on the dynamics in the asteroids belt. part ii: detailed study of the main resonances

O N T H E D Y N A M I C S IN T H E A S T E R O I D S BELT.

P A R T II : D E T A I L E D S T U D Y OF T H E M A I N R E S O N A N C E S .

ALESSANDRO MORBIDELLI Departement of mathematics, FUNDP

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

ANTONIO GIORGILLI Dipartimento di matematica dell' Universitd

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

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

Abs t rac t . The general theory exposed in the first part of this paper is applied to the following resonances with Jupiter's motion : 3/2, 2/1, 5/2, 3/1, 7/2, 4/1; these are the most relevant resonances for the asteroids. The whole analysis is performed in the framework of the spatial problem of three bodies, both in the circular and in the elliptic case. The results are also compared with the observed distribution of the asteroids.

K e y w o r d s : resonances, perturbation method, Kirkwood gaps, periodic orbits.

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

In the first part of this paper we have developed a general theoretical framework for

studying the dynamics of orbits in the restricted three body problem. The method applies to bo th the resonant and the nonresonant case; however, its application is particularly interesting in investigating the topology and the stability of the families of resonant periodic orbits.

In the present second part we apply some of the methods developed there to the study of the main resonances which are of interest for the asteroid belt.

The main differences with respect to the previous works can be summarized as follows :

i) we are able to produce an extensive analysis of the main resonances on the basis of a general theoretical framework;

ii) such an analysis is performed for the spatial (three dimensional) case.

As illustrated in part I, the main idea is to expand the Hamiltonian of the problem of three bodies in the neighbourhood of a given orbit, and then to apply the known methods of per turbat ion theory. The reliability of these expansions to a quite low order (never exceeding 4) can be justified in view of the methods of

Celestial Mechanics and Dynamical Astronomy 47:173-204, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

174 A. MORBIDELLI AND A. GIORGILLI

rigorous perturbative theory leading to stability estimates of Nekhoroshev's type (see Giorgilli and Galgani (1978, 1985) and Colombo and Giorgilli (1989)). This turns out to be in fact impractical up to now. Thus we limited ourselves to give estimates which look quite reasonable in the light of the rigorous results. These estimates are based on a numerical computation of the coefficients of the expansions up to the required orders.

For what concerns the planar periodic orbits we obtain results which are consistent with those already produced by several authors, quoted in the references below; only in a few cases the consideration of the third dimension, namely the motion orthogonal to Jupiter's plane, changes the two--dimensional picture, by introducing instability. The consideration of the spatial periodic orbits produces instead a lot of new informations which are in principle interesting, since they produce a full picture of the dynamics associated to the resonances. As a matter of fact, however, it turns out that such new results do not substantially change the known picture of the asteroidal dynamics, at least for values of the eccentricity and inclination which are important for a comparison with the observed distribution of the asteroids. This is mainly due to the fact that the spatial periodic families typically exist for high eccentricities and/or inclinations; this is clearly in conflict with the commonly accepted evolutionary model of the Solar System, which excludes the presence of asteroids at too high inclination, or with the very nature of the three body problem, which ignores the existence of the internal planets.

The results are mostly presented in graphic form, with an attempt to give a brief presentation. Sect. 2 describes the common features of the graphics, in order to shorten the discussion of the single resonances. Sect. 3 to 8 contain the separate discussion of the resonances 3/2, 2/1, 5/2, 3/1, 7/2 and 4/1, in that order. In particular, sect. 4, which refers to the resonance 2/1, contains two subsections where tile possible relevance of the secondary resonances due to the third dimension is investigated. This requires some extensions of the analytic considerations of part I. Let us stress, however, that such an extension is quite straightforward; this is indeed, in our opinion, the main advantage offered by a general approach : it can be readily extended to cover new cases.

2. General se t t ing

In order to simplify the discussion of the single resonances, we collect in this section some general informations, also making explicit reference to the formulae given in part I. The results are represented, as far as possible, in graphic form.

For each resonance we start by analyzing the periodic orbits on the plane in the framework of the circular problem. For what concerns the topology of the

DYNAMICS IN THE ASTEROIDS BELT (PART II) 175

orbits in the phase-space of the averaged Hamiltonian and the displacement of the periodic orbits we surely refer to the works of Schubart (1964) and LemMtre (1984). In the first graphic of each figure, referred to as graphic (a), we then report a comparison between the estimated region of libration and the observed position of tile asteroids. The coordinates used are the averaged square root of the semimajor axis and the eccentricity; the unit on the horizontal axis is so chosen that 1 is the semimajor axis of Jupiter's orbit. The solid line represents the family of the periodic orbits which are stable for the averaged circular problem, and the dashed line is the separatrix which bounds the region of libration, as given by the averaged pendulum-like Hamiltonian. This kind of computation is carried on by using the planar averaged model, which is an integrable one; then the perturbative scheme developed in the first part of this paper is not involved. Indeed this kind of study is not an original one; a similar work has been done by Dermott and Murray (1981, 1983) on all the mmn resonances. In our case, however, the use of a local expansion should allow to achieve more correct results, as proved by Ferraz-Mello (1988) in the case of the 3/2 resonance.

On graphic (a) the crosses and the circles represent the observed asteroids, as given by Marsden's 1987 catalogue, and reduced to the plane of Jupiter's orbit. The crosses are related to the asteroids with inclination lower than 15°; we choose this value, as 15 ° is approximatively the border of the neighbourhood R < e when we expand the Hamiltonian around an orbit on the plane (R0 = 0). Here, some remarks are in order. The first one is that, besides reducing the data to the plane of Jupiter's orbit, one should average over a full period its orbital elements defined in the rotating frame, also taking into account that the Sun is not at the origin of the reference frame. This, of course, could be done. We note however that the averaged orbital elements defined in the rotating frame adopted are essentially the averaged orbital elements defined with respect to the Sun, being the averaged position of the Sun in the barycenter. On the other hand, the elements of an asteroid measured with respect to the Sun have a much smaller fluctuation than those measured with respect to the center of mass. For these reasons we compare directly the orbital elements as given by the catalogues with the values given by the theory, also taking into account that we are not interested in the single asteroid, but in their statistical distribution. A second, possibly more relevant remark concerns the fact that we compare the eccentricity and the semimajor axis of an observed asteroid, which has typically an inclination different from zero, with the size of the region of libration computed in the reference plane. In order to check the consistency of such a choice, we have averaged the maximal amplitude of libration, corresponding to a libration of a, observed in the diagram L, e over the oscillation of the inclination, and we have found that it slowly decreases as the value of the averaged inclination becomes higher; then we choose to consider the values for zero-inclination as a reliable estimate, which allows us to quickly distinguish between a resonance populated by asteroids and an empty one.

