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A Note on the Spin–Orbit, Spin–Spin, and Spin–Orbit–Spin Resonances in the Binary System

Article in The Astronomical Journal · November 2017 DOI: 10.3847/1538-3881/aa96ab

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Xiyun Hou Xiaosheng Xin Nanjing University Nanjing University

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A Note on the Spin–Orbit, Spin–Spin, and Spin–Orbit–Spin Resonances in the Binary Minor Planet System

Xiyun Hou and Xiaosheng Xin School of Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing, Jiangsu 210023, China; [email protected] Received 2017 May 8; revised 2017 October 23; accepted 2017 October 24; published 2017 November 29

Abstract By considering a varying mutual orbit between the two bodies in a binary minor planet system, modified models for the spin–orbit, spin–spin, and spin–orbit–spin resonances are given. For the spin–orbit resonances, our study shows that the resonance center changes with the mass ratio and the mutual distance between the two bodies, and the size of the body in the resonance. The 1:1, 3:2, and 1:2 resonances are taken as examples to show the results. For the spin–spin and spin–orbit–spin resonances, our studies show that the resonance center changes with the rotation states of the two minor planets. The 1:1 spin–spin resonance and the 1:2:1 spin–orbit–spin resonance are discussed in detail. Simple analytical criteria are given to identify the resonance centers, and numerical simulations were ran in order to verify the analytical results. Key words: celestial mechanics – methods: numerical – minor planets, : general

1. Introduction coupling problems in these systems (Fang & Margot 2012; Naidu & Margto 2015; Xin & Hou 2017). Spin–orbit resonance is a problem with a long history in the In the conventional model for spin–orbit resonances, the solar system (Goldreich & Peale 1966). It happens when the mutual orbit is assumed as an invariant circular or eccentric spinning period of one body and its with respect Keplerian orbit (Goldreich & Peale 1966; Celletti 1990,Celletti to another body are incommensurate. The most well-known & Chierchia 2000, 2008, Lhotka 2013; Gkolias et al. 2016). This and ancient example is the Moon, which is in 1:1 spin–orbit assumption is reasonable when treating the spin–orbit resonance resonance, causing only one side to perpetually face the Earth ( ) of natural satellites for major planets because the satellite Peale 1969; Beletskii 1972 . Except for the Moon, it is found generally has a much smaller mass and size compared to those of that 1:1 spin–orbit resonance is widespread in the satellite fl ( theplanet.Asaresult,thein uence of its rotational motion on system of the planets Murray & Dermott 1999; Kouprianov & the mutual orbit is negligible. However, in the binary minor Shevchenko 2005; Antognini et al. 2014). Mercury was the – planet systems, size and mass difference between the two bodies sole object found, until now, for which the spin orbit is usually not as obvious as that of the planet–satellite systems, resonance is not 1:1, but instead 3:2. This peculiarity and sometimes the two minor planets even have comparable ( has drawn the attention of many researchers Correia & sizes and masses. Moreover, the primary and the satellite usually ’ Laskar 2004; Rambaux & Bois 2004, Lamaître & D Hoedt have a much closer distance (several to tens of radii of the ) 2006, Noyelles et al. 2013 . Except for the regular rotators, primary), which makes the spin–orbit coupling in these systems there are also natural satellites that are in chaotic rotations, such much stronger (Fahnestock & Scheeres 2006, Scheeres et al. ( ) as Hyperion Wisdom et al. 1984; Klavetter 1989 . Nyx and 2006, Compère & Lamaître 2014, Hou et al. 2017).Thismakes ( Hydra were also speculated to rotate chaotically Correia et al. the assumption of an invariant mutual orbit no longer valid (Xin ) 2015; Showalter & Hamilton 2015 . However, it seems that &Hou2017). Usually, the orbital motion and rotational motion ( these two bodies actually rotate regularly Weaver et al. 2016; have to be simultaneously considered. Even for the simplest Quillen et al. 2017), as their rotation speeds are much faster planar case where the minor planet’s equatorial plane coincides than was previously anticipated. Recently, with the confirma- with its orbital plane, it is a Hamiltonian system of 3 degrees of tion of the existence of binary minor planet systems (Margot freedom (DOF), which is more complex than the conventional et al. 2002; Merline et al. 2002; Richardson & Walsh 2006; spin–orbit resonance model. Numerical approaches can be taken Margot et al. 2015), spin–orbit resonance is also found in these (Xin & Hou 2017), but in this work, we aim at an analytical systems (Scheeres 2006; Pravec et al. 2015). Due to limitations analysis and use simple criteria to describe the stability of these of observations, currently we are only able to identify those that spin–orbit resonances. To simplify our study, we only focus are in 1:1 spin–orbit resonance (i.e., the synchronous state). But on the 1:1, 3:2, and 1:2 spin–orbit resonances. Due to tidal due to their large populations, more irregular shapes of dissipations and thermal effects, both satellite and primary minor the components, much shorter tidal evolution timescales planets may change their rotation states and enter spin–orbit (Goldreich & Sari 2009; Jacobson & Scheeres 2011), and resonances. This makes the study of spin–orbit resonances for thermal effects that can change the rotation states of the both bodies reasonable. For the satellite, studies show that the near-Earth populations (Ćuk & Burns 2005; Bottke et al. 2006; resonance center of our model is the same as the conventional Scheeres 2007; MacMahon & Scheeres 2010), spin–orbit model; however, the resonance width is different. For the resonances are suspected to be widespread in the population of primary, depending on the mass ratio and the mutual distance binary minor planet systems, and more interesting phenomena between the two bodies, and the size of the primary, a qualitative than those predicted by the conventional spin–orbit theories can difference between our model and the conventional model be expected. There already exists some work on the spin–orbit may exist.

