The cometary nucleus as an aggregate of planetesimals

ICARUS 7 5 , 5 4 0 - 5 5 1

(1988)

The Cometary Nucleus as an Aggregate of Planetesimals TETSUO YAMAMOTO* AND TAKASHI KOZASA~"A *Institute of Space and Astronautical Science, Yoshinodai 3-1-1, Sagamihara, Kanagawa 229, Japan, and tInstitute for Cosmic Ray Research, University of Tokyo, Tanashi, Tokyo 188, Japan

Received July 6, 1987; revised December 14, 1987 Properties of planetesimals and their aggregates that formed in the primordial solar nebula and have survived up to the present time are studied in relation to the origin of comets. The result that a planetesimal of small mass is formed by nonhomologous sedimentation of grains as pointed out by Greenberg et al. (1984 Icarus 59, 87-113) is taken into account in determining the initial mass of a planetesimal. Analytic formulae are presented for the mass of the aggregates, their size, and the number of planetesimals that compose the aggregate; each is expressed as a function of the heliocentric distance. It is pointed out that the mass of a planetesimal formed by gravitational instability of the dust layer is related to the total number of comets. From a comparison of the total numbers of comets and aggregates of planetesimals, it is shown that the initial mass of a planetesimal is 10 - 4 t o 10 -7 times smaller than that estimated by assuming complete sedimentation of grains. The spatial distribution and mass spectrum of the aggregates are calculated. The spatial distribution shows that most of the aggregates are populated outside the planetary region and has a maximum around a few hundred AU. It is shown that, if the surface density of grains in the solar nebula is expressed by a power law of the heliocentric distance, the differential mass spectrum is approximated by a power law with the exponent - ~ for a wide range of the mass, regardless of the power index of the grain surface density. The presence of a cometary cloud is suggested beyond the planetary region but not so far as the Oort cloud. Discussion is given on the cloud properties. ©1988Academic Press, Inc.

1. INTRODUCTION The origin of c o m e t s is one of the most important p r o b l e m s not only for the study of comets but also for revealing the evolution of the Solar System, especially of its outer region. A variety of theories has been presented (see recent reviews by Weissman 1985, 1986a, Bailey et al. 1986, Clube and N a p i e r 1986, Oort 1986); one group of these theories is m o r e or less related to the events that t o o k place at the time of the formation of the Solar S y s t e m , and the other to the events that took place later in the evolution of the Solar S y s t e m , e.g., capture of come-

tary bodies in massive molecular clouds during encounters with the Solar S y s t e m (Clube and N a p i e r 1982, 1984). The theories in the latter group bring interesting aspects to the history of the Solar System, but seem to remain speculative at present, although it has b e e n shown that the o b s e r v e d orbital energy distribution of long-period comets can be realized within a time scale of the order of 10 6 t o 10 7 years (Yabushita 1979). On the other hand, the theories in the f o r m e r group have been stimulated by progress in the study of the origin of the Solar System. H o w e v e r , there still remains c o n t r o v e r s y regarding the formation region of comets, varying f r o m the asteroidal zone originally p r o p o s e d by Oort (1950) to the outlying f r a g m e n t a r y cloud of the solar nebula ( C a m e r o n 1973, 1978, Biermann and

Present address: Department of Physics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606, Japan. 54o 0019-1035/88 $3.00 Copyright © 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

THE COMET AS AN AGGREGATE OF PLANETESIMALS Michel 1978) and nearby star-forming clouds at the time of the formation of the Sun (Donn 1976). Many of those who are studying dynamical evolution of comets consider the Uranus-Neptune zone the favorable formation region since these planets are the most efficient ejectors of comets to the Oort cloud (e.g., Fernandez 1982). Assuming that this zone is the formation region, Greenberg et al. (1984) have studied the growth of planetesimals. They have shown that Neptune formed well within the time of Solar System formation and that comets are planetesimals that did not accrete to form the planet. The key to the origin of comets exists in a cometary nucleus. On the basis of the observed behaviors of comets and of the icy conglomerate model (Whipple 1950), detailed models of a cometary nucleus have been proposed: the planetesimal-aggregate model (Donn 1981), the primordial rubble pile model (Weissman 1986b), and the fractal model (Donn and Hughes 1986). The pictures of Comet Halley's nucleus taken by spacecraft (Keller et al. 1986, Sagdeev et al. 1986) prompted Gombosi and Houpis (1986) to propose the icy-glue model of the nucleus. Although these models may explain various characteristics of the observed cometary phenomena, the cosmogonical grounds are not sufficiently clear, and fundamental properties of the nucleus are overlooked; for instance, the models do not explain why the nucleus is of the size observed. In this paper, we assume that cometary nuclei are remnant planetesimals and their aggregates that did not accrete onto the planets and investigate quantitatively their properties expected from this hypothesis by applying the theory of the formation of planetary systems. In Section 2 the basic equations describing the growth and random velocity of planetesimals are given, and the time scale for reaching a kinematic steady state is estimated. Section 3 presents analytic formulae and results describing the properties of aggregates of

541

planetesimals and those of their spatial distribution and mass spectrum. Discussion is presented in Section 4 on the existence of a cometary cloud outside the planetary region and on its properties. 2. GROWTHOF PLANETESIMALSAND TIME SCALE FOR REACHINGA KINEMATIC STEADY STATE In developing a theory of the formation of planetary systems around stars, Nakano (1987a,b) has investigated the kinematic state and growth of planetesimals in the Solar System. Following him, we assume for simplicity that the mass, size, and random velocity are expressed by representative values, each of which is determined as a function of the heliocentric distance r and the time t. A planetesimal moves in a nearly circular orbit in the midplane of the solar nebula, but tends to have nonzero eccentricity and inclination by mutal gravitational scatterings as it grows. The deviation of the velocity from the circular Keplerian velocity is called random velocity, which is denoted by v. Planetesimals grow by coalescence due to mutual collisions with relative velocity v. If we ignore disruption due to collision, growth of a representative mass m of an aggregate of planetesimals is described (Nakano 1987b) by 1 dm

m dt

1 -

tc'

