on the dynamics in the asteroids belt. part i : general theory

O N T H E D Y N A M I C S I N T H E A S T E R O I D S B E L T .

P A R T I : G E N E R A L T H E O R Y .

ALESSANDRO MORBIDELLI Department of Mathematics, FUNDP

Rempart de la Vierge 8, B-5000 Namur, Belgium E-mail AMORB [email protected]

ANTONIO GIORGILLI Dipartimento di Matematica dell'Universit~

Via Saldini 50, 20133 Milano, Italy E-mail [email protected]

(Received : 24 July, 1989; accepted : 21 September, 1989)

Abst rac t . The classical problem of the dynamics in the asteroids belt is revisited in the light of recently developed perturbation methods. We consider the spatial problem of three bodies both in the circular and in the elliptic case, looking for families of periodic or quasi periodic orbits. Some criteria for deciding the stability of these families are also indicated.

Keywords : resonance, perturbation method, Kirkwood gaps, periodic orbits

1. I n t r o d u c t i o n

Since the discovery of gaps in the asteroid belt by Kirkwood (1866), a lot of work has been devoted to the mechanical explanation of such a phenomenon, both by analytic and by numerical methods based on suitably simplified models. The analytic framework of all these works is usually the classical three body problem in the planar (both elliptic and circular) case. The aim of the present paper is to produce a theoretical analysis of the same problem in the spatial case. In particular we look for general methods to find families of resonant periodic orbits and to study their stability properties.

The guiding idea of almost all the at tempts to explain the existence of gaps is essentially to exploit the fact that they are apparently connected with resonances between the mean motion of the asteroid and that of Jupiter. In this context, let's consider in particular two different approaches; on the one hand, one has the beautiful analytic works by Moser (1958) and Brjuno (1970), who, in the framework of the stability theory of periodic orbits, find general results which could be possibly applied to the asteroids model; on the other hand, one finds several authors, who, starting from the work of Poincard (1892) on the evolution of the eccentricity for

Celestial Mechanics and Dynamical Astronomy 47: 145-172, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

146 A. MORBIDELLI AND A. GIORGILLI

orbits close to the circular one, use the averaging method of perturbation theory in order to develop quite simple models.

In fact, the actual application of the analytic results by Moser and Brjuno is not so straightforward : one has to determine a (family of) periodic orbit(s) in the frame rotating with Jupiter's mean motion, and to perform a number of transformations which are in principle possible, but can hardly be explicitly determined. For example, families of periodic orbits have been numerically determined by Colombo (1968), but an attempt of ours to apply Moser's methods to these orbits turned out to be impractical. Thus, it is not so surprising that most authors preferred to develop simpler models, applying averaging methods to a power expansion of the Hamiltonian in the eccentricity up to a quite low order, in the spirit of Poincar@'s theory. To quote only recent works, let's refer to the papers by Wisdom (1983, 1985) and Henrard et al. (1983-a, 1986), where extensive references to previous works can be found. The first of these authors stresses the possible relevance of chaotic dynamics which occurs in the elliptic case, since it can raise the eccentricity of an orbit up to values high enough to produce close encounters with Mars; the same approach was also considered by Henrard and Lema~tre (1987, 1989) and by Henrard and Caranicolas (1988), pointing out that such a model can be successfully applied to the 3/1 resonance, but seems to be insufficient to explain the gap at the 2/1 resonance. In the latter case, a possible explanation may be given by the evolutionary model proposed by Henrard and Lemaltre (1983-b). Thus a quite wide gap appears to exist between the analytical approach of Brjuno and Moser and the second approach. So, it seemed to us to be worthwhile trying to fill such a gap in two aspects : i) the consideration of the spatial motion, and ii) the application of general perturbation techniques to the complete three body model.

A reason for considering the spatial model is that a careful examination of the distribution of the asteroids clearly shows that many of them have a quite high inclination : for example, using the data in Marsden's 1987 catalog, we found that the inclination of the orbit with respect to Jupiter's plane is larger than 15 degrees for 528 asteroids, which is about 15 percent of the total number (3495). Moreover, by separately plotting the distribution for inclinations below and above 15 degrees respectively, one clearly sees different peaks, and this suggests the possible relevance of the inclination. So it seems worthwhile trying to look for stable regions outside Jupiter's plane. Moreover, until now, to our knowledge, the analytical methods were applied only to investigate the planar stability of orbits lying on Jupiter's plane, while it's interesting to study also their stability with respect to the transversal motion.

Concerning the model, we avoid to introduce any simplification in the Hamilto- nian of the spatial restricted problem of three bodies, at the same time trying to use the peculiarities of that model. To this end, we directly apply the perturbative methods to the complete Hamiltonian, thus allowing the perturbative scheme itself to indicate which are the relevant terms in the expansions. As will be described later, we expand the Hamiltonian in the neighbourhood of a given, but arbitrary,

DYNAMICS IN THE ASTEROIDS BELT (PART I) 147

orbit, and then obtain equations for the family of orbits satisfying a given resonance or nonresonance condition. Next, the stability of the orbits belonging to a given family is investigated with the aid of the normal form of the Hamiltonian in the neighbourhood of that family. This requires in general the construction of the normalized Hamiltonian by successive steps, up to order 4. A rigorous analysis of the stability could in principle be given, for example along the lines of Giorgilli et al. (1989), but it would require hard work in determining some constants related to the analyticity properties of the original Hamiltonian.

In the present paper we develop the formal theory, leaving the actual application of our methods to a forthcoming paper. In particular we pay at tent ion to producing formulae where all the constants can be explicitly computed by numerical methods. The paper is organized as follows. In sect. 2 we briefly recall the essential lines of the normal form theory we are going to use. In sect. 3 we give a complete exposition of our method in the case of the circular three body problem; the elliptic case is discussed in sect. 4. The conclusions follow.

2. T h e p e r t u r b a t i o n s c h e m e

The per turbat ion scheme we are going to use is strictly related to the known Lie transform algorithms, which are commonly used in Celestial Mechanics. However, for reasons that will be evident below, we prefer a completely algebraic approach, since it removes the need for an explicit per turbat ion parameter. Such an approach has been already used by Giorgilli et al. (1978, 1985, 1989). A complete general discussion can be found in Colombo and Giorgilli (1989), and we recall here just the essential lines of the algorithm.

We deal with a many parameter expansion of the Hamiltonian, namely we consider as per turbat ive parameters both a small parameter, E say, like in Von Zeipel's normal form method , and the size of the domain where the canonical coordinates are allowed to vary. More precisely, if some of the canonical variables are allowed to vary only in a small neighbourhood of a fixed point (for example, an elliptic equilibrium), we consider the polynomial expansion of the Hamiltonian in both the parameter and the coordinates.

A near to identi ty canonical transformation is defined making use of an operator

T x acting on phase space functions. Given a generating sequence X = {XS}s>l , with X, of order s, for any function f on the phase space, we introduce the operator T x via the recursive definition

s>O

(2.1) f o = f , f , = ~{Xj f,-.i} ,

j = l


where {., .} denotes the Poisson brackets. This is seen to define a near to iden- t i ty canonical transformation, because if f is any of the canonical coordinates, say p or q, and one considers the transformation defined by (p,q) = (Txp',T×q'), then the coordinate transformation turns out to be canonical. Moreover one has f(TxP', Txq' ) = (Txf)(q',p'), which is nothing but the exchange theorem for Lie transforms (see for reference Gr6bner (1967)).

