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On the formation of the outer satellite groups of Jupiter
.'ARCS 15, 18ti -189 (1971)
On the Formation of the Outer Satellite Groups of Jupiter ( a. C,()IA)MB() l ",iver.viQl qf l'wlova, l'adoca. Ital!l (ltl,']
Nm~th.s'o~ia'~ A,s'trophff.s'ical ()b.s'er~atory, ('ambridffe, Ma.s.sach usett,~ 0213S AND
F. A. F R A N K L I N Nmithso~dan ,4 stropbysical Ob,s'er~:atory and Harvard College Obsercatory. ( 'ambridge. 31a.~sacb~setlx 0o138 I~ ~(mvod M a r c h 9, 1971 (
"
T h i s p a p e r p r e s e n t s o x i d e n e e s u g g e s t i n g tht~t J u p i t e r ' s s e v e n o u t e r s a t e l l i t e s , w h i c h e x i s t in t w o d i s t r a c t g r o u p s , w e r e f l w m o d I)y a s i n g l e collision o f a n a s t e v . i d a n d a l a r g e r st~tellite.
Kuiper (1961), ill discussing the two groups of J o v i a n satellites, J V1, ,I V I I , and a X, which we shall call g r o u p A, and J V I I I , , I I X , ,l X I , and ,1 XI1, which we shall eall group I3, has suggested, because of evident similarities within eaeh group o f s e m i m a j o r a x e s a n d i n c l i n a t i o n s , t,hat, t h e s e o b j e e t s m a y h a v e been f o r m e d b y t w o r a t h e r t h a n s e v e n e v e n t s . I n t h i s note, we wish t o explore the possibility o f a
f'urther reduction in the n u m b e r of events, fi'om t w o t,o one. Specifically, we wish to ask if' a single collision could aecmmr for both satellite ~,,'roups. There are at least three situat.ions i n v o l v i n g collisions t, hat, might lead t,o tilt' Iwoduetion of several small bodies e w m t u ally f'orn~ing satellite t h m i l i e s (a) a collision of t w o ast, eroids w i t h i n ,lUlfitcr's sphere of i l ] f l u e n e e , (h) a collision between two bound satellites, (el a (,ollision between an asteroid and a satellite. Belin't' undertakin~ a more detailed s t u d y xw, shouhl like to present a few general a r g u m e n t s that seem to make the eollisiomd origin, a~M l m r t i e u l a r l y (('), {luite reasomd)h'. The l'aut t h a i no asteroids are currentJy ,A~serw,d to penetrate the sl~hero of inttu enee ot',l Ulfiter cannel, because of the very short lift,times of such bodies, he used as ;ul a r g u m e n t against ('ither {(') or (a). \V¢'
now know t h a t Jupit, er can deplete the I)opulation ill the asteroid belt, within a region - 1 a.u. from the planet. The small n u m b e r of (non-Trojan) asteroids remaining relatively elose t,o JuI)iter are able to persist there only beeause they are near a resonance 2/3 or 3/4 of' J u p i t e r ' s period), are librating, and hence are prevented from making close apl)roaches to the planet. It therefore seems exeee(lingly likely that ()f the m a n y asteroids t h a t ()nee existed near ,lulfiter, only those with this imrticular t y p e of permanence remain t o d a y , while most of'the initial popuhttion in this region has t>een removed. (kmsequently we wouhl argue that the <,ollision that we postuhtte ()ceurred early in the history of the solar system when the density of asteroidal bodies lle;H' J u p i t e r was hivher. It. seems very probable that, in members of the H i r ~ y a m a o r 1 3 r o u w e r families, u e are witnessing the result of collisions between asteroids. 1n at least t xv4~respects. there is a distinct similarity between these families and Nroups A and B. First t l . ' observed ranoe in certain orbital eh, ments a m o n g t'amih" or vroul~ members shams t h a t the vehwit 5 dispersion of holh is ¢,[' lhe order of I0~ m~sec.. A second simi]arit 3 is eml}hasized hv the results of .\riders (I.qli5). who showed that for m a n \ o f ' t l . '
OUTER
SATELLITE
H i r a y a m a families, the bulk of the mass is contained in one b o d y whose d i a m e t e r is of the order of 100 kin. P r e s u m a b l y such families are the result of a single collision between p a r e n t bodies of r a t h e r different masses. This situation is closely paralleled b y the example of groups A and B, where the largest b o d y ( J V I ) has a d i a m e t e r ~120 km and all the others lie in the 20- to 40-km range. These preliminary t h o u g h t s suggest t h a t the process of f o r m a t i o n of the outer satellite groups of J u p i t e r might resemble collisional effects now a p p a r e n t l y seen in the asteroidal belt and t h a t an analysis of the possible