Nearly-integrable dissipative systems and celestial mechanics

Eur. Phys. J. Special Topics 186, 33–66 (2010)c© EDP Sciences, Springer-Verlag 2010DOI: 10.1140/epjst/e2010-01259-2

THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Review

Nearly-integrable dissipative systems and celestialmechanics

A. Celletti1,a, S. Di Ruzza2,b, C. Lhotka1,c, and L. Stefanelli1,d

1 Dipartimento di Matematica, Universita di Roma Tor Vergata, Italy2 Dipartimento di Matematica, Universita di Roma La Sapienza, Italy

Received 12 July 2010 / Received in final form 3 August 2010Published online 14 September 2010

Abstract. The influence of dissipative effects on classical dynamical models ofCelestial Mechanics is of basic importance. We introduce the reader to the subject,giving classical examples found in the literature, like the standard map, the Henonmap, the logistic mapping. In the framework of the dissipative standard map, weinvestigate the existence of periodic orbits as a function of the parameters. We alsoprovide some techniques to compute the breakdown threshold of quasi-periodicattractors. Next, we review a simple model of Celestial Mechanics, known as thespin-orbit problem which is closely linked to the dissipative standard map. In thiscontext we present the conservative and dissipative KAM theorems to prove theexistence of quasi-periodic tori and invariant attractors. We conclude by reviewingsome dissipative models of Celestial Mechanics. Among the rotational dynamicswe consider the Yarkovsky and YORP effects; within the three-body problem weintroduce the so-called Stokes and Poynting–Robertson effects.

1 Introduction

The trajectory of the celestial bodies is often modeled by a conservative problem, most notablythe so-called three-body problem. It is certainly a very important approximation, which oftenprovides a realistic solution of the dynamics. The same holds for the rotation of a celestialbody, whose approximate solution can be found by integrating the equations of motion in theconservative setting. However, dissipative effects may act on the system and they can providea more realistic solution of the equations of motion. For example, concerning the three-bodyproblem dissipative effects like the radiation pressure or the Poynting–Robertson effect aredefinitely important, especially to understand the dynamics in the early stage of formation ofthe solar system. As far as rotational dynamics is concerned, the tidal torque in a non-rigidplanet or the Yarkovsky effect may become very important in order to explain the evolution ofthe dynamics.In this work we are mainly interested to the restricted case of the three–body problem and

to the spin-orbit model of rotational dynamics. Thanks to the interest raised by these prob-lems, we review the behavior of dissipative systems with particular attention to the case ofnearly–integrable, dissipative models, since the restricted three-body problem or the spin-orbitproblem belong to the class of nearly-integrable systems. In particular, we investigate analytical

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34 The European Physical Journal Special Topics

and numerical techniques to understand the behavior of periodic and quasi-periodic orbits aswell as their interplay in the formation of resonances. To this end, we start by introducingthe reader to discrete and continuous systems, periodic orbits, quasi-periodic motions, chaos,strange attractors, etc. As a paradigmatic example, we mainly refer to the standard mapping,both in the conservative and in the dissipative settings. We also review Henon map and the lo-gistic mapping in order to introduce fractal structures and period doubling bifurcations (Sec. 2).Referring to the standard map, we show the existence of periodic orbits by means of two

different methods: a constructive version of the implicit function theorem and a suitable parame-terization of the solution. These results provide that for each periodic orbit the drift parameter,entering the equations defining the dissipative standard map, can vary in an interval in whichthe existence of the periodic orbit is ensured; the variation of the drift as a function of theperturbing parameter shows the typical shape of an Arnold’s tongue. Based on the existenceof the periodic orbits, we review some methods for the computation of the breakdown thresh-old of quasi-periodic attractors; these techniques are founded on the analytic representation ofquasi-periodic attractors, on the behavior of the norms defining the parameterizing function oron the convergence of the approximating periodic orbits. In the latter case, we also study theconvergence of the drift parameter associated to the approximating periodic orbits to that ofthe quasi-periodic attractor. We conclude Sec. 3 by introducing some numerical experiments,aimed to investigate the variation of the drift as a function of the frequency and to understandhow the number of quasi–periodic attractors changes as the perturbing parameter varies.As a continuous paradigmatic model we select the spin-orbit problem in Celestial Mechanics

(both in the conservative and dissipative frameworks), which is closely related to the standardmap. The dissipation is identified with the tidal torque provoked by the internal non-rigidityof the celestial body (Sec. 4). In particular, we are interested to analyze the phenomenon ofthe resonances and their interplay with invariant attractors. To this end, we briefly review aseminal mathematical theory, known as KAM theory, to prove the existence of quasi-periodictori – in the conservative case – and of invariant attractors – in the dissipative model (Sec. 5).We also present different kinds of dissipative effects in Celestial Mechanics. We start by

illustrating the Yarkovsky and YORP effects, which typically affect asteroid-size bodies; thefirst one describes the influence of the solar radiation on the dynamics of the body, while thelatter is a variant of the Yarkovsky effect, once the irregular shape of the body is taken intoaccount. Concerning the restricted, three-body problem, we consider the planar, circular casewith the following dissipations: the Stokes case, which is a viscous drag acting, for example,on the primordial solar nebula, and the Poynting–Robertson case, which is a radiation forcemodeling, for example, the effect of the solar radiation in the early solar system. As a motiva-tion for having shakered the reader with many different discrete and continuous systems, weconclude with a short review of some methods to construct suitable maps, apt to describe thedissipative N -body problem (Sec. 6).

2 Phenomenology of conservative and dissipative systems

This Section introduces the reader to the phenomenology of conservative and dissipative sys-tems, based on simple but still instructive mapping models: the conservative and dissipativestandard maps, the circle map, the Arnold circle map, the Henon mapping and the logistic map.We give examples of the different kinds of motions, found in dynamical systems: periodic andquasi-periodic orbits as well as chaotic motions close to the separatrix in the conservative case;periodic and quasi-periodic attractors as well as strange attractors and basins of attraction inthe dissipative case.

2.1 The conservative standard map

The standard map (sometimes also called the Chirikov–Taylor map, see [16]) is given by thediscrete system:

yj+1 = yj + εf (xj)

xj+1 = xj + yj+1 , j ∈ N,(1)

Diffusion and Dissipation in Quasi-Integrable Systems 35

where yj ∈ R and xj ∈ T, f is a periodic function and ε > 0 is called the perturbing parameter.The mapping is symplectic and it can be derived from a generating function

W (yj+1, xj) = xjyj+1 +y2j+1

2+ εF (xj)

for a suitable periodic function F = F (xj), via the equations

yj =∂W

∂xj

xj+1 =∂W

∂yj+1.

The mapping is conservative, since the determinant of the Jacobian matrix is equal to unity:

det J = det

(1 ε ∂f /∂xj

1 1 + ε ∂f /∂xj

)= 1.

For ε = 0 the mapping reduces to the circle map

yj+1 = yj

xj+1 = xj + yj+1, j ∈ N,

where yj = y0 is a constant of motion and xj = x0 + jy0 for any j ∈ N.The fixed points at order N of the mapping can be derived by yj+N = yj and xj+N = xj .

For N = 1 they are defined via the equations:

f (xj) = 0

yj+1 = y0 = 0.

For the classical standard map with f(x) = sinx, the first order fixed points are located at(y0, x0) = (0, 0) and (y0, x0) = (0, π). Their linear stability is given by the variational equations

(δyj+1

δxj+1

)= J(x0, y0)

(δyj

δxj

), j ∈ N,

where the Jacobian J is now computed at the fixed points. The corresponding eigenvalues aredetermined by solving the characteristic polynomial

λ2 − (2± ε)λ+ 1 = 0,

where the positive sign holds for (0, 0) and the negative sign for (0, π). The former turns outto be unstable, due to the presence of an eigenvalue greater than one, the latter is stable, sincethe eigenvalues are complex conjugate with real part less than one.

The classical standard map is associated to the time dependent Hamiltonian

H(p, q, t) =p2

2+ ε cos qδ1(t),

where δ1(t) is a periodic δ-function of period 1. The dynamics is therefore given by a sequence offree propagations in terms of pendulum dynamics, interleaved with periodic kicks. The discretesystem (1) can be seen as the evolution in phase space in between successive kicks.


Fig. 1. Left: Iteration of the integrable conservative standard map for rational initial conditions 2πp/q.Each point in the web of resonant initial conditions labels an invariant, periodic orbit. Right: Iterationof the integrable conservative standard map for irrational initial conditions y ∈ R \ Q. The dynamicstakes place on invariant, quasi-periodic curves.

2.1.1 The integrable case

For ε = 0 the standard map is integrable: all motions are periodic or quasi-periodic, since forall j, yj = y0 and xj+1 = xj + y0. Assuming y0 = 2πp/q being a rational multiple of 2π, thetrajectory {(xj , yj), j ∈ Z} is a periodic orbit with period q, while p measures how many timesthe interval [0, 2π) is run before coming back to the starting position (see Fig. 1 left, for initialconditions y0 = 2πp/q, p, q < 30). The second kind of motion, called quasi-periodic, is linkedto irrational initial conditions, where again yj remains constant, but with increasing number ofiterations the line yj = y0 is filled densely; this motion will come arbitrarily close to its initialdatum (in terms of xj), but never reach it (compare with Fig. 1, right). To distinguish betweenperiodic and quasi-periodic motions, it is common to introduce the rotation number:

ω = limj→∞

xj − x0j.

For ε = 0 the limit ω tends to p/q for periodic orbits and ω ∈ R \Q otherwise.

2.1.2 Slightly perturbed case

In general it is not possible to find a closed form solution for ε �= 0. Nevertheless, for ε sufficientlysmall, it is possible that periodic and quasi–periodic motions persist under the perturbation.Invariant curves are only slightly displaced and deformed, periodic orbits become surroundedby closed invariants of motion, called librational curves. In addition, chains of islands appeararound periodic orbits with periods multiple of the order of the resonance (see Fig. 2, leftpanel, for ε = 0.5). Increasing ε the rotational curves become more and more deformed anddistorted, while the librational curves increase their amplitudes. Close to separatrices of higherorder resonant islands, chaotic motion appears and starts to fill an increasing region in phasespace. Nevertheless, the existing invariant curves still restrict the variation of momentum tobe bounded. Close to the breakdown, invariant tori are replaced by cantori, which are stillinvariant sets, but they are graphs of a Cantor set.

2.1.3 Strong perturbed case

For large values of the parameters (see Fig. 2, right panel, for ε = 1) the chaotic regime is large,and the motion can even reach unbounded domains of phase space (see Fig. 3 for ε equal to

the golden ratio ωg ≡√5−12 ). In particular it has been shown [20] that the last invariant torus

of the classical standard map disappears at ε = 0.971635.


Fig. 2. Left: iteration of the classical standard map for ε = 0.5. The phase portrait is composed byperiodic orbits, invariant rotational and librational curves, as well as higher order resonant islandsand locally confined chaotic regions. Right: iteration of the classical standard map for ε = 1. All theinvariant curves break up; the size of the chaotic domain increases as ε increases. Higher order resonantpoints are surrounded by ghost curves.

Fig. 3. Iteration of the classical standard map for ε equal to the golden ratio ωg. For ε large enough,chaos dominates the dynamical behavior of the system.

