SP-345 Evolution of the Solar System





[161] Celestial mechanics applied to the motion of planets and satellites shows that of the orbital parameters the longitude mathematical symbol
of the pericenter and the longitudemathematical symbol of the ascending node vary monotonically, whereas the eccentricity e and the inclination i exhibit secular variations within certain limits. The most constant parameter is the semimajor axis a. There is a famous theorem by Lagrange and Poisson which states there are no secular perturbations in a to the first and second approximation. Surveys of the orbital variation and the stability of the solar system treated within the framework of celestial mechanics are given, for example, by Brouwer and Clemence (1961a) and by Hagihara (1961).

From a physical point of view, the constancy of a is connected with the constancy of the orbital angular momentum C = [a(1-e2)]1/2. It is difficult to change the orbital momentum of a body because momentum must then he transferred either to another body or to the interplanetary medium. As the density in interplanetary space is very low, the latter process is not very efficient. A transfer of angular momentum by tidal action seems to be the only important mechanism by which a considerable change can take place.

Angular momentum can also be exchanged through resonance effects. These may be very important, but only when bodies are locked in the resonance. In general, resonances conserve, rather than change, the structure.

A possible change in the solar rotation resulting from the solar wind flow will be discussed in ch. 25.

The authors cited above express, rather vaguely, the opinion that the solar system probably is more stable than can be proven by ordinary celestial-mechanics methods. The effects of resonances have not been included in these discussions. The study of resonance effects provides criteria for a high degree of stability.



Under present conditions, bodies locked in resonances are likely to remain in that state for an indefinite time. However, a breaking of a resonance [162] capture is possible under certain conditions (sec. 8.9). The amplitude of the librations is a measure of resonance stability. If the librations increase to an amplitude of 180°, the bodies break loose from the resonance. In many cases the librations are very small (see table 8.5.1), indicating a high degree of stability.

A proportional change in the periods of all the orbiting bodies in a satellite system or in the planetary system will not alter the resonances in that system. Such a change can be produced by an increase or decrease in the mass of the central body. Consequently, little cam be learned about such mass variations from a study of the resonance pattern. As discussed in sec. 10.3, we can make more definite conclusions concerning changes in the relative positions of the orbits of the secondary bodies.


10.2.1. Argument for Stability From Near Commensurabilities

We assume with Goldreich (1965) that, if once an exact resonance is established, the bodies will remain in resonance indefinitely. Thus the existence today of near-commensurabilities establishes limits upon the amount the orbits in question could have changed since hetegonic times. As table 10.2.1 shows, the period of Jupiter is intermediate between the 2/5 resonance of Saturn and the 1/7 resonance of Uranus. Similarly, the period of Uranus is intermediate between the 3/1 resonance of Saturn and the 1/2 resonance of Neptune. Hence, if we assume that the period of Saturn mathematical symbol
and the period of Uranus mathematical symbol
have been constant, we can conclude that the period of Jupiter mathematical symbol
can never have been as much as 0.67 percent shorter because then it would have been trapped in 2/5 resonance with Saturn, nor can it have been as much as 1.18 percent longer, because of the 1/7 resonance with Uranus.


Table 10.2.1. Limits on Possible Change in Orbital Period for Jupiter and Uranus As Indicated by Near-Commensurabilities With Adjacent Planets





Possible resonance

2/5mathematical symbol

mathematical symbol

1/7mathematical symbol

3mathematical symbol

mathematical symbol

1/2mathematical symbol

Orbital period (yr)







Deviation from present period







[163] Similarly, if mathematical symbol
and mathematical symbol
have been constant, mathematical symbol
cannot have been 2.0 percent shorter because of the 1/2 resonance with Neptune, nor 5.2 percent longer because of the 3/1 resonance with Saturn. Similar arguments can be applied to the near-commensurabilities in the satellite systems.

The conclusion to be drawn from this discussion is that the orbital periods in the solar system are likely to have varied less than a few percent since hetegonic times. The only exceptions are the Earth-Moon and the Neptune-Triton systems.

