r ( \ R t s 4 8 . 1 9 4 8 ( l 9 8 l)
D e r i v a t i oonf t h e C o l l i s i o nP r o b a b i l i tbye t w e e nO r b i t i n gO b j e c t s :
T h e L i f e t i m eosf J u p i t e r ' sO u t e rM o o n s
DONALD J. KESSLER
N A . S A . . / o / l r t s r r nS n u c c C c n t e r / S N - l ' H t t t t s l r t t t ' 7 e - r u s7 7 0 5 8
R e c e i v e dM a y 2 6 , l 9 8 l l r e v i s e d A u g u s t 3 , l 9 8 l '
Equations are derived which relate orbital parametersto the probability of collision between
o r b i t i n g o b j e c t s . T h e s e e q u a t i o n sf o l l o w f r o m a n e w c o n c e p t u a l a p p r o a c h . a n d a r e i n a f o r m t o b e
easily applied to a variety oforbital collision problems. The equations are used in this paper to
c a l c u l a t e t h e c o l l i s i o n l i f e t i m e o f J u p i t e r ' s e i g h t o u l e r s a t e l l i t e s .T h e a v e r a g e t i m e b e t w e e n c o l l i sions for anl of the four retrograde moons was calculated to be 270 billion years. while lhe
c o r r e s p o n d i n gt i m e f o r t h e f o u r p o s i g r a d em o o n s w a s 0 . 9 b i l l i o n y e a r s . T h i s r e l a t i v e l y s h o r t t i m e f o r
t h e p o s i g r a d e m o o n s i s s t r o n g l y s u g g e s t i v eo f a p a s t c o l l i s i o n h i s t o r y . T h e c o n s e q u e n c e so f t h e s e
c o l l i s i o n s a n d t h e p o s s i b l e r e l a t i o n s h i p1 o t h e P i o n e e r l 0 a n d I I p e n e t r a t i o nd a t a i s d i s c u s s e d .
determine the collisional lifetime of Jupiter's outer eight moons.
I N ' fR O D U C T I O N
0 p i k t 1 9 - 5 1d) e r i v e d a n d p u b l i s h e de q u a t i o n s w h i c h r e l a t e dt h e p r o b a b i l i t yo f c o l l i sion between two orbiting objects to their
o r b i t a l e l e m e n t s .T h e s e e q u a t i o n sh a v e b e "cosmic
w e i g h t i n gf a c c o m e t h e b a s i so f a
tor" which has been used both to correct
for observation selection effects and to pred i c t e x p e r i m e n t a l m e a s u r e m e n t sf r o m o r . o w e v e r .O p i k
b i t a l e l e m e n td i s t r i b u t i o n s H
assumedone object to have zero eccentrici t y a n d i n c l i n a t i o n . W e t h e r i l l ( 1 9 6 7 )e l i m i nated this assumption and generalized
6 p i k ' s e q u a t i o n si n a f o r m w h i c h r e q u i r e s
integration over the orbital path and varying angle between the orbital planes. A
m o r e c o n c i s e d e r i v a t i o no f t h e s e e q u a t i o n s
"spatial
denfollows from a concept of
s i t y . " o r a v e r a g en u m b e r o f o b j e c t s f o u n d
in a unit volume. It is the purposeof this
paper to derive a general form of Opik's
e q u a t i o n su s i n g t h i s c o n c e p t . T h e r e s u l t i n g
equations are in a form to be easily applied
to varity orbital collision problems. They
were used in a previous paper ( Kessler and
Cour-Palais, 1978)to calculate the collision
lifetime of artificial Earth-orbiting satellites. They will be used in this paper to
CONCE[rf
A s s u m et h a t o n e w i s h e s t o k n o w t h e c o l l i s i o n p r o b a b i l i t i e sw i t h i n a v o l u m e o f s p a c e
which is small comparedto any uncertainties in the orbital paths of objects which
p a s st h r o u g h t h a t v o l u m e . T h e p o s i t i o n so f
the objects in the volume can then be assumed to be random. and moving in a
straight line. The situation is then analogous to the kinetic models of a gas where
the flux within the volume is given by
r': sy,
il)
where F is the number of impactsper unit
c r o s s - s e c t i o n aal r e a p e r u n i t t i m e . S i s t h e
spatial density, or the number of objects
found within a unit volume. and V is the
velocity of the objects relative to the detection area. The average number of collisions, .Ay',
on an object of collisional crosssectional area o in time t would then be
given by
N : Fot.
(2)
The collision cross-sectional area between two randomly oriented objects of av-
19
00r9-1035/8
l / l 00039I 0$02.00/0
Copyright O l98l b) Academic Press. Inc.
All rights of reproduction in an) form reserved.
C O L L I S I O NL I F E T I M E S J: U P I T E R . SM O O N S
erage radii rr and 12 ma.ssesmr and m2, a n d w i l l o n l y b e a f u n c t i o n o f d i s t a n c ef r o m
r e s p e c t i v e l y .i s g i v e n b y O p i k ( l 9 - 5 1 )a s
the body and latitude.The latitudedependence is a function of the inclination of the
o : r ( r , t , r ) , ( l + V " z fW ) ,
(3)
o r b i t , w h i l e t h e d i s t a n c e d e p e n d e n c ei s a
w h e r e % , t h e e s c a p e v e l o c i t y i s g i v e n b y function of the pericentron and apocentron
l 2 ( m , + m r ) G / ( r , + r ) 1 1 t 2 ,a n d G i s t h e d i s t a n c e sT. h u s i t i s c o n v e n i e n t o w r i t e :
u n i v e r s a lg r a v i t y c o n s t a n t . N o t c t h a t i f t h c
(-5)
s ( R ,B ) : f l P ) ' s ( R ) ,
escape velocity is small compared to exp e c t e d i n t e r s e c t i o nv e l o c i t i e s ,t h e v e l o c i t , w h e r e s ( R ) i s t h e s p a t i a l d e n s i t y a t a d i s t e r m s i n E q . ( 3 ) m a y b e i g n o r e d . E q u a t i o n s tance R from the central body averaged
( l ) a n d ( 2 ) a r e v a l i d e v e n i f o n l y o n e o b j e c t o v e r a l l l a t i t u d e s ,a n d l F ) i s r h e r a r i o o f t h e
p a s s e st h r o u g h t h e u n i t v o l u m e a n d s p e n d s spatial density at latitude
B to the spatial
o n l y a s m a l l f r a c t i o n o f t i m e w i t h i n t h e v o l - d e n s i t ya v e r a g eo v e r a l l l a t i t u d e s .T h i s r a t i o
u m e . I n t h i s c a s eS m a y b e t h o u g h to f a s t h e i s i n d e p e n d e n to f R f o r a p a r t i c u l a r o r b i t .