As discussed in sect. I-4, the eccentricity of Jupiter's orbit forces a change in the


behaviour of the periodic orbits of the planar case. The main effect is a fluctuation of the eccentricity of the asteroid's orbit. The bars in graphic (a) represent the extrema of such a fluctuation, while the dot inside the bar gives the eccentricity of the corresponding periodic orbit of the circular case. The extrema are computed by studying the one-degree of freedom simplified Hamiltonian

- - O K 1 ,~ K I ( E , ~ = O , R = O , S = O , & = O , ejup =O)+eao~jup t~ , r ,#=O,R=O,S=O,6"=O ) (2.1)

This is a pretty suitable model which comes directly from (I-4.6); it includes the non-linear effects, but, on the other hand it sets the inclination equal to zero and neglects each part that depends on S and ~r which are considered constants and equal to zero. The results given by this model can be considered pretty correct in tile case where the longitude of perihelion r is rotating and, correspondingly, the angle a has only a small fluctuation. This happens in all the cases which are reported. The longitude of Jupiter's perihelion is assumed to be fixed and, in the frame adopted, equal to 90 ° .

The next graphics introduce the new results that are obtained by studying the 3-dimensional problem along the lines exposed in the part I.

Graphic (d) refers to the stability of the family studied in graphic (a) with respect to the motion orthogonal to the plane. More precisely, still referring to the stability criteria discussed in sect. I-3.3, in particular eq. (I-3.13), we plot the

significant quantity a = ~-~/e q (in log scale) as a function of the eccentricity. Such a quantity will be referred to as stability ratio. The orbit is unstable if a < 1 and stable for a > 1, but in order to expect stability over very long times we have to ask such a ratio to be large enough.

Still on the same family of periodic orbits, graphic (b) reports the main secondary resonances. Moving bottom-up, the first line reports the resonances between the

terms W and e - ~ , namely between the period Pg of g for an orbit close to the plane, and that of libration, P~ say; the second line reports the resonances between

W and e (~-~ - °0~) , namely between the sidereal period of revolution of the pe-

riastron, P,~, and that of libration, P~. Finally the third line reports the resonances

between eP-g~ and e ( - ~ - ~ - ) , namely between P,, and Pg; all these resonances

may be relevant for the stability of the orbit. Moreover for each resonance the coordinates of the periodic orbits of the elliptic problem and their stability properties will be indicated in the text, along the lines of sect. 4.5 of Part I.

Graphic (c) (right hand upper corner) reports the displacement of the families of periodic orbits with non-zero inclination. These families are computed by numerically solving the equations (I-3.14). Here the symbology is the following :


• The solid line represents a family of stable orbits which satisfy the stability

condition of sect. I-3.4. • The dashed line is a family of unstable orbits. These orbits correspond to fixed

points for the Hamiltonian; their instability is revealed only by the sign of the determinant of the quadratic form in R2 and g2 given by (I-3.19).

• The dot ted line is a family of orbits which are unstable, because they are saddle points already for the variables S and b.

The coordinates used are the eccentricity e and the inclination i; the various inter- sections among the families are only apparent, because the corresponding angles g and a are different. We recall here that g is the argument of the perihelion and G the longitude of the perihelion measured in the rotat ing frame when the asteroid is in between Jupiter and the Sun (the angular position of Jupi ter in the frame adopted is 90 °, so, when G = 90 ° the perihelion is aligned with the planet). The

corresponding values of c~ and g for each family are reported on the graphic itself. Tile reader must always remember that , thanks to the resonant condition and to the symmetry with respect to the invariant plane, these values are "multiplied". Indeed, if a given family is given by angles G0 and g0, a corresponding family with the same properties exists for

27rk Go,k ,n "~- GO "~- i ' gO,k ,n = go + Vt7~

where k E [0, . . . , i - 1], n E [0, 1] and i is the order of the resonance.

3 . T h e 3 / 2 r e s o n a n c e

In graphic 1-(a) we s tudy the pericentric branch of the family of periodic orbits and its neighbourhood of libration. On this subject a detailed study has been recently

done by Ferraz-Mello (1988); we then report here only some very brief information and compare, as a check, a few numerical results.

The solid line is the family itself, which is curved in the semi-major axis as a function of the eccentricity. The comparison between the data reported by Ferraz- Mello and our data shows a good agreement : the difference never exceeds 10 -4 times the distance between Jupiter and the Sun.

The dashed line delimits the region of libration on the plane; for an average inclination of 7 ° the amplitude of libration in L is 94% for e = 0.1 and 90% for e = 0.2 of that computed on the plane. Almost all the asteroids are inside, thus appearing associated to the resonance. Graphics 2-(e) and 2-(g) confirm this fact. The former shows the small inclination of the asteroids, which thus can all be studied as in a neighbourhood of orbits on the plane; the lat ter is an histograxn which reports the number of objects as a function of [G - G0[ = I~1 and proves


the existence of a non uniform distribution, as one should expect for a resonant population of asteroids.

On the right hand corner of graphic 1-(a) the amplitude of the forced fluctuation of the eccentricity of an asteroid due to the ellipticity of Jupiter 's orbit is indicated. Here the difference between Ferraz-Mello's and ours values for e is about 10 -3.

Graphic 2-(f) provides sorts of proper elements for the asteroids : here the lines are level curves for (2.1) and thus report the oscillation of the eccentricity with the rotation of the longitude of the perihelion in a sidereal frame. As explained in sect. 9, such a model is correct as long as the perihelion ~- rotates and cr has only a faint oscillation. This is no more true if e < .06, where we have a librating r and a circulating ~r. In the region populated by asteroids, however, the study performed is significant, and shows that the asteroids are much more clustered if observed in their "proper eccentricity" rather than in their instantaneous eccentricity. Moreover it shows how the asteroids avoid the high eccentric region, dominated by the presence of two periodic orbits of the elliptic problem. For the pericentric branch these are s i t u a t e d at (e = 0 .49 , T = 90 o) a n d (e = 0 .42 , T = 270 o) ( t h e s a m e v a l u e s are f o u n d

by Ferraz-Mello (1989). A perturbative study along the lines of sect. 4.5 of Part I shows the instability of the former and the planar stability of the latter. Indeed the study of the three dimensional problem reveals very interesting aspects. As shown in graph. 1-(d), the plane is unstable for eccentricities which range between 0.387 and 0.432; then the periodic orbit of the elliptic problem must be considered as an unstable one. The stability ratio ~ is large (> 10) only for eccentricities lower than 0.3, a value which bounds the observed distribution of the asteroids.

