SP-345 Evolution of the Solar System

 

9. SPIN AND TIDES

 

[147] 9.1. TIDES

The spins of the celestial bodies contain information that is important for the study of the formation and evolutionary history of the solar system. When the celestial bodies formed by accretion this process gave them a certain spin; this will be discussed in ch. 13. There are reasons to believe that many bodies (e.g., the asteroids and the giant planets) still have essentially the same spin as they did immediately after their accretion. In many other cases the spin has been more or less drastically changed. This applies to all satellites, to the Earth and, to some extent, to Neptune also.

The main effect producing changes in the spins is likely to be tidal action by which the spinning of a body is braked. The theory of the terrestrial tides, as produced by the Moon and the Sun, has been developed especially by Jeffreys (1962) and by Munk and MacDonald (1960). The latter authors state (p. 15) that "there are few problems in geophysics in which less progress has been made." Even if this statement overestimates progress in other fields, it shows what diffficult problems the tddes present.

For our purpose we are interested not only in the terrestrial tides but also in the tides on other celestial bodies. The internal structure of celestial bodies is almost unknown and therefore very little about tidal effects upon these bodies can be theoretically established. We have to look for possible effects on the orbits of satellites to make any conclusions.

 

9.2. AMPLITUDE OF TIDES

Let us first discuss an idealized case of two homogeneous fluid bodies. Suppose that a secondary or companion body with radius Rsc is orbiting around a central or primary body with radius Rc. The densities of the bodies are [Greek letter] capital theta subscript c
and [Greek letter] capital theta subscript sc
, the masses are mathematical equation andmathematical equation
, and the distance between their centers of gravity is r. The gravitational attraction of Msc deforms the spherical shape of Mc so that its oblateness becomes

 

mathematical equation
(9.2.1)

 

[148] a formula which is a good approximation for r >> Rc (far outside the Roche limit). The height hc of the tides is

 

mathematical equation
(9.2.2)

 

Similar expressions hold for Msc:

 

 

mathematical equation
(9.2.3)

 

mathematical equation
(9.2.4)

 

Table 9.2.1 shows some typical examples. For the satellites of Jupiter and Saturn, mathematical symbol is put equal to 1.

 


TABLE 9. 2. 1. Tidal Effects Between Central Bodies and Their Secondary Bodies, in Terms of Oblateness mathematical symbol and Height of Tide
h for Each Body (Idealized Case)

Central body

Secondary

Central body

Secondary

mathematical symbol

hc (cm)

mathematical symbol

hsc (cm)

.

Earth

Moon

21 X 10-8

67

2.8 X 10-5

.25 X 104

Jupiter

Io

8.0 X 10-8

290

4.9 X 10-3

44 X 104

Europa

1.2 X 10-8

47

1.2 X 10-3

9.5 X 104

Ganymede

1.5 X 10-8

54

.30 X 10-3

4.0 X 104

Callisto

.23 X 10-8

8.4

.055 X 10-3

.69 X 104

Saturn

Mimas

.20 X 10-8

6.1

34 X 10-3

39 X 104

Enceladus

.15 X 10-8

4.3

16 X 10-3

22 X 104

Tethys

.81 X 10-8

24

8.5 X 10-3

26 X 104

Dione

.13 X 10-8

3.8

4.0 X 10-3

8.3 X 104

Titan

.78 X 10-8

23

.12 X 10-3

1.4 X 104

Neptune

Triton

170 X 10-8

2100

4.2 X 10-4

3.9 X 104

Sun

Mercury

11 X 10-13

38 X 10-3

.17 X 10-7

200

Venus

2.4 X 10-13

8.5 X 10-3

.027 X 10-7

81

Earth

1.1 X 10-13

3.9 X 10-3

.96 X 10-7

3.1

Jupiter

2.6 X 10-13

8.9 X 10-3

.029 X 10-7

10


 

Calculations based on eqs. (9.2.1-9.2.4); data from tables 20.5.1 and 2.1.2 and Newburn and Gulkis (1973). For the satellites of Jupiter and Saturn mathematical symbol
is set equal to one.

[149] As shown by these examples, the tides produced by a secondary body on a primary body are very small. In fact the oblateness mathematical symbol
never exceeds 10-6. In contrast, the satellites are strongly deformed, withmathematical symbol
of the order 10-3. If they are close to the Roche limit, eq. (9.2.3) does not hold. At the Roche limit, the tides become infinite.