history of groups A and B might be of some i m p o r t a n c e in the s t u d y of collisional processes in the solar system. I f our point of view has merit, it must answer the following questions: (a) Is it possible t h a t the jovicentric distances of the members of groups A and B were ever equal? (b) I f a collision did occur, can the various energy and m o m e n t u m requirements be fulfilled? (c) W h y , subsequent to impact, did the several members of each group have such low relative velocities t h a t their inclinations and semimajor axes have remained v e r y similar? We shall deal with these questions in turn. Thus we first ask whether, in their present or recently past state, the individual orbits of the two groups o v e r l a p - are the peri-jovia of the outer group ever comparable to the apo-jovia of the inner? The answer to this question has a real meaning in the sense t h a t a p a r t i c u l a r response to it has a v e r y high p r o b a b i l i t y of remaining valid for a long time. This outlook, anticipated on the basis of Poisson's theorem, has been established numerically for the outer jovian satellites b y the integrations of H e r g e t (1968). His work shows t h a t variations in the semim a j o r axis, ~a/a, are much less t h a n similar variations in other orbital elements. I f members of the two satellite groups are the result of one i m p a c t some time in the past, it is reasonable to e x p e c t t h a t values of a just after the collision and as observed t o d a y are essentially the same and t h a t an overlap of jovicentric distances would still be found t o d a y , provided only t h a t inte-
GROUPS
OF JUPITER
187
grations are sufficiently extensive to reflect the full range of possible jovicentric distances. Members of group A in general show smaller osculating eccentricities t h a n those of group B ; J X has the smallest average eccentricity of the three inner group members. Calculations of H e r g e t show t h a t the m a x i m u m jovicentric distance of J X in the time interval 1966 to 2000 is 0.0880 a.u. W e shall a d o p t this value as a likely r e p r e s e n t a t i v e m a x i m u m jovicentric distance for members of group A, with the proviso t h a t a longer integration might show s o m e w h a t larger values. We now t u r n to the outer group in which, owing to the greater i m p o r t a n c e of solar p e r t u r b a tions, a greater range in osculating orbital p a r a m e t e r s is likely. H e r g e t ' s results show t h a t two members of the outer group, J V I I I and J X I I , do p e n e t r a t e t o d a y into the region of group A. F o r the f o r m e r satellite, a m i n i m u m jovicentric distance of 0.0685 a.u. is reached in the time interval 1966 to 2000; for the latter, the value is 0.0889 during the same interval. F o r the other two, J I X and J XI, the m i n i m u m jovicentric distances given by H e r g e t in the 34-year interval are 0.1067 and 0.1104 a.u., respectively. A longer integration (~1910 to 2000) for J X I , also given b y H e r g e t (1968), does not indicate lower values. I t is clear, however, t h a t even this longer integration does not span all possible ranges in jovicentric distance. Two v e r y long-term effects t h a t do not a p p e a r in Herget's calculation are changes in J u p i t e r ' s eccentricity (Brouwer and Clemence, 1961) and the rotation of the apse of a satellite's orbit over a comt)lete period. In order to include these two effects and to look more carefully at the minimum j ovicentric distances of these two satellites, we have carried out some new integrations for the case least favorable to our hvpothesis, t h a t of J u p i t e r XI. We have considered several different orientations of the orbit of J X 1 with respect to the S u n - J u p i t e r radius v e c t o r at a given epoch. F o r the case in which we have set the eccentricity of J u p i t e r equal to 0.062 and, at t 0, placed J X I at a p o c e n t e r in its orbit a b o u t the planet and at superior conjunction, we
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G. C O L O M B O A N D F. A. F R A N K L I N
