SP-345 Evolution of the Solar System

 

21. MASS DISTRIBUTION AND THE CRITICAL VELOCITY

 

21.1. MASS DISTRIBUTION IN THE SOLAR SYSTEM

21.1.1. Inadequacy of the Homogeneous Disc Theory

[369] In theories of the Laplacian type it is assumed that the matter that formed the planets originally was distributed as a more or less uniform disc. The inadequacies of this type of approach have been discussed in secs. 2.4 and 11.2. For completeness a Laplacian-type theory applicable to the planetary system must also prove applicable to the satellite systems. Hence let us turn our attention to the empirical aspects of Laplacian theories as applied to the satellite systems.

As has been discussed in sec. 18.10, the distributed density (see secs. 2.4-2.5) for the group of inner Saturnian satellites (fig. 2.5.3) is reasonably uniform from the ring system out to Rhea, and within this group a uniform disc theory might be acceptable. But outside Rhea there is a wide region devoid of matter, followed by the giant satellite Titan, the very small Hyperion, and the medium-sized Iapetus. An even greater discrepancy between the homogeneous disc picture and the observed mass distribution is found in the Jovian satellite system (fig. 2.5.2). Although the Galilean satellite region is of reasonably uniform density there are void regions both inside and outside it. This same general density pattern also holds for the Uranian satellite system (fig. 2.5.4).

Thus the distributed densities of the satellite systems of Jupiter, Saturn, and Uranus do not substantiate the homogeneous disc theory. Obviously the planetary system does not show a uniform distribution of density. In fact the distributed density varies by a factor 107 (fig. 2.5.1).

In spite of this there are many astrophysicists who believe in a homogeneous disc as the precursor medium for the planetary system. The low density in the asteroid region is then thought of as a "secondary" effect, presumably arising from some kind of "instability" caused by Jupiter. However, under present conditions several big planets (e.g., of 10 to 100 times the mass of Mars) moving between Mars and Jupiter would be just as perfectly stable in all respects as are the orbits of the present asteroids. And no [370] credible mechanism has been proposed explaining how Jupiter could have prevented the formation of planets in this region.

In addition to these obvious discrepancies between the implied uniform and the actually observed distributions of mass in the solar system, the whole disc idea is tied to the theoretical concept of a contracting mass of gas which could collapse to form both the central body and the surrounding secondaries via the intermediate formation of the disc. As has been pointed out in sec. 11.2, small bodies cannot be formed in this way and it is questionable whether even Jupiter is large enough to have been formed by such a collapse process. Another compelling argument against a gravitational collapse of a gas cloud is found in the isochronism of spins (secs. 9.7-9.8 and ch. 13). We have also found in ch. 20 that the chemical composition of the celestial bodies speaks against a Laplacian homogeneous disc. Other arguments against it are found in the detailed structure of the Saturnian rings and the asteroidal belt (see secs. 18.6 and 18.8). It is very unlikely that these features can be explained by a Laplacian model or by gravitational collapse,

 

21.1.2. Origin and Emplacement of Mass: Ejection of Mass

Since the concept of the homogeneous disc consequently is unrealistic when applied to any of the actual systems of central bodies with orbiting secondaries, we must look for other explanations of how the mass which now forms the planets and satellites could have been emplaced in the environment of the central bodies.

In principle, the mass which now constitutes the planets and satellites could either have been ejected from the central body or could have fallen in toward the central body from outside the region of formation. It is difficult to see how a satellite could have been ejected from its planet and placed in its present orbit. Such processes have been suggested many times, but have always encountered devastating objections. Most recently it has been proposed as an explanation of the origin of the Moon, but has been shown to be unacceptable (see Kaula, 1971; and ch. 24).

This process would be still less attractive as an explanation of e.g., the origin of the Uranian satellites. In fact, to place the Uranian satellites in their present (almost coplanar circular) orbits would require all the trajectory control sophistication of modern space technology. It is unlikely that any natural phenomenon, involving bodies emitted from Uranus, could have achieved this result.

An ejection of a dispersed medium which is subsequently brought into partial corotation is somewhat less unnatural, but it also requires a very powerful source of energy, which is hardly available on Uranus, to use the same example. Moreover, even in this case, the launch must be cleverly [371] adjusted so that the matter is not ejected to infinity but is placed in orbit at the required distances. Seen with the Uranian surface as launch pad, the outermost satellites have gravitational energies which are more than 99 percent of the energy required for escape to infinity.

 

21.1.3. Origin and Emplacement of Mass: Infall of Matter

Hence it is more attractive to turn to the alternative that the secondary bodies derive from matter falling in from "infinity" (a distance large compared to. the satellite orbit). This matter (after being stopped and given sufficient angular momentum) accumulates at specific distances from the central body. Such a process may take place when atoms or molecules in free fall reach a kinetic energy equal to their ionization energy. At this stage, the gas can become ionized by the process discussed in sec. 21.4; the ionized gas can then be stopped by the magnetic field of the central body and receive angular momentum by transfer from the central body as described in sec. 16.3.

 

21.2. THE BANDS OF SECONDARY BODIES AS A FUNCTION OF GRAVITATION. POTENTIAL ENERGY

If the hypothesis assuming infall of matter is correct, then the matter that has fallen into the solar system would have accumulated at predictable distances from the central body. This distance is a function of the kinetic energy acquired by the matter during free fall under the gravitational attraction of the central body. Let us consider the positions of a group of secondary bodies as a function of their specific gravitational potential, mathematical symbol
, where

 

mathematical equation(21.2.1)

 

and G is the gravitational constant, Mc is the mass of a central body, and rorb is the orbital radius of a secondary body. The gravitational potential r determines the velocity of free fall and thus the kinetic energy of infalling matter at a distance rorb from the central body. In fig. 21.2.1, we have plotted this energy as a function of Mc for the Sun-planet system as well as for all the planet-satellite systems.

We see from fig. 21.2.1 that:

 


[
372]

FIGURE 21.2.1

FIGURE 21.2.1. Structure of the solar system in terms of the mass of the central bodies and the gravitational potential energy of the bodies orbiting around them. For a detailed analysis refer to secs. 21.2, 21.3, and 23.9.2. (From Alfvén and Arrhenius, 1972.)

 

[373] (1) The secondary bodies of the solar system fall into three main bands.

(2) Whenever a band is located far enough above the surface of a central body there is a formation of secondary bodies in the region. These two important observational facts will be discussed in this and the following chapter.

Although there are some apparent exceptions to the general validity of these conclusions, cogent explanations can be offered for each discrepancy. Venus has no satellites, probably because of its extremely slow rotation and lack of a magnetic field. Both these properties, rotation and magnetic field of the central body, are the prerequisites for formation of secondary bodies, as was discussed in sec. 16.1. Further, we find no satellite systems of the normal type around Neptune and the Earth. The reason for this seems to be straightforward; both these bodies might very well have once produced normal satellite systems, but they have been destroyed by the capture of Triton (McCord, 1966) and of the Moon (ch. 24). Mercury has a very slow rotation and a weak magnetic field and is perhaps not massive enough for satellite formation. Whether Pluto has any satellites is not known.

We have not yet discussed the Martian satellites which fall far outside the three bands. From a formal point of view they may be thought to indicate a fourth band. However, the Martian satellite system is rudimentary compared to the well-developed satellite systems of Jupiter, Saturn, and Uranus, and the Martian satellites are the smallest satellites we know. In view of the rudimentary character of the Martian satellite system, we do not include this in our discussion of systems of secondary bodies. This question is further discussed by Alfvén and Arrhenius (1972) and in ch. 24.

In fig. 21.2.1 the satellite systems are arranged along the horizontal axis according to the mass of the central body. Groups of secondary bodies belonging to a particular band are generally located somewhat lower if the central body is less massive, thus giving the bands a slight downward slope. As a first approximation, however, we can consider the bands to be horizontal. (The reason for the slope is discussed in sec. 23.9.2).