If now the Hamiltonian K is also expanded, in the form

K(p,q) = Z Ks ' s > o

with Ks of order s and/Co integrable, the normal form is obtained by solving the equation TxK = Z, with respect to the unknown X and Z, and with the further condition for Z to be in normal form. This means typically that some angles do not appear in the transformed Hamiltonian, so that there exists one or several formal integrals. In practice, one looks for a normal form up to a finite order, r say, by introducing in the equation above the expansion Z = ~s>_o Zs, and obtaining the recursive system

Z0 =/Co (2.2) {Ko, xs} + Zs = ,rs ,

where kvs is known, namely

s - -1 .

j = l j----1

(2.3)

Here, the second index in Kt,j denotes the j - t h term generated by the application of T x to Kt, as in the very definition (2.1). The solution of this system up to order r, allows to reduce the non normalized part of the Hamiltonian (the remainder) to order at least r + 1.

From a rigorous viewpoint, one could produce explicit estimates on the size of the remainder, for example along the lines of Giorgilli et al. (1989), and choose an optimal normalization order, in the sense that the size of the remainder is close to a minimum. In general, by such a procedure one can conclude that the time evolution of the formal integrals of the normalized Hamiltonian is so slow that it can only be detected over a very large time scale, possibly exceeding, for example, the age of the Solar System. Of course, this could hardly be done by explicit computations, due to the great technical difficulties. In fact, we apply the formal normalization procedure, and expect that one should be able, in principle, to prove that the formal results hold for a very large time : in this sense we shall speak of "formal stability".

DYNAMICS IN THE ASTEROIDS BELT (PART I)

3. T h e c i r c u l a r case

149

In the present section we consider the Hamiltonian of the circular restricted three body problem, and develop a general scheme which allows to find resonant periodic orbits both out and on Jupiter 's plane, and to study their stability. Next, a similar work will be done for nonresonant orbits too.

3. 1. THE HAMILTONIAN OF THE MODEL

We consider a rotating rectangular frame with origin at the center of mass of the Sun-Jupiter system, with the y axis pointing towards Jupiter, and the z axis directed as the angular momentum. Jupiter is assumed to revolve on a circular orbit with constant angular velocity n. The mass and length units are chosen so that

M S = 1 , d®-jup = 1 ,

where M® is the mass of the Sun, and d®-jup the distance between Jupiter and the Sun; moreover the gravitational constant is taken as one, thus determining also the unit of time. The Jupiter 's mass will be denoted by #. Then, with r = X/x 2 + y2 + z 2, the Hamiltonian takes the form

1 2 1 K =~(P= + P~ + P]) + n(p .y - p ,x) -

~/.~ + (y _ y®)= + z2

= 7 ( v ~ + p , + p~) + (p~y - p ~ x ) - - ; - v - # k - , ( p ~ y _ p . ~ )

1 (--k½) ( 2y ~ k 1 ~- -- E #k--1 [l

r (1+#)2 r2 + ( l + ~ ) r 27 + x/x 2 + ( y - y j . p ) 2 + z 2 k_>l

tt , / x ~ + (u - uj°p) ~ + z2

so that it can be given the form K = K0 + #Ka, with K0 independent of #. In the formula above we have adopted the notation

(ak) a(a-1). . . (a-k+l) = ~r. ,

where a is an arbitrary real number; P*,Pv and pz are the momenta which are conjugated to x, y and z respectively.

We then introduce the Delaunay variables

L = ~ , l = m(t - to)

G = x / a ( 1 - e 2) , 9 = w

H = x / a ( 1 - e 2 ) c o s i , h = f t ,


through the osculating variables a (semimajor axis), e (eccentricity), i (inclination), (longitude of the ascending node), w (argument of the perihelion) and m (mean

motion). The ascending node is measured with respect to the x axis of our reference frame, so that it turns out to be a fast variable, because of the rotat ion of the frame. The unper turbed Hamiltonian K0 is then

1 K 0 - 2L 2 H , (3.1)

while the per turbat ion K1 takes the form

1 1

( , / ) ( - - G 2 G 2 n>l L 2 1 - 1 - ~ eosE ,,,>__1 (1 -,[- # ) 2 L 4 1 -- - eosE

G ~ 1 - L 4 1 - 1 - - ~ cosE -4- (1 +/~)----------- ~

2 (sinheosg+ Hcoshsing)L2 e o s E - , ~ sing)GLsinE 1+. v (a

(:

The Delaunay variables are surely appropriate for the kind of s tudy we are going to perform; however, in order to achieve a simpler scheme and to allow an easier comparison with the previous known results of many authors, we prefer to use the following modified variables

L = L , )~=l+g+h S ---- L - H , p = - g - h (3.3)

R = G - H , g=g;

A is then the mean longitude of the asteroid and p is the opposite of the longitude of the perihelion, which is here a fast variable as the perihelion is observed in a rotat ing frame. The expressions (3.1) and (3.2) can be easily translated in these new variables.

3. 2. T H E R E S O N A N T C A S E : A FIRST FORMAL REDUCTION OF THE PROBLEM

Starting with the Hamiltonian above of the spatial restricted three body problem we expand it in Taylor series with respect to # around the value 0, and with respect


to the action variables L, S, R around an arbitrary point L0, So, R0. Since we want

to s tudy a resonant case, L0 is fixed at the value of resonance. The choice of So and R0 is instead totally arbitrary; here the only condition is that So - R0 ~ 0, in order to avoid the circular orbits, which are singular points for the Hamiltonian in Delaunay's variables. This local expansion, which is similar in principle to the one recently introduced by Ferraz-Mello (1988), allows us to s tudy the neighbourhood of each point of the six-dimensional phase space, except for those points repre- senting circular orbits, the neighbourhood of which is a sort of black zone for our theory; the problems raised by the presence of the singularities will be discussed fur ther in the text.

Let us introduce the parameter e = v/-fi and the variables L = L - L0, S =

S - So a n d / ~ = R - R0, namely the variations of the actions with respect to the point of expansion. We consider L, S,/~ to be of order e, so that we can expand the Hamiltonian in orders of magnitude (that is orders of e) based on both the

small parameter # and the radius of the neighbourhood we intend to study. In the neighbourhood of a circular orbit, due to the singularity of the Hamiltonian in Delaunay variables at that point, we are forced to assume the more stringent condition S,/~ ,,~ e(So - Ro), which implies e - e0 -~ x/Te0, so that , as is easily checked, the expansion of the Hamiltonian in orders of e is preserved.

The term of the first order comes directly from the unper turbed Hamiltonian, and is obviously

Since we consider a resonant orbit, I / L a must be a rational number, (i + j ) / j say, with i , j integer; i is commonly called "order of the resonance". Because of the presence of zero denominators, it 's impossible to find even a formal canonical t ransformation that eliminates the angles A and p altogether. So we perform the canonical t ransformation

z

Z

(3.5)

cr is now a slow variable; when the asteroid is in between Jupiter and the Sun,

once each i synodic revolutions, a coincides with the longitude of the perihelion measured in the rotat ing frame. It 's an easy mat te r to check that the variable a defined here is exactly the typical critical argument introduced by Poincard (1892) and Schubart (1964) and generally used also in more recent works (see for example Lemaltre 's paper (1984)). By (3.5), the first order term (3.4) obviously becomes

i :A . (3.6)

152 A, MORBIDELLI AND A. GIORGILLI

Proceeding now as illustrated in section 2, it 's possible to find a formM canonical ^ ^

t ransformation (A, S, R, A, a, g) = Tx(A1, S1, R1, ,kl, o'1, gl ), which gives the Hamil- tonian a normal form with respect to A1, so that A1 turns out to be a formal integral.

