Evolution of comet orbits under the perturbing influence of the giant planets and nearby stars

nCARUS 42, 406--421 (1980)

Evolution of Comet Orbits under the Perturbing Influence of the Giant Planets and Nearby Stars J U L I O A. F E R N A N D E Z * Obsert'atorio Astronomico Nacional, Alfonso XII. 3. Madrid-7, Spain Received N o v e m b e r 29, 1979: revised March 19, 1980 The orbital evolution of 500 hypothetical comets during 10 ~ years is studied numerically. It is assumed that the birthplace of such comets was the region of Uranus and Neptune from where they were deflected into very elongated orbits by perturbations of these planets. Then, we adopted the following initial orbital elements: perihelion distances between 20 and 30 AU, inclinations to the ecliptic plane smaller than 20°, and semimajor axes from 5 x 10 a to 5 x 104 AU. Gravitational perturbations by the four giant planets and by hypothetical stars passing at distances from the Sun smaller than 5 × 10"~AU are considered. During the simulation, somewhat more than 50% of the comets were lost from the solar system due to planetary or stellar perturbations. The survivors were removed from the planetary region and left as members of what is generally known as the cometary cloud. At the end of the studied period, the semimajor axes of the surviving comets tend to be concentrated in the interval 2 x 10~ < a < 3 x 10~ AU. The orbital planes of the comets with initial a ->_ 3 × 104 AU acquired a complete randomization while the others still maintain a slight predominance of direct orbits. In addition, comet orbits with final a < 6 × 10 ~ AU preserve high eccentricities with an average value greater than 0.8. Most " n e w " comets from the sample entering the region interior to Jupiter's orbit had already registered earlier passages through the planetary region. By scaling up the rate of apparitions of hypothetical new comets with the observed one, the number of members of the cometary cloud is estimated to be about 7 × 10" and the conclusion is drawn that Uranus and Neptune had to remove a number of comets ten times greater. I N'FRODU CTION

T w o basic questions about c o m e t s concern the place where they come from and where they were originally formed. It has long been proposed that comets are members of the solar system (see, e.g., Oort, 1950; Opik, 1973). Alternatively, an interstellar c o m e t a r y origin has been suggested (see, e.g., Radzievskii and T o m a n o v , 1977; Yabushita and Hasegawa, 1978), but this hypothesis meets a severe difficulty in the lack of observations o f c o m e t s with markedly hyperbolic orbits as we should expect if they c o m e from interstellar space (Sekanina, 1976). In effect, no original hyperbolic orbits have so far been o b s e r v e d in c o m e t s with perihelion distances q > 3 AU (Marsden et al., 1978). Original slightly hyperbolic orbits have only been c o m p u t e d * Present address: Max-Planck-lnstitut fiJr Aeronomie, D-3411 Katlenburg-Lindau 3, Germany.

in some c o m e t s with smaller perihelion distances. This strongly suggests that all c o m p u t e d original orbits would be elliptical if nongravitational forces were taken into account. Such lbrces originate from the jet reactions associated with the vaporization of c o m e t a r y ices, mainly w a t e r snow (Marsden et al., 1973), whose effects are most important within heliocentric distances o f 2-3 AU (Marsden and Sekanina, 1973). According to O o r t ' s view, c o m e t s are found in storage at huge distances from the Sun and form a sphericaJ cloud. Presumably these c o m e t s were formed with the planets and then placed into the cloud by the combined action of stellar and planetary perturbations. Thus, the study of comets acquires a more general interest in connection with the origin of the solar system, since they could be the residual building blocks after the formation of planets. According to Opik (1973) and others,

4116 0019-1035/80/060406- 16S02.00/0 Copyright (c! 19811 by Academic Prc~s, Inc All rights of reproduction in any lorm reserved.

EVOLUTION OF COMET ORBITS Jupiter was the main ejector of c o m e t s to the Oort cloud. H o w e v e r , the lower gravitational binding energy per unit mass o f bodies in the region of Uranus and Neptune convert these planets into the more suitable ejectors of solid matter to the cloud region (Kuiper, 1951; Safronov, 1972; Fern~.ndez, 1978, which hereafter we will refer to as Paper I). In contrast, the much greater gravitational binding energy of bodies in the Jupiter region would m a k e more probable that they were ejected hyperbolically out of the solar system in the strong interactions with this massive planet. Through a numeric',d e x p e r i m e n t we will attempt to reach conclusions about the probability that a comet, r e m o v e d from the planetary region, can be finally incorporated into the c o m e t a r y cloud, and to analyze the probabilities that such a c o m e t reenters the planetary region by stellar perturbations. We will accept the following starting picture: the c o m e t s were the residuals left after the formation of planets which, b e c a u s e of perturbations chiefly by Uranus and Neptune, changed their rather circular orbits into very elongated ones. We will then analyze the later evolution of such c o m e t orbits taking into account, besides the perturbations of planets, those of stars passing near the Sun.

407

lo 5 10

E c

2o

15 10

10

40

70

100

aor,g( 103AU ] Ft(;. I. O b s e r v e d distribution o f the original semimajor a x e s o f the long-period c o m e t s with 300 < aor,g < l0 s A U . The black histogram s h o w s the partial sample o f c o m e t s with periheliaq :- 2 A U , presumably little affected by the action o f the nongravitational forces (Marsden and Sekanina. 1973). aorig ~ 104 A U t o t h e planetary region; these are " o l d " c o m e t s that are repeating their passages. In a passage through the region interior to the orbit of Jupiter a c o m e t undergoes a change in its reciprocal semimajor axis A(l/a) of about +-7 x

10 -4

AU -t

(Everhart

and

Raghavan,

1970). Therefore, it is very probable that most comets with a,,r~, ~> 104 AU, that is with (I/a),,rig ~< 10 -4 AU -~, pass through the region interior to the orbit of Jupiter OBSERVED DISTRIBUTION OF THE ORIGINAl. for the first time, for which they are genS E M I M A J O R A X E S IN L O N G - P E R I O D C O M E T S erally known as " n e w " comets. The a,,r,~ values of new c o m e t s fall mainly We understand as the original semimajor axis of a c o m e t orbit (a,,~J the one it has within the range 104< a,,r~< 6 x 104AU before entering the planetary region. The with a remarkable p r e d o m i n a n c e in the n a r r o w e r range 2 x 104 < a~,r,~ < 3 x 104 study of the a,,r,~-distribution is important AU, that is, aphelion distances Q in the with regard to a better understanding about r a n g e 4 x 104< Q < 6 × 104AU. A more the place where the c o m e t s c o m e from. Recently Marsden e t al. (1978) published remarkable concentration in the same intera list of 200 c o m e t s with orbital elements of val is verified when the partial sample of very good or good quality. The a,,r~K-distri- c o m e t s with perihelion distances q > 2 AU bution corresponding to 128 comets from is considered. The original aphelion distances of new such a list is shown in Fig. 1. T h o s e c o m e t s with original hyperbolic orbits, a,,r~ > 10 ~ comets could be indicators of the region occupied by the c o m e t cloud. Hence we AU or a,,r~, < 300 AU were left out. As we will see later, stellar perturbations can expect that such a cloud would have a are ineffective in bringing comets with radius o f about 4 - 6 x 104 AU in good

