The Disturbing Function for Polar Centaurs and Transneptunian Objects

MNRAS 471, 2097–2110 (2017) doi:10.1093/mnras/stx1714 Advance Access publication 2017 July 8

The disturbing function for polar Centaurs and transneptunian objects Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019

F. Namouni1‹ andM.H.M.Morais2‹ 1UniversiteC´ oteˆ d’Azur, CNRS, Observatoire de la Coteˆ d’Azur, CS 34229, F-06304 Nice, France 2Universidade Estadual Paulista (UNESP), Instituto de Geocienciasˆ e Cienciasˆ Exatas, Av. 24-A, 1515 13506-900 Rio Claro, SP, Brazil

Accepted 2017 July 6. Received 2017 July 6; in original form 2017 April 12

ABSTRACT The classical disturbing function of the three-body problem is based on an expansion of the gravitational interaction in the vicinity of nearly coplanar orbits. Consequently, it is not suitable for the identification and study of resonances of the Centaurs and transneptunian objects on nearly polar orbits with the Solar system planets. Here, we provide a series expansion algorithm of the gravitational interaction in the vicinity of polar orbits and produce explicitly the disturbing function to fourth order in eccentricity and inclination cosine. The properties of the polar series differ significantly from those of the classical disturbing function: the polar series can model any resonance, as the expansion order is not related to the resonance order. The powers of eccentricity and inclination of the force amplitude of a p:q resonance do not depend on the value of the resonance order |p − q| but only on its parity. Thus, all even resonance order eccentricity amplitudes are ∝e2 and odd ones ∝e to lowest order in eccentricity e. With the new findings on the structure of the polar disturbing function and the possible resonant critical arguments, we illustrate the dynamics of the polar resonances 1:3, 3:1, 2:9 and 7:9 where transneptunian object 471325 could currently be locked. Key words: celestial mechanics – comets: general – Kuiper belt: general – minor planets, asteroids: general – Oort Cloud.

the planet’s perihelion and node because the solar system planets’ 1 INTRODUCTION eccentricities and inclinations with respect to the invariable plane The increasing detections of Centaurs and transneptunian ob- are small. As the number of integer combinations is infinite, one jects (TNOs) on nearly polar orbits (Gladman et al. 2009;Chen usually seeks and checks only the strongest resonances: those with et al. 2016) raises the question of their origin and relationship to a force amplitude that implies a sizable resonance width within the Solar system planets. Amongst the dynamical processes that which to capture the Centaur or TNO. This choice is as reasonable govern the evolution of such objects are mean motion resonances. as it is rewarding provided that one remembers that the force am- In this context, it was shown recently, through intensive numerical plitudes associated with a candidate resonance φ are obtained from simulations, that mean motion resonances are efficient at polar orbit the classical disturbing function that is an expansion in powers of capture (Namouni & Morais 2015, 2017). It is, therefore, important eccentricity and inclination of the planet–object gravitational inter- to have a thorough understanding of the processes of resonance action for nearly circular and coplanar orbits. Thus, our intuition crossing and capture for nearly polar Centaurs and TNOs so that regarding the angle combination φ and its dynamical suitability for we can identify the pathways that led to such orbits and ultimately resonance is based on the assumption of near-coplanarity. It is the uncover their origin. object of this work to remedy this shortcoming in the dynamical Identifying a mean motion resonance for a Centaur or a analysis of polar Centaurs and TNOs by deriving a disturbing func- TNO with a Solar system planet is a fundamentally simple tion for nearly polar orbits and studying the properties of its force task. One has to search for angle combinations of the form amplitudes. φ = qλ − pλ − k + (p − q + k) that can be stationary or The history of the classical disturbing function is intertwined oscillating around the equilibrium defined by the condition φ˙ = 0. with that of celestial mechanics. For a historical perspective, we In the previous expression, λ and λ are, respectively, the mean lon- refer the reader to Brouwer & Clemence (1961). For the purposes gitudes of the object and the planet, and are, respectively, the of this work, we note that the disturbing function of the three- object’s longitudes of perihelion and ascending node, and p, q and body problem takes two different forms. The first form is a power k are integer coefficients. In the angle combination φ, we ignored series in terms of eccentricity e and sin2(I/2), where I is the inclination. This form therefore assumes that the object’s orbit does not depart significantly from prograde coplanar motion. It is used E-mail: [email protected] (FN); [email protected] (MHMM) widely to study the formation and dynamics of planetary systems,

C 2017 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society 2098 F. Namouni and M. H. M. Morais the formation and evolution of planetary rings and the formation shall denote R¯ d . The second term, that we denote R¯ i , is the indirect and resonance capture of planetary satellite systems (Murray & perturbation that comes from the reflex motion of the star under the Dermott 1999; Ellis & Murray 2000). The second form is a power influence of the mass m as the standard coordinate system is chosen Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019 expansion in terms of the ratio α = a/a where a and a are the to be centred on the star. In the following, we use the notations and semimajor axes of the object and planet, respectively. This form is steps of the literal expansions for nearly coplanar prograde orbits used mainly to study the dynamics of artificial and natural planetary (Murray & Dermott 1999) and nearly coplanar retrograde orbits by satellites that have large inclinations as they are influenced by the Morais & Namouni (2013a) so that the reader is able to see the distant Sun or the Moon. It was recently revisited for the study of similarities and differences of the three expansions. the secular evolution of hierarchical planetary systems (Laskar & The classical series of the disturbing function is expanded in Boue´ 2010). In the case of Centaurs and TNOs, the second form powers of e and sin2(I/2), which is adequate for nearly coplanar is not particularly useful as such objects can be quite close to the prograde motion since sin2(I/2) vanishes for I = 0. The classical planets’ orbits but unlike satellites they revolve around the Sun not series can also be used for nearly coplanar retrograde orbits af- the planet. Reasonable order expansions with respect to α cannot ter having applied the procedure devised by Morais & Namouni model the dynamics when the semimajor axes’ ratio does not sat- (2013a) that allows one to get retrograde resonant terms from pro- isfy α 1. We will therefore seek a disturbing function that is grade ones. In essence, the retrograde series is an expansion in terms not expanded with respect to α but is written as a power series of of e and cos2(I/2) where the latter vanishes for I = 180◦.However, eccentricity and some function of the inclination that vanishes if the neither the prograde series nor its retrograde counterpart can be object’s orbit is exactly polar. We find in Section 2 that the natural used for polar orbits. Instead, inspection of the expressions of cos ψ function is simply cos I. The classical disturbing function and its (2) and reveals that a polar expansion has to be done with respect zero reference inclination can also be transformed into a disturbing to e and cos I that vanishes for I = 90◦. We therefore write function for nearly coplanar retrograde orbits, that is with 180◦ ref- 2 = 1 + r2 − 2r cos( − λ)cos(f + ω) − 2r , (3) erence inclination, to study the dynamics of retrograde resonances (Morais & Namouni 2013b). The retrograde disturbing function where is helped to identify the first Centaurs and Damocloids in retrograde =−sin( − λ) sin(f + ω)cosI. (4) resonance with Jupiter and Saturn (Morais & Namouni 2013a). The plan of the paper is as follows. In Section 2, we write down the Expanding the direct perturbation term −1 in the vicinity of = 0, explicit steps of the literal expansion of the gravitational interaction we have of a planet with a particle on a nearly polar orbit in powers of ∞ − (2i)! i − i+ eccentricity and inclination cosine. The reader who is not interested 1 = (r ) (2 1), (5) 2i (i!)2 0 in the details of the expansion algorithm can skip this part and find i=0 the resulting disturbing function in Section 3. The properties of 2 = + r2 − r − λ f + ω = where 0 1 2 cos( )cos( ). Defining − i+ the polar disturbing function are compared to those of the classical r/α − 1 = O(e) and expanding (2 1) around = 0, we get disturbing function of nearly coplanar prograde orbits as well as 0 ∞ k k k that of nearly coplanar retrograde orbits in Section 4. The validity − i+ α d (2 1) = 1 + ρ−(2i+1), (6) domain of the polar disturbing function is linked to secular evolution 0 k dαk k= ! and discussed in Section 5. Examples of polar resonances are found 1 −(2i+1) 2 −(i+1/2) in Section 6. Section 7 contains concluding remarks. ρ = [1 + α − 2 α cos( − λ )cos(f + ω)] . (7) The validity of the expansion with respect to zero eccentricity will 2 LITERAL EXPANSION FOR NEARLY POLAR be discussed in Section 5, where we examine the secular coupling ORBITS of eccentricity and inclination. In particular, a maximum value of eccentricity will be determined for the polar disturbing function of Consider a test particle of negligible mass that moves under the fourth order. gravitational influence of the sun of mass M and a planet of mass The next step in the literal expansion is to develop the function m M. The motion of m with respect to M is a circular orbit of ρ−(2i + 1) into a two-dimensional Fourier series with respect to the λ radius a and longitude angle . The reference plane is defined by angles f + ω and − λ as follows: the sun–planet orbit. The test particle’s osculating Keplerian orbit 1 jk with respect to M has semimajor axis a, eccentricity e, inclination I, ρ−(2i+1) = b α k f + ω + j − λ , i+1/2( )cos[ ( ) ( )] (8) true anomaly f, argument of pericentre ω, and longitude of ascending 4 −∞