2.2 The dissipative standard map

The generalization of the standard map to the dissipative setting is given by the discrete system

yj+1 = byj + c+ εf(xj)

xj+1 = xj + yj+1, j ∈ N

where again, yj ∈ R and xj ∈ T, ε > 0 and f is a periodic function. The quantities b, c ∈ R+are called the dissipative and drift parameters, respectively. If b = 1 and c = 0 one recovers theconservative standard mapping, while for b = 0 one obtains the one–dimensional mapping

xj+1 = xj + c+ εf(xj),

sometimes called the Arnold circle map.The mapping is contractive for 0 < b < 1, since the determinant of the Jacobian J is equal

to b:

det J = det

(b ε∂f /∂xj

b 1 + ε∂f /∂xj

)= b.


Fig. 4. Left: iteration of the dissipative standard map for b = 0.9, c = 2π · 0.027, ε = 0.9. A periodicattractor of period 23 together with a periodic attractor of period 1. Right: iteration of the dissipativestandard map for b = 0.9, c = 2π · 0.062, ε = 0.2. The attractor has the form of an invariant curve.

It is common to introduce the quantity

α =c

1− b ,

since for ε = 0 the trajectory {y = α} × T is an invariant object. In fact, the conditionyj+1 = yj = byj + c

implies immediatelyα = bα+ c

and therefore c = α(1 − b). The latter relation also implies that the parameter c vanishes inthe conservative case b = 1. The dynamics associated to the dissipative standard map admitsattracting periodic orbits, invariant curve attractors (limit cycles) as well as strange attractorsof complex geometrical structure; introducing a suitable definition of dimension, the strangeattractors are shown to have a non–integer dimension (namely a fractal one). The differentkinds of attractors found in the dissipative standard map are shown in Fig. 4–Fig. 6: periodicorbits for f(x) = sinx, ε = 0.9, b = 0.9, c = 2π · 0.027 (in Fig. 4, left); an invariant attractorfor the same f and b, but ε = 0.2 and c = 2π · 0.062 (in Fig. 4, right); the case of coexistenceof a limit cycle and a periodic attractor for ε = 0.7, b = 0.9 and drift parameter c = 2π · 0.062(in Fig. 5, left); finally, a strange attractor for f(x) = sinx + sin 3x with ε = 0.6, b = 0.8,c = 2π · 0.1198 (in Fig. 5, right).The basin of attraction of an attracting set is defined as the set of initial conditions falling

on the attractor as j →∞. In general, however, the time needed to reach the attractor is finite,say j → N ∈ Z+ and it may also depend on the initial conditions (x0, y0).To be more formal, we define recursively an attracting set by:

XA = {(xj , yj) : (byj + c+ εf (xj) , xj + yj+1) ∈ XA for any j ≥ N},namely XA is the set of points which after N iterations of the mapping equations remain onthe attractor forever. Let us furthermore define the distance map according to

rj = d0 ((xj , yj) ,XA) , j ∈ N,where the distance d0 between a point ξ ∈ Y and a generic set Z is defined via

d0(ξ, Z) = minz∈Zd(ξ, z),

i.e. the minimum distance to any point of the set Z, being d the Euclidean distance. It is clear,that d0 ((xj , yj) ,XA) = 0 implies (xj , yj) ∈ XA. The definition is useful also to unveil the timeneeded to fall on the attractor, which is connected to the box counting dimension.The basin of attraction corresponding to Fig. 4 (right panel) is shown in Fig. 6 (left panel),

while the rates of attraction is given in Fig. 6 (right panel). The color scale indicates the timeneeded to reach the attractor for different initial conditions (x0, y0) ∈ [0, 2π).


Fig. 5. Left: iteration of the dissipative standard map for b = 0.9, c = 2π ·0.062, ε = 0.7. Coexistence ofa periodic attractor and an invariant curve attractor. Right: iteration of the dissipative standard mapfor f(x) = sinx + sin 3x, b = 0.8, c = 2π · 0.1198, ε = 0.6. The strange attractor of complex geometricform has fractal dimension and it is self similar on different scales (see also Henon map).

Fig. 6. Left: basin of attraction for Fig. 4 right, with b = 0.9, c = 2π · 0.062, ε = 0.2. Right: rates ofattraction for Fig. 4 right, with b = 0.9, c = 2π · 0.062, ε = 0.2. The lighter the color, the faster theattractor is reached; the scale corresponds to the linear slope of the distance map rj versus j on alogarithmic plot. The attractor is marked in black. The interplay between the two attracting sets givesrise to a complex geometric structure of the phase space.

2.3 Other examples of dissipative mappings

We close this Section with two other famous examples of dissipative mappings: i) the Henonmapping [22] and ii) the logistic map [34], the former being two dimensional, the latter a onedimensional polynomial mapping. The mapping equations for the Henon map are given by:

xj+1 = yj + 1− αxj2

yj+1 = βxj , j ∈ Z,

where α, β ∈ R are real parameters and xj , yj ∈ R. The Jacobian matrix of the discrete systemis given by:

J =

(−2αxj 1β 0

).

The mapping is therefore contractive as far as β ∈ (−1, 1) for which the determinant turns outto be −β. The meaning of the map is the following: each region in the phase plane (x, y) is


Fig. 7. Left: Henon map for x0 = 0.5, y0 = 0.2, α = 1.4, β = 0.3. The box magnifies parts of theattractor, indicating the self similar nature of the fractal geometrical object. Right: diagram of thelogistic map for different values of the parameter r. Increasing the parameter, period doubling occurs,leading to transient regimes of periodic and chaotic final states of the system.

shrunk by a factor |β| at each iteration step of the mapping. The final state of the system turnsout to be an attracting set, with fractal, selfsimilar structure. The phase portrait for α = 1.4and β = 0.3 is shown in Fig. 7, left. The rectangle is a magnification of the attractor, showingthe selfsimilarity of the fractal object.The final example is the logistic map. The mapping equation is given by the polynomial:

yj+1 = ryj (1− yj) , j ∈ Z,

where r > 0 is a parameter and yj ∈ (0, 1). This mapping is the most simple example of apopulation model, where the initial datum y0 is the initial normalized population and yj givesthe population at year j. The role of the parameter r can be summarized as follows: for 0 < r < 1the population will not survive, independently from the initial condition y0. For 1 < r ≤ 2 thesystem stabilizes around (r− 1)/r independently of the initial population. The same holds truefor 2 < r ≤ 3 with a transient oscillating behavior. From r > 3 on, the attracting state ofthe system doubles its period for successive values of the parameter r at which bifurcationoccurs. Periodic attractors interleaved with chaotic regimes fill the parameter space until chaosis reached for r within 3.57 and 4 (compare with Fig. 7, right).

3 On the dynamics of the dissipative standard map

In this Section we introduce the concept of parameterization of periodic orbits for the dissipativestandard map. We provide information about the values of the drift parameter allowing theexistence of periodic orbits. We also review some techniques to compute the breakdown ofinvariant attractors. In this Section, we will deal with the dissipative standard map introducedin the previous Section. The study of this map is particularly interesting because it is closelyrelated to the spin-orbit resonances, which will be discussed later. In particular, the existence ofperiodic orbits for the mapping allows us to have some enlightening results about the captureof a body in a spin-orbit resonance.

3.1 Periodic orbits of the dissipative standard map

Let us introduce again the equations defining the dissipative standard map as

{yn+1 = byn + c+ ε sinxnxn+1 = xn + yn+1,

(2)


where xn ∈ T = R/(2πZ), yn ∈ R, ε > 0 is the perturbing parameter, 0 < b < 1 is the dissipativeparameter and c > 0 is the drift parameter. The periodic solutions of (2) can be found providedsome conditions on the drift parameter c are satisfied, meaning that the existence of periodicorbits of a given period can be proved within a whole interval of the drift c.A periodic solution (xn, yn) of frequency ω = 2πp/q for the mapping (2), must satisfy the

following periodicity conditions: {xn+q = xn + 2πpyn+q = yn.

(3)

If a solution (xn, yn) satisfies (3), then the point (xn, yn) comes back to itself after q iterationsof the map; the integer p tells us how many times the torus is run before the point comes backto itself. The following result about the existence of periodic orbits was proved in [14].

Proposition 1 ([14]) Let M : R× T→ R× T be the map defined in (2) and let p, q ∈ Z+ becoprime. Then, for fixed values of b, ε, say b = b, ε = ε, one can find an interval Ipq ⊂ R, suchthat for any c ∈ Ipq, the map M admits at least a periodic orbit of period p/q.This result can be proved through two methods and both of them provide explicitly the driftinterval Ipq for which periodic orbits can be found (see [14] for complete details). The firstmethod consists of an application of the implicit function theorem, which can be used in orderto prove the existence of an interval of the drift in which the existence of periodic orbits of agiven period is ensured. To this end, we define the vectorial function F = (F1, F2) as

F (x0, y0; b, ε, c) =Mq(x0, y0; b, ε, c)− (x0, y0)− (2πp, 0),

where (x0, y0) are the initial conditions and Mq is the qth iteration of the mapping (2). If the

function F (x0, y0; b, ε, c) = 0, then the point (x0, y0) belongs to a periodic orbit of frequency2πp/q for some b, ε, c, since condition (3) is satisfied. Under suitable conditions on F and onthe Jacobian of F , and under smallness conditions on the parameters, the implicit functiontheorem can be implemented in order to find a whole interval of the drift in which the existenceof periodic orbits is guaranteed.The second method is constructive; it allows us to write explicitly both the solution of the

periodic orbit and the interval Ipq. Indeed, we can write the solution xn through a suitableparametric representation as

xn = θn + u(θn), θn ∈ T, (4)

where u(θn) is a continuous, periodic function and the parametric coordinate θn evolves linearlyas θn+1 = θn + ω, where ω = 2πp/q is the frequency of the periodic orbit which we are lookingfor. By means of some easy computations and inserting the parameterization (4) in Eqs. (2),the following proposition can be proved.

Proposition 2 ([14]) Defining the quantity

γ ≡ (1− b)ω − c, (5)

then the function u = u(θn) associated to the solution (4) of the dissipative standard map (2)must satisfy the equation

u(θn + ω)− (1 + b)u(θn) + bu(θn − ω) + γ = ε sin(θn + u(θn)). (6)

In order to find a periodic solution of the dissipative standard map we have to look for a solutionof Eq. (6) satisfying condition (3). To this end, we expand the function u(θ) in Taylor series as

u(θn) =

+∞∑j=1

uj(θn)εj , (7)

for some real functions uj which can be expanded in Fourier series as

uj(θn) =

+∞∑�=1

(a�j sin( θn) + b�j cos( θn)

), (8)


where the unknown real coefficients a�j , b�j can be explicitly computed. We also expand thedrift c and the term γ in Taylor series as

c =

∞∑j=0

cjεj , γ =

∞∑j=1

γjεj , (9)

where cj and γj are suitable real coefficients. Inserting the expansions (7), (8), (9) in Eq. (6),we obtain a new equation in the unknowns a�j , b�j , γj of the form

∞∑j=1

εj∞∑�=1

{sin( θn)

[(1 + b)a�jα

(p,q)� − (1− b)b�jβ(p,q)�

]

+cos( θn)[(1− b)a�jβ(p,q)� + (1 + b)b�jα

(p,q)�

]+ γj

}

=

∞∑j=1

εj

{j∑�=1

[S(p,q)�j sin( θn) + C

(p,q)�j cos( θn)

]+D

(p,q)j

}, (10)

where the real constants α(p,q)� , β

(p,q)� are given by

α(p,q)� = cos

(2π · p

q

)− 1

β(p,q)� = sin

(2π · p

q

),

while the quantities S(p,q)�j , C

(p,q)�j and D

(p,q)j depend on the terms a�i, b�i for i = 1, . . . , j − 1.

