Inverse modeling of argon step-release data from lunar impact spherules

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi:1.129/25je2646, 26 Inverse modeling of argon sep-release daa from lunar impac spherules Jonahan Levine 1,2,3 and Rober A. Rohde 1 Received 23 November 25; revised 3 March 26; acceped 9 March 26; published 23 June 26. [1] We use argon sep-release measuremens o model he iniial disribuion of argon isoopes in 177 lunar impac spherules. The speed of modern compuers permis us o approach his inverse problem in new ways, and he echniques we develop may be exended o sudy a wide range of samples, diffusing species, and geomeries. Lunar spherules, by virue of heir simple shapes and hisories, seem o be excellen candidaes for inverse modeling. Neverheless, impac spherules preserve chemical or maerial heerogeneiies ha are relics of heir paren maerials. As a resul, we find ha he disribuion of argon isoopes in mos impac spherules is more complex han can be meaningfully consrained by a pracical number of precise measuremens. The spaial disribuion of argon from differen sources, such as solar implanaion, would be beer probed in spherules by oher echniques, such as sepwise eching. Ciaion: Levine, J., and R. A. Rohde (26), Inverse modeling of argon sep-release daa from lunar impac spherules, J. Geophys. Res., 111,, doi:1.129/25je Inroducion [2] Argon isoopes are measured exensively in radioisoopic daing experimens on boh erresrial and exraerresrial samples. In lunar samples, he sable isoopes 36,38,4 Ar are derived from several sources, including decay of 4 K, solar implanaion, and spallaion reacions induced by cosmic radiaion [Turner e al., 1971]. There may also be conaminaion from erresrial amospheric argon. Shorlived 37 Ar and 39 Ar are arificially creaed by neuron irradiaion, primarily of 4 Ca and 39 K [Merrihue and Turner, 1966; Turner, 197; Turner e al., 1971]. (Neuron irradiaion creaes small amouns of he sable argon isoopes as well, bu we monior and correc for hese inerfering nuclear reacions. Likewise, small amouns of cosmogenic 37,39 Ar are creaed on he Moon, bu he sauraion concenraions of hese isoopes are negligible, and moreover many 37 Ar half-lives have elapsed since he lunar samples were brough o Earh.) Raios of argon isoopes in several parial releases may be used o evaluae he relaive imporance of hese sources, and o calculae he isoopic composiions of some of hese componens. [3] Albarède [1978] was he firs o use isoopic daa from sepwise heaing experimens o model he iniial disribuion of each isoope in a sample. If he iniial disribuion of argon isoopes in lunar samples can be known, i would be of grea imporance for consraining he energy specrum of 1 Deparmen of Physics, Universiy of California, Berkeley, Berkeley, California, USA. 2 Formerly a Laboraorio TANDAR, Comisión Nacional de Energía Aómica, Buenos Aires, Argenina. 3 Now a Chicago Cener for Cosmochemisry and Deparmen of Geophysical Sciences, Universiy of Chicago, Chicago, Illinois, USA. Copyrigh 26 by he American Geophysical Union /6/25JE2646 implaned solar paricles, idenifying erresrial conaminaion, deducing hermal hisories, or recognizing argon losses due o nuclear recoil during neuron irradiaion. [4] Lunar impac spherules [Culler e al., 2; Levine e al., 25] (see Figure 1) are glass droples ha quench from meled or vaporized rock in he afermah of a meeorie impac on he Moon. To deermine spherule formaion ages wih he 4 Ar/ 39 Ar isochron echnique, Culler e al. [2] and Levine e al. [25] degassed individual spherules by sepwise heaing wih infrared lasers. Impac spherules can be disinguished from lunar volcanic spherules by several crieria [e.g., Delano and Livi, 1981], and Culler e al. [2] and Levine e al. [25] argued for impac origin of heir spherules chiefly on he basis of heir chemical composiions, chemical heerogeneiy, and young ages. In his paper, we use he argon isoopic daa acquired in hese geochronology experimens o follow Albarède [1978] in asking wheher and how sepwise release daa can meaningfully consrain he iniial disribuion of argon isoopes in a sample. Lunar spherules provide a paricularly simple geomery in which o aemp his analysis. [5] Argon is ranspored wihin specimens by diffusion [e.g., Turner e al., 1973], and aoms are released when hey diffuse hrough he surface. Each parial release herefore includes mixures of argon from many differen iniial locaions in he spherule. Forward modeling of he diffusion process consiss of calculaing how much argon would be released in each sep of a specified heaing program, given a cerain iniial disribuion of argon isoopes. We review approaches o he forward problem in secion 2. Because he diffusion equaion is linear in he concenraion, we may imagine he iniial concenraion o be a linear superposiion of several funcions, he evoluions of which we can calculae separaely. In subsequen secions, we follow Albarède [1978] in aemping he inverse of his problem, 1of13

2 consraining he iniial disribuion of argon wih he measured parial releases in each heaing sep. The lineariy of he diffusion equaion makes his a linear inverse problem. [6] Our work akes advanage of he speed of conemporary digial compuers o perform a more complee search for inverse soluions han could have been achieved earlier. We are herefore able o generalize and exend he mehod of Albarède [1978], and develop new algorihms wih which o approach he inverse diffusion problem. In he remainder of his paper, we presen soluions o he forward and inverse problems, and apply hese o he argon daa acquired from lunar impac spherules by Culler e al. [2] and Levine e al. [25]. 