Tilting Styx and Nix but Not Uranus with a Spin-Precession-Mean-Motion Resonance

MNRAS 000, 000–000 (0000) Preprint 2 June 2017 Compiled using MNRAS LATEX style ﬁle v3.0

Tilting Styx and Nix but not Uranus with a Spin-Precession-Mean-motion resonance

Alice C. Quillen1, & Yuan-Yuan Chen1, and? order and author list to be determined 1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 USA

2 June 2017

ABSTRACT A Hamiltonian model is constructed for the spin axis of a planet perturbed by a nearby planet and both are in orbit about star. We expand the perturbation to first order in inclination and eccentricity, finding terms describing spin resonances involving the spin precession rate and the two planetary mean motions. Convergent planetary migration allows the spinning planet to be captured into spin resonance. With the spinning body near zero initial obliquity, the spin resonance can lift its obliquity to near 90 degrees if the resonance to first order in inclination dominates, whereas it could be lifted near 180 degrees if zeroth order or first order terms in eccentricity dominate. Past capture into such a spin resonance could give an alternative non-collisional scenario accounting for Uranus’s high obliquity. However we find that the planet migration rate must be so slow that this mechanism cannot be responsible for Uranus’s high obliquity. Our Hamiltonian model explains how Styx and Nix can be tilted to high obliquity via outward migration of Charon, a phenomenon previously seen in numerical simulations. Keywords: Planets and satellites: dynamical evolution Charon’s minor satellites, showed another type of spin res- and stability, celestial mechanics, Planets and satellites: in- onance (Quillen et al. 2017). We found that a commensura- dividual: Styx, Planets and satellites: individual: Nix bility involving a mean motion resonance between Charon To do: Drift within MMR. Why is the spin resonance and a minor satellite and the satellite’s spin precession rate drifting with evolution in MMR? could influence its obliquity (Quillen et al. 2017). For satellite Styx, near a 3:1 mean motion resonance with Charon, we saw obliquity variations when the angles 1 INTRODUCTION φs1 = 3λStyx − λCharon − φStyx − ΩStyx

Resonances involving planet or satellite spin can cause φs2 = 3λStyx − λCharon − 2φStyx (1) chaotic tumbling, prevent a body from tidally despinning or aﬀect obliquity. Mercury was captured into a spin-orbit where librating about constant values rather than circulat- resonance, with spin a half integer multiple of its mean mo- ing. Here λStyx and λCharon are the mean longitudes of Styx tion, (e.g., Goldreich & Peale 1966; Noyelles et al. 2014) and Charon and ΩStyx is the longitude of the ascending node whereas Hyperion is chaotically tumbling due to spin-orbit of Styx, and orbital elements are measured with respect to resonance overlap (Wisdom et al. 1984). Secular spin-orbit Pluto or the center of mass of the Pluto/Charon binary, not resonances occur when the period of precession of the spin the Sun. The precession angle φStyx describes the orientation axis of a planet is commensurate with one of the periods of Styx’s spin axis. The New Horizons Mission found that in the secular variation of the orbit (Ward 1974). These Pluto and Charon’s minor satellites, Styx, Nix, Kerberos cause chaotic obliquity variations in Mars (Ward 1973, 1974; and Hydra have not tidally spun down to near synchronous Laskar & Robutel 1993; Touma & Wisdom 1993). Capture rotation (Weaver et al. 2016). Quillen et al.(2017) proposed into the secular spin resonance connected to the vertical that a spin resonance involving a mean motion resonance be- secular eigenfrequency associated with Neptune may have tween Charon and a minor satellite and the satellite’s spin tilted Saturn’s obliquity to its current value of 26.7◦ (Ward precession rate, when drifting due to an outwards migrating & Hamilton 2004; Hamilton & Ward 2004). For an object in Charon, might account for the high, near 90◦ obliquities dis- orbit about a binary, spin-binary resonance involves a com- covered by the New Horizons Mission (Weaver et al. 2016) mensurability between the binary mean motion, the orbital of all four of Pluto and Charon’s minor satellites, Styx, Nix, mean motion and the body’s spin (Correia et al. 2015). Kerberos and Hydra. Quillen et al.(2017) suggested that the Our numerical study of obliquity evolution of Pluto and minor satellite current obliquities need not be primordial.

c 0000 The Authors 2 Quillen et al.

orbit about a star, however we keep in mind that we can also consider a spinning satellite in orbit about a planet, asteroid Table 1. List of Symbols or Kuiper belt object. We assume that the planet is rapidly spinning about its principal inertial axis and this is known a semi-major axis as the gyroscopic approximation. The planet’s moments of e orbital eccentricity inertia are A, C with A > C and the planet’s spin angular I orbital inclination momentum is Ls = wCˆs where w is the spin angular rotation Ω longitude of the ascending node rate and ˆs is a unit vector. The planet’s spin axis satisfies ω argument of pericenter M mean anomaly dˆs = α (ˆs · nˆ)(ˆs × nˆ) (2) $ = ω + Ω longitude of pericenter dt s λ = M + $ mean longitude M∗ mass of central star (Colombo 1966), with time derivatives taken with respect to n mean motion the inertial frame. Here nˆ is a unit vector perpendicular to nˆ orbit normal unit vector the orbit plane, aligned with the orbital angular momentum ˆs spin direction unit vector vector. The precession rate C,A moments of inertia of oblate planet 2 w spin angular rotation rate of planet 3 (C − A) n αs = 3 , (3) αs spin precession rate 2 Cw (1 − e2) 2 α ratio of semi-major axes T torque vector where the orbital mean motion is n and the orbital eccen- r orbital radius tricity is e. φ spin precession angle MacCullagh’s formula gives the instantaneous torque on θ obliquity angle an oblate planet due to point mass M∗ s ≈ I/2 used in low inclination expansions GM R spinning planet’s equatorial radius ∗ ˆ ˆ ˆ ˆ T = 3(C − A) 3 (r · s)(r × s) (4) J2 second zonal gravity harmonic for the spinning planet r qs normalized quadrupole coefficient of satellite system where r = rˆr is the vector between the planet’s center of ls normalized angular momentum of satellite system mass and M∗. Equation2 can be derived using MacCullagh’s λC normalized moment of inertia about principal axis p canonical momentum variable, a function of obliquity formula for the instantaneous torque by averaging over the 1 R ∆ distance between perturber and spinning body orbit or computing hTi = P Tdt where the orbital period ψ, Ψ angles used in expansion of disturbing function is P = 2π/n, and assuming that the planet remains spin- (j) bs (α) Laplace coefficient ning nearly about its principal axis (Colombo 1966). Thus j resonance index equation2 is consistent with resonance strength in a Hamiltonian model dˆs 1 ν distance to resonance in a Hamiltonian model = hTi. (5) j j j dt Cw c0, cs, cs0 coefficients used to compute spin resonance strengths j j ce1, ce01 ” Due to secular perturbations arising from other plan- j j ets, the orbit normal nˆ can be a function of time (e.g., ce3, ce03 ” β ” Colombo 1966; Ward 1975). A time dependent equation2 has been used to study tidal evolution into Cassini states (Colombo 1966; Ward 1975) and obliquity evolution of Mars As we were lacking a model for this type of spin reso- (Ward 1973, 1979; Bills 1990) and Saturn (Ward & Hamilton nance strength, we were unable to assess its strength or even 2004). Phenomena discovered and explored include capture identify which type of resonant angle was likely to be most into spin-secular resonance states (Saturn; Ward & Hamil- important for each of Pluto and Charon’s minor satellites. ton 2004) and chaotic obliquity evolution (Mars; Touma & We address this issue here with the development of a Hamil- Wisdom 1993). tonian model for this spin resonance in section2. In section 3 we explore resonance capture by allowing the resonance in our Hamiltonian model to drift. In section4 we apply our 2.1 A Hamiltonian model for spin about a model to Pluto and Charon’s minor satellites. principal axis Uranus also has a high obliquity, of 98◦. Could a similar spin resonance have tilted Uranus during a previous time Using angular spherical coordinates φ, θ in an inertial refer- when Uranus was in or near a mean motion resonance with ence frame to specify the spin axis another giant planet? Using the Hamiltonian model of sec- ˆs = (cos φ sin θ, sin φ sin θ, cos θ), (6) tions2 and3 we answer this question in section5. To aid the reader a list of symbols is included in Table1. equation2 can be written as a Hamiltonian dynamical system with canonical momentum p, conjugate to the precession angle φ

