The low-temperature electrical properties of carbonaceous meteorites

Earth and Planetary Science Letters, 28 (1975) 37- 45

© Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

~a~ LY~

THE LOW-TEMPERATURE ELECTRICAL PROPERTIES OF CARBONACEOUS METEORITES AVIVA BRECHER, PETER L. BRIGGS and GENE SIMMONS Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge. Mass. (USA)

Received July 1, 1975 Revised version received August 13, 1975

Electrical conductivities and dielectric constants have been measured over the temperature range 90-300°K on several carbonaceous chondrites and some terrestriM analogues. The conductivities of meteorites of different petrologic subtypes range over many orders of magnitude and the low-temperature activation energies are typically much smaller than those observed in terrestrial materials at higher temperatures. The electrical properties of carbonaceous chondrites vary systematically with chenfical-mineralogical characteristics in that: (l) activation energy at low temperature is greater in the more volatile-rich meteorites containing hydrated silicates, and (2) conductivity is greater in the more reduced meteorites of higher petrologic subtype. These new data on the electrical properties of chondrites hold important implications for both tile thermal and magnetic histories of small bodies in the early solar system.

1. Introduction The carbonaceous chondrites are recognized to contain primordial solar nebula condensates for the temperature range of 1 0 0 - 5 0 0 ° K , and thus represent more primitive material than any other solar system solids now available for laboratory analysis. Although the chemistry and mineralogy of meteorites have been extensively studied, and the resulting data have been applied to detailed chemical models o f solar system condensation [ 1 - 4 ] , their physical properties have not been studied to the same extent, even though such data may provide important constraints on the models. Most chemical models o f the condensation sequence are consistent with the presence o f carbonaceous chondrite material in the asteroid belt. In addition, recent spectral reflectivity observations o f mainbelt asteroids indicate that carbonaceous chondritic surfaces are relatively common (e.g. Pallas, and Ceres, the largest asteroid, both with density about 2.5 g/cm a [ 5 - 7 ] . However, some other large asteroids spectrally resemble the highly differentiated basaltic achondrites (e.g. Vesta with density about 3.5 g/cm a [5]), enstatite, or iron-rich chondrites [7].

Since the initial composition o f bodies forming in a narrow range of heliocentric distances (e.g. Ceres, 2.4 A.U.; Vesta, 2.6 A.U.) is not expected to vary greatly, the variety o f asteroidal surface compositions suggests important differences in their thermal evolution. The concentrations o f long-lived radionuclides available at formation could not have provided appreciable heating o f any asteroid-sized objects [8]. Nor could fossil short-lived radionuclides account for drastically different thermal histories of asteroids of similar size and initial composition [8]. Accretional gravitational energy release as heat is small even for instantaneous collapse o f asteroid-sized objects. Thus, an external heating mechanism probably operated on some o f these bodies in the various degrees of differentiation observed. Two possible external heat sources are collisional impact heating [9], and radiant heating by the sun during a highly luminous Hayashi phase o f evolution, both apparently inadequate [10]. Sonett et al. [11,12] suggested that solar-wind induction heating may have been significant for some asteroidsized bodies if their surface electrical conductivities allowed electromagnetic coupling to a T-Tauri solar wind during the early history o f the solar system.

38 Since the models they developed were based on electrical conductivity functions of terrestrial rocks, they needed to postulate initially elevated surface temperatures (~773°K) for asteroids, whose present blackbody temperatures are about 160°K in order to achieve significant heating rates. We therefore set out to determine the electrical properties of meteorites in order to provide realistic initial parameters for such thermal evolution models and to determine if initially cool, undifferentiated bodies might be similarly affected by an early solar wind. In principle, by judicious choice of meteorite types, we believe that it will be possible to simulate the evolution of electrical conductivity with progressive thermal processing, from carbonaceous to basaltic-achondritic material. Variations of conductivity within each chemical group could afford insight into its dependence on metal content, degree of oxidation, petrographic texture, etc. Although it is agreed that ordinary chondrite types 4 - 6 represent a metamorphic sequence, the issue in the case of carbonaceous chondrites is not settled [ 10,13,14]. The fact that volatile abundances in carbonaceous meteorites correlate with petrographic subtype [14] justifies our approach, that the sequence C-1 -+ C-4 approximates increasing metamorphic grade [ 13], even though some consider the carbonaceous chondrites to be complex mixtures of high- and low-temperature condensates [4,151. Recent evidence that the carbonaceous chondrites, which appear never to have been heated above 450°K [1 ], possess stable paleomagnetization suggests a further connection of electrical conductivity with the thermal and magnetic history of small bodies, as magnetic fields of the strength required (% 1 0 e ) are likely by-products of solar-wind induction heating of asteroids, although some paleoremanence may have been acquired in the condensation-accretion stages [16,17]. The scant early data on the room temperature conductivity of a few stony meteorites, obtained by a variety of experimental techniques [ 18,19], has been summarized by Wood [20]. More recently, Schwerer et al. [21 ] have measured the high-temperature conductivity of a carbonaceous chondrite, Allende, which we have also studied. Data from previous work is listed along with results from the present study in Table 1. This investigation expands a recent first look at the