o b t a i n e d a minimum jovicentric distance of 0.089 a.u. Starting positions and velocities for this case correspond, at t - 0, to a semimajor axis of 0.151 a.u. and an eccentricity of 0.21. B o t h these values are o b t a i n e d from H e r g e t ' s analysis and are decidedly conservative representative values t a k e n from osculating ranges. The results are not sensitive to the eccentricity of J u p i t e r . Other integrations, in which the alignment of J X I ' s orbit is comparable to its present value, closely reproduce H e r g e t ' s results. All integrations cover at least seven complete revolutions of J u p i t e r a b o u t the Sun. I t is clear t h a t no f u r t h e r claim can be made for these sample calculations b e y o n d the simple fact t h a t , under certain normally occurring conditions, it is possible for all members of group B to p e n e t r a t e into the region of group A. This knowledge, however, is all t h a t is c u r r e n t l y required b y our hypothesis. We begin a discussion of the second question b y pointing out t h a t a collision of the types m e n t i o n e d in the second paragraph has enough energy to f r a g m e n t the participating bodies. Anders (1965) argues t h a t the crushing strength of asteroidal material is at most a b o u t l0 ~ ergs/gm. F o r two bodies with a relative velocity even as low as a few kilometers per second, the available energy, 1 m a r e B (VA-VB)", 2 m A 4- m a is easily sufficient to f r a g m e n t both. T h a t the outer satellite group is stable in the present dynamical e n v i r o n m e n t seems v e r y likely (Hunter, 1967; Hdnon, 1970). H o w close group B is to escape from J u p i t e r is more difficult to ascertain, but we can make the following estimate. The sphere of influence of J u p i t e r at its mean distance is, according to Tisserand's definition, 0.32 a.u. Since the m a x i m u m eccentricity of J u p i t e r ' s orbit is 0.062, we shall lower this value to 0.3 a.u. and assume t h a t a n y object t h a t once ventures outside this sphere will e v e n t u a l l y be lost from the planet's system. L e t us suppose now t h a t the collision occurred at a distance ro = 0.088 a.u. from J u p i t e r and let us c o m p u t e
the velocities necessary to inject members of' the two groups into orbits with their currently observed semimajor axes. Let I'0 be the circular velocity at the distance r0. Then, in order to have a semimajor axis a N, the required " i n j e c t i o n " velocity, VN, of a particle at r0 is
• /2aNa'N--r0) ~ "'"
l"N = 1 0 [
F o r the members of group A, the velocities I'N show a range of a b o u t 60 m/sec, from 2.86 kin/see ( J V I ) t o 2.92 kin/see ( J X ) ; for group B, a range of nearly 100 m/sec, from 3.64 km/sec ( J X I I ) to 3.73 km/sec ( J I X ) . A velocity of 4.05 km/sec corresponds to an a N of 0.3 a.u. However, bodies with a u substantially less t h a n 0.3 a.u. would still be greatly p e r t u r b e d by the Sun, would acquire apo-jovia greater t h a n this figure, and would consequently escape. Thus, the condition I'N < 3.9 kin/see, or a N < 0.25 a.u., p r o b a b l y represents an upper limit if a b o d y is to remain a jovian satellite. F r o m these brief considerations, we can reasonably infer t h a t the members of group B represent the low-energy tail of the t o t a l i t y of collision fragments and t h a t the mass of the original b o d y entering the collision m a y have been much larger t h a n the sum of the four observed bodies. These remarks a p p e a r to leave two alternatives. First, we m a y suppose t h a t the b o d y responsible for group B had a mass roughly equal to the sum of its four observed members, i.e., a b o u t 1/30 t h a t of group A. In this case, since likely m a x i m u m and m i n i m u m velocities, relative to J u p i t e r at the proposed point of impact, of an incoming asteroid in a direct orbit are ~15 and 5 km/sec, respectively, the COlTeSponding energies are easily sufficient to f r a g m e n t both bodies and, particularly for velocities near 5 km/sec, pose no problem for the bodies of group B to achieve an orbit b o u n d to Jupiter. However, if the parent b o d y of group A initially had a near-zero inclination to J u p i t e r ' s equator, there is not, in such a collision, sufficient m o m e n t u m perpendicular to the orbital vector of b o d y A to produce the present inclination of 28 + for the three members of t h a t group. Hence, the second alternative
OUTER SATELLITE GROUPS OF J U P I T E R