1 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin The conventional model for spin–orbit resonances also Aʼs barycenter to Bʼs barycenter, and θ denotes the polar angle assumes a spherical planet and considers only the satellite’s of r in the inertial frame. O is the barycenter of the system. qA spin–orbit coupling by truncating its gravity at the second and qB denote rotation angles of A and B with respect to the order.1 But in binary minor planet systems, both components inertial frame. may be irregular, and due to their comparable sizes and masses, Supposing the semimajor axis of the unperturbed mutual it also makes sense to consider the direct and indirect orbit is a¯, we use the following units throughout this work interaction between the non-spherical parts of the two bodies, 3 which leads to the unique and interesting phenomenon []LaMmmT==+=¯,,[]AB [] GML [][] .1 () of the so-called spin–spin resonance. Recently, some interesting papers (Batygin & Morbidelli 2015; Nadoushan & There is no closed-form of the mutual potential between two Assadian 2016) on this phenomenon have been published. ellipsoids. Because we are dealing with the planar case, we use However, in these works, the mutual orbit is again assumed the explicit expression of the 4OD mutual potential given by invariant and only the direct interaction term between the non- Hou et al. (2017). It reads spherical parts of the two bodies is considered. In this 3 5 contribution, we will consider the indirect interaction term, UU=++=-+012 U U[1,2 rVrVr˜˜ 2 +4 ]() i.e., the term contributed by a varying mutual orbit. We will see that the strength of the indirect term is comparable to that of the where ( ( ) ( ) direct term see Equations 44 and 48 and the text following ˜ them). Depending on the rotational states of the two bodies, the VAA212=+cos()() 2qq - 2AB + A 3 cos 2 qq - 2 , introduction of the indirect term can change the resonance VBB˜412=+cos() 2qq - 2AA + B 3 cos () 4 qq - 4 – – – center of the spin spin resonance or the spin orbit spin +-+-BB45cos() 2qq 2BB cos () 4 qq 4 resonance. Because we consider both the direct and indirect +-+--BB67cos()( 2qqAB 2 cos 4 qqq 2 AB 2 )() . 3 terms, our results fit the physical system better and are demonstrated by numerical simulations. The coefficients Ai and Bi are same as those in Hou et al. (2017) In this study, we separate the spin–orbit resonance (where where they are expressed by generalized inertia integrals. Here, the rotation of only one body and the mutual orbit motion is we choose to express them as involved) and the spin–spin resonance (where only the ) rotations of the two bodies are involved from the general AJ=+1 ()Aaa2 JB 2 ,3,3, AJ =A a2 AJ =B a2 spin–orbit–spin resonances (where the rotations of both bodies 1 2 2 A 2 B 2 22 A 3 22 B ) 3 A 4 B 4 9 AB 22 and the orbital motion are involved . This paper is organized as BJJ1 =-()4 aaA +4 B + JJ22 aaAB, follows. Section 2 introduces the force model and equations of 8 4 BJJJBJ=-15 ()A aaaa4 -AB22, = 105A 4 , motion used in this work. We use two ellipsoids to simulate the 2 2 42 A 22 2 AB 3 42 A two bodies of the binary minor planet system. To simplify BJJJBJ=-15 ()B aaaa4 -B A 22, = 105B 4 , the discussion, only the planar case is studied in this work. The 4 2 42 B 22 A AB 5 42 B mutual potential is truncated at the 4th order and 4th degree 9 AB 22 105 AB 22 BJJ6 ==aa,, B7 JJ aa (4OD). Section 3 introduces the modified spin–orbit resonance 2 22 22 AB 2 22 22 AB model by considering the contribution from a varying mutual where (Balmino 1994) orbit. This part of the work is carried out in the 2nd order and 2nd degree (2OD) mutual potential. Using the 1:1, 3:2, and 1:2 a ab22+-2 c 2 ab22 - a ==¯* ,,,JJ**** * =** – * a¯ 2 10a¯¯2 22 20a 2 spin orbit resonances as examples, this study shows that * * depending on the mass ratio and mutual distance of the two 15 5 5 JJJJJJJJ********=-(( )2 +2, ( )2 ) =- , = ( )2 , bodies, the resonance center of specific spin–orbit resonances 40 7 2 22 42 7 222 44 28 22 fi – change. Section 4 introduces the modi ed spin spin resonance ∗ model by considering both the indirect term and the direct term. for which indicates A or B, and a¯* indicates the reference ’ Then, focusing on only one spin–orbit–spin resonance, radius for A or B, which takes the value of the ellipsoid s ˜ Section 5 introduces the modified spin–orbit–spin resonance longest axis. In this work, we call the terms given by V2 in model, which also considers both the direct and indirect terms. Equation (2) the second order and second degree (2OD) terms, Section 6 concludes the study. and the terms given by V˜4 in Equation (2) the fourth order and fourth degree (4OD) terms. 2. Model Description Equations of motion for the system are We use two tri-axial ellipsoids A and B to approximate the ⎧ ˙˙2 ¶U ¨ r˙ 1 ¶U ( ) ⎪rr¨,-=-+qqq2 =-2 two bodies. The shape parameter for A B is aAAAbc ⎨ ¶r rr¶q -3 m ¶U m ¶U ,4() (aBBBbc). A density of 2 gcm is used in this work. ⎪¨ ¨ ⎩qqA =-A , B =- B Only the planar case is considered in this study, i.e., the Iz ¶qA Iz ¶qB ellipsoids are assumed to rotate along their shortest axis, with their equatorial planes coincident with their orbital plane. An where illustrative picture of the relative geometry is given by mm ma()22+ b Figure 1(a). The vector r denotes the position vector from m = AB , I A = A AA, 2 z 2 ()mmAB+ 5()mmaAB+ ¯ 1 A tri-axial ellipsoid shape model may be used for the satellite. However, as ma()22+ b the torque on the satellite by the planet is usually only computed to the lowest I B = B BB.5() z 2 order, it is identical to stating that the satellite’s gravity is truncated at the 5()mmaAB+ ¯ second order.

2 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin

Figure 1. Left: relative geometry between two ellipsoids. This model will be used for the spin–spin resonance and spin–orbit–spin resonance. Right: relative geometry between a sphere and an ellipsoid. This model will be used for the spin–orbit resonance of the ellipsoid B. Please be aware that B does not necessarily have to be the satellite. It can also be the primary.

Substituting Equation (3) in, we have terms of the mutual potential, the system’s Hamiltonian is

B ⎧ 1 I 2 rr¨cos22-=--qqq˙2 13[() A + A - Hrr=+(˙22q˙2) +z q˙ ⎪ rr2412 A B ⎪ 22m +-A3 cos()] 2qq 2 B ⎡ ⎤ ⎪ 1 A1 A3 5 -++⎢ cos() 2qq - 2⎥ . () 8 ⎪ -+[()()BBcos 2qq -+ 2 B cos 4 qq - 4 ⎣ 3 3 B ⎦ 6 12AA 3 r r r ⎪ r ⎪ +-B4 cos()] 2qq 2 B ⎪ Using the following Delaunay variables ⎨ +-+-BB56cos() 4qq 4BAB cos ( 2 q 2 q ) ()6 2 ⎪ +--B7 cos()] 4qq 2AB 2 q LaGaelMg==-==,1,,()w () 9 ⎪ q¨ +=-2sin22sin22r˙ q˙ 2 [(AA qq - ) + ( qq - )] B ˙ ( ) ⎪ r r 5 23ABand pIB = z qm, we can rewrite Equation 8 as ⎪ 1 --+-[(2BB sin 2qq 2 ) 4 sin ( 4 qq 4 ) 1 m A1 A3 ⎪ r 7 23AAH =- +p2 - -cos()() 2qq - 2 . 10 ⎪ 2 B B 3 3 B +-2sin22B4 ()]qqB 2L 2Iz r r ⎩⎪ +-+--4BB57 sin() 4qq 4BAB 4 sin ( 4 qq 2 2 q )] For the traditional spin–orbit resonance model (Wisdom et al. 1984), the mutual orbit is considered as invariant so for the orbital motion, and only the bold-faced terms in Equation (10) appear. But here we ⎧ simultaneously consider the variations in the orbit, so the ¨ 2sin22mqqA2 ()- A ⎪ qA = A 3 contribution from the orbital motion, i.e., the terms not bold- Irz ⎪ m faced in Equation (10), are also included. This difference ⎪ +-+-A 5 [(2BB23 sin 2qq 2AA ) 4 sin ( 4 qq 4 ) Irz expands the DOF of the Hamiltonian from 1 to 3, with the two ⎪ --+--2sin2BB67()qqAB 2 2sin42 ( qqq AB 2 )] additional DOF associated with the orbital motion. Expanding ⎨ ()7 ( ) fi e ⎪ ¨ 2sin22mqqA3 ()- B Equation 10 to the rst order of , we have qB = B 3 ⎪ Irz 1 m 2 A1 3A1 m HpeM=- + - - cos ⎪ 2L2 I B B L6 L6 +-+-B 5 [(2sin22BB45qqBB ) 4sin44 ( qq ) 2 z ⎪ Irz ⎪ A ⎡ e ⎩ -+--+-3 ⎣cos()() 2lg 2 2qq cos lg 2 2 +-+--2sin2BB67()qqAB 2 2sin42 ( qqq AB 2 )] L6 B 2 B ⎤ ++-7e cos() 3lg 2 2q ⎦ . for the rotational motions. Equations (6) and (7) will be used in 2 B the numerical simulations for the spin–spin resonance and the ()11 spin–orbit–spin resonance. Their simplified version, where the By following canonical transformation 4OD terms are omitted, will be used in the numerical simulations for the spin–orbit resonance. LLGGL¢=, ¢= - llggg¢= +,, ¢= () 12

