On the alleged collisional origin of the Kirkwood gaps

IC~RUS 26, 367--376 (1975)

'On the Alleged Collisional Origin of the Kirkwood Gaps 1 T. A. H E P P E N H E I M E R Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California 91125 Received March 5, 1975; revised April 14, 1975 This paper examines two proposed mechanisms whereby asteroidal collisions and close approaches may have given rise to the Kirkwood Gaps. The first hypothesis is that asteroids in near-resonant orbits have markedly increased collision probabilities and so are preferentially destroyed, or suffer decay in population density, within the resonance zones. A simple order-of-magnitude analysis shows that this hypothesis is untenable since it leads to conclusions which are either unrealistic or not in accord with present understanding of asteroidal physics. The second hypothesis is the Brouwer~lefferys theory that collisions would smooth an asteroidal distribution function, as a function of Jacobi constant, thus forming resonance gaps. This hypothesis is examined by direct numerical integration of 50 asteroid orbits near the 2 : 1 resonance, with collisions simulated by random variables. No tendency to form a gap was observed. I. INTRODUCTION T h e origin o f t h e K i r k w o o d G a p s r e p r e s e n t s a p r o b l e m in a s t r o p h y s i c s r a t h e r t h a n a p r o b l e m in celestial mechanics. F o r o v e r a c e n t u r y , t h e association of t h e K i r k w o o d G a p s w i t h orbital c o m m e n s u r a b i l i t i e s h a s b e e n recognized. B u t t h e r e is no generally a c c e p t e d p h e n o m e n o l o g i c a l t h e o r y w h e r e b y celestial-mechanical r e s o n a n c e p h e n o m e n a are s h o w n to give rise to gaps. I n r e c e n t years, a n u m b e r o f a u t h o r s h a v e p r o p o s e d t h a t collisions b e t w e e n asteroids m i g h t s u p p l y t h e a s t r o p h y s i c a l m e c h a n i s m leading to gaps. Such collisional h y p o t h e s e s are of t w o types. T h e first t y p e holds t h a t n e a r - c o m m e n s u r a b l e orbits lead t o m a r k e d l y increased probabilities of collision a n d d e s t r u c t i o n o f asteroids. This h y p o t h e s i s is associated w i t h t h e o b s e r v a t i o n t h a t n e a r - c o m m e n s u r a b l e orbits are subject to markedly greater perturbational excursions in orbital e l e m e n t s (particularly eccentricity) t h a n are n o n c o m m e n s u r a b l e orbits. T h e second t y p e o f h y p o t h e s i s is due t o B r o u w e r (1963) a n d J e f f e r y s (1967). This h y p o t h e s i s holds t h a t collisions will s m o o t h

a n a r b i t r a r y asteroidal d i s t r i b u t i o n function so as to a p p r o a c h a n e q n i p a r t i t i o n o f J a c o b i integral. I f t h e d i s t r i b u t i o n f u n c t i o n is in f a c t s m o o t h e d w i t h respect to J a c o b i integral, t h e n a V - s h a p e d g a p a t a comm e n s u r a b i l i t y m u s t necessarily follow. This p a p e r involves a direct e x a m i n a t i o n of b o t h t y p e s of hypothesis. I t will be seen t h a t t h e first t y p e leads to conclusions which are i n s u p p o r t a b l e , a n d t h a t no evidence is f o u n d t o s u p p o r t t h e second. I n w h a t follows, we are p r i n c i p a l l y concerned w i t h t h e 2:1 resonance in t h e S u n - J u p i t e r s y s t e m , which is t h e location of t h e H e c u b a Gap. This is t h e m o s t pron o u n c e d of t h e gaps, a n d is associated w i t h t h e s t r o n g e s t resonance effects. Thus, a n y h y p o t h e s i s of g a p origin which fails for t h e 2:1 resonance can h a r d l y suceed for t h e o t h e r K i r k w o o d resonances. I I . THEORY OF COLLISIONS

T h e general c h a r a c t e r of n e a r - c o m m e n surable orbits has b e e n discussed b y n u m e r ous a u t h o r s (Greenberg et al., 1972; L e c a r a n d F r a n k l i n , 1973; Giffen, 1973; Scholl a n d x Contribution No. 2585 of the Division of Froeschl~, 1974). Such orbits are s u b j e c t Geological and Planetary Sciences, California to long-period p e r t u r b a t i o n s leading t o Institute of Technology, Pasadena, Calif. 91125. v a r i a t i o n s of several p e r c e n t in s e m i m a j o r Copyright ~) 1975 by Academic Press, Inc. 367 All rights of reproduction in any form reserved. Printed in Great Britain