p r o b a b i l i t yo f s e e i n gt h e o b j e c t w i t h i n a u n i t a n d i s o n l y a f u n c t i o n o f i n c l i n a t i o n .
volume if one looked at random times. and
'V becomes the probability of collision.
R A D I A LD T ] P E N D E N C E
H o w e v e r , s t r i c t l y s p e a k i n g ,S i s s t i l l t h e
average spatial density. and N the average
Assume a spherical shell of radius R,
n u t h b e r o f c o l l i s i o n s . I n g e n e r a l ,t h e v a l u e t h i c k n e s s A R , b e t w e e n t h e p e r i c e n t r o n( / ,
of S for a single object is found from
a n d a p o c e n t r o n ,. / ' . d i s t a n c e so f a n o r b i t .
The volume of the shell is
(4)
S : AI/TL^U.
(6)
LU : 4rR2AR.
where At is the time spent in the volume
element A U over time T. Time 7 is as- For each revolutionof an object in the ors u m e d t o b e l o n g e n o u g h t h a t a l l r e g i o n so f b i t , t h e o b j e c t p a s s e s t h r o u g h t h e s h e l l
space accessible to the object have been t w i c e . T h u s , a s s e e n i n F i g . 1 , t h e t i m e i n
t r a v e r s e d . T h u s , t h e c o l l i s i o n p r o b a b i l i t y t h e s h e l lf o r o n e p e r i o d o f t h e o r b i t i s g i v e n
for orbiting objects is reduced to defining b v
these parameters in terms of orbital ele(7)
A , t: 2 A ^ R / V . ,
ments.
DERIVATION
A s O p i k ( 1 9 - 5 1p) o i n t e d o u t , s e c u l a r p e r turbations lead to progressive changes in
the longitude of the node and of the argument of pericentron. The time for these two
parameters to complete one revolution is
usually small compared to the time interval
of interest. Consequently, the distribution
of these two parameters for sporadic meteors, asteroids, comets, or artificial satellites around the Earth are nearlv random.
T h u s . b o t h 0 p i k r t e Sl ) a n d W e t h e r i l t( 1 9 6 7 )
assumed that all values for these two parameters were equally probable. That assumption will also be made here. Under
this assumption, spatial density around a
central body will not vary with longitude,
F t c . l . R a d i a l d e p e n d e n c eo f s p a t i a l d e n s i t y
DONALD J. KESSLER
A I
a l
where V. is the radial velocity component
LATITUDE DEPENDENCE
of the object velocity vector relativeto the
The latitude dependence of spatial dencentral body. The orbital period is given by s i t y i s d u e
e n t i r e l y t o t h e i n c l i n a t i o n ,i . o f a n
and
the secular change in the arguorbit,
(8)
T : 2 n ( a 3I p " ) 1 1 2
m e n t o f p e r i c e n t r o n .I m a g i n ea n o b j e c t a t a
w h e r e t h e s e m i m a j o r a x i s , r r , i s e q u a l t o d i s t a n c eb e t w e e nR , a n d R r + A R f r o m t h e
(q * cl')12, and p is the universal gravita- c e n t r a l b o d y . l m a g i n e a l s o t h a t i t h a s
t i o n a l c o n s t a n tt i m e s t h e m a s so f t h e c e n t r a l stopped moving in its orbit. but the argub o d y . T h e v e l o c i t y o f a n o r b i t i n g o b j e c t m e n t o f p e r i c e n t r o nc h a n g e sa t a c o n s t a n t
rate (i.e., all values for the argument of
relativeto its central body is
p e r i c e n t r o na r e e q u a l l y p r o b a b l e ) .S u c h a n
(9) object would sweep out a circle of radius
v" - ptt'(2lR - lf a)tt2.
R , . T h e p l a n eo f t h e c i r c l e w o u l d b e i d e n t i The value of V. is then
cal to the plane of the orbit, so that the
:
(
V,. sin 7
V,
r0) a n g l e b e t w e e n t h e c i r c l e a n d t h e e q u a t o r i a l
p l a n e i s t h e o r b i t i n c l i n a t i o n .r . T h e s p a t i a l
w h e r e 7 i s t h e a n g l e b e t w e e n t h e o b j e c t d e n s i t ya v e r a g e do v e r a l l l a t i t u d e si s s i m p l y
velocity vector and the tangent of the I divided by the volume of the spherical
s p h e r e s h o w n i n F i g . l . C o n s e r v a t i o n o f s h e l l .o r
a n g u l a r m o m e n t u m r e q u i r e st h a t ,
. s ' ( R , :) l l 4 t R 1 2 L , R .
il4)
:
(l l)
c c ' r s72 q q ' l R ( 2 a R ) .
