SP-345 Evolution of the Solar System

 

23. THE STRUCTURES OF THE GROUPS OF SECONDARY BODIES

 

[429] 23.1. IONIZATION DURING THE EMPLACEMENT OF PLASMA

In the preceding chapter we discussed the hypothesis that the location of the different groups of secondary bodies is determined by the critical velocity phenomenon. However, the internal structures of the groups differ in the respect that in some of them (e.g., the giant planets) the mass of the bodies decreases rapidly with increasing distance from the central body, whereas in other groups (e.g., the inner Saturnian satellites) the reverse is true. In this chapter we shall show that this difference in structure among the groups probably is related to the total energy dissipated in the process of emplacement of the plasma. This leads to the conclusion that the structure of a group depends on the ratio T/tau between the typical orbital period T of the secondary bodies of the group and the spin period [Greek letter] tau of the central body. There is observational support of this dependence (see secs. 23.5-23.6). In fact the mass distribution in the groups is evidently a function of T/tau
.

As in some of the earlier chapters we are obviously far from a detailed theory, and the aim of our discussion is essentially to call attention to what may be the basic phenomena determining the structure of the groups.

According to our model, a gas of mass m, originally at rest at "infinity," falls in to the ionization distance rion where it becomes partially ionized (fig. 23.2.1). By transfer of angular momentum from the central body this mass is brought into partial corotation (ch. 17). It condenses and through processes discussed in secs. 18.2 and 18.10 it is eventually placed in a circular orbit with the radius r. In sec. 17.6 we found that the total release of energy during this process is

 

mathematical equation(23.1.1)

 

where [Greek letter] omega
= (GMc/r3)1/2 is the angular orbital velocity of m. As mathematical equation
and as within a group r does not vary by more than a factor of 6 (see table 2.5.1), [430] we do not introduce a very large error if in our order-of-magnitude calculation we approximate eq. (23.1.1) by

 

mathematical equation
(23.1.2)

 

where Tion is the orbital period of a fictitious body orbiting at the ionization distance rion, [Greek letter] tau
is the spin period of the central body, mathematical equation
, and mathematical equation
.

If we equate m to the mass of an atom ma and let rion = GMcma/eVion (from eq. 21.10.2) we have

 

mathematical equation
(23.1.3)

 

Part of this energy will be dissipated in the central body or in its ionosphere and part of it in the plasma which is brought into partial corotation. Without a detailed analysis it is reasonable to guess that these parts are about equal. The energy is delivered to the plasma by the electric currents which transfer the momentum and then primarily is converted to an increase in the electron temperature. When this has reached a certain value, most of the energy is radiated, but a fraction [Greek letter] zeta is used for ionization.

In laboratory studies of electric currents in gases it has been shown that [Greek letter] zeta
seldom exceeds 5 percent. For example, in a glow discharge the minimum voltage Vc between the electrodes (which actually equals the cathode potential drop) is usually 200-300 V (essentially only pure noble gases have lower values). This holds, for example, for H2, N2, and air (V. Engel, 1955, p. 202), for which the voltage needed to produce ionization is in the range 10-15 V. Hence this ratio [Greek letter] zeta
= Vion / Vc, which gives the fraction of the energy which goes into ionization, is about 0.05. Even if the discharge in our case differs in certain respects, we should not expect [Greek letter] zeta
to be drastically different. Taking account of the fact that only a fraction of W is dissipated in the plasma we should expect [Greek letter] zeta
to be less than 0.05.

Hence, even without making any detailed model of the process we may conclude that if [Greek letter] zeta
W denotes the energy that goes into ionization of the plasma, [Greek letter] zeta
is not likely to exceed 0.05. This means that it is impossible to produce a complete ionization of the plasma if mathematical symbol is of the order 10 or less. A considerably higher value is probably needed for complete ionization to occur.

[431] We then conclude:

(1) Other things being equal, the degree of ionization during emplacement is a function of mathematical symbol
.

(2) We may have complete ionization if is, for example, 100 or more, but probably not if it is of the order of 10 or less.

In sec. 23.2 we shall treat the case

 

mathematical equation
(23.1.4)

 

which indicates complete ionization, reserving the case of incomplete ionization

 

mathematical equation
(23.1.5)

 

for sec. 23.3.

