SP-345 Evolution of the Solar System

 

PART B

The Accretion of Celestial Bodies

 

11. ACCRETIONAL PROCESSES

 

11.1. SURVEY OF PART B

[171] In part A we have reviewed the observed features of the solar system and the general laws of physics that govern it. Relying upon these observations and laws, it was found possible to reconstruct with some confidence the state at the end of the hetegonic era.

In part B we shall try to determine what processes are responsible for producing the structure of the solar system that prevailed at that specific time. For reasons that have been outlined in chs. 1 and 7 and which will be discussed further in the following, the formation of the planets and satellites existing now and at the end of the hetegonic era must be due to accretion of smaller bodies, which in their turn ultimately must have accreted from single grains. This conclusion is in principle straightforward, mainly because the other types of processes proposed prove to be impossible. The concept of planetesimal accretion has been drawn upon as a qualitative basis many times in the past (Alfven, 1942, 1943a, 1946; Schmidt, 1945; Safronov, 1954). However, to be thoroughly convincing it must also be supported by quantitative explanation of how orbiting grains with high relative velocities can interact to form larger bodies. We find that jet streams form an important intermediate stage in this evolution. The conditions for their development places important constraints on the conditions under which the original grains could have formed. The analysis of this earliest phase will consequently be treated (in parts D and C) only after the accretional evolution has been investigated in detail in the present part (B); this is in keeping with the actualistic principle (ch. 1) designed to keep us in as close contact with reality as possible.

This chapter contains a general analysis of accretion, which considers what lines of approach are of interest to follow up and which ones can be ruled out immediately. This analysis also defines the boundary conditions for the grain-producing plasma processes.

With the theory of jet streams (ch. 6) and of accretional processes studied in this chapter as starting points, a general theory of the accretion of planets and satellites is given in ch. 12. Some results of this theory can be checked by future space experiments that are within the present state of the art.

[172] In ch. 13 the accretional theory of spins is presented and compared with the observed spin isochronism (sec. 9.7). This leads to an explanation of the spin periods of the different planets.

Chapter 14 treats the comet-meteoroid complex and considers how celestial bodies are accreted today. From this study of accretion we obtain important knowledge of accretional processes in general. The possibility of observing accretional processes that occur today in our neighborhood reduces the speculative element in our study of accretional processes in the hetegonic era.

Although part B concerns accretional problems in general, the formation of the Saturnian rings and the asteroids is not included. They represent a stage that has evolved very little from that established at the end of the preceding era of grain formation by condensation. Hence, they will conveniently be treated separately in part C (ch. 18).

 

11.2. GRAVITATIONAL COLLAPSE OF A GAS CLOUD

As we have seen in ch. 6, viscosity-perturbation of the Kepler motion produces an "apparent attraction" that may produce contractions in cosmic clouds. Before this was recognized, however, the only effect that could produce a contraction was believed to be the self-gravitation of the cloud. For this reason it is generally believed that stars are formed by gravitational contraction of vast interstellar clouds. The condition for contraction is given by the Virial Theorem, which requires that the potential energy of the cloud (assumed to be a uniform sphere of radius R) must exceed twice the thermal energy; i.e.:

 

mathematical equation(11.2.1)

 

where M is the mass of the cloud; N = M/ma is the number of atoms with average mass ma; k is Boltzmann's constant; and T is the temperature. If the average atomic weight is ma / mH we have

 

mathematical equation
(11.2.2)

 

with [Greek letter] chi = 2 X 10-16 (cm K)/g.

 

[173] As pointed out (e.g., by Spitzer (1968)), there are serious difficulties in understanding the formation of stars by this model. In particular, a large rotational momentum and magnetic flux oppose the contraction. It is far from certain that the model is appropriate.

However, we shall not discuss the problem of star formation here (it will be reserved for ch. 25), but rather the formation of planets and satellites. Laplace made the suggestion, admittedly qualitative and speculative, that these bodies were formed from gas clouds that contracted gravitationally. This idea has been adopted by a number of subsequent workers, without realization of its inherent inadequacy.

