A Perturbative Treatment of Motion Near the 3/1 Commensurability

ICARUS 63, 272--289 11985)

A Perturbative Treatment of Motion near the 3/1 Commensurability

JACK WISDOM Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute ~1" Technology, Cambridge, Massachusetts 02139

Received January 26, 1985: revised May 10, 1985

A semianalytic perturbation theory for motion near the 3/1 commensurability in the planar elliptic restricted three-body problem is presented. The predictions of the theory are in good agreement with the features found on numerically generated surfaces of section; a global understanding of the phase space is achieved. The unusual features of the motion discovered by J. Wisdom (1982, Astron. J. 87, 577-593; 1983a, Icarus 56, 51-74) are explained. The principal cause of the large chaotic zone near the 3/1 commensurability is identified, and a new criterion for the existence of large-scale chaotic behavior is presented. ~, 1985 Academic Press. Inc.

I. INTRODUCTION in a large chaotic zone near the 3/1 commensurability. When the effects of phase That there are gaps in the distribution of mixing are removed the boundary of the 3/1 asteroids near low-order mean-motion Kirkwood gap is sharp and corresponds commensurabilities with Jupiter was dis- precisely to the boundary of the 3/I chaotic covered in the last century, yet there is still zone, within the errors of the asteroid or- no generally accepted explanation of their bital elements. Furthermore, the essential origin. A significant obstacle is that the features of this picture have been verified long-term behavior of trajectories near with conventional numerical integrations of commensurabilities in the elliptic-restricted the exact equations of motion. These fea- three-body problem is not understood. tures include the unusual behavior of the There is no analytic theory for this motion eccentricity and the existence and width of and the numerical experiments have been the chaotic zone. Thus the possibility that too limited (by expense) to give significant these unusual features are artifacts of the insight. The numerical problems were mapping approximation is ruled out (Wis- largely overcome with the development of a dom, 1983a). Since the time scale for the new "mapping" method for studying the removal of material on these trajectories is long-term behavior near commensurabili- short compared to the age of the Solar Sys- ties that is approximately 1000 times faster tem (Wetherill, 1975), the existence of the than previously used methods (Wisdom, 3/1 Kirkwood gap is entirely explained by 1982). When followed with the mapping, this mechanism. More recently, 1 have trajectories near the 3/1 commensurability shown by direct numerical integration that display some very curious behavior, viz, chaotic trajectories can actually reach such they may spend as much as a million years large eccentricities that they cross Earth's with eccentricities below 0. l, and then sud- orbit as well as the orbit of Mars (Wisdom, denly jump to eccentricities larger than 0.3. 1985). Thus this chaotic zone not only ac- If this behavior were common the 3/I counts for the 3/! Kirkwood gap, but is Kirkwood gap could be swept clear by probably the long sought route by which close encounters and collisions with Mars. meteorites are transported from the aster- I have shown that in fact such behavior is oid belt to Earth (Wisdom, 1983b, 1985; common and is characteristic of trajectories Wetherill, 1985). 272 0l) 19-1035/85 $3.00 Copyright ~t~ 1985 by Academic Press. Inc. Atl rights of reproduction in any form reserved. MOTION NEAR THE 3/1 COMMENSURABILITY 273

While it is now known that there is a served. Section V compares the resulting large and significant chaotic zone near the solutions to numerically generated surfaces 3/1 commensurability, it is dissatisfying of section, and some qualitative features of that there is still no analytic theory for this the solutions are described. The principal behavior. There are still many unanswered cause of the chaotic behavior is identified in questions. Why does the chaotic zone have Section VI, and a simple new criterion for the shape it has? Why do chaotic trajecto- the existence of large-scale chaotic behav- ries seem to have a high-eccentricity mode ior is presented. The limitations of the ap- and a low-eccentricity mode? Why does proximations used in this paper are dis- quasiperiodic libration lead to such large cussed in Section VII, and a summary is variations in eccentricity? Why do there ap- presented in Section VIII. pear to be three distinct zones of quasiperiodic libration on the representative plane II. REVIEW OF THE QUALITATIVE [cf. Fig. 6 in Wisdom (1983a) and Fig. 4 in BEHAVIOR Murray and Fox (1984)]? What governs the In traditional descriptions of the phenom- limit to the eccentricity growth in the planar enon of libration the resonant argument o- elliptic problem? In fact, why is there a cha- = pl + q~ - (p + q)lj plays a central role. otic zone at all? Is it due to the short-period Here I is the mean longitude, tb is the longi- terms in the disturbing function, the over- tude of perihelion, and p and q are integers. lap of second-order resonances, or some- The subscript J refers to Jupiter. Near reso- thing more complicated? It is still an open nance tr varies slowly compared to the question whether or not chaotic behavior mean motions of the asteroid and Jupiter. played a role in the formation of the other Those terms in the disturbing function Kirkwood gaps, but before the even more which contain o- produce large effects rela- complicated dynamics of these resonances tive to the short-period terms whose contri- is challenged it seems wise to try to under- butions tend to average to zero. A good ap- stand the dynamics of the 3/1 resonance proximation to the motion is obtain by more thoroughly and answer these lingering considering only those terms in the disturb- questions. That is the purpose of this paper. ing function which contain o-. This ap- Most "numerical integrations" in this pa- proach was introduced by Poincar6 (1902). per use the mapping. It has already been For the circular restricted problem the re- adequately shown that the mapping com- sulting system of equations is integrable, pares well with the exact differential equa- and an extra integral may be identified. It is tions (Wisdom, 1983a; Murray and Fox, customary to plot several trajectories with 1984). The goal here is not to further dem- the same value of this extra integral to- onstrate this fact, but to understand the gether using e and cr as polar coordinates. very complicated behavior which has been Such plots of e and tr are often called Poin- found. In the next section, I review some of car6 diagrams. While the figures in Section the basic characteristics of the long-term IV bear some superficial resemblance to motion near the 3/1 commensurability. In Poincar6 diagrams, it is important to keep Section III, I describe the perturbative ap- in mind that they are in fact not related. In proach used in this paper, which involves the elliptic restricted problem considered the identification of two timescales in the here, the Poincar6 diagram is of limited use- long-period problem and the analytic solu- fulness. In the averaged circular problem tion of the motion on the shorter of these near the 3/1 commensurability o- either li- two time scales. The very-long-period be- brates about ~" or circulates, depending on havior is computed in Section IV under the the initial conditions. In the unaveraged cir- assumption that the action of the motion on cular problem there is in addition a small the shorter time scale is adiabatically con- chaotic zone whenever an infinite period 274 JACK WISDOM

