- Email: [email protected]
Meteorite flaws and scaling for atmospheric entry
Accepted Manuscript Meteorite flaws and scaling for atmospheric entry Kathryn L. Bryson, Daniel R. Ostrowski, Aline Blasizzo PII:
S0032-0633(17)30411-7
DOI:
10.1016/j.pss.2018.06.018
Reference:
PSS 4569
To appear in:
Planetary and Space Science
Received Date: 3 November 2017 Revised Date:
26 June 2018
Accepted Date: 26 June 2018
Please cite this article as: Bryson, K.L., Ostrowski, D.R., Blasizzo, A., Meteorite flaws and scaling for atmospheric entry, Planetary and Space Science (2018), doi: 10.1016/j.pss.2018.06.018. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Meteorite Flaws and Scaling for Atmospheric Entry
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Kathryn L. Brysona,b,∗, Daniel R. Ostrowskia,b , Aline Blasizzoc a
NASA Ames Research Center Bay Area Environmental Research Institute c Texas Tech University
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Abstract
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Research is being conducted to understand the behavior of asteroids entering the atmosphere in order to help quantify their impact hazard. The strength of the body plays a critical role in determining the outcome of their impact events and is needed for many asteroid mitigation options. Meteorites are the physical material we have here to understand the larger parent body. Our objective is to scale flaw parameters in meteorites to their parent body, and therefore providing a way to scale strength from the smaller meteorites to the larger asteroids.
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The Chelyabinsk meteorite fall early February 15th, 2013 increased the awareness of asteroids impacting Earth. In response there has been an increased focus on the detection of threatening asteroids, deflecting them away from Earth if possible, and the effects of entry and impact should deflection fail. Within the NASA Ames Asteroid Threat Assessment Project (ATAP), multiple analytic models of asteroid breakup have been proposed [1, 2]. Using meteor fall data they have investigated the effects different parameters have on atmospheric breakup. The strength of the body plays a critical role in determining the outcome of their impact events [3, 4] and is needed for many asteroid mitigation options [5]. Meteorites are our only samples here
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1. Introduction
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Keywords: Meteorites, Asteroids, Near-Earth Objects, Weibull, Scaling
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Corresponding author Email address: [email protected] (Kathryn L. Bryson)
Preprint submitted to Planetary and Space Science
June 27, 2018
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2. Experimental
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Meteorites in the Natural History Museums of Vienna and London were examined. In Vienna we examined the ∼1,100 meteorites on public display which includes all meteorite classes, and 35 meteorites with obvious visible flaws were imaged. In London, due to time limitations, we examined their 567 H and 481 L chondrites, and imaged 34 meteorites with obvious visible flaws. The focus in both collections was to examine falls over finds to limit the effects of weathering, so 75% of the meteorites imaged are fall. The flaw patterns in selected individuals were imaged with a Nikon D7100, with an image size of 4000 by 6000 pixels. Flaws include cracks, shock veins, and slickensides, all of which have been shown in stress testing to be points of failure [9]. In addition 6 meteorites on loan from the Antarctic meteorite collection were also imaged.
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2.1. Microscope Imaging Thin sections of selected samples were borrowed from the Natural History Museum, London and from the Antarctic meteorite collection. The selected thin sections matched up to larger imaged meteorites. Photomicrographs were prepared under transmitted light with a Leitz Orthoplan microscope fitted with a 3MP digital camera. Selections of the thin section with images of 2048 by 1536 pixels were stitched together to form a complete image of the thin section.
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on earth to represent their parent bodies. The need is to scale the strength from the meteorite to the larger parent body. Scaling factors for strength vary greatly in terrestrial rocks [6] while a scaling factor of 0.167 has been favored by nature [7]. This factor has been used from some of the earliest meteoroid entry models [8]. A range of scaling factors has been determined previously from light curve fireball data [3], but factors vary greatly between different events and even within a single event. Only limited meteorite stress testing based scaling factors exist [9]. The limited stress testing is based on the quantities of samples needed and the destructive nature of the testing. Our study focuses on quantifying the flaw concentrations in multiple types of meteorites to use to scale to their parent body, therefore providing a scaling factor for meteoroid entry models that is nondestructive.
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Figure 1: Flaws (red tracings) in this slab of Bluff (a) (left) are insensitive to meteorite texture and have no point of origin. Flaws (yellow tracings) in thin section of Bluff (a) (right). The surface area is outlined in red in both images.
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3. Results
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3.1. Images In this study of over a thousand meteorite fragments (mostly hand-sized, some 40 or 50 cm across), we identified six kinds of flaw patterns. Not all fragments examined contained obvious surface or cut face flaws, and were not imaged. As not all fragments were imaged, quantitative statistics are not available from this study.
