Geodynamics and state of the earth's interior

JOURNAL OF GEODYNAMICS 1, 79--100 (1984) 79

G E O D Y N A M I C S A N D STATE OF THE EARTH'S INTERIOR

HEINZ STILLER i SIEGFRIED FRANCK ~ and DIEDRICH MOHLMANN 2

Zentralinstitut fiir Pkysik der Erde, 15 Potsdam, G.D.R.

2 lnstitut fiir Kosmosforschung, I 199 Berlin, Rudower Chaussee 5, G.D.R.

(Received December 28, 1983: accepted January 18, 1984)

ABSTRACT

Stiller, H.. Franck, S. and M6hlmann, D., 1984. Geodynamics and state of the earth's interior. Journal of Geodynamics, 1: 79-100.

The paper is concerned with the connection of planetary geodynamics and properties of the material

inside the Earth. The notion "state" is considered both in the local and in the global sense. We discuss

various interpretations of phase boundaries in the Earth's mantle and propose a new idea on the

description of seismic discontinuities. The present thermal state of the Earth's core is investigated by

using data about the attenuation of seismic waves. Because of some peculiarities in the phase diagram of

the favoured core material Fe-S we come to the conclusion that the dynamics in the outer core may be

characterized by the motions of small droplets. The corresponding rate of release of gravitational

potential energy is enough to drive a geodynamo with an average toroidal field of mean size. According

to our point of view, global planetary evolution started from an originally homogeneous Earth. The

differentiation of core and mantle started in upper layers. It is an important source of energy for all

following processes.

INTRODUCTION

The investigation of the properties of planetary matter at high pressures and temperatures, as they occur in the interior of the planets, is a modern field of science, that utilizes results and methods from geo- and space-sciences, physics and chemistry. It is the only way to come to direct information about the situation in the deep interior of our planet.

Another source of information are the observations of seismic waves (including normal vibrations), gravity, geomagnetism and heat flow at the surface of the Earth. The interpretation of these phenomena makes it necessary to have an idea about their origin in the interior. This is also denoted as the so-called inverse problem. It follows that even the combination of the most modern results of all these branches makes it not possible to find a unique picture about the deep interior, but the possible distributions of

0264-3707/84/$3.00 c 1984 Geophysical Press Ltd.

80 STILLER, F R A N C K A N D M O H L M A N N

Global Tectonics [

Earth's Interior Investigations Terrestrial Planets

fracture mechanics mu[ticomp, systems non- hydrostatic pressure partial melting

po[ymorph, phase transitions, convection, bonding

equation of state thermal profi(e melting meta[tization dynamo

© core formation

Fig. I. Schematic representation of the relation between physics of the Earth's interior, high pressure research and planetary evolution in connection with global tectonics.

physical parameters may be restricted to a narrow range and are denoted as mechanical Earth models.

The International Lithosphere Project is concentrated on investigations of the uppermost parts of the Earth. But we know that all important driving driving mechanisms of processes in the crust have their origin in the deeper interior. Therefore, also for such aims we have to study the interaction between crust and mantle and mantle and core, respectively.

Another aspect of this problem is the planetary evolution and its connection with global tectonics. In Fig. 1 we have tried to represent all these lines schematically. In the present paper we want to review some results on this subject. The review is not complete, but contains mainly our own work in this field.

PHASE TRANSITIONS IN THE EARTH'S INTERIOR

General considerations

Phase transitions belong to the most interesting phenomena used for the interpretation of geophysical observations. Discontinuities of physical

G E O D Y N A M I C S AND STATE OF THE EARTH'S INTERIOR 81

parameters at chemical boundaries are only determined by the properties of the adjoining media, but at phase boundaries there may be completely new effects connected with singularities of thermodynamical potentials.

At phase transitions in the Earth's interior we find always jumps from a lower to a higher density and sometimes also to higher co-ordination numbers of certain cations. If one tries to correlate results of experimental high pressure investigations with seismological observations (e.g. velocity distributions), one has to find theoretical models that describe the relation between thermodynamical properties (stability of a phase, density jump) and elastical properties, such as wave velocities. Such a relation should, in principle, follow from lattice dynamics if one knows exactly the behaviour of the lattice particles during the transition. But there are a lot of unknown parameters and therefore we use phenomenological models.

