assessing the implications of k isotope cosmochemistry for

Assessing the implications of K isotope cosmochemistry forevaporation in the preplanetary solar nebula

Edward D. Young *Department of Earth Sciences, University of Oxford, Parks Road, Oxford OX1 3PR, UK

Received 26 May 2000; received in revised form 14 September 2000; accepted 24 September 2000

Abstract

Homogeneity in 41K/39K among Earth, Moon, chondrites, and achondrites has been cited as evidence thatevaporation of protoplanetary dust could not have caused variability in the mass of moderately volatile elements amongSolar System planets and meteorite parent bodies. Herein, consideration of a simple physical model for evaporation ofmineral grains with finite K diffusivities shows that measured 41K/39K do not preclude extensive mass loss byevaporation of solid mineral grains in the preplanetary nebula. Instead the data constrain the grain size, temperature,and duration attending grain evaporation through the effects that these factors have on the evaporative^diffusive Pecletnumber. It is proposed that evaporation of solid dust comprising the precursors to planets can not be ruled out as afirst-order mechanism for controlling the elemental composition of the rocky planets and asteroids. ß 2000 ElsevierScience B.V. All rights reserved.

Keywords: potassium; isotopes; early solar nebula; mineral evaporation

1. Introduction

The ¢nding that 41K/39K among the terrestrialplanets, meteorites, and the Moon is uniform towithin approximately 2x or less (Humayun andClayton [1]) has been cited as evidence that evap-oration could not have played an important rolein determining the compositions of planets andmeteorite parent bodies (Humayun and Clayton[1] ; cf. Esat [2]). Condensation rather than evap-oration1, it is argued, was the principal mecha-nism for elemental fractionation between terres-

trial planets and the asteroids. Here thisconclusion is questioned on the basis that: (1)major rock-forming elements, including K, en-tered the circumstellar accretion disk comprisingthe early solar nebula in solid form; (2) evapora-tion of a heterogeneous mixture of crystalline sol-ids could have caused elemental depletion pat-terns similar to those observed in Solar Systembodies; and (3) evaporation of crystalline solidswould result in uniform K isotope ratios likethose observed.

0012-821X / 00 / $ ^ see front matter ß 2000 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 0 0 ) 0 0 2 7 6 - 4

* Tel. : +44-1965-282-121; Fax: +44-1865-272-072;E-mail: [email protected]

1 Evaporation is used as a general term encompassing volati-lization of a condensed phase either from the liquid or solidstate, in keeping with usage of the term in the astronomicaland cosmochemical literature.

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2. Origin of rock-forming elements in the SolarSystem

Astronomical observations show that the majorrock-forming elements most likely entered thenascent Solar System in solid form and that atleast some of these solids were crystalline. Thesolid provenance of these elements places con-straints on the processes that could have a¡ectedelemental abundances in Solar System bodies.

The major rock-forming elements (lithophile el-ements) are depleted in the gas phase of the dif-fuse interstellar medium (ISM) relative to solarabundances [3]. The paucity of these elements inthe gaseous ISM is usually expressed as a frac-tional depletion N(x)2 for element x :

N �x� � 1310D�x� �1�

where D(x) is the depletion index:

D�x� � xH

h i� log

Nx

NH

� �ISM gas

3logNx

NH

� �h

�2�

and Nx and NH refer to numbers of atoms forelement x and hydrogen, respectively. The missingmaterial is thought to reside in interstellar dust

[4]. With the assumption that the interstellar en-vironment in total has cosmic (i.e., solar) abun-dances of the elements, the depletion index, N(x),speci¢es the fractional amount of a given elementsequestered in dust in the di¡use ISM:

N �x� �Nx

NH

� �ISM dust

Nx

NH

� �h

�3�

When N(x) is 0, there is no depletion and theelement x is entirely in the gas phase. When N(x)is 1, the element x is absent from the gas phaseand must reside entirely in the dust. Table 1shows fractional depletions for rock-forming ele-ments along the well-studied line of sight towardthe luminous star j Ophiuchi [5,6] as well as de-pletions for a composite of several sight lines.Depletions for Si, Mg, Ca, Al, Fe, Na, and Kare all v 0.89 and for most the values exceed0:96, indicating that these elements are presentprimarily as solids in the ISM. The extent towhich solar abundances of elements are truly rep-resentative of `cosmic' abundances has been chal-lenged, but use of alternative cosmic abundancesdoes not substantially change the fractional deple-tions shown in Table 1 [7].