The secondary resonances present on the pericentric branch for e < 0.5 are reported in graphic 1-(b). We observe that those of the first series are very clustered, namely there are a lot of resonances in a small range of the eccentricities. Note

however that , because both W and ~-~R have the same sign (their ratio is positive), and because both (~2 + #2) a n d / ~ are positive quantities, the stability is guaranteed also in a resonant condition. Instead the resonances between P~ and P~ are less clustered, but may give origin to some local instability. Finally the resonances between P~ and Pg, which are typical of the spatial elliptic problem, are of pret ty high order and are significantly different from 1 as the eccentricity becomes greater than 0.125; thus they should not give origin to any relevant phenomenon of instability.

Finally in graphic 1-(c) we report the families of periodic orbits with non-zero

inclination and their stability. The families A and B bifurcate from the pericentric branch on the plane in the two points where it becomes spatially unstable. The size of the neighbourhood of stability of the family A can be evaluated by the formal theory as explained in sect. 1-3.4. We find that for the variables ,9 and

such a size is pret ty wide for low inclination and becomes smaller and smaller with increasing values of the inclination; eventually, the family becomes unstable as


shown in graphic 1-(c). On the contrary the size of the neighbourhood of stability for the variables R2 and g2 is very small for low inclination (it disappears on the plane) and becomes wider for higher inclination. So, one can expect stability only for values of inclination approximately in the range [15 ° , 40°]. No asteroids are observed in this region.

The family D bifurcates from the apocentric branch. In the planar circular problem the apocentric branch is made up of orbits with the main feature that the asteroid is at the aphelion while it is in between Jupiter and the Sun. If the eccentricity is larger than 0.05 the orbit is unstable (see Ferraz-Mello (1988) for more details). Still in the framework of the planar circular problem, we have found that the apocentric branch has a singularity for e = 0.31, which corresponds to an orbit which collides with Jupiter. For e > 0.31 the apocentric branch becomes planarly stable. In the diagram S, cr of the planar circular averaged problem the pericentric and the apocentric branches are separated by a family of orbits with intermediate a which all collide with Jupiter. Introducing the third degree of freedom, namely taking into account the effects of the inclination of the asteroid, we have found that such a planarly stable part of the apocentric branch is unstable in the space up to e = 0.96. This is the point of bifurcation of the spatial family D as seen in graphic >(c).

4. T h e 2 /1 r e s o n a n c e

We come now to the problem of the 2/1 Kirkwood gap. Fig. 3-(a) reports the pericentric branch of the family of the stable periodic orbits (solid line) and the separatrix, which limits the region of libration in the averaged circular problem (dashed line). Here the fact that the size of the region of libration has been computed only for zero-inclination has a truly negligible effect; indeed, for i = 7 ° one has a region of libration which is 98% and 97% of that computed on the plane, for e = 0.1 and e = 0.2 respectively. As we see, the region of libration is almost empty; many asteroids come very close to the dashed lines but all stay at the border of the region of libration. Only a few asteroids appear to be inside but their inclination is large. Griqua is the most famous, and its inclination is about 22.8 °. These objects can eventually be associated to a planar periodic orbit, but probably at the limit of the neighbourhood of long-time stability.

The fluctuation of the eccentricity due to Jupiter 's elliptic orbit does not seem to give any relevant informations on the origin of the gap. Indeed, as seen in graphic 3- (a), the fluctuation is very small; moreover, for (e = 0.76,7- = 270 °) and (e = 0.71, r = 90 °) there are two periodic orbits of the elliptic problem. A perturbative s tudy shows that the former is unstable, while the latter is planarly stable. The

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study of the vertical stability, summarized in graphic 3-(d), shows that the ratio of stability ~ is very high for all the orbits of the family with e < 0.5. The family becomes spatially unstable only for eccentricities in the range [0.67, 0.80], just where the periodic orbits of the elliptic problem are situated.

Still studying the properties of the pericentrie branch, we refer to the upper part of graphic 3-(b), which shows the secondary resonances among the frequencies of the angles, which may be present in the Hamiltonian given by (I-4.8). As in the previous section we find that the resonances between Pg and P~ can not give origin

to any instability, because the formal integral Tx(W (32 + ?r~)+ c ~ R) bounds the variation of both ~2 + ~2 and R. The resonances between P~ and P~, have already been studied, by Henrard et al. (1987, 1988) and Murray (1986), and do not seem sufficient to explain the gap. The resonances between P~ and P9 are instead very interesting. They are of very high order, but the ratio between these two periods is very close to 1, ranging from 0.98 to 0.927 for eccentricities that go from 0.025 to 0.3. Thus we are very close to a 1 to 1 secondary resonance, and this, in principle, could give origin to very interesting phenomena. Indeed the normalization scheme of sect. 4.3 of Part I can not be applied because of the quasi-resonance which would give origin to small denominators; this implies that the stability with respect to the motion orthogonal to the reference plane can not be concluded any more by simply looking at the stability ratio c~ given in graphic 3-(d). Sect. 4.1 and 4.2 are all devoted to a detailed study of this situation. We anticipate here that such a study finally shows that the existence of this quasi-resonance does not significantly affect the stability of the orbits.

Graphic 3-(e) finally shows the families of periodic orbits with non zero inclination, which are present in the 2/1 resonance. The families A and B bifurcate from the pericentrie branch in the points where its spatial stability changes. The family A is stable and the size of the neighbourhood of stability seems to be pretty large, expecially for values of the inclination larger than 20 degrees. Of course the eccentricity is so large that we are not surprised by the absence of the asteroids. On the contrary no families bifurcate from the apocentrie branch; indeed, like in the 3/2 resonance, for e > 0.5874 (a value which corresponds to a collision with Jupiter), the apoeentric branch becomes planarly stable, but, in this ease, it is spatially unstable up to e = 1.