shows a transition from low-eccentricity behavior to high-eccentricity behavior. Typical chaotic trajectories show bursts of low-eccentricity behavior and bursts of high-eccentricity behavior. Each mode of behavior has apparently random duration, and has been observed to last from essentially no time at all (less than a few thousand years) to several million years. The behavior of the eccentricity is correlated with the behavior of the resonant argument, o-. During the low-eccentricity mode the [) resonant argument circulates or librates, (] l()f] :2(][~ :~(',l~ but does not appear to be particularly co- herent. On the other hand, during the high- FIG. I. The usual resonant argumenl ~r : / + 20~ eccentricity peaks, cr librates about a 36 versus time measured in millennia for a typical cha- slowly varying center which goes through 7r otic trajectory. This trajectory has AH .... 4.22 x when e is at a maximum. 10 ~'. The semimajor axis for this same trajectory is shown in Fig. 3. The behavior of the trajectory separates libration from circula- semimajor axis is also correlated with the tion (Chirikov, 1979; Wisdom, 1982, resonant argument. During the high-eccen- 1983a). The behavior of o- in the elliptic tricity peaks the semimajor axis oscillates problem is much more complicated. An ex- back and forth across the resonance zone. ample of that behavior for a typical chaotic During the low-eccentricity mode the semi- trajectory is shown in Fig. 1. Sometimes o- major axis sometimes crosses the reso- circulates and sometimes it librates. Some- nance center, but sometimes stays on only times it librates about a slowly varying one side for a period of time. During those center. periods when the semimajor axis remains Figure 2 shows the eccentricity versus on one side of the resonance the resonant time for the same chaotic trajectory. It argument (r circulates. Thus short numeri-

I :2 f~:-' [

:! 5) 1

;11 [ L' ,tf~ 'i l] :) (] :~H] J. .... ]()(] i_'()() H'~() (:i ] ()() :~()() ::l ()() t

FIG. 2. The eccentricity versus time (in millennia) FIG. 3. The semimajor axis (in AU) versus time (in for the same trajectory as in Fig. 1. millennia) for the same trajectory as in Figs. 1 and 2. MOTION NEAR THE 3/1 COMMENSURABILITY 275 cal integrations (less than 10,000 years) 1047.355 is the mass of Jupiter. Units are might identify the trajectory as a nonreso- chosen such that aj = 1 and Ij = t. In these nant, (r-circulating trajectory. units the period of Jupiter is 27r. The mo- menta L and p may be written in terms of the usual osculating elliptic elements: L = lI1. SOLUTION ON THE L1BRATION TIME and p = ~ [1 - (1 - e2)t/2], where SCALE a is the semimajor axis and e is the eccen- There are three natural time scales. The tricity. The conjugate coordinates are the shortest time scale is the orbital period (a mean longitude I and minus the longitude of few years). The intermediate time scale is perihelion co, respectively. The disturbing that of the libration of the resonant argu- functions are ment (typically a few hundred years). This time scale corresponds to the fastest oscil- Rsec = - 2pF - ejGV~p cos(co) lations visible in Figs. 1-3. The longest time scale is associated with the motion of and the longitude of perihelion (typically sev- Rres = 2pC cos(/ - 2co - 3/j) eral thousand years). This is the time scale for the slow evolution of the "guiding cen- + ejDN/~p cos(/ - to - 3/j) ters" of (r and e. The existence of several + eZE cos(/ - 3/j), well-separated time scales could simplify where those terms beyond second order in the analytic treatment, especially if the fast- the eccentricity are ignored. The coeffi- est oscillations could be approximated ana- cients are evaluated at the resonant value of lytically. In that case the longer period vari- the semimajor axis and have the values F = ations could be found by averaging the -0.2050694, G = 0.1987054, C = equations of motion over the fast oscilla- 0.8631579, D = -2.656407, and E = tions, and, in turn, the changes in the solu- 0.3629536. The eccentricity of Jupiter is tion for the fast oscillations could be found given its current value ej = 0.048. This under the assumption that the action of the Hamiltonian is a good representation of the fast oscillations is adiabatically conserved dynamics near the 3/1 resonance. during the slow evolution. Unfortunately, The analytic development is considera- the latter two time scales are not always bly simplified if the difference of mean lon- distinct. Nevertheless, in this section, I as- gitudes ~b = l - 3/j is used to describe the sume that the libration of the resonant argu- motion on the libration time scale rather ment is fast compared to the variation of than (r. Of course, ~b and o- are essentially the eccentricity and longitude of perihelion, the same when the variation of co is frozen. and analytic approximations to the motion The momentum conjugate to ¢h is • = L. At on the libration time scale are derived. the same time it is convenient to change to Those situations where this assumption the Poincar6 momentum x = N/~p cos(co) fails are identified in subsequent sections. and its conjugate coordinate y = ~pp The resonant Hamiltonian for the planar- sin(co). In these variables the Hamiltonian is elliptic restricted three-body problem is (see Wisdom, 1982) H - 2qb2 3~ + txF(x 2 + y2)

/4- 2L 2 /zRsec (p, co) + ejI.zGx - /z[C((x 2 - y2) + ejDx

-- /zRres(L, 1, p, co), + ej2E]cos ~b - I,z[C2xy + ejDy] sin ~b. (1) where Rs~c is the secular part of the disturbing function, Rres is the resonant part of the Equivalently, this may be rewritten in the disturbing function,/Zl = 1 -/x, and/x = 1/ form 276 JACK WISDOM

:27 quadratic terms, the Hamiltonian governing the variation of ~ and A is

1 H' = ~ o~A 2 + /*A cos ?,,

the ordinary pendulum Hamiltonian. It is helpful to keep in mind that the coefficient of the quadratic term a = -3/*j/q~,e~2 4 = - 12.98851 is negative. The form of the solutions depends on the value of H': for H' < -A the angle ,X circulates, for -A < H' < A the angle ), librates

0 about zero, and the separatrix has H' = (! If)t) :2{)0 :!~)iI -A. The analytic solutions for the pendulum are well known, so 1 do not present FIG. 4. The deviation of the angle 4, - l 3Is from their derivations. In the case of libration the guiding phase P(x, y) versus time (in millennia) tk)r the same trajectory as in Figs. 1-3. The slow drily has sin(2n- 1)ot been removed, and 4) P(x, y) now circulates or libra- = 4 ,• = (2n - 1)~[~ --- 7~)rrK'/K]" tes with varying frequency and amplitude about rr. where K(k) and K'(k) are the complete elliptic integrals of the first kind with modulus /xi k[. - ((/,A - H')/2/*A) I/2, and 60 = a-o0/ H - 2~ 2 3~ + /*F(x 2 + 3'2) + ej/*Gx (2K). The frequency of small-amplitude oscillations is o0 = (-e~/*A) I/2. The period of - /*A(x, y) cos(~ - P(x, y)), (2) these small-amplitude oscillations depends where on the eccentricity and longitude of perihelion through A, but is of order 150 years. A(x, y) = [(C(x 2 - y2) + ejDx + e~E) 2 The motion on the separatrix is + (2xyC + ejDy)q ~- ~, = 4tan-I(e ~0,) - 7r. and In the case of circulation 2xyC + ejDv tan P(x, y) = C(x2 _ y2) + ejDx + e]E ~, = ot + 9 ~ sin not - ~= n cosh(nrrK'/K)' In this last form the "guiding center" of the libration is evident. To illustrate this Fig. with o = rroc/K and oc = (-c~(/,A - H')/ 4 shows a plot of 4~ - P(x, y) for the same 2) v2. Here the elliptic modulus is kc = (2/*A/ trajectory used in Fig. 1. There is no longer (/*A -H')) 1/2. In each case the behavior of a slow drift, only a circulation or oscillation A may be determined through the relation of varying amplitude and frequency dMdt = uA. For completeness, about ~-.

