SP-345 Evolution of the Solar System

 

12. ON THE ACCRETION OF PLANETS AND SATELLITES

 

12.1. PLANETESIMAL ACCRETION

[189] According to the planetesimal (embryo) model of accretion, all planets and satellites have been formed by accretion of smaller bodies. The craters of the Moon, of Mercury, and of Mars and its satellites give clear evidence that an accretion of smaller bodies has been of major importance at least during the last phase of their formation. Theories of the spin of planets (see ch. 13) indicate that the planetesimal model is useful for the explanation of the rotation of planets. The isochronism of spin periods discussed in secs. 9.7 and 9.8 indicates that both planets and asteroids are likely to have been formed in this way.

The planetesimal accretion theory encounters some apparent difficulties. One of these is that, if planets are accreting by capturing grains moving in elliptic orbits in their neighborhood, one can calculate how long a time is needed before most of the grains are accreted to a planet or satellite. As shown by Safronov (1960), the time which Neptune and Pluto require to capture most of the grains in their environment is several times the age of the solar system. Safronov concludes from this that Neptune, for example, has only captured a small fraction of the matter accumulated in its neighborhood, and the rest is assumed to remain dispersed. This is not very likely. Although the matter in the asteroidal region has not accreted to a big planet, it is not dispersed. By analogy, if Neptune had not yet captured all the mass in its environment, one should expect the rest to be found as asteroid-like bodies. According to Safronov the "missing mass" must be some orders of magnitude larger than Neptune's mass. So much mass could not possibly be stored as asteroids because it should produce detectable perturbations of the orbits of the outer planets.

What appeared earlier to be a difficulty is that, according to practically all models of the embryonic state, it must have resembled the present state in the asteroidal region. In fact, if any embryo should be growing by accretion, it is necessary that a large number of asteroid-size bodies would be moving in Kepler orbits in its surrounding. But the relative velocities between visual asteroids can be as high as 5 km/sec. It is known that collisions at such hypervelocities usually lead to disruption or erosion so that larger bodies are fragmented into smaller bodies. Collisions are not likely to [190] lead to an accretion of smaller bodies to larger bodies unless the relative velocity is below a certain limit vLm which is not very well known, but may be about 0.5 km/sec (see sec. 7.4 and also compare Gault et al., 1963).

As was shown in the preceding chapter, however, it is likely that in asteroidal jet streams the relative velocities go down to very low values. We conclude that for the subvisual bodies in the asteroidal region, low velocities may predominate leading to accretion.

 

12.2. A JET STREAM AS AN INTERMEDIATE STEP IN FORMATION OF PLANETS AND SATELLITES

The jet stream concept discussed in ch. 6 seems to resolve these difficulties. We shall devote this chapter to a study of this possibility.

There is strong indication (although perhaps not a rigorous proof) that a large number of grains in Kepler orbits constitute an unstable state (ch. 6). Even if the mutual gravitation between them is negligible (so that a gravitational collapse is excluded), mutual collisions tend to make the orbits of the colliding grains similar. Hence the "viscosity" of an assembly of grains in Kepler orbits introduces an "apparent attraction" that tends to focus the grains into a number of jet streams.

The general structure of the jet streams we are considering should resemble the jet streams found in the asteroidal region (ch. 4). There is also a similarity with meteor streams, although their eccentricities are usually very large. Although there is strong indication that the jet-stream mechanism (ch. 6) is producing asteroidal and meteor streams, this is not yet proven with such certainty that our discussion here should be dependent upon these phenomena. Hence, in this chapter, we shall treat the hetegonic jet streams independently of present-day observations of meteor and asteroidal streams; but later we will use such data to a certain extent.

