Journal of Geophysical Research: Planets

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

Authors

  • Jonathan Levine,

    1. Department of Physics, University of California, Berkeley, Berkeley, California, USA
    2. Laboratorio TANDAR, Comisión Nacional de Energía Atómica, Buenos Aires, Argentina.
    3. Now at Chicago Center for Cosmochemistry and Department of Geophysical Sciences, University of Chicago, Chicago, Illinois, USA.
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  • Robert A. Rohde

    1. Department of Physics, University of California, Berkeley, Berkeley, California, USA
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Abstract

[1] We use argon step-release measurements to model the initial distribution of argon isotopes in 177 lunar impact spherules. The speed of modern computers permits us to approach this inverse problem in new ways, and the techniques we develop may be extended to study a wide range of samples, diffusing species, and geometries. Lunar spherules, by virtue of their simple shapes and histories, seem to be excellent candidates for inverse modeling. Nevertheless, impact spherules preserve chemical or material heterogeneities that are relics of their parent materials. As a result, we find that the distribution of argon isotopes in most impact spherules is more complex than can be meaningfully constrained by a practical number of precise measurements. The spatial distribution of argon from different sources, such as solar implantation, would be better probed in spherules by other techniques, such as stepwise etching.

1. Introduction

[2] Argon isotopes are measured extensively in radioisotopic dating experiments on both terrestrial and extraterrestrial samples. In lunar samples, the stable isotopes 36,38,40Ar are derived from several sources, including decay of 40K, solar implantation, and spallation reactions induced by cosmic radiation [Turner et al., 1971]. There may also be contamination from terrestrial atmospheric argon. Short-lived 37Ar and 39Ar are artificially created by neutron irradiation, primarily of 40Ca and 39K [Merrihue and Turner, 1966; Turner, 1970; Turner et al., 1971]. (Neutron irradiation creates small amounts of the stable argon isotopes as well, but we monitor and correct for these interfering nuclear reactions. Likewise, small amounts of cosmogenic 37,39Ar are created on the Moon, but the saturation concentrations of these isotopes are negligible, and moreover many 37Ar half-lives have elapsed since the lunar samples were brought to Earth.) Ratios of argon isotopes in several partial releases may be used to evaluate the relative importance of these sources, and to calculate the isotopic compositions of some of these components.

[3] Albarède [1978] was the first to use isotopic data from stepwise heating experiments to model the initial distribution of each isotope in a sample. If the initial distribution of argon isotopes in lunar samples can be known, it would be of great importance for constraining the energy spectrum of implanted solar particles, identifying terrestrial contamination, deducing thermal histories, or recognizing argon losses due to nuclear recoil during neutron irradiation.

[4] Lunar impact spherules [Culler et al., 2000; Levine et al., 2005] (see Figure 1) are glass droplets that quench from melted or vaporized rock in the aftermath of a meteorite impact on the Moon. To determine spherule formation ages with the 40Ar/39Ar isochron technique, Culler et al. [2000] and Levine et al. [2005] degassed individual spherules by stepwise heating with infrared lasers. Impact spherules can be distinguished from lunar volcanic spherules by several criteria [e.g., Delano and Livi, 1981], and Culler et al. [2000] and Levine et al. [2005] argued for impact origin of their spherules chiefly on the basis of their chemical compositions, chemical heterogeneity, and young ages. In this paper, we use the argon isotopic data acquired in these geochronology experiments to follow Albarède [1978] in asking whether and how stepwise release data can meaningfully constrain the initial distribution of argon isotopes in a sample. Lunar spherules provide a particularly simple geometry in which to attempt this analysis.

Figure 1.

Scanning electron micrographs of impact spherules taken from Apollo 12 soil 12023. The nearly perfect spherical shape and smooth surface texture of spherules (a) 50, (b) 71, and (c) 40 are shared by roughly half of the lunar impact spherules. (d) Spherule 8, which is nearly spherical but has a blocky surface texture. Other impact spherules are vitreous but ellipsoidal, including spherules (e) 68 and (f) 6. We observed both oblate and prolate ellipsoids. Finally, this study included a small number of impact spherule fragments, including (g) spherule 45 with conchoidal fracture and (h) spherule 49, which reveals the surfaces of bubbles in the fractured face of the spherule. Scale bar in each panel represents 100 μm.

[5] Argon is transported within specimens by diffusion [e.g., Turner et al., 1973], and atoms are released when they diffuse through the surface. Each partial release therefore includes mixtures of argon from many different initial locations in the spherule. Forward modeling of the diffusion process consists of calculating how much argon would be released in each step of a specified heating program, given a certain initial distribution of argon isotopes. We review approaches to the forward problem in section 2. Because the diffusion equation is linear in the concentration, we may imagine the initial concentration to be a linear superposition of several functions, the evolutions of which we can calculate separately. In subsequent sections, we follow Albarède [1978] in attempting the inverse of this problem, constraining the initial distribution of argon with the measured partial releases in each heating step. The linearity of the diffusion equation makes this a linear inverse problem.

[6] Our work takes advantage of the speed of contemporary digital computers to perform a more complete search for inverse solutions than could have been achieved earlier. We are therefore able to generalize and extend the method of Albarède [1978], and develop new algorithms with which to approach the inverse diffusion problem. In the remainder of this paper, we present solutions to the forward and inverse problems, and apply these to the argon data acquired from lunar impact spherules by Culler et al. [2000] and Levine et al. [2005].

2. Forward Problem

2.1. Formulation of the Problem

[7] To mathematically describe the release of argon from a lunar spherule, we model the specimen as a perfect sphere, and we assume that argon diffuses within the spherule with an effective diffusivity D that might depend on temperature but not on position. We represent the initial (i.e. t = 0) concentration of a certain isotope at position r inside the spherule by c(r, t = 0). The concentration anywhere within the spherule at any later time can be determined by solving the diffusion equation,

equation image

The diffusion equation relates spatial derivatives of the concentration (on the left side of equation (1)) with its time derivative (on the right side). Given the geometry of a spherule, it is most convenient to express the concentration in spherical polar coordinates (a radial coordinate, which we denote by r, and azimuthal and polar angular coordinates), with the origin at the center of the spherule.

