SP-345 Evolution of the Solar System

 

4. THE SMALL BODIES

 

[51] 4.1 SURVEY AND CLASSIFICATION

The small bodies are asteroids, comets, and meteoroids (down to the size of subvisual grains). In our analysis we shall concentrate our attention on the small bodies because they contain so much important information about the early periods of the evolution of the solar system. Furthermore, the embryonic (planetesimal) state leading to the formation of planets and satellites must necessarily have been similar, at least in some respects, to the present dynamic state of the small bodies, and we can learn much about the former by studying the latter.

In the satellite systems there may be bodies corresponding to the asteroids and meteoroids, and perhaps even to comets, but since we cannot observe such bodies we know nothing about them. The only exception is the Saturnian rings, which are known to consist of very small bodies (see sec. 18.6).

In the planetary system practically all the observed small bodies have at least part of their orbits inside the orbit of Jupiter. However, there is no reason to assume that small bodies, as yet unobserved, are not abundant in orbits beyond Jupiter.

There is a vast gap of about two orders of magnitude between the mass of the smallest planets (Mercury with M = 33 X 1026 g and the Moon (being a captured planet) with M = 7.3 X 1025 g) on the one hand and the largest "small bodies" (namely, the asteroids Ceres with Mapproximately12 X 1023 g and Pallas with Mapproximately
3 X 1023 g) on the other (Schubert, 1971). The mass distribution among visual asteroids is relatively continuous over 10 orders of magnitude, down to 1014 g for Adonis and Hermes, and probably is continued by a population of subvisual asteroids, down to what we may call asteroidal grains. We know very little about the latter groups. Neither the micrometeroid detectors nor the optical asteroid/meteoroid detector on the Pioneer 10 flyby mission to Jupiter registered any increase in space density of particles in the range 1O-3 to 0.15 cm while passing through the asteroid belt. A significant increase in the abundance of larger particles (0.15-1.5 cm) [52] was, however, observed in the belt (Soberman et al., 1974; Kinard et al., 1974).

The term "meteoroid" was originally used for a body moving in space which, upon entering the Earth's atmosphere, produces a meteor and, in very rare cases, may be retrieved on the ground as a meteorite. However, the term meteoroid is now used for any small piece of matter moving in space.

The comets differ in appearance from the asteroids in having a diffuse region, the coma, and, at least during some part of their orbit, dust and plasma tails. They are often, but not always, observed to have one or more nuclei. Cometary mass is not very well known but probably falls within a range similar to that for a small asteroid (1015-1019 g).

The orbits of asteroids, comets, and meteoroids share, in part, the same region of interplanetary space, but they are located in vastly different regions of velocity space. We may describe their orbital motion by three parameters: the semimajor axis a , the eccentricity e , and the inclination i of the orbit. From ch. 3 we find that a is a measure of the average distance from the central body, e is a measure of the radial oscillation, and i is a measure of the axial oscillation about the average distance.

The orbital data for about 1800 asteroids are listed in Ephemerides of Minor Planets. Recently the Palomar-Leiden survey has added 2000 new asteroidal orbits (van Houten et al., 1970). Orbital data for comets are found in the Catalogue of Cometary Orbits (Porter, 1961).

If we classify the small bodies by their values of a, e, and i, we find that almost all of them belong to one of six populations (figs. 4.3.3, 4.4.1 and 4.6.1), three of which are large, and three, small. The three large populations are

 

(1) Main - belt asteroids:

 

e < 1/3, i < 20°, 2.1 < a < 3.5 AU

 

(2) Short-period comets and meteoroids (including Apollo-Amor asteroids):

 

1/3 < e < 0.95, i < 30°, a < 15 AU

 

(3) Long-period comets and meteoroids:

 

e > 0.95, i is random, a > 15 AU

 

The three small populations are

 

[53] (4) The Trojans (captured in and oscillating about the Lagrangian points behind and ahead of Jupiter):

 

a ~ 5.2 AU

 

(5) The Hilda asteroids:

 

e ~ 0.2, i ~ 10°, a ~ 3.95 AU

 

(6) The Hungaria asteroids:

 

e ~ 0.1, i ~ 25°, a ~ 1.9 AU

 

The asteroid groups (4), (5), and (6) do not lie within the main belt. The reason for choosing e = 1/3 as a limit will become obvious in ch. 17. The choice of the limit between short and long-period comets and meteoroids is a question of semantics. In lists of comets the border is usually taken to be T = 200 yr, corresponding to a = 34 AU, whereas for meteors the limit is taken as T = 12 yr (a = 5.2 AU). Our choice of 15 AU is intermediate. For an orbit of eccentricity 0.95, perihelion is at 0.75 AU.