2. Forward Problem 2.1. Formulaion of he Problem [7] To mahemaically describe he release of argon from a lunar spherule, we model he specimen as a perfec sphere, and we assume ha argon diffuses wihin he spherule wih an effecive diffusiviy D ha migh depend on emperaure bu no on posiion. We represen he iniial (i.e. =) concenraion of a cerain isoope a posiion r inside he spherule by c(r, = ). The concenraion anywhere wihin he spherule a any laer ime can be deermined by solving he diffusion equaion, r 2 cðr; Þ ¼ 1 Þ : Figure 1. Scanning elecron micrographs of impac spherules aken from Apollo 12 soil The nearly perfec spherical shape and smooh surface exure of spherules (a) 5, (b) 71, and (c) 4 are shared by roughly half of he lunar impac spherules. (d) Spherule 8, which is nearly spherical bu has a blocky surface exure. Oher impac spherules are vireous bu ellipsoidal, including spherules (e) 68 and (f) 6. We observed boh oblae and prolae ellipsoids. Finally, his sudy included a small number of impac spherule fragmens, including (g) spherule 45 wih conchoidal fracure and (h) spherule 49, which reveals he surfaces of bubbles in he fracured face of he spherule. Scale bar in each panel represens 1 mm. The diffusion equaion relaes spaial derivaives of he concenraion (on he lef side of equaion (1)) wih is ime derivaive (on he righ side). Given he geomery of a spherule, i is mos convenien o express he concenraion in spherical polar coordinaes (a radial coordinae, which we denoe by r, and azimuhal and polar angular coordinaes), wih he origin a he cener of he spherule. [8] The boundary condiion appropriae for his problem comes from he fac ha he spherules of Culler e al. [2] and Levine e al. [25] were degassed wihin he exracion line of a noble gas mass specromeer evacuaed o a pressure of 1 9 am. We model he ulra-high vacuum surrounding he spherule by imposing a zero-concenraion boundary condiion on he argon a r = a, where a, he spherule radius, is ypically 125 mm. [9] Individual spherules were incremenally degassed by infrared laser heaing (wih an argon ion laser in he case of Culler e al. [2] and a CO 2 laser in he case of Levine e al. [25]), which allows for lower argon backgrounds and more rapid heaing and cooling han ypically available in a furnace; however, he mahemaical reamen we presen is independen of how he samples are heaed. One shorcoming of he laser heaing sysem used in hese experimens is ha he emperaure aained by he spherule in each sep is no measured. Spherules in hese sudies were held a high emperaure for 15 s (long compared wih he 1 2 s required for hea o diffuse approximaely one spherule s radius hrough basalic glass), hen lef unheaed while he gas released during each heaing pulse was analyzed by mass specromery. A soluion o he forward problem consiss of evolving he iniial concenraion c(r, = ) forward in ime under he diffusion equaion, subjec o he 2of13

3 zero-concenraion condiion imposed on he boundary, and calculaing how much of each argon isoope would have diffused hrough he surface during each heaing sep. [1] The lunar impac spherules of Culler e al. [2] and Levine e al. [25] ypically released a oal of 1 14 mol of 4 Ar and 36 Ar, wih smaller amouns of oher isoopes. The argon yield was divided among 7 14 heaing seps, hough some heaing seps released argon a levels indisinguishable from insrumenal blanks. (Blank levels are differen for each isoope, and depend on he hisory of he mass specromeer, bu are ypically a few 1 18 mol of Ar, and 1 17 mol 4 Ar.) This relaively small number of measuremens implies ha we have only a small number of inpus for he inverse problem, or, in oher words, a small number of useful consrains on he iniial disribuion of argon in he spherules. In fac, no finie number of measuremens could compleely deermine he iniial argon concenraion if arbirarily complicaed disribuions are permied. We can make progress in he inverse problem only if we forgo uniquely specifying he iniial disribuion, and insead search for iniial concenraion profiles ha are consisen wih he measured daa. Any concenraion profile ha is consisen wih all he measured daa is considered a soluion o he inverse problem; we may favor one soluion over ohers only if we have addiional informaion abou wha a plausible iniial concenraion profile migh look like Iniial Condiion of he Impac Spherules [11] We assume ha he iniial disribuion of argon in lunar impac spherules is spherically symmeric. This assumpion is parly moivaed on physical grounds ha we discuss below. Addiional moivaion for assuming spherical symmery comes from he fac ha even he larges number of measured parial releases on individual impac spherules would poorly consrain he iniial argon disribuion if i varied in all hree dimensions. In such a case, N measuremens would permi he modeling of he argon disribuion wih a spaial resoluion characerized by a wavelengh l a/n 1/3. By conras, if he iniial concenraion were spherically symmeric, hen he same number of measuremens permi a much finer spaial resoluion, of order l a/n. In he case of spherical symmery, even seven measuremens allow us o describe an iniial radial concenraion profile raher well, for example by esimaing he firs seven erms in is Taylor series expansion. [12] There is good reason o believe ha several componens of he iniial argon endowmen of some lunar impac spherules ough o be spherically symmeric. Mos spherules are quie nearly spherical in shape (Figure 1), wih long and shor axes ha differ in lengh by less han 1% [e.g., Fulchignoni e al., 1971]. Oher impac spherules form oblae or prolae ellipsoids, hough irregular spherule fragmens exis as well. Furhermore, since he peneraion deph of primary galacic cosmic rays (energies >1 MeV) grealy exceeds he diameer of a spherule, he disribuion of spallaion 36,38 Ar is uniform in a chemically well-mixed spherule. By conras, implaned solar argon and parenless 4 Ar [e.g., Manka and Michel, 1971; Wieler and Heber, 23] mus be preferenially implaned in he upward-facing surface of a spherule. If impacs frequenly reorien spherules in he lunar soil, he accumulaion of implaned gases will approximae a symmeric ouer coaing. However, even if he implaned argon is heavily biased oward one par of he surface, he diffusion of hese componens will a firs be similar o ha of argon disribued symmerically. Because solar argon is implaned in he ouer 2 nm of lunar grains [Eberhard e al., 197], he radial concenraion gradien iniially exceeds laeral gradiens by a large facor, and hence he ne flow will be nearly radial in early heaing seps. Laer in he heaing program, when radial and laeral concenraion gradiens are nearly equal, spherical symmery may be a poorer approximaion for he behavior of hese isoopes. [13] In addiion, all impac spherules were formed a high emperaure from molen or vaporized rock. We resriced his sudy o hose spherules for which isochron ages were deermined and which lacked evidence for undegassed class, because hese spherules remained ho for long enough o degas