 

11.2.1. Objection to Gravitational Collapse As a Mechanism for the Formation of Planets and Satellites

11.2.1.1. Insufficient gravitation. If for an order of magnitude estimate we put ma / mH = 10 and T = 100K for formation of planets and satellites, we find

 

b < KM (11.2.3)

 

with K = 10-17 cm/g. For the biggest planets with Mapproximately1030 g we find R < 1013 cm, indicating, from these considerations alone, that Jupiter and Saturn may have been formed by this mechanism. But even in the mass range of Uranus and Neptune (M1029 g) we run into difficulties because gravitational effects do not become important unless the clouds by some other means have been caused to contract to 1012 cm, which is less than 1 percent of the distance between the bodies. Going to the satellite systems or a hypothetical body consisting of all of the matter in the asteroid belt, we see immediately that gravitational contraction is out of the question. For a typical satellite mass (say, 1023 g), we find R < 106 cm (which means that the dimension of the gas cloud should be comparable to that of the present body). Hence, we conclude that the gravitational contraction of gas clouds is inadequate as a general model for the formation of the bodies in the solar system.

As another example that shows how negligible the gravitational attraction is in forming a satellite system, let us consider the inner part of the Saturnian satellite system. This system of secondary bodies is certainly one of the most regular with respect to systematic spacing of bodies and small inclinations and eccentricities of orbits. The masses of Mimas and Enceladus are of the order 10-7 of the mass of Saturn. At an orbital distance intermediate between Mimas and Enceladus, the gravitational attraction due to these bodies is less than 10-5 of the gravitational attraction of Saturn.

[174] (Before the formation of the satellites the matter now forming them is likely to have been spread out over the whole orbit, which makes the ratios still smaller by one or more orders of magnitudes.)

A somewhat different way to express what is essentially the same objection is the following. The distance

mathematical equation
(11.2.4)

to the interior and/or exterior Lagrangian points is a measure of the extension of the gravitational field of a secondary body with mass M in an orbit r around a central body of mass Mc. Only if the original extension of a gas cloud of mass M is smaller than rL is a gravitational collapse possible. Table 11.2.1 gives the distance to Lagrangian points for the planets. Figure 11.2.1 shows the maximum possible extensions of gas clouds that could gravitationally collapse to form Mimas, Enceladus, and the terres-.....

 


TABLE 11.2.1. Distance to the Lagrangian Points of the Planets and Selected Satellites Indicating Sphere of Gravitational Dominance

Planet

Orbital radius a

1012 cm

Mass a

1027 g

Distance to Lagrangian point c

1010 cm

.

Mercury

5.791

0.3299

2.206

Venus

10.821

4.870

10.11

Earth

14.960

5.967

14.96

Mars

22.794

0.6424

10.35

Jupiter

77.837

1899.

531.6

Saturn

142.70

568.

652.1

Uranus

286.96

87.2

700.2

Neptune

449.67

102.

1160.

Pluto

590

1.1

335.7

.

Satellite (planet)

1010cm

1024 g

108 cm

.

Amalthea (Jupiter)

1.81

b 0.005

0.173

lo (Jupiter)

4.22

89.2

10.6

Mimas (Saturn)

1.86

0.04

0.532

Titan (Saturn)

12.2

137.

52.7

lapetus (Saturn)

35.6

2.24

38.8

 
a Data taken from tables 2.1.1-2.1.2.
b Mass estimated using radius of 7 X 106 cm and density of 3.5 g/cm3
c Calculated using eq. (11.2.4).

 


[
175]

FIGURE 11.2.1.&emdash;The inner region of the Saturnian satellite system (below).

FIGURE 11.2.1.&emdash;The inner region of the Saturnian satellite system (below). The small, filled circles (almost points) show the regions within which the gravitational fields of Mimas and Enceladus predominate. The regions of gravitational dominance of the terrestrial planets are shown above. The figure illustrates that gravitational collapse is not a reasonable mechanism for the formation of these bodies because of the minimal extension of their gravitational fields. The same conclusion holds for all satellites and planets, with the possible exception of Jupiter.

 

...-trial planets. It is obvious that the geometrical extensions of the gravitational fields of these bodies are much too small to make formation by collapse a viable suggestion.

Kumar (1972) also shows that, because of the limited extension of the Lagrangian points of Jupiter, the influence of solar tides would prevent any [176] gravitational collapse of the gas cloud from which Jupiter could be assumed to have formed. Since gravitational collapse can be excluded as a theory of Jovian origin, surely it must be excluded on similar grounds for all other secondary bodies in the solar system.

The Laplacian approach cannot be saved by assuming that the present satellites once were much larger ("protoplanets" and "protosatellites" as in Kuiper's theory (Kuiper, 1951)). As shown above, there are discrepancies of too many orders of magnitude to overcome in such a theory.