According to the simplest model, a jet stream is a toroid with a large radius r0 (equal to the orbital radius of a grain moving in a circular orbit around a central body) and a small radius mathematical equation. The stream consists of a large number of grains moving in Kepler orbits with semimajor axes close to r0 and with eccentricities e and inclinations i of the order of[Greek letter] beta or less. If a particle moving in the circle r0 has an orbital velocity v0, for other particles in the jet stream this velocity is modulated by a randomly distributed velocity v (| v | << | v0 |). We will denote the average of | v | by u and call it the internal velocity (approximately the average relative velocity) of the jet stream. This is the vector sum of differential velocities of the order v0 e, v0 i, and (v0deltaa)/2a produced by the eccentricities, inclinations, and differences in semimajor axes a of the individual orbits.

In our qualitative model we put

 

[191] mathematical equation(12.2.1)

and assume [Greek letter] beta
to be constant. Hence

mathematical equation
(12.2.2)

The "characteristic volume" U of the jet stream is

 

mathematical equation
(12.2.3)

or

mathematical equation
(12.2.4)

where

mathematical equation
(12.2.5)

 

and mathematical equation
is the Kepler orbital period.

This structure of a stream should be compared with Danielsson's observed "profile" of an asteroidal jet stream (fig. 4.3.6). The cross section of our model, applied to a jet stream in the asteroid belt with a = 2.2, u = 0.5 km/sec, and v = 20 km/sec should, from eq. (12.2.2), have the radius x = 0.055 AU. As the figure shows, this is in fair agreement with observations.

 

12.3. ACCRETION OF AN EMBRYO

According to our model, the accretion of large bodies takes place in two steps. The grains condensed in or were captured by a partially corotating plasma (chs. 16 and 17). The process results in grains in elliptic orbits. The precession of the ellipses will sooner or later bring them to collide with [192] a jet stream in the region where they move. This will eventually lead to incorporation of the grains in the stream. Before incorporation has taken place, or in connection with this, an extraneous grain may make a hypervelocity collision with a grain in the jet stream and hence be vaporized, melted, or fragmented. Even if the grain is thus modified or loses its identity, the ultimate result is that its mass is added to the jet stream. The subsequent collisions will reduce the relative velocity of the grain, its fragments, or its recondensation products until they reach the internal velocity of the jet stream.

The result of a collision may either be fragmentation-erosion, leading to a decrease in the size of at least the largest of the colliding bodies, or accretion leading to larger bodies. These processes have not been studied very much in the laboratory, especially not for the type of bodies with which we are concerned. The processes depend very much upon impact velocity and the bodies' chemical composition, size, and physical properties (whether they are brittle or fluffy). We know from the studies of Gault and others that impact at supersonic velocities results in melting, vaporization, and fragmentation of a total mass of the order of 102-104 times that of the projectile. However, in the subsonic range these effects decrease rapidly with decreasing impact velocities.

At velocities exceeding the equivalent of the crushing energies of brittle solid bodies, collisions between such bodies still result in comminution of projectile and target. Below this range, of the order of 10-100 m/sec in the most common brittle solar-system materials, as many particles exist after the collision as existed before.

For accretion to take place, a force has to act between the particles which exceeds the rebound after collision. Such force can be supplied by electric and magnetic dipoles. The latter are restricted to ferromagnetic components; the effect of magnetic clustering can be seen in meteorites (fig. 22.7.1).

Adhesion and clustering due to electric polarization is probably the most important process for initial accretion in a jet stream; it also determines the persistent clustering and particle adhesion on the lunar surface (Arrhenius and Asunmaa, 1973, 1974; Asunmaa et al., 1970; Asunmaa and Arrhenius, 1974). The equivalent relative particle velocities below which accretion by this process can take place are estimated at 1-10 m/sec.

Once electret clusters, such as in the lunar dust, have formed in a jet stream, capture of subsonic particles in such clusters would probably become effective. Ballistic observations indicate that projectiles in the velocity range of a few hundred m/sec effectively dissipate their energy within fluffy targets. Hence we assume here that 0.5 km/sec is a reasonable value for the limiting velocity vLm below which particles can add mass to fluffy aggregates. It would be important to clarify such capture phenomena in a more quantitative fashion by appropriate experiments.