any inheried argon. I is reasonable o begin, herefore, wih he assumpion ha many impac spherules are well-mixed wih respec o a leas some chemical elemens, so ha heir radiogenic 4 Ar, cosmogenic 36 Ar and 38 Ar, and reacor-produced 37,39 Ar are uniformly disribued. [14] Surprisingly, he assumpion of chemical uniformiy is relaively weak. Measured 37 Ar/ 39 Ar raios vary by a facor of 3 among heaing seps on individual impac spherules, implying ha he raio of heir paren isoopes, 4 Ca and 39 K, varies by a leas his facor over he spherule volumes [Levine e al., 26]. Nearly all impac spherules released argon wih a relaively low 37 Ar/ 39 Ar raio early in he heaing program, and his raio increased as heaing coninued (Figure 2). Levine e al. [26] infer from his behavior ha impac spherules have relaively calcic cores and relaively poassic exeriors. This sense of zoning is conrary o ha expeced if spherules quenched from impac mel: If a mel drople were iniially uniform, i would develop a poassium depleion in he ouermos porion by preferenial evaporaion of volaile elemens [Yu e al., 23]. [15] The variable 37 Ar/ 39 Ar raios do no specify wheher he K concenraion, he Ca concenraion, or boh are nonuniform. The relaively calcic cores migh arise from undigesed class of high-ca (and presumably high meling emperaure) phases being presen inside mos impac spherules. Any such class mus have escaped chemical assimilaion even hough hey were degassed of heir inheried argon when he impac spherules enclosing hem were formed (hese class, if hey are indeed presen, do no have sysemaically higher radiogenic 4 Ar conens). Even if impac spherules have undigesed class a heir ceners, heir poassium concenraions may be nearly uniform. Alernaively, impac spherules could have formed by condensaion of impac vapor, wih more refracory elemens condensing firs and hen being manled by less refracory elemens. In his case, he concenraions of boh poassium and calcium migh be nonuniform. Finally, poassium could be enriched in he surfaces of impac spherules by an unidenified process, leaving calcium concenraions nearly uniform. [16] For reasons ha we discuss in secion 3.1, i is necessary for he inverse problem o assume ha a reacor-produced argon isoope derived from eiher poassium or calcium has an approximaely uniform disribuion in each 3of13

4 Figure 2. Evoluions of 37 Ar and 39 Ar releases from (lef) Apollo 12 and (righ) Apollo 14 impac spherules. For each spherule, he cumulaive fracions of 37 Ar (derived from 4 Ca) and 39 Ar (derived from 39 K) released afer each sep are ploed. A rack connecs he daa for each spherule. A chemically uniform spherule would release 37 Ar and 39 Ar in he same proporion during each heaing sep, and such a rack would lie along he dashed diagonal line. Mos impac spherule racks are enirely below his line, indicaing ha impac spherules release 39 Ar before he corresponding fracion of 37 Ar. This implies ha impac spherule rims are generally more poassic, and ha cores are relaively calcic. The field above he main diagonal, wih few impac spherule racks, would correspond o chemical zoning in he opposie sense, wih relaively calcic exeriors and poassic ineriors. Uncerainies shown represen 1s. spherule. We performed our analysis under he assumpion of uniform 39 Ar (from poassium) and uniform 37 Ar (from calcium), each in urn. We canno direcly es he possibiliy ha boh elemens were disribued nonuniformly in he impac spherules of Culler e al. [2] and Levine e al. [25], excep by considering boh of he alernaives. I would be ineresing o disinguish among hypohesized modes of spherule formaion based on he disribuion of calcium and poassium by measuring Ca/K raios in siu in a new se of impac spherules [Levine e al., 26], hough he scarciy of poassium in lunar maerials makes his a challenge for many analyical echniques Solving he Forward Problem [17] The assumpion of spherical symmery allows us o ignore he angular dependence of he Laplacian operaor r 2 in equaion (1), and o cas he diffusion equaion as r; r 2 r ð Þ ¼ Þ The concenraion c(r,) is found by solving equaion (2) subjec o he boundary condiion of zero concenraion a r = a. The oal amoun of any isoope presen a ime is calculaed by inegraing c(r,) over he volume of he spherule. The amoun of each isoope released by he imes ={ i } a which measuremens are made may be calculaed by subracing he amoun of he isoope remaining inside he spherule a i from he amoun presen a =. [18] There are several mahemaical approaches o solve he diffusion equaion. Here we presen wo differen echniques. The Fourier series soluion and he Green s funcion soluion are boh expressed as infinie sums, and sricly are equivalen o one anoher. However, hey have differen compuaional advanages in pracical cases, where he infinie sums are necessarily runcaed afer a small number of erms. [19] In he Fourier series mehod, he unknown concenraion c(r,) is se equal o he produc of a funcion of r and a funcion of. The diffusion equaion (2) separaes ino ordinary differenial equaions in each variable, boh of which are elemenary. The soluion ha obeys he boundary condiions is cr; ð Þ ¼ 2 X 1 ar n¼1 sin npr e n2 p 2 a Z a cr ð ; Þr sin npr dr : ð3þ a Here, o simplify noaion, we have wrien he soluion in erms of he dimensionless ime = R DT ð Þ a d [Albarède, ; Shuser and Farley, 24], where we explicily noe he dependence of he diffusiviy on emperaure T. The dimensionless ime is scaled in such a way ha = 1 4 is he average ime a which a paricle saring a he cener of he spherule escapes hrough he surface. From equaion (3), we can calculae he oal amoun G of he argon isoope ha remains in he sphere a any ime by inegraing he concenraion over he volume, GðÞ ¼ X1 n¼1 ð 1 n Þ nþ1 Z a e n2 p 2 8ar cr ð ; Þsin npr dr : a The amoun released in a heaing sep ha lass from i o i+1 is simply ð4þ R i ¼ Gð i Þ Gð iþ1 Þ: ð5þ 4of13

