gle, fl at in trin sic m ean plan e ,q0,p0are con stan tforall ob je cts •the fre e in clin ation vectorcom pon en ts q ( 1 ,p1) are selected from on e of the tw o Gaussian distribution s that con trib ute to the classical b elt in clin ation distrib ution (see Section 5.1 an d E quation 5.1).

•the com plete in clin ation vectoris then calculated accord in gto 9 •the in clin ation,i, an d lon gitude of ascen din g n ode, Ω , are determ in ed from the in clin ation vector com pon en ts.

The ecliptic latitude an d lon gitude of the ob je ct is then calculated.Foreach β ,λ pairin the re al,ob served population ,w e re peatedly gen erate sim ulated ob je ctsun til on e fallsin the ran ge β±1◦ an d λ±5◦,an d then the appare n tm ean plan e of the setof sim ulated ob served ob je ctsiscalculated;in cre asin gordecre asin gthe allow ed β ( ,λ ) ran gesw ithin factorof ab out tw o does n ot sign ifican tly chan ge the fin al un certain ty estim ates re ported in this w ork . A fter re peatin g this procedure 40,000 tim es, w e use the R statistical pack age to com pute ellipsesin q ( ,p) thaten close 68.2 %,95.4%,an d 99.7% (1,2 ,an d 3-σ) of the sim ulated ob served m ean plan es.The 1-σerrorb arsin Figure 3 corre spon d to the m axim um an d m in im um valesof ian d Ω that fall alon g those ellipsesforeach population .

Oursim ulation s of the high-apopulation are sim ilarto those forthe classical Kuiperbelt.The orbital elem en ts of the un derlyin g p opulation are chosen in a slightly m odified m an n er: •sem i-m ajoraxisan d perihelion distan ce are assign ed ran dom ly from the ran ge a obs,i(0.95,1.05)an d qobs,i(0.95,1.05),w here a obs,i an d qobs,i are the observed sem i-m ajoraxis an d perihelion distan ce of a ran dom ly selected re al observed ob ject.

•m ean an om aly an d argum en tof pericen terare selected ran dom ly from the ran ge (0,2 π) •the in clin ation vectorcom pon en ts q ( ,p) are selected from the distribution fun ction described in Equation 9 (assum in g either a Gaussian distrib u tion ab out the m ean plan e or an em pirically fit distrib u tion ab out the m ean plan e, see Section 5.2 ).

20 •the in clin ation,i, an d lon gitude of ascen din g n ode, Ω , are determ in ed from

the in clin ation vector com pon en ts.

W e again gen erate sim ulated ob je ctsun til on e fallsn eareach observed β ,λ pair,howeverwe also re quire thatthe obje ct’sheliocen tric distan ce fall w ithin 10% of an observed ob je ct’sheliocen tric distan ce.W e add the heliocen tric distan ce con strain tto accoun t forthe fact that the high-aKBOs are m ore stron gly b iased toward discovery at perihelion than the low er e ccen tricity classical b e lt ob je cts. In practice this addition al con strain t has on ly a very sm all affect on the error e llipses (b ecause w e are assum in garan dom distrib ution of the argum en t of perihelion ),b utw e in clude itforcom pleten ess.

D . L IN E A R SE C U L A R

TH E OR Y FOR TH E FOR C E D IN C L IN A TIO N IN TH E KU IPE R BE LT W ITH A N A DDITION A L DIS TA N T,L OW -MA SS PE R TU R BE R

W e con sider the forced in clin ation vector,q ( 0 ,p0 )=sin I0 (cosΩ 0 ,sin Ω 0 ), of a m asslesstestparticle in the distan tKuiperb elt,asdeterm in ed b y the k n ow n eightplan ets(M ercury through N eptun e) plusa distan tplan etof m assm9 ,sem i-m ajoraxisa 9 ,w hose orb ital plan e hasin clin ation I9to the in variab le plan e of the k n ow n plan ets.(Throughoutthissection ,w e use “in variable plan e” tore ferto the plan e w hich isn orm alto the totalorbitalan gularm om en tum vectorof the k n ow n plan etsab outthe Sun .) In the lin ear approxim ation ,the forced in clin ation vectorisstraightforw ardly given b y the L aplace-Lagrange seculartheory M ( urray & Derm ott 1999), 9 µi cos(fit+ γi),sin(fit+ γi) ,(D1) B−fi i=1 (q0,p0)=− w here 9