2 SPIN EVOLUTION p = (1 − cos θ). (7)

A spinning oblate planet in orbit about a central mass M∗ The Hamiltonian that has spin axis, ˆs tilted with respect to the orbit plane α H (p, φ) = − s (ˆs · nˆ)2 (8) precesses. We refer to the spinning object as a planet in n 2

MNRAS 000, 000–000 (0000) 3 with ˆs a function of p, φ,(Goldreich & Toomre 1969; Peale to that directly exerted on the planet. Here θ is the planet’s 1974). Hamilton’s equations are obliquity and Ij the inclination of the j-th satellite with respect to the planet’s equatorial plane. Without satellites ∂Hn p˙ = − (9) q = l = 0 and with J = (C − A)/(mR2) equation 13 ∂φ s s 2 reduces to equation3. ∂H φ˙ = n (10) ∂p and are equivalent to the equations of motion for the spin 2.3 A Perturbed Hamiltonian Model axis in equation2. The angle φ ∈ [0, 2π] describes spin precession. With We consider a spinning planet in orbit about a star at zero orbit normal nˆ in the z direction, θ ∈ [0, π] is the planet’s orbital inclination. When averaged over the orbit period and obliquity. The canonical momentum p ∈ [0, 2] with p = 0 at over the longitude of the ascending node the Hamiltonian θ = 0. With the addition of a third angle describing body describing the planet’s spin (equation8) orientation about the spin axis, φ, θ are Euler angles. αs H (p, φ) = − p(2 − p) (16) Equation2 resembles equation4 but with r replacing nˆ. 0 2 Because r is independent of θ, φ the instantaneous spin vec- and giving spin precession rate tor (prior to averaging over the orbit) can also be described with a Hamiltonian system with φ˙ = −αs cos θ (17) 3 (C − A) GM H(p, φ) = (ˆs · ˆr)2, (11) with αs as given in equation 13. 2 Cw r3 We consider a Hamiltonian model in the form again with ˆs a function of p, φ. Hamilton’s equations are equations of motion equivalent to H(p, φ, t) = H0(p) + H1(p, φ, t) (18) dˆs 1 where H1 is a time dependent perturbation. = T. (12) dt Cw MacCullagh’s formula gives the torque on our spinning When averaged over the orbit period this equation is consis- planet due to the perturbing planet with mass mp. The tent with equation2. The Hamiltonian in equation 11 can torque dependent upon the radial vector between the two be averaged by writing r and ˆr in terms the mean anomaly planets and mean longitude and taking the average over one or both Gm T = 3(C − A) p ((r − r ) · ˆs) ((r − r ) × ˆs) , (19) of these angles. The gyroscopic approximation should be a 5 p p |r − rp| good one as long as the orbital period is much larger than the spin rotation period. For rigorous derivations with aver- where r is the radial vector to the spinning planet and rp aging see Bou´e& Laskar(2006). the radial vector to the perturbing planet. The associated Hamiltonian perturbation term is

2 2.2 Precessional Constant with Satellites 3(C − A) Gmp ((r − rp) · ˆs) H1(p, φ, t) = 5 (20) Cw |r − rp| 2 An spinning oblate planet locks its satellites to its equa- tor plane so that the system precesses as a unit (Goldreich and it is a time dependent perturbation. H1 H0 because 1965). The precession rate in equation3 can be modified to the mass of the perturbing planet is much less than the mass take into account the satellites with of a star; mp M∗. We describe the orbits in terms orbital elements 3 n2 J + q α = 2 s , (13) a, e, i, I, Ω,M which are semi-major axis, eccentricity, incli- s 2 w λ + l C s nation, longitude of the ascending node, argument of peri- (Ward 1975; French et al. 1993; Ward & Hamilton 2004) but center and mean anomaly, respectively. We also use the mean neglecting the orbital eccentricity. Here J2 is the coefficient longitude λ = Ω + ω + M and the longitude of pericenter of the second zonal gravity harmonic (from the quadrupole $ = Ω+ω. Our spinning and perturbing planets orbit a star moment) of the planet’s gravitational potential field and with mass M∗. 2 λC = C/mR is the planet’s moment of inertia about its We now depart from the notation used above with r, rp principal axis normalized to the product of planet mass and referring to spinning and perturbing planet, respectively. Or- the square of the planet’s equatorial radius. The parameter bital elements for the object with the larger semi-major axis 0 0 0 0 0 0 0 2 X mj aj nj will be referred to with a prime (a , e ,I , Ω ,M , λ , $ ) and l ≡ (14) s m R w those with the smaller semi-major axis without a prime. The j ratio of semi-major axes α ≡ a/a0. With radial vectors r is the angular momentum of the satellite system normalized and r0 for inner and outer orbiting mass, equation 20 for the 2 to mR w where mj , aj , nj are the masses, semi-major axes spinning planet becomes (for the orbit about the planet) and mean motions of each 2 0 2 satellite. The parameter (C − A) 2 mp as ((r − r ) · ˆs) H1(p, φ, t) = 3 n 0 5 (21) Cw M∗ |r − r | 2 2 1 X mj aj sin(θ − Ij ) qs ≡ (15) 2 m R sin θ where as is the semi-major axis of the spinning object. j a0 for external spinning body is the effective quadrupole coefficient of the satellite system a ≡ (22) s a for internal spinning body. with qs/J2 being the ratio of the solar torque on the satellites

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When the spinning body is external, we mean that it is per- and turbed by the mass mp that has orbit interior to our spinning (r · ˆs)2 sin2 θ ≈ [1 + cos(2(λ − φ))] + body. a2 2 Taking into account a satellite system around the spin- 2s sin θ cos θ [sin(2λ − Ω − φ) ning planet + sin(φ − Ω)] . (30) (C − A) (J + q ) → 2 s One can average over the orbital period by integrating this C (λ + l ) C s square over 2π in mean longitude λ. The terms proportional (comparing equation3 with 13) and deﬁning to cos(2(λ − φ)) and sin(2λ − Ω − φ) average to zero. The secular term proportional to sin(φ − Ω) only disappears if ∆ ≡ |r − r0|, (23) an additional average over Ω or φ is performed. This leaves we can write equation 21 as only sin2 θ/2, consistent with the Hamiltonians in equations

03 8 and 16. m a 3 a 2 p s 0 ˆ The cross term in ((r − r0) · ˆs) H1(p, φ, t) = αs 0 5 (r − r ) · s . (24) M∗ a ∆ (r · ˆs)(r0 · ˆs) sin2 θ It is convenient to write time in terms of the preces- ≈ cos(λ + λ0 − 2φ) + cos(λ − λ0) aa0 2 sion constant with unitless τ = α t. The total Hamiltonian s + sin θ cos θs sin(λ + λ0 − Ω − φ) including perturbation (using equations 18, 16, 24) + sin(λ − λ0 − Ω + φ) 03 p a 2 H(p, φ, τ) = − (2 − p) + β (r − r0) · ˆs (25) + sin θ cos θs0 sin(λ + λ0 − Ω0 − φ) 2 ∆5 − sin(λ − λ0 − Ω0 + φ) . (31) with unitless coefficient 3 In celestial mechanics the disturbing function, arising mp as β ≡ 0 (26) from gravitational interactions between point masses, con- M∗ a tains primarily dependent on the ratio mp/M∗ of perturbing 1 1 2 02 0 − 1 = = (r + r − 2rr cos ψ) 2 planet and stellar masses. ∆ |r − r0| with cos ψ ≡ (ˆr · rˆ0). As is common in expansions of the 2.4 Evaluating the perturbation term in the disturbing function we define Hamiltonian to first order in inclination Ψ ≡ cos ψ − cos($ + f − $0 − f 0) (32)

