The plane of the Edgeworth–Kuiper belt

Simon J. Collander-Brown,a Mario D. Melita,b,* Iwan P. Williams,b and Alan Fitzsimmonsa

a Department of Physics, Queens University Belfast, Belfast BT7 1NN, United Kingdomb Astronomy Unit, School of Mathematical Sciences, Queen Mary, University of London, E1 4NS, United Kingdom

Received 13 May 2002; revised 1 October 2002

Abstract

We examine possible locations for the primordial disk of the Edgeworth–Kuiper Belt (EKB), using several subsets of the known objectsas markers of the total mass distribution. Using a secular perturbation theory, we find that the primordial plane of the EKB could haveremained thin enough to escape detection only if it is clustered very closely about the invariable plane of the Solar System.© 2003 Elsevier Science (USA). All rights reserved.

Keywords: Transneptunian objects; Planetary dynamics; Orbits; Origin, Solar System

1. Introduction

Over the past few years there has been a dramaticincrease in the numbers of Edgeworth–Kuiper belt ob-jects (EKBO) discovered (581 at the time of writing, 133of these in 2001). These discoveries make it clear that thestructure of the Edgeworth–Kuiper belt (EKB) is morecomplicated than had been first thought. After the firstfew discoveries it was clear that the EKBOs could besplit into two distinct types, those trapped in resonancewith Neptune and those on near circular orbits beyond 42AU. The discovery of 1996 TL66 (Luu et al., 1997)introduced a third class of EKBO, the so-called scatteredobjects. However, as Fig. 1 shows the strangest feature ofthe distribution of EKBOs is not where they are, butrather where they are not. At the time of writing noobjects on low eccentricity orbits have been found be-yond 50 AU. A number of deep searches that should havedetected such objects have been carried out (e.g., Allen etal., 2001; Chiang and Brown, 1999; Gladman et al., 1998;Luu and Jewitt, 1998). It should be noted that all of thesesearches did find classical EKBOs.

It has been suggested (Hahn, 2000) that the objectsbeyond 50 AU orbit in a dynamically cold disk less than 1°across about the invariable plane of the Solar System (IP).If this is the case then the deep searches that have for themost part been carried out in the ecliptic could well havemissed most bodies. There is also a further possibility. If theprimordial plane of the solar nebula did not coincide withthe currently observed IP and the EKBOs where in thenebula plane, how thin would the distribution have re-mained? The answer must depend on the location of thisplane.

The IP is the plane defined by the total angular momen-tum of the system (excluding the Sun1). The orbital plane ofthe Earth is inclined with respect to the IP by�1.6°. Henceif the cold disk is less than�1.0° across and in the IP,searches in the ecliptic may not find it.

Since the angular momentum is conserved, if the systemis left in isolation the present IP of the system wouldcoincide with the primordial IP at formation. However, theangular momentum vector could have changed direction fora number of different reasons, for example, through pertur-bations from an external body or through a change in the

* Corresponding author. Fax:�44-20-8983-3522.E-mail address: m.d.melita@qmul.ac.uk (M.D. Melita).

1 Its inclusion would change the direction of the angular momentum by�0.08°.

R

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mass of the system (planets or massive planetesimals beingexpelled in hyperbolic orbits).

The plane of the Sun’s rotation is angled at approxi-mately 7.35° to the plane of the ecliptic. If this were theplane of the nebula then this would imply the opposite of thenebula being in the IP, that at some point in its history, theSolar System has gained or lost a substantial amount ofangular momentum, either by the loss of a large body or bythe interaction with a passing star.

There is of course no reason to assume that the primor-dial thin disk should have been aligned exactly with eitherof these two planes. A reasonable assumption is that theplane of the known EKB bears some information aboutthe plane of the primordial disk. Several attempts are madeto estimate its location, based on the available data inSection 2.

In Section 3 we present a simple dynamical model tostudy the evolution of a distant thin disk. Our goal is toinvestigate whether planetary perturbations over the age ofthe Solar System could prevent it from remaining thinenough to avoid detection.

Finally, our results are discussed in Section 4.

2. The plane of the EKB

In theory, the calculation of the plane of any set ofobjects is simple: it is orthogonal to the average angularmomentum vector of all the objects that make up the disk.However, for the EKB there are two problems: we do notknow all of the objects that are part the disk and we do notknow the masses of the known objects. We overcome thefirst problem by assuming that the currently known objectsare representative of the population as a whole. Since mostof the mass lies in the large mass objects (Gladman et al.,2001) we may overcome the second problem by taking thelarge mass objects as markers. We note that this hypothesismay not be correct if there is a Pluto-sized object as yetundiscovered.