At this point I assume that x and y are 2O0kl. A cn(o0t) frozen, and may thus be treated merely as O~ parameters in the equations of motion. De- fine h = 4~ - P(x, y) - 7r and, in this approx- in the case of libration, and imation, its conjugate momentum A qb - 2o C @ ..... where ~,-e~ = (/*~/3) I/3 is the center of A = -- dn(oct) OL the resonance defined by aH/a~ 0. Ex- panding H about ~e~ and keeping only the for circulation, where cn and dn are Jacobi MOTION NEAR THE 3/1 COMMENSURABILITY 277 elliptic functions. Th e elliptic modulus for dX _ ddp dP OP dx OP dy - aA each case was given above. For circulation, dt dt dt Ox dt Oy dt' an independent solution is obtained by mul- tiplying h and A by -1. and Several other properties of the pendulum dA dO are required in subsequent sections. For li- - /zA sin h. bration dt dt

cos h = 1 - 2~sn2(oJ0t) Since both dx/dt and dy/dt are proportional and to p~ the time scale for significant evolution ofx and y is proportional to 1/~. This is long sin h = 2kLSn(w0t)(1 -- ~snZ(oot)) j/z, compared to the period of small-amplitude while for circulation oscillations of h which is proportional to /~ i/2. Heuristically, only the average effect cos h = cn2(coct) - sn2(oct), of the rapidly oscillating terms containing and cos h and sin h will contribute to the large variations ofx and y. This suggests that ifx sin X = 2sn(oJct)cn(oct), and y are divided into long-period parts and short-period parts, x -- £ + ~: and y = f + ~7, where sn is another Jacobi elliptic function. the long-period variations The action is defined by the integral. may be determined from the equations

1 =- ~ Adh. d£ OA(£, y-) d-t = - 2uF~ - i~ --cgy (cos x) The action is the libration case is OP(£, y-) - uA(£, y-)-- 1 8 /za 1/2 O~ (sin h) = 7r a [E(kt.) - (1 - /~L)K(kL)], and and the action in the circulation case is dy aA(X, ~) I= 4 _~ i/2 E(kc) d---t = 2~F£ + ej~G + ~ 0£ (cos h) kc OP(£, y-) E(k) is the complete elliptic integral of the + ~A(£, y-) O~ (sin h), second kind. where the angle brackets indicate an average over one oscillation period of h. These iv. AVERAGING AND LONG-TERM averages are EVOLUTION 1 f[ 2E(k0 Using Hamiltonian (2) the complete (cos h) -= -~ cos hdt - K(ke~ 1 equations of motion are and dx OA = -21~Fy - ~ -~y cos h 1 ff d-7 (sin h) -= ~ , sin hdt = O, OP - ~A ~ sin h, for libration, and dy oa d---t = 2txFx + ejIxG + t~ -~x cos h (cos h) ~ ~ill' cos hdt OP 2E(kc) 2 + /xA -~x sin h, - k~K(kc~ + I k~ 278 JACK WISDOM and possible then to plot a number of trajectories of 2 and V with (roughly) the same (sin X) --- ~I f r sin Xdt O, value of H to get a global view of the nature of such trajectories. In so far as the result- for circulation. In the evaluation of these ing trajectories all have the same value of expressions the table of integrals of squares the resonant Hamiltonian, the resulting plot of Jacobi elliptic functions in Abramowitz is similar to the Poincar6 diagram. Keep in and Stegun (1970) is helpful. mind though that the period of oscillation The equations of motion for x and f are on the Poincar6 diagram is the libration pe- then riod, while here oscillations on the libration time scale have been removed by averaging (1£ OA to reveal a much longer period motion, dt - - 21aFfi - Ix ~ {cos X) (3a) Examples of these plots ~ versus £ for d~ OA given values of H are shown in Figs. 5-7. dt = 2~£f, + ixGel + tx~-f (cos X). (3b) These correspond, respectively, to curves 3-5 shown in Fig. 10 of Wisdom (1983a). Of course these equations are subject to the When A -- 0 the numerical values of H and constraint that as £ and f vary the elliptic H' differ by a constant term which was ig- modulus used in evaluating the expressions nored in writing H'. That constant term is A for (cos X) and (sin X) is chosen so that the - -35/3/x~/3/2. Defining AH ~= H - A, Figs. appropriate action remains invariant. An al- 5,6, and7 have AH--1.21 x 10 S, AH= ternate, more rigorous derivation of these 4.22 × 10 6 and AH = -3.04 x 10 ~', equations is given in the appendix. respectively. Each trajectory is followed by The advantage of the Poincar6 diagram is an ordinary numerical integration of the that it gives at a glance a complete view of equations of motion for £ and f given the motion for any given value of the extra above, subject to the constraint that the integral. For the circular restricted problem action lk~r the ,k motion keep its initial that extra integral is just the Hamiltonian with all terms which do not involve cr removed by averaging. That Hamiltonian has (3:] only one degree of freedom. In the unaveraged circular problem, that "integral" exhibits short-period variations. However, it still restricts the motion and in many cases it is probably a first approximation to a true integral. I call such quantities "quasi- V 0 0 integrals." Hamiltonian (2) is the analogous quasi-integral for the elliptic restricted problem. This Hamiltonian has two degrees of freedom and is not integrable. The simplicity of the Poincar6 diagram is thus lost. On the other hand, the fact that there are l) i/ often two time scales allows much of this simplicity to be restored. The equations of motion for £ and ~ are two dimensional, and FIG. 5. The guiding trajectories in the 2, ~ plane for must at least approximately conserve H. (It AH - - 1.21 × 10 ~. The shaded region is the "zone of uncertainty," where the action of the X oscillations is is not precisely conserved because of the no longer conserved. X circulates for guiding trajecto- adiabatic approximation and removal of the ries enclosed by the zone of uncertainty, and librates oscillations on the libration time scale.) It is about 7r for all other trajectories. MOTION NEAR THE 3/1 COMMENSURABILITY 279