 

4.2. EVOLUTIONARY DIFFERENCES BETWEEN LARGE AND SMALL BODIES

The present-day evolution of the planets and satellites differs very much from that of the small bodies. As we will discuss further in ch. 10, there seems to have been very little change in the dynamic structure of the bigbody population during the last few billion years. Two exceptions are known: the Moon and Triton, which are likely to have been planets. During and after their capture by the Earth and Neptune, respectively, their orbits were altered by tidal effects.

All satellites and planets have small periodic changes in their orbits due to "perturbations," but there is no certain evidence of any major systematic change in the orbits. Thus the motions of the planets and satellites are likely to have been governed exclusively by the laws of classical celestial mechanics. The large-body system probably reached a "final" state very similar to the present one as early as 4 or even 4.5 Gyr ago.

In contrast to this stability of the big-body population, the small-body populations are in a state of evolution. This evolution is very rapid for comets, which may change their appearance from one day to the next, and which have a total lifetime of the order of a few hundred years.

[54] The asteroidal rate of evolution falls within the extreme values for cometary and planetary evolution. If we calculate the collision probability for asteroids we find that collisions necessarily must occur, resulting in orbital changes, fragmentation, and accretion. However, these processes have never been directly observed, and as we do not know the physical properties of the asteroids, we cannot with certainty predict whether these collisions predominantly lead to accretion or to fragmentation.

As we shall see later (sec. 18.8), there are some features of the asteroidal belt which cannot have changed very much since formation. For the visual asteroids, evolutionary effects have a time constant of millions or billions of years. On the other hand, the subvisual asteroids must necessarily interact so much as to produce a more rapid evolution.

The mutual collisions between small bodies affect their orbits. From a theoretical point of view we are confronted with the problem of an interaction among a large number of bodies which is similar to a basic problem in plasma physics. The treatment of celestial mechanics in ch. 3, based on the guiding-center method, is actually designed to facilitate the contact with plasma physics which is necessary for the understanding of the evolution of the asteroid population and also of the precursor states of planets and satellites. The orbital evolution of the small bodies will be discussed in chs. 5, 13, 14, 18, and 19.

 

4.3. MAIN-BELT ASTEROIDS

The main-belt asteroids, of which more than 1700 are tabulated in Ephemerides of Minor Planets and another 1800 in the Palomar-Leiden survey, move in the region between Mars and Jupiter. (The lists of asteroids also include some bodies which, according to our classification, are not main-belt asteroids.) The orbits of main-belt asteroids have, on the average, higher eccentricities and inclinations than those of the major planets. Data on asteroidal eccentricities, as given in the Ephemerides of Minor Planets for 1968, are shown in fig. 4.3.1. The average eccentricity is -0.14. There are few asteroids with eccentricities higher than 0.25.

Figure 4.3.2 shows the number of asteroids as a function of inclination; data are from the 1968 Ephemerides. The average inclination is 9.7 deg, and there are few asteroids with inclinations above 25 deg. Graphs showing statistical correlations between various orbital elements of the asteroids have been published by Brown et al. (1967).

If we plot the number N of known asteroids as a function of the semimajor axis a we obtain fig. 4.3.3 (the N, a diagram). We see that most of the asteroids are located between 2.1 and 3.5 AU, constituting the main belt. The diagram shows a series of sharp gaps where very few, if any, [55] asteroids are found. The location of these gaps agrees well with the distances at which resonance effects from Jupiter should occur. As the period T is proportional to a3/2, all bodies with a certain a value have the same period. The gaps correspond to mathematical symbol1/2, 1/3, 2/5, and 3/7, the gap for 1/2 being very pronounced. Gaps corresponding to 2/7, 3/8, 3/10, 4/11, 5/12, 6/13 have also been traced....

 


FIGURE 4.3.1.- Number of asteroids as a function of eccentricity.

FIGURE 4.3.1.- Number of asteroids as a function of eccentricity. Data from the Ephemerides of Minor planets for 1968 are shown for 1670 asteroids.


 


FIGURE 4.3.2.- Number of asteroids as a function of inclination.

FIGURE 4.3.2.- Number of asteroids as a function of inclination. Data from the Ephemerides of Minor Planets for 1968 are shown for 1670 asteroids.



