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<ul><li><p>ISSN 00271349, Moscow University Physics Bulletin, 2010, Vol. 65, No. 1, pp. 5964. Allerton Press, Inc., 2010.Original Russian Text B.R. Mushailov, V.S. Teplitskaya, 2010, published in Vestnik Moskovskogo Universiteta. Fizika, 2010, No. 1, pp. 5964.</p><p>59</p><p>INTRODUCTION</p><p>The existence of a transNeptunian belt was predicted by K. Edgeworth (1949) and G. Kuiper (1951).The Kuiper belt is located 3050 AU from the Sun.</p><p>The possibility of the existence of several Kuiperbelts was predicted in [1, 2]. These transNeptunianbelts may exist in the region 100 a < 1000 AU. Thecurrent discoveries of large objects (with diameters ofabout 1000 km) beyond Neptunes orbit show thatlarge planets might also be found there. In particular,[3] predicted the presence of transNeptunian regionswith hypothetical large planets and calculated themean motion resonances with these hypotheticalplanets.</p><p>There are a great number of studies devoted toCentaurs and transNeptunian 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 conventionally divided into classical KBOs and several subclasses of resonant objects [4]. The resonant objectsare understood here as objects with librational motion[5, 6]. The evolution of their orbits and modern structure of the transNeptunian belt are considered to beshaped by Neptunes dynamic evolution [79].</p><p>However, the majority of the studies on TNOs evolution use integration over time periods (millions ofyears) for which the dynamic instability effects are significant. 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 threebody problem, including gravitational perturbations from Ura</p><p>nus, Saturn and Jupiter [1012] enables one to build areliable analytical model, predict the evolution of theorbital parameters of TNOs, and find the most probable regions in which to search for new objects. Basedon the results in [10], the authors of [13] discovered anumber of objects in these regions.</p><p>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 circulation), the resonance effects are determined by thecommensurability of the motions of gravitatingobjects. Hence, contrary to some works, some of thesocalled 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 distributions of KBOs [49], 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 purpose of our paper.</p><p>RESONANCES</p><p>The condition for twofrequency resonance is therational quasicommensurability of frequencies of the</p><p>type [(k + l)n kn'] O( ), where l is the order andk is the multiplicity of the resonance; l and k are natural numbers; n < n' are frequencies or mean dailymotions of gravitating bodies, and is the reduced</p><p>mass. The resonance band with the order 2n' /(k + l)</p><p>On the Role of Resonance Effects in Statistical Distributions of Centaurs and TransNeptunian Objects</p><p>B. R. Mushailov and V. S. TeplitskayaAstrometry Department, Sternberg Astronomical Institute, Moscow State University, </p><p>Universitetskii pr., Moscow, 119992 Russiaemail: verateplic@yandex.ru</p><p>Received July 8, 2009; in final form, October 13, 2009</p><p>AbstractThis paper discusses patterns in the distribution of Centaurs, scattered disks, and Kuiper beltobjects according to the semimajor 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 semimajor axis distribution with an exact orbital resonance determinedwithin the classical threebody problem.</p><p>Key words: Kuiper belt, scattered disk objects, Centaurs, orbital resonances, transNeptunian objects, statistical distributions.</p><p>DOI: 10.3103/S0027134910010145</p></li><li><p>60</p><p>MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010</p><p>MUSHAILOV, TEPLITSKAYA</p><p>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 resonance effect. Given a fixed l, the maximum amplitudeof the resonance effect is achieved at k = 1. Orbital resonances affect the motions of some large planets, satellites, asteroids, comets, and meteor flows in theSolar System and exoplanetary systems.</p><p>The dynamic evolution of the orbits in which manyof the aforementioned bodies move can be correctlyinterpreted using the resonance version of the classicalthreebody problem with regard to secular perturbations from exterior bodies.