0.2 is near 7r, for then the pendulum period tends to infinity. In terms of the elliptic moduli, infinite periods occur when k = 1. 0.1 This condition can be solved explicitly for the infinite period line ~(x-) for any given value of H. In particular, k = 1 occurs when H' = -/~. Substituting H' = AH -/zF(£ 2 -t- ~2) _ i.Laej.~ ' and squaring both sides yields a quadratic equation for ~2. The so- 0.1 lutions of this equation are shown as lines in the middle of the shaded regions. It is clear that the approximation will also break -0.2 down in some neighborhood of these 0.1 0.0 0.1 0.2 0.3 curves as well. The shaded regions represent this neighborhood, but its actual extent FIG. 6. The guiding trajectories for AH = -4.22 x 10-6. The zone of uncertainty is again shaded, and has remains undetermined. This region is called almost divided into two separate zones, h circulates the "zone of uncertainty." It is in this zone for guiding trajectories inside each lobe of the zone of that the actual trajectories can no longer be uncertainty, and librates about ~- for all others. followed by the averaging method. The zone of uncertainty separates regions value. The Bulirsch-Stoer algorithm with where h oscillates from those where it cir- relative accuracy e = 10- I' was used to per- culates. In each case h circulates in the re- form the integration. gion enclosed by the zone and librates in Now, the assumption of adiabatic invari- the region outside the zone. Trajectories ance of the action is only valid when the follow a guiding trajectory until they reach libration period is short compared to the the zone of uncertainty where they exhibit periods associated with the variations of £ some complicated motions until finally they and ~. Obviously, this assumption breaks reappear on some neighboring guiding tra- down when the amplitude of oscillation of h jectory (with a different action) and the process is repeated. 0.2 ' An example of the actual appearance of a chaotic trajectory in the x, y plane is shown

0.E I I I

0.1

y o.o

O.2 I -o.1 0.1 0.0 0.1 0.2 0.3

FIG. 7. The guiding trajectories for AH = -3.04 x --0.2 I I 1 10-6 . The zone of uncertainty has now divided into -0. 0.0 0.1 0.2 0.3 two separate zones. Still, h circulates for guiding tra- x jectories inside each zone of uncertainty; and librates FIG. 8. The actual appearance in the £, 37plane of the about ~r for all others. chaotic trajectory represented in Figs. 1-4. 280 JACK WISDOM in Fig. 8. This is the same trajectory that minimum of a. This section is similar to the was used in Figs. 1-4. However, the plot is section 1 use in this paper. complicated and the correspondence with Figure 4 shows that ~b regularly crosses the guiding trajectories (in particular Fig. 6) its guiding center P(x, y) + rr. To generate is not transparent. It is evident that an alter- surfaces of section, I plot the canonically nate method of comparison must be em- conjugate long-period variables x and y ployed. whenever ~b - P(x, y) crosses 7r in a positive sense. This section removes the libra- v. SURFACES OF SECTION tion variable 4) and its canonical momentum The fact that Hamiltonian (2) has only ~. Points on the resulting surface of section two degrees of freedom immediately sug- are in one to one correspondence with tra- gests that it be studied with surfaces of sec- jectories which pierce the section. The per- tion (see H6non and Heiles, 1964). It is not, turbation analysis shows that this condition however, immediately apparent what the for section is almost the same as a passing conditions for that section should be. One through a minimum. For the numerical choice might be to plot x and y whenever computations, I use the mapping. Strictly = ~res (i.e., a = are~), but I have illustrated speaking the mapping differs from Hamilto- how • avoids this value during much of the nian (2) by short-period terms and pos- low-eccentricity mode. Another possibility sesses too many degrees of freedom to gen- is to plot x and y for some particular value erate surfaces of section. However, this is of tr or 4~, but Fig. 1 reveals that no choice also true of the planar-elliptic problem. In of o- will adequately represent the high- this sense the mapping is closer to the origi- eccentricity spikes. Giffen (1973) chose to nal problem than the averaged Hamilto- plot a and o- versus e when a was at a maxi- nian, and in any case these short-period mum. Scholl and Froeschl6 (1974, 1975, terms are largely unimportant. Computa- 1976, 1977, 1981) followed Giffen in this tional economics dictates that I use the choice. This choice is satisfactory in that a mapping. The typical length of the "inte- regularly passes through a maximum during grations" used to produce the following both the low-eccentricity mode and the surfaces of sections was 10 7 years per cha- high-eccentricity mode. However, two fea- otic trajectory, an unpleasantly long time tures make this choice unsatisfactory. even for the numerical integration of the First, a point on such a surface of section averaged Hamiltonian (2). does not uniquely determine the trajectory The surfaces of section for the three val- to which it belongs. As a consequence in- ues of AH used in the last section are shown variant curves can cross one another! This in Figs. 9-11. Because the mapping is not can be seen by examining Fig. 2 in Froes- continuous in the time, I have plotted inter- chl6 and Scholl (1976). The two invariant polated values of x and y for each crossing. curves with initial eccentricities 0.18 and The overall correspondance with the per- 0.20 belong to a family of similar invariant turbation method is now evident. The two curves, yet the loops which they form on sets of figures do not quite superimpose be- the surface of section are not concentric. cause x and y are not equal to Y and )- when Two invariant curves with more nearly ~b crosses P + ~-. The differences between equal starting eccentricities would be these two sets of variables are computed in forced to cross. Also, this surface of sec- the appendix. The curves in Fig. 12 corre- tion mixes the long-period evolution with spond to guiding trajectories in Fig. 6, but the evolution on the iibration time scale. the correction from Y, ~ to x, y for 4~ = P + The long-period evolution is not illuminated 7r has been applied. The structure of each by such a plot. A much better choice would plot is now identical. The agreement is ex- have been to plot the long-period variables cellent in the other two cases as well, x and y (or e and d~) at every maximum or though the curves become distorted if they MOTION NEAR THE 3/1 COMMENSURABILITY 281

0.3 conjectured to be representative of the phase space in the sense that almost all trajectories would eventually assume these angles. It is interesting to examine whether or not this conjecture is correct for the three surfaces of section. All of the quasi- y 0.0 periodic trajectories fill invariant curves which cross the x axis, and all but those in the small island to the left of the origin cross the positive x axis. When ~ librates the guiding phase on this axis is such that it librates about either 0 or 7r. For x between