368

T.A. HEPPENHEIMER

axis. The eceentrieity is highly variable, and for suitable initial conditions an initial value e 0 = 0 . 1 may increase to 0.3 or greater. B u t the asteroid avoids close approaches to Jupiter, so that it is not subject to ejection. Thus, in contrast to the small perturbations associated with noncommensurable orbits, the large variations in eccentricity are a characteristic feature of orbits near resonance. Lecar and Franldin (1973), Giffen (1973), and Scholl and Froschl6 (1974) have suggested that near-commensurable asteroids would then have markedly shorter lifetimes against destruction by collision. This hypothesis m a y be studied through use of Dohnanyi's (1969) theory of collisions. The effect of systematically higher eccentricity may be considered through Wetherill's (1967) collision theory. Dohnanyi considers collisions among a population of asteroids defined b y the McDonald: and Palomar-Leiden surveys. His nominal value of asteroidal albedo is 0.2, with a range of uncertainty of x3 +l This uncertainty gives rise to a concomitant uncertainty of collisional lifetime, ×4 ±~ However, recent observational work (Chapman et al., 1975) has shown that many ~asteroids are of carbonaceous-chondritic :composition and have very low albedos. This work indicates a nominal asteroidal albedo close to Dohnanyi's lower limit of 0.067. This value will be used and results given b y Dohnanyi modified accordingly. The asteroidal population considered is defined by the mass-distribution law N(m) = 1.24 × 1017m -°'837,

(1)

~here m is asteroid mass in kilograms, '~(m) is the number of asteroids with mass ~>m. Equation (1) is plotted as the downward-sloping curve in Fig. 1. The number density of this population;per meter cubed in the asteroid belt, in the mass range of to m + dmkg, is given b y f ( m ) (/m = 9.92

×

lO-19m-l'S37dm/meter 3 - A m -~ din. (2)

~rhe aster0id~ure on eccentric and inclined b r b i t g ~ d ::::~ e w i t h i~ae~n c011i~on

L06 10 qPJ%D|US (~ERS)

6

13 12 1I

/

9 7

N

SOLAR SYSTEM 3

-i

.~

t~ ~ 4 e ~

l i i 13 1~ 15 16 17 18 19 20 21 22 23 24 25 LOG 3.0 MASS (KILOGRAMS)

Fro. 1. Cumulative number and collisional lifetime of asteroids as functions of mass, following Dohnanyi's (1969) theory.

velocity V = 5km/sec (Piotrowski, 1953). Also, there is the normalization factor l, used subsequently, I = (37rl/2/4p) 2/3, (3) where p = density of the asteroidal material ~3g/cm 3. Finally, there is /~', defined as follows. An asteroid of mass m is completely shattered b y an impacting body, striking at velocity V, if the body is of mass >~m/F'. In Dohnanyi's work, / " is derived from hypervelocity impact experiments. However, / " m a y also be associated with mechanical considerations. Thus, let B be the mechanical binding energy of the asteroidal material. Then the condition for shattering the asteroid m a y be given, B = ½V21F '.

(4)

In Dohnanyi's work, V = V = 5km/sec, / " = 6250; then B = 2000J/kg. For a massive body, B is increased b y the body self-gravitation. Let B 0 be the binding energy due to mechanical cohesion, r the body radius in meters; then, for p ~ 3g/cm 3, B = B 0 + 10 "~r 2

(5)

~nd from (3), r 2 = lm2/3Llr. T h e n (4) and (5) ~ g e t h e r give F' ="P'(F,m). According to

369

COLLISION ORIGIN OF KIRKWOOD GAPS

TABLE I

Dohnanyi the lifetime 7 against collisional shattering of an asteroid of mass m i s

LIFETIMEFACTORW -~- W (a, e, i), AI~TERWETHERIT,L(1967)

(6)

~" = m " - 5 / 3 / I V A ~ ),

where A, ~ are given by (2) and F'~--I

¢~ =

~

./"1'~--4/3 F'~--5/3 +

-- 1

2 -

-

a -- 4/3

+

-

-

i (deg) 0.39 0.44 0.5

(7)

~ -- 5/3"