[193] If u < vLm, grains inside the jet stream will accrete. Their size will be statistically distributed. In our model we choose the biggest embryo and study how it accretes by capturing smaller grains. We assume it to be spherical with radius R. This is a reasonable assumption for the later stages of accretion, but probably not very adequate for the earlier stages. However, no major error is likely to be introduced by this assumption.

In case such an embryo is immersed in a stream of infinitely small particles, which have the pre-accretional velocity u in relation to the embryo, the capture cross section is, according to sec. 7.3,

 

mathematical equation
(12.3.1)

 

where ves is the escape velocity. From eq. (7.3.2) we find that the "time of escape"

 

mathematical equation
(12.3.2)

 

is independent of R. Hence for ves >> vLm the capture cross section is proportional to R4.

We cannot be sure that eq. (12.3.1) holds for the case where the embryo is moving in a Kepler orbit in a gravitational field. As shown by Giuli (1968a,b), an embryo moving in a circular orbit will accrete grains under certain conditions. His calculations are confined to the two-dimensional case when all grains move in the same orbital plane as the embryo. As shown by Dole (1962), if the grains also move in circles (far away from the embryo), there are 14 different "bands" of orbits which lead to capture. Of these only four are broad enough to be of importance. Hence eq. (12.3.1) can at best be approximately true. Unfortunately the three-dimensional case of Giuli's problem has not yet been solved; hence a qualitative comparison between eq. (12.3.1) and his exact calculations is not possible. A quantitative comparison seems to indicate that eq. (12.3.1) gives reasonable values for the capture cross section. We shall therefore use it until a more precise relationship has been developed.

We denote by [Greek letter] rho
the space density of condensable substances. The jet stream may also contain volatile substances that are not condensing to grains, but according to sec. 11.4 these are also accreted by an embryo as soon as the escape velocity becomes much larger than the thermal velocity.

[194] The plasma condensation of grains and the plasma capture (sec. 21.12) of preexisting grains takes place essentially outside the jet streams, and the orbiting grains resulting from this (sec. 17.5) are captured by the jetstreams. Also, the noncondensable substances may partly be brought into the jet streams. It is not necessary to make any specific assumption here about the amount of volatile substances. (Indirectly they may contribute to the damping of the internal velocities and help to dissipate the kinetic energy.)

The growth of the embryo is, from eq. (7.3.5)

 

mathematical equation
(12.3.3)

 

When the embryo has grown large enough so that ves, equals u, gravitational accretion becomes important. The value of the radius of the embryo at this transition state between nongravitational and gravitational accretion is

 

RG = tesu (12.3.4)

 

Substituting eqs. (12.3.2) and (12.3.4) into eq. (12.3.3) we have

 

mathematical equation
(12.3.5)

 

Integration yields

 

mathematical equation
(12.3.6)

 

We now define a time ta when accretion would produce an embryo of infinite radius if the supply of grains were continuous. Setting R/RG = infinity we have

 

mathematical equation
(12.3.7)

[195] and

 

mathematical equation
(12.3.8)

 

Setting R = RG in eq. (12.3.6) we obtain

mathematical equation
(12.3.9)

 

Hence, in a medium of constant density and constant u an embryo increases in diameter from zero to infinity in the finite time ta. Half this time is needed for reaching RG, the size at which the gravitation of the embryo becomes important. As t-t0 approaches ta, dR/dt approaches infinity, and the increase becomes catastrophic.