MNRAS 471, 2097–2110 (2017) Disturbing function for nearly polar orbits 2099 the ± signs are for prograde and retrograde orbits, respectively Furthermore for p = 0andq = 0, one needs to account for the (Morais & Namouni 2013a). The two-dimensional Laplace coeffi- resonant terms that are generated by T1 and T2 under the change cients also satisfy the following relations: p →−p and q →−q as the series (13) is summed over positive Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019 jk kj (−j)k j(−k) (−j)(−k) as well as negative k and j. This transformation produces two new bs = bs = bs = bs = bs , (10) terms, T3 and T4, that correspond to the same resonance: s jk (j+1)(k+1) (j+1)(k−1) (j−1)(k+1) e2 Db = b + b + b 2 s s+1 s+1 s+1 T =− q A ,p,q, cos[qλ − pλ + (p − q)] = T , (20) 2 3 4 0 0 1 j− k− jk + b( 1)( 1) − 2αsb , (11) s+1 s+1 e2 T = A qλ − pλ + p − q + − . s 4 0,p,q−2,2 cos[ ( 2) 2 ] (21) Dn bjk = Dn−1 b(j+1)(k+1) + Dn−1 b(j+1)(k−1) 32 s s+1 s+1 2 In the indices of A0, p, q,0 and A0, p, q − 2, 2, we use the properties (10) + Dn−1 b(j−1)(k+1) + Dn−1 b(j−1)(k−1) s+1 s+1 of the two-dimensional Laplace coefficients. The secular terms can = = jk jk be obtained by setting p 0andq 0inT1 and T2 but not in T3 and − 2αsDn−1 b − 2(n − 1)sDn−2 b , (12) s+1 s+1 T4 because the same term would be counted twice. Another way of → where the operator D = d/dα. Substituting the series (8) into the seeing this is that (0, 0) is a fixed point of the transformation (p − →− expression (6) and the latter into the expansion (5), the direct part p, q q). R¯ of the perturbation is written as the following series: For the indirect part of the disturbing function, i , one requires only the use of the elliptic expansions (15) to transform true anoma- i 1 (2 )! i l lies into mean anomalies and perform the eccentricity expansion. R¯ d = (r ) Ai,j,k,l cos[k(f + ω) + j( − λ )], l! 2i+2(i!)2 The resulting expressions of the direct and indirect parts of the polar 0≤i,l<∞ −∞

MNRAS 471, 2097–2110 (2017) 2100 F. Namouni and M. H. M. Morais

0 m n 1 m n Table 1. Force coefﬁcients cmn(p, q, α) of the term e cos I cos φ. Table 2. Force coefﬁcients cmn(p, q, α)oftheterme cos I cos (φ − ω).

