This is a tiny contribution to Meteoritics/Planetology, which was never published in this form. We sent it to the 1999 LPS Conference, but they declined to have it as a poster. Some 6 other, where I also was coauthor, was accepted, so I do not know what was their problem with this one; my guess is that the organizers simply overlooked the serious thermodynamic problems behind.
There are serious problems, indeed. Take e.g. the Callen axioms. One tells that extensive S is homogeneous linear function of the extensives. If not, there is no Thermodynamics. Going from axioms to explanations, homogeneous linearity is cca. equivalent with the requirement that "for a system in equilibrium be the same to describe it as one system, or two half-systems in equilibrium". If so, then V1+V2=V, N1+N2=N, E1+E2=E,… and s=S1(V1,N1,E1)+S2(V2,N2,E2).
But, of course, the entropy of a subsystem can depend only on the data of that respective subsystems, not all the surrounding ones too.
It is possible that, by some miracle, a thermodynamic description arises even without this; but in the usual derivation of Thermodynamics for a macro system it is usual to assume that all interactions are of short range. Then you can divide a macro system to several subsystems; they are still macro ones, and only a very narrow surface region feels the neighbouring subsystems. Going with N or V to infinity, the ratioof these border layers go to 0, and indeed S5 depends only on V5, N5 and E5.
A Pb nucleus contains 208 nucleons, so quite fair multiparticle descriptions are possible for it. Still, pressure P (the thermodynamic intensive, canonical pair of –V) does not want to be 0, albeit the nucleus is in equilibrium with vacuum. This is generally ignored in Heavy Ion Physics, and what else to be done; but this is a symptom. Reason: nuclear forces are not surface terms in a nucleus.
Now: gravity is never a surface phenomenon. In the Newtonian limit it is an infinite range force with positive feedback. We are rather lucky if such a system has a thermodynamic description at all. Plus see the technical problems below.
Later the idea became a part of the paper N. Fáy, B. Lukács, K. Martinás & Sz. Bérczi: On Cosmic Spherule Composition: Gravitation+Thermodynamics, PIECE’99, ed. Miura Y., Yamaguchi, 1999, p. 25. But I would like to preserve the original form here.
ON THERMODYNAMICS OF ASTEROID MANTLES
B.Lukács1 & K. Martinás2
1 CRIP RMKI H-1525 Bp. 114. Pf. 49., Budapest, Hungary
2 R. Eötvös University, H-1117 Pázmány P. sétány 1a., Budapest, Hungary
ABSTRACT
The possibility of a thermodynamic formalism is discussed for mantles of asteroids as sources of basaltic meteorites.
INTRODUCTION
Majority of eucrite, howardite and diogenite meteorites are interpreted as fragments of basalts originally on the surfaces of asteroids, mainly on 4Vesta and vestoids. On the other hand, very probably the asteroids, and also the major planets, had been accumulated from a pool of matter rather similar to the chondrites. The final product, therefore, underwent two subsequent fractional distillations. First the smaller planets lost some of the volatiles; then not the average molten matter appeared on the surface as basalt. Therefore if we compare the basaltic meteorites to chondrites, simi\larities are seen as well as differences.
The problem discussed here is demonstrated by a Figure, and one more will be displayed later. Both contain bulk data from the NIPR Catalog [1], 441 chondrites (all of them, except for one G and 4 unique) and all the diogenites, howardites and eucrites. The first Figure is Al/Si vs. Mg/Si.
As it is seen, the chondrites form a compact group, roughly around the cosmic abundance, but the basalts form a chain, resembling a straight line, starting almost from the chondrite cluster and pointing away in the sequence diogenites Þ howardites Þ eucrites. We obviously see here the product of an evolutionary process [2]. However now we want to concentrate on the origin of the straight line. The line starts from a point definitely less magnesian than any possible chondritic composition. Either HED parent bodies were fairly different from chondrites, or something transformed the distribution. Briefly: even most "chondritic" HED basalts were slightly "andesitic" compared to the primordial composition.