[
56]

FIGURE 4.3.3.- The (N, a) diagram (number of asteroids as a function of semimajor axis).

FIGURE 4.3.3.- The (N, a) diagram (number of asteroids as a function of semimajor axis). Most asteroids are located between 2.1 and 8.3 AU, constituting the main belt. The groups at 1.9, 3.9, and 5.2 AU are the Hungaria, Hilda, and Trojan asteroids, respectively. The sharp minima in the main belt are the Kirkwood gaps which occur at the 1/3, 2/5, 3/7, and 1/2 resonances of Jupiter.

 

.....(Hirayama, 1918). Perhaps Mars also produces a resonance at mathematical symbol
(Dermott and Lenham, 1972), but no resonance with the period of Saturn or any other planet has been found. Although the location of these gaps, which are known as the Kirkwood gaps,* at the resonance points leaves no doubt that they are due to a resonance; the mechanism producing the gaps is not understood (sec. 8.6).

As pointed out by Burkenroad (Alfvén et al., 1974), the number-density distribution (N, a) diagram does not give a very good picture of the real mass distribution (M, a) in the asteroidal belt. For example, some families contain a large number of very small bodies. As asteroid masses have not been measured directly we use eq. (4.3.3) to calculate the asteroid mass and plot the (M, a) diagram (fig. 4.3.4). The diagram shows that practically all mass is located in the main belt between 2.1 and 3.5 AU. Of the asteroids outside this region, only the Hildas (at 3.95 AU) have a considerable mass.

The Kirkwood gaps are more pronounced in the (M, a) diagram than....

 


[
57]

FIGURE 4.3.4.- The (M, a) diagram (mass of asteroids in g per 0.01 AU as a function of semimajor axis a in AU).

FIGURE 4.3.4.- The (M, a) diagram (mass of asteroids in g per 0.01 AU as a function of semimajor axis a in AU). To stress the logarithmic scale, the region of greatest mass density is shaded. These regions contain practically all the mass in the asteroid belt. Mass was calculated from magnitude using eq. (4.3.3). The diagram includes data for all asteroids of a < 5.0 AU listed in the Ephemerides of Minor Planets for 1968. The resonances are indicated as the ratio of the orbital period of a body at a specific value of a to the orbital period of Jupiter. The Kirkwood gaps correspond to the 1/3, 2/5, 3/7, and 1/2 resonances. (From Alfvén et al, 1974.)

 

....in the (N, a) diagram, especially in the case of the 1/2 resonance gap. In contrast to the (N, a) diagram, the (M, a) plot is not likely to change very much as new asteroids are discovered since these new asteroids will necessarily be small.

The masses and radii of Ceres, Vesta, and Juno have been measured (see Schubart, 1971; Morrison, 1973), but the values are probably not definitive (table 4.3.1). The diameters of other asteroids are too small to be measured, and their masses cannot be determined directly. Therefore sizes and masses are estimated from their apparent magnitudes with reasonable assumptions about the albedo and the average density. Following Allen (1963), we use

 

mathematical equation(4.3.1)

 

where R is the radius of the asteroid in km, p is the albedo, and g is the absolute visual magnitude (defined as the apparent magnitude at a distance of 1 AU). Putting p = 0.135 (Ceres albedo) we obtain

 

log R = 3.385-0.2g (4.3.2)

 


[
58] TABLE 4.3.1. Physical Properties and Orbital Parameters of Selected Asteroids.

Number and name

Radius a

Mass b

Magnitude c

Spin Period h

Orbital parameters

T

a

e

i

km

g

.

hr

days

AU

.

°

.

1 Ceres

d 567

f 1.2 x 1024

4.0

9.07

1681

2.767

0.079

10.6

2 Pallas

e 350

g 3.0 x 1023

5.1

.

1684

2.767

0.235

34.8

3 Juno

e 98

f 1.4 x 1022

6.3

7.21

1594

2.670

0.256

13.0

4 Vesta

d 285

f 2.4 x 1023

4.2

5.34

1325

2.361

0.088

7.1

6 Hebe

116

2.4 x 1022

6.6

7.74

1380

2.426

0.203

14.8

7 Iris

110

2.1 x 1022

6.7

.

1344

2.385

0.230

5.5

10 Hygiea

127

3.1 x 1022

6.4

.