408

JULIO A. FERNANDEZ

agreement with a previous determination by Marsden and Sekanina (1973) based on a more reduced sample. The number of observed long-period comets decreases notably for aor~ > 6 x 104 A U ; this may be due to a real decrease in the number of cloud comets a n d / o r a lower probability that stellar perturbations bring comets into the planetary region.

THE MODEL A sample of 500 hypothetical comets has been considered with the following initial orbital elements: perihelia (qmit) and inclinations (iinit) taken at random within the intervals 20 < qinit < 30 AU and 0 ° < iinit < 20°. The sample was divided into 10 groups of 50 comets each with the following starting semimajor axis (ainit): 0.5 x 104, 104, 1.5 × 104. . . . . 5 × 104 AU. With these starting orbital elements an attempt has been made to represent a group of comets, formed in the region o f Uranus and Neptune with low-inclination orbits, which acquired their very elongated orbits through planetary perturbations. The orbital evolution o f each comet was followed during a period of l0 '~ years, except when ejected out of the solar system by a planetary or stellar perturbation. When a < 250 AU, the computation was also stopped in order to avoid excessive consumption of computer time in repeated passages through the planetary region. The perturbations of the four giant planets were taken into account for each comet passage through the planetary region. The equations of motion were integrated numerically using Cowell's method. The comet mass was neglected and, for simplicity, the orbits o f the giant planets were taken as circular and coplanar. Different values of planetary true anomalies. taken at random, were adopted for each c o m e t a r y passage. In each passage, the integration o f the cometary orbit was started when the comet entered a sphere o f radius 50 AU centered

at the Sun and was stopped when it left this sphere. The original orbital elements of the comet were the q, a, and i values resulting from the last perturbation undergone by the comet, either stellar or planetary. Moreover, for all the passages, a longitude of ascending node 1) = 0 was adopted and the longitude of perihelion to was taken at random in the interval 0 < to < 2~r. The computation of the planetary perturbations was carried out in those cases where the incoming comet had a q < 35 AU. When the comet attained q > 35 AU because of stellar perturbations, it was assumed that the comet was no longer perturbed by the planets. I f a comet came very close to a planet the barycentric coordinates were changed into planetocentric ones. The best compromise between precision and computer time was searched, taking into account that the requirement of a very small error in a statistical study of this kind was not essential. As a check of the precision of the obtained results, the energetic changes, or changes in the reciprocal semimajor axes A( 1/a), o f 200 random passages for comets w i t h a = l 0 4 A U , ! < q < 5 AU and all the orientations were calculated. The resultant A(I/a)-distribution is practically symmetrical about zero with a standard deviation o f 6.4 × l0 -~ AU -~, which is close to the value obtained by Everhart and Raghavan (1970). The distribution of the absolute values of A(l/a) is in good agreement with the distributions shown by Everhart (1968) for 2000 random comets and 70 observed comets. For another sample o f 200 random comets with similar characteristics but perihelia in the interval 20 < q < 30 AU, a standard deviation o f 1.1 × l0 -~ AU -~ was obtained for the A( I/a)-distribution. The perturbing action of hypothetical nearby stars, passing at distances from the Sun less than DM = 5 × l0 ~ A U = 2.5 pc, was considered throughout the computation. This value is ten times greater than the largest a,~t used in the model. Such a criterion has attempted to account not only

EVOLUTION OF COMET ORBITS for the closest stellar passages but also for somewhat distant ones. For the relative velocity V of each star referred to the Sun a value o f 30 km s e c - ' was adopted which, in accordance with Rickman (1976), corresponds to the observed mean value for nearby stars. As this velocity is much greater than that corresponding to a comet during most o f its orbital path, the comet can be considered at rest with reference to the Sun during the stellar encounter. The impulses per unit mass received by the comet and the Sun in the passage o f a star o f mass M will be given by --->

lo=K~, (1) --*

D

I~=K-~,

where K = 2 G M / V and De, D are the closest distances from the Sun and the comet to the passing star (Fig. 2a). D.., D are both unit vectors. With reference to the Sun the comet will receive an impulse I=

/~-

I~

(2)

The number of stellar encounters N for the studied period o f T = l09 years was obtained from N = noDM"T, where n0 is the stellar flux in the neighborhood o f the Sun, for which the value given by Rickman of l0 stars pc -2 l0 -6 years was adopted. Times t~ taken at random in the interval 0 < t~ < T and set in ascending order were assigned to the N stellar encounters. A random mass Mi was also assigned to each one of the N stars in such a way that the resultant Mcdistribution followed the observed mass distribution for nearby stars. Such a mass distribution was deduced for the luminosity function of nearby stars taken from Wielen (1974). Values o f 0. I M~: and l0 1t4.-..,were taken as lower and upper limits of the M,-distribution. It was also assumed that 3 5 ~ were double stars. For

409

each star/, the closest Sun distance D ~ was taken such that the star had the same probability to pass through any point of the target circle o f radius DM centered at the Sun and normal to the stellar path. Five subgroups of l0 comets were considered within each group with a given aimt, with a different set o f values (t~, M,, D~;,) for the N stellar encounters being applied to each one of them. The geometry o f a stellar encounter is shown in Figs. 2a, b. The star was supposed to move along an unperturbed straight path with constant velocity V with reference to the Sun. A set of Cartesian axes centered at the Sun was taken, with the x-axis parallel to the star's path and the y-axis on the plane formed by the Sun and the star's path. The S u n - c o m e t distance r was deduced for encounter i from its relation with (t~ - tq), t~ and tq being the times o f the encounter with star i and the last comet passage through its perihelion, respectively. The comet coordinates in a stellar encounter were given by: x~ = ( - 1 + 2ct~)p with similar equations for y,., z,., where the numbers (a . . . . ) were taken at random in y

Ca) 0e

st°r "s/.) ,

Xc

izz /

X

[b) It, . . . . .