In the following we present a model that was applied by Stiller and Franck (1980a) to the interpretation of seismic discontinuities.

Our starting point is a thermodynamical potential, e.g. free enthalpy G(T, p): G(T, p) = F(T, V) + p V (1)

G(T, p) is also sometimes called 'Gibbs' free energy. Besides temperature (T) and pressure (p) we introduce also the so-called order parameter r /and strain tensor ui i G = G(T, p, rh u~j). We assume that G may be divided into several parts: a part from the phase transition, contribution from elastical properties, interaction between elastical degrees of freedom and the phase transition, and a rest term which contains other contributions, such as electronic terms and the electron-phonon interaction

G(T, p, tl, uii) = Gph(T, p, rl) + G,,l(u~i) + Gi,,t(rh uii) + Go(T, p) (2)

0 ( 1 ] )

l

G (r I )

P = Po ~< Pc

G(T I )

P>>Pc

o o

Fig. 2. Stability analysis with help of the order parameter for a pressure induced phase transition. At

P - P0 the phases are in equilibrium, Pc is the limit of stability of the low-pressure phase.

82 STILLER, FRANCK AND MOHLMANN

The term Gpk(T, p, q) must be constructed in such a way that it contains information about the stability of both phases during a change of thermodynamical conditions. The order parameter r/ was introduced by Landau (see Landau and Lifschitz, 1971) to describe the order of the lattice particles on one side of the transition and it may be handled as a thermodynamic quantity like temperature and pressure. In figure 2 we show an example of a stability analysis with help of the order parameter.

Seismic discontinuities at phase boundaries

The density dependence of elastic velocities is often discussed with the aid of Birch's empirical law (Birch, 1961):

Vp = a(]~) + bp (3)

where M is the mean atomic weight and p is the density. However, Birch himself has given some experimental data for phase transitions at which the denser phase has the lower velocity Vp and so (3) is not valid. In our model we wish to investigate in principle the possible behaviour of vp and v, at a pressure- or temperature-induced phase transition. As can be found in detail elsewhere (Stiller and Franck, 1980a), it is possible to calculate correction terms AC,m to the elastic constants with help of (2). The corresponding seismic velocities can be easy calculated by the wellknown formulae

2 =-_ (C,? q_ 2e44)/fl (4) Up

~2 l;s = C44/P (5)

Vp , V s

--• (b~

(a)

[ =

Po P

Fig. 3. Three types of possible seismic discontinuities at phase transitions.

GEODYNAMICS AND STATE OF TttE EARTH'S INTERIOR 83

The resulting possible seismic discontinuities at a phase transition are represented in figure 3. With help of the 3 types of seismic discontinuities found in our investigation, it is possible to interpret both velocity distributions in the Earth's interior and experimental results of ultrasonic investigations (see Stiller and Franck, 1980a). All attempts to make quantitative estimates with help of the presented model arrive at the problem of the definition of an order parameter for the various geophysically relevant phase transitions. There is hope that experimental investigations and theoretical studies will help to handle this problem.

On the problem of ferroelectricity in the mantle

According to the prediction of Ringwood (1975) the major components of the Earth's mantle, olivine and pyroxene will assume the perovskite structure in then deeper mantle as a form of highest density.

Based on this idea and on experimental high pressure investigations it was proposed (see e.g. Schloessin and Timco, 1977; Litov and Anderson, 1978) that the perovskite structure in the mantle is connected with ferroelectric phenomena causing elastic, dielectric, and electric anomalies. The possibility of the occurence of ferroelectricity at high pressures and temperatures within the framework of microscopic theory was investigated by Franck and Stiller (1980).

The ABO 3 perovskite unit cell is shown in figure 4. A ferroelectric state may arise if the positive ions (e.g. A 2+, B 4+ ) are shifted in opposite direction

/

Q) A

®B • 0

Fig. 4. Displacement of atoms at the ferroelectric phase transition in perovskites.


to the oxygens so that the centres of positive and negative charges are different. In this way each unit cells has a nonvanishing dipole moment and one may observe a macroscopic spontaneous polarization.

