the main secular resonances ν6, vs and ν16 in the asteroid belt

THE MAIN SECULAR RESONANCES u6, us AND 1,'16 IN THE ASTEROID

BELT

ALESSANDRO MORBIDELLI AND JACQUES HENRARD Ddpartement de mathdmatique FUNDP

8, Rempart de la Vierge, B-5000 Namur, Belgique

(Received 18 July, 1990; accepted 17 February, 1991)

Abstract. In this paper an analytical model, suitable for a global description of the dynamics in a secular resonance of order 1, is derived from the general perturbation study developed in a previous paper (Morbidelli and Henrard (1991)). Such a model is then used to study the secular resonances ur, u5 and u16, and pictures illustrating the secular motion are obtained. The peculiarities of the ~'5 resonances are discussed in detail. The results are compared with those obtained by the theories of Yoshikawa and Nakai-Kinoshita. Some numerical simulations performed by Ch. Froeschld and H. Scholl are discussed in the light of the new theoretical results. New numerical experiments on the ~'6 resonance are also presented.

Key words: asteroid belt-secular resonance-action angle variables-surface of section-numerical simulation

1. Introduction

It is well known that, as a consequence of the mutual gravitational interaction among the planets, the elliptic elements of their orbit (eccentricity, longitude of perihelion, inclination, longitude of node, respectively indicated by e, ~ , i, ~9 in the notations of Poincart) are not constant, but vary following a law which is, in a first approximation, the superimposition of harmonics with different constant frequencies. Among these, the most important ones are those of the Sun-Jupiter- Saturn system, called 95, Y6 and s6. The first two are related to the variation of their eccentricity and longitude of perihelion; the latter to the variation of the inclination and longitude of node.

The presence of the planets has also a perturbing effect on the motion of a massless asteroid, inducing, particularly, a secular precession of the longitudes of perihelion and node. When the frequencies of these precession are commensurable with the frequencies of the planets, which we mentioned above, a secular resonance Occurs .

In a previous paper (Morbidelli and Henrard (1991)), we have developed a perturbation scheme by which we have avoided any expansion of the Hamiltonian with respect to the eccentricity or the inclination of the asteroid, in order to have results valid for any value of these variables. Moreover, by introducing suitable action-angle variables, we have avoided any operation of averaging not supported by a coherent formal perturbation scheme. As an application, we have discussed the location of the strongest secular resonances in the main asteroid belt.

The present paper is a continuation of the previous one. Here the aim is to discuss in detail the dynamics associated to the three main secular resonances

Celestial Mechanics and Dynamical Astronomy 51" 169-197, 1991.

(~) 1991 Kluwer Academic Publishers. Printed in the Netherlands.

170 ALESSANDRO MORBIDELLI AND JACQUES HENRARD

us, v6 and v16. The first two occur when the precession rate of the longitude of perihelion of the asteroid is equal to Y5 or 96 respectively; the latter is given by the 1/1 commensurability between the precession rate of the longitude of node and the frequency s6. For this purpose we derive from the theory developed in the previous paper an analytic model suitable for a semi-numerical study. This gives a global description of the resonant and quasi-resonant dynamics.

The main recent analytic results on the dynamics in secular resonances are those of Yoshikawa (1987) on the v6 resonance and of Nakai and Kinoshita (1985) on the b,16 resonance. The basic ideas of these two works are similar: the Hamiltonian of the problem is averaged with respect to the mean anomalies of the asteroid and of the planet and over the argument of perihelion y = ~ - ~. Then, assuming constant the inclination (for the v6 resonance) or the eccentricity (for the/.t16 resonance) the authors get a one--degree of freedom Hamiltonian which can be studied in detail. Moreover, in the theory of Yoshikawa, the original Hamiltonian is expanded in series of e and i and truncated at degree 4. As a consequence his theory can not be easily applied for the L,5 resonance because of its high inclination (,,~ 30°). In the following we will refer to these theories for a comparison of the results.

On the same problem of the dynamics in secular resonance, a lot of experimental work has been done by Ch. Froeschl6 and H. Scholl (1986a, 1986b, 1987a, 1987b, 1988a, 1988b, 1989, 1990). Their numerical simulations give an invaluable source of data which reveals the complexity and the charm of the problem, and constitute a fundamental test for the analytical theories. In this paper we will discuss their results and reproduce some of their figures.

The present work is structured as follows: first we derive from the perturbation theory the analytic model for the description of the dynamics associated to a 1/1 resonance involving the frequency of the longitude of perihelion. This is done in section 2. Then, in Sect. 3 and 4, these theoretical results are applied for the investigation of the v6 and ~'5 resonances. Finally we briefly describe an analogous model for the I}16 resonance and discuss the results so obtained (sections 5 and 6).

2. A global model for a 1/1 secular resonance of the perihelion

In the previous paper (Morbidelli and Henrard (1991)) we have developed a local perturbation study of the resonant dynamics, namely we have been able to describe the phase space in the neighbourhood of the resonant orbit which, in the proper variables, is represented as an equilibrium point. This study has been successful in pointing out many features of the resonant dynamics, like the strength of the resonance itself, the stability of the resonant orbit, the banana-shape form of the neighbourhood of libration in case of stability, and so on, but, unfortunately, it has not been able to show the global picture of the phase space. To achieve this result, in this section we deduce from that study an analytical model which can be studied with semi-numerical techniques.

We have seen in section 2 of the previous paper that, after the normalization

THE MAIN SECULAR RESONANCES IN THE ASTEROID BELT 171

with respect to the mean longitudes, the Hamiltonian of the problem of the secular motion of an asteroid under the effects of the perturbations of the planets, is given by:

- - 1 K = 2A 2 + njAj + ¢2vgiAgj + ¢2vssA8~ + e2Ko(2g) (1)

¢4 --{- ¢ 3 R 1 (h, g, Ags, ,k,~ ) + ~ 4 R 2 (h, g, .kg¢, A~ ) + ~ {X2, Ko } (2g) + . . .

In this formula A = ~ (where a is the mean semi major axis of the asteroid), and h = - ~ . The quantity e is a small parameter, such that its square is equal to the ratio between the mass of Jupiter and that of the Sun. Moreover j is an index which refers to the planets from Mercury (1) to Neptune (8). The actions Aj are conjugated to the mean longitudes of the planets, with frequencies n j; the actions Ag~ and Asj are respectively conjugated to the secular angles Agj = - 9 i t - aj and A,~ = - s i t - /3j with frequencies e2~,g~ = - g j and e2t, sj = - s j . Finally the perturbation K P has been expanded in Taylor series with respect to the Poincar6 variables of the planets, thus getting the series )-'ira_>0 em+2Km where rn designates the degree of the polynomial Kin. The bar over each term indicates that a numerical average over the mean longitudes has been performed. X2 is the generating function of lowest order of the Lie's algorithm of normalization and {.,. } is the Poisson bracket. All the dependencies on the angular variables have been explicitly indicated.