Equating terms of the same order of ε in Eq. (10), we are able to compute the unknowns upto the order q − 1 where, we recall, q is the period of the periodic orbit we are looking for. Inparticular, the function u(θn) associated to a periodic orbit of period p/q can be computed upto the order q−1 of its Taylor expansion. We denote by u(q−1)(θn) the truncations to the orderq − 1 of the ε–expansions of u(θn):

u(q−1)(θn) =q−1∑j=1

uj(θn)εj ,

where the functions uj(θn) are given as in (8). The real coefficients a�j , b�j are explicitlydetermined for all j = 1, . . . , q − 1 and for = 1, . . . , j. Equating terms of order q, the realconstants α

(p,q)q , β

(p,q)q turn out to be zero and the coefficients aqq, bqq cannot be determined.

As a consequence, the coefficients a�j , b�j remain undetermined for j ≥ q. Therefore, theapproximation u(q−1)(θn) of the solution u(θn) is well determined and the complete solution isgiven with an error of the order εq.In a similar way, we denote by γ(q) the truncation of γ up to the order q:

γ(q) ≡q∑j=1

γjεj .

Similarly, we compute the expansion of the drift up to the order q as

c(q) ≡q−1∑j=0

cjεj ,

where the relation between c and γ is given by Eq. (5) so that c0 = (1 − b)ω and cj = −γj .The real coefficients γj are explicitly computed for all j = 1, . . . , q − 1. On the other hand, theterm γq turns out to be a function of θ, namely it can be written as

γq = S(p,q)qq sin(qθn) + C

(p,q)qq cos(qθn) +D

(p,q)q , (11)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

eps

c

Fig. 8. Arnold’s tongues providing the drift c as a function of ε. The figure shows the tongues associatedto three different periodic orbits, precisely with frequencies 2π · 1/3 (left), 2π · 2/3 (center), 2π · 1/1(right), for b = 0.9.

where S(p,q)qq , C

(p,q)qq , D

(p,q)q are real known constants. This implies that the term γq, or equiv-

alently the term cq, can take a whole interval of values as θ varies in [0, 2π), being γq(θn) abounded periodic function of period 2π. Therefore, the expansion of the drift up to the order qis given by

c(q) ≡ c(q)(θn) = c0 + εc1 + ε2c2 + · · ·+ εq−1cq−1 + εqcq(θn), (12)

where c0, . . . , cq−1 are constants (depending on b, p, q), while cq = cq(θn) belongs to an intervalIpq ≡ [c(p,q)− , c

(p,q)+ ] whose endpoints are given by the minimum and the maximum of the function

(11) (depending also on b, p, q). As a consequence, the term c(q) belongs to a connected intervalIpq ⊂ R.

3.2 Arnold’s tongues

From Eq. (12) it is clear that the drift interval Ipq depends on the perturbing and the dissipativeparameters. If we fix the dissipative parameter b, it can be seen that the amplitude of the intervalIpq decreases as the perturbing parameter ε gets smaller. If we consider the behavior of the driftc as a function of the perturbing parameter ε, we obtain a typical trend defining the Arnold’stongues in which the existence of periodic orbits is ensured.Arnold’s tongues are well known structures in the literature (compare with [7]) and they

are defined as the subset of the parameters in which periodic orbits exist. More precisely, weconsider a rational number p/q with p, q ∈ Z+ coprime, we fix the dissipative parameter b andwe define the Arnold’s tongue A as

A = {(c, ε) ∈ R× R : M q(xn, yn) = (xn + 2πp, yn), for some (xn, yn)},

where Mq is the qth iteration of the map M defined by Eq. (2). Through the parameterizationdescribed in Sec. 3.1, we are able to find explicitly the set A for a given period and a givenvalue of the dissipative parameter. In Fig. 8, the variation of the drift is plotted as a functionof the perturbing parameter; the figure shows the Arnold’s tongues for the periodic orbits withfrequency 2π·1/3, 2π·2/3, 2π·1/1 and for the dissipative parameter equal to b = 0.9. We observethat the interval Ipq corresponds to the section of the set A for a fixed value of the perturbingparameter ε. We evaluate the stability of the periodic orbit by computing the eigenvalues ofthe monodromy matrix along a full cycle of the periodic orbit: within each Arnold’s tonguethe periodic orbit is stable. The stable periodic orbits become unstable through a saddle-nodebifurcation, when approaching the boundary of the Arnold’s tongue.The possibility to write explicitly the boundary of the Arnold’s tongues provides important

information about the occurrence of spin–orbit resonances, being the dissipative standard map


closely related to the spin-orbit problem [12]. The drift c corresponds to the eccentricity e ofthe Keplerian orbit on which the satellite moves around the planet, so that the existence ofa periodic orbit within a whole interval of the drift corresponds to the occurrence of a givenresonance within specific values of the eccentricity.

3.3 Breakdown of quasi-periodic attractors

In this Section we review some methods to evaluate the break–down threshold for invariantattractors. In the conservative case there are several well–established techniques which allow usto compute with great accuracy the value of the perturbing parameter at which a given invariantsurface disappears. In the dissipative setting we discuss three different techniques based onan analytical representation of the invariant attractor, on the behavior of the parameterizingfunction and on the approximation through periodic orbits (see [9,14]). For sake of simplicity,we consider the dissipative standard mapping (see (2)), though the extension of these methodsto the dissipative spin–orbit problem is quite straightforward.

3.3.1 Analytical representation

Combining the standard map Eq. (2) with the parametric representation (4), one obtains theequation

u(θ + ω)− (1 + b)u(θ) + bu(θ − ω)− εg(θ + u(θ)) + γ = 0, (13)

where γ = ω(1− b)− c. Defining the operators D1 and Db as

D1u(θ) = u(θ +ω

2

)− u

(θ − ω2

), Dbu(θ) = u

(θ +ω

2

)− bu

(θ − ω2

)

and setting Δb ≡ DbD1 = D1Db, Eq. (13) can be written as

Δbu(θ)− εg(θ + u(θ)) + γ = 0 . (14)

The solution of (14) is found provided γ satisfies the compatibility condition

γ = −〈uθΔbu〉.

Next, we expand u and γ in Taylor series around ε = 0 as

u(θ) =∞∑j=1

uj(θ)εj , γ =

∞∑j=1

γjεj ;

equating same orders of ε, one obtains explicit expressions for uj , γj . Once these terms arecomputed, we assume that the solution is explicitly known up to the order N , namely

u(N)(θ) =

N∑j=1

uj(θ)εj , γ(N) =

N∑j=1

γjεj ;

these functions satisfy (14) up to an error term χ(N)(θ; ε) ∼ O(εN+1):

Δbu(N)(θ)− εg(θ + u(N)(θ); ε) + γ(N) = χ(N)(θ).

In order to find a better approximate solution, we can implement Newton’s method startingfrom the order N for some corrections u(c)(θ) and γ(c), such that

u(2N)(θ) = u(N)(θ) + u(c)(θ) , γ(2N) = γ(N) + γ(c).


An explicit expression for u(c) and γ(c) is given by the following set of formulae (see, e.g., [13]),

where E = 〈E〉 and E = E − E:

V = 1 +∂u(N)

∂θ

W = V(θ +ω

2

)V(θ − ω2

)

E = V χ(N)

a =〈W−1D−1b E − E〈W−1D

−1b u

(N)θ 〉

〈W−1〉 − (b− 1)〈W−1D−1b u(N)θ 〉

(15)E1 = D

−1b E + a(−1 + (b− 1)D

−1b u

(N)θ )− ED−1b u

(N)θ

w = −V D−11 (W−1E1)u(c) = w − V 〈w〉γ(c) = γ(N) − E + (b− 1)a.

Having computed the solution up to the order N = 7, we implement (15) in order to determineu(14); denoting by u(k) the ε–expansion of u(14) up to the order k, we back transform to theoriginal variables through the expressions

x = θ + u(k)(θ)

y = ω + u(k)(θ)− u(k)(θ − ω).

As an example, we compute the approximate solution at different orders and for different valuesof ε for the invariant attractor with frequency 2πω1, where ω1 is defined through its continuedfraction expansion as

ω1 = [0; 2, 5, 3, 1∞] =

1

2 + 15+ 1

3+ωg

,

having denoted the golden ratio by ωg = [0; 1∞] =

√5−12 . In Fig. 9 we consider several orders

of approximations with k = 4, 6, 8, 10, 12, 14 (N = 2 was discarded, since it is too low toprovide meaningful results). We also consider different values of the perturbing parameter, i.e.ε = 0.5, 0.7, 0.9. Each graph is obtained using a grid of 500 values in θ ∈ [0, 2π]. As far asthe invariant attractor exists, the graphs obtained varying the order of the approximation arealmost overlapping; when the perturbing parameter increases, the different graphs show moreevident displacements and deformations, which will lead to a loss of regularity of the invariantattractor. Such method allows us to have a good description of the invariant attractor in theregular regime; when increasing the perturbing parameter a refined description requires higherorders of approximation and consequently a much higher computational effort.

3.3.2 Sobolev’s norms

The behavior of the Sobolev’s norms of the function parameterizing the invariant attractorprovides a method to compute the breakdown threshold as the perturbing parameter is increased[9]; indeed, this criterion is based on the remark that close to the breakdown the smooth normsmust blow up. More precisely, let us consider the Fourier series of u truncated to the order Nas

u(N)(θ) =∑|k|≤N

uke2πikθ;

denoting the L2–norm as ‖u‖L2 ≡∑k∈Z |uk|2, we define

‖u‖r = ‖∂rθu‖L2 + |〈u〉|.


1 2 3 4 5 6x

2.8

2.9

3

3.1

3.2y a

1 2 3 4 5 6x

2.6

2.8

3

3.2

y b

1 2 3 4 5 6x

2.4

2.6

2.8

3

3.2

3.4

y c

1 2 3 4 5 6x

2.8

2.9

3

3.1

3.2y d

1 2 3 4 5 6x

2.6

2.8

3

3.2

y e

1 2 3 4 5 6x

2.4

2.6

2.8

3

3.2

3.4

y f

Fig. 9. Graph of the approximations of the invariant attractor with ω = 2πω1, b = 0.9. a) ε = 0.5,k = 14; b) ε = 0.7, k = 14; c) ε = 0.9, k = 14; d) ε = 0.5, all orders k = 4, 6, 8, 10, 12, 14; e) ε = 0.7,all orders k = 4, 6, 8, 10, 12, 14; f) ε = 0.9, all orders k = 4, 6, 8, 10, 12, 14.

Table 1. Critical value of εcrit for different b and for ω1 = [0; 2, 5, 3, 1∞], ωg =

√5−12(after [9]).

ω1 b εcrit ωg b εcrit0.5 0.91968 0.5 0.9792150.8 0.859174 0.8 0.9732490.9 0.846356 0.9 0.972088

For trigonometric polynomials we consider the Sobolev’s norm defined as:

‖u(N)‖r =

⎛⎝ ∑|k|≤N

(2πk)r|uk|2⎞⎠12

.