The spatial density between latitude B
C o m b i n i n gE q s . ( 9 ) . ( 1 0 ) .a n d ( l l ) g i v e s a n d B + A B i s g i v e n b y ( 4 ) . w h e r e A r i s n o w
t h e t i m e t h e o b j e c t i s b e t w e e nB a n d B + A B
the radial velocitv to be
during one revolution of the argument of
p e r i c e n t r o n ,f i s t h e t i m e f o r t h e a r g u m e n t
a/
t o m a k e o n e r e v o l u t i o n .a n d A L / i s t h e v o l umebetween and + Lp.lf the argument
( t - = = o -o ' = ) - l " '(. l l ) o f p e r i c e n t r oBni s m oBv i n g a t a n g u l a rv e l o c i t y
U
Rt2u R)l l
<r.r,
then from Fig. 2,
u ' -[ ' ( i _ 1 )
(l-s)
L,r, - l\,Blor sin a.
T h u s . c o m b i n i n gE q s . ( 6 ) . ( 7 ) .( U ) ,a n d ( l 2 )
l l i t h E q . t 4 ) . i r n t lr c u r r i r n g i n g
lerms.
w h e r e a i s t h e i . r n g l eb e t w e e nt h e p a t h o f t h e
o
b j e c t a n d t h e l i n e o f c o n s t a n t l a t i t u d e .T h e
:
l/4#Ro[(R
.r(R)
R)]','. (13)
4)(q'
factor of 2 results fiom the object going
w h e r e q < . R < r 7 ' .W h e n R < q o r R > r 7 ' , t h r o u g h t h e v o l u m e t w i c e f o r e a c h r e v o l u . s ( R ): 0 . s ( R ) i s t h e s p a t i a l d e n s i t y a v e r - t i o n o f t h e a r g u m e n t . F r o m t h e g e o m e t r y .
a g e d o v e r a l l l a t i t u d e sa t a d i s t a n c e1 l f r o m
(l6)
T'r - 2nla
t h e c e n t r a l f o r c e . I t m a y a l s o b e t h o u g h to f
a s t h e p r o b a b i l i t y o f f i n d i n g a n o r b i t i n g o b - and
j e c t w i t h i n a s h e l l o f r a d i u sR a n d u n i t v o l L U t : 2 n ' R , 2c o s B A B A R
(ll)
u m e . T h e l a t i t u d ea t w h i c h t h i s o b j e c t m a y
so that
be found is derived next.
s'(R,,
B): #e:
Ztr"Rrt sin a cosBAll
(l8)
42
DONALD J. KESSLER
where 0 < B < i. S'(Rr, B) is rhe spatial (20) would not changeif the object were
densityat R, and B, given unit probability allowedto movein its orbit (i.e.. haveless
that the objectis betweenR, andRr + AR. than unit probability of being founcl beBy definition,/(F)is the ratioof Eq. (18)to tween R, and Rr + AR). In this general
E q . ( 1 4 )o r
case,both Eqs.(t+) and il8) wouldcontain
a multiplierwhich reflecredthe probabilitv
(19)
s i na c o sB
f(Bl:2lr
of findingthe objectberweenRi and R, i
wherethe valueof a is found1lomspherical AR. This multiplierwould be divideclout.
trigonometryin Fig. 2:
and not appearin Eqs. ( l9) or (20).
Thus,the spatialdensityat any particular
cosd:cosi/cosB.
( 20)
distance
and latitudeis foundby combining
N o t e t h a t t h e g e n e r a l i t yo f E q s . ( 1 9 ) a n d E q s . ( S )(, 1 3 ) t, t q ) , a n d( 2 0 ) o
,r
s(R,B):
I
2f Ral(sin2i- sin?XR - q)(q' - R)lttr
w h e r eq < R < q ' a n d 0 < p < i . W h e n R <
q, R > q'. or B > i,S(R, B) : 0. The
s i n g u l a r i t i e sw h i c h o c c u r a t R : a , R : q '
and B : i are integrable, as shown in Appendix A, where numerical techniques are
developed.
T h e g e n e r a l i t i e se x p r e s s e db y E q s . ( l ) ,
(2), and (21) provide a convenient framework for a variety of collision problems.
For each application, the relative velocity
is required: Appendix B contains a derivation of the relative velocity between two
orbiting objects. The following examples il-
lustratethe utilityof theseequations.
COLLISION
P R O B A B I L I TBYE T W E E N
A
S P A C E C R A FATN D A N O R B I T I N G
OBJECT
Assume a spacecraft is located for a time
/, at a distanceR from the central force and
a t l a t i t u d eB . A s s u m e a l s o t h a t a n o r b i t i n g
object has orbital parameters t1. t1'. i, an
equally probable longitude of the node and
a r g u m e n t o f p e r i c e n t r o n ,a n d a v e l o c i t y V
relative to the spacecraft. The probability
of collision follows from combining Eqs.
( l ) , ( 2 ) , a n d ( 2 1 ) .o r
Vot
2 n 3 R a l ( s i n z i- s i n ' B ) ( R - q ) ( q ' -
C O M P A R I S O N T O O P I K ' S ( I 9 . 5I ) R E S U L T S
6pik
essentially assumed that the
"spacecraft"
in the previous example was
in the equatorial plane and in a circular
o r b i t . S e t t i n CB : 0 i n E q . ( 2 2 )g i v e sO p i k ' s
results of
(:l)
R)lttz
(22\
t h e e x p r e s s i o ni s i n t e g r a t e do v e r t i m e . B o t h
R and V would vary with time as the obiect
i n t h e e q u a t o r i a l p l a n e p r o g r e s s e di n i t s
orbit.