We conclude from the gravitational energy diagram that groups of bodies are formed in regions where the specific gravitational energy has values in certain discrete ranges.

In fig. 21.2.1 we have also plotted the positions of synchronous secondary orbits as well as those of the Lagrangian points for the satellite systems. The position of a synchronous secondary orbit around a primary body is a natural inner limit to a system of secondary bodies, since any secondary body located inside this position would have to orbit faster than the central body spins. (As discussed in sec. 23.9.1, bodies may, under special conditions, orbit inside the limit.) Of all the secondary bodies in the solar system only Phobos orbits within the synchronous limit,

[374] A natural outer limit for a satellite system is the Lagrangian point situated at a distance rL from the planet, given by

mathematical equation
(21.2.2)

where rsc is the planetary distance from the Sun, Msc is the planetary mass, and mathematical symbol
is the solar mass. Table 11.2.1 gives the distances to the Lagrangian points. Outside rL, the gravitational attraction on a satellite due to the Sun exceeds that due to the planet. Hence a satellite must orbit at a distance much smaller than rL in order not to be seriously perturbed by solar gravitation.

 

21.3. COMPARATIVE STUDY OF THE GROUPS OF SECONDARY BODIES

We have found in table 2.5.1 and sec. 18.10.1 that the regular bodies in the solar system belong to certain groups. Accepting the conclusions of sec. 21.2 we shall now attempt a more detailed study of these groups. Physical data for both the planetary and satellite systems are given in tables 2.1.1-2.1.3. Our general method is to compare each group of secondary bodies with its neighbors to the left and right within the same baud of the gravitational potential energy diagram (fig. 21.2.1).

We start with the Jovian system which should be compared with the planetary system and the Saturnian system. The giant planets, the Galilean satellites of Jupiter, and the inner satellites of Saturn (Janus through Rhea) fall in the same energy interval (allowing for the general slope discussed earlier). There is a conspicuous similarity between the group of the four big bodies in the planetary and in the Jovian systems, the four giant planets corresponding to the four Galilean satellites. However, there is also a difference; whereas in the planetary system the innermost body in this group, Jupiter, is by far the largest one, the mass of the bodies in the Galilean satellite group slightly increases outward. In this respect the Jovian system is intermediate between the group of giant planets and the inner Saturnian satellites, where the mass of the bodies rapidly increases outward. The latter group consists of six satellites and the rings. (The difference in mass distribution among the inner Saturnian satellites, the Galilean group, and the giant planets is discussed in secs. 23.6-23.8).

[375] The fifth Jovian satellite, Amalthea, orbits far inside the Galilean satellites. It falls in the same energy band as the group of terrestrial planets. We may regard it as an analog to Mars while the other terrestrial planets have no correspondents probably because of the closeness to the surface of Jupiter. The mass of Amalthea is unknown. Its diameter is estimated to be about 160 km. As the diameter of Io is about 3730 km its volume is about 104 times that of Amalthea. The mass ratio of these two satellites is unknown. The volume ratio of Io to Amalthea is of the same order as the volume of Jupiter to Mars, which is 9000, but the close agreement is likely to be accidental.

The outermost group of Jovian satellites, Jupiter 6, 7, and 10, is rudimentary. One may attribute the rudimentary character of this group to its closeness to the outer stability limit rL for satellite formation, which is closer to this group than to any other group in the diagram. Although this group of Jovian satellites falls within the band including the outer Saturnian satellites and the Uranian satellites, it has no other similarity with these two groups.

In the planetary system the same band may have given rise to Pluto and Triton (the latter being later captured by Neptune in a similar way as the Moon was captured by the Earth).

The Uranian satellites form the most regular of all the groups of secondary bodies in the sense that all orbital inclinations and eccentricities are almost zero, and the spacings between the bodies are almost proportional to their orbital radii (q = rn+1/rnapproximately constant). The group is situated far outside the synchronous orbit and far inside the Lagrangian point. It should be noted that this is also the case for the Galilean satellites which also form a very regular group. In fact, these two groups should be studied as typical examples of satellite formation in the absence of disturbing factors.

Titan, Hyperion, and lapetus are considered as a separate group which we refer to as the "outer Saturnian satellites." The assignment of these three bodies to one group is not altogether convincing and the group is the most irregular of all with regard to the sequence of satellite orbital radii and masses. However, it occupies a range of [Greek letter] capital gammavalues which closely coincides with that of the Uranian satellites. Furthermore, if we compare the group with both its horizontal neighbors, we find that the irregular group in the Saturnian system constitutes a transition between the rudimentary group in the Jovian system and the regular group in the Uranian system. In this respect there is an analogy with the Galilean group in which the almost equal size of the bodies is an intermediate case between the rapid decrease in size away from the central body in the giant planets and the rapid increase in size away from the central body in the inner Saturnian system. The probable reason for these systematic trends is discussed in secs. 23.6-23.8.

 

[376] 21.4. THEORETICAL BACKGROUND FOR THE BAND FORMATION

Attempts to clarify the mechanism which produces the gravitational energy bands (sec. 21.2) should start with an analysis of the infall of the gas cloud toward a central body. To avoid the difficulties inherent in all theories about the primitive Sun, we should, as stated in ch. 1 and sec. 16.9, base our discussion primarily on the formation of satellites around a planet.

The gas cloud we envisage in the process of satellite formation is a local cloud at a large distance from a magnetized, gravitating central body. This cloud, called the source cloud (see sec. 21.11.1) is located within the jet stream in which the central body has formed or is forming and is thus part of the gas content of the jet stream itself (see fig. 21.4.1). This cloud also contains grains from which the central body is accreting. For the sake of simplicity let us assume that initially the cloud is at rest at such a low.....

 


FIGURE 21.4.1.- Qualitative picture of the infall of gas from a jet stream toward a planet.

FIGURE 21.4.1.- Qualitative picture of the infall of gas from a jet stream toward a planet. The gas becomes ionized, is brought into partial corotation, and eventually forms satellite jet streams.

 

[377] ....temperature that the thermal velocity of the particles can be neglected compared to their free-fall velocity. Then every atom of the cloud will fall radially toward the center of the gravitating body. If the gas cloud is partially ionized, the ions and electrons, which necessarily have a Larmor radius which is small compared to the distance to the central body, will be affected by the magnetic field even at great distances from the central body, with the general result that their free fall will be prevented. Hence in our idealized case only the neutral component, the grains and gas, will fall in. The infall of grains is the basic process for the formation and growth of the central body (which acquires spin as a result of the asymmetry of this infall; for a detailed discussion see ch. 13).

Let us now consider the infall of gas in an idealized case in which the gas is not disturbed by the infall of grains. Probably such a situation occurs when the accretion of the central body is near completion. Hence we assume that for a certain period of time there is a constant infall of gas toward the central body which is assumed to be in approximately its present state.

Suppose that at some distance r from the central body there is a very thin cloud of plasma which also has negligible thermal velocity and which, due to the action of the magnetic field, is at rest. (The effect of rotation of the body is neglected here; it will be introduced in sec. 21.13.) The plasma density is assumed to be so low that the mean free path of the atoms exceeds the dimensions of the cloud. (For densities smaller or equivalent to103-104, the mean free path is larger than the dimension of the satellite formation regions; i.e., smaller or equal to planet-outermost satellite distance. However, the dimension of the cloud in question may be an order of magnitude or so smaller, allowing somewhat higher densities, but the mean free path would still be much greater than the dimension of the cloud.)

When the infalling atoms reach the plasma cloud, some will pass through it without colliding and some will make nonionizing collisions and be deflected, but neither of these processes will affect the conditions in the plasma cloud very much.