 

23.2. COMPLETE IONIZATION

We shall now discuss the extreme case mathematical equation
, implying that the plasma is completely ionized. The gas which falls in is stopped at the critical velocity sphere, which is defined by mathematical equation
, where it immediately becomes partially ionized (see fig. 23.2.1). The transfer of angular momentum gives it an azimuthal velocity which increases until partial corotation is achieved. The energy release associated with this process ionizes the plasma completely.

As stressed earlier, it is important to note that homogeneous models are obsolete in cosmic plasma physics. To reduce the speculative element which hetegonic theories necessarily include, it is essential to connect the models as far as possible with laboratory experiments and such cosmic phenomena as we observe today. For the discussion, references to magnetosphere and especially for solar phenomena are essential. The transfer of angular momentum through a set of "superprominences," as discussed in sec. 16.7 and by De (1973), is the background for our present treatment (see fig. 16.6.1).

Hence we should consider the infall of gas as taking place in a series of intermittent events with a finite extension and a finite lifetime. Several infalls could very well take place simultaneously. The gas which arrives [432] at the critical velocity sphere rion and becomes partially ionized is rapidly incorporated in a superprominence which is almost completely ionized because mathematical equation
guarantees that in the long run there is enough energy for ionization. The processes to which the infalling gas is subject at rion confine the gas to a magnetic flux tube. Its final destiny is either to fall along this flux tube to the central body or to attain an increasing angular momentum so that it is brought to the neighborhood of the equatorial plane. There are regions around the axis of the central body where the former process takes place, whereas the latter process occurs in a band near the equatorial plane.

Figure 23.2.1 is a projection on a meridional plane and should be interpreted with what is said above as a background.

As the average mass distribution is uniform over the surface of the sphere rion, the mass dM between the latitude circles at [Greek letter] lambdaand [Greek letter] lambda + d 
[Greek letter] lambda
amounts to

 

mathematical equation
(23.2.1)

 


FIGURE 23.2.1.&emdash;Complete ionization of infalling gas.

FIGURE 23.2.1.- Complete ionization of infalling gas. Gas falling in from infinity reaches the critical velocity at rion (the critical velocity sphere) and becomes partially ionized. It is rapidly included in "superprominences" which, if mathematical equation
, are almost completely ionized. Matter falling in at low latitudes (a, b, and c) will be emplaced near the equatorial plane and condense there. Matter arriving at the critical velocity sphere at high latitudes (d) will be drawn into the central body. Note that the processes a, b, and c do not necessarily interfere because they may occur at different times or even simultaneously at different longitudes.

 

[433]

mathematical equation
(23.2.2)

 

where rB is the distance to the central body from a point on the line of force and r is the value of rB at the equatorial plane. Putting rB = rion we obtain by differentiating eq. (23.2.2)

 

mathematical equation
(23.2.3)

 

and

 

mathematical equation
(23.2.4)

 

This function is plotted in fig. 23.2.2.

 


FIGURE 23.2.2.- Matter stopped at the critical velocity sphere is displaced outward along the magnetic field lines and condenses in the region of the equatorial plane.

FIGURE 23.2.2.- Matter stopped at the critical velocity sphere is displaced outward along the magnetic field lines and condenses in the region of the equatorial plane. For a rough estimate it is assumed, rather arbitrarily, that all matter between the present orbits of Jupiter and Saturn is now included in Jupiter, etc. As shown by table 23.2.1, this gives roughly the observed mass distribution. The essence of the analysis is that the distributed density in the region of the giant planets is compatible with the model of sec. 23.2. (From Alfvén, 1962.)

 

[434] Let us now see whether it is possible that the outer planets have originated from a gas having the mass distribution given by eq. (23.2.4).