supposes, as we saw above was plausible, t h a t the parent of the members of group B was a more massive body and t h a t most of the collision fragments have either escaped or are present as smaller undetected bodies in the jovian environment. The momentum and energy requirements can all be satisfied (and the inclination changed from 0 ° to 28 °) if the mass of the parent of group B was ~ 1/8 the mass of the group A parent. The spread in semimajor axis, a, among members of group A implies, as we saw above, a velocity range of ~60 m/see. This velocity range corresponds to an inclination range of ~ l °, and this value is in fact the observed spread among members of group A. The same argument when applied to members of group B suggests t h a t the range in inclination should be ~2 °, whereas the actual value is ~15 °. This difference, which m ay in part be due to solar perturbations, suggests t h a t the range in a, or v, t h a t the four members of group B now exhibit m a y not be representative of the total range produced by the collision ; i.e., some of the mass of body B did not go into orbit about Jupiter. The third q u e s t i o n - - w h y members of the two groups show such small velocity ranges t h a t each group is well characterized by a single semimajor axis--is the most difficult for our hypothesis to answer precisely. What makes the question so difficult is the vast range of possibilities inherent in collisional processes. I t is conceivable t h a t a collision could, in addition to producing a partial fragmentation, impart a rotation to the two colliding bodies at the time of impact. A sample calculation shows t h a t a spin rate of ~0.5 rev/hr would, for a 60-kin (radius) body suddenly cracked into several pieces, produce among the fragments a velocity dispersion similar to t h a t shown by members of group A. (Because the rotation rate depends so critically on radius, such a mechanism would require t h a t the parent of group B, if spherical, had a mass only a little less than t h a t of group A in order t h a t the velocity range of both groups be comparable.) On the other hand, a collision between two bodies of similar mass could produce fragmentation
189
without necessarily introducing a great velocity dispersion among the largest fragments. Perhaps the most appropriate comment can be made by returning to the observational evidence provided by asteroids. We have already mentioned the accumulating evidence t h a t the H i r a y a m a families are the consequence of collisions. The existence of such families, as collisional products t h a t have maintained a certain identity, seems to suggest the real possibility of the same phenomenon occurring within the jovian system. We wish to conclude this outline of a possible history of the outer jovian satellites with the following remarks: (a) We expect the outer satellites to show light fluctuations, possibly of a periodic short-term n a t u r e . / b ) The possibility t hat there are m any smaller bodies in orbits somewhat similar to those of the known outer satellites must be considered highly likely. Such objects might be observed by and prove a hazard to future space missions to Jupiter. ACKNOWLEDGMENTS I t is a p l e a s u r e to t h a n k Dr. I. I. S h a p i r o a n d Dr. F. L. W h i p p l e for v a h m b l e criticism a n d useful discussions. I~EFERENCES ANDERS, E. (1965). F r a g m e n t a t i o n h i s t o r y of asteroids. Icarus 4, 399 408. BROUWER, D., AND CLEMENCE, G. M. (1961). O r b i t s a n d m a s s e s of p l a n e t s a n d satellites. I n " P l a n e t s a n d S a t e l l i t e s " (G. P. K u i p e r an(t B. M. M i d d l e h u r s t , Eds.), pp. 31-94. U n i v . of Chicago Press, Chicago, Illinois. H~NON, M. (] 970). N u m e r i c a l e x p l o r a t i o n of t h e r e s t r i c t e d p r o b l e m . VI. H i l l ' s case: Nonperiodic orbits. Astron. Astrophys. 9, 24-36. HERGET, P. (1968). E p h e m e r i d e s of C o m e t Schwassmann-Wachmann I and the outer satellites of J u p i t e r . Publ. Cincinnati Obs. No. 23. HUNTER, R . B. {1967). M o t i o n s of satellites a n d a s t e r o i d s u n d e r t h e influence of J u p i t e r a n d t h e sun. Monthly Notices Roy. Astron. Soc. 136, 245-265. KUIPER, G. P. (1961). L i m i t s of c o m p l e t e n e s s . I n " P l a n e t s a n d S a t e l l i t e s " (G. P. K u i p e r a n d B. M. M i d d l e h u r s t , Eds.), pp. 575-591. U n i v . of Chicago Press, Chicago, Illinois.