From the Hamiltonian in equation 25 we evaluate − 1 2 02 0 0 0 2 ∆0 ≡ r + r − 2rr cos($ + f − $ − f ) (33) 03 a 2 and (r − r0) · ˆs , giving ∆5 h i− 1 keeping terms that are zero-th order in orbital eccentricity ∆−1 = r2 + r02 − 2rr0(cos($ + f − $0 − f 0) + Ψ) 2 and first order in inclination. With radial vector r = (x, y, z), (34) in terms of orbital elements (see page 235 by Murray & Dermott 1999). To first order in  cos Ω cos(ω + f) − sin Ω sin(ω + f) cos I  Ψ, r = r  sin Ω cos(ω + f) + cos Ω sin(ω + f) cos I  , (27) −1 −1 0 −3 sin(ω + f) sin I ∆ ≈ ∆0 + rr ∆0 Ψ. (35) and likewise for the other mass at r0 using primed orbital Likewise to first order in Ψ, −5 −5 0 −7 elements. To zero-th order in eccentricity (or for e = 0) we ∆ ≈ ∆0 + 5rr ∆0 Ψ. (36) can replace the true anomaly f with the mean anomaly M Our Hamiltonian term depends on ∆−5 rather than the in- in the above expression. As is customary at low inclination verse of ∆ as is true for the disturbing function. To second (Murray & Dermott 1999) we let order in inclinations 2 2 cos I ≈ 1 − I /2 ≈ 1 − 2s Ψ ≈ s2 cos(λ + λ0 − 2Ω) − cos(λ − λ0) sin I ≈ I ≈ 2s. + ss0 cos(λ − λ0 − Ω + Ω0) − cos(λ + λ0 − Ω − Ω0) To zero-th order in e but first order in s equation 27 can be + s02 cos(λ + λ0 − 2Ω0) − cos(λ − λ0) (37) written as (see Murray & Dermott 1999 page 240). As Ψ lacks zeroth x x ≈ ≈ cos(ω + Ω + M) ≈ cos λ and first order terms in inclination, we need only take the r a y y first term in equation 36 to carry out an expansion to first ≈ ≈ sin(ω + Ω + M) ≈ sin λ r a order in inclination. Expanding ∆0 z z ∞ −(2i+1) 1 X 0 0 ≈ ≈ 2s sin(ω + M) ≈ 2s sin(λ − Ω). (28) ∆ = cos[j($ + f − $ − f )]× r a 0 2 We compute the dot product of r with the spin vector ˆs as j=−∞ " ∞ l l−k # given in equation6 to first order in inclination variable s X 1 X l r k r0 − 1 − 1 Ai,j,k,l−k r · ˆs l! k a a0 ≈ sin θ cos(λ − φ) + cos θ(2s) sin(λ − Ω), (29) l=0 k=0 a (38)

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(see equation 6.65 by Murray & Dermott 1999) with coeffi- than the internal one; n0 < n recalling that n = λ˙ . The slow cients arguments must be those containing λ − (j + 2)λ0 and so ∂m+n are associated with second order mean motion resonances. A (α) ≡ ama0n a0−(2i+1)b(j) (α) , (39) i,j,m,n m 0n i+ 1 Retaining only those three arguments in equation 43 for a ∂a ∂a 2 single j and using equations 42 and 43 we can write a near (j) where bs (α) is a Laplace coefficient dependent on the ratio resonance Hamiltonian (equation 25) as of semi-major axes α = a/a0 < 1. j:j+2 p 0 H(p, φ,τ) = − (2 − p) To zero-th order in eccentricities (e = e = 0) we need 2 only evaluate j 0 + βc0p(2 − p) cos(jλ − (j + 2)λ + 2φ) 1 (j) p A2,j,0,0(α) = b (α), (40) + (1 − p) p(2 − p)× a05 5/2 h 0 j 0 giving to first order in inclination (and with e = e = 0) βcss sin(jλ − (j + 2)λ + Ω + φ)

∞ j 0 0 0 i −5 −5 1 X 0 (j) + βc 0 s sin(jλ − (j + 2)λ + Ω + φ) , (44) ∆ ≈ ∆ ≈ cos[j(λ − λ )]b (α). (41) s 0 2a05 5/2 j=−∞ where we have replaced θ with p using sin θ cos θ = (1 − p 2 Using equations 30, 31 and 41 to zero-th order in eccen- p) p(2 − p) and sin θ = p(2 − p). The unitless coefficients tricity and first order in inclination j 1 2 (j+2) (j) (j+1) c0(α) ≡ α b (α) + b (α) − 2αb (α) a03 4 5/2 5/2 5/2 ((r − r0) · ˆs)2 ≈ ∆5 j (j+1) 2 (j+2) cs(α) ≡ αb5/2 (α) − α b5/2 (α) ( 2 sin θ h 2 1 + α cj (α) ≡ αb(j+1)(α) − b(j) (α) . (45) 2 s0 5/2 5/2 + α2 cos(2(λ − φ)) + cos(2(λ0 − φ)) These coefficients are twice those in equation 43 because we have taken positive and negative j terms that give the same 0 0 i − 2α cos(λ + λ − 2φ) − 2α cos(λ − λ ) argument. We recall that time is in units of αs as defined h in equation 13 and the coefficient β depends on the mass + 2 sin θ cos θ × ratio mp/M∗ (equation 26). To aid in applications we have 2 2 computed these coefficients for j = 1 to 6 and their values sα sin(2λ − Ω − φ) + sα sin(φ − Ω) 0 0 0 0 0 are listed in Table2. + s sin(2λ − Ω − φ) + s sin(φ − Ω ) Our Hamiltonian (equation 44) contains terms that are − sα sin(λ + λ0 − Ω − φ) first order in orbital inclination. This is to be compared to − sα sin(λ − λ0 − Ω + φ) first order mean motion orbital resonances that lack first order terms (in s) in an expansion of the disturbing function − s0α sin(λ + λ0 − Ω0 − φ) and second order inclination mean motion resonances that ) 2 0 0 0 i by definition are proportional to s . Previous calculations of + s α sin(λ − λ − Ω + φ) spin perturbations have considered the role of secular fre- quencies on planet spin orientation by considering how the ∞ 1 X (j) 0 × b (α) cos(j(λ − λ )). (42) orbit variations affect the torque from the star. In contrast 2 5/2 j=−∞ here we have directly evaluated the torque from a nearby planet. The non-secular terms with arguments that are not multiples We could similarly consider how a nearby planet induces 0 of λ − λ can be rewritten in terms of a single cosine or sine perturbations on the orbit of our spinning planet and then of orbital elements and φ; expand the equation for the torque from the star taking into sin2 θ h account these perturbations. The orbit perturbations arising cos(jλ − (j − 2)λ0 − 2φ) α2b(j−2) + b(j) − 2αb(j−1) 8 5/2 5/2 5/2 from the perturbing planet depends on the ratio of mp/M∗ i as does our β, but here our Hamiltonian perturbation con- + cos(jλ − (j + 2)λ0 + 2φ) α2b(j+2) + b(j) − 2αb(j+1) 5/2 5/2 5/2 tains both zeroth and first order terms in s. In contrast near sin θ cos θ h a second order resonance orbital perturbations are likely to + × 2 be second order in e and s. Because it contains zeroth and sin(jλ − (j − 2)λ0 − Ω0 − φ)(b(j) − αb(j−1))s0 first order terms in the expansion, the torque directly ex- 5/2 5/2 erted onto the spinning planet from a nearby planet might be 0 0 (j) (j+1) 0 − sin(jλ − (j + 2)λ + Ω + φ)(b5/2 − αb5/2 )s stronger than variations on the torque from the star caused by orbital perturbations from a perturbing planet. We have + sin(jλ − (j − 2)λ0 − Ω − φ)(α2b(j−2) − αb(j−1))s 5/2 5/2 neglected these orbital perturbations, but future work could 0 2 (j+2) (j+1) i take them into account. − sin(jλ − (j + 2)λ + Ω + φ)(α b5/2 − αb5/2 )s . In our numerical exploration of Styx we found two (43) slowly moving angles, φs1, φs2 (defined in equation1) when Arguments that are rapidly varying will not strongly there were obliquity variations. These angles can be recog- perturb the spinning planet as they effectively average to nized as arguments in the Hamiltonian in equation 44 with 0 zero. Only slowly varying arguments give resonantly strong index j = 1, and identifying λ = λStyx and λ = λCharon. perturbations. The external body has a slower mean motion Our perturbation computation gives terms with arguments

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Equation 30 gains (r · ˆs)2 sin2 θ h =+ e cos(3λ − $ − 2φ) a2 2 i − 3 cos(λ + $ − 2φ) − 2 cos(λ − $) . (48)