 

12.4. MASS BALANCE OF THE JET STREAM

Let us assume that in a certain region there is a constant infall and ionization of gas and solid particles during a time tinf resulting in production of grains that are all captured in a jet stream. In the jet stream an embryo is accreting, so that finally all the emplaced mass is accumulated to one secondary body-a planet if the region we consider is interplanetary space or a satellite if it is space around a planet. The final mass of the accreted body is denoted by Msc (mass of final secondary body). Hence the rate of mass injection into the jet stream is Msc/tinf. We assume that this mass is uniformly distributed over the volume U of the jet stream. The jet stream loses mass to the embryo which is accreting according to

 

mathematical equation
(12.4.1)

 

Hence we have

mathematical equation
(12.4.2)

 

[196] Incorporating eqs. (12.3.5) and (12.3.8), we find

mathematical equation
(12.4.3)

 

12.5. ENERGY BALANCE IN A JET STREAM

The jet stream we consider is fed by am infall of condensed grains, each having a relative velocity v in relation to the jet stream. The rate of energy input to the jet stream is Msc v/2tinf . On the other hand, the jet stream loses energy through internal collisions. In our qualitative model we assume that the mass is distributed in N identical spherical grains, each with radius Rgn, a cross section mathematical equation
, and a massmathematical equation
. Their number density is

 

mathematical equation
(12.5.1)

 

They collide mutually with the frequency mathematical equation
, where u is the internal velocity of the jet stream. We assume that at each collision a fraction [Greek letter] alpha
of the kinetic energy Wgn= 1/2mgnu2 is lost. Hence the energy loss rate per grain is

 

mathematical equation
(12.5.2)

which gives

 

mathematical equation
(12.5.3)

 

or

 

mathematical equation
(12.5.4)

 

where Mj is the total mass of the jet stream.

[197] According to our assumption in sec. 12.3 there is a limiting velocity vLm such that, if u > vLm collisions result in net fragmentation and thus a decrease in grain size Rgn accompanied by an increase in the loss of kinetic energy of the jet stream.

The conclusion is that within wide limits a jet stream will adjust itself in such a way that the losses due to collisions in the stream are balanced by the injected energy. The process will tend to make u = vLm. Hence the volume U of the jet stream is likely to remain constant, and the energy balance is produced by a change in the size of the grains in the stream.

When injection stops, there is no energy input to the jet stream. Collisions will decrease the internal velocity. As mathematical equation
we have

 

mathematical equation
(12.5.5)

 

Eventually all the mass in the jet stream is accreted to one spherical, homogeneous body, the radius of which is Rsc. Assuming the density of this body to be mathematical symbol, we put

 

mathematical equation
(12.5.6)

 

and we find

 

mathematical equation
(12.5.7)

 

If Rgn is constant, the thickness of the stream will decrease linearly and reach zero after a time

 

mathematical equation
(12.5.8)

 

[198] 12.6. ACCRETION WHEN THE INFALL INTO THE JET STREAM IS CONSTANT

We have found reasons for putting u = constant, and hence U = constant. If the injection starts at t = 0, and we neglect the mass accreted by the embryo, we have the average density of the jet stream

mathematical equation
(12.6.1)

 

Introducing this into eq. (12.3.5) we obtain

mathematical equation
(12.6.2)

or after integration

 

mathematical equation
(12.6.3)

 

Equating [Greek letter] rho
to Mj/U, eq. (12.3.8) becomes

mathematical equation
(12.6.4)

Substituting eq (12.6.4) into eq. (12.6.3) gives

mathematical equation
(12.6.5)

These equations are valid only for t smaller or equal totinf. To obtain an approximate value [199] for the time tc after which there is a catastrophic increase of the embryo, we allow R/RG to approach infinity and substitute t = tc, giving

 

tc = (2 tinf ta )1/2 (12.6.6)

 

We have two typical cases, both of which are illustrated in fig. 12.6.1.

(1) tc << tinf. The density in the jet stream increases in the beginning linearly, and the radius of the embryo increases as t2, its mass, as t6. The linear increase in jet-stream density continues until the embryo rather suddenly consumes most of the mass in the stream. The catastrophic growth of the embryo stops even more rapidly than it has started, and for t > tc the embryo accretes mass at about the rate it is injected (dMem/dt) approximately (Msc/tinf).

(2) tc >> tinf. The injection stops before any appreciable accretion has taken place. The jet stream begins to contract because no more energy is fed into it to compensate the loss due to collisions. When it has contracted so much that its density is large enough, accretion sets in.