(2.1)

where tc is the collision mean free time given by _= 1 tc

( 3 m ~ 2/3 4~ \4-~pJ nav(I + 20),

(2.2)

in which Ps is material density of the aggregate, na the number density, and 0 the Safronov parameter for mutual gravitational scattering of the aggregates defined in the case of equal masses by

G m (4zrps] v3, 0 -- - v- 2 \ 3 m /

(2.3)

where G is the gravitational constant. In the

542

YAMAMOTO AND KOZASA

expression of the coalescence cross section involved in Eq. (2.2) we have put f = 1, where f is a factor for measuring the increase of the coalescence cross section due to tidal disruption during a close encounter of solid bodies (Hayashi et al. 1985). The number density na is related to the surface density 0-j of the dust layer in the solar nebula as

0-dUK (2.4)

na - 2 m r v '

where VK = ( G M U r ) 1/2 is the circular Keplerian velocity (3//o being the solar mass) and v / V ~ v K equals the mean inclination (Hayashi et al. 1985). The random velocity of the aggregates is excited by mutual gravitational scatterings and is dissipated by collisions. Since aggregates have spent most of their life up to the present time in a gas-free state, we ignore dissipation of the random velocity due to gas friction. Then the time variation of v is expressed (Nakano 1987b) by 1 dv 2

1

v 2 d~ - tE

1

to'

(2.5)

where te is a time scale for excitation of the random velocity given by Chandrasekhar's relaxation time (Chandrasekhar 1942) and is expressed by 1

rcfeGZmZn~ln A

tE

V3

,

(2.6)

where fE --~ 6.49 in the case of equal masses, and In A is a factor coming from the integration o v e r the impact parameter; the expression of A is given by Eq. (17) in N a k a n o ' s paper (1987b). The mass of a planetesimal formed by gravitational fragmentation of the dust layer if it occurred after complete grain sedimentation toward the midplane of the solar nebula is expressed (Safronov 1969, Hayashi 1972, Goldreich and Ward 1973) by

where fp is a constant coming from the effects of perturbed motion of the aggregates perpendicular to the nebular disk and the external pressure acting on the outer boundary of the dust layer and is equal to 1.057 (Sekiya 1983). Taking account of nonhomologous sedimentation of grains, Greenberg et al. (1984) have pointed out that, before complete sedimentation of grains, the dust layer reaches a critical density of about twice the local Roche density and becomes unstable. In consequence they have found that the mass of a planetesimal is much smaller than that estimated from Eq. (2.7). In view of their results, we introduce a parameter "/(0 < Y < 1) and express the initial mean mass m0 by m0 = yml.

For an accurate evaluation of Y, which would depend on the heliocentric distance and the time, we need to study the processes of sedimentation and gravitational clumping of grains in a combined manner. Let us assume, in first approximation, that 3, is constant. We scale the time t as 1 ( 3 f p M o ] 2/3 t

X = 7 5 \ Ps r3 /

(2.7)

t-K'

(2.9)

where tK = 2 r r ( r 3 / G M o ) 1/2 is the circular Keplerian period, and take the variables g = m / m o , the number of planetesimals composing the aggregate, and 0, instead of m and v. Then the basic Eqs. (2.1) and (2.5) are transformed, respectively, into l dg g dx -

1+20 (2.10)

gl/3

and 1 + 20 (5

1 dO 0 dx

-

gl/3

"3

/3 -

fEln A 4

/302 t 1 + 20/'

(2.11)

where

(27r2r3~ 2 m, = \ f - 7 ~ o / 0-3,

(2.8)

(2.12)

THE COMET AS AN AGGREGATE OF PLANETESIMALS F r o m Eq. (2.11) we can see that 0 app r o a c h e s a constant 0~ with increasing x. We call this state a steady state. It can be shown (see Appendix) that the steady state is reached at x - 53. As a result the ratio of the time scale t~t for reaching the steady state to the age of the Solar S y s t e m ts ( ~ 4 . 6 × l0 9 years) is evaluated to be

tst

3T 1/3 ( ps r3 ~2/3 tK

5

3s-77 /

Z'

(FAun7/2 ( ,~ ~1/3 ( -~0.18 \1--~/

\10 5/

Ps

\gcm

]2/3 3/

• (2.13)

Thus the aggregates are considered to reach the steady state before the present time in the region of r ~< 160 A U for y ~ 10 -5 , a typical value of y as estimated in the next section. Since/3 is kept almost constant (/3 --- 16), 0~ is calculated to be 0.44 (Nakano 1987b) f r o m Eq. (2.11).

3. CHARACTERISTICS OF PLANETESTIMAL AGGREGATES In the following calculation, we assume the surface density O-d of grains in the solar nebula to be e x p r e s s e d by a p o w e r law:

O'd -----O'dlrA~j,

(3.1)

where O-d~is a constant, and rAU is the heliocentric distance in astronomical units. We a s s u m e this law for r m u c h larger than the size of the present planetary region in view of recent o b s e r v a t i o n s of extended dust disks around a substantial fraction of young main-sequence stars in the solar neighborhood (e.g., A u m a n n et al. 1984, Smith and Terrile 1984) and of protostellar disks associated with bipolar flow sources in starforming regions (e.g., H a r v e y 1985). The outer limit for our solar nebula is discussed in Section 4. W h e n numerical evaluations are n e c e s s a r y in the following, we adopt o-d given by the H a y a s h i model (1981), in which O-d is given by putting n = 23and O ' d l = 30 g cm 2 for 2.7 < rAu < 36. This O'dl includes the m a s s of r o c k y and metallic grains plus icy grains and results f r o m the