However, under the condition that the atoms arrive at the cloud with a sufficiently high velocity, atoms may become ionized at some of the collisions. Due to the magnetic field, the ions and electrons thus produced will rapidly be stopped and become incorporated in the plasma cloud. Hence the density of the plasma cloud will grow, with the result that it will capture infalling atoms at an increased rate. In an extreme case the density may become so high that the mean free path of atoms is smaller than the size of the cloud, resulting in a complete stopping of the infalling gas. (We assume that the magnetic field is strong enough to support the resulting dense plasma cloud; see secs. 16.3-16.5.)

Basic theoretical analysis of electric breakdown in a gas treats the conditions under which the electric field will give sufficient energy to an electron [378] to produce new electrons so that an avalanche may start. The "original" existence of free electrons can be taken for granted. Our case is essentially similar. The existence of thin plasma clouds anywhere in space can be taken for granted. The question we should ask ourselves is this: What are the conditions under which the infalling atoms get ionized so frequently that the density of the original plasma cloud will grow like an avalanche? It is likely that the infall velocity is the crucial parameter. In our simple model the infalling gas cloud will be stopped at the distance rion where its velocity of fall reaches the value vion such that

 

mathematical equation
(21.4.1)

 

where ma is the mass of an atom. At this distance the specific gravitational potential energy mathematical symbol
will have the value mathematical symbol
with

 

mathematical equation
(21.4.2)

 

Hence mathematical symbol
is a function only of vion. Because vion is the parameter which sets the lower limit for ionization of the infalling gas, vion may be considered as an analog to the breakdown electric field in the theory of electrical discharges.

The analogy between the stopping of an infalling cloud and the electric breakdown is in reality still closer. In fact, seen from the coordinate system of the infalling cloud, there is an electric field

 

E = -v X B (21.4 3)

 

which increases during the fall because both the velocity of fall v and the dipole magnetic field B increase. If the electric field exceeds a certain critical value Eion, a discharge will start via some (yet unspecified) mechanism for energy transfer to the electrons. This will lead to at least partial ionization of the falling gas cloud. In situations where the collision rate for the electrons is low the mechanism for transfer of energy from the electric field (i.e., from the falling gas) to the electrons is very complicated and not yet quite clarified (the electric field -v X B in the coordinate system of the gas [379] cloud cannot directly accelerate the electrons; in a magnetic field the electron cannot gain more than the potential difference over a Larmor radius for every collision and this is very small). Nonetheless this mechanism has empirically been demonstrated and proven to be highly efficient in a variety of plasma experiments (see sec. 21.8 and references). Under certain (rather general) conditions, this will lead to a braking of the velocity of the cloud and possibly to stopping it. The discharge will occur when v exceeds the value vion which is connected with Eion by

 

Eion = -vion X B (21.4.4)

 

Hence the ionization of the infalling cloud may also be due to the electric field's exceeding Eion.

 

21.5. ATTEMPTS TO INTERPRET THE BAND STRUCTURE

If we equate the ionization energy eVion of an atom of mass ma to its gravitational energy in the presence of a central body of mass Mc, we have

 

mathematical equation
(21.5.1)

 

or

 

mathematical equation
(21.5.2)

 

As we will see later there is a mechanism which converts the kinetic energy of am atom falling toward a central body to ionization energy. Hence, eq. (21.5.2) allows one to determine for an atom of known mass and ionization potential the orbital radius from the central body at which ionization can take place.

In table 21.5.1 we list a number of elements of cosmochemical importance along with their estimated relative abundance, average atomic mass, ionization potential, eVion, gravitational energy as given by eq. (21.5.2), and [380] critical velocity which will be discussed in a later section. Just as the[Greek letter] capital gamma
values for the bodies in the solar system, as given by eq. (21.2.1), were plotted in fig. 21.2.1, so the [Greek letter] capital gamma
values for the elements as given by eq. 21.5.2 are plotted in fig. 21.5.1. Looking at this plot of gravitational potential

 


TABLE 21.5.1. Parameters Determining the Gravitational Energy Band Structure

 

Element a

lonization potential Vion (V)

Average atomic mass (amu)

Log gravitational potential energy log[Greek letter] capital gamma
                                                      (g/cm)

Atomic abundance b relative to Si = 106

Critical velocity c Vcrit (105 cm/sec)

Band

.

H

13.5

1.0

20.29

2 X 1010

50.9

I

He

24.5

4.0

19.94

2 X 109

34.3

I

Ne

21.5

20.2

19.18

2 X 106

14.3

II

N

14.5

14.0

19.18

2 X 106

14.1

II

C

11.2

12.0

19.11

1 X 107

13.4

II

O

13.5

16.0

19.08

2 X 107

12.7

II

(F)

17.42

19.0

19.11

4 X 103

13.3

II

(B)

8.3

10.8

19.08

1 X 102

12 1

II

[Be|

9.32

9.0

19.18

8 X 10-1

14.1

II

[Li]

5.39

6.9

19.04

5 X 101

12.2

II

Ar

15.8

40.0

18.78

1 X 105

8.7

III

P

10.5

31.0

18.70

1 X 104

8.1

III

S

10.3

32.1

18.70

5 X 105

7.8

III

Mg

7.6

24.3

18.60

1 X 106

7.7

III

Si

8.1

28.1

18.60

1 X 106

7.4

III

Na

5.12

23.0

18.30

8 X 104

6 s

III

Al

5.97

27.0

18.48

7 X 104

6 s

III

Ca

6.09

40.1

18.30

9 X 104

5 4

III

Fe

7.8

5s 8

18.30

9 X 105

5.2

III

Mn

7.4

54.9

18.30

1 X 104

5.1

III

Cr

6.8

52.1

18.30

1 X 104

5.0

III

Ni

7.6

s8.7

18.30

5 X 104

5.0

III

(Cl)

13.0

35.5

18.70

2 X 103

8.4

III

(K)

4.3

39.1

18.30

2 X 103

4.6

III

a Minor elements (abundance 102 -104) are indicated by parentheses; trace elements (abundance < 102) are indicated by brackets.

b The very fact that separation processes are active in interstellar and circumstellar space makes it difficult to specify relative abundances of elements except by order of magnitude and for specific environments (such as the solar photosphere, the solar wind at a given point in time, the lunar crust). This is further discussed in ch. 20. The abundances are the averages estimated by Urey (1972). Most values are based on carbonaceous chondrites of Type II which form a particularly well analyzed set, apparently unaffected by the type of differentiation which is characteristic of planetary interiors. Supplementary data for volatile elements are based on estimates for the solar photosphere and trapped solar wind. All data are normalized to silicon, arbitrarily set at 106.

c All values are calculated from eq. (21.6.1) using the data presented in this table.


 


[
381]

FIGURE 21.5.1

FIGURE 21.5.1.- The gravitational energy [Greek letter] capital gamma
and ionization potential of the most abundant elements. Roman numerals refer to row in the periodic table, with "III" including the fourth row. All elements in a band have approximately the same gravitational energy and vion, as discussed in secs. 21.4 and 21.5. Minor and trace elements are indicated, respectively, by parentheses and brackets.

 

....energy versus ionization potential we find that all the elements fall in one of three bands. Hydrogen and helium give a value for [Greek letter] capital gamma
which falls in the region of the lowest band (which will be referred to as Band I, since this is comprised of elements of the first row of the periodic table) All the elements in the second row of the periodic table (Li-F), including C, N, and O, have values around [Greek letter] capital gamma
approximately1019 falling in the intermediate band (Band II), whereas all the Common heavier elements found in the third and fourth row of the periodic table fall in the upper band (Band III). This means that if a gas dominated by any one of the most abundant elements falls in toward the central body, its kinetic energy will just suffice to ionize it when its gravitational potential energy reaches the values indicated by its appropriate band. For our discussion it is decisive that the value of the cosmically most abundant elements fall in a number of discrete bands rather than forming a random distribution.

