Migration of a Moonlet in a Ring of Solid Particles: Theory And

Home , Moonlet

arXiv:1006.1573v1 [astro-ph.EP] 8 Jun 2010 aoaor asoé nvri´ eNc Sophia-antip Nice Cassiopée, UniversitéLaboratoire de aoaor I-M 18 E/NSUiestePrsDid CEA/CNRS/Université Paris 7158, AIM-UMR Laboratoire aoaor I-M 18 E/NSUiestePrsDid CEA/CNRS/Université Paris 7158, AIM-UMR Laboratoire o10mlinyas eedn ntedmnn irto regime. migration m kg dominant headings: 700 the Subject of on order depending the years, of million be 100 far should so to observations ring by A provided the rates of migration the case, that In niiul(aun lnt n aelts ig plane — rings satellites: and planets — (Saturn) individual efidta ntm clso eea er h irto fpropelle of migration migration non-stochastic the former, years the o several (while depending of effects observations, scales stochastic the time by for on simu account that Our to find We moonlet. enough a large on torque be stochastic can a A- exert Saturn’s may A-ring) in stable orbit propeller’s the migr of corresponding variations the fla observed that with t the find particles, We compute explain solid pr numerically. we of and Thus, in disk analytically two-dimensional planets pressureless. or a both is of in some it migration embedded fact, as moonlet I system, in a type ring ; the of ring to theory the applied standard with The interaction gravitational detected. ring the their in to create they due features “propeller” so-called the through oee,lcldniyflcutos(u ogaiywksi h ma the in wakes gravity to (due fluctuations density local However, sp Cassini the by identified been have objects sized meter Hundred eateto ple ahmtc n hoeia Physics Theoretical and Mathematics Applied of Department eateto ple ahmtc n hoeia Physics Theoretical and Mathematics Applied of Department eateto ple ahmtc n hoeia Physics Theoretical and Mathematics Applied of Department irto famolti igo oi atce : particles solid of ring a in moonlet a of Migration etefrMteaia cecs ibroc od Cambr Road, Wilberforce Sciences, Mathematical for Centre etefrMteaia cecs ibroc od Cambr Road, Wilberforce Sciences, Mathematical for Centre etefrMteaia cecs ibroc od Cambr Road, Wilberforce Sciences, Mathematical for Centre hoyadapiaint aunspropellers. Saturn’s to application and Theory lnt n aelts yaia vlto n stability and evolution dynamical satellites: and planets ..42,034Nc ee 4, Cedex Nice 06304 4229, B.P. E/aly 19 i-u-vteCedex, Gif-sur-Yvette 91191 CEA/Saclay, E/aly 19 i-u-vteCedex, Gif-sur-Yvette 91191 CEA/Saclay, onC B. C. John Sébastien Aurélien Julien ABSTRACT Hanno Salmon Papaloizou 1 − Charnoz Rein 2 Crida h g ftepoelr hud’ xed1 exceed shouldn’t propellers the of age The . France ls/CR bevtied aCôte d’Azur, la de Observatoire / CNRS / olis -ikinteractions t-disk rt RUSried’Astrophysique, IRFU/Service erot, rt RUSried’Astrophysique, IRFU/Service erot, [email protected] hs onessol migrate, should moonlets These . ugssta h ufc density surface the that suggests oiae after dominates nvriyo Cambridge, of University , nvriyo Cambridge, of University , nvriyo Cambridge, of University , France France h aaeeso h rings. the of parameters the n ain hwta hstorque this that show lations edffrniltru etby felt torque differential he deC30A UK 0WA, CB3 idge deC30A UK 0WA, CB3 idge deC30A UK 0WA, CB3 idge tpaeaydsscntbe can’t disks otoplanetary si ieyt edominated be to likely is rs ring. ccati aunsAring A Saturn’s in acecraft ia aito aebeen have variation bital to aei o ml to small too is rate ation ufc est profile, density surface t lnt n satellites: and planets — gnlygravitationally rginally ∼ 10 4 − 5 years). 1. Introduction tudinal extent about 3 km (Tiscareno et al. 2006, 2008; Sremˇcevićet al. 2007). They are most prob- The theory of disk-satellite interactions has ably caused by the presence of moonlets about for the most part been developed after Voy- hundred meters in size, embedded in the ring, and ager’s encounter with Saturn. The satellites scattering ring particles (Spahn and Sremˇcević that orbit beyond the outer edge of the rings 2000). As they are embedded in the ring, these perturb the dynamics of the particles compos- small bodies should exchange angular momentum ing the rings. This leads to an exchange of with the ring, and migrate (Crida et al. 2009). angular momentum between the rings and the This migration could be detected by Cassini ob- satellites, and to the formation of density waves servations through the cumulative lag, or advance in the rings. Lin and Papaloizou (1979) and with time t of the orbital longitude φ induced by Goldreich and Tremaine (1979, 1980) have cal- a small variation of the semi-major axis and the culated, using two different methods, the total angular velocity Ω (δφ = δΩ δt), offering for torque exerted by a satellite on a disk interior the first time the possibility to× confront directly (or exterior) to its orbit. This torque is called the planetary migration theory with observations, the one-sided Lindblad torque, because it can be and to give insights and constrains on the physical computed as the sum of the torques exerted at properties of the rings and of the moonlets. Lindblad resonances with the secondary body. In In this paper, we address the question of the the lowest order, local approximation, the inner theoretical migration rate of these propellers, us- and outer torques are equal and opposite: the ing both numerical and analytical approaches. torque exerted by a satellite on a disk located The theory is then confronted to observations. In inside its orbit is negative, with the same abso- Sect. 2, we review the standard theory of type I lute value as the positive torque exerted on a disk migration ; the differences between migration in located outside the orbit. protoplanetary disks and in Saturn’s rings are ex- Reciprocally, a disk exerts a torque on the sec- plained, showing the need for a new calculation of ondary body. When the strictly local approxi- the migration rate of embedded moonlets. This mation is relaxed, the inner and outer torques rate is given in Sect. 3 for an homogeneous, ax- are not exactly equal and opposite (Ward 1986). isymmetric disk with a flat surface density profile, Their sum, called the differential Lindblad torque, as a result of numerical computation in Sect. 3.1, is generally negative. As a consequence, the or- and analytical calculation in Sect. 3.2. In Sect. 4, bital angular momentum of a body embedded in we consider the effect of density fluctuations in the a disk decreases, and so does its semi-major axis rings, in particular the role of short-lived gravitat- (on circular Keplerian orbits, the orbital angular ing clumps, also called gravity wakes, which are momentum is proportional to the square root of known to be numerous in the A ring (Colwell et al. the semi-major axis). This is planetary migration 2006). We then conclude in Sect. 5 on what our of type I (Ward 1997). So far, this phenomenon model tells us on the properties of the rings, given has been mainly studied in the frame of plan- the observed migration rates. ets embedded in protoplanetary gaseous disks (see Papaloizou et al. 2007, for a review). 2. Review of Type I migration and the dif- The Cassini spacecraft has been orbiting the ferential Lindblad torque Saturnian ring system since 2004, offering the possibility to observe the coupled evolution of the In protoplanetary gaseous disks, the perturba- ring system and the satellites. Due to short orbital tion caused by a terrestrial planet leads to the for- timescales (1 year is equivalent to about 700 orbits mation of a one armed spiral density wave, leading of the A ring) it may be possible to observe the ex- the planet in the inner disk, and trailing behind change of angular momentum between the two sys- the planet in the outer disk. This wave is pressure tems. One of the most striking discoveries of the supported and generally called the wake, but it has Cassini spacecraft is the observation of propeller nothing to do with the gravity wakes mentioned shaped features in the A ring (located between above : the latter are local features, while the 122000 and 137000 km from Saturn), with longi- planet wake spirals through the whole disk. The