2042

3.151

0.099

3.8

15 Eunomia

140

4.1 x 1022

6.2

6.08

1569

2.645

0.185

11.8

16 Psyche

106

1.8 x 1022

6.8

4.30

1826

2.923

0.135

3.1

51 Nemausa

46

1.5 x 1021

8.6

7.78

1330

2.366

0.065

9.9

433 Eros

8

1.8 x 1022

12.3

5.27

642

1.458

0.223

10.8

511 Davida

97

1.4 x 1022

7.0

5.17

2072

3.182

0.177

15.7

1566 Icarus

0.7

5.2 x 1015

17.7

2.27

408

1.077

0.827

23.0

1620 Geographos

1.6

6.2 x 1015

15.9

5.22

507

1.244

0.335

13.3

1932 HA Apollo

0.5

2.0 x 1015

18

.

662

1.486

0.566

6.4

1936 CA Adonis

0.15

5.0 x 1013

21

.

1008

1.969

0.779

1.5

1937 UB Hermes

0.3

4.0 x 1014

19

.

535

1.290

0.475

4.7

a Calculated as a function of magnitude g using eq. (4.3.2): log R = 3.85 - 0.2g. Albedo of 0.135 has been assumed.
b Calculated as a function of magnitude g from eq (4.3.3): log M = 26.4 - 0.6g. Spherical shape, albedo of 0.135, and average density of 3.6 g cm-3 have been arsumed.
c Allen, 1963.
d Morrisson. 1973.
e Dollfus, 1971.
f Schubart, 1971.
g Calculated using the tabulated value of the radius and arsuming spherical shape.
h Gehrels, 1971.


 

With the assumption of an average density of 3.6 g/cm3, we find

 

log M = 26.4 - 0.6g (4.3.3)

 

for M in grams (see table 4.3.1).

 

4.3.1. Subvisual Asteroids

There are good reasons to suppose that the asteroid population is continuous and includes very small bodies which we may call "asteroidal grains." From Earth-based observations we know nothing about the size [59] spectra of subvisual asteroids. Extrapolations of the size spectra of visual asteroids have been made, for example, by Dohnanyi (1969 ), who has treated all known asteroids as one single distribution. This is a rather dangerous procedure because the (M, N) relation differs among the populations and hence varies with a (as is obvious from the difference between the (N, a) and (M, a) diagrams (figs. 4.3.3 and 4.3.4)).

The subvisual asteroids may be of decisive importance in keeping jet streams (ch. 6.) together. They may also be important for other viscosity effects in interplanetary space. The only way of getting information about them is probably from space probes sent to the asteroid belt. The micrometeoroid impact experiment (Kinard et al., 1974) on the Pioneer 10 flyby mission to Jupiter demonstrated that there is no substantial increase in the asteroid belt of particles of about 10-3 cm. For larger particles, which have smaller number densities, impact instrumentation does not provide statistically significant information. For the size range 10-2 to 15 cm, data were first obtained by the optical telescope experiment on Pioneer 10 (Soberman et al., 1974). These measurements show an increase in the largest particles (1.5 to 15 cm in size) in the asteroidal belt.

 

4.3.2. Hirayama Families

Hirayama (1918) discovered the grouping of some asteroids in families. The members of one family have almost the same values of a, i, and e. As Brouwer (1951) has pointed out, both i and e are subject to secular variations with periods of the order 104 to 105 yr. From a hetegonic point of view, we want to eliminate these. This can be done by introducing the "proper elements."

The eccentricity e and the longitudemathematical symbol of the perihelion of a Kepler orbit are subject to secular variations. The same is the case for the inclination i and the longitude of the ascending node mathematical symbol
. Following Brouwer (1951) and Brouwer and Clemence (1961b) we write:

 

mathematical equation
(4.3.4)

mathematical equation
(4.3.5)

mathematical equation
(4.3.6)

mathematical equation
(4.3.7)

 

[60] For a given asteroid the proper eccentricity E and the proper inclination I are constants. The longitude of the proper perihelionmathematical symbol
increases and the longitude of the proper node mathematical symbol
decreases at the same uniform rate, with one cycle occurring in the period mathematical symbol
. The quantities p0, q0, P0, and Q0 are the forced oscillations produced by planetary perturbations, p0 and q0 being functions of the planetary eccentricities and perihelia and P0 and Q0 being functions of the planetary inclinations and nodes. The period as well as the forced oscillations are all functions of the mean orbital distance a; the sample values in table 4.3.2 are taken from Brouwer (1951 ) and Brouwer and van Woerkom (1950). For further detail see Kiang (1966).