From the view-point of lattice dynamics the phenomenon of ferroelectricity in perovskites may be explained with help of a dipole-active soft mode at zero wave vector. Approaching the phase transition temperature T,. (Curie temperature) from the paraelectric phase the soft mode vibration freezes in and thus causes the corresponding shifts of positive and negative ions. Starting from a microscopic Hamiltonian it has been shown (see Vaks, 1973) that the Curie temperature T,. is given by

K~ T c = F Va - V~ (6) v,

where V~, stands for the long-range dipolar attraction between ions in various unit cells and Vr describes the short-range repulsion between ions within the unit cell. K R is Boltzmann's constant. The quantity F depends on certain force constants and may be approximated by a simple relation to mean atomic mass in the unit cell and a characteristic elastic velocity.

Franck and Stiller (1980) found out that according to microscopic theory ferroelectricity should not exist at thermodynamic conditions such as those below the 670 km~discont inui ty of the mantle (T~> 2 × 103 K, p ~> 25 GPa). For reaching Curie temperatures in this range, the long-range interaction must predominate over short-range interaction much more than in all cases observed up to now. At high pressures it is even much more unlikely to reach such conditions because high Curie temperatures may not arise by compression but by extension of the lattice. Therefore we prefer the usual interpretation of mantle discontinuities as given for example by Zharkov and Yrubitsyn (1980).

STATE OF P L A N E T A R Y MATTER IN THE EARTH'S CORE

Core anelast ic i ty a n d thermal s tate

According to our general view the Earth's core may be divided into a liquid outer core and a solid inner core. This fact results from the observations that transverse seismic waves vanish in the outer core but not in the inner one. Beside such basic knowledge derived from the distribution of seismic velocities also data of seismic wave attenuation may provide us in formation on the state of matter in the core of the Earth.

The seismic wave attenuation coefficient Q ~ describes the dissipating part of mechanical wave energy per wave length. From seismological observations

G E O D Y N A M I C S A N D STATE OF THE EARTH'S INTERIOR 85

Doornbos (1974) found the distribution of the seismic P-wave attenuation coefficient in the Earth's core. His results are shown in figure 5, curve 1.

It is interesting that there is an appreciable asymmetric peak at the inner core boundary (ICB) and that the attenuation is higher in the solid inner core than in the liquid outer core. An explanation of such a behavior was given by Stiller et al. (1980).

In the framework of Landau theory melting at the ICB can be characterized by the vanishing of a long range order parameter. A P-wave, running through the inner core, leads to periodic compressions and dilatations. So the order parameter comes from its equilibrium value to non-equilibrium states and relaxes back. The relaxation time is proportional to ( T m - - T ) ' , where T,,, is the melting point. Every relexation process to the thermodynamical equilibrium is an irreversible process with increase of entropy and consequently energy dissipation. For a quantitative analysis one needs a certain distribution of temperature and melting temperature within the inner core. Curve 2 of figure 5 was computed with help of the Higgins and Kennedy

Q-I 10 3

5

4

I

2 ~

\

1 I I I I 1 1 1 I 1 [ I - t I 1200 1000 800 600 400 2o0 JCB JCB

,- 1 /

3 J 0

Fig. 5. Seismic attenuation coefficient Q 1 in the Earth's core: 1 data of Doornbos (1974). 2 relaxation

model with temperature distribution of Higgins and Kennedy (1971), 3-relaxation model with

temperature distribution of Stacey (1977). 4-contribution of attenuation in inhomogeneous matter.


(1971) temperature distribution while curve 3 refers to the thermal model of Stacey (1977). According to Higgins and Kennedy (1971)the temperature in the centre of the Earth is only about 15 deg. below the corresponding melting temperature. We see that the corresponding curve 2 approximates the data of Doornbos (1974) very well.

From our investigation we may conclude that the whole inner core is in a thermodynamical state very near to its melting point. The inner core is so to say already "soft" and this is why seismic wave attenuation is so high.

Dynamics of the outer core

There have been discussed at least three energy sources for the geodynamo: precession, thermal convection (driven by radioactivity and/or latent heat) and compositional convection. Compositional or chemical convection was proposed by Braginsky (1963). He assumed that gravitational energy release by the solidification of the inner core is the primary source of energy which sustains the geodynamo. This idea was confirmed in recent studies, especially

one tiquicl

-1--,

4 - ,

/ /

two tiquids

solid iron + liquid

solid

I I I 5 10 15

Fe S ( wt. °/o )

\ \

\ \ \

k/

Fig. 6. Phase diagram of the system Fe S at high pressures and temperatures.