</p><p>It follows from [11] that if the analytical expressionfor the semimajor 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:</p><p>(1)</p><p>where a0 is the semimajor axis in the absence of perturbation from the giant planets;</p><p>(2)</p><p> is an integral constant; is Neptunes mass; </p><p>and are the masses and semimajor axes of the</p><p>orbits of Jupiter, Saturn, and Uranus respectively. TheSun is taken as the unit mass; the semimajor axis of</p><p>Neptunes orbit, as the unit distance; is therespective Laplace coefficient.</p><p>The semimajor axis of a TNOs orbit corresponding to an exact orbital resonance with regard to perturbations from giant planets has the form:</p><p>(3)</p><p>a a0 ,+=</p><p> 0miaii L1/2</p><p>0( ) i( ) iDL1/20( ) i( )+[ ],</p><p>i 1=</p><p>3</p><p>=</p><p>013 k</p><p>k l+( ) 1 +( ) </p><p> 2/3, i ai/,= =</p><p>m1 3,</p><p>a1 3,</p><p>L 1/2( )0</p><p>a k l+</p><p>k </p><p> 1/3</p><p>2</p><p>,=</p><p>where a is Neptunes orbit semimajor axis.For the first order resonance (l = 1), let us intro</p><p>duce the variables: x = cos(), y = sin();the values and were determined in [11]; the solution of the limited elliptic threebody problem withregard to secular perturbations from giant planets isreduced to integration of an autonomous canonicalequation system with one degree of freedom of the</p><p>form = , = , with the Hamiltonian:</p><p>(4)where </p><p>(5)</p><p>I and are integrals of the problem; a, e, i are the semimajor axis, eccentricity and inclination of the orbit ofthe TNO. The expression for the coefficient B is alsogiven in [11].</p><p>Stationary solutions for the variables x, y are determined by a system of algebraic equations</p><p>(6)</p><p>The phaseplane configuration is shown in Fig. 1.Based on the integrals of problem (5), we can</p><p>present instability areas in a diagram (e, a) and compare them to the observed data (Fig. 2).</p><p>TNOs may live for a long time if their orbital elements are located outside the shaded areas and concentrate near the boundaries of these areas. Considering 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).</p><p>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 susceptible to migration.</p><p>For objects in orbital commensurabilities (4 : 3, 3 : 2,and 2 : 1) with Neptune, we used the orbital parameters 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</p><p>2 2</p><p>dxd F</p><p>y dy</p><p>d F</p><p>x</p><p>F x2 y2+( )2</p><p>A x2 y2+( ) Bx,+ +=</p><p>A 4/ k 1+( )[ ] E 2 +{ },=</p><p>E kk 1+ 1 +</p><p>1/6</p><p>,=</p><p> a 1 k 1 k+( ) 1 e2 icos+[ ]2,=</p><p>I2 4 a 1 e2 i/2( ),sin2=</p><p>4x x2 y2+( ) 2Ax B+ + 0,=</p><p>4y x2 y2+( ) 2Ay+ 0.=</p><p>x3 x2 x1 x</p><p>y</p><p>1</p><p>2</p><p>3</p><p>Fig. 1. Phaseplane configuration. The trajectories crossing the stationary point (x1, 0) are separatrices delimitingthree characteristic areas. Because of symmetry, only theupper half plane is given.</p></li><li><p>MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010</p><p>ON THE ROLE OF RESONANCE EFFECTS 61</p><p>(Area 3); for the 3 : 2 commensurability, 83% ofTNOs; for the 2 : 1 resonance with regard to secularperturbations, 65% of TNOs.</p><p>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 perturbing factors characterized by the independentparameters = (1, 2, , n). The probability of a perturbationaffected trajectory transition from Area iinto Area j on the phase plane is determined by the following expression [11]:</p><p>where ' = arcsin1 + 12,</p><p>Wij = 1( )j 2</p><p>4 j ' j 3( )</p><p>2+</p><p>1( )i2 '</p><p>, i = 1 2, j, = 2 3,,</p><p>112</p><p>21 1 2( )3sgn+ </p><p> 1/4,=</p><p>21</p><p>3 1 43</p><p>2( )</p><p>1/2, 3</p><p> 1 2arccos+3</p><p> .cos= =</p><p>Here, 1 and 2 are independent parameters whichare in the following relationship with the coefficients</p><p>of Hamiltonian (4): 1 = , 2 = .</p><p>The coefficient B (given a fixed ) depends on theresonance multiplicity and order; A is the system statefunction characterized by integral constants.</p><p>The probabilities of TNO transition from Area i</p><p>into Area j depending on () are given in Fig. 3.For the objects under study, the probabilities of transition from one area on the phase plane into another are</p><p>ABB/A/ 27</p><p>4B</p><p>2</p><p>A3</p><p> 'cot</p><p>Fig. 2. Instability areas in the ea diagram in case of firstorder resonance for various multiplicities k.