ISSN 0027�1349, Moscow University Physics Bulletin, 2010, Vol. 65, No. 1, pp. 59–64. © Allerton Press, Inc., 2010.Original Russian Text © B.R. Mushailov, V.S. Teplitskaya, 2010, published in Vestnik Moskovskogo Universiteta. Fizika, 2010, No. 1, pp. 59–64.

59

INTRODUCTION

The existence of a trans�Neptunian belt was pre�dicted by K. Edgeworth (1949) and G. Kuiper (1951).The Kuiper belt is located 30–50 AU from the Sun.

The possibility of the existence of several Kuiperbelts was predicted in [1, 2]. These trans�Neptunianbelts may exist in the region 100 ≤ a < 1000 AU. Thecurrent discoveries of large objects (with diameters ofabout 1000 km) beyond Neptune’s orbit show thatlarge planets might also be found there. In particular,[3] predicted the presence of trans�Neptunian regionswith hypothetical large planets and calculated themean motion resonances with these hypotheticalplanets.

There are a great number of studies devoted toCentaurs and trans�Neptunian objects (TNOs). Someworks distinguish between two main dynamic classes:(1) Centaurs and scattered disc objects (SDOs) and(2) Kuiper belt objects (KBOs). The latter are conven�tionally divided into classical KBOs and several sub�classes of resonant objects [4]. The resonant objectsare understood here as objects with librational motion[5, 6]. The evolution of their orbits and modern struc�ture of the trans�Neptunian belt are considered to beshaped by Neptune’s dynamic evolution [7–9].

However, the majority of the studies on TNOs evo�lution use integration over time periods (millions ofyears) for which the dynamic instability effects are sig�nificant. Therefore, the reliability of results obtainedunder the gravitational model is doubtful. The conceptof the partial determinacy of the dynamic evolution ofTNOs based on the limited elliptic three�body prob�lem, including gravitational perturbations from Ura�

nus, Saturn and Jupiter [10–12] enables one to build areliable analytical model, predict the evolution of theorbital parameters of TNOs, and find the most proba�ble regions in which to search for new objects. Basedon the results in [10], the authors of [13] discovered anumber of objects in these regions.

Because of the steady growth of the discoveredTNO population, it is still relevant to systematize theobserved TNO distributions by orbital parameters.Regardless of the motion pattern (libration or circula�tion), the resonance effects are determined by thecommensurability of the motions of gravitatingobjects. Hence, contrary to some works, some of theso�called classical KBOs are in fact resonant objects,because they display orbital commensurabilities withthe large planets of the Solar System. Although thereare a large number of works on the statistical distribu�tions of KBOs [4–9], up to now there has not beenenough attention paid to uncovering and providing atheoretical framework for the patterns in the observeddistributions of TNOs and Centaurs, which is the pur�pose of our paper.

RESONANCES

The condition for two�frequency resonance is therational quasi�commensurability of frequencies of the

type [(k + l)n – kn'] ≤ O( ), where l is the order andk is the multiplicity of the resonance; l and k are natu�ral numbers; n < n' are frequencies or mean dailymotions of gravitating bodies, and μ is the reduced

mass. The resonance band with the order 2n' /(k + l)

μ

μ

On the Role of Resonance Effects in Statistical Distributions of Centaurs and Trans�Neptunian Objects

B. R. Mushailov and V. S. TeplitskayaAstrometry Department, Sternberg Astronomical Institute, Moscow State University,

Universitetskii pr., Moscow, 119992 Russiae�mail: verateplic@yandex.ru

Received July 8, 2009; in final form, October 13, 2009

Abstract—This paper discusses patterns in the distribution of Centaurs, scattered disks, and Kuiper beltobjects according to the semi�major axes, eccentricities and inclinations of their orbits. Over half of them arefound to move in resonant orbits and these were predicted earlier. An interpretation is given for the divergenceof the maximum in the observed semi�major axis distribution with an exact orbital resonance determinedwithin the classical three�body problem.

Key words: Kuiper belt, scattered disk objects, Centaurs, orbital resonances, trans�Neptunian objects, statis�tical distributions.

DOI: 10.3103/S0027134910010145

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MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010

MUSHAILOV, TEPLITSKAYA

is localized near the point of exact commensurability:n/n' = k/(k + l) in case n < n', or n/n' = (k + l)/k in casen > n'. Commensurabilities of the first order (Lindbladresonances) are characterized by the maximum reso�nance effect. Given a fixed l, the maximum amplitudeof the resonance effect is achieved at k = 1. Orbital res�onances affect the motions of some large planets, sat�ellites, asteroids, comets, and meteor flows in theSolar System and exoplanetary systems.