We assume that rion coincides roughly with the present value of the orbital radius of Jupiter (mathematical symbol) and that all gas situated between mathematical symbol
and the orbital radius of Saturn (mathematical symbol
) is used to build up Jupiter. (The fact that, according to ch. 17, all distances are likely to decrease by a factor of 2/3 is not crucial in this respect.) In the same way we assume that all matter between mathematical symbol
andmathematical symbol
(Uranus) is condensed to Saturn, etc. Thus we should expect the following masses of the planets:

 

Jupiter:

 

mathematical equation
(23.2.5)

Neptune:

 

mathematical equation
(23.2.6)

where mathematical symbol
is the orbital radius of Pluto and A is defined by

 

mathematical equation
(23.2.7)

 

The relative masses of the planets calculated from equations of the form (23.2.7) and the observed masses are given in table 23.2.1. The calculated values agree with observations within a factor of 2. (The integral from Pluto to infinity is 32 units, but, as this mass has become ionized near the axial region of the Sun, it is likely to have fallen directly into the Sun; note "d" in fig. 23.2.1.)


[
435] TABLE 23.2.1. Mass Distribution Among Giant Planets Calculated for mathematical equation
.

.

Mass (Earth = 1)

Planet

Calculated

Observed

.

Jupiter

320

317

Saturn

88

95

Uranus

26

15

Neptune

10

17

 

The assumption that the gas is divided exactly at the present distances of the planets is, of course, arbitrary, and a more refined calculation has been given elsewhere (Alfvén, 1954, ch. V). But if we go in the opposite direction, we can interpret the result as follows. Suppose that we distribute the masses of the outer planets so that we obtain a continuous mass distribution in the equatorial plane. A projection of this along the magnetic lines of force upon a sphere gives us an almost uniform mass distribution. Consequently, the mass distribution obtained in this way shows a reasonable agreement with the mass distribution among the giant planets.

We now turn our attention to the outer Saturnian satellites. This is a group which also has a very high value of mathematical symbol
. The group is irregular (see sec. 23.8) and it is difficult to deduce the original mass distribution from the three existing bodies. However, it is evident that in this group also most of the mass is concentrated in the innermost body, Titan, which is situated somewhat below the ionization limit.

 

23.3. PARTIAL IONIZATION

It is only in two groups, the giant planets and the outer Saturnian satellites, that the innermost body is the biggest one. In all other groups there is a slow or rapid decrease in size inward. The reason for this is probably that the value of mathematical symbol
is too small to satisfy eq. (23.1.4); this is discussed in detail in secs. 23.5-23.7.

A small value of mathematical symbol
can be expected to have two different effects (see fig. 23.3.1):

(1) On the critical velocity sphere there is a limit between the region close to the axis from which the matter is drawn in to the central body and the region from which matter is brought to the equatorial plane. When mathematical symbol
decreases, this limit is displaced away from the axis. The result of this is that...

 


[
436]

FIGURE 23.31.- Partial ionization of infalling gas.

FIGURE 23.31.- Partial ionization of infalling gas. Small values of mathematical symbol
( <20) imply an increase of the region near the axis of the central body from which matter is drawn into the central body. incomplete ionization at rion is also implied and diffusion of neutral gas toward the central body will take place. The result is a displacement inward of the region of plasma emplacement and a change in the mass distribution within a group of secondary bodies.

 

...no matter is brought down toward the equatorial plane at a large distance from the critical velocity sphere. Hence, in comparison with the case of very largemathematical symbol
, the outer limit of the region where bodies are produced will be displaced inward.

(2) As all the gas is not ionized at the critical velocity sphere, part of it will fall closer to the central body, where sooner or later a considerable part of its condensates are collected in jet streams. Hence mass is collected even far inside the critical velocity sphere. These two effects are further discussed in sec. 23.7.

 

23.4. CHANGE OF SPIN DURING THE FORMATION OF SECONDARY BODIES

From this discussion we would expect the mass distribution within a group of bodies to depend on the value of mathematical symbol
. However, the value of this quantity would not be the present value but the value at the time of formation. The angular momentum which Jupiter, Saturn, and Uranus have transferred to orbital momenta of their satellites is small (of the order of 1 percent; see table 2.1.2) compared with the spin momenta of these planets, and no other mechanism by which they can lose a large fraction of their momenta is known (see sec. 10.4). Hence, it is reasonable to suppose that they possessed about their present angular momenta at the time of formation of their satellite systems.

[437] Their moments of inertia may have changed somewhat during the planetary evolution, but this change is likely to be rather small. Hence, it seems reasonable to state that the axial rotations of these planets had approximately their present angular velocity at the time when their satellite systems were formed.