Figure 4.3.5 shows the relationship between the "osculating" elements (referring to the present orbits) and the "proper" elements according to Kiang (1966). The vectors E and I rotate around a center O' with periods given in table 4.3.2. The distance of the vector from origin gives the numerical value of e and i, and the angles these lines make with their respective horizontal axes give the longitudes of the perihelion and the node. The position of the center O' is essentially given by the eccentricity and inclination of Jupiter and varies with a period of 300 000 yr.

Brouwer (1951) has given the values of E, I,mathematical symbol
, and mathematical symbol
for 1537 asteroids. Based on this material, he treats the problem of Hirayama families. He demonstrates that in an (E, I) diagram the points belonging to a Hirayama family show a somewhat higher concentration than in an (e, i) diagram. This enhanced concentration made it possible for him to detect a number....

 


FIGURE 4.3.5

FIGURE 4.3.5. Geometric illustration of the relationship of the osculating elements (e,mathematical symbol
, i,mathematical symbol
) to the proper elements (E,mathematical symbol
, I,mathematical symbol
) of motion for asteroids. As the representative point b describes a circle of typical period of 20 000 yr about point O, point O' migrates such that E and I remain constant. The vectors give the values for e, E, sin i and I, and the angle each vector makes with the horizontal axis gives the appropriate longitude of perihelion or ascending node, function of the forced oscillation p0, q0, P0 and Q0 is predominately determined by the eccentricity and (From Kiang, 1966.)

 


[
61] Table 4.3.2. Typical Values of the Periodic Variation in the Proper Elements of Asteroid Orbital Motion.

a (AU)

mathematical symbol
                                    (yr)

q

p0

Q0

P0

.

2.15

41 400

+0.0567

- 0.0363

+0.0108

+0.0031

2.60

26 300

+0.0302

- 0.0056

- 0.0006

+0.0181

3.15

14 400

+0.0374

+0.0058

- 0.0029

+0.0210

4.00

4 400

+0.0421

+0.0093

- 0.0038

+0.0222

Tabulated values from Brouwer (1951) and Brouwer and van Woerkom (1950).

 

 

....of new families. For example, it is evident that the largest of the families, the Flora family, consists of at least two, and possibly four families, called Flora I, II, III, and IV.

For orbital motion adequately described by celestial mechanics, the sum mathematical symbol
+ mathematical symbol
is an invariant to a first approximation (eq. (3.6.7)). Brouwer shows that for some families there is a maximum of mathematical symbol
+ mathematical symbol
characteristic of families or groups. Subjecting all asteroid data to computer analysis, Arnold (1969) has revised Brouwer's analysis of asteroid families. He has confirmed the existence of all the Hirayama families and of some but not all the Brouwer families. Further, he has discovered a number of new families.

Lindblad and Southworth (1971 ) have made a similar study using another statistical method to discriminate between real families and those which are due to statistical fluctuations. They confirm Hirayama's and essential parts of Brouwer's families and also some, but not all, of Arnold's new families. They conclude that about 40 percent of all numbered asteroids belong to families. They have also subjected the new asteroids discovered by the Palomar-Leiden survey to similar tests (Lindblad and Southworth, 1971).

 

4.3.3. Asteroidal Jet Streams

Members of the same family generally have different values of mathematical symbol
and mathematical symbol
. This means that the space orientation of their orbits differs. In some cases, however, there are a number of orbits with the same mathematical symbol
and mathematical symbol
, so that all five orbital parameters (a, i, e, mathematical symbol
, mathematical symbol
) are similar. Hence, their orbits almost coincide, and the asteroids are said to be members of a "jet stream" (Alfvén, 1969; Arnold, 1969; Danielsson, 1969a). Using a method that has proved successful in detecting meteor streams, Lindblad and Southworth (1971) have made searches independently in the numbered asteroid population (Ephemerides of Minor Planets) and in Palomar-Leiden data. They find 13 jet streams with at least 7 members. The largest [62] jet stream has 19 members. Their streams only partially overlap with those defined by Arnold.