The dynamic evolution of the orbits in which manyof the aforementioned bodies move can be correctlyinterpreted using the resonance version of the classicalthree�body problem with regard to secular perturba�tions from exterior bodies.

It follows from [11] that if the analytical expressionfor the semi�major axis (a) of the orbit of a passivelygravitating body (TNO) includes secular perturbationsfrom Jupiter, Saturn, and Uranus (Pi = 1, 2, 3), then ithas the form:

(1)

where a0 is the semi�major axis in the absence of per�turbation from the giant planets;

(2)

γ is an integral constant; μψ is Neptune’s mass;

and are the masses and semi�major axes of the

orbits of Jupiter, Saturn, and Uranus respectively. TheSun is taken as the unit mass; the semi�major axis of

Neptune’s orbit, as the unit distance; is therespective Laplace coefficient.

The semi�major axis of a TNO’s orbit correspond�ing to an exact orbital resonance with regard to pertur�bations from giant planets has the form:

(3)

a a0 Δ,+=

Δ Δ0mi

ai

����βi L1/20( ) βi( ) βiDL1/2

0( ) βi( )+[ ],i 1=

3

∑=

Δ013�� k

k l+( ) 1 μψ+( )������������������������������⎝ ⎠⎛ ⎞ 2/3

, βi ai/γ,= =

m1 3,

a1 3,

L 1/2( )

0

a αψk l+

k��������⎝ ⎠⎛ ⎞

1/3

Δ–2

,=

where aψ is Neptune’s orbit semi�major axis.For the first order resonance (l = 1), let us intro�

duce the variables: x = cos(η), y = sin(η);the values ξ and η were determined in [11]; the solu�tion of the limited elliptic three�body problem withregard to secular perturbations from giant planets isreduced to integration of an autonomous canonicalequation system with one degree of freedom of the

form = , = – , with the Hamiltonian:

(4)where

(5)

I and γ are integrals of the problem; a, e, i are the semi�major axis, eccentricity and inclination of the orbit ofthe TNO. The expression for the coefficient B is alsogiven in [11].

Stationary solutions for the variables x, y are deter�mined by a system of algebraic equations

(6)

The phase�plane configuration is shown in Fig. 1.Based on the integrals of problem (5), we can

present instability areas in a diagram (e, a) and com�pare them to the observed data (Fig. 2).

TNOs may live for a long time if their orbital ele�ments are located outside the shaded areas and con�centrate near the boundaries of these areas. Consider�ing the asymmetry of the theoretical “instabilityareas” about the exact commensurability points n0(k),it is easy to see that, other conditions being equal, theresonance regions are more likely to contain TNOs(with higher eccentricities as well) at n < n0(k) than atn > n0(k).

Figure 2 shows that the discovered TNOs arelocated on the plane defined by the orbital parameters(e, a) consistent with the theoretical results; all ofthem are outside the instability areas except for 3.7%of objects for the 2 : 1 resonance and 0.55%, for the3 : 2 resonance with Neptune, which are the most sus�ceptible to migration.

For objects in orbital commensurabilities (4 : 3, 3 : 2,and 2 : 1) with Neptune, we used the orbital parame�ters given in [14] to calculate the positions of therespective image points on the phase plane. It turnedout that for the 4 : 3 commensurability 90% of all thediscovered objects were located in the libration area

2ξ 2ξ

dxdτ���� ∂F

∂y����� dy

dτ���� ∂F

∂x�����

F x2 y2+( )2

A x2 y2+( ) Bx,+ +=

A 4/ k 1+( )[ ] γ E 2–– Δ+{ },=

E kk 1+���������� 1 μψ+

1/6

,=

γ a 1– k– 1 k+( ) 1 e2– icos+[ ]2–,=

I2 4 a 1 e2– i/2( ),sin2

=

4x x2 y2+( ) 2Ax B+ + 0,=

4y x2 y2+( ) 2Ay+ 0.=

x3 x2 x1 x

y

1

2

3

Fig. 1. Phase�plane configuration. The trajectories cross�ing the stationary point (x1, 0) are separatrices delimitingthree characteristic areas. Because of symmetry, only theupper half plane is given.

MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010

ON THE ROLE OF RESONANCE EFFECTS 61

(Area 3); for the 3 : 2 commensurability, 83% ofTNOs; for the 2 : 1 resonance with regard to secularperturbations, 65% of TNOs.