COLLISION
P R O B A B I L I TBYE T W E E N
TWO
ORBITINC
O B J E C T(SG E N E R A L
CASE)
Vot
A s s u m eS , [ g i v e nb y E q . ( 2 1 ) ]i s r h e s p a ( 23)
ZnxRa sin i l(R - q)(q' - R)ytz
tial density from an object with orbital eleNote that Eq. (23) need not be restricted ments qr, c1'r,andi,. The probability that the
to circular orbits if r is replaced withr/r, and object will be found within the volume eleN :
C O L L I S I O N L I F E T I M E S :J U P I T E R . SM O O N S
P A T H O F O S J E C TW I T H
CONSTANR
T AND
ARGUMENT
OF PERICENTRON
VELOCITY
ANGULAR
FIc. l. Latitudedependence.
ment riU in any one instant is SrtIU. Assume also a second object can pass through
the same volume. The spatial density from
t h i s o b j e c ti s S r , a l s og i v e n b y E q . ( 2 1 ) ,a n d
i t h a s o r b i t a l e l e m e n t s4 2 , 4 1 , a n d i r . W h e n
the first object is in the volume element, the
probability of collision per unit time with
t h e s e c o n do b j e c t f o l l o w s f r o m E q s . ( l ) a n d
(2) and is S2Vo, where V is the relative
velocity between the two objects and o is
t h e i r c o l l i s i o nc r o s s s e c t i o n .T h u s , t h e c o l l i sion frequency within the volume element
is S2VoS,rlU, and the collision frequency
i n a l l v o l u m e e l e m e n t s a c c e s s i b l et o b o t h
objects is
.
Nlt:l
l
S$zVodU.
(24)
J volumr
Equation (24) may be integrated numeric a l l y b y a t e c h n i q u eu s e d i n t h e a p p e n d i c e s .
This technique averages spatial density
w i t h i n a l a r g e v o l u m e e l e m e n t .T h u s , f e w e r
volume elements are required to perform
the integration. Having developed the necessary equations, the collision frequency
b e t w e e nJ u p i t e r ' so u t e r m o o n s w i l l n o F b e
calculated.
x 106to 32 x 106km. The otherfour have
between24.8and 29'.with disinclinations
tancesrangingfrom 9.3 x 106to 14.2x 106
km. Note that the two groups seemto be
independentin that they do not intersect
with one another.The orbital and physical
datafor thesesatellitesare givenby Morrisonand Burns(1976)andshowinin TableI.
The radii of the two larger satellitesare
knownwithin l0 km: however.the radiifor
the remainingsatellitesare uncertain by
about a factor of 2. Solar perturbations
dominatethe motionof thesemoons.Szethat the orbitsof
behely( 1978)determined
the four retrogrademoons are unstable,
a captureorigin. However, the
suggesting
orbits of the four posigradewere found to
be stable.The time requiredfor secularperturbationsto rotate the argumentof pericentron and longitude of node is very
short-a few hundredyears(Kozai, l98l),
hence,the assumptionthat valuesfor these
two parameterare equallyprobableseems
justified.
The orbital parametersand diametersin
TableI were usedin Eq. (3) and (24)using
the numericaltechniquesexpressedin Eq.
) f Appendix
( 3 A ) , ( l 4 A ) , a n d ( 8 A ) - ( l 3 . Ao
(
2
B
)a n d ( 3 8 ) i n
A , a n d t h e v e l o c i t yE q s .
AppendixB. Valuesfor AR were variedto
testsensitivityand rangedfrom 0.2 x 106to
in
2 x 106km, with only minorfluctuations
the collisionprobabilities.A valueof AB :
3' was used.
The rate of collisionsbetweenthe posiTABLE I
O n s t r n r e . l o P t t v s t c a rD A r A F o RJ u p l r E ns
O u r e n S e .E
r r r t t es
L I F E T I M EO F J U P I T E R 'O
SU T E R
COLLISION
MOONS
Eight of Jupiter's moons are in irregular
orbits. These moons are the most distant
from Jupiter and can be divided into two
groups. One group consists of four retrograde orbits, with inclination ranging from
l4-5to 16f , and distancesranging from 14.4
+J
6 Himalia
7 Elara
l0 Lysithea
l3 Leda
8 Pasiphae
9 Sinope
I I Carme
l2 Ananke
Pericentron
( lff km)
Apocentron
( 1 0 6k m )
lnclination
(deg.)
Radius
(km;
9.66
9.ll
1 0 .I 9
9.49
t4.4
t7.l
t 7. 7
t7.2
13.28
t4.t7
13.23
t2.'73
32.2
30.3
27.0
24.2
27.6
24.8
29.0
26.7
14,5
1 53
164
t47
8,s
40
8
.l !^ t
9
l0
1
44
D O N A L DJ . K E S S L E R
g r a d e m o o n s w a s c a l c u l a t e dt o b e l . 1 3 x
l 0 - e y e a r . o r I c o l l i s i o ne v e r y 9 0 0 m i l l i o n
years. The retrograde group has a much
smaller collision rate of 3.7 x l0-r2lyear, or
I collision every 270 billion years. These
t i m e s a r e c o n s i s t e n t w i t h t h e e s t i m a t e so f
P o f l a c kc t u l . ( 1 9 7 9 ) T
. h u s . u n l e s st h e r e i s a
significant population of retrograde moons
just below the threshold of detecrability,
the retrograde population could have been
fairly stable for the last 4._sbillion years.
However, this is apparently not true for the
p o s i g r a d em o o n s .