543

gas surface density o- = 1.7 x 103 rA~ 2 g cm -2 (0.35 < rAU < 36). The r-dependence of o- (n -- ~) is derived from the transport of angular m o m e n t u m of the nebular gas due to magnetic and mechanical turbulent viscosities and is nearly the same d e p e n d e n c e as that obtained b y spreading the present masses of the planets o v e r the area of the planetary region (Hayashi 1981). It is, of course, not certain that o-d ~ rA~2 outside the planetary region as well. H o w e v e r , it is worth noting that the distribution of the surface brightness of/3 Pic disk, whose radius is at least 400 A U , is very close to that derived by assuming this r-dependence of o-d (Nakano 1987a). U n d e r the steady-state condition we integrate Eq. (2.10) by putting 0 = 0N and obtain g at the present time ts as

with l ( 3 f p M o ~ 2j3 ts (1 + 20~) tK---l A = 3 \~sr~ /

= 4.4 X 105 ,

(3.3)

where rl = 1 A U and tK1 =-- tK(r = rO = 1 year. N o t e that g is independent of o-d. F r o m numerical integrations of Eqs. (2.10) and (2.11) for a wide range of initial values of 0, Eq. (3.2) is confirmed to hold well for large r as well where the steady state is not reached. The m a s s of the aggregate is e x p r e s s e d by rn = g m o ,

(3.4)

where m0 is given by Eq. (2.8) with Eq. (2.7). Figure 1 shows m as a function of r for y = 1 to 1 × 10 -9 and n = ~. In general, m varies as r -9/2-3n and is independent of y in the inner region where A/yV3rv/A2 ~> 1, whereas in the outer region m ~ m0 and varies as y • r 6-3n. The m a s s m b e c o m e s minimum at r = [(3 + 2 n ) A / ( 4 - 2n)yl/3] 2/7 A U (--= rmin) for n < 2, and the minimum mass mmin is given by

544

YAMAMOTO AND KOZASA

I 024 01log i'

1022i .-. . . . . . . . i . . . . . . I'''"i~]iii~~ ~......-'.... . "~i::::I 102

1020

.

10 ~:~

101a

Ntot =

°L_):: .... 1014 lO

102 heliocentric

has little effect o n c h a n g i n g the orbital radius for the aggregates outside the planetary region. T h u s Eq. (3.6) holds for r > 30 A U , say, w h e r e m o s t o f the aggregates are populated. T h e total n u m b e r o f the aggregates is calculated f r o m

103

distance/AU

FIG. 1. Mass and equivalent radius of the aggregates of planetesimals as a function of the heliocentric distance r for n = ~, Y = 1 to 1 x l 0 -9, and the material density of the aggregate ps = 1 g cm 3 See the text for the explanation of n and 3'. The dashed lines indicate the mass and radius of initial planetesimals.

f2

rain

Nrdr,

(3.8)

w h e r e Rmin is the inner limit for the prese n c e o f icy planetesimals and R is the o u t e r limit. Since Nr d e c r e a s e s v e r y rapidly with d e c r e a s i n g r, we can practically p u t R r n i n = 0. O n the o t h e r hand, w e tentatively put R to be infinity for a r o u g h estimate o f 3'. In this a p p r o x i m a t i o n , Ntot is calculated to be 2

2

gtot -- f p M o bnA-((8 4 2 2rr 3rlo-dl

4n)/7)

3"-((13+4n)/21) (3.9)

mmin =

m0(rmin),

(3.5)

f r o m Eqs. (3.4), (3.2), (2.8), and (2.7). T h e radius o f an equivalent sphere having the s a m e v o l u m e as the aggregate o f mass m is e x p r e s s e d by a = (3m/4¢rps) 1/3 and is s h o w n in the right o r d i n a t e o f Fig. 1. If we a s s u m e that all the grains h a v e b e e n e x h a u s t e d to f o r m planetesimals, the surface d e n s i t y (in n u m b e r ) o f the aggregates is given b y o-a/m. T h e n the spatial distribution o f the aggregates is e x p r e s s e d by

Nrdr = 2rrr(o-d/m)dr,

(3.6)

w h i c h is s h o w n in Fig. 2. Nr d e c r e a s e s v e r y rapidly with d e c r e a s i n g r as 3"0 . rlla+2n in the inner region and d e c r e a s e s as 3"-J • r -5+2n in the o u t e r region. Nr has a maxim u m f o r n < 5/2 at heliocentric distance r m given b y rm =

l l + 4n 2(5 - 2n)

A

with 2

bn-~B

(13 + 4n 8 - 4n'~ J 7 ' 2(2n + 3)(4n - 1)Tr =

,

(3.10)

w h e r e B is the B-function. It is to be pointed out that n m u s t be less than 2 in o r d e r for Ntot to be finite.

1013

"~

109

.~-107 I

]2/7

" 3"1/33

AU,

(3.7)

w h i c h is i n d e p e n d e n t o f o'al. Migration o f the aggregates in a radial direction during g r o w t h due to r a n d o m m o t i o n m a y be neglected unless m > 1025 g ( H a y a s h i et al. 1985). F u r t h e r m o r e , p l a n e t a r y p e r t u r b a t i o n

103]0L~..... 102

1(] 3

heliocentric distance/AU Flo. 2. Spatial distribution N. of the aggregates for n = ~ and3, = 1 to 1 x 10 9.