This conclusion does not hold for the Sun. Its present angular momentum is only 0.6 percent of the total angular momentum of the solar system. Hence, if the Sun has lost angular momentum only through transfer to planets, it has transferred 99.4 percent of its original angular momentum to the orbital momenta of the giant planets. This effect would have made the value of mathematical symbol
about 180 times larger at the beginning of the formation of the planetary system. However, the Sun may also have lost angular momentum to the solar wind. Whether this has been an appreciable amount or not is uncertain (see sec. 25.4), but it is possible that this factor of 180 should be still larger.

On the other hand, the moment of inertia of the Sun may have changed. If, at a very early stage, the Sun was burning its deuterium, its radius would be about 16 times larger than now (sec. 25.6). If the planets were formed around a deuterium-burning Sun, these two effects would approximately compensate each other, and the present values of mathematical symbol
would be valid.

These considerations are not very important for the formation of the giant planets because this group would, for either extreme value of [Greek letter] tau, have values of mathematical symbol
which satisfy eq. (23.1.4). On the other hand, it does not seem legitimate to use the present values of mathematical symbol
for the terrestrial planets. Hence we exclude them from our analysis.

23.5. OBSERVATIONAL VALUES OFmathematical symbol

Before calculating theoretically the values of mathematical symbol
for the different groups, we shall plot the observational values of the ratio mathematical symbol
between the Kepler period TK of a secondary body and the period [Greek letter] tau
of the axial rotation of its central body. This gives us fig. 23.5.1.

It appears that for the giant planets the value of mathematical symbol
of the order of several hundred and for the outer Saturnian satellites about one hundred. The Galilean satellites and the Uranian satellites have similar values, ranging from about 5 up to about 50. The inner Saturnian satellites have values between 2 and 10. (The values for the terrestrial planets, which should not be included in our analysis, lie between 3 and 30.)

To characterize each group by a certain value of mathematical symbol
we could take some sort of mean of the values for its members. From a theoretical point of view the least arbitrary way of doing so is to use the value Tion of the Kepler motion of a mass moving at the ionization distance, as we have done in...

 


[
438]

FIGURE 23.5.1.- Ratio between the orbital period TK of secondary bodies and the spin period  of the central body

FIGURE 23.5.1.- Ratio between the orbital period TK of secondary bodies and the spin period [Greek letter] tau
of the central body. The latter quantity may have changed for the Sun, but not for the planets. The secondaries are grouped according to the cloud in, which they formed. From left to right are the terrestrial planets, the giant planets, outer Saturnian satellites, Galilean satellites of Jupiter. the Uranian satellites, and the inner satellites of Saturn (From Alfvén, 1962.)

 

....sec. 23.1. Referring to fig. 21.11.2 we see that each group falls into one of me clouds surrounding its central body. To analyze a group in terms of mathematical symbol
we must choose the ionization distance rion for the group as a whole. In this treatment we shall use the rion which corresponds to the critical velocity vcrit of each cloud as denoted in fig. 21.11.2.

Setting r = rion, we have

 

 

mathematical equation
(23.5.1)

 

and from eq. (21.10.2)

 

mathematical equation
(23.5.2)

 

[439] It follows that

mathematical equation
(23.5.3)

 

where vcrit is the velocity characterizing the cloud.

 

23.6. MASS DISTRIBUTION AS A FUNCTION OF mathematical symbol
.

In fig. 23.6.1 the masses of the bodies are plotted as a function of the orbital distances. The distances are normalized with the ionization distance rion as unit: [Greek letter] delta= r/rion. This value for each body is called the "normalized distance." The normalized distances for the planets and their satellites are given in table 23.6.1.

 

The values of the normalized distance are not rigorously obtained. As rion is a function of vcrit the uncertainty introduced in assigning a characteristic vcrit to a specific cloud (see sec. 21.11-21.12) also pertains to the values of the normalized distance. Further (see sec. 21.13), one ought to reduce the rion to 0.89 rion to take account of the 2/3 falldown process of condensation (see sec. 17.5) and the corotation of the plasma. However, we attempt only a general understanding of the relationship of mathematical symbol
to the mass distribution. Thus the inaccuracy introduced in choosing rion and hence Tion for each group does not diminish the validity of the trends observed in each group.