This accretion, is also catastrophic.

 

12.7. DISCUSSION

Our derivation of the accretion of celestial bodies in jet streams is based on a number of simplifying assumptions: There is not, as yet, any detailed general theory of jet streams; further, the relation between volatile and less volatile substances is far from clear. Such an approach is usually dangerous in astrophysics and it is likely that the present theory will have to be revised when sufficient observational facts become available. We may, however, receive some observational support from a comparison with asteroidal jet streams (sec. 4.3.3), to some extent with meteor streams (secs. 14.2 through 14.4), and with the record in meteorites (sec. 22.6).

In Danielsson's profile of the asteroidal jet stream Flora A (fig. 4.3.6), the cross section of the stream is approximately[Greek letter] sigma subscript j= 0.04 AU2. All orbits are confined within this surface, but most of them within about half of it ([Greek letter] sigma subscript j
= 0.02 AU2). As mathematical equation
(sec. 12.2) and the semimajor axis for the Flora A jet stream is 2.2 AU, setting r0 = 2.2 AU we find for [Greek letter] sigma subscript j
= 0.04 AU2, [Greek letter] beta
= 0.052, and for [Greek letter] sigma subscript j
= 0.02 AU2, [Greek letter] beta
= 0.036. As the orbital velocity of the Flora A jet stream is 20 km/sec, the [Greek letter] beta
value should be, from eq. (12.2.1), 0.025 assuming a u of 0.5 km/sec. This is in any case the right order of magnitude. With our present knowledge of the collisional properties of the grains (and with our qualitative treatment) we cannot expect a better agreement.

 


[
200]

FIGURE 12.6.1. Schematic representation of the accretion of an embryo from a jet stream.

FIGURE 12.6.1. Schematic representation of the accretion of an embryo from a jet stream. Plasma emplacement and infall of grains to the jet stream occurs during the time tinf. The accreting embryo at first acquires mass slowly hut then reaches catastrophic accretion at time tc when all mass present m the jet stream is accreted by the embryo. For me case tinf > tc, a slow rate of accretion continues after the runaway accretion occurs. The slow accretion continues until plasma emplacement has ceased. For the case tinf < tc, as illustrated in the lower graph, after emplacement ceases and contraction of the jet-stream volume by negative diffusion increases the density in the jet stream, accretion commences and culminates in catastrophic accretion.

 

It should be observed that Danielsson also has found other jet streams with much larger spread. This does not necessarily contradict our conclusions because these could be interpreted as jet streams in formation from a number of bodies that initially had a larger spread. In order to check our conclusions a much more detailed study is obviously necessary.

 

[201] 12.8. NUMERICAL VALUES

Table 12.8.1 presents calculated values of ta, tc, and [Greek letter] rho
for the planets. Equations (12.6.4) and (12.2.4) were used to obtain ta:

mathematical equation
(12.8.1)

The large radius of the jet stream r0 is approximated by the present semimajor axis of each planet. To evaluate the constant term in eq. (12.8.1) we have used u = O.5 km/sec and the solar mass as Mc. The values of tc are calculated from eq. (12.6.6) assuming an infall time tinf of 3 X 108 yr.

The table also contains information on the semimajor axis, mass, and density of each planet as well as the volume and density for each planetary jet stream. To facilitate intercomparison these values, excepting planetary....

 


FIGURE 12.9.1.-The growth of planetary radii with respect to time. Runaway accretion occurs early for Mercury, Venus, Earth, and Jupiter.

FIGURE 12.9.1.-The growth of planetary radii with respect to time. Runaway accretion occurs early for Mercury, Venus, Earth, and Jupiter. The time of runaway accretion approaches that of the duration of mass infall for Saturn, Mars, and the Moon. (From Ip, 1974c.) For Uranus, Neptune, Pluto, and Triton, runaway accretion occurs only after infall has ceased and the jet stream has contracted due to negative diffusion; this growth is schematically represented by the dashed curve.