Retaining in (2) only the terms up to order 3 in e, and dropping the constants - 1 / ( 2 A 2) and Aj we have

K = ¢ 2 (vgsAgs + v,sA,s) + e2Ko(2g) + e3K, (h ,g , Ag, ,Asj) . (2)

We assume now to be in the region of the phase space where g is a circulating variable, and we introduce suitable action-angle variables in order to take the integrable "Hamiltonian" K0(29) to be dependent only on the actions. This is done via the canonical transformation

P = P h Z = Z(P,.7, ¢) g = g(P, . l , ¢ ) , (3)

where Z, g and ~ are periodic in ¢. Moreover P = x/-a(1 - v/1 - e 2 cos i) is the

conjugated to h and Z = ~/a(1 - e 2)(1 - c o s i) is conjugated to g. The angle action

¢ is proportional to the time spent on the orbits of Ko. The action J , conjugated to ¢, turns out to be

J = ~ Z(g)dg = P - ~ (P - Z)(g)dg = P - --27r ' (4)


where 7 -- (Z(g) ,g) is the trajectory solution of K0, and so can be computed in a numerical way. The quantity A has a very simple geometrical meaning: it is the area enclosed by the trajectory described by K0 in the coordinates 9 and P - Z. This is in some sense a natural choice for the coordinates, as P - Z ,-~ e, for small e. The "Harniltonian" K0, written in the new variables, is a function of P and ,/ only, and so the dynamical system described by it is characterized by two constant frequencies

m

OKo OKo and u ~ - (5)

u h , - OP OY

All these steps, briefly summarized above, have been explained in detail in section 2 of our above mentioned paper.

We assume now to be in presence of a 1/1 commensurability between uh, and one of the frequencies of the Solar System vg~, which we will denote with vg~0.

Then we drop from K1 all the terms which depend on the angles A,j, for all j , and all the terms which depend on Ag~, for all k ¢ k0, where Agh0 is the angle (of frequency t,,g%) which is resonant with the perihelion if; this operation can

be performed in practice without difficulties, as K1 can be easily expanded in Fourier series of Ag~ and Ask. (Making reference to the theory exposed in the previous paper, this is roughly equivalent to studying the Hamiltonian in the proper

.r ~ M M defined there in section 3). As we have eliminated the variables P ~ , _ , "-9~,--*~ conjugated angles, all the actions Asj and the Ag h, with k ~ k0 are now constants, and we can drop them. Finally, we perform the canonical transformation

P = P o" = h I-Agko

A~h ° = Agko + P Agko = Agko , (6)

which introduces the resonant angle o-, conjugated to P , and eliminates the depen- dence on Agh0 in our approximated Hamiltonian. As a consequence, the action A~h °

is a new constant of motion and quits our problem. The new Hamiltonian becomes:

K = e2K0(P , o r) - ~2vg%P + e3K1 (if, ~b, P, or). (7)

The Hamiltonian (7) is our global model for the 1 to 1 secular resonance between the longitude of perihelion h ~ and the angle Ag h . To study this model we strictly follow Henrard (1990). As the "energy" K is ~ae only exact constant of motion of (7), we choose one of its (three-dimensional) level surfaces; then we section it with the hyperplane ~b = 0 (which corresponds to g = 0), getting in this way a two-dimensional manifold. On this manifold we choose as polar coordinates the proper eccentricity e and the proper resonant angle or; we call them proper as they have been filtered of all the oscillations given by the fast angles (mean anomalies), by the non resonant secular angles A,~ and Agk, (k 5~ k0), and computed for 9 = 0,


so that they correspond to the definition of proper elements given by Williams and Faulkner (1981) and in sections 2 and 3 of the previous paper. Now, for each point of the manifold (described by a couple (er, e)), we compute the proper inclination i by using the condition that K(e, a, i, ¢ = 0) =const., and then, by considering e, i at g = 0 as a starting point for a trajectory of the integrable Hamiltonian K0(2g), we compute the actions P and J. If the term e3Ki were not dependent on ¢, J would be a second constant of motion, and its level curves, drawn on the manifold, would describe completely the secular dynamics in the proper variables e and tr. Unfortunately this is not the case, as J is an approximate constant of motion up to order e only. If this approximation is not good enough, we can compute from J a new action J~, which is a constant of motion up to order e 2, and use its level curves to describe the secular motion. Neglecting the terms of order ~,~/v2L(v~, is the frequency of tr and u¢ is the frequency of the cycle of g described by K0, and this ratio is generally small), such an action J~ is given by the following formula (see Henrard (1990)):

j , = j + ~"--~1 ( K 1 ( ¢ , t r ) - < K1 >¢ (tr)) (8)

m

where with < K1 >,p we denote the average of K l with respect to ~b. The computation of J~ is straightforward in principle: indeed one already knows

KI(0 , er), u,p and J , and so one should "just" evaluate < K1 >~; as a matter of fact, however, although being possible, this turns out to be a very long operation, as it requires to compute an integral along a trajectory of a numerically doubly averaged function (KI). We will discuss this problem in more details in the next section.

3. Dynamics in the u~ secular resonance

In this section we expose the results obtained on the dynamics in the u6 resonance. Our main tool is the model described in the previous section. We have applied it for two values of the semi major axis of the asteroid, equal respectively to 0.4 and 0.5 times that of Jupiter (equal respectively to 2.08 and 2.60 A.U.). The value assumed for the frequency 96 is that given by Bretagnon (1974) (i.e. 26.2167"/year). We have chosen this value, as it is the closest to the one obtained by our numerical simulation of the Sun-Jupiter-Saturn system, which will be used later for a numerical verification of the results. Moreover we believe that a different choice of the frequency of #6 would not give a different description of the resonant dynamics, but just slight quantitative modifications, for example on the inclination of the resonant orbit for a given eccentricity. In the previous paper we have discussed the problem of the location of the u6 resonance considering also the more accurate (for the full Solar System) value of 96 given by Nobili et al. (1989) (i.e. 28.246"/year).


3.1. DYNAMICS AT 2.08 A.U.

The first problem in the application of the methods exposed in section 2, is the computation of the trajectory 7 described by K0 which is necessary to compute the value of d. Indeed, the Hamiltonian Ko is not known explicitly, but can just be computed for any value of e, i and g through the numerical evaluation of double integrals. As a consequence, the direct method, which consists in numerically solving the Hamiltonian equations given by K0 would not be practical as it would be a very slow algorithm. For a value of the semi major axis of 2.08 A.U., the /,'6 resonance is present at very low inclination (lower than 10°), so that we know from the theory that the trajectories in polar coordinates Z, g are close to circles. For this reason, the method that we have adopted is that of the local expansion of K0 already wexplained in detail in section 5 of our previous paper. In a few words, we expand K0 in Taylor series with respect to Z = Z - Z0 (where Z0 is given by e, i at g = ¢ = 0) up to order two in Z and in Fourier series of g up to the harmonic cos 4g. With this approximation we get a new analytical Hamiltonian in Z and g with constant coefficients, and the trajectory -y can be computed through an explicit formula. The computation of the area enclosed by such a trajectory is then straightforward.

The evaluation of d ~, starting from d is a much more difficult and long operation. First of all it's not enough to know the geometrical shape of 7, but one needs to know it as a function of time, namely the couple (Z(g(t), g(t)). Then one has to integrate along this trajectory parametrized by the time the function K1 (see (8)), which, similarly to K0, is not known explicitly but can be computed at any point through a double numerical integral.

The difference between the results obtained by considering constant d or d ~ has a precise physical meaning. The results obtained through the computation of d ~ are equivalent to those obtainable studying the one degree of freedom model

K = e2K0(P, J) - e2Ueho P + e3ccos (tr) , (9)

where e is the average over the trajectory 7 of K1 valued at cr = 0, namely, from (6) and (3):

Xf~ c = ~ K I ( P , Z(9(t)), 9(t), h - Agk0 = O(t)) dt = < K1 >9 (or = 0), (10)

where 7' is the period of the trajectory 7. Conversely the results obtained through the consideration of d as a constant of motion are equivalent to those obtainable studying the same model (9) where c is the value of K1 at 9 = 0, Z(9 = 0) and cr = 0(0) = 0, namely

c =-KI(P,Z(O), O = O, cr = 0 ) . (11)

As Z(9), K1 (9) and o(t) differ from constants by terms of the order of i 2, the two results are equivalent on the invariable plane i = O. Therefore, the difference has


to be negligible as long as the inclination is small, like in our case. For this reason, we have disregarded the computat ion o f J t and simply plotted the level curves of J to describe the secular motion.