electrical properties of primitive meteorites in the low temperature range corresponding to asteroidal surface temperatures at heliocentric distances of 2--5 A.U. [221.

2. Experimental technique The samples studied were between 0.5 and 0.75 cm thick, with surface areas of several c m 2, so that small inhomogeneities might not unduly affect the data. An accuracy of about 10% could be expected for such large areas. To measure the DC electrical conductivity of our samples, a three-electrode pulse method was used [23]. This method yields values of both the steady-state bulk conductivity and the dielectric constants (Fig. la). Current flowing between silver electrodes painted on opposite faces of a sample slab is measured as a voltage across a small resistance; a guard electrode removes surface currents from the measuring circuit. A typical oscilloscope trace exhibiting the time decay of initial polarization to a steady-state value of bulk conductivity is shown in Fig. 1b. To avoid the effects of slowly decaying accumulated polarization charges on conductivity, sample faces were kept shorted between measurements. Conductivity measured with the pulse technique at low voltages can vary with applied voltage as intrinsic carriers are gradually mobilized. The measurements were made at voltages between 40 and 100 V/cm, values sufficient to saturate intrinsic conduction; these voltages were selected by verification of Ohm's law at each temperature. After prolonged drying at 45°C in vacuum to remove atmospheric pore water, samples were immersed in a flowing atmosphere of dry argon in a glass enclosure cooled by liquid nitrogen circulating in a double aluminum jacket for low-temperature conductivity measurements. It was not deemed necessary to control the oxygen fugacity of the ambient atmosphere, not only because the carbonaceous chondrites are nonequilibrium assemblages, but also because chemical-mineralogical reactions do not take place at T < 300°K. Several two-terminal measurements of conductivity at room temperature were also made with an electrometer to compare the two techniques; in the electrometer measurements, we used electrodes of both silver paint and thin, adherent nickel foil pressed onto the

39

3-TERMINAL PULSE METHOD

Ein t (PG)

" I [ L sAMPL (RI ,OR) - -

R2 /Omsec ii n

RI

l

~Ee~__. Jl

4.

20/zsec

A =

Eout I

8 msec R2<< RI

'~

Eout

ii n

Eout : R2

Z : El." = lin o"---A--"

o"

r~ RICR ~ T

Fig. 1. (a) Typical input signal from the pulse technique, a t0-ms voltage pulse applied to Mighei. Upper trace (scale 2 ms/cm X 0.1 V/cm, grid squares arc 1 cm X 1 cm): rapid polarization of the sample decays tA) to indicate DC conductivity (B). Lower trace (20 us/cm X 1.0 V/cm): with expanded time scale, the decay of polarization transient tA) is exhibited in detail (A'). (b) Schematic of experimental arrangement and data reduction.

sample with sponge-lined clamps until optimum contact was obtained, as indicated by no further increase in measured conductivity. The electrometer measurements appear much less reproducible and reliable than three terminal pulse measurements, probably because of contact resistance and surface conduction effects.