3. Spin–Orbit Resonance Equation (11) is transformed into For the spin–orbit resonance, we simplify the model in 1 m 2 A1 3A1 HpeM=-2 +B B -6 - 6 cos Figure 1(a) by assuming A as a sphere, as shown in 2L¢ 2Iz L¢¢L ( ) ʼ A3 ⎡ e Figure 1 b . In this case, A s rotation is decoupled from the -¢--¢-+¢6 ⎣cos()( 2llg 2qqB cos 2 B ) – L¢ 2 system, and we study the spin orbit coupling problem of the ⎤ ( ) ( ) +¢--¢7e cos() 3lg 2q ⎦ . () 13 ellipsoid B. From Equations 2 and 3 truncated at the 2OD 2 B

3 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin

Figure 2. Two possible configurations of B with respect to A, for the 1:1 spin–orbit resonance during the whole orbital period, and for the 3:2 and 1:2 spin–orbit resonances when B is at its orbit periapsis.

The angle l¢ is actually the angle θ in Figure 1. In the following, If we only consider the first term in the square bracket of we focus on the three spin–orbit resonance angles in the Equation (18), we get the conventional model for the 1:1 spin– bracket, i.e., 1:1, 1:2, and 3:2. It is important to note that if orbit resonance, which assumes an invariant orbit between the we truncate the mutual potential at higher orders of e or two bodies. The resonance center for the 1:1 spin–orbit normalize the Hamiltonian to high orders, we can get other resonance in the conventional model is always for the resonance terms. But here we only take the three resonance ellipsoid’s long axis to point at the sphere, but this is no terms in Equation (13) as examples. longer the case in Equation (18). Considering Equations (9), (12), and (15), neglecting the relatively small variations in y1 3.1. The 1:1 Resonance and taking it as a constant, we can rewrite Equation (18) as First, we study the 0th order resonance—the 1:1 resonance. ⎛ ⎞ 236A qm˙ In this case, neglecting other short-period terms in x¨ =-3 ⎜ -B + ⎟ sin 2x ,() 19 1 32 B 1 Equation (13) and only keeping the resonance term, we have aa¯¯⎝ a¯ Iz ⎠

1 m 2 A1 A3 where a¯ is the mean semimajor axis of the mutual orbit. If B is H =- +-p - cos()() 2l¢- 2qB . 14 -32 2 B B 6 6 trapped in 1:1 resonance, q˙B = a¯ . Substituting into 2L¢ 2Iz L¢ L¢ Equation (19), and paying attention to the fact that a¯ = 1 in Introduce the following transformation the units of our work (see Equation (1)), we can further rewrite Equation (19) as xl123=¢-qB,, x =¢ lxg = ¢ xASx¨2sin2,1311=- () 20 ypyLpyG12=-BB,,,15 = ¢+ 3 = ¢ () where Equation (14) is transformed into 2 m 5maA ¯ S1 =-=3 - 3. 1 m 2 A1 B 22 H =- +-y I ()()mmabAB++ 2 B 1 6 z BB 2()yy12+ 2Iz ()yy12+ Depending on the mass ratio between mA and mB, the mutual A - 3 cos 2x .() 16 B S 6 1 distance a¯, and the size of , the sign of 1 can change. If ()yy12+ S1 > 0, the resonance center is x1 = 0 and Bʼs long axis is ( ) This is a 1-DOF system with xy, as variables and y as a always pointing at A, as shown in Figure 2 a .IfS1 < 0, the 1 1 2 ʼ parameter. From Equation (16), we have resonance center is x1 =p 2 and B s short axis is always pointing at A, as shown in Figure 2(b). For the case of a ⎧ 1 m 6A xy˙ =++1 satellite around a major planet where mmBA and ⎪ 1 ()yy++3 B 1 ()yy7 12 Iz 12 ’ ⎪ a¯  ()abBB, , we have S1  0. The satellite s long axis is ⎨ + 6A3 cos 2x pointing at the planet, such as the Earth–Moon system. In ()yy+ 7 1 .17() ⎪ 12 binary minor planet systems, the size difference between the ⎪ 2A3 yx˙1 =- 6 sin 2 1 two bodies is usually not as large as the planet–satellite ⎩ ()yy12+ systems. Except for the satellite, the primary may also be Differentiating the first equation of Equation (17) with respect captured in spin–orbit resonances due to tidal dissipations or to time, substituting the second equation in, and neglecting thermal effects. It makes sense to discuss the spin–orbit high-order terms, we have resonance for both the satellite and the primary. ( ) ⎡ ⎤ 1 If B is the satellite, it means ⎢ m 3 6my1 ⎥ 2A3 x¨1 =- + + .18() mA ⎣ I B ()yy+ 4 IyB ()+ y⎦ ()yy+ 6  0.5. z 12 z 12 12 ()mmAB+

4 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin Assuming the density of A is the same as that of B, then we from which we have have aAB a . Because a¯ ()aaAB+ 2 a B, we have ⎡ ⎢ 427m 72my1 a¯2 a¯2 x¨1 =- - - 2. ⎣ I B ()yy+ 3 4 IyB ()+ 3 y ()ab22+ ()2a2 z 21 z 21 BB B ⎛ ⎞ ⎤ 3 4my1 17¶e ⎥ A3 +⎜ + ⎟ ´ exsin1 . Substituting the above two relations into the expression for S1, 3 B ⎥ 6 ⎝ ()yy21+ 3 Iz ⎠ e ¶y1 ⎦ 23 ()yy21+ we have S1  2. This means the satellite always has its long axis pointing at the primary. There is no qualitative difference ()25 between our improved model and the conventional model. fi However, it is easy to show that the resonance width is changed Again, if we only consider the rst term in the square bracket of ( ) – from Equation 25 , we get the conventional model for the 3:2 spin orbit resonance, which assumes an invariant orbit between the two bodies. The resonance center for the 3:2 spin–orbit 23AS31()+ resonance in the conventional model is always for the for the conventional model to ellipsoid’s long axis to point at the sphere at the periapsis of the orbit, but this is no longer the case when the two bodies are