In conclusion the 2/1 resonance exhibits stability characteristics which are similar, or even better, than the 3/2 resonance, and so looks more suitable to guest asteroids. Our opinion is that the restricted problem of the three bodies does not give any mechanism which can be able to explain the absence of the asteroids, and some kind of external mechanism is necessary in order to explain the existence of the gap. A possible explanation has been proposed by Henrard and LemMtre (1983); it lies on the effects produced by the resonance sweeping which should have occurred in the first ages of the Solar System, during the expulsion of the primor- dial nebula. Nowadays, to our knowledge, no other possible explanations of the 2/1

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onan

ce

2/1.

The

sam

e as

Fig

ure

1.

,.<

Z > E

oo

0.9

m

> e ,-t


Kirkwood gap have been found. We only remark that the strong stability of this region can be an active mechanism in keeping the resonance region empty, despite of the high density of asteroids in the neighbourhood.

4.1. THE 1/1 RESONANCE BETWEEN Pe AND P~v : THEORETICAL STUDY

In this section our aim is to extend the analytic theory exposed in Par t I to the case where Pg and P,, are close to a 1/1 resonance, in order to investigate any possible behaviour of the orbits. In the next section the theory will be applied to determine whether the periodic orbits of the 2/1 resonance are really stable or whether one can find here a possible explanation of the origin of the 2/1 Kirkwood gap. The expansions of the Hamiltonian involved in this kind of study have been obtained by the aid of the MACSYMA algebraic manipulator.

We report first of all the first three orders of the Hamiltonian that we have to study; in the notations adopted in sect. I-3.3 and I-4.3, these are

z3 = w 2 + b(nl

Z{ =e]fl(gl)Rl+ 6x 3V~ (93K15"a0cr3 + ¢~ a ¢92K1 5. + __e~ f O'-~l, _ o2g-------~ ) OejupOa(rl)x x ~k o--R~ (gl)R1 08(9o'E1 5"

(1-ff~-K1" " R 2 ~ (gl )EI R1 10--~1E2"~ Z4-- 103K15.424x 4 (9o 4 +~2 ~ - - ~ - ~ ( g l ) 1 - + 1]

+¢3dE1+¢4f2(T1)+¢2('(92Kl\(gS(90" J(921~1) 5 . ~ + i (9A(90" ~ z 2¢ ( 03Kl~ (gl ,R1 (93(90 "2(93K1 E1 )5.2

\ a~v [ (92K1 ( 9 ~ (T1)E'q(92K1 z , ~ /) . 4 2 pju(92K1 z ,, 2x ~*2a (gejup (90.2(92K1 (T1)~r2 +eaa ( ~ ( T l , g l ) R 1 "t-~ a (ge,2--1TiJ-'[- . (4.1)

Here

OK1 OK1 OK1 b - b + ~ d - - -

OR ' OE OS (4.2)

COK1 ~ , c3K1 OK1 ~ f l ( g , ) = - - ~ ( g l ) OR ' ~f~(~") = a0~----~(rl) '

that is we have considered the specific case we want to study where both ~ (gl) -

01el and a ~°~---~. g (T1) are small enough to be considered as perturbat ions of order V~ OR ~ j u p

and c respectively. In this case the canonical transformation (I-4.7) is no more necessary and will be included in the deformation given by the transformation of normalization. So one easily identifies Z3 with the expression given by (I-4.8); here the coefficient of E1 has been expanded in such a way that it is equal to that of R1 plus a small difference (of order ~). This is the correct approach to s tudy a situation close to a 1/1 resonance; in fact, looking for a normalization with respect to R, + E l only, we avoid the presence of small denominators. Then the Hamiltonian becomes that of a one-degree of freedom problem, which can be easily studied.

DYNAMICS IN THE ASTEROIDS BELT (PART II)

To this end, we have to perform the canonical transformation

E1 ---- E ' - R' ,

R 1 = R ! , 1 -

¢ = - - ~ ( S + i~) , V2z

7" 1 ..~_ T t

gl = gl + rl 1 .

Vzz

187

(4.3)

such a transformation, and the division of the hamiltonian by e, gives Z3 the form iW~r I + ebE'. Now it's possible to normalize the Hamiltonian with respect to Z3; the technical scheme is the one explained in case ii of sect. I-3.3. It consists, first of all, in normalizing the Hamiltonian with respect to ~2r/2 by the use of a suitable canonical transformation of the kind (~, r], E ' , r ' , R', g') = Tx(~2 , r/2, E2, r2, R2, g2); then, being ~2r/2 a formal constant of motion, one gets rid of it, divides the Hamil- tonian again by e, which corresponds to a rescaling of the time, and finally obtains an Hamiltonian whose term of lowest order is

F1 = bE2 •

Now it's possible to give the Hamiltonian a normal form with respect to E2, by a canonical transformation (E2, r2, R2, g2) = Tx(E3, r3, R3, g3). The new Hamilto- nian is that of a one-degree of freedom problem, with variables R3 and g3. The lowest order term of such a Hamiltonian turns out to be

1 02K1 R2 02K1 ~E 1 02K1 ~E F3--2"~- ~ 3-- ~---~'~[ 3-Ra)R3+~'-~-~-k 3-R3)2+ed(Ea-R3)

f , "R i [ 0agl R 0aK1 (E3 - R3) ~rl2 + ¢a k ~ g a ) 3

) 1 + I{x , ,Z , } 02Kl (E3 - R3) + ~ { x 1 , Z { } OSOejup

(4.4)

where

i f 2 i a e [ 02K1 02K1, ~R

and

02 K1 02K1 ~ E ] 0eju~0a 0--S-O~ t a - Ra)

.I

+-~ L\OROa ] R~3-20ROo.-- ~ 3[ 3 - R 3 ) + k0--ff~// (Ea-Ra) 2 ]

{ x ' , z ' } - Ra¢--]2* b~ J0 (fl(gl))2dgl

Here each term must be considered averaged over r2 (we neglect the bars for sim-

ev

~ ¢v

d d

I ,

, ,

' -I00

-0.0

100

' -I00

-0.0

100

' -100

-0.0

100

g

g

g

Fig.

4.

Res

onan

ce 2

/1.