GEODYNAMICS AND STATE OF THE EARTH'S INTERIOR 87

by Loper and Roberts (1981). The solidification of the inner core results from a gradual cooling of the entire Earth. It is generally accepted that the Earth's core is not pure iron but contains a significant fraction of lighter material. There are geochemical and cosmochemical investigations that justify the presence of sulfur in the core of the Earth and other terrestrial planets. According to Verhoogen (1973) the Fe-S phase diagram at high pressure should be similar to that of the system Sb-S at normal pressure. Therefore, our investigation is based on the phase diagram given in figure 6. The most interesting feature of this diagram is the existence of a liquid immiscibility field between the Fe-endmember and the eutectic. Above the liquidus curve there exist two liquis: a heavy liquid with low sulphur content of 2.6 wt. % S and a lighter liquid with 15 wt. % S.

In the following we present a model of the dynamics in the outer core, first proposed in this form by Franck (1982): At more than 4 X 09 y (4Ga) ago the whole Earth's core was in a liquid state. But after that the slow cooling of the entire Earth led to freezing of core material. As explained in detail, for example, by Loper and Roberts (1981), the solidification started in the centre of the Earth and so time inner core has been growing slowly to its present size. The core material is an alloy and the solid inner core that freezes from such an alloy has a smaller mass fraction of light material than the outer core. Referring to figure 6, we conclude that the inner core is pure metal because it freezes from the heavy liquid with only about 2.6 wt. % S. This property requires an upward flux of sulphur at the inner core boundary. Because of liquid immiscibility above the inner core boundary the sulphur may not alloy with the heavy liquid. The only possibility is nucleation of liquid droplets with sulphur content of about 15 wt. %.

The permanent flux of sulphur causes the droplets to grow up to such a size where buoyancy is strong enough to overcome drag forces. It was computed by Franck (1982) that the corresponding radius is R >_/6 × 10 5 m and that these droplets are also stable because of the stability condition R ~ 1 m. For simplicity we assume that ascension of the droplets occurs from the inner core boundary near to the core-mantle boundary with possible stable stratification at the top of the outer core. The transport of light material (droplets) up to the top of the outer core (see figure 7) leads to a rearrangement of core material connected with a release of gravitational potential energy at a rate of about P = 2.26 × 10 j~ W. This power is available to drive a dynamo.

There are at least three necessary conditions of hydromagnetic dynamo action: motions of liquid electrically conducting material in a rotating system. According to our knowledge, all these conditions may be also fulfilled in an outer core with ascending Fe-S droplets. As, for example, described by Stiller et al. (1975) the electrical conductivity of Fe-S changes from that of a low gap semiconductor to metallic already at low pressures of the order of kilobars.


A rough estimation of the size of the magnetic field can be found by equating P to the ohmic power dissipation P0 and using a formula given by Loper and Roberts (1981)

P0 i015 W = ~ X (Bo) 2 (7)

The result is an average toroidal field B a of about 1.5 × 10 ~ 2 T (or 150 G). Such a field could be classified as one of mean size.

stabty stratified tayer

ascending droplet

Fig. 7. Simple illustration of the process of inner core growing and ascending droplets in the outer core.

The influence of the Earth's rotation on the droplet motion is not taken into account.


PLANETARY EVOLUTION

The problem of the present internal structure of the Earth is directly connected with the field of planetary cosmogony. In the framework of so- called cold origin models, the Earth grows by accumulation of planetesimals. In the standard model of Schmidt (1959) planetesimals contain both silicates and iron in chondritic abundances and so the primitive Earth is homogeneous with inhomogenities on a scale of the order of magnitude of planetesimals (10 ~ ... 102km). The following process of differentation depends on the original temperature distribution in the Earth because it is very important if any component of planetary matter is liquid or solid at the corresponding pressure and what is the viscosity of the unmelted material. The original temperature distribution of the planet Earth which grows by accumulation of planetesimals my be calculated from the condition of energy conservation (Zharkov and Trubitsyn, 1980):

p G M ( r ) _ eo[ T ( r ) 4 - - T~] d t + pep[ T ( r ) - - TR] dr (8) r

with p-density of the accumulating bodies, G-constant of gravity, M(r)-mass of the growing Earth with radius r, a-Stefan-Boltzmann's constant, e-emissivity, Cp-Specific heat, TR-basic temperature of accumulating bodies. Equation (8)