~

• |

c

FIG. 2. Geometry of a stellar encounter: S: Sun and C: comet.

410

"••

JULIO A. FERNANDEZ

the interval [0, 11 and p is a scale factor such that x,," + y c 2 + z,. z = r " . Finally, another important geometrical element is the angle /3 formed by the transverse impulse It and the transverse velocity vt of the comet (Fig. 2b). A random value in the interval 0 -
31: I

~, ~D >ty

~

,

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7

,

_~m

~!

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~j

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.,.r

m

3

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;

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i

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i

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i

i-"

RESULTS OF TH!4 NUMERICAL EXPERIMENT Figure 3 shows the distribution of perihelia, inclinations, and semimajor axes o f the sample comets at five different times. To analyze the influence of the a.,~, values in the later evolution o f the c o m e t a r y orbits. two groups of comets were distinguished: those with am. <-- 25,000 AU (Group 1) and those with a.,. _-> 30,000 AU (Group 2) A shift in the perihelia toward greater values is observed with time. an expected result from a theoretical point of view (see Appendix IIL At the end of the studied period, the values o f the perihelion distances spread from some tens of AU to more than l0 s AU, and 31 of the surviving comets, that is 16e~, had a q < I()a AU. These comets are the most suitable for coming again into the planetary region because of stellar perturbations. As expected, Group I (shaded histogram) was less influenced by stellar perturbations and consequently underwent a smaller increase in the q ' s (in this case the comets with q < 1() :~ AU at l0 "~years were 34qf). At the end of the studied period, there was a predominance of direct orbits in Group l, whereas Group 2 were completely randomized in the orientations of their orbital planes.

1

3

5

LOGq(AU)

gO"

(.

180'] 0

5

10

15

El. (104AU)

FIG. 3. D i s t r i b u t i o n o f p e r i h e l i a , i n c l i n a t i o n s , and s e m i m a j o r a x e s o f the c o m p u t e d c o m e t s a m p l e ( G r o u p I + G r o u p 2L T h e s h a d e d p a r t s of the h i s t o g r a m s c o r r e s p o n d to the partial s a m p l e o f c o m e t s w i t h initial a _-< 25000 A U ( G r o u p It. T h e c o m e t s in e l l i p t i c a l o r b i t s that a c q u i r e d an a - 1.5 × l0 s A U (14 at 10 ~* y e a r s ) w e r e not r e p r e s e n t e d in the a - d i s t r i b u t i o n diag r a m s . At u p p e r right, the t i m e in y e a r s is i n d i c a t e d .

A little more than 50% of the surviving comets maintained semimajor axes within the i n t e r v a l 2 x l0 t < a < 5 x l0 ~AU. For a < 5 x l0 ~ AU, a small fraction of comets is found with widely spread values of their semimajor axes which extend beyond 1.5 x l05 AU, particularly after 5 x l0 ~ years. Nonetheless, for a ~>10s AU the ejection probability of a comet by stellar perturbations is strikingly increased. In effect, as can be seen in Table 1, the number of ejected comets by stellar perturbations as compared with the surviving number is small for a ~ l0 s AU and such ejections were mainly due to very close stellar passages. This ratio

EVOLUTION OF COMET ORBITS TABLE I RATIO (R) oF EJECTED COMETS BY STELLAR PERrURBATIONS T O SURVIVING COMETS IN 10 9 YEARS Range of semimajor axes (l& AU)

50
a < < < a >

R

50 100 150 150

0. l l 0.16 0.50 1.86

increases quickly for a ~ 10 '~ AU, where the cumulative effect of more distant stellar passages b e c o m e s the principal cause of c o m e t a r y ejections. At distances of about 2 x 10 '~ AU the action of the galactic tide, which was not considered in this numerical experiment, is an additional cause of instability in comet orbits (Lyttleton, 1974). The orbital eccentricities averaged o v e r intervals of Aa = 2 × 104 AU are shown in Fig. 4 for different times. For 108 years, they still maintain values very close to unity, independently o f a. For 109 years, there is still a strong predominance o f the very elongated orbits for a ~>6 × 104 AU, where the average e is greater than 0.8. The predominance o f high eccentricities had already been pointed out b), Opik (1973) and it is o f great importance, since the

probability that a c o m e t enters the planetary region because of stellar perturbations is increased. An important n u m b e r o f c o m e t s , a total of 205, were lost to hyperbolic orbits because of planetary perturbations. The cumulative n u m b e r o f ejected c o m e t s as a function of time is shown in Fig. 5. In group I as well as in G r o u p 2, the majority of the ejections occurred within some tens of millions of years, when most c o m e t orbits kept their perihelia within or very near the planetary region. H o w e v e r , ejections by planetary perturbations were registered throughout the studied period. The perihelion distances o f the comets o f smaller semimaj o r axes undergo smaller changes because o f stellar perturbations, implying a greater n u m b e r o f passages through the planetary region before leaving it. Instead, m a n y comets o f larger ainit ( 4 . 5 and 5 × 104 AU) were r e m o v e d from the planetary region in the first orbital revolution and registered no passages throughout the studied period. This different behavior is reflected in different percentages o f c o m e t s ejected hyperbolically: nearly 50% for G r o u p I and only - 3 5 % for G r o u p 2. A theoretical analysis o f the perihelion changes by stellar perturbations Aq per orbital revolution as a function of the semimajor axis (cf. Appendix II) confirms the c o m p u t e r results. This analysis is an extent

120

10'

1.0

411

F

m

;

GROUP I

,o" e

/f/

/

60

108 ~ .

:

....

/

o.8

II

J'-

I

~. ....

/

2

//f

I

\

0.6

A.f-

i

, .....

i

i

i

2

4

6

8

10

B

_ ~-'- ~ 12

[ 104AU1

FIG. 4. A v e r a g e e c c e n t r i c i t i e s o f c o m e t o r b i t s w i t h i n i n t e r v a l s Aa = 2 × 10 ~ A U as a function o f t h e i r s e m i m a j o r a x e s for different t i m e s in y e a r s .

6.0

6.5

A

*

,

7.0

7.5

_.*. I1.0

L .... 8.5

/ 9.0

LOGt ( y r s )

FIG. 5. C u m u l a t i v e n u m b e r o f c o m e t s e j e c t e d into h y p e r b o l i c o r b i t s by p l a n e t a r y p e r t u r b a t i o n s as a function o f time.