[8] The boundary condition appropriate for this problem comes from the fact that the spherules of Culler et al. [2000] and Levine et al. [2005] were degassed within the extraction line of a noble gas mass spectrometer evacuated to a pressure of 10−9 atm. We model the ultra-high vacuum surrounding the spherule by imposing a zero-concentration boundary condition on the argon at r = a, where a, the spherule radius, is typically ∼125 μm.

[9] Individual spherules were incrementally degassed by infrared laser heating (with an argon ion laser in the case of Culler et al. [2000] and a CO2 laser in the case of Levine et al. [2005]), which allows for lower argon backgrounds and more rapid heating and cooling than typically available in a furnace; however, the mathematical treatment we present is independent of how the samples are heated. One shortcoming of the laser heating system used in these experiments is that the temperature attained by the spherule in each step is not measured. Spherules in these studies were held at high temperature for ∼15 s (long compared with the ∼10−2 s required for heat to diffuse approximately one spherule's radius through basaltic glass), then left unheated while the gas released during each heating pulse was analyzed by mass spectrometry. A solution to the forward problem consists of evolving the initial concentration c(r, t = 0) forward in time under the diffusion equation, subject to the zero-concentration condition imposed on the boundary, and calculating how much of each argon isotope would have diffused through the surface during each heating step.

[10] The lunar impact spherules of Culler et al. [2000] and Levine et al. [2005] typically released a total of ∼10−14 mol of 40Ar and 36Ar, with smaller amounts of other isotopes. The argon yield was divided among ∼7–14 heating steps, though some heating steps released argon at levels indistinguishable from instrumental blanks. (Blank levels are different for each isotope, and depend on the history of the mass spectrometer, but are typically a few 10−18 mol of 36–39Ar, and ∼10−17 mol 40Ar.) This relatively small number of measurements implies that we have only a small number of inputs for the inverse problem, or, in other words, a small number of useful constraints on the initial distribution of argon in the spherules. In fact, no finite number of measurements could completely determine the initial argon concentration if arbitrarily complicated distributions are permitted. We can make progress in the inverse problem only if we forgo uniquely specifying the initial distribution, and instead search for initial concentration profiles that are consistent with the measured data. Any concentration profile that is consistent with all the measured data is considered a solution to the inverse problem; we may favor one solution over others only if we have additional information about what a plausible initial concentration profile might look like.

2.2. Initial Condition of the Impact Spherules

[11] We assume that the initial distribution of argon in lunar impact spherules is spherically symmetric. This assumption is partly motivated on physical grounds that we discuss below. Additional motivation for assuming spherical symmetry comes from the fact that even the largest number of measured partial releases on individual impact spherules would poorly constrain the initial argon distribution if it varied in all three dimensions. In such a case, N measurements would permit the modeling of the argon distribution with a spatial resolution characterized by a wavelength λ ≈ a/N1/3. By contrast, if the initial concentration were spherically symmetric, then the same number of measurements permit a much finer spatial resolution, of order λ ≈ a/N. In the case of spherical symmetry, even seven measurements allow us to describe an initial radial concentration profile rather well, for example by estimating the first seven terms in its Taylor series expansion.

[12] There is good reason to believe that several components of the initial argon endowment of some lunar impact spherules ought to be spherically symmetric. Most spherules are quite nearly spherical in shape (Figure 1), with long and short axes that differ in length by less than 10% [e.g., Fulchignoni et al., 1971]. Other impact spherules form oblate or prolate ellipsoids, though irregular spherule fragments exist as well. Furthermore, since the penetration depth of primary galactic cosmic rays (energies >1 MeV) greatly exceeds the diameter of a spherule, the distribution of spallation 36,38Ar is uniform in a chemically well-mixed spherule. By contrast, implanted solar argon and “parentless” 40Ar [e.g., Manka and Michel, 1971; Wieler and Heber, 2003] must be preferentially implanted in the upward-facing surface of a spherule. If impacts frequently reorient spherules in the lunar soil, the accumulation of implanted gases will approximate a symmetric outer coating. However, even if the implanted argon is heavily biased toward one part of the surface, the diffusion of these components will at first be similar to that of argon distributed symmetrically. Because solar argon is implanted in the outer ∼200 nm of lunar grains [Eberhardt et al., 1970], the radial concentration gradient initially exceeds lateral gradients by a large factor, and hence the net flow will be nearly radial in early heating steps. Later in the heating program, when radial and lateral concentration gradients are nearly equal, spherical symmetry may be a poorer approximation for the behavior of these isotopes.

[13] In addition, all impact spherules were formed at high temperature from molten or vaporized rock. We restricted this study to those spherules for which isochron ages were determined and which lacked evidence for undegassed clasts, because these spherules remained hot for long enough to degas any inherited argon. It is reasonable to begin, therefore, with the assumption that many impact spherules are well-mixed with respect to at least some chemical elements, so that their radiogenic 40Ar, cosmogenic 36Ar and 38Ar, and reactor-produced 37,39Ar are uniformly distributed.

[14] Surprisingly, the assumption of chemical uniformity is relatively weak. Measured 37Ar/39Ar ratios vary by a factor of ∼3 among heating steps on individual impact spherules, implying that the ratio of their parent isotopes, 40Ca and 39K, varies by at least this factor over the spherule volumes [Levine et al., 2006]. Nearly all impact spherules released argon with a relatively low 37Ar/39Ar ratio early in the heating program, and this ratio increased as heating continued (Figure 2). Levine et al. [2006] infer from this behavior that impact spherules have relatively calcic cores and relatively potassic exteriors. This sense of zoning is contrary to that expected if spherules quenched from impact melt: If a melt droplet were initially uniform, it would develop a potassium depletion in the outermost portion by preferential evaporation of volatile elements [Yu et al., 2003].

Figure 2.