For each group a straight line is drawn in fig. 23.6.1, and the slope of this line gives a picture of the variation of the average mass density of the gas from which the bodies condensed. Such a line can, in general, be drawn in such a way that the individual dots fall rather close to the line (mass difference less that a factor of 2). An exception is found in the outer Saturnian group, where Hyperion falls very much below the line connecting Titan and lapetus.

The figure shows that the mass distribution within the groups depends in a systematic way on the value of mathematical symbol
. Among the giant planets (mathematical symbol
= 520) the masses decrease outward, as discussed in detail in sec. 23.2. The Jovian (Galilean) satellites with mathematical symbol
= 29 have almost equal masses. In the Uranian group (mathematical symbol
= 12) the masses increase outward, on the average, whereas the inner Saturnian satellites (mathematical symbol
=8) show a rapid and monotonic increase outward. The outer Saturnian satellite group which has mathematical symbol
= 80 should be intermediate between the giant planets and the Jovian satellites. If a straight line is drawn between the dots representing Titan....

 


[
440] TABLE 23.6.1. Normalized Distance for Secondary Bodies in the Solar System.

Primary

Cloud

Secondary

Normalized distance

rorb/rion

.

Sun

B

Mercury

0.56

Venus

1.05

Earth

1.46

A

Moon

0.67

Mars

1.01

C

Jupiter

0.49

Saturn

0.89

Uranus

1.79

Neptune

2.81

D

Triton

0.63

Pluto

0.83

Jupiter

A

Amalthea

0.84

C

Io

0.28

Europa

0.44

Ganymede

0.70

Callisto

1.24

D

Rudimentary

-

Saturn

C

Mimas

0.41

Enceladus

0.52

Tethys

0.65

Dione

0.83

Rhea

1.16

D

Titan

0.60

Hyperion

0.73

lapetus

1.75

Uranus

D

Miranda

0.42

Ariel

0.61

Umbriel

0.85

Titania

1.40

Oberon

1.87


 

....and Iapetus, the slope of this line is steeper than we would expect. However, Hyperion falls very far from this line, which hence does not represent the mass distribution within the group in a correct way. For reasons we shall discuss later, this group is not so regular as the other groups (see sec. 23.8). Furthermore, the mathematical symbol
value for the giant planets is uncertain because we do not know the spin period of the primeval Sun, which indeed must have changed when it transferred most of its angular momentum to the giant....

 


[
441]

FIGURE 23.6.1.

FIGURE 23.6.1.- Mass distribution within the groups of secondary bodies as a function of their normalized distances [Greek letter] delta= r/rion. The figure shows that within a group characterized by a large value of mathematical symbol
, the mass decreases outward. For s value of mathematical symbol
which is small, the mass decreases inward. (From Alfvén, 1962.)

 

....planets. An evolution of the solar size and spin as suggested by Alfvén (1963) should give an average value of mathematical symbol
for the giant planets which may be smaller than the value for the outer Saturnian satellites. This would eliminate the only exception to the systematic trend in fig. 23.6.1. It was suggested above that the Mercury-Venus-Earth group should not be included in the analysis because we could not be sure that the Sun has [442] the same angular velocity now as when this group was formed, which means that its mathematical symbol
value may not be the correct one. The present value is mathematical symbol
= 8.5, close to the value of the inner Saturnian group. The mass distribution is also similar to the conditions in this Saturnian group (see fig. 23.6.1). Hence, if the present value of mathematical symbol
for this group is used, the terrestrial planets fit, though probably coincidentally, in the sequence of fig. 23.6.1. Likewise, the Moon and Mars are deleted from the discussion because of the uncertainty of the Sun's spin period in the formative era.

 

23.7. DISCUSSION OF THE STRUCTURE OF THE GROUPS OF SECONDARY BODIES

In an earlier treatise (Alfvén, 1954) an attempt was made to develop a detailed theory of the variation of the mass distribution as a function of mathematical symbol
. As this was done before experimental and theoretical investigations had clarified the properties of the critical velocity, the discussion must now be revised to some extent. We shall not try here to treat this problem quantitatively but confine ourselves to a qualitative discussion of the two effects which, according to sec. 23.3, should be important. These are best studied for the C cloud (sec. 21.11.1 and fig. 21.11.2) because this has produced three groups with very different values of mathematical symbol
(giant planets with mathematical symbol
= 520, Galilean satellites with mathematical symbol
= 29, and inner Saturnian satellites with mathematical symbol
= 8).