A regular behavior of the Sobolev’s norm of u(N), as the parameter increases, yields evidenceof the existence of the invariant attractor; on the contrary, a blow–up of the norm of u(N) isan indication of the breakdown of the invariant attractor. Indeed, it suffices that the Sobolev’snorm reaches a given threshold, whose choice does not affect much the final result. Based onthis criterion, one can compute the breakdown values as a function of the dissipative parameter;some results for ω1 and ωg are reported in Table 1 (see [9]).

3.3.3 Approximation through periodic orbits

As in the conservative case, an invariant attractor with frequency ω can be approximatedthrough a sequence of periodic attractors, say with periods

pjqjsuch that

limj→∞

pj

qj= ω.

Expanding ω in continued fraction representation as

ω = [a0, a1, a2, a3] = a0 +1

a1 +1

a2+1

a3+...

,

the sequence of the approximating periodic orbits has the following periods:

p0

q0= a0,

p1

q1= a0 +

1

a1,

p2

q2= a0 +

1

a1 +1a2

,p3

q3= a0 +

1

a1 +1

a2+1a3

.


1.48

1.5

1.52

1.54

1.56

1.58

1.6

1.62

1.64

1.66

1.68

0 1 2 3 4 5 6

y

x

1.614

1.615

1.616

1.617

1.618

1.619

1.62

1.621

1.622

0 1 2 3 4 5 6

y

x

Fig. 10. Left: the golden attractor and the periodic orbits 3/2, 5/3, 8/5, 13/8, 21/13. Right: the goldenattractor and the periodic orbits 34/21, 55/34, 89/55, 144/89.

We recall that for each set of values of the parameters, there exists a whole interval of the driftparameter c corresponding to periodic orbits with fixed frequency (see, e.g., [14]). An exampleof periodic orbits approximating the golden attractor is reported in Fig. 10.In the conservative framework, Greene’s method [20], originally developed for the conser-

vative standard map, is based on the conjecture that the breakdown of an invariant curve isstrictly related to the stability character of the approximating periodic orbits. The same holdsin the dissipative context, provided the periodic orbit is chosen within the drift interval whichadmits the periodic orbit with given period p/q. The stability of such periodic orbit can be eval-uated by computing the monodromy matrix associated to (2) along the periodic orbit. Moreprecisely, let us denote by Tp,q and Dp,q the trace and the determinant of the monodromymatrix and let us introduce the residue Rp,q as

Rp,q ≡1 +Dp,q − Tp,q2(1 +Dp,q)

.

The eigenvalues λ(p,q)1,2 of the monodromy matrix are linked to the residue through

λ(p,q)1,2 =

1

2(1− 2Rp,q)(1 +Dp,q)±

1

2

√4Rp,q(Rp,q − 1)(1 +Dp,q)2 + (1−Dp,q)2 .

The stability of the p/q–periodic orbit depends on the value of the residue; precisely, the orbitis stable whenever

R(p,q)− < Rp,q < R

(p,q)+ ,

where R(p,q)± ≡ 1

2±12

√1− (1−Dp,q)2(1+Dp,q)2

. For a fixed value of b, let us denote by εp,q(b) the maximal

value of the perturbing parameter ε within the interval [0, 1), such that the periodic orbit withfrequency p

qis stable. An example of the application of this strategy to the golden attractor

is reported in Table 2, where three sample values of the dissipative parameter b have beenconsidered. This table shows that the stability thresholds seem to converge to a limit as theorder of the periodic approximant grows, thus defining a breakdown threshold of the invariantattractor consistent with Sobolev’s norms (compare with Table 1).

3.4 Drift convergence

The parametric representation of periodic orbits (Sec. 3.1) and the construction of quasi–periodic attractors (Sec. 3.3.1) allows us to understand the relation between the drift of periodicand quasi–periodic orbits. We recall that a whole interval Ipq of the drift is associated with aperiodic orbit of period p/q, while a unique constant c = c(ω) is associated with a quasi–periodicorbit with frequency ω.


Table 2. Stability threshold εp,q(b) of the periodic orbits approximating ωg =√5−12(after [9]).

p/q εp,q(b = 0.9) εp,q(b = 0.8) εp,q(b = 0.5)8/13 0.999 0.993 0.99913/21 0.999 0.999 0.99921/34 0.999 0.999 0.99934/55 0.993 0.994 0.99255/89 0.981 0.986 0.98789/144 0.980 0.980 0.983144/233 0.976 0.977 0.980233/377 0.975 0.978 0.979377/610 0.974 0.975 0.979

Table 3. The drift parameter for several periodic orbits approximating the golden ratio for b = 0.9,ε = 0.9. The value of the drift for the golden ratio amounts to c = 0.386077 (after [14]).

p/q c p/q c

1/2 0.322143 21/34 0.3858622/3 0.421515 34/55 0.3861593/5 0.375734 55/89 0.3860465/8 0.390044 89/144 0.3860898/13 0.384586 144/233 0.38607213/21 0.386640 233/377 0.386079

As we have seen in Sec. 3.3.3, we have numerical evidence that a quasi–periodic attractor canbe approximated by periodic attractors. It can be shown that also the drift can be approximatedin the same way. Let us consider a sequence of rational numbers pj/qj converging to an irrationalω as j tends to infinity. Through the parametric representation, it can be proved that theamplitude of the drift interval decreases as qj increases and, in particular, the sequence ofintervals Ipjqj tends to the constant c = c(ω) as j tends to infinity.As an example, let us consider the golden number ωg ≡

√5−12 ; we take a sequence of

periodic orbits Pj with frequencies 2πpj/qj , where pj/qj is a rational approximant of ωg. Then,we compute a 6th order approximation of the expansion of the drift of the orbits Pj and of the

orbit associated to the golden frequency 2πωg, and we denote by γ(6)pj/qj

and γ(6)ωg the respective

approximate drifts. We notice a fast convergence of the drifts of the rational approximants tothe drift of the invariant golden attractor as j tends to infinity. In Table 3 we show an averagevalue of the drift within each interval Ipjqj found through a numerical evaluation; the resultsshow that the drifts of the rational approximants tend to the drift of the invariant goldenattractor with frequency 2πωg.

3.5 Some numerical experiments on the dissipative standard map

We investigate the behavior of the frequency of invariant attractors as the drift varies. Tothis end, we fix b and ε and, through the parametric representation, we compute the driftc = c(ωg) corresponding to the invariant attractor with frequency 2πωg, where ωg is againthe golden number. Then we consider 200 values of the drift around c(ωg) within the interval[c(ωg) − 0.3, c(ωg) + 0.3]. For each of these values, we take random initial condition (x, y); weperform a preliminary set of 106 iterations of the map (2) and then we determine the resultingfrequency of motion 2πω. In Fig. 11, we plot the frequency as function of the drift for b = 0.9and for different values of ε. The dynamical interpretation of these graphs agrees with thestandard frequency analysis developed in the conservative framework (see [26]): a monotonicincrease corresponds to invariant attractors, while plateaus indicate regions of periodic orbits.We observe that for small values of the perturbing parameter, the curve ω = ω(c) is regular,while, as ε increases, a larger number of plateaus can be observed, making less probable theexistence of invariant attractors.


2.8 3

3.2 3.4 3.6 3.8

4 4.2 4.4 4.6 4.8

5

0.3 0.35 0.4 0.45

freq

uenc

y

c

a)

2.5

3

3.5

4

4.5

5

0.3 0.35 0.4 0.45

freq

uenc

y

c

b)

3 3.2 3.4 3.6 3.8

4 4.2 4.4 4.6 4.8

5 5.2

0.3 0.35 0.4 0.45

freq

uenc

y

c

c)

3 3.2 3.4 3.6 3.8

4 4.2 4.4 4.6 4.8

5 5.2

0.3 0.35 0.4 0.45

freq

uenc

y

c

d)

Fig. 11. Varying the drift parameter for b = 0.9 and a) ε = 0.1, b) ε = 0.5, c) ε = 0.9, d) ε = 0.98.

0

1

2

3

4

5

6

0 1 2 3 4 5 6

y

x

a)

0.5 1

1.5 2

2.5 3

3.5 4

4.5 5

5.5

0 1 2 3 4 5 6

y

x

b)

1

1.5

2

2.5

3

3.5

4

4.5

5

0 1 2 3 4 5 6

y

x

c)

Fig. 12. Random choice of 100 frequencies in the interval (0, 1) for b = 0.9 and a) ε = 0.5, b) ε = 0.7,c) ε = 0.9.

A further investigation is done by choosing several values of the frequency and studying thebehavior of different invariant attractors (if they exist) as the perturbing parameter is varied.We generate 100 random frequencies ω1, · · · , ω100 within the interval (0, 1); the numbers aregenerated with 16 decimal digits and up to this precision it is checked that they do not belongto the set of rational numbers. Then we fix b and ε, and for any ωj (j = 1, · · · , 100), we computethe corresponding value of the drift parameter c = c(ωj), so that the corresponding orbit hasfrequency 2πωj up to an error of 10

−8. It could be possible that the drift parameter c(ωj)cannot be computed, since the attractor with frequency 2πωj might not exist for the givenvalues of b and ε. In Fig. 12, we plot the existing curves for b = 0.9 and for different values of ε.We choose the initial conditions as the points (x, y) = (0, 2πωj); then we perform a preliminarynumber of 106 iterations in order to look for the drift. Finally we plot each curve iterating 103

times the Eq. (2). The results show that the number of invariant attractors decreases as theperturbing parameter increases; a similar behavior can be observed as b varies.


4 An example from Celestial Mechanics: The spin–orbit model

A resonance in Celestial Mechanics occurs when some celestial bodies are in particular config-urations that recur periodically in time. A celestial body is characterized by two fundamentalmotions: the revolution around a central body and the rotation about its spin-axis. Let us in-troduce an important resonance involving both of these motions. We consider a body orbitingaround a central body, for example a satellite around a planet. Let Trev and Trot be, respec-tively, the period of revolution of the satellite around the planet, and the period of rotation ofthe satellite around its spin-axis. We say that the two bodies are in a p : q spin-orbit resonanceif the ratio between Trev and Trot equals to the ratio of two small integers, namely

Trev

Trot=p

q,

with p, q ∈ Z+.In the solar system, most of the biggest satellites are in a synchronous resonance (namely a

1 : 1 resonance): the period of revolution of the satellite around the planet is the same as theperiod of rotation about its spin-axis. For this reason, these satellites always show the sameface to their host planet, as it is well known for the Moon. In our solar system, there exists onlyone body trapped in a resonance which is not synchronous; it is the case of Mercury, which isin a 3 : 2 resonance with the Sun: during two orbital revolutions around the Sun, the planetmakes three rotations about its spin-axis. We show below the mathematical model describingthe spin-orbit dynamics.We proceed to introduce the spin-orbit model, which allows us to describe the spin–orbit

resonances. Let us consider a satellite S orbiting around a central planet P and rotating aboutits spin-axis under the following assumptions:

i) the satellite is a triaxial ellipsoid and it orbits around the planet on a Keplerian ellipse(we neglect the perturbations due to other bodies);

ii) the direction of the spin-axis coincides with that of the smallest physical axis of theellipsoid;

iii) the spin-axis is perpendicular to the orbital plane (hypothesis of zero obliquity);iv) dissipative effects are neglected (conservative model).