5 [2] We noe hree feaures of he soluion in equaions (3) (5). Firs, he measured quaniies are expressed as infinie sums of inegrals, which may, in general, be impossible o solve analyically. Therefore, for pracical purposes, we will need boh o approximae he inegrals and o runcae he infinie sums afer a reasonable number of erms. Second, for larger han abou.1, only a few erms are necessary o approximae he infinie sum, because of he rapidly decreasing facor e n2 p 2. This is advanageous, insofar as he series converges rapidly. Third, for very small values of, he series converges exceedingly slowly, and many erms need o be reained in he sum. [21] The Green s funcion approach is an alernaive mehod for solving he diffusion equaion. Insead of represening he concenraion of argon by a sum of sines, we decompose he iniial disribuion ino Dirac d-funcions. The d-funcion has uni area and infiniesimal widh; i evolves under he diffusion equaion ino a Gaussian whose widh broadens wih ime. The relaive weigh of each Gaussian in he concenraion a ime is he same as he weigh of he corresponding d-funcion in he iniial concenraion. The Green s funcion expansion, hough i appears more cumbersome han he Fourier series, offers compuaional advanages ha we describe below. [22] Expressed as a sum of broadening Gaussians, he concenraion of argon inside he spherule is given by cr; ð Þ ¼ Z a cr ð ; Þ r ar 1 pffiffiffiffiffiffiffiffi 4p X 1 n¼ 1 e ðrþ2na rþ 2 =4a 2 ðrþ2naþrþ e 2 =4a 2 dr : ð6þ The infinie sum of posiive and negaive Gaussians, spaced every 2a apar, is required o mach he boundary condiions a r = and r = a; he spreading of he unphysical image sources exacly cancels he spreading of he real argon a he boundaries, for all imes. Though he Green s funcion soluion, like he Fourier series soluion, is obviously unphysical ouside he sphere, i does obey he diffusion equaion inside he region < r a, and i obeys he boundary condiions. The fac ha he diffusion equaion (2) is linear and homogeneous is sufficien o ensure ha a soluion obeying paricular boundary condiions is unique, and hus ha equaion (6) is equivalen o he Fourier series soluion in equaion (3). [23] From equaion (6), we may develop alernaive expressions for he oal amoun of an argon isoope remaining in he spherule as a funcion of ime, and for he amoun released in a paricular heaing sep. Inegraing he concenraion over he volume of he spherule, we find p GðÞ ¼ ffiffiffiffiffi 4p a 3 Z 1 xcðx; Þdx X1 n¼ 1 p ffi 2 2 e 8 2 p 2 ffiffiffi 2n x 1þ2n x p ffi 2 < 4 e 2 þ e 1þ2nþx ffi 2 p 2nþx 2 p 2 e ffi : þ ffiffiffi p 1 þ 2n x p ðx 2nÞ erf p 2 ffiffiffi 2n x erf p 2 ffiffiffi ) 1 þ 2n þ x þ ðx þ 2nÞ erf p 2 ffiffiffi 2n þ x erf p 2 ffiffiffi : ð7þ Here we use a dimensionless radial variable x = r/a, and erf represens he sandard error funcion. As before, he amoun of he isoope released in a heaing sep from i o i+1 is given by R i ¼ Gð i Þ Gð iþ1 Þ: ð5 Þ [24] Unlike he Fourier series soluion (3) (4), which requires many erms of he infinie sum o be reained a early imes bu which quickly converges for large, he sum over image sources in he Green s funcion soluion (6) (7) requires few erms iniially, and more as ime advances. A early dimensionless imes (e.g., 1 4 ), he Fourier series expansion in equaion (4) requires a number of erms n which exceeds 1/2 in order o make a good approximaion; he Green s funcion expansion, on he oher hand, is saisfacory wih fewer han 2 erms even a lae dimensionless imes (.5). Because of his, we generally prefer o employ he Green s funcion soluion where pracical. [25] For he simple case of an iniially uniform concenraion, he inegrals in equaions (6) and (7) may be calculaed analyically, and he fracion of he isoope released by any ime may be quickly obained. The expression for he fracion released by dimensionless ime is f uniform ðþ ¼ 1 X1 n¼ 1 rffiffiffi h 2n 2 þ 2n 2 þ 1 p e ð n2 þ2nþ1 Þ= þ 2n 2 þ 2 3 e n 2 = 2n 2 þ n þ 2 1 e ð4n 2 4nþ1Þ=4 þ 2n 2 n þ 2 1 e ð i 4n 2 þ4nþ1þ=4 þ 1 4n 3 3n þ 6n þ 1 : 2 2n þ 1 erf p 2 ffiffiffi n þ 1 þ 2erf p ffiffiffi þ 4n 3 6n þ 6n n n þ 1 erf p ffiffiffi erf p ffiffiffi þ 4n 3 3n þ 6n 1 2n þ 1 erf pffiffiffi : ð8þ Employing equaion (8) where possible removes a poenial source of error from numerically inegraing over he Green s funcions in equaion (7). Wih hese explici soluions o he forward problem, we now proceed o develop he mahemaical ools for he inversion. 3. Tools for he Inversion 3.1. Deermining he Time [26] An imporan firs sep in he inversion process is deermining he dimensionless imes a which measuremens were made. The definiion = R DT ð Þ a d [Albarède, ; Shuser and Farley, 24] is ofen no useful for calculaing direcly, as he diffusiviy during each heaing sep may no be known. This is he case for he daa of Culler e al. [2] and Levine e al. [25]. Wihou known values of, i is impossible o inver eiher equaions (4) or (7) from measured releases. Albarède [1978] circumvened his problem by assuming ha reacor-produced 37 Ar had been uniformly disribued and using parial releases of his 5of13

6 isoope o calculae for each heaing sep. Because he parial release daa from lunar impac spherules show ha he iniial concenraions of 37 Ar and 39 Ar canno boh have been uniform (Figure 2) and because we do no have an a priori reason o favor one of hese as a uniform isoope, we performed all of our analysis wice, firs wih he assumpion of uniform 39 Ar concenraion and hen wih he alernaive assumpion of uniform of 37 Ar. Our resuls are quie similar in he wo cases. [27] Boh reacor-produced isoopes are radioacive. Especially in he case of shor-lived 37 Ar (half-life 35 days), one mus correc measured parial releases for he effec of radioacive decay, so as no o underesimae he fracion of he spherule degassed in laer heaing seps. In pracice, we use parial release values ha are correced for insrumenal background, mass discriminaion (i.e., isoopic fracionaion wihin he gas exracion sysem, mass specromeer, and deecor), radioacive decay, and inerfering nuclear reacions during irradiaion. The las correcion allows us o inerpre our resuls more easily: our values for 39 Ar, for example, are correced for a small amoun derived from 42 Ca(n,a) 39 Ar reacions, and herefore represen only he quaniy of his isoope ha is derived from 39 K. [28] Following Albarède [1978], we used he fracional releases of he assumed-uniform isoope o inver equaion (8) and o compue a he end of each heaing sep. Equaion (8) is no inverible in closed form, bu because i is a monoonic funcion of ime, we can find wih arbirary precision by a mehod of successive approximaions. If all he argon isoopes have he same diffusiviy, hen he values of hus deermined apply o all he isoopes. If he diffusiviy of argon isoopes in a spherule imposes a mass-dependen fracionaion, he