The probabilities of transitions into the librationarea can be estimated by analyzing the probabilities oftrajectories transitions from one area on the phaseplane into another under the influence of various per�turbing factors characterized by the independentparameters δ = (δ1, δ2, …, δn). The probability of a per�turbation�affected trajectory transition from Area iinto Area j on the phase plane is determined by the fol�lowing expression [11]:

where Δ' = arcsinε1 + ψ1ε2,

Wij = 1–( )j 2

4 j–��������Δ ' j 3–( )π

2��+

1–( )iπ2�� Δ '–

��������������������������������������������

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

, i = 1 2, j, = 2 3,,

ε112��

ψ2

1 1 ψ2–( )ε3sgn+�����������������������������������⎝ ⎠⎛ ⎞ 1/4

,=

ε21

3����� 1 4ε3

2–( )1/2

, ε3π 1 ψ2–arccos+

3������������������������������������⎝ ⎠⎛ ⎞ .cos= =

Here, ψ1 and ψ2 are independent parameters whichare in the following relationship with the coefficients

of Hamiltonian (4): ψ1 = – , ψ2 = – .

The coefficient B (given a fixed μψ) depends on theresonance multiplicity and order; A is the system statefunction characterized by integral constants.

The probabilities of TNO transition from Area i

into Area j depending on (δ) are given in Fig. 3.For the objects under study, the probabilities of transi�tion from one area on the phase plane into another are

AB���∂B/∂δ∂A/∂δ������������� 27

4����B2

A3����

Δ 'cot

Fig. 2. Instability areas in the e–a diagram in case of first�order resonance for various multiplicities k.

0−π/4−π/2 π/4 π/2 Δ

Wij

W32 W33 W13 W31

W12 W32W21W31

1

Fig. 3. Probabilities Wij of TNOs transitions from Area iinto Area j depending on cotΔ'(δ) = .Δ

0.40

0.35

0.30

0.25

0.20

0.10

0.05

0

e

30 31 32 33 34 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50a, a.e.

k = 1

k = 2k = 3k = 4

0.15

36

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MUSHAILOV, TEPLITSKAYA

significantly dependent on the parameter ρ =(∂B/∂δ)/(∂A/∂δ). For example, the probability oftransition into Area 3 for the 4 : 3 and 3 : 2 resonances:W13 > W12 at ρ from –0.05 to –0.03, and W23 > W21 atρ from 0.03 to 0.05. For the 2 : 1 resonance: W23 > W21

at ρ from –0.018 to 0.011 and W13 > W12 at ρ from 0.01to 0.02.

Using different variants of the three�body problemand given some special conditions, the orbital param�eters of the hypothetical large planets and the reso�nance regions related to these planets where hypothet�ical minor bodies may be located at cosmogonic timeintervals have been predicted [3]. The presence of res�onant objects between the orbits of giant planets hasalso been predicted [15]; in [11], the authors calcu�lated the regions of resonant TNOs and studied theevolution of their orbital parameters. A summary tableof the resonance regions from [2, 11, 15] has the fol�lowing form:

Here, Planet 1 and Planet 2 are hypothetical largeplanets for which the table also cites the related com�mensurability areas (the area width was on the order

of ) and the quantity of objects in these areas. Themarked line stands for the case that includes the secu�

μ

lar perturbations (3). As to Centaurs and SDOs, over26.3% of these objects are located in the indicatedarea; as to KBOs, 79% with regard to (3). The totalband comprising all these areas covers 17.65% of thearea 5–125 AU for Centaurs and SDOs and 32.7% ofthe area 30–50 AU for KBOs.

PATTERNS IN THE OBSERVED DISTRIBUTIONS

The population of the Centaurs and TNOs thus fardiscovered allows one to analyze statistical significantpatterns in the distributions of these objects by someorbital parameters. Thus, we used the data in [14](May 2009 est.) to form distributions of Centaurs andTNOs by semi�major axes (Fig. 4) as well as the eccen�tricities and inclinations of their orbits and comparedthese data with the theoretical results.