Equation (6) is plotted in Fig. 1, for two values of B 0 : 2000 and 200J/kg. This latter value may be typical of the more friable materials which actually comprise m a n y asteroids. I t is apparent t h a t for asteroids with collisional lifetimes ~>109yr, the actual value of B0 is unimportant since the principal contribution to B arises from the gravitational field. This is the case for bodies of diameter > 100km. Further, only ~10 bodies m a y be expected to have eollisional lifetimes greater t h a n the age of the solar system. Such bodies have diameter >~200km; Chapman et. al. (1975) count some 15 such bodies. Thus, the population of asteroids must be regarded as predominantly the fragments of catastrophic collisions, with only a few of the largest asteroids remaining as unfragmented primordial bodies. The lifetimes of Fig. 1 are derived from a simple physical theory which does not take account of the detailed structure of the asteroid belt, or of the effect on collision lifetime of the orbital elements of a particular asteroid. This information is given by Table I, adapted from Wetherill (1967). Thus, the lifetime given by (6) or by Fig. 1, for an asteroid of given mass, is to be multiplied by the appropriate constant W of Table I, I t is seen t h a t for asteroids in the region of interest, even if near-resonant asteroids are regarded as having eccentricity e = 0.5 while nonresonant asteroids have e = 0.1, the latter have lifetimes tess t h a n twice those of the former. Consider first whether the asteroid belt m a y be in a highly fragmented state, with Kirkwood-Gap asteroids fragmented to small sizes. An asteroid destroyed in collision is not pulverized into small fragments but rather gives rise to a power-law spect r u m of fragmental masses similar to (1). Let ~ < 1 be the fractional mass of the

0.0001

0.1 0.2 0.5

0.95 0.99

0.6 30 90 6 12 155 0.6 30 90 0.6 0.6 30

3.61 5.18 3.93 2.65 2.96

1.58 2.67 2.03 1.52 1.93

0.88 2.91 2.88 0.63

0.98 3.28 4.35 0.84

1.42 2.27 1.73 1.40 1.64 0.52 1.21 4.26 3.22 1.06 0.98 4.69

0.62 0.73 2.42 3.20 2.42 2.64 2.96 1.76 6.05 6.12 1.53

2.24 2.04

largest fragment. Let ~ = (lifetime in commensurable orbit)/(lifetime in a noncommensurable orbit); from the foregoing argument, ~ > 0.5. Finally, let us assume t h a t the Kirkwood Gaps once contained asteroids, but t h a t these asteroids have been fragmented below the level of visibility. To escape detection in the PalomarLeiden survey, these fragments could not have exceeded HI kilometer in diameter. L e t m K be the mass of such a fragment, which barely escapes detection. Next we consider t h a t the Kirkwood Gaps span the order of a few percent of the asteroidal range of a (semimajor axis). Hence, if these gaps were normally populated, from Fig. 1 there should be at least a few asteroids of diameter ~ 1 0 0 k m , in the gaps. Let m a be the mass of such an asteroid. We now consider t h a t the asteroids were initially populated by bodies of maximum mass M ; t h a t in Kirkwood-sized regions away from the gaps these bodies were fragmented to maximum mass ms; but t h a t within the gaps the fragmentation was more extensive and gave a population of maximum mass m ~ . The non-Kirkwood bodies suffered at least n a collisions; hence, M ~ " . = ms,

(8)

370

T. A. H E P P E N H E I M E R

while the K i r k w o o d bodies suffered a t least hal 7 collisions a n d M jz"o/" = m K.

(9)

Combining (8) a n d (9) gives the result

A(t) .o.-Kl,kwood = Al-~ul-"o

(7}-I __ 1)

(u1-. ~ - l _ i)"

(10) which is i n d e p e n d e n t of n~ and also of #. I t is c o n v e n i e n t to express (10) in t e r m s of diameters: d K = 1kin for the K i r k w o o d bodies, d~ = 100km for the non-Kirkwoods, and D for the primordial bodies of mass M. Then, (ma/M) = (mK/M)",

(acid)) = (adD)".

so t h a t

(11)