2AS31 close to each other and have comparable sizes and masses. Considering Equations (9), (12), and (22), neglecting the in our model. This difference in resonance width is not obvious variations in y1 and taking it as a constant, we can when S1  0 but is obvious when S1 is not very large. approximately rewrite Equation (25) as (2) If B is the primary, mmAA( + m B) can be very small. This makes S1 < 0 possible for the primary minor planet (see ⎡ ⎢ 427m 72my1 Section 3.5 for an example—624 Hektor). In this case, for the x¨1 =- - - ⎣ B 2 B same system, there is a qualitative difference between our Iz a¯ Iaz ¯ model and the conventional model—the resonance center is ⎛ ⎞ ⎤ 3 4my 17¶e A different. ++⎜ 1 ⎟ ⎥ 3 exsin .() 26 32 B ⎥ 3 1 ⎝ a¯¯Iz ⎠ e ¶y1 ⎦ 2a 3.2. The 3:2 Resonance For the 3:2 resonance, also neglecting other short-period In the units of this paper, a¯ = 1. Noticing that B ˙ ˙ -32 terms in Equation (13) and only keeping the resonance term, yppI1 =-BB2, = z qmB and qB = 32a¯ for the 3:2 we have resonance, the last term in the square bracket vanishes and Equation (26) becomes 1 m 2 A1 7A3 H =- +-p - elcos() 3¢- 2qB - g ¢ . L¢2 B B L¢6 L¢6 7A3 2 2Iz 2 x¨1 =- Se21sin x ,() 27 ()21 2 where Introducing the following transformation 2 4m 20maA ¯ xl123=¢-32qB -¢ gxlxg , =¢ , =¢ S =-= - 2 B 27 2227. Iz ()()mmabAB++BB yp12=-BBB2, yLp = ¢+ 3 2, yGp 3 = ¢- 2,() 22 – we transform Equation (21) into Similar to the case of 1:1 spin orbit resonance, depending on the mass ratio between mA and mB, the mutual distance a¯, and S 1 2m A the size of B, the sign of 2 can change. When S2 > 0, the H =- +-y 2 1 23()yy+ 2 I B 1 ()yy+ 3 6 resonance center is at x1 = 0. The physical interpretation is that 21 z 21 Bʼs long axis is pointing at A at the periapsis of the mutual 7A3 ( ) - excos1 .() 23 orbit, as shown in Figure 2 a . This is exactly the case for 23()yy+ 6 21 Mercury. When S2 < 0, the resonance center is at x1 =p. The physical interpretation is that Bʼs short axis is pointing at This is a 1-DOF system, with xy1, 1 as variables and yy23, as A at the periapsis of the mutual orbit, as shown in Figure 2(b). ( ) parameters. From Equation 23 , we have Similar to the analysis for the 1:1 resonance, when treating the – ⎧ spin orbit resonance for the primary in the binary minor planet 3 4my1 18A1 ⎪ x˙1 =++3 B 6 system, a qualitative difference between our model and the ()yy21+ 3 Iz ()yy21+ 3 ⎪ ⎡ ⎤ conventional model may appear. ⎨ +-7A3 ex⎢ 18 1 ¶e ⎥ cos ,24() 23()yy+ 6 ⎣ ()yy+ 3 e¶y ⎦ 1 ⎪ 21 21 1 3.3. The 1:2 Resonance ⎪ 7A3 yex˙1 =- 6 sin 1 ⎩ 23()yy21+ For the 1:2 resonance, also neglecting other short-period terms in Equation (13) and only keeping the resonance term,

5 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin

Figure 3. Time history curves of the resonance angles for the (a) 1:1 resonance; (b) 3:2 resonance, and (c) 1:2 resonance. With different a¯ values, the resonance centers change. we have In the units of this paper, a¯ = 1. Noticing that y1 = B ˙ -32 -ppI2, = qmB and q˙B = a¯ 2 for the 1:2 resonance, 1 m 2 A1 A3 BBz H =- +-p + elcos()¢- 2qB + g ¢ . the last term in the square bracket vanishes and Equation (33) 2L¢2 2I B B L¢6 2L¢6 z becomes ()28 xASex¨2=- sin,() 34 Introducing the following transformation 1331 where xl123=¢-2,qB + gx ¢ =¢ lx , = g ¢ 2 yp12=-BB2, yLpyGp = ¢+ 2, 3 = ¢+ B 2,() 29 3 m 3 5ma¯ S =- =- A 3 B 22. ( ) 4 I 4 ()()mmabAB++ noticing that L¢=yy21 + , we transform Equation 28 into z BB 1 2m A m m H =- +-y 2 1 Again, depending on the mass ratio between A and B, the 2 B 1 6 S 2()yy21+ Iz ()yy21+ mutual distance a¯, and the size of B, the sign of 3 can change. A When S3 > 0, the resonance center is at x1 = 0. The physical + 3 excos .() 30 6 1 meaning is that Bʼs long axis is pointing at A at the periapsis of 2()yy21+ the mutual orbit. When S3 < 0, the resonance center is at ( ) The same as Equation 23 , this is also a 1-degree of freedom x1 =p. The physical meaning is that Bʼs short axis is ( ) DOF system, with xy1, 1 as variables and yy23, as parameters. pointing at A at the periapsis of the mutual orbit. The From Equation (30), we have illustrative maps are the same as those in Figure 2. Also similar ⎧ to the analysis for the 1:1 resonance, when treating the spin– 3 4my1 6A1 ⎪ x˙1 =++3 B 6 orbit resonance for the primary in the binary minor planet ()yy21+ Iz ()yy21+ ⎪ ⎡ ⎤ system, a qualitative difference between our model and the ⎨ A3 61¶e +-6 ex⎣⎢ ⎦⎥ cos 1,31() conventional model may appear. ⎪ 2()yy21+ ()yy21+ e¶y 1