Hence we reach the conclusion that the self-gravitation of a cloud is, at least in many cases, much too small to produce a gravitational collapse. Much more important than the self-attraction is the "apparent attraction" which, according to sec. 6.4, is a result of a viscosity-perturbed Kepler motion and leads to a formation of jet streams as an intermediate stage in the accretion of celestial bodies.

11.2.1.2. Gravitational contraction and angular momentum. The formation of planets and satellites by gravitational contraction of a gas cloud also meets with the same angular momentum difficulty as does star formation. If a gas cloud with dimensions R is rotating with the period [Greek letter] tau
, its average angular momentum per unit mass is of the order of mathematical equation
. If it contracts, this quantity is conserved. If the present mass of say, Jupiter once filled a volume with the linear dimensions [Greek letter] beta
times Jupiter's present radius, its rotational period must have been of the order mathematical equation
. where mathematical symbol is the present spin period of Jupiter. The maximum value of [Greek letter] tau
is defined by the orbital period, which for Jupiter is about 104 times the present spin period. Hence, we find [Greek letter]beta
<100, which means that the cloud which contracted to form Jupiter must be less than 1012 cm in radius. This is only 1 or 2 percent of half the distance between Jupiter and Saturn, which should be approximately the separation boundary between the gas forming Jupiter and the gas forming Saturn. (It is only 10 percent of the distance to the libration or Lagrangian point, which could also be of importance.) Hence, in order to account for the present spin period of Jupiter if formed by contraction of a gas cloud, one has to invent some braking mechanism. Such a mechanism, however, must have the property of producing the spin isochronism (sec. 9.7). No such mechanism is known.

 

11.3. PLANETESIMAL ACCRETION: ACCRETION BY CAPTURE OF GRAINS OR GAS

We have shown that the formation of planets and satellites by collapse of a gas cloud is unacceptable. This directs our attention to the alternative; namely, a gradual accretion of solid bodies (embryos or planetesimals) from [177] dispersed matter (grains and gas). This process is often called planetesimal accretion and is a qualitative concept that can be traced back to the 18th century; for complete references see Herczeg (1968). Planets and satellites are assumed to have grown from such bodies as a rain of embryos and grains hit their surface, continuing until the bodies had reached their present size.

A number of direct observations support this concept. The saturation of the surfaces of the Moon, Mars, the Martian satellites, and Mercury with craters testifies to the importance of accretion by impact, at least in the terminal stages of growth of these bodies. Although now largely obliterated by geological processes, impact craters may have also been a common feature of the Earth's primeval surface.

Second, the spin isochronism (sec. 9.7) can be understood at least qualitatively as a result of embryonic accretion. The observed isochronism of spin periods requires that the same process act over the entire observed mass range of planets and asteroids, covering 12 orders of magnitude. Consequently, all seriously considered theories of planetary spin (Marcus, 1967; Giuli, 1968a and b) are based on the embryonic (planetesimal) growth concept.

Finally, the directly observable record in grain aggregates from space (now in meteorites) demonstrates that many of the grains, now preserved as parts of meteorites, condensed as isolated particles in space. After such initial existence as single particles, clusters of loosely (presumably electrostatically) bonded grains can be shown, by means of irradiation doses, to have existed over substantial time periods. These aggregates in their turn show evidence of alternating disruption and accretion before arriving at the most recent precursor states of meteorites; i.e., bodies several meters in size or possibly even larger.

This observational evidence, which is discussed in more detail in ch. 22, lends support to the concept that aggregation of freely orbiting grains into larger embryos constituted an important part of the hetegonic accretion process.

 

11.4. GRAVITATIONAL ACCRETION

As we have learned from sec. 7.3.1, the accretion process consists of two phases, nongravitational accretion and gravitational accretion.1 We shall first discuss the latter phase.

When a particle hits the embryo, it causes secondary effects at its impact site. If the impacting particle is a solid body, it produces a number of ejecta, [178] most of which are emitted with velocities predominantly smaller than the impact velocity. If the particle is large enough, it may split the embryo into two or more fragments. If the embryo is large enough, the escape velocity is almost the same as the impact velocity (see sec. 7.3.1 ) and we can be sure that only a small fraction of the ejecta can leave the embryo.