[
202] Table 12.8.1. Values of ta, tc, and [Greek letter] rho
Related Parameters Characterizing the Accretion of the Planetsa

.

mathematical symbol

mathematical symbol

mathematical symbol

mathematical symbol

mathematical symbol

ta (yr)

tc (yr)

.

Mercury

0.386

0.0222

0.0568

5.46

2.54

0.78 x 106

22 x 106

Venus

0.720

0.269

0.826

5.23

3.03

0.63 x 106

20 x 106

Earth

1.00

1.00

1.00

5.52

1.00

2.0 x 106

34 x 106

Moon

1.00

1.00

1.0123

3.35

0.0123

130 x 106

270 x 106

Mars

1.52

5.34

0.108

3.92

0.0202

82 x 106

220 x 106

Jupiter

5.19

726

318

1.31

0.438

2.2 x 106

36 x 106

Saturn

9.53

8 250

95.1

0.7

0.0115

61 x 106

190 x 106

Uranus

19.1

133 000

14.6

1.30

1.09 x 10-4

8.8 x 109

>300 x 106

Neptune

30.0

810 000

17.1

1.66

2.12 x 10-5

51 x 109

>300 x 106

a Values for orbital radius, mass, and average density are taken from tables 2.1.1 and 2.1.3.

 

[203] ....density, are given relative to Earth. As calculated from eq. (12.2.4), the jet stream volume for the Earth mathematical symbol
is 1.9 X 1037 cm3.

 

12.9. CONCLUSIONS ABOUT THE DIFFERENT TYPES OF ACCRETION

Table 12.8.1 shows that the values of ta fall into three groups (fig. 12.9.1). Mercury, Venus, Earth, and Jupiter all have values around 106 yr, which must be much shorter than tinf. Uranus and Neptune have values that are larger than the age of the solar system; hence ta > tinf . There is an intermediate group, consisting of the Moon, Mars, and Saturn with taapproximately108 yr. This is probably of the same order of magnitude as tinf. In any case we cannot be sure whether tinf or ta is the larger quantity.

Our conclusion is that there are three different pathways of accretion.

(1) Early runaway accretion. For Mercury, Venus, Earth, and Jupiter the catastrophic growth of the embryo took place early in the time period of infall of matter into the circumsolar region.
 
(2) Late runaway accretion. For the Moon, Mars, and Saturn the catastrophic growth took place near the end of the infall.
 
(3) Delayed runaway accretion. Uranus and Neptune cannot have accreted until, after the end of the infall, their jet streams eventually contracted so that[Greek letter] beta had decreased considerably from its original value.

 

12.10. EARLY TEMPERATURE PROFILE OF ACCRETED BODY

When a grain in the jet stream is brought to rest on the surface of a growing embryo, the impact velocity is

 

vimp = (u2 + ves2)1/2 (12.10.1)

 

Upon impact the kinetic energy of the grain is almost entirely converted into heat energy. In order to study the temperature of the accreting body we calculate the thermal power per unit surface area WT delivered by impacting grains

 

mathematical equation
(12.10.2)

 

[204] where the mass of the impacting grain is equated to dMem. Defining u and ves as in eqs. (12.3.4) and (12.3.2) we have

 

mathematical equation
(12.10.3)

 

or assuming the density of the embryo to remain constant we have, from eq. (12.3.5),

 

mathematical equation
(12.10.4)

 

which shows that for R >> RG the heat delivered per cm2 sec is proportional to the mass increase of the whole embryo. The function dMem/dt is shown in fig. 12.6.1 and dR/dt in fig. 12.9.1. Hence wT has a maximum at tapproximatelytc. If wT is balanced by radiation from the surface of the accreting body, its surface temperature should vary similarly to wT. This means that the maximum temperature is reached when a fraction [Greek letter] gamma
of the mass is accumulated:

 

mathematical equation
(12.10.5)

 

Hence in an accreted body the region at a radial distance mathematical equation
(where Rsc is the final radius) has received most heat:

 

mathematical equation
(12.10.6)

 

For the Earth ta = 2 X 106 yr. If as above we tentatively put tinf = 3 X 108 yr, we have

 

mathematical equation
(12.10.7)

 

[205] Hence the different layers were accreted with different temperatures: The innermost part was cold, the layers for which [Greek letter] delta = 0.5 were hot, and again the outer parts were cold. The value [Greek letter] delta
= 0.5 depends on a guess for tinf, but is rather insensitive to this value. If we choose for example tinf = 108 or 109 yr, [Greek letter] delta
is changed to 0.58 or 0.40, respectively.