Evolutions of 37Ar and 39Ar releases from (left) Apollo 12 and (right) Apollo 14 impact spherules. For each spherule, the cumulative fractions of 37Ar (derived from 40Ca) and 39Ar (derived from 39K) released after each step are plotted. A “track” connects the data for each spherule. A chemically uniform spherule would release 37Ar and 39Ar in the same proportion during each heating step, and such a “track” would lie along the dashed diagonal line. Most impact spherule “tracks” are entirely below this line, indicating that impact spherules release 39Ar before the corresponding fraction of 37Ar. This implies that impact spherule rims are generally more potassic, and that cores are relatively calcic. The field above the main diagonal, with few impact spherule tracks, would correspond to chemical zoning in the opposite sense, with relatively calcic exteriors and potassic interiors. Uncertainties shown represent 1σ.

[15] The variable 37Ar/39Ar ratios do not specify whether the K concentration, the Ca concentration, or both are non-uniform. The relatively calcic cores might arise from undigested clasts of high-Ca (and presumably high melting temperature) phases being present inside most impact spherules. Any such clasts must have escaped chemical assimilation even though they were degassed of their inherited argon when the impact spherules enclosing them were formed (these clasts, if they are indeed present, do not have systematically higher radiogenic 40Ar contents). Even if impact spherules have undigested clasts at their centers, their potassium concentrations may be nearly uniform. Alternatively, impact spherules could have formed by condensation of impact vapor, with more refractory elements condensing first and then being mantled by less refractory elements. In this case, the concentrations of both potassium and calcium might be nonuniform. Finally, potassium could be enriched in the surfaces of impact spherules by an unidentified process, leaving calcium concentrations nearly uniform.

[16] For reasons that we discuss in section 3.1, it is necessary for the inverse problem to assume that a reactor-produced argon isotope derived from either potassium or calcium has an approximately uniform distribution in each spherule. We performed our analysis under the assumption of uniform 39Ar (from potassium) and uniform 37Ar (from calcium), each in turn. We cannot directly test the possibility that both elements were distributed nonuniformly in the impact spherules of Culler et al. [2000] and Levine et al. [2005], except by considering both of the alternatives. It would be interesting to distinguish among hypothesized modes of spherule formation based on the distribution of calcium and potassium by measuring Ca/K ratios in situ in a new set of impact spherules [Levine et al., 2006], though the scarcity of potassium in lunar materials makes this a challenge for many analytical techniques.

2.3. Solving the Forward Problem

[17] The assumption of spherical symmetry allows us to ignore the angular dependence of the Laplacian operator ∇2 in equation (1), and to cast the diffusion equation as

equation image

The concentration c(r,t) is found by solving equation (2) subject to the boundary condition of zero concentration at r = a. The total amount of any isotope present at time t is calculated by integrating c(r,t) over the volume of the spherule. The amount of each isotope released by the times t = {ti} at which measurements are made may be calculated by subtracting the amount of the isotope remaining inside the spherule at ti from the amount present at t = 0.

[18] There are several mathematical approaches to solve the diffusion equation. Here we present two different techniques. The Fourier series solution and the Green's function solution are both expressed as infinite sums, and strictly are equivalent to one another. However, they have different computational advantages in practical cases, where the infinite sums are necessarily truncated after a small number of terms.

[19] In the Fourier series method, the unknown concentration c(r,t) is set equal to the product of a function of r and a function of t. The diffusion equation (2) separates into ordinary differential equations in each variable, both of which are elementary. The solution that obeys the boundary conditions is

equation image

Here, to simplify notation, we have written the solution in terms of the dimensionless time τ = equation imagedt′ [Albarède, 1978; Shuster and Farley, 2004], where we explicitly note the dependence of the diffusivity on temperature T. The dimensionless time is scaled in such a way that τ = equation image is the average time at which a particle starting at the center of the spherule escapes through the surface. From equation (3), we can calculate the total amount G of the argon isotope that remains in the sphere at any time τ by integrating the concentration over the volume,

equation image

The amount released in a heating step that lasts from τi to τi+1 is simply

equation image

[20] We note three features of the solution in equations (3)(5). First, the measured quantities are expressed as infinite sums of integrals, which may, in general, be impossible to solve analytically. Therefore, for practical purposes, we will need both to approximate the integrals and to truncate the infinite sums after a reasonable number of terms. Second, for τ larger than about 0.1, only a few terms are necessary to approximate the infinite sum, because of the rapidly decreasing factor image This is advantageous, insofar as the series converges rapidly. Third, for very small values of τ, the series converges exceedingly slowly, and many terms need to be retained in the sum.

[21] The Green's function approach is an alternative method for solving the diffusion equation. Instead of representing the concentration of argon by a sum of sines, we decompose the initial distribution into Dirac δ-functions. The δ-function has unit area and infinitesimal width; it evolves under the diffusion equation into a Gaussian whose width broadens with time. The relative weight of each Gaussian in the concentration at time τ is the same as the weight of the corresponding δ-function in the initial concentration. The Green's function expansion, though it appears more cumbersome than the Fourier series, offers computational advantages that we describe below.

[22] Expressed as a sum of broadening Gaussians, the concentration of argon inside the spherule is given by

equation image

The infinite sum of positive and negative Gaussians, spaced every 2a apart, is required to match the boundary conditions at r = 0 and r = a; the spreading of the unphysical “image” sources exactly cancels the spreading of the real argon at the boundaries, for all times. Though the Green's function solution, like the Fourier series solution, is obviously unphysical outside the sphere, it does obey the diffusion equation inside the region 0 < ra, and it obeys the boundary conditions. The fact that the diffusion equation (2) is linear and homogeneous is sufficient to ensure that a solution obeying particular boundary conditions is unique, and thus that equation (6) is equivalent to the Fourier series solution in equation (3).