The results are shown in Fig. 1, which is made up of 6 pictures. The computations have been carried on by taking into account only the perturbations by Jupiter and Saturn. Each picture represents a level surface o f the Hamiltonian K of (7) in polar coordinates e and - t r (we choose - t r instead o f or so that the direction of circulation/libration in these coordinates corresponds to that in e, to - ws used in the numerical simulations that we will describe later). Remember that the section is taken at ~b = g = 0 so that the eccentricity e is actually the "proper eccentr ici ty" as defined by Williams and Faulkner (1981) and denoted as ex in our previous paper. This is actually the min imum eccentricity during the rotation of the argument o f perihelion g. The value o f e on the circular border is equal to 0.6 On the upper fight comer of each picture the inclination o f the orbit with e = 0 is reported. This value characterizes the value of K which is constant all over the Surface. In the lower fight comer, the value of the proper inclination of the orbit with e = .6, tr = 90 ° is reported, to give an idea of its variation on the surface K = constant. The lines drawn on each picture are equidistant level curves of J ; in the approximation described above, we expect that the secular motion of e, cr evolves along these lines.

The six pictures are drawn for decreasing values o f the inclination. In the first one (i = 8 °) we have one stable equilibrium point with librating orbits around it. For some of these orbits the angle ~r shows a libration around 0, for others it can be considered as a circulating angle. Indeed all the orbits have here the same nature as no separatrix is present. This shows the great difficulty that one may have trying to give an interpretation to the numerical simulations, as orbits with the same nature may have an apparently completely different behaviour with respect to the motion of the variable tr.

Picture 2 (i = 6.5 °) is drawn for a critical value of K , which corresponds to the presence o f an angular critical point for tr = rc The equilibrium point at tr = 0 is moved to a higher value of the eccentricity.

In picture 3 (i = 5.5 °) we have the presence of three equil ibrium points: one at tr = 0, which is stable, and two at tr = zr, o f which one is a saddle and one a center. In this case we have the presence of a separatrix. We will call " l ibrator" an orbit that librates around the stable equilibrium at tr ---- 0, " inner circulator" an orbit which librates around the center at tr ---- 7r (even if tr may appear as a librating variable around re), and "outer circulator" an orbit which circulates outside of the separatfix. One can easily compute that the "libration" and the "outer circulation" are clockwise (in the coordinates e, - a ) .

For decreasing values o f the inclination, the stable equilibrium at a ---- 0 and the saddle point migrate to higher values o f the eccentricity, while the center at tr = rc approaches the circular orbit. This is shown in the last three pictures.

An analogous scenario of evolution o f the dynamics with respect to the variation


i =8.0 i=6.5

I .2) 9.8)

t~5.5 i=5.0

.9) 8.5)

4.0 "=3.0

~ (7.9) :7.2)

Fig . 1. The v6 resonance for a = 2.08 A.U. The s t ra ight l ine on each c i rc le des igna tes the poin ts (e, cr = 0).


of one particular parameter has been pointed out by Henrard and Lemaitre (1983) for the 1/2 mean motion resonance. We refer to their work for a more detailed discussion.

It is interesting now to say a few words about the relationship with the results obtained by the local theory presented in our previous paper. As a first step such a theory is able to localize the resonance by looking at the proper free frequency vh, given by K0. This is analogous to localize the extremal point of K0 - z/gh0 P

as a function of the eccentricity. Since K0 is independent on or, this gives, on a picture as those illustrated above, one circle of equilibrium points. According to the theorem of Poincar6 (see Poincar6 (1892)), this circle has to be destroyed when the angular part K1 (or) is considered. Then, in our theory, we expect the presence of two equilibrium points, one at cr = 0 and the other at cr = 7r, of which one has to be stable, and the other unstable. These two points are related to the saddle and to the center (at cr = 0) shown on the pictures drawn for i < 6.5 °. The theory foresees also that the eccentricities of these two orbits has to be slightly different because of the contribution of the term K1 in the final determination of their location. Moreover the librating orbits around the stable equilibrium point are expected to have a banana-shape form, if observed in polar coordinates. These two results can be easily seen in Fig. 1. The local perturbation approach developed in our previous paper is adequate to study these two points.

However, at the end of Section 4 of our previous paper, we point out that something else must be present in the picture, although not properly described by our theory. This is related to the fact that the function KI in the action- angle variables adopted, is not analytic at e = 0. As a consequence the term OKI/OP + O-K1 ~O J, which appears in the final equation for the localization of the equilibrium points, is divergent for e ~ 0. Then, on the opposite side of the stable equilibrium point, such equation can have only 0 solutions or 2 solutions. Cases where we have 2 solutions are shown in the pictures with i < 6.5 °, and the second solution is related to the existence of the stable equilibrium point at the center of the inner circulators. The local theory of the previous paper is not adequate to study in detail this point: it is too close to the singularity of the coordinate system used, and the assumption on the order of magnitude of some quantities should be modified. Conversely a case where no solution exists is that showed in the picture with i = 8 °. The transition between the situation where two solutions are present to that where no solution exists, is given by the case were the two solutions are coincident. This case is shown in the second picture (i = 6.5°).

We come now to the comparison of our results with those obtained by Yoshikawa. As a matter of fact, a comparison is not straightforward. Indeed in the paper of Yoshikawa the pictures of secular motion are given in cartesian coordinates e, or; moreover Yoshikawa plots the level curves of the energy over the surface given by the condition i =const., which is an approximation of our choice J =const., while we plot the level curves of J over the surface given by K =constant. Anyway,


although the comparison can be just qualitative, we can conclude that the results are roughly equivalent. For a discussion of the numerical simulations performed for this value of the semi major axis, we then refer to Yoshikawa's paper.

3.2. DYNAMICS AT 2.60 A.U.

We have performed the same kind of study of the v6 resonance also for a value of the semi major axis of the asteroid equal to 0.5 times that of Jupiter (2.60 A.U.). The difference is that, for this value of a, the v6 resonance is present at a much larger inclination (,,~ 20°), and so some of the approximations adopted in the previous case are no longer adequate.

The local expansion of the Hamiltonian K---0 with respect to Z up to degree 2, in order to achieve a simple computation of the trajectory 7 becomes worse and worse with increasing inclination. This happens because, at high inclination, the variation of Z on the cycle -~ is too large, so that the error (which is of order of ~3) becomes relevant. In this case, then, we have adopted the method of the local expansion only for the orbits of K0 with initial condition (at g = 0) e < 0.2. For e > 0.2 a different method has been adopted: as K0 is an integrable Hamiltonian, we look for the trajectory 7 as a level curve of the "energy" K0 itself. This can be done by a Newton-Raphson method, for instance, and does not require so many computations of K0. We recall that each of these computations implies the numerical evaluation of a double integral: this explains why this algorithm is much slower than the previous one. Anyway the required computing time is still reasonable, and this method has the advantage to be correct at all orders in e and i.