-0.3 ej(-D - ~/D 2 - 4CE)/(2C) = 0.01 and 0.2 0.0 0.2 0.4 ej(-D + ~/D 2 - 4CE)/(2C) = 0.14 the x guiding phase puts ~b near zero when y is FrG. 9. Surface of section for AH = -1.21 x 10-5. near 0, but outside this range ~b librates An interpolated point is plotted when th crosses P(x, y) about 7r near y = 0. Now consider the out- + rr in a positive sense. This figure corresponds to ermost invariant curves. These curves all Fig. 5. cross the positive x axis (~ -- 0) with tb librating near 7r. Since the libration frequency is in general incommensurate with come too close to the points where A = 0 the precession frequency the representative (see appendix). The quasiperiodic trajecto- phases are approached arbitrarily closely. ries are well represented by the adiabatic The innermost invariant curves also all approximation. This is also true of the cha- cross the positive x axis, but this time ~b otic trajectories when they are not near the circulates. Again, the generally incommen- zone of uncertainty. The chaotic zones are surate frequencies eventually give the rep- mainly filled with long arcs of dots which resentative angles. The same conclusion follow paths similar to those computed in follows for the quasiperiodic zones to the Section IV. This behavior is most apparent fight of the origin in Figs. 9 and 1 l, except in Fig. 9, which corresponds to Fig. 5. that in Fig. 9 two representative crossings Several qualitatively different types of quasiperiodic trajectories are present. In all the figures or librates and t~ circulates on the 0.2 outermost invariant curves, and both or and tb circulate on the invariant curves near the origin. In Figs. 9 and I l, there are quasipe- 0.1 riodic zones to the right of the quasiperiodic zone near the origin. In these quasiperiodic zones both or and t~ librate (about ~r y 0.0 and 0, respectively). Finally, there are small islands to the left of the origin in Figs. l0 and l l, and here again both or and tb -0.1 librate (about 70. The existence of these island explains the appearance of several zones of quasiperiodic libration in Fig. 6 in -0.2 Wisdom (1983a). All of these features are -0.1 0.0 0.1 0.2 0.3 predicted by the perturbation analysis. x In Wisdom (1983a) the particular set of FIG. 10. Surface of section for AH = -4.22 × 10-6. initial angles ~b = zr and dJ = ~bj = 0 were This figure corresponds to Fig. 6. 282 JACK WISDOM

0,2 ~ i i tes. For this zone ¢ librates about 0 near y = 0 and avoids values near 7r. Thus no points appear on the representative plane. 0.1 There is a forbidden region on the representative plane, but on the surface of section the chaotic zone is continuous.

y O0 Finally, Fig. 11 makes clear the odd "in- termittent" behavior shown in Fig. 13 of Wisdom (1983a). That trajectory belongs to

01 the chaotic zone surrounding the origin. As described qualitatively in Wisdom (1983a) the chaotic zone has two parts: a low-ec-

0.2 centricity zone and a "narrow path to high 0.1 0,0 0.1 0.8 03 eccentricity." As a trajectory wanders in X this chaotic zone, it occasionally enters this