Illu

stra

ting

the

poss

ible

pha

se d

iagr

ams

of th

e H

amil

toni

an (

4.5)

. S

ee S

ecti

on 4

.1.

o r F r-_


plicity), so that Ea is a formal constant of motion; all the terms which depend on

ga are explicitly indicated. The Hamiltonian F3 has the general form

['3 = o~R] + [fl + f (g3 )] R3 (4.5)

where fl depends parametrically on Ea and (2r/2 , and f is a periodic function of g3. We recall that (2r/2 is the radius of libration around the quasi periodic orbit, and that the value of Ea is related with the starting value of the inclination of the orbit over the plane of Jupiter 's motion. In fact from (4.3) one gets E ' = R + E and Ea is essentially the average of E I over rl; neglecting now any secular variations of R and E with respect to the period of r ', which are described by the final Hamiltonian

F3, we may consider the average of E to be equal to zero, and thus we get that Ea is equal to the starting value of R.

We intend now to report a brief general study of an Hamiltonian of the kind (4.5). First of all we look for equilibrium points, which can be found by solving the

system of

2c~R + /3 + f ( g ) = 0 , --~g ( g ) R = 0

Assume now that ~ , as a function of g, has only two zero-points in A and B, so that the lat ter equation has two solution points for g. Looking then for the solution of the former one for R, one has 3 cases, according to the possible sign of R, which can be resumed by the phase-diagrams of fig 4. In the first two cases we have the presence of a separatrix; in particular, in the second case the plane given by R = 0 is unstable. In both cases a stable zone is formed out of the plane; the t ime-scale of a libration around the stable equilibrium point can be valued to be about 1

million years. Note however that , if R becomes much greater than e, the expansion in orders of ~ adopted to s tudy the Hamiltonian would be no more consistent. In case iii, instead, there are no equilibrium points; in this case the system is actually non-resonant and one has proved its formal stability.

4. 2, THE 1/1 RESONANCE BETWEEN P~ AND Pg; DETAILED STUDY

The application of the general theory exposed in the previous section is straightforward; indeed one just has to numerically evaluate the various terms appearing in (4.4). In this way one gets an Hamiltonian of the kind (4.5) where each term is known.

In our case we have studied two orbits, with mean eccentricity equal to 0.15 and 0.25 respectively, considering them representative of the behaviour of the oth- ers. First of all we have set both ~2r/2 and E3 equal to zero; this is equivalent to studying only orbits which are non-l ibrat ing and close to the plane. One can prove analytically that f (ga) is identically equal to zero; indeed f(ga) is given by the average over r of terms of the kind

(o)

"0 ++

looo ...

. +Oo

o o

: ++

1 c:

; ..

o

* ,

,

h ~

, °+

+ °'

°

°°"

I /

+ o

"~

IO

0 4'

-I~

4"

4.

4.

0

"t o

-~

+ J

t-

4.

+ 0

~2 ~

.. ~

.~ ~

+-.+

+A++

++

o++

.I

c,i

p K

:,%

~I+

.IIT

.~T

~,:

~.o

:+'I

,+.,

I

: o+

',

* ,o

o +

- I

mm

..+

.fi[

t~.,

',,~

),y

.~t~

.i.

~,+P

r+

*o

+ ""

--'+

8 ~~

.~,+

+P-+

~:¢~

/o.+

,++?

't+.Z

(~

+ .+

_ -I

t+ ,~

+~ ,

~+ +

+ +%pJ

+'+:++

~,+

+

°:+

l C

) .

E +

+~,'q

~"

~.?"

+

--

c;,"

0.71

0.

72

0.73

L

O. 7

/-+

0.75

0

.76

(b)

t~

P/P

0 t'

'

l'

ll

lt

''

:l

':

l:

''

:l

ll

l

6 I

':

:'

1'

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'1

I '1

-0

.0

0.1

0.2

0.3

0.~

0.5

LO.O

(c) I

; I

l tl

I I

~ I

i I

' I

' I

r I

r I

l:h 0=

90,

9=90

B, 0

=150

, 9=

PO

, I

i I

I J.

i J

I I

, I

, L

l I

l I

,

0.2

0.~

0.6

0.8

e

1.0

.>

o E

3>

:z

.>

5 c~

F

Fig

s. 5

a-c.

%

ll-

~9

0 E

)

-0.0

(d) ..

..

'

..

..

I

..

..

l

i"

''

l

..

..

v

''

~'

l

..

..

'

''

'~

I

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'

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I

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v

..

..

l

'~

''

v

..

..

I

''

=~

..

..

i

..

..

I

..

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~

..

..

I

..

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i

, ,

~ ~

I .

..

.

= .

..

.

I .

..

.

i .

..

.

1 .

..

.

= .

..

.

I .

..

.

= .

..

.

I .

..

.

O. 1

0 0

.20

0

.30

0

.40

0

.50

0

.60

0.

70

e

Fig.

5d.

Figs

. 5a

-d.

Res

onan

ce 5

/2.

The

sam

e as

Fig

ure

1.

K Z >, ~r

N

N

m

>

m

t~


d~j .pi k cos ( r ~ + (,~ + 1)T + ng3)

with l = m :F (1 - n ) + an even number and k = n + an even number. These terms are not zero only if n 4-1 = 0, namely for n = 4-1; this cannot be the ease as k must be an even number (since in the restricted three body problem only even powers of i are present). The terms which depend on g3 in the new Hamiltonian are then present only at a higher order, as they earl be only coefficients of even powers of ejup. Because of this fact, if the coefficient/3 which appears in (4.5) were equal to zero, then the analysis of the terms of higher order would become necessary; in our case we have verified that , since {X', Z~/2} is negligible,/3 is essentially equal to the coefficient -ed; d is defined in (4.2), and, because the resonance between Pg and P~ is not a perfect one, it is different from zero. Indeed d is pre t ty great, being about -11 for the orbit with e = 0.15 and -6 for the one with e = 0.25. The coefficient oL which appears in (4.5), results instead -0.43 for e = 0.15 and -0.33 for e = 0.25. It 's easy to check, then, that the phase-space diagram of our Hamiltonian is qualitatively similar to that shown in the first graphic of fig. 4; however in our cases the amplitude of libration in R3 is very narrow, because f(93) has only a small variation as a function of 93 (as such a variation is given only by the terms of higher order). Moreover the equilibrium points in Ra and g3 do exist only for large values of R3, namely about 25e for e = 0.15 and 18e for e = 0.25; so we can conclude that , for a non-l ibrat ing orbit close to the plane, the Hamiltonian is that of a typical non-resonant system.