A shortfall in the £ux of dust from stellar sour-ces relative to the rate of destruction in the di¡useISM implies that grains grow and regrow in in-terstellar space in a process of cyclical conversionfrom solid to vapor and back again [8]. Much of

Table 1Elemental abundances and fractional depletions in the di¡use ISM

Element (Nx/NH)ah (Nx/NH)b

jOph N(x)hjOphc (Nx/NH)d

comp N(x)hcompc

O 7.4U1034 1.7U1034 0.77 5.4U1034 0.27Si 3.5U1035 8.9U1037 0.97 8.2U1037 0.98Mg 3.8U1035 1.2U1036 0.97 1.0U1036 0.97Fe 3.2U1035 3.0U1037 0.99 2.7U1037 0.99Ca 2.3U1036 4.6U10310 1.00 ^ ^Al 3.0U1036 4.6U1039 1.00 ^ ^Na 2.1U1036 2.4U1037 0.89 2.1U1037 0.90K 1.3U1037 8.0U1039 0.94 1.0U1038 0.92aSolar data from Anders and Grevesse [45].bj Ophiuchi sight line, data from [5,6].cFractional depletion as de¢ned in the text.dComposite of several sight lines, data from Duley [46].

2 In this section the symbol N refers to fractional depletion ofelemental abundances. Throughout the remainder of the paperN is used to express isotopic abundances. The two meanings ofthe symbols should be clear from the context.

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the growth apparently occurs in the denser molec-ular clouds [9,10]. Once solids of the ISM enter amolecular cloud they are protected from exposureto the destructive forces like shock waves andsputtering that prevail in the di¡use phases ofthe ISM [8,9]. As a result, rock-forming elementsshould be retained as solids in molecular clouds.

The Solar System formed from an accretionarydisk of rotating gas and dust that collapsed froma molecular cloud [11]. Support for this descrip-tion of the early solar nebula comes from obser-vations of circumstellar disks associated withyoung stars [12]. Astronomical observations ofmolecular cloud cores and accretion disks suggestthat grains (including crystalline olivine and py-roxene) in these environments are larger on aver-age than typical grains in the di¡use ISM [13].Grain sizes in disks range from 9 1 Wm to s 10cm [13,14]. New infrared spectroscopic data con-¢rm that silicates and ices are important constit-uents of accretion disks during the earliest stagesof low-mass star formation [15]. Grain size andcrystallinity in protoplanetary accretion disks ap-parently increase by coagulation and annealing attemperatures below 250 K [16].

The implication of these observations is thatrock-forming elements persist in solid form inthe di¡use ISM, in molecular clouds, and in cir-cumstellar accretion disks. Although the nature ofthe solid grains may be altered to varying degreesin each environment by cyclical growth, anneal-ing, and devitri¢cation, the preponderance of evi-dence is that elements like Si, Mg, Ca, Al, Fe, Na,and K entered the protoplanetary disk surround-ing the nascent Sun as solids.

3. Processing solids in the early solar nebula

Elements comprising rocky objects of the SolarSystem were separated according to their volatil-ities [17]. Since the rock-forming elements appar-ently entered the nebula in solid form, transfer ofthese elements between condensed and vaporousphases must have occurred primarily within thenebula. Separation of elements according to vol-atility can occur by either incomplete evaporationor incomplete condensation. In general, partial

condensation suggests that the elements evolvedfrom high to low temperatures while partial evap-oration suggests the material evolved from low tohigh temperatures. The meaning of `high' and`low' in this context depends upon the elementsinvolved in the partitioning (Section 4).

Temperatures and pressures in the accretiondisk were highest near its center and decreasedwith radial distance [18]. Astronomical observa-tions of young stellar objects and mathematicalsimulations suggest that early in its evolutionthe protosun accreted at a rate of V1U1035

Mh yr31 [11]. The disk is expected to have actedas a steady-state source for the growing nebularcore for tens of thousands of years [18]. Materialwithin 2 AU of the growing star would have beenconsumed every 80 years based on a steady-statecanonical disk mass of 0.02 Mh. As the protosunentered its T Tauri phase, accretion is expected tohave slowed, by analogy with T Tauri stars, even-tually settling to a rate of V1U1038 Mh yr31

that persisted for perhaps several million years[11]. The material comprising the inner 2 AU ofa 0.02 Mh disk would have been consumed every80 000 years during this interval. These estimatesfor accretion rates suggest that the material fromwhich planets were made came from distal regionsof the accretion disk (heliocentric distances be-yond the present-day positions of planets) as aresult of radial drift.

Hydrodynamical calculations provide estimatesfor pressures and temperatures in the accretiondisk that gave rise to our Solar System [18^20].Based on these models, one can infer that themaximum midplane temperature attained at thepresent-day median position of the asteroidmain belt (2.7 AU) was approximately 1000 K.At 1 AU temperatures may have been as highas 1500 K. Accretion could have sustained suchhigh temperatures for perhaps 20 000 years [20].Pressures are expected to have been 9 1035 bar atpositions v 1 AU.