T a k i n g 7/= i gives D = 10akm ; even larger values are f o u n d for larger 7, e.g., D = 10Skm for 7 = 0.6. B u t such planet-sized bodies, from Fig. 1, clearly are stable against fragmentation, even within an asteroid belt containing m a n y more large bodies t h a n exist t o d a y . Thus, the K i r k w o o d Gaps could not h a v e arisen from a more extensive f r a g m e n t a t i o n o f bodies. Then, consider a n o t h e r h y p o thesis: t h a t the asteroidal p o p u l a t i o n has suffered decay, w i t h the p o p u l a t i o n in the K i r k w o o d Gaps suffering more extensive d e c a y t h a n t h e rest of the asteroids. Consider a p o p u l a t i o n distribution A = A ( m , t ) , the n u m b e r of asteroids per u n i t v o l u m e in the mass range m to m + dm, at t i m e t. L e t this p o p u l a t i o n be subject to d e c a y according to a power law, OA/~t = - K A " , (12) where we m a y h a v e K = K ( m ) . A t time t = t o, A = A o and the solution to (12) is (n -- 1) K ( t - to)]. A ( t ) l-" = Aol-"[1 q ~ot_---~ (13) N o w as before, let 7 = K i r k w o o d - G a p lifetime)/(non-Kirkwood lifetime), so t h a t KKlrkwood = Knon_Kirkwood/TJ ; note that

A~ -I K has dimensions of 1/time. Also, let u--(asteroid density outside Kirkwood Gaps)/(asteroid density in Kirkwood Gaps); u ~ I00. Then, A(t)non_Kirkwood= uA(t)Kirkw¢~d. L e t us assume t h a t at t = t o, A = A 0 in b o t h regions. T h e n it is seen t h a t (n - 1)K(t - to)/A~-" = - - ( u ' - - n - 1)/( u ' - n 7 - ' -- 1) (14)

(15)

W e now restrict the admissible range of n b y insisting t h a t the r i g h t - h a n d side of (15) be positive; this leads t o u l - " / > V, (16) where we h a v e used V < 1. F o r u = 100, 7 = 0 . 5 , n~< 1.15. F o r o t h e r plausible values of u and 7 it is also t r u e t h a t n c a n n o t be m u c h larger t h a n unity. Then, the associated admissible d e c a y laws, (12), m a y be described as " q u a s i - e x p o n e n t i a l . " The ratio A o / A ( t ) t h e n m a y be quite large. F o r a n exponential law, n = 1, for example, A 0 ~ UAnon_Kirkwood ~ U 2A Kirkwood" W e t h u s are led to conclude t h a t if the K i r k w o o d Gaps arose t h r o u g h differential d e c a y of an initial population, the d e c a y law would be quasiexponential and would most likely h a v e involved large-scale depopulations of an initially more or less uniform distribution. I t is a m a t t e r of current c o n t r o v e r s y w h e t h e r the asteroids suffered large-scale collisional mass loss, which implies dep o p u l a t i o n of a t least t h e larger bodies. However, t h r e e points b e a r on the question of w h e t h e r a d e c a y process such as described m a y in fact h a v e occurred. First, D o h n a n y i ' s (1969) work shows t h a t ff population d e c a y occurs at all, it occurs with e x p o n e n t n = 2. Second, for bodies smaller t h a n ~ 1 0 0 k m diameter, the effect of p o p u l a t i o n evolution is to p r o m o t e r e l a x a t i o n of an a r b i t r a r y initial distribution into a power-law dist r i b u t i o n similar to (2). The resulting distribution is in a s t e a d y s t a t e : in a given mass range, as m a n y bodies are lost t h r o u g h collisions as are gained from f r a g m e n t a t i o n of larger bodies. This process of relaxation is qualitatively different from a d e c a y process. I t involves a r e d i s t r i b u t i o n o f m a s s s u c h t h a t for large m, A ( m , t ) decreases b u t for smaller m, corresponding to diameters less t h a n a few tens o f kilometers, A ( m , t ) m a y actually increase. Third, C h a p m a n (1975) a n d C h a p m a n et al. (1975) h a v e n o t e d t h a t the asteroidal

COLLISION ORIGIN OF KIRKWOOD GAPS

population m a y be regarded as involving two families: C-type (carbonaceous) and S-type (siliceous), with S-types predominating near the Kirkwood Gaps. Chapman has further noted t h a t the C-type population is fully relaxed and does not show distribution anomalies, whereas the S-type population is not relaxed and is anomalously abundant at ~150km diameter. The S-type population, following arguments due to Anders (1965), thus cannot be regarded as highly fragmented or evolved. But if initially there were m a n y S-types in what is now the Kirkwood Gaps, then it is difficult to see how they would have been preferentially removed while the S-type population as a whole failed to evolve to a relaxed state. This last point can be treated quantitatively, using a variation of the argument of Eqs. (8)-(11). Consider a population of asteroids with initial diameters ~100km, such as may have constituted an initial S-type population. To allow some evolution but not complete relaxation of the nonKirkwood population, let n a = 1, i.e., each body has suffered an average of one collision. In reality, some will have suffered no collisions and remain as the anomalously abundant population observed by Chapman, while others will have suffered multiple collisions so as to produce a smallfragment population with diameters less than 100km. For the Kirkwood bodies, however, we require n K ~ -6/log/z since the bodies are to be ground down by at least a factor 106 in mass. Let T be the time for population evolution, r 0 the mean collisional lifetime of the original 100 km bodies; since n a = 1, T = r 0, in agreement with Fig. 1. But in the Kirkwood Gaps the lifetime initially is ~/r0 ; for the largest first-generation fragment, in accord with (6), the lifetime is approximately ftl/6T]T0; for the largest n K th generation fragment, /z'J6Vz 0. Then, T = ~/v0[1 +/~1/6 +/~2/6 + . . . + ~6(ng--l)/6], where the quantity in brackets is (1 -- p~/6)/ (1 - ~i/6). Then, 6 ng = ~-7:-- log [1 - T(1 -/zl/6)1~¢0]. log~