In the group of the giant planets the bodies have normalized distances [Greek letter] delta
= r/rion, with a maximum of 2.81 (see fig. 23.6.1 and table 23.6.1). In the two other groups the maximum value of [Greek letter] delta
is 1.24 for the Galilean and almost the same (1.16) for the inner Saturnian satellites. The decrease in outward extension may be caused by the first effect discussed in sec. 23.3. Of the matter stopped at distance rion, that found in a larger region around the axis is drawn down to the central body (compare figs. 23.2.1 and 23 3.1). In this situation no matter is brought to the equatorial plane along those lines of force which intersect this plane at a large distance.

Further, the second effect discussed in sec. 23.3 allows matter to become ionized closer to the central body because not all the matter is ionized and stopped at the ionization distance rion. A result of this is that the innermost body of the Galilean group has a normalized distance of only [Greek letter] delta
= 0.28, compared to 0.49 for the giant planets. In the inner Saturnian group this effect is even more pronounced because of the smaller value of mathematical symbol
. Certainly, the innermost body (Mimas) of this group has a [Greek letter] delta
value of 0.41, but the satellite group continues inside the Roche limit in the form of the ring system. Here we find matter collected almost down to the surface of Saturn, corresponding to a [Greek letter] delta
value as low as 0.1.

 


[
443]

TABLE 23.8.1

TABLE 23.8.1. Values of mathematical symbol
Where Tion Is the Kepler Period of a Body at the Ionization Distance and [Greek letter] tau Is the Period of Axial Rotation of the Central Body.

According to the theory, bodies are produced only in the groups above the line in the table.

 

A similar effect, although less pronounced, is indicated in the D cloud by the fact that the [Greek letter] delta
value of Miranda in the Uranian system is 0.42, and Titan, the innermost body of the outer Saturnian group, has [Greek letter] delta
= 0.60. However, there is no similar difference between the outer limits.

 

23.8. COMPLETE LIST OF mathematical symbol
FOR ALL BODIES.

Table 23.8.1 presents all the mathematical symbol
values above unity for the A, B, C, and D clouds captured around the largest bodies in the solar system (see fig. 21.11.2). Also some values slightly below unity are given for comparison.

The six groups represented in figs. 23.5.1 and 23.6.1 all have mathematical symbol
values greater or equal to8. As the process we have discussed has a general validity, we should expect similar groups to be produced in all cases where we find the same values of mathematical symbol
, unless special phenomena occur which prevent their formation. In addition to these six groups, we also find high values of mathematical symbol
in three more cases. This means that we would also expect groups of bodies in these cases:

[444] (1) D cloud around the Sun: We would expect a group of planets outside the giant planets. Pluto and probably also Triton may belong to this group. (Like the Moon, Triton was initially a planet which later was captured; see McCord, 1966.) As the D cloud should contain heavy elements (see sec. 21.11), the high density of Pluto, and possibly Triton (see sec. 20.5), may be explained. According to ch.19 the extremely large distance to the Sun has made the hydromagnetic transfer of momentum inefficient because the transplanetary magnetic field has interfered with the solar field. This group has only these two members. But there may also be as yet undiscovered members of this group.

(2) D cloud around Jupiter: The absence of regular D cloud satellites around Jupiter may appear surprising. However, as has been shown elsewhere (Alfvén, 1954, p. 161), the solar magnetic field, if it is strong enough, should prevent, or interfere with, the production of satellites. The region which is most sensitive to this interference is the D cloud region around Jupiter; next is the D cloud region around Saturn. Hence, the solar magnetic held may have prevented the D cloud satellites around Jupiter and at the same time made the outer Saturnian satellites as irregular as they are with regard to the sequence of masses and orbital radii.

Another possibility is that the D cloud region is too close to the Lagrangian points to allow the formation of a regular group.