To complete this study, the computat ion of the terms of (4.4) which are linear in ~2r/2 or E3 is necessary. These terms change the te rm/3 in (4.5) if the values of ~27/2 or E3 are not equal to zero; these values correspond to librating or inclined orbits respectively. We have computed that both for a librating orbit and for an inclined one, the equilibrium points in R3 and g3 are shifted to higher values of R3.

This means in conclusion that all the orbits must be considered as non-resonant and formally stable; the quasi-resonance between Pg and P,, does not produce any effect on the stability of the orbits.

5. T h e 5 / 2 r e s o n a n c e

In graphic 5-(a) we summarize, as usual,the main results tha t can be obtained by a planar s tudy of the stable family of periodic orbits; the family itself (solid line), and the limit of the region of libration (dashed line) are shown. The existence of the gap is absolutely evident; also in this case, like for the 2/1 resonance, the dependence on the inclination of the maximai amplitude of libration is very faint. Indeed, for example, for i = 7 °, the region of libration in L is 97% that computed on the plane. As one clearly sees, the resonant region is much narrower here than in the 2/1 or 3/2 cases, and has a relevant size only for e > 0.1.


The effects produced by the ellipticity of Jupiter 's orbit are very important in the 5/2 resonance. Here the periodic orbits of the elliptic problem are present for low eccentricity. These are characterized by the following values of the eccentricity a~d of the longitude of the perihelion : (e = 0.014, ~- = 90°), (e = 0.045, 7- = 270 °) and (e = 0.27, ~- = 90°). The first one is unstable and is reduced to the origin of the reference frame (F, f ) by the introduction of Sessin's variables (see part I for reference). The second one is unstable too, while the last one is stable.

The error bars reported on graphic 5-(a) refer to orbits with circulating perihe- lions; the eccentricity has a strong fluctuation, and periodically reaches the critical value 0.417, which allows a possible collision with Mars. For what concerns the vertical stability of the family, we see in graphic 5-(d) that the plane is always stable; however such a stability is very faint for eccentricities higher than 0.35, becomes pret ty strong for 0.275 < e < 0.35, is faint again for eccentricities down to 0.1, and finally becomes strong again for eccentricities below 0.1. Because of this, we believe the size of the long-time vertical stability to be pret ty smM1, at least for eccentricities which range from 0.1 to 0.275.

The displacement of the secondary resonances between Pg and P~ is given by graphic 5-(b). We see here that these resonances are of pret ty low order, and, differently from what happens in the 3/2 and 2/1 cases, may cause some local

instability, because the coefficients W and -~R have an opposite sign.

The topology of the families of periodic orbits with non-zero inclination is given in graphic 5-(c). In this case none of the families bifurcates from a family on the plane as it happens instead in the 3/2 and 2/1 resonances. The family A is stable only for eccentricities higher than 0.35. The neighbourhood of stability does not seem much wide, especially for low values of the inclination. The family B is instead very interesting, as it originates from a circular orbit, with a low value of the inclination, namely 8.6 degrees only. B is stable up to 37 ° of inclination. However the size of the neighbourhood of stability is very small also in this case; indeed the

second derivatives of the perturbation 0~ff_t and ~ are very small. This deter- Og2 Oa2

mines, for example, that the maximal amplitude of libration for S, corresponding to a maximal libration of a of 60 °, is 4.5 • 10 -3 for the orbit with i = 15 ° and 4 .6 .10 -3 for the orbit with i = 31%

6. T h e 3 / 1 r e s o n a n c e

In graphic 6-(a) a detailed study of the planarly stable family of the periodic orbits is given. As usual, the solid line signs the family of the periodic orbits, and the dashed lines delimit the region of libration. The existence of the gap is evident; indeed the asteroids are all outside or very close to the separatrix. The amplitude

(a)

"o °I

''

'

' '

"i;

: '

' '

' '

' +o

o

° ~o

,~ :1

- j

t [

r:

+ o

° '~

o .

~ o,

,, ,,.

,<, o

,,O

',

I :

., ~>

.,. o

'.,.

T

"+°

+ +1

' ++

"°"

I

6 l%

.i r.,.

,.~~`

, I :

;~

~

• 'f,

~. ,,.'~

+,=...v

~f~. g

~ + il

' ++

o+"~

.+ I-+,

+++;

~+

_:

+

,+*

+ +

+;

++ L

* ~,

+ -

~+

,~*~

t

o 0

el,i,

m

#i" i

~l

I I

I h"

I

i !

m

i+i

i P

mil

l i

m'l"

rt

m

m

F [

(~

i m

"i" I

' v.67

0.6

8

0.6

9

0.7

0

0.71

0.

72

L

(b)

p,,'p

m

l II i

I I

: :

i '

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I i

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: :

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t t

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I -0

.0

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0.2

0.3

0.4

0.5

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o "0.0

(c}

' i

I w

I '

I i

I i

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I i

I i

I i

I I

" FI

,o=1

80,

a=PO

~.

B,o

=IE)

, 9=

1E)

C, o

=90,

a=9

0 ....

......

......

. w.

. ..

. ,,

.

..........

..........

....... -,~

,.-~,i~

I-,, L

J I

~ l

i l

i I

i I

J If

, I

i 'I

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0.4

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m r-

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.0

Fig

s. 6

a--c

.

%

T--

la %

,ir

,-

d) I

I 1

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r I

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-0.0

0.

1 0.

2 0.

3 0.

4 0.

5 0.

6 0.

7 0.

8 e

Fig.

6d.

Fig

s. 6

a~l.

R

eson

ance

3/1

. T

he s

ame

as F

igur

e 1.

K

Z

>

o.~

rn

m

(el

. ",

.

+ :1'

, +

o ,,,

o

,I,

o i

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,,li,

0 ~

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* .

~.

oeqD

I.

lil

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*.

t tr

~

.i.*

* .

,i ,i,

.,.,~

qll

o,

T o+

'**+

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q,'i

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lili

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:£-1

0.

64

0.65

0.

66

0.6,

7 0.

68

L

c;

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04

<5

(:3

(b)

P/P

o

II

P/P

o

II

.,,q .,,

c, .,&

_,,,q

.~

.<,&

_~ -

,o&

Il

ll

ll

l_

l,

ll

il

ll

li

,

llll

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ll

I i

= w

I I

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T I

w r

i ~

I

-0.0

0.