The semi�major axis distribution of KBOs isobserved to have two pronounced maximums, with thesemi�major axes corresponding to the observed distri�bution maximums of 39–40 and 43–44 AU beingconsistent within computational accuracy with thevalues 38.50–40.37 and 42.60–44.48 AU predicted in[11, 15]. The eccentricities and inclinations of the

The distribution of the objects by resonance areas

Planet Commensurability Interval Δa, AUQuantity

KBOs Centaurs and SDOs

Saturn 2 : 1 14.826 15.681 – 2

Saturn 5 : 2 17.181 17.858 – 4

Saturn 3 : 1 19.845 21.207 – 9

Saturn 4 : 1 23.662 24.518 – 7

Uranus 5 : 3 27.265 27.412 – 1

Neptune 5 : 4 34.624 35.165 6 0

Neptune 4 : 3 36.062 36.800 10 0

Neptune 3 : 2 38.499 40.371 225 1

Uranus 3 : 1 39.280 40.686 191 1

Neptune 5 : 3 41.901 42.577 49 2

Neptune 7 : 4 43.374 43.894 140 0

Neptune 2 : 1 46.482 48.836 76 0

Neptune 2 : 1 42.599 44.48 433 0

Planet 1 1 : 2 47.185 47.416 5 0

Uranus 4 : 1 47.761 49.432 20 1

Neptune 5 : 2 54.669 55.430 – 8

Uranus 5 : 1 56.263 58.123 – 8

Planet 1 2 : 3 57.207 57.394 – 3

Neptune 3 : 1 61.004 64.023 – 14

Neptune 4 : 1 73.726 77.459 – 4

Planet 1 4 : 3 90.852 91.149 – 1

Planet 2 3 : 4 123.740 124.222 – 1

MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010

ON THE ROLE OF RESONANCE EFFECTS 63

TNO orbits localized near the maximum correspond�ing to the semi�major axis 43.5 AU are small: e ~ 0.0–0.17, i ~ 0°–10°, which is consistent with the results in[3, 15]. For the maximum corresponding to the exactcommensurability of 3 : 2 with Neptune, the eccen�tricities are mostly located in the area 0.15–0.30, andthe orbit inclinations, from 0° to 20°. It follows from(3) that correct estimation of the secular perturbationsfrom Uranus, Saturn and Jupiter leads to correlations ofthe observed distribution maximum (at a = 43.5 AU)

with the 2 : 1 resonance with Neptune. The semi�major axis distributions of Centaurs and SDOs are alsoobserved to have pronounced maximums correspond�ing to the 4 : 1, 5 : 2 resonances with Saturn and 3 : 1,5 : 2 resonances with Neptune. The eccentricities ofCentaurs and SDOs are mostly located in the interval0.2 < e < 0.7, and the osculating inclinations of theirorbits generally do not exceed 30°.

We found (from the data for 2002, 2006, and 2008)that refinement of the orbital parameters and increasein the populations of the discovered objects does notaffect the form of the above distributions. The correla�tion coefficients between the similar orbital parame�ters from the data for 2006 and 2009 are 0.99, and forthe respective values of the distribution series, from0.91 to 0.99, depending on the sample size. Whenchanging the semi�major axis diagram interval from 5to 1 AU for Centaurs and SDOs and from 1 to 0.1 AUfor KBOs, the revealed tendencies persist. Reducingthe sample interval enables one to see minor details ofthe distribution.

Other conditions being equal, the statistical distri�butions under study are influenced by selective effects(e.g., more distant and/or less bright objects are moredifficult to discover). To smooth out the selectiveeffects, we conducted wavelet analysis of the distribu�tions, which confirmed the invariance of the distribu�tion maximum pattern (Fig. 5).

CONCLUSIONS

The maximums in the semi�major axis distribu�tions of KBOs and irregular bodies, i.e., Centaurs andSDOs, correlate with the orbital resonance regionscorresponding to the maximum amplitude of the reso�

9590858075706560555045403530252015105a, a.e.

300

250

200

150

100

50

0

Quantity

Fig. 4. General distribution of Centaurs and TNOs by semi�major axes.

504642383430

504642383430

Observed distribution

Smoothed distribution

Fig. 5. Smoothing the selective effects (noises) in the KBOdistribution with the Daubechies wavelet.

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MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010

MUSHAILOV, TEPLITSKAYA

nance effects. It is possible to eliminate the divergenceof the maximum in the observed KBO distribution bytheir semi�major axes from the theoretical estimatecorresponding to the exact orbital resonance if the sec�ular perturbations from giant planets are correctly esti�mated to account for the uncertainty of integral con�stants. Statistical significant patterns in the distribu�tions of the studied objects by their orbital parameterspersist regardless of the population size and orbitalparameter refinement. In the neighborhood of theobserved maximums (a0) in the semi�major axis distri�butions after the elimination of selective effects, onecan see, in full agreements with the theoretical predic�tions, an asymmetry of the distributions; i.e., at a > a0there are more objects than at a < a0. The majority ofthe objects studied in the present paper (Centaurs,KBOs) move in resonant orbits and were predictedearlier.

ACKNOWLEDGEMENTS

This work was supported by the Russian Founda�tion for Basic Research, project no. 06�02�16795a.

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