values of for each pisoope ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are differen, bu obey he relaionship (i) / (uniform) = m uniform =m i for isoopes of mass m i. This effec would be small for argon isoopes, and i has been repored in only one experimen [Trieloff e al., 25]. Such a fracionaion of argon has never been unambiguously observed in glass; moreover, a suie of helium diffusion sudies in minerals [Trull and Kurz, 1993; Shuser e al., 24] and glass [Trull and Kurz, 1999] have yielded apparenly differen mass-fracionaion laws, and so i is uncerain wha expecaion one should have for lunar glass spherules. We repeaed our analysis wih mass-dependen and massindependen values of, and found ha changes his small in (<6% for 36 4 Ar) do no affec our conclusions. Frequenly, successive values of are separaed by a facor of 2 1, which is much larger han he offses ha could be due o mass-dependen fracionaion Inversion Techniques [29] We nex ouline a general scheme for invering parial release daa o find plausible iniial concenraion profiles of argon isoopes. The inversion begins by decomposing he iniial concenraion of all isoopes ino N componens and assuming funcional forms f i (x) for each. In erms of hese componens, he iniial concenraion is expressed by cx; ð Þ ¼ b i f i ðþ; x where he undeermined parameers b i represen he weighs of each componen. In equaion (9) and in wha follows, ð9þ summaion over repeaed indices is implied. The componens f i (x) may, in general, be any funcions over he inerval x 1. One uses equaions (3) or (6) o calculae he amoun of each argon isoope every componen would release from he spherule during inervals defined by he measuremen imes j. Leing he marix elemen W ij denoe he calculaed release from componen i beween imes j and j+1, and represening he se of measured isoopic releases by M j, we hen have M j ¼ b i W ij : ð1þ If he number N of componens is chosen o be equal o he number of parial releases, hen W is a square marix. If is inverse exiss, an iniial concenraion profile is given by cx; ð Þ ¼ M j W 1 ji f i ðþ: x ð11þ If he funcion c(x,) found in his way is physically reasonable, hen i represens a plausible iniial concenraion funcion consisen wih he measured sep releases. [3] In general, here is no guaranee ha an inverse exiss o W, he marix of parial releases from each componen, or ha he iniial concenraion given by equaion (11) is physically plausible. For example, if wo or more heaing seps begin a already large values of, such ha he Fourier series in equaion (4) can be approximaed by a single erm, he corresponding rows of he W marix will be proporional o one anoher and he marix mus be singular. The runcaion of he Fourier series afer only one erm is a valid approximaion for heaing seps afer.25, when he facor e n2 p 2 drops by 1, imes beween he leading erm and he n = 2 erm, making he second erm negligible. Physically, he singulariy of W means ha diffusion blurs he iniial concenraion by lae dimensionless imes, so ha, o he exen ha he second erms in he Fourier series may be negleced, no new spaial informaion can be gleaned from he final heaing sep. If he vecor of measuremens M (and is associaed uncerainy envelope) lies ouside he range of marix W, hen no inverse soluion exiss. This is sufficien o show ha he se of parial releases M is inconsisen wih any spherically symmeric iniial concenraion ha may be expressed as a linear combinaion of he chosen f i (x). [31] I is clear ha differen choices of he funcions f i (x) in equaion (9) will give rise o differen invered iniial concenraion profiles. This nonuniqueness is a consequence of rying o consrain a coninuous funcion (i.e. he iniial concenraion) wih only a small number of measuremens. Some of he nonunique soluions may be dismissed on physical grounds, and ohers, perhaps soluions ha have relaively smooh concenraion profiles, may be favored. The sronges physical consrain ha we may apply a priori is ha he iniial concenraion of argon be non-negaive everywhere. Some choices of f i (x) may resul in a nonnegaive c(x,) even as oher ses of expansion funcions yield unphysical negaive concenraions while sill being mahemaically consisen wih he observed sep releases. Sabaier [1977a, 1977b] describes how convex analysis may be used o impose he consrain of nonnegaiviy on he oherwise linear inverse problem. In 6of13

7 he presen work, we insead rely on an exhausive search algorihm ha we have wrien o resric our soluions o nonnegaive iniial concenraions. [32] Albarède [1978] used a paricular se of decomposiion funcions f i (x), for which he marix elemens W ij are proporional o he fracion of an iniially uniform isoope ha would remain in he spherule afer a ime i + j. The advanage of his se of funcions is ha an algorihm opimized for rapid implemenaion of equaion (8) (or is Fourier series equivalen) could also rapidly compue he marix elemens W ij. This advanage is miigaed by he speed of modern compuers, which can quickly calculae parial releases from arbirary iniial concenraions; our Macinosh Powerbook G4 running Malab calculaes parial argon releases in.2 s, carrying he infinie sums as far as needed for par-per-billion accuracy. This allows us o explore oher choices of componens f i (x). We examined he lunar spherule daa of Culler e al. [2] and Levine e al. [25] using he mehod of Albarède [1978] and many oher ses of f i (x), and never found any iniial concenraions ha were nonnegaive everywhere and consisen wih all he measuremens on any impac spherule Inversion by Exhausive Searching [33] Our failure o find nonnegaive iniial concenraion profiles using equaion (11) ha are consisen wih he parial releases of argon from lunar impac spherules led us o moun an exhausive search. In order o deermine wheher any plausible spherically symmeric iniial concenraions could have yielded he measured argon releases, we allow ourselves many more degrees of freedom han we have measuremens. Of course, using a larger number of adjusable parameers emphasizes he fac ha any soluions we find are no unique; wih he exhausive search, we only seek o learn wheher any nonnegaive soluions exis o he inverse problem, no wheher a paricular inverse soluion represens he rue iniial disribuion of argon isoopes in an impac spherule. [34] The requiremen of nonnegaiviy is difficul o map ono he weighs of an