Equation 31 gains

Table 2. Resonance coefficients 0 2 ( (r · ˆs)(r · ˆs) + sin θ 0 = j j j aa 4 Resonance j α c (α) cs(α) c 0 (α) 0 s h 3:1 1 0.481 0.765 1.782 -4.844 e cos(2λ + λ0 − $ − 2φ) + cos(2λ − λ0 − $) 4:2 2 0.630 1.312 6.173 -11.423 i 5:3 3 0.711 2.027 14.270 -22.378 − 3 cos(λ0 + $ − 2φ) − 3 cos(λ0 − $) + 6:4 4 0.763 2.904 27.179 -38.793 h 7:5 5 0.799 3.941 45.999 -61.763 e0 cos(2λ0 + λ − $0 − 2φ) + cos(2λ0 − λ − $0) 8:6 6 0.825 5.139 71.831 -92.386 ) j j 0 0 i Resonance j α ce1(α) ce01(α) − 3 cos(λ + $ − 2φ) − 3 cos(λ − $ ) . 2:1 1 0.630 -26.751 -0.384 3:2 2 0.763 -168.337 -0.179 (49) 4:3 3 0.825 -584.812 0.799 5:4 4 0.862 -1505.813 2.943 Using 6:5 5 0.886 -3231.011 6.647 ∂ 1 ∂ α 7:6 6 0.902 -6130.115 12.304 = D and = − D (50) ∂a a0 α ∂a0 a0 α j j Resonance j α ce3(α) ce03(α) d 4:1 1 0.397 -1.238 2.997 for derivatives, with Dα ≡ dα , we compute coefficients (as 5:2 2 0.543 -3.255 5.831 defined in equation 39) 6:3 3 0.630 -6.546 10.170 ∂ 7:4 4 0.689 -11.425 16.304 0−5 (j) 0 (j) A2,j,0,1 = a −5b5/2(α) + a 0 b5/2(α) 8:5 5 0.731 -18.201 24.534 ∂a 1 (j) These are coefficients defined in equations 45, 58, and 59. We used = − [5 + αDα]b (α) a05 5/2 series expansions for the Laplace coefficients to compute them. ∂ 1 A = a0−5a b(j) (α) = αD b(j) (α). (51) 2,j,1,0 ∂a 5/2 a05 α 5/2 0 consistent with the form we guessed from the slow moving To first order in e, e angles we had seen in our simulations (see Quillen et al. 0 0 0 cos[j($ + f − $ −f )] ≈ cos[j(λ − λ )] 2017). Our Hamiltonian model effectively describes those 0 spin-resonances. + je cos[(1 + j)λ − jλ − $)] − je cos[(1 − j)λ + jλ0 − $)] − je0 cos[jλ + (1 − j)λ0 − $0)] + je0 cos[jλ − (1 + j)λ0 + $0)] (52) 2.5 Perturbation terms to first order in eccentricity using equation 6.90 by Murray & Dermott(1999). Taking first order terms from equation 52, the coeffi- From the Hamiltonian in equation 25 we evaluate r cients 51 and noting that in equation 38 the factor a − 1 ≈ a03 −e cos M = −e cos(λ−$) to first order in e, we can compute and (r − r0) · ˆs2 , ∆5 first order (in eccentricity) terms from equation 38 ∞ ( but keeping terms that are first order in orbital eccentric- −5 + 1 X ∆ = ity and zeroth-order in inclination. To first order in orbital 0 2a05 eccentricity e equation 28 gains these terms j=−∞ " 0 α (j) x 1 3 e cos[(j + 1)λ − jλ − $] − Dα + j b (α) =+ e( cos(2λ − $) − cos $) 2 5/2 a 2 2 # y + 1 3 α = e( sin(2λ − $) − sin $)) + cos[(j − 1)λ − jλ0 + $] − D − j b(j) (α) + a 2 2 2 α 5/2 z + = 0. (46) " a 0 0 0 α 5 (j) e cos[jλ − (j − 1)λ − $ ] Dα + − j b5/2(α) We have neglected terms that are proportional to es. Equa- 2 2 tion 29 gains these terms #) 0 0 α 5 (j) + cos[jλ − (j + 1)λ + $ ] Dα + + j b (α) . r · ˆs e 2 2 5/2 =+ sin θcos(2λ − $ − φ) − 3 cos($ − φ). (47) a 2 (53)

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−5 These terms are added to equation 41 giving ∆ to ﬁrst 2 2 0 order in e, e0. + 1 + α + α cos(2λ − 2φ) + cos(2λ − 2φ) Using equations 30, 31, 41, 48, 49, and 53 we compute ! the ﬁrst order in eccentricity terms that are added to equa- 0 0 − 2α cos(λ + λ − 2φ) − 2α cos(λ − λ ) tion 42;

03 0 2 2 ( ∞ " a ((r − r ) · ˆs) + sin θ ≈ X ∆5 4 × j=−∞ 2h α eα cos(3λ − $ − 2φ) e cos[(j + 1)λ − jλ0 − $] − D + j b(j) (α) 2 α 5/2 i α − 3 cos(λ + $ − 2φ) − 2 cos(λ − $) + e cos[(j − 1)λ − jλ0 + $] − D − j b(j) (α)+ 2 α 5/2 h 0 0 − eα cos(2λ + λ − $ − 2φ) + cos(2λ − λ − $) 0 0 0 α 5 (j) + e cos[jλ + (1 − j)λ − $ ] Dα + − j b5/2(α) 0 0 i 2 2 − 3 cos(λ + $ − 2φ) − 3 cos(λ − $) #) α 5 h + e0 cos[jλ − (1 + j)λ0 + $0] D + + j b(j) (α) − e0α cos(2λ0 + λ − $0 − 2φ) + cos(2λ0 − λ − $0) 2 α 2 5/2 i (54) − 3 cos(λ + $0 − 2φ) − 3 cos(λ − $0) Combining arguments and taking only argu- 0h 0 0 + e cos(3λ − $ − 2φ) ments that contain φ these terms can be written i − 3 cos(λ0 + $0 − 2φ) − 2 cos(λ0 − $0)

∞ X (j) 0 × b5/2(α) cos(jλ − jλ )... j=−∞

sin2 θ h α α i cos[jλ − (j − 3)λ0 − $ − 2φ] e α2 − D + j − 2 b(j−3) + α (αD − 2j + 3) b(j−2) + − D + j − 1 b(j−1) + 8 2 α 5/2 α 5/2 2 α 5/2 sin2 θ h α α i cos[jλ − (j + 3)λ0 + $ + 2φ] e α2 − D − j − 2 b(j+3) + α (αD + 2j + 3) b(j+2) + − D − j − 1 b(j+1) + 8 2 α 5/2 α 5/2 2 α 5/2 sin2 θ h α 9 α 7 i cos[jλ − (j − 3)λ0 − $0 − 2φ] e0 α2 D − j + b(j−2) + α (−αD + 2j − 8) b(j−1) + D − j + b(j) + 8 2 α 2 5/2 α 5/2 2 α 2 5/2 sin2 θ h α 9 α 7 i cos[jλ − (j + 3)λ0 + $0 + 2φ] e0 α2 D + j + b(j+2) + α (−αD − 2j − 8) b(j+1) + D + j + b(j) + 8 2 α 2 5/2 α 5/2 2 α 2 5/2 sin2 θ h α α i cos[jλ − (j − 1)λ0 + $ − 2φ] e − D − j − 1 b(j+1) + α (αD + 2j − 3) b(j) + α2 − D − j − 2 b(j−1) + 8 2 α 5/2 α 5/2 2 α 5/2 sin2 θ h α α i cos[jλ − (j + 1)λ0 − $ + 2φ] e α2 − D + j − 2 b(j+1) + α (αD − 2j − 3) b(j) + − D + j − 1 b(j−1) + 8 2 α 5/2 α 5/2 2 α 5/2 sin2 θ h α 1 α 1 i cos[jλ − (j − 1)λ0 − $0 + 2φ] e0 α2 D + j + b(j−2) + α (−αD − 2j) b(j−1) + D + j − b(j) + 8 2 α 2 5/2 α 5/2 2 α 2 5/2 sin2 θ h α 1 α 1 i cos[jλ − (j + 1)λ0 + $0 − 2φ] e0 α2 D − j + b(j+2) + α (−αD + 2j) b(j+1) + D − j − b(j) (55) 8 2 α 2 5/2 α 5/2 2 α 2 5/2

the terms we previously computed (zero-th order in e, e0 and Inspection of the arguments in equation 55 implies that first order in s, s0) because they are only important near resonant terms will be important (with slowly varying ar- second order mean motion resonances. However, at higher guments) near first order mean motion resonances, where order in e, e0, s, s0 additional terms will contribute near all jλ − (j + 1)λ0 is slowly varying (with positive j) and near of these mean motion resonances. third order mean motion resonances, where jλ − (j + 3)λ0) is slowly varying. Near a third order mean motion resonance (and to first order in eccentricities and inclinations) we can consider a near resonance Hamiltonian (similar to equation 44) Near a third order mean motion resonance, equation 56 p shows that the torque directly exerted by the planet is first H(p, φ, τ)j:j+3 ≈ − (2 − p)+ 2 order in eccentricity. The perturbing planet should cause j 0 orbital perturbations depending upon the third order of the βce3ep(2 − p) cos[jλ − (j + 3)λ + $ + 2φ]+ eccentricity. So the variations in the torque from the star due βcj e0p(2 − p) cos[jλ − (j + 3)λ0 + $0 + 2φ]. (56) e03 to the orbit variations are likely to be smaller than the those Near a third order mean motion resonance, we can neglect caused directly from the torque of the perturbing planet.