Evidently, for perhaps tens of thousands ofyears, solid silicates and oxides should havebeen subject to partial evaporation upon driftingfrom cool distal regions of the nebula to thewarmer region that lay between 3 and 1 AUfrom the protosun [21,22]. The bulk of the dust

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was ferromagnesian in composition [13]. There islittle evidence that temperatures were su¤cientlyhigh in this region to cause complete evaporationof ferromagnesian solids (s 1500 K, [21]). Parti-ally evaporated grains that eventually drifted clos-er to the nebular core would have been replacedby fresh in-falling solids coming from cooler re-gions beyond 3 AU and from above and belowthe midplane of the disk. Separation of elementsin the gas and solid phases is thought to haveoccurred by a net outward £ow of pressure-sup-ported gas relative to the in-falling grains [23].Chondrules in chondrites record temperatures inexcess of 1500 K but are widely acknowledged tobe the products of transient heating events such aslocalized collisions or collision-induced shockwaves [24].

4. Partitioning of elements by volatility

Elemental abundances normalized to Ca andCI carbonaceous chondrite are used throughoutthis paper. Concentrations normalized to Si andCI are used in cosmochemistry by convention.Normalization to Ca is preferred because sili-con-normalized concentrations obfuscate thenature of volatilization reactions. Silicon is notconserved in the condensed phase during volatili-zation at the temperatures of interest [25]. Ca, onthe other hand, is retained in the condensed phaseduring volatilization [25].

Normalization to Ca and CI yields quantitiesthat are equivalent to masses relative to a CI pre-cursor if volatilization controlled elemental con-centrations. This useful property is revealed by¢rst considering the ratio of the concentration ofelement x in an object before and after volatiliza-tion:

cx

cx0

� �� mx

mx0

M0

M�4�

where cx is the concentration of element x in thecondensed phase, mx is the mass of element x inthe condensed phase, M is the total mass of thecondensed phase, and the subscript 0 refers to theinitial quantities. From Eq. 4 it is evident that

changes in concentration arise from changes intotal mass as well as from changes in the massof the element of interest. Concentration ratiosalone do not distinguish one e¡ect from the other.It is for this reason that results of some evapora-tion experiments have been reported in terms ofextensive mass loss rather than as changes in con-centrations [25].

For unambiguous comparisons between exper-imentally determined volatilities of the elementsand the elemental abundances in Solar Systemobjects, it is useful to remove the e¡ects of totalmass on the concentration data and instead exam-ine the masses of the elements relative to theirinitial values. This is possible by recognizingthat Ca, a refractory lithophile element, is e¡ec-tively non-volatile under the relevant conditions[25^27]. Assuming that the compositions of CImeteorites are representative of the original com-positions of planetary and asteroidal precursors(i.e., solar abundances apply), the ratio of theCa concentration in CI meteorites (cCa

CI ) to theconcentration of Ca in a given object j (cCa

j) isequal to the total mass of the object relative to itsmass when it had a CI composition, or Mj/MCI,so long as volatility controlled fractionation:

cCaCI

cCaj

!� mCa

CI

mCaj

Mj

MCIv

Mj

MCI

� ��5�

Multiplying Eq. 4 by Eq. 5 yields:

cxj

cxCI

� �cCa

CI

cCaj

!� mx

j

mxCI

MCI

Mj

Mj

MCI

mCaCI

mCajv

mxj

mxCI

� ��6�

where mxj is the mass of element x in reservoir j.

Eq. 6 permits calculation of the mass of an ele-ment x in reservoir j representing a planet or par-ent body relative to the mass of x in the originalCI-like precursor. Concentrations transformed inthis way are extensive in so far as the originalmass of Ca (i.e., numbers of Ca atoms) was un-changed during elemental fractionation. It is im-portant to realize that the right hand side of Eq. 6applies to an aliquot of the CI reservoir what everthe original size of that aliquot. The only assump-tion is that reservoir j began with relative elemen-tal abundances equal to those of the CI reservoir.

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By applying Eqs. 5 and 6 it is possible to com-pare directly the elemental abundances of SolarSystem objects with abundances in residues fromevaporation experiments. Fig. 1 shows the massesof Si, Mg, Fe, and K for Earth, Mars (SNC me-teorites and the assumption that all elements lessvolatile than Na are present in CI concentrations[28]), Moon, L ordinary chondrite, and CV car-bonaceous chondrites relative to CI plottedagainst the total mass ratio on the basis of con-stant Ca abundance. Also shown are publishedexperimental data obtained by evaporating mol-ten samples of the Allende CV3 meteorite [25] andmolten synthetic samples with CI abundances ofSi, Mg, Al, Ca, Fe, and Ti [26].

The rocky bodies of the Solar System have el-emental depletion patterns similar to the residuesfrom the evaporation experiments (Fig. 1),although the data for Fe are systematically dis-

placed from the experimental results. The similar-ity con¢rms that volatility was the ¢rst-order con-trol on the relative abundances of the major rock-forming elements and provides a benchmark forassessing the magnitudes of the depletions.