arbirary se of componens f i (x), so we use nonoverlapping op-ha or boxcar shaped funcions cx; ð Þ ¼ X i 8 >< < x p i b i p i < x q i ; >: q i < x < 1: ð12þ where he parameers p i and q i are he boundaries of he ih op ha, which rises o a value of b i. The requiremen ha he iniial concenraion be non-negaive is imposed by resricing he search space o b i for all i. We decomposed he iniial concenraion ino 118 op-has, and ried o adjus he weigh of each in order o mach he 7 14 measuremens on each spherule. Because we expec ha implaned solar argon is concenraed near he surface of he spherules, we chose he boundaries of our op has o be more closely spaced near x = 1. In he limi of arbirarily many op has, one approximaes a coninuous iniial concenraion; we found ha 118 op has performed equally well as wice ha number, in half he compuing ime. [35] The difficuly in our mehod is ha a sraighforward marix inversion canno enforce he requiremen ha all he op has have nonnegaive weighs. Insead, we have devised a search algorihm ha seeks o minimize he errorweighed misfi beween he measuremens and a model calculaion, given by h 2 ¼ 1 N X N j¼1 M j P 2 b i W ij i ; ð13þ while demanding ha all of he op-ha weighs b i obey b i. The se of s j are he uncerainies in each measuremen. Our search algorihm (auxiliary maerial 1 Tex S1) cycles hrough hree differen sraegies o minimize he misfi. Beginning wih a saring guess of he b i, we firs ake advanage of he fac ha equaion (13) is a quadraic, concave-up funcion of each of he b i, and we ierae a process of adjusing each elemen sequenially so ha h 2 converges o is local minimum. If a negaive elemen of b i would lower h 2 furher, i is assigned a value of zero (his minimizes h 2 subjec o he physical consrain of nonnegaiviy). Second, afer a fixed number of ieraions, we rescale he enire se of b i in an effor o minimize h 2. This helps o converge on he opimal soluion more rapidly han adjusing each of he b i individually. Third, o avoid becoming rapped in local minima of h 2 raher han is global minimum, we repeaedly le he se of b i ake random jumps o oher areas of he parameer space, o see wheher he misfi is smaller elsewhere. The size of he random jumps is gradually made o decrease as he algorihm converges on a soluion; his is characerisic of so-called simulaed annealing opimizaion algorihms [e.g., Press e al., 1989], which mimic he flucuaions in an annealing crysal as i cools o a low-energy configuraion. [36] We esed our algorihm on forward-modeled parial releases calculaed from iniial argon disribuions ha were spherically symmeric, wih radial concenraion profiles given by polynomial, exponenial, logarihmic, and rigonomeric funcions, and combinaions of hese. The calculaed releases were assigned uncerainies of a few percen, which is ypical of he measuremen precision of he lunar spherule daa. In all cases, our search algorihm ook less han one minue o converge on a plausible spherically symmeric iniial concenraion profile ha mached he model releases. The misfis in hese inversions were usually smaller han h 2 =1 2. The performance of our search algorihm on simulaed releases calculaed from idealized spherically symmeric iniial concenraions shows ha, in cases where a nonnegaive soluion o he inverse problem is known o exis, he algorihm is able o find one. However, we sress ha he invered concenraions were no he same as he concenraion profiles we used o calculae he model releases, and our algorihm insead found anoher iniial concenraion profile ha was consisen wih he same parial releases. [37] The saisical properies of h 2 defined by equaion (13) are somewha unusual and difficul o assess rigorously, 1 Auxiliary maerial is available a fp://fp.agu.org/apend/je/ 25je2646. s 2 j 7of13

8 because he search space for he inversion is consrained by he requiremen ha b i. We herefore examined he behavior of our search algorihm in Mone Carlo simulaions o address he quesion of how small a residual should be considered a good fi beween measuremens and calculaed parial releases from a paricular iniial concenraion profile. We discuss hese simulaions in deail in he following secion. 4. Analysis of Lunar Spherule Daa [38] We applied our exhausive search algorihm o he parial release daa from a subse of he lunar impac spherules sudied by Culler e al. [2] and Levine e al. [25]. Culler e al. [2] measured parial argon releases from 179 spherules colleced during he Apollo 14 mission o he Fra Mauro Peninsula, of which 19 yielded saisically accepable 4 Ar/ 39 Ar isochron ages [Levine, 24]. Levine e al. [25] examined 178 spherules from an Apollo 12 soil colleced in he Procellarum Basin, finding saisically accepable 4 Ar/ 39 Ar isochrons for 81 of he spherules. A 4 Ar/ 39 Ar isochron ha is saisically accepable indicaes binary mixing of poassium-derived argon (i.e., reacor-produced 39 Ar derived from 39 K and radiogenic 4 Ar from in siu decay of 4 Ar) and surfaceimplaned argon (in his case parenless 4 Ar and solar 36 Ar). If he parial releases canno be resolved ino hese wo componens, no unique age may be deermined, and his likely implies eiher incomplee degassing during spherule formaion or parial loss or redisribuion of argon due o heaing (and possibly shock) from subsequen nearby impacs. Any mechanism for inroducing addiional componens of argon can violae he assumpion of spherical symmery, eiher by affecing he disribuion of argon isoopes or by esablishing a nonuniform diffusiviy inside he spherule. Therefore spherules wihou isochrons are no good candidae physical sysems for our inverse modeling of he iniial disribuion of argon isoopes. We eliminaed all such spherules from our analysis. We also excluded he small number of impac spherules whose argon release paerns showed evidence for undegassed class inside he spherules, which we recognized from a endency oward more radiogenic 4 Ar in he highes-emperaure seps han indicaed by he apparen age of he res of he impac spherule. However, impac spherules wih class ha were degassed of heir argon bu no chemically assimilaed migh remain among hose we analyzed. Such class could hamper our inversions, by inroducing nonuniform chemical composiions or diffusiviies. The oal populaion of analyzed impac spherules consiss of 14 from he Apollo 14 sample and 73 from he Apollo 12 sample. [39] In a number of