0 1 1 1 c A0,p,q,0, c − [2(1 − q)A0,p,q−1,0 + A0,p,q−1,1], 00 2 10 4 Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019 α α c0 A − A − A c1 q − A ,p− ,q− , − A ,p− ,q, + A ,p+ ,q, 01 8 ( 1,p−1,q−1,0 1,p−1,q+1,0 1,p+1,q−1,0 11 16 [(2 3)( 1 1 2 0 1 1 0 1 1 0 − + − − − − + − + A1,p+1,q+1,0), A1, p 1, q 2, 0) A1, p 1, q 2, 1 A1, p 1, q,1 c0 1 − q2A + A + A , + A1, p + 1, q − 2, 1 − A1, p + 1, q,1], 20 8 ( 4 0,p,q,0 2 0,p,q,1 0,p,q,2) α2 1 3α2 c0 3 A − A + A c − [2(2 − q)(A2,p−2,q−3,0 − 2A2,p−2,q−1,0 02 64 ( 2,p−2,q−2,0 2 2,p−2,q,0 2,p−2,q+2,0 12 128 + A − + − 2A − + 4A − − 2A2, p, q − 2, 0 + 4A2, p, q,0 − 2A2, p, q + 2, 0 2, p 2, q 1, 0 2, p, q 3, 0 2, p, q 1, 0 + A + − − 2A + − 2A + − + A2, p + 2, q − 2, 0 − 2A2, p + 2, q,0 + A2, p + 2, q + 2, 0) 2, p 2, q 3, 0 2, p, q 1, 0 2, p 2, q 1, 0 + + − c0 − α q2 − A − A A2, p + 2, q + 1, 0) A2, p − 2, q − 3, 1 2A2, p − 2, q − 1, 1 21 32 [(4 2)( 1,p−1,q−1,0 1,p−1,q+1,0 + A2, p − 2, q + 1, 1 − 2A2, p, q − 3, 1 + 4A2, p, q − 1, 1 − A1, p + 1, q − 1, 0 + A1, p + 1, q + 1, 0) − 4(A1, p − 1, q − 1, 1 − 2A2, p, q + 1, 1 + A2, p + 2, q − 3, 1 − 2A2, p + 2, q − 1, 1 + A1, p − 1, q + 1, 1 + A1, p + 1, q − 1, 1 − A1, p + 1, q + 1, 1) + A2, p + 2, q + 1, 1] − A1, p − 1, q − 1, 2 + A1, p − 1, q + 1, 2 + A1, p + 1, q − 1, 2 1 c1 q q − q2 − A ,p,q− , + q q − A ,p,q− , − A1, p + 1, q + 1, 2] 30 32 [2 (7 4 3) 0 1 0 (4 1) 0 1 1 − + − 0 5α3 4A0, p, q − 1, 2 2qA0, p, q − 1, 2 A0, p, q − 1, 3] c [3(A3,p−3,q+1,0 − A3,p−3,q−1,0 − A3,p−1,q−3,0 03 256 5α3 c1 q − A ,p− ,q− , − A ,p− ,q− , + A3, p − 1, q + 3, 0 + A3, p + 1, q − 3, 0 − A3, p + 1, q + 3, 0 13 512 [(2 5)( 3 3 4 0 3 3 3 2 0 + − − − + − − − + A3, p + 3, q − 1, 0 − A3, p + 3, q + 1, 0) + 9(A3, p − 1, q − 1, 0 3A3, p 3, q,0 A3, p 3, q 2, 0 3A3, p 1, q 4, 0 + − − − − + − + − A3, p − 1, q + 1, 0 − A3, p + 1, q − 1, 0 + A3, p + 1, q + 1, 0) 9A3, p 1, q 2, 0 9A3, p 1, q,0 3A3, p 1, q 2, 0 + + − − + − + + − A3, p + 3, q − 3, 0 − A3, p − 3, q + 3, 0 + A3, p − 3, q − 3, 0 3A3, p 1, q 4, 0 9A3, p 1, q 2, 0 9A3, p 1, q,0 − + + − + − + + − + A3, p + 3, q + 3, 0] 3A3, p 1, q 2, 0 A3, p 3, q 4, 0 3A3, p 3, q 2, 0 − + − c0 1 q2 q2 − A − q2 A + A 3A3, p + 3, q,0 A3, p + 3, q + 2, 0) A3, p − 3, q − 4, 1 40 128 [ (16 9) 0,p,q,0 8 ( 0,p,q,1 0,p,q,2) + A3, p + 3, q − 4, 1 − A3, p + 3, q + 2, 1 + A3, p − 3, q + 2, 1 + 4A0, p, q,3 + A0, p, q,4] + 3(A3, p − 3, q − 2, 1 − A3, p − 3, q,1 + A3, p − 1, q − 4, 1 35α4 c0 A ,p− ,q+ , + A ,p− ,q− , + A ,p+ ,q− , 04 4096 [ 4 4 4 0 4 4 4 0 4 4 4 0 − A3, p − 1, q + 2, 1 − A3, p + 1, q − 4, 1 + A3, p + 1, q + 2, 1 + + + − − − + − + A4, p 4, q 4, 0 4(A4, p 4, q 2, 0 A4, p 4, q 2, 0 − A3, p + 3, q − 2, 1 + A3, p + 3, q,1) + 9(A3, p − 1, q,1 + − − + − + + + − + A4, p 2, q 4, 0 A4, p 2, q 4, 0 A4, p 2, q 4, 0 − A3, p − 1, q − 2, 1 + A3, p + 1, q − 2, 1 − A3, p + 1, q,1)] + + + A4, p + 2, q + 4, 0 A4, p + 4, q − 2, 0 A4, p + 4, q + 2, 0) c1 − α q − q + q2 A − A 31 128 [ (7 18 8 )( 1,p−1,q−2,0 1,p−1,q,0 + 6(A4, p − 4, q,0 + A4, p, q − 4, 0 + A4, p, q + 4, 0 − A1, p + 1, q − 2, 0 + A1, p + 1, q,0) + A4, p + 4, q,0) + 16(A4, p − 2, q − 2, 0 + A4, p − 2, q + 2, 0 2 + (8 − 3q − 4q )(A1, p − 1, q − 2, 1 − A1, p − 1, q,1 + A4, p + 2, q + 2, 0 + A4, p + 2, q − 2, 0) − 24(A4, p − 2, q,0 − A1, p + 1, q − 2, 1 + A1, p + 1, q,1) + (7 − 2q)(A1, p − 1, q − 2, 2 − A4, p, q − 2, 0 − A4, p, q + 2, 0 − A4, p + 2, q,0) − A1, p − 1, q,2 − A1, p + 1, q − 2, 2 + A1, p + 1, q,2) + 36A4, p, q,0] + A1, p − 1, q − 2, 3 − A1, p − 1, q,3 − A1, p + 1, q − 2, 3 c0 − 3α2 q2 − A − A + 22 256 [2(2 3)( 2,p−2,q−2,0 2 2,p−2,q,0 A1, p + 1, q,3] + A2, p − 2, q + 2, 0 − 2A2, p, q − 2, 0 − 2A2, p, q + 2, 0 + 4A2, p, q,0 − 2A2, p + 2, q,0 + A2, p + 2, q − 2, 0 c2 p, q, α m n φ − ω + A2, p + 2, q + 2, 0) − 6A2, p − 2, q − 2, 1 − A2, p − 2, q − 2, 2 Table 3. Force coefﬁcients mn( )oftheterme cos I cos ( 2 ). + 12A2, p − 2, q,1 + 2A2, p − 2, q,2 − 6A2, p − 2, q + 2, 1 1 − − + + − + − c2 − q + q2 A + − q A A2, p 2, q 2, 2 12A2, p, q 2, 1 2A2, p, q 2, 2 20 16 [(6 11 4 ) 0,p,q−2,0 (6 4 ) 0,p,q−2,1 − 24A2, p, q,1 − 4A2, p, q,2 + 12A2, p, q + 2, 1 + A0, p, q − 2, 2], α + 2A2, p, q + 2, 2 − 6A2, p + 2, q − 2, 1 − A2, p + 2, q − 2, 2 c2 q2 − q + A − A 21 64 [(4 15 12)( 1,p−1,q−3,0 1,p−1,q−1,0 + + + + − + + 12A2, p 2, q,1 2A2, p 2, q,2 6A2, p 2, q 2, 1 − A1, p + 1, q − 3, 0 + A1, p + 1, q − 1, 0) − + + A2, p 2, q 2, 2] − 4(q − 2)(A1, p − 1, q − 3, 1 + A1, p − 1, q − 1, 1 + A1, p + 1, q − 3, 1 − A1, p + 1, q − 1, 1) + A1, p − 1, q − 3, 2 − A1, p − 1, q − 1, 2 − A1, p + 1, q − 3, 2 + A1, p + 1, q − 1, 2] c2 1 + q − q2 + q3 − q4 A To illustrate this property, we shall consider the even order inner 40 192 [(12 26 88 68 16 ) 0,p,q−2,0 2 3 5:1 resonance and the odd order outer 2:9 resonance and write − 2(6 − 23q + 24q − 8q )A0, p, q − 2, 1 + − + − down the corresponding series to second order in eccentricity and (6 9q)A0, p, q − 2, 2 4(2 q)A0, p, q − 2, 3 + inclination cosine. Using Tables 1 and 3 for the 5:1 resonance, we A0, p, q − 2, 4] c2 3α2 − q + q2 A get 22 512 [(20 19 4 )( 2,p−2,q−4,0 − 2A2, p − 2, q − 2, 0 + A2, p − 2, q,0 − 2A2, p, q − 4, 0 R¯ 5:1 = c0 , ,α + c0 , ,α I + c0 , ,α e2 + d [ 00(5 1 ) 01(5 1 )cos 20(5 1 ) + 4A2, p, q − 2, 0 − 2A2, p, q,0 + A2, p + 2, q − 4, 0 + c0 , ,α 2 I λ − λ + − 2A2, p + 2, q − 2, 0 + A2, p + 2, q,0) 02(5 1 )cos ]cos( 5 4 ) + (10 − 4q)(A2, p − 2, q − 4, 1 − 2A2, p − 2, q − 2, 1 + A2, p − 2, q,1 + c2 (5, 1,α)e2 cos(λ − 5λ + 6 − 2 ) 20 − 2A2, p, q − 4, 1 + 4A2, p, q − 2, 1 − 2A2, p, q,1 + c2 , − ,α e2 λ − λ + + . + + − − + − + + 20(5 1 ) cos( 5 2 2 ) (23) A2, p 2, q 4, 1 2A2, p 2, q 2, 1 A2, p 2, q,1) + A2, p − 2, q − 4, 2 − 2A2, p − 2, q − 2, 2 + A2, p − 2, q,2 λ − λ + − The resonant terms cos ( 5 5 )and − 2A2, p, q − 4, 2 + 4A2, p, q − 2, 2 − 2A2, p, q,2 cos (5λ − λ + 3 + ) whose force amplitudes are proportional + A2, p + 2, q − 4, 2 − 2A2, p + 2, q − 2, 2 + A2, p + 2, q,2] to e and to cannot appear as the corresponding force coefﬁcients all vanish because or = 4 is even. They are written as α 1 1 1 c (5, 1,α) =− A0,5,0,0 = 0, (24) c (5, 1,α) = [−(A , , , − A , , , + A , , , 10 4 11 16 1 4 1 0 1 4 1 0 1 6 1 0 − A1,6,1,0) − A1,4,1,1 + A1,4,1,1 1 1 c (5, −1,α) =− (4A , , , + A , , , ) = 0, (25) + A − A = , 10 4 0 5 2 0 0 5 2 1 1,6,1,1 1,6,1,1] 0 (26)

MNRAS 471, 2097–2110 (2017) Disturbing function for nearly polar orbits 2101

3 m n Table 4. Force coefﬁcients cmn(p, q, α)oftheterme cos I cos (φ − 3ω). Table 6. Force amplitudes and cosine arguments of the indirect part.