THE COMET AS AN AGGREGATE OF PLANETESIMALS Theoretically the value of 3, is to be determined from the study of formation of planetesimals. As Greenberg et al. (1984) point out, the lower limit to the planetesimal size is imposed by the time required for gravitational instabilities to develop, which is a few times the Kepler period (Safronov 1969). Taking account of this result and nonhomologous sedimentation of grains, they obtained the radius of initial planetesimals (at the N e p t u n e zone) of about 3 km from the condition that the dust layer becomes gravitationally unstable when the dust density reaches a density of the order of the local Roche density (-3MU2~rr3). The value of 3, corresponding to this size is 2 × I0 7. It is expected that this size is near the lower limit, since it is the size resulting from the dust density reached during about the time interval stated above (Greenberg et al. 1984) and since the condition upon the dust density is a necessary condition for planetesimal formation. Consequently the 3,-value of the order 10 -7 may be regarded as the lower limit. Another way to estimate a possible range of 3' is provided by setting Ntot in Eq. (3.9) to the original total number of comets. According to estimations of the present population of comets in the Oort cloud and the fraction of original comets that survived against various loss mechanisms reviewed by Weissman (1985), the original total number of comets is estimated to be 1011 to 1013. These values, however, should be regarded as the underestimate unless there are efficient mechanisms to transfer comets in the region around rm to the Oort cloud. The population of comets in the inner region is not well known at present. Weissman (1986c) estimates the total number of comets in the " i n n e r c l o u d " to be 1013to 1014 . In view of the large uncertainty, we take roughly Ntot -- 1011 to 1014. F o r this range of N t o t , w e obtain 3' - 1 0 - 4 to 10 -7 for o'01 = 30 g cm -2 and n = ~ from Eq. (3.9) with Eq. (3.10). It is interesting to note that the lower value o f 3' estimated in this way is in good agreement with the possible lower limit es-

545

TABLE I MASS m AND RADIUS a OF THE AGGREGATE AND THE INITIAL PLANETESIMAL (DENOTED BY THE SUBSCRIPT 0) AT rm, THE HELIOCENTRIC DISTANCE AT A MAXIMUM OF THE SPATIAL DISTRIBUTION N r 3~

1 10 -1 10 -2 10 -3 10 4 10 5 10 6 10 -7 10 8 10 -9 10 -1°

rm (AU) 62 77 96 120 150 190 230 290 360 450 550

m (g)

a (km)

m0 (g)

a0 (km)

2.4 × 10z2 180

1.3 × 102z 150

3.4 4.7 6.6 9.1 1.3 1.8 2.4 3.4 4.7 6.6

1.8 2.5 3.5 4.8 6.7 9.4 1.3 1.8 2.5 3.5

× × × × x × X × × ×

10 zl 10z° 10 ~9 10 TM 10 TM 1017 10 I6 1015 1014 1013

93 48 25 13 6.7 3.5 1.8 0.93 0.48 0.25

× x × × x × X × × X

102j 76 10z° 39 10 ~9 20 1018 10 1017 5.4 1016 2.8 1016 1.5 1015 0.76 1014 0.39 1013 0.20

Note. The parameter values adopted are n = 1.5, o'd~ - 30 g cm 2, a n d p s = l g c m 3.

timated above from the size of planetesimals. It is worthwhile to calculate g, m, and a at r = rm. F r o m Eqs. (3.2) and (3.7) we have ( g(rm) =

21 . ; 11 + 4 '

(3.11)

which depends only on n. The mass and equivalent radius of the aggregates at rm are calculated by using g(rm). Table I lists the values of rm, m(rm), and a(rm) with those of the initial planetesimal mass m0(rm) and their equivalent radius a0(rm) for n -- ~. For n = ~ and 10 7 < 3' ~< 10-4, we obtain from Table I the following picture of the aggregate at rm: (1) The aggregate is composed of approximately two planetesimals [g(rm; n = ]) -~ 1.9]. (2) Consequently the shape of the aggregate is expected to vary between the two extreme cases: (a) spherical shape (i.e., two planetesimals completely merged) with the diameter 2a (4 ~< 2a ~< 26 km for p~ = 1 g cm-3), and (b) dumbbell-like shape (i.e., two planetesimals barely stuck on each other) with the long axis of 4a0 (6 ~< 4a0 ~< 40 km) and the short axis of 2a0 (3 ~< 2a0 ~< 20 km), if the initial planetesimals are assumed to be spherical and to suffer no

546

YAMAMOTO AND KOZASA note that the size and shape of Comet Halley's nucleus revealed by the Giotto (Keller et al. 1986) and Vega (Sagdeev et al. 1986) spacecraft are close to those expected from the aggregate of two partially merged planetesimals for 7 - 10 5. Using Nr given by Eq. (3.6) we obtain a size distribution of the aggregates in terms of g, the number of planetesimals composing the aggregate. F r o m Eqs. (3.6), (3.2), and (3.9), the cumulative distribution function ~ g ( g ) is given by

0.8 c3 ~

1. I:n 1.6

.~ 0.4 0,2

0.g

1.7

10

102

number of ptanetesimals

FIG. 3. Cumulative size distribution qbg of the aggregates in terms of the number of planetesimals g composing the aggregate for n = 1 to 1.9, where n is the power index of the grain surface density in the solar nebula, @g indicates the fraction of aggregates larger than g and is normalized such that qbg(l) = 1.

significant change in the density at coalescence. The actual shape would fall between these two extreme cases depending on the strength of the planetesimal material and the collision velocity. It is interesting to

2 f~

- 21bn

Nr/(dg/dr) dg dg gs/3(gl/3 -- 1)(4n-1)/7,

(3.12)

where @g is normalized such that qbg(1) = 1 and is illustrated in Fig. 3. Note that (I)g is independent of y and o-d~. As can be seen from this figure, the aggregates having g 1, i.e., initial planetesimals, occupy the largest population unless the nebular radius is so small that, say, a minimum g ~> 2. The cumulative mass spectrum is expressed by

rl

l

(~m(m) ~ f2max

qbg(g) = ~tot

~m(m)dm~ N~ttot(fr~Nrd?'-[- fRmin Nrd?') bZI 14 ~-~nl

( m \3--@-~](42.~( 8 4n)/7 (3 7 2/'/)2 \(mmin]2/3{1--\-~l-'('~]m /

J

2n}(13+4n)/(3+2n)"~"m-/(mmin)(13+4n)/(9+6n)