E I ~ ' v ~ f

depth =

Fig. 8. Initial temperature distribution in the Earth (see Vitjazev and Majeva, 1980).


describes how the gravitational energy of infalling bodies is partly reemitted into space and partly converted to thermal energy which heats the material. A calculation with help of (8) leads to temperature distributions as shown in Figure 8. In the centre the temperature is relatively low. Melting temperature is reached in the upper mantle and in outer layers temperature is low because a nearly complete planet grows only very slowly (see also Coradini et al., 1983). Looking at figure 8, it is evident that the process of differentiation between iron and silicates starts in the upper mantle.

According to Elsasser (1963), at the beginning great blobs of iron (that is iron compounds as Fe-S) originate to a radius of some hundred kilometers. Because of a Rayleigh-Taylor-instability these blobs sink through the mantle and cause core formation (see figure 9). Vitjazev and Majeva (1977) assumed that the melted material (Fe-S) collects in a spherical shell that migrates down slowly by melting partly the underlying material (see figure 10). The silicates do not melt and form the mantle. The velocity of motion of a solid-liquid

Fig. 9. Model of core formation according to Elsasser.


/

\

Fig. 10. Model of core formulation according to Vitjazev and Majeva.

phase boundary is determined by the process of thermal diffusion. Therefore this process keeps on a long time, about 109 a. Taking into account that the ages of the oldest magnetized rocks are about 3.5 X 109 a, one should favour models with somewhat faster core formation. A third model of core formation is the scenario of Stevenson (1981) starting similar to the model of Vitjazev and Majeva but involving a spontaneous symmetry breaking with can occur on a time scale of hours (see figure 11). This rapid event is followed by a steady-state process of downward migration of iron blobs with typical sizes 1 ... 10 km. Up to now it is not possible to decide between the three scenarios shown in figures 9, 10, 11. But one can estimate the energy that is set free because of the core formation in an originally homogeneous Earth. Kalinin and Sergeeva (1977) calculated this energy as 2.68 X 10 38 erg, which would be enough to let rise the temperature of the whole Earth about some thousands degrees. But as can be shown by detailed investigations, the energy resulting from core formation is not equally distributed in the planetary interior but concentrated near the core. Therefore the temperature profiles immediately

92 STILLER. F R A N C K AND M O H L M A N N

following core formation show very steep gradients in the mantle. So it follows that there was a very vigorous thermal convection in the young Earth.

Also in the present state most of the heat in the Earth's core is the result of core formation. Therefore the geodynamics which is now observed results mainly from these early parts of the planetary evolution.

iron compounds

Jot mantte with convection

Fig. 11. Early stages of core formation in the model of Stevenson (1981).

GEODYNAM1CS AND STATE OF THE EARTH'S INTERIOR 93

PLANETOGONIC ASPECTS

As has been discussed in the foregoing part, there are great uncertainities in the time-scale of the accretional growth of the Earth. Therefore, results of models of the cosmogony of the planetary system have to be taken into account, to formulate appropriated initial and boundary conditions for the description of the evolution of the Earth. Vice versa geophysical results can give additional arguments and constraints to select among the various and numerous models of origin and evolution of the Solar system.

Further information about the planetogenic processes can be found from results of comparative planetology. Here it is of great importance that the Solar system has four developed systems of central body and satellites, namely the planetary system and the satellite systems of Jupiter, Saturn and Uranus. The postulate, that especially common and comparable properties and structures of these systems are of cosmogonic relevance, can be used then as a criterion to identify characteristic parameters of the formation processes and to separate them from peculiar or "individual" properties. Vice versa, the verification of the existence of such common and comparable characteristics is a strong argument in favour of the assumption of a comparable evolution of preplanetary and presatellite discs. Characteristics of the above described type will be used as "reconstruction characteristica'. They combine the ideas of the "hetegony principle" and the "actualistic principle" of Alfven and Arrhenius (1976) with the assumption of a "common formation principle" (Stiller. Yreder, M6hlmann, 1981).