⎪ A3 yex˙1 =- 6 sin 1 3.4. Numerical Simulation ⎩ 2()yy21+ In this subsection, we numerically verify the analytical from which we have criteria given by Equations (20), (27), and (34).Wefix Aʼs ⎡ radius as aA = 0.1 km and Bʼs shape parameters as ⎢ 49m 24my1 x¨1 =- - a :b :c = 1.0 km:0.6 km:0.3 km; thus, we have a fixed value ⎣ B 4 B B B B Iz ()yy21+ Iyz ()21+ y of m = 0.005494337779677. In this simulation, A is the ⎛ ⎞ ⎤ satellite and B is the primary, and we study the spin–orbit 1 4my 1 ¶e A + ⎜ + 1 ⎟ ⎥ 3 exsin . resonance of the primary. We vary the mutual distance a¯ to ⎝ 3 B ⎠ ⎥ 6 1 B ()yy21+ Iz e ¶+y1 ⎦ 2 ()yy21 adjust the value of Iz . Figures 3(a) shows the time history curve ()32 of the resonance angle f1 =¢-l qB for a¯ =+11(aaAB) and a¯ =+12(aaAB). Figure 3(b) shows the time history curve of fi Yet again, if we only consider the rst term in the square the resonance angle f2 =¢--32lgqB for a¯ =+15(aaAB) bracket of Equation (32), we get the conventional model for the and a¯ =+17(aaAB). Figure 3(c) shows the time history curve 2:1 spin–orbit resonance. With similar treatment to the above of the resonance angle f3 =¢-lg2qB + for a¯ =+7()aaAB treatment, we have and a¯ =+5()aaAB. The values of Si ()i = 1, 2, 3 for these tests are given in Table 1. From these simulations, one can see ⎡ ⎤ ⎛ ⎞ that the numerical results agree with the analysis by ⎢ 49m 24mmy1 ⎜ 1 4 y1 ⎟ 1 ¶e ⎥ x¨1 =-- + + Equations (20), (27), and (34). The initial conditions of ⎣⎢ I B a¯2 IaB ¯ ⎝ a¯32 I B ⎠ e ¶y ⎦⎥ z z z 1 r,,,qqr˙ ˙ for all of the panels in Figure 3 are A3 ´ exsin1 .() 33 ˙ 2a¯3 r0000====1,qq 0,r˙ 0, 1,

6 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin

Table 1 ˙ Initial Conditions of qB, qB for Figure 3 and the Values of Si (i = 1, 2, 3) Figure 3a 3a 3b 3b 3c 3c a¯ 11(aaAB+ ) 12(aaAB+ ) 15(aaAB+ ) 17(aaAB+ ) 5()aaAB+ 7(aaAB+ ) qB0 p 2 0 p 2 00p 2 ˙ qB0 1 1 1.5 1.5 0.5 0.5 S1,2,3 S1 =-0.026 S1=0.539 S2 =-1.220 S2=0.353 S3=0.136 S3 =-0.454

Table 2 S1 Value for the Primary in 14 Doubly Synchronous Systems 90 Antiope 617 Patroclus 809 Lundia 854 Frostia 1089 Tama 1139 Atami 1313 Berna 15.11 195.73 16.31 70.86 18.54 27.48 44.46

2478 Tokia 4492 Debussy 4951 Iwamoto 7369 Gavrilin 624 Hektor 3169 Ostro 69230 Hermes

33.24 43.71 278.99 44.61 −2.92 11.63 25.56

and the initial conditions of qBB, q˙ for each curve are given in appears as an equilibrium point at Bʼs short axis or long axis. Table 1. The equations of motion used for numerical Studying the stability of the equilibrium point is identical to integration are Equations (6) and (7) but without the 4OD studying the stability of the 1:1 spin–orbit resonance (Scheeres ) ( terms, i.e., B ~=B 0. Moreover, as A is a sphere, A = 0. 1994; Feng & Hou 2017 . Using the same physical model i.e., 17 2 + ) The first equation of Equation (7) is not used and the rotational an ellipse a sphere but with the closed-form of the mutual gravity, studies by Gabern et al. (2006) show that the stability motion of A is decoupled from the system. of the equilibrium point at the ellipsoid’s short axis and long axis changes with the mass ratio of the system and the mutual 3.5. Remarks distance between the two bodies. Using the actual asteroid fi fi 4179 Toutatis and assuming ctitious natural satellites around The analysis above is strict and demonstrated by arti cially it, Scheeres (2006) studied the stability of these fictitious designed numerical simulations. It would be more convincing fi fi – satellites in the body- xed frame of Toutatis and arrived at if we could nd examples of such spin orbit resonances in the similar conclusions as those given by Gabern et al. (2006). real world. However, due to the limitation of observations, Conclusions in these works agree with the conclusion of our currently we can only identify whether the binary minor planet study, which uses the Hamiltonian approach to treat the 1:1 system is in synchronous rotation or asynchronous rotation spin–orbit resonance. Extending the study from equilibrium (Pravec et al. 2015). As a result, currently we can only discuss fi – points to special periodic orbits in the body- xed frame of B, the 1:1 spin orbit resonance. Also, as explained in Section 3.1, we are also able to study other spin–orbit resonances (Xin & generally only for the primary minor planet, we can expect Hou 2017). qualitatively different results between our improved model and As far as the authors know, except for the 1:1 spin–orbit the conventional model. For the current database of binary resonance, no other spin–orbit resonances or spin–spin and minor planet systems (http://www.asu.cas.cz/asteroid/ – – ( ) ) fi – spin orbit spin resonances see following sections in the real binastdata.htm , we can con rm the 1:1 spin orbit resonance binary minor planet systems are reported until now. Never- of the primary only for doubly synchronous systems where – theless, with such a large population and various physical both bodies are in 1:1 spin orbit resonance. This is also mechanisms that can change the status of the binary systems understandable, because the satellite usually enters the (for example, the YORP effect can change the rotation states of synchronous state earlier than the primary. For the 14 doubly a single minor planet (Bottke et al. 2006), and the BYORP synchronous binary minor planet systems found (Pravec (Ć ) ) effect uk & Burns 2005; MacMahon & Scheeres 2010 and et al. 2015; Shang et al. 2015 , we can only approximately tidal dissipation can change the mutual orbit (Murray & determine the size of the two components but not their shapes. Dermott 1999)), it is possible that binary minor planet systems That means, even if we were able to confirm that both bodies ( ) – – ’ at least temporarily captured in the spin orbit resonances are in 1:1 spin orbit resonance in these systems, we can t other than the 1:1 one exist. observe and can only speculate the relative configuration. Take B as the primary in these systems. Assume 4. Spin–Spin Resonance bBBB»»=bcD p2 where Dp is the diameter of the primary, and assume both bodies have same density. Using the data In order to consider spin–spin resonance, we use the listed in Table 1 of Shang et al. (2015), Table 2 shows the S1 ellipsoid+ellipsoid model depicted in Figure 1(a). Truncated value for the primary in these doubly synchronous systems. at the 4OD terms of the mutual potential, EOMs for this system Judging from this table, it seems that only in the binary system are already given by Equations (6) and (7). In the mutual 624 Hektor that the primary has its short axis pointing at its potential given by Equations (2) and (3), the terms satellite. Because the value of Dp for this system is only a poor B2 cos() 2qq- 2 A , B4 cos() 2qq- 2 B , B6 cos() 2qqAB- 2 , and estimate (Shang et al. 2015), this configuration still remains to B7 cos() 4qq-- 2AB 2 qare direct interaction terms between be confirmed. the non-spherical parts of the two bodies. Generally, the In Figure 1(b),ifB is in exact 1:1 spin–orbit resonance, the coefficients B2 and B4 are much smaller compared with A2 and trajectory of A, when viewed in the body-fixed frame of B, A3, which share the same resonance angles, so they can be