2 planet wake carries angular momentum, so that discussed above, the horseshoe drag –exerted on the angular momentum given by the planet to the the planet by the gas on horseshoe orbits around disk is not deposited locally (e.g. Crida et al. 2006, the planetary orbit – plays a significant role in Appendix C). Therefore, the disk profile is hardly type I migration (see e.g. Ward 1991; Masset 2001; modified in this linear regime. Still, the negative Baruteau and Masset 2008; Kley and Crida 2008; torque exerted by the outer disk on the planet Paardekooper and Papaloizou 2009; Paardekooper et al. through the wake is larger in absolute value than 2009). the positive torque from the inner disk. Without In contrast to gaseous protoplanetary disks, going into the details (for which the reader is ref- pressure effects in Saturn’s rings are not impor- ereed to Ward 1997), the main reason the outer tant. The aspect ratio h is of the order of 10−7. disk wins over the inner disk lies in pressure ef- The spiral density waves that are observed in the fects: from the dispersion equation of a pressure A ring are gravity supported, not pressure sup- supported wave, one finds that the location rL,m ported. Thus, the standard theory of type I migra- where the wave with azimuthal mode number m, tion does not apply. In particular, the spiral planet corresponding to the mth Lindblad resonance with wake doesn’t appear. The interaction of the moon- the planet, is launched, is not exactly the location let responsible for the propeller structure with the of the resonance given by Kepler’s laws. The shift disk is observed to take place within a few hundred is not symmetrical with respect to the planet posi- kilometers. Resonances with m & 103 are located tion for inner and outer resonances, but favors the within this distance and should play a significant outer ones. As a consequence, the planet feels a role. However, in the standard type I migration negative total torque, called the differential Lind- the important resonances have m 1/h 107. blad toque, and given by Tanaka et al. (2002) : Therefore, Eq. (1) can’t be directly∼ applied∼ to a 2 4 2 −2 moonlet in Saturn’s rings. A new approach is Tdiff = Cq Σr Ω h , (1) − p p needed, adapted to the two main characteristics where the index p refers to the planet, rp being of the problem : the fact that the rings are made the radius of its orbit and Ωp its angular velocity, of solid particles, and the fact that the interaction q is the planet to primary mass ratio, Σ is the is taking place very close to the moonlet. surface density of the disk in the neighborhood of the planetary orbit. Finally, h is the aspect 3. The ring-moonlet interaction ratio of the disk, being the ratio between its scale In this section, we compute the interaction be- height and the distance to the central body, being tween a moonlet and a ring test particle. In this proportional to the square root of the gas pressure. analysis, the gravity of the other ring particles is In a typical protoplanetary disk, h 0.05. The neglected. This leads to the torque exerted on numerical coefficient C is given by C≈ = 2.340 the moonlet by an initially unperturbed, homoge- 0.099ξ, where ξ is the index of the power law of− neous ring. In subsection 3.1, the computation is the density profile : Σ r−ξ. ∝ performed numerically. In subsection 3.2, it is de- This result is robust. In particular, the value of rived analytically. Both results are in agreement, the negative torque is almost independent on the and a corresponding migration rate for the moon- slope of the density profile ξ. This is due to the so- let is given and discussed in subsection 3.3. called pressure buffer : the resonances are shifted From now on, m denotes the mass of the moon- when the density gradient varies (Ward 1997). If let (and not anymore the order of a resonance). the disk were pressureless, then the expression of The moonlet, has a circular orbit of radius r C would be completely different. Also, the aspect m around the central planet of mass M. The grav- ratio h in Eq. (1) appears because r doesn’t L,m itational potential due to the moonlet is Ψ. The converge towards r when m tends to infinity, but p radial and azimuthal components of the equation towards r (1 2h/3), due to pressure effects. To p of motion of a ring particle in two dimensional sum up, the gas± pressure plays a fundamental role cylindrical polar coordinates (r, φ) are in type I migration. It should be mentioned for completeness that, d2r dφ 2 ∂Ψ GM r = (2) in addition to the differential Lindblad torque dt2 − dt − ∂r − r2