MNRAS 000, 000–000 (0000) 8 Quillen et al.

Near a first order mean motion resonance p H(p, φ,τ)j:j+1 ≈ − (2 − p)+ 2 p βc2j (2 − p) cos[2(λ − (j + 1)λ0) + 2φ]+ 0 2 2j p 0 βcs s(1 − p) p(2 − p) sin[2(λ − (j + 1)λ ) + Ω + φ]+ 2j 0 p 0 0 βcs0 s (1 − p) p(2 − p) sin[2(λ − (j + 1)λ ) + Ω + φ]+ p βcj e (2 − p) cos[jλ − (j + 1)λ0 − $ + 2φ]+ e1 2 p βcj e0 (2 − p) cos[jλ − (j + 1)λ0 − $0 + 2φ]. (57) e01 2 Here zero-th order and first order in inclination terms contribute but they are indexed by 2j rather than j. Near a first order mean motion resonance, equation 57 shows that the torque directly exerted by the planet is first order in eccentricity. The perturbing planet would also cause orbital perturbations that are first order in eccentricity. Vari- ations in the torque from the star due to the orbit variations could be similar in size to those caused directly from the torque of the perturbing planet and these could be computed in future work. The coefficients for Hamiltonians in equation 56 and 57 are twice those listed in equation 55 so as to include the contribution from a corresponding negative j term that gives the same argument;

j 1 h 2 α (j+3) ce3 ≡ α − Dα − j − 2 b5/2 + Figure 1. Level contours of the Hamiltonian in equation 64 with 4 2 2 α i a perturbation proportional to sin θ with resonance strength α (αD + 2j + 3) b(j+2) + − D − j − 1 b(j+1) βcj = 0.05. Each subplot shows level contours of the Hamilto- α 5/2 2 α 5/2 0 nian with a different value of distance to resonance ν2 and with h j 1 2 α 9 (j+2) the value of ν2 labelled on the upper right. For |ν2| < 2 there are ce03 ≡ α Dα + j + b5/2 + 4 2 2 two stable and two unstable1 fixed points. For |ν2| > 2 there are α 7 i no fixed points. α (−αD − 2j − 8) b(j+1) + D + j + b(j) α 5/2 2 α 2 5/2 (58) shown in the Hamiltonians in equations 44 and 57. Perturba- and tions that are first order in eccentricity are also proportional j 1 h 2 α (j+1) to sin2 θ (see equation 55) and these would be important ce1 ≡ α − Dα + j − 2 b + 4 2 5/2 near first and third order mean motion resonances (as in the i (j) α (j−1) Hamiltonians given in equation 56 and 57). In contrast per- α (αDα − 2j − 3) b5/2 + − Dα + j − 1 b5/2 2 turbations that are first order in inclination are proportional j 1 h 2 α 1 (j+2) to sin θ cos θ (see equation 43). These are relevant for first ce01 ≡ α Dα − j + b5/2 + 4 2 2 and second order mean motion resonances (as in the Hamil- α 1 i tonians given in equations 44 and 57). We have two types of α (−αD + 2j) b(j+1) + D − j − b(j) . α 5/2 2 α 2 5/2 perturbations, those proportional to sin2 θ and those propor- (59) tional to sin θ cos θ. In terms of p, the canonical momentum, sin2 θ = p(2−p) and sin θ cos θ = (1−p)pp(2 − p). The two For low j we computed these coefficients and list them in types of perturbation terms give Hamiltonians with differ- Table2. ent shapes (as a function of p) and this affects the resonance Near a second order mean motion resonance (j : j + 2) strength as a function of obliquity. with j an odd integer, the first order in eccentricity terms do not contribute. Consequently equation 44 remains accurate to first order in eccentricity for odd j second order mean motion resonances. 3.1 Hamiltonian model for a perturbation proportional to sin2 θ Taking the Hamiltonian in equation 44, appropriate for a 3 DRIFTING TOY HAMILTONIAN MODELS second order mean motion resonance we define a frequency

The perturbations to the Hamiltonian arising from terms −1 0 ν2 ≡ αs (jn − (j + 2)n ). (60) that are zero-th order in inclination and eccentricity are pro- 2 portional to sin θ (see equation 43). These would be impor- Here the spin precession frequency αs makes ν2 unitless. tant near ﬁrst and second order mean motion resonances, as Retaining a single j term and zero-th order (in inclination)

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with

j ≡ βc0. (65)

The Hamiltonian is time independent as long as ν2 is fixed, and ν2 sets the distance to the center of resonance. We have neglected the phase c as it can be removed via shifting ϕ or τ. We can write the frequency (equation 60) α a 3/2 ν s = j − (j + 2) . (66) 2 n a0 During an epoch of planet migration the semi-major axes a, a0 may drift. When two planets approach each other either 0 a increases or a decreases so the frequency ν2 decreases. If the planet orbits separate, ν2 increases. For various values of ν2 level curves for the Hamiltonian in equation 64 are shown in Figure1. There are no fixed points for |ν2| > 2. For |ν2| < 2 there are single stable fixed points at $ = 0, π with p or θ value increasing with ν2. The p value for the fixed points range from near 0 to near 2 corresponding to obliquity ranging from near 0 to near 180◦. We can mimic planet or satellite migration by allowing ν2 to slowly vary. We let ν2 be linearly dependent on time. We note that α , setting our unit of time and the strength Figure 2. Level contours of the Hamiltonian in equation 70 with s a term first order in inclination proportional to sin θ cos θ, with of the coefficients, also depends on the semi-major axes. For j the moment we regard them as constants and allow only resonance strength βcss = 0.05 and for different values of distance to resonance ν2. This Figure is similar to Figure1. For 0 < ν2 < 1 ν2 to vary. As ν2 decreases, corresponding to the planets there is a single stable fixed point at ϕ = π. For −1 < ν2 < 0 migrating so that they approach one another, a planet ini- there is a single stable fixed point at ϕ = 0. For |ν2| > 1 and tially at low obliquity could be captured into a stable fixed ν2 = 0 there are no fixed points. point and lifted in obliquity. Using Hamilton’s equations for Hamiltonian of equation 64 and = 0.01, we integrated a planet or satellite, within initial low p = 0.001 (correspond- perturbation term alone the Hamiltonian in equation 44 1 ing to an obliquity of θ = 2.6◦) and withν ˙ = −0.001. The p 2 H(p, φ, τ) = − (2 − p) time evolution of θ, ϕ are shown in Figure3. The planet is c0j 2 j captured into a resonant region near a fixed point and lifted + βc0p(2 − p) cos(ν2τ + 2φ + c) (61) to an obliquity of near 180◦ and then it escapes resonance. with c a constant phase. The resonance strength we used is small. Even when the A similar Hamiltonian would be derived near a first resonance is weak and narrow, a planet or satellite could be order mean motion resonance and retaining only a sin- captured into it and have its obliquity lifted to high values gle first order in eccentricity term using the Hamiltonian as the resonance frequency drifts. in equation 57. In this case the relevant frequency would −1 0 ˙ be one of the following ν = αs (2jn − 2(j + 1)n + Ω), −1 0 ˙ 0 −1 0 3.2 Hamiltonian model for a perturbation αs (2jn − 2(j + 1)n + Ω ), αs (jn − (j + 1)n +$ ˙ ) or −1 0 0 proportional to sin θ cos θ αs (jn − (j + 1)n +$ ˙ ) depending upon the term. A Hamiltonian similar to 61 can also be derived near a third We now consider a Hamiltonian model with perturbation order mean motion resonance retaining only a single first proportional to sin θ cos θ. We retain a single j term first order in eccentricity term and using the Hamiltonian in order in inclination in the Hamiltonian 44. Using a frequency equation 56. In this case the relevant frequency would be −1 0 −1 0 0 ν = αs (jn − (j + 3)n + $) or αs (jn − (j + 3)n + $ ). −1 0 Performing a canonical transformation with time depen- ν2 ≡ αs (jn − (j + 2)n + Ω) (67) dent generating function that is a function of old coordinates and a canonical coordinate transformation with generating and a new momentum function 0 1 0 F2(φ, p , τ) = (ν2τ + 2φ)p , (62) 0 1 0 2 F2(φ, p , τ) = (ν2τ + φ)p , (68) 2 giving new momentum p0 = p equal to the old one and a we derive a new angle new angle 1 ϕ = ν2τ + φ (69) ϕ = (ν2τ + 2φ). (63) 2 and a new Hamiltonian Transforming the Hamiltonian in equation 61 we find a new p Hamiltonian in these new coordinates K(p, ϕ, τ)csj = − (2 − p) + ν2p p ν p 2 K(p, ϕ, τ) = − (2 − p) + 2 + p(2 − p) cos(2ϕ) (64) p c0j 2 2 + (1 − p) p(2 − p) sin(ν2τ + φ) (70)