µj =

9

BjIji,B=− i=1

1 n Bj,Bj = 4

j=1

mj (1) α jα ¯ jb 3/2(α j),(D2 ) m

fi an d Iji are the fre quen ciesan d am plitudesof the n odal secularm odesof the plan ets,γi isa phase determ in ed b y in itial con dition s,m isthe m assof the sun,mj an d a j are the m assan d sem i-m ajoraxisof the j-th plan et,n isthe m ean m otion of the testparticle, aa j α j =m in { , }<1,(D3) aj a

1 if a<aj , α¯ j =

a /aif a <a, j

(D 4)

j

(1)

an d b3/2(α ) is a L aplace coeffi cien t, (1)

b3/2(α )=

2

1 π

dψ 0

cosψ .(D5) (1 + 2 α cosψ + α 2 )3/2 (1)

Forthe calculation spre sen ted b elow ,w e n ote the follow in guseful approxim ation sforb

b

(1)3/2(α )

3α forα 1, 2

π (1−α )2forα − →1.

3/2(α ),

(D 6)

A distan t low m ass plan et,a 9 a 8 an d m9 m8 , w ould n ot sign ifican tly affect the lin ear secular solution of the k n ow n plan ets.There fore ,w e assum e that the secularm odes 1–8 are un perturbed b y such a plan et.W e also assum e that the hypothetical “plan et-9”’s in clin ation re lative to the in variab le plan e w ell exceeds the am plitude of the secularperturb ation s that w ould b e in flicted on it b y the k n ow n plan ets. In this case, the secularly forced in clin ation of an ob je ctin the Kuiperb eltw ould departfrom the in variable plan e on ly forKBO’s w ith sem i-m ajoraxesaclose to a 9 ,an d w e can w rite (q0 ,p0 ) (q0 ,p0 )0 + S forced,9 cos(f9 t+ γ9 ),sin(f9 t+ γ9 ) ,(D7)

2 w here the first term , (p0 ,q0)0, describ e s the forced plan e defin e d b y the k n ow n plan e ts, an d the last term is the deviation from thatplan e ow ed to plan et-9. ForKBOsw ith a>50au q ( 0 ,p0)0asym ptotically m erges w ith the in variab le plan e of the solarsystem ’s k n ow n plan ets.The am plitude of the perturb ation of the forced plan e due to plan et-9 is given b y µ9 S f orced ,9 = B9sin I9 = .(D8) 8 −B+ f9 j=1 Bj + B9 + f9 The n odal pre cession fre quen cy,f9,of plan et-9 isdeterm in ed b y the orb it-averaged quadru polargravity of all the otherplan ets,an d can b e approxim ated as f9 −

8

n9 4

i=1

3 mj (1) α j9 b 3/2(α j9) − m

w here α j9 =a j/a 9 1,an d w e used the approxim ation b Fora testparticle w ith a a 8 ,w e can approxim ate 8

(1 )

j=1

7/2 8

a8 a9

i=1

mj m

aj a8

2

,(D9)

3/2(α j9 )≈3α j9 .