heaing seps, he amoun of argon released from paricular impac spherules was consisen wih insrumenal background levels. In hese cases, we mahemaically combined parial releases and associaed uncerainies wih daa from he following heaing seps, unil he amoun of argon observed in each combined sep was unambiguously greaer han zero. Therefore inversions of daa from some impac spherules need o mach as few as hree above-background parial argon releases. More ypically, however, six o eigh measuremens need o be mached. [4] We used he exhausive search algorihm o find he nonnegaive spherically symmeric iniial disribuion of argon isoopes which, when evolved under he diffusion equaion, yielded calculaed releases ha agreed mos closely wih he real measuremens on each impac spherule, in he sense of minimizing he h 2 residuals. Apollo 12 resuls are shown in Figures 3 and 4, and Apollo 14 resuls are depiced in Figures 5 and 6. Figure 3 for he Apollo 12 impac spherules and Figure 5 for he Apollo 14 specimens represen daa calculaed under he assumpion of iniially uniform 39 Ar (i.e., parial releases of 39 Ar were used o deermine he value of afer each heaing sep); for Figures 4 and 6, i was assumed insead ha 37 Ar was iniially uniform. Figures 3 6 are logarihmic in heir verical axes, as he range of h 2 exends over many orders of magniude. Summary informaion relaed o Figures 3 6 are given in auxiliary Tables S1 and S2. [41] For he isoope ha was assumed uniform in order o calculae afer each heaing sep, we are guaraneed ha an exac inverse soluion (i.e., he uniform iniial concenraion soluion) exiss. However, given he saring guess we used in he inversion algorihm (we began wih all of he b i =) and he limis we imposed on he number of ieraions, he algorihm generally did no converge on his exac soluion. Insead, i seled for an approximae soluion, whose misfi wih he daa (ypically 1 8 < h 2 <1 3 ) provides one esimae of he value of h 2 o be expeced in cases where parial release daa is ruly inverible. [42] For oher isoopes, here is no guaranee ha an inverse soluion exiss, so we mus address he quesion of wha values of h 2 imply a good fi, and wha values imply ha no spherically symmeric iniial disribuion could accoun for he daa. We used a Mone Carlo approach o deermine he values of h 2 ha could be expeced from our algorihm when rying o inver uncerain daa. For each argon isoope measured from each impac spherule, we found he se of parial releases ha mos closely mached he measured values bu which was exacly consisen wih spherical symmery; we perurbed his se of parial releases by amouns consisen wih he experimenal errors, and used our algorihm o inver he perurbed daa. A leas one inverse soluion mus exis in hese Mone Carlo ess: The spherically symmeric iniial concenraion ha yielded he unperurbed daa will be consisen wih he perurbed daa, wih a misfi of h 2 1. A much smaller misfi migh be found if he perurbed se of parial releases agrees more closely wih a differen spherically symmeric iniial concenraion profile. Figures 3 6 show he range of h 2 values obained in he Mone Carlo ess. On he basis of he Mone Carlo resuls, we adop an approximae crierion of h 2 >2 for concluding, wih 95% confidence, ha a paricular se of measuremens has no spherically symmeric inverse soluion. However, he precise saisical confidence in his conclusion depends weakly on he number of measuremens being fi and on he number of adjusable parameers ha are independen in our inversion. [43] The differen ranges of h 2 found in simulaions corresponding o disinc impac spherules are mosly due o he differen number of measuremens being fi, wih ses of hree parial releases (e.g., Apollo 12 spherule 39) ypically yielding misfis wih h 2 <1 15. Misfis his small were hardly ever observed in simulaions of six or more 8of13

9 Figure 3. Misfis beween observed parial releases for 73 Apollo 12 impac spherules [Levine e al., 25] and bes fiing spherically symmeric models wih 39 Ar assumed uniform. The hick dark lines in each panel show he h 2 difference beween he observed parial releases of he named argon isoope and he calculaed releases from he bes fiing spherically symmeric iniial concenraion. Noe ha he verical axis is logarihmic, and ha beer fis are implied by lower h 2. The dimensionless imes are compued under he assumpion of iniially uniform 39 Ar; herefore a model concenraion wih spherical symmery can always be found o mach he observed 39 Ar parial releases. Shaded bands represen he range of h 2 values from 21 Mone Carlo simulaions (darker shading denoes middle 5%), in which he spherically symmeric model ha mos closely mached he measured parial releases was perurbed by simulaed experimenal errors and hen reinvered. Horizonal dashed lines represen h 2 = 2; as described in he ex, successful inversions, which are lised in auxiliary Table S1, have misfis ha plo below his line. parial releases. All simulaions used he same number of adjusable parameers (118) o mach he parial releases, regardless of he number of measuremens being fi. However, doubling or rebling he number of op-ha funcions did no improve he fis in simulaions wih 6 parial releases. [44] We noe he following feaures of Figures 3 6. Firs, regardless of which isoope was assumed o have had he iniially uniform concenraion, inversions of 36 Ar and 38 Ar parial release daa yield he larges values of h 2, and he smalles number of successful inversions. Misfis characerized by h 2 < 2 are found in inversions of 36 Ar and 38 Ar daa from 15 3% of he impac spherules. A leas 9% of he 36 Ar and 38 Ar in mos impac spherules is implaned, so he scarciy of successful inversions for hese isoopes suggess ha asymmeric solar irradiaion is parly o blame for deviaions from spherical symmery. [45] Second, successful inversions (i.e., h 2 <2)of 4 Ar daa are found for approximaely 5% of he impac spherules, regardless of which reacor-produced isoope was assumed uniform in order o calculae. I is somewha surprising ha more inversions of 4 Ar daa were no successful under he assumpion of uniform 39 Ar, since he disribuion of radiogenic 4 Ar should be idenical o he assumed-uniform disribuion of 39 Ar [Merrihue and Turner, 1966]. Perhaps asymmeric implanaion of parenless 4 Ar [Manka and Michel, 1971; Wieler and Heber, 23], whose disribuion is similar o ha of solar 36 Ar and 38 Ar, makes parial releases of 4 Ar uninverible. However, we observe no simple relaionship beween he fracion of 4 Ar ha is radiogenic and h 2. [46] Third, we consider he reacor-produced isoopes. Under he assumpion of uniform calcium and 37 Ar (Figures 4 and 6), we are able o inver parial releases of 9of13