FIG. 11. Surface of section for AH = -3.04 x 10 6. narrow high-eccentricity path. It was ob- This figure corresponds to Fig. 7. There are two large served that more generally there are two chaotic zones for this value of AH. A trajectory in the modes of behavior for a chaotic trajectory: chaotic zone surrounding the origin enters the narrow a low-eccentricity mode and a high-eccen- part of the chaotic zone which extends to high eccentricities at irregular intervals. tricity mode. These two modes correspond to different regions of the chaotic zone. In the low-eccentricty mode points appear occur. (That this sometimes happens was around the quasiperiodic zone surrounding noted in Wisdom (1983a).) It is obvious the origin. In the high-eccentricity mode though that the small quasiperiodic zone to points appear on the long arcs to the right the left of the origin will never reach the side of the figures. The zone of uncertainty representative angles, but this is evidently separates these two modes of behavior. quite a small part of the phase space. While Each time the trajectory emerges from this the representative plane is almost but not quite representative, as it happens the two planes of initial conditions presented in 0.3 Wisdom (1983a) form complimentary pairs and are representative in the sense that almost all trajectories cross at least one of O1 them. It is possible now to identify the mysteri- ous process whereby trajectories pass from y 0.0 the lower curves to the upper curves in Figs. ll and 12 of Wisdom (1983a). Figure 10 corresponds to curve 4 in that paper. As (} 1 a chaotic trajectory evolves in Fig. 10 it may spend some time near the inner invariant curves. During this time points appear 0.2 01 00 0, I 02 03 on the lower part of curve 4. When the tra- x jectory follows a (h-librating) guiding trajectory to large eccentricity, points appear FIG. 12. Trajectories on the surface of section computed from the perturbation theory. These curves have on the upper part of curve 4. Notice though AH = -4.22 x 10 6, and correspond to Figs. 6 and 10. that on Fig. 6 there is a small zone between The region in which guiding trajectories terminate in the two zones of uncertainty where h libra- the zone of uncertainty (k > 0.99) has been shaded. MOTION NEAR THE 3/1 COMMENSURABILITY 283 zone it has a new X action. Sometimes this nar-elliptic problem the most important action will place the trajectory in the low- source of chaotic behavior seems to be as- eccentricity mode and sometimes in the sociated with the zones of uncertainty. high-eccentricity mode. (It should be ex- Comparison of Figs. 5-7 with Figs. 9-11 plicitly mentioned that Fig. 11 shows two shows a close correspondence between the disjoint chaotic zones, one of which ex- chaotic zones and the guiding trajectories hibits only the high-eccentricity mode.) which terminate near the zone of uncertainty. (The trajectories were terminated VI. ORIGIN OF THE CHAOTIC BEHAVIOR when k > 0.99.) This comparison for Figs. 6 One of the most useful criteria for pre- and 10 is made explicit in Fig. 12. Trajecto- dicting the onset of large-scale chaotic be- ries on the surface of section for AH = havior is the resonance overlap criterion -4.22 x 10 -6 have been computed using (see Chirikov, 1979). In this criterion, the the correction from x, y to x, y for q~ = P + location and widths of low-order reso- 7r derived in the appendix, and the region in nances are analytically computed, and which guiding trajectories terminate in the whenever the separation of two resonances zone of uncertainty has been shaded. It is is smaller than the sum of their half-widths evident that a good prediction of the extent a large zone of chaotic behavior is ex- and shape of the chaotic zone can be ob- pected. This criterion is ideally suited to tained in this way. some problems. For example, the chaotic Why is it that entering these zones of un- rotation of Hyperion was predicted using certainty leads to chaotic behavior? Away this criterion before it was verified numeri- from them the action constitutes an approx- cally (Wisdom et al., 1984). The resonance imate integral, and the trajectories behave overlap criterion has also been used by predictably. Close to the zones of uncer- Chirikov (1979) to determine the width of tainty the action is no longer conserved the chaotic separatrix in the presence of since the periods are not sufficiently differ- high-frequency perturbations. In other situ- ent. However, the loss of the approximate ations the application of the resonance integral is not in itself sufficient cause for overlap criterion is obscure, and it is not chaotic behavior. As the trajectory enters very useful. One example is the original the zone of uncertainty, the X motion is problem of Hrnon and Heiles (1964). An- forced near its separatrix, and motion near other is the problem considered in this pa- a separatrix is generally chaotic. It is possi- per! Now I have shown (Wisdom, 1980) ble that these temporary excursions into that chaotic behavior does occur when res- the chaotic X separatrix are responsible for onances corresponding to different mean- the large-scale chaotic behavior. The X se- motion commensurabilities overlap. Super- paratrix could become chaotic either as a ficially, the problem of motion near the 3/1 result of the presence of the short-period commensurability looks quite similar (see terms in the disturbing function or the pres- Goldreich, 1982; Dermott and Murray, ence of the long-period evolution of the 1983). There is a large chaotic zone and at guiding trajectories, in the first case the the same time there are three mean-motion chaotic behavior may be considered to resonances that strongly overlap. How- result from the overlap of high-order reso- ever, the fact that these resonances belong nances between the short-period terms and to the same commensurability is important. the much longer period X oscillations near In this section, I show that the chaotic zone the separatrix (see Chirikov, 1979). In the near the 3/1 commensurability is not a con- latter case, as the period of the X oscilla- sequence of the overlap of these mean-motions grows near the separatrix it must at tion resonances. some point become comparable to the peri- Near the 3/1 commensurability in the pla- ods associated with the x, y motion. The 284 JACK WISDOM strengths of resonances between these two cff OA = - ~ ~ degrees of freedom grow as the periods be- d-~- oy come comparable and at some point reso- and nances begin to overlap. The latter mechanism is undoubtedly more important, but to dS OA properly exhibit it the perturbation theory d--t = /~ O-f (cos ,k). presented in the appendix would have to be carried further. The x, y variables would Since depends only on A and some themselves have to be converted to action- constants these equations are derivable angle variables, and then the resonance from the Hamiltonian H -- /4(A), where structure analyzed for resonance overlap. /I(A) is defined by dkl/dA =/~(cos ?,). Thus Such an analysis should be able to predict A(£ f) is a constant of the guiding motion, the critical value of k above which the mo- and consequently so is the elliptic modulus. tion is chaotic. Of course, any guiding tra- This means that to the extent that the action jectories which terminate cross all values of remains invariant the amplitude of the h k, so the actual critical value of k is irrele- motion does not change and is never forced vant for them. To a first approximation the across its separatrix. There will thus be no observed zones of chaotic behavior coin- large chaotic zone associated with trajecto- cide with these terminating guiding trajec- ries forced to enter a zone of uncertainty. I tories. In Fig. 12, the shaded region corre- have verified this by studying the surface of sponds to those guiding trajectories which section for AH = - 1.0 x 10 -5 with F = G = reach k > 0.99. This choice was arbitrary. 0. The zone of uncertainty still exists and The actual critical value of k seems to be there is a small chaotic zone associated closer to 0.9. For this value all of the com- with it, but this zone is so small that it is puted chaotic zones are in essentially exact barely visible on the scale of Figs. 9-11. agreement with the surfaces of section. In There is no large chaotic zone when trajec- Fig. 12 the difference between these two tories are no longer forced to enter the zone critical values of k is slight, but with k > 0.9 of uncertainty. I did not find chaotic zones the predicted chaotic zone surrounds the near the unstable equilibria of the guiding small island to the left of the figure as is trajectories, but I expect that with sufficient observed in Fig. 10. While I have not dem- magnification this would be possible. How- onstrated in terms of resonance overlap ex- ever, such small chaotic zones are unim- actly why it happens, the fact remains that portant here. This example not only dem- chaotic behavior seems to occur whenever, onstrates the importance of entering the as a consequence of the adiabatic invari- zone of uncertainty, but it shows that the ance of the action, the zone of uncertainty broadened guiding separatrix is not an im- is entered. portant source of chaotic behavior. (The It is desirable then to identify those fea- width of the chaotic guiding separatrix is tures of the problem which lead to the very small because the ratio of the libration crossing of the zone of uncertainty. That it frequency to the frequency of small-ampli- is not a simple or necessary consequence of tude oscillations of £, )7 is large and the the overlap of mean-motion resonances width depends exponentially on this fac- may be demonstrated by considering the ef- tor.) fect of removing the secular terms. The re- The secular terms have been shown to be sulting problem still has three strongly essential for large-scale chaotic behavior. 