However, this conclusion rests on the rather uncertain assumption mat resonance locking cannot be broken. This is probably true under present conditions. It was probably not valid during me hetegonic era when viscous effects were more important. The tentative conclusion we have drawn here is not in conflict with the suggestion, also very tentative, in sec. 8.8.1 mat me near-commensurabilities are broken resonances.



Another argument for a high degree of stability of me solar system comes from me relationships between Mimas and Cassini's division. From me conclusions reached in sec. 18.6 we see mat me maximum increase in Mimas' orbital distance since me formation of me rings is a few percent. Similar and even more convincing conclusions follow from the study of me asteroid belt in relation to Jupiter (sec. 18.8). Also in this case we find what is obviously a product of the hetegonic processes conserved to our time with an accuracy of better than 1 percent.

Hence, we have to accept mat at least in certain respects me orbital dynamics of me solar system have a very high degree of stability.



As stated in chs. 8 and 9, mere are also good reasons to believe that for most planets me spin has not changed much since they were formed. (As me asteroids are in a state of evolution, this does not mean that their spins have remained unchanged for 4.5 Gyr.) However, for all satellites me spin has been braked greatly by tidal effects, making me spin periods equal to me orbital periods.

Much of me primeval spin of me Earth has been transferred to the Moon, and to a smaller extent me same is true in me Neptune-Triton system. The other giant planets have probably not been braked appreciciably after their satellite systems formed. Even me transfer of angular momentum during satellite formation did not change their spins by more [164] than a few percent. In fact, for all the giant planets the total orbital momentum of the satellites is more than one order of magnitude smaller than the spin of the primary (table 2.1.2).

The spin isochronism (ch. 9) holds for bodies as different as small asteroids (mass~1018 g) and the giant planets (mass~1030 g). The conclusion from this is that the spin of most of the asteroids has not changed very much, at least not in a systematic way, since their formation.

To what extent the spins of the terrestrial planets have been braked is uncertain. The very slow rotations of Mercury and Venus may be due to a braking produced by solar tides (in combination with resonance effects; see ch. 13). The spin of Mars is unexpectedly slow. This cannot be due to tidal effects from its satellites because they are too small to take up an appreciable momentum. The large solar distance makes it unlikely that solar tides could be very efficient, but perhaps such an effect cannot be ruled out.

Pluto is reported to have a very slow rotation (6 days). We know too little about this planet to speculate about the factors influencing its spin.

The spin of the planets is discussed further in ch. 13.



We have reasons to believe that a series of dramatic events between 4 and 5 billion years ago produced the solar system. To reconstruct these events it is necessary to determine how the system has changed since its origin. Unless we are able to compensate for changes in the solar system after its formation, we have little chance of understanding the primordial processes. As we shall see later, there is a rapidly increasing body of chemical information relating to the formation of the solar system. But also from a dynamic point of view there is, as discussed above, a surprisingly large amount of data referring to the initial formation. With a few notable exceptions we find that the large bodies of the solar system (planets and satellites) are, at present, in a state that is not very different from that after they had just formed.

In the literature there are numerous suggestions of changes in the structure of the solar system. In some instances dramatic changes in the orbits of planets and satellites are proposed. Most of these suggestions would never have been published if the authors had investigated the dynamic implications.

Summing up, there is no indication of any major change in the planetary orbits. Of the satellites, only the Moon and Triton have undergone large orbital changes. Probably both were initially independent planets which [165] were later captured and brought to their present orbital position by tidal effects. There is no evidence that any of the normal (prograde) satellites have had their orbits appreciably changed.

Concerning the small bodies (asteroids, comets, meteoroids), the conclusion is different. As we have found, viscous effects including collisions are of importance in many cases, and this implies change in orbital elements. The retrograde satellites Jupiter 8, 9, 11, and 12 and the Saturnian satellite Phoebe belong to this category. Their capture into their present orbits may have taken place during the post-accretional phase, although it is perhaps more likely that it occurred late during accretion.

Suggestions have been made that the Martian satellites are recently captured asteroids. As they are the only bodies in the solar system that do not fit into the general matrix of ch. 23, it would certainly be agreeable from a theoretical standpoint to explain them in this way. This seems difficult, however. The presumably captured satellites mentioned above move in retrograde and highly eccentric orbits, drastically different from the low eccentricity and low-inclination orbits of the Martian satellites.