W i t h a n a v e r a g ec o l l i s i o n r a t e o f l . l 3 x
l O - e / y e a r ,t h e r e i s o n l y a 0 . 0 0 6 p r o b a b i l i t y
t h a t t h e f o u r p o s i g r a d es a t e l l i t e sc o u l d h a v e
e x i s t e d f o r 4 . _ 5b i l l i o n y e a r s w i t h o u t c o l l i d i n g . T h u s , u n l e s ss o m e r e s o n a n c ee x i s t s b e tween the orbits of these four moons. or
u n l e s st h e m o o n s h a v e o n l y r e c e n t l y b e e n
captured, or recently originated from the
breakupof a singleparent, there probably
h a v e b e e n c o l l i s i o n sb e t w e e n m o o n s i n t h i s
r e g i o n .T h e i m p l i c a t i o n sa r e t h a t t h e r e m u s t
h a v e b e e n o t h e r o b s e r v a b l es a t e l l i t e si n t h e
past. Collisions reduced the number of
l a r g e r s a t e l l i t e s .b u t i n c r e a s e ds i g n i f i c a n t l y
a population of smaller objects resulting
from the fragmentation of these larger obj e c t s . T h e s a t e l l i t e sc u r r e n t l y o b s e r v e da r e
e i t h e r t h o s e w h i c h j u s t h a p p e n e dt o h a v e
n o t c o l l i d e d ,o r a r e t h e r e m a n t so f c o l l i s i o n s
in the past.
The averagecollision velocity between
t h e p o s i g r a d es a t e l l i t e sw a s c a l c u l a t e dt o b e
about 2 km/sec. At this velocity,according
t o D o h n a n y i( 1 9 7l ) t h e c o l l i s i o nw o u l d b e
catastrophic if the ratio of the largest satell i t e r a d i u st o t h e s m a l l e s tr a d i u s w e r e a b o u t
l 0 o r l e s s , a n d i f t h e s t r u c t u r eo f t h e s a t e l lites were similar to basalt. Davis cl zr1.
(1979\ point out that for objects as large as
Himalia and Elara, gravity is important, reducing this ratio to 4 and 6, respectively.
Table II breaks the collision rates down to
the individual satellites and denotes
whether or not the collision would be catastrophic or not. The uncertainty shown in
the catastrophic vs noncatastrophicpredic-
TABLE II
C - ' o r , r - r s rRonl tl e s t r w , r . t - .I tN! o r v r o u l r p o s r c n a o r
S , qt r . t . L t t l _ s
Collision pair
6 Himalia-7 Elam
6 Himalia-10 Lysithea
6 Himalia-13 Leda
7 Elara-10 Lysithea
7 Elara-ll Leda
l0 Lysithea-ll Leda
'Iotals
Collision
rate/year
4.3 x
1.8 x
3 . 1x
5.1x
5.7 r
1.9 x
Ll3 x
10-10
l0-ro
16to
l0-tt
1g tt
lo-''
l0 e
C a r a s t r o p h (i cC )
or noncatis
r r o p h i c( N )
C
N
N
cl
N l
('l
4.9 i I0-'o((').
6 . - 5i l 0 - r 0 ( N )
t i o n r e s u l t s f r o m t h e u n c e r t a i n t yi n t h e a c tual size of the satellites.An additional
uncertainty would result from the uncertainties in composition. impact mechanicsa
, n d a c t u a lc o l l i s i o nv e l o c i t y .H o w ever, note that approximately half of the
c o l l i s i o n s w o u l d b e c a t a s t r o p h i ca n d h a l f
noncatastrophic.
T h u s , o n e c o u l d i m a g i n ea n o r i g i n a l p o p ulation of several times its current size.
Catastrophic collisions would have occ u r r e d o v e r t h e p a s t 4 . - 5b i l l i o n y e a r s .T h e s e
collisions may have produced large fiagments which would have been observable
f r o m E , a r t h ;h o w e v e r , m o s t o f t h e s e f r a g m e n t s w o u l d h a v e d i s a p p e a r e db y n o n c a tastrophiccollisions with the larger satell i t e s . I t i s p o s s i b l et h a t l 3 L e d a r e p r e s e n t s
the largest fragment produced from such a
c a t a s t r o p h i cc o l l i s i o n : h o w e v e r . n o t e t h a t
i t s m o s t p r o b a b l ef a t e i s t o c o l l i d e w i t h e i ther 6 Himalera or 7 Elara in a noncatast r o p h i c c o l l i s i o n .T h e s e t w o l a r g e r s a t e l l i t e s
must contain some very largecraters.The
disappearanceof these larger fragments by
n o n c a t a s t r o p h i cc o l l i s i o n s c o u l d a c c o u n t
for not observing the larger number of fragments usually expected from collisional
t h e o r y ( G e h r e l s ,l 9 7 l ) .
H o w e v e r , j u s t b e l o w t h e d e t e c t a b l el i m i t
from Earth there may be an increasing
n u m b e r o f f r a g m e n t sw i t h d e c r e a s i n gs i z e .
In fact, if the detectable limit were at the
s a m e s i z e i n t h e a s t e r o i db e l t a s a t J u p i t e r .
regionsof the asteroid belt around 2 AU
C O L L I S I O NL I F E T I M E S J: U P I T E R . SM O O N S
would appearto have a similar size distribution as Jupiter's moons. However, for
s i z e s s m a l l e r t h a n 4 t o 11 k m r a d i u s ( d e pending on albedo). the number of astero i d s a t I A U i s o b s e r v e dt o i n c r e a s er a p i d l y
w i t h t l c c r e . u s i n sgi z c . ( s c e K e s s l e r , l 9 7 l )
lhis larger numher of smaller objects
w o u l d c a u s ea n e v e r l a r g e r n u m b e r o f c a t a strophic and noncatastrophic collisions
among an unobserved population. These
c o l l i s i o n sw o u l d p r o d u c e a n a l m o s t c o n t i n u o u s s o u r c eo f d u s t . T h e P o y n t i n g - R o b e r t son effect would cause this dust to spiral
out of the collisionregioninto lower orbits
a r o u n d J u p i t e r . E v e n t u a l l y t h e s ed u s t p a r t i cles would either destroy each other
t h r o u g h m u t u a l c o l l i s i o n so r e n t e r t h e a t m o s p h e r eo f J u p i t e r . I f a s u f f i c i e n tp o p u l a t i o n
o f t h e s e p a r t i c l e ss u r v i v e m u t u a l c o l l i s i o n s ,
the particle flux from this orbiting dust
cloud could easily exceedthe flux fiom interplanetary' meteoroids. especially at
s m a l l e rr a d i a l d i s t a n c e sf i o m J u p i t e r . S i n c e
v o l u m e d e c r e a s e sw i t h d e c r e a s i n g r a d i a l
d i s t a n c e .b o t h s p a t i a l d e n s i t y a n d v e l o c i t y
w o u l d i n c r e a s e c. a u s i n gt h e o r b i t i n g p a r t i c l e
f l u x t o i n c r e a s ea s J u p i t e r i s a p p r o a c h e d .