The second approximation adopted at a = 2.08 A.U. was to consider J , instead of ,P, as a constant of motion. For low inclination the difference is negligible, as pointed out earlier, but not in our present case. However, tests performed by computing both J and J~ along the axis cr = 0 or <r = 7r have revealed that the difference between the results is just quantitative (for example the region of libration turns out to be somewhat narrower if it is computed through J~), but never qualitative (see Fig. 2 for an example). As the aim now is to discuss the essential features of the resonant dynamics, and not the secular quantitative behaviour of one single asteroid, we have decided, nevertheless, to assume again J as constant of motion, thus saving a lot of computer time.

The results obtained are shown in Figure 3, which is exactly the analogous of Figure 1 for a value a = 2.6 A.U. As one sees, the essential features of the dynamics at a = 2.08 and a = 2.60 A.U. are the same.

This result is surprising. Indeed it is completely different from that reported in Yoshikawa's work. Yoshikawa found that the saddle point is placed (when present) at <r = 0 and the center of libration at cr = 7r: consequently all our pictures appear to be rotated by 180 degrees!

In a situation like this the role of numerical simulations becomes really crucial. As a matter of fact it is not easy to discriminate between the two theories with


04

XT

"~T

(D

x:r

o,,

co

oo

co

rx_

09

inner c i r c . ~ . - ~ .

- - '".. inner circ.

t

s / : / /.."

i /

, / /

. / -

/ "

-0 .4 -0 .2 0.0 0.2 0.4 0.6

e cos(a)

Fig. 2. The value of J (dashed line) and of J' (dotted line) on the axis a = ~r, a = 0, on the level surface of the energy with a circular orbit of 18.8 degrees of inclination. From left to right the first maximum corresponds to the location of the saddle point, the minimum to the stable equilibrium point at the center of the inner circulators, and the last maximum to the stable point with banana-shape librators. For what concerns the location of the equilibrium points, the results obtained by considering j , or J are then in good agreement. More important quantitative differences concern the amplitude of the regions of libration and circulation which can be deduced by looking at the curves above and assuming J' or J as constants of motion. For example, in the case J' =const. the region of inner circulation ranges from -0.5 to 0.2; in the case J =const. the same region ranges from -0.45 to 0.05.

a s ingle numerical s imulation. Indeed many trajectories could be interpreted as cons is tent with both the descriptions, as w e wil l see. So, as w e want to determine the real pos i t ion o f the saddle point, w e have to l o o k for s o m e particular trajectory, l ike, for example , a banana-shape librating orbit, which must be opposi te to it.

3.3. NUMERICAL SIMULATIONS AT 2 .60 A.U.

The first numerical s imulat ion that w e consider is that o f two real asteroids, Vinifera and Tina, which have been s h o w n to be in secular resonance by Ch. Froeschl6 and H. Schol l (1987b). We report here two pictures about Vinifera's mot ion taken from


(33.d) 33.0)

18 5 18.8 "= .

.8) :32.5)

18.0 "=17.5

I .1) :31.8)

Fig. 3. The v6 resonance for a --- 2 .6 A .U . The straight l ine on each circle des ignates the points (e , cr = 0) .


0 rID CO

,W ca

v

~a o

Q

MODEL 9 BODY. I0 (759) VINIFERA [ I

- 1

i . , . .

.I : " ¢ . . ' : : s ~ '~ I * " - - " " i f ' ' ' . ~ " : i o • . • • ' • , ; * " • • t d • , ~ ' , . : . , - . .~.~ • .~ ~ . : : . : ' , . . . ' J

:~ : ~ " , : ~ ' ; " . ~ - . : i , ' : : ~ "'; : ' 1 - - o • . . i i • • . o • " 1 . r o • I ~ . . , . •

; "~ ;,~ :" '.~ "~, ,: ' ~ . i . ' ".~ i • -~ ~ ~ "J "; d ~" ~ ~ ~ ~ ~ i,~ ~ '.. -,!d ;~ :~ • ~ ~ , "~ ',l

. . . . . ~ ~ ¢ F .," ~ ' "

(a) , , , I , ' ' , 1

5 10 TIME (I0 ~ YEARS)

MODEL 9 BODY 10 i

[ I -0.2 0

( " / 5 9 ) VINIFERA i

(b) I

0.2

Fig. 4. Numerical simulations on 759 Vinifera. On the left (4a) ~ - ~ s is given as a function of time; on the right (4b) the motion is illustrated in polar coordinates e, uJ - c o s . (Ch. Froeschld and H. Scholl, 1987b).

the above quoted paper; the case of Tina appears to be analogous. In Figure 4a the difference between the longitude of perihelion of Vinifera and that of Saturn is reported as a function of time, for 1 million years. A libration around 180 degrees, with a period of about 50 thousand years is evident. This libration is meaningless for our purposes. Indeed this is due to the fact that the circulation of the perihelion of Saturn is not a linear function of the time; #6 is just its average frequency, and if we plot the quantity ~ s - g 6 t we will obtain a graphic almost identical to that of Fig. 4a, with oscillations of the same amplitude and period. The theory exposed in our previous paper shows that the critical angle (r, which has a dynamical meaning, is

- #6t, not ~ - ~s . The oscillations shown in Fig. 4a must then be considered as a noise. Analogously the banana-shape form of the orbit, shown in polar coordinates in Fig. 4b is just due to the noise.

If we analyze Fig. 4a on a longer time scale (106 years), we see that w - c o s

has no significative evolution: there is an oscillation (with period of about 800,000 years) of maximal amplitude of 15 degrees. We can conclude, then, that Vinifera is placed almost exactly in a (stable) equilibrium point (of eccentricity ,-, 0.2). But is this equilibrium point the center of the inner circulators, or the center of the banana-shape librators? The numerical simulation is not able to tell us the answer. So neither the results of Yoshikawa nor our results are contradicted, at this stage, by the experiments.

As we have not found numerical simulations in the literature enabling us to show where the saddle point is placed (we will speak later of Yoshikawa experiments) we have decided to perform new numerical experiments. Our program computes the motion of the Sun-Jupiter-Satum system together with the motion of the asteroid. The integrator adopted is RA15, designed by Everhart. For a better analysis, we


(a) (b)

Fig. 5. Numerical simulations for a ----- 2.6 A.U.; the polar coordinates are the eccentricity e and the critical angle tr; e is ranging from 0 to .9. The initial conditions are e = .35, tr = 0 and i = 26 ° in (a) and 25.5 in (b).

have plotted the results in polar coordinates e and ~ - #6t, after having filtered them of all the oscillations with period less than 5 x 105 years, in order to take in evidence the secular motion associated to 0-. The digital filtering of the data has been performed by using a suitable program designed by M. Carpino, whose principles are explained in Carpino et al. (1987).

In Figure 5 we report two significative results. In Fig. 5a an orbit computed with initial conditions of the asteroid tr = 0, g = 0, a = 2.6A.U., e = .35, i = 26 °, A = 0. The planets are placed at their perihelia, which are aligned. The eccentricity of Saturn is maximal and that of Jupiter minimal. The obtained orbit shows a libration around an equilibrium point placed at cr = 0 and e ,,~ .45 with a period of about 7 x 106 years. The shape of this orbit is surely not circular, but looks like a sort of "D". This is indeed the shape of the librating orbits which are closest to the equilibrium point, as shown by our theoretical computations presented in Fig. 3. If this is true, then, going further from the equilibrium point, the orbit should assume the classical banana-shape form.

To verify this hypothesis, we have then performed a second numerical integrations, changing only the value in the initial inclination (to 25.5°). By doing so, from a theoretical point of view, we change the surface of "energy". This is analogous to passing from one picture of Fig. 3 to the following. The stable equilibrium point then moves to a higher eccentricity, and so the orbit that we are simulating now turns out to be further from the equilibrium point than the previous one. This new simulation is shown in Fig. 5b, and the banana-shape is this time evident. The maximal eccentricity reached during the libration is also larger, exceeding 0.6.