[382] This ionization is a collective phenomenon dependent upon the gas mixture in the source cloud. The gas as a whole will tend to be stopped in one band. In the light of the above discussion, we note that because of the discrete regions where the [Greek letter] capital gamma
values of the most abundant elements fall, the discrete bands of gravitational energy discussed in sec. 21.2 may be explained by the hypothesis that they are related to these [Greek letter] capital gamma
values. This relationship is discussed in detail in secs. 21.7-21.13.

 

21.6. THREE OBJECTIONS

When the preceding analysis was first made (Alfvén, 1942, 1943a, 1946) there were three objections to the ensuing hypothesis:

(1) There was no obvious mechanism for the transfer of the kinetic energy into ionization. The requirement that [Greek letter] capital gamma subscript ion
of eq. (21.4.2) and [Greek letter] capital gamma
of eq. (21.5.2) should be equal; i.e.,

 

mathematical equation
(21.6.1)

 

was crucial to the hypothesis, but no reason was known for this equality to be true.

(2) There was no empirical support for the hypothesis that masses of gas falling in toward central bodies would have different chemical compositions.

(3) The chemical compositions of the bodies found in each gravitational potential energy band are not characterized by the elements giving rise to those bands. For example, the terrestrial planets fall in a band which corresponds to the [Greek letter] capital gamma
value for hydrogen and helium, but they contain very little of these elements. The band of the giant planets corresponds to C, N, and O, but these planets were believed to consist mainly of hydrogen and helium.

However, the above situation has changed drastically over three decades of theoretical studies and empirical findings. Although we are still far from a final theory, it is fair to state that objection (1) has been eliminated by the discovery of the critical velocity phenomenon as discussed in sec. 21.721.10. With reference to (2), we now know that separation of elements by plasma processes is a common phenomenon in space. We shall discuss such separation and variation of chemical compositions in sec. 21.11. In sec. 21.12 we shall consider objection (3) in light of the dependence of chemical composition on gravitational potential.

In the meantime, no alternative theory has been proposed which in terms [383] of known physical processes explains the positions of the groups of bodies and which at the same time is consistent with the total body of facts describing the present state of the solar system.

 

21.7. SEARCH FOR A "CRITICAL VELOCITY"

Early attempts to theoretically analyze the stopping of an infalling cloud were not very encouraging. Equating the gravitational and ionization energies has no meaning unless there is a process by which the gravitational energy can be transferred into ionization. Further, in an electric discharge the energy needed to actually ionize an atom is often more than one order of magnitude greater than the ionization energy of that atom, because in a discharge most of the energy is radiated and often less than 10 percent is used for ionization.

In view of the fact that, as stated in ch. 15, all theoretical treatments of plasma processes are very precarious unless supported by experiments, it was realized that further advance depended on studying the process experimentally. As soon as the advance of thermonuclear technology made it possible, experiments were designed to investigate the interaction between a magnetized plasma and a nonionized gas in relative motion. Experimental investigations have now proceeded for more than a decade. Surveys have been made by Danielsson (1973) and Lehnert (1971).

 

21.8. EXPERIMENTS ON THE CRITICAL VELOCITY

Many experimental measurements of the burning voltage in magnetic fields were made independently. They demonstrated the existence of a limiting voltage VLm which if introduced into eq. (21.4.4) with Eion = VLm /d (d being the electrode distance) gives almost the same values of v jon as are calculated from eq. (21.6.1). This upper limit of the burning voltage is directly proportional to the magnetic field strength but independent of gas pressures and current in very broad regions. The presence of neutral gas, however, is a necessity for this effect to occur; once a state of complete ionization is achieved these limiting phenomena no longer appear.

Of the first observations most were accidental. Indeed the effect sometimes appeared as an unwanted limitation on the energy storage in various plasma devices, such as thermonuclear machines like the Ixion, the early homopolars, and the F-machines (Lehnert, 1966).

 

21.8.1. Homopolar Experiments

One of the earliest experiments which was especially designed to clarify [384] the phenomena occurring when a neutral gas moves in relation to an ionized gas was performed by Angerth, Block, Fahleson, and Soop (1962). The experimental apparatus, a homopolar device, is shown in fig. 21.8.1. In a vessel containing a gas at a pressure of the order of 5 X 10-3 0.2 torr, or 1014-1016 atoms/cm3, a radial electric field is established by connecting a capacitor bank between two concentric cylindrical electrodes. There is an almost homogeneous magnetic field of up to 10 000 G perpendicular to the plane of the lower figure. To have any reference to our problem, the gas density in the experiment should be scaled down in the same relation as the linear dimension is scaled up. As the densities during the formation of the planetary system should have been of the order of 101-105 atoms/cm3, and the scaling factor is 1010-1013, the experiment is relevant to the astrophysical problem. The temperatures are determined by the plasma process both in the experiment and in the astrophysical problem and should therefore be equal.

A portion of the gas is ionized by an electric discharge. This ionized component is acted upon by a tangential force, resulting from the magnetic held and the radial electric field, and begins to rotate about the central....

 


FIGURE 21.8.1.- Homopolar apparatus.

FIGURE 21.8.1.- Homopolar apparatus. A voltage V is applied across an inner electrode (8) and an outer electrode (4) to give a radial electric field E. The electric field, in the presence of am axial magnetic field B, acts on the ionized portion of the gas to set it into rotation (7). The interaction between the rotating magnetized plasma and the nonionized, nonrotating gas (in contact with the wall) produces a voltage limitation indicating that the relative velocity of the two components attains a critical velocity vcrit. (From Angerth et al, 1962.)

 


[
385]

FIGURE 21.8.2

FIGURE 21.8.2.- The limiting value VLm of the burning voltage as a function of gas pressure for hydrogen in the homopolar experiment. VLm is independent of pressure, but proportional to the magnetic field B. (From Angerth et al, 1962.)

 

....axis. The nonionized component remains essentially at rest because of the friction with the walls. Hence there is a relative motion between the ionized part of the gas and the nonionized gas If the relative motion is regarded from a frame moving with the plasma, there is a magnetized ionized gas at rest which is hit by nonionized gas. We can expect phenomena of the same general kind as when a nonionized gas falls toward a central body through a magnetized ionized gas (a plasma).

The experiment shows that the ionized component is easily accelerated until a certain velocity, the "critical" velocity vcrit, is attained. This critical velocity cannot be surpassed as long as there is still nonionized gas. Any attempt to increase the burning voltage Vb above the limiting value VLm in order to accelerate the plasma results in an increased rate of ionization of the gas, but not in an increase in the relative velocity between the ionized and nonionized components From a theoretical point of view the phenomenon is rather complicated. The essential mechanism seems to be that kinetic energy is transferred to electrons in the plasma and these electrons produce the ionization (see sec. 21.9).

The limiting value of the burning voltage was found to be independent of the gas pressure in the whole range measured (fig. 21.8.2) but dependent on the magnetic field (fig. 21.8.3), as one would infer from eq. (21.4.4). Further, the burning voltage was independent of the applied current; i.e.,...

 


[
386]

FIGURE 21.8.3.

FIGURE 21.8.3.- Limiting voltage VLm versus the magnetic field B in the homopolar experiment. VLm is proportional to B and depends also on the chemical composition (O, D, H) of the gas being studied. (From Angerth et al., 1962.)

 


FIGURE 21.8.4.

FIGURE 21.8.4.- Burning voltage Vb versus applied current I, for hydrogen and nitrogen in the homopolar experiment. Vb is independent of current (degree of ionization) up to a maximum value related to the complete ionization. The plateau defines the limiting voltage VLm related to the critical velocity. (From Angerth et al., 1962.)

 


[
387]

FIGURE 21.8.5.- Critical velocity vcrit versus applied current for seven gases studied in the homopolar experiment.