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Figure 3. Integration of the Hamiltonian in equation 64 with a Figure 4. Integration of the Hamiltonian in equation 70 with a zero-th order perturbation (in inclination) ∝ sin2 θ and resonance first order perturbation (in inclination) proportional to sin θ cos θ strength = 0.01. Here the resonance drifts withν ˙2 = −0.001 cor- and resonance strength = 0.01. Here the resonance drifts with responding to the planets slowly migrating closer together. Initial ν˙2 = −0.001 for planets slowly migrating closer together. Initial conditions are (ϕ, p) = (1.5, 0.001), and ν2 = 2.4. The top panel conditions are (ϕ, p) = (1.5, 0.001), and ν2 = 1.4. The top panel shows obliquity as a function of time whereas the bottom panel shows obliquity as a function of time whereas the bottom panel shows the precession angle. The integrated body is captured into shows the precession angle. The integrated body is captured into resonance at low obliquity and exits near an obliquity of 180◦. resonance at low obliquity and exits near an obliquity of 90◦. with at high probability. This critical drift rate is approximately j equal to the square of the resonance libration frequency ≡ βcss. (71) (Quillen 2006). There is a critical initial eccentricity, below The Hamiltonian would look the same if a first order term which capture into mean motion resonance is assured when 0 −1 proportional to s were used with frequency ν2 = αs (jn − drift is adiabatic (Borderies & Goldreich 1984). The drift 0 0 −1 0 (j + 2)n + Ω ) instead of αs (jn − (j + 2)n + Ω), and the rate defining the adiabatic limit and the limiting eccentric- j 0 perturbation strength, , is proportional to βcs0 s instead of ity ensuring capture can be estimated from the Hamiltonian j βcss. A similar Hamiltonian would be derived near a first by considering the dimensions of the coefficients (Quillen order mean motion resonance and using a single j term that 2006). By considering the form of the Hamiltonians in equa- is first order in inclination from the Hamiltonian in equation tions 64, 70 at low p we can similarly estimate the drift rate 57. required for capture into spin resonance. Level curves for the Hamiltonian in equation 70 are At low p (low obliquity) the perturbation term in the 2 shown in Figure2 for different values of ν2. For this Hamil- Hamiltonian of equation 64 (with perturbation ∝ sin θ) is tonian for −1 < ν2 < 0 a single fixed point is near ϕ = 0 and proportional to 2p cos 2φ, so the drifting resonance behaves it has p < 1. However at ν2 = 0 the fixed point disappears like a second order mean motion resonance that is propor- 2 and there are no fixed points. For 0 < ν2 < 1 there is again tional to e or the Poincarémomentum variable Γ. The res- a single fixed point but with p > 1. onance libration frequency at low p is ωlib ∝ . In analogy As before we integrate the equations of motion for a to the Hamiltonian for mean motion resonance, we expect slowly drifting system. We plot a planet trajectory in Fig- that capture into resonance for the drifting Hamiltonian of 2 ure4 using the Hamiltonian in equation 70, = 0.01, equation 64 is likely for drift rates |ν˙2| . and for initial ν˙2 = −0.001, initial conditions (ϕ, p) = (1.5, 0.001) and momentum p . . ν2 = 1.4. Again we find that resonance capture is possible In contrast the resonant term for the Hamiltonian in for a particle initially at low obliquity. However because the 70, with perturbation ∝ sin θ cos θ, at low p is proportional √ fixed point disappears near an obliquity of 90◦ the planet to p and so this resembles a first order mean motion reso- must escape resonance near this value rather than near 180◦ nance, proportional to e or the square root of the Poincaré 2 √ 2 as for the perturbation that is proportional to sin θ (ex- momentum variable, Γ. The libration frequency ωlib ∝ 3 . plored in subsection 3.1). 4 The resonance should capture at a drift rate |ν˙2| . 3 and 2 for initial p . 3 . To estimate coefficients for the limits we show capture 3.3 Adiabatic Limits for Resonance Capture probabilities for a range of drift rates for the Hamiltonians When the drift from migration is too fast or not adiabatic, in equations 64 and 70 in Figure5. In both cases we set a particle in orbit will jump across mean motion resonance perturbation strength = 0.01. We computed the capture rather than capture into resonance (Quillen 2006). At drift probabilities for three different initial p values, 0.001, 0.01, rates below a critical drift rate, the resonance can capture and 0.1 corresponding to initial obliquities of 2.6, 8.1 and

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26◦. Each point shown in Figure5 was computed from an average of 30 integrations where each integration was begun outside of resonance at a randomly chosen initial φ. To re- duce dependence on phase we also chose random initial ν2 values (but ensuring that we began outside of resonance). For those integrations that captured into resonance (as determined by a large increase in obliquity or p) we computed the mean ﬁnal obliquity (of the integrations that captured) and these are shown as points on the bottom panels in Fig- ure5. The solid lines show ﬁts of a tangent function to the capture probability. Figure5 shows that for low initial p capture takes place −3 2 at |ν˙2| ∼ 10 for the sin θ resonance with Hamiltonian in equation 64. Using the dependence on 2 we estimate that

2 |ν˙2|sin2 θ . 10 (72) for capture into the sin2 θ resonance. −3 Capture takes place at |ν˙2| ∼ 4 × 10 for the sin θ cos θ resonance (Hamiltonian of equation 70) giving

4 |ν˙2|sin θ cos θ . 2 3 (73) for capture into the sin θ cos θ resonance. The sin θ cos θ resonance (Figure5b) exhibits a sharper sensitivity to drift rate than the sin2 θ one (Figure5a), consistent with previous studies showing that first order mean motion resonances have a more abrupt transition in capture probability near the adiabatic limit (Quillen 2006). Following resonance capture, Figure5 shows that after escaping resonance the final obliquity is somewhat sensitive to initial obliquity and drift rate. For the sin2 θ resonance, higher initial obliquity gave lower final obliquity after resonance capture. For both types of resonances, nearer the adiabatic limit and at higher drift rates, the final obliquities are lower. How long does it take to lift the obliquity in one of these spin resonances? For the sin2 θ resonance, once captured into resonance, p must drift from 0 to 2 but the coefficient ∝ p in the Hamiltonian in equation 64 contains a factor of 1/2. So the total time in resonance depends on 1/|ν˙2|. For the sin θ cos θ resonance the total time in resonance is equivalent. Figure 5. a) The top panel shows capture probabilities for the 2 Restoring units, once captured at low obliquity, the time to Hamiltonian in equation 64 with resonant term ∝ sin θ with lift the obliquity to near 90◦ or near 180◦ for the sin2 θ or resonance strength = 0.01 as a function of drift rate for three different initial obliquities or p values. The initial p values 0.001, sin θ cos θ resonances, respectively is 0.01, 0.1, shown in the key on the lower left, correspond to ini- 1 tial obliquities of 2.6, 8.1 and 26◦. The bottom panel shows the Tlift ≈ . (74) |ν˙2|αs average final obliquities (in degrees) when capture took place for the same initial p values. The x axes show the log10 of the ab- solute value of the drift rate |ν˙2|. The drift rate is higher on the left hand side and gives lower capture probabilities. Each point 4 APPLICATION TO PLUTO AND CHARON’S is computed from an average of 30 integrations. b) Similar to a) MINOR SATELLITES except for the Hamiltonian in equation 70 and with resonant term ∝ sin θ cos θ. We consider spinning minor satellites Styx and Nix, that have orbits exterior to Charon. With an exterior spinning 1 2 body, the unitless coefficient β (defined in equation 26) is c0(α) = 0.76 giving a resonance term ∝ sin θ. There are two 1 approximately equal to the mass ratio of Charon and Pluto, terms first order in orbital inclination that larger, cs = 1.782 1 β ∼ 0.1. Styx is near the 3:1 resonance with Charon and Nix and cs0 = −4.84 and giving a resonance term ∝ sin θ cos θ. near the 4:1 resonance. The c0 term need only be multiplied by β giving resonance We first consider Styx, with an orbital period of 20 days. strength = 0.1 × 0.76 = 0.08 for the Hamiltonian in equa- 1 1 As Styx is near the 3:1 second order mean motion resonance tion 64. The cs and cs0 coefficients should be multiplied by we consider the Hamiltonian in equation 44 with j = 1. the orbital inclinations of Styx and Charon, respectively to Inspection of Table2 shows that there is a zero-th order estimate the strength of the sin θ cos θ resonance. Working in (in inclination and eccentricity) perturbation with coefficient a coordinate frame aligned with Styx’s orbit (and with the