T h i s i s e x a c t l y w h a t w a s o b s e r v e db y P i o neers l0 and I I as they flew past Jupiter.
A n i n c r e a s ei n f l u x c a n b e c a u s e db y g r a v i t a t i o n a l c o n c e n t r a t i o no f t h e i n t e r p l a n e t a r y
flux. However. Humes (l9tt0) has recently
determined that the near circular orbits
a r o u n d t h e S u n w h i c h w o u l d b e r e q u i r e dt o
c a u s e t h e l a r g e g r a v i t a t i o n a li n c r e a s e do b s e r v e d b y P i o n e e r a r e n o t c o n s i s t e n tw i t h
t h e f l u x m e a s u r e m e n t sm a d e j u s t p r i o r t o
and after the Jupiter flyby. Thus, an orbiti n g p o p u l a t i o na r o u n d J u p i t e r i s r e q u i r e dt o
e x p l a i n t h e m e a s u r e m e n t sT. h e s o u r c e o f
t h e s ep a r t i c l e sc o u l d b e i n d i r e c t l yf r o m p a s t
collisions between Jupiter's posigrade
outer moons.
C]ONCLUSIONS
General equations were derived which
relate orbital parametersto collision probabilities between orbiting objects. These
equations predict a relatively long colli-
45
sion lifetime for Jupiter's four retrograde
moons. and a shorter collision lifetime for
the four irregular posigrade moons. Past
c o l l i s i o n sb e t w e e n t h e m o o n s i n t h i s r e g i o n
may have produced an orbiting fragment
p o p u l a t i o nt o o s m a l l t o b e s e e nf r o m E a r t h .
C l o l l i s i o n sb e t w e e n t h e s e f r a g m e n t sm a y b e
r e s p o n s i b l ef o r t h e o r b i t i n g d u s t d e t e c t e d
b y P i o n e e r sl 0 a n d 1 1 .
A P P E N I )XI A
A v E R n , c eS p , q , ' r ' r ,D
q ,et N
. stlv wl.rHIN A
F
V o l u v E E l e n a n N r : D e v E L o p t r , l E No T
i ' J i $ { t P . (i 4 L / a t i t ' , 6 ; r '
Periodic perturbations will cause the
p e r i c e n t r o n ,a p o c e n t r o n a n d i n c l i n a t i o n t o
m a k e s m a l l c h a n g e ss o t h a t t h e s e d i s t a n c e s
are never exactly the sameover any interval of time. Even if they were. it is physic a l l y i m p o s s i b l et o p u t a s e c o n d o b j e c t a t
exactly that distance-there will always be
s o m e u n c e r t a i n t yi n t h e l o c a t i c l n .T h u s . i t i s
n o t o n l y m o r e m e a n i n g f u lt o d e t e r m i n et h e
a v e r a g es p a t i a l d e n s i t y w i t h i n s o m e f i n i t e
v o l u m e , c o m p u t e r i n t e g r a t i c l n so v e r v o l ume are much quicker by taking large volu m e e l e m e n t s .I n t h i s a p p e n d i x .e q u a t i o n s
will be developed which average spatial
density over a finite volume. These equat i o n s a r e u s e f u l i n t h e a p p l i c a t i o no f n u m e r ical techniques.
The average number of objects found
within a finite volume is simply the integral
of spatial density over that volume. The
a v e r a g es p a t i a l d e n s i t y i s t h e n s i m p l y
s-JsdulJdt_t.
ilA)
E q . ( 1 A ) w i t h ( - s )a n d b r e a k i n g
Combining
dU into the componentsof dR and dB las in
E q .( l 7 ) 1 .
'e : @ . r a A )
I I R 2c o sB , l R l B
Thus, it becomesconvenientto define
S ( R , R , P . P ' ) : / ( 8 , B ' ) i ( R .R ' ) . ( 3 A )
46
D O N A L DJ . K E S S L E R
where
Sinces : 0 outsidetheselimits,
ln s(R)R' dR
i ( R .R ' ) : ' ' n ,
(44)
J(R. R'
4n2aA,RR
| , n - (, 2 R ,-' ? a )
(eA)
L t ' \ ( . t- q t - T
z l)
li'n'an
and
v^
['' nB, cosBdB.
fl?.p' 1: qJ,-,cosBdB
when R '< q and R' = q'
(_5A) . r ( R . R ' :)
I
4r2aARR
./)R - ),,\'l
ltr -
sin-'
\?;))
Li
S ( R , R ' . B , p ' ) i s t h e n t h e a v e r a g es p a r i a l
when R, :- q, and R > (1
d e n s i t y b e t w e e nR a n d R , . a n d b e t w e e n
B
a n d B ' . i ( R . R ' ) i s t h e a v e r a g es p a t i a ld e n and
s i t y b e t w e e n R a n d R ' , a v e r a g e do v e r a l l
latitudes. and.llF, p') is the ratio of the
i(R, R') : |/4rR2AR
a v e r a g es p a t i a ld e n s i t y b e t w e e n a n d
B
B, to when R -: q and R, > q,.
the average spatial density over all latiE q u a t i o n ( - 5 A )i n t e g r a r e st o
t u d e s . I t i s c o n v e n i e n tt o d e f i n e
AR:R'_R
and
^B:B'-8,
(64)
iloA)
illA)
t t B' . B:' )zr(srn
.-j-
B' - sin6)
- sin-l(ff)],,ror
[,'"-'(+s,J
O A ) w h e nB ' < i .