FIGURE 21.8.5.- Critical velocity vcrit versus applied current for seven gases studied in the homopolar experiment. (The slope of the Ar curve is related to the magnetic field's being too weak to make the ion gyro-radii small enough). The theoretical vcrit, for each gas, as calculated from eq. (216.1), is indicated on the ordinate. (From Angerth et al., 1962.)

 

....was equal to VLm until this exceeded a certain value (which is related to the degree of ionization; see fig. 21.8.4). Given the relationship of the burning voltage to the radial electric field and the value of the axial magnetic field, one can, from eq. 21.4.4, determine the critical velocity from the measured value of the limiting voltage VLm. The dependence of the critical velocity on the chemical composition of the gas was also investigated and found to agree with eq. 21.6.1. Within the accuracy of the experiment, this equation has been checked experimentally for H, D, He, O, and Ne (and also for Ar, but with less accuracy). The experimental results are shown in fig. 21.8.5, where one can observe that the plasma velocity remains rather constant while the applied current (and thus the energy input and degree of ionization) is changed over almost two orders of magnitude.

 

21.8.2. Plasma Beam Hitting Neutral Gas Cloud

The experiment most directly related to the cosmic situation was carried out by Danielsson (1970) and Danielsson and Brenning (1975). The experimental arrangement is shown in fig. 21.8.6. The hydrogen plasma is...

 


[
388]

FIGURE 21.8.6.- Experimental arrangement for critical velocity measurement used by Danielsson.

FIGURE 21.8.6.- Experimental arrangement for critical velocity measurement used by Danielsson. The left part is a plasma gun, emitting a magnetized plasma with a velocity v0. In a long drift tube, the longitudinal magnetization is changed to transverse magnetization A thin cloud of gas is injected through the gas valve. If v0 is below the critical velocity, the plasma beam passes through the gas cloud with very little interaction be cause the mean free path is long. If v0 is above the critical velocity, there is a strong interaction, bringing the velocity to near the critical value At the same time, the gas cloud becomes partially ionized. (From Danielsson, 1969b.)

 

....generated and accelerated in an electrodeless plasma gun (a conical theta pinch) and flows into a drift tube along a magnetic field. The direction of the magnetic field changes gradually from axial to transverse along the path of the plasma. As the plasma flows along the drift tube much of it is lost by recombination at the walls. A polarization electric field is developed and a plasma with a density of about 1011-1012 cm-3 proceeds drifting across the magnetic field with a velocity up to 5 X 107 cm/sec. In the region of the transverse magnetic field the plasma penetrates into a small cloud of gas, released from an electromagnetic valve. This gas cloud has an axial depth of 5 cm and a density of 1014 cm-3 at the time of the arrival of the plasma. The remainder of the system is under high vacuum. Under these conditions the mean free path for direct, binary collisions is much longer than 5 cm so that the interaction according to common terminology is collisionless.

In the experiment it was observed that the velocity of the plasma was substantially reduced over a typical distance of only 1 cm in the gas cloud (see fig. 21.8.7). It was also found that this reduction in plasma velocity depends on the impinging velocity as shown in fig. 21.8.8. If the neutral gas was helium there was no change in velocity for the smallest impinging velocities (below ~ 4 X 106 cm/sec) as the plasma penetrated the gas. For higher impinging velocities there was a relatively increasing deceleration of the plasma.

By investigation of radiation emission from the plasma and neutral gas it was found that the electron energy distribution changed drastically at the penetration of the plasma into the gas and that the ionization of the....

 


[
389]

FIGURE 21.8.7.- Velocity retardation to near the critical value in the Danielsson experiment.

FIGURE 21.8.7.- Velocity retardation to near the critical value in the Danielsson experiment. Plasma deceleration with depth of penetration z in a neutral gas cloud of helium is shown. The front of the cloud is located at z = -5 cm, and the center, at z = 0 cm. The plasma undergoes deceleration from the impinging velocity v0 to near the critical velocity vcrit of helium. Data for two values of the magnetic field B are shown. (From Danielsson, 1969b.)

 


FIGURE 218.8.- Plasma deceleration as a function of impinging velocity in the Danielsson experiment.

FIGURE 21.8.8.- Plasma deceleration as a function of impinging velocity in the Danielsson experiment. Plasma velocity vz=1 in the neutral gas cloud of helium, 1 cm beyond the cloud center, as a function of the initial plasma vacuum velocity v0 is shown. The critical velocity vcrit for helium is indicated. For v0 less than vcrit there was no change in velocity;vz=1 = v0 . For v0 greater than vcrit deceleration was marked; vz=1 remained close to vcrit (From Danielsson, 1969b.)

 

[390] ...gas atoms was two orders of magnitude faster than anticipated from the parameters of the free plasma stream. The characteristic electron energy was found to jump from about 5 eV to about 100 eV at least locally in the gas cloud. This was inferred to be the cause of the ionization and deceleration of the plasma.

So far Danielsson's experiment has demonstrated that even in a situation where the primary collisions are negligibly few there may be a very strong interaction between a moving plasma and a stationary gas. In helium this interaction is active above an impinging velocity of 3.5 X 106 cm/sec and it leads to:

 

(1) Local heating of the electrons,
(2) Ionization of the neutral gas.
(3) Deceleration of the plasma stream,

 

21.8.3. Other Experiments

Analysis of a number of other experiments confirms these conclusions. In some of the experiments the critical velocity is much more sharply defined and hence better suited for a detailed study of the phenomenon. The experiment described above has the pedagogic advantage of referring most directly to the cosmic situation.

 

21.8.4. Conclusions

Experiments investigating the critical velocity or voltage limitation phenomenon have been conducted under a wide variety of experimental conditions (see Danielsson, 1973), These experiments have demonstrated that as the relative velocity increases a critical velocity vcrit is reached. When v < vcrit there is a small and often negligible interaction between gas and plasma. With v > vcrit very strong interaction sets in, leading to ionization of the gas. The onset of ionization is abrupt and discontinuous. The value of vcrit for a number of gases has been measured. Although under certain conditions there are deviations up to perhaps 50 percent, the general result is that vcrit is the same as vion, as given by eq. (21.6.1).

 

21.8.5. Possible Space Experiments

Experiments on the critical velocity phenomenon carried out in space are of particular interest since they give more certain scaling to large dimensions. The upper atmosphere provides a region where plasma-gas interaction of this kind could suitably be studied in the Earth's magnetosphere. The first observation of the critical velocity effect under cosmic conditions was reported by Manka et al. (1972) from the Moon. When an abandoned lunar [391] excursion module was made to impact on the dark side of the Moon not very far from the terminator, a gas cloud was produced which when it had expanded so that it was hit by the solar wind gave rise to superthermal electrons.

 

21.9. THEORY OF THE CRITICAL VELOCITY

A considerable number of experiments representing a wide variety of experimental conditions have each demonstrated an enhanced interaction between a plasma and a neutral gas in a magnetic field. However, the theoretical understanding of the process is not yet complete although much progress has been made; a review is given by Sherman (1973). An initial theoretical consideration might reasonably suggest that an ionizing interaction between a gas and a plasma should become appreciable when the relative velocity reaches a value of (2eVion/ma)1/2 (as noted in sec. 21.6, eq. (21.6.1)) because the colliding particles then have enough energy for ionization. However, two serious difficulties soon become apparent: (1) The kinetic energy of an electron with the above velocity in the plasma is only (me/ma) eVIon (where me is the electron mass), or just a few millivolts, and (2) ionizing collisions between the ions and the neutrals will not occur unless the ion kinetic energy in the frame of reference of the neutral gas exceeds 2eVIon. This second difficulty follows from the fact that, assuming equal ion and neutral masses and negligible random motion of the neutrals, the maximum inelastic energy transfer equals the kinetic energy in the center-of-mass system of the colliding particles. It is then evident that any theoretical justification of the critical-velocity hypothesis must explain how the ion and/or electron random velocities are increased.