It 's an easy mat te r to check that the first order term X1 of the generating function is independent of the actions, so that the first three orders of the new Haxniltonian are just the average over X of the corresponding old ones. The second order term X2 is instead linear in the actions, and thus the 4 th order in the new Hamiltonian

is the average of the old one plus a function of the angles gl and cq only.

The next step of our work is to look for the equilibrium points of the new Hamiltonian and to study their stability. Observe that these points correspond to the periodic orbits in the original rotating frame which have the "natural period", namely which close after i synodic revolutions, just like a resonant unper turbed

Keplerian orbit observed in the rotating frame. In sect. 3.3 and 3.4 the periodic orbits on the plane and out of the plane will be separately found and studied.

3.3. THE RESONANT CASE : FAMILIES OF PERIODIC ORBITS ON JUPITER'S PLANE

A local expansion around a given orbit on the plane is performed by setting R0 = 0;

it 's an easy mat te r to check that in the obtained expansion the only functions which depend on g are coefficients of powers of /~. This means that 0 is an equilibrium value for /~ as it obviously had to be, since the plane of Jupiter 's motion is invariant for the restricted three body problem. By setting /~ = 0 we obtain exactly the same equations that characterize the two degrees of freedom study of the circular restricted three body problem. But we won't reduce ourselves to this case, since our goal is to explore the stability of the equilibrium plane.

We expand now the Hamiltonian around a fixed but arbi t rary point al,0, denot-

ing by 6 = al - al,0. We then perform the following canonical t ransformation

(3.7) s , = a =

and consider both :~ and fir to be of order of¢, so that the quadratic terms in S and are of the same order. It 's important to observe that through these transformations the order of magni tude of each term of the Hamiltonian is never decreased; we


report then all its terms up to order 4 :

153

• cqK1 5 Z{ = -2c2v, re~ii S ~ + e~ - ~ a (., = e + + ( g i ) n ,

f l O a K , - a J ) = . -

_ 1 i)4K154 e~fl"~'-gl, ,R ~ (O2K, JO2KI"~&S+ O~K1 (gl)A, R1) Z, 24 cga 4 + ~-~-~(g , ) , + \O--ffOa ~ ~-'-A-Oa) OAOR /

+ e o-R-b--~ail, g~) ~ + ~ oaAo%~-~ A~2 + 3c3~A~S ~ .

(3.8) Here, all the derivatives must be evaluated at Lo,So,Ro = 0, al,0 and # = 0,

and the overbar means average over A. The only functions depending on gl are explicitly indicated• The coefficients c2 and c3 come directly from the expansion of 1/(2L 2) in L0, and thus result equal to -3 / (2L 4) and 2/L~ respectively.

The search of the periodic orbits on the plane is very simple; indeed, as the plane is invariant and gl is meaningless on such a plane, one has to look for equilibrium points in S and 5. Recalling that both S and 5 are variations, one just has to find the point of expansion So and al,0 such that the origin is a fixed point for and 5. Considering only terms of order not greater than the 4 th, this gives the two equations

OK1 02K1 - - + A 1 - - - 0 , Oa OAOa

~2 (OK1 J (~K1 ~ 2c2~ hi J 2 - - 3ca-A 1 -- 0 i - i f - A / , '

(3.9)

where A1 must be considered as a function of So, being A1 = X - L0 -// 'So with a constant A; indeed the motion is possible only on the surface where A is constant.

The system (3.9) is made of two equations in two unknown variables (the values So and al,0) and in general gives 0,1 or more solution points, depending on the value of A; these points form one-dimensional families in the parameter A. An analytical study of such a system for low values of the eccentricity has been already done by Poincar6 (1892) and Schubart (1964), for the resonances of the first order, and by LemMtre (1984), for the resonances of order 2, 3 and 4. An alternative approach, which allows to achieve results also for higher values of the eccentricity, could be to solve the system (3.9) by numerical methods.

Our aim is now to study the spatial stability of the planar periodic orbits just

found. To this end we consider only the families with the property that ~ < 0 (the


only ones which are stable in the averaged planar circular problem) and perform the canonical transformation

= xS', 5 = -,~ (3 .10) x

where x is so chosen that

- ~ = W . (3.11)

The coefficient x v ~ which relates the original S, and & to S and 5 is a scale factor that takes the shape of the region of libration into a circle. By this transformation, the lowest order term takes the form

OK1, z , = ~ ( # ~ + ~ ) + ~ --5-y~gl)Ri. (3.12)

So we have the three following possibilities :

i) there exist values of g, for which ~IR (gl) = 0;

ii) ~ ( g ~ ) as a function of g~ has only a small fluctuation with respect to its mean value;

iii) °0J~ (gl) as a function of gl has a quite large fluctuation with respect to its mean v a l u e .

In the case i) the orbit is unstable; this is easily seen by drawing the phase diagram of Z2 for the variables Ra and gl. It seems then quite unreasonable to believe that a per turbat ion can turn such an unstable diagram into a stable one.

In the case ii) let's write

OK1 OlQ (gl) = - - + ~qA(gl) (3.13) OR OR

with supg 1 IA(gl)] = 1 and a suitable q (the double bar indicating a second average o v e r g l ) . The gl dependent function )~ can thus be considered as a higher-order perturbat ion. So, after having divided the Hamiltonian by e, which corresponds to a rescaling of the time, the lowest order term of the Hamiltonian is independent of gl and takes the form

W OK1 ,.~ z2 = ($2 + ~2) + ~_5_k_~1 .

If there is no resonance between 7 w and ~ we have formal stability of the orbit. The technical scheme to reduce the Hamiltonian into normal form is slightly pecu- liar and runs as follows : after having introduced the complex canonical variables

v

= --~. ( s + i~) Vzz

v

= --~. ( s - i ~ ) , Vzz


one looks for a canonical transformation ((, r/, Ra, gl) = Tx(~2,72, R2, g2) in order to reduce the Hamiltonian to normal form with respect to the term {2r/2 only. Then, since ~2r/2 is a formal constant of motion one gets rid of it; finally, one rescales the t ime by dividing the Hamiltonian by c, and this gives the Hamiltonian a normal form with respect to R2. Having now bounded the variations of all the action variables, the spatial stability of the planar periodic orbit is formally proved. On

the contrary, if there is a resonance between ~-w and -~R, the study of the stability of the orbit is much more complicated and one has to consider also higher order terms of the Hamiltonian. In particular we cannot exclude the possible existence of a family of spatial periodic orbits with larger periods, bifurcating from the planar one.

The case iii) lies in between cases i) and ii). The function -~R (gl) in fact has no zero-points , but the part dependent on gl cannot be considered as a small per turbat ion as in case ii). Formally the normalization scheme runs exactly as in case ii), but it is likely that a rigorous study would reveal that diffusion may happen in shorter times. So, although we don't find a strong instability as in case i), we expect this case to be practically unstable over the long time scale of our problem.