Fig. 1 shows that the rocky bodies for whichbulk compositions are known are depleted notonly in moderately volatile elements like K, butalso in the more refractory lithophiles Si and Mg(Si and Mg are transitional in volatility betweenrefractory lithophiles and moderately volatile lith-ophiles), and that the extent of depletion is to beexpected based on the total mass loss for thesebodies relative to CI progenitors. Depletions inthe moderately refractory lithophiles are not evi-dent when normalizing to Si by de¢nition (e.g.,[28]). The ambiguity that arises when using Si isevident from Eq. 6 when Ca is replaced by Si. Inthis case the relation mj

Si = mSiCI can not be as-

Fig. 1. Fractional mass depletions in Si, Mg, Fe, and K plotted against fractional reductions in total mass for Solar Systembodies (black symbols) and residues of partial evaporation in the laboratory (gray symbols). Values for Solar System bodies werecalculated using Eqs. 5 and 6. Standard symbols are used for Earth, Mars (SNC), and Moon. Additional symbols are: ¢lled tri-angle = CI carbonaceous chondrite, upside down ¢lled triangle = L ordinary chondrite, and open triangle = CV carbonaceouschondrite. Data for Earth, Mars, and chondrites are from Wa«nke and Dreibus [28], Wasson and Kallemeyn [41], and Mc-Donough and Sun [29]. Those for the Moon are from Taylor [42]. Gray crosses show residues of evaporated molten CV3 chon-drite [25]. Gray circles represent residues of evaporated melts with CI compositions [26]. Since K was not reported for the experi-ments, K in the Solar System objects is compared with Fe from experiments because the two are crudely comparable in theirvolatilities with respect to refractory lithophile elements.

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sumed to hold (cf. Fig. 1) and di¡erences in con-centrations normalized to Si re£ect di¡erences inthe mass of Si as well as di¡erences in the mass ofthe element of interest (x). If Ca was not com-pletely conserved by virtue of its proven refrac-tory nature, the points representing Solar Systembodies in Fig. 1 would be shifted somewhat alongthe positive slope of the depletion trend (bothabscissa and ordinate are similarly a¡ected), butthe overall pattern is robust. Depletion of Si andMg in bulk Earth relative to CI chondrites hasbeen observed previously [29]. When Ca is treatedas a conserved element, it appears that rockybodies in the Solar System are depleted in notonly the moderately volatile lithophiles like K,but in the more refractory rock-forming elementslike Si and Mg as well.

Depletion in the more refractory elements con-strains the thermal regime in which vapor andsolids coexisted in the vicinity of terrestrial planetand asteroid accretion. In order for the pattern inFig. 1 to result from partial condensation, temper-atures would have to have been higher thancondensation temperatures for the prevalent fer-romagnesian silicates (s 1500 K, [21]) so thatthe moderately refractory lithophile elementsthat comprise these phases could have been pri-marily in the gas phase. Otherwise these elementscould not have condensed from the gas to formthe solids because they would not have beenpresent in the gas in the ¢rst place. Conversely,if the pattern in Fig. 1 is the result of partialevaporation, temperatures must have approached,but not exceeded, at least for long, ferromagne-sian silicate condensation temperatures (9 1500K) in the region of terrestrial planet formationto allow for an appreciable rate of Si and Mgevaporation (evaporation rates are proportionalto equilibrium vapor pressures) at the present-day position of Earth and Moon (1 AU) withoutcompletely volatilizing the solids. As described inSection 3, the temperature structure implied byevaporation of moderately refractory lithophileelements is consistent with our present under-standing of the thermal structure of protoplanet-ary disks of Solar System mass while condensa-tion of these elements from a gas implies a hotterenvironment.

5. K isotope fractionation ^ previous work

On the basis that evaporation would lead tokinetic K isotope fractionation according to theRayleigh fractionation law, Humayun and Clay-ton [1] concluded that their measurements show-ing uniformity in 41K/39K for terrestrial and me-teoritical materials within approximately 2xrequires that evaporation was not an e¡ectiveagent for volatile element depletion relative toCI chondrite. This conclusion has in£uenced sub-sequent work on the evolution of the early solarnebula. For example, Cassen [30] cited the con-clusions of Humayun and Clayton as importantevidence that the inner portion of the solar cir-cumstellar disk must have been su¤ciently hot forcomplete vaporization of silicate minerals. Basedon this premise, Cassen showed that partial con-densation triggered by a slowing of stellar accre-tion rate and reductions in disk opacity attendinggrain aggregation could have caused depletions inmoderately volatile elements resembling those ex-hibited by some carbonaceous chondrites in theregion between 1 and 3 AU.