3 d2φ dr dφ 1 ∂Ψ and r +2 = . (3) merical integrations with various b in the case of dt2 dt dt − r ∂φ a moonlet of mass m =3 10−12M, starting the × particle at an azimuth φ0 = 3000 r /r = 0.3 The angular velocity of the moonlet is ω = | | H m 3 (where the moonlet is at φ = 0). This angle is GM/rm . Let r0 be the radius of the initially large enough so that at this location the influence circularp orbit of a test particle and Ω = GM/r 3 0 of the moonlet is negligible and the orbital pa- its angular velocity. We note b = r0 pr is the m rameters of the test particle are not disturbed, as impact parameter, and ˆb = b/r the− normalized H will be checked later. We find that the horseshoe m 1/3 impact parameter, where rH = rm 3M is the regime occurs for ˆb< 1.8, and the scattered regime Hill radius of the moonlet. occurs for ˆb > 2.5. For 1.774 < ˆb < 2.503, however, the trajectory approaches the center of the 3.1. Numerical computation of the ring moonlet to within a distance smaller than 0.95 r . moonlet interaction H In that case, if one assumes the moonlet is a point In this section, the numerical integration of the mass, the test particle eventually leaves the Hill above equations of motion is performed, in order sphere, either on a horseshoe or a circulating tra- to find the trajectories of ring particles in the pres- jectory, but the outcome changes several times ence of a perturbing moonlet in the frame corotat- with increasing ˆb. In the case we are concerned ing with the moonlet. To measure the tiny asym- about here, the moonlet most likely almost fills metry between the inner and the outer part of the its Roche lobe, and therefore we stop the inte- ring, the full equations are integrated, without lin- gration of the trajectory as soon as the distance earization or simplification. A Bulirsch-Stoer al- between the test particle and the moonlet is less gorithm (Press et al. 1992) is used, and a Taylor than 0.95 rH , assuming a collision. expansion is performed in the code when neces- The specific orbital angular momentum J = sary to subtract accurately large numbers, in order r2(dφ/dt) of the test particles is computed along to achieve machine double precision (10−16). We the trajectories. Angular momentum is exchanged have checked that the Jacobi constant is conserved during the close encounter with the moonlet. For to this precision along the trajectories. Examples ˆb > 2.503, the test particle is scattered onto an of obtained trajectories are given in Fig. 1. eccentric orbit of larger angular momentum than It is well known within the framework of the re- initially, which results in a gain in angular mo- stricted 3-body problem that if ˆb is small enough, mentum. The variation of orbital angular momen- the test particle has a horseshoe| | shaped orbit, tum along the trajectory is shown in the bottom while if ˆb is larger than 2.5, the test particle is panel of Fig. 2 for the case ˆb = 3, where the top circulating,| | and scattered∼ into an eccentric orbit. panel is the trajectory. The difference in angu- This can be seen in Fig. 1. We perform many nu- lar momentum between the initial circular orbit at φ0 =0.3 sgn(b) and the end of the integration, when φ = 0.3 again, is noted ∆J. In the figure, only the| | interval 0.05 <φ< 0.05 is displayed, 1.001 for convenience. Most− of the exchange of angular

φ momentum occurs when φ < 0.01. 1 | | ) cos

m Figure 3 shows ∆J (top thick curve) as a func- | |

(r / r ˆ 0.999 tion of b, in units of the speciﬁc angular momen- 2 ˆ tum of the moonlet Jm = rmω. For0 < b 6 1.774, -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 the horseshoe trajectory corresponds to a U-turn (r / r ) sin φ m towards the central planet, and to a loss of angular momentum for the test particle. More precisely, as 1 2 Fig. 1.— Trajectories of test particles perturbed for circular orbits J r / , one expects for such a −12 1 ∆r∝ by a moonlet of mass m = 3 10 M, located U-turn ∆J/J = 2 = b/rm ; this is indeed the × r − at (r = rm, φ = 0) (that is at (0, 1) in the plot), case for ˆb< 1.3. In the case where the test particle in the frame corotating with the moonlet. Dashed collides with the moonlet, we assume that it gives circle : orbit of the moonlet. all its orbital angular momentum to the moon-

4 2 let : ∆J = r0 Ω Jm, so that ∆J/J b/(2rm). This also appears− in Fig. 3. The opposite≈ holds 5 for b< 0.

Computing ∆J as a function of b to numer- H 4 ) / r m ical precision enables us to also compute the 3 diﬀerence between the inner and outer disk : ( r - 2 δJ(b) = ∆J(b) + ∆J( b). This quantity is − -0.04 -0.02 0 0.02 0.04 small with respect to ∆J(b), but nonetheless well 2 determined and converged in our simulations :

] 1 ω 2 m

∆J(b) + ∆J( b) is constant after the encounter r 0

− -5 to a precision better than 0.5% for all φ > 0.02. -1

| | J [10 This validates our choice of φ0. In Fig. 3, the ∆ -2 bottom thick dashed curve shows δJ in the same -3 scale as ∆J . We see that δJ > 0 for all b > 0 -0.04 -0.02 0 0.02 0.04 φ and that|δJ | ∆J, with ≪ Fig. 2.— Top panel: trajectory of a test particle −4 δJ/∆J 5 10 ˆb (4) with impact parameter ˆb = 3 ; the motion of the ≈ × particle is toward negative φ. Bottom panel: vari- for circulating trajectories, and ation of the speciﬁc orbital angular momentum J − of the same particle along its trajectory. δJ/ ∆J 1.17 10 4 ˆb | |≈ × for horseshoe orbits. In the following subsection, the empirically found Eq. (4) is derived analytically and justiﬁed.

0.001 ∆ | J|num 3.2. Analytic model for the ring moonlet ∆ ( J)Eq. (32) 0.0001 δ ( J)num interaction ∆ 4.92(b/rm)( J)num (b/r ) J 1e-05 m m In this section, we consider only circulating tra- (b/2rm) Jm

1e-06 jectories. Developing to second order the exchange ] ω 2 of angular momentum during an encounter with m [ r the moonlet ∆J, we can ﬁnd the asymmetry δJ. 1e-07

1e-08 3.2.1. Solution for the perturbed moonlet orbit 1e-09 Let us start again from Eq. (2) and (3). The 1e-10 ring particle is assumed to be on an unperturbed 1 10 b / rH circular orbit of radius r0 = rm + b. It orbits with angular velocity Ω = GM/r 3 such that φ =Ωt, 0 Fig. 3.— Angular momentum exchanges during where without loss ofp generality we have deﬁned one close encounter, as a function of the impact pa- the origin of time t = 0 to be when the particle rameter. Top, thick, red curve : ∆J(b), from nu- is at φ = 0. Under the perturbation induced by merical simulations. Green, thin, dashed, straight Ψ, the particle moves to r = r + x, and φ = 0 line : ∆J(b), as given by Eq. (32). Bottom, thick, Ωt+y/r , where x and y are assumed to be small. m dark blue, long-dashed curve : δJ(b), from nu- Linearizing Eq. (2) and (3) about the circular orbit merical simulations. Thin, light blue, dash-dotted state, we obtain equations for x and y in the form line : δJ(b) as given by Eq. (34), taking ∆J from d2x dy ∂Ψ the simulations. Orange, thin, double- and triple- 2Ω 3Ω2x = and (5) dt2 − dt − − ∂r dashed lines : (b/rm)Jm, and (b/2rm)Jm, respec- 0 tively, to compare with ∆J . | | d2y dx 1 ∂ Ψ + 2Ω = (6) dt2 dt − r ∂φ 0 0

5 Here the subscript 0 denotes evaluation on the un- where it is implicit that the real parts of such com- perturbed particle orbit. plex expressions is to be taken, and