10 Figure 4. As in Figure 3, bu dimensionless imes of each heaing sep were calculaed on he assumpion of iniially uniform 37 Ar raher han 39 Ar. Therefore a model concenraion wih spherical symmery can always be found o mach he measured 37 Ar parial releases. Successful inversions (i.e., h 2 < 2) are lised in auxiliary Table S1. 39 Ar for approximaely 8% of he spherules. By conras, we obain inversions of 37 Ar releases from only 5% of he spherules under he assumpion of iniially uniform poassium and 39 Ar (Figures 3 and 5). I is unclear wha his discrepancy implies abou he argon conen and chemical composiion of lunar impac spherules. In paricular, we hesiae o conclude on his basis ha he assumpion of uniform 37 Ar is he beer one. In siu spo analyses of K and Ca on a new suie of spherules would show definiively wheher eiher of he reacor-produced isoopes could be expeced o have a uniform iniial concenraion. Such measuremens could also consrain he modes of spherule formaion and growh in he afermah of meeoroid impacs. 5. Discussion and Conclusions [47] Lunar impac spherules seemed a priori o be ideal candidaes for inverse modeling. Their spherical shape makes he diffusion process paricularly easy o describe mahemaically, and he fac ha hey were formed a high emperaures offered he hope of effecive chemical mixing, so ha a leas one of he reacor-produced argon isoopes would be uniformly disribued. The assumpion of spherical symmery, which we adoped in he hope of meaningfully consraining iniial concenraions wih only a few measuremens, mus be reconsidered given he failure of our exhausive search algorihm o find any spherically symmeric iniial concenraions consisen wih parial argon releases for he majoriy of impac spherules. The assumpion ha one isoope had a uniform concenraion, which we made so ha could be deermined for each heaing sep, mus also be reevaluaed. Figures 3 6 show ha, given our assumpions, fewer han one fourh of he impac spherules can have nonnegaive spherically symmeric iniial concenraion profiles ha accoun for he parial releases of all isoopes. [48] Spherical asymmery could arise from shape anisoropy, asymmeric exposure o solar paricle irradiaion, and some kinds of chemical heerogeneiy. We noed from Figures 3 6 ha parial release daa of 36 Ar and 38 Ar, he isoopes dominaed by solar implanaion, were successfully invered by our exhausive search algorihm in he smalles number of cases. One possible explanaion for his is ha asymmeric solar implanaion of hese isoopes rouinely makes parial release daa of 36 Ar and 38 Ar uninverible, given he limiaions of our model. [49] Asymmeric exposure o solar irradiaion would imply ha impac spherules are rarely reoriened a he lunar surface. Since every observed impac spherule has implaned solar 36 Ar and 38 Ar, each spherule mus be brough o he op of he regolih afer is formaion, ye asymmeric exposure requires ha repeaed impacs no 1 of 13

11 Figure 5. As in Figure 3, bu for 14 Apollo 14 impac spherules [Culler e al., 2]. Dimensionless imes were calculaed assuming ha 39 Ar was iniially uniform. The many downward spikes in his figure are an arifac of he smaller number of measuremens on many spherules, compared wih Apollo 12 analyses. Our search algorihm frequenly maches ses of hree measured values o wihin h 2 =1 15. Successful inversions (i.e., h 2 < 2) are lised in auxiliary Table S2. bring spherules back o he surface many imes, wih random orienaions. Because ejeca deposis are spread over a very large area, one possibiliy is ha spherules, or, more precisely, he impac spherules ha lose all heir argon, are deposied upon formaion a he very op of he lunar surface, where hey are exposed o solar irradiaion from only one side. Subsequen impacs in which preexising spherules are no grealy heaed migh be more likely o bury spherules raher han expose hem o furher solar irradiaion, in a new orienaion. Unforunaely, lile is presenly known abou he size of spherule-producing impacs nor he geographical disribuion of spherules in ejeca. An alernaive possibiliy is ha our selecion of impac spherules wih isochron ages may have excluded spherules ha suffered repeaed impacs and reorienaions. [5] We know ha mos impac spherules are chemically heerogeneous on he basis of heir varying 37 Ar/ 39 Ar raios, shown in Figure 2, and from energy-dispersive x-ray analysis on spherule surfaces [Levine e al., 24]. There are hree ways in which chemical heerogeneiy can preven he inversion of parial release daa o consrain iniial concenraion profiles. Firs, if neiher reacor-produced isoope had a uniform iniial concenraion, hen he values of used for all he isoopes are misaken. Using synheic daa (i.e., calculaed parial releases from spherically symmeric iniial concenraion profiles), we found ha errors in of a facor of 2 or larger are likely o preven us from finding inverse soluions. This problem could be alleviaed in fuure experimens by pyromerically measuring he emperaure of he specimen during each heaing sep, o aid in calculaing direcly from he duraion of heaing, he spherule radius, and an esimaed diffusiviy (he las quaniy canno be deermined independenly during sepwise heaing of impac spherules, and would need o be assumed). Second, anomalous concenraions of calcium ha are posiioned away from he cener of he spherules would give rise o an asymmeric disribuion of cosmogenic 36 Ar and 38 Ar, jus as variaions in poassium concenraions would, in general, imply spherical asymmery in he disribuion of radiogenic 4 Ar. Third, if he chemical heerogeneiy is due o he presence of class or voids inside he spherules, he diffusiviy of argon would be nonuniform. In general, chemical heerogeneiy deparing from spherical symmery does no presen a fundamenal barrier o inversion of sep-release daa, bu i would necessiae full hree-dimensional modeling, requiring many more measuremens on individual specimens han have been made on lunar spherules. [51] The work of Albarède [1978] offers hins a which of he assumpions in our model are mos imporan o he 11 of 13