If the impacting particle is an ion, atom, or molecule it may be absorbed by the embryo, increasing its mass. However, it may also be reemitted either immediately or after some time delay with a velocity equal to its thermal velocity at the temperature of the embryo. As in typical situations in space, the temperature of a grain (or embryo) is much smaller than that of the surrounding plasma; the emission velocity is normally considerably smaller than the impacting velocity. Hence, gas will also be accreted when the escape velocity of the embryo is greater than the thermal velocity of the gas.

Gravitational accretion becomes increasingly rapid as the gravitational cross section of the embryo increases; eventually this leads to a runaway accretion. To distinguish this from the gravitational collapse with which it is totally unrelated, we shall call it "accretional catastrophe." A quantitative discussion of gravitational accretion, including the runaway process, is given in ch. 12.

 

11.5. NONGRAVITATIONAL ACCRETION

Gravitational accretion is rather straightforward, but nongravitational accretion is more difficult to understand. When an embryo is hit by a particle with a velocity much larger than the escape velocity, the ejecta at the collision may in principle have velocities in excess of the escape velocity and hence leave the embryo. At least at hypervelocity impacts the total mass of the ejecta may be much larger than the mass of the impinging particle. Hence, the impact may lead to a decrease in the mass of the embryo. Moreover, upon impact, the embryo may be fragmented.

For such reasons it is sometimes suggested that nongravitational accretion cannot take place. However, there seems to be no other process by which it is possible to generate bodies large enough to accrete further (by the help of gravitation). Hence, the existence of large (planet-sized) celestial bodies makes it necessary to postulate a nongravitational accretion.

To return to the example of the inner Saturnian satellites (fig. 11.2.1), the rings and the inner satellites must have been produced in closely related processes (see sec. 18.6). The ring has an outer limit because particles farther out have accreted to form the satellites instead of remaining in a dispersed state. Their incipient accretion must have been nongravitational. Also, as we shall see in sec. 18.8, conditions in the asteroid belt give further insight into the planetesimal accretion process.

[179] The only small bodies we have been able to study more closely are Deimos and Phobos. They are completely saturated with craters that must have been produced by impacts which have not broken them up. As their escape velocities cannot have exceeded some 10 m/see, they must have accreted essentially without the help of gravitation.

 

11.5.1. Objections to the Nongravitational Accretion Process

In the past, the major obstacle to understanding the incipient accretion process was the difficulty in visualizing how collisions could result in net accretion rather than in fragmentation. These difficulties have largely been eliminated by the first-hand data on collision processes in space obtained from studies of the lunar surface, the record in meteorites, and the grain velocity distribution in jet streams.

As pointed out by many authors (e.g., Whipple, 1968), the relative velocities between particles considered typical (for example, colliding asteroids) are of the order 5 km/sec, and, hence, collisions would be expected to result largely in fragmentation of the colliding bodies. At such velocities a small body colliding with a larger body will eject fragments with a total mass of several thousand times the mass of the small body. The probability of accretion would, under these circumstances, appear to be much smaller than the probability of fragmentation.

This is the apparent difficulty in all theories based on the embryonic accretion concept. Indeed, as will be shown in sec. 11.7.4, such accretion requires that the orbits of the grains have eccentricities of at least e = 0.1, and in some cases above e = 0.3. The relative velocity at collision between grains in such orbits is of the order

 

uapproximatelyvorbe(11.5.1)

 

where vorb is the orbital velocity. Since vorb is of the order 10 to 40 km/see, u necessarily often exceeds 1 km/sec so that the collisions fall in the hypervelocity range.

 

11.5.2. Accretion in Jet Streams

The solution to this problem lies in the change of orbits that occurs as a result of repeated collisions between grains. This process has been analyzed in detail in ch. 6. The net result is a focusing in velocity space of the [180] orbits and equipartition of energy between participating grains leading to relative velocities continuously approaching zero at the same time the particle population contracts into a jet stream. The process can be considered as a result of the "apparent attraction" caused by the viscosity-perturbation of Kepler motion.

An observational example of how such a reduction of relative velocities takes place in a jet stream has been given by Danielsson. In his study of the "profile" of some asteroidal jet streams (see fig. 4.3.6), he found that in certain focal points the relative velocities are as low as 0.2 to 1 km/sec. At such velocities collisions need not necessarily lead to mass loss, especially not if the surface layers of the bodies are fluffy. Furthermore, the velocities refer to visual asteroids, but, as the subvisual asteroids have a stronger mutual interaction, their relative velocities may be much smaller.