In both cases the libration is clockwise, as foreseen by the theory.

T H E M A I N S E C U L A R R E S O N A N C E S IN T H E A S T E R O I D B E L T 183

0 . 9 ' ' I ' ' I ' ' I ' '

- ,---t

-,-.4

(1)

0 . 6

0 . 3

, ; . . ~

~ . o , •

~.o.

'°'~°i

~!~[~-,. ~.;. -"

,~ ' . . . .

" ' ~ -',, . " . . " , : • " " : ' ~ " ..*-. ~ .T~ . , . ." , . . . . m.'.." "'~ "t- .'" • "'F?'~.;~'.~",:. "~" ~:" .~-~ o ~ 71o o ~ . # ~ o~ " o . "., ".o o'~B oo o ". ° Oo i;.~

• ".~_ .~Z o'~o" "''I'." ~ • "~ " "°I 6 ".~'o r. ~ , , Z ~ . ~ . . . . . . . . . . . . , , . , ~ . , . , a I.... . . . . . ~ ' ~ _1" -'-~'~,,~.,7." -~-" -'--":'.~r('~L--. -~ ' , . --,r-'." " ". -.

• . . ~ ~ , ~ : ~ 2 . . . . ' - .- . ..-.,,~- --,;.~;,- . ,.

• -.~o • .'Z .t~,,.~;, , •

O.O ' , I l i I I t I I i

0 90 180 270 360

6o - ¢ a S (degrees) Fig. 6. The evolution of the eccentricity as a function of ~ - ~ s for a fictitious asteroid. (Yoshikawa, 1987). The continuous line, drawn by us, indicates the evolution of the secular motion.

Curiously, another experimental indication that the saddle point should be at tr = rc can be found in Yoshikawa's paper. We report one o f his results in Figure 6. Unfortunately the results are not plotted in polar coordinates, and the critical angle used is w - o 's , so that there is a great dispersion of the points due to the "noise" , as already discussed with Fig. 4.

In Fig. 6 the continuous line drawn by us shows the evolution of the orbit. The initial conditions for the asteroid are e ,-, 0.3 and w - o,s = 0. The asteroid starts to librate clockwise around 0, the eccentricity increases up to --, 0.6, and then it becomes a clockwise circulator, of high eccentricity. Yoshikawa himsel f writes that this behaviour "is not predicted by the analytical model. Since the inclination of this body becomes too high and the amplitude o f the inclination variation is too large, the accuracy of this model cannot be maintained". Conversely, such a behaviour can be easily interpreted with the aid of the pictures o f Fig. 3. In that case the asteroid would start as a banana-shape librator around tr = 0 (which is clockwise), it would pass the separatrix somewhere and finally become an outer circulator (still clockwise) that reaches its maximal eccentricity again when or = 0.

There is a possible misunderstanding that may occur while reading different papers on the secular resonances. Yoshikawa and Froeschl6 and Scholl often write in the papers about their numerical simulations that the inclination is approximately constant during the secular evolution. On the contrary, looking at the pictures o f

184

o 25.0 .=J

¢0 ¢-.

.~ 2 0 . 0

~ 15.0

ALESSANDRO MORBIDELLI AND JACQUES HENRARD

I ! !

0 200000 400000 600000

Time ( y e a r )

0 . 8 ' ' '

0 .,-I t . 0 4 . I J "

t.) ~ V I I ' ~ ' '" ' l ' - ' 1 ~

0.2 m i I

0 2DO000 400000 600000

Time ( y e a r ) Fig. 7. The evolution of the inclination and of the eccentricity as functions of time (Yoshikawa, 1987). The oscillations of shortest period are associated to the circulation of g. The proper eccentricity and the proper inclination (as defined by Williams) correspond to the minimum and the maximum reached, respectively, by the eccentricity and the inclination during these oscillations.

Fig. 3, one sees that the proper inclination of the surface, on which the motion occurs, varies very much passing from e = 0 to e ----- .6. So we expect an evolution of the proper inclination together with the proper eccentricity. The contradiction here is just apparent, and the explanation is in the definition of "proper inclination". In our previous paper the "proper inclination" and the "proper eccentricity" were defined as the inclination and the eccentricity reached when g = 0 during one cycle 3' described by K0 (whose period is about 8000 years). This corresponds to the maximum of the inclination and the minimum of the eccentricity during the cycle. This choice has been done as, in this way, the "proper" eccentricity and inclination determine univocally the action variables P and J . For a better explanation of the apparent contradiction, we report in Figure 7 two pictures of Yoshikawa where the evolution of the inclination and of the eccentricity is separately shown as a function of the time. The oscillations of shortest period are those given by the cycle 7- If we consider the "proper inclination", namely the maximal point of each of these oscillations, we have a variation from 20 ° to more than 25 ° together with the variation of the eccentricity from 0.2 to ~ 0.5, which confirms our results. On the other hand, Yoshikawa, Froeschl6 and Scholl refer themselves to the average inclination, which is really almost constant.

O9

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

C )

c5

c o

o~

i I i I I I

0.)

0

.4

I r I I I I I I ' I v 6

Co

I I t I l I I I I I I I I I I I I

-0.0 0.2 -0.0 0.2 0.~ 0.6

e e

185

Fig. 8. The value of J as a function of e at tr = 90 °. On the left (8a) the computation concerns the u5 resonance, on the right (8b) the v6.

4 . D y n a m i c s i n t h e u s s e c u l a r r e s o n a n c e

In this section we come to the problem of the dynamics in the u5 secular resonance. Up to now, this problem has been scarcely explored, because of the high inclination which cannot be handled with accuracy by truncated models. In our case, as the theory has been developed with a particular care not to loose accuracy for large inclination or eccentricity, we could hope to be able to study the u5 resonance along the lines fol lowed for the u6. We will see that, unfortunately, this is not the case.

We have started our analysis o f the u5 resonance by using the same analytic model illustrated in the previous section. We have computed, for a semi major axis equal to 2.6 A.U., the action J as a function o f e for a value of ~r = 90 ° on different level surfaces of the energy. This has been done as, at tr = 90 °, K1 (9 - 0) is equal to zero, as well as < K1 >¢ . So, in this case, J and J~ are equal, and the computat ion is fast and precise. The value o f J has been computed with the same algorithm adopted for the u6 resonance at a = 2.6 for e > 0.2 (as explained in the previous section), which is correct, in principle, for any value of i. In Figure 8a we report the results obtained on the level surface o f the energy characterized by the orbit e ---- 0, i ---- 31 o. In Figure 8b an analogous picture for the u6 resonance is shown (on the surface of energy of the correspondent model characterized by e = 0, i = 19°). The points where the function J(e) has a maximum or a minimum, give the locations of the resonances as described by the Hamiltonian function K0 alone. As one sees, the resonance u5 occurs at a minimum of J(e), while the u6 occurs at a maximum. This is a very important qualitative difference between the two resonances. It means that the banana-shape libration around the stable equil ibrium point is counter clockwise in the u5 resonance, while it is clockwise in the u6. The result obtained for a ---- 2.08 A.U. is analogous.

To get the complete picture o f the resonant dynamics, as we did in Fig. 1 and Fig.