3. Results and discussion The observed conductivities of the samples are shown in Fig. 2 as a function of temperature over the range 9 0 - 3 0 0 ° K . The error bars correspond to 5 ° in

temperature and to the blur in individual oscilloscope traces, and appear to be realistic. The reproducibility of measurements with the pulse technique is excellent. The electrical conductivity of silicates may be expressed in the form:

o(7') = ~ o i e x p ( - E i / k T ) i where each term represents a different conduction mechanism with a particular activation energy, Ei, and temperature-independent term, o i. The values of the constants, OO, and activation energies, E 0, corresponding to the linear portion of log a(T) vs. l I T

40 T (°K) ~000

300

200

150

I00

I

I

1

I

g:

i

-E

l l I

l Allende

~+-..+~+

_

5"

Y -E b

~N,~

_A

Mighe=

_c

-I(

-I

-~:o

4

L

I

I

6 I000/

8 T

I

I0

|

are given in Table 1. Also shown in Fig. 2 are yalues of conductivity measured at low temperature on close terrestrial analogues to carbonaceous chondrites, a serpentine (antigorite), and a serpentine marble;both have layer lattice structure containing bound water. Serpentine-like phyllosilicates are expected to be abundant low-temperature condensation/alteration products in equilibrium condensation models of the solar nebula [ 1,2,15], and comprise most of the matrix material of type I and 1I carbonaceous chondrites. For purposes of comparison with previous data, Schwerer et al.'s [21 ] data on the high-temperature conductivity of the meteorite Allende during initial heating (before thermal cycling) is included in Fig. 2 (dashed curve). Note that the value of conductivity of the low-temperature end of their curve agrees well with our present observations of Allende. Also traced in Fig. 2 is the conductivity behavior of a terrestrial olivine determined by Rikitake [24]. The latter has been used by Sonett et al. in modelling the steady-state induction heating of asteroids [ 12]. Over the low-temperature range, the conductivities of chondrites vary slowly and are orders of magnitude greater than conductivities of terrestrial materials such as the mineralogically similar serpentines in the same temperature range, and olivine at much higher temperatures. At 300°K, conductivities of these carbonaceous chondrite samples range from 10-s (ohmcm)-1 for Lance, to 5 X 10-11 (ohm.cm)-i for Orgueil. Activation energy values (Table 1) are in the range 0.016-0.037 eV, much smaller than room temperature activation energies observed by Schwe.rer et al. for a lunar basalt (0.17 eV), a lunar gabbro (0.51 eV), and a terrestrial olivine (0.92 eV) [21 ]. Although lunar rocks, as measured, exhibit generally much lower conductivities and higher activation energies than chondrites, a semiconducting Moon with activation energy ~<0.14 eV is quite compatible with global electromagnetic data [26]. The low activation energies measured are indicative of hopping-type impurity conduction or semiconduction.

Fig. 2. Electrical conductivifiesof four carbonaceous chondrites and two terrestrial serpentines as functions of temperature. Also shown are high-temperature data on Allende [21] (dashed line), and the conductivity function for olivine used by Sonett et al. [12] in modelling the induction heating of asteroids.

41 TABLE 1 Electrical co nductivity in meteorites and terrestrial rocks Sample

Type

Carbonaceous chondrites Orgueil C-1 Cold Bokkeveld C-2 Mighei C-2

Murchison Lanc~

C-2 C-30

Allende

C-3V

Ornans

C-4,50

Other meteorites Kelly Arriba Leedey Plainview Richardson Rose City Pinto Mts. Eleno vka Nor to n Cty. Goalpara

LL-4 L-5 L-6 H-5 H-5 H-6 (L) (L) Ach. Ach.

Terrestrial r o c k s Serpentine Serpentine marble Olivine

o(300°K) (ohm-cm) -1

Low-temperature a 0 (ohm-cm) -1

Low-temperature E i ~ (eV)

Reference

Notes*

3 - 8 × 10 -11 4 - 7 . 5 X 10 - l l 0 . 4 - 4 X 10 -9 3 X 10 -9 2 X 10 -8 5 X 10 -9 0 . 2 - 2 X 10 -s 1 . 4 - 3 . 5 × 10 -5 2 X 10 -5 1.2 X 10 -5 0 . 2 - 5 × 10-6 4 - 5 . 5 X 10 -7 2.7 X 10 -7 2.7 × 10 -7 1 - 4 × 10 -11

--

--

2.8 3.8 2.2 4.8 -

0.037 0.023 0.022 0.016 -

[22] [22] [22] [22] -[22] [22] [22] [21] [22]

El El El PIFC PF PF EF EF PI PF EIFC EF PF EI

2 × 10 -9 1 - 8 × 10 -6 4 × 10-7 1 . 6 - 1 1 × 10 -4 1 . 8 - 6 × 10 -7 7 - 1 5 × 10 -2 7 × 10 -9 8 X 10-6 1.2 X 10-8 1 - 4 × 10 -3 8.5 × 10-4