1 overlapping mean-motion resonances, but now show that in fact several overlapping trajectories, are no longer driven across the resonant terms are not important. Consider zone of uncertainty. This may be seen by first the circular problem, i.e., set ej -: 0. considering Eqs. (3) with F = G := 0: Hamiltonian (1) then contains a term pro- MOTION NEAR THE 3/1 COMMENSURABILITY 285 portional to F and a resonant term propor- much higher eccentricities the differences tional to C. In this case A = C(x 2 + y2). will become more serious and there will Since the secular part of the Hamiltonian is probably begin to be qualitative differences proportional to A, an argument similar to as well. At present these regions can only the one used in the last paragraph shows be studied numerically. that A is a constant of the motion and the As a model for the motion of asteroids zone of uncertainty is not crossed. How- the planar-elliptic problem is itself deficient ever, this crossing can be forced by adding on two counts. It neglects the inclinations, an additional secular term. To illustrate this and it neglects the long-period variations of consider Hamiltonian (1) with D -- E = 0 Jupiter's orbital elements. The perturbation and ej = 0.5. There is only one resonant method I have presented in this paper can term, but there are two secular terms. It is accomodate both of these complications. no longer true that the secular part of the The shortest of the Solar System eigenpe- Hamiltonian is proportional to A. Conse- riods is 46,000 years (Brouwer and van quently, A is no longer a constant of the Woerkom, 1950). Obviously the averaging motion and many guiding trajectories are over the libration time scale is still valid. forced to terminate in the zone of uncer- The complicating factor is that the guiding tainty. A surface of section for this problem trajectories can no longer be understood on with M-/= -1.0 × 10 -5 shows a large cha- a two-dimensional plot and can themselves otic zone. The overlap of mean-motion res- be chaotic. Considering that the typical onances is a complicating factor, but not time scale for the precession of the perihe- the cause of the chaotic behavior. The es- lion is less than 20,000 years the Solar Sys- sential feature that gives rise to chaotic be- tem secular variations may not have any havior is the crossing of the zone of uncer- major effect. (Remember though that in the tainty. For this to happen it is necessary inclined problem the "secular resonance" that the secular Hamiltonian and A be inde- associated with this 46,000-year period ap- pendent of each other. It is not hard to show proaches the 3/1 commensurability for that in terms of Poisson brackets this condi- "proper" inclinations near 15° (Williams tion for large-scale chaotic behavior may be and Faulkner, 1981). This shift in the posi- stated [Rsoc, A] :~ 0. tion of the secular resonance with inclina- The overlap of mean-motion resonances tion depends on the mixed fourth-order is not responsible for the 3/1 chaotic zone. terms in the secular disturbing function, Rather, the chaotic zone arises because tra- i.e., terms of order e2i2.) Probably the more jectories are forced to enter the zone of un- serious deficiency is the restriction to the certainty. plane. I have found that chaotic trajectories seem to exhibit more freedom in the three- VII. DISCUSSION dimensional restricted problem (reach As a representation of the planar-elliptic higher eccentricities) and that the chaotic restricted three-body problem, the most se- zone widens as the inclination to Jupiter's rious deficiency of Hamiltonian (1) is the orbit plane increases (Wisdom, 1983a). truncation of the disturbing function to There is no problem in carrying out the av- terms of order e 2. This neglects terms of eraging for the three-dimensional problem, order e 4. At large eccentricities the pertur- but now the averaged equations have two bation theory presented in this paper will be degrees of freedom. Thus again the guiding quantitatively incorrect. However, com- trajectories can themselves be chaotic. It parisons with ordinary numerical integra- may be the case that there are several cha- tions show that it does a pretty good job otic zones, some associated with a zone of even at eccentricities as high as 0.4 (Wis- uncertainty and some associated with cha- dom, 1983a; Murray and Fox, 1984). At otic guiding trajectories. Perhaps the 286 JACK WISDOM proper context to understand the effects of havior near the 3/1 commensurability in the the inclinations and the variations in Jupi- planar-elliptic problem. The condition that ter's orbital elements will be that of Arnold this will happen takes the remarkably sim- diffusion (see Chirikov, 1979). ple form [Rsec, A] 4: 0. I am not familiar with a simpler criterion for the existence of viii. SUMMARY large-scale chaotic behavior. The general picture developed in this pa- Recently a number of authors (e.g., Tor- per for motion near the 3/1 commensurabil- bett and Smoluchowski, 1982; Henrard and ity in the planar-elliptic restricted three- Lemaitre, 1983) have proposed cosmogonic body problem is the following. Most of the theories to explain the existence of the time there exist three distinct time scales. Kirkwood gaps. Indeed, apart from its in- Variations on the orbital time scale are gen- trinsic interest as a classic problem in celes- erally ignorable. The libration time scale is tial mechanics, the primary goal in studying usually so short compared to the precession the distribution of asteroids is to under- time scale that analytic approximations for stand more about how the Solar System the variations on the libration time scale formed, to place constraints on cosmog- can be derived. With the assumption that ony. It is clear though that before this goal the action of the libration motion is adiabat- can be realized the dynamical mechanisms ically preserved, the effect of the long-pe- which could have modified that distribution riod evolution on the libration amplitude over the billions of years since it was ini- can be determined. In turn the variations on tially laid down must be properly under- the libration time scale can be analytically stood. I have shown (Wisdom, 1982, 1983a) averaged to obtain their effect on the long- that in fact there exist important dynamical period evolution of the eccentricity and lon- mechanisms for modifying the distribution gitude of perihelion. However, as a conse- of asteroids which were previously over- quence of the preservation of the action the looked or discounted (Scholl and Froes- amplitude of the libration motion is some- chl6, 1974). In particular, the 3/1 commen- times driven near rr, where the libration pe- surability is accompanied by a large chaotic riod becomes large. At this point the time zone, and trajectories within this chaotic scales are no longer distinct, and the action zone cross planetary orbits. The existence is no longer conserved. In this "zone of and extent of the chaotic zone and the uncertainty" the analytic approximations planet-crossing nature of the trajectories can no longer be used to predict the motion. within it have been verified by conventional After some short time in the zone of uncer- numerical integration (Wisdom, 1983a, tainty, trajectories reemerge to conserve 1985). The precise size and shape of the 3/I some new value of the action until the zone Kirkwood gap is explained by the removal of uncertainty is reached again. This pic- of the chaotic and quasiperiodic planet ture of the motion is quite successful. It crossers. No cosmogonic mechanisms are accurately predicts the "guiding centers" required. This paper has presented a way of of cr and e, as well as numerous features of understanding why this chaotic zone is the surfaces of section. In particular the lo- present and of predicting its general struc- cations and shapes of the chaotic zones are ture, at least in the planar-elliptic approxi- accurately predicted by identifying chaotic mation. The general method will also work trajectories with those guiding trajectories when the inclinations and secular variations which are forced to enter the zone of uncer- of the planets are included, but a thorough tainty. That many trajectories are forced to investigation of the consequences is be- enter the zone of uncertainty is recognized yond the scope of this paper. Whether or as the most important cause of chaotic be- not the inclinations and secular variations MOTION NEAR THE 3/1 COMMENSURABILITY 287 introduce important qualitative changes in lower limit is chosen so that the angle con- the dynamics (such as Arnold diffusion), jugate to I is zero when ~b' --- P + 7r. This quantitatively they are important in that integral could be evaluated in terms of ellip- trajectories in the 3/1 chaotic zone are tic integrals, but it is not required. The gen- Earth-crossing with them, but only Mars- erating function for the full problem is then crossing in the planar-elliptic approxima- taken to be tion. The method presented in this paper should also work well for the other principal commensurabilities, to the extent that their dynamics are well represented by truncated disturbing functions. It seems + tz.,i cos(4, - P) d~ + xy, that a good understanding of the long-term where the identity transformation from x evolution of trajectories near commensurabilities in the Solar System is close at hand. and y and ~f and y has been added, and H', Soon it may be possible to untangle those A(x, y) and P(x, y) have been replaced by cosmogonic clues which lie dormant in the H(I, A) = H'(I, A), distribution of asteroids. A(x, y-) = A(x, y), and APPENDIX P(x, y-) = P(x, y), This appendix presents an alternate derivation of Eqs. (3) using canonical perturba- respectively. The minus sign is chosen so tion theory. First, to prepare for the pendu- that d~b'/dt is positive when ~b' = /~ + 7r. lum approximation a canonical transfor- Since this generating function is time inde- mation to the variables ~b' = ~b and 4~' = pendent the new Hamiltonian is equal to 4p - 4~re~ is made and only the quadratic the old Hamiltonian expressed in terms of terms in ~' are kept. Since ~' is of order the new variables. These new variables are tz ~/2, this approximation ignores terms of or- obtained as derivatives of F: der/x 3/2. The next step is to transform the OF I~ {OH OA ~', ~' variables to action-angle variables. g= O--~=x+ ~(_[d.~A)l/2 -~ --~ I~ This is accomplished by first solving the Hamilton-Jacobi equation for the pendulum +-~°X/2) + ~'(~') -3~a~ aF Odp' and OF I~ (OH OA = ___{2 [H'+ p.A cos(~b' - p)]}u2 Y= Ox Y+ to obtain the generating function +~~a t2) + ~'(~,') ~"a'~ : + 4Y The two integrals in these expressions are