1 8 6 ALESSANDRO MORBIDELLI AND JACQUES HENRARD

3 for the 116 resonance, we should study the analytic model for all the values of a. In this case, the computation of the more accurate constant J~, instead of J , would be necessary. Indeed we have verified that the function < K1 >¢, considered as a function of the inclination for a fixed value of e and ~r, changes sign at a value i0(e) which is very close to the location of the resonance itself. So, the difference between studying the dynamics through the computation of J~ or through the computation of J (which, we recall, is equivalent to considering < Ki >¢ or K1 (¢ = 0)) can be really dramatic. This is indeed the case at a = 2.6 A.U., where, considered as functions of cr on the level surface of the energy considered in Fig. 8a, J(cr) has a minimum for cr = 0 while J~(~r) has a maximum. This means that in the first case we foresee the existence of the saddle point at cr = rr, in the second at cr = 0. Then, the picture we get by drawing the level lines of J~, is analogous to one of those of Figure 3, turned by 180 degrees, and with a much narrower banana-shape region of libration.

The question now is whether or not this is a good picture of the real dynamics. Indeed the model considered for these computations has been obtained by retaining only the terms K0 and K1 of the full secular Hamiltonian (see (2)). In particular

~4 we have neglected the quadratic terms in the masses, namely ~-{X2, K0} which

65 ~5 plays a role analogous to Ko, and T{X3, K0} + ~-{X2, K1 }, which contains terms in cos o" as K1. These terms, in principle, give small contributions to the location of the resonance and to the width of the angular part (provided that we are not too close to a mean motion resonance), but in this case, where the location of the resonance and of the change of sign of < K1 >¢ are very close, they can easily reverse their relative position, changing completely the dynamical picture.

As we have already explained in the previous paper discussing the problems related to the precise location of the secular resonances, we are not able to compute the quadratic terms in the masses in a practical way which is correct for all the values of the inclination. The numerical approach, which, in principle, would allow to avoid any expansion in series o f / and e, is impracticable in the reality. So, in order to have at least an estimation of the quadratic terms in the masses, we have followed the classical approach of the series expansion. Starting from the Hamiltonian e2Ko + e3K1 expanded up to the terms of degree 4 in the inclination and 5 in the eccentricity, as given by a program designed by S. Effinier, we have computed X2

C4 ~5 ~5 and X3 and then the poisson brackets 2-{X2, K0} and T{X3, K0} + ~-{X2, K1}. This has been done by using the algebraic series manipulator MINIMS designed by M. Moons. Finally we have dropped all the harmonics in the argument of perihelion g and in the mean longitudes of the asteroid and of the planet. This is a very rough estimate, even if the final result contains all the terms up to i 4. Indeed an inclination of 30 ° is equal to ,,0 1/2 in radians, and so the convergence in terms of i )- is very slow. The not encouraging results are shown in figure 9. The first picture is drawn for e = 0.1, the second for e = 0.2. The three dashed regions are related to the main mean motion resonances 3/1, 2/5 and 2/1 from left to right. The theory is not

T H E M A I N S E C U L A R R E S O N A N C E S IN T H E A S T E R O I D B E L T 187

o "-.T

. x

o

' ' ' ' i ! i ! i i : ' i : ' : : : ' " 1 ' , " ' : : : : " . . . . . . " . : 1 : : . , " : : : : : .

i i i i i : " i i i i i : ~ ' " i i i i i L;,5 . . . . . . . . . . . . ~d . . . . . .

. . . . . ii!i ii!!!F : iiii: - ~,2"'--,,3 " . i i : : i - . [ 2 , ~ i ! " . - ~ . ~ i i i b

. . . . . . ~.iiii.- _L - i ! i i i - - ( a ) . . . . . . . . . . . I . . . . " ' 1 " ' ; , " , ' " , " I , , 5 : " : - ,

2 . 0 2 . 5 3 . 0

a ( A . U . )

(e = . 1 ] x

(:D x a-

x

c ) oo

J " : : ! : : : ' : : ' : : : ' . I 'i " : : : : ' .

- d , 5 I : 2 i i " : ! ! ! i i " ' [ i i i i i - -

-- . . . . . . . ~ _ - - - ~ ! : : : : : 3 : : : : : ' : ~ . : : : : : - . - i - - - - ' ~ ' ~ : :- : - . ~i i i i - ~ , ,~i i i i ~----------~' :l i i i i -

~,~,~ ° : ? P i i i - : : i ! i i - - ~ - - : ~ i i i i - . . . . . . - . ~ . . . . . Z . . . . ;

" : : : : : . " : : : : : . • . . . . ._

. . . . I . . . . " ' 1 " ' ; , " ' ; ' ' , " I , ,

2 . 0 2 . 5 3 . 0

a(A.U.)

(e =.2) x

Fig. 9. The location of the ~'5 resonance and of the line where the angular part in tr changes the sign, on a plane (a, proper i), for a proper eccentricity of 0.1 (9a, above) and 0.2 (9b, below). Solid lines 1, 2, and 3 give the location of the v5 resonance computed by considering, respectively, the quadratic term in the masses up to i 4, i 2, or neglecting it at all. The dashed lines 4 and 5 designate the change of sign of the angular part in tr, computed by considering the quadratic term in the masses up to i 4 o r neglecting it.

adequate to study the neighbourhood o f a mean motion resonance, and that is why these regions are considered as blank zones. The amplitude of these zones is given by the size of the grid used for the computations, which is 0.1 A.U. for the semi major axis, and so does not have a direct dynamical meaning (for the details of this kind of computat ion see the previous paper). The other mean motion resonances, like the 4/1 or the 7/3, for example, have a narrower region of influence, which can not be detected by our grid. So we have passed over them drawing continuous line. The presence of the 4/1 resonance, is however responsible o f the bumps on the lines at a = 2.08 A.U.


In Figure 9 the solid lines 1, 2 and 3 represent the location of the v5 resonance. The first one is obtained by considering all the components of the quadratic terms of the masses up to i4, the second by considering only the components up to i2, and the third not considering the quadratic terms in the masses at all. The dashed lines 4 and 5 localize the inclination where the coefficient of the angular part in cos ~r changes the sign. Line 4 is computed by retaining all the components of the quadratic terms in the masses up to degree 4 in i and line 5 by considering K1 alone.

As one clearly sees, the agreement among all these computations is good for a < 2.1 A.U.: this means that the quadratic term in the masses gives a really negligible contribution. The resonance is located at an inclination which is lower than the inclination related to the change of sign, and so we expect the presence of the main stable equilibrium point at tr = 0 and the saddle at cr = 7r. On the contrary, for a > 2.1 A.U., and especially for a > 2.6 A.U., the lines are well separated. This indicates that the computation is very sensitive to the degree of truncation of the series, and does not give reliable results. In particular the relative position of the v5 resonance and of the change of sign of the angular part cannot be determined. Moreover, even if the quadratic terms in the masses were computed with the highest accuracy, their large size makes us expect that also terms of higher order in the masses, coming from the Lie algorithm of normalization with respect to mean anomalies, would play a crucial role in the determination of this relative position.

So, at the present state, we are not able to foresee which is the dynamics at the v5 resonance. We can only qualitatively outline the three possible scenarios:

- (i) the resonance is located at a lower inclination than the change of sign of the angular part in cr of the secular Hamiltonian. The saddle point is at ~r = 7r (when it exists) and the equilibrium point, center of the banana-shape librating orbit, is at tr ---- 0. The pictures then are similar to those reported in Fig. 3.

- (ii) the resonance is located at a larger inclination that the change of sign. The dynamical pictures are then turned by 180 degrees with respect to those of Fig. 3.