[23] From equation (6), we may develop alternative expressions for the total amount of an argon isotope remaining in the spherule as a function of time, and for the amount released in a particular heating step. Integrating the concentration over the volume of the spherule, we find

equation image

Here we use a dimensionless radial variable x = r/a, and erf represents the standard error function. As before, the amount of the isotope released in a heating step from τi to τi+1 is given by

equation image

[24] Unlike the Fourier series solution (3)(4), which requires many terms of the infinite sum to be retained at early times but which quickly converges for large τ, the sum over image sources in the Green's function solution (6)(7) requires few terms initially, and more as time advances. At early dimensionless times (e.g., τ ≈ 10−4), the Fourier series expansion in equation (4) requires a number of terms n which exceeds τ−1/2 in order to make a good approximation; the Green's function expansion, on the other hand, is satisfactory with fewer than 20 terms even at late dimensionless times (τ ≈ 0.5). Because of this, we generally prefer to employ the Green's function solution where practical.

[25] For the simple case of an initially uniform concentration, the integrals in equations (6) and (7) may be calculated analytically, and the fraction of the isotope released by any time may be quickly obtained. The expression for the fraction released by dimensionless time τ is

equation image

Employing equation (8) where possible removes a potential source of error from numerically integrating over the Green's functions in equation (7). With these explicit solutions to the forward problem, we now proceed to develop the mathematical tools for the inversion.

3. Tools for the Inversion

3.1. Determining the Time

[26] An important first step in the inversion process is determining the dimensionless times τ at which measurements were made. The definition τ = equation imagedt′ [Albarède, 1978; Shuster and Farley, 2004] is often not useful for calculating τ directly, as the diffusivity during each heating step may not be known. This is the case for the data of Culler et al. [2000] and Levine et al. [2005]. Without known values of τ, it is impossible to invert either equations (4) or (7) from measured releases. Albarède [1978] circumvented this problem by assuming that reactor-produced 37Ar had been uniformly distributed and using partial releases of this isotope to calculate τ for each heating step. Because the partial release data from lunar impact spherules show that the initial concentrations of 37Ar and 39Ar cannot both have been uniform (Figure 2) and because we do not have an a priori reason to favor one of these as a uniform isotope, we performed all of our analysis twice, first with the assumption of uniform 39Ar concentration and then with the alternative assumption of uniform of 37Ar. Our results are quite similar in the two cases.

[27] Both reactor-produced isotopes are radioactive. Especially in the case of short-lived 37Ar (half-life 35 days), one must correct measured partial releases for the effect of radioactive decay, so as not to underestimate the fraction of the spherule degassed in later heating steps. In practice, we use partial release values that are corrected for instrumental background, mass discrimination (i.e., isotopic fractionation within the gas extraction system, mass spectrometer, and detector), radioactive decay, and interfering nuclear reactions during irradiation. The last correction allows us to interpret our results more easily: our values for 39Ar, for example, are corrected for a small amount derived from 42Ca(n,α)39Ar reactions, and therefore represent only the quantity of this isotope that is derived from 39K.

[28] Following Albarède [1978], we used the fractional releases of the assumed-uniform isotope to invert equation (8) and to compute τ at the end of each heating step. Equation (8) is not invertible in closed form, but because it is a monotonic function of time, we can find τ with arbitrary precision by a method of successive approximations. If all the argon isotopes have the same diffusivity, then the values of τ thus determined apply to all the isotopes. If the diffusivity of argon isotopes in a spherule imposes a mass-dependent fractionation, the values of τ for each isotope are different, but obey the relationship τ(i)(uniform) = equation image for isotopes of mass mi. This effect would be small for argon isotopes, and it has been reported in only one experiment [Trieloff et al., 2005]. Such a fractionation of argon has never been unambiguously observed in glass; moreover, a suite of helium diffusion studies in minerals [Trull and Kurz, 1993; Shuster et al., 2004] and glass [Trull and Kurz, 1999] have yielded apparently different mass-fractionation laws, and so it is uncertain what expectation one should have for lunar glass spherules. We repeated our analysis with mass-dependent and mass-independent values of τ, and found that changes this small in τ (<6% for 36–40Ar) do not affect our conclusions. Frequently, successive values of τ are separated by a factor of 2–10, which is much larger than the offsets that could be due to mass-dependent fractionation.

3.2. Inversion Techniques

[29] We next outline a general scheme for inverting partial release data to find plausible initial concentration profiles of argon isotopes. The inversion begins by decomposing the initial concentration of all isotopes into N components and assuming functional forms fi(x) for each. In terms of these components, the initial concentration is expressed by

equation image

where the undetermined parameters bi represent the weights of each component. In equation (9) and in what follows, summation over repeated indices is implied. The components fi(x) may, in general, be any functions over the interval 0 ≤ x ≤ 1. One uses equations (3) or (6) to calculate the amount of each argon isotope every component would release from the spherule during intervals defined by the measurement times τj. Letting the matrix element Wij denote the calculated release from component i between times τj and τj+1, and representing the set of measured isotopic releases by Mj, we then have

equation image

If the number N of components is chosen to be equal to the number of partial releases, then W is a square matrix. If its inverse exists, an initial concentration profile is given by

equation image

If the function c(x,0) found in this way is physically reasonable, then it represents a plausible initial concentration function consistent with the measured step releases.

[30] In general, there is no guarantee that an inverse exists to W, the matrix of partial releases from each component, or that the initial concentration given by equation (11) is physically plausible. For example, if two or more heating steps begin at already large values of τ, such that the Fourier series in equation (4) can be approximated by a single term, the corresponding rows of the W matrix will be proportional to one another and the matrix must be singular. The truncation of the Fourier series after only one term is a valid approximation for heating steps after τ ≈ 0.25, when the factor image drops by 10,000 times between the leading term and the n = 2 term, making the second term negligible. Physically, the singularity of W means that diffusion blurs the initial concentration by late dimensionless times, so that, to the extent that the second terms in the Fourier series may be neglected, no new spatial information can be gleaned from the final heating step. If the vector of measurements M (and its associated uncertainty envelope) lies outside the range of matrix W, then no inverse solution exists. This is sufficient to show that the set of partial releases M is inconsistent with any spherically symmetric initial concentration that may be expressed as a linear combination of the chosen fi(x).