371

But n K ~ - 6 / l o g p so, since T/T 0 - - 1 , 1 ~ [1 - (1 - / ~ / ~ / ~ ] ~ 0.9 and ft < (1 -- 0.9~/)6. We have taken ~ > 0.5 so t h e n / z < 0.027; for more plausible values, e.g., 7 = 0.8, /z < 0.0004. Then, the cited point can be upheld only if the result of a collision is indeed a pulverization into tiny fragments. But in fact we must expect/z to be of the order of a few tenths. I t is the hypothesis and not the asteroids which is pulverized. The hypothesis of differential decay of populations, as a cause of the Kirkwood Gaps, thus is open to attack on several grounds and appears highly dubious. Consider now still another hypothesis: t h a t collisions influenced the orbits of the asteroids in such a way as to give rise to the Kirkwood Gaps. This concept constitutes the core of t h e Brouwer-Jefferys theory. I I I . THE BROUWER-JEFFERYS THEORY

Consider the motion of an asteroid perturbed by Jupiter, in the vicinity of a commensurability (i0 + q ) / p . Let a = asteroid semimajor axis (Jupiter = 1), L = a ~/2 ; let R be the disturbing function, R* the disturbing function after elimination of short-period terms. Then the following integral exists as was shown by Brouwer (1963). 2 ½L-2+(p+q)L/p+R*=F. (17) A distribution of asteroids which is a smooth function of P in the vicinity of the commensurability then is V-shaped as a function of a with zero value at the commensurability. Brouwer further showed t h a t the observed Kirkwood Gaps were V-shaped as functions of a, but smooth as functions of P. But whereas Brouwer's work gives an excellent physical description of the gaps, it is not a phenomenological theory of origin since it fails to provide a mechanism for smoothing an initially singular distribution function of /'. Such a singular distribution would be smooth (free of gaps) as a function of a. Jefferys (1967) sought 2 The integral (17) was first found by Brown and Shook but was independently rediscovered by Brouwer period.

372

T. A. H E P P E N H E I M E R

to complete the logic of Brouwer's theory by proposing t h a t asteroidal collisions and close approaches would cause the distribution function to become smooth with respect to F. This hypothesis can be tested directly. In what follows, we consider a population of 50 asteroids, initially distributed uniformly in a. Each asteroid orbit is permitted to evolve under Jupiter perturbations, represented by a second-order disturbing function. At randomly chosen times, the motion of each asteroid is subjected to a random change in velocity, simulating a collision, and the new orbital elements resulting from the collision are propagated forward in time to the next collision. The evolution of the distribution function thus may be directly observed. For simplicity, and with no loss of generality, the motion is treated as a circular coplanar restricted three-body problem. Further, the limitation to second order of the disturbing function has no effect upon the mathematics of the problem. I f Jefferys' hypothesis is correct, than a gap will arise through collisions even though the disturbing function is truncated. Strictly speaking, the simulated collisions are in fact random variations applied to the individual asteroids. A true asteroidal collision is a binary process, wherein both asteroidal motions suffer change, with conservation of angular momentum. I t will be subsequently noted how results obtained with the simulated unary collisions can be extended to a case of simulated binary collisions. Let t be the time, A' the mean longitude of Jupiter; by convention, A' = t. Let e be the eccentricity of the asteroid, ~ its longitude of perihelion, A its mean longitude. Then the disturbing function R* in the vicinity of the 2:1 resonance is given (Brouwer and Clemence, 1961) by R*= (0) + T i e 2 + T2e cos ce + T3 e2 COS3ce], x [½ bl/2