3 1 3 2 Cosine argument Force amplitude c [(8q − 42q + 62q − 24)A0,p,q−3,0 30 96 Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019 2 − 3(12 − 15q + 4q )A − + (6q − 12)A − α 0, p, q 3, 1 0, p, q 3, 2 λ − 3 (1 + cos I)e − A − ] 4 0, p, q 3, 3 λ − λ − α + I − 1 e2 − 1 e2 I − 1 e4 3 α 2 3 2 (1 cos 2 2 cos 64 ) c [(60 − 107q + 54q − 8q )(A1,p−1,q−2,0 − A1,p−1,q−4,0 31 384 λ − λ − 2 + 2 − α e2(3 + e2 − 3cos I) + A1, p + 1, q − 4, 0 − A1, p + 1, q − 2, 0) 48 − − + 2 − λ − λ − α + I e3 − e 3(20 19q 4q )(A1, p − 1, q − 4, 1 A1, p − 1, q − 2, 1 2 16 (1 cos )(3 4 ) α − A + − + A + − ) + (15 − 6q)(A − − λ − λ − + I − e3 1, p 1, q 4, 1 1, p 1, q 2, 1 1, p 1, q 2, 2 2 3 2 48 (cos 1) − A1, p − 1, q − 4, 2 + A1, p + 1, q − 4, 2 − A1, p + 1, q − 2, 2) λ − λ − 3α e2 e2 − I − 3 2 16 ( cos 1) − A − − + A − − + A + − 1, p 1, q 4, 3 1, p 1, q 2, 3 1, p 1, q 4, 3 3λ − λ − 4 + 2 − 3α e4 − A1, p + 1, q − 2, 3] 256 λ − λ − − α + I e3 4 3 6 (1 cos ) λ − λ − − 125α e4 c4 p, q, α 5 4 768 Table 5. Force coefﬁcients mn( ) of the term α m n λ + − − 3 I − e e cos I cos (φ − 4ω). 2 4 (cos 1) λ + λ − − α − e2 − e4 + e2 − I 2 128 (64 32 32( 2) cos ) c4 1 − q + q2 − q3 + q4 A λ + λ − − α e2 + e2 + I 40 768 [(120 394 379 136 16 ) 0,p,q−4,0 2 48 (3 3cos − − + − 2 + 3 α 4( 60 107q 54q 8q )A0, p, q − 4, 1 2λ + λ − 3 − (1 + cos I)e3 + − + 2 48 (120 114q 24q )A0, p, q − 4, 2 λ + λ − − − α I − e e2 − 2 2 16 (cos 1) (3 4) + (20 − 8q)A − + A − ] 0, p, q 4, 3 0, p, q 4, 4 λ + λ − − 3α e4 3 4 256 λ + λ − − − 3α e2 − e2 − I 3 2 2 16 (1 cos ) λ + λ − − α I − e3 α 4 3 2 6 (cos 1) c1 , − ,α = − A − A + A α 11(5 1 ) [ 5( 1,4,3,0 1,4,1,0 1,6,1,0 λ + λ − − − 125 e4 16 5 4 2 768 − A1,6,3,0) − A1,4,3,1 + A1,4,1,1

+ A1,6,3,1 − A1,6,1,1] = 0, (27) to occur in the e-resonant term (k = 1) but also in higher order terms such as e3 (k = 3) and e5 (k = 5) that like (28) would not exist in where we have used the two-dimensional Laplace coefficient rela- the classical disturbing function of nearly coplanar orbits for the tions (10) and the property A = 0whenj + k is odd. We remark i, j, k, l seventh order 2:9 resonance. that unlike the classical disturbing function for nearly coplanar orbits, the resonance order does not appear in the powers of eccentricity and inclination cosine of the force amplitudes. Moreover, such 3.2 Indirect part terms as (23) would not exist in the classical disturbing function as the lowest order pure eccentricity term would be proportional The arguments and force amplitudes up to and including fourth order to e4. To dispel doubt on the existence of an unknown symmetry of the disturbing function’s indirect part for nearly polar orbits are that would make the force coefficients of (23) vanish, we list their given in Table 6. The terms present in the expansion concern only ≤ ≤ non-zero numerical values for nominal resonance α = 0.341 995: resonances of the type 1:q with 0 q 5. These terms therefore c0 , = . c0 , = . c0 , = concern only perturbers located inside the object’s orbit. 00(5 1) 0 000 696 76, 01(5 1) 0 000 586 162, 20(5 1) . c0 , =− . c2 , = . 0 004 322 25, 02(5 1) 0 001 117 03, 20(5 1) 0 002 422 41 c2 , − = . and 20(5 1) 0 003 970 14. Furthermore, in Section 6 we show 4 COMPARING THE DISTURBING examples of capture in high-order resonances for low values of FUNCTIONS OF NEARLY COPLANAR ORBITS the integer k that are smaller than the resonance order or (see also AND NEARLY POLAR ORBITS the next paragraph). This more general fundamental difference between the two disturbing functions of nearly coplanar and nearly The first main difference between the disturbing functions of nearly polar orbits will be discussed in Section 4. coplanar orbits and nearly polar orbits is the fact that the expan- The next example is given by the second order in eccentricity and sion order is not related to the resonance orders. To understand inclination cosine series of the 2:9 resonance that is free of second- this difference, recall that a literal expansion of the classical dis- order eccentricity terms because or = 7 is odd. The corresponding turbing function (of nearly coplanar orbits) to order N in eccentric- expressions are obtained from Table 2 and read: ity and inclination produces cosine terms that represent at most resonances of order N. For instance, the fourth order series of R¯ 2:9 = e c1 , ,α + c1 , ,α I d [ 10(2 9 ) 11(2 9 )cos ] Murray & Dermott (1999) applied to a particle perturbed by a planet × cos(9λ − 2λ − 6 − ) on a nearly coplanar prograde circular orbit produces only the co- λ − λ + λ − − λ + + e[c1 (2, −9,α) + c1 (2, −9,α)cosI] sine terms: cos [j( ) f0( , )], cos [j (j 1) f1( , 10 11 )], cos [jλ − (j − 2)λ + f2( , )], cos [jλ − (j − 3)λ + f3( , × cos(9λ − 2λ − 8 + ). (28) )] and cos [jλ − (j − 4)λ + f4( , )], where the functions fi The values of the various force coefficients evaluated at nominal res- represent the correct combinations of the longitudes of pericentre α = c1 , = . c1 , = onance, 2.725 68, are 10(2 9) 0 000 227 527, 11(2 9) and ascending node that we do not reproduce explicitly to avoid . c1 , − = . × −6 c1 , − = 0 000 149 595, 10(2 9) 8 680 79 10 and 11(2 9) cumbersome notation. This shows that the possible resonances are 15.6653 × 10−6. Similarly to the previous example, the terms in of order zero to four but no more. The literal expansion of the dis- (28) would not exist in the classical disturbing function as the low- turbing function of nearly polar orbits produces cosine terms for est order pure eccentricity term would be proportional to e7.Using any type of resonance p:q with no restriction on the resonance order the numerical integration of the full equations of motion, examples or =|p − q|. A close inspection of the expansion shows that this of the 2:9 resonance are given in Section 6 where libration is shown property is related to the presence of the two independent angles

MNRAS 471, 2097–2110 (2017) 2102 F. Namouni and M. H. M. Morais f + ω and − λ that require the use of two-dimensional Laplace coplanar and prograde. The Poincare´ canonical variables are given coefﬁcients unlike the classical disturbing function that makes use as the three pairs: of one-dimensional Laplace coefﬁcients. Therefore, whereas the 2 = mna ,λ= M + ω + , Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019 nearly polar disturbing function of order 4 can be used for the 2:9 2 2 1/2 resonance discussed in the previous section to study the e-terms = mna 1 − (1 − e ) ,γ=−ω − ,