Danielsson (1971) points out some of the limitations of the earlier work and introduces a new method to find "the profile of a jet stream." He calculates the distance between the intersections of two orbits with a heliocentric meridian plane as a function of the longitude and takes the mean quadratic value of the quantity as a measure of the "distance" between the orbits. This distance is a measure of how closely the orbits are associated. Applying this method to three of the jet streams, he concludes that the orbits of all the members of the jet stream are well collimated everywhere along the path. As an example, the profile of the Flora A jet stream is shown in fig. 4.3.6. Furthermore, two of the streams show marked focusing regions where a majority of the orbits come very close together and where the relative velocities are an order of magnitude smaller than those between randomly coinciding asteroid orbits. In fact, the relative velocities are as low as 0.2 to 1 km/sec. This should be compared to the orbital velocities of about 20 km/sec and the average collision velocity of two arbitrary asteroids, which is in the range 2 to 5 km/sec (Danielsson, 1971). As we shall see in chs. 11 and 12, this result is important for me theory of accretion.

 


FIGURE 4.3.6.- Profile Of the Flora A jet stream

FIGURE 4.3.6.- Profile of the Flora A jet stream. Diagram shows the intersections of the individual orbits of these asteroids with a heliocentric meridional plane as this plane is rotated one cycle around the ecliptic polar axis. The positions of the orbits are shown relative to the mean orbit of the jet stream. The dotted circle shows the cross section of the jet stream as theoretically calculated in secs. 12.2 and 12.7. Most of the asteroid orbits in Flora A fall within the dotted circle Profiles of other asteroidal jet streams show less concentration. (From Danielsson, 1971.)

 

[63] 4.3.4. Evolution of the Main Belt

The main-belt asteroids were earlier thought to be debris of one or more "exploded planets." As we shall find (secs. 11.8 and 18.8), there are decisive arguments against this view. Instead, we should consider them as a large number of "planetesimals," accreted from small grains that have condensed from a plasma. They are in a state of evolution that eventually may concentrate most of their mass into one or a few bodies. Even now almost 80 percent of the total mass in the asteroid belt is contained in the four biggest bodies.

The study of the main-belt asteroids is of hetegonic importance because the state in this region is likely to be similar in certain respects to a state of accretion through which all planet and satellite groups once have passed. Whereas this evolutionary period required perhaps 108 yr for the formation of planets and satellites (ch. 12), the time scale for a corresponding evolution in the asteroidal belt is longer than the age of the solar system. The reason is the extremely low density of matter in the asteroidal region (see ch. 2), which in fact is 10-5 of the distributed density in the adjacent planetary regions. The evolution of the main belt will be discussed later, especially in sec. 18.8.

 

4.4. THE HILDA AND HUNGARIA ASTEROIDS

Outside the main belt there is a small group, the Hilda asteroids, aapproximately3.95 AU. These are captured in resonance by Jupiter so that their periods are (averaged over a very long time) 2/3 of Jupiter's period (see sec. 8.5.4). There is a single asteroid, Thule, not very far from the Hildas which is also captured in a similar way, but its period is 3/4 of Jupiter's. These will be discussed in connection with the theory of resonances (ch. 8).

The Hungaria asteroids, at a approximately
1.9 AU, are orbiting just inside the inner boundary of the main belt. They have been believed to be in 2/9 resonance with Jupiter, but this seems not to be the case (Ip, 1974b). Their inclinations are usually high (i approximately
25°), but they have an eccentricity < 0.2 (see fig. 4.4.1).

The existence of groups of bodies at the Jupiter resonance points 3/4 and 2/3 (perhaps 2/9) constitutes an analogy to the resonance captures in the satellite systems and also to the Neptune-Pluto resonance. At the same time, the positions of these bodies present a puzzling contrast to the absence of bodies at the Kirkwood gaps. This will be discussed in ch. 8.

 


[
64]

FIGURE 4.4.1.- The inner region of the asteroid belt (a < 2.2 AU).

FIGURE 4.4.1.- The inner region of the asteroid belt (a < 2.2 AU). The main-belt asteroids (2.0 < a < 2.2) have small eccentricities and inclinations: the Apollo-Amor asteroids, higher inclinations and eccentricities; and the Hungaria asteroids (1.8 < a < 2.0), high inclinations but small eccentricities. Data from the Ephemerides of Minor Planets for 1968.

 

4.5. THE TROJANS

In the orbit of Jupiter there are two points, one 60° behind and one 60° ahead of Jupiter, at which points a body can move in a fixed position with regard to Jupiter and the Sun (see fig. 8.5.3). In the neighborhood of these points- the Lagrangian points- there are a number of small bodies, the Trojans, which usually are included in tables of asteroids. They oscillate** about these points. Their period, averaged over a long time, is necessarily the same as Jupiter's. Their origin is probably different from that of all other groups of asteroids. In fact, they are likely to be remnants of the planetesimals from which Jupiter once accreted. It is possible that the retrograde satellites of Jupiter, which are likely to be captured, have a genetic connection with the Trojans.