7 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin neglected. Only the two terms B6 cos() 2qqAB- 2 and from which we solve B7 cos() 4qq-- 2AB 2 qshould be considered. These two direct interaction terms are also listed in Batygin & Morbidelli WhAlgfAl11=-sin() ¢- ¢ -1 2 sin ( 2 ¢- 2qA ) ( ) 2015 where a different physical model is used. In this section, -¢-+¢-¢--¢fAsin()() l 2qqAA g fA sin 3 l 2 g 2 2 3 2 , we concentrate on the resonance angle qAB- q , and call it -¢--¢-+¢gAsin() 2 l 2qq gA sin ( l 2 g ) spin–spin resonance because only the rotations of two bodies 1 3 BB2 3 are involved in this resonance. In the next section, we consider -¢--¢gA3 3 sin() 3 l 2qB g the resonance angle 4qq--22A qB, and call it spin–orbit– ()39 spin resonance not only because it involves two rotating bodies but also because their mutual orbits are involved in the where resonance. -1 Before we proceed to more details on spin–spin resonance, 3e 1 1 m hf==666,,1 -A pA LLL¢¢¢2 ()Iz we must exclude the situation where both angles q - qA and -1 q - qB are in spin–orbit resonance, i.e., a double-synchronous e 1 2m fp2 =-66 - A A , state for the binary minor planet system. In this case, 2LL¢¢()Iz q˙˙AB-=q 0 is just a simple mathematical outcome of the -1 -1 ˙˙ ˙˙ 7e 3 2mm1 1 two spin–orbit resonances q -=qA 0 and q -=qB 0, but not fpgp3 =-66A A ,,1 =- 66 B B 2LL¢¢()Iz 2LL¢¢ ()Iz the spin–spin resonance with physical interpretations we are -1 -1 going to deal with here. e 1 2mm7e 3 2 gpgp2 =-66 -B B ,.3 = 66 - B B Truncated at the fourth order of the mutual potential, the 2LL¢¢()Iz 2LL¢¢ ()Iz Hamiltonian of the system is Normalizing the Hamiltonian to the second order, we have 1 HH=++01 H H 2,35() KH22= +á HWH 111,,, ñ+á +á HW 01 ñ 2! WHW102ñ+á,. ñ () 40 where As we only deal with 1–1 spin resonance, we only need to keep the term with the angle 2qq- 2 and remove all other short- 1 mm22 AB H0 =- ++p p , W 2L¢2 22I A A I B B period terms by a proper generating function 2. With some z z mathematical efforts, only keeping the resonance term, we have V˜˜2 V4 H1 =-,2.H2 =- () 36 ⎡ ¶g ¶f r 3 r 5 Kgf=+-+⎢ 6AA23()AA23 11 2 ⎣ L¢¢6 11 L 6 ()¶¢L ¶¢L H corresponds to the unperturbed system where the two AAe ¶g ¶f 7AAe ¶g ¶f 0 ++++23 22 23 33 bodies’s rotations are decoupled from their mutual orbit. 4L¢6 ()()¶¢G ¶¢G 4L¢6 ¶¢G ¶¢G fi e H ⎤ Truncated at the rst order of the orbit eccentricity , 1 is ++++AA23¶e ()gf7AA23¶e () gf 4L¢6 ¶¢G 22 4L¢6 ¶¢G 33⎦ A 3A Helg=-1 -1 cos() ¢- ¢ ´-cos() 2qqAB 2 . () 41 1 L¢¢6 L 6 -¢-+¢-+¢A2 cos() 2llg 2qqA2 e cos ( 2 ) Introducing L¢¢6 A L 6 2 A A2 7e ˙ ˙ ˙ ˙ -¢--¢cos() 3lg 2q w12=-=-22,22,( qq¯ ˙AA)( qw¯ )( w =-=- qq¯ ˙ BB)( qw¯ ) L¢6 2 A A3 A3 e -¢-+¢-+¢6 cos() 2llg 2qqB 6 cos ( 2 B ) and noticing that a¯ = 1 and the canonical transformations L¢¢L 2 ( ) ( ) A ˙˙B Equations 9 , 12 , and pI==z qmA , pIz qmB , we have -¢--¢A3 7e cos() 3lg 2q , () 37 A B 6 2 B L¢ ¶f fg==1 ,,1611 =()- w1 , 1 ww1 ¶¢L 2 where 12 w1 2 ¶g1 61()-¶w2 e 1- e e 1 ==-=-2 ,,,f2 ¶¢L w ¶¢G e 2 w1 - 1 ⎛ GL¢+ ¢⎞2 ¶e 1 -¢e2 G 2 e =-1,⎜ ⎟ = , 7ee1 1 7 1 ⎝ ⎠ fg3 ==-=,,,2 g3 L¢ ¶¢L e L¢ 2 www122+-+1 2 1 2 1 ¶f 2 ¶f 2 ¶e 1 - e2 2 ==-1 - e ,,3 71- e =- .38() ¶¢G 21e()ww1 - ¶¢G 21e ()1 + ¶¢G eL¢ ¶g 2 ¶g 2 2 ==-1 - e ,.423 71- e () ¶¢G 21e()ww2 - ¶¢G 21e ()1 + The generating function W1 satisfies (where á**ñ, indicates the Poisson bracket) Substituting these relations into Equation (41), noticing that

www12== ¶H01¶W KH11=+áñ=- HWH 011, ¶¢L ¶¢l for the 1:1 spin–spin resonance, we can reduce Equation (41) in ¶H01¶WH¶ 01¶WA 1 a much simpler form as - - =- , 6 ¶p ¶qqA ¶p ¶ B L¢ A B KS24=-cos() 2qqAB 2 , () 43