= = 1/2 associated with k 1andk 3, a literal expansion of order 7 of Z = mna2 1 − e2 (1 − cos I),z=−, (29) the classical disturbing function is required to get the first possible resonant term. This property motivated Ellis & Murray (2000)to where m is the particle’s mass, n its mean motion and M its mean come up with an algorithm that produces the force amplitude of a longitude. The appropriate variables for polar motion must have given cosine term of any resonance order without having to expand the property that the actions related to eccentricity and inclination the classical disturbing function literally. vanish when motion is exactly polar and circular. The Poincare´ The second main difference between the two disturbing functions action satisfies this condition but not Z. The latter can be replaced / is the fact that the powers of eccentricity and inclination cosine in the by Z = − − Z = mna2(1 − e2)1 2 cos I, the normal component force amplitudes of the polar disturbing function are independent of of angular momentum. The remaining variables are obtained from the value of the resonance order or (except its parity discussed in the the following generating function: next paragraph). In the classical disturbing function, the lowest order F = (λ + z) + (γ − z) − zZ . (30) eccentricity and inclination power of the force amplitude of a given cosine term is or. Indeed for any resonance of order or, the force Using Z = ∂zF , z = ∂Z F etc., we find amplitude of the cosine term is proportional to eor −2k sin(I/2)2k to = , λ = λ − = M + ω, the lowest order in e and I where the integer k satisfies 0 ≤ 2k ≤ or. The examples of the 5:1 and 2:9 resonance in the previous section = , γ =−ω,

1/2 showed that their force amplitudes to lowest order in e and cos I Z = mna2 1 − e2 cos I, z = . (31) were affine with respect to cos I as c0 + c0 cos I and quadratic in 00 01 e through the terms, c0 e2, c2 e2 and c−2e2 for 5:1 (equation 23). It can be seen that the choice of the correct variable Z modifies 20 20 20 To lowest order in e and cos I, the 2:9 force amplitude is linear in the mean longitude λ , the longitude of pericentre γ and the an- c1 e c−1e gle z associated with the longitude of ascending node showing e through the terms 10 and 10 (equation 28). In the classical disturbing function, if we seek a linear dependence in eccentricity that the argument of pericentre is one of the natural angles that for 2:9, we must carry along the inclination to the sixth power. The should describe polar motion. For comparison, when we modified only inclination-free force amplitude is proportional to e7. the Poincare´ canonical variables in Namouni & Morais (2015)to When the algorithm of Morais & Namouni (2013a) is applied study retrograde resonances by choosing the new inclination action to the classical disturbing function (of prograde orbits) to produce as Zr = 2( − ) − Z so that Zr vanishes for exactly coplanar a series for nearly coplanar retrograde orbits, the force amplitude retrograde motion, the mean longitude and longitude of pericentre of a p:q retrograde resonance to lowest order in e and cos (I/2) were modified to λr = M + ω − and γ r = − ω, thus producing is proportional to e|p + q|−2kcos (I/2)2k,where0≤ 2k ≤|p + q|. the natural angles with which retrograde resonances can be studied. This gives even to a first-order resonance (i.e. |p − q|=1), the We note that our choice of the third canonical action is not unique. / dynamical structure of a high-order resonance. For instance, the For instance instead of mna2(1 − e2)1 2 cos I, one could employ / planar 1:2 resonance is equivalent to the third-order 1:4 resonance mna2(1 − e2)1 2 (1 − sin I) that has the added advantage of being (Morais & Giuppone 2012). Therefore, unlike the polar case, retro- positive regardless of inclination. The corresponding new angles, grade force amplitudes involve a retrograde resonance order defined however, are no longer function of the old angles; they will also as o¯r =|p + q|. depend on , and Z . The third main difference between the classical disturbing func- Using the new polar canonical angles, the argument of the cosine p:q tion and that of nearly polar orbits is the dependence on the parity terms in the disturbing function (22) is transformed as φk = φ − of the resonance order and the corresponding universal binarity of kω = pλ − q(λ − z ) + kγ implying a simple physical meaning the force amplitudes of resonant terms. To lowest order in e and that polar mean longitude need only be measured as if motion cos I, all resonances p:q with an even or =|p − q| have force am- were two-dimensional. The particle’s longitude of ascending node plitudes that are quadratic with respect to eccentricity and constant must be used as a reference line to measure the mean longitude with respect to inclination whereas all resonances p:q with an odd of the planet as the latter two angles lie in the same plane. The γ or =|p − q| have force amplitudes that are linear with respect to remaining term k gives the kth-harmonic that could be excited eccentricity. As was mentioned in the previous section, this curious by the planet. Lastly, we also mention that the new polar canonical behaviour stems from the presence of the two independent angles variables are related to the classical Delauney variables given as f + ω and − λ in the relative distance . We suspect, but can- (L = , l = λ + γ ), (G = − , g =−γ )and(H = Z , not prove, that the property is related to the fact that for a given h = z ). These variables were used by Kozai (1962) to study the resonance, one must have prograde as well as retrograde arguments secular evolution at large eccentricity and inclination in the three- in the same polar series unlike the classical disturbing functions of body problem that will be discussed in the next section. nearly coplanar prograde or retrograde orbits. We gain further insight into the structure of the disturbing func- 5 SECULAR POTENTIAL AND VALIDITY tion for nearly polar orbits by seeking the natural variables with DOMAIN OF THE DISTURBING FUNCTION which polar motion can be studied. To do this we recall that instead of using the standard orbital elements, Poincare´ devised canoni- The validity domain of the disturbing function of nearly polar or- cal action-angle variables for studying the three-body problem that bits is related to the secular potential that governs the long-term have the property of including two actions related to eccentric- dynamics of the particle. The reason is the large inclination of ity and inclination that vanish when motion is exactly circular, the particle’s orbital plane relative to the planet’s that could lead