+ 13 +24~ L{(4\3---47-2nnn/ 2n~,4-2,,,/7 3 +7 x {1-

(

w h e r e mmax = m a x i m ( R ) , m(Rmin)], mmin is

given by Eq. (3.5), and rl and r2 (rj -< r2) are the heliocentric distances where the aggregate mass equals m (see Fig. 1). The first term should vanish for m > m(R), and the second term for m > m(Rmin). The analytic approximation holds well for m/mmi, >~ 5. For mmin <~ m <~ re(R), the first term dominates the second one, and ~m reduces to a power law given by ~ m o¢ m -2/3 (or ~m oc

m

),13+4,,,/0+6n)}] j

for m >> mmin,

(3.13)

m-5/3). It is remarkable that in this mass range the exponent of the mass spectrum is independent of n (<2) as well as y and o-d~ and that its value is close to the exponent of the observed mass spectra of long-period comets (Whipple 1975, Hughes and Daniels 1980, Fernandez 1982, Weissman 1983). We discuss the mass spectrum in more detail in the next section.

THE COMET AS AN AGGREGATE OF PLANETESIMALS

547

mation distance of the largest population of comets. The above rm is about twice the distance inferred from the chemical compoThe presence of a cometary cloud (the sition of the ice of a cometary nucleus by inner Oort cloud) has been suggested from the study of the orbital evolution of comets Yamamoto (1985), who adopted two types (e.g., Weissman 1985) as a feed source of of temperature distributions (radiative equicomets to the Oort cloud against removal of librium and adiabatic) in the solar nebula in comets in the Oort cloud at encounters of deriving the formation distance. The differgiant molecular clouds (GMCs) and as one ence may arise, in regard to the physical of the sources of short-period comets. study, partially from the uncertainty of the From a comparison of the observed orbital temperature distribution in the solar nebula energy distribution of long-period comets and complicated nature of sublimation of with the results of his theory of the dynami- the ices and their mixture (Bar-Nun et al. cal evolution of a hypothetical cometary 1985, 1987, Grim and Greenberg 1986, cloud surrounding the Solar System, Bailey 1987, Kouchi 1987) and of other process(1983a) has suggested the presence of a pri- ings. In any case, however, it is plausible mordial cometary cloud well inside the that the formation region is outside the present Oort cloud. The origin of the cloud planetary region. More detailed compleis not clear at present. Cameron (1978) pro- mentary studies from both dynamical and posed a scenario that a cometary cloud physical viewpoints are required to allow formed around 1000 AU in a subdisk of the for more quantitative conclusions. massive solar nebular and was later ejected Next let us discuss the mass spectrum of to the Oort cloud at the time of the nebular the cloud. By numerical simulation of growth of planetesimals in the Neptune mass loss. It is clearly seen from Fig. 2 that most of zone, Greenberg et al. (1984) have shown the aggregates are populated beyond the that Neptune formed within the age of the planetary region (r > 30 AU), indicating the Solar System and at the same time that the presence of a cometary cloud outside the mass spectrum of observed comets is replanetary region. Our results, which are in produced, when the planetesimals are asprinciple based on a low-mass nebula sumed to have had the same mass spectrum model, implies that the cometary cloud as that of observed comets. They suggest originates from a remnant of planetesimals that the mass spectrum of planetesimals reand their aggregates consisting of a few sulted from the gravitational clumping proplanetesimals or less that did not accrete to cess of the dust layer. Our results provide form the planets during the age of the Solar another and quantitative explanation of the System because of the low density at large size distribution of comets. We have shown that the different masses of the aggregates heliocentric distances. We now discuss properties of the cloud formed at different heliocentric distances of planetesimals and their aggregates on the leads to the differential mass spectrum ~m approximately proportional to m -5/3 for m basis of the results obtained in Section 3. One of the notable characteristics of the m ( R ) [Eq. (3.13)], regardless of the valcloud is that the spatial distribution of the ues of n and y. The feature of the mass aggregates has a maximum at the distance spectra of observed long-period comets is rm, which depends on n (< 2) and y. For n roughly in accordance with this mass spec= ~ and 10 4 ~ .~ ~ 10-7, we have 150 ~< rm trum. Although we have ignored disruption ~< 290 AU (Table I), well beyond the plane- of planetesimals in collisional growth, the tary region but not as far as the Oort cloud mass spectrum changes little even if the disand Cameron's cometary cloud. This dis- ruption process is taken into account, since tance is considered to be the probable for- most of the fraction contributing to the 4. DISCUSSION