_ -

_ -

--

--

-

-

[18] [ 18] [ 19 ] [19] [18] [18] [20] [20] [20] [22] [22]

EI PF

1 - 1 0 × 10 -1° 4 × 10 -9 1 - 2 0 × 10 -11 5 × 10 -1°

3 × 10 -9 _ 1.2 × 10 -1°

0.057 _ 0.042

-

EF PF EF PF

1 X

-

-

[24]

-

10 -12

x 10 -8 X 10 -9

X 10 -s

× 10 -7

* S u m m a r y of electrical c o n d u c t i v i t y in meteorites from the present w o r k and previously published data. El indicates e l e c t rome t e l m e a s u r e m e n t on irregularly shaped sample; EF, e l e c t r o m e t e r m e a s u r e m e n t of irregular sample w i t h fusion crust. PI, PIFC, PF indicate pulse m e a s u r e m e n t s on irregular, irregular with fusion crust, and flat samples.

The increase in conductivity and decrease in activa-

p r o g r e s s i v e l y l o w e r 2 ~ N e g a s l o s s as g r a i n s i z e i n -

t i o n e n e r g y w i t h p e t r o l o g i c s u b t y p e c o u l d a l s o b e as-

creases with petrologic subtype

cribed to a higher density of defects introduced by

t i o n p o p u l a t i o n o f d e f e c t s ( t r a c k s ) is still p r e s e n t in

cosmic-ray bombardment

(T. Shankland, personal

~1% of olivine grains in the matrix of C-2's [27],

1975), since cosmic-ray exposure

b u t t h e i r c o n t r i b u t i o n t o c o n d u c t i v i t y is p r o b a b l y

communication,

[10]. A pre-aggrega-

a g e s d o i n d e e d i n c r e a s e f r o m C-1 ~ C-4 [ 1 0 ] . H o w -

minor. The fact that these tracks have not been an-

ever, this may be a specious correlation due to

nealed confirms that at least the C-2's aggregated at

42 low temperatures and have remained "cold" throughout the age of the solar system. Conductivity is expected to correlate strongly with the available amounts of free carriers, i.e. metal content. The petrologic sequence (C-I to C-4) parallels the increase in iron-metal content and decrease in the amounts of volatiles, H, C, N, S, H2 O, etc. [28]. Indeed, the more conductive meteorites are those of higher petrologic subtype, which also display a weaker temperature dependence of o, and lower activation energies. The low-temperature activation energies of meteorite and serpentine samples decrease with both decreasing total iron content of the sample and with the content of bound water associated with hydrated silicates [14]. In the meteorites, higher activation energies also correlate with higher volatile contents, indicating the relative difficulty of mobilizing charge carriers from water and organic compounds compared to transition metals and their oxides. As might have been expected from relative pore water contents, the effect of the ice-water transition is quite pronounced in the more primitive C-2 meteorites, and even more so in the terrestrial serpentines. The gradual increase in both o and e with temperature in the range 100 250°K, and their more abrupt increase in the range - 2 0 to -40°C are features similar to those reported for a variety of terrestrial rocks and soils at temperatures below 0°C [29]. The presence of pore or adsorbed water can affect markedly measured conductivity, because water is an electrolyte when liquid and a semiconductor when solid [29]. Alvarez [30] has demonstrated that the effect of water on conductivity cannot be removed completely by drying rock samples, but the effect of residual water is negligible at T < 250°K for these highly conducting meteorite samples. If H2 O were a principal contributor to conduction, the fine-grained, porous C-2 meteorites should have had higher conductivities than the C-3 specimens: the opposite relation is observed. The reproducibility of the data on electrical conductivity over the low-temperature range after different drying intervals is further evidence that residual moisture is unimportant. DuFresne and Anders [31 ] suggested that carbonaceous chondrites have been exposed to and altered by water in an extraterrestrial environment. Although the free water presently in these meteorites may be terrestrial or indigenous (D/H ratios of both meteorites and

I

I

r

~

J

-II

•

E u

I

l-once

-12

AI Iende . ~ = ~ _ _ ~ -

o v

~~ -13 -I

Mur~k ~ 40n

Ser

-15

50

J

I00

t"1

t

150 200 T (°K)