G i v e n s ( R ) a n d f ( 9 i n E q s . ( 1 3 ) a n d ( 1 9 ) , S i n c e / : 0 w h e n B- 1
r e s p e c t i v e l y .b o t h E q s . ( 4 , A )a n d ( - 5 { ) g 3 n
2
b e i n t e g r a t e de x a c t l y . H o w e v e r , t h e i n t e _ n a .a ' t:
'
zr(sinB' - sin B)
g r a l t o E q . ( 4 A ) i s n e e d l e s s l yl o n g a n d c o m _
p l e x , i f A R i s s m a l l c o m p a r e dt o R . I f t h i s i s
[t - .,.-,/yF\'l ,,,^
t r u e , s o m e a v e r a g ev a l u e o f d i s t a n c e .R c a n
Lz-s'n \affi/l {rrA)
be definedbetween R and R, and can be when
B' > i.
brought out of the integral.Equation (4,{)
Thus, Eq. (3A) can be used with Eqs.
t h e n i n t e g r a t e st o
( 6 , 4 ) - (l 3 A ) r o d e r e r m i n e a v e n r g e s p a t i a l
d e n s i t i e sn e a r p e r i c e n t r o n .a p o c e n t r o n ,o r
a t l a t i t u d e sn e a r t h e i n c l i n a t i o no f t h e o r b i t .
j(R, R' ) =
These equations are extremely useful in
4n2uLRR
c a l c u l a t i n gt h e c o l l i s i o n p r o b a b i l i t i e sw h e r e
- zo)
fL r i n - '\( 2q R-, q /
it is necessaryto numerically integrateover
the volume of space where the two orbits
- sin , (2t: t'\l ,ro, may intersect. Values of AR and Ap can
be
\ q - q / J
f a i r l y l a . r g e .s i n c e S i s a l r e a d y t h e r e s u l t o f
an integration over volume: one constraint
when R > q and R' < q,.When R : *
is that the volume elements cannot be so
( R + R ' ) a n d A R < 0 . I R , t h e e r r o r i n t h i s large
that velocity varies significantly
equation was found to be much less than within the element.
Such a constraint also
l7c when compared with the exact inteeral ensures that AR is small
compared to R, as
of Eq. (4,{).
requiredfor Eqs. (8A) through (llA). An-
C O L L I S I O N L I F E T I M E S :J U P I T E R . SM O O N S
47
other constraint results when calculating
the collision probability between two obj e c t s a s i n E q . ( 2 4 ) , w h e r e t h e s p a t i a ld e n sity of botlt cannot significantly vary within
the volume element. However, if only
one oI the spatial densities is varying
s i g n i f i c a n t l yb e t w e e n R a n d R ' o r b e t w e e n
B and B'. then an average of the other can
be brought out of the integral, along with
V o ( a s s u m i n gt h e i r v a r i a t i o n i s a l s o s m a l l ) and the integral reduces to one spatial density over volume. This is the same integral
e v a l u a t e di n t h i s a p p e n d i x , s o t h a t E q . ( 2 4 )
could be written
.\'t = rr
j:
:
5,,SrrVr-\Ur, (l4A)
Frc. Bl. Velocity vectorrelationships.
r olunt'
where eitherS, or 5, can vary significantly
within the volume element. but not both.
These conditions would occur near the
maximum latitude if both objects had
nearly the same inclination, or near the
p e r i c e n t r o no r a p o c e n t r o ni f e i t h e r o f t h e s e
t w o d i s t a n c e sf o r o n e o b j e c t w a s n e a r l y t h e
s a m e a s e i t h e r d i s t a n c ef o r t h e o b i e c t .
cos 4t : sin 7, sin 7,
+ c o s 7 r c o s7 , c o s ( a r - d ) .
APPENDIX B
RerRrrvp Verocrry
Since the relative velocity is requiredto
c a l c u l a t e c o l l i s i o n p r o b a b i l i t i e s ,t h e e q u a tions giving the relative velocity between
two orbiting objects will be derived here.