We know neither the chemical composition nor the heat conductivity of the Earth's interior very well (sec. 20.5.1). Also, the content of radioactive substances, which could contribute to the heating of the interior, is unknown. We are not in conflict with any facts or plausible conclusions if we assume that neither the radioactive heating nor the thermal conductivity has changed the temperature structure in a drastic way. Hence our results may give a simple explanation for the fact that only an intermediate part of the Earth is melted, whereas both the inner core and the mantle are solid. According to our result, the outer core was heated most intensely, whereas both the central region and the outer layers were formed cool.

As the heat per unit surface is proportional to dMem/dt, the average formation temperature of a celestial body is proportional to Msc/tinf. If we assume tinf to be similar for the different bodies, the formation temperature (under the condition of similar accretion processes) is proportional to their present masses.

 

12.11. CONCLUSIONS ABOUT THE TEMPERATURE PROFILE OF PLANETS

The equations for wT have been integrated numerically by Ip (1974c). His results for the different planets are shown in fig. 12.11.1. From this we can draw the following general conclusions about the internal temperatures of the planets.

(1) The giant planets were formed with a hot region in the interior. The heat structures of these planets differ in the respect that, while the heat maximum of Jupiter occurs at about half the radius, this maximum for Saturn occurs somewhat further out. In both cases there is a cold accretional phase later. For both Uranus and Neptune substantial heat was delivered also to the outermost layers.

If the primeval heat profile of these planets is conserved at least to some extent, it may be an essential factor affecting their average density. To what extent such a conservation is possible depends on the thermal conductivity in the interior, which is unknown (sec. 20.2).

(2) Venus should have about the same heat structure as the Earth, but with the melted region closer to the center (see fig. 12.11.1). Mercury should have a temperature maximum still closer to its center, but due to its smallness the temperature is much lower.

 


[
206]

FIGURE 12.11.1.- Thermal profiles of the growing planets. (From Ip, 1974c.)

FIGURE 12.11.1.- Thermal profiles of the growing planets. (From Ip, 1974c.)

 

(3) The average heating power on Mars should have been one order of magnitude less than for the Earth. The temperature maximum should be rather close to the surface (perhaps at 0.9 of its radius), where, if at all, a liquid region may have existed. The Moon has a similar heat profile.

In all cases subsequent radioactive heating and thermal conduction may have modified the early heat profile, as is indeed indicated by the fact that the interior of the Moon now appears partially molten and that local volcanism has occurred on Mars.

 

12.12. THE ACCRETIONAL HOT-SPOT FRONT

Our conclusions about the low average formation temperature of some celestial bodies or specific zones in them should not be interpreted as meaning that their constituent matter has never been melted. On the contrary, for large celestial bodies every part, with the exception of the central cores, has been heated above the melting point repeatedly. One can attribute [207] these processes to a front of "accretional hot spots" which sweeps through the body outwards.

Suppose that an energy Wm is needed to melt a mass M of a certain substance. We define a velocity vm by the condition

 

 

mathematical equation
(12.12.1)

 

As soon as a body of this substance has a velocity v > vm its kinetic energy suffices to melt it if converted into heat. For most substances vm is of the order of 105 cm/sec.

If a body with mass M1 and velocity v hits a target of the same composition its kinetic energy suffices to melt a mass

 

mathematical equation
(12.12.2)

where

mathematical equation
(12.12.3)

 

A fraction of its energy will be used for the production of shock waves, the ejection of fragments from the place of collision, and the emission of radiation, but eq. (12.12.2) gives the correct order of magnitude.