We suppose that the perturbation is induced by 2π/β a moonlet of mass m that is on a circular orbit of 1 Sn = S(t) exp( inβt)dt . (13) radius rm and has an angular velocity ω. Then its 2π Z0 − azimuthal coordinate φ = ωt + φ , with φ m m,0 m,0 The periodic solution of Eq. (11) is now readily being a constant. The perturbing potential written down as Gm Ψ= (7) n=∞ 2 2 Sn exp(inβt) r + r 2rmr0 cos(φ φm) x = . (14) 0 m − − (Ω2 n2β2 + iγnβ) p n=X−∞ becomes a function of time through substituting − φ φm = (Ω ω)t φm,0 therein. We may write this in terms of a Green’s function − − − Thus we have deﬁned through 1 ∂Ψ 1 ∂Ψ n=∞ . (8) 1 exp(inβτ) r ∂φ ≡ r (Ω ω) ∂t G(τ)= . (15) 0 0 0 0 2π (Ω2 n2β2 + iγnβ) − n=X−∞ − Using this in Eq. (6) and integrating with respect to time, we obtain Then the solution for x may be written

2π/β dy Ψ ′ ′ ′ + 2Ωx = , (9) x = β S(t t )G(t )dt . (16) dt − r (Ω ω) 0 0 Z0 − − which when combined with Eq. (5) gives an equa- Note that as the orbit of a ring particle relative tion for x in the form to the planet is periodic, the solution given by 2 Eq. (16) includes the effects of infinite numbers of d x 2 ∂Ψ 2ΩΨ +Ω x = + = S (10) repeating encounters. However, we wish to con- dt2 − ∂r r (Ω ω) 0 0 sider the case when dissipative effects, although − 3.2.2. Solution of the linearized equations weak, are strong enough to recircularize orbits between encounters in which case they will be in- To solve Eq. (10), we note that the perturbing dependent of each other. This condition requires potential Eq. (7) evaluated on the unperturbed that γ/ ω Ω = γ/β 1. This is equivalent to re- orbits is a periodic function of time with period quiring| that− the| damping≫ time scale be short com- 2π/ ω Ω =2π/β. Thus we should look for a peri- | − | pared to the relative orbital period between moon- odic response. In order to do this we have to intro- let and ring particle. On account of the length duce a small frictional term into Eq. (10) to enable scale of the encounters of interest being compara- transients to decay and a net torque on the moon- ble to the Hill radius of the moonlet, this is much let to be set up. When the frictional term is small longer than the orbital period itself, so that we it is expected that the resulting torque should not may adopt the ordering depend on it (e.g. Goldreich and Tremaine 1980). Hence we add a frictional term γ(dx/dt) to the γ/ ω Ω = γ/β 1 γ/Ω. (17) left hand side of Eq. (10), where γ/Ω is a small | − | ≫ ≫ constant parameter so that it now reads In order to make use of the above ordering we write down the form of the Green’s function derived in d2x dx ∂Ψ 2ΩΨ +γ +Ω2x = + = S. the appendix (see Eq. (A7) ) valid for 0

6 where ω = Ω2 γ2/4. The function is de- Thus A and B are constants representing epicyclic γ − ﬁned elsewherep through its periodicity with pe- oscillation amplitudes induced after the close ap- riod 2π/β. Making use of the inequality Eq. (17) proach of the ring particle to the moonlet. we may replace the Green’s function Eq. (18) by We remark that the approximations made in the simple expression obtaining Eq. (23) and Eq. (24) relate to how dissipation is treated. There has been no assump- exp( γτ/2) sin(ω τ) G(τ)= − γ . (19) tion that the particle trajectories are symmetric ωγβ on opposite sides of the moonlet so that curvature eﬀects remain fully incorporated during particle Then the solution Eq. (16) gives moonlet encounters. When dissipation is negligi- −γt/2 −γt/2 ble during the encounter, then immediately after- x = A(t)e sin(ωγ t)+ B(t)e cos(ωγ t), (20) ward an epicyclic oscillation is established. The where assumption that dissipation circularizes orbits between encounters implies that we should consider t 1 ′ ′ ′ ′ approaching ring particles to be on circular orbits. A(t)= S(t ) exp(γt /2) cos(ωγ t )dt ωγ Zt−2π/β The above discussion indicates that errors associ- (21) ated with this assumption are exponentially small. and

t 1 ′ ′ ′ ′ 3.2.3. Angular momentum transfer B(t)= S(t ) exp(γt /2) sin(ωγt )dt . −ωγ Zt−2π/β For the set up considered here, symmetry con- (22) siderations imply that S(t) is an even function of To make use of the above expressions, we con- time (see below), so that B = 0. The generation sider the situation when the ring particle has a of the epicyclic oscillation is associated with an close encounter with the moonlet at time t = 0, angular momentum transfer between the moonlet thus we take φm.0 = 0 (we note that a non zero and particle. To find this we firstly note that φm,0 can be dealt with by rotating the coordinate system and shifting the origin of time). The 2 ∆J = √GM af (1 e ) √r0 , (25) source term S is then expected to be highly peaked q − − around t′ =0, and almost all of the contributions to the above integrals will occur for t′ < 2π/Ω. where af and e are the post encounter semi-major Furthermore, during this dynamical| | interaction,∼ axis and eccentricity of the particle. We also note dissipation will be negligible. Thus, if we are in- that the Jacobi constant implies that the change of terested in times after the main interaction, but the particle orbital energy and angular momentum are related by ∆E = GM[1/(2r0) 1/(2a )] = before significant dissipation takes place, we may − f set γ = 0 and extend the limits of the integra- ω ∆J. This can be used to eliminate af in Eq. (25) tion to . However, in practice one may have to after which ∆J may be found correct to second apply a±∞ cut off to the potential at large distances order in e A/r0 with the result that ∆J = Ω2A2/[2(ω ≡Ω)]. This in turn may be simply de- from the moonlet in order to do that (see below). − But this should not matter if the important inter- termined after evaluating A. Note that ∆J < 0 action occurs when the moonlet and ring particle for particles interior to the moonlet which have are close. Ω >ω and conversely ∆J > 0 for particles orbiting exterior to the moonlet. Then we simply have ∞ 1 ′ ′ ′ A = S(t ) cos(Ωt )dt (23) 3.2.4. Development of the perturbing potential Ω Z−∞ We now consider

and ∞ 1 ′ ′ ′ ∂Ψ 2ΩΨ B = S(t ) sin(Ωt )dt . (24) S(t)= + . (26) − ∂r r0(Ω ω) −Ω Z−∞ 0 −