In our calculations we use only orbits that correspond toobjects observed at more than one opposition, taken fromthe MPC WWW site (Marsden, 2002).

The resonant and the scattered objects are not consid-ered, mainly because they have exchanged a great deal ofangular momentum with Neptune. This leaves the classicalbelt objects, which we define as those with orbits with 40 �a � 47 AU, e � 0.3, and I � 0.25 rad. The limits in thesemi-major axis remove any objects that may be in the 3:2or 2:1 mean motion resonances with Neptune. This selec-tion criterion leave us with 141 objects.

Levison and Stern (2001) suggest that there may be twoseparate inclination distributions within the classical EKB:the first is dynamically hot with inclinations up to 35° madeup of objects with H � 6.5, and the second a dynamicallycold distribution with inclinations less than 5° made ofobjects with H � 6.5. It seems reasonable to assume that the“cold” disk better represents the original plane of the EKB.Doing this reduces the population to 122 objects.

In Fig. 2 we plot the normalized angular momentumvectors, projected onto the plane of the ecliptic for the realEKB objects. Also shown are the locations of various meanvalues calculated as discussed above. The distance from theorigin is equal to the sine of the inclination of the planerelative to the ecliptic, and the azimuthal angle defineswhere the planes cross. Also plotted is the position of the IPand the position of the average plane of the EKB if aPluto-size object were placed at the extreme edge of theorbital element space encompassed by the selection criterialisted above. The position of the plane of the solar equator,which would be at (�0.122, 0.32), is outside the range ofthis plot. It is apparent that the plane of the ecliptic is not thecenter of the distribution of EKBOs, and the position of thesolar equator is nowhere near. Unfortunately each of thepossible methods for calculating the plane of the EKB givessignificantly different results. The IP is consistent with thedistribution and it is noticeable that the average plane of thelow mass objects is very close to the IP. The most reason-able conclusion from these data is that if there is a plane ofthe EKB then it is very close to the IP.

Fig. 1. Plots of eccentricity and inclination against the semi-major axis forthe known Edgeworth–Kuiper belt objects. Objects observed in more thanone opposition are indicated by a solid marker.

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3. The evolution of the thin disk

We investigated the evolution of distant thin disks atvarious locations with respect to the IP to determine howthin will they remain over the age of the Solar System underplanetary perturbations.

Beyond 50 AU no first-order secular or mean motionresonances exist in the EKB (Knezevic et al., 1991). Sincewe are interested in a low eccentricity and inclination dis-tribution of objects, a low-order secular theory (see forexample Murray and Dermott, 1999) is an accurate modelfor the dynamical evolution of the distant EKB. To test thismodel we carried out a series of one hundred million yearintegrations using a Runge–Kutta–Nystrom 12(10)17 inte-grator (Dormand et al., 1987). These integrations confirmedthe validity of the model.

The evolution of the complex eccentricity (h � e sin(�),h � e cos(�)) and the complex inclination (p � i sin(�), q� i cos(�)) of the EKB objects is given by

h � efree sin(gt � �) � h0,

k � efree cos(gt � �) � k0, (1)

p � ifree sin( ft � �) � p0,

q � ifree cos(ft � �) � q0, (2)

where efree, ifree, �, and � are constants determined by theinitial conditions of the object, and (h0, k0) and (p0, q0)define the forced eccentricity and inclination

eforced � �h02 � k0

2, (3)

iforced � �p02 � q0

2, (4)

where (h0, k0) and (p0, q0) depend on the proper elementsand frequencies of the planets (Brouwer and van Woerkom,1950). The perihelion and node proper frequencies of theobject g and f are given by

g �n

4 �i�1

N mi

MJ

�ib3/ 2�1 ��i, (5)

f � �n

4 �i�1

N mi

MJ

�ib3/ 2�1 ��i, (6)

where n is the mean motion of the object (note that g � �fin this approximation), mi and MJ are the masses of theplanets and the Sun respectively, �i � ai/a are the ratiosbetween the semi-major axis of the planets and the objects,and b3/2

(1)(�i) is the corresponding Laplace coefficient (Mur-ray and Dermott, 1999). It should be noted that these ex-pressions are valid only for objects with orbits exterior tothat of the planets, inclinations and nodes are referred to theIP, and since the constants given by Brouwer and vanWoerkom (1950) were used, then the all 9 planets wereincluded for consistency, although the contributions fromthe inner planets are negligible.