Let us choose the units of measure such that the frequency of revolution is equal to one;then, under these assumptions, the equation of motion of the spin–orbit problem is given by(compare with [10]):

x+ ε(ar

)3sin(2x− 2f) = 0, (16)

where a, r, f are, respectively, the semimajor axis, the orbital radius and the true anomaly ofthe Keplerian ellipse of the satellite. In particular, a is a constant, while r and f are knownfunctions of the time and they depend on the eccentricity e of the Keplerian orbit; the unknownvariable x represents the angle between the direction of the largest physical axis, which belongsto the orbital plane, and a references axis, for example the perihelion line. The parameter εmeasures the oblateness of the satellite and it is given by ε = 3

2B−AC, where A < B < C denote

the principal moments of inertia (see Fig. 13).We can remove assumption iv) in order to add dissipative effects, for example a tidal torque

due to the internal non-rigidity of the satellite. In particular, we can replace the fourth assump-tion by the following:

iv) we consider a dissipative tidal torque depending linearly on the angular velocity of rotation,say x.

Under the new assumption iv), Eq. (16) can be modified as

x+ ε(ar

)3sin(2x− 2f) = −μ(x− η), (17)


Fig. 13. The spin–orbit model of a satellite S moving around a central planet P. Here r is the orbitalradius, f denotes the true anomaly, a is the semimajor axis and x is the rotation angle.

where μ and η are suitable terms depending on the eccentricity and on the physical propertiesof the system ([33]). Technical details on the last equation are provided in Appendix A.From a mathematical point of view, the spin–orbit resonances correspond to periodic solu-

tions of the spin–orbit problem, where a solution x(t) is said to be periodic if there exist twointegers p, q with q �= 0, such that the following holds:

x(t+ 2πq) = x(t) + 2πp.

We remark that the equations of the dissipative standard map correspond to the Poincare mapat time 2π of Eq. (17), obtained integrating it with a leap–frog method and through a suitablechange of variables.

5 The conservative and dissipative KAM theorem

We devote this Section to the proof of the existence of invariant tori for the spin–orbit problem.We first discuss the conservative KAM theorem and then we review the existence of invariantattractors by means of a dissipative KAM theorem.

5.1 Quasi–periodic (conservative) invariant tori

We consider a continuous, nearly–integrable n–dimensional Hamiltonian system, which is rep-resented by a Hamiltonian function of the form

H(I, ϕ) = h(I) + εf(I, ϕ), (18)

defined in the phase spaceM≡ V ×Tn, where V is an open bounded region of Rn, for regularfunctions h and f , being ε the perturbing parameter, such that for ε = 0 Hamilton’s equationsassociated to (18) are integrable.

Definition 1 A KAM torus with frequency ω associated to H is an n-dimensional invariantsurface described parametrically by θ ∈ Tn, such that the corresponding flow is linear, i.e. aftera time t one has θ ∈ Tn → θ+ωt, where ω ∈ Bn is assumed to be a diophantine vector, namelythere exist C > 0 and τ > 0 such that

|ω ·m|−1 ≤ C|m|τ , ∀ m ∈ Zn {0}. (19)


KAM theorem [1,24,30] ensures the persistence of invariant tori with diophantine frequency,provided the perturbing parameter ε is sufficiently small and provided the unperturbed Hamil-tonian is non–degenerate, i.e. for a given torus {I0} × Tn ⊂M one has

deth′′(I0) ≡ det(∂2h

∂Ii∂Ij(I0)

)i,j=1,...,n

�= 0.

Alternatively, one can replace the above condition with the so–called Arnold isoenergetic non–degeneracy condition:

det

(h′′(I0) h′(I0)h′(I0) 0

)�= 0,

ensuring the existence of KAM tori on the energy level corresponding to h(I0), namelyM0 ≡{(I, ϕ) ∈M : H(I, ϕ) = h(I0)}.The first application of KAM theory concerned the three–body problem [21]; the original

versions of the theorem gave very unrealistic constraints about the size of the perturbation,ensuring the existence of invariant tori. However, the implementation of interval arithmetic andthe construction of refined KAM estimates allowed us to obtain results which are consistentwith the physical expectation. For example, we quote here the construction of invariant surfacesfor the spin–orbit problem described in Sec. 4. Let us consider Eq. (16) where we expand theKeplerian quantities r and f as powers of the eccentricity and as trigonometric functions ofthe time. Taking a suitable truncation in terms of the eccentricity, one obtains the followingone–dimensional, time–dependent Hamiltonian function:

H(y, x, t) ≡ y2

2− ε

[(−e4+e3

32− 5

768e5)cos(2x− t)

+

(1

2− 54e2 +

13

32e4)cos(2x− 2t)

+

(7

4e− 123

32e3 +

489

256e5)cos(2x− 3t)

+

(17

4e2 − 115

12e4)cos(2x− 4t)

+

(845

96e3 − 32525

1536e5)cos(2x− 5t)

+533

32e4 cos(2x− 6t) + 228347

7680e5 cos(2x− 7t)

]. (20)

Being the phase space 3-dimensional, the 2-dimensional KAM tori separate the phase spaceinto invariant regions; in particular, any pair of invariant tori will provide a strong stabilityproperty of the trapped orbits in the sense of confinement in a definite region of the phasespace. More precisely, let P(p

q) be a periodic orbit of the spin–orbit problem, associated to the

p : q resonance. Its stability can be obtained by proving the existence of trapping rotationaltori, say T (ω1) and T (ω2) with ω1 < p

q< ω2. In order to meet the diophantine condition (19),

one can select ω1 and ω2, respectively, as

Γ(p/q)k ≡ p

q− 1

k + ωg, Δ

(p/q)k ≡ p

q+

1

k + ωg, k ∈ Z , k ≥ 2,

with ωg ≡√5−12 . A computer–assisted KAM proof allows to obtain the following results [10,11].

Result on the Moon–Earth system. Recall that the Moon is trapped in a 1 : 1 spin-orbitresonance and that the observed values of its parameters are εMoon = 3.45·10−4, eMoon = 0.0549.By implementing a computer-assisted KAM proof for the Hamiltonian (20) with e = eMoon, onecan prove the existence of the invariant tori T (1 ± 1

2+ωg) trapping the synchronous resonance

for all values of the perturbing parameter ε ≤ 0.09.


Result on the Mercury–Sun system. Recall that Mercury is trapped in a 3 : 2 spin-orbit resonance and that the observed values of its parameters are εMercury = 1.5 · 10−4,eMercury = 0.2056. By implementing a computer–assisted KAM proof for the Hamiltonian (20)with e = eMercury, one can prove the existence of the invariant tori T ( 32 ±

12+ωg

) trapping the

3 : 2 resonance for all values of the perturbing parameter ε ≤ 0.007.We proceed to give a sketch of the KAM proof. For a given function V = V (x, t) periodic

in x and t, let us write the equation of motion associated to (20) as

x+ εVx(x, t) = 0. (21)

Let us fix a frequency ω satisfying the diophantine condition

|ω − pq|−1 ≤ C|q|2 , ∀p, q ∈ Z , q �= 0, C > 0.

Definition 2 A rotational KAM torus with frequency ω is an invariant surface described para-metrically by

x = θ + u(θ, t), (θ, t) ∈ T2

y = v(θ, t),(22)

where u, v are periodic continuous functions depending on ω, ε, such that the flow in thecoordinates (θ, t) is linear, namely after a time s, the coordinates evolve as (θ, t)→ (θ+ωs, t+s).

We remark that (22) is a diffeomorphism provided 1 + ∂u∂θ�= 0 for all (θ, t) ∈ T2. Let us define

the operator D as

D ≡ ω ∂∂θ+∂

∂t.

Inserting (22) in (21), the following equation must be satisfied:

D2u(θ, t) + εVx(θ + u(θ, t), t) = 0. (23)

Notice that the inversion of D provokes the appearance of small divisors; in fact, expanding uin Fourier series as u =

∑n,m unme

i(nθ+mt), then

D−1u =∑n,m

unm

i(nω +m)ei(nθ+mt),

where we refer to the quantities nω +m as small divisors. We next proceed to solve (23) by aNewton iteration procedure. More precisely, let u0 be an approximate solution of (23) up to anerror term e0:

D2u0 + εVx(θ + u0, t) = e0. (24)

By a Newton’s method we construct a sequence of approximate solutions {uj}, such that theysatisfy the equation

D2uj + εVx(θ + uj , t) = ej ,

with an error term quadratically smaller, namely |ej | = O(|ej−1|2) = O(|e0|2j

). A set of accurateKAM estimates provides a complete control on the parameters and ensures the convergence ofthe method, so that, under a smallness requirement on the perturbing parameter ε, we obtainthat the error terms tend to zero, i.e. limj→∞ |ej | = 0.The success of the method strongly depends on the choice of the initial approximation; to

this end, using the analyticity of u in the perturbing parameter, we expand it in Taylor seriesas

u(θ, t; ε) =

∞∑j=1

u(j)(θ, t)εj ; (25)


the functions u(j) can be iteratively constructed by inserting (25) in (23) and equating sameorders of ε [10]. Then we can choose the initial approximation as a truncation to a suitableorder N of the Taylor expansion (25), say

u0 ≡ u0(θ, t; ε) =N∑j=1

u(j)(θ, t)εj ;

this function satisfies (24) with an error term e0 = O(εN+1), showing that the larger is N ,

the better is the initial approximation. We conclude by remarking that the computation ofu0 for large N involves a lot of computations. A computer is therefore necessary to reachthe desired result. In order to control the rounding-off and propagation errors introduced bythe machine, interval arithmetic [25] was implemented [10]. Roughly speaking, the idea ofinterval arithmetic is the following. We know that the computer stores any real number using asign–exponent–fraction representation, with a precision varying on the machine. Having fixedsuch precision, we can construct a lower and upper quantity which bounds the real numberfrom below and above; this means to construct an interval around any real number. Let usrefer to sum, subtraction, multiplication and division as elementary operations. We proceedto reduce any other operation to a sequence of elementary operations; for example, we usethe Taylor expansion of exponential, logarithm, trigonometric functions, square root, etc. Wenext construct functions which provide elementary operations between intervals (rather thanbetween real numbers). The implementation of this strategy allows us to obtain a computer-assisted KAM proof.

5.2 Quasi–periodic (dissipative) attractors

We look for invariant attractors of the dissipative spin–orbit problem, which is described bythe equation

x+ εVx(x, t) + μ(x− η) = 0, (26)

where we recall that the function V depends on the orbital radius and the true anomaly bymeans of the relation

Vx(x, t) =

(a

r(t)

)3sin(2x− 2f(t)).

The expressions for μ and η are given in Appendix A. We immediately remark that for μ �= 0,but ε = 0, the torus T0 ≡ {y = η} × {(θ, t) ∈ T2} is a global attractor with frequencyη(e) = 1 + 6e2 + O(e4). In fact, it is easy to show that the general solution is given by x(t) =

x0 + η (t− τ) + 1−exp(−μ(t−τ))μ(v0 − η).

For ε �= 0 we parameterize the solution as

x(t) = θ + u(θ, t), (27)

where θ = ω and u = u(θ, t) is a real analytic function on T2, such that 1 + ∂u∂θ�= 0 for all

(θ, t) ∈ T2. The following result has been proven in [13] (we recall that 〈·〉 denotes the averageon θ and t).