MNRAS 000, 000–000 (0000) 12 Quillen et al. obliquity measured with respect to Styx’s orbit normal), we comparison of terms in Table2 suggests that the coefficients 1 1 1 use the inclination of Charon and need only the cs0 coef- would be a few times larger than the ce3 or ce03 coefficients. ficient. Taking into account β = 0.1, we estimate a reso- We have checked that the expansion to first order in es0 (or 1 0 0 nance coefficient strength = 0.1 × cs0 = 0.48ICharon where e s ) would give terms proportional to sin θ cos θ and with ICharon is the inclination of Charon’s orbit relative to Styx’s. argument similar to a third-order mean motion resonance. ◦ With an inclination of ICharon = 6 = 0.1 radians we find For the 4:1 resonance, dependence on inclination and eccen- = 0.05. The resonant perturbation term is ∝ sin θ cos θ and tricity for the sin θ cos θ term implies that the spin resonance the relevant Hamiltonian is equation 70. strength would be weaker for Nix than Styx and so a slower The two resonance strengths, ∝ sin2 θ and that ∝ migration rate and longer time would be needed to tilt Nix sin θ cos θ, have similar strengths when Charon’s inclination than Styx. However, the constraints on these quantities for ◦ 1 is ∼ 8 . With a moderate inclination, the cs0 resonance Styx were not difficult to satisfy, so the mechanism for tilt- 1 might be more important than the c0 one. After capture, ing Nix is also likely to be effective. If these spin resonances Styx should exit the sin θ cos θ resonance with an obliquity operated on Styx and Nix, the near 90◦ obliquities imply near 90◦, as we saw in our simulations (section 5 by Quillen that Charon’s orbit was relatively inclined during the mi- et al. 2017) and as illustrated in Figure4. In contrast after gration as the strongest perturbation terms ∝ sin θ cos θ are 2 capture into the c0 resonance, ∝ sin θ, Styx should exit the first order in orbital inclination. resonance near an obliquity of 180◦. Our toy model considers each resonant term separately, Which spin resonance is encountered by Styx first as but likely both sin2 θ and sin θ cos θ terms are present and si- Charon drifts away from Pluto? The two resonances are multaneously important for these spin resonances. We have not on top of each other as the sin2 θ resonance has argu- not yet explored higher inclinations or slower migration rates ment with frequency ν2αs = nStyx − 3nCharon, whereas for in our mass-spring model simulations, leaving this for future the sin θ cos θ resonance the argument frequency is ν2αs = work. At moderate inclination both Styx and Nix might ini- nStyx −3nCharon −ΩCharon. As Ω˙ Charon < 0, for an outward tially be captured into a sin θ cos θ resonance but escape or drifting Charon, Styx should encounter the sin θ cos θ reso- could be subsequently pushed higher by a sin2 θ resonance, nance first and that may explain why the obliquity was lifted possibly accounting for Nix’s 123◦ (Weaver et al. 2016) and to near 90◦ in our simulations rather than 180◦ (see section higher than 90◦ obliquity. 5 by Quillen et al. 2017). At high obliquity, the sin2 θ res- Kerberos is near a 5:1 mean motion resonance with onance could still be important. Perhaps this resonance, or Charon and Hydra near a 6:1 mean motion resonance. Sim- evolution involving both perturbation terms could account ilar spin resonances could have operated on both of these for the higher than 90◦ obliquities of Nix and Hydra mea- satellites. Our Hamiltonian models in sections 2.4 and 2.5 sured by Weaver et al.(2016). imply that the spin resonances near the 5:1 and 6:1 mean With resonance strength ∼ 0.05 the drift rate in the motion resonances would be greater than first order in in- sin θ cos θ resonance (using equation 73) |ν˙2| . 0.04 and giv- clination and eccentricity. They would be weaker, require 2 ing a constraint on the migration rate |n˙ | . 0.04αs or migra- slower migration rates for capture and longer time to evolve 2 n −1 within resonance to lift the obliquities. tion on a timescale tmig & 30 n . Here either Charon αs We previously speculated that a mean motion reso- can move away from Pluto or equivalently Styx could mi- nance between Nix and Hydra could affect Hydra’s obliq- grate inward. As discussed in section 2.4 of Quillen et al. uity (Quillen et al. 2017). However the strength of such a (2017), the precession rate for all the minor satellites is resonance depends on the mass ratio of Nix and Pluto and n Ps Ps αs ∼ where is the ratio of spin period to orbital 2 Po Po this would make the spin resonances 4 to 5 orders of mag- period. Currently Po ∼ 6.2, 13.6 for Styx and Nix, respec- Ps nitude weaker than those involving Charon. The resonance tively. This gives a constraint on the migration timescale strengths computed here arise from the direct torque applied from a orbiting point mass, in this setting from Nix 2 directly onto Hydra. Indirectly Nix could excite the orbit of a Ps tmig = 30 Po (75) Hydra due to a 3:2 first order mean motion resonance be- a˙ & P o tween them and the torque from Pluto and Charon arising or a few thousand times greater than the orbital period. This from the orbit perturbations of Hydra might affect Hydra’s limiting migration rate is not slow and would be satisfied spin. We have not yet computed this type of perturbation during an epoch of circumbinary disk evolution (e.g., Kenyon but suspect it would be most effective near a first order & Bromley 2014) . The total time required for the lift in mean motion resonance (first order in eccentricity and be- obliquity to take place (using equation 74) would be only tween Nix and Hydra), as discussed at the end of section of order a hundred orbital periods or a few thousand years, 2.5. easily satisfied, and consistent with the rapid obliquity lifts seen in our simulation. Nix is near the third order 4:1 mean motion resonance 5 APPLICATION TO URANUS and the Hamiltonian we consider is that in equation 56. The terms that are important are the coefficients that are first or- The successful Nice model (Tsiganis et al. 2005) and its vari- 1 1 der in eccentricity, ce3 and ce03 and with j = 1. However this ants (e.g., Morbidelli et al. 2009; Nesvorny 2011; Nesvorny & resonance is ∝ sin2 θ and our simulation showed resonance Morbidelli 2012; Nesvorny 2015; Deienno et al. 2017), pos- escape near an obliquity of ∼ 90◦. We would attribute this tulate an epoch or epochs of planetary migration beginning behavior to higher order terms, proportional to es0 that we with planets near or in mean motion resonance (Morbidelli neglected from our computations in section 2.5. However a et al. 2007). During planet migration, secular resonances and