It is possible that there are similar groups of small bodies in the Lagrangian points of other planetary bodies, but these have not yet been discovered.

Clouds of small bodies in the Moon's Lagrangian points an its orbit around the Earth) were first reported by Kordylevsky. Recent observations from spacecraft in transit to the Moon are claimed to verify their existence (Roach, 1975).

 

[65] 4.6. THE COMETARY-METEOROID POPULATIONS

In the same region of space as the asteroids we have discussed, there is another popuIation of bodies, the comets and meteoroids. Due to their high eccentricities, (e is greater or equivalent to 1/3), comets and meteoroids occupy a different region in velocity space than do the asteroids. A transition between the two regions can be achieved only by a change of the velocity vector by at least a few km/sec. This seems to be a rather unlikely process because a high-velocity impact usually results in fragmentation, melting, and vaporization, but only to a limited extent in a change in the velocity vector. In principle, a transition could be achieved by planetary perturbations of the orbits, but such processes are probably important only in special cases (Zimmerman and Wetherill, 1973) if at all. Hence there seems to be a rather clear distinction between the comet-meteoroid populations and those asteroid populations which we have discussed.

 

4.6.1. Comets and Apollo-Amor Asteroids

The origin of the high-eccentricity population is likely to be different from that of the asteroid population. The former will be discussed in chs. 14 and 19 and the latter in ch. 18. Most of the visible members of these populations are comets, but there are also other visible bodies in essentially similar orbits which do not have the appearance of comets but look like ordinary asteroids. They are called, after prominent members of their groups, "Amor asteroids" if their orbits cross Mars' orbit, but not the Earth's orbit, and "Apollo asteroids" if their orbits cross both. Sometimes both groups are referred to as "cometary asteroids." Figure 4.4.1 shows that these asteroids occupy a region in velocity space distinct from that occupied by the main-belt asteroids. As we shall see later, there are good reasons to suppose that the Apollo-Amor asteroids are genetically associated with the comets; they are thus sometimes (with a somewhat misleading metaphor) referred to as burned-out comets.

 

4.6.2. Meteor Streams

Comets are closely related to meteor streams. In accordance with the definition in sec. 4.1, a meteor stream in a strict sense is a stream of meteoroids in space that is observable because it is intercepted by the Earth's atmosphere where the meteoroids give rise to luminous phenomena (meteors). There must obviously be many meteoroid streams that never come sufficiently close to Earth to be called meteor streams. To simplify terminology, we will refer to all elliptic streams (table 19.8.1) as meteor streams. The [66] orbital elements of some stream meteors are the same as those of certain comets (fig. 4.6.1), indicating a genetic relationship. We would expect that a large number of meteor streams, as yet undetected, exist in interplanetary and transplanetary space. (Micrometeoroid impact detectors on space probes are now in operation, but their cross sections are very small; optical detectors (see Soberman et al., 1974) promise improved data.)

Not all meteors belong to a meteor stream. The Earth's atmosphere is also hit by "sporadic meteors" in random orbits; however, they might belong to yet undiscovered meteor streams.

The long-period and short-period comets/meteoroids have such different dynamic properties that it is practical to divide them into two populations. The boundary between these populations is somewhat arbitrary. If we....

 


FIGURE 4.6.1.- Meteor streams, short-period comets, and long-period comets.

FIGURE 4.6.1.- Meteor streams, short-period comets, and long-period comets. Retrograde bodies are only found in almost parabolic orbits (e > 0.85). Data from Porter (1961).

 

[67] ....classify them according to their periods, we find that for T > T1 the orbital inclinations are random, varying from +180° to -180°, and we define this population as long-period comets/meteoroids. On the other hand, for T< T2, all the bodies have prograde orbits. We call this population short-period comets/meteoroids. This leaves us with a transition region of medium-period bodies (T2 < T < T1) in which the prograde dominance becomes more marked with decreasing T. The observational values are T1 = 200 and T2 = 15 yr, corresponding to aphelion distances of about 70 and 10 AU.

 

4.6.3. Long-Period Comets

Of the 525 comets with accurately determined orbits, 199 are elliptic, 274 almost parabolic, and 52 slightly hyperbolic (Vsekhsvyatsky, 1958, p. 2; see also fig. 4.6.1). However, if the orbits of the hyperbolic comets are corrected for planetary disturbances, all of them seem to become nearly parabolic. Hence, there is no certain evidence that comets come from interstellar space. As far as we know, all comets seem to belong to the solar system. Planetary disturbances, however, change the orbits of some comets so that they are ejected from the solar system into interstellar space.