~Te come now to the problem of the presence of the singularity in correspondence of the circular orbit. As is well known such a singularity can be eliminated by using

the polynomial Poincar~ variables; however in these variables it's not easy at all to determine a general scheme in order to apply the Lie transform method we exposed in sect. 2. The modified Delaunay variables (3.3) turn out to be more handsome. Unfortunately they have also a singularity at the origin, and we avoid it by making a corresponding restriction of the domains which excludes the circular orbits. Such a restriction is not so bad as it could seem; it is well known, indeed, that for the resonances of higher order than the first, the region of libration on the plane around a stable equilibrium never includes the circular orbit. Moreover, if the order of the resonance is an even number, this is true also for small values of the inclination. The restriction of the domain is then suitable for this kind of resonances; in the other cases, on the contrary, one loses some information on the dynamics as the orbits

with circulating a can not be described. Note however that the critical argument of these orbits is not librating; thus, in some sense, they could be considered as non-resonant .

3. 4. T H E R E S O N A N T CASE : FAMILIES OF P E R I O D I C O R B I T S W I T H N O N - Z E R O

I N C L I N A T I O N

The search for periodic orbits with non-zero inclination is a bit more complicated as the argument of the perihelion g must be considered too. Moreover, by setting R0 # 0 all the terms of the expansion are in general gl dependent. So one has to look for equilibrium points in the variables $1, R1, al and gl altogether. This gives


the system OK1 0 2 K1 0--~ + A10hOa = 0,

OK1 0 2 K1 + A, o-- n - o ,

OK1 02K1 (3.14) + A1 0---~q - - 0 , o-7

) • ~2 (OK' j OK, -2c2~A, J 2 - 3c3-A 1 = 0

\O-S i OA , "

This system is made of 4 equations in 4 unknown variables (S0, R0, a l , gl); again A1 must be considered as a function of So, being A1 = A - L0 - {'S0 where A is a constant of motion. The system can be solved numerically and the solutions will form families parametrized by the value of A.

Considering now a solution (S0, R0, al,0,gl,0) of the system (3.14), we expand the Hamiltonian in Taylor series with respect to 5 = al - al,0 and ~ = gl - gl,0. We then perform the canonical transformation

5 (3.15) S, : V ~ , ~ - v~'

and consider both #, S and ~ to be of order e. By doing so, we achieve the following expansion of the Hamiltonian which we report up to order 4.

c~. +~--ff~-~ )

~1 03Kl -3 + ~-~ f 02K' -^ , o-b-~KT- -~

o.<,.. ,o:.,.= (o.<, -- 1 (94K1 0"4 "+ E2 t ~ ~ 1 "~'- 0---~g lg"t"~----~gg "PtOSO0" iOAOer Z4 24 Oa 4

+ 3c3 ~-f A1 D 2 --I- t-~ O-f f~g + 20RO~r 20AOa (3.16)

In this expression all the derivatives must be evaluated in Lo,So,Ro, gl,0,al,0, and # : 0, and the bar over each of them denotes an average over A, so that they are just numerical coefficients.

The further step of our work is to discuss the stability of the periodic orbits just found. First of all, since the coefficient of ~2 in Za is negative, we have an elliptic

equilibrium point in S and 5 only if ~ < 0; otherwise the point is a saddle, and 0o2 its instability is not worthwhile of any further discussion.

In the first case, we perform the canonical transformation

= x S ' , a = - a (3.17) X


with x given by (3.11), such that the Hamiltonian is brought into the form of a

pcr turbed harmonic oscillator (with angular velocity denoted by W). In order to apply the algebraic algorithm to reduce the Hamiltonian in normal

form with respect to the harmonic oscillator, we need two more technical transformations. First of all we divide the Hamiltonian by e, which corresponds to a rescaling of the time, and then introduce the complex conjugated variables

~/zz x/zz (3.1s)

w(~2 + ~2) is transformed into iW~?. by which the harmonic oscillator term y Following again the scheme of section 2, we can now define a formal canonical

t ransformation (~, ~, R~, g) = Tx(~2, ~72, R2, g2) in order to reduce the Hamiltonian in normal form with respect to the product ~2~2; thus 02 = i~2~2 turns out to be a formal constant of motion.

Thus the Hamiltonian is formally reduced to that of a one degree of freedom

problem and its lowest order term is

F3 =

_ _ _

3 04K1 2 2 1 03 K1 R ~ 1 03 K1 48ex 4 0 a 4 ~2 ~2 2ix20---~--~a i 2~2~2 2ix20gOa 2 g2~2~2

e O 2KI R~ + 02K1~ ~ e O 2K1 2 1 N; 2 OR 2 ~O---~gl~2u2+-2-~g292"]--2 {X~'Z~}

(3.19)

here the term {X{, Z{ }N comes from the algorithm (2.3), because

{x{, Z~ }N _ 1 0 2 K 1 R 2 + _ _ _

iW OROcr x OgOo" x 00 -3 x 3 (3.20)

02K1 g2 ) 21 ~ 02K1 + - - - + f(~2~2) R2 +ie \ OROa x OgOo x

Consider now 02 = i~2q2 as a parameter. For 02 = 0 the Hamiltonian above has an obvious equilibrium point for R2 = g2 = 0; this represents the periodic orbit we were looking for. For positive values of 02 one has instead a family of

equilibrium points for the same Hamiltonian (3.19), which corresponds to a family of librating orbits for the original system. The stability of such a family, at least for small values of 02, is controlled by the determinant of the quadratic form in R2, g2, which does not change with 02. We note however that we cannot guarantee a priori the existence of equilibrium points for which this determinant is positive.

The last consideration we have to make, concerns the size of the stability region around a stable equilibrium point. In principle, one should estimate such a size by a rigorous approach, as explained in sect. 2. This is perfectly compatible with our

158 A. MORBIDELLI A N D A. GIORGILLI

scheme, but the technical difficulties make such a computat ion impractical, up to now. So, we give an heuristic estimate as follows. Coherently with our approach to the problem, none of the variables appearing in our expansion may be of order greater than e. This gives an a priori bound. For exaznple ~ < e implies ~ < :re, which in turn means 5' < :re and thus S < :r2e~. From these considerations and from (3.5) we have that , for the orbit of a resonant asteroid, the maximum libration of the averaged square root of the semimajor axis is x2ie~. So the smaller is the value of :r, the smaller is the probabili ty of finding asteroids on orbits librating around the resonant periodic one we have just studied. So, it 's evident tha t the formal theory points out which are the parameters that play a fundamental role in the study of the periodic orbits. These parameters may be easily computed numerically for each orbit one is interested in. In such a way, the theory, although having a general significance, is ready to be adapted to each particular case.

3. 5. NOTE ON THE NON RESONANT CASE

The study of the non resonant case in the restricted three body problem is generally not performed, since in two degrees of freedom it's almost trivial. Indeed, thanks to the non resonance condition, it 's easy to show that all orbits are quasi-periodic and formally stable, as the Hamiltonian can be readily reduced to normal form. This is no more true if we approach the problem in three degrees of freedom. In fact, because of the degeneration of the Hamiltonian itself (only two actions are present in the unper turbed part) the third degree of freedom always enters the problem as a resonant one. Thus it's surely interesting to explore the displacement of formally stable quasi-periodic orbits with non-zero inclination, as well as the spatial stability of the orbits lying on the reference plane. Note finally that the most inclined asteroids (up to 35 degrees) are typically non-resonant ones.