8 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin

Figure 4. (a): The curve of S4 with respect to wA. When wA is smaller than 0.5 (or between 1.5 and 3.25), S4 > 0 and the resonance center of the resonance qA - qB is around p 2. When wA is between 0.5 and 1.0 (or between 1.0 and 1.5, or larger than 3.25), S4 < 0 and the resonance center of the resonance qA - qB is around 0. (b)–(f): Time history curves of the resonance angle qA - qB lasting 1000 orbital periods, for different values of wA. where Table 3 Initial Conditions for Figures 4–5 and Values of S45, S ()24ww-- 132 12 S423= AA- 2. B6 Figure 4b 4c 4d 4e 4f 5b 5c 5d ()1 - ww22 0 qB p 2 0.02 0.02 p 2 0.02 p 2 0.02 0.02 As a result, the Hamiltonian for this resonance can be written as wA 0.10 0.77 1.35 2.15 3.95 1.39 1.76 3.15 wB 0.10 0.77 1.35 2.15 3.95 0.61 0.24 −1.15 1 1 m KK=++ K K =- + p2 01 2 2 A A 2! 2L¢ Iz S (= ) ( )–( ) m A 1 the curve of 4 with respect to wA wB . Figures 4 b f shows +-p2 1 +-S cos()() 2qq 2 . 44 B B 6 4 AB the time history curve of the resonance angle qAB- q for 2Iz L¢ 2 different values of wA. The initial conditions for these maps are fi ( ) ( ) Obviously, the sign of S determines the resonance center of given by the rst order solution to Equations 6 and 7 . 4 Neglecting the details (Hou & Xin 2017), the solution are the 1:1 spin–spin resonance. If S < 0, the resonance center is 4 directly given here. qAB-=q 0.IfS4 > 0, the resonance center is q -=qp2. From the expression of S , for fixed minor ⎧ ¯¯ ¯¯ AB 4 ⎪r =+1cos22cos22aa01 +( qq -AB)( + a 2 qq - ) planets A and B, we know that its sign can be positive or 0 ⎪qqq=+¯ + b12sin( 2 q¯¯ - 2 qAB)( + b sin 2 q¯¯ - 2 q) , negative, depending on the rotation speed of A and B. Because ⎪ qw¯ =+D()1,t we are studying 1:1 spin–spin resonance, w = w . One point ⎨ q BA ¯¯ ()45 ( ) S ⎪ ¯¯0 2sin22mqqA2 ( - A) to note here is that the second term the -2B6 term in 4 comes qqqAA=+A -2 A , qwAA =t ⎪ w1 Iz from the direct interaction between the non-spherical parts of A ⎪ ¯¯ ⎪ ¯¯0 2sin22mqqA3 ( - B) and B, and the first term in S4 comes from the indirect term, qqqBB=+-B 2 B , qwBB =t ⎩ w2Iz which disappears if we assume an invariant mutual orbit. From 0 00 the expression of A2, A3, and B6 (see Equation (3)), we know where q ,,qqABare initial phase angles, and that the indirect term is of the same strength as the direct term ¯˙ and should be taken into consideration. If we only consider the D=wwqwwwqq3,A11 = 2( -AA)( = 21 +D- ) , ˙ 32A1 -Dwq direct -2B6 term in S4, it leads us to the conclusion that the wqw=-221,(¯ )( =+D- wwa ) = , 20BBq 3 resonance center is always at qAB-=q 0, as in the previous 2 ()43- w1 23()ww1 --1 3 models of Batygin & Morbidelli (2015), Nadoushan & ab1 ==2 AA21,,2 2 2 ww1()1 - 1 ww1 ()1 - 1 Assadian (2016). 2 ()43- w2 23()ww2 --2 3 ab1 ==2 AA31,.2 2 3 Numerical simulations are given to support the above ww2()1 - 2 ww2()1 - 2 analysis. We consider two ellipsoids with same size and shape – 0 0 a*:b*:c* = 1.0 km:0.8 km:0.6 km. The semimajor axis a¯ of the For the 1:1 spin spin resonance, by setting q ==0,q A 0, unperturbed circular orbit is 20()aaAB+ . Figure 4(a) shows and using the values given in Table 3, the initial conditions at

9 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin

Figure 5. (a): The curve of S5 with respect to wA. When wA is smaller than 0.5 (or larger than 1.5), S5 < 0 and the resonance center of the resonance 2l¢-qqA - B is around 0. When wA is between 0.5 and 1.0 (or between 1.0 and 1.5), S5 > 0 and the resonance center of the resonance 2l¢-qqA - B is around p 2. (b)–(d): Time history curves of the resonance angle 2l¢-qqA - B lasting 1000 orbital periods, for different values of wA. t=0 for the orbits in Figure 4 can be calculated by Substituting Equation (42) in, we have Equation (45). From these figures, we know that the resonance KS25=¢--cos()() 4 l 2qqAB 2 , 47 center of 1:1 spin–spin resonance does change with respect to different spin statuses of the two bodies, and the numerical where simulations agree with the analytical criterion of Equation (44). ⎡ ⎤ 512w2 - 512w2 - Although the initial conditions of the orbits in Figure 4 are S = AA⎢ 1 + 2 ⎥ - 2. B 523⎣ 2 2 2 2 ⎦ 7 provided by Equation (45), which is the first order solution to 21ww1 ()1 - 21ww2 ()- Equations (6) and (7), these orbits are numerically integrated by As a result, the Hamiltonian for this resonance can be written as Equations (6) and (7). 1 1 m KK=++ K K =- + p2 – – 01 2 2 A A 5. Spin Orbit Spin Resonance 2! 2L¢ Iz – – m A 1 With similar treatment as the spin spin orbit resonance, but +-p2 1 +¢--Slcos()() 4 2qq 2 . 48 B B 6 5 AB only keeping the term with the resonance angle 2Iz L¢ 2 4l¢-22qqA - B, we have Obviously, the sign of S5 determines the resonance center of ⎡ 6AA AA ¶g f – – Kgf=+++⎢ 23()23 11 the 1:2:1 spin orbit spin resonance. If S5 < 0, the resonance 2 ⎣ L¢7 11 L¢6 ()¶¢L ¶¢L center is 2l¢-qqAB - =0.IfS5 > 0, the resonance center is AAe¶g 23 3 22l¢-qqAB - = p. According to the expression of S5,we - 6 4L¢ ¶¢G know that its sign can be positive or negative, depending on the ¶g ¶f ¶f ---7AAe23 2 AAe23 3 7AAe23 2 – 44L¢6 ¶¢G L¢6 ¶¢G 4L¢6 ¶¢G rotation speed of A and B. As we are studying the 1:2:1 spin – ˙¯ AA23¶e 7AA23¶e AA23¶e orbit spin resonance, the restriction 2qw--=AB w 0 holds. --6 ggf3 6 2 -6 3 4L¢ ¶¢G 4L¢ ¶¢G 4L¢ ¶¢G For a fixed q˙¯, if we change the value of w , w is changed ⎤ A B +¢--7AA23¶e flcos()() 4 2qq 2 . 46 accordingly. The second term (the -2B term) in S comes 4L¢6 ¶¢G 2 ⎦ AB 7 5 from the direct interaction between the non-spherical parts of A