1 0

.2

0.3

0

.4

0.5

(:3 "0

.0

(c) =

I 1

I •

I ~

I f

l ~

I •

I e

| ~

• e

R,o

--K

),9=

P0

Bio

=2

,70

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~O

B

I I

J I

l I

I I

• I

1 L

i t

J I

• I

•

0.2

0

.4

0.6

0

.8

8

1.0

P ! E >.

z .>

c~

c~ E

Figs

. 7a-

c.

"o

,%

T Q "

-0

.1

(d) .

..

.

i,

J ~

J !

..

..

i

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v

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..

.

w .

..

.

i .

..

.

• .

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.

1 .

..

.

f .

..

.

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• '

~

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..

.

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..

.

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~ ,

i .

..

.

I .

..

.

t .

..

.

I .

..

.

i .

..

.

I ,

. .

, t

..

..

.

L~

,

, i

..

..

I

..

..

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..

..

I

..

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0.1

0

0.2

0

0.3

0

O. {

-,0

0.5

0

0.6

0

0.7

0

e

Fig

. 7d

.

Fig

s. 7

a-d.

R

eson

ance

7/2

. T

he s

ame

as F

igur

e 1.

,.<

Z N

m

> m

m


of libration has only a faint dependence on the inclination also in this case; for example, if e = 0.2 and i = 7 ° it is 98% that computed on the plane for the same eccentricity. Fig. 6-(a), then makes a correct comparison between the position of each asteroid and the size of the resonant zone. As a matter of fact, to be precise, one asteroid, named Alinda, is present nearby the periodic orbits, but its eccentricity is so high (0.6) that it crosses the orbit of Mars; then the study of the evolution of its orbit cannot be done by using the model of three bodies.

For what concerns the periodic orbits of the elliptic problem, these are given by the following values of eccentricity and longitude of perihelion : (e = 0.1, r = 90°), (e = 0.82, r = 90°), (e = 0.79, r = 270°); only the last one is stable.

The planarly stable family turns out to be formally stable also with respect to the vertical motion, except for eccentricities which range between about 0.615 and 0.77. However, one may see in graphic 6-(d) that the stability ratio ~ is always small (it is less than 4 for e < .3). Thus we believe the size of the neighbourhood of vertical stability to be very small, namely to find asteroids associated to the family only if very close to the reference plane.

Graphic 6-(b) reports the main secondary commensurabilities between the periods of the angles. The situation here is similar to that of the 2/1 resonance : the commensurabilities between P~ and P~ are of rapidly increasing order and the first onc can not give origin to any instability as the ratio is positive; the resonances between P~ and Pg are of pret ty high order. However, for all the values of the ecccntricity, the ratio between these two periods is always about 0.68, a number pret ty close to 2/3=0.66

The topology of the families of the periodic orbits with inclination different from zero is given in graphic 6-(c). Three fmnilies have been found. The family A is stable but its eccentricity is too great to be populated by asteroids; it bifurcates from the stable family on the plane in the point (e = 0.615) where the latter becomes spatially unstable. The family B is unstable and also bifurcates from the same family on the plane, but in the point with e = 0.77. Finally C is an unstable family with very high inclination.

7. T h e 7 / 2 resonance

Graphic 7-(a) gives in the usual notations the amplitude of the region of libration around the stable family on the plane. As one clearly sees this is extremely small, and eventually disappears, for e < 0.2. almost all the asteroids present in the region have a lower eccentricity , and thus must be considered as non-resonant even if their mean motion is very close to the commensurability with that of Jupiter. If we just look to the distribution of the asteroids with respect to their mean motion,


then, we don't find any gap corresponding to the 7/2 resonance. Because of this fact , in some cases, this resonance has been considered as populated by asteroids, just like the 3/2. Graphic 7-(a) reveals the misunderstanding : indeed the true resonant region is empty also in this case, but the gap is not present because the region of libration does not divide the L, e diagram in two disconnected parts.

The explanation of the absence of asteroids in the resonant region seems evident in this case : in fact, because of its shape, the size of the resonant region is relevant only for eccentricities higher than 0.3; but if the eccentricity is higher than 0.26, the asteroid crosses the orbit of Mars. A close encounter with the red planet, not described by the problem of the three bodies, can surely take the asteroid away from its original orbit. For what concerns the pure three bodies problem, we have computed that the fluctuation of the eccentricity of the orbit of a resonant asteroid, given by the ellipticity of Jupiter 's one, is really small, not being higher than 0.025 for orbits with ~ > 0.2. Only one periodic orbit of the elliptic problem is present, for e = 0.026 and r = 90 °, and it is unstable.

As in the case of the 5/2 resonance, the planar family of periodic orbits is always stable with respect to the vertical motion, however, as revealed by graph. 7-(d), the spatial stability is strong only for e < 0.3

Finally, graphic 7-(c) shows the topology of the periodic orbits with non-zero inclination. These form two families, A and B; the inclination is higher than 30 degrees for all the values of the eccentricity. Some of the orbits of these families are stable, but, as in the case of the 5/2 resonance, the region of libration in the action variables is actually very small.

8 . T h e 4 / 1 r e s o n a n c e

Graphic 8-(a) shows in detail the family of the stable periodic orbits which lie on the reference plane. There are no asteroids in this region. For this value of the semimajor axis, an orbit with e > 0.193 crosses that of Mars and this, probably, is enough to get rid of both the resonant and the non-resonant asteroids.

By studying the elliptic problem one discovers, like in the case of the 5/2 resonance, the existence of three periodic orbits. These are given by the following values of e and ~-: (e = 0.011, T = 90°), (e = 0.085, r = 270°), (e = 0.45, r = 90°). Only the last one is stable; its high eccentricity means in practice that each orbit which is librating around the stable periodic one has its eccentricity periodically increased to values larger than 0.455, that is much over the value which allows a collision with Mars.

Also in this case the planar family turns out to be stable with respect to the vertical motion for all the values of the eccentricity. Such a stability should be pret ty strong for e < 0.4 (see graphic 8-(d)).