The four "classical" (kinematic) reconstruction characteristica of the above defined regular satellites are used as guidelines in nearly all cosmogenic models for the planetary system: - -coplanar i ty of orbits in every system --circulari ty of orbits --parallelism of the spin of the central body and the orbital angular

momentum of its satellites --parallelism of orbital momentum and spin of the satellites

The great diversity in planetary cosmogonic models shows, that these classical characteristics are not a sufficient basis. Probably, no more can be concluded from these characteristics, than the existence of a remarkable thin preplanetary- or presatellite disc. Further "reconstruction characteristica'" are necessary to get more detailed information about thin discs (Stiller, Treder, M6hlmann, 1981). These shall be discussed now under the viewpoints of existing radial structures, properties of planetary rotation and necessary time scales.

One of the most suggestive radial structures in the planetary and the satellite systems is the mass-distribution m(r) or equivalently, the derived


"averaged density" (see Alfven, Arrhenius, 1976). It is remarkable that the mass-distribution in the different systems exhibits several analogies such as the separation of an inner and outer group with a clear division between them, and the position of the greatest individual mass in the inner part of the outer group. This can be understood as being caused by structural processes in the disc.

Evidence for a possible evolution to at least radial structures (instead of a "smooth" disc) can be found also in the existence of (structural) bands or rings in the Solar system, which eventually could give information about processes being able to structure a disc.

Further support for an early separation process results from the inter pretation of "isotope-anomalies" in extra-terrestrial matter, which teach, that the matter in the preplanetary disc can not have been mixed effectively, since a mixing would result, at least, in similar isotope distributions throughout the system.

The isotope anomalies can be understood by assuming (as we do) a very early separation of the yet inhomogeneous disc into non-interacting substructures, or by the action of a strong large-scale magnetic field, giving constraints to an effective mixing, or by local injections of preplanetary matter. It is interesting to note, that in the case of an early separated structure each substructure could be characterizable by its specific isotope-rtios, which, of course, can be modified by a further differentiating evolution.

Furthermore, the space-research proven fact, that the early lunar surface was molten, is an indication for a very rapid accretional growth (and not from sporadically moving planetesimals). This can be shown by estimating the lunar accretional mass flow m = 4nr2pv from the temperature of the molten lunar surface (M6hlmann, 1982) and assuming an equilibrium between gravitational energy gain 7Mrh/R and the heating of the impacting matter mcT and the radiative energy loss 47rar2T (M-lunar mass with radius r, M47rr~). For an order of magnitude estimation any initial relative velocity of the accreting matter can be neglected, giving for L, = (27M/r) ~2. Then the mass density p of the impacting mass flow can be shown to be between

10 5 K g m 3~<p~<10 4 K g m 3

This density of the directly accreting matter is by orders of magnitudes greater then the discussed above "averaged densities". This seems not to be realistic. Therefore, the necessary relatively high accretional densities can have existed only in substructures with higher than the averaged densities.

In connection with the possible evolution of a disc to the above described discrete structures (rings, bands) mathematical "laws" or "schemes" should be mentioned, giving algorithms for finding the approximate distances of the satellites from their central bodies (Nieto, 1972). These relations may be of


importance if they help to find physical processes leading to these assumed discrete structures. The value of such ordering schemes increases, if the postulated existence of yet unknown satellites can be veryfied. The object 1979 J2 in the Jovian system has been predicted on this way (M6hlmann, 1980). On the other side, these schemes become meaningless if they produce too many "missing satellites". Up to now, there exists no consequent derivation, based on a sound physical basis for one of these schemes, there is only a "fitting" of different physical scenarios, especially to the "Titius-Bode-law'" (v. Weizsficker, 1944, Prentice, 1978; Nieto, 1972), but the existence of common structure-schemes for the four systems strengthens the assumption of comparable formation processes.

Summarizing, it shall be noted that there is clear evidence for an early ordered structure of the preplanetary matter instead of a stochastic evolution towards the actually observable structures in the Solar system and its satellite systems. Further information about the preplanetary phases can be found from properties of the rotation of planets.