Different theories have been suggested by Sockol (1968), Petschek (1960), Hassan (1966), Lin (1961), Drobyshevskii (1964), Lehnert (1966, 1967), and Sherman (1969, 1972). They all refer to different mechanisms of transfer of energy from the atoms/ions to the electrons. We shall not discuss these theories here, but only cite the rather remarkable conclusion which Sherman (1973) draws from his review. He states that for the most part the theories discussed are internally self-consistent. The different theories give a good description of those situations which satisfy the assumptions on which the theories are based. It is remarkable that several widely different theoretical models should all predict the values of E/B near to (2eVion/ma)1/2 Correspondingly, the experiments show that values of E/B near the critical value are observed over a wide range of experiments. The critical velocity hypothesis is then an experimentally confirmed relationship which is valid over a wide range of conditions, but it seems likely that more than one theoretical model is necessary to explain it.

[392] If the atomic mass in eq. (21.6.1) is replaced by the electron mass me, we have a result which is a formal analog to the well-known law discovered by Franck and Hertz: mathematical equation
. The experimental and theoretical investigations demonstrate that a number of mechanisms exist which make the results of the Franck and Hertz experiment for pure electron interaction valid also for a plasma. The only difference is that here one additional step in the interaction is needed which transfers the energy from the atoms (or the ions, depending on the choice of coordinate system) to the electrons (Danielsson and Brenning, 1975).

Hence the critical velocity experiment may be considered as the "plasma version" of the classical Franck-Hertz experiment.

 

21.10. CONCLUSIONS ABOUT THE CRITICAL VELOCITY

We conclude from the survey of relevant experimental and theoretical investigations that the critical velocity vcrit at which a neutral gas interacts strongly with a magnetized plasma is

 

mathematical equation
(21.10.1)

 

Hence if a gas of a certain chemical composition is falling toward a magnetized central body from a cloud at rest at infinity, it will become ionized when [Greek letter] capital gamma
has reached the value

 

mathematical equation
(21.10.2)

 

Consequently objection (1) of sec. 21.6 does not apply and eq. (21.6.1) is validated.

 

21.11. CHEMICAL COMPOSITION OF INFALLING GAS

Objection (2) of sec. 21.6 states that there is no empirical support for the hypothesis that masses of gas falling toward a central body would have different chemical compositions. In this section we shall discuss the theoretical conditions under which such chemical differentiation and fractionation could occur.

 

[393] 21.11.1. The Basic Model

Let us return to the simple model of sec. 21.4 which refers to a jet stream, partially ionized either by radiation or, more importantly, by hydromagnetic effects. We assume that the source cloud which contains all elements (e.g., in an abundance relationship more similar to some average "galactic" composition than now found in the satellites and planets) is partially ionized to such an extent that all elements with ionization potential higher than a certain value VIon (t) are neutral, but all with an ionization potential lower than VIon (t) are ionized. The Larmor radii of electrons and ions are all assumed to be negligible, but all mean free paths are assumed to be larger than the source-cloud dimension. The region which we call "source cloud" may be so defined. All neutral atoms will begin to fall toward the central body.

Let VIon (t) decrease slowly with time (for example, through a general cooling of the plasma by radiation or a change in current such as that discussed by De (1973) in the case of solar prominences). When it has fallen below the ionization potential of helium, helium ions will begin to recombine to form a neutral gas which falls in toward the gravitating central body. Helium reaches its critical velocity of 34.4 X 105 cm/sec at a [Greek letter] capital gamma subscript ion
value (which we now realize, recalling eq. 21.10.2, to be equivalent to the Ii value) of 0.9 X 1020 g/cm (the upper region of Band I of fig. 21.11.1). The gas will at this point become ionized, forming a plasma cloud which will be referred to as the A cloud.

When Vion (t) decreases further, and has passed the ionization potential of hydrogen (which is nearly equal to the ionization potentials of oxygen and nitrogen), hydrogen, oxygen, and nitrogen will start falling out from the source cloud. Because hydrogen is by far the most abundant element, we can expect the infalling gas to behave as hydrogen and to be stopped at[Greek letter] capital gamma
= 1.9 X 1020 g/cm (the lower region of Band I) forming what we shall call the B cloud. In a gas consisting mainly of H, the elements O and N will not be stopped at their critical velocities because of the quenching effect of the hydrogen on the acceleration of electrons that would lead to ionization in pure oxygen or nitrogen gas.

Next will follow an infall dominated by carbon, which is stopped at a vcrit of 13.5 X 105 cm/sec and a [Greek letter] capital gamma
value of 0.1 X 1019 g/cm (Band II), forming the C cloud; and finally the heavier elements, mainly silicon, magnesium, and iron, will fall in to [Greek letter] capital gamma
= 0.3 X 1018 g/cm (Band III), producing the D cloud with a weighted mean critical velocity of 6.5 X 105 cm/sec.

 

21.11.2. The A, B, C, and D Clouds in the Solar System

From the above discussion one can consider the solar system as forming from four plasma clouds. The planets would form by accretion of planetesi-....

 


[
394]

FIGURE 21 11.1. Critical velocity and ionization potential of the most abundant elements.

FIGURE 21 11.1. Critical velocity and ionization potential of the most abundant elements. The left-hand ordinate showing gravitational potential energy [Greek letter] capital gamma
and the right-hand ordinate showing the critical velocities of the controlling elements. of the A, B, C, and D clouds allow a comparison of [Greek letter] capital gamma
values and vcrit values.

 

...-mals and grains, the matter condensing from the plasma cloud in the specific region of gravitational potential of each planet. The location of each plasma cloud is determined by the critical velocity of its controlling elements as depicted in fig. 21.11.1. Hence each plasma cloud can be characterized by a dominant critical velocity. Figure 21.11.2 shows the gravitational potential energy bands labeled as plasma clouds A, B, C, and D with their respective critical velocities indicated.

We see from fig. 21.11.2 and the discussion in the previous section that Mercury, Venus, and Earth formed from the B cloud, while Moon and Mars accreted within the A cloud. As indicated in fig. 21.11.2, there was probably an overlap and possibly an interchange of matter between the A and B clouds in the region of the Earth and the Moon. The giant planets formed within the C cloud, while Pluto and perhaps Triton accreted within the D cloud. Referring to fig. 20.7.1a, we can see that, although there is a wide range of densities in the solar system, the bodies which formed in the same cloud have similar densities. This pattern can be understood on the basis of relatively constant composition within each cloud, but variance of composition among the A, B, C, and D clouds.

[395] Returning to fig. 21.11.2 we see that there were plasma clouds formed around each of the planets shown. Our hetegonic principle stresses that the same processes which formed the planetary system should also prove capable of forming the satellite systems. As depicted in fig. 21.4.1, the jet stream formed within a plasma cloud will provide material for a planet and will function as the source cloud for a series of plasma clouds that will form around that planet by the processes discussed in sec. 21.11.1. Thus, each planet with sufficient magnetization and spin will act as the central body around which A, B, C, and D clouds will form.

Formation of the plasma clouds depends upon attainment of critical velocity by the element determining the orbital distance of the cloud to the....

 


FIGURE 21.11.2.

FIGURE 21.11.2. Gravitational potential energy [Greek letter] capital gamma
as a function of the mass of the central body for the planetary and satellite systems. The right-hand ordinate showing critical velocity affords comparison of [Greek letter] capital gamma
values for the planets and satellites with vcrit values for abundant elements and the A, B, C, and D clouds..

 

[396] ....central body. For planets of less mass, the inner clouds cannot form due to inadequate acceleration of the infalling gas from the source clouds. We see in fig. 21.11.2 that Jupiter is massive enough for an A cloud to form, but not for a B cloud to form. The Galilean satellites of Jupiter formed in the Jovian C cloud. The Saturnian inner satellites formed in the Saturnian C cloud, while the outer satellites formed in the D cloud around Saturn. The satellites of Uranus accreted in the Uranian D cloud.