MNRAS 000, 000–000 (0000) 13 encounters between planets can alter planet spin orientation 6 SUMMARY AND DISCUSSION or obliquity (Ward & Hamilton 2004). Likewise the current obliquities of the giant planets could give clues about the ex- We have explored a Hamiltonian model for the dynamics of tent and speed of planetary migration during early epochs the principal axis of rotation of an orbiting planet or satel- of Solar system evolution (e.g., Boué& Laskar 2010; Brasser lite assuming that the spinning body remains rapidly spin- & Lee 2015). ning about its principal inertial axis. We have computed the torque exerted on the spinning body from a point mass The large obliquity of Uranus has primarily been at- also in orbit about the central mass (a star if the our spin- tributed to a tangential or grazing collision with an Earth- ning object is a planet or a planet if our spinning object sized proto-planet at the end of the epoch of accretion (e.g., is a satellite). Unlike many previous studies (e.g., Colombo Safronov 1969; Korycansky et al. 1990; Slattery et al. 1992; 1966), we do not average over the orbital period. We have Lee et al. 2007; Parisi et al. 2008; Parisi 2011). However, computed perturbations from the point mass to first order in Boué& Laskar(2010) proposed a collisionless scenario in- orbital inclination and eccentricity (but not their product). volving an additional, but now absent, satellite. Proposed The perturbation terms are either proportional to sin2 θ or is that a close encounter at the end of the era of migration to sin θ cos θ. Taking a single Fourier component (of the per- ejected this satellite while Uranus was at high orbital incli- turbation) we derive the Hamiltonians for the spin orien- nation, leaving the planet at high obliquity following orbital tation shown in equations 44, 56, and 57 that are relevant inclination damping. near second order, third order and first order mean motion resonances, respectively, but affect obliquity and spin preces- We consider the possibility that during an early epoch sion rate. The resonant arguments for the resonance near a of planetary migration Uranus was captured into a mean second order mean motion resonance in equation 44 are con- motion resonance with another giant planet and while in sistent with slowly moving angles (equation1) we previously it, its obliquity was lifted due to spin resonance. The spin saw in numerical simulations of Styx when its obliquity var- resonance strengths (see equation 26) depend on the mass ied (Quillen et al. 2017). Our Hamiltonian model provides ratio of perturbing planet to that of the central star and a framework for estimating the strengths of spin resonances this is at most the ratio of Jupiter’s mass to that of the Sun involving a mean motion resonance and the precession angle or 10−3 . With Uranus external to Jupiter our parameter of a spinning body. β is also decreased by the cube of the ratio of semi-major Numerical integrations of one of these one-dimensional axes. The spin resonance strengths are likely to be 100 times Hamiltonians (equations 44, 56, or 57) containing a single weaker than we considered in the Pluto-Charon system, even resonant perturbation term show that if the resonance drifts, taking into account the larger coefficients for first order mean a spinning body initially at low obliquity can be captured motion resonances listed in Table2. Furthermore equation into spin resonance and its obliquity lifted to near 180◦ 74 implies that the time required to lift Uranus would be of or 90◦ depending upon whether the perturbation term is order 1000 times its precession period. ∝ sin2 θ or sin θ cos θ. The sin θ cos θ requires non-zero orbital inclination of the spinning body with respected to the Taking into account its satellite system, Uranus’s pre- perturbing one. We estimate the maximum drift rate allow- cession period is currently approximately 108 years (Ward ing spin resonance capture and the timescale required to 1975). The precession period is so slow that there is not reach the maximum obliquity when the body escapes res- enough time during a Nice model epoch of planetary migra- onance. Resonance capture into these spin resonances only tion for the spin resonances considered here to tilt the planet. takes place if the perturbing mass and spinning body have If Uranus had a much heavier and more extended satellite approaching orbits (similar to capture into mean motion or- system (Mosqueira & Estrada 2003a,b) then its precession bital resonance). period would be reduced. In this case we could consider capture into a 3:2 or 4:3 resonance with Saturn and a spin reso- We applied our Hamiltonian model to a migrating nance associated with one of these mean motion resonances. Pluto-Charon satellite system. The spin resonance seems However the resonance strengths depend on eccentricity and capable of accounting for the large obliquity variations we inclination, making them even weaker. The time required to previously saw in our simulations of the spin evolution of lift the planet remains long, of order the age of the Solar sys- satellites Styx and Nix near the 3:1 and 4:1 mean motion tem, requiring a timescale longer than the postulated era of resonances with Charon. Outward migration of Charon or migration. Uranus and Saturn migrating in such proximity inward migration of Styx and Nix would have allowed ini- are unlikely to remain stable and the spin resonance would tially low obliquity Styx and Nix to be captured in to spin resonance and lift their obliquities. As Styx and Nix have require hundreds of spin precession periods to tilt Uranus. ◦ ◦ We conclude that the type of spin resonance explored here obliquities near 90 and not near 180 , the capture would cannot account for Uranus’s high obliquity. have involved a resonance proportional to sin θ cos θ and so requires Charon’s orbit to be inclined by a few degrees with The spin secular resonance mechanism explored by respect to the orbits of Styx and Nix. Due to Charon’s large Ward & Hamilton(2004) involves drift of secular resonance. mass, the resonances are sufficiently strong that the con- Uranus currently is near a secular resonance, the vertical straints on the migration rate for resonance capture are loose secular eigenfrequency associated with Neptune, so the spin- and the time needed to tilt the satellites is short, of order secular resonance could have operated on a timescale of bil- only a thousand years. Similar spin resonances could operate lions of years. For a mean-motion/spin precession to operate on Kerberos and Hydra, though they would be higher order the planet would need to be near a mean motion resonance in eccentricity and inclination and so weaker. for billions of years but Uranus is no longer near one. We explored whether Uranus could be tilted by a similar

MNRAS 000, 000–000 (0000) 14 Quillen et al. mechanism during an early epoch when Uranus might have Constraining the Giant Planets? Initial Configuration from been in a first order mean motion resonance with another gi- Their Evolution: Implications for the Timing of the Planetary ant planet such as Saturn. However Uranus’s spin precession Instability, Astronomical Journal, 153, 153–166. period is long enough that tilting the planet would require Estrada, P. R., & Mosqueira, I. 2006, A gas-poor planetesimal billions of years. Since Uranus probably did not spend much capture model for the formation of giant planet satellite systems, Icarus, 181, 486–509. time in or near mean motion resonance with another planet, French, R. G., et al. 1993, Geometry of the Saturn system from the this type of spin resonance is unlikely to account for Uranus’s 3 July 1989 occultation of 28 SGR and Voyager observations, current high obliquity. Icarus, 103, 163–214. The type of spin-resonance explored here is weak for Frouard, J., Quillen, A. C., Efroimsky, M., & Giannella, D. 2016, planets in the Solar system and so it was justifiably neglected Numerical Simulation of Tidal Evolution of a Viscoelastic from previous studies. However the spin-resonances dis- Body Modelled with a Mass-Spring Network, Monthly No- cussed here may be important in migrating rapidly spinning tices of the Royal Astronomical Society, 458, 2890–2901. satellite and planetary systems, such as Pluto and Charon’s Goldreich, P. 1965, Inclination of satellite orbits about an oblate and possibly migrating compact exoplanet or satellite sys- precessing planet, Astronomical J., 70, 5–9. tems prior to tidal spin down. The approximate Hamiltonian Goldreich, P. and Toomre, A. 1969, Some Remarks on Polar Wan- model presented here could aid in interpretation of future dering, J. Geophys. Res., 74, 2555. Goldreich, P., & S. J. Peale, S. J. 1966, Spin-orbit coupling in the simulations of spin evolution in such settings. solar system, Astronomical Journal, 71, 425–438. We explored toy Hamiltonian models containing a sin- Hamilton, D. P., & Ward, W. R. 2004, Tilting Saturn. II. Numer- gle resonant argument. However real systems are likely to ical Model, Astronomical J., 128, 2510—2517. be affected by multiple terms, each dependent on an ar- Kenyon, S. J., & Bromley, B. C. 2014, The Formation of Pluto’s gument that varies at a similar frequency. Spin evolution Low-Mass Satellites, Astronomical Journal, 147, 8–25. could be chaotic due to these nearby resonances. Interaction Korycansky, D. G., Bodenheimer, P., Cassen, P., & Pollak, J. between the terms might allow resonance escape at obliqui- B. 1990, One-dimensional calculations of a large impact on ties between 90 and 180◦ possibly accounting for the 123◦ Uranus, Icarus, 84, 528–541. obliquity of Nix (Weaver et al. 2016). The spin resonances Laskar, J., & Robutel, P. 1993, The chaotic obliquity of the plan- are important near mean motion resonances, and so future ets, Nature, 361, 608-612. study of spin evolution where these spin resonances are im- Lee, M. H., Peale, S. J., Pfahl, E., & Ward, W. R. 2007, Evolution of the obliquities of the giant planets in encounters during portant should also consider the orbital dynamics in or near migration, Icarus, 190, 103–109. mean motion resonance (e.g., Voyatzis et al. 2014). Morbidelli, A., Tsiganis, K., Crida, A., Levison, H. F., & Gomes, —————- R. 2007, Dynamics of the Giant Planets of the Solar System in the Gaseous Protoplanetary Disk and Their Relationship to the Current Orbital Architecture, Astronomical J., 134, 1790–1798. Morbidelli, A., Brasser, R., Tsiganis, K., Gomes, R., & Levison, H. F., 2009, Constructing the secular architecture of the solar system I. The giant planets, Astron. & Astroph., 507, 1041– 1052. 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