Most of the planets and smaller bodies in the Solar system have comparable rotational properties. There is, at first, the above mentioned prograde orien- tation; spin and orbital angular momentum are parallel. Secondly, the rotation periods of most of the free rotating bodies vary only in a relatively small intervall (about 10hours). This second fact has been called Spin Isochronismus (Alfven, Arrbenius, 1976).

These properties can be explained within the frame of "gravitationally focussed accretion" (M6hlmann, 1983). These results, at least, for the value of the angular velocity ~o = K/~,2, where K is a numerical constant being really of that order of magnitude to reproduce the observed values of planetary rotation, 17 is the density of the planet. This angular velocity does not depend on the mass or the radius of the planet. It has nearly the same value for all objects since/~ does not change essentially.

In this connection it must be noted, that this process of gravitationally focussed accretion must have been relatively fast. This is due to the spreading of the orbits of all those particles which were disturbed but not accreated. To avoid this dispersion of the particles throughout the preplanetary disc, the above described process must be effective on a short time scale. This conclusion agrees with the scales, derived for the lunar growth from the discussed above fact of the existence of an early molten lunar surface. With the approximations, described above, there follows from the accretional mass flow ~l=4npR2v=p(24nT/fi)~/2M a characteristic time scale for lunar growth of about 104-105 years. With other values, the existence of an early molten lunar surface can not be explained by the heating due to accretional every: but other sources for this heating are rather questionable.

As will be discussed later, also the (azimuthal) collapse time for


preplanetary disc matter, given by T = 27r/(47ryp) 1/2 corresponds to about 104 years.

Therefore, it can be concluded, that the processes of gravitationally focussed structurization and accretion in early preplanetary phases proceeded rather quickly within the given above time-scales. Of course, these effective processes could come into action only in those late preplanetary phses, when friction, pressure, tidal disruption and/or collisional effects became less efficient than the self-gravity of the disc-matter.

The role of friction in early preplanetary evolution has been studied by L/ist (1952) and Lynden-Bell and Pringle (1974). As has to be expected, any mass distribution loses kinetic energy due to frictional dissipation, and structures must decay. Consequently it tends to fall on the central body. On the other side, the conservation of the total angular momentum can be realized only if simultaneously a part of the disc matter is transported into the outer regions of the system. Consequently, any structurizing processes can become effective only if viscous interactions, as the radial momentum transport due to radial differential rotation can be neglected.

This is an essential restriction for the particle density and demonstrates, that the structurizing processes became essential not in the early plasma-gas phases but in the later particle-phases (of grains and planetesimals). The order of magnitude of the resulting characteristic particle number densities can be shown to be N ~ 10 8m 3. A further restriction for structurizing processes can be found from the action of the tidal force 2y(Mc/R3)r of the central mass M c on a radially extended structure around R with diameter 2r. The destructive tidal force can be overcome by the internal gravitative self- attraction of this structure with density ~5 if

4 r3p 2Mcr 3 M~. ~ - ~ r ~ - > / R~-5---, p/> 27r R~ =-Pc (9)

It is interesting to note here, that the condition p >Pc throughout a preplanetary disc corresponds to Md~c >~ M c. Therefore, "thick" preplanetary discs with Mms c ~>M c may evolve probably by further local gravitational collapses, giving giant protoplanets or multiple stars.

As can be seen by numerical comparison, the averaged density, typical for the Solar system preplanetary disc is of the order of or smaller than I0 -~° kg/m 3. Therefore, p ) p c could have been satisfied only at distances >-0~3m behind the presently known planetary system. So, the inner preplanetary disc had a too low mass-density, or equivalently Mdi~ < M,., and its evolution was not characterized by the discussed above direct gravitational collapse of a thick disc. In this connection it is interesting to take account of the reconstruction characteristicum, that the outer bodies in all the four systems discussed have higher orbital inclinations. So they might have had


their origin in a direct gravitational collapse in the originally, also vertically extended, thick disc, permitting higher inclinations. For the two-dimensional case of a thin (plane) disc with height H and a surface density Z = i" dhp ~ Hp, the given above order of magnitude discussion of the importance of tidal effects modifies to

Mc ZTrr2 (10) 2 ~7-3 r 4 r 2

Here, r ~> H has to be assumed. Otherwise the given above 3-dimensional model has to be used. From (4) follows with

7[ r <~ ~ y Z ~ k 2 ~ rc (l 1)

what, in principle, corresponds to the results of Goldreich and Ward (1973). Consequently, regions with H ~< r ~< r c could collapse, if their internal pressure (which was not taken into account here) is sufficiently small. With Z = l ip, eqn. (4) gives a mass-density, necessary for the collaps, being of the same order of magnitude as in the three-dimensional case, leading to the same conclusion, that this collapse might have been possible only in the outer regions of the Solar system.