Therefore all discussion of band formation, gravitational potential energy bands, and the plasma clouds A, B, C, and D refer to both planetary and satellite systems.

 

21.11.3. Refinement of the Basic Model

This is the simplest model that can produce chemically differentiated mass accumulation in the observed gravitational potential energy bands. Of course it is much too simple to be realistic. When discussing and developing it we have to take into account the following facts:

(1) There are a number of plasma processes which could produce chemical separation in a cosmic cloud (see Arrhenius and Alfvén, 1971).

(2) The critical velocity of a gas mixture has not yet been thoroughly studied. We expect that the value vcrit is determined by the most abundant constituent in the cloud.

(3) Other charged species besides single atomic ions have been neglected. The more complete picture including the expected distribution of molecular compounds is discussed below.

 

21.11.4. Effect of Interstellar Molecules

The elementary treatment given above suggests only the gross features of the emplacement band structure. This is modified to some extent by the fact that the elements in the source regions are likely to prevail not only as monatomic species but also, at least to some extent, as molecules and molecular ions. The experiments carried out with diatomic molecular gases (sec. 21.8) indicate that ionization at the critical velocity limit is preceded by dissociation and therefore that the limit is determined by the atomic mass and ionization potential. Only homonuclear molecules (H2, D2, N2) have so far been investigated, but it is reasonable to assume that, in the case of heteronuclear molecules such as CH, CH4, OH, and the multitude of other polynuclear molecules observed in dark clouds in space, the element with the lower ionization potential will determine the critical limit. The main effect expected from the presence of molecular precursors would therefore be transport and emplacement of stoichiometric amounts of hydrogen, accompanying carbon, oxygen, and nitrogen into the C cloud.

[397] In the case of the commonly observed simple hydrides (CH, NH, OH, OH2, CH2, NH2, NH3), the ligated hydrogen contributes relatively little to the mass of the molecule. Furthermore, the molecular ionization potential is similar to or slightly lower than that of the core atom. Hence, even if there is an, as yet, undetermined effect of the molecular state, we would expect the critical velocity to remain close to that of the core atom.

In the case of molecules containing elements from rows 2 and 3 (SiO, AlO, MgO), the ionization potential of the molecule is substantially increased over that of the metal atom. Critical velocities (which are entirely hypothetical) calculated from mass and ionization potential of such molecules place them in the same band as the metals (the increased ionization potential is balanced by the mass increase). The effect, if any, would consequently be to contribute oxygen to the D cloud.

It is important to notice that in no case does molecular formation from abundant species of atoms lead to such an increase in ionization potential that penetration inside the C cloud is possible by this mechanism. Deposition in the A and B clouds therefore would depend entirely on transport of impurities, together with major amounts of helium and hydrogen, and on evaporation of solid grains falling toward the Sun, as discussed in sec. 21.12.

One can conclude from the above discussion that, although direct empirical evidence of source cloud composition during the formative era of the solar system is indeed lacking, there are many cogent theoretical possibilities to account for differing composition of the gravitational potential energy bands resulting from infall into the circumsolar region. Therefore objection (2) of sec. 21.6 is relevant only in its emphasis on the need for continued observation and experimentation.

 

21.12. THE CHEMICAL COMPOSITION OF THE SOLAR SYSTEM AND INHOMOGENEOUS PLASMA EMPLACEMENT

Objection (3) of sec. 21.6 states that the chemical compositions of the bodies found in each gravitational potential energy band are not characterized by those elements which theoretically give rise to each specific band. In this section we shall consider a more detailed theoretical model of band formation.

 

21.12.1. A Model of Band Formation

We are certainly far from a consistent model of the infall of plasma. The discussion here will therefore be confined to some basic principles.

[398] As stated in chs. 15 and 16, homogeneous models are of little value in astrophysics. Hence if we assume that the source cloud is a homogeneous shell from which there is a symmetric and time-constant infall of gas (the simple model of the previous section), we may go completely astray. Inhomogeneous models are necessarily rather arbitrary, and the final choice between possible models can be made only after extensive experiments in the laboratory and in space.

In almost any type of inhomogeneous model one should envisage a number...

 


FIGURE 21.12.1.- infall pattern for source clouds active during different time periods.

FIGURE 21.12.1.- infall pattern for source clouds active during different time periods. The Infalling gas from one source cloud will be dominated by one element. The mass of infalling gas will be stopped when the dominant element is ionized; i.e., in the cloud corresponding to the critical velocity value of that element's band. For example, a Band II element reaches its critical velocity at a value rion which falls within the C cloud.

 


[
399]

FIGURE 21.12.2.- infall pattern for source clouds active during the same time period.

FIGURE 21.12.2.- infall pattern for source clouds active during the same time period. A gas infall from a sand II element-dominated source cloud will be ionized and stopped in the C cloud. If this plasma has not had time to condense, it will interact with any infall from a sand I element-dominated source cloud The sand I gas infall will be trapped in the C cloud by the plasma there and not reach its own rion in the B cloud.

 

....of source clouds from which a gas is falling down during finite periods (see fig. 21.12.1). At a certain instant one or several clouds may be active. The chemical composition of the gas falling in from a certain source cloud may vary. For our model the most important question to ask is which element dominates in such a way that it determines the value of the critical velocity vcrit and hence the arresting value of the gravitational potential energy [Greek letter] capital gamma subscript ion
Suppose that, after there has been no infall for a long time, gas with a certain value of [Greek letter] capital gamma subscript ion
begins to fall in from one source cloud. This gas will then accumulate in the band characterized by [Greek letter] capital gamma subscript ion
. If another cloud with [400] a different characteristic [Greek letter] capital gamma subscript ion
begins to yield gas, this will accumulate in the correspondingly different region, under the condition that when the infall of the second cloud starts the first infall has already ceased, and there has been enough time for the accumulated plasma to condense. However, if this condition is not satisfied, the plasma from the first infall may interfere with the second infall.

Suppose, for example, that the first infall produced a plasma cloud in the C-cloud band, and that the second gas infall has a [Greek letter] capital gamma
value of the B band. Then it can reach the B region only if there is no plasma in the C region, because, if there is, the infalling gas, normally penetrating to the B region, will interact with the plasma in the C region (if it is dense enough; mean free path shorter than C-cloud thickness) and become ionized and hence stopped prematurely. Under certain conditions most of the new cloud will be trapped in the C region. See fig. 21.12.2.

Hence we see that an infall of hydrogen-rich material may be trapped in any of the bands. It arrives at the B cloud only if it is not hindered by plasma in any of the upper bands, but if a recent infall of gas into, e.g., the C cloud, has taken place, most of the gas that subsequently falls in may be trapped there. Then under certain conditions there may be, for example, more B-cloud gas trapped in the C region than there is C-cloud gas.

From this we can draw the important conclusion that although the stopping of infalling gas in a certain band depends on the vcrit value of a controlling element, an inhomogeneous model need not necessarily predict that this element should dominate the ultimate chemical composition of the cloud. Although the trigger element would be enriched to some extent, the ultimate chemical composition of the band need not necessarily deviate drastically from that of the source clouds.

 

21.12.2. Effects of Transplanetary Condensation on the Composition of Planets and Satellites

We have seen (ch. 19) that most of the condensates forming in the transplanetary region must have assumed highly eccentric orbits around the Sun. When penetrating through the regions where plasma is accumulated, these solids may partially evaporate and inject part of their mass into the plasma cloud. This ablation effect would become important when the infalling grains have been accelerated to high velocities relative to the plasma clouds and in regions where the plasma density is comparatively high. Hence one would expect contamination by grain ablation to be most pronounced in the A (helium) and B (hydrogen) regions, and a major fraction of the condensable ions gathered there may be such ablation products.

