[15] 2.1 ORBITAL PROPERTIES OF PLANETS AND SATELLITES

The most important invariants of the motion of a celestial body are the absolute values of the spin angular momentum and the orbital angular momentum. Although the space orientation of these vectors is not constant with time, but changes with a period ranging from a few years to 10⁶ years (the first figure referring to close satellites and the latter to the outer planets), there are reasons to believe that, with noteworthy exceptions, the absolute values of these vectors have remained essentially constant since the formation of the bodies (see chs. 10, 17, and 21).

There are exceptions to this general rule. Tidal effects have changed the orbital momentum and spin of the Moon and the spin of the Earth in a drastic way (see ch. 24) and have produced a somewhat similar change in the Neptune-Triton system (see ch. 9). It is possible that the spin of Mercury was slowed by solar tides until a spin-orbit resonance stabilized the system, but it is also possible that Mercury was produced with its present spin. The spins of all satellites have been braked to synchronism with the orbital motion. To what extent the orbits of satellites other than the Moon and Triton have been changed by tides remains a controversial question. As we shall see in ch. 9, the changes have probably been very small.

In tables 2.1.1, 2.1.2, and 2.1.3, we list for planets and satellites the physical properties and orbital elements that are relevant to our discussion. Of particular importance is the specific orbital angular momentum C (that is, the angular momentum per unit mass) of the orbiting body, defined by

where r_orb is the radius vector from the central body (ideally from the common center of gravity) and v_orb is the orbital velocity. The absolute.....

[16-17] Table 2.1.1. Orbital and Physical Parameters for the Planets.

[18-20] Table 2.1.2. Orbital and Physical Parameters for the Satellite Systems.

[21] Table 2.1.3. Orbital and Physical Parameters for the Retrograde Satellites and the Moon.

[22] ....values of C are listed, as are those of the total angular momentum C_M = M_scC, where M_sc is the mass of the secondary body.

If M_c is the mass of the central body and G is the gravitational constant, then the semimajor axis a and the eccentricity e of the orbital ellipse are connected with C through

All the planets and the prograde satellites, with the exception of Nereid, have e < O.25. Most of them, in fact, have e < O.1 (exceptions are the planets Mercury and Pluto; the satellites Jupiter 6, 7, and 10; and Saturn's satellite Hyperion). Hence

(2.1.3)

is usually a good approximation and is correct within 3.1 percent for e < O.25 and within 0.5 percent for e < O.1.

The sidereal period of revolution T is calculated from the value of the semimajor axis:

(2.1.4)

or approximately

(2.1.5)

and the average orbital velocity v_orb is calculated from

(2.1.6)

[23] In table 2.1.1., the orbital inclination of the planets i refers to the orbital plane of the Earth (the ecliptic plane). It would be more appropriate to reference it to the invariant plane of the solar system, the so-called Laplacian plane. However, the difference is small and will not seriously affect our treatment.

For the satellites, the orbital inclination is referred to the most relevant reference plane. For close satellites, this is the equatorial plane of the planet because the precession of the orbital plane is determined with reference to this plane. For some distant satellites, the influence from the Son's gravitational field is more important; hence the orbital plane of the planet is more relevant.

2.2. PHYSICAL PROPERTIES OF PLANETS AND SATELLITES

Having dealt with the orbital characteristics, we devote the remainder of each table to the secondary body itself. Given the mass M_sc and the radius R_sc of the body, its mean density is calculated from

(2.2.1)

From the observed periods of axial rotation (spin periods), , the planetary normalized moments of inertia are tabulated. If is the radius of gyration and R the radius of the body, the ratio is a measure of the mass distribution inside the body. The moment of inertia per unit mass and unit R², , of a homogeneous sphere is 0.4. A smaller value indicates that the density is higher in the central region than in the outer layers of the body.

Next, the inclination i_eq of equator to orbital plane is tabulated for each planet in table 2.1.1.

The velocity necessary for shooting a particle from the surface of a celestial body of radius R to infinity is the escape velocity v_es. This is also the velocity at which a particle hits the body if falling from rest at infinity. We have

(2.2.2)

[24] If a satellite is orbiting very close to the surface of the planet, such a "grazing satellite" has a = R. Its orbital velocity is v_es /2^1/2.

A convenient scale for time is provided by the quantity

(2.2.3)

referred to as the "time of escape." It follows from eqs. (2.2.1-2.2.2) that

(2.2.4)

It is easily shown that if a particle is shot vertically from a body of radius R with velocity v_es, it reaches a height

(2.2.5)

after the time t_es. This time is related to the period T_gz of a "grazing satellite" in table 2.1.2 through

(2.2.6)

For the Earth ( = 5.5 gcm^-3), we have t_es = 10 min and T_gz = 89 min.

There is also a column listing the value of q_n, the ratio of the orbital distances of adjacent bodies, q_n = a_n+1 /a_n. The quantity q_n takes the place of the number magic of Titius-Bode's "law" (see sec. 2.6).