On the Formation of the Outer Satellite Groups of Jupiter ( a. C,()IA)MB() l ",iver.viQl qf l'wlova, l'adoca. Ital!l (ltl,']
Nm~th.s'o~ia'~ A,s'trophff.s'ical ()b.s'er~atory, ('ambridffe, Ma.s.sach usett,~ 0213S AND
F. A. F R A N K L I N Nmithso~dan ,4 stropbysical Ob,s'er~:atory and Harvard College Obsercatory. ( 'ambridge. 31a.~sacb~setlx 0o138 I~ ~(mvod M a r c h 9, 1971 (
"
T h i s p a p e r p r e s e n t s o x i d e n e e s u g g e s t i n g tht~t J u p i t e r ' s s e v e n o u t e r s a t e l l i t e s , w h i c h e x i s t in t w o d i s t r a c t g r o u p s , w e r e f l w m o d I)y a s i n g l e collision o f a n a s t e v . i d a n d a l a r g e r st~tellite.
Kuiper (1961), ill discussing the two groups of J o v i a n satellites, J V1, ,I V I I , and a X, which we shall call g r o u p A, and J V I I I , , I I X , ,l X I , and ,1 XI1, which we shall eall group I3, has suggested, because of evident similarities within eaeh group o f s e m i m a j o r a x e s a n d i n c l i n a t i o n s , t,hat, t h e s e o b j e e t s m a y h a v e been f o r m e d b y t w o r a t h e r t h a n s e v e n e v e n t s . I n t h i s note, we wish t o explore the possibility o f a
f'urther reduction in the n u m b e r of events, fi'om t w o t,o one. Specifically, we wish to ask if' a single collision could aecmmr for both satellite ~,,'roups. There are at least three situat.ions i n v o l v i n g collisions t, hat, might lead t,o tilt' Iwoduetion of several small bodies e w m t u ally f'orn~ing satellite t h m i l i e s (a) a collision of t w o ast, eroids w i t h i n ,lUlfitcr's sphere of i l ] f l u e n e e , (h) a collision between two bound satellites, (el a (,ollision between an asteroid and a satellite. Belin't' undertakin~ a more detailed s t u d y xw, shouhl like to present a few general a r g u m e n t s that seem to make the eollisiomd origin, a~M l m r t i e u l a r l y (('), {luite reasomd)h'. The l'aut t h a i no asteroids are currentJy ,A~serw,d to penetrate the sl~hero of inttu enee ot',l Ulfiter cannel, because of the very short lift,times of such bodies, he used as ;ul a r g u m e n t against ('ither {(') or (a). \V¢'
now know t h a t Jupit, er can deplete the I)opulation ill the asteroid belt, within a region - 1 a.u. from the planet. The small n u m b e r of (non-Trojan) asteroids remaining relatively elose t,o JuI)iter are able to persist there only beeause they are near a resonance 2/3 or 3/4 of' J u p i t e r ' s period), are librating, and hence are prevented from making close apl)roaches to the planet. It therefore seems exeee(lingly likely that ()f the m a n y asteroids t h a t ()nee existed near ,lulfiter, only those with this imrticular t y p e of permanence remain t o d a y , while most of'the initial popuhttion in this region has t>een removed. (kmsequently we wouhl argue that the <,ollision that we postuhtte ()ceurred early in the history of the solar system when the density of asteroidal bodies lle;H' J u p i t e r was hivher. It. seems very probable that, in members of the H i r ~ y a m a o r 1 3 r o u w e r families, u e are witnessing the result of collisions between asteroids. 1n at least t xv4~respects. there is a distinct similarity between these families and Nroups A and B. First t l . ' observed ranoe in certain orbital eh, ments a m o n g t'amih" or vroul~ members shams t h a t the vehwit 5 dispersion of holh is ¢,[' lhe order of I0~ m~sec.. A second simi]arit 3 is eml}hasized hv the results of .\riders (I.qli5). who showed that for m a n \ o f ' t l . '
OUTER
SATELLITE
H i r a y a m a families, the bulk of the mass is contained in one b o d y whose d i a m e t e r is of the order of 100 kin. P r e s u m a b l y such families are the result of a single collision between p a r e n t bodies of r a t h e r different masses. This situation is closely paralleled b y the example of groups A and B, where the largest b o d y ( J V I ) has a d i a m e t e r ~120 km and all the others lie in the 20- to 40-km range. These preliminary t h o u g h t s suggest t h a t the process of f o r m a t i o n of the outer satellite groups of J u p i t e r might resemble collisional effects now a p p a r e n t l y seen in the asteroidal belt and t h a t an analysis of the possible history of groups A and B might be of some i m p o r t a n c e in the s t u d y of collisional processes in the