However, Esat [2] questioned the conclusionthat K isotope homogeneity is evidence for partialcondensation. Esat argued that Rayleigh fraction-ation is an ideal circumstance that may not havecharacterized evaporation in the early solar neb-ula. He also asserted that condensation, like evap-oration, should lead to kinetic isotope fractiona-tion. Esat and Williams [31] later demonstratedthe veracity of the latter argument by showingthat condensation of gaseous K derived from vol-atilization of K-rich glass results in enrichments in41K relative to 39K of 1^11x in the laboratory.

Humayun and Clayton expressed their resultsin terms of mass of K relative to initial mass.As a result, their analysis of the range of possiblemechanisms for evaporative K loss and 41K/39Kfractionation was incomplete. Implicit in the con-clusion reached by Humayun and Clayton is theassumption that the di¡usivity of K in the evap-orating material was su¤ciently rapid that the41K/39K was e¡ectively uniform in the residualcondensed phase, as noted by Esat [2]. By com-paring measured N41K with a Rayleigh fractiona-tion model, Humayun and Clayton made the tacit

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assumption that separation of elements and iso-topomers by evaporation of protoplanetary mate-rials could only have taken place if the latter weremolten.

Young et al. [32] showed that oxygen isotopefractionation can be detected in the bulk residuesof evaporated crystalline silica and that the mag-nitude of observable fractionation depends uponthe rate of sublimation relative to the rate of dif-fusion. The mathematical model describing evap-oration in terms of relative rates of ablation anddi¡usion used by Young et al. has proven appli-cable to other systems. For example, Davis et al.[33] measured the fractionations of Si, Mg, and Oisotopes attending evaporation of molten forster-ite and found that purely kinetic fractionationfactors, expected to apply during free (Langmuir)evaporation, could not be used in combinationwith a Rayleigh fractionation model to accountfor the data. The authors noted that the Rayleighmodel may not be valid because di¡usivities of Si,Mg, and O in the melt are insu¤cient to maintaina homogeneous residue. The data of Davis et al.are modeled successfully using purely kinetic frac-tionation factors when the relative rates of abla-tion and di¡usion are considered [32]. Competingrates also explain the di¡erent behaviors of dis-similar solid materials during evaporation. Crys-talline SiO2 (cristobalite), unlike crystallineMg2SiO4 (forsterite), exhibits discernible oxygenisotope fractionation upon free evaporation inthe laboratory because it volatilizes more slowlythan forsterite. The fact that di¡erent crystallinesolids evaporate at di¡erent rates has not beenincluded in evaluations of the signi¢cance of theK isotope data until now.

6. Physical model for evaporation

6.1. Isotope fractionation

The present work addresses the possibility thatvolatilization of K occurred by evaporation ofsolids with limited capacity for K di¡usion ratherthan the in¢nite K di¡usivity implied by the Ray-leigh model. The mathematical treatment hasbeen described previously [32].

The isotopic and elemental e¡ects of evapora-tion of mineral dust are simulated by equationsfor kinetic fractionation at the surface of a shrink-ing sphere coupled with those for di¡usionaltransport within the sphere. The model sphere istaken as an analogue for a mineral grain evapo-rating in the early solar nebula.

Kinetic theory predicts that there will be frac-tionation of isotopes at the surface of a volatiliz-ing mineral grain according to the equation:

K ��mm0

r�7�

where K is the isotope fractionation factor, m isthe mass of the light isotopomer, mP is the mass ofthe heavy (rare) isotopomer, and negligible quan-tum e¡ects have been ignored. The fractionationfactor describes the reaction rate for the heavyisotopomer relative to that for the light. In thecase of volatilization of K, K is 0.9753 [1]. Alter-native formulations for K involve evaluation ofvibrational frequencies in the crystalline phase[32].

The Peclet number for ablative and di¡usivetransport, L, controls the degree to which isotopefractionation at the surface of a condensed phaseis recorded in evaporative residues. It is de¢ned interms of the initial radius of the volatilizing object(assuming a spherical symmetry) r³, the rate ofshrinkage due to mass loss rç (Dr/Dt, where t istime), and the di¡usivity of the element of interestD such that

L � 3r�_r

D�8�

In general, values s 50 preclude detectable iso-tope fractionation signals in the bulk material be-cause fractionation at the ablating grain surface isnot transported into the interior of the grain bydi¡usion before it is removed by ablation. In thisformulation, rç is constant, in accord with exper-imental data for free evaporation [32,34].

Equations based on K and L [32] were solvednumerically to obtain radial pro¢les of N41K inevaporating materials. The pro¢les were inte-grated to obtain bulk N41K values at di¡erentfractional weight losses.