412

J U L I O A. FERN,~NDEZ

o f R i c k m a n ' s (1976) w o r k w h e r e he f o c u s e d his a t t e n t i o n o n the r e m o v a l o f c o m e t s f r o m J u p i t e r ' s influence z o n e . F o r a c o m e t o r b i t o f q = 30 A U a p r a c t i c a l l y null m e a n c h a n g e p e r o r b i t a l r e v o l u t i o n a n d a stand a r d d e v i a t i o n ~<1 A U w e r e o b t a i n e d for a < l04 A U ( F i g . 6a). T h e p e r i h e l i o n dist a n c e s o f such c o m e t s a r e little a f f e c t e d b y s t e l l a r p e r t u r b a t i o n s for w h i c h in the g r e a t m a j o r i t y o f c a s e s t h e y will r e t u r n to t h e p l a n e t a r y r e g i o n . F o r a = 2.5 x l04 A U Aq is still negligible w h e r e a s the s t a n d a r d d e v i a t i o n re,a c h e s a v a l u e o f a b o u t 10 A U , so it is to b e e x p e c t e d that in t h e s u c c e s s i v e

100

!

75

!

(_ o~ t~

E

I

so

c

25 ~.'--zzE72L 0

10

20

_ .~-/~ 30

40

50

>SO

number of passages Fl(;. 7. Distribution of the number of former passages through the planetary region of the comets ejected hyperbolically by planetary perturbations. The shaded histogram correpsonds to Group I comets.

300

,'

ca) / /

200

/

tS]r 100

-100

,

,

,

1

t°~

=

,

L

,

3 5 a ( 104AU )

~

7

(b)

c_

.E)

E

10 z

c

to I

1

I

1

0.5

1.0

1.5

2.0

a

(104

AU)

FiG. 6. Perihelion changes of a comet orbit because of stellar perturbations as a function of its semimajor axis. (a) Mean change for a comet with initial perihelion q - 30 AU in an orbital revolution. The dashed curves correspond to the mean _+ standard deviation. (b) N u m b e r of revolutions that such a comet needs to experience a mean change of + 5 AU in its perihelion distance.

p a s s a g e s s m a l l i n c r e a s e s and d e c r e a s e s in q will o c c u r , a l t h o u g h t h e a v e r a g e v a l u e will s t a y a l m o s t c o n s t a n t . H o w e v e r , s u c h small r a n d o m c h a n g e s in the p e r i h e l i o n d i s t a n c e s will be sufficient to r e m o v e c o m e t s f r o m the p l a n e t a r y r e g i o n a f t e r a few r e v o l u t i o n s . F r o m a = 3 x l04 A U t h e m e a n c h a n g e Aq i n c r e a s e s r a p i d l y . F o r a -'= 6 x l04 A U w e h a v e Aq --= 80 -+ 70 A U , that is, t h e r e will be a p r o b a b l e p e r i h e l i o n c h a n g e b e t w e e n + l0 and - 150 A U in an o r b i t a l r e v o l u t i o n w h i c h will r e m o v e t h e c o m e t a w a y f r o m the p l a n e t a r y r e g i o n . T h e r e f o r e , as w a s s e e n in the numerical experiment, we can expect that m o s t c o m e t s with ainit > 5 - 6 x l04 A U will n e v e r r e t u r n to the p l a n e t a r y region. T h e n u m b e r o f r e v o l u t i o n s that a h y p o t h e t i c a l c o m e t with q = 30 A U a n d a ~< 104 A U n e e d s for u n d e r g o i n g a m e a n c h a n g e in the p e r i h e l i o n o f Aq = + 5 A U is g r e a t e r t h a n 100 (Fig. 6b). M o r e p r o b a b l y , b e f o r e l e a v i n g the p l a n e t a r y r e g i o n , s u c h a c o m e t w o u l d r e c e i v e a s t r o n g p e r t u r b a t i o n from Neptune which would change completely its a - v a l u e . C o n f i r m i n g this, few c o m e t s in the g r o u p with a.,,t = 5 - 1 0 × 10 :~ A U s u r v i v e d with r a t h e r u n c h a n g e d s e m i m a j o r a x e s d u r i n g the s t u d i e d p e r i o d o f 109 y e a r s . Only occasional very close passing stars c o u l d p l a c e s u c h c o m e t s in v e r y s t a b l e o r b i t s w i t h q ~> 35 A U a n d a <~ 104 A U . M o s t c o m e t s e j e c t e d h y p e r b o l i c a l l y in

EVOLUTION OF COMET ORBITS 16

interior to Jupiter's orbit (we will call it the " o b s e r v a b l e region"). Such new c o m e t s were deflected f r o m the c o m e t a r y cloud to the inner planetary region by stellar perturbations. Only one new c o m e t had an aorlK s o m e w h a t smaller than 10' AU, confirming the a b o v e assertion that stellar perturbations are ineffective in bringing c o m e t s with a < 10' AU into the planetary region, unless a very close stellar e n c o u n t e r takes place. Many of the new c o m e t s repeated their passages through the o b s e r v a b l e region as " o l d " comets, in general with an ao~ of several hundreds or some thousands astronomical units. The ao~,g-distribution of the hypothetical incoming comets, seen in Fig. 9, can be c o m p a r e d with the o b s e r v e d one seen in Fig. I. In both cases there are few comets with aorig > 6 × 10' AU. In the experimental sample the aort, values o f the new c o m e t s concentrate in the interval

group I

E

group 2

._! 10

I-~ 20

rtO

q (AU) FIG. 8. Perihelion distribution of comets ejected in hyperbolic orbits by planetary perturbations. Perihelion distances are taken from their last passage before

ejection. G r o u p 2 registered less than five passages before being ejected. On the other hand, the n u m b e r o f previous passages was on average much greater for G r o u p I (Fig. 7). The perihelion distributions o f both groups also show remarkable differences (Fig. 8). For G r o u p 1, the strong concentrations around the Saturn and Neptune distances suggest that these planets were the principal ejectors of comets. Instead for G r o u p 2, the greater concentration is shifted towards the region of Jupiter and Saturn, since the larger perihelion changes due to stellar perturbations make m a n y o f these c o m e t s fall under the strong perturbing influence o f these two planets after few revolutions. Then, in this case we can expect the ejection to o c c u r in the first revolutions. As a result o f the c o m b i n e d action o f stellar and planetary perturbations, the population o f studied c o m e t s decreased with time. Such a decrease was greater in G r o u p I wherein the comets were bound to the planetary region for a longer time. The n u m b e r o f surviving c o m e t s in the sample at time t is a p p r o x i m a t e l y proportional to t -~, where r/ = 0.2 for Group I and "0 ~0.13 for G r o u p 2. A total of 52 passages o f " n e w " c o m e t s were registered within the planetary region

413

20 15 10 5 15

E

~

s I

3

5

7

g

20

15

5 rHA 10

40

70

100

uor,g( 103AU ) ° FnG. 9. Computed distribution o f original semimajor axes o f all the comets entering the observable region w i t h 300 < ~,,~. < l0 s A U . A l l the " n e w °" comets but one had ~r,= > 10' A U and o n l y o n e - - n o t p l o t t e d - had ~ , ~ > l0 s A U . On the o t h e r hand, there were six " o l d " comets with e,,,, > 10' A U . The new comets o f

Group I were distinguished from the remaining ones (new comets of Group 2 + old comets) by the shaded area.