MNRAS 471, 2097–2110 (2017) Disturbing function for nearly polar orbits 2103

k m n to large eccentricity and inclination oscillations. In the three-body Table 7. Force coefficients smn(α) of the secular term e cos I cos (kω). problem with a planet on a circular orbit, secular evolution of a 0 1 particle with a non-resonant orbit is given by the Kozai–Lidov po- k = 0 s (2A0,0,0,1 + A0,0,0,2) 20 16 Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019 tential (Kozai 1962;Lidov1962). Its integral expression (Quinn, s0 3α2 A − A + A 02 32 ( 2,0,0,0 2 2,2,0,0 2,2,2,0) Tremaine & Duncan 1990) is written with our notations as 2 s0 3α A + A + A 22 128 (6 2,0,0,0 6 2,0,0,1 2,0,0,2 Gm 2π −1/2 2 − 12A − 12A − 2A RKL = R kK(k)r df, 2, 2, 0, 0 2, 2, 0, 1 2, 2, 0, 2 2 2 1/2 + + + πa α (1 − e ) 0 6A2, 2, 2, 0 6A2, 2, 2, 1 A2, 2, 2, 2) R s0 1 A + A 2 4 40 256 (4 0,0,0,3 0,0,0,4) k = , 4 R + 2 + z2 s0 35α A − A + A ( 1) 04 2048 (9 4,0,0,0 24 4,0,2,0 3 4,0,4,0 R2 = r2 − z2, + 16A4, 2, 2, 0 + 3A4, 4, 0, 0 − 8A4, 4, 2, 0 + A ) z = r sin I sin(f + ω), (32) 4, 4, 4, 0 = s2 1 A + A + A k 2 20 16 (6 0,0,2,0 6 0,0,2,1 0,0,2,2) where K is the complete elliptic integral of the first kind and 2 3α2 s − (20A2,0,0,0 + 10A2,0,0,1 + A2,0,0,2 r = α − e2 / + e f 22 256 (1 ) (1 cos ) is the particle’s orbital radius defined in − 60A2, 2, 0, 0 − 30A2, 2, 0, 1 − 3A2, 2, 0, 2 Section 2. The Kozai–Lidov potential is the doubly averaged grav- + 40A2, 2, 2, 0 + 20A2, 2, 2, 1 + 2A2, 2, 2, 2 itational potential with respect to the particle’s and planet’s mean − 20A2, 2, 4, 0 − 10A2, 2, 4, 1 − A2, 2, 4, 2 longitudes and does not involve any expansion with respect to ec- + 20A2, 4, 0, 0 + 10A2, 4, 0, 1 + A2, 4, 0, 2) ω s2 1 A − A + A centricity and inclination. As it depends on the sole angle , one can 40 192 (12 0,0,2,0 12 0,0,2,1 6 0,0,2,2 use the Delauney variables and find that both the semimajor axis α + 8A0, 0, 2, 3 + A0, 0, 2, 4) and the normal component of angular momentum (1 − e2)1/2 cos I = s4 1 A + A k 4 40 768 (120 0,0,4,0 240 0,0,4,1 are constants of secular evolution. Motion generated by the Kozai– + 120A0, 0, 4, 2 + 20A0, 0, 4, 3 + A0, 0, 4, 4) Lidov potential thus occurs in the eccentricity-argument of pericentre plane. In writing the expression (32), no assumption was made ◦ on the inclination; therefore, it can equally be ≤ or ≥90 .Close The initial inclinations are taken as I(e = 0) = 85◦,75◦,65◦ examination of (32) and noting the manner in which the normal and 55◦ (Fig. 1). Owing to the symmetry with respect to the po- coordinate, z, enters its expression reveal that the secular structures lar plane, these values produce exactly the same eω–portraits as in the eω–plane for prograde and retrograde orbits are identical, I(e = 0) = 95◦, 105◦, 115◦ and 125◦, respectively. The only differ- and one can study the former and deduce the latter because of the ence is the inclination range that reads [I(e = 0):180◦] for retrograde potential’s reflection symmetry with respect to the polar plane. orbits instead of [I(e = 0):0◦] for prograde orbits. The various level We study the validity domain of the disturbing function by com- curves in each panel correspond to additional orbits with a normal paring the secular potential it produces with the Kozai–Lidov poten- angular momentum Z equal to that of the reference particle namely tial as in essence the former is an expansion of order N of the latter (1 − e2)1/2 cos I = cos [I(e = 0)]. It is seen on the full Kozai–Lidov with respect to eccentricity and inclination cosine for nearly polar potential that particles located inside the planet’s orbit (α = 0.5) are orbits. This comparison is valid only in the absence of resonance unstable in the sense that they inevitably reach a near-unit eccentric- libration but should illustrate the typical values of eccentricity and ity corresponding to an orbit that is nearly coplanar with the planet. inclination cosine where the fourth-order series can be used. The In the absence of mean motion resonance libration, the time it takes literal expansion of Section 2 shows that the secular potential to for a particle on a nearly polar orbit with a moderate eccentricity order N in eccentricity and inclination cosine is given as to reach a nearly coplanar orbit is long (see e.g. in Section 6). The 1 k m n fourth-order secular potential is found to reproduce the dynamics R¯ = b00 α + s α e I kω , ◦ s 1 ( ) mn( ) cos cos( ) (33) e ≤ I e = ≥ 2 2 quite well for 0.5 and ( 0) 65 in the case of an external 0≤k,n≤N perturber (i.e. α = 0.5). The potential R¯ s can be used to follow k≤m≤N the dynamics on time-scales shorter than the libration around the ◦ k,m,n even Kozai–Lidov resonance at ω = 90 . For time-scales comparable to m+n=N the libration time, one needs to push the expansion order to larger values so as to improve the dynamics’ rendition. Particles outside sk α where the corresponding force coefficients mn( )aregivenin the planet’s orbit (i.e. α = 2) fall into two types of motion. Up to = Table 7 for N 4. an eccentricity e = 0.5, the argument of pericentre circulates and We assess the possible large variations of eccentricity and incli- eccentricity and therefore also inclination have small amplitude nation produced by the secular potentials for nearly polar orbits by variations. The validity domain of the fourth-order polar disturbing ω / R¯ plotting in the e –plane the level curves of RKLa Gm and s for function is e ≤ 0.5 for I(e = 0) ≥ 75◦ and e ≤ 0.2 for 55◦ ≤ I(e = 0) α = α = two initially circular orbits located at 2and 0.5, so as ≤ 65◦. Above e = 0.5, eccentricity can be made close to unity by to illustrate the effects of internal and external perturbers, respec- the two Kozai–Lidov resonances at ω = 0 and 90◦ (with the obvi- tively. Since these locations are near the 1:3 and 3:1 resonances, ous exception of the resonance centres vicinity) and the use of the respectively, both secular potentials reflect the dynamics of circu- disturbing function would require a larger (than 4) expansion order 2 lating orbits to first order in the perturber’s mass. We will see in like in the case of the external perturber. Section 6 the evolution of eccentricity and inclination of resonant In order to understand the secular evolution of the line of nodes orbits is somewhat modified by the critical arguments’ libration. that was absent from the previous analysis, we use the secular potential (33) and apply it to the motion of a massless particle 2 Near-mean motion resonances introduce an additional secular potential perturbed by an internal planet (i.e. α>1) when the particle’s orbit effect for circulating orbits whose amplitude is of the second order in the is far from the Kozai–Lidov resonances. The corresponding eω– perturber’s mass (Hagihara 1972). curve is therefore located in the bottom part of the second-row plots

MNRAS 471, 2097–2110 (2017) 2104 F. Namouni and M. H. M. Morais Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019

Figure 1. Level curves of the secular potentials for an initially circular orbit with semimajor axis ratio α = 0.5 (top panels) and α = 2 (bottom panels) for four inclination values I = 85◦,75◦,65◦,55◦. The solid blue lines represent the Kozai–Lidov potential (32) and the dashed red lines the fourth-order polar secular potential (33). The thin lines represent the collision singularity. of Fig. 1. The secular potential (33) restricted to second order in 6 EXAMPLES OF POLAR RESONANCE eccentricity and inclination cosine will suffice to follow the motion In this section, we provide an illustration of how the disturbing of the longitude of ascending node. It is written as function helps us to identify the correct resonant arguments as- 0 2 2 2 0 2 R¯ s = s e + s e cos 2ω + s cos I, (34) sociated with the polar motion of a particle that interacts with a 20 20 02 −5 Neptune-mass planet on a circular orbit (m /M = 5.12 × 10 ). where we removed the first term of (33) as it does not influence We will not develop a comparison of the analytical polar disturbing eccentricity and inclination. The Lagrange planetary equations can function using the Lagrange equations with numerical integrations, be written in terms of e, ω, I and (Brouwer & Clemence 1961, as it is beyond the scope of this work. Instead, we integrate the full page 289) and truncated for nearly polar orbits to lowest order in equations of motion only to follow the evolution of the particle’s eccentricity and inclination cosine to give orbit and show a variety of polar resonances. We shall consider the following resonances 1:3, 3:1, 2:9 and 7:9. 2 −1 2 −1 e˙ =−(na e) ∂ωRs , ω˙ = (na e) ∂eRs , (35) We learned in Section 3 that the arguments that enter p:q the disturbing function are of the form φk = φ − kω,where φ = qλ − pλ + (p − q). The fundamental mode k = 0 occurs I˙ =−na2 −1∂ R , ˙ = na2 −1∂ R , ( ) s ( ) I s (36) only for even-order resonances. It is a pure-inclination term that in = −3 1/2 R = principle could librate regardless of eccentricity, as its first-order where n (GMa ) is the particle’s mean motion and s 0 0 −1 force amplitude is c + c cos I giving the resonance a pendulum- Gm a R¯ s the fully dimensional secular potential. We note how the 00 01 like dynamical structure almost independent of inclination for polar- I–equations (36) differ from those of nearly coplanar orbits in that like orbits I ∼ 90◦. However, nearly polar orbits have a large relative the variation rates are not inversely proportional to inclination unlike inclination with respect to the planet’s orbit and that in turn can force the eω–equations (35) that keep their classical form. Furthermore, a coupling of eccentricity and inclination variations similar to what by using the potential (34) in the Lagrange equations, it is found was seen in the previous section with the Kozai–Lidov resonance. that the inclination I is a constant of motion implying far smaller We therefore illustrate the fundamental mode k = 0 by placing the variations for I than for e when resonant terms are included. The particle directly in the Kozai–Lidov resonance that is coupled to the longitude of ascending node’s variation rate for nearly polar orbits outer 1:3 mean motion resonance, thus ensuring that the argument is also constant, as it depends only on I and given as of perihelion ω is stationary and allowing the k = 0 mode to librate. nαms0 I Fig. 2 shows as functions of time, normalized to the planet’s or- 02 sin 2 ˙ =− . (37) bital period T, the evolution of the orbital elements along with the M 1:3 resonant angles φk = 3λ − λ − 2 − kω for k = 0and−2. The ◦ ◦ The secular force coefficient that enters the expression of ˙ can be particle’s initial orbital elements are e = 0.2, ω = 90 , I = 120 , s0 ∼ α − . + α − 2.1 −1 <α = ◦ α = 2/3 approximated by 02 11( 1)(6 5 370( 1) ) for 1 0 and 3 is the nominal resonance location. The argu- ≤ s0 > φ1:3 5. Therefore, as 02 0, the line of nodes of prograde nearly ment 0 librates with a variable period starting from a maximum of polar orbits regresses, whereas that of retrograde nearly polar orbits 4000 T and evolving to a minimum of 1000 T . A similar behaviour φ1:3 precesses. The nearer to the polar motion, the smaller the variation is forced on the −2 argument because of secular resonance. The 1:3 rate of the longitude of ascending node. These results are confirmed resonance also modifies the eω–secular structure in that it allows ◦ in the next section. the Kozai–Lidov resonance at ω = 90 to occur at a much lower