Theorem 1 Let ω be diophantine; there exists ε0 > 0 such that for any 0 < ε ≤ ε0 and0 ≤ μ < 1, there exists a function u = u(θ, t) with zero average and with 1 + uθ �= 0, such that(27) is a solution of (26), provided that

η = ω(1 + 〈(uθ)2〉). (28)

We remark that condition (28) relates the drift parameter to the frequency of the invariantattractor. In Appendix A we show that η is a function of the orbital eccentricity e and that μdepends on the dissipation constant K as well as on the eccentricity; then, the above theorem


can be rephrased by saying that there exists an invariant attractor, provided the oblateness εis sufficiently small for any 0 ≤ K < 1 and provided that the eccentricity is related to K andto the frequency of the invariant attractor by (28).The proof of the theorem is perturbative in ε, but in fact it is uniform in μ; since u depends

continuously on μ as μ tends to zero, then the conservative KAM torus bifurcates on theattractor for μ tending to zero. The idea of the proof is the following. We start by defining theoperators

∂ω ≡ ω∂

∂θ+∂

∂t, Dμ,ωv ≡ ∂ωv + μv , Δμ,ω ≡ Dμ,ω∂ω = ∂ωDμ,ω;

setting γ ≡ μ(ω − η), then u and γ must satisfy the equation:

Fμ,ω(u, γ) ≡ Δμ,ωu+ εVx(θ + u, t) + γ = 0. (29)

The solution of (29) is obtained using a Newton’s method. We start from an approximate solu-tion (v0, β0) = (v0(θ, t;μ, ω), β0(μ, ω)) which satisfies (29) with an error term χ0 = χ0(θ, t;μ, ω):

Fμ,ω(v0, β0) ≡ Δμ,ωv0 + εVx(θ + v0, t) + β0 = χ0.

Then we compute a new solution (v1, β1), which satisfies (29) with an error term quadraticallysmaller. Iterating this procedure, one constructs a sequence (vj , βj) such that the error term isalways quadratically smaller. Finally, under a suitable smallness condition on the parameters,the solution is obtained as the limit (u, γ) = limj→∞(vj , βj). The proof is constructive andit can be implemented provided the compatibility condition 〈(1 + uθ)Fμ,ω(u, γ)〉 = 0 holds,which in turn it results equal to require that η satisfies (28). The basic steps of the proofare described as follows (we refer to [13] for complete details), where we define the norm of afunction u = u(θ, ε) for ξ > 0 as

‖u‖ξ ≡∑

(n,m)∈Z2|unm|e(|n|+|m|)ξ.

Step 1. Give some properties of the operators Dμ,ω, Δμ,ω, as well as of their derivatives andinverse functions. In particular, we can bound the inverse of the operator Dμ,ω (combined withderivatives of the function) as

‖D−sμ,ω ∂pθu‖ξ−δ ≤ σp,s(δ) ‖u‖ξ,

for some 0 < δ < ξ and for p, s ∈ Z+, where

σp,s(δ) ≡ sup(n,m)∈Z2\{0}

(|i(ωn+m) + μ|−s|n|pe−δ(|n|+|m|));

the above quantity can be bounded as

σp,s(δ) ≤(sτ + p

e

)sτ+pCsδ−(sτ+p).

Notice that the inversion of the operator Dμ,ω produces divisors of the form i(ωn +m) + μ,which can be small as far as μ is close to zero. Since 〈(1 + uθ)Fμ,ω(u; γ)〉 = μω〈(uθ)2〉 + γ, ifFμ,ω(u; γ) = 0 one recovers the compatibility condition (28).Step 2. Given an approximate solution (v, β) of Fμ,ω(u; γ) = 0, we construct a quadraticallysmaller approximation (v′, β′) by a Newton iteration scheme. More precisely, starting from

Fμ,ω(v;β) ≡ Δμ,ωv + εVx(θ + v, t) + β = χ,

we look for a solutionv′ = v + v, β′ = β + β,


such that the corrections v, β are orders of ‖χ‖, while Fμ,ω(v′;β′) = O(‖χ‖2). In order to findv and β, we define W ≡ 1 + vθ and we introduce the quantities

Q1 ≡ ε[Vx(θ + v + v, t)− Vx(θ + v, t)− Vxx(θ + v, t)v] , Q2 ≡W−1χθ v;

it follows that

Fμ,ω(v′;β′) ≡ Fμ,ω(v + v;β + β) = χ+ β +Aμ,ω,v v +Q1 +Q2

with Aμ,ω,v v ≡W−1 Dμ,ω(W 2D0(W−1v)). One can find explicit expressions for v, β, such thatthey satisfy the relation

χ+ β +Aμ,ω,v v = 0,

providing that χ′ ≡ Fμ,ω(v + v, β + β) = Q1 +Q2 is quadratically smaller.

Step 3. Given estimates on the norms of v, vθ, v, vθ, β, a KAM algorithm is implemented tocompute an estimate on the norm of the error function χ′. The result provides that the normof χ′ can be bounded as

‖χ′‖ξ−δ ≤ C1δ−s‖χ‖2ξ ,for some positive constants C1 and s.

Step 4. Implement a KAM condition which provides that, under smallness requirements onthe parameters, there exists a sequence (vj , βj) of approximate solutions, converging to the truesolution:

(u, γ) ≡ limj→∞(vj , βj),

where (u, γ) satisfy Fμ,ω(u; γ) = 0.Step 5. A local uniqueness is shown by proving that if there exists a solution ξ(t) = θ+w(θ, t)

with θ = ω and 〈w〉 = 0, then w ≡ u, while η satisfies the compatibility condition η =ω (1 + 〈(uθ)2〉).

6 Dissipative models in Celestial Mechanics

In this Section we present a review of classical dissipative models in Celestial Mechanics, con-cerning rotational and orbital dynamics and, precisely, the Yarkovsky effect and the YORPeffect (Sec. 6.1), the gas drag and the radiation forces in the case of the restricted three–bodyproblem (Sec. 6.2).

6.1 The Yarkovsky and YORP effects

The motion of the asteroids is influenced by the combined action of the solar radiation andtheir thermal inertia. In this Section we describe the effect of the solar radiation on the motionof the asteroids, known as the Yarkovsky effect, and we introduce also its variant called YORPeffect.

6.1.1 The Yarkovsky effect

The Yarkovsky effect is a non-gravitational force that affects asteroid–size objects orbitingaround the Sun; it was discovered at the beginning of the 20th century by the Russian civilengineer Ivan Osipovich Yarkovsky. This effect is due to the reemission of the solar radiationabsorbed by the body, that reradiates the solar energy in the infrared in an amount proportionalto the fourth power of the temperature in each point of the surface [35]. The reemission is notdirected towards the Sun, and has a component along the orbit of the asteroid, causing a dragon the motion; in fact there is a lag between the directions of absorption and reemission of theenergy, due to the finite thermal inertia of the surface of the body.


Fig. 14. Left: the diurnal Yarkovsky effect for a body with spin-axis perpendicular to the orbit. Right:the seasonal Yarkovsky effect for a body whose spin-axis lies in the orbital plane.

The Yarkovsky effect causes a drift of the semimajor axis with the consequent displacementof the asteroid from its orbit. We can distinguish a diurnal component, the effect discoveredby Yarkovsky, and a seasonal component, discovered at the end of the 20th century [35] andsometimes called thermal drag. The diurnal component is linked to the rotation of the asteroidabout its axis and causes either shrinking or expanding orbits, depending on the sense of rotationof the asteroid; the seasonal effect, due to the revolution of the asteroid around the Sun, causesonly the shrinking of the orbit (the asteroid will be driven toward the Sun); it can induce anincrease of the inclination and a decrease of the eccentricity. In the following we describe thetwo components in more detail [5].

Diurnal component.We consider the case where the asteroid moves on a circular orbit aroundthe Sun and its spin-axis is normal to the orbital plane, so that the Sun is always aligned withits equator. The solar radiation heats up the sunward side and this energy is reradiated inthe infrared. More energy is reradiated from the hottest side than from the other one; as aconsequence the asteroid feels a force in the direction away from the hottest side, and the effectof this force is to weaken the Sun’s grip on the body. Since all celestial bodies have thermalinertia, the hottest part is the afternoon side; therefore the force felt by the asteroid also hasan along-track component, that causes a change in the semimajor axis. If the asteroid rotatesin a prograde sense, the orbit will expand; if the rotation is retrograde, the orbit will shrink.The effect vanishes if the spin-axis is in the orbital plane. The amplitude of the force dependson the physical characteristics of the body, such as composition, size and shape, as well as itsdistance from the Sun and the spin-axis inclination.

Seasonal component. We assume again that the orbit is circular, but the spin-axis nowbelongs to the orbital plane. This hypothesis implies that the Sun heats up one hemisphere,e.g. the northern one, more than the other one. This gives rise to a force which pushes awayin the direction opposite to the northern hemisphere; the thermal inertia of the body impliesthat there is an along-track force that tends to shrink the orbit. In this case, the effect doesnot depend on the sense of rotation, so the orbit can only shrink. If the eccentricity of theasteroid is small, this force is in competition with the motion and it causes an orbital decay,thus behaving as a dissipative force. The amplitude of the effect depends on the inclination ofthe spin–axis, and it vanishes if the spin-axis is normal to the orbital plane.Assuming isotropic reemission and that the surface of the body is smooth, the force df felt

by the body per unit of mass is given by [5]

df = −23

εσ

mcT 4n⊥dS,


where T is the surface temperature, m is the mass of the body, c is the velocity of light, ε isthe thermal conductivity, σ is the Stefan–Boltzmann constant, n⊥ is the the normal vector tothe body surface element dS.Using a local coordinate system (x, y, z) with the z-axis aligned with the spin-axis and the x

and y axes lying in the equatorial plane, we can separate the two components described above:the diurnal component (fx, fy), depending mainly on the rotation frequency and the seasonalcomponent fz, depending only on the mean motion n of the asteroid.The main effect on the motion is the drift of the semimajor axis; considering a spheric

asteroid with radius R and averaging over one revolution around the Sun, the mean variationof the semimajor axis is given by [5]:

(da

dt

)diurnal

= −89

β

nFω(R,Θ) cos γ +O(e)

(da

dt

)seasonal

= −49

β

nFn(R,Θ) sin

2 γ +O(e),

where β is a parameter depending on the physical characteristics, like the albedo factor andthe radiation pressure coefficient, γ is the obliquity of the spin–axis, Fω and Fn are functions ofthe body radius R and of the thermal parameter Θ, related to the ratio between the absorptionand reemission of the radiation.The theory of the Yarkovsky effect has been applied to the study of the evolution of small

bodies in the solar system and their dynamics. For example, we refer to [6] and [39] for thestudy of the role played by the Yarkovsky effect in the spreading of asteroid families, like theKoronis and Eos families, or to [29] for a study of the mechanisms leading to the escape ofasteroids from the main belt due to the Yarkovsky effect.

6.1.2 The YORP effect

The YORP (Yarkovsky–O’Keefe–Radzievskii–Paddack) effect is a torque due to the reemissionand reflection of the solar radiation in the case where a rotating asteroid has an irregular shape.The name was invented in 2000 by Rubincam [36], but the effect was observed for the first timein 2007 [37,38].In the case of a symmetrical asteroid, the reaction force due to the reemission and reflection

from any given surface element will be normal to the surface, so that no torque is produced,while an irregular shape implies that the asteroid reflects the absorbed energy in differentdirections, producing a torque.The first example was given by Rubincam [36] through a model consisting of a sphere with

two wedges attached to its equator; the energy reemitted from the wedges produces a torque,because the faces of the wedges are not coplanar. We refer to [37] for a detailed description ofthe YORP effect.The main effect of this torque on the dynamics is to induce changes in the spin vector and

in the obliquity of the asteroids. We suppose that the asteroid rotates around its shortest axisof inertia; we denote by C the associated moment of inertia, s the unit vector of the spin-axisand ω the angular velocity of the asteroid. Then the angular momentum of the asteroid is givenby L = Cωs. The evolution of the angular momentum caused by an acting torque T is givenby:

dL

dt= T , (30)

where the term T includes the YORP torque and the tidal torques. If C is constant, from (30)we can separate the evolution of the angular velocity and of the spin-axis

dω

dt=T · sC,

ds

dt=T − (T · s)sCω

.