Equations (9.2.1) and (9.2.3) can be generalized to rigid bodies by the introduction of a correction factor containing the rigidity (see, for example, Jeffreys, 1962; Munk and MacDonald, 1960).

 

9.3. TIDAL BRAKING OF A CENTRAL BODY'S SPIN

If a homogeneous fluid body of negligible viscosity is a spinning central body, its secondary will produce tidal bulges located on the line McMsc (fig. 9.3.1). If the viscosity of Mc is finite, the tidal bulges are displaced through an angle [Greek letter] epsilon, due to the time lag caused by viscous effects. The internal motions in the body are associated with an energy dissipation w (ergs/sec). The energy is drawn from the spin of the body (i.e., the spin is braked). As no change is produced in the total angular momentum of the system consisting of the spinning central body and the tide-producing secondary body, spin angular momentum is transferred to the orbital angular momentum of the secondary body.

The value of w depends on the physical state of the body and on the amplitude of the tides.

Suppose that the tidal bulge is displaced at an angle [Greek letter] epsilon
in relation to the tide-producing body (see fig. 9.3.1). A quantity Q defined by mathematical equation
(in analogy with what is customary in treating losses in electric circuits), is then often used. This formalism is misleading because it....

 


FIGURE 3.1.- Classical but inadequate model of momentum transfer due to tides.

FIGURE 3.1.- Classical but inadequate model of momentum transfer due to tides. The force of attraction between the satellite Msc and the near tidal bulge a exceeds that between Msc and b; a component of the net torque retards the rotation of the planet Mc and accelerates the satellite in its orbit. The actual situation in the case of the Earth is illustrated in fig. 9.4.1. In the case of Mars, Jupiter, Saturn, and Uranus, the angle e is probably negligible.

 

[150] ....gives the impression that each body has a characteristic constant Q. In reality Q (as well as [Greek letter] epsilon
) depends both on the frequency and on the amplitude. The amplitude dependence of the tidal braking is in general very large (Jeffreys, 1962) so that Q decreases rapidly with the height of the tides. Hence, it is not correct to assign a certain Q value to each celestial body. As shown by Jeffreys (1962), the relation between the solar tides and lunar tides on the Earth is very complicated, and the Q value of the Earth is different for these two tides. This difference is even greater if the tidal amplitudes are very different.

 

9.3.1. Fluid Body

Seen from the coordinate system of the spinning central body, the tidal deformation corresponds to a standing wave. The fluid motion, which in a nonstructured body is associated with this wave, is of the order

 

mathematical equation
(9.3.1)

 

where [Greek letter] capital omega
is the angular velocity of the central body and, for a spin period of mathematical equation
. For the case of tides produced on one of the giant planets by a satellite, we have [Greek letter] tau subscript c
= 10 hr = 3.6 X 104 sec; mathematical symbol = 10-7 and Rc = O.5 X 1010 cm, and, consequently, v~0.1 cm/sec. It seems highly unlikely that such low velocities can produce any appreciable dissipation of energy even over a very long period of time. (The order of magnitude of the energy dissipation with laminar flow is mathematical equation
ergs/sec where the viscosity,[Greek letter] eta
~10-2 poise for water. With R = 0.5 X 1010 cm and v = 0.1 cm/sec we obtain w = 5 X 105 ergs/sec.)

If instead we evaluate eq. (9.3.1) for the case of a satellite of a giant planet (= 1O hr, = 10-3, Rsc =. 5 X 108 cm), we find v~20 cm/sec.

 

9.3.2. Solid Body

In a small solid body (asteroid-sized), only elastic deformations are produced with minimum of energy dissipation. In satellites which are so large that their rigidity does not prevent deformations (lunar-sized bodies) these may often be nonelastic, and, hence, associated with big energy losses.

As far as is known, all satellites have spin periods equal to their orbital periods. If a planet is a fairly homogeneous solid body, it probably experiences negligible tidal braking. The deformations are of the order mathematical equation
~10-7 and may be purely elastic. In this range, deformation forces are far below the yield limit of most materials.

 

[151] 9.3.3. Structured Bodies

The most difficult case occurs when the body has a complicated structure involving fluid layers of different densities. The Earth is characterized by this type of layering, and in spite of all investigations we still are far from complete understanding of tidal braking of the terrestrial spin. Most of the dissipation of energy takes place in shallow seas, at beaches and regions near the shores. Hence, a knowledge of the detailed structure of a body is necessary in order to reach any conclusions about the tidal retardation of its spin velocity.