We start by expanding the Hamiltonian in Taylor series as in section 3.2. From a pure mathematical viewpoint, we can speak of non-resonance only if the frequencies of A and p are incommensurable; in practice, however, we may consider as non- resonant also the ease where the resonant harmonics which would give origin to zero-denominators appear at a very high order and so can be easily neglected.

Thanks to the non-resonance condition, we can find a canonical t ransformation (L, R, S, A, g, p) = Tx(L1 , R1, $1, A1, gl, pl ) which gives the Hamiltonian a normal form with respect to L1 and S1 altogether. So L1 and $1 are formal constants of motion, and we may consider them to be equal to 0; indeed, having values of L1 or $1 different from 0 is the same as expanding the Hamiltonian around different

points L0 or So. Thus one is reduced to a one degree of freedom Hamiltonian which can be easily

studied, since the problem is formally reduced to an integrable one. In part icular the equilibrium points in R1 and gl, which correspond to quasi-periodic orbits in the rotat ing frame, can be found by solving the system

OK1 OK1 - - 0 , - - 0 . ( 3 . 2 1 )

Og OR


For each fixed non-resonant value L0 this system is made of two equations, in two unknown variables (R0 and gl), with So as a parameter; this gives a family of equilibrium points. The stability of each equilibrium point is given by the determinant of the quadratic form

) ' (3.22)

where 0 = gl - gl,0 and gl,0 is a solution of (3.21); moreover the bar over each coefficient means a double average over ~ and p.

For what concerns the circular orbits, a simple study in Poincard variables shows that they are always equilibrium points for the averaged Hamiltonian. A simple study of the level curves of the Hamiltonian in Delaunay's variables is then sufficient to describe the whole phase-space. A proof of the three-dimenslonal stability of the planar circular orbit can be found in Arnold's paper (1983).

4. T h e el l ipt ic c a s e

Nowadays almost all works about the motion in the asteroid belt are carried on using the elliptic restricted three body model. Jupiter 's eccentricity, in fact, although very small, introduces a new degree of freedom in the problem, thus giving raise to new phenomena. Such phenomena were in some cases related to the origin of some of the Kirkwood gaps as in (Wisdom, 1983), (Henrard and Lemaitre, 1987), (Henrard and Caranicolas, 1988) and (Lema~tre and Henrard, 1989).

In the present section we give a further contribution to many works on the elliptic problem, by studying it in the case of three degrees of freedom, following the same guiding ideas which characterized our approach to the circular problem.

4. 1. THE HAMILTONIAN OF THE MODEL

Because of the eccentricity of Jupiter 's orbit it is not possible to find a coordinate frame such that the Hamiltonian of the problem is still time independent.

Taking as a basis the circular case, we plan to introduce the eccentricity of Jupiter 's orbit as a further perturbative parameter. To this end, we choose a rotat ing frame with an angular velocity equal to Jupiter 's mean one, with the y axis pointing towards Jupiter 's mean position. In such a frame both Jupiter and the Sun are moving, describing small trajectories around the positions they would have if the eccentricity of their orbit were exactly zero. The equations of such trajectories


axe

Xju p = ajup(COS Ejup -- ejup)Sin ( A j u p ) - ajup v/X -- e2up sin Ej.p c o s ( A i ~ p )

yjup = ajup(COS Ejup - ejup) cos (Ajup) + aiup ~/1 - e~u p sin Ej.p sin (Aj.p)

&® = - a ® ( c o s Ej.p - ejup)sin (Ajup) + a® ~/1 - e~p sin El . p cos (Aj.p)

~® = - . ® ( c o s S j . . - ej~,) cos (~j~.) - .® ~/1 - e~up sin Ej~, sin (~j~,)

(4.1)

where 1 #

a® - - Ajup =- n t = v / l + p • t a j u p - 1 + # ' 1 + # '

We shall use the variables x®, y® defined by x® = - & ® / a ® , y® = - ~ ® / a ® . Thus the Hamiltonian system is no more autonomous; in order to s tudy it by the same methods as in section 3, we introduce a new variable E, conjugated to Ajup; so E is related to the variation of the energy of the asteroid in the rotat ing frame, which is no more constant.

Proceeding now as in section 3.1 the Hamiltonian becomes

K = K0 + #K1 , (4.2)

with

1 ~ 2 _l_p2z)T(pxy_pyx)WE_ 1 Ko =~(Px +vy 7 , 1

I~l : E (~.).j--1 ((PxY- pyT,)+ E) 1>1

1 ' E ( - i f )p , -1 (tt(z~) + Y~)) 2(xx® +__yy®)~' _ 1

- 7 \(--i-7.)~r~ ~ (l+.)r~ ) ~/(~_~j°pl~+(~_yjupl~+z~ j_>l (4.3/

Then we introduce again, besides E and Ajup, the modified Delaunay variables L , S , R , A , a , g as in sect. 3.1, so that one gets

1 Ko - L + S + E ,

2L 2

while K I ( L , S, R, E , A, a, g, Ajup) can be easily determined. Note that the eccentricity of Jupiter is of order e, being nowadays .0485 (and

never exceeding .06); so it's natural to expand the Hamiltonian K in Taylor series with respect to ejup around the value 0, writing eiu p = a¢. The Hamiltonian evaluated at ejup = 0 is exactly the one previously studied in section 3, while the fur ther terms of the expansion can be considered as a new per turbat ion (globally of order ¢).


Proceeding as in sect. 3.2, we expand the Hamiltonian in Taylor series in ¢ = v/-fi

and in the variables L = L - L0, S = S - So a n d / ~ = R - R0, where L0 is fixed

by the resonance condition, while So and Ro are arbitrary; L, S, R and E are all considered to be of order ¢, thus obtaining an expansion of the Hamiltonian in orders of magnitude. Observing now that the first order term, namely ~. L -4- S + E, contains a double resonance, we perform the canonical t ransformation :

a = L + J ( , ~ + E ) , ) ~ = ~

(4.41 $

E = E , - / / = )tju p - J ) t . z

This transformation gets rid of the double resonance between L, 5 and ]_,, E , and introduces a slow-varying variable v; v is exactly the same critical argument generally introduced in the elliptic problem. The angle - a - v is nothing but the longitude of the perihelion of the asteroid measured in a sidereal frame. This angle can be introduced by the following canonical transformation, which gets rid of the

remaining resonance between S and E :

S ' = S + E , a = a (4.5) E = - E , T = - - a - - V .

Still following the scheme of section 3.2 we perform a first canonical transfor- mat ion (h, S' , /~, E , A, a, g, r ) = Tx(A1 , $1, R1, El , At, al, g~, 7" 1 ) which removes the

angle A1.