[31] It is clear that different choices of the functions fi(x) in equation (9) will give rise to different inverted initial concentration profiles. This nonuniqueness is a consequence of trying to constrain a continuous function (i.e. the initial concentration) with only a small number of measurements. Some of the nonunique solutions may be dismissed on physical grounds, and others, perhaps solutions that have relatively smooth concentration profiles, may be favored. The strongest physical constraint that we may apply a priori is that the initial concentration of argon be non-negative everywhere. Some choices of fi(x) may result in a nonnegative c(x,0) even as other sets of expansion functions yield unphysical negative concentrations while still being mathematically consistent with the observed step releases. Sabatier [1977a, 1977b] describes how convex analysis may be used to impose the constraint of nonnegativity on the otherwise linear inverse problem. In the present work, we instead rely on an exhaustive search algorithm that we have written to restrict our solutions to nonnegative initial concentrations.

[32] Albarède [1978] used a particular set of decomposition functions fi(x), for which the matrix elements Wij are proportional to the fraction of an initially uniform isotope that would remain in the spherule after a time τi + τj. The advantage of this set of functions is that an algorithm optimized for rapid implementation of equation (8) (or its Fourier series equivalent) could also rapidly compute the matrix elements Wij. This advantage is mitigated by the speed of modern computers, which can quickly calculate partial releases from arbitrary initial concentrations; our Macintosh Powerbook G4 running Matlab calculates partial argon releases in 0.2 s, carrying the infinite sums as far as needed for part-per-billion accuracy. This allows us to explore other choices of components fi(x). We examined the lunar spherule data of Culler et al. [2000] and Levine et al. [2005] using the method of Albarède [1978] and many other sets of fi(x), and never found any initial concentrations that were nonnegative everywhere and consistent with all the measurements on any impact spherule.

3.3. Inversion by Exhaustive Searching

[33] Our failure to find nonnegative initial concentration profiles using equation (11) that are consistent with the partial releases of argon from lunar impact spherules led us to mount an exhaustive search. In order to determine whether any plausible spherically symmetric initial concentrations could have yielded the measured argon releases, we allow ourselves many more degrees of freedom than we have measurements. Of course, using a larger number of adjustable parameters emphasizes the fact that any solutions we find are not unique; with the exhaustive search, we only seek to learn whether any nonnegative solutions exist to the inverse problem, not whether a particular inverse solution represents the true initial distribution of argon isotopes in an impact spherule.

[34] The requirement of nonnegativity is difficult to map onto the weights of an arbitrary set of components fi(x), so we use nonoverlapping top-hat or boxcar shaped functions

equation image

where the parameters pi and qi are the boundaries of the ith top hat, which rises to a value of bi. The requirement that the initial concentration be non-negative is imposed by restricting the search space to bi ≥ 0 for all i. We decomposed the initial concentration into 118 top-hats, and tried to adjust the weight of each in order to match the 7–14 measurements on each spherule. Because we expect that implanted solar argon is concentrated near the surface of the spherules, we chose the boundaries of our top hats to be more closely spaced near x = 1. In the limit of arbitrarily many top hats, one approximates a continuous initial concentration; we found that 118 top hats performed equally well as twice that number, in half the computing time.

[35] The difficulty in our method is that a straightforward matrix inversion cannot enforce the requirement that all the top hats have nonnegative weights. Instead, we have devised a search algorithm that seeks to minimize the error-weighted misfit between the measurements and a model calculation, given by

equation image

while demanding that all of the top-hat weights bi obey bi ≥ 0. The set of σj are the uncertainties in each measurement. Our search algorithm (auxiliary material Text S1) cycles through three different strategies to minimize the misfit. Beginning with a starting guess of the bi, we first take advantage of the fact that equation (13) is a quadratic, concave-up function of each of the bi, and we iterate a process of adjusting each element sequentially so that η2 converges to its local minimum. If a negative element of bi would lower η2 further, it is assigned a value of zero (this minimizes η2 subject to the physical constraint of nonnegativity). Second, after a fixed number of iterations, we rescale the entire set of bi in an effort to minimize η2. This helps to converge on the optimal solution more rapidly than adjusting each of the bi individually. Third, to avoid becoming trapped in local minima of η2 rather than its global minimum, we repeatedly let the set of bi take random jumps to other areas of the parameter space, to see whether the misfit is smaller elsewhere. The size of the random jumps is gradually made to decrease as the algorithm converges on a solution; this is characteristic of so-called “simulated annealing” optimization algorithms [e.g., Press et al., 1989], which mimic the fluctuations in an annealing crystal as it cools to a low-energy configuration.

[36] We tested our algorithm on forward-modeled partial releases calculated from initial argon distributions that were spherically symmetric, with radial concentration profiles given by polynomial, exponential, logarithmic, and trigonometric functions, and combinations of these. The calculated releases were assigned uncertainties of a few percent, which is typical of the measurement precision of the lunar spherule data. In all cases, our search algorithm took less than one minute to converge on a plausible spherically symmetric initial concentration profile that matched the model releases. The misfits in these inversions were usually smaller than η2 = 10−2. The performance of our search algorithm on simulated releases calculated from idealized spherically symmetric initial concentrations shows that, in cases where a nonnegative solution to the inverse problem is known to exist, the algorithm is able to find one. However, we stress that the inverted concentrations were not the same as the concentration profiles we used to calculate the model releases, and our algorithm instead found another initial concentration profile that was consistent with the same partial releases.

[37] The statistical properties of η2 defined by equation (13) are somewhat unusual and difficult to assess rigorously, because the search space for the inversion is constrained by the requirement that bi ≥ 0. We therefore examined the behavior of our search algorithm in Monte Carlo simulations to address the question of how small a residual should be considered a “good fit” between measurements and calculated partial releases from a particular initial concentration profile. We discuss these simulations in detail in the following section.