- (iii) part of the level surface of the "energy" K is in the region where the coefficient of the angular part in cos tr is positive, and part in the region where it is negative. A possible consequence is that the two stable equilibrium points have the same value of tr. A picture like this should be similar to one of Fig. 3, with the difference that the centre of the inner circulators is also placed at tr ----- 0. The 180 degrees rotation of this picture represents another possibility.

We leave to the numerical simulations the task of determining which of these scenarios is the real one. Anyway, the fact that amplitude of the angular part of the Hamiltonian (7) is close to zero should imply that the banana-shape resonant region is very narrow.


(a) i

MODEL 7

(b) i i

MODEL 7 BODY 19 !

i

o ~I,1 J

simulations of fictitious Fig. 10. Numerical asteroids (Ch.

BODY 22

0

in the v5 resonance. Froeschld and H. Scholl, 1986b). The initial conditions are: (on the left) a = 2.6 A.U., e = 0.14, i = 31.87, w - w j = 359°; (on the r igh t )a = 2.6 A.U., e = 0.12, i = 30.43, w - wj = 259 °

4.1. NUMERICAL SIMULATIONS OF THE Z/5 RESONANCE

We come now to analyze some numerical experiments on the dynamics in the v5 resonance performed by Ch. Froeschld and H. Scholl and published in theft papers 1986b and 1990. In Figures 10a and 10b we report their results about two fictitious asteroids with semi major axis equal to 2.6 A.U. The data are presented in cartesian coordinates 9 J = [2(1 - x/(1 - e 2 ) ] 1/2 C O S ( ~ - - ~ J ) and ff12 J =

[2(1 -- V/(1 - e2)] U2 sin (~ - ~ j ) on a square of side (-0.4,0.4). ~ j is the longitude of perihelion of Jupiter. The two asteroids are librating, along orbits with a well defined banana-shape, around an equilibrium point at (r = 0. The eccentricity of these equilibrium points should not be larger than 0.2. The librations are counter clockwise, as foreseen by the theory. These simulations show unequivocally that, at least for e ~ 0.15 and a ,-~ 2.6 A.U. the v5 resonance is located at a lower inclination than the change of sign of the angular part in tr.

In the graphs reported in Fig. 10, superimposed to the banana-shape libration, one can see many cycles of almost circular shape. These cycles are given by the variation of the eccentricity and by the non uniform precession rate of the longitude of perihelion, which are associated to the trajectory 7 described by K0. So, if we want to see the secular variations of the "proper eccentricity", we have to look only at the points of minimal eccentricity of each of these circular cycles. In this way we get something which is comparable with what is described by the theory.

The first remark that we can do looking at the graphs in this way is that the amplitude of the variation of the proper eccentricity is, in the two cases, relatively small. This means that the amplitude of the angular part in tr of the secular Hamiltonian must be very small; so, we should not be far from the point where it changes the sign. To investigate this, let us compare, for example Figures


o

(945) BARCELONA

la? 0

,l,i 'w

~ o

(945) BARCELONA

(b)

0

Fig. 11. Numerical simulation of the real asteroid 945 Barcelona. On the left (1 la) the evolution is clockwise, on the right (1 lb) counter clockwise. (Ch. Froeschl6 and H. Scholl, 1990).

10a and 10b with Figure 5b, which are drawn for the same value of the semi major axis. In Figure 5b we see that, for a libration of 90 degrees of amplitude in tr, the amplitude of the variation of the eccentricity is about 0.35. In Figure 10a, where the libration in tr has also about 90 degrees in amplitude, the associated variation of the proper eccentricity is ,,~ 0.05. In Fig. 10b, which shows the maximal possible libration (180°), the variation of the proper eccentricity is 0.1.

This fact is even more evident in the simulation of the motion of the real asteroid 945 Barcelona (a ,-~ 2.63 A.U.), as performed by Froeschl6 and Scholl (1990). Their results are reproduced in Figure 1 la and 1 lb. In Fig. 1 l a t r evolves from ,-~ 90 ° to ,-~ - 9 0 °, in Fig 1 lb from ,,~ - 9 0 ° to ~ 90 °. Here the variation of the proper eccentricity associated to the libration is so negligible that the authors have been compelled to show the two pictures separately.

Another interesting simulation reported in Froeschl6 and Scholl, 1990, is that of the real asteroid 1580 Betulia (a ,,~ 2.1979A.U., i ,,~ 52°e ,,~ 0.6). In Figure 12a we report the evolution of F - F j as a function of the time for 5 million years. The striking feature is that this angle stays in a neighbourhood of 180 degrees for a period of more than two million years, then makes one clockwise circulation in about one million years, and this behaviour seems to be periodical. The evolution of the eccentricity as a function o f f - F j is shown in Fig. 12b.

We can try to hazard an interpretation of these strange phenomena on the basis of the three possible scenarios illustrated above. The basic idea is that, as the evolution o f F - F j is very slow at 180 degrees, the asteroid should be close to an equilibrium point there. Looking at Fig. 12 we can conclude that such an equilibrium point is unstable. So, the behaviour of Betulia should be that of an asteroid approaching asymptotically the saddle point along an homoclinic curve. As the circulation is clockwise, we may conclude that Betulia evolves along the inner homoclinic curve

T H E M A I N S E C U L A R R E S O N A N C E S IN T H E A S T E R O I D BELT 191

o (158o) Bm'uu~

: t • I ,

(a) |

o I , ~ | , - 4 0 - 2 0

TI~E (tO ~ YEARS)

1580) BE'rULIA

~ . . _ , ~ ? ~

I:~+::~:~ Im~W/ - -~ , '~ ,:,.~-~...'.J ,, ;:.-. 4~.~.2:~ .x. ,. r-.~,.:..:.~

(b} , , . . ! ! . .

0 90 180 270 360 - ~ (degrees)

Fig. 12. Numerical simulation of the real asteroid 1580 Betulia. (Ch. Froeschld and H. Scholl, 1990).

and so it can be classified as an inner circulator. Indeed we have seen (and verified on the numerical simulations) that the banana-shape libration is counter clockwise, and from this we can deduce the direction of the circulation on the two critical curves. We look now at Figure 12b, where the evolution of the eccentricity is shown. We see that the eccentricity (the "proper" one as well as the average) is larger when w - wj = 0 than when ~ - ~ j = 7r (the difference is about 0.25). So the equilibrium point which is the center of the inner circulations should be placed at a = 0, as well as the center of the banana-shape librations, which must be opposite to the saddle point. The conclusion then seems to be: the value of o- related to the two stable equilibrium points is the same. Betulia is a candidate to show that the scenario (iii) is really possible.

5. A g l o b a l m o d e l f o r a 1/1 s e c u l a r r e s o n a n c e o f t h e n o d e

In this and in the following section we study another secular resonance which seems to play an important role in the explanation of the distribution of the asteroids: the Vl6. In order to achieve a global description of the dynamics connected to this resonance, we need first to develop a global model, analogous to that used for the 1.s 6 and v5 resonances.

Starting from the Hamiltonian (2) and assuming that we are in the region where # is a circulating variable, we introduce again the action-angle variables P, h', J and ¢,defined in (3). As the critical argument is now of the kind h' + ¢ - A86 (remember that h ~ + ¢ is the opposite of the proper longitude of node), we change to variables

P - - P ~' = h ' + ¢ (12) S = J - P ¢ = ¢ ,

192

and then to

P = P Ats6 : A s 6 + P

ALESSANDRO MORBIDELLI AND JACQUES HENRARD

O" = ~ l - - A,S 6

As6 = As6 • (13)

In this way, after having dropped all the harmonics in As~ with j # 6 and Ag h , we get

K = sZK0(P, S) - 6211s6 P --]- 63K1 (0 ", ~b, P, S) , (14)

which is the analogous of (7); (14) is the Hamiltonian of our global model for a 1 to 1 secular resonance of the node.