548

YAMAMOTO AND KOZASA

m a s s s p e c t r u m is the aggregates having suffered no collision up to now (g -~ 1). F r o m m o r e detailed c o m p a r i s o n of the mass spectra of c o m e t s and aggregates of planetesimals, we can derive a constraint on the outer b o u n d a r y of the solar nebula. We a d o p t the o b s e r v e d mass spectrum of long-period c o m e t s calculated b y Weissman (1983), who used the intrinsic distribution of c o m e t a r y absolute magnitude obtained by Everhart (1967). His mass spectrum has an intermediate slope in the range of the smaller mass (the exponent of the differential spectrum equals about 1.7) c o m p a r e d with those of other authors (Whipple 1975, H u g h e s and Daniels 1980, Fermindez 1982) and has a steeper slope (a -~ 2.5) in the range of the larger mass. As a result, the m a s s s p e c t r u m has a knee at the mass corresponding to H10 = 6, where Hi0 is the brightness of a c o m e t with c o m a corrected to unit heliocentric and geocentric distances. The m a s s s p e c t r u m (I) m for n = ~and y -- 10 -5 is c o m p a r e d with the o b s e r v e d s p e c t r u m in Fig. 4 by taking the solar nebula radius R as a parameter. The mass is

f

5 XIO

~~-~'-->--o 10-4 xlo t

10 .6 I0

10 2

10 3

10 4

m/m FIG. 4. Comparison of cumulative mass spectra of planetesimal aggregates (solid curve) and long-period comets (dashed curve). The mass spectra of the aggregates are s h o w n by taking the radius RAu of the solar nebula as a parameter; all of them are for n = ~ and 3' = 10-5. Both spectra are normalized to be unity at m~ mmin = 1. See the text for the explanation of the minimum mass mmin,

e x p r e s s e d in terms of m/mmin. F o r ~m, mmin is given by Eq. (3.5), and for the o b s e r v e d spectrum, we h a v e set a m i n i m u m mass to be the mass corresponding to H10 = 11.5. Both spectra are normalized to be unity at m/mrnin = 1. (I)m varies a s m -2/3 for m m(R) as stated before, rapidly decreases as m a p p r o a c h e s re(R), and decreases for larger rn a s m -19/18, which part of the mass spectrum c o m e s f r o m a contribution of the aggregates at the inner region where r < rffm(R)). The rapid decrease near m -m(R) is due to the cutoff of the solar nebula. The mass at the knee in the o b s e r v e d s p e c t r u m is given by m/mmin = 160. The heliocentric distance corresponding to this mass is calculated to be rknee = 8200(7/ 10-5) -2/z~ AU, which is about a few 10ths of the inner radius of the present Oort cloud. It can be seen f r o m Fig. 4 that, in order to r e p r o d u c e the o b s e r v e d spectrum, R should be larger than rknee at least. 2 The steeper slope for m/mmin > 160 in the o b s e r v e d spectrum can be interpreted as follows: We note that the time required for formation of planetesimals is limited by the sedimentation time of grains. The time for the substantial fraction of grains to sediment is estimated to be of the order of 1000 tK (Weidenschilling 1980, N a k a g a w a et al. 1981). Thus s o m e portion of the grains is left without forming planetesimals at a r o u n d 10 4 A U e v e n after the formation of the Solar System, and the n u m b e r of planetesimals around this distance b e c o m e s less than that estimated by Eq. (3.6). The portion of the grains left is e x p e c t e d to increase with r. This results in the steeper slope in the mass spectrum. The distance rknee is regarded as the distance inside which grains completed sedimentation. One p r o b l e m is the absolute value of the mass. If we a s s u m e the mass of the comets with Hi0 = 11.5, the faintest long-period comets listed in E v e r h a r t ' s data, to be the minimum mass, we have mmin(obs) = 7.1 x 2 The error of N~ot given by Eq. (3.9) is less than 1.5% i f R > r k ....

THE COMET AS AN AGGREGATE OF PLANETESIMALS 1014(Av/0.3) -3/2 g according to Weissman (1983, 1985), where Av is the visual albedo of a nucleus surface. The actual minimum mass would be smaller. On the other hand, the minimum mass of the aggregates is given by mmin = 1.3 x 1018(~/10-5) 6/7 g (or amin = 6.7(y/10-5) 2/7 km) from Eq. (3.5). Thus the mass is to be scaled by a factor of 1.8 X 103 ( ' y / 1 0 - 5 ) 6/7 (Av/0.3) 3/2 t o reconcile cometary mass with aggregate mass. This factor is much larger than unity unless y < 10-s or Av < 0.3. The reason of this disagreement is not clear, but might be ascribed partially to the uncertainities in estimating cometary mass from the observed brightness. From the condition that R > rknee, the mass of the solar nebula is estimated to be larger than 0.22(y/l0 -5)- VZIMe if we extrapolate the Hayashi model (1981). Accordingly the total mass of the aggregates is estimated by substracting the solid mass required to form the planets ( - I 0 0 Me) to be around 1.2 × 103 (y/lO -s) 1/21Me, where Me is the Earth mass. Note that this value is regarded to be the initial total mass, since we do not take into account various loss processes, for example, loss due to encounters of GMCs. We have discussed the properties of the cloud of planetesimal aggregates beyond the planetary region and have examined the hypothesis that cometary nuclei are the remnant planetesimals and their aggregates. One of the observational clues available at present to verify the hypothesis is the mass spectra of comets, which we have discussed above, although the cometary mass spectra allow for other interpretations. Detection of thermal emission from the clouds will provide another clue. Bailey (1983b) calculated thermal emission flux from the cloud assuming a spherical shell structure having power-law density distribution and mass spectrum. His cloud model includes arbitrary parameters to be determined in principle by the study of the origin and evolution of the cloud, such as the power indices of the mass spectrum and

549

density distribution, total mass, upper and lower limits to the mass, and inner and outer radii of the cloud. The properties of the cloud of aggregates of planetesimals described in this paper provide the initial conditions for the cloud evolution (Bailey 1983a). APPENDIX

We derive the time scale xst for reaching the steady state. Let K = f102/(1 + 20) and du = dx/gl/3; then we have d d--~ In K =dulnB+2 =duln/3+2(1

1

1 +~

~u l n 0

+ 0) ~ - K ,

(A.I)

where we have used Eq. (2.11) for obtaining the last expression. Substitution of 0 obtained from Eq. (2.10) into Eq. (A.I) yields dK ldg =---+ K(5/3 - K) du g du l

1,

(A.2)

where the first term in the right-hand side of Eq. (A.1) is neglected, since, as inferred from the definition offl [Eq. (2.12)1, In fl is a very slowly varying function of u, which is confirmed to be valid from numerical calculations as well. Integrating Eq. (A.2), we obtain K=

5/3 1 + ( 5 / 3 K 0 - 1) , exp{-(5/3)(u + In g)}

(A.3)

where Ko =- K ( u -- 0). We define xst to be the time where (5/3)(u + In g) = 1, i.e., x~t is given by a solution of , dx

f~ gl/3 -

3 5

In g(Xst).