 

11.5.3. Electrostatically Polarized Grains

Charging and persistent internal electric polarization are found to be characteristic of lunar dust (Arrhenius et al., 1970; Arrhenius and Asunmaa, 1973). As a result, lunar grains adhere to each other with forces up to a few hundred dynes and form persistent clusters. This is probably a phenomenon common to all solids exposed to radiation in space. Hence, electrostatic forces were probably of similar importance during accretion. The nongravitational accretion in the hetegonic era may have been largely caused by electrostatic attraction (sec. 12.3).

 

11.5.4. Fluffy Aggregates

Meteorites provide evidence of the relative importance of various processes of disruption and accretion. The decisive importance of loosely coherent powder aggregates in absorbing impact energy is indicated by the high proportion of fine-grained material in chondrites, which form by far the largest group of meteorites. The low original packing density of this material is also suggested by evidence from meteors. Such fluffy aggregates probably represent the state of matter in jet streams at the stage when a substantial portion of the collision debris of the original grains has, through inelastic collisions, reached low relative velocities so that they can adhere electrostatically.

Hypervelocity impact of single grains on fluffy aggregates results in large excess mass loss (Vedder, 1972). In the subsonic range, however, it is possible and likely that an impinging particle will lose its energy gradually in penetrating the fluffy embryo so that few or no ejecta are thrown out. The impinging particle may partially evaporate in the interior of the embryo and hence preserve the fluffy structure.

[181] The early stage of accretion can be considered to be at an end when an aggregate reaches a mass such that gravitational acceleration begins to control the terminal impact velocities. The catastrophic growth process the follows and leads to the accretion of planets and satellites has already been discussed in sec. 11.4 and will be discussed further in ch. 12.

 

11.6. ACCRETION OF RESONANCE-CAPTURED GRAINS

There are some regions in our solar system where planetesimal accretion may be in progress at the present time; namely, at some of the resonance points. We know three different regions in which several bodies are gravitationally captured in permanent resonances. These are

(1) and (2) The two libration or Lagrangian points ahead of and behind Jupiter where the Trojans are moving.

(3) The Hilda asteroids (20 asteroids), which are in 2/3 resonance with Jupiter.

In each of these three groups, the bodies are confined to movement in certain regions of space (sees. 8.5.3 through 8.5.4). Each of these groups probably includes a large number of smaller bodies. Some energy is pumped into these groups of bodies because the gravitational field is perturbed, in part due to the noncircular orbital motion of Jupiter and in part due to perturbations from other planets. Furthermore, other asteroids (and comets and meteoroids) pass the region and may collide with the members of the group, thereby feeding energy into it.

However, these sources of energy input are probably comparatively unimportant; consequently, we neglect them in the following idealized model. Hence, the only significant change in the energy of the group of bodies is due to mutual collisions, if such occur. If these collisions take place at hypervelocities, they lead to fragmentation. The number of bodies increases, but as the collisions are at least partially inelastic the total internal kinetic energy of the assembly decreases. According to our assumptions, there are no effects increasing the internal energy significantly, so the result will be that the relative velocities decrease until collisions occur only in the range in which accretion predominates. The result will be a net accretion. We would expect that all the matter in each of the groups would eventually accrete to form one body.

Therefore, if we treat the case where initially a large number of small grains (e.g., resulting from primordial condensation) were injected into the velocity space of one of our idealized groups, we could expect to follow in detail the accretional process from grains to planets. In the Hilda group, most of the mass is found in one object (Hilda herself). From this we may conclude that the accretional process is already far advanced.

[182] There are a number of other resonances where only one small body is found to be trapped in resonance by a large body (ch. 8). Such cases are Thule (3/4 resonance with Jupiter), Pluto (3/2 resonance with Neptune), and Hyperion (4/3 resonance with Titan). These cases may represent a still more advanced state than that in the Trojan groups and the Hilda group with all of the observable mass gathered into one body. (There may, however, be small, still unknown companions.) We may also consider Mimas to be trapped by Dione, and Enceladus, trapped by Tethys, as similar cases.