7 We begin by recalling that The associated angular momentum exchanged is then given by Gm Ψ= 2 2 2 − r0 + rm 2r0rm cos(φ φm) 2(Gm) 1 2Ω − − ∆J = K0(ξ0) p r3r (ω Ω)3 × 2 − (Ω ω) Gm 0 m − − = . (27) 2 2 2 − r0 + rm 2r0rm cos(βt) 1 rm − +K1(ξ0)ξ0 + . (31) In order to evaluatep the Fourier transform as spec- 2 (r0 rm) − ified by Eq. (23), which was derived under the In a strictly local approximation under which assumption that the interaction occurs only near the inner and outer sides are symmetric, the con- closest approach, we must truncate the potential tributions from orbits equidistant from the moon- at large t . As the encounter takes place over a let would cancel, leaving the net result to be de- time |1/β,| this can be achieved by replacing termined by the surface density profile. However, cos(βt≪) in Eq. (27) by 1 β2t2/2. Note that a although we have assumed the interactions are lo- dimensionless estimate of− the error involved is of cal, we did not assume symmetry between the ex- order (β/ω)2 (r /r )2, where r is the Hill ra- H m H terior and interior orbits. Accordingly we evalu- dius of the moonlet.∼ This is small enough that the ate the difference in the magnitude of ∆J eval- leading order asymmetry in the angular momen- uated from orbits equidistant from the moonlet : tum transferred to orbits with the same impact r = r b. The leading order contribution to ∆J parameter on either side of the disk can be esti- 0 m is symmetric± in b. The lowest order contribution mated. is antisymmetric and accordingly leads to cancel- As the first stage in evaluating the Fourier lation between the two sides. We make use of the transform of S specified in Eq. (23) that gives the 2 expansions ξ0 =2/3 b/(2rm)+ O((b/rm) ), and epicyclic amplitude, we evaluate − 2Ω/(Ω ω)= 4rm/(3b)(1 b/(4rm)) + O(b/rm) ∞ − − − 1 together with standard properties of Bessel func- C = Ψ cos(Ωt)dt tions to write Ω Z−∞ ∞ 1 Gm cos(φ) dφ = . 2 β 2 2 2 2 64(Gm) rm 2 b −Ω Z−∞ (r r ) Ω /β + r r φ Ω ∆J = (2K0(2/3)+ K1(2/3)) 1+ α , Ω 0 m 0 m 243ω3b5 r − (28) m p (32) This can also be expressed as where 2GmK0(ξ0) C = , (29) 3 (6K1(2/3)+3K0(2/3)) − Ωβ√r0rm α = + =2.46 . (33) 4 (4K0(2/3)+2K1(2/3) where ξ0 = (Ω r0 r )/(β√r0r ), and K de- | − m| m j The first order term of Eq. (32) was already given notes the modified Bessel function of the second by Goldreich and Tremaine (1980). It is plotted kind of order j. as a straight green dashed line in Fig. 3. Our expansion to second order enables us to go further, 3.2.5. Total angular momentum exchange and to give the expression of the magnitude of the We may now use the above expression together asymmetry between the two sides of the disk : with Eq. (26) to evaluate the epicyclic amplitude δJ Eq. (23) (noting that the radial derivative is with =2α b /r =4.92 b /r . (34) ∆J | | m | | m respect to r0 with other quantities held fixed) so obtaining It is such that for an orbit with a given impact parameter, the angular momentum exchanged in 2Gm 1 2Ω A = K0(ξ0) the outer disk is the larger. −Ωβr0√r0rm × 2 − (Ω ω) − In the case studied numerically, we had rH = −4 −4ˆ 1 rm 10 , so that b /rm = 10 b. Then, Eq. (34) + K1(ξ0)ξ0 + . (30) | | 2 (r0 r ) remarkably agrees with the numerical fit Eq. (4). − m

8 The light blue dot-dashed curve in Fig. 3 displays material executing horseshoe turns, called the 4.92 10−4 ˆb ∆J. horseshoe drag, is for a Keplerian disk with flat × In the context of the above, we note that ap- density profile : proximations made in obtaining equation Eq. (31) 9 4 2 such as effectively starting and truncating the in- THS = Σw ω , (37) 8 teraction at some finite though large distance from the moonlet could conceivably lead to changes where w is the half-width of the horseshoe region. comparable to those given by Eq. (34). However, In our case, w = 1.774 rH , which gives THS = −2 4/3 2 2 such changes are again approximately symmetric 2.6 (Σ/Mrm )(m/M) Mrm ω . The agreement for trajectories on both sides of the moonlet and is good because Ward’s analysis is based only on thus approximately cancel so we do not expect geometrical effects and angular momentum vari- such effects to significantly alter Eq. (34). ation in a Keplerian disk, without any pressure effect. Therefore, it also applies in Saturn’s ring. 3.3. Migration rate and discussion We remark that taking w =2rH in Eq. (37) gives a perfect match with what we find numerically for If the surface density of ring particles is Σ, the the total horseshoe drag. total rate of angular momentum transferred to the moonlet is In conclusion, from Fig. 4, the total torque felt by a moonlet of mass m on a circular orbit of ra- dJ ω Ω dius r around a planet of mass M can be written = Σ ∆J | − |dr rdφ , (35) m dt − Z Zdisk 2π as 4/3 where the integral is taken over the disk. The Σ m 2 2 T = 17.8 −2 Mrm ω . (38) particles exterior to the moonlet contribute neg- − Mrm M atively while those interior contribute positively. Note that to get the same dependency of the type I The cumulative torque exerted by the moonlet on torque in the parameters of the system, one has the region of the ring located within a distance b to assume h r /r in Eq. (1) ; however, in a to its orbit reads then : H m protoplanetary∝ disk, h is fixed and independent b ′ ′ ′ of the mass of the secondary body, so that this Tc(b)= Σ(rm + b )(∆J(b )) ω Ω db (36) proportionality would not be justified. Z−b | − | The torque is related to the migration speed The normalized cumulative torque through T = 0.5 m rmΩ(drm/dt). Hence we de-