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2.2. Image Analysis A small initial subsection of meteorites where both a hand sample and thin section were available to image, further image analysis was performed. The digital images were analyzed using the ImageJ software to trace flaw features. In the thin section images flaws were traced using the Ridge Detection plugin [10] with larger flaws traced using the line segment tool (Fig. 1 right). The plugin does detect some noise as features at the level of the wavelength of visible light, but will not be fit in the analysis. In the fragment images, all flaws were traced using the line segment tool (Fig. 1 left). The line segment traced flaws include shock veins and cracks. The density of the flaws in both groups of images was then determined based on the measured surface area in the image.
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3.1.1. Patterns (1) Chondrites usually showed random flaws with no particular sensitivity to meteorite texture. Approximately 80% of these imaged indicated 3
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no point of origin (Fig. 2a), while 20% show a vertex indicating a point of origin for the flaws (Fig. 2b). (2) Approximately 10% of the chondrites imaged, have a distinct and strong network of flaws making an orthogonal (T-intersections) (Fig. 2c) or triple intersections (Y-intersections) (Fig. 2d) structure. The Chelyabinsk meteorite has the many triple intersection of flaws allowing many locations for fragmentation, which explains the very large number of centimeter-sized fragments that showered the Earth. (3) Fine irons with large crystal boundaries show flaws along the crystal boundaries (Fig. 2e). (4) Coarse irons contain flaws along kamacite grain boundaries (Fig. 2f), while other (5) fine irons have random flaws (Fig. 2g), c.f. chondrites. Finally, (6) CM chondrites showed that water-rich meteorites contained flaws around clasts (Fig. 2h).
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Figure 2: The six flaw patterns described in section 3.1.1: a. Pervomaisky (Pattern 1 no point of origin), b. Chandpur (Pattern 1 point of origin), c. Futtehpur (Pattern 2 orthogonal) d. Pacula (Pattern 2 triple-intersections), e. Arispe (Pattern 3), f. Canyon Diablo (Pattern 4), g. Coahuila (Pattern 5), h. Sutters Mill (Pattern 6)
3.2. Density Distributions The density and trace length for the flaws in both a fragment and a thin section of selected sample were measured. For the antarctic meteorites where multiple samples were present in our lab and images more images from different orientations could be taken, 20 images of the hand samples were measured. We assume that flaws follow the Weibull distribution [11], where flaws are assumed to be randomly distributed through the body and 4
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the likelihood of encountering a flaw increases with distance. The strength of the object can then be scaled based on: σl = σs (ns /nl )α
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where σs and σl refers to stress in the small and large object, ns and nl refer to the number of flaws per unit volume of the small and large object, and α is the shape parameter or scaling factor called the Weibull coefficient. The value for α is unclear [6] and a large range in α has been determined from light curve data [3]. Materials with a smaller α are relatively homogeneous, and as α approaches zero the material is more metallic (uniform). The images collected provide a two-dimensional view of the flaws. A relationship exists between the distributions of measured trace length and actual flaw size [12]:
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NL = kL L−1/(2α)+1
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where NL is the number per unit area of 2D flaw traces greater than length, L, and kL is a constant. Therefore, the slope of a log-log plot of trace length versus flaw density is proportional to α. This technique has been shown by others such as Housen and Holsapple, 1999, to provide a value for α for static strength that does vary slightly from a scaling factor determined for dynamic strength. However the non-destructive nature of this technique can still help to limit ranges and determine trends. 3.2.1. Density Plots A small number of meteorites were examined with this technique as the first attempt to try this on meteorites. Figure 3 plots the flaw lengths and densities measured of both the thin section (blue) and the hand sample (red) of four meteorites from the collections of meteorites imaged in Vienna and London. In these plots, it shows that the flaw distribution is not a power law over the entire range of sizes displayed, this flattening has been observed in other terrestrial samples [7] but is also due to the limitations in counts at the smaller end and limitations of the sample size at the larger end for both the thin section and hand samples independently. A power law can still be fit to the data and an α determined (Table 1). The error shaded is based on the uncertainty of where the lower limit in the thin section measurements is part of the power law fit. From the fit, the lower limit of the fit was varied by ±9 × 10−5 − 1.5 × 10−3 cm depending on the ridge detection plugins distribution of flaw sizes. The line fit is based on the median lower limit
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α = 0.186 ± 0.013 100
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value and includes all measured traces above that value. The error shaded in the figure allows for both kL to vary in the fit and held constant, as we are not examining kL properties of the meteorites at this time as they are not a constraint for our current meteoroid entry models. The line is fit to all the data presented from the beginning of the error shaded area to the largest flaws. In the case of a fully cracked stone described by Housen and Holsapple (1999) with an α of 0.167 equation 2 becomes:
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Figure 3: Distribution of flaw trace length for meteorites measured in thin and hand sections. The black line is based on the relationship between trace density and length, with a slope providing α. The grey shaded area displays the error on α. The blue points are the measurements from the thin section, while the red points are from the hand sample. The dotted line represents the fully cracked stone [7].