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6.2. Elemental fractionation

The mathematical treatment of elemental frac-tionation is identical to that of isotope fractiona-tion if K is rede¢ned. For elemental fractionationK is the ratio of K concentration in vapor to thatin evaporating solid, a re£ection of the volatilityof K relative to the remainder of the body. Theprecise composition and state of the material thatcomprised protoplanetary dust is uncertain, butestimates of the vapor pressure of K relative toother more refractory components in chondritesand basalts suggest K is greater than 10 and per-haps as high as 1000 [35]. The precise value for Kis not critical, however. From asymptotic solu-tions to the equations of Young et al. [32] wheretime (t) is su¤ciently long, it can be shown thatdepletion at the surface of the evaporating sphereis limited by movement of the boundary accord-ing to the relation

limt!r

cK

cK� �1K

�9�

such that di¡erences in depletion with K= 10 andK= 1000 are negligible. A value of 100 was usedfor the calculations presented below.

7. Results

Four curves re£ecting a range of possible e¡ectson K and 41K/39K upon evaporation are shown inFig. 2, upper panel. The plot is entirely analogousto ¢gure 5 of Humayun and Clayton [1] and thetwo can be compared directly. The abscissa is themass of K relative to the initial mass of K as itmust be in order for comparisons with Rayleighfractionation. The ordinate is the calculated shiftin 41K/39K expressed as per mil deviations fromthe initial ratio. The largest e¡ects occur for Ray-leigh (or perfect) fractionation, corresponding to aPeclet number (L) of 0. With larger L the isotopicshifts for a given mass loss of K decrease sharply.Fig. 2, lower panel is a magni¢cation along theordinate of the plot in Fig. 2, upper panel, inwhich some of the Solar System bodies analyzedby Humayun and Clayton [1] are compared with

the evaporation curves. In constructing Fig. 2,lower panel, it was assumed that the initial chem-ical composition of the planetary precursor mate-rial was that of CI chondrite. The mass ratio ofK relative to the CI abundance was obtained us-ing Eq. 6. The mass of K relative to CI and the41K/39K measurements of Humayun and Clayton[1] were then used to place each of the Solar Sys-tem bodies on the plot. Humayun and Claytonused Si and La, rather than Ca, as normalizingelements. Lanthanum provides an appropriatemeans for calculating mass ratios of K and yields

Fig. 2. (Upper panel) Plot of N41K (relative to initial 41K/39K) vs. mass fraction of K remaining upon evaporation. Curvesre£ect di¡erent values for the evaporative^di¡usive Pecletnumber L de¢ned in the text. Rayleigh fractionation (uppercurve) corresponds to L= 0. Note suppression of isotope ef-fects with increasing L. The kinetic isotope fractionation fac-tor is 0.9573 for all curves. (Lower panel) Magni¢ed scaleshowing high-L curves relative to Solar System bodies. TheN41K values for the bodies are those of Humayun and Clay-ton [1]. The abscissa values for Earth, Mars, and chondrites(CV, CI, L) were calculated as described in the text fromdata in Wa«nke and Dreibus [28], Wasson and Kallemeyn[41], and McDonough and Sun [29]. The eucrite datum (IJ)represents the median of data for Ibitira (high Ca, highN41K) and Juvinas (low Ca, low N41K). The abscissa valuesfor the eucrites were calculated from data in Wa«anke et al.[43] and Morgan et al. [44] and those for the Moon fromTaylor [42].

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values comparable to those based on conservationof Ca (Ca and La are both highly refractory [26])while use of Si introduces errors as described inSection 4.

The calculations show that evaporation charac-terized by L values of approximately 200 þ 100 areconsistent with the measured N41K values for eu-crites, bulk Earth, SNC meteorites (Mars), CVcarbonaceous chondrites, ordinary chondrites(e.g. L), and CI carbonaceous chondrites. Thecurves in Fig. 2, lower panel, apply to the ratiosof r³, rç, and D regardless of the actual values forthe parameters.

8. Discussion

8.1. K isotope fractionation

The simple physical model for mineral evapo-ration shows that homogeneity in 41K/39K amongEarth, Moon, chondrites, and achondrites doesnot preclude volatilization of K in the early solarnebula. Homogeneity in 41K/39K does place con-straints on the evaporative^di¡usive Peclet num-ber, and thus grain size, temperature, and dura-tion attending volatilization by virtue of thee¡ects that these factors have on L.

High L values apply to all objects, solid or liq-uid, once they reach asteroid size (r³ dominates).The constraint that L was v100 therefore hasgreatest signi¢cance for the processing of con-densed phases in the early solar nebula prior toaccretion of asteroidal-sized objects when thecharacteristic size was on the order of centimetersor less [23].

8.2. K concentrations

High-L evaporation could have occurred eitherby evaporation of solids or, under the right con-ditions, by evaporation of liquids. In the case ofliquids there are two ways to achieve high L num-bers. One is if the initial sizes of the liquid objectswere on the order of many tens of centimeters, aconstraint imposed by the di¡usivities of elementsin silicate melts. This would require that the ob-jects were much larger than chondrules and cal-cium aluminum-rich inclusions (CAIs) in chon-drites. Chondrules and some CAIs are the onlyobjects known to have been molten in space inthe early solar nebula and there is no evidencethat they were ever this large. The other waythat silicate liquids might have evaporated athigh L is if the objects were similar in size tochondrules and CAIs but evaporated at ratesthat exceed free evaporation rates by a factor of100 or more. This occurs in the laboratory whenambient pressures approach 100 kPa [36].