4. Analysis of Lunar Spherule Data

[38] We applied our exhaustive search algorithm to the partial release data from a subset of the lunar impact spherules studied by Culler et al. [2000] and Levine et al. [2005]. Culler et al. [2000] measured partial argon releases from 179 spherules collected during the Apollo 14 mission to the Fra Mauro Peninsula, of which 109 yielded statistically acceptable 40Ar/39Ar isochron ages [Levine, 2004]. Levine et al. [2005] examined 178 spherules from an Apollo 12 soil collected in the Procellarum Basin, finding statistically acceptable 40Ar/39Ar isochrons for 81 of the spherules. A 40Ar/39Ar isochron that is statistically acceptable indicates binary mixing of potassium-derived argon (i.e., reactor-produced 39Ar derived from 39K and radiogenic 40Ar from in situ decay of 40Ar) and surface-implanted argon (in this case parentless 40Ar and solar 36Ar). If the partial releases cannot be resolved into these two components, no unique age may be determined, and this likely implies either incomplete degassing during spherule formation or partial loss or redistribution of argon due to heating (and possibly shock) from subsequent nearby impacts. Any mechanism for introducing additional components of argon can violate the assumption of spherical symmetry, either by affecting the distribution of argon isotopes or by establishing a nonuniform diffusivity inside the spherule. Therefore spherules without isochrons are not good candidate physical systems for our inverse modeling of the initial distribution of argon isotopes. We eliminated all such spherules from our analysis. We also excluded the small number of impact spherules whose argon release patterns showed evidence for undegassed clasts inside the spherules, which we recognized from a tendency toward more radiogenic 40Ar in the highest-temperature steps than indicated by the apparent age of the rest of the impact spherule. However, impact spherules with clasts that were degassed of their argon but not chemically assimilated might remain among those we analyzed. Such clasts could hamper our inversions, by introducing nonuniform chemical compositions or diffusivities. The total population of analyzed impact spherules consists of 104 from the Apollo 14 sample and 73 from the Apollo 12 sample.

[39] In a number of heating steps, the amount of argon released from particular impact spherules was consistent with instrumental background levels. In these cases, we mathematically combined partial releases and associated uncertainties with data from the following heating steps, until the amount of argon observed in each “combined step” was unambiguously greater than zero. Therefore inversions of data from some impact spherules need to match as few as three above-background partial argon releases. More typically, however, six to eight measurements need to be matched.

[40] We used the exhaustive search algorithm to find the nonnegative spherically symmetric initial distribution of argon isotopes which, when evolved under the diffusion equation, yielded calculated “releases” that agreed most closely with the real measurements on each impact spherule, in the sense of minimizing the η2 residuals. Apollo 12 results are shown in Figures 3 and 4, and Apollo 14 results are depicted in Figures 5 and 6. Figure 3 for the Apollo 12 impact spherules and Figure 5 for the Apollo 14 specimens represent data calculated under the assumption of initially uniform 39Ar (i.e., partial releases of 39Ar were used to determine the value of τ after each heating step); for Figures 4 and 6, it was assumed instead that 37Ar was initially uniform. Figures 36 are logarithmic in their vertical axes, as the range of η2 extends over many orders of magnitude. Summary information related to Figures 36 are given in auxiliary Tables S1 and S2.

Figure 3.

Misfits between observed partial releases for 73 Apollo 12 impact spherules [Levine et al., 2005] and best fitting spherically symmetric models with 39Ar assumed uniform. The thick dark lines in each panel show the η2 difference between the observed partial releases of the named argon isotope and the calculated “releases” from the best fitting spherically symmetric initial concentration. Note that the vertical axis is logarithmic, and that better fits are implied by lower η2. The dimensionless times τ are computed under the assumption of initially uniform 39Ar; therefore a model concentration with spherical symmetry can always be found to match the observed 39Ar partial releases. Shaded bands represent the range of η2 values from 21 Monte Carlo simulations (darker shading denotes middle 50%), in which the spherically symmetric model that most closely matched the measured partial releases was perturbed by simulated experimental errors and then reinverted. Horizontal dashed lines represent η2 = 2; as described in the text, “successful” inversions, which are listed in auxiliary Table S1, have misfits that plot below this line.

Figure 4.

As in Figure 3, but dimensionless times τ of each heating step were calculated on the assumption of initially uniform 37Ar rather than 39Ar. Therefore a model concentration with spherical symmetry can always be found to match the measured 37Ar partial releases. Successful inversions (i.e., η2 < 2) are listed in auxiliary Table S1.

Figure 5.

As in Figure 3, but for 104 Apollo 14 impact spherules [Culler et al., 2000]. Dimensionless times τ were calculated assuming that 39Ar was initially uniform. The many downward spikes in this figure are an artifact of the smaller number of measurements on many spherules, compared with Apollo 12 analyses. Our search algorithm frequently matches sets of three measured values to within η2 = 10−15. Successful inversions (i.e., η2 < 2) are listed in auxiliary Table S2.

Figure 6.

As in Figure 5, but with dimensionless times τ computed under the assumption of uniform 37Ar. Successful inversions (i.e., η2 < 2) are listed in auxiliary Table S2.

[41] For the isotope that was assumed uniform in order to calculate τ after each heating step, we are guaranteed that an exact inverse solution (i.e., the uniform initial concentration solution) exists. However, given the starting guess we used in the inversion algorithm (we began with all of the bi = 0) and the limits we imposed on the number of iterations, the algorithm generally did not converge on this exact solution. Instead, it settled for an approximate solution, whose misfit with the data (typically 10−8 < η2 < 10−3) provides one estimate of the value of η2 to be expected in cases where partial release data is truly invertible.

[42] For other isotopes, there is no guarantee that an inverse solution exists, so we must address the question of what values of η2 imply a “good fit,” and what values imply that no spherically symmetric initial distribution could account for the data. We used a Monte Carlo approach to determine the values of η2 that could be expected from our algorithm when trying to invert uncertain data. For each argon isotope measured from each impact spherule, we found the set of partial releases that most closely matched the measured values but which was exactly consistent with spherical symmetry; we perturbed this set of partial releases by amounts consistent with the experimental errors, and used our algorithm to invert the perturbed data. At least one inverse solution must exist in these Monte Carlo tests: The spherically symmetric initial concentration that yielded the unperturbed data will be consistent with the perturbed data, with a misfit of η2 ≈ 1. A much smaller misfit might be found if the perturbed set of partial releases agrees more closely with a different spherically symmetric initial concentration profile. Figures 36 show the range of η2 values obtained in the Monte Carlo tests. On the basis of the Monte Carlo results, we adopt an approximate criterion of η2 > 2 for concluding, with ∼95% confidence, that a particular set of measurements has no spherically symmetric inverse solution. However, the precise statistical confidence in this conclusion depends weakly on the number of measurements being fit and on the number of adjustable parameters that are independent in our inversion.