 

9.4. SATELLITE TIDAL BRAKING OF PLANETARY SPINS

The Earth-Moon system is the only system where we can be sure that a significant tidal braking has taken place and is still taking place. According to the elementary theory, the Moon should produce tidal bulges in the oceans (as in fig. 9.3.1); when the Earth rotates, these would remain stationary. Because of the viscosity of the water, the relative motion produces an energy release that brakes the spin of the Earth. At the same time, the tidal bulge is displaced a phase angle [Greek letter] epsilon
in relation to the radius vector to the Moon. This produces a force that acts in the direction of the lunar orbital motion: Hence, one would expect the Moon to be accelerated. However, since the force transfers angular momentum to the Moon, the lunar orbital radius increases, with the result that the lunar orbital period also increases. The paradoxical result is that the accelerating force slows down the lunar orbital velocity.

The theory of tidal bulges which is presented in all textbooks has very little to do with reality. The observed tides do not behave at all as they should according to the theory. Instead, the tidal waves one observes have the character of standing waves excited in the different oceans and seas which act somewhat like resonance cavities (fig. 9.4.1).

Even if the tidal pattern on the Earth is very far from what the simple theory predicts, there is no doubt that a momentum transfer takes place between the Earth and the Moon. The effect of this has been calculated by Gerstenkorn (1955), MacDonald (1966), and Singer (1970). According to these and other theories (Alfven, 1942, 1954), the Moon was originally an independent planet that was captured either in a retrograde or in a prograde orbit.

There is considerable doubt as to the extent to which the models are applicable to the Earth-Moon system (see Alfvén and Arrhenius, 1969, and ch. 24). Resonance effects may invalidate many details of the models.

 


[
152]

FIGURE 9.4.1a.- Phase relations of tides in the Pacific and Atlantic oceans.

FIGURE 9.4.1a.- Phase relations of tides in the Pacific and Atlantic oceans. The map shows the cotidal lines of the semidiurnal tide referred to the culmination of the Moon in Greenwich. The tidal amplitude approaches zero where the cotidal lines run parallel (such as between Japan and New Guinea). Much of the tidal motion has the character of rotary waves. In the south and equatorial Atlantic Ocean the tide mainly takes the form of north-south oscillation on east-west lines. This complex reality should be compared to the simple concept which is the basis for existing calculations of the lunar orbital evolution and which pictures the tide as a sinusoidal wave progressing around the Earth in the easterly direction (dot-and-dashed curve in fig. 9.4.1b). (From Defant, 1961.)

 


[
153]

FIGURE 9.4.1b.- Tidal amplitude on the Atlantic coasts as an example of the actual amplitude distribution in comparison with the simple Laplacian tide concept.

FIGURE 9.4.1b.- Tidal amplitude on the Atlantic coasts as an example of the actual amplitude distribution in comparison with the simple Laplacian tide concept. The curves show the average range at spring tide of the semidiurnal tide as a function of latitude. The solid curve represents the tide on the western side of the Atlantic Ocean; the dashed curve, the eastern side of the Atlantic Ocean and the dot-and-dashed curve, the Laplacian tide. In the comparison with the (much less known) open ocean amplitudes, the coastal amplitudes are increased by cooscillation with the oceanic regions over the continental shelves. The distribution illustrates further the facts that tidal dissipation is governed by a series of complex local phenomena depending on the configuration of continents, shelves, and ocean basins, and that the theoretical Laplacian tide obviously cannot serve even as a first-order approximation. (From Defant, 1961.)

 

[154] There seems, however, to be little reason to question the main result; namely, that the Moon is a captured planet, brought to its present orbit by tidal action. Whether this capture implies a very close approach to the Earth is unresolved. This problem will be discussed in more detail in ch. 24.

The Neptune-Triton system is probably an analog to the Earth-Moon system. The only explanation for Neptune's having a retrograde satellite with an unusually large mass seems to be that Triton was captured in an eccentric retrograde orbit that, due to tidal effects, has shrunk and become more circular (McCord, 1966).

As Neptune has a mass and a spin period similar to those of Uranus, it is likely to have had a satellite system similar to that of Uranus (see sec. 23.8). The capture of Triton and the later evolution of its orbit probably made Triton pass close to the small primeval satellites, either colliding with them or throwing them out of orbit. Nereid may be an example of the latter process (McCord, 1966).

The satellites of Mars, Jupiter, Saturn, and Uranus cannot possibly have braked these planets by more than a few percent of the planetary spin momenta. The total orbital angular momentum of all the satellites of Jupiter, for example, is only 1 percent of the spin momentum of Jupiter (see table 2.1.2). This is obviously an upper limit to any change the satellites can possibly have produced. As we shall find in ch. 10, the real effect is much smaller, probably completely negligible.