It should be remembered that the libration amplitudes in some of the cases cited are small, in some cases less than 1°. As we have found in sees. 8.1 and 9.6, this is difficult to reconcile with the tidal theory of resonance capture because very efficient damping of the librations is needed. Our model of planetesimal accretion, on the other hand, provides a mechanism for energy loss through mutual collisions between the accreting bodies, which may result in a small libration. In fact, in the accretional state we have a number of bodies librating with different phase and amplitude. Their mutual collisions will decrease the libration of the finally accreted body.

A detailed analysis of the proposed model is desirable to demonstrate its applicability to real cases.

 

11.7. NECESSARY PROPERTIES OF AN ACCRETIONAL PROCESS

We shall now discuss the more general case of accretion. We start from the assumption that plasma containing a large number of grains is distributed in different regions around a central body. We require that the accretion of these grains shall finally lead to the formation of the celestial bodies we observe. From this requirement we can draw certain conclusions about the properties of the grains and about their dynamic state. We shall in this section confine ourselves to a discussion of the latter question.

We find that the celestial mechanical data that should be explained by a theory of accretion are as follows.

 

11.7.1. The Orbital Elements of the Bodies

The total angular momentum CM of a celestial body should be the sum of the orbital momenta of all the grains that have contributed to the formation of the body. The eccentricity e and the inclination i of the orbits of the accreting grains change during the accretion because of collisions. The values of e and i of the resulting body depend upon the details of the mechanism of accretion, but are generally less than those of the grains.

 

[183] 11.7.2. The Spacing; of the Bodies

The spacing ratio qn = (rn+1)/rn between two consecutive planetary or satellite orbits is given in tables 2.1.1 and 2.1.2. In the different groups it usually varies between about 1.18 (Mimas-Enceladus) and 2.01 (Saturn-Uranus). A theory of accretion should explain the values of q.

Of special interest is the fact that, with the exception of the group of the very small bodies, Jupiter 6, 10, and 7, there are no q values smaller than 1.15. It is important to clarify why the matter accumulated in, for example, the region of the Uranian satellites has accreted to form large bodies instead of, say, 100 satellites with spacings q = 1.01 or 1.02. If such a state were established, it would be just as stable as the present state with four bodies.

Hence, the gathering of primordial matter into a small number of bodies is an important fact for which the accretional model should account.

 

11.7.3. The Spin of the Bodies

The accretional mechanism should leave the bodies with the spins they had before tidal braking. Because all satellites and a few planets have been severely braked, the observational data we can use for checking a theory consist of the spin values of asteroids and the tidally unaffected planets. In particular, we have to explain the spin isochronism (sec. 9.7).

 

11.7.4. The Eccentricity of the Grain Orbits

From sec. 11.7.2 we can derive an interesting property of the orbits of the grains that form the raw material for the accretional process.

Suppose that the processes of grain capture and condensation have resulted in a large number of grains all moving around the central body in exactly circular orbits in the equatorial plane. Two spherical grains with radii R1 and R2 moving in orbits a1 and a2 can collide only if

 

deltaa = a2 - a1 < R1+ R2 (11.7.1)

 

Since in the solar system R1 and R2 are usually very small compared with the spacing of the orbits, R1+R2<<deltaa, which means that we can have a large number of grains in consecutive circular orbits. At least in systems (e.g., the Uranian system) where the total mass of the satellites is very small compared to the mass of the central body, such a system would be perfectly stable from a celestial-mechanics point of view. Such a state would resemble the Saturnian rings and is conceivable even outside the Roche limit.

[184] Hence, the fact that in each different group of bodies (see table 2.1.5) there are only a small number (3 to 6) of bodies shows that the grains out of which the bodies were formed cannot have orbited originally in circles in the equatorial plane.

Suppose next that we allow the original grains to orbit in circles with certain inclinations i. Then grains with the same angular momenta C but with different values of i will collide, but there will be no collisions between grains with different a values. In case the collisions are perfectly inelastic, they will result in grains with the C values unchanged but all with the same i values. Such a state is again dynamically stable but irreconcilable with the present state of the solar system.

Hence, we find that the original grains must necessarily move in eccentric orbits. (Originally circular orbits with different i values would result in eccentric orbits in the case where the collisions are not perfectly inelastic. This case is probably not important.)

An estimate of the minimum eccentricity is possible, but not without certain assumptions. Let us make the assumption (which later turns out to be unrealistic in certain respects) that a satellite or planet accretes by direct capture of grains.