The extent to which high-L evaporation ofliquids could have altered K concentrations inprotoplanetary material depends on complicatedinteractions among L and the K values for all ofthe elements involved. In general, extreme valuesfor K, of order 0.1 or less for a refractory elementlike Ca for example, can produce large e¡ects forconcentrations even at a L of v 100. Modelingsuch e¡ects requires variable K and rç (to see thisconsider what happens when concentrations ofthe most refractory elements approach unity).The range of possibilities involving di¡usion-lim-ited evaporation of liquids is currently under in-vestigation [37].

Table 2Total masses and elemental masses relative to CI precursors

Body j Mi/MaCI mj

Si/mSiCI

b mjMg/mMg

CIb mj

Fe/mMgCI

b mjRb/mRb

CIb mj

K/mKCI

b

Moon 0.29 0.56 0.57 0.16 0.04 0.04Earth 0.54 0.73 0.85 0.97 0.12 0.15Mars (SNC) 0.53c 0.82c 0.78c 0.80c 0.20c 0.23c

CV 0.48 0.72 0.72 0.63 0.27 0.27L 0.70 1.24 1.08 0.83 0.98 1.03aEq. 5 and data from [28,29,41,42].bEq. 6 and data as above.cBased on the assumption that Ca, Si, and Mg concentrations are identical to CI [28].

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Although high-L evaporation of liquids is fea-sible under special circumstances, it is normal forevaporation of crystalline solids. If the lithophileelements in the early solar nebula resided in aheterogeneous population of solids, then elemen-tal fractionation by evaporation should have oc-curred since solids with di¡erent chemical compo-sitions and di¡erent structures evaporate atdi¡erent rates. If this heterogeneous materialwas crystalline, it would have evaporated withhigh L, causing elemental fractionation but notisotope fractionation.

9. Evaporation of heterogeneous crystalline solids

The progenitor planetary material that hostedK in the early solar nebula is unknown. However,alkalis could have been present in solids otherthan the abundant ferromagnesian phases evi-denced in astronomical observations. Solids inthe di¡use ISM are largely amorphous and evap-oration of feldspathic glass is almost certainly alow-L process with respect to K (L6 1) because ofthe high di¡usivities of alkalis in the glass [38].However, there is evidence for annealing of amor-phous material to produce crystalline phases inprotoplanetary disks [16] and crystalline feldsparscan evaporate with high L. The e¡ects of volatil-izing a multimineralic reservoir of crystalline sol-ids can be illustrated by comparing the evapora-tion rates of feldspar, an analogue for theunknown K host in the early solar nebula, andforsterite, representing the ferromagnesian phasesin the nebula. The choice of feldspar as the ana-logue for the K carrier is based on the observa-tion that some primitive materials like interplan-etary dust particles contain feldspar as a host foralkali elements [39]. The actual source of the po-tassium is of course speculative in the absence ofmore detailed astronomical data bearing on theidentity of mineral grains surrounding young stel-lar objects.

Precise measurements of feldspar evaporationrates have not been made, but the maximumrate of free evaporation (Langmuir evaporation)can be estimated from Knudsen cell measure-ments of plagioclase vapor pressures [27,35] using

the expression:

Jevap � 1NA

P��2ZmkT�p �10�

where Jevap is the evaporative £ux in mol/(m2 s),NA is Avagadro's constant, P is the vapor pres-sure (Pa), m is the mass of the evaporating species(kg), k is Boltzmann's constant, and T is temper-ature (K). Nagahara and Kushiro [27] found thatplagioclase evaporates (sublimes) incongruently toyield a non-stoichiometric product rich in Ca andSi at grain margins. Based on their analysis, thereaction for plagioclase evaporation is:

�Ca13xNaxAl23xSi2�xO8�sBxNav � xAlv�

xSiOv � 3x2

O2v � �Ca13xAl232xSi2O834x�s �11�

where subscripts s and v signify solid and vapor,respectively. The gas phase in this reaction has thestoichiometry of nepheline. An Arrhenius-typerate equation for plagioclase evaporation (Table3) is de¢ned by the average mass for the gaseousspecies in Eq. 11, the vapor pressures of 1U1033

Fig. 3. Contours (solid lines) of log L for K in crystallinefeldspar as a function of temperature and grain size basedon parameters in Table 3. Also shown are contours (graylines) for logd where d (s) is the life span of an evaporatinggrain of feldspar assuming a spherical geometry.

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and 0.1 Pa at 1373 and 1773 K, respectively citedby Nagahara and Kushiro [27] (consistent withthe data in [35]), Eq. 10, and the slope and inter-cept for a plot of ln Jevap vs. 1/T.

