Chin.Astron.Astrophgs.(1989)13/4,408-416 @ Pergamon Press plc. a translation of Printed in Great Britain Acta Astron.Sin. (1989)30/2,11?-125 0275-1062/89$10.00+.00

THE SPACE DISTRIBUTION AND ORBITAL

RESONANCES OF ASTJiROIDS1

LIU Lin LIAO Xin-hao Department of Astronomy, Nbnjing Uni versi ty

Received 1988 September 8

Abstract We have made a detailed numerical analysis of the various orbital resonances of the asteroids and examined the variation in the distribution of trajectories near the equilibrium points in the phase plane, and hence arrived at a preliminary explanation of the features (gaps and clusters) in their distribution in space.

1. INTRODUCTION

The space distribution of the asteroids of the solar system has the following two remarkable features:

1. Concentration between the orbits of Mars and Jupiter, forming the band of asteroids. The orbital semi-major axes of Mars and Jupiter are 1.524 and 5.204AU while those of the asteroids are mostly between 2 and 3.5 AU.

2. There are many gaps in the distribution of the mean motion n, at values n/nl=p/q=3/1, 5/2, 7/3, 2/l, . . . (nl is the mean motion of Jupiter). Contrariwise, there are also some clusters in the outer part, e.g., at p/q= 3/2 (the Hilda group).

The first feature is easily understood. For an asteroid cannot be too close to Mars or Jupiter, otherwise it will be captured by either. The second feature is more striking: why is it that there exist gaps (the Kirkwood gaps) between 2.0 and 3.5AU and concentrations in the outer parts nearer to Jupiter ? Many explanations have been offered, but none can be considered to be satisfactory Ill.

Although the Kirkwood gaps may be closely connected with the formation and evolution of the solar system, it is still useful to investigate the evolution of the asteroid orbits from a purely gravitational point of view in order to reveal the cause of the gaps. And this will necessarily involve the problem of orbital resonances at the conmensurabilities n/n1 = p/q, The effect of orbital resonance on the evolution of the asteroid orbits and the differences between different values of p/q are two main topics here, We have obtained some useful results as regards both topics.

1 Program supported by the National Natural Science Foundation

Resonant Asteroids 409

2. DYNAMICAL MODEL

It is obviously convenient, for our purpose, to adopt the restricted problem of three bodies consisting of the Sun, Jupiter and an asteroid with a negligible mass. Perturbations by Mars need not be considered. Considering that the orbital inclinations of the asteroids are not large and taking into account the results of studies in Refs. [2. 31. it is appropriate to use the plane model. Hence we have simplified-the problem to the planar restricted problem of three bodies. The hamiltonian for this simple model is

F- L+ R( Gl. c, 0, M; 01). (11

where a, e, 6, Mare the Kepler elements of the asteroid orbit and the o1 appearing in the disturbing function R is any orbital element of Jupiter and is a known function of time t. Equation (1) is a non-autonomous (non-stationary) system. Since we assume the orbit of Jupiter to be an invariable ellipse, only one of ol involves t, namely, the mean anomaly 4. Hence equation (1) can be reduced into a Sdimensional autonomous system, or generalized into a 6-dimensional hamiltonian system.

Adopting the variables customarily used in the study of orbital resonance, [3], L, e, f, 6, we have

F - F(L, c, 7, 3, .\I,) 1 - _ + R(t, c, 1. d, IV,).

XqL)’ (2)

The variables E and f are the main resonance parameters, they are defined by

z - ,/:/q, 1 - qM - phi, + ~(3 - 13,). (31

We used the units defined in Ref.[S], in which the mean motions of the nsteroid and Jupiter are, respectively,

n q a-3/2 = p-3 9

q = (1 t n)*’ q const. (41

ml being the mass of Jupiter in solar units.

The expression (21 represents a complicated nonlinear dynamic system and it is difficult to study it directly, for the present available mathematical tools are not enough. Accordingly, following the usual way of tackling the perturbed problem of two bodies we divide (2) into two portions,

where

F - F;,+ AF. (51

Fir - Fo(2) + F,(Z,r)cosl.