4.2. THE RESONANT CASE : FAMILIES OF QUASI-PERIODIC ORBITS LYING ON THE REFERENCE PLANE

In this section we develop a general scheme to study the stability of a given stable periodic orbit of the circular problem under the effect of the eccentricity of Jupiter; So, al,0 and A1 are then one solution of (3.9). Adopting now the same notations as in sect. 3.3, the terms of lowest order of the new Hamiltonian, which has now one

162 A. MORBIDELLI AND A. GIORG1LLI

more degree of freedom, are

f j 2 l ~ 6 . 2 " ~ + ~ ( g , ) R t + e 2 ( c 9 K 1 0 K 1 ) E I + e a a O K 1

Z{- 6 O~ a oqejupOO " OSOa r~la)

1 03I~1 ~r4 .j_ ~2 ( 1 02Kl (~Q2K1 jO2K1)~ .~

0 2 K t , "E R I ~ E ~ _ 02Kt A E 0 ~K~, " t -'O-S"O~ {,gl ) 1 1 + -~ cq S OAOS 1 1 + ~ [gl } 1 R1 /

+ e ( 1 0 R O a 2oaK1 (gt)Rl~. 2

[ 0 2 K1 "'r "R + ~ , ~ - - ~ ~ , g ~

1 0aK1 E1#2 + 1 03K1 . -2 ~ / 2 cqScgtr 2 - ~ , t a + 3c3 .A ,S 2

- - ) 02 K1 . 02 K1 O~jupOS(n)E1 + ~ ( n ) A ,

(rl)

4a2~O2K1 ¢2a ~ 2 . ( r l ) ~ . 2 . +e Oej~u p (rl) -[ 2 COejupaCr

(4.6) Here the only functions which depend on rl and gl are explicitly indicated; the other functions are evaluated at So and oh,0 giving just numerical coefficients.

We perform now the canonical transformation

E~ = E1 + ~ ( r l ) , (4.7) T 1 ~ T 1 ,

where e( r l ) is chosen in such a way that the last term in Z3 disappears. Then, we apply again the transformation (3.17), and divide the Hamiltonian by e , which corresponds to a rescaling of time, so that the lowest order of the Hamiltonian becomes

z3 = w ( ~ 2 + ~2) + e -~ - (g l )Rl + ~ \ 0E OK1 "~ ~-~ ] E;. (4.S)

This is possible only if the expression aK~ _ OK~ does not vanish; otherwise a OE OS different s tudy is necessary, as is illustrated in sect. 4.5.

It seems here that , if there are no resonances among the coefficients, the stability of the orbit is not affected by Jupiter 's eccentricity. We note however, that one should carefully consider the domain where the expansion of the per turbat ion is performed. Indeed, if [e~(A)[ becomes larger than So, the domain includes the singularity which corresponds to the circular orbit. This case is discussed in the next section. Suppose instead that this is not the case, and that we can prove the formal stability of the orbit; then, going back to our original variables, we discover by (4.7) and (4.5) that our equilibrium orbit is characterized by a periodic fluctuation of S with the same period as r l . Such a fluctuation has in general a


period incommensurable with that of the mean anomaly of the asteroid. We can thus speak of quasi-periodic orbits only, while in the circular ease we could find periodic ones. This is precisely due to Jupiter 's eccentricity. Indeed for these orbits one has a constant, while T changes. Recalling that at and r are the longitudes of the node in the rotat ing frame and in the sidereal frame respectively, this means that the sidereal period is in general incommensurable with Jupiter 's one. Such a phenomenon has no effect in the circular case, due to the obvious symmetry of Jupiter 's orbit, but is clearly relevant in the elliptic case.

The study of the stability may be complicated by the existence of three possible resonances among the coefficients in (4.8). The resonance between w and

( ~ 0Kt ) already in the problem and has already been stud- planar C - - OS may occur

led in some cases; in the last section we will discuss the ease where ~ E -- -~S = 0. Only the three dimensional study of the problem, however, may reveal the exis-

tence of the other two; the resonance between -7-w and cP-~R is typical of the circular

problem, while the one Uetween and (%' - ) (the lowest order one) may be only found in the elliptic case. A specific study of each of these cases could be done with methods similar to those described above.

4. 3. NORMALIZATION OF THE ELLIPTIC PROBLEM FOR SMALL VALUES OF THE

MEAN ECCENTRICITY AND INCLINATION

As mentioned in the previous section, in the variables adopted the presence of the singularity which corresponds to the plane circular orbit raises new difficulties in the elliptic problem. In the circular case, in fact, as discussed in sect. 3.3, one is allowed to choose a neighbourhood which is small enough not to include the singularity and perform the normalization there. In some cases one loses information about the behaviour of the orbits which lay outside such a domain; in this sense the study may not be an optimal one, but it is surely significant to investigate the local

stability of the periodic orbit in subject. In the elliptic problem, on the contrary, the size of the domain in which the normalization must be carried on is, in some sense, chosen by the system. Indeed the ellipticity of Jupiter 's orbit may force the eccentricity of an asteroid to reach values very close to zero, or even to pass through zero. From a mathematical viewpoint, one has to expect the expansion of the previous section not to be convergent if leo(rl)l is greater than So. On the other hand one can prove analytically that this case must occur for each resonance. Indeed, for the resonances of first order, setting the inclination equal to zero, the Hamiltonian starts with the terms

a x / ~ cos o" A- bejup cos t~ (4.9)

and thus e~(v) ,-0 v/2-Sej,pcos u which is greater than S for small S. For the resonances of order i greater than 1, instead, the Hamiltonian for small e turns out to


b e

+ b(u'5 ) cos (io) + cv' ej.p cos + .) + d ( y r ~ ) i - a ej,~p cos ((i - 1)a - u) + e (V~) i -2e2 , , v cos ((i - 2)a - 2u) + . . .

(4.10) and thus e~(u) ,-~ x/r~ejup cos (a + u) which is again greater than S for small S.

The use of Poincar~ variables turns out not to be helpful to overcome this problem. Indeed, it removes the singularity by introducing the polynomial variables, but one is no more allowed to redefine the angle variables in order to take into account

the resonance. More useful seems to be the use of Sessin's variables (Sessin, 1981), defined by the canonical transformation

X/2(F - R) cos f =

~ 2 ( F - R) sin f =

M =

V ~

R = gt

f f 2 ( S - R) c o s a + flejnp c o s u

X/2(S - R) sin a - flejup sin v

1 2 2 Z + ~(fl ejup) - f lejupX/2(r - R) cos ( f + v)

v

R

g + a - f

(4.11)

where fl is a free parameter to be determined on purpose; R = G - H is zero on the plane. Such a transformation has the relevant properties

S + E = F + M

S = r - X/2(S - R)flejup cos (a + v) + const,

so that its application is straightforward; indeed, for a suitable fl one can t ransform (4.9) into

a ' V ~ F cos f (4.12)

and, if i > 2, t ransform (4.10) into

a~F + higher order terms (4.13)

In both cases, then, the large term which determines the function ~O(v) disappears; cS(g) is now given only by higher order terms and thus turns out to be smaller than F already for very small values of this variable. For resonances of order two there are still some consistence problems. Indeed Sessin's t ransformation allows to get rid only either of the term v/2Scos (a + g) or of v/2-Scos (a - g) in (4.10), so that , even at low order, the Hamiltonian is still dependent on y. However the transformation can still be advantageous to reduce the size of ~.

If one considers now a small value of the inclination, thus suitably changing (4.9) and (4.10), the situation is very similar; indeed, working close to the plane


the terms which depend on the inclination are automatically of higher order. At low order, then, (4.9) and (4.10) can be just rewritten with S - R instead of S.