As most cometary orbits are very eccentric, the approximation methods that were developed in ch. 3 are not applicable. The following relations between semimajor axis a, specific orbital angular momentum C, perihelion rP, aphelion rA, and velocities vA at rA and vP at rP are useful. We have

 

C2 = GMca(1- e2) (4.6.1)

rA = a (1 + e) (4.6.2)

rP =a (1 - e) (4.6.3)

 

and

 

mathematical equation(4.6.4)

 

where mathematical symbol
= 3 X 106 cm/sec is the orbital velocity of the Earth and mathematical symbol its orbital radius. Similarly

[68] mathematical equation
(4.6.5)

 

As e approaches unity, we have approximately

mathematical equation
(4.6.6)

 

It is often impossible to ascertain definitely whether the highly eccentric orbits of long-period comets are ellipitical or parabolic; we shall refer to these comets as "almost parabolic." The almost-parabolic comets may in reality be elliptical but with their aphelia situated in what Oort (1963) calls the "cometary reservoir," a region extending out to at least 1017 cm (0.1 light-yr). Their orbital periods range from 103 up to perhaps 106 yr (see Oort, 1963). This theory has further been discussed by Lyttleton (1968). The long-period comets spend most of their lifetime near their aphelia, but at regular intervals they make a quick visit to the regions close to the Sun. It is only in the special case where the comet's perihelion is less than a few times 1013 cm that it can be observed. Even the order of magnitude of the total number of comets in the solar system is unknown, but one would guess that it is very large.

The space orientation of the orbits of long-period comets appears random, the number of such comets in prograde orbits being almost the same as the number in retrograde orbits. From this we tend to conclude that on the average the comets in the reservoir are at rest in relation to the Sun, or, in other words, share the solar motion in the galaxy. From eq. (4.6.6) a comet whose perihelion is at 1013 cm will at its aphelion have a tangential velocity of 5 X 104 cm/sec if rA = 1015 cm, and 5 X 102 cm/sec if rA = 1017 cm. As the solar velocity in relation to neighboring stars is of the order of several km/sec, these low velocities in the cometary reservoir clearly indicate that this reservoir is a part of the solar system. However, it is not quite clear whether this conclusion is valid because the comets are selected; only those which have perihelia of less than a few AU cam be observed from the Earth.

If comets originate in the environment of other stars or in a random region in interstellar space, their orbits should be hyperbolas easily distinguishable from the nearly parabolic orbits observed. Hence we have confirming evidence that the comets are true members of our solar system and that the cometary reservoir is an important part of the solar system.

Oort (1963) has suggested that the comets originally were formed near Jupiter and then ejected into the cometary reservoir by encounters with [69] Jupiter. This seems very unlikely. As we shall see in the following it is more likely that the long-period comets were accreted out in the cometary reservoir. Objections to such a process by Opik (1963, 1966) and others are not valid because they are based on homogeneous models of the transplanetary medium (see chs. 15 and 19).

 

4.6.4. Short-Period Comets

The short-period comets differ from the long-period comets in that their orbits are predominantly prograde. In fact, there is not a single retrograde comet with a period of less than 15 yr (Porter, 1963, pp. 556, 557). The short-period comets have long been thought to be long-period comets that accidentally have come very close to Jupiter, with the result that their orbits have been changed (Everhart, 1969). This process is qualitatively possible, but its probability is several orders of magnitude too small to account for the observed number of short-period comets (sec. 19.6) (unless we make the ad hoc assumption that there is a special "reservoir" supplying comets to be captured by Jupiter, an assumption that leads to other difficulties).

As we shall see in chs. 14 and 19, it is more likely that short-period comets are generated by accretion in short-period meteor streams. After a certain period of activity, the comet may end its life span as an Apollo-Amor asteroid (Opik, 1961). Hence, the similarity in orbits between short-period stream meteors, comets, and Apollo-Amor asteroids could be due to a genetic relationship between them, which suggests that they ought to be treated as one single population (see sec. 19.6). A similar process may also account for the formation of long-period comets in long-period meteor streams.


* As Kopal (1973) has noted, their existence was first pointed out by K. Hornstein.

** What in other branches of science is called "oscillation" is in celestial mechanics traditionally termed "libration."


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