(3) D cloud around Neptune: We should also expect a D cloud group around Neptune. If a group was once formed from such a cloud, it is likely to have been largely destroyed by the retrograde giant satellite Triton, when it was captured. The evolution of the Neptune-Triton system is likely to have been similar, in certain respects, to that of the Earth-Moon system (see ch. 24). This implies that Nereid is the only residual member of an initial group of satellites, most of which may have impacted on Triton in the same way as the Earth's original satellites presumably impacted on the Moon, forming the maria relatively late in lunar history.

It should be added that the A cloud around the Sun probably has produced Mars and also the Moon as an independent planet, which was later captured (ch.24).

So far we have discussed all the cases in which mathematical symbol
has a value in the same range as the six groups of fig.23.6.1. It is of interest to see what happens if mathematical symbol
is smaller than this. From table 23.8.1 we find that the next value (mathematical symbol
=1.6) belongs to the A cloud around Jupiter. In the region where we expect this group, we find only one tiny satellite, the fifth satellite of Jupiter, which has a reduced distance [Greek letter] delta
= r/rion = 0.84. This body may be identified as the only member of a group which is rudimentary because of its small mathematical symbol
value. If we proceed to the next value, which is mathematical symbol
= 1.3 for the C cloud around Uranus, we find no satellites at all.

Hence, the theoretical prediction that no satellite formation is possible [445] when mathematical symbol
approaches unity is confirmed by the observational material. The transition from the groups of fig. 23.6.1 to the absence of satellites is represented by Jupiter's lone A cloud satellite, Amalthea.

 

23.9. COMPLETENESS

Summarizing the results of our analysis we may state that they justify our original assumption; namely, that it makes sense to plot the secondary bodies as a function of [Greek letter] capital gamma
. In fact, according to the diagram (fig. 21.2.1), a necessary condition for the existence of a group of secondary bodies is that the gravitational potential in those regions of space have specific values, and, whenever this condition is fulfilled, bodies are present.

All the known regular bodies (with a possible uncertainty in the identification of Pluto and Triton) fall within three horizontal bands- with a possible addition of a fourth band for the Martian satellites. Groups of bodies are found wherever a band falls within the natural limits of formation of secondary bodies.

There is no obvious exception to this rule but there are three doubtful cases:

(1) The band producing the Uranian, the outer Saturnian, and outermost Jovian satellites may also have produced bodies in the planetary system. It is possible that Pluto and Triton, whose densities seem to be higher than those of the giant planets, are examples of such a group.

(2) From only looking at the observational diagram (fig. 21.2.1) we may expect a correspondence to Martian satellites in the outermost region of the Uranian system, and possibly also in the outskirts of the Saturnian system. However, we see from fig. 20.11.2 that no critical velocity is sufficiently small for infalling matter to be stopped in these regions; hence there is no theoretical reason to expect such bodies.

(3) It is likely that a group of natural satellites originally was formed around the primeval Earth but was destroyed during the capture of the Moon. Before the capture of the Moon the Earth had a much more rapid spin. A reasonable value for the spin period is 4 hr. With a D cloud around the Earth this gives T/ [Greek letter] tau = 2.2. This value is intermediate between Amalthea and the inner Saturnian satellites. Hence we should expect that the Earth originally had a satellite system somewhat intermediate between Amalthea and the inner Saturnian satellites. The satellites were necessarily very small, and all were swallowed up or ejected by the much bigger Moon (see ch. 24).

 

23.9.1. Note on the Inner Limit of a Satellite System.

As derived in sec. 17.3 the state of partial corotation is given by

[446] mathematical equation
(23.9.1)

with

mathematical equation
(23.9.2)

 

As [Greek letter] omega, the angular velocity of the orbiting body, cannot surpass the angular velocity [Greek letter] capital omega of the spinning central body, we cannot have equilibrium unless r > r0 with r0 defined by

 

 

mathematical equation
(23.9.3)

Introducing the synchronous radius rsyn for a Kepler orbit when [Greek letter] omega
=[Greek letter] capital omega

 

mathematical equation
(23.9.4)

we find

mathematical equation
(23.9.5)

 

The minimum distance rmin of condensed matter in circular orbit given by me 2/3 law (sec. 17.5) is

 

mathematical equation
(23.9.6)

 

and

mathematical equation
(23.9.7)

 

[447] Hence, within an order of magnitude, the synchronous orbit gives the inferior limit to the position of a satellite. Due to the nature of the condensation process (sec 17.5), cos[Greek letter] lambda
approaches unity.