From now onwards we apply exactly the same techniques we used to study the model (7). On the manifold given by the surface of energy of K sectioned by the hyperplane ~ = 0, we choose now as polar coordinates the proper inclination i and the proper resonant angle a. For each point i, ¢r the condition K(e , i, a, ~ = 0) =const., allows us to determine the proper eccentricity e; knowing now e and i for ~ = g = 0, the study of the integrable Hamiltonian K0 allows us to determine S. The action S is a second integral of (14) up to order 6, whose level curves describe the secular motion in the variables i and a. A better approximation of the second integral, which is a constant of motion up to order 6 2 is given by

S' = S(~) + ~--~¢1 (Kl('~',o')- < "K1 >~ (o')) , (15)

where, again, < KI >,p denotes the average of K1 with respect to ~b.

6. Dynamics in the//16 secular resonance

We have applied the model designed in the previous section to get the global picture of the dynamics in the/-/16 resonance. For the same reasons explained while studying the I/6 for a semi major axis equal to 2.08 A.U., we have used the approach of the local expansion of K0 in order to compute the cycle .y and the value of S. Namely we have expanded K---0 in Taylor series up to degree 2 in S = S - So (where So is given by the values of e and i when # = ~b = 0), and in Fourier series up to the harmonic cos 49. Moreover we have used the level lines of S to describe the secular motion (instead of those of St).

The results are illustrated in Figure 13 and Figure 14. In Figure 13 the dynamics is shown on 6 different level surfaces of the energy K in (14). Each surface is characterized by the value of the eccentricity of the orbit with inclination equal to zero, which is reported in the upper right comer of each picture. Each picture has been drawn in polar coordinates i, -cr. i is zero at the center and equal to 30 degrees on the limit border. The straight line designates the points (i, cr = 0). The semi major axis is the same for all the pictures, and equal to 0.38 times that of Jupiter (1.976 A.U.).


.1 125

.15 • 175

.2 . 23

Fig. 13. The 1/16 resonance for a = .38 times that o f Jupiter (1.976 A.U.).


. 3 8 . 3 9

• z;O . ,::;1

Fig. 14. The v16 resonance for different values of the semi major axis indicated on the graph (the unit is the semi major axis of Jupiter). The proper eccentricity of the planar (i = 0) orbit is 0.1

If we forget for a moment the meaning of the coordinates, these pictures are completely analogous to those of Fig. 1 and 3. Indeed there are no principal differences between a secular resonance of the node and of the perihelion. That is why we have been able to study the two cases together in the perturbation study presented in our previous paper.

In Figure 14 we show the evolution of the dynamics in the v16 resonance through a variation of the semi major axis. Each of the four pictures has been drawn for a different semi major axis, indicated in the upper right comer. The level surface of the energy K chosen for each value of the semi major axis is always that correspondent to a plane orbit with eccentricity equal to 0.1. As one sees, the evolutions of the dynamics through the variation of the eccentricity or through the variation of the semi major axis are analogous. The eccentricity and the semi major axis are then both transverse parameters with respect to the resonance.


We have compared our results with those of Nakai and Kinoshita (1985). Again the comparison is not so straightforward, as they draw the level lines of the energy on a surface given by e =const. However our results are in qualitative agreement. A possible misunderstanding may have its origin in the choice of the coordinates. Nakai and Kinoshita use the inclination and the difference between the longitudes of node of the asteroid and of Jupiter. In Figures 13 and 14 we use the inclination and - t r . This last is equivalent to the difference between the longitudes of node of the asteroid and of Saturn. As the difference between the longitudes of the nodes of Jupiter and Saturn is of 180 degrees, our pictures are turned by 180 ° with respect to those of Nakai and Kinoshita.

A detailed comparison between the analytic theory and the numerical simulations has already been done in Nakai and Kinoshita (1985) and in Froeschl6 and Scholl (1986b). We refer to those articles for a discussion of the numerical results.

6.1. THE 1/16 RESONANCE FOR LIBRATING ARGUMENT OF PERIHELION

In our previous paper, studying the location of the secular resonances in the asteroid belt, we pointed out that the 1/16 resonance may occur also for asteroids with librating argument of perihelion and we have plotted, in a a, e, i diagram, the location of the equilibrium points of the argument of perihelion where the proper frequency of node is equal to s6. Moreover we have developed a local theory of the secular resonant dynamics in case of librating argument of perihelion.

The existence of the vl6 resonance for orbits with librating argument of perihelion was not known before, to our knowledge, and no studies about the main features of its dynamics exist. For this reason we have applied our local theory to this case achieving the following results:

- (i) the main stable equilibrium point, center of the banana-shape librating orbits, is placed at tr = 180 ° (i.e. the difference between the longitude of node of the asteroid and of Jupiter is 0), while, in case of circulating #, it is placed at tr = 0, as shown in Fig. 13 and 14.

- (ii) in the variables i, ~9 - ~gj (where ~gj is the longitude of node of Jupiter), the libration is clockwise, while it is counter clockwise when g is a circulating variable.

The numerical simulation of the real asteroid 2335 James can be explained in this context. This simulation has been performed by Ch. Froeschl6 and H. Scholl and reveal a fantastic complexity of behaviours which make it a real jewel among the numerical experiments on the dynamics in secular resonances. The results have been partially published in Froeschl6 and Scholl, 1988a. A complete discussion of the behaviour of James would be very long. So we will dedicate to it a full further paper (see Froeschl6, Morbidelli, Scholl (1991)).


7. Conclusions

In this paper we have derived, from the general perturbation study exposed in a previous paper, two analytical models for a global description of the dynamics associated to a secular resonance of order 1 involving, respectively, the longitude of perihelion, and the longitude of node. These models have been studied with semi-numerical techniques, in order to explore the three main secular resonances I/6,//5 and//16.

Our results on the v6 resonance confirm those of Yoshikawa only at low inclination. Conversely, at high inclination, our pictures of the global dynamics are rotated by 180 ° with respect to those of Yoshikawa. Numerical simulations performed by us and by Yoshikawa himself seem to indicate that our results are more accurate.

For what regards the //5 resonance, we have pointed out the critical vicinity of the surface on which the coefficient of cos (tr) in the averaged Hamiltonian vanishes. This means that the resonant region is narrow, and that the location of the stable equilibrium may change from tr = 0 to ~r = 180. Because of the poor precision of the estimation of the quadratic terms in the masses, we have not been able to determine the relative position of such a surface with respect to the location of the//5 resonance. Nevertheless, we have outlined three possible scenarios of the resonant dynamics, which guided us in the analysis and interpretation of the numerical experiments performed by Ch. Froeschl6 and H. Scholl.

Finally, our results on the//16 resonance are in qualitative agreement with those of Nakai and Kinoshita. Moreover, we have performed a local study of the //16 resonance close to the equilibrium point of the argument of perihelion. We have shown, in this way, that the stable and the unstable equilibrium points exchange their position with respect to the case where the argument of perihelion is a circulating angle.

Acknowledgements

We wish to thank M. Carpino for his precious technical aid. Some discussions with Ch. Froeschl6 and H. Scholl were very useful, and we appreciate their cooperation. The first author is grateful to the Commission of the European Communities, whose grant has largely contributed to the realization of this work.

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the main secular resonances ν6, vs and ν16 in the asteroid belt

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