Applications of the YORP theory apply mostly to the study of the evolution of small bodies inthe solar system. As an example, we refer to [40] for the study of the influence of the YORPeffect on changes of the spin vectors of the Koronis family of asteroids and to [19] for an analysisof the evolution of rubble piles under the combined effect of tidal and radiation torques.

6.2 The restricted, planar, circular three-body problem with dissipation

The motion of an asteroid around the Sun, as a first approximation, can be described byKepler’s laws. However, in the real solar system, the Sun–asteroid system is not isolated, beinggravitationally influenced by the presence of the other planets, in a measure which is propor-tional to the mass of each planet: this means that the motion of the asteroid is perturbed fromits Keplerian orbit. As a first approximation, we can take into account the force exerted byJupiter, the largest planet in the solar system, although its influence remains “small” whencompared to the gravitational action of the Sun, since the mass ratio between Jupiter and theSun is of the order of 10−3. Therefore, the Sun–Jupiter–asteroid system can be studied as aperturbation of the Sun–asteroid system, known as the three-body problem.If, instead of the asteroid, we consider a dust particle, such a model can be used to study

the early stages of the formation of the planets, which can be described as a protoplanet (theprimordial core of a planet) rotating around the Sun in a cloud of particles and gas. In such ascenario, we are led to consider some additional forces, which are in general non-conservative;they played a very important role in the process of formation of the solar system. Here we willconsider two examples, widely used in the literature [2–4,15], to study the capture in orbitalresonance of planetesimals during the formation of the planets: the Stokes drag, which is a non–gravitational effect experienced by a small particle moving around a star in a nebula of gas andit is due to the collisions of the particle with the molecules of the gas; the Poynting–Robertsondrag force, which is a drag force acting on a small particle moving around the Sun and it is dueto the radiation incident on the particle.In this Section we first recall the main features of the conservative restricted, planar, circular

three-body problem (Sec. 6.2.1) and then we review some dissipative contributions (Sec. 6.2.3).In particular, we take into account the Poynting–Robertson and Stokes dissipations, which aredescribed in details in Sec. 6.2.2.

6.2.1 The conservative restricted, planar, circular three-body problem

The conservative three-body problem describes the dynamics of a system composed by threebodies interacting only through the gravitational force. The problem is said to be restricted, ifthe mass of one body (the asteroid or the particle in our examples) is negligible with respectto the masses of the two others, which are called primaries. This means that, in the restrictedproblem, the motion of the primaries is not influenced by the presence of the third body; theprimaries keep moving on Keplerian orbits. In this Section we deal with the circular and planarcase, assuming that the primaries move on circular orbits and that the three orbits lie all onthe same plane.We denote by P1 and P2 the primaries and by P the third body, with masses m1 ,m2 ,m,

respectively, where m1 ≥ m2 and m� m1,m2. Let us introduce the coordinates of P , P1, P2 inan inertial planar frame as follows: P = (ξ, η), P1 = (ξ1, η1), P2 = (ξ2, η2). We choose suitableunits of mass, length and time such that the gravitational constant, the distance between theprimaries, and the total mass of the system are equal to 1; then in the sidereal coordinatesystem the equations of motion of the third body read as

ξ = (1− μ)ξ1 − ξr13

+ μξ2 − ξr23

,

η = (1− μ)η1 − ηr23

+ μη2 − ηr23

,


Fig. 15. The three bodies in the synodic frame: the two primaries P1 and P2 lie on the x-axis at fixedpositions while the third body P is orbiting around them with respective distances r1 and r2 to P1and P2.

where μ is the reduced mass given by

μ =m2

m1 +m2,

while r1 and r2 are the distances of P from P1 and P2, respectively, defined as

r21 = (ξ1 − ξ)2 + (η1 − η)2,r22 = (ξ2 − ξ)2 + (η2 − η)2.

We switch to a synodic (rotating) reference frame, having the same origin as the sidereal oneand rotating synchronously with the primaries. Then, the primaries stay fixed with coordinatesP1 = (−μ, 0), P2 = (1 − μ, 0). Denoting by (x, y) the coordinates of the third body in thesynodic frame, the equations of motion become:

x = 2y + x− (1− μ)x+ μr13

− μx− 1 + μr23

y = −2x+ y − (1− μ) yr13− μ yr23,

(31)

withr21 = (x+ μ)

2 + y2, r22 = (x− 1 + μ)2 + y2.We recall that the three–body problem admits 5 equilibrium points: 3 unstable, L1, L2, L3,located on the x-axis of the synodic frame, and two stable (below Gascheau’s value), L4, L5,called Lagrangian or triangular, because each of them forms an equilateral triangle with theprimaries.

6.2.2 Stokes drag and Poynting–Roberson drag force

In this Section we provide a physical description of the two main dissipative forces that affectsmall particles in the solar system and in the primitive nebula.

Stokes drag. A particle moving in a viscous fluid experiences a viscous drag force, due tothe collisions with the fluid molecules; as a consequence the particle velocity decreases. Theexpression of this force depends on the size of the particle and on its relative velocity withrespect to the fluid, in particular on the so-called dimensionless Reynolds number, that gives ameasure of the ratio between the inertial force and the viscous force.If the particle is small or it moves very slowly, meaning low Reynolds number, the drag is

due to the difference between the number of collisions experienced by the particle in the forward


and in the backward direction; the drag fs is linearly proportional to the difference u = v−vgbetween the particle velocity v and the velocity of the gas vg and it is given by

f s = −ku,

where k is proportional to the viscosity of the gas and to the size of the particle. This force isknown as Stokes drag.If the particle is large or it moves faster through the fluid, i.e. high Reynolds number, the

fluid becomes turbulent and vortices are created around the particle, causing the drag f t tobecome quadratic in the relative velocity (the case sometimes referred to as Epstein drag):

f t = −kquu,

where the coefficient kq now depends on the size and density of the particle and on the gasdensity, but it is independent of the viscosity.In the solar nebula, the gas is moving circularly around the Sun, with a velocity vg slightly

smaller by a factor α than the local Keplerian velocity vkep, i.e.

vg = αvkep(r),

withvkep(r) = Ω(r)× r = r−3/2Iz × r,

where Ω is the angular velocity, Iz is the unit vector perpendicular to the orbital plane. Typicalvalues of α used in modeling the solar nebula are α ≥ 0.99; in particular α = 0.995 is used bothin the linear (Stokes) [2] and quadratic (Epstein) case [27].Since these drags are purely dissipative, they can only cause the orbit of the particle to cir-

cularize and shrink. Their contribution has been identified as a possible mechanism of resonancetrapping in the solar nebula [2] .

Poynting–Robertson drag force. A particle orbiting around the Sun is continuously bom-barded by the solar photons and therefore is subjected to a force due to the difference betweenthe impulse carried out by the absorbed and reemitted photons. This force is composed of anoutward radial component, due to the absorption of the direct solar radiation, and of a tangen-tial component directed along the particle velocity, due to the difference between the momentaof the photons reemitted by the particle in the forward and backward directions. The globaleffect on the motion is to slow down the particle, whose orbit tends to spiral towards the Sun.The radial component f rad, aligned in the direction away from the Sun, is proportional to

the amount of light hitting the particle, which is given by the product of the solar intensity Wreceived by the particle and the geometrical cross section A of the particle

f rad =WA

cr,

where r is the unit vector along r. If the velocity of the particle has a radial component, thenthe intensity W is given by the solar flux W0 reduced by a factor 1− r/c:

W =W0

(1− rc

).

We recall that the flux W0 is proportional to the inverse of the square of the distance fromthe Sun. This component can only increase the distance of the particle from the Sun. Notethat if the orbit of the particle is circular, this force is orthogonal to the velocity and it is notdissipative.The tangential component, always opposed to the velocity, is caused by the combination of

the motion of the particle with the reradiation of the solar energy. Being in thermal equilibrium,the particle reemits the radiation received from the Sun in all directions; however its motioncauses the photons reemitted in the forward direction to have higher momenta than those


Fig. 16. A particle orbiting around the Sun with velocity v is hit by the solar radiation, which isalways aligned in the radial direction r. The radiation pressure pushes the particle away from the Sun(orange arrow). The particle reemits in all directions but with higher momenta in the forward directioncausing a negative push, equivalent to a drag (blue arrow).

reemitted in the backward direction (because of the Doppler-effect). As a consequence theparticle experiences a drag fPR given by

fPR = −WA

c

v

c.

The total force f felt by the particle is finally the sum of the two components f rad and fPR:

f =W0A

c

(1− rc

)[r − vc

]� W0A

c

[(1− rc

)r − vc

],

where in the last expression we just kept terms up to the first order in v/c.We remark that the constant component of the radial term, independent of the velocity, is

conservative; it is usually called the radiation pressure. The other two terms give a dissipativeforce known as the Poynting–Robertson drag [8]. Already Robertson observed that the dissipa-tive forces linked to the solar radiation can cause a slow decay of the semimajor axis. Note thatsometimes the whole radial term is called radiation pressure, while only the term aligned withthe velocity is called Poynting–Robertson drag.

6.2.3 The dissipative, restricted, planar, circular three-body problem

In this Section we introduce the Stokes and Poynting–Robertson dissipations in the frameworkof the three-body problem; adding the components (Fx, Fy) of the dissipative forces, Stokes orPoynting–Robertson drag force, the equations of motion (31) become

x = 2y + x− (1− μ)x+ μr13

− μx− 1 + μr23

+ Fx

y = −2x+ y − (1− μ) yr13− μ yr23+ Fy.

In the case of the Stokes drag the gas is assumed to have a constant density and to be in circularmotion around the barycenter of the primaries. Let Ω = Ω(r) ≡ r−3/2 be the Keplerian angularvelocity at distance r =

√x2 + y2 from the origin of the synodic frame and let α ∈ [0, 1) be

the ratio between the gas and Keplerian velocities [31]. In this reference frame the Stokes drag


0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.35 0.4 0.45 0.5 0.55

k = 0

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.35 0.4 0.45 0.5 0.55

k = 1e-4

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

k = 1e-3

Fig. 17. Deformation and destabilization of a libration orbit around the L4 Lagrangian equilibriumpoint. Left: conservative case (k = 0). Center: moderate dissipation (k = 10−4), the orbit is displacedand extended. Right: for larger dissipation (k = 10−3) the orbit ends on the Sun.

can be described as a linear function of the relative velocity of the particle with respect to thegas with components (Fx, Fy) in synodic coordinates [32] given by

(Fx, Fy) = −k(x− y + αΩy, y + x− αΩx),

where k is the positive dissipative constant depending on the viscosity of the gas, on the radiusand on the mass of the particle [2].In the case of the Poynting–Robertson dissipation, we consider only the dissipative compo-

nents of the force. The tangential component of the drag (tx, ty) can be described as a forceproportional and opposed to the velocity of the particle in the synodic frame, with a magnitudeinversely proportional to r21, since the amount of light hitting the particle is proportional to theinverse of the square of the distance from the Sun:

(tx, ty) = −k

r12(x− y, y + x) .