T h e r e l a t i v e v e l o c i t y f o l l o w s f i - o mt h e v e c tor relationship
V-V",-V"r,
(lB)
where V", and V", are the respective velocit i e s o f t h e t w o o b j e c t s r e l a t i v et o t h e c e n t r a l
b o d y . T h e m a g n i t u d eo f t h e s e t w o v e l o c i t i e s i s g i v e n b y E q . ( 9 ) . T h e m a g n i t u d eo f
the relativevelocity is
W :
V " r 2 + V " z 2- 2 V " r V " , c o s f ,
f i g u r e ,t h e x a x i s p o i n t s t o w a r d t h e c e n t r a l
body, while the ,y axis is along a line of
constant latitude. This makes the ,r'-z plane
to be tangent to the spherical surface shown
i n F i g . l . T h u s , f r o m t h e d e f i n i t i o n so f r r
a n d 7 , t h e y c a n b e a d d e dt o F i g . B l . a n d
(2B)
where { is the angle between the vectors
V " , a n d V " r . F i g u r e B l i l l u s t r a t e st h e s e t w o
vectors in a stationary coordinate system
centered at the point of intersection. In the
(3B;
where values for 7 and a are found from
E q . ( t l ) a n d ( 2 0 ) , r e s p e c t i v e l y .V a l u e s f o r
e i t h e r 7 o r a m a y b e e i t h e r p o s i t i v eo r n e g a t i v e , r e s u l t i n gi n l 6 c o m b i n a t i o n so f a , , a r ,
7 , , a n d 7 , f o u n d w i t h i n a p a r t i c u l a rv o l u m e
e l e m e n t .H o w e v e r , a s f o u n d f r o m E q s . ( 2 8 )
a n d ( 3 B ) , t h e n u m b e r o f p o s s i b l ei n t e r s e c t i o n v e l o c i t i e sw i t h i n a v o l u m e e l e m e n t i s
r e d u c e dt o f o u r . E a c h o f t h e s ef o u r p o s s i b l e
velocities are equally probable to be found
w i t h i n t h e v o l u m e e l e m e n t .T h u s . f o r c a l c u l a t i n g c o l l i s i o n p r o b a b i l i t i e s ,i t i s v a l i d t o
use an average of these four velocities in
Eqs. (1). (22), (23), (24), or (l4A). Howe v e r , t h i s i s r r o l t h e s a m e a s s a y i n gt h a t t h e
p r o b a b i l i t yo f c o l l i s i o nw i t h e a c h v e l o c i t y i s
equal. As noted in Eq. (l), rhe probability
of collision is proportional to the relative
velocity within the volume element.Thus,
the higher velocities are the more probable
collision velocities.This becomes an im-
48
D O N A I - DJ . K T S S L E R
portantconsideration
when one beginsto
evaluatethe effectsof a collision.
AC KNOWT-EDGMENTS
T h e a u t h o r w i s h e s t o t h a n k H e r b e r t A . Z , o o kf i r r h i s
t h o r o u g h r c v i e w , a n d c o n s t r u c t i v es u g g e s t i o n si n t h c
meteoroid experiments: Interpliinetarl, and ne:rr
S a t u r n . . / . ( i t ' o p l t r s . / l r , . r .t l 5 , 5 t i 4 l - 5 8 5 1 .
K r . s s l - E nD
, . . 1 .( l 9 7 l ) . E s t i m a t e o f p a r t i c i e d e n s i t i e s
and collision danger for spacecraftmoving through
t h e a s t e r o i d b e l t . I n I ) l t t . t i tu l . \ ; t u l i ( \ ( ) l . l l i t t t t r
I ' l u t t t ' t . t . p p . 5 9 - s 6 1 1 5N
. ASA SP-167.
K r , s s l . r , RI .) . . 1. . . . r l D ( ' o t r n - [ ) . r r. r r s .B . ( i . ( l r ) 7 1 { )(.' ( } l -
l i s i o n f r e q u en c y o f u r t i f i c i u ls l r t c l l i t c s T
: hc crcution
o f a d e b r i s b e l t . J . O r t t p l t t r . / ? r , r . [ 1 3 .2 6 3 7 - 2 6 4 6 .
K < r z n r .Y . t l 9 t l l ) . S a t e l l i t et h e o r y ' .( t l t t r . l l t t l t . 2 3 ,
36-s 371;1.
M o n n r s o N .D . . . , r r D B r r R N S. .l. A . ( 1 9 7 6 ) T. h e J o v i a n
s a t e f f i t e s .l n . l t t J t i r c r t ' f . G e h r e l s . E d . ) . p p . 9 9 1 1 0 3 4 .I h e U n i v . o f A r i z o n a P r e s s .T u c s o n .
RtsFERENCES
( ) p r x . E . J . ( l 9 - sl ) . C o l l i s i o n p r o b a b i l i t i e sw , i t h r h c
[ ) e v r s .l ) . R . . C u q p v . q r . C . R . . G n r r . N r l r n < R
p l a n e t sa n d t h e d i s t r i b u t i o no f i n t e r p l a n e t a r \m. a t t e r .
; ,. . e N l r
W r - r o r , x sH
t r r r r N r ; .S . J . ( 1 9 7 9 ) C
. o l l i s i o n ael v o l u _
I ' n t c . l l r t . r ' .I r i . s l tA t u t l . - 5 4 , 1 6 5 1 9 9 .
t i o n o f a s t e r o i d s :P o p u l a t i o n .r o t a t i o n s ,a n d v el o c i - P < l r .er , <r , . 1 . 8 . . B u n s s . . l. A . . r r o ' f , r u e r , n . N { . E .
t i e s . I n A . \ t ( r t ) i ( 1 .(\T . C e h r e l s . E d . ) , p p . 5 : t t _ 5 _ 5 7
( 1 9 7 9 ) .C a s d r a g i n p r i m o r d i a l c i r c u n t p l a n e t a r \c n 'l'ucson.
The Univ. of Arizona Press.
velopes: A mechanism lor satellite capture. l rlrrr.r
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37, -s87-6 | | .
o f a s t e r o i d s . l n l ' l t t s i t u l . \ t t t d i t . s r t l . l l i n t t r I ' l u n t , t . s , S z r - - s r - - H E rV. .y (. l 9 7 t l ) .S t a b i l i t i o f t h e s o l a rs v s t e m .l n
p p . 1 6 3 - 1 9 . sN
. ASA SP-167.
I ) t ' n u n r i c . rs2 l t l r t S r t l u r. ! t r l l r r r ( R . L . l ) u n c o m b e .
G I - . H n r s, t. 1 ' . ( 1 9 7 0 ) .l n . l t t l t i r t , r ( T . C i c h r e l s .E d . ) . p .
E d . ) . p p - 7 - 1 5 . R e i d e l .B o s t o n .
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b e l l . . l . ( i t r t l t l t . r ' .R
o r g a n i z a t i o n o l t h i s p a p e r . I n a d c l i t i o n .s i g n i f i c a n ls u g gestlonswere received from the reftrees. George W.
Wetherilland William [:. Wiesel. which were gratef u l l y i n c o r p o r a t e di n t o t h e p a p e r .