(18) T 1 = -~(D + 1)2) b}~, T 2 = -- ~ (4 "[-"n'~/~(2) .a.~] U,l/2~ T3 = ~ (44 + 13D + D2) b,%

Here/~ = (mass of Jupiter)/(mass of Jupiter + S u n ) and we take p = 1/1000. Also blJ/)2- b~)2 (a) is the Laplace coefficient, Db (s) ~ ~ is 1 / 2 --- ac~(S) 1/2/lda , D2b (s) 1 / 2 - D I~D b (s) 112l" W the critical or resonance variable, ce = 3t ~-~. The disturbing function (18) is valid both in the 2:1 resonance and in the adjacent resonance-free region. The reason is t h a t where resonance effects grow unimportant, the resonance terms in (18) are reduced in magnitude so t h a t (18) approaches the simpler form appropriate to a resonancefree region. The Lagrange's planetary equations then are given by -

da dt de dU =

de

-

3 aR* n a ace ¢2)1/2

(1

[2 -- (1 - - e2) 1/2] aR*,

-~

n a 2e

-~ = - ~

3

(1 DR*

- - e 2 ) 1/2

na2e

+

(19)

x [1 -- (1 -- e2~1/2]aR*, i

d~ dt

(1 - -

J

ae

e2) 1/2 a R *

~t,2e

ae

where n --- asteroid mean motion. Also, e is defined by 2 = ]ndt + e.

Then, = 2 -- n

dt

dr"

(30)

We also have ~ = M + ~, where M is the mean anomaly. At any time t, let a, e, ce, be given from numerical integration. Then M is known and from standard expansions in elliptic motion, the true anomaly f is found. Now define a cartesian coordinate system (x,y) with origin at the Sun; by convention, let the angle ~ be measured from the positive x-axis, taken as zero. Then we have x = r cos (f + ~), y = r sin (f + ~), :~ = ~ cos (f + ~) -- r/sin (f + ~), = ~ sin (f ~) + r/cos (f + ~), (21)

:373

COT,I,TSIONORIGIN OF KIRKWOOD GAPS and r = a(1 -- eZ)] (I + e cos/), f = (1 + ecos f)2/[a(l -- e2)]3/2, T h e r e is also the v e l o c i t y : V 2 = ~2 + ?~2. T o simulate a collision, we t a k e (22)

~ = :~ -F O.OI25 V . R V , , , 1) = 1)+ 0.0125 V ' R V 2 ,

where = is used in the c o m p u t a t i o n a l sense of "is replaced b y . " I n (22), RV, ,2 is a r a n d o m variable having equally probable values between - 1 a n d +1. Thus, the simulated collision does not change the asteroid position b u t adds small perturbations, r a n d o m in m a g n i t u d e a n d sign, t o the c o m p o n e n t s of velocity. I n (22), the factor 0.0125 V is selected so t h a t the change in a due t o a collision is no g ~ t e r t h a n app r o x i m a t e l y Aa = 0.01. Then, to r e t u r n from coordinates (x, y, 5, Y) to coordinates (e, e,'~; ~), we h a v e a = 1 / ( 2 / r - V2), e = [1 - - (x?~ yx)2/a]l/2 -

n e x t set of initial conditions (ao,%) was used t o begin s t u d y of the n e x t asteroid. The c o m p u t a t i o n was p e r f o r m e d in single precision on Caltech's I B M 370-158 computer. T h e B r o u w e r integral (17) was c o m p u t e d at each time step a n d / ' w a s held c o n s t a n t to five decimal places for a given initialization. (Of course, / , changed in general following a collision.) C o m p u t a t i o n was facilitated b y first c o m p u t i n g a m a t r i x o f the required Laplace coefficients a n d their derivatives, a t values of a from 0.50 to 0.75 in intervals o f Aa = 0.01. Thereafter, at a n y a, required values were e x t r a p o l a t e d from t a b u l a t e d values using a t h r e e - t e r m T a y l o r series. Some 5.5sec of CP U t i m e were required to establish t h e matrix, using recursion formulas given b y B r o u w e r a n d Clemence (1961). The complet e c o m p u t a tion t h e n required some 2 0 m i n o f C P U time, using a p a c k a g e d R o m b e r g extrapolation integration routine. Figure 2 shows the initial values (a0, %) for the a r r a y o f asteroids studied. Figures 3, 4, and 5 show the s u b s e q u e n t coordinates

-

O.tl

+ f = t a n -I (y/x), f = cos-t '([a(1 -- e2)/r] --

(23) 1}

/e.