(A.4)

Since the left-hand side should be positive, g < e 3/5 (= 1.82) (or g~/3 < 1.22) for x < xst, and so we can put g = 1 in a first approximation. In consequence we obtain x~t 3/5, which value is proved to be a good

550

YAMAMOTO AND KOZASA

approximation from a comparison with the results of numerical integration of Eqs. (2.10) and (2.11). ACKNOWLEDGMENTS The authors thank Dr. J. A. FernS.ndez for valuable comments on improving the manuscript and Dr. S. Sato for discussion on dust disks around stars. One of the authors (T.Y.) acknowledges financial support by a Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan (61302025) and by a Special Project on the Evolution of Matter, University of Tsukuba. Numerical computations were performed by FACOM 380 computers at the Institute of Space and Astronautical Science and the Institute for Nuclear Study, University of Tokyo. REFERENCES

AUMANN,H. H., F. C. GILLETT,C. A. BEICHMAN,T. DE JONG, J. R. HOUCK, F. Low, G. NEUGEBAUER, R. G. WALKER, AND P. WESSELIUS 1984. Discovery of a shell around Alpha Lylae. Astrophys. J. 278, L23-L27. BALLEY, M. E. 1983a. The structure and evolution of the Solar System comet cloud. Mon. Not. R. Astron. Soc. 204, 603-633. BAILEY, M. E. 1983b. Theories of cometary origin and the brightness of infrared sky. Mon. Not. R. Astron. Soc. 205, 47P-52P. BAILEY, M. E., S. V. M. CLUBE, AND W. M. NAPIER 1986. The origin of comets. Vistas Astron. 29, 53112. BIERMANN, L., AND K. W. MICHEL 1978. On the origin of cometary nuclei in the presolar nebula. Moon Planets 18, 447-464. BAR-NuN, A., J. DROR, E. KOCHAVI,AND D. LAUFER 1987. Amorphous water ice and its ability to trap gases. Phys. Rev. B 35, 2427-2435. BAR-NuN, A., G. HERMAN, AND D. LAUEER 1985. Trapping and release of gases by water ice and implications for icy bodies. Icarus 63, 317-332. CAMERON, A. G. W. 1973. Accumulation processes in the primitive solar nebula. Icarus 18, 407-450. CAMERON, A. G. W. 1978. Physics of the primitive solar accretion disk. Moon Planets 18, 5-40. CHANDRASEKHAR~S. 1942. Principles of Stellar Dynamics, Chap. II. Univ. of Chicago Press, Chicago. CLURE, S. V. M., AND W. M. NAPIER 1982. Spiral arms, comets, and terrestrial catastrophism. Q. J. R. Astron. Soc. 23, 45-66. CLUBE, S. V. M., AND W. M. NAPIER 1984. Comet capture from molecular clouds: A dynamical contraint on star an planet formation. Mon. Not. R. Astron. Soc, 208, 575-588. CLUBE, S. V. M., AND W. M. NAPIER 1986. Giant comets and the galaxy: Implications of the terrestrial record. In The Galaxy and the Solar System (R.

Smoluchowksi, J. N. Bahcall, and M. S. Matthews, Eds.), pp. 260-285. Univ. of Arizona Press, Tucson. DONS, B. 1976. Comets, interstellar clouds, and star clusters. In The Study of Comets, pp. 663-672. NASA SP-393. DONN, B. 1981. Comet nucleus: Some characteristics and a hypothesis on origin and structure. In Comets and Origin o f Life (C. Ponnamperuma, Ed.), pp. 2129. Reidel, Dordrecht. DONN, B., AND D. HUGHES 1986. A fractal model of a cometary nucleus by random accretion. In Proceedings o f the 20th ESLAB Symposium on the Exploration o f Halley's Comet, Vol. 3, pp. 523-524. ESA SP-250. EVERHARX, E. 1967. Intrinsic distributions of cometary periherlia and magnitude. Astron. J. 72, 10021011. FERNANDEZ, J. A. 1982. Dynamical aspects of the origin of comets. Astron. J. 87, 1318-1332. GOLDREICH, P., AND W. R. WARD 1973. The formation of planetesimals. Astrophys. J. 183, 1051-1061. GOMBOSl, T. I., AND H. L. HOUPIS 1986. An icy-glue model of cometary nuclei. Nature 324, 43-44. GREENBERG, R., S. J. WEIDENSCHILL1NG, C. R. CHAPMAN, AND D. R. DAVIS 1984. From icy planetesimals to outer planets and comets. Icarus 59, 87-113. GRIM, R. J. A., AND J. M. GREENBERG 1986. Photochemical SE: Evidence of interstellar grains in comets. In Proceedings o f the 20th ESLAB Symposium on the Exploration o f Halley's Comet, Vol. 3, pp. 21-27. ESA SP-250. GRIM, R. J. A., AND J. M. GREENBERG 1987. Processing of H2S in interstellar grain mantles as an explanation for $2 in comets. Astron. Astrophys. 181, 155-168. HARVEY, P. M. 1985. Observational evidence for disks around young stars. In Protostars and Planets H (D. C. Black and M. S. Matthews, Eds.), pp.484-492. Univ. of Arizona Press, Tucson. HAYASHI, C. 1972. Origin of the Solar System. Proc. Lunar Planet. Syrnp. 5th, 13-18. ISAS, Tokyo. [in Japanese] HAYASHI, C. 1981. Structure of the solar nebula, growth and decay of magnetic fields and effects of magnetic and turbulent viscosities on the nebula. Suppl. Prog. Theor. Phys. 70, 35-53. HAYASHI, C., K. NAKAZAWA, AND Y. NAKAGAWA 1985. Formation of the Solar System. In Protostars and Planets lI (D. C. Black and M. S. Matthews, Eds.), pp. 1100-1153. Univ. of Arizona Press, Tucson. HU~HES, D. W., AND P. A. DANIELS 1980. The magnitude distribution of comets. Mon. Not. R. Astron. Soc. 191, 511-520. KELLER, H. U., C. ARPIGNY, C. BARBIERI, R. M. BONNET, C. CAZES, M. CORADINI, C. B. COSMOVICI, W. A. DELAMERE, W. F. HUEBNER, D. W.