If two adjacent embryos during the late stages of the accretional process move in circles with radii a1 and a2 and the spacing ratio is q = a2/a1, all grains must have orbits which intersect either a1 or a2. If not, there would be grains that are captured neither by a1 nor by a2, and these would finally accrete to a body between a1 and a2, contrary to our assumptions. As the ratio between the apocentric and pericentric distances is (1 + e)/(1 - e) we find

 

mathematical equation
(11.7.2)

 

mathematical equation
(11.7.3)

 

Since in some cases (e.g., in the giant-planet group) q = 2.0, we find that at least for some groups e greater or equal to1/3. For smaller q values such as q = 1.2 On the inner Saturnian satellite group), we obtain e > 0.09. These results are not necessarily correct for a more complicated model of the accretion. However, as we shall find in ch. 12, it is essentially valid also for the two-step accretional process considered there.

 

[185] 11.8. THE PRESENT STATE OF ASTEROIDS, METEOROIDS AND COMETS, AND THE EXPLODED-PLANET HYPOTHESIS

We have found that the planetesimal approach requires a state characterized by a number of bodies moving in eccentric Kepler orbits. This state has a striking resemblance to the present state in the asteroidal belt. In fact, as shown in fig. 4.3.1, the eccentricities of the asteroid orbits vary up to about 0.30 or 0.35. There are very few asteroids with higher eccentricities. Thus, from this point of view it is tempting to identify the present state in the asteroidal region with the intermediate stage in a planetesimal accretion.

This is contrary to the common view that the asteroids are fragments of one or several planets, exploded by collisions. There are a number of other arguments against the explosion hypothesis:

(1) There is no doubt that collisions occur between asteroids. Arguments have been developed, particularly by Anders (1965), that the resulting fragmentation contributes to the observed size distribution of asteroids. However, Anders also points out that only the small-size part of the distribution is explained by fragmentation and that the large-size asteroids show another distribution which he attributes to "initial accretion" but which could equally well be explained as concurrent with the fragmentation.

(2) As discussed in sec. 11.5, it has been believed that collision of small objects could not lead to accretion; in this situation it appeared necessary first to postulate the formation of one or several large parent bodies by some undefined ad hoc process and then to decompose these to generate the wide size range of objects now observed. Obviously this approach does not solve the problem of accretion, which is only ignored or relegated to the realm of untenable hypotheses.

(3) It was long thought that meteorites could be produced from one or several parent "planets" of lunar size or larger that could have been located in the asteroid belt. The reasons for this assumption were mainly that several types of meteorites show evidence of heating of the accreted components. One way of interpreting this would be that they came from the interior of a planet, where heat would accumulate due to radioactive decay. The observed heating affects are, however, equally well, or better, explained by external sources (Wasson, 1972). The most obvious heating process is the dissipation of orbital energy by gas friction, as discussed in ch. 19. The monotonic decrease in power of the orbital thermal pulses would explain the diffusion profiles observed in the [Greek letter] gamma
phase of nickel-iron meteorites (Wood 1964, 1967).

(4) The occurrence of microcrystalline diamond in meteorites, at one [186] time suggested to be due to high static pressure inside a planet, has been shown to be associated with and most likely caused by shock effects (Anders and Lipschutz, 1966); diamond can also grow metastably at low pressure from the gas phases (Angus et al., 1968).

(5) Finally, it was long thought that planetary-sized bodies with a fractionated atmosphere were needed to generate oxidized and hydrated minerals and some of the organic components observed in carbonaceous meteorites. It is now known, however, that extensive fractionation can occur in the pre-accretionary stages. This is illustrated by the variation in composition of comets, which have much higher oxygen/hydrogen ratios than, for example, the solar photosphere (sec. 21.6). Hydroxysilicates (such as chlorite) and ferriferrous iron oxide (magnetite) can form by direct condensation, and the classes of organic compounds observed in meteorites are readily synthesized in plasmas of the type observed in space and likely to have prevailed in the hetegonic era.

Hence, there appears to be nothing in the structure and composition of meteorites that indicates that their precursor bodies were ever larger than a few meters. (Further discussion of their possible maximum size is given in sec. 22.4).

In summary, there is no conceptual need for large bodies as predecessors of asteroids and meteorites. Furthermore, an assumption of such large bodies cannot be reconciled with the present dynamic state of the asteroids and with physically acceptable models for their formation. Most likely, the asteroids are generated by an ongoing planetesimal collision interaction process, where competing


1 Gravitational accretion should not be confused with "Gravitational collapse," which is a completely different process.


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