414

JULIO

A. F E R N , ~ , N D E Z

104 < aorig < 5 X 104 AU. But, when only G r o u p 1 c o m e t s are considered, the strong concentration is limited to the narrow range from 2 to 3 x 104AU, in good accordance with the m a x i m u m concentration for the o b s e r v e d new comets. The experimental a-distribution for the old comets closely follows the law n(a)da :x a -Zda, that is, an equal n u m b e r o f c o m e t s in equal intervals o f l / a in accordance with theoretical predictions (van W o e r k o m , 1948). In the o b s e r v e d c o m e t a r y sample the fraction of old comets in relation to the new ones is smaller than expected, but this is due to a selection effect, since old c o m e t s are more likely to pass u n o b s e r v e d due to the wastage that they have undergone in their previous passages. The variation in the rate of c o m e t a r y passages through the planetary region as a function o f time was also analyzed. The n u m b e r of passages per 10~ years within the observable region was considered (Fig. 10a). Passages within 35 AU were also considered, provided that in each "'new p a s s a g e " the c o m e t had previously attained a perihelion q > 50 AU (Fig. 10b). In both

,oh ........ "

"~ ~ s

i

o

~

10

0

2

4

6

8

10

t (10Syrs) Fl(i. 10. Histograms showing the n u m b e r of comet passages through the planetary region as a function of time: (a) within 5.2 AU from the Sun, (b) within 35 AU. The shaded histograms correspond to Group I comets.

--

0

is0' /----v,~.~

6

ir.~, 0

200

/~ j

//~-q

/---i ,

400

600

800

1000

t [106 yrs) FI6. I I. Evolution o f the orbital elements o f one of the c o m p u t e d comets.

cases a clear p r e d o m i n a n c e o f passages in the first l08 years can be noticed, that is, when most comets kept their perihelia close to the planetary region. A n u m b e r of passages proportional to the considered perihelion intervals (16 f o r q < 5.2 AU and 107 for q < 35 AU, within the period 10~ < t < 10"~ years) was registered in accordance with theoretical predictions (Opik, 1966) and other numerical e x p e r i m e n t s (Weissman, 1977). According to our results, the rate of passages through the planetary region decreases a p p r o x i m a t e l y as t -], for t > 10~ years. This result is applicable to both the G r o u p I and the G r o u p 2 comets. The evolution of the orbital elements of a particular c o m e t is shown in Fig. ! 1. After acquiring perihelion distances greater than 2000 AU, the c o m e t entered the planetary region for the first time at 4.3 × 10~ years. It later registered a new passage and finally, at 8.52 × 10s years, came into the observable region with q -- 0.72 AU. Next, it entered the obsevable region ten successive times as an old comet, afterwards leaving the planetary region. The orbital inclination registered a large variation during the studied period, whereas the semimajor axis oscillated within the interval 3.5 × 10 4 < a < 6.5 × 104 AU. The preceding example illustrates a characteristic which was verified in most of our

EVOLUTION OF COMET ORBITS experimental new comets. Prior to their passages through the o b s e r v a b l e region they had registered passages through the planetary region b e y o n d Jupiter, exceeding 100 passages in some cases. In addition, 6 out o f the 52 new c o m e t s again passed through the o b s e r v a b l e region with an aor~g > l04 AU. Thus, it is to be e x p e c t e d that a small part of the c o m e t s we call " n e w " have in reality already passed through the o b s e r v a b l e region, as has been noted by Bailey (1977). There is a p r e d o m i n a n c e o f direct orbits in the incoming new comets within the observable region (Fig. 12a). H o w e v e r , when only the passages f o r t > l0 ~ years are considered, the /-distribution is in rather good a c c o r d a n c e with random orientations o f the orbital planes, as o b s e r v e d in the long-period comets. The q-distribution of the experimental new c o m e t s more or less fits a uniform distribution (Fig. 12b). DISCUSSION

It is generally accepted that planets were formed from the accumulation of small bodies or planetesimals. The composition of the giant planets suggests that planetesimals formed in the outer part o f the planetary region would be c o m p o s e d

.c/

O°

900

180°

L

E c

12-

[--

]

8-

I

I

0

1

(b)

.... 2 3

4

q (AU)

FiG. 12. ( a ) Inclination distribution of the incoming new comets from the computed sample where the shaded part corresponds to the passages for t ;- 10~ years. (b) Distribution of their perihelia.

415

mainly of ices like the comets. Then, the presence o f comet-like objects in the early planetary zone was a natural by-product of the formation of the giant planets. It is reasonable to suppose that any residual bodies not integrated into the giant planets were r e m o v e d by planetary perturbations and part o f them would follow very elongated ellipses like those which were the starting point o f this study. Uranus and Neptune were the principal ejectors o f c o m e t s along near-parabolic orbits (we will call them NP comets), corresponding to them about 75% o f all N P comets r e m o v e d from the planetary region, while the remainig 25% corresponded to Jupiter and Saturn in accordance with previous studies (Safronov, 1972, Paper I). It has usually been assumed that comets were r e m o v e d from the planetary zone to the cloud immediately after the formation of planets. H o w e v e r , assuming that Uranus and Neptune were the main contributors we would expect long time scales for the removal of matter, a process continuing perhaps at present. For example, lp (1977) has found from a numerical study that an important fraction of the Uranus and Neptune crossing bodies have lifetimes greate r 10"~years. Opik (1973) also e m p h a s i z e s the long lifetimes o f planetesimais in the Uranus and Neptune region, although his proposed value o f 20 × 10a years seems to be overestimated (Paper I). Finally, Uranus and Neptune could have been capturing until now material coming from a belt located beyond Neptune (Fern~indez, 1980). H o w e v e r , according to the results of our numerical experiment, it is to be stressed that a very recent evolution (say, t < 10 ~ years) for most o f the comets we o b s e r v e at present has to be rejected. A very recent evolution of c o m e t s from their birthplace in the outer planetary zone would imply a p r e d o m i n a n c e of o b s e r v e d c o m e t s in direct orbits, unless one hypothesizes an alreadyestablished r a n d o m orientation of the orbital planes of bodies r e m o v e d from the planetary zone.