The velocity dependent part of the radial component (rx, ry) decreases much faster than thetangential component, being proportional to the inverse of the fourth power of the distancefrom the Sun:

(rx, ry) = −k

r14((xx+ yy)x, (xx+ yy)y).

Then, the total dissipative force, known as Poynting–Robertson drag force, in the synodic frametakes the following form [32]:

(Fx, Fy) = −k

r12

(x− y + x

r12(xx+ yy), y + x+

y

r12(xx+ yy)

),

where k is the dissipative constant, depending on the size of the particle and on the brightnessof the source.An analysis of the effect of these kinds of dissipation on the dynamics of the three-body

problem can be found in [31], where the stability of the Lagrangian points is investigated;several types of dissipations are considered, among which the Poynting–Robertson drag force,that causes the Lagrangian points L4 and L5 to become linearly unstable.The quadratic drag was studied in [27], where the capture in resonance of planetesimals

orbiting in the solar nebula in a simplified model is investigated: namely a three-body problemcomposed of Sun-protoplanet-planetesimal, with the protoplanet in a circular orbit. It was foundthat resonances can counteract the orbital decay due to the dissipation, as it was also observedin [2] for the Stokes drag in a similar model. However, the capture is in general temporary, and itonly slows down the drift of the orbit towards the Sun (see also [3] for the Poynting–Robertsoncase).In Fig. 17 we can see the effect of the Poynting–Robertson drag dissipation (given by (6.2.3))

on the motion by comparing the integration of a libration orbit around the Lagrangian pointL4 (the so-called tadpole orbits) with different strengths of the dissipation. As the dissipation


0

500

1000

1500

2000

2500

0 100 200 300 400 500 600

colli

ding

orb

its

t

k=1e-4k=1e-3

Fig. 18. Number of orbits colliding with one of the primaries as a function of the time over a total of10201 orbits, for two values of the dissipation coefficient: k = 10−3 and k = 10−4.

increases, the orbit is displaced and it extends more and more, until it becomes unstable andspirals towards the Sun for large value of the dissipation. In Fig. 18 we give another example,showing that the main effect of the dissipation is to lead the orbits to collide with the primaries,and that the number of colliding orbits increases as the strength of the dissipation and integra-tion time increase. The figure refers to the Sun-Jupiter-particle system in presence of Stokesdissipation; we plot the number of orbits ending on one of the two primaries as a function oftime for two values of the dissipation coefficient.

6.3 Dissipative mappings

We conclude by providing some results on the construction of mappings describing dissipa-tive dynamics. When studying an N -body problem, the computational time for the numericalintegration of the associated system of differential equations can be quite long; for this rea-son mapping methods have been developed, since a mapping can describe the behavior of thedynamics with a large gain on the computational time. A well-known mapping for the conser-vative N -body problem was developed by Wisdom and Holman [41], but their algorithm usesa symplectic integration scheme, which is not suitable for the dissipative case.As we have seen, for a more realistic study of some planetary problems, as well as in

modeling the development of the Solar System, it is very important to take into account theeffect of dissipative forces. An algorithm for including dissipative effects by means of a mappingtechnique was provided in [28], using an algorithm based on an extension of [41]. This algorithmuses a second-order integration scheme and it was applied in [28] to the restricted, planar,circular three-body problem with a gas-drag dissipation (quadratic model). The results werecompared with the numerical integration of the equations of motion using a fourth-order Runge–Kutta integrator. It turns out that the integration error using the mapping grows only linearlywith the integration time, while with Runge–Kutta the dependence is quadratic. Moreover themethod is at least one order of magnitude faster than Runge–Kutta for the same error levels.Another method for the construction of mappings for non-conservative systems describing

the planetary dynamics was developed in [17], which uses a second order method, applied to themotion of a massless particle orbiting around a central body in the presence of quadratic gasdrag and Poynting–Robertson dissipations. Their results were compared with the integration ofthe corresponding differential system carried out using a 15th order Radau integration scheme;it was shown that the mapping follows quite well the overall integration.In a more recent work [23], adapting the algorithm developed in [28], a mapping for the

dissipative N -body problem was developed for the interaction between massless particles withone dominant central mass. The effects of the Poynting–Robertson drag, the radiation pressure


and the solar wind were considered. The algorithm was implemented on a two-body problemand on a restricted planar circular three-body problem; the tests gave good results, showing thatthe mapping provides a good approximation for the motion of a particle under the combinedeffect of these dissipative forces.We remark that the low order as well as the fixed step size (a feature common to all

the techniques based on the Wisdom and Holman algorithm) are typical limitations for all themethods quoted here. Another limit of these techniques is the impossibility of a good integrationof close encounters, which is relevant in the case of dissipative systems whenever the effect ofthe dissipation can cause the particle to spiral towards the Sun.

The authors are indebted to Daniel Benest, Claude Froeschle and Elena Lega for the organization of theschool “La dynamique des systemes gravitationnels: defis et perspectives”. The authors acknowledge thegrants ASI “Studi di Esplorazione del Sistema Solare” and PRIN 2007B3RBEY “Dynamical Systemsand applications” of MIUR (Italy).

A Appendix: The tidal torque

The motion of a satellite around a planet, under the gravitational interaction, is described byEq. (16) in Sec. 4. However, an important contribution comes from dissipative forces, whichappear if we take into account the internal non-rigidity of the satellite.In fact the side of the satellite towards the planet feels a force stronger than the other

side, and this produces a tidal bulge due to the non-rigidity of the satellite; because of internalfriction, the maximum tide height lags behind the direction of the planet. This shifted bulgecauses a tidal torque that will act as a dissipative force and will slow down the rotation of thesatellite.The expression of this torque is given by (see [33])

T =3k2M

2R5

2r6(r · rT )(rT × r),

where M is the mass of the planet, R is the satellite’s radius, k2 is the tidal Love number, rand rT are unitary vectors aligned towards the planet and the tidal maximum, respectively.We introduce the tidal dissipation in our model by adding the averaged torque to the righthand side of equation (16). We refer to [33] for the derivation of the following expression of themagnitude of the tidal torque T (x, t) as a linear function in the relative angular velocity x:

T (x, t) = −Kd [L(e, t)x−N(e, t)] ,

where

L(e, t) =(ar

)6, N(e, t) =

(ar

)6f ,

with a, r defined as in Section 4 and e being the orbital eccentricity, while Kd is the dissipativeconstant, which is of the order of 10−8 for bodies like the Moon or Mercury. This constant isgiven by [12]

Kd = 3nk2

ξQ

(Re

2

)2M

m,

where n is the mean motion, Q is the damping quality factor (comparing the frequency ofoscillations of the system to the rate of dissipation of energy), ξ is a relative moment of inertiasuch that C = ξmR2e, Re being the equatorial radius of the satellite and m is the mass of thesatellite S.Averaging over one orbital period [18] we obtain a tidal torque depending only on L(e) and

N(e), where the bar stands for the average over the time; the corresponding equation of motionreads as [12]

x+3

2

B −AC

(ar

)3sin(2x− 2f) = −Kd

(L(e)x−N(e)

)


with

L(e) =1

(1− e2)9/2

(1 + 3e2 +

3

8e4), N(e) =

1

(1− e2)6(1 +15

2e2 +

45

8e4 +

5

16e6).

With reference to (17) we set

μ = KdL(e), η =N(e)

L(e).

References

1. V.I. Arnold, Russ. Math. Surveys 18, 13 (1963)2. C. Beauge, S. Ferraz-Mello, Icarus 103, 301 (1993)3. C. Beauge, S. Ferraz-Mello, Icarus 110, 239 (1994)4. C. Beauge, S.J. Aarseth, S. Ferraz-Mello, Monthly Notices Royal Astron. Soc. 270, 21 (1994)5. W.F. Bottke Jr., D. Vokrouhlicky, D.P. Rubincam, D. Nesvorny, Annu. Rev. Earth Planet Sci. 34,157 (2006)

6. W.F. Bottke, D. Vokrouhlicky, M. Broz, D. Nesvorny, A. Morbidelli, Science 294, 1693 (2001)7. H.W. Broer, C. Simo, J.C. Tatjer, Nonlinearity 11, 667 (1998)8. J.A. Burns, P.L. Lamy, S. Soter, Icarus 40, 1 (1979)9. R. Calleja, A. Celletti, CHAOS 20, issue 1, 013121 (2010)10. A. Celletti, J. Appl. Math. Phys. (ZAMP) 41, 174 (1990)11. A. Celletti, J. Appl. Math. Phys. (ZAMP) 41, 453 (1990)12. A. Celletti, Stability and Chaos in Celestial Mechanics (Springer-Praxis, 2010), XVI, p. 26413. A. Celletti, L. Chierchia, Arch. Rat. Mech. Anal. 191, 311 (2009)14. A. Celletti, S. Di Ruzza, Periodic and quasi-periodic attractors of the dissipative standard map(preprint) (2010)

15. A. Celletti, L. Stefanelli, E. Lega, C. Froeschle, Some results on the global dynamics of theregularized restricted three-body problem with dissipation (preprint) (2010)

16. B.V. Chirikov, Phys. Rep. 52, 264 (1979)17. R.R. Cordeiro, R.S. Gomes, R. Vieira Martins, Cel. Mech. Dyn. Astron. 65, 407 (1997)18. H. Curtis, Orbital Mechanics (Butterworth-Heinemann, Elsevier, 2005)19. P. Goldreich, R. Sari, Astrophys. J. 691, 54 (2009)20. J.M. Greene, J. Math. Phys 20, 1183 (1979)21. M. Henon, Bull. Astron. 3, 49 (1966)22. M. Henon, Comm. Math. Phys. 50, 69 (1976)23. T.J. Kehoe, C.D. Murray, C.C. Porco, Astron. J. 126, 3108 (2003)24. A.N. Kolmogorov, Dokl. Akad. Nauk. SSR 98, 527 (1954)25. O.E. Lanford III, Physics A 124, 465 (1984)26. J. Laskar, C. Froeschle, A. Celletti, Physica D 56, 253 (1992)27. R. Malhotra, Icarus 106, 264 (1993)28. R. Malhotra, Cel. Mech. Dyn. Astron. 60, 373 (1994)29. A. Morbidelli, D. Vokrouhlicky, Icarus 163, 20 (2003)30. J. Moser, Nach. Akad. Wiss. Gottingen, Math. Phys. Kl. II 1, 1 (1962)31. C.D. Murray, Icarus 112, 465 (1994)32. C.D. Murray, S.F. Dermott, Solar System Dynamics (Cambridge University Press, 1999)33. S.J. Peale, Icarus 178, 4 (2005)34. H.O. Peitgen, H. Juergens, D. Saupe, Chaos and Fractals: New Frontiers of Science (New York,Springer-Verlag 1992), p. 585

35. D.P. Rubincam, J. Geophys. Res. 100, 1585 (1995)36. D.P. Rubincam, Icarus 148, 2 (2000)37. D.J. Scheeres, Icarus 188, 430 (2007)38. P.A. Taylor, et al., Science 316, 274 (2007)39. K. Tsiganis, H. Varvoglis, A. Morbidelli, Icarus 166, 131 (2003)40. D. Vokrouhlicky, D. Nesvorny, W.F. Bottke, Nature 425, 147 (2003)41. J. Wisdom, M. Holman, Astron. J. 102, 1528 (1991)

Nearly-integrable dissipative systems and celestial mechanics

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