3 a8 n8 a 4

Bj

4

n8

7/2 8 i=1

mj m

aj a8

2

.(D10)

A n d,fora in the sem i-m ajoraxis n eighb orhood of plan et-92,w e can approxim ate B9

1 2π

n

a8 a

3/2 m 9

α 9α ¯9 .(D11) m (1 −α 9)2

W e n ote thatf9hasa n egative value,in dicatin ga n egative rate of n odal pre cession (i.e .,n odal re gre ssion ).W e also n ote that 8 j=1 Bj + f9 is gre aterthan zero w hen a<a 9 an d less than zero w hen a>a 9 ;it van ishes w hen a=a 9 ,at w hich location the forced in clin ation of the test particle approaches the in clin ation of plan et-9.A ddition ally,w e n ote that the den om in atorin E quation D8 has a sin gularity w hen B=f9.Fora low m ass plan et-9,this sin gularity (a n odal secularre son an ce) occurs ata sem i-m ajoraxis value a ν9>a 9 .The righthan d side of E quation D8 is positive fora<a ν9an d becom esn egative fora>a ν9;w e can absorb thissign change asa phase change of πin the forced in clin ation vectorb y re placin gγ9 w ith γ9 + πfora>a ν 9,so S forced ,9 = sin Iforced can be in terp re ted as a positive quan tity forall values of a.

Fora<a 9 ,usin gE quation sD9–D11 in E quation D8, w e fin d the forced in clin ation , sin If orced = 1 +

3π(1 −α 2

7/2 2 9)(1 −α 9 )

α9

8 2 i=1mja j m9 a2

−1

sin I9.(D12 )

W e can re -arran ge the ab ove equation to derive an expre ssion forthe m assof plan et-9, m9

3π

(1 −α 9 )2 1−(a/a 9 )7/2 α 9α ¯9 2

8 2 i=1mja j a2

sinIf or ced sin I9 −sin Iforced

.(D13)

The forced in clin ation s calculated usin gthe ab ove approxim ation forthe lin earseculartheory (E q.D 12 ) agre e w ell w ith forced in clin ation s calculated from the full lin earseculartheory equation s.F igure 7in Section 6 illustrates a few possib le com b in ation s of plan et-9 param etersm , 9 ,a 9an d I9,that could explain the ob served ~ 4◦ –7◦ deviation of the m ean plan e aw ay from the in variab le plan e forKBOs w ith a~ 50−80au.F igure 11 show s the forced in clin ation s fortest particles w ith a= 30−100au given b y the full lin earsecularsolution forthe kn ow n gian t plan ets plus an addition al plan etw ith a 9 =62 au,I9 =7◦ an d m9 =0.1,1,2 mm a rs; the ran ge of KBO sem i­m ajor axes affected b y the addition al plan etdepen dson itsm ass.A M ars-m ass plan etcan gen erate a ~ 15au w ide w arp in the m ean plan e,b ut even a lun ar-m ass ob je ctcan gen erate a ~ 5−10au w ide w arp in the m ean plan e.

2W e note that for secular theory to be valid , a test particle m ust be suffi ciently far from the planet to avoid the eff ects of overlappin g m ean m otio n resonances. For a M ars­m ass planet at the distances w e consid er, this only requires a sem im ajor axis diff erence of 1au.

2

forced inclination (deg)

extra perturber with i=7, a=62 au 14

0.1 Mars masses 1 Mars mass 2 Mars masses

12 10 8 6 4 2 0 30

40

50

60

70

80

90

100

a (au)

Figu re11.The forced inclinations for test particles as afunction of sem i-m ajor axis from the fullinear secular solu tion for the know n giant planets plu s an additionalplanet w ith a9 = 62 au,I9 = 7â&#x2014;Śand m 9 = 0 .1m mar s (purple),1m mar s (green),and 2 m mars (blue).

Astronomer Biography of The Month Written By Brian Jones

8th of July 2017 is the anniversary of the death, at The Hague on 8 Jul 1695, of the Dutch astronomer Christiaan Huygens. He was a founding member of the French Academy of Sciences and made significant contributions to mathematics and clock design. However, he is probably best remembered for the discovery, on 25 Mar 1655, of Saturnâ&#x20AC;&#x2122;s largest moon Titan and for correctly interpreting (rather than actually discovering) the true shape and nature of Saturnâ&#x20AC;&#x2122;s rings.

Back Cover Image - Alex Sanders Image Taken From a Hotel In Sicily Taken for the Sanderphil Urban Observatory