Furthermore, transplanetary dust accelerated through the B region would interact chemically with the hydrogen characteristic of that region. The [401] physical ablation process would thus be accompanied by selective vaporization of species of SiO, MgO, OH, and SH, leaving the infalling solid grains with an increasing concentration of metallic iron, vaporizing toward the end of the trail, near the central body. Comparable chemical effects are observed in the interaction of the Moon (and possibly of comets) with the solar wind and in laboratory experiments exposing silicates and oxides to proton beams (sec. 14.6), atomic hydrogen, or molecular hydrogen in the temperature range above ~1200K.

This ablation process is physically analogous to the observed ablation of meteors in the Earth's upper atmosphere- the velocity and composition of the particles being the same, the density of the medium being lower (ionospheric), its extension much larger, and its chemical effect reducing rather than oxidizing.

Transplanetary material must also have collided with the grains and embryos in the jet streams, adding material to these.

The total effect of the interaction of transplanetary bodies with interplanetary material (fig. 21.12.3) would thus include complete vaporization of some of the grains, capture and transfer of angular momentum to small dust particles in the plasma clouds, and, furthermore, collision, vaporization, fragmentation and ultimately accretion of some transplanetary material in the planetary jet streams. Larger meteoroid aggregates may have been heated and slowed down by friction in each perihelion passage in interplanetary space with gradually decreasing peak temperature in each Kepler period because of the deceleration. Ultimately such objects would be captured by a jet stream.

Chemical fractionation at ablation of transplanetary dust in the inner solar and planetary nebulae (A-B clouds) may be the explanation for the increasing density of secondary bodies toward the central bodies in the Jovian and planetary systems (sec. 20.7).

 

21.12.3. Fractionation Associated With or Following Condensation

All the fractionation processes so far discussed precede condensation of the solids from which the bodies in the solar system subsequently accumulated. In addition, it is likely that fractionation processes associated with the condensation and later evolution have influenced the chemical composition of the present bodies. We do not know much about the state of the early Sun, for example, if it had a radiation field as intense as today (sec. 25.5). If this were so, volatile compounds such as water ice may have been prevented from condensing and accumulating in the inner part of the planetary system, as pointed out, for example, by von Weizsäcker (1944), Berlage (1948), and Urey (1952). In close proximity to the Sun, a high....

 


[
402-403]

FIGURE 21.12.3.- Simplified diagram of the effect of transplanetary condensation on the chemical composition of the planets.

FIGURE 21.12.3.- Simplified diagram of the effect of transplanetary condensation on the chemical composition of the planets. The primary emplacement of plasma controlled by the critical velocity produced the A, B, C, and D clouds. The A and B clouds were enriched in He and H. The C cloud composition was enriched in C, 0, N, and Ne, while the D cloud contained an excess of heavy elements. Part of the D cloud condensation took place outside the solar magnetic field and resulted in particles and embryos in almost parabolic orbits, passing through the interplanetary clouds. Due to ablation of these bodies in the massive A and B clouds, and chemical interaction with the hydrogen in the latter, substantial amounts of transplanetary D-type material were deposited in the inner part of the solar system (and in the corresponding regions of the satellite systems). The distribution after transplanetary falldown and H-He diffusion indicates the possible redistribution of heavy elements into the A- and B-cloud regions and the diffusion of H and He to the C region, where light gases could be partially accreted by giant planetary embryos. The observed density distribution among the planets (see fig. 20.7.1a) reflects the compositions of the respective clouds. The terrestrial planets, forming in the A and B clouds, have higher densities than the giant planets which formed in the C cloud. Pluto and Triton have higher densities indicative of the composition of the D cloud in which they accreted.

 

[404] ...temperature of the radiation field could perhaps decrease the condensation rate of oxygen compounds with silicon and magnesium, which have high vapor pressures relative to iron. However, it is also possible that the solar radiation was negligible, and we must look for another explanation of some of the quoted facts. This is suggested by the similar trends of the density distribution (sec. 20.7) in the inner part of the solar system and in the Jovian satellite system, where effects of the radiation field can hardly be held responsible.

Another late fractionation effect is the gravitational retention of increasingly light gases by planets and satellites of increasing size. The embryos accreting to form the giant planets may, after having reached a few Earth masses, have been able to collect and retain substantial amounts of hydrogen and helium.

 

21.12.4 Conclusions About the Chemical Composition of Celestial Bodies

We are necessarily dealing with highly hypothetical phenomena which do not allow us to draw very specific conclusions. However, we here summarize the processes most likely to influence the bulk composition of the accreted bodies:

(1) The critical velocities of the element groups corresponding to clouds A, B, C, and D; this effect would also be responsible for the spacing of the groups of secondary bodies around their primaries.

(2) The vapor pressure of the solids that can form from the gases controlling the cloud formation; since hydrogen and helium are not condensable, the bodies formed in the A and B clouds consist largely of "impurities".

(3) The fractional vaporization of infalling transplanetary material in the dense A and B clouds, preferentially depositing refractory elements such as iron in the central reactive hydrogen cloud (B cloud).

(4) Trapping of infalling gases with high critical velocities in already established clouds.

(5) Fractionation at condensation, due to the gradient in the solar radiation field.

(6) Gravitational accumulation of hydrogen and helium by the giant planets.

It will require much work before we can decide between models giving similar composition to all the bands and models in which there are appreciable chemical differences among the regions. Such work should include interaction and fractionation experiments in hydrogen and in mixed plasmas as well as the sampling and analysis of comets and asteroids which possibly consist of materials representative of the primordial states.

What has been said in ch. 20 and sec. 21.12 shows the complexity of the [405] problems relating to chemical composition of the celestial bodies. Although objection (3) (sec. 21.6) is no longer valid, we are still far from a detailed theory of chemical composition.

 

21.13. MODIFICATION OF CRITICAL VELOCITY IONIZATION DISTANCE DUE TO INTERACT WITH A PARTIALLY COROTATING PLASMA

The simple model of sec. 21.4 could be developed in different directions. The falling gas need not necessarily interact with a plasma at rest. If, for example, the plasma is in the state of partial corotation (see ch. 17), its tangential velocity is (from table 17.3.1)

 

mathematical equation
(21.13.1)

 

Adding this vectorially to the velocity of fall

 

 

mathematical equation
(21.13.2)

 

we get a resulting relative velocity vrel

 

mathematical equation
(21.13.3)

 

When vrel reaches the critical velocity vcrit the infalling gas can become ionized. Let us determine the orbital radius rrel at which ionization can take place.

From eq. (21.4.2) we have

 

mathematical equation
(21.13.4)

 

[406] Equating vcrit to vrel we obtain

 

mathematical equation
(21.13.5)

 

This relative velocity due to the corotation of the magnetized plasma attains the critical value vion at 4/3 the orbital radius at which ionization would occur if the plasma were not in a state of partial corotation with the central body.

There is yet another effect seen when the interacting plasma is in a state of partial corotation. Condensation and accretion of matter reduces the orbital radius by a factor 2/3 (as explained in ch. 17). Combining the effects of the tangential velocity and the condensation characteristic of a corotating plasma, we obtain the value for the effective ionization radius r'ion for a gas falling through a corotating plasma:

 

mathematical equations
(21.13.6)

 

Therefore in fig. 21.11.2 the critical velocity scale should be displaced downward along the gravitational energy scale so that the value of rion is decreased to 0.89rion and corresponds to r'ion for the case of corotation of the plasma.

Yet another correction may be of some importance. If the central body is accreting mass during a period of plasma accumulation, the angular momentum of the grains condensing in its environment will change during the accretion. Our present calculations are valid in the case that practically all the gas infall takes place when the central body is close to its final state of accretion. A refinement of the theory in this respect cannot be made before the variation of the gas content in the jet stream can be estimated. It should also be remembered that the formation of secondary bodies cannot start before the central body has grown sufficiently large to acquire a magnetic field which makes transfer of angular momentum possible.


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