2.3. PROGRADE AND RETROGRADE SATELLITES

All the planets and most of the satellites orbit in the same sense ("prograde") as the spin of their respective central body. This is probably the result of a transfer of angular momentum from the spin of the central [25] body to the orbital motion of the secondary bodies at the time when the system was formed (chs. 16-17).

However, there are a few satellites which orbit in a "retrograde" direction. With the exception of Triton, their orbits differ from those of prograde satellites also in the respect that their eccentricities and inclinations are much larger. As their origin is likely to be different (they are probably captured bodies), they are listed separately (table 2.1.3). Since the Moon is likely to be a captured planet, it also is included in table 2.1.3 (see ch. 24).

The heading "grazing planet (satellite)" refers to the dynamic properties of a fictitious body moving in a Kepler orbit grazing the surface of the central body. Similarly, "synchronous planet (satellite)" refers to a fictitious body orbiting with a period equal to the spin period of the central body. The data for such bodies provide useful references for the orbital parameters of the system.

Some of the relations given in tables 2.1.1 and 2.1.2 are plotted in the diagrams of figures 2.3.1 through 2.3.4.

2.4. THE LAPLACIAN MODEL AND THE DISTRIBUTED-DENSITY FUNCTION

Discussion of the origin of the solar system has been dominated for centuries by the Laplacian model. Laplace himself presented this model only as a qualitative suggestion. In spite of many later efforts, it has not been possible to formulate theories of this type in a quantitatively satisfactory way.

According to models of this type, a primeval nebula somehow formed from interstellar matter and assumed the shape of a uniform disc of gas which contracted and, in this process, threw off a series of rings that collapsed to form planets. The model idealizes the planetary system as consisting of a uniform sequence of bodies, the orbital radii of which obey a simple exponential law (or Titius-Bode's "law").

A consequence of the Laplacian model would be that the planetary masses obey a simple function of the solar distance; however, this conclusion is so obviously in disagreement with observations that this aspect has been avoided. In a more realistic version of this approach, it is necessary to assume that the density varied in a way that reflects the mass variation of the planets. This mass distribution of the Laplacian nebula may be called the "distributed density" obtained by conceptually smearing out the mass of the present bodies.

As we shall see in the following (especially chs. 11-13, 16, 18), there is yet another serious objection to the Laplacian concept. We shall find that at any given time a gas or plasma with this distributed density could not.....

[26] Figure 2.3.1. Specific angular momentum of the Sun and planets. (From Alfvén and Arrhenius, 1970a.)

Figure 2.3.2. Specific angular momentum of Jupiter and its prograde satellites.

[27] Figure 2.3.3. Specific angular momentum of Saturn and its prograde satellites.

Figure 2.3.4. Specific angular momentum of Uranus and its prograde satellites.

[28] .....have existed. Instead, there must have been an emplacement of plasma over a long period. However, the density distribution of this emplacement is correlated with the "distributed density," which hence is an important function, even if it should not be taken literally.

To reconstruct the distributed density in the solar system, some rather arbitrary assumptions must be made. However, as the density varies by several orders of magnitude from one region to another, a certain arbitrariness would still preserve the gross features of the distribution. For the present discussion, we assume that the mass M_n of a planet or satellite was initially distributed over a toroidal volume around the present orbit of the body. We further assume that the small diameter of the toroid is defined by the intermediate distances to adjacent orbiting bodies; that is, the diameter will be the sum of half the distance to the orbit of the adjacent body closer to the central body and half the distance to the orbit of the body farther from it. We find

(2.4.1)

where r_n is the orbital distance of the n^th body from the central body , is the distributed density, and q_n =r_n+1/r_n. Numerical values of q_n are given in tables 2.1.1 and 2.1.2, where we note that secondary bodies tend to occur in groups bordered by large expanses of empty space. The q_n value for a body on the edge of such a gap is enclosed in parentheses.

Equation (2.4.1) has no physical meaning at the inner or outer edge of a group of bodies, but in these cases we tentatively put the small radius of the torus equal to one-half the distance to the one adjacent body. inside the groups q_n is about 1.2-1.6, which means that the square of the term in brackets varies between 0.1 and 1.0. Hence, in order to calculate an order of magnitude value, we can put

(2.4.2)

[29] which is the formula employed for the "distributed density" column in tables 2.1.1 and 2.1.2. These values are plotted in figs. 2.5.1 through 2.5.4; smooth curves are drawn to suggest a possible primeval mass density distribution.

It should be kept in mind, however, that the terrestrial planets, for example, contain mostly nonvolatile substances, presumably because volatile substances could not condense in this region of space or on bodies as small as these planets. As the primeval plasma probably contained mainly volatile substances, its density in this region may have been systematically a few orders of magnitude larger than indicated in the diagram.

2.5. DISCUSSION OF THE DISTRIBUTED-DENSITY DIAGRAMS

It is natural that there should be an outer limit to the sequence of planets, presumably determined by the outer limit of an original disc. Furthermore, it is conceivable that no matter could condense very close to the Sun if the radiation temperature were prohibitively high in this region. But unless a number of ad hoc hypotheses are introduced, theories of the Laplacian type do not predict that the distributed density should vary in a nonmonotonic way inside these limits.