Summarizing, it must be stated, that in the very early "plasmathermal regime" a preplanetary disc maY evolve, but thermal pressure and friction are able to destroy all large-scale structures. Therefore, the main constructive process, characterizing this regime is the condensation of the disc of plasma- gas-dust, giving droplets and grains, growing to sizes of the order of centimeters. Then the "gravitational regime" starts, which is characterizable by possible gravitational collapses in the disc, which can be prevented only by the tidal action of the central body. Only at sufficient high local densities the collapse may occur.

The discrepancy between the necessary and relatively short time-scale for structurization by gravitational collapses in the preplanetary disc and the non- sufficient mass-density of the axisymmetric disc can be overcome by taking into account non-axisymmetric processes, which are able to concentrate, as an example, the matter of a band in a small part of this axisymmetric structure, producing there a greater mass-density, which is able to counteract the destruction by tidal forces.

It can be shown, that in the "gravitational regime" there is indeed the possibility for a non-axisymmetric "azimuthal" gravitational instability, which is able then to generate sufficient high densities to permit the evolution of large-scale radial structures (M6hlmann, 1984).

For the conditions in the preplanetary disc (Pc ~ 10 10 kg/m 3) there follows then a characteristic time scale for the instability of thousand years, which can


be prolonged by the action of the velocity field of the collapsing matter. Furthermore, it can be shown, that in this (linearized) description, the contraction tends to be a non-local effect. This results from the velocity field, coming from neighbouring regions and from the growing gravitational action of the contracting region. Therefore, and probably similar to density-waves and spiral-arms in the galactic disc, this azimuthal perturbation involves a large-scale radial extension. The resulting radially-extended region of sufficient high density can decay then, since it has the appropriated conditions for a further gravitational collapse, generating on this way larger bodies on an time scale of 104-105 years. Of course also smaller bodies may accumulate during these contractive and collapse processes. It is interesting to note here, that a proposed elongated radial structure as a predecessor of the planetary system, has been discussed even by N61ke (1930).

This process cannot be locally restricted. The perturbation evolves the disc to a radially extended structure of higher density, which is able to be fragmented by the mentioned above direct gravitational collapse.

The proposed scenario for the cosmogony of the Solar system (see Fig. 12) starts therefore with an also vertically extended preplanetary disc, which by the action of friction or magnetic fields evolves to a thin (plane) disc. Even tually, gravitational collapses are able to produce km-sized or even greater bodies in the outer regions of the thick disc. The thin disc is characterized in

--young central body with thick gas plasma envelope

A W

--central body with thin disc of gas-plasma and condensating particles

" / ~ - - rea ing o axisymmetry by / ~ the "azimuthal instability"

/ ~ and gravitationally focussed flow of particles into a region with increasing density

Fig. 12. Planetogonic evolutionary scenario.

\

/ - \

\ --. _~ / \ / / " /

/

central body with isolatcd protoplanetar_~ collapsing fragments


this "thermal phase" by condensation to droplets and grains, which open the "gravitational phase" if they grow to cm-sized or greater bodies. The axisymmetrical thin disc of these bodies is unstable against an "azimuthal instability", leading to a non-axisymmetrical extended radial structure (density wave) of higher density which may fragment by gravitational collapses, producing at last the planets.

C O N C L U D I N G R E M A R K S

According to our point of view, there is a deep connection between planetogony, geodynamics and the state of matter in the planetary interiors. This comes from the simple consideration that important forces acting on the Earth's matter result not only from the Moon and the Sun but from the Earth's interior itself. Our way of thinking is sketched in Figures I and 12.

We hope that further research, as well in geodynamics and in planetary high pressure physics as in planetogony, will help to make the relation between these fields closer and so result in a better understanding of the real causes of geodynamic phenomena.

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Geodynamics and state of the earth's interior