416

JULIO A. FERN,~NDEZ

Let us see then what model of cometary cloud we must expect on the basis of our numerical results. It will be assumed that comets have been incorporated into the cloud throughout the solar system's lifetime, so we will place our average "'time of r e m o v a l " 2 × 10'j years ago. We can expect that most removed N P comets which have fallen under the influence o f stellar perturbations, say w i t h a > 104 AU, had values of semimajor axes close to this lower limit since their frequency is proportional to a -z. Therefore, we will stress the results obtained for Group I of comets when they differ from those of Group 2. When considering the results of Group 1, we find a remarkable concentration of original semimajor axes o f new comets passing through the observable region in the narrow range 2 x 104 < a,,rig < 3 x 104 A U. For very elongated orbits this implies a concentration o f aphelion distances in the range 4 x 104< Q < 6 x 10*AU, in good agreement with the observed maximum concentration of original aphelion distances which defines the "radius of the Oort c l o u d . " This maximum concentration can be accounted for by one of the two following possibilities: a great number of cloud comets is found really at this range of distances a n d / o r it corresponds to the place wherein stellar perturbations are more effective in bringing comets within the planetary region. The numerical experiment has shown that the greatest number o f comets appear effectively in the range of semimajor axes 2 x 104 .< t'l < 3 x 104 AU at the end of the studied period, verifying that there is a close correspondence between the defined radius of the Oort cloud and the range of aphelion distances at which the greatest number of cloud comets is found. For comets with a ~ 2 x 104 AU, stellar perturbations are strong enough to remove a great fraction of them from the planetary region before they undergo a strong planetary perturbation. But the situation changes quickly for a <~ 2 × 104 AU, where planetary perturbations are the principal cause

for depletion of cloud comets. For a ~< 104 AU, only occasional very close stellar passages could put comet orbits passing through the planetary region into other very stable ones with perihelia outside the planetary region and small a. There are two reasons for believing that the n u m b e r o f c o m e t s decreases f o r a ~> 3 × 104 AU. As a increases, there is a lower probability of getting N P comets because of planetary perturbations. On the other hand, the increasingly strong action of stellar perturbations causes comet orbits to evolve more rapidly. As it was seen, for a ~> I0:' AU (Q ~> 2 x l0 ~ AU) the depletion of comets due to stellar perturbations increases notoriously, and this can be taken as an upper limit for the cometary cloud. This result is in good agreement with the early value quoted by Oort (1950) and is remarkably smaller than the radius value of 6.7 × 10:' AU given by Opik (1973). It is to be expected that most comet orbits with a <-- 6 × 104 AU preserve high eccentricities for time scales of the order of the solar system age. It was seen that a great number of comets maintain at the end of the studied period their perihelia close to the planetary zone with some hundreds or thousands astronomical units. This is of great importance with regard to the amount of cloud comets needed to maintain the rate of observed new comets at a steady state, since the probability that a comet reenters the planetary region because of stellar perturbations is enlarged. This result removes one o f Lyttleton's (1974) objections to the cometary cloud, namely, the very low probability that stellar perturbations bring comets into the planetary region, which he infers from the consideration o f a cloud o f comets in more or less circular orbits. A rough estimate about the number of comets needed to form the cloud can be established by scaling up the results of our numerical experiment. For the sample of 500 comets, a rate of - 1 comet 10-~ year within the observable region was obtained at t = I0 "~years. Then, we should expect a

EVOLUTION OF COMET ORBITS rate o f - 0 . 5 comet 10-~ year at t = 2 × 109 years, according to a diminution law at t-L Therefore, if we assume t h a t No comets were placed in the cloud an average o f 2 x 109 years ago, at present 10 -'~ No o f them would pass through the observable region in 108 years. Delsemme (1973) pointed out that the rate of passages o f observed comets is 0.3 years -~ 10 -~ A U -~ per Aq = 1 AU, that is, this gives a rate o f - i.6 x I0 ~ passages 10-s year within the observable region. 0 p i k (1973) estimated a rate o f 1.3 annual apparitions o f new comets within the region interior to Mars orbit, leading to 4 x 10s passages 10-~ year. We will adopt an average value of - 2 . 8 × 108 passages 10 -s year. Applying the above result, we would have an amount o f 2.8 × 101' comets removed by the giant planets from which about 75% would have corresponded to removals by Uranus and Neptune, that is about 2 x 10 ~. It was seen in Paper I that the fraction of NPcomets(saywith0< ( l / a ) < 2 × 10 -4 AU -~) as compared to those ejected in hyperbolic orbits because o f close encounters with Uranus and Neptune is about 0.4. Therefore, the total number o f comets removed by Uranus and Neptune would be of 2 x 10 ~ x 1 . 4 / 0 . 4 = 7 x 10 ~ . T h e e s t i m a tion of the total mass is difficult due to uncertainites in the knowledge of individual comet masses. For example, assigning an average mass o f 10 ~r g to each comet, we would have an ejected total mass of 7 x I 0 z~ g -~ 12 M÷ (Earth masses), that is, somewhat smaller than the Uranus mass, which appears to be reasonable. Within the uncertainty o f the latter result, one can consider as feasible that Uranus and Neptune have removed several tens of Ms without undergoing a significant modification in the shape and size o f their orbits. This is so because the energy lost by the planet, and carried away by the removed bodies, would have been compensated by the bodies deflected to the inner regions which would then be subject to the gravitational influence of Jupiter and Saturn