</p><p>0/4/2 /4 /2 </p><p>Wij</p><p>W32 W33 W13 W31</p><p>W12 W32W21W31</p><p>1</p><p>Fig. 3. Probabilities Wij of TNOs transitions from Area iinto Area j depending on cot'() = .</p><p>0.40</p><p>0.35</p><p>0.30</p><p>0.25</p><p>0.20</p><p>0.10</p><p>0.05</p><p>0</p><p>e</p><p>30 31 32 33 34 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50a, a.e.</p><p>k = 1</p><p>k = 2k = 3k = 4</p><p>0.15</p><p>36</p></li><li><p>62</p><p>MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010</p><p>MUSHAILOV, TEPLITSKAYA</p><p>significantly dependent on the parameter =(B/)/(A/). For example, the probability oftransition into Area 3 for the 4 : 3 and 3 : 2 resonances:W13 > W12 at from 0.05 to 0.03, and W23 > W21 at from 0.03 to 0.05. For the 2 : 1 resonance: W23 > W21at from 0.018 to 0.011 and W13 > W12 at from 0.01to 0.02.</p><p>Using different variants of the threebody problemand given some special conditions, the orbital parameters of the hypothetical large planets and the resonance regions related to these planets where hypothetical minor bodies may be located at cosmogonic timeintervals have been predicted [3]. The presence of resonant objects between the orbits of giant planets hasalso been predicted [15]; in [11], the authors calculated the regions of resonant TNOs and studied theevolution of their orbital parameters. A summary tableof the resonance regions from [2, 11, 15] has the following form:</p><p>Here, Planet 1 and Planet 2 are hypothetical largeplanets for which the table also cites the related commensurability areas (the area width was on the order</p><p>of ) and the quantity of objects in these areas. Themarked line stands for the case that includes the secu</p><p>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 5125 AU for Centaurs and SDOs and 32.7% ofthe area 3050 AU for KBOs.</p><p>PATTERNS IN THE OBSERVED DISTRIBUTIONS</p><p>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 semimajor axes (Fig. 4) as well as the eccentricities and inclinations of their orbits and comparedthese data with the theoretical results.</p><p>The semimajor axis distribution of KBOs isobserved to have two pronounced maximums, with thesemimajor axes corresponding to the observed distribution maximums of 3940 and 4344 AU beingconsistent within computational accuracy with thevalues 38.5040.37 and 42.6044.48 AU predicted in[11, 15]. The eccentricities and inclinations of the</p><p>The distribution of the objects by resonance areas</p><p>Planet Commensurability Interval a, AUQuantity</p><p>KBOs Centaurs and SDOs</p><p>Saturn 2 : 1 14.826 15.681 2</p><p>Saturn 5 : 2 17.181 17.858 4</p><p>Saturn 3 : 1 19.845 21.207 9</p><p>Saturn 4 : 1 23.662 24.518 7</p><p>Uranus 5 : 3 27.265 27.412 1</p><p>Neptune 5 : 4 34.624 35.165 6 0</p><p>Neptune 4 : 3 36.062 36.800 10 0</p><p>Neptune 3 : 2 38.499 40.371 225 1</p><p>Uranus 3 : 1 39.280 40.686 191 1</p><p>Neptune 5 : 3 41.901 42.577 49 2</p><p>Neptune 7 : 4 43.374 43.894 140 0</p><p>Neptune 2 : 1 46.482 48.836 76 0</p><p>Neptune 2 : 1 42.599 44.48 433 0</p><p>Planet 1 1 : 2 47.185 47.416 5 0</p><p>Uranus 4 : 1 47.761 49.432 20 1</p><p>Neptune 5 : 2 54.669 55.430 8</p><p>Uranus 5 : 1 56.263 58.123 8</p><p>Planet 1 2 : 3 57.207 57.394 3</p><p>Neptune 3 : 1 61.004 64.023 14</p><p>Neptune 4 : 1 73.726 77.459 4</p><p>Planet 1 4 : 3 90.852 91.149 1</p><p>Planet 2 3 : 4 123.740 124.222 1</p></li><li><p>MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 65 No. 1 2010</p><p>ON THE ROLE OF RESONANCE EFFECTS 63</p><p>TNO orbits localized near the maximum corresponding to the semimajor axis 43.5 AU are small: e ~ 0.00.17, i ~ 010, which is consistent with the results in[3, 15]. For the maximum corresponding to the exactcommensurability of 3 : 2 with Neptune, the eccentricities are mostly located in the area 0.150.30, 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)</p><p>with the 2 : 1 resonance with Neptune. The semimajor axis distributions of Centaurs and SDOs are alsoobserved to have pronounced maximums corresponding 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.</p><p>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 correlation coefficients between the similar orbital parameters 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 semimajor 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.</p><p>Other conditions being equal, the statistical distributions under study are influenced by selective effects(e.g.,...</p></li></ul>