&l(Z) - A- XqZ)’

+ R.(Z, cl.

F,(L, r)corl - R.(z, c, I).

IFiIIIFol -O(g).

(6)

(7)

(8) (9)

410 LIU Lin & LIAO Xin-hao

Here, e= e(r), & and Rr are, respectively, the non-periodic terms and the main resonance term of the original disturbing function R, the small parameter E characterizing their relative size. AF represent the omitted small terms.

The hamiltonian defined by (6) is an autonomous system with just one degree of freedom in (L, f). Even though Fo( L) and Fl( L,e) are still rather complicated, they are integrable. In the E-1 phase plane, for different values of p/q, the equilibrium points and distribution of trajectories near them are clear [4]. This model, though simple, does reflect the basic properties of orbital resonance. Near the equilibrium point at the centre of the phase plane we have closed trajectories, and for the asteroids in the resonance band, their semi-major axis a (or r) and resonance angle 1 (a kind of conjunction longitude) can vary only within certain limits, so we have a situation similar to the libration of simple pendulum. Some authors have labelled the orbital resonance so determined as the Ideal Resonance [5]. However, the basic features of orbital resonance implied in the integrable expression (6) can only predict that there should be a concentration of asteroids for each p/q resonance band, and cannot say anything about their essential differences (that is, the degree of stability of the corresponding trajectories), and thus fails to explain the gaps in the asteroid band. Therefore, the key question is how the addition of the A~term will change the distribution of the trajectories. At this point, the intention of adopting the dynamical model (5) should be clear: it is the same as when we study the effect of perturbation on the Kepler orbits in the perturbed problem of two bodies.

AF can be taken in different ways , and some exploration was made in Ref.[2]. The preliminary results indicated the main factor that changes the resonance features of the basic system (6) is the term involving the eccentricity of Jupiter q, the other periodic terms will not change those features, unless at the edge of the resonance band (which case is just what we shall not consider). Accordingly, to simplify the mathematical model as much as possible, people have often used the averaged model [6] in place of the actual model (2), by restricting AF to non-periodic and resonant terms (including higher frequency resonances, that is kf terms with kk2). Even so, the number of terms to be taken remains a problem, otherwise the averaged model cannot adequately reflect the evolution of the resonant orbits, and answer the question posed at the beginning of this paper. We therefore first made an extensive numerical study of the actual model with the following three aims in view:

1. To understand the appearance of the basic dynamical features of the reference system (6) in the actual motion and the effect of orbital resonance in the evloution of the asteroid orbit.

2. To understand the differences shown in the actual motion of the basic features of systems (6) for different values of p/q, and to provide the necessary numerical information for the explanation of the space distribution.

3. To use as standard of comparison for the averaged model, and to provide a basis for simplifying the mathematical model, which in turn provides a possiblity of revealing more rigorously the stability character of the resonance state structure of different values of p/q.

Resonant Asteroids 411

3. PRELIMINARY RESULTS OF NUMERICAL STUDY

The above p/q resonance problem, the equations of motion corresponding to (2) are

d--f, 0 - (Z e 1 5 M,)‘, f - Cfi fi f3 k f5Y.

Here, T represents transpose. The five components of f are

(10)

(111

where

’

(12)

x, y and xl, y1 are the coordinates of the asteroid and Jupiter. We have x

I- 0

- u(cosE -c) .P-tu Y

r2 - ( I I2 - x’ + y’, r: - x: + Y:, A* - (2 - r,): + (Y - Ya)‘,

E-e&E-M,

G - d.(l - ~‘1. (13)

The calculation of rl is similar to that of r. The expressions for the other auxiliary quantities are

H--+(q&2 -p)H,, K----(qJE - P)KI, D

HI-&(wE+r),K, --~(l++nE~ (14)

The set of equations (10) is a 5-dimensional autonomous (or stationary) system. We made a numerical study of it in Ref.[Bl and gave the main features of the resonant motion, the variation of the resonance angle 1. However, just using this one variation to describe the evolution of the resonance orbit is not ideal. We therefore calculated at the same time the Lyapunov characteristic number (LCN), [Tl, which shows more clearly the asymptotic behaviour of the system (10). Since we are concerned here only with the orbital stability of the asteroid motion, we need only