417

(see, e.g., Everhart's (1977) numerical study of the transfer of comets from the Uranus and Neptune region to the inner planetary region). The early population of comets incorporated into the cloud has been depleted by the combined action of stellar and planetary perturbations. For the 250 starting comets o f Group 1, 71 remained at t = 10"~ years and an extrapolation o f this result according to a diminution law as t-°'" would lead to about 60 comets at t = 2 x 109 years, that is 25% o f the early population. As was seen, the number o f comets removed into nearparabolic orbits by the giant planets was estimated to be about 2.8 x 10 ~. Then, after 2 x 109 years we would expect a p o p u l a t i o n o f 2 . 8 x 10 ~t x 0.25 = 7 x 10 t° comets. This result is somewhat smaller than that previously given by Oort (2 x 10 ~ comets), due to the predominance of very elongated orbits in the cloud, which increases the probability that comets pass through the planetary zone. Again, for an average cometary mass o f 1017 g the cometary cloud would have a total mass o f 7 x I02r g ~ 1.2 M..:~. CONCLUSIONS Summing up, it has been stressed that the comet cloud was formed with material coming mainly from the region of Uranus and Neptune, comprising the adjoining region outside it. The long time scales to reach near-parabolic orbits for the bodies in this region would have as a consequence that comets were not incorporated into the cloud right after the formation o f planets, but they would have been incorporated into it through all the solar s y s t e m ' s lifetime. This fact has its effects in the properties of the comet cloud. Thus, we can expect a predominance o f very elongated orbits and a certain tendency to direct orbits. The comets tend to predominate in the narrow range o f semimajor axes 2 x 104 < a < 3 x 104 AU. This can be defined as the optimum range for which stellar perturbations favor the maintenance o f comets

418

JULIO A. FERN,~NDEZ

within the c l o u d . T h e y a r e s t r o n g e n o u g h to r e m o v e c o m e t s f r o m the p l a n e t a r y r e g i o n before planetary perturbations change s t r o n g l y t h e i r s e m i m a j o r a x e s but t h e y a r e not so s t r o n g as to p r o v o k e a r a p i d o r b i t a l e v o l u t i o n l e a d i n g to t h e e j e c t i o n o f c o m e t s in h y p e r b o l i c o r b i t s . In a d d i t i o n , w e c a n define a r a n g e o f s e m i m a j o r a x e s o f 104 < a < 5 × 104 A U in w h i c h b o t h s t e l l a r and p l a n e t a r y p e r t u r b a tions influence the o r b i t a l e v o l u t i o n o f c o m e t s in a g r e a t e r o r s m a l l e r d e g r e e . F o r t'/ .~> 5 X 10 4 A U , it is tO be e x p e c t e d that s t e l l a r p e r t u r b a t i o n s r e m o v e c o m e t s from the p l a n e t a r y r e g i o n d u r i n g the first r e v o l u tion, so t h e y w o u l d b e c o m e the o n l y f a c t o r o f l a t e r o r b i t a l e v o l u t i o n . I n s t e a d , for a ~< 10 1 A U , c o m e t s are a l m o s t e x c l u s i v e l y subj e c t to p l a n e t a r y p e r t u r b a t i o n s and o n l y s e l d o m v e r y c l o s e s t e l l a r p a s s a g e s c a n affect t h e i r o r b i t s s u b s t a n t i a l l y . It is verified that a g r e a t m a j o r i t y o f " ' n e w " c o m e t s e n t e r i n g the o b s e r v a b l e region had a l r e a d y r e g i s t e r e d p a s s a g e s t h r o u g h t h e p l a n e t a r y region. T h e n u m b e r o f c l o u d c o m e t s is e s t i m a t e d to be a b o u t 7 × 10 z°. T h i s w o u l d i m p l y a removal of one order of magnitude more comets away from the planetary region by U r a n u s a n d N e p t u n e . A s s u m i n g an a v e r a g e c o m e t a r y m a s s o f l0 ~Tg, that n u m b e r w o u l d c o r r e s p o n d to a total r e m o v e d m a s s o f - 1 2 M::., w h i c h a p p e a r s to be r e a s o n a b l e . T h u s , the r e s u l t s o b t a i n e d l e n d s u p p o r t to the t h e o r y that an O o r t - t y p e c l o u d , f o r m e d with t h e r e s i d u a l m a t e r i a l left a f t e r f o r m a tion o f the g i a n t p l a n e t s , is the o n l y , o r at least the p r i n c i p a l , s o u r c e for the o b s e r v e d long-period comets. Only a drastic upwards r e v i s i o n in the c o m e t a r y m a s s e s o r in the rate o f a p p a r i t i o n s o f n e w c o m e t s w o u l d m a k e it n e c e s s a r y to t u r n o u r a t t e n t i o n to alternative sources of comets. APPENDIX I CHANGES

approximately given by

vt 2 = 2tazl /r 2,

(AI)

w h e r e p, = G M D u e to the s t e l l a r e n c o u n t e r , w e will h a v e a n e w t r a n s v e r s e v e l o c i t y v; given b y ( s e e Fig. 2b)

v~2_

21,zq' r2

- -

It" + v, z + 21, vt c o s /3

(A2)

from which

aq : q '

-q /.2

= 2---~(It" + 21tvt c o s / 3 ) .

(A3)

F r o m Fig. A I w e d e d u c e c o s i' = c o s i • c o s il -

sini-sin

i l c o s a,

(A4)

w h e r e a w a s t a k e n at r a n d o m in the i n t e r v a l 0 < a < rr/2 for e a c h e n c o u n t e r . B e s i d e s , we have (sin iO/It = (sin/3)/v~.

(A5)

By c o m b i n i n g Eqs. (A4) and (A5) we o b tain: Ai = i' - i. F i n a l l y , the c h a n g e in the o r b i t a l e n e r g y will be g i v e n by

/',(I/a)

=

lla'-

=

I/u,(v ~ -

Ila

lluXv"- v'")

=

(vr + L)" -

(vt

+

It)Z),

(A6)

w h e r e v2 = vrz + vt2 a n d 1 2 = / 2 + It" a n d

IN T H E O R B I T A L

ELEMENTS ~u,/, z~i AND ~(l/a) OF A COMET IN A STELLAR ENCOUNTER

L e t vl be the t r a n s v e r s e v e l o c i t y o f the c o m e t b e f o r e the s t e l l a r e n c o u n t e r , w h i c h is

Fici. AI. Representation on the celestial sphere of the orbital planes of comet C before and after stellar pert urbation.

EVOLUTION OF COMET ORBITS after some calculations we obtain

A(l/a) :

-- I/P(,(/2 + 2Vr/r + 2Vt/t COS /3).

(A7)

A P P E N D I X 11 M E A N C H A N G E IN T H E P E R I H E L I O N D I S T A N C E ( q ) OF A C O M E T IN N E A R - P A R A B O L I C O R B I T AS A C O N S E Q U E N C E O F P E R T U R B A T I O N S BY N E A R B Y STARS

Let us consider the perturbation caused by a star o f mass M passing at a minimum distance D.:~ from the Sun and D from the comet (Fig. 2a). Let 0 be the angle formed by the impulse I and the comet radius (Fig. 2b). Defining ~ = l,/vt and being It = I sin 0, we will have from Eq. (A3) Aq = q(8" sin" 0 + 28 sin 0 cos /3).