solar system. I f our point of view has merit, it must answer the following questions: (a) Is it possible t h a t the jovicentric distances of the members of groups A and B were ever equal? (b) I f a collision did occur, can the various energy and m o m e n t u m requirements be fulfilled? (c) W h y , subsequent to impact, did the several members of each group have such low relative velocities t h a t their inclinations and semimajor axes have remained v e r y similar? We shall deal with these questions in turn. Thus we first ask whether, in their present or recently past state, the individual orbits of the two groups o v e r l a p - are the peri-jovia of the outer group ever comparable to the apo-jovia of the inner? The answer to this question has a real meaning in the sense t h a t a p a r t i c u l a r response to it has a v e r y high p r o b a b i l i t y of remaining valid for a long time. This outlook, anticipated on the basis of Poisson's theorem, has been established numerically for the outer jovian satellites b y the integrations of H e r g e t (1968). His work shows t h a t variations in the semim a j o r axis, ~a/a, are much less t h a n similar variations in other orbital elements. I f members of the two satellite groups are the result of one i m p a c t some time in the past, it is reasonable to e x p e c t t h a t values of a just after the collision and as observed t o d a y are essentially the same and t h a t an overlap of jovicentric distances would still be found t o d a y , provided only t h a t inte-
GROUPS
OF JUPITER
187
grations are sufficiently extensive to reflect the full range of possible jovicentric distances. Members of group A in general show smaller osculating eccentricities t h a n those of group B ; J X has the smallest average eccentricity of the three inner group members. Calculations of H e r g e t show t h a t the m a x i m u m jovicentric distance of J X in the time interval 1966 to 2000 is 0.0880 a.u. W e shall a d o p t this value as a likely r e p r e s e n t a t i v e m a x i m u m jovicentric distance for members of group A, with the proviso t h a t a longer integration might show s o m e w h a t larger values. We now t u r n to the outer group in which, owing to the greater i m p o r t a n c e of solar p e r t u r b a tions, a greater range in osculating orbital p a r a m e t e r s is likely. H e r g e t ' s results show t h a t two members of the outer group, J V I I I and J X I I , do p e n e t r a t e t o d a y into the region of group A. F o r the f o r m e r satellite, a m i n i m u m jovicentric distance of 0.0685 a.u. is reached in the time interval 1966 to 2000; for the latter, the value is 0.0889 during the same interval. F o r the other two, J I X and J XI, the m i n i m u m jovicentric distances given by H e r g e t in the 34-year interval are 0.1067 and 0.1104 a.u., respectively. A longer integration (~1910 to 2000) for J X I , also given b y H e r g e t (1968), does not indicate lower values. I t is clear, however, t h a t even this longer integration does not span all possible ranges in jovicentric distance. Two v e r y long-term effects t h a t do not a p p e a r in Herget's calculation are changes in J u p i t e r ' s eccentricity (Brouwer and Clemence, 1961) and the rotation of the apse of a satellite's orbit over a comt)lete period. In order to include these two effects and to look more carefully at the minimum j ovicentric distances of these two satellites, we have carried out some new integrations for the case least favorable to our hvpothesis, t h a t of J u p i t e r XI. We have considered several different orientations of the orbit of J X 1 with respect to the S u n - J u p i t e r radius v e c t o r at a given epoch. F o r the case in which we have set the eccentricity of J u p i t e r equal to 0.062 and, at t 0, placed J X I at a p o c e n t e r in its orbit a b o u t the planet and at superior conjunction, we
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G. C O L O M B O A N D F. A. F R A N K L I N