S t a n d a r d expansions in eliptic m o t i o n again give M as a function o f f , e, a n d so ;~ is k n o w n ; since t is known, ~0is found. T h e n (19) a n d (20) m a y be reinitialized and the integration continued. F i f t y asteroids were followed, each t h r o u g h 20 simulated collisions. Initial values were (a 0, %, q% = 0 ) w h e r e a 0 = 0.61, 0.615, 0:62;>,.., 0,655 and % = 0 . 0 5 , 0.1, 0.15, 0.2, 0.25. A t i m e step At = 10.0 was used t h r o u g h o u t , where At = 2~r is the period of J u p i t e r . F o r each asteroid, the c o m p u t a t i o n was begun b y p r o p a g a t i n g (19) and (20) t h r o u g h two cycles of q0 in circulation or libration, or four passes through q0 = 0 m o d u l o 7r. T h e n a collision was simulated, (19) a n d (20) were reinitialized, and a r a n d o m time interval t~ ~< 400 was generated. Thereafter, w h e n the t i m e elapsed since the collision was within A t of t~, a n o t h e r collision was simulated a n d a new value of t c was generated. The c o m p u t a t i o n p r o c e e d e d until t h e 20th collision, w h e r e u p o n t h e

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as is to be expected from a random walk in the (a, e) domain. The variations in a are rather less pronounced (Fig. 7), but are still substantial. No tendency is seen to remove ~z 0,2 asteroids from the region of the Kirkwood • ;. Gap, which extends approximately from 0.i ' ' " a = 0 . 6 2 to a = 0 . 6 4 . On the contrary, there are numerous cases of an asteroid leaving the gap due to a collision, but i i i i i i i i , ', i i , , , , o.y~.~ o.ss o.~ o.~s '0.7 o.Ts returning to the gap in a subsequent SEMIMAJOR AXIS (JUPITER = 1 ) collision. The effect of collisions on the distribuFIG. 4. Same as Fig. 3, after l0 collisions. tion of asteroids is summarized in Fig. 8. (a, e) for each of the 50 asteroids, respec- Four histograms are presented, showing, tively, after 1, 10, and 20 simulated respectively, the distribution in a following collisions. In Fig. 3, it is seen t h a t even one collision numbers 1-5, 6-10, 11-15, and collision suffices to destroy the orderly 16-20. The distributions are given per 0.01 character of the initial array. Figures 4 and ofa. Some 200 data points are presented in 5 show t h a t under further collisions, the each histogram. The first histogram shows a marked general character of the distribution is to expand through a larger extent of the decrease in asteroid density between (a, e) domain, as by a random-walk process a = 0.61 and a = 0.62, almost precisely for each asteroid. No tendency is seen to where one would expect a Kirkwood Gap. form a gap near the 2:1 resonance, at No such decrease is observed after the a = 0.0630. first collision (Fig. 3), so this decrease is Figures 6 and 7 show the evolution of the consequence of collisions 2-5. But in the five selected asteroids, whose initial con- second histogram, this decrease has disditions are, respectively, as follows. (a0, e0) appeared and has been replaced by the =(0.61, 0.05); (0.62, 0.1); (0.63, 0.15); maximum of the histogram, which indi(0.64, 0.2); (0.65, 0.25). :Figure 6 shows the cates t h a t it is a statistical artifact and not evolution in e for these asteroids; the evidence of a gap in formation. This is different styles of line used are solely for confirmed by the last two histograms, the purpose of clarity. I t is seen t h a t three wherein the small fluctuations in asteroidal asteroids show an overall increase in e density are of an evidently statistical between the initial e0 and e after the 20th character. collision, while two show a decrease. The major features of the histograms of Substantial increases in e are seen to occur, Fig. 8 are their approximately Gaussian 0,~

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overall shape and the tendency of successive histograms to show overall broadenand lowering. These features are consistent with diffusion of the asteroids in the (a, e) domain. No tendency is seen to form a gap in the vicinity o f a = 0.63, or ix) develop an