ON MANTLE FORMATION
Consider a chondritic aggregate just in the early Solar System. The short-living radioisotopes as Al26 or Pu244 were still present, so even asteroids could have been melted. If we concentrate on 4Vesta, it is of radius almost 300 km, one of the largest. We do know that many smaller has been internally segregated, therefore we may assume that there was a stage when all Vesta, except for a thin crust, was molten. Then its average composition must have been very close to the chondritic one. The metallic iron dropped out of the silicate and went to the center, but SiO2, FeO, MgO, Al2O3, CaO and the minor constituents remained in the mantle material. As told, probably the melting was not partial, fractional crystallisation was a minor effect because the crust was rather thin, so one would expect the chondritic Mg/Si ratio in the products of the first volcanism. Such diogenites are not seen.
With cooling the crust becomes thicker and thicker, the lava crosses it in more and more slowly, fractional crystallisation becomes more and more substantial. In addition, chemical equilibration starts between the ascending lava and the crust. The result is a more and more aluminous and calciferous basalt, and our guess is that this is seen in the diogenite Þ howardite Þ eucrite sequence, see also the second Figure.
Now the HED line crosses the chondritic hub, but goes beyond it. At the starting point the basalt contains, then, relatively less Ca, Al and Mg, than the primordial composition. Since Fe/Si (not shown) does not compensate this, the "starting basalt" seems simmply "richer in Si", than chondrites. See also Ref. 3.
THE EFFECT OF GRAVITY
Something then happened with the ancient mantle material. The minimal assumption would be gravity, being present. We show immediately, however, that unresolved problems are still present, not in Gravity+Thermodynamics, but about the knowledge of actual equations of state of silicates.
Gravity acts by correcting the conditions for equilibrium throughout the body. Let us follow here Guggenheim [4]. On his p. 328 he gives an equation for equilibrium in gravitational field as
mi + MiF= const (in space)
where mi is the chemical potential of the ith material component, Mi is a corresponding mass, and F is the gravity potential. The structure of this equation remains unchanged even in General Relativity [5]. For thin gases, as well known, the equation leads to the familiar
n1/n2 µ exp(-(m1-m2)U(r)/kT)
where U is the potential.
PROBLEMS WITH SILICATES
Now it would seem that we are ready to calculate the dependence of mantle composition on depth in the young Vesta. In truth it is not yet possible, but the remaining problems root not in thermodynamics but in the unsatisfactory knowledge about molten silicate mixtures. We would need an equation of state S=S(V,E,NI) for many components. Now let us think, as a model, in an Mg and an Al silicate (as terrestrial SiMa and SiAl). The above equation cannot be applied. The main problem is not the special form; it follows from the gnlnT term of the entropy density, which is much more general than for ideal gases and can appear with substantial interactions too [6,7]. However now the „molecular weight" of an Mg-silicate is smaller than that of an Al-bearing one, and still the density of SiMa is higher than that of SiAl. A correct equation of state would give automatically the multiplicator of U in the exponent; until that we only note that the form is a density difference times the average volume of the elementary units in the molten silicate. That unit may be roughly one Si, one or two metal atom + the necessary O’s, but the strong interactions suggest that neighbours will not fluctuate independently, and a multiplicative number factor will appear in the exponent.
However, such problems can be solved, if desired, by laboratory experiments with simple molten silicates.
THE MAGNESIUM GAP
Now we ignore all the said technical problems for a moment and take an estimation by substituting the mass difference with the density difference times the average volume of one „unit". The temperature is the melting point, cca. 1500 K. Then for a body R=r*107 cm, near to the surface, one gets
n1/n2 µ exp(-10-2r2)
which is already can be observed for 4Vesta, r»3. Therefore the upper layers of the mantle are not of the average chondrite composition.
REFERENCES
[1] Yanai K. & Kojima H.: The Catalog of Antarctic Meteorites. NIPR, Tokyo, 1995
[2] Lukács B. & Bérczi Sz.: Antarctic Meteorites XXII, 94-96 (1997)
[3] Bérczi Sz. & Lukács B.: in KFKI-1998-07, pp. 6-12
[4] Guggenheim E.A.: Thermodynamics. North Holland, Amsterdam, 1985
[5] Balazs N.L.: in Re;ativity and Gravitation, ed. by. Ch. G. Kuper & A. Peres, Gordon & Breach, NY, 1971
[6] Diósi L. & Lukács B.: J. Chem. Phys. 84, 5081 (1986)
[7] Zimányi J. & al.: Nucl. Phys. 484A, 647-660 (1988)
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