o b t a i n e d a minimum jovicentric distance of 0.089 a.u. Starting positions and velocities for this case correspond, at t - 0, to a semimajor axis of 0.151 a.u. and an eccentricity of 0.21. B o t h these values are o b t a i n e d from H e r g e t ' s analysis and are decidedly conservative representative values t a k e n from osculating ranges. The results are not sensitive to the eccentricity of J u p i t e r . Other integrations, in which the alignment of J X I ' s orbit is comparable to its present value, closely reproduce H e r g e t ' s results. All integrations cover at least seven complete revolutions of J u p i t e r a b o u t the Sun. I t is clear t h a t no f u r t h e r claim can be made for these sample calculations b e y o n d the simple fact t h a t , under certain normally occurring conditions, it is possible for all members of group B to p e n e t r a t e into the region of group A. This knowledge, however, is all t h a t is c u r r e n t l y required b y our hypothesis. We begin a discussion of the second question b y pointing out t h a t a collision of the types m e n t i o n e d in the second paragraph has enough energy to f r a g m e n t the participating bodies. Anders (1965) argues t h a t the crushing strength of asteroidal material is at most a b o u t l0 ~ ergs/gm. F o r two bodies with a relative velocity even as low as a few kilometers per second, the available energy, 1 m a r e B (VA-VB)", 2 m A 4- m a is easily sufficient to f r a g m e n t both. T h a t the outer satellite group is stable in the present dynamical e n v i r o n m e n t seems v e r y likely (Hunter, 1967; Hdnon, 1970). H o w close group B is to escape from J u p i t e r is more difficult to ascertain, but we can make the following estimate. The sphere of influence of J u p i t e r at its mean distance is, according to Tisserand's definition, 0.32 a.u. Since the m a x i m u m eccentricity of J u p i t e r ' s orbit is 0.062, we shall lower this value to 0.3 a.u. and assume t h a t a n y object t h a t once ventures outside this sphere will e v e n t u a l l y be lost from the planet's system. L e t us suppose now t h a t the collision occurred at a distance ro = 0.088 a.u. from J u p i t e r and let us c o m p u t e
the velocities necessary to inject members of' the two groups into orbits with their currently observed semimajor axes. Let I'0 be the circular velocity at the distance r0. Then, in order to have a semimajor axis a N, the required " i n j e c t i o n " velocity, VN, of a particle at r0 is
• /2aNa'N--r0) ~ "'"
l"N = 1 0 [
F o r the members of group A, the velocities I'N show a range of a b o u t 60 m/sec, from 2.86 kin/see ( J V I ) t o 2.92 kin/see ( J X ) ; for group B, a range of nearly 100 m/sec, from 3.64 km/sec ( J X I I ) to 3.73 km/sec ( J I X ) . A velocity of 4.05 km/sec corresponds to an a N of 0.3 a.u. However, bodies with a u substantially less t h a n 0.3 a.u. would still be greatly p e r t u r b e d by the Sun, would acquire apo-jovia greater t h a n this figure, and would consequently escape. Thus, the condition I'N < 3.9 kin/see, or a N < 0.25 a.u., p r o b a b l y represents an upper limit if a b o d y is to remain a jovian satellite. F r o m these brief considerations, we can reasonably infer t h a t the members of group B represent the low-energy tail of the t o t a l i t y of collision fragments and t h a t the mass of the original b o d y entering the collision m a y have been much larger t h a n the sum of the four observed bodies. These remarks a p p e a r to leave two alternatives. First, we m a y suppose t h a t the b o d y responsible for group B had a mass roughly equal to the sum of its four observed members, i.e., a b o u t 1/30 t h a t of group A. In this case, since likely m a x i m u m and m i n i m u m velocities, relative to J u p i t e r at the proposed point of impact, of an incoming asteroid in a direct orbit are ~15 and 5 km/sec, respectively, the COlTeSponding energies are easily sufficient to f r a g m e n t both bodies and, particularly for velocities near 5 km/sec, pose no problem for the bodies of group B to achieve an orbit b o u n d to Jupiter. However, if the parent b o d y of group A initially had a near-zero inclination to J u p i t e r ' s equator, there is not, in such a collision, sufficient m o m e n t u m perpendicular to the orbital vector of b o d y A to produce the present inclination of 28 + for the three members of t h a t group. Hence, the second alternative
OUTER SATELLITE GROUPS OF J U P I T E R