MNRAS 471, 2097–2110 (2017) Disturbing function for nearly polar orbits 2105 Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019

α ω φ1:3 φ1:3 Figure 2. Orbital elements , e, , I, and resonant arguments 0 , −2 as a function of time for a particle at the outer 1:3 resonance with a Neptune-mass planet. Initial parameters are eccentricity e = 0.2, inclination I = 120◦, longitude of ascending node = 0◦, argument of pericentre ω = 90◦ and relative mean λ − λ = ◦ φ1:3 ≤ / ≤ 4 longitude 0 . The bottom right panel is a zoom of 0 for 0 t T 10 . eccentricity than that of non-resonant orbits considered in Section 5. rapidly at maximum eccentricity. In a complementary way, the ar- φ3:1 φ3:1 However, whereas the eccentricity’s variations are moderate, the in- gument 0 circulates during 4 ’s libration and briefly librates at clination’s are quite small and the line of nodes precesses linearly maximum eccentricity. as the orbit is retrograde confirming the results of Section 5. With the next example, we illustrate how a resonance of order On the subject of the effect of the Kozai–Lidov secular potential or 1 can display librations with |k| < or and consequently a on polar orbits, we also examine the evolution of a particle that force amplitude ∝e|k|, a property inherent to the polar disturbing librates in the inner 3:1 resonance located at α = 3−2/3 to give an function first encountered in Section 3 and discussed in Section 4. example with an external perturber and ascertain the similarities and We therefore examine the outer 2:9 resonance example (28) located 2/3 differences with the secular evolution of the non-resonant orbits dis- at α = (9/2) that has an odd resonance order, or = 7, and possible 2:9 cussed in Section 5. The possible resonant critical arguments now resonant arguments φk = 9λ − 2λ − 7 − kω,where|k|≥1is φ3:1 = λ − λ + − kω ck , e|k| read k 3 2 ,wherek is an even integer. Plac- an odd integer, and force amplitudes |k|0(2 9) to lowest order ing a particle at the bottom of the eω-plane of the first top panel in eccentricity. of Fig. 1 with e = 0.1, ω = 0, = 0 and choosing an inclination Figs 4–6 show the evolution of the orbital elements as well as the ◦ 2:9 I = 95 whose secular structure away from resonance is identical arguments φk for k = 1, 3 and 5 for three different initial condi- to that of I = 85◦ (as explained at the beginning of Section 5), we tions. With the initial parameters I = 70◦, e = 0.05, = 0◦ and = ◦ φ2:9 show in Fig. 3 the evolution of orbital elements as a function of 0 , the particle librates with the critical argument 1 while time. As expected from the effects of the secular potential of an ex- the other arguments circulate (Fig. 4). More precisely, the libration ternal perturber, the argument of pericentre circulates periodically involves two periods: a short one 9800 T and a longer modulation × 6 φ2:9 in the narrow strip adjacent to the Kozai–Lidov resonances where 2 10 T . The first period is from the fundamental mode 0 the eccentricity reaches a maximum value em = 0.995. However, that circulates on the second time-scale. We know this because the the minimum inclination that should have been 180◦ at maximum fundamental mode is not related to eccentricity and consequently eccentricity if the particle were circulating instead of librating in is not influenced by its secular evolution. The longer period cor- mean motion resonance is now reduced to 150◦, indicating that responds to the time necessary to counter the fundamental mode’s the conserved quantity involving inclination is no longer the nor- long-term slow circulation with the slow drift of the argument of ω φ2:9 mal component of angular momentum like that of the Kozai–Lidov pericentre , giving rise to the resonant angle 1 . We note that φ2:9 φ2:9 potential of non-resonant orbits. In effect, of the three critical ar- the resonant arguments 3 and 5 display the fast libration of the φ3:1 φ3:1 φ3:1 guments 0 , 2 , 4 the particle is found to stably librate in fundamental mode’s short period, yet each is circulating with a slow = ◦ 5 φ2:9 φ2:9 the k 4 resonance with a 120 -amplitude and 10 T -period ex- rate for 3 and slightly faster for 5 . This shows that when iden- plaining why its secular evolution is not completely described by tifying Centaurs and TNOs in polar resonance, one must integrate the Kozai–Lidov potential. It is also interesting to note how other their orbits over long time spans so as not to misinterpret evolu- φ3:1 φ2:9 φ2:9 arguments display quasi-librations. For instance, 2 follows the tions such as those described for 3 and 5 as true librations in φ3:1 libration of 4 over half the libration period only to circulate resonance.

MNRAS 471, 2097–2110 (2017) 2106 F. Namouni and M. H. M. Morais Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019

α ω φ3:1 φ3:1 φ3:1 Figure 3. Orbital elements , e, , I, and resonant arguments 0 , 2 and 4 as a function of time for a particle at the inner 3:1 resonance with a Neptune-mass planet. Initial parameters are eccentricity e = 0.1, inclination I = 95◦, longitude of ascending node = 0◦, argument of pericentre ω = 0◦ and relative mean longitude λ − λ = 180◦.

φ2:9 Figure 4. Orbital elements’ and resonant angles’ evolution of particle at the outer 2:9 resonance with a Neptune-mass planet. Libration occurs for 1 . Initial parameters are eccentricity e = 0.05, inclination I = 70◦, longitude of ascending node = 0◦, argument of pericentre ω = 0◦ and relative mean longitude λ − λ = 90◦.

MNRAS 471, 2097–2110 (2017) Disturbing function for nearly polar orbits 2107 Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019

φ2:9 Figure 5. Orbital elements’ and resonant angles’ evolution of particle at the outer 2:9 resonance with a Neptune-mass planet. Libration occurs for 3 . Initial parameters are eccentricity e = 0.1, inclination I = 70◦, longitude of ascending node = 0◦, argument of pericentre ω = 0◦ and relative mean longitude λ − λ = 90◦.

φ2:9 Figure 6. Orbital elements’ and resonant angles’ evolution of particle at the outer 2:9 resonance with a Neptune-mass planet. Libration occurs for 5 . Initial parameters are eccentricity e = 0.2, inclination I = 85◦, longitude of ascending node = 0◦, argument of pericentre ω = 180◦ and relative mean longitude λ − λ = 0◦.

MNRAS 471, 2097–2110 (2017) 2108 F. Namouni and M. H. M. Morais Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019

Figure 7. Orbital elements and resonant angles’ evolution of TNO 471325 at the outer 7:9 resonance with a Neptune-mass planet in the context of the three-body problem. Initial parameters are eccentricity e = 0.3, inclination I = 110◦, longitude of ascending node = 0◦, argument of pericentre ω = 90◦ and relative mean longitude λ − λ = 180◦.