Sessin's variables are also useful for another reason : from (4.10), one can easily realize the existence of an equilibrium point of the longitude of perihelion in the sidereal frame, for a fixed small value of the eccentricity; indeed there must exist a point So, a0, v0 such that the derivative of the expression (4.10) with respect to S is zero. This implies that an equilibrium point for the perihelion a + v always exists; moreover, in case of exact resonance, there exist a periodic orbit of the elliptic problem with both a and v constants. In the scheme of the previous section this is revealed by the fact that one has

~ Otk*I c~tk*Ic~s ) 02 K1 i - - e a ~ l , vl,O) = 0

for some value rl,0; thus the latter term e a ~ ( r l 0), which in the previous ~jup~

scheme was placed in Z4, turns out to be of the same order as the former one 0I<~ _ 0 E -~S; the scheme then clearly fails and must be modified, for example along the lines of sect. 4.5. As we have already seen, however, Sessin's transformation is able to kill the large term of order v/~ejup in (4.10); then in the new variables F, f the problem exposed disappears and the perturbative scheme can be restored. In this sense Sessin's variables must be considered as the correct variables to explore the elliptic case for small values of the eccentricity.

The application of Sessin's transformation in the scheme adopted is in fact very simple. One expands the Hamiltonian in Taylor series with respect to L around the fixed value L0. By doing so the unperturbed Hamiltonian becomes

_KL + s + E - + . . . (4.14) j 2L~

The following transformation then eliminates the resonance which is present in the linear part of (4.14)

a = L + J ( s + E ) ,

S = S ,

E = E ,

the unperturbed Hamiltonian is turned into

a=p--JA Z

- - V ---- -~jup - - j ) ~ ; Z

(4.15)

( _ )2 ! A _ a . . . j ~-L~ A - E) +

Now one introduces Sessin's variables F, f , M, v with a suitable parameter /3 in

order to minimize the relevant term of the development a--~-g (v~ 0~j,p v J, as explained before.


Thanks to one of the relevant properties of Sessin's transformation, the unperturbed Hamiltonian is not changed in form, being now

The approach now closely follows that of the previous section, by just replacing S, a and E with F, f and M respectively : one expands the Hamiltonian in Taylor series with respect to F and A around fixed values F0 and A0 = Foi/j and with respect to R around 0 (since the orbit lies on the plane); the expression so found is consistent, since one gets a new function ee(r) which is small, thus removing the problems discussed above. Finally one performs the normalization of the Hamiltonian in a domain which does not include the new singularity which is present for F = 0.

To describe the motion in the original variables one has to perform the inverse of Sessin's transformation. For example let's imagine we have found on the plane an equilibrium point Fo,fo, with, for simplicity, f0 = 0 (as a mat ter of fact, since the dependence on u is not removed from the whole Hamiltonian, the equilibrium points do not exist for F0 # 0; they are replaced by small periodic orbits). As v

rotates (recall that 0K~ _ 0K~ is now different from zero) in the original variables O M OF

S, g the system follows an orbit which can be classified in one of the 4 following classes : 1) F0 = 0 : this implies x / ~ = 151 ej,p and a+v =const. This is the periodic orbit

of the elliptic problem found by analyzing the expansion of the Hamiltonian at low order in the eccentricity for resonances of order higher than 1. For resonances of order 1 this orbit does no more exist, since in this case F0 --- 0 is never an equilibrium point.

2) ~ < Ifl[ ejup : in this case the argument a circulates with the same period as v. This implies the libration of the longitude of the perihelion in the sidereal frame. The action S varies in between the values

1(2Fo + 52e~.p - 2181 ej.p 2v~--~o) &~=~ SM = 1(2F0 +/~2e2up "~ 2 I/~1 ejup 2V/~00)

(4.16)

where S m > 0 and S M < 2 Ifll ejup.

3) 2x/~0 = [ill ejup : this is a critical orbit; when a = 90 °, S is 0. This orbit signs the passage from class 2 to class 4.

4) ~ > 151 ej~p : the argument ~ is now librating around 0 while S varies in between Sm and SM given by (4.16), with SM > 2151 ej,p. The longitude of the perihelion measured in a sidereal frame is now circulating. The scheme developed using the original variables in the previous section is able

in practice to describe only the orbits of class 4 which have an amplitude of libration in ~r which is less than 7r/i, where i is the order of the resonance.


4. 4. THE RESONANT CASE : FAMILIES OF QUASI-PERIODIC ORBITS WITH NON

ZERO INCLINATION

We concentrate now on the study of the neighbourhood of a given stable periodic orbit of the circular problem with non-zero inclination, corresponding to the solutions So, R0, al,0, gl,0 and A1 of (3.14). For sake of simplicity we perform the study in the modified Delaunay's variables, which are suitable for large eccentricity; next a note on the use of Sessin's variables for low eccentricity and large inclination will be provided. Here we closely follow the procedures of sect. 3.4 and just outline the

differences. Considering the new additional degree of freedom introduced by the ellipticity

of Jupiter 's orbit, the Hamiltonian, instead of (3.16), but in the same notations, turns out to be

5 02K'-------~ s ~ -^ . c02K 1 n - ¢92K1 ElO'~ Z~ --V '~ c93Kl ~3 Oa 3 OejupOCr(Vl)& +¢-2 ~ r g + 0 - - ~ - ~ / ~ l O ' - cgScOa

/

1 03K1&4+e2 (10--~1R2, 0 2 1 ¢ 1 - * , 1 0 2 K 1 - 2 ~

(1 03I~1 ~2 1 03[(1 R1~2 1 03I~1 E1~2 + 1 03I~1 A1~2 +3c3~A1~2/ +e ~ + -2 cORc% 2 2 OSOa ~ 2 0AcOa - - - - - ~ "

% ]

{ c92K1 rA ' R 02K1 , , ~ c92K 1 . . . . cg~K-;--, ,, '~ +e 3a ~ ~ , 1) l O ~ S tv,)/:.,1 -t- ~tVl)goejupog * ~ t ' r l )Al)

4 2 0 2 K 1 , , ~2a O~K1 ( n ) b 2 a 0---~---.2 ('rl) + 2 oqejupOqC r2 jup

(4.17) Here the only functions which depend on rl are explicitly indicated.

Now, after the transformation (3.17) on S and b we perform the (4.7) on E1 with a suitable e, such that the term of lowest order of the Hamiltonian, divided by c, is reduced to

--£ + + \ OE ] E~. (4.18)

Now, if there is no resonance between w and e( °0/~ OK1 -5- - o s ) we Call go on normalizing the Hamiltonian.

The normalization scheme is analogous to the one followed in case ii) in section 3.3, and in such a way, one gets a one-degree of freedom Hamiltonian, the lowest


order term of which is :

1 03K1 R2(2rl 2 1 03K1 g2,~2rl 2 + e 02K1 O2K1 n e O2K1 2 F3 - 21~2 0 RO~r~ : i ~ OgO~' ~ - - ~ - R] + %---yNg ~,2g~ + ~ - - 0 7 g~ +

ea Oej u p Og OgO~ (E2 -- e-'~) g2 -b s ea OejupOR -O--~ t , 2 ~-~) R2 "4- 2

(4.19)

where

i W OROa x + OgO-----a Oa 3 x a ] +

0 2 K 1 02K1 R2 + - - +e2fl(R~,g2)+eE2f2(R2,g~) +ie OROa x OgOa

(4.20)

and the double overbar denotes a double average over A and r; finally, } is the

average of ~ over T, while f l , f2 are linear functions in (R2, g2) that can be explicitly computed; one can prove that both ~ and f l are identically equal to zero.