250

Fig. 3. Dielectric constants of four carbonaceous chondrites

and two terrestrial serpentines as functions of temperature.

terrestrial water vary greatly within similar ranges and cannot be diagnostic) [32] ; estimates of the amount of H2 O incorporated into carbonaceous bodies upon formation are rather uncertain. Because the contribution of H2 O to conduction is small, we believe that our laboratory measurements are representative of the bulk electrical behavior of primitive carbonaceous bodies. Estimates of the dielectric constant, e(T), may be inferred from observations of the decay time 0f polar ization, r, in response to a voltage impulse: r !T = e(T)/o(T). Dielectric constants, e(T), are shown as a function of temperature in Fig. 3. As with conductivity, the variation in dielectric constant with temperature below 250°K is generally steeper in the mor~ primitive meteorites. Polarization decay times, r, of 2-20/as and implied dielectric constants of 10-11 to 10-~4 farad/cm for carbonaceous chondrites may be compared with reported decay times for lunar electromagnetic transients of about 20 seconds [33]. For a lunar surface conductivity of 10-11 (ohm.cm)-1, this decay time corresponds to e = 2 × 10-l° farad/cm, whereas for a

43 conductivity of lff 4 (ohm-cm)-l appropriate to 1000 km depth in the moon, the corresponding e = 2 X 10-~ farad/cm. If these carbonaceous meteorite samples are representative of asteroid surfaces, then cool asteroids may be expected to respond very rapidly to transient electromagnetic phenomena, so that transverse-electric induction may be less important for heating than the steady-state (transverse magnetic) induction.

4. Implications o f results

Estimates of the conductivity function, o(T), are necessary for modeling the thermal history of a planetoid via its electromagnetic interaction with a paleosolar wind. For bodies with neither an atmosphereionosphere system nor an intrinsic magnetic field strong enough to stand off the solar wind, the heating due to currents induced in a body by the solar wind is dependent on the coupling between the body and the impinging plasma. For steady-state induction due to the motional electric field of the solar wind (Em ~ X if), Sonett et al. [12] derived two limiting rates of heating,/~, by ohmic losses as induced curren ts: 121= 8nmv 2 /oor 2 for strong coupling to the solar wind, and:

f l = o 17XYl 2 for weak coupling. Here (nmv 2) is the solar plasma energy flux;/J is magnetic permeability, known to be only slightly greater than unity for carbonaceous chondrites and ordinary chondrites from available data on their magnetic susceptibility [16,34] ;B is the magnetic field associated with the solar wind; o is DC bulk conductivity; and r is the planetoid radius. Sonett et al.'s trial models of asteroid sized objects all involve induction heating rates appropriate to weak coupling to the active T-Tauri phase assumed for the sun [12]. However, since terrestrial olivine was used as the model solid in these calculations, a hot, opaque "hohlraum" (gas-dust enclosure) around the sun was hypothesized so as to reach temperatures near 775°K in the asteroid belt, thereby raising the surface conductivity of test objects to a threshold value for initiation of induction heating ('10-9

(ohm-cm)-1 ). This ad hoc postulate appears to be in conflict with the various temperatures inferred for the asteroid belt region, as well as with evidence that carbonaceous chondrites have not been subjected to high temperatures at or since their formation [I--3, 351. A C-2 asteroid, with a surface at temperatures appropriate to present day radiative equilibrium in the asteroid belt ( T ~ 160°K) and with an interior only slightly warmer might have a conductivity function similar to that of Murchison, ranging from 0.5 × l0 -° ( o h m - o n 0 -1 at the surface to 5 X 10-9 or more in the interior. With such a conductivity profile, there appears to be no need for a hot hohlraum in order to make some initially cool asteroids susceptible to electromagnetic heating. Thus, in light of our data, the heating mechanism of Sonett et al. now appears to be more nearly compatible with evidence from the early solar system and, we believe, quite plausible. The rate and duration of heating is expected to vary with the characteristics of individual bodies, such as their exact sizes and conductivity profiles, previous thermochemical history, and compositional homogeneity. Detailed numerical models of the thermal histories of primitive sub-planetary bodies are now being developed, taking into account both heat sources such as radionuclides and solar-wind induction and heat sinks such as decomposition of hydrous minerals and possible outgassing of a body. It is interesting that the increase in conductivity and decrease in activation energy through the petrologic sequence from C-1 to C-4 parallel the effect observed by Schwerer et al. [21] as they heated a single sample of Allende to successively higher temperatures. This seems to reinforce the idea that C-1 C-4 can approximate a metamorphic sequence. Thermal cycling of Allende between room temperature and successively higher maximum temperatures produced a continual decrease in the activation energy at temperatures below the maximum temperature in a cycle (new type of carriers generated), and simultaneously an increase in room temperature conductivity over its initial value (additional carriers produced) by an amount related to the maximum temperature attained in a cycle. Our preliminary data on high-temperature o(T) in Mighei show a similar trend. The new low-temperature conductivity data, together with the thermal hysteresis observed at T > 300°K, suggest a possible qualitative scenario