T H E C O M E T AS AN A G G R E G A T E OF P L A N E T E S I M A L S HUGHES, C. JAMAR, D. MALAISE, H. J. REITSEMA, H. U. SCHMIDT,W. K. H. SCHMIDT,P. SEIGE, F. L. WHIPPLE, AND K. WILHELM 1986. First Halley multicolor camera imaging results from Giotto. Nature 321, 320-326. KOUCHI, A. 1987. Vapour pressure of amorphous H20 ice and its astrophysical implications. Nature 330, 550-552. NAKAGAWA, Y., K. NAKAZAWA, AND C. HAYASHI ! 981. Growth and sedimentation of dust grains in the primordial solar nebula. Icarus 45, 517-528. NAKANO, T. 1987a. The formation of planets around stars of various masses and the origin and the evolution of circumstellar dust clouds. In Star Forming Regions (Proc. IA U Symposium No. 115) (M. Peimbert and J. Jugaku, Eds.) pp. 301-313. Reidel, Dordrecht. NAKANO, T. 1987b. Formation of planets around stars of various masses. I. Formulation and a star of one solar mass. Mon. Not. R. Astron. Soc. 224, 107130. OORT, J. H. 1950. The structure of the cloud of comets surrounding the Solar System, and a hypothesis concerning its origin. Bull. Astron. Inst. Netherlands 14, 91-110. OORT, J. H. 1986. The origin and dissolution of comets. Observatory 106, 186-193. SAFRONOV, V. S. 1969. Evolution o f the Protoplanetary Cloud and Formation o f the Earth and Planets. Nauka, Moskow. English translation: NASA TTF677 (1972), U.S. Department of Commerce, Springfield, VA. SAGDEEV, R. Z., F. SZABO, G. A. AVANESOV, P. CRUVELLIER, L. SZAB6, K. SZEG(), A. ABERGEL, A. BALAZS, I. V. BARINOV, J.-L. BERTAUX,J. BEAMONT, M. DETAILLE, E. DEMARELIS, G. N. DUL'NEV, G. ENDRO(~ZY, M. GARDOS, M. KANYO, V. I. KOSTENKO, V. A. KARASIKOV, T. NGUYNETRONG, Z. NYITRAI, I. RENY, P. RUSZNYAK, V. A. SHAMIS, B. SMITH, K. G. SUKHANOV,F. SZAB6, S. SZALAI, V. I. TARNOPOLSKY, I. TOTH, G. TSU-

551

KANOVA, B. I. VALNiC~K, L. VARHALMI, YU, K. ZAIKO, S. I. ZATSEPIN, YA. L. ZIMAN, M. ZSENEI, AND B. S. ZHUKOV 1986. Television observation of Comet Halley from Vega spacecraft. Nature 321, 262-266. SEKIYA, M. 1983. Gravitational instabilities in a dustgas layer and formation of planetesimals in the solar nebula. Prog. Theor. Phys. 69, 1116-1130. SMITH, B. A. AND TERRILE, R. J. 1984. A circumstellar disk around/3 Pictoris. Science 226, 1421-1424. WEIDENSCHILLING, S. J. 1980. Dust to planetesimals: Settling and coagulation in the solar nebula. Icarus 44, 172-189. WEISSMAN,P. R. 1983. The mass of the Oort cloud. Astron. Astrophys. 118, 90-94. WEISSMAN,P. R. 1985. The origin of comets: Implications for planetary formation. In Protostars and Planets lI (D. C. Black and M. S. Matthews, Eds.), pp. 895-919. Univ. of Arizona Press, Tucson. WEISSMAN,P. R. 1986a. The Oort cloud and the galaxy: Dynamical interactions. In The Galaxy and the Solar System (R. Smoluchowski, J. N. Bahcall, and M. S. Matthews, Eds.), pp. 204-237. Univ. of Arizona Press, Tucson. WEISSMAN, P. R. 1986b. Are cometary nuclei primordial rubble piles? Nature 320, 242-244. WEISSMAN, P. R. 1986c. The Oort cloud in transition. In Asteroids, Comets, Meteors II (C.-I. Lagerkvist, B. A. Lindblad, H. Lundstedt, and H. Rickman, Eds.), pp. 197-206. Uppsala University, Uppsala. WHIPPLE, F. 1950. A comet model. I. Acceleration of Comet Encke. Astrophys. J. 111, 374-394. WHIPPLE, F. L. 1975. Do comets play a role in galactic chemistry and y-ray bursts? Astron. J. 80, 525-531. YABUSHITA, S. 1979. A statistical study of the evolution of the orbits of long-period comets. Mon. Not. R. Astron. Soc. 187, 445-462. YAMAMOTO, T. 1985. Formation environment of cometary nuclei in the primordial solar nebula. Astron. Astrophys. 142, 31-36.