412 LIU Lin & LIAO Xin-hao

calculate the largest LCN, and not the entire spectrum. Taking advantage of a feature of system (10) we can simplify it into 8 4-dimensional systen, namely.,

where

k=- f, w - J(#)W. 115)

@f J-- aa’

The specific form of the matrix (11). The corresponding largest expression:

(161

J(o) can be dervied from the expressions LCN, A, is given by the following

(17)

Obviously, k’(t) can only be found from a numerical solution of (15). In the calculation, ortho-normalization was applied and the initial value of wft) was taken to be

Wo) - ( 1 0 0 0): (18)

TABLE 1 Uelevant Orbital Data of 24 Asteroids (Real and Fictitious)

PI9

312

413

3/3

NO. z c &‘)

1362 0.7889 0.3640 -81.6 23.6 f94.5

1362‘4 0.7889 0.3640 0.0 23.6 i38.8

03 0.4378 0.14SO 24.8 271.7 *35.7

153A 0.4378 0.1450 91.0 271.7 *94.1

153B 0.4378 0.1450 110.0 271.7 $=112.8

153c 0.4373 0.1450 133.0 271.7 *135.5

1530 0.4378 0.1450 147.0 271.7 *150.0

1269 0.4330 0.0950 -52.2 159.7 sircul

1269A 0.4360 0.0950 --52.2 159.7 f52.4

12698 0.4360 0.0950 -91.0 159.7 f93.0

1941 0.4380 0.2770 -33.3 365.4 *37.4

194lh 0.4380 0.2770 -91.0 365.4 *93.8

19418 0.4480 0.2770 -110.0 365.4 f113.1

194lC 0.4380 0.2770 -122.0 365.4 l 125.0

279 0.30267 0.0067 -37.2 155.9 f75.4

239h 0.30267 0.0067 -80.0 155.9 f85.2

279B 0.30267 0.0700 -37.2 155.9 i39.t

2796 0.30267 0.0700 -95.0 155.9 i95.1

2790 0.30267 0.1000 -37.2 155.9 f37.5

2798 0.30267 0.1000 -100.0 155.9 *100.0

887 0.6931 0.5550 45.4 99.8 f134.7

887h 0.6931 0.5550 143.0 99.8 f37.0

1381 0.6914 0.1azo lf9.4 20.8 f120.2

138lA 0.6914 0.1820 180.0 20.8 f33.6

Resonant Asteroids 413

Then, the value of X calculated according to (17) will be the largest LCN, to be denoted by Xl. If X1=0, then the orbit is stable; if 11 >O, then the orbit is unstable. The relation between 11 and the orbital resonance will be clarified in the results of the actual calculations below.

We selected a set of asteroids (real and fictitious) in the four low order resonance bands, p/q= 2/l, 3/2, 4/3 and 3/l, where resonance effects are the strongest and calculated for each their orbital evolution and the largest LCN 11. The relevant data and the results are given in TABLES l-3.

The fictitious asteroids are distinguished by having a letter after their numbers. TABLE 1 lists the initial values of L, e, f, ii used in the integrations and AI is the range of variation in 1 given by the reference system (6). The Af values explain our selection of the fictitious asteroids: --they were chosen so as to provide a more uniform distribution in space, covering both deep and shallow resonances, for a more effective investigation of our problem.

TABLE 2 Orbital Changes in One Million Years of 24 Asteroids

PI4 -- NO.