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Outside of the plane there are other two families of periodic orbits, as shown in graphic 8-(c). The orbits of the family A are stable for e < .35, and end into a circular orbit with inclination of 17 °. The size of the region of libration around this family is very small, like in the case of the 5/2 resonance. Indeed, for a maximal

libration of 60 ° in a, the corresponding libration of S is 2.4.10 -3 for the orbit with i = 26 ° and 4 .7 .10 -3 for i = 30 °. B is stable only for high eccentricity.

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

Using the methods developed in part I, we have studied in the framework of the

circular problem the families of periodic orbits corresponding to the resonances 3/2, 2/1, 5/2, 3/1, 7/2 and 4/1 in the spatial case. The stability of all these families has been investigated, and the size of the stability neighbourhood has been estimated; then the results have been compared with the observed distribution of the asteroids. For the families lying in the Jupiter 's plane, the results are consistent with the previous ones that can be found in the papers quoted in the references. Moreover, the study has been extended to the spatial elliptic case, and the possible relevance

of the secondary resonances for the stability has been investigated.

The conclusions can be summarized as follows. i. In the framework of the circular problem, all the resonances admit a stability

region associated to a certain family of periodic orbits in the plane. Such a region turns out to be filled up with asteroids in the case of the resonance 3/2, while is

empty for all other resonances. This is in contrast with the conclusions of some authors that some resonances, like for instance 7/2, contain asteroids.

ii. Spatial families of periodic orbits with a significant stability neighbourhood exist

also for nonzero inclinations, in particular for the resonances 3/2, 2/1 and 3/1; however these families have very high eccentricity, so that it is unlikely that they

can contain asteroids. iii. The ellipticity of Jupiter 's orbit forces an oscillation of the eccentricity of the

periodic orbits of the circular case. For the resonances 5/2, 3/1, 7/2 and 4/1 such an oscillation can cause an asteroid either to reach the limit of the stability region or to intersect the orbit of Mars, so that it could be invoked as a possible mechanism to deplete the resonant region. This does not apply to the resonances 3/2 and 2/1, so that the depletion of the 2/1 Kirkwood gap seems to have no mechanical explanation in the framework of the complete restricted problem of

three bodies.


R e f e r e n c e s

Brjuno, A. D., (1970) : "Instability in a Hamiltonian system and the distribution of the asteroids.", Mat. USSR Sbornik, 12,271-310.

Colombo, G. and Franklin, F. A. and Munford, C. M., (1968) : "On a family of periodic orbits of the restricted three body problem and the question of the gaps in the asteroid belt and in Saturn's rings.", Astron. J. 73, 111-123.

Colombo, R. and Giorgilli, A., (1989) : "A rigorous algebraic approach to Lie transform in ttamiltonian mechanics.", preprint.

Dermott, F. S. and Murray, C. D., (1981) : "Resonant structure of the asteroid belt.", Nature, 290,664-668.

Dermott, F. S. and Murray, C. D., (1983) : "Nature of the Kirkwood gaps in the asteroid belt.", Nature, 301,201-205.

Ferraz-Mello, S., (1988) : "The high eccentricity libration of the Hildas.", Astron. J. 96, 400-408.

Ferraz-Mello, S., (1989) : "A theory of planar planetary corotations.", preprint. Froeschl(!, C. and Scholl, H., (1974) : "Asteroidal motion at the 3/1 commensurability.",

Astron. Astroph., 33,455-458. Froeschld, C. and Scholl, H., (1975) : "Asteroidal motion at the 5/2, 7/3 and 2/1 reso-

nances.", Astron. Astroph., 42,457-463. Froeschld, C. and Scholl, H., (1976) : "On the dynamical topology of the Kirkwood gaps.",

Astron. Astroph., 48, 389-393. Froeschld, C. and Scholl, H., (1982) : "A systematic exploration of the three-dimensional

asteroidal motion at the 2/1 resonance.", Astron. Astroph., 111,346-356. Giffen, R., (1973) : "A study of commensurable motion in the asteroid belt.", Astron.

Astroph., 23, 387-403. Giorgilli, A. and Galgani, L., (1978) : "Formal integrals for an autonomous Hamiltonian

system near an equilibrium point.", Celest. Mech., 1T, 267-280. Giorgilli, A. and Galgani, L., (1985) : "Rigorous estimates for the series expansions of

Hamiltonian perturbation theory.", Celest. Mech., 37, 95-112. Henrard, J., and Lema[tre, A., (1983) : "A mechanism of formation for the Kirkwood

gaps.", Icarus, 55,482-494. Henrard, J., and Lemaitre, A., (1987) : "A perturbative treatment of the 2/1 Jovian

resonance.", Icarus, 69, 266-279. Henrard, J., and Caranicolas, N., (1988) : "Motion near the 3/1 resonance of the planar

elliptic restricted three body problem.", preprint. LemMtre, A., (1984) : "High order resonances in the restricted three body problem.",

Celest. Mech., 32, 109-126. LemMtre, A. and Henrard, J., (1989) : "On the origin of chaotic motion in the 2/1 Jovian

resonance.", accepted by Icarus. Liu, L. and Innanen, K. A., (1985) : "Studies on orbital resonance.", Astron. J., 90,

877-891. Milani, A., and Nobili, A., (1984) : "Resonant structure of the outer asteroid belt.",

Celest. Mech., 34, 343-355. Milani, A., and Nobili, A., (1985) : "The depletion of the outer asteroid belt.", Astron.

Astrophys., 144, 261-274. Milani, A., Murray. C.D., and Nobili, A., (1986) : "The Hilda group and the Hecuba

gap." in Asteroids, Comets, Meteors, Reprocentralen HSC, Uppsala. Morbidelli, A. and Giorgilli, A., (1989) : "On the dynamics in the asteroids' belt. Part

I : general theory.", Celest. Mech., in print. Moser, J., (1958) : "Stability of the Asteroids.", Astr. J., 63, 439-443 (1958) Murray, C. D., (1986) : "Structure of the 2/1 and 3/2 Jovian resonances.", Icarus, 65,

70-82. Schubart, J., (1964) : Special Report, Smithsonian Astr. Obs. 149.


Wisdom, J., (1983) : "Chaotic behavior and the origin of the 3/1 Kirkwood gap.", Icarus, 56, 51-74.

Wisdom, J., (1985) : "A perturbative treatment of the motion near the 3/1 commensurability.", Icarus, 63, 272-289.

on the dynamics in the asteroids belt. part ii: detailed study of the main resonances

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