supposes, as we saw above was plausible, t h a t the parent of the members of group B was a more massive body and t h a t most of the collision fragments have either escaped or are present as smaller undetected bodies in the jovian environment. The momentum and energy requirements can all be satisfied (and the inclination changed from 0 ° to 28 °) if the mass of the parent of group B was ~ 1/8 the mass of the group A parent. The spread in semimajor axis, a, among members of group A implies, as we saw above, a velocity range of ~60 m/see. This velocity range corresponds to an inclination range of ~ l °, and this value is in fact the observed spread among members of group A. The same argument when applied to members of group B suggests t h a t the range in inclination should be ~2 °, whereas the actual value is ~15 °. This difference, which m ay in part be due to solar perturbations, suggests t h a t the range in a, or v, t h a t the four members of group B now exhibit m a y not be representative of the total range produced by the collision ; i.e., some of the mass of body B did not go into orbit about Jupiter. The third q u e s t i o n - - w h y members of the two groups show such small velocity ranges t h a t each group is well characterized by a single semimajor axis--is the most difficult for our hypothesis to answer precisely. What makes the question so difficult is the vast range of possibilities inherent in collisional processes. I t is conceivable t h a t a collision could, in addition to producing a partial fragmentation, impart a rotation to the two colliding bodies at the time of impact. A sample calculation shows t h a t a spin rate of ~0.5 rev/hr would, for a 60-kin (radius) body suddenly cracked into several pieces, produce among the fragments a velocity dispersion similar to t h a t shown by members of group A. (Because the rotation rate depends so critically on radius, such a mechanism would require t h a t the parent of group B, if spherical, had a mass only a little less than t h a t of group A in order t h a t the velocity range of both groups be comparable.) On the other hand, a collision between two bodies of similar mass could produce fragmentation
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without necessarily introducing a great velocity dispersion among the largest fragments. Perhaps the most appropriate comment can be made by returning to the observational evidence provided by asteroids. We have already mentioned the accumulating evidence t h a t the H i r a y a m a families are the consequence of collisions. The existence of such families, as collisional products t h a t have maintained a certain identity, seems to suggest the real possibility of the same phenomenon occurring within the jovian system. We wish to conclude this outline of a possible history of the outer jovian satellites with the following remarks: (a) We expect the outer satellites to show light fluctuations, possibly of a periodic short-term n a t u r e . / b ) The possibility t hat there are m any smaller bodies in orbits somewhat similar to those of the known outer satellites must be considered highly likely. Such objects might be observed by and prove a hazard to future space missions to Jupiter. ACKNOWLEDGMENTS I t is a p l e a s u r e to t h a n k Dr. I. I. S h a p i r o a n d Dr. F. L. W h i p p l e for v a h m b l e criticism a n d useful discussions. I~EFERENCES ANDERS, E. (1965). F r a g m e n t a t i o n h i s t o r y of asteroids. Icarus 4, 399 408. BROUWER, D., AND CLEMENCE, G. M. (1961). O r b i t s a n d m a s s e s of p l a n e t s a n d satellites. I n " P l a n e t s a n d S a t e l l i t e s " (G. P. K u i p e r an(t B. M. M i d d l e h u r s t , Eds.), pp. 31-94. U n i v . of Chicago Press, Chicago, Illinois. H~NON, M. (] 970). N u m e r i c a l e x p l o r a t i o n of t h e r e s t r i c t e d p r o b l e m . VI. H i l l ' s case: Nonperiodic orbits. Astron. Astrophys. 9, 24-36. HERGET, P. (1968). E p h e m e r i d e s of C o m e t Schwassmann-Wachmann I and the outer satellites of J u p i t e r . Publ. Cincinnati Obs. No. 23. HUNTER, R . B. {1967). M o t i o n s of satellites a n d a s t e r o i d s u n d e r t h e influence of J u p i t e r a n d t h e sun. Monthly Notices Roy. Astron. Soc. 136, 245-265. KUIPER, G. P. (1961). L i m i t s of c o m p l e t e n e s s . I n " P l a n e t s a n d S a t e l l i t e s " (G. P. K u i p e r a n d B. M. M i d d l e h u r s t , Eds.), pp. 575-591. U n i v . of Chicago Press, Chicago, Illinois.