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initially gaplike statistical fluctuation (as in the first histogram) into a deep, Vshaped feature. Thus, on direct test, the Brouwer-Jefferys theory finds no support. The results of this section m a y be extended to a simple simulation of binary collisions. Consider a set of asteroids of equal masses. At time t let asteroid n have elements (a,, e,); let the "nearest neighbor" of asteroid n be asteroid i for which [ ( a , - a ) t 2 + ( e , - et)'-] is the minimum value among all asteroids. Let asteroid n suffer a simulated collision as in (22), and, to conserve momentum, let the nearest neighbor be given equal but opposite perturbations in @, y). Thus the simulated unary collisions of this paper are replaced b y simulated binary collisions. B u t for each asteroid, the statistics of the sequence of perturbations suffered differs in no respect from the statistics of the simulated unary collisions of this paper; hence such a simulation of binary collisions also fails to give rise to Kirkwood-type gaps. The foregoing work may be objected to as being a highly unrealistic model for asteroid evolution b y collisions. The asteroids are not of equal masses; they would be fragmented b y collisions of far less magnitude than are treated here; the random variable R V in (22) has a Gaussian rather than uniform distribution. Such objections are irrelevant because the Brouwer-Jefferys theory is independent of asteroidal mass distribution, collision mechanics, etc. Indeed, such objections strengthen the foregoing findings since they imply that even collisions strong enough to completely shatter the asteroids will nevertheless fail ~o give rise to a Kirkwood Gap. I V . CONCLUSIONS

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Two types of hypotheses have been investigated, both proposing that the Kirkwood Gaps arose through collisions between asteroids. The first holds that asteroids in resonance are preferentially destroyed or broken up in collisions. An examination of this t y p e of hypothesis leads to conclusions which cannot be supported. The second hypothesis holds

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FIG. 8. Histograms of asteroid distribution in semimajor axis. No tendency t o form a gap is observed. t h a t collisions m o d i f y a n a s t e r o i d a l dist r i b u t i o n so as t o give rise t o g a p s a t commensurabilities. But direct simulation of a p o p u l a t i o n s u b j e c t to collisions shows i n s t e a d t h a t t h e i n i t i a l d i s t r i b u t i o n diffuses o u t w a r d in t h e (a, e) d o m a i n ; no t e n d e n c y t o f o r m a g a p is e v i d e n t . Thus, t h e first t y p e of h y p o t h e s i s a p p e a r s i n s u p p o r t a b l e , a n d t h e s e c o n d h y p o t h e s i s finds no s u p p o r t f r o m these investigations. These findings do n o t rule o u t t h e poss i b i l i t y t h a t collisions m a y h a v e p l a y e d a role in f o r m a t i o n of t h e K i r k w o o d Gaps. B u t t h e y i n d i c a t e t h a t collisional effects, if important, must have been more subtle t h a n t h o s e t r e a t e d here. ACKNOWLEDGMENTS It is a pleasure to thank Dr. Peter Goldreich for valuable discussions and helpful criticism. This work was performed under NASA grant NGL 05-002-003. REFERENCES

A.NT)ERS,E. (1965). Fragmentation history of asteroids. Icarus 4, 398-408. BROUWER, D. (1963). The problem of the Kirkwood Gaps in the asteroid belt. Astron. J. 68, 152-159. BEOUWER, D., AND CLE~ENCE, G. M. (1961). MethodsofCelestial Mechanics:AcademiePress, New York.

CHAPMAN,C. R. (1975). The collisional evolution of carbonaceous and chemically-differentiated asteroids. Presented at American Astronomical Society Division for Planetary Sciences 6th Annual Meeting, Columbia, Md., February 17-21, 1975. CHAPMAN, C. R., MORRISON,D., AND ZELLNER, B. (1975). Surface properties of asteroids: A synthesis of polarimetry, radiometry, and spectrophotomctry. Icarus 25, 104-135. DOH~A~rI, J. S. (1969). Collisional model of asteroids and their debris. J. Geophys. Res. 74, 2531-2554. GIFFEN, R. (1973). A study of commensurable motion in the asteroid belt. Astron. Astrophys. 23, 387-403. GREENBERG, R. J., COUNSELMAN, C. C., ANT) SHAPIRO, I. I. (1972). Orbit-orbit resonance capture in the solar system. ~cience 178, 747749. JEFFERYS, W. H. (1967). N0ngravitational forces and resonances in the solar system. Astron. J. 72, 872-875. L•CAR, M., AND FRANKLIN,F. A. (1973). On the original distribution of the asteroids. Icarus 20, 422-436. PIOT~OWSKI,S. (1953). The collisionsof asteroids. Acta Astron., Ser. A 5, 115-138. ScHoLL, H., ANDFRO~.SCHL~,C. (1974). Asteroidal motion at the 3:1 commensurability. Astron. Astrophys. 33, 455-458. W~THERII~, G. W. (1967). Collisions in the asteroid belt. J. (leophys. Res. 72, 2429-2444.