The di¡usivity of K in plagioclase can be com-bined with the estimated rate of plagioclase evap-oration to yield values for L as a function of grainsize and temperature. Results (Fig. 3) show thatgrains of crystalline feldspar with diameters of20 Wm or more will evaporate with L high enoughto preclude K isotope fractionation (Lv 100) attemperatures corresponding to the region of plan-et and asteroid formation in the early solar neb-ula. The minimum grain size consistent with high-L evaporation can be extended downward toabout 1 Wm if evaporation took place furtherout in the nebula at temperatures of 600 K orless. High-L evaporation of K from plagioclaseis supported by the lack of K depletion in plagio-clase residues following evaporation [27].

Comparing the life span of grains in Fig. 3 withthe time scales for transport in the nebula (Section3) one ¢nds that felspar grains less than severalmm in diameter could have evaporated com-pletely at heliocentric distances as great as 3 AU

(i.e., as far out as the outer limit of the present-day asteroid main belt). In this same region andunder the same conditions (i.e., Langmuir evapo-ration) the amount of crystalline forsterite lost tothe vapor phase would be minimal (Fig. 4). Fur-ther out in the nebula, beyond 3 AU, the rate offeldspar evaporation allows for signi¢cant massloss at temperatures as low as 600 K while theelements comprising forsterite would remain en-tirely in the solid. It follows that appreciableamounts of alkali elements residing in crystallinefeldspathic materials would have been lost to va-por in the nebula while moderately refractory el-ements like Si and Mg would have been retainedin solids until the dust reached the region 6 1.5AU from the Sun (Figs. 3 and 4).

Figs. 3 and 4 suggest that evaporation of ferro-magnesian and feldspathic crystalline solids at dis-tances from the protosun of V3 AU would haveresulted in loss of K and other alkalis (e.g., Rb,Table 2) relative to Ca, Si, Mg, and other morerefractory lithophile elements without fractiona-tion of the isotopes of K. Volatilization of feld-spar would have moved Al from solid to vaporwhile leaving Ca in the solid (Eq. 11). Mobility ofAl relative to Ca is a signature of feldspar^vaporreaction since it contrasts with the behavior ofthese elements during evaporation of moltenchondritic material where Al and Ca are bothretained by the condensed phase [25,33]. It hasbeen suggested that Al may be depleted by asmuch as 15% relative to Ca in at least the upperportion of Earth's mantle [40]. Depletions of Alrelative to Ca in mantle rocks are usually attrib-uted to the e¡ects of fractionation during igneousprocesses rather than to the composition of bulkEarth [29], but an alternative explanation could

Fig. 4. Contours (solid lines) of logL for Mg in forsterite asa function of temperature and grain size based on parame-ters in Table 3. Also shown are contours (gray lines) for logdwhere d (s) is the life span of an evaporating grain of forster-ite assuming a spherical geometry.

Table 3Arrhenius rate equations k = A exp(3Ea/(RT))

k A Ea Notes(kJ/mol)

DalbiteK , cm2/s 3.0U1033 242.0 [38]

rç plagioclase, cm/s 82.3 226.4 calculationdescribed in text

DforsteriteMg , cm2/s 1.3U106 608.0 [47]

rç forsterite, cm/s 4.8U108 574.4 [22,34]

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be incongruent vaporization of feldspathic plane-tary precursors.

10. Conclusions

Consideration of a simple physical model inwhich mineral grains with ¢nite K di¡usivitiesshed mass by evaporation shows that homogene-ity in 41K/39K among rocky bodies of the SolarSystem does not preclude extensive mass loss byevaporation relative to a CI reservoir in the pro-toplanetary solar nebula. Instead the data con-strain the grain size, temperature, and durationattending mineral evaporation through the e¡ectsthat these factors have on the evaporative^di¡u-sive Peclet number. Signi¢cant depletion in K andother alkali elements without changes in 41K/39Kamong rocky bodies of the Solar System is ex-plained if the progenitor K hosts were crystallinemineral grains that evaporate more rapidly thanferromagnesian crystalline phases. Low masses ofmoderately refractory rock-forming elements likeSi and Mg relative to a CI chondrite reservoir insome Solar System bodies support the proposedevaporation model.

It is suggested that transfer of elements betweena gaseous and condensed state varied across theearly solar nebula in response to temperature andthat evaporation of solid dust comprising the pre-cursors to planets can not be ruled out as a mech-anism for this transfer.

Acknowledgements

The author wishes to thank Denis Marc Audetfor his help in the numerical calculations, andDon Porcelli, S. Ross Taylor, Robert N. Clayton,and an anonymous reviewer for thoughtfulcriticisms and suggestions that greatly improvedthe ¢nal product. Funding was provided by anaward from PPARC, UK.[AH]

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assessing the implications of k isotope cosmochemistry for

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