As we see in fig. 2.5.1, the Laplacian model of a disc with uniform density is very far from a good description of reality. The density in the region between Mars and Jupiter is lower by five orders of magnitude than the density....

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[30] ....in adjacent regions. The existence of one or more broken-up planets, the fragments of which should now be the asteroids, is often postulated. Even if this assumption were correct, it could not explain the very low density of matter in this region. Within the framework of the Laplacian nebular model, this low-density region would require a systematic transport of mass outward or inward, and no plausible mechanism to achieve this has been proposed. (The difficulties inherent in this view are discussed further in sec. 11.8.)

If we try to look at fig. 2.5.1 without the prejudice of centuries of bias toward Laplacian models, we find ourselves inclined to describe the mass distribution in the planetary system in the following way.

There have been two clouds of matter, one associated with the terrestrial (or inner) planets and a second with the giant (or outer) planets. These clouds were separated by a vast, almost empty region. The inner cloud covered a radial distance ratio of q(/ ) = 3.9 (where q is the ratio between the orbital radii of the innermost and outermost bodies within one group). For the outer cloud, the corresponding distance ratio is q(/)=5.8, or, if Pluto is taken into account, q(/) = 7.6 (see table 2.5.1). The clouds were separated by a gap with a distance ratio of q(/) = 3.4. The bodies deriving from each of the two clouds differ very much in chemical composition (ch. 20).

As always, the analysis of a single specimen like the planetary system is necessarily inconclusive; thus it is important to study the satellite system to corroborate our conclusions. We find in the Jovian system (fig. 2.5.2 that the four Galilean satellites form a group with q = 4.5. Similarly, the group of five Uranian satellites (fig. 2.5.3) have a q value of 4.6. These values fall within the range of those in the planetary system.

In the case of the planetary system, one could argue that there are no planets inside the orbit of Mercury because solar heat prevented condensation very close to the Sun. This argument is invalid for the inner limit of the Galilean satellites, as well a8 for the Uranian satellites. Neither Jupiter nor Uranus can be expected to have been so hot as to prevent a formation of satellites close to the surface. We see that Saturn, which both in solar distance and in size is intermediate between Jupiter and Uranus, has satellites (including the ring system) virtually all the way to its surface. Hence, the Saturnian system inside Rhea would be reconcilable with a Laplacian uniform disc picture, but neither the Jovian nor the Uranian systems are in agreement with such a picture.

Further, in the Saturnian system (fig. 2.5.4), the fairly homogeneous sequence of satellites out to Rhea is broken by a large void (between Rhea and Titan q = 2.3). Titan, Hyperion, and possibly also Iapetus may be considered as one group (q = 2.9). The inner satellites including the ring should be counted as a group with q(Rhea/Janus)=3.3.

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Thus we find that the celestial bodies in the solar system occur in widely separated groups, each having three to six members. The planet and satellite groups are listed with their orbital ratios in table 2.5.1. A more thorough consideration of this grouping is undertaken in ch. 21. It is reasonable that the outer Jovian (prograde) satellites should be considered as one group consisting of closely spaced small members.

Other prograde satellites include Amalthea, Nereid, Phobos, and Deimos. The band structure, discussed in ch. 21, suggests that Amalthea is the only observed member of another less massive group of Jovian satellites. Nereid is perhaps the only remaining member of a regular group of Neptunian.....

....satellites that was destroyed by the retrograde giant satellite Triton during the evolution of its orbit (sec. 24.3). Phobos and Deimos form a group of extremely small Martian satellites.

2.6. TITIUS-BODE'S "LAW"

Titius-Bode's "law" has been almost as misleading as the Laplacian model. In spite of the criticism of this theory by Schmidt (1946a), it still seems to be sacrosanct in all textbooks. In its original formulation it is acceptable as a mnemonic for memorizing the inner planetary distances. It is not applicable to Neptune and Pluto, and, had they been discovered at the time, the "law" would probably never have been formulated. It is now usually interpreted as implying that the ratio q_n between consecutive orbital distances should be a constant. It is obvious from table 2.1.1 that this is usually not the case. Attempts have been made to find similar "laws" for the satellite systems. This is possible only by postulating a distressingly large number of "missing satellites."

As we shall find in chs. 11, 13, 17, 19, and 21, the orbital distances of planets and satellites are determined mainly by the capture of condensed grains by jet streams. In many cases, resonance effects are also important, as discussed in ch. 8. Both these effects give some regularity in the sequence of bodies, and, in certain limited regions, an exponential law of the Titius-Bode type may be a fairly good approximation, as shown by the fact that the value of q_n in some groups is fairly constant. But neither in its original nor in its later versions does the "law" have any deeper significance.

To try to find numerical relations between a number of observed quantities is an important scientific activity if it is regarded as a first step toward finding the physical laws connecting the quantities (Nieto, 1972). No such connection to known physical laws has emerged from the swelling Titius-Bode literature, which consequently has no demonstrated scientific value.

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