We note that the stability of such an orbit is not affected by Jupiter 's eccentricity, because the quadratic form in R2, g2 coincides with the one of the circular case. Anyway, the fluctuation of E1 over the period of rl determines by (4.5) a f luctuation of S, namely of the eccentricity. So the periodic orbits of the circular problem are changed into quasi-periodic orbits by the eccentricity of Jupiter. In case of a

resonance between w and ~( OK~ 0K~ ), the study of the stability of the asteroid's Y OE -- OS orbit is much more complicated. We skip here the general study of such a situation.

The scheme we have just developed can be applied only for large eccentricity for the same reasons exposed when studying the planar case in sect. 4.3 and 4.4. As a mat te r of fact in most cases the presence of periodic orbits with non-zero inclination is found only for very high eccentricity, as we will show in the second part of this paper. Only in a few cases, for resonances of higher order than the first, periodic orbits with small eccentricity are found, and then the use of Sessin's

variables is precious again; for a resonance of order i > 2, one has an Hamiltonian where the relevant terms, for low eccentricity, are :

a 1 (i, g ) ( S -- R) + a~ (i, g ) M f ~ - R)eju p cos (a + v) + aa (i, g ) ~ R)eju p sin (a + v)

By the use of two Sessin's transformations the two large terms which depend on u can be removed from the Hamiltonian (one can easily determine a transformation in the spirit of Sessins's one such that S = F - ~ejup ~¢/2(S - R) sin (a -t- u)). In the new Sessin's variables the scheme of study exposed in this section can be restored, formally replacing F, f and M by S, a and E.


4. 5. PERIODIC ORBITS ON THE PLANE IN THE ELLIPTIC PROBLEM

As exposed in sect. (4.2), the rotation of the longitude of the perihelion of the asteroid measured in a sidereal frame introduces a long-periodic oscillation of the energy and, mainly, of the eccentricity. This oscillation vanishes if the dynamics allows the longitude of perihelion to stay fixed. In this case the orbit turns out to be periodic, both in the sidereal and in the rotating frame, with a period which is exactly commensurable with that of Jupiter.

One periodic orbit of this kind was found in sect. 4.3 for a small eccentricity for each resonance of order greater than 1. This is surely particular, as the longitude of the perihelion rotates in the circular problem and is stopped only by the contribution of the terms coming from Jupiter's eccentricity thanks to their divergence for e ---* 0. If we decrease the eccentricity of Jupiter to zero, the periodic orbit degenerates into the circular one. Such an orbit, at low order, is reduced to the origin of the reference frame introduced by Sessin's variables.

As we will show in the second part of this paper, more interesting periodic orbits can be found in all the main resonances for high values of the eccentricity. These orbits are of a different nature, as the longitude of perihelion is fixed already in

the circular problem; indeed it results ~ - -~s = 0. Sessin's transformation can not be defined in this case, as the parameter/3 must go to infinity approaching the periodic orbit.

In this section our aim is to develop a general scheme to study the stability of these orbits; the frame adopted is that of sect. 4.2.

Starting from the Hamiltonian (4.6) we look for an equilibrium point in the variables E1 and vl for E1 = 0. By setting S = 5 = R1 = 0, this leads to the system

OK1 OK1 02K-------~1 02K1;A1 = 0

- - O--ff + - ~ -- CaoejupOS OAOS (4.21)

02 K1 03 K1 03 K1 OejupOT + 0ejup0A0,r A1 + eaOe2upO v = 0 ,

Each term of (4.21) must be considered parametrical in So, namely in the eccentricity. Because of the periodicity in 71, then, the system (4.21) gives in general two solution points, (S0,1,vl,1) and (S0,2,rl,2) say; in the approximation of the 4 th order these points correspond to the periodic orbits of the elliptic problem. We consider now all the terms in (4.6) evaluated in S0,1 and expand all the terms depending on 7"1 in Taylor series around the fixed point rl,1 (the same of course could be done for S0,2 and 7-1,2). We denote by ~ = rl - r1,1 and consider both E1 and ÷ to be of order e. Then we divide the Hamiltonian by e and perform the transformation (3.17) with x such that

j2 °n2K1 g a ~ 2C2_~X2 = 0 a2 + ~ j u p ~

X2 ~ W

170 A. MORBIDELLI AND A. GIORGILL1

For sake of symmetry we consider the following expansion of the Hamiltonian in orders of magnitude, where some terms are considered at a lower order with respect to their natural one

W Z3 -----.~ (,~2 + 5.2) , 2 0I~1, "?-e "-0-~-- I, gl ) RI

1 _ e~ 02K1 1 03K153+e -~a 0 2 K ~ 5 + - (gl)Rxe

Z{ 6x/~x a 0o .3 x OejupOCr x OROtr

V~ ( ~ +*a 03K1 ~ e{ 03K1 -. -x \ OSO~r OejupOSOe ] E16"+ --~-aOejupOrOara

1 03Kl 6. 4 + • { I " ~ K 1 , ,R2 ( 02 gl

1 03K1 (g1)R15. 2 +e2a 02KI (gl)R1 + 2x 20ROo "2 OejupOtf

1 ae 0 aK1 +2 1 (O'-O-~l +ca 03K1 ~ 03K1 f'El+-~

, 02K1 , , 1 03Ka E15. 2 .ve..~.~tgl)A1Ra +3c3x~3A1,~2 2x 20SOo "2 2

02 K~ 0SOR (gl)R1 E1

JO2K1) 8 ~ ) i cOAOo" (4.22)

If the fluctuation of -~n (gl) over the period of gl is small with respect to its mean

value -~R, and there is no resonance between the latter and W, the normalization scheme is that of the circular problem (see sect. 3.3). At the end one formally gets a one-degree of freedom Hamiltonian in E2 and r2, where (El, ?) = Tx(E2, T2) and T x is the canonical operator of normalization, where the term of lowest order, is

1 (-02K1 03K, ) 03K1 ac 03K, F3 = ~ Ik OS 2 +ca 2 E~ - ca E272+ r 2

0 e j u p 0 S 0Cjup0S0T 2 0e jup0T 2

[ ] 02K1 03 K1 03K1 -26a ~ + ga OejupOSOo. OejupOvOaE2v2

where fl(R2,~2q2) and f2(R2,~2~12) are zero for R2 = ~2~72 = 0. The sign of the determinant of this quadratic form gives the stability of the equilibrium point according to Liapunov's criterion.


5. C o n c l u s i o n s

171

We have developed a general per turbat ion scheme for the problem of the three bodies, with part icular at tention to the dynamics of asteroids. The main tool con- sists in making a local expansion of the Hamiltonian of the problem in a suitable neighbourhood and then in applying the general methods of per turbat ion theory to that expansion. Such a method has been successfully applied to the study of the asteroidal dynamics in space, both in the circular and in the elliptic case; moreover it appears that it could be extended, without substantial modifications, to further cases involving more degrees of freedom.

Acknowledgements

We wish to thank L. Galgani for his constant support and encouragement during the preparat ion of this work. Some discussions with J. Henrard were illuminating, and his suggestions were invaluable. We also thank M. Carpino for his technical aid, and G. Contopoulos for very useful discussions.

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on the dynamics in the asteroids belt. part i : general theory

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