10 The Astronomical Journal, 154:257 (12pp), 2017 December Hou & Xin and B, and the first term in S5 comes from the indirect term, simulations are performed. The validity of our model is which disappears if we assume an invariant mutual orbit. From demonstrated by the agreement between the analytical the expression of A2, A3, and B7 (see Equation (3)), we know analysis and numerical simulations. Due to the limitation of that the indirect term is of the same strength as the direct term observations, currently we do not have any real examples that – – – and should be taken into consideration. If we only consider the are in spin spin or spin orbit spin resonances. However, direct -2B term in S , it leads us to the wrong conclusion that considering the large population of the binary minor planet 7 5 systems, and the fact that they are small and can easily change the resonance center is always at 2l¢-qq - =0. AB their rotation status by tidal dissipation and thermal effects, Numerical simulations are given to support the above the spin–spin resonance and spin–orbit–spin resonances are analysis. We consider same two ellipsoids as those in expected to (at least temporarily) exist in the binary minor Section 4. The semimajor axis a of the unperturbed circular ¯ planet systems. orbit is 20()aaAB+ . Figure 5(a) shows the curve of S5 with respect to w . In this special case, for a specific value of w = v, A A X.Y.H. wishes to thank the support from National Natural ˙¯ the case is symmetric with respect to the case wA =-2q v. Science Foundation of China (11773017, 11703013, 11673072). ( )–( ) Figures 5 b d show the time history of 4l¢-22qqA - B for The authors greatly thank the anonymous reviewer for his/her different values of wA and wB. The initial conditions are also valuable comments that greatly improved this paper. ( ) 0 0 given by Equation 45 , by setting q ==0,q A 0 and using fi the values in Table 3. From these gures, we know that the ORCID iDs resonance center of the 4l¢-22qqA - B resonance does change with respect to different spin status of the two bodies, Xiaosheng Xin https://orcid.org/0000-0002-5806-3739 and the numerical simulations agree with Equation (48). One remark is that B is actually in retrograde rotation in Figure 5(d). References The equations of motion used for integration are Equations (6) and (7). Antognini, F., Biasco, L., & Chierchia, L. 2014, JNS, 24, 473 Balmino, G. 1994, CeMDA, 60, 331 Batygin, K., & Morbidelli, A. 2015, AJ, 810, 110 Beletskii, V. V. 1972, CeMDA, 6, 356 6. Discussion and Conclusion Bottke, W. F. B., Jr, Vokrouhlický, D., Rubincam, D. P., & Nesvorný, D. 2006, Compared with the conventional spin–orbit resonance AREPS, 34, 157 fi Celletti, A. 1990, ZaMP, 41, 174 model, which uses an invariant orbit, a modi ed model is Celletti, A., & Chierchia, L. 2000, CeMDA, 76, 229 proposed that considers the variations in the mutual orbit Celletti, A., & Chierchia, L. 2008, CeMDA, 101, 159 caused by rotational motions. This modification is unnecessary Compère, A., & Lamaître, A. 2014, CeMDA, 119, 313 when treating the spin–orbit resonance of a small satellite, as Correia, A. C. M., & Laskar, J. 2004, Natur, 429, 848 Correia, A. C. M., Leleu, A., Rambaux, N., & Robutel, P. 2015, A&A, the effects of its rotational motion on the mutual orbit are 580, L14 negligible. However, in a binary minor planet system where Ćuk, M., & Burns, J. A. 2005, Icar, 176, 418 the difference in sizes and masses between the primary and Fahnestock, E. G., & Scheeres, D. J. 2006, CeMDA, 96, 317 the satellite are not so large and the two bodies are close to each Fang, J., & Margot, J. L. 2012, AJ, 143, 24 other, it may be inappropriate to neglect the effects of the Feng, J.-L., & Hou, X.-Y. 2017, AJ, 154, 21 Gabern, F., Koon, W. S., Marsden, J. E., & Scheeres, D. J. 2006, SIADS, rotational motions on the mutual orbit. Thus, our model is more 5, 252 applicable. For the satellite, there is no qualitative difference Gkolias, I., Celletti, A., Efthymiopoulos, C., & Pucacco, G. 2016, MNRAS, between our model and the conventional model, but the 459, 1327 resonance width in two models is different. More importantly, Goldreich, P., & Peale, S. 1966, AJ, 71, 425 – Goldreich, P., & Sari, R. 2009, ApJ, 691, 54 if we study the spin orbit resonance of the primary, a Hou, X.-Y., Scheeres, D. J., & Xin, X. 2017, CeMDA, 127, 369 qualitative difference (i.e., the resonance center may be Hou, X.-Y., & Xin, X.-S. 2017, AsDyn,2,39 different) appears, depending on the mass ratio and the mutual Jacobson, S. A., & Scheeres, D. J. 2011, ApJL, 736, L19 distance between the two bodies and the size of the primary. Klavetter, J. J. 1989, BAAS, 21, 983 Using 1:1, 3:2, and 3:2 spin–orbit resonances as examples, Kouprianov, V. V., & Shevchenko, I. I. 2005, Icar, 176, 224 Lamaître, A., & D’Hoedt, S. 2006, CeMDA, 95, 213 simple analytical expressions of the resonance width are given, Lhotka, C. 2013, CeMDA, 115, 405 and numerical simulations are presented to demonstrate the MacMahon, J., & Scheeres, D. J. 2010, CeMDA, 106, 261 analysis. For 14 real binary asteroid systems that are in doubly Margot, J. L., Nolan, M. C., Benner, L. A., et al. 2002, Sci, 296, 1445 synchronous rotation, we apply our theory to the primary. Margot, J. L., Pravec, P., & Taylor, P. 2015, in Asteroids IV, ed. P. Michel, F. E. DeMeo, & W. F. Bottke (Tucson, AZ: Univ. Arizona Press), 355 Thirteen of them have their long axis pointing at the satellite, Merline, W. J., Weidenschilling, S. L., Durda, D. D., et al. 2002, in Asteroids while only one of them (624 Hektor) has its short axis pointing III, ed. W. F. Bottke et al. (Tucson, AZ: Univ. Arizona Press), 289 at its satellite. Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics (Cambridge: Compared with the models in Batygin & Morbidelli (2015) Cambridge Univ. Press) and Nadoushan & Assadian (2016), which use an invariant Nadoushan, M. J., & Assadian, N. 2016, Icar, 265, 175 fi – Naidu, S. P., & Margto, J. L. 2015, AJ, 149, 80 orbit, a modi ed model for the spin spin resonance and the Noyelles, B., Frouard, J., Makarov, V. V., & Efroimsky, M. 2013, Icar, 241, 26 spin–orbit–spin resonance using a varying mutual orbit is Peale, S. J. 1969, AJ, 74, 483 proposed. Depending on the rotation status of the two bodies, Pravec, P., Scheirich, P., Kušnirák, P., et al. 2015, Icar, 267, 267 the resonance center changes. This is a qualitative difference Quillen, A. C., Nichols-Fleming, F., Chen, Y.-Y., & Noyelles, B. 2017, Icar, – 293, 94 between our model and previous models. Using 1:1 spin spin Rambaux, N., & Bois, E. 2004, A&A, 413, 381 resonance and 1:2:1 spin–orbit–spin resonance as examples, Richardson, D. C., & Walsh, K. J. 2006, AREPS, 34, 47 simple analytical expressions are given, and numerical Scheeres, D. J. 1994, Icar, 110, 225

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Scheeres, D. J. 2006, CeMDA, 94, 317 Showalter, M. R., & Hamilton, D. P. 2015, Natur, 522, 45 Scheeres, D. J. 2007, Icar, 188, 430 Weaver, H. A., Buie, M. W., & Buratti, B. J. 2016, Sci, 351, aae0030 Scheeres, D. J., Fahnestock, E. G., Ostro, S. J., et al. 2006, Sci, 314, 1280 Wisdom, J., Peale, S. J., & Mignard, F. 1984, Icar, 58, 137 Shang, H.-B., Wu, X.-Y., & Cui, P. Y. 2015, Ap&SS, 355, 69 Xin, X.-S., & Hou, X.-Y. 2017, MNRAS, submitted

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