Increasing the particle’s eccentricity to e = 0.1 has two effects: the three-body problem, these simulations do not reﬂect the actual the argument of perihelion’s regression is faster (see below) lead- evolution of the object but will give us an idea on the possible φ2:9 ing to resonance with the critical argument 3 (Fig. 5). The two resonant arguments that might be involved. With this in mind, the libration periods are decreased: the short one to 4500 T and the initial orbital elements are: semimajor axis α = 1.182, eccentric- long one to 1.66 × 106 T. The cautionary note that we pointed ity is e = 0.3 and inclination I = 110◦. We choose = 0◦ and out in the previous example is still valid for the new eccentricity ω = 90◦. As an even order resonance, the permissible resonant φ2:9 φ2:9 φ7:9 = λ − λ − − kω as the arguments 1 and 5 circulate but display the librations arguments are k 9 7 2 ,wherek is an even in- associated with the short period. The situation is even more mis- teger. Fig. 7 shows the evolution of the orbital elements as well φ2:9 φ7:9 = leading for 5 because if the integration time span were restricted as the arguments k for k 2, 4 and 6. It is seen that the ar- × 5 φ7:9 to the interval [7,10] 10 T , the argument would seem to be gument 4 for TNO 471325 in the three-body problem librates genuinely librating. around 180◦ with an amplitude and a period of 68◦ and 15550 T, For the example in Fig. 6, we increased the inclination to I = 85◦ respectively. The argument of pericentre ω circulates rapidly, the and eccentricity to e = 0.2 , resulting in the libration of the critical longitude of ascending node precesses linearly and the incli- φ2:9 argument 5 with a small amplitude and the two periods 1400 T nation has moderate variations as predicted in Section 5. The = × 5 φ7:9 φ7:9 (fast one associated with the fundamental mode k 0) and 5 10 T both resonant arguments 2 and 6 circulate but only the lat- (the slower one associated with k = 5). With this eccentricity, the ter displays temporary librations similar to those of the previous φ2:9 φ2:9 behaviour of the arguments 1 , 3 is less misleading regarding resonance. the importance of the short period librations discussed with the Increasing the eccentricity to e = 0.4 and initializing ω at 180◦ φ7:9 previous examples and their evolution is more clearly circulating. makes the orbit librate with the critical argument 2 with an ampli- The reason is the small amplitude of the short period libration seen tude of 120◦ and a period of 22 000 T (Fig. 8). Thus, the trend noted φ2:9 in 5 . in the previous example, about how a larger k could be associated The 2:9 resonance examples also illustrate how the inclination’s with a larger e, is not conﬁrmed. The remaining arguments circulate φ7:9 φ7:9 relative variation is small and how the line of nodes of subpo- but now it is 4 that displays temporary librations whereas 6 lar orbits regresses linearly at a rate that decreases as the incli- circulates with the fastest rate. nation approaches 90◦. We also note that when the eccentricity When the eccentricity is increased further to e = 0.64 (ω = 180◦), φ7:9 is increased, the librating critical argument’s integer k increases. the particle librates with the critical argument 6 of amplitude ◦ φ7:9 φ7:9 This trend however is not generally valid, as the next example 70 and period of 16 000 T (Fig. 9). The arguments 2 and 4 will show. circulate rapidly without temporary librations. With the last example we examine the outer 7:9 resonance in We conclude that for the same location and inclination as TNO which TNO 471325 (Chen et al. 2016) could currently be captured. 471325, the resonant argument depends strongly on the observed φ7:9 Since our interest is the polar disturbing function in the context of eccentricity. It is likely that 4 is the correct librating critical

MNRAS 471, 2097–2110 (2017) Disturbing function for nearly polar orbits 2109 Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019

Figure 8. Orbital elements and resonant angles’ evolution of particle at the outer 7:9 resonance with a Neptune-mass planet. Initial parameters are eccentricity e = 0.4, inclination I = 110◦, longitude of ascending node = 0◦, argument of pericentre ω = 180◦ and relative mean longitude λ − λ = 180◦.

Figure 9. Orbital elements and resonant angles’ evolution of particle at the outer 7:9 resonance with a Neptune-mass planet. Initial parameters are: eccentricity e = 0.64, inclination I = 110◦, longitude of ascending node = 0◦, argument of pericentre ω = 180◦ and relative mean longitude λ − λ = 180◦. argument as the effect of the other planets on resonant polar asteroids 7 CONCLUDING REMARKS must be reduced because of their peculiar orbital geometry unless The main technical results of this work consist of (i) an ex- there is strong interaction between the planets such as a mean motion plicit algorithm for generating a literal expansion of the disturbing resonance that carries over to the motion of the polar asteroid.

MNRAS 471, 2097–2110 (2017) 2110 F. Namouni and M. H. M. Morais function for nearly polar orbits of general order N in eccentricity ACKNOWLEDGEMENTS and inclination cosine (ii) the explicit form of the fourth-order polar disturbing function through the direct part (22) with its force coef- The authors thank Zoran Knezeviˇ c´ for his thorough and helpful ficients given in Tables 1–5, the indirect part written explicitly in review of the manuscript. FN thanks the University of Rio Claro Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/2097/3939747 by Universidade Estadual Paulista Jï¿½lio de Mesquita Filho user on 25 July 2019 Table 6, and the secular potential (33) whose force coefficients are (UNESP) for their hospitality during his stay when part of this work giveninTable7. Beyond the technical results, our original motiva- was performed. This work was supported by grant no. 2015/17962-5 tion for deriving a literal expansion of the disturbing function for of Sao˜ Paulo Research Foundation (FAPESP). nearly polar orbits is the realization that general attitude regarding resonance identification for polar Centaurs and TNOs is based on REFERENCES decades, and for some aspects more than a century, of use in planetary dynamics of the classical disturbing function derived for nearly Brouwer D., Clemence G. M., 1961, Celestial Mechanics. Academic Press, coplanar prograde orbits. It is therefore not surprising that we have New York revealed new features unseen in the classical disturbing function, Chen Y.-T. et al., 2016, ApJ, 827, L24 Ellis K. M., Murray C. D., 2000, Icarus, 147, 129 especially the structure of the force amplitudes that define resonance Gladman B. et al., 2009, ApJ, 697, L91 strength. In particular, the fact that regardless of resonance order, a Hagihara Y., 1972, Celestial mechanics. Vol. 2, Perturbation Theory. Mas- particle can librate in the lowest harmonics (small k in equation 22) sachusetts Institute of Technology (MIT), Cambridge, MA of the disturbing function is interesting, as it explains an important Kozai Y., 1962, AJ, 67, 591 observation that was made in our numerical studies of resonance Laskar J., Boue´ G., 2010, A&A, 522, A60 capture at arbitrary inclination (Namouni & Morais 2015, 2017) Lidov M. L., 1962, Planet. Space Sci., 9, 719 that resonance order is not a good indicator of resonance strength Morais M. H. M., Giuppone C. A., 2012, MNRAS, 424, 52 nor capture efficiency. This observation was particularly striking Morais M. H. M., Namouni F., 2013a, Celest. Mech. Dyn. Astron., 117, 405 for the outer 1:5 resonance that exceeded 80 per cent capture Morais M. H. M., Namouni F., 2013b, MNRAS, 436, L30 efficiency for the most eccentric nearly polar orbits (fig. 6 of Murray C. D., Dermott S. F., 1999, Solar System Dynamics. Cambridge Univ. Press, Cambridge Namouni & Morais 2017). TNO 471325 provides a good exam- Namouni F., Morais M. H. M., 2015, MNRAS, 446, 1998 ple of a near polar asteroid locked in resonance. Our three-body Namouni F., Morais M. H. M., 2017, MNRAS, 467, 2673 φ7:9 simulations suggest that the resonant critical argument is 4 .Fur- Quinn T., Tremaine S., Duncan M., 1990, ApJ, 355, 667 ther simulations including all Solar system planets are required to Williams J. G., 1969, PhD thesis, University of California at Los Angeles confirm this possibility and discover yet more Centaurs and TNOs in polar resonance with the giant planets. This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS 471, 2097–2110 (2017)