4/3 −2 duce that Tc(b)/ (m/M) (Σ/Mrm ) 2 1/3 h i drm Σrm m = 35.6 rmΩ . (39) is plotted in Fig. 4. The proportionality to Σ is dt − M M 4/3 obvious ; that Tc (m/M) is numerically ver- −15∝ −9 The migration rate is here proportional to the iﬁed for 3 10 6 m/M 6 3 10 , and has mass of the moonlet to the power 1/3, in con- been already× found analytically× by Ward (1991) trast to standard type I migration where drm/dt for the horseshoe drag in a similar context. m. A numerical application to the case of an∝ −18 8 Most of the total torque comes from scattered, m = 10 MSaturn = 5.68 10 kg moonlet in circulating particles, in particular the ones with orbit in the A-ring of density× Σ = 400 kg m−2 smallest impact parameter ˆb 2.5. This makes ≈ at rm = 130000 km from Saturn gives drm/dt = the total torque sensitive to the physical size of the 0.23 myr−1. Increasing the mass by two orders − moonlet (taken as 0.95 rH here), as some particles of magnitude to correspond to a radius of 200 m colliding with the moonlet could be circulating if speeds up the migration rate by a factor∼ of only it were smaller. 4.5 to 1 myr−1. ∼ ∼− The role of the horseshoe drag appears to be After time t, a migrating propeller will be non negligible, amounting to shifted longitudinally with respect to a corre- −2 4/3 2 2 4.1(Σ/Mrm )(m/M) Mrm ω . The expres- sponding non migrating one by a distance rm δφ = sion∼ of Ward (1991) for the torque arising from 3Ω dr /dt t2/4. For the above parameters, this | m |

9 gives 713 [t/(1 year)]2 m. A shift of this magni- density fluctuations. The effect of these density tude is potentially detectable on a timescale of a fluctuations on the moonlet is studied in next sec- year to a few years (Porco et al. 2004 and note tion. also Burns et al. 2009). Actually, migration of propellers has already been detected (Burns et al. 4. The role of density fluctuations and re- 2009; Tiscareno et al. 2010). During one time sulting stochastic migration period of nearly a year, a particular propeller has been seen moving outward at a rate of The analytic calculations and numerical simu- 110 myr−1 ; and during a later similar time pe-∼ lations in the previous chapters assume an inflow riod, the same propeller has been seen moving in- of particles on circular orbits only perturbed by ward at a rate of 40 myr−1 (Tiscareno et al. the nearby moonlet. However, we know that Sat- 2010, and personal∼ communication). urn’s A ring is marginally gravitationally stable (Daisaka et al. 2001). The Toomre Q parameter These observations are not compatible with the (Toomre 1964), which is a measure of the impor- above theory. But we recall that the process of tance of self-gravity, is expected to be of the order migration of a moonlet described above assumed a of 2 7, indicating that the ring particles’ mutual smooth particle disk with constant surface density. gravity∼ is indeed a strong effect. It leads to the Here we note that there are features and mech- regular formation and dispersion of gravity wakes, anisms that might produce a significantly faster which are local density enhancements elongated in migration rate, possibly in both directions inward parallel directions by the Keplerian shear. Those and outward, and non constant in time. One can over-densities give rise to stochastic forces which first think of a radial density gradient : as there is act on the embedded moonlet. no pressure buffer here, this would directly affect the balance between the torques from the inner A very similar effect is expected to occur in and outer parts of the ring. This would also affect protoplanetary disks. These disks are thought to the torque from the horseshoe region, which could be turbulent due to the magneto-rotational in- turn positive. However, if the migration is gov- stability (MRI, Balbus and Hawley 1991). The erned by the gradient of some quantity, it seems turbulent fluctuations create over-densities which likely that the moonlet would have approached an interact gravitationally with embedded small extremum in that quantity, and thus should have mass planets. The stochastic forces make the attained a migration rate comparable to that esti- planet undergo a random walk. An analytic mated for a constant surface density. model of this random walk has been derived by Rein and Papaloizou (2009). In the following, we Another possibility resulting in the moonlet apply this model to moonlets embedded in Sat- migrating faster than what the previous calcu- urn’s rings. To do that, we need to get an estimate lation indicates, and possibly outward, is a run- of the amplitude of the stochastic forces. away migration in a planetesimals disk (Ida et al. 2000; Levison et al. 2007, for a review), similar to 4.1. Numerical calculations the type III migration of planets in protoplanetary disks (Masset and Papaloizou 2003). In this We perform three-dimensional simulations of regime, the migration of the moonlet in the disk ring particles, in a shearing box, similarly to Salo leads to a positive feedback on its migration rate, (1995). The simulations are done in a local cube because of the material of the inner (resp. outer) of size H with shear periodic boundary conditions, disk making horseshoe U-turns to the outer (resp. and the origin of the box is fixed at a semi ma- inner) disk. This speeds up the migration, pos- jor axis of a = 130000 km. A BH tree code sibly leading to a runaway. However, this leads (Barnes and Hut 1986) is used to calculate the inevitably to an asymmetry in the horseshoe re- self-gravity between ring particles and resolve in- gion, while the propeller structures observed are elastic collisions. Collisions between particles are rather symmetrical. resolved using the instantaneous collision model Finally, the A-ring of Saturn is not homoge- and a velocity dependent coefficient of restitution neous. It is close to gravitational instability, which should lead to the formation of gravity wakes and

10 given by Bridges et al. (1984) :

v −0.234 ǫ(v) = min 0.34 −1 , 1 , (40) × 1 cm.s 18 where v is the impact speed projected on the vec- 16

tor joining the centers of the two particles. The ] 2

ω 14 2 code is described in more detail in Rein et al. m 12

(2010). ) [Mr -2 m 10

The size of ring particles is not well constrained. /Mr Σ Therefore and to be able to scale to diﬀerent loca- /( 8 -4/3 tions in Saturn’s rings, we perform multiple sim- 6

ulations. For a given simulation, all the particles (b)*(m/M) 4 c are assumed to be spherical and have the same size T 2 (or radius), which varies from simulation to simu- 0 lation from 0.52 to 13 meters. The simulation pa- 0 1 2 3 4 5 6 7 8 9 10 rameters are listed in Table 1. The nomenclature b / rH and physical parameters are, for easy comparison, Fig. 4.— Cumulative torque given by Eq. (36), the same as in Lewis and Stewart (2009), as our exerted by a moonlet on the region of the ring simulations are similar to theirs. rm b

Gmpart -4e-08 ˆf = ˆr, (41) − ˆr 2 + d2 -8e-08 | | 1e-09 where ˆr is the vector linking the origin to the 5e-10 0 S3 particle and mpart is the mass of the particle. -5e-10 The smoothing length d is set equal to the moon- -1e-09 let’s size. In a self-consistent simulation, one 1e-10 should include the moonlet with it’s real physi- 5e-11 0 S1 cal size. However, this goes beyond the scope of -5e-11 this paper and will be considered in future work -1e-10 0 5 10 15 20 (Rein and Papaloizou 2010). Our purpose here is time [orbits] to estimate the underlying stochastic ﬂuctuations in the migration rate that occur independently of Fig. 5.— Azimuthal component of the speciﬁc − the moonlet. This procedure is reasonable as long gravitational force felt by the moonlet in m s 2 for as the moonlet is in a steady state, namely if it simulations L2, S3 and S1 (from top to bottom). doesn’t accumulate or lose a large amount of mass over one orbit. In all our simulations we assume a moonlet size of d = 200 m.