The forced eccentricities and inclinations in the Kuiperbelt beyond 60 AU are smaller than 10�3 and 0.25°, respec-tively and the magnitudes of the proper frequencies rangefrom 0.1 arcsec/year and below.

According to Eqs. (1) and (2) the evolution of the com-plex eccentricity and inclination is defined as an epicycleabout the forced values with frequencies g and f, respec-tively. Hence the closer the initial conditions to the forcedvalues, the smaller the free values, i.e., the smaller theamplitude of the variation.

Inclinations librate about the IP, but nodes rotate. Thus,if the initial distribution is not in the IP, after a timescalemuch greater that the node precession period, it would loosememory of its initial shape and it will distribute about the IP,with a mean inclination about that of the initial plane.

3.1. Simulations

First we study a primordial distribution of distant disksclosely clustered about the IP (node � 107.65°, inclination� 1.57°). A distribution of 1000 orbital elements was se-lected with a and e randomly distributed in the ranges 50 �a � 100 AU and e � 0.05 and with inclinations normallydistributed about the IP with a dispersion �i � 0.5°. Theother angular elements are randomly distributed in the range0°–360°.

This distribution has been evolved according to Eqs. (1)and (2). Note that the semi-major axis do not suffer secularvariation in this approximation, and since the secular evo-

Fig. 2. A plot showing the projection of the normalized angular momentumvectors projected onto the plane of the ecliptic. The dots represent the realEdgeworth–Kuiper belt objects, Q is the IP, * is the plane defined using theobjects with H � 6.5, is the plane defined by assuming all the bodieshave equal mass, � is the plane defined by the average angular momentum,and � is the plane defined by adding a Pluto to the plane defined by theaverage angular momentum.

24 S.J. Collander-Brown et al. / Icarus 162 (2003) 22–26

Fig. 3. (Bottom) Sky distribution with respect to the ecliptic of a thin primordial plane about the IP and its corresponding evolved distribution after 4.5 Gyr.(Top) Sky distribution with respect to the ecliptic of a primordial plane inclined with respect to the IP. The initial and final distributions after 4.5 Gyr areshown. The location of some deep sky searches performed so far are also plotted (Gladman et al., 1998; Chiang and Brown, 1999; Allen et al., 2001; Fletcheret al., 2000). The IP is indicated.

25S.J. Collander-Brown et al. / Icarus 162 (2003) 22–26

lution does not depend on the mean longitudes, this has beengenerated randomly to plot the distribution in the sky. Theinitial and final distribution, together with the location of thedeep sky surveys, are shown in Fig. 3 (bottom). Since thefree values of the inclination of these objects are very small,then there is little departure from its initial values. Such adisk would be undetected by most of the surveys performedalong the ecliptic.

A second distribution in which the orbits are clusteredabout a plane with a mean inclination of i0 � 2° (�i � 1.5°)and a node �0 � 50° (�� � 30°) was also investigated. Theresults are shown in Fig. 3 (top). It is apparent that over theage of the Solar System the distribution has lost memory ofits original plane and its thickness about the IP is such thatit should have been detected.

4. Conclusions

If there is a thin disk of objects orbiting beyond 50 AUit will only remain thin if it is close to the invariable plane.However, if it is not close, it will loose memory of its initialstage and evolve into a much thicker disk about the invari-able plane. This conclusion has a number of consequences.If a thin disk is ever found, this means that the invariableplane is the primordial plane of the Solar System and thiswould put very stringent limits on the evolutionary historyof the Solar System. It would rule out the possibility of anylarge lost planets or close encounters with stars. On theother hand any searches for this distant disk should becarried out in the invariable plane. The observations ofAllen et al. (2002) would suggest that there is no such thindisk at red magnitudes smaller than 25.7. Some explanationwill have to be found for the absence of large objectsbetween 50 and 76 AU. However, at present, smaller objectsthat escape detection cannot be ruled out. Several hypoth-eses can be put forward to explain such an edge. If the diskresumes after �76 AU, it would imply the presence of anagent that has cleared a gap in the distribution (Brunini andMelita, 2002). If the disk is effectively truncated at 50 AU,then a close stellar passage would be a plausible hypothesis(Ida et al., 2000). The outer limit of the solid distribution inthe primordial solar nebula may have been located at thosedistances or the accretion process is truncated at thosedistances.

Acknowledgments

S.J.C.-B. and M.D.M. acknowledge PPARC for provid-ing the funding that enabled this work to be carried out.

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