 

9.5. SOLAR TIDAL BRAKING OF PLANETARY SPINS

Again, the Earth is the only case for which we can be sure that solar tides have produced, and are producing, an appreciable change in spin. How large this change is seems to be an open question. The effect depends on the behavior of the tides on beaches and in shallow seas, as do the effects of lunar tides on Earth.

It has been suggested that tides have braked the spins of Mercury and perhaps Venus so much that they eventually have been captured in the present resonances (see sec. 8.8 and Goldreich and Peale, 1966 and 1967). This is a definite possibility and implies that initially these planets were accreted with an angular velocity that was larger than their present angular velocity, perhaps of the same order as other planets (fig. 9.7.1).

However, as discussed in ch. 8, the orbit-orbit resonances are probably not due to tidal capture, but are more likely to have been produced at the time when the bodies were accreting. In view of this, the question also arises whether the spin-orbit resonances of Mercury, and of Venus, if it is in resonance, were produced during their accretion. It seems at present impossible to decide between this possibility and the tidal alternative. The latter [155] would be favored if there had ever been shallow seas on these planets. We have yet no way of knowing this in the case of Venus; for Mercury the apparently preserved primordial cratered surface would seem to rule this out.

It seems unlikely that solar tides have braked the spins of the asteroids or of the giant planets to an appreciable extent.

 

9.6. TIDAL EVOLUTION OF SATELLITE ORBITS

Goldreich and Soter (1966) have investigated the possible tidal evolutions of the satellite systems. They have pointed out that, where pairs of satellites are captured in orbit-orbit resonances, both the satellites must change their orbits in the same proportion. They have further calculated the maximum values of the tidal dissipation of energy (in their terminology the minimum (Q values) that are reconcilable with the present structure of the satellite systems. There seems to be no objection to these conclusions.

Goldreich and Soter have further suggested that the maximum values of energy dissipation are not far from the real values and that tidal effects have been the reason for satellites being captured in resonances. This problem has already been discussed in ch. 8. The conclusion drawn is that small librations in some of the resonances cannot be understood as tidal effects.

Further, we observe resonances in the planetary system that certainly cannot have been produced in this way, so that it is in any case necessary to assume a hetegonic mechanism for production of some resonance captures. Finally, the structure of the Saturnian rings demonstrates that Mimas' orbit cannot have changed by more than 1 or 2 percent since the formation of the Saturnian system (sec. 18.6).

Hence, present evidence seems to speak in favor of the view that, with the exception of the Moon and Triton, no satellite orbits have been appreciably changed by tidal action.

 

9.7. ISOCHRONISM OF SPINS

Photometric registrations of asteroids show intensity variations that must be interpreted as due to rotation of a body with nonuniform albedo or nonspherical shape. Several investigators (e.g., Taylor, 1971) have measured the periods of axial rotation of some 30 to 40 asteroids and have found no systematic dependence on the magnitudes of the asteroids. In fact, as is shown in fig. 9.7.1 and table 9.7.1, almost all asteroids have periods that deviate by less than 50 percent from an average of 8 or 9 hr. It appears that this result is not due to observational selection.

Regarding the planets, we find that the giant planets as well have about the same period. It has always struck students of astronomy that the axial....

 


[
156]

FIGURE 9.71.- Periods of axial rotation for some asteroids and some of the planets in relation to their masses. (

FIGURE 9.71.- Periods of axial rotation for some asteroids and some of the planets in relation to their masses. (From Alfvén, 1964.) The rotation period of Pluto is not well known and the rotation periods of Mercury and Venus are influenced by resonance effects; these three planets are thus not represented in the figure. The value of rotation period for the Earth is that prior to capture of the Moon. Data for asteroids is taken from table 9.7.1 and data for the planets from table 21.1. From the graph one concludes that spin period is not a function of mass. indeed, most of the spin periods all fall within a factor of two of 8 hr. We refer to this similarity of periods of rotation as the law of isochronous rotation or the isochronism of spins.

 

....rotations of Jupiter, Saturn, and Uranus are almost equal. The period of Neptune is somewhat longer (15 hr), but a correction for the tidal braking of its retrograde satellite reduces the period at least somewhat (see McCord, 1966). For the Earth we should use the period before the capture of the Moon; according to Gerstenkorn, that period was most likely 5 or 6 hr (see Alfvén, 1964).