11 Name ra τ ρp Σp H N L2 13 m 0.1 0.5 g cm−3 885 kgm−2 5000m 4808 S1 0.52 m 0.1 0.7 g cm−3 49.7 kg m−2 1000 m 120548 S3 1.3 m 0.2 0.7 g cm−3 246 kgm−2 1000m 38188

Table 1: Simulation parameters. The first column identifies the simulation, following the convention of Lewis and Stewart (2009). The second and third column give the size (or radius) of the particles and their density. The fourth and fifth column give the surface density and the size of the computational domain, respectively. The last column lists the number of particles.

4.2. Results walks with standard deviation given as a function of time by ∆a = 0.27 t/(1 year) m, ∆a = We measure the amplitude and the correlation 2.7 t/(1 year) m, and ∆ap= 270 t/(1 year) m time of the stochastic forces in all simulations. respectively. The results are listed in Table 2. We also plot p p the time evolution of the azimuthal (y) force com- 4.3. Discussion ponent in Fig. 5. Whereas the correlation time in all simulations is almost the same, the amplitude From the above, it can be seen that increasing of the stochastic fluctuations varies by almost a the surface density by factors 4 5 changes the factor of 103. The forces in the vertical direction migration rate by two orders of magnitude.− Thus, are negligible and not presented here. The diffu- the results show clearly that the surface density Σ sion coefficient, being a measure of the strength is much more important than in the regular, type of stochastic forces, is defined as D =2 f 2 τ (see I like migration, presented in Sect. 3. This can h i Rein and Papaloizou 2009), where f 2 1/2 and τ be easily understood with a toy model. The crit- h i are the root mean square value and the approx- ical unstable wavelength λ scales linearly with Σ imate correlation time of the specific stochastic (Toomre 1964). If we assume a fixed moonlet size, force in one direction. the ratio of moonlet size to λ therefore changes The change in semi major axis a due to the with Σ. In the limit where λ is much smaller than effect of stochastic forces with diffusion coefficient the moonlet radius, the stochastic forces are negli- D after time t is given by gible as the density distribution is approximately homogeneous on the relevant scales. This is the 2 ∆a = √Dt, (42) case in simulation S1. In the other limit where λ ω is larger than the moonlet, the moonlet undergoes where ω is the mean motion of the moonlet a random walk that is similar to that of individual (Rein and Papaloizou 2009). Note that in this ring particles, as seen in simulation L2. regime, the acceleration of the moonlet doesn’t The range in migration rates found in sim- depend on its mass (as can be seen in Eq. (41)). ulations shows that over time scales of several Therefore, the migration rate and the diffusion co- years, the migration of a moonlet of mass m efficient are independent of the mass of the moon- −16 ∼ 10 MSaturn may be dominated by a random let ; this might be an observational indication for walk in some situations (eg. those in simulations this migration regime. S3 and L2, the latter carried out with particles Assuming an initial semi-major axis of a = of radius 13 m). However, in regions of the rings 130000 km and D 10−17 m2 s−3 as found in ∼ where the surface density is small (eg. simula- simulation S1, one can calculate the expected tion S1), the moonlet may be in a regular, non- difference in semi major axis after one orbit stochastic migration regime. In that case, the due to stochastic forces which turns out to be model from Sect. 3 can be applied. ∆a = 0.01m. For simulation S3, assuming This dependence offers an exciting possibility D 10−15 m2 s−3, one finds ∆a =0.11m. For the ∼ −11 2 −3 to constrain the nature of the ring particles and simulation L2, assuming D 10 m s , one the physical processes occurring in the rings by finds ∆a = 10.5 m. These∼ translate to random

12 Simulation Correlation time [s] Diffusion coefficient [m2 s−3] Q τx τy Dx Dy L2 cutoff 5000 7000 9.61 10−12 14.97 10−12 4.1 smooth 6000 9000 5.34 × 10−12 9.11 ×10−12 × − × − S1 cutoff 2000 4000 1.92 10 17 2.26 10 17 7.2 smooth 3000 8000 1.59 × 10−17 1.69 × 10−17 × − × − S3 cutoff 2000 6000 7.50 10 16 30.24 10 16 2.5 smooth 4000 10000 8.87 × 10−16 20.48 × 10−16 × × Table 2: Simulation results. The first column gives the name of the simulation, as defined in Table 1. The second and third columns give the correlation time in the x (radial) and y (azimuthal) direction, respectively. The fourth and fifth columns list the diffusion coefficients. The sixth column is the Toomre Q parameter, as measured in the simulation.