44

of heating. If solar-wind induction were to begin heating a body, its interior conductivity would tend to increase, perhaps even permanently, upon cooling. Eventually, the deep interior might become so conducting, that ohmic losses would deposit substantial energy only in a shallower, less conducting zone of the asteroid. Depending on the detailed initial characteristics of a body, induction heating might be insignificant at all times or important for one or more episodes during a T-Tauri phase of the sun (~107 years). Episodes of induction heating at various depths could provide the brief transient heat source long sought to explain the apparent cooling rates and high temperatures associated with iron meteorites and with ordinary chondrites [36], both formed at low pressures. In addition to thermal history considerations, the electrical conductivity of carbonaceous chondrites may partially constrain models of the paleosolar wind. The magnetic field strength required to stand off the solar wind is: B ~ n 1/5 v/r

(r is heliocentric distance, v is solar-wind velocity, n is solar-wind number density) [e.g. 37]. One may constrain either the solar-wind parameters or the induced magnetic field. An asteroidal intrinsic magnetic field of 3 0 - 4 0 gamma would suffice to stand off the present solar-wind flux of 2 X 10a (cm 2-s)-1 at 2 A.U. The carbonaceous chondrites studied appear to have experienced magnetic fields ~1 gauss [16,17]. In the early solar system, such an induced field could have stood off a solar wind with a high particle number density of 2.5 × 107/cm a , if the velocity were a normal 400 km/s, but only "~4 × 106/cm a for v = 1000 km/s. For a solar-wind velocity of 100 km/s, the number density implied is 4 × l0 s/cm a . This range of possible maximum solarwind proton fluxes ( 1 - 4 × 101 s/cm 2 -s) implied for the solar system by the remanent magnetization of carbonaceous chondrites is comparable with the stellar-wind parameters inferred for young, active stars [38]. To date, the conductivity behavior of only Allende [21 ] and Mighei (Briggs and Brecher, unpublished data) have been studied at temperatures above 300°K. Further studies, particularly at high temperature, of meteorite electrical conductivities are needed to

complement existing data. For more realistic modelling of planets, information about the effect of pressure and thermal history on conductivity should be obtained, as well as thermal conductivities representative of primitive rocky material. During this study, a single measurement of the thermal conductivity of Allende was obtained at room temperature, 1.6 X 10-3 cal/cm-s-°K. This result is quite different from values typical of terrestrial samples; it is about 1/3 of the value for serpentines. The unexpectedly high electrical conductivities of carbonaceous chondrites reported here indicate a need for new models of planetoid interaction with the solar wind to determine how bodies of different types and sizes might be expected to evolve and populate the classes: (a) never heated, (b) warmed sufficiently to lose some volatiles, (c) substantially heated and differentiated. Detailed modelling should be useful both for insights into the genesis of meteorites and the early history of the solar system, and for realistic estimates of the electric and magnetic response of small chondritic bodies to the solar wind, to be verified in future missions to the asteroids and rocky satellites.

Acknowledgements We are grateful to Drs. Frondel and Olsen for meteorite samples, and to F. Miller and W.B. Westphal for valuable technical advice and assistance in measuring electrical properties. Thermal conductivity measurements were made under the supervision of M. Chessman. We thank Drs. J.S. Lewis, C.P. Sonett, and A. Duba for useful discussions. The revised version has benefited from helpful reviews by Drs. T. Shankland, J.F. Kerridge, and an unnamed referee. This work has been supported in part by NASA grant NSG-7077.

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