Hence we find the very remarkable fact that the axial period is of the same order of magnitude for a number of bodies with very different masses. In fact, when the mass varies by a factor of more than 1011-i.e., from less than 1019 g (for small asteroids) up to more than 1030 g (for Jupiter)- the axial period does not show any systematic variation. We may call this remarkable similarity of rotational periods the spin isochronism.

Obviously this law cannot be applied to bodies whose present rotation is regulated by tidal action (planetary satellites) or captured resonances (Mercury and perhaps Venus; see sec. 9.3). Excepting such bodies, the only body with a rotation known to be far from the order of 8 hr is Pluto, which rotates in 6 days. Mars ([Greek letter] tau
= 25 hr) and Icarus ([Greek letter] tau
= 2 hr) each deviate by a factor of three.

[157] In ch. 13 mechanisms producing the isochronous rotation are discussed; with this as background a more detailed analysis of planetary spins will be given (sec. 13.6).

 


TABLE 9.7.1. Periods and Magnitudes of Asteroids.

Asteroid

Name

Magnitude

Rotation period (hr)

.

1

Ceres

4.11

9.07

3

Jumo

6.43

7.21

4

Vesta

4.31

5.34

5

Astraea

8.00

16.80

6

Hebe

6.70

7.74

7

Iris

6.84

7.13

8

Flora

7.48

13.6

9

Metis

7.27

5.06

11

Parthenope

7.78

10.67

13

Eeeria

7.97

7 04

15

Eunomia

6.29

6.08

16

Psyche

6.89

4.30

17

Thetis

8.69

12.27

18

Melpomene

7.79

14

19

Fortuna

8.35

7.46

20

Massalia

7.48

8. 09

21

Lutetia

8.68

6 13

22

Kalliope

7.48

4.14

23

Thalia

8.34

6.15

24

Themis

8.18

8. 5

27

Euterpe

8. 56

8. 50

28

Bellona

8.15

15.7

29

Amphitrite

7.26

5 38

30

Urania

8.78

13.66

39

Laetitia

7.41

5 13

40

Harmonia

8.45

9.13

43

Ariadne

9.18

5 .75

44

Ngsa

8.02

6.41

51

Nemausa

8.66

7.78

54

Alexandra

8.82

7.05

61

Danae

8.77

11.45

110

Lydia

8.80

10.92

321

Florentina

11.38

2.87

349

Demhowska

7.29

4.70

354

Eleonora

7. 56

4.27

433

Eros

12.40

5.27

511

Davida

7.13

5.17

532

Herculira

7.98

18.81

624

Hektor

8.67

6.92

1566

Icarus

17.55

2.27

1620

Geographos

15.97

5.22

(Data from Gehrels, 1971.)

 

[158] 9.8. CONCLUSIONS FROM THE ISOCHRONISM OF SPINS

Concerning the mechanism that produces the similarity of spin periods in most of the tidally unmodified bodies, the following conclusions can be drawn:

(1) The similarity of the spin periods cannot be produced by any process acting today. For example, we cannot reasonably expect that the rotation of Jupiter is affected very much by any forces acting now.

(2) The equality of the spin periods cannot have anything to do with the rotational stability of the bodies. The giant planets, for example, are very far from rotational instability. It is unlikely that one could find a mechanism by which the present isochronism of spins can be connected with rotational instability during the prehistory of bodies as different as a small asteroid and a giant planet.

(3) Hence, the spin isochronism must be of hetegonic origin. All the bodies must have been accreted by a process with the characteristic feature of making their spin periods about equal, no matter how much mass is acquired. There are accretion processes that have this property (see ch. 13).

(4) The spin isochronism further shows that the asteroids cannot derive from a broken-up planet. If a planet explodes (or if it is disrupted in some other way), we should expect an equipartition of the rotational energy among the parts. This means that, on the average, the periods of axial rotation of the smallest asteroids should be much smaller than those of the larger asteroids. This is in conflict with the observed statistical distribution.

(5) The braking of the axial rotation of celestial bodies has not been very significant since their accretion. A braking produced by an ambient uniform viscous medium ought to lengthen the period of a small body much more than the period of a larger body. The fact that asteroids as small as some 10 kilometers rotate with the same period as the largest planets indicates that even such small bodies have not been braked very much since they were formed. In this essential respect, the solar system seems to be in the same state now as it was when it was formed. Thus, detailed analysis of the present state of the solar system can yield insight into hetegonic processes.


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