measuring the migration of moonlets. But note propellers in Saturn’s rings. Nonetheless, density that because regular migration gives a decrease in fluctuations in the rings, due to their proximity the semi-major axis that is linear in time, provided to gravitational instability, can lead to stochastic it continues to operate, it will always ultimately torques on a moonlet, that may dominate on the dominate the behavior for large time because the timescale of the Cassini mission, to an extent that spreading of the semi-major axis associated with depends mainly on the surface density of the rings. stochastic migration increases only as the square These stochastic torques may account for the ob- root of time. servations. At the present day, the number of observed mi- The possibility that the migration of propellers gration rates is not sufficient to draw a statisti- is induced by stochastic processes rather than by cally significant conclusion. However, the fact that a regular type I like migration is therefore very the migration varies in rate and direction clearly exciting : this may help to infer the local surface favors the stochastic migration model presented density, and therefore the size of the ring parti- in this section. Considering a migration rate of cles. Indeed, both quantities are linked through ∆a = 100 m in t = 1 year in Eq. (42), one finds the optical depth, which is observationally well D| =| 1.4 10−12 m2 s−3. Taking ∆a = 40m constrained. Our estimate of Σ 700 kgm−2 is in t = 1× year gives D = 2.2 |10−|13 m2 s−3. equivalent to 10 m size particles∼ in the A ring This is in the range obtained in the× simulations, for a single-sized∼ population. This is in good and tends to favor the case of simulation L2, and agreement with available estimates from stellar oc- Σ 700 kgm−2 in the A ring. cultations. For the A ring, Voyager occultations ∼ (Zebker et al. 1985) find that the radius ra of par- 5. Conclusion ticles follows : 0.1 m (assumed) < ra < 11 m, with a power index 3. For the 28 Sgr occul- In this paper we have calculated the differen- tation (French and Nicholson∼ − 2000), a range 1m tial torque exerted on a moonlet by the outer and < ra < 20m is found, with a power index between the inner disk with a smooth, flat surface den- 2.7 and 3, and an effective size (the single aver- sity profile. We performed both an accurate nu- −age size accounting− for the fluctuations in photon merical integration and a second order analytical count) being about 7m (Cuzzi et al. 2009). calculation. These approaches were found to be Fortunately the Cassini mission has been ex- in excellent agreement where their domains of va- tended to 2017. In the meantime, numerous ob- lidity overlap. The migration rate found in this servations of the propellers will hopefully give a case is proportional to the mass of the moonlet clear picture of their orbital evolution for a pe- to the power 1/3. It is about 1 myr−1 for a riod of 10 years, representing about 10000 orbits. 200 m radius moonlet in the A-ring.− This is way This will allow us to test the hypothesis presented too low to explain the observed migration of the in this paper, and in particular to check whether

13 the propellers really are in stochastic migration, the other hand, an estimate of the lifetime of a whereas first results seem to favor the stochastic Pan size moon ( 14 km in radius) against the hypothesis. today’s cometary∼ flux is provided by Dones et al. Among the questions that still need to be (2009) and gives a range between 100 Myr and addressed is why all the propellers seem to be 16 Gyr, depending on the size distribution of im- gathered in the A ring, in places apparently de- pactors. Therefore, the recent occurrence of such void of density waves ? Indeed the propellers an event, about 4 to 4.5 billion years after solar seem gathered in a couple of narrow radial bands system formation, is possible. Note also that an of only about 1000 km width (Tiscareno et al. age of about 100 Myr is coherent with some esti- 2008). This is especially surprising since the mates of Saturn’s ring age despite of the lack of A ring is densely populated by numerous density fully satisfactory explanation for their origin (see waves launched by the nearby small moons (At- Charnoz et al. 2009a, for a review). las, Prometheus, Pandora, Janus, Epimetheus). We see that the question of propeller’s migra- Is there a systematic mechanism that would eject tion is inextricably linked to the issue of the ori- the propellers away from density waves ? Or does gin of Saturn’s moons embedded in the rings, the present location of propellers just reflect the which is still a mystery. Porco et al. (2007) and initial location of the parent body, assuming that Charnoz et al. (2007) have jointly proposed that the propeller population comprises the fragments small moons embedded in the rings could be ag- resulting from the destruction of an ancient moon gregates of material on an initial shard denser than orbiting within the rings ? ice. When destroyed by meteoritic bombardment, If the moonlets really undergo stochastic mi- these could release dense chunks of material that gration, then Eq. (42) may strongly constrain the could explain the origin of the propellers. How- age of the propellers, which can’t be larger than ever, the origin of Saturn’s ring system is still the time needed to diffuse over ∆a > 1000 km. a matter of debate (Harris 1984; Charnoz et al. Unfortunately, as long as D is unknown Eq. (42) 2009b). Knowledge of the age of the propellers doesn’t provide any useful information. However, could provide important constraints on the age of considering that ∆a is proportional to the square the main ring system and its embedded moons, as root of the time, and assuming that a moonlet there are strong indications that these could have migrates about 100 m in 1 year, one finds that about the same age, provided these moonlets hide ∆a = 1000 km for t = 100 million years. Assum- a dense shard (Charnoz et al. 2007; Porco et al. ing ∆a = 40 m in 1 year, we find that it requires 2007). Understanding the migration rate of the 625| million| years to diffuse over 1000 km. Note propellers is therefore an important piece of this that∼ for 200 m radius moonlets, the spreading due puzzle. to stochastic migration equates to the contraction of the semi-major axes occurring as a result of We thank J. Burns for stimulating discussions smooth, regular migration after 104 yr, and and the organisers of the “Dynamics of Discs and ∼ then ∆a 10 km. Thus it would take about a Planets” workshop at the Isaac Newton Institute ≈ million years to migrate through ∆a = 1000 km in Cambridge where these took place, as well as M. in this case, largely through the action of the non- Tiscareno for providing us with migration rates. stochastic, regular migration process, if that can Hanno Rein was supported by an Isaac Newton be assumed to operate smoothly and simultane- Studentship, STFC, and St John’s College, Cam- ously with the stochastic migration process. A bridge. 100 m radius moonlet would migrate through only 500 km in the same period, so that the smooth, regular migration process spreads a population of moonlets of various sizes over 1000 km in one to two million years. These could be the times since the catastrophic disruption of a small moon orbiting at 130 000 km from Saturn, that was broken into smaller moonlets by a meteoritic impact. On

14 A. Evaluation of the Green’s function Here we evaluate the Green’s function deﬁned by Eq. (15) as

n=∞ 1 exp(inβτ) G(τ)= . (A1) 2π (Ω2 n2β2 + iγnβ) n=X−∞ − To perform the summation we use the general result that if for a general periodic function

n=∞ g(τ)= b(n) exp(inβt), (A2) n=X−∞

with period 2π/β, and b(n) being deﬁned as an integrable function, we set

∞ 1 F (τ)= b(n) exp(inβτ)dn, (A3) 2π Z−∞ then n=∞ g(τ)=2π F (τ +2πn/β). (A4) n=X−∞ We set 1 b(n)= . (A5) 2π(Ω2 n2β2 + iγnβ) − Then the integral Eq. (A3) deﬁning F (τ) is readily performed by contour integration with the result that for t> 0, exp( γτ/2) sin(ω τ) F (τ)= − γ , (A6) 2πωγβ otherwise F (τ)=0. Here ω = Ω2 γ2/4. Using the above to evaluate the sum Eq. (A4) as a geometric γ − progression yields g(τ) G(τ) forp 0 <τ < 2π/β as ≡ exp( γτ/2) sin(ω τ) exp( γπ/β) sin(ω (τ 2π/β)) G(τ)= − γ − − γ − , (A7) ω β [1 + exp( 2γπ/β) 2 exp( γπ/β) cos(2πω /β)] γ − − − γ the function is determined elsewhere by its periodicity with period 2π/β.

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