I362

I362A

153

153A

1536

l53C

l53D

1269

1269A

126YB

1941

I94IA

19418

l941C

279

279A

2790

279c

279D

279E

687

B87A

1381

1381A

AL

0.7853-U.nU18

0.7887-0.7984

0.4355-0.1361

0.4332-U.4IU5

0.4324-0.1114

0.1312-0.1127

0.4325-0.1413

0.4347-0.43SY

0.4330-0.4407

0.43++-0.4393

0.1316-0.4422

0.3011-0.3051

0.3001-0.3057

0.3018-0.30~6

0.3017-U.JU16

0.6840-0.7021

0.6908-0.6927

0.6907-0.6927

PC W) 0.2372-0.3754 *113.1

0.3344-0.4053 *74.4

0.1036-0.2473 *40.9

U.UYI4-U.2735 *99.s

0.0890-0.2813 kll7.6

0.0783-0.2949 5142.0

r--+1 circul

0.0230-0.2206 f125.5

0.0745-0.2268 f63.3

O.OIZi-0.2351 *Ion.3

0.1325-0.2988 f70.2

U.1052-U.3160 fl31.5

r-1 circul

e--r* circul

u.o044-0.1385 f91.1

o.oooo-0.l301 circul

0.0594-0.1920 f48.3

e-1 sired

0.0870-0.2266 k46.4

r-1 circul

0.3822-0.8356 cirsul

r-1 cirsul

0.0829-0.1828 circul

0.0846-0.1628 circul

1,

0 0 0 0 0 + + 0 0 0 0 ? + + 0 0 0 + 0 + + + 0 0

TABLE 2 gives the results of the actual model (2). A comparison of the values of AI given in this TABLE and in TABLE 1 shows that the characteristic librating of 1 (the main resonance feature) begins to disappear in the shallow resonance regions , while in the deep resonance region there is almost no change in the distribution of trajectories. The situation is not quite the same for different resonances. In the 3/2

414 LIlJ Lin 8 LIAO Xin-hao

case, libration is kept over a wide zone, while in the 3/l case, the opposite is true, libration disappears almost throughout. For the 2/l case, a small modification in the mathematical model will also make the libration of No. 1362 vanish, so that the zone where libration persists is probably small also. For the 4/3 case, if the eccentricity is small then the libration is small and is hard to persist, while if the eccentricity is large then because of its nearness to Jupiter, resonance will lead to e-1 1, then not only is libration soon destroyed, the orbit will be unstable and the asteroid will leave the resonance band altogether.

The last column of TABLE 2 gives the values of Xl, with t representing 1, >O. Comparing with the values of Af in the preceding2column we see easily that, when libration is preserved, we have Xl=0 , the orbit is stable and the asteroid is kept in the resonance band; when libration disappears and circulation takes place then we have either 11 >O or X1=0, the former is accompanied by e+l and the asteroid wandering off the resonance band, the latter still corresponds to a stable situation where although the asteroid is not in the resonance band, it still remains in its vicinity.

TABLE 3 Values of 1, at Various Times

P/4 --

-

No.

1362

1362A

153

153A

1538

I53C

153D

1269

1269A

12690

1941

1941A

194lB

1941c

279

279A

2,Y"

279C

279D

279E

887

887A

1381

1381.4

lO’(yr) 0.0110

0.0062

O."liO

0.0131

O.Ul25

0.0120

0.2

0.0140

0.0113

0.0130

0.0118

0.0119

0.0098

0.1

0.0127

"."155

"."LZ‘

0.2

u.0112

0.3

0.0100

0.0105

0.0157

U.01‘7

10'

0.0016

0.0016

0.00*4

0.0017

0.0017

0.0014

0.2

0.0019

O.OOL5

0.0016

0.0014

0.0020

0.0031

0.0019

0.0020

U.OUlL

0.2

0.0015

cl.3

0.0014

0.0008

0.0022

0.0021

-r

_-

-

LO'

0.0002

0.0002

0.0001

0.0002

0.0""2

0.0005

0.2

0.0002

0.0002

0.0002

0.0002

o.oo,o

0.01

0.0002

O.""O?

O.OO"2

0.2

0.0002

0.3

0.0012

0.0006

0.0002

0.0002

5X10'

0.00004

0.00004

0.00003

0.00003

0.00004

0.00020

0.00004

0.00003

0.00004

0.00003

0.00090

0.00004

O.OOQO5

O.OOUO4

0.00003

0.0015

0.0005

0.00005

0.00005

_-

-

10'

0.00002

0.00002

0.00002

0.00002

0.0"""2

0.0002Q

0.00002

0.00002

0.00002

0.00002

0.0007L

0.0000?.