There are only two cases known where matter is orbiting inside the synchronous orbit:

(1) Phobos: The orbital radius of Phobos is 0.44 of the synchronous orbit. Matter could be brought into circular orbit at this distance only if cos [Greek letter] lambda
= (0.44/0.58)3/5 or cos [Greek letter] lambda
< 0.85 and [Greek letter] lambda
> 31°. There is no apparent reason why condensation should have taken place exclusively so far from the equatorial plane of Mars. Possible explanations for the small orbital radius of Phobos are (a) Mars might have slowed down its spin after the generation of Phobos. This is compatible with the fact that according to the law of isochronism Mars should have had an initial spin period of the order of 5 hr (as with the Earth before the capture of the Moon). This would leave Phobos far outside the synchronous orbit. However, it is difficult to see how the required slowdown could have occurred. (b) Phobos might have been generated when Mars was much smaller than it is today. Even if the mass of a central body increases, the angular momentum of its orbiting body remains constant. Hence the mass must have increased at least in the proportion (0.58/0.44)3 = 2.29. (c) It has sometimes been suggested that Phobos might be a captured satellite. Phobos' small eccentricity and inclination make this suggestion highly unlikely.

(2) Saturnian rings: The synchronous orbit is situated in the outer part of the B ring. The minimum value 0.58rsyn is very close to Saturn, being only 7 percent of Saturn's radius above the surface of the planet. The density in the C ring, which begins at 0.8 of the synchronous orbit, is very small, but this is due to the "ring's own shadow" (see sec. 18.6) and is not likely to be connected with the synchronous orbit. Hence in the Saturnian rings we see a confirmation that matter can also be accreted at some distance inside the synchronous orbit.

 

23.9.2. Slope of the Bands in the Gravitational Potential Energy Diagram.

In ch. 21 we expected theoretically that the bands in which the secondary bodies are located should be horizontal; i.e., independent of the mass of the central body. In the diagram of fig. 21.2.1 we observe a slight slope of the bands. In fact, the gravitational energy at which the C groups are located is larger for Jupiter than for the Sun, and larger for Saturn than for Jupiter. From what has been discussed above, this slope is likely to be due to the fact that mathematical symbol values for these three groups differ. The similar difference between the D cloud groups of Saturn and Uranus may be attributed to the same effect.

 

[448] 23.9.3. Further Regularity of the Groups

Besides the regularity of the group structures as a function of mathematical symbol
, the total mass of the secondary bodies depends in a regular way on the mass of the central body. This is shown in fig. 24.3.1.

Furthermore, it seems that the number of satellites is a unique function of mathematical symbol
(fig. 24.3.2). These empirical regularities have not yet been analyzed theoretically. At present we must confine ourselves to stating that our way of analyzing the solar system leads to discoveries of a number of regularities that may be important for the formulation of future theories.

 

23.10. CONCLUSIONS ABOUT THE MODEL OF PLASMA EMPLACEMENT

The model of plasma emplacement which we have treated in chs. 21 and 23 must necessarily be more speculative than the theories in earlier chapters. The basic phenomenon, ionization at the critical velocity, although well established, is not yet so well understood in detail that we know the behavior of gas mixtures in this respect. Specifically it remains to be clarified what excess of a particular element is necessary to make the critical velocity of this element decisive for the stopping and ionization of the gas. Nor is the distribution of elements between molecular ions sufficiently known. In connection with what has been found in sec. 21.12, this means that we cannot predict the chemical composition of the bodies in a specific group.

Moreover, such predictions cannot yet be verified since the chemical composition of celestial bodies belonging to different clouds is not yet known. We are far from the days when it was claimed with certainty that Jupiter consisted almost entirely of pure solid hydrogen. It is now generally admitted that we do not know with certainty the bulk composition of the Earth and, still less, of any other body (see sec. 20.2-20.5). Hence, detailed, precise predictions will not be possible until the theory is refined under the influence of more adequate experimental and observational data.

The success of the model in giving a virtually complete and nonarbitrary classification of the bodies in the solar system qualifies it as a framework for future theories.


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