1"I I12 + ~ cos(~> - e)Jl d~>. i1= , fL

H' is considered to be a function of the ,t4) action. Fortunately, the explicit inversion ofl(H', A) for H'(I, A) can be avoided. The 288 JACK WISDOM and of/l are of order e. The derivatives of fi are of order e/A. Thus the two sets of variables seem to differ by terms of order #z 1/~ as de- 12 = -- O£ +~r sired. However, each correction contains cos(d) -/S)dd) some power of X in the denominator, and _ )1/2 while A is typically of order e 2 near (x, y) + ~2 cos(d) - P)][ equal to (0.01, 0) or (0.14, 0) it is smaller. For libration Denoting the distance in the x, y plane to these critical points by 8, the perturbation 11 = F(T, ki) method breaks down when 8 is less than a and number of order e/xl/2. This restriction turns out not to be very serious. Such a restric- /2 = F(T, kc) - 2E(T, ki), tion might have been expected on intuitive grounds. In the ordinary circular problem where F(T, kL) and E(T, kL) are elliptic inte- at large eccentricities the libration mecha- grals of the first and second kind, respec- nism primarily involves the mean longi- tively, Y = sin l{/.tA[/ + cos(d)' - P)]/(IxA tudes. On the other hand, at very low ec- - /1)} and kL = ((p,A - /4)/(2/zA)) I/2. The centricity the libration mechanism is mainly integrals have similar forms for circulation. controlled by the motion of the perihelion. The derivative OH/aA is now determined by The approximation used here is more like differentiating the formulae for the action the first case in that the variation of the with respect to A. For libration mean longitudes is considered to dominate OH(I, A) 2E(kL) the motion on the libration time scale. It - I a,X, K(ki) might be expected then that there would be regions where this assumption fails. It is and for circulation perhaps surprising though that here there a/4(l, ,4) 2E(kc) 2 are two critical points, one at low eccentric- ~di ~,K(kc) + 1 k~ ity and one at moderate eccentricity (x - 0.14). Here E(k) and K(k) are complete elliptic in- To order e/zi/2/3 the relation between the tegrals and the elliptic moduli are assumed coordinates can be found by replacing/1, to be written in terms of / and A(x, y-). Col- A, and fiby H', A(£, y-), and P(£, 37), respec- lecting terms tively, in the expressions above. The explicit conversion from £ and 37 to the sur- F(T, kL) £ = x + --~ 2E(kL) faces of section is 6o0 K(kL) 1,'2 OP ~P 2 (H' - #xA) 3--~ - 2E(T, k~)] + @'(d)') a-7, x _f + and and F(T, kL) 2 ] 1/2 3P Y = 37+ /x 2E(kL) COo K(kL) v Y- s(H' - #~A) ~.

- 2E(T, kL)] -~ + di)'id)') -~, Finally, the new Hamiltonian is H - H'(I, A(£, y~) + txF(£ 2 + y~-) for libration only. + ejlxG£ + o(ep,3/2/8), The brackets containing elliptic functions are of order unity; qb' is of order/z~/2; 6o0 is Since the angle conjugate to I does not ap- of order elzvZ; A is of order e2; derivatives pear, / is invariant to this order. The equa- MOTION NEAR THE 3/1 COMMENSURABILITY 289 tions of motion for • and 37 are then GIFFEN, R. (1973). A study of commensurable motion in the asteroid belt. Astron. Astrophys. 23, 387-403. dY OH OH' OA GOLDREICH, P. (1982). Cosmology and Astrophysics d--~ = 037 - 21~F37 oa 037 (Essay in Honor of Thomas Gold), p. 121. Cornell Univ. Press, Ithaca, N.Y. and HI'NON, M., AND C. HEILES (1964). The applicability of the third integral of motion: Some numerical ex- d37_ OH OH' OA amples. Astron. J. 69, 73-79. dt 0,~ - 21~FY + ejl~G + O---A 0---~ HENRARD, J., AND A. LEMAITRE (1983). A mechanism of formation for the Kirkwood gaps. Icarus 55, 482- Since the expressions for OH'/OA evaluated 494. above are identical to the expressions for MURRAY, C. D., AND K. FOX (1984). Structure of the (cos h) evaluated in Section IV, these equa- 3 : 1 Jovian resonance: A comparison of numerical tions are the same as Eqs. (3). methods. Icarus 59, 221-233. POINCARI~, H. (1902). Sur les planetes du type d'He- cube. Bull. Astron. 19, 289-310. ACKNOWLEDGMENTS SCHOLL, H., AND C. FROESCHL# (1974). Asteroidal motion at the 3/1 commensurability. Astron. As- The basic approach of this paper, the separation of trophys. 33, 455-458. time scales and the use of the adiabatic invariance of SCHOLL, H., AND C. FROESCHLI~ (1975). Asteroidal the action to approximate the motion on the libration motion at the 5/2, 7/3 and 2/I resonances. Astron. time scale, was suggested by Peter Goldreich. This Astrophys. 42, 457-463. work was supported in part by the NASA Planetary TORBETT, M., AND R. SMOLUCHOWSKI(1982). Motion Geology and Geophysics Program under Grant of the Jovian commensurability resonances and the NAGW 706. character of celestial mechanics in the asteroid zone: Implications for kinematics and structure. As- tron. Astrophys. ll0, 43-49. REFERENCES WETHERILL, G. W. (1975). Late heavy bombardment ABRAMOWITZ, M., AND I. A. STEGUN (1970). Hand- of the Moon and terrestrial planets. Proc. Lunar Sci. book of Mathematical Functions. Dover, New Conf. 6th, 1539-1561. York. WETHERILL, G. W. (1985). Asteroidal source of ordi- BROUWER, D., AND A. J. J. VAN WOERKOM (1950). nary chondrites. Meteoritics 18, 1-22. The Secular Variations of the Orbital Elements of WILLIAMS, J. G., AND J. FAULKNER (1981). The posi- the Principal Planets. Astronomical Paper of the tions of secular resonance surfaces. Icarus 46, 390- American Ephemeris XII, II. 399. CHmlKOV, B. V. (1979). A universal instability of WISDOM, J. (1980). The resonance overlap criterion many dimensional oscillator systems. Phys. Rep. and the onset of stochastic behavior in the restricted 52, 263-379. three-body problem. Astron. J. 85, 1122-1133. DERMOTT, S. F., AND C. D. MURRAY (1983). Nature WISDOM, J. (1982). The origin of the Kirkwood gaps: of the Kirkwood gaps in the asteroid belt, Nature A mapping for asteroidal motion near the 3/1 com- 301, 201-205. mensurability. Astron. J. 87, 577-593. FROESCHLI~, C., AND H. SCHOLL (1976). On the dy- WISDOM, J. (1983a). Chaotic behavior and the origin of namical topology of the Kirkwood gaps. Astron. As- the 3/1 Kirkwood gap. Icarus 56, 51-74. trophys. 48, 389-393. WISDOM, J. (1983b). Chaotic behavior near the 3/1 FROESCHLI~, C., AND H. SCHOLL (1977). A qualitative commensurability as a source of Earth crossing as- comparison between the circular and elliptic Sun- teroids and meteorites. Meteoritics 18, 422-423. Jupiter-asteroid problem at commensurabilities. WtSDOM, J. (1985). Meteorites may follow a chaotic Astron. Astrophys. 57, 33-39. route to Earth. Nature 315, 731-733. FROESCHLI~, C., AND H. SCHOLL (1981). The stochas- WISDOM, J., S. J. PEALE, AND F. MIGNARD (1984). ticity of peculiar orbits in the 2/1 Kirkwood gap. The chaotic rotation of Hyperion. Icarus 58, 137- Astron. Astrophys. 93, 62-66. 152.