[43] The different ranges of η2 found in simulations corresponding to distinct impact spherules are mostly due to the different number of measurements being fit, with sets of three partial releases (e.g., Apollo 12 spherule 39) typically yielding misfits with η2 < 10−15. Misfits this small were hardly ever observed in simulations of six or more partial releases. All simulations used the same number of adjustable parameters (118) to match the partial releases, regardless of the number of measurements being fit. However, doubling or trebling the number of top-hat functions did not improve the fits in simulations with ≥6 partial releases.

[44] We note the following features of Figures 36. First, regardless of which isotope was assumed to have had the initially uniform concentration, inversions of 36Ar and 38Ar partial release data yield the largest values of η2, and the smallest number of successful inversions. Misfits characterized by η2 < 2 are found in inversions of 36Ar and 38Ar data from 15–30% of the impact spherules. At least ∼90% of the 36Ar and 38Ar in most impact spherules is implanted, so the scarcity of successful inversions for these isotopes suggests that asymmetric solar irradiation is partly to blame for deviations from spherical symmetry.

[45] Second, successful inversions (i.e., η2 < 2) of 40Ar data are found for approximately 50% of the impact spherules, regardless of which reactor-produced isotope was assumed uniform in order to calculate τ. It is somewhat surprising that more inversions of 40Ar data were not successful under the assumption of uniform 39Ar, since the distribution of radiogenic 40Ar should be identical to the assumed-uniform distribution of 39Ar [Merrihue and Turner, 1966]. Perhaps asymmetric implantation of “parentless” 40Ar [Manka and Michel, 1971; Wieler and Heber, 2003], whose distribution is similar to that of solar 36Ar and 38Ar, makes partial releases of 40Ar uninvertible. However, we observe no simple relationship between the fraction of 40Ar that is radiogenic and η2.

[46] Third, we consider the reactor-produced isotopes. Under the assumption of uniform calcium and 37Ar (Figures 4 and 6), we are able to invert partial releases of 39Ar for approximately 80% of the spherules. By contrast, we obtain inversions of 37Ar releases from only ∼50% of the spherules under the assumption of initially uniform potassium and 39Ar (Figures 3 and 5). It is unclear what this discrepancy implies about the argon content and chemical composition of lunar impact spherules. In particular, we hesitate to conclude on this basis that the assumption of uniform 37Ar is the better one. In situ spot analyses of K and Ca on a new suite of spherules would show definitively whether either of the reactor-produced isotopes could be expected to have a uniform initial concentration. Such measurements could also constrain the modes of spherule formation and growth in the aftermath of meteoroid impacts.

5. Discussion and Conclusions

[47] Lunar impact spherules seemed a priori to be ideal candidates for inverse modeling. Their spherical shape makes the diffusion process particularly easy to describe mathematically, and the fact that they were formed at high temperatures offered the hope of effective chemical mixing, so that at least one of the reactor-produced argon isotopes would be uniformly distributed. The assumption of spherical symmetry, which we adopted in the hope of meaningfully constraining initial concentrations with only a few measurements, must be reconsidered given the failure of our exhaustive search algorithm to find any spherically symmetric initial concentrations consistent with partial argon releases for the majority of impact spherules. The assumption that one isotope had a uniform concentration, which we made so that τ could be determined for each heating step, must also be reevaluated. Figures 36 show that, given our assumptions, fewer than one fourth of the impact spherules can have nonnegative spherically symmetric initial concentration profiles that account for the partial releases of all isotopes.

[48] Spherical asymmetry could arise from shape anisotropy, asymmetric exposure to solar particle irradiation, and some kinds of chemical heterogeneity. We noted from Figures 36 that partial release data of 36Ar and 38Ar, the isotopes dominated by solar implantation, were successfully inverted by our exhaustive search algorithm in the smallest number of cases. One possible explanation for this is that asymmetric solar implantation of these isotopes routinely makes partial release data of 36Ar and 38Ar uninvertible, given the limitations of our model.

[49] Asymmetric exposure to solar irradiation would imply that impact spherules are rarely reoriented at the lunar surface. Since every observed impact spherule has implanted solar 36Ar and 38Ar, each spherule must be brought to the top of the regolith after its formation, yet asymmetric exposure requires that repeated impacts not bring spherules back to the surface many times, with random orientations. Because ejecta deposits are spread over a very large area, one possibility is that spherules, or, more precisely, the impact spherules that lose all their argon, are deposited upon formation at the very top of the lunar surface, where they are exposed to solar irradiation from only one side. Subsequent impacts in which preexisting spherules are not greatly heated might be more likely to bury spherules rather than expose them to further solar irradiation, in a new orientation. Unfortunately, little is presently known about the size of spherule-producing impacts nor the geographical distribution of spherules in ejecta. An alternative possibility is that our selection of impact spherules with isochron ages may have excluded spherules that suffered repeated impacts and reorientations.