We see that it is doubtful whether in a body as small as the Moon [Greek letter] xi has become much larger than unity. For planets like the Earth it may be 10 to 100 for the last phase of accretion.

When the matter melted by an impacting body has cooled down it may be remelted many times by the impact of other bodies in its close neighborhood. The impacting matter will, however, increase the radius of the embryo, and finally the volume we are considering, originally located at the surface, is buried so deeply that no new impact will be able to melt it. Before this is achieved it is likely to be molten [Greek letter] xi
times (because all impacting matter melts [Greek letter] xi
times its own mass).

In retrospect we can picture this as am accretional front of hot spots, discontinuous in time and space and moving outward with the surface layer of the growing protoplanet. All matter is heated [Greek letter] xi
times before the front has [208] passed. The factor [Greek letter] xi
increases in proportion to Rsc2 (Rsc is the radius of the growing protoplanet). The front is able to melt all material as soon as [Greek letter] xi
>>1, which probably occurs at about 108 cm from the center. As long as the impact frequency is low the impacts produce locally heated regions that radiate their heat differentially (sec. 12.13) and cool down again. The accretional heat front will leave a cool region behind it. This is what is likely to have been the case in the Earth's central core and in the mantle, and also in the entire Moon. If, on the other hand, the impact frequency is large, the heated regions have no time to cool. The accretional heat front will leave a hot region behind it. According to the interpretation in sec. 12.11 this is how the Earth's outer core became molten.

 

12.13. DIFFERENTIATION EFFECT OF THE ACCRETIONAL HEAT FRONT

In a volume of matter melted by impact of large embryos, a chemical separation would be expected to take place due to the heavy components' sinking and the light components' floating in the reservoir of liquid or liquids generated by melting. This phenomenon is common in the interior of the present crust of the Earth, where the heating, however, comes from sources other than impact. Gravitative differentiation in a planetary accretional heat front, as suggested here and by Alfvén and Arrhenius (1970b) has been observed on the Moon (see, for example, Urey et al., 1971).

Furthermore, in melt systems of this kind, ions with liquid-solid distribution coefficients favoring their concentration in the liquid will remain in the light residual melt and become removed to the top of the reservoir. Particularly, ions with large radii are included in this process; examples are potassium, barium, the rare-earth elements (particularly in divalent state), and the actinides, including the (next to potassium) most important radioactive heat sources, uranium, thorium, and plutonium.

By reiteration of differentiation every time a new impact occurs in the same region, the accretional hot-spot front will produce a differentiated crust on a global basis. In this way a limited amount of differentiated material may be brought the entire distance from the interior of the body to the surface. The lower limit at which this effect occurs is given by [Greek letter] xi
approximately1.

The heavy components, such as dense magnesium silicates, transition metal oxides, sulfides, and metal, will sink down in the locally heated regions, but if the heat front leaves a solidified region below it the heavy component cannot sink more than the thickness of the heated region. This thickness will depend on the size of the impinging embryo and the impact rate. Typically it seldom exceeds a few kilometers.

Hence the accretional heat front may bring light components and [209] associated heavy ions from the interior to the surface, but it will not bring dense components downward more than a very small distance. The change in the proportion of dense materials from that in the accreting planetesimals is thus mainly a secondary effect of the displacement of the light component.

The gravitative differentiation in the accretional hot-spot front explains why the outer layers of both the Earth and the Moon contain unusually large amounts of low-density components and radioactive elements. It is well known that the interior of the bodies must have a much lower content of such elements because, otherwise, the total heating of the bodies would be very large. Since in the Moon's core [Greek letter] xi
< 1, it should not be effectively depleted of radioactive elements. This may explain why the lunar interior is partially melted.

The constraints imposed by the accretional phenomena on the evolution of the Earth are discussed in more detail in ch. 26.

 


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