O.O""O2.

".""""T.

0.00002

0.0020

0.00002

O.OUOO?

TABLE 3 shows the convergence of 11 in our calculations. In the case where libration is kept, the Al >O convergence is fast, and the contrary

2 except in the case of No.153C. -translator

Resonant Asteroids 415

case of h i=O is also very obvious. For the intermediate case where 1 passes from libration to circulation, the convergence of 11 is slow; this corresponds to the system being close to the critical bifurcation point.

The above statements can be summarized as follows:

1. Libration in f corresponds to 11 =O. This means that resonance exerts a kind of stabilizing effect on the orbit evolution.

2. The 3/l resonance band inside the asteroid belt has an unstable structure, and the 2/l band at the edge is also not quite stable. When there are other perturbing factors present, the resonance characteristics will not be preserved even in the deep resonance zone and hence gaps will be formed. On the other hand, the 3/2 resonance band outside the main belt towards Jupiter has a more stable structure, hence a concentration of asteroids. As for the 4/3 case, because of the requirement of e being small, the resonance effect is not strong and the original ideal resonance structure will be difficult to maintain.

3. For the asteroids in the interior resonance bands like the 3/l and 2/l bands, even if the resonance characteristic vanishes, the orbit is stable (that is, X,=0); whereas for the 3/2 and 4/3 cases, once that happens we shall have Xi >O and e+l and the asteroid not only leaves the resonance band, but will wander away from the Sun. This explains why there are many asteroids around the 3/l and 2/l gaps forming the V-shape frequency distribution. These gaps are surrounded by high order resonances with resonance effects so weak that they are indistinguishable from the case of no resonance, hence as long as the orbit is stable it will remain in the corresponding space and no further gaps will form. Similarly, at the 3/2 resonance, the concentration of the asteroids will be an isolated group. And these are just the characteristic features of the space distribution of asteroids.

4. CONCLUDING REMARKS

Strictly speaking our numerical study has not solved completely the problem of structural stability of the orbital resonance bands, for it has not given a criterion to distinqguish any essential difference between different p/q resonances. However, when the mathematical model is complicated and the mathematical tools are inadequate, the calculation of both the orbital evolution and the LCN can still lead to some useful results. In one aspect, such an exercise shows that it is possible to explain the gaps and concentrations of the asteroids in space from a purely gravitational calculation of the stability character of the resonance bands. Low order resonances can cause large changes in the orbit making the asteroid leave the resonance band and form gaps in the distribution; they can also stabilize the orbit through libration and cause the asteroid to remain in the resonance band, forming a concentration. The key question is, for a given value of p/q, whether the structure is stable or unstable.

To solve this problem of stability strictly by theory, we must simplify the mathematical model. The first place we can simplify is the averaging of (5), making it not an autonomous system. This procedure is convenient even from the viewpoint of numerical study, for it means that we can at

416 LIU Lin & LIAO Xin-hao

least increase the steplength of integration and make the problem amenable to more widely available machines. Further simplification can be made on the basis of the averaged model, Of course, while we want to have a simple model we must not change the properties of the original dynamical system, that is, we mat preserve the orbital resonance properties of the latter and any essential differences between different values of p/q. In this respect, the numerical results on the actual model given in this paper can be used as a check on the reliability of any simplified scheme. After a suitable simplification is found, it may be possible to make use of some of the results gained in recent studies of nonlinear dynamical systems to tackle the question of structural stability of orbital resonance.

REFERENCES

[l] Greenberg, R. and Scholl, H., Asteroids, (Univ.Arizona: Tucson), 1979, p.30.

[21 Liu, L. and Innanen, K.A., A.J. 90 (1985), 887. [31 LIU Lin and LIAO Xin-hao, Scientia Sinica, Ser. A., 12 (1964) 149. (41 Liu, L., et al., A.J., 98 (1985), 877. [5J Garfinkel, B., A.J., 71 (19661, 657. [6] Schubart, J., Smithsonian Astrophys. Obs. Special Report 149 (1964). 171 Benettin, G. et al., Meccanica 15 (1980), 9.