[50] We know that most impact spherules are chemically heterogeneous on the basis of their varying 37Ar/39Ar ratios, shown in Figure 2, and from energy-dispersive x-ray analysis on spherule surfaces [Levine et al., 2004]. There are three ways in which chemical heterogeneity can prevent the inversion of partial release data to constrain initial concentration profiles. First, if neither reactor-produced isotope had a uniform initial concentration, then the values of τ used for all the isotopes are mistaken. Using synthetic data (i.e., calculated “partial releases” from spherically symmetric initial concentration profiles), we found that errors in τ of a factor of ∼2 or larger are likely to prevent us from finding inverse solutions. This problem could be alleviated in future experiments by pyrometrically measuring the temperature of the specimen during each heating step, to aid in calculating τ directly from the duration of heating, the spherule radius, and an estimated diffusivity (the last quantity cannot be determined independently during stepwise heating of impact spherules, and would need to be assumed). Second, anomalous concentrations of calcium that are positioned away from the center of the spherules would give rise to an asymmetric distribution of cosmogenic 36Ar and 38Ar, just as variations in potassium concentrations would, in general, imply spherical asymmetry in the distribution of radiogenic 40Ar. Third, if the chemical heterogeneity is due to the presence of clasts or voids inside the spherules, the diffusivity of argon would be nonuniform. In general, chemical heterogeneity departing from spherical symmetry does not present a fundamental barrier to inversion of step-release data, but it would necessitate full three-dimensional modeling, requiring many more measurements on individual specimens than have been made on lunar spherules.

[51] The work of Albarède [1978] offers hints at which of the assumptions in our model are most important to the success of the inversion method. Albarède [1978] used his inversion technique to study argon isotopic data acquired by Turner [1972] from an ensemble of lunar plagioclase grains. The samples were not likely to have been spheres, yet Albarède [1978] obtained nonnegative spherically symmetric concentration profiles that were consistent with the first five (of seven) measured partial releases (the final partial releases were combined for the sake of mathematical robustness in his inversion). This demonstrates that, in at least some cases, partial releases of argon from a nonspherical sample can be modeled as though they came from a perfect sphere, so deviations from spherical shape are not likely to have caused the failure of our inversion of lunar spherule data. The 36Ar and 38Ar in the plagioclase grains were primarily cosmogenic rather than implanted [Turner, 1972], so the question of whether exposure to solar irradiation was symmetric is relatively unimportant for those samples. One way in which the mineral grains studied by Albarède [1978] were better suited for inverse analysis than glass impact spherules is their more uniform chemical compositions and argon diffusivities. This seems essential for the success of inverse modeling. Future candidates for an inversion study include lunar volcanic spherules [Reid et al., 1973], which are noted for their compositional uniformity [Delano and Livi, 1981]. Volcanic spherules were intentionally avoided by Culler et al. [2000] and Levine et al. [2005], who sought to determine the ages of meteorite impacts on the Moon. Some volcanic spherules incorporate crystals of olivine, and these should be avoided in an inversion study, as these spherules could have nonuniformities of composition and diffusivity, just as impact spherules do.

[52] The fact that the majority of lunar impact spherules are chemically heterogeneous imposes constraints on models of spherule formation. Evidently, impact spherules form too fast to become chemically well mixed, leaving some spots richer in calcium while others, <100 μm away, are richer in potassium. Clasts, even when they are completely degassed, may still be mineralogically and chemically unassimilated. Richter et al. [2003] measured the diffusivity of potassium and calcium in basaltic melts, and their results suggest that ∼250-μm spherules would be chemically homogenized had they remained molten for a few tens of seconds or longer. This places a lower limit on the cooling rate of the spherules. A complementary upper bound on the cooling rate is derived from the fact that the spherules in this study lost all their argon during formation. The diffusion of argon in basaltic melt [Lux, 1987] is ∼10 times faster than that of calcium or potassium [Richter et al., 2003], so argon would require at least a few seconds to escape from the melted droplet, though argon loss would continue more slowly once the spherule quenched.

[53] The mathematical tools we have developed to approach the forward and inverse diffusion problems are of general utility. Though many authors prefer to use the Fourier series solution (indeed, it was to solve the problem of heat diffusion that Fourier series were invented), the alternative expansion in Green's functions often converges more quickly. Because the two solutions are equivalent to one another, either of them could be employed, depending on what is most convenient in a particular circumstance. Similarly, there is a great deal of freedom in choosing a set of expansion functions for the inverse problem. The advantage of the particular set chosen by Albarède [1978] is largely mitigated by the speed of modern computers. Unfortunately, no set of expansion functions is guaranteed to perform best in all situations. Moreover, though the determination of a continuous concentration profile from a finite number of measurements is an underconstrained problem, no physically plausible solution is guaranteed to exist. The algorithm we have developed conducts a quick, reliable, and exhaustive search for nonnegative inverse solutions under the same set of assumptions used by Albarède [1978].

[54] Because of the necessity of assuming spherical symmetry and chemical homogeneity, and because of the shortcomings of these assumptions as they apply to lunar impact spherules, modeling the diffusion process is not the most effective way to determine the implantation depth of solar particles. Eberhardt et al. [1970] etched layers of different depths on a large number of lunar regolith grains, and from the absence of solar wind gas in the most deeply etched samples, determined that the solar wind noble gases were concentrated in the outer ∼200 nm. Wieler et al. [1986] developed a system of closed-system stepwise etching by nitric acid vapor, which they used to study the so-called solar energetic particle component implanted in lunar grains. These experiments directly probe the concentration of noble gas isotopes as a function of depth in a sample, though etch rate must be assumed uniform over the surface. Lunar spherules would be good candidates for study with this technique, as the glass of the spherule does not present lattice orientations with variable resistance to acid attack, and spherical geometry makes the assumption of uniform etching more straightforward to apply. In this way, it may be possible to identify and isolate any contamination from the terrestrial atmosphere, to measure the implantation depth of solar wind gas, and potentially to distinguish normal solar wind from solar energetic particle implantation. Whereas stepwise heating forces some gas to diffuse inward even as some escapes from the surface, distributing implanted argon over many partial releases, etching may concentrate more of the solar argon in early steps, allowing for more efficient detection and isotopic characterization.

Acknowledgments

[55] We are grateful to P. Renne for furnishing the Apollo 14 data, as well as his encouragement and assistance, and to R. Muller for several recommendations which furthered this study. The manuscript was improved by the suggestions of D. Shuster and V. A. Fernandes. Our research was supported by National Science Foundation through a Graduate Research Fellowship and grant 0502034 (to J. L.), and by the Ann and Gordon Getty Foundation and the Folger Foundation.

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