(A8)

For stars passing very close to the Sun as compared to the S u n - c o m e t distance r(say D~ < 2-3r), we can admit as a rough estimate that in such stellar encounters the impulse L~ predominates over 1,. and then from Eqs. (i) and (2) we will have

2GM I VD.:: '

(A9)

419

the very close encounters, 0 values smaller than 7r/2 will predominate. Taking this effect into account, it would be reasonable to adopt for 0 a pdf o f po(O) dO = sin 0 (1 0 / ~ ) dO, 0 <= 0 <- rr. By substituting the pdf's of 0 and /3 in Eq. (AI0) and by integrating we obtain Aqt = ~SZq.

(All)

The variance o f Aql will be given by O-i 2

p•(fl) d/3 fo-' (AqO2po(O) dO - (~-0",

(AI2)

which after integration leads to 0-1z = iiq28" (1 + 82/15).

(AI3)

For more distant stellar encounters ( D : > 2-3r), we can admit as a rough approximation that the vectors D and D,.~are parallel, in this case we will have

I = 2GMr cos O/VD:~" = 1' cos 0

(AI4)

that is, 1 depends on 0 in this case. Then, defining 8' I ' / v t and applying a similar procedure to the above we will obtain =

where V = 30 km sec -~ and where we adopt f o r M an average mass of 0.7 M,~ (single and double stars). In order to simplify the calculations we discard the separate consideration o f the very close stellar encounters to the comet (D < 2-3r) which, at any rate, are also in a great part encounters very close to the Sun. The mean change in the perihelion Aq~ caused by a star passing very near the Sun will be given by

Aql =

f,jpo(/3)d/3 I

dO,

(AlO)

where Aq~ is obtained from Eq. (A8) with 1 taken from Eq. (A9). A uniform probability density function (pdf) for /3 can be taken as a reasonable one. The election o f a simple and realistic pdf for 0 is not so easy. Generally, except in

A q 2 = i~8 ~ ,2q,

0-z2 = ~ t o

(AI5) tl + 8'2/35).

(AI6)

During an orbital revolution the comet will be perturbed by many stars. These perturbing stars will meet the comet at different distances r from the Sun. For simplicity, let us adopt for r a single value averaged in time of ~ = 1.7a, such that the comet will have r > F during half its sidereal period. Let n(D~,) dD:~ be the number of stars passing at distances from the Sun within the interval (D.:,, D + d D ) per unit time; it will be given by

n(D;.) dD.; = noD~dD-,,

(AI7)

where no --- 10 stars pc -z 10-6 year. The total mean change in the perihelion (Aq,)r during an orbital revolution, caused by very close stellar encounters, will be

420

JULIO A. FERN,ANDEZ

then given by (Aq~)r = ~ q T fi'~ 8Zn(D~) dD<:~ = .~qr~2noTlog(DL/Dm),

(AI8)

w h e r e ~5~ = 8Dc~ = 2 G M / V v , , T = a a~z is the sidereal period o f the c o m e t , DE = 2 - 3 pc is the a d o p t e d limit d i s t a n c e b e t w e e n the "'very close" and " c l o s e " encounters, and Dm is the m i n i m u m d i s t a n c e to the S u n at w h i c h a star is e x p e c t e d to pass w i t h i n the period T. This d i s t a n c e has b e e n t a k e n with the c r i t e r i o n that the p r o b a b i l i t y that a star passes at a d i s t a n c e D:: < D m from the S u n d u r i n g a period T is ~, that is n(D:.~ < Dm) = n,,DmZT = ~, wherefrom D m = (i/2r~,T) t~z.

(AI9)

In a s i m i l a r w a y we will o b t a i n for the variance (cr,").r = 2q(Aq07 + &qZS~'n,,T(I/D~" -

I/DL~).

(A21)

( cr~2)v = 2q( Aqz) r + T~-'rsq28'14noT/Dt. ~,

(A22)

w h e r e 8'~ = 8'D,:~ a n d the u p p e r limit o f the i n t e g r a t i o n is t a k e n as infinite. T h e n , the m e a n c h a n g e in the p e r i h e l i o n per orbital r e v o l u t i o n will be g i v e n by ( A q ) r = ( A q 0 r + (Aq2)r

ACKNOWLEDGMENTS I wish to thank Drs. S. J. Weindenschilling and E. Everhart who. as referees, made useful comments on the manuscript and Eng. A. Barcia for revising the English text. The computations were carried out on the CDC CYBER 172 system of the lnstituto Nacional de Investigaciones Agrarias. This research has been supported by a Grant from the Ministry of Education of Spain under the International Cooperation with lberoamerica Program.

(A20)

By a p p l y i n g a n a n a l o g o u s p r o c e d u r e to the m o r e d i s t a n t e n c o u n t e r s ( D . > DL) we obtain ( A q D r = toqS~t2 n~,T/Di. 2 ,

(2r a n d 3r) did not yield i m p o r t a n t differe n c e s in the results. W e h a v e t h e n a d o p t e d as v a l u e s for the m e a n a n d s t a n d a r d d e v i a tion o f Aq the a v e r a g e s o f the r e s u l t s obt a i n e d with b o t h v a l u e s o f DE. W e c a n r e v e r s e the p r o b l e m a n d calculate from Eq. (A23) the time t that m u s t e l a p s e before a g i v e n m e a n c h a n g e in the p e r i h e l i o n d i s t a n c e o f a c o m e t t a k e s place, a s s u m i n g in this case that the s e m i m a j o r axis o f its orbit r e m a i n s u n c h a n g e d d u r i n g t. This was applied to the results o b t a i n e d in Fig. 7b.

(A23)

a n d the s t a n d a r d d e v i a t i o n

REFERENCES BAILEY, M. E. (1977). Some comments on the Oort

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O'T = ((Orl2.)T -~- (Or22)T) 1/2.

(A24)

It is to be n o t e d that (Aq)r is i n d e p e n d e n t ofq. T h e a d o p t i o n o f a d i s t a n c e Ol, s e p a r a t i n g the v e r y close e n c o u n t e r s from the m o d e r ately close o n e s is more or less a r b i t r a r y . H o w e v e r , the c a l c u l a t i o n s o f Eqs. (A23) and (A24) with t w o different v a l u e s o f DE

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