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The dotted lines indicate the fully cracked stone [7], indicating that these samples were not fully cracked. In comparison figure 4 contains plots of five antarctic meteorites. Similarly to the plots in figure 3, the flaw distribution is not a power law over the 6
DOM 10004
α = 0.172 ± 0.032
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Figure 4: Distribution of flaw trace length for Antarctic meteorites measured in thin and hand sections. The black line is based on the relationship between trace density and length, with a slope providing α. The grey shaded area displays the error on α. The blue points are the measurements from the thin section, while the red points are from the hand sample. The dotted line represents the fully cracked stone [7].
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Table 1: Calculated Weibull coefficients
Bluff (a) Pervomaisky Futtehpur Tamdakht ALHA77294 DOM 10004 GRA 06116 MIL 07036 WSG 95300 LAR 06286
L5 L5 L6 H5 H5 L5 H5 H5 H3.3 H6
0.013 0.021 0.021 0.025 0.032 0.026 0.017 0.018 0.016 0.032
0.186 0.216 0.205 0.188 0.172 0.174 0.189 0.201 0.196 0.154
No. of Hand Samples 1 1 1 1 9 5 5 5 5 4
Flaw Pattern
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Pattern Pattern Pattern Pattern Pattern Pattern Pattern Pattern Pattern Pattern
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1 1 2 1 1 1 1 1 1 1
no origin no origin orthogonal no origin no origin no origin no origin no origin no origin no origin
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4. Discussion
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Figure 5 compares the varying Weibull coefficients of terrestrial samples to meteorites by meteorite group. The ordinary chondrites examined have similar coefficients. Both the random and orthogonal flaw patterns were measured in the ordinary chondrites, and no correlation is observed to flaw pattern. The α values are slightly higher then the commonly used 0.167, which is granite, and the fully cracked stone value possibly favored by nature [7]. This provides a range for the Weibull coefficient for entry modeling of asteroids that match to ordinary chondrites. In addition, the terrestrial samples may not provide representative values. The Yakuno basalt [13] and pyrophyllite [14], though helpful for high-velocity impact testing, do not match well to the values observed in meteorites. As previously mentioned very few Weibull coefficients have been determined for meteorites and their parent bodies. In 2011, Popova et al determined that there was a large range of α based on their estimated bulk strength from fireball data at multiple breakups of nine cases of meteoroids. From that data, using a modified version of equation 1 comparing masses
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entire range, but a power law can still be fit. Multiple fragments of antarctic meteorites allow for multiple hand sample traces causing the scatter seen in the hand sample traces (red). In comparison to the fully cracked stone (dotted line), all of the antarctic samples appear to not be fully cracked.
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instead number of flaws:
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Figure 5: Weibull values plotted by meteorite group in comparison to terrestrial rocks. The ordinary chondrites have similar values for α yet offset from terrestrial samples.
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you can fit a value for α for each case and not just the range for all (Table 2). None of these fireballs match to the currently analyzed meteorites from this study. Due to the limited number of falls observed, figure 6 presents the values for α and error converted into a Gaussian distribution presented in dark blue. You can see that it covers a large range of values likely because of the varying meteorite types (Table 2), but also the limited number of fireballs observed. In comparison the two published values for α from meteorite stress testing by Cotto-Figueroa et al [9] are plotted similarly in light blue. Also in figure 6 are the same commonly used values based on terrestrial rocks from figure 5. These are in comparison to our calculated values of α based on flaw image analysis in red. Similar to figure 5 our values do overlap with the nature favored 0.167, but do vary out from there. From stress testing the value of α for Allende overlaps with our range, but the value for their sample of Tamdakht does not. The cutting of samples during preparation for
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Table 2: Calculated values for α based on Popova et al’s [3] estimated bulk strength values for fireball breakups
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σ 0.181 0.145 0.325 0.492 0.316 0.150 0.0595 0.207 0.150
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Meteorite Type Almahata Sitta Ureilite-an Bunburra Rockhole Eucrite Jesenice L6 Lost City H5 Mor´avka H5 Tagish Lake C2-ung Villalbeto de la Pe˜ na L6 Innisfree L5 Peekskill H6
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Based on the meteorites examined in our study, six flaw patterns have been observed. The majority of the meteorites displayed no particular sensitivity to meteorite texture. The ordinary chondrites analyzed had either this flaw behavior or the orthogonal pattern, and all but one had a Weibull coefficients higher than the commonly used 0.167. Within this small sampling, there was not a correlation of flaw pattern to α, but this study will continue to examine a larger selection of meteorites to see if there is a correlation between flaw pattern and α. The H and L ordinary chondrites examined here have an overlapping range of α, and this study will also examine more meteorite classes. Varying meteorite types and flaw patterns may explain the variations in α determined from fireball data [3]. If no correlation is found a reasonable range may be determined from these results. Values of α will be used in models created by the Asteroid Threat Assessment Project (ATAP) to try to determine the behavior of asteroids
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Cotto-Figueroa et al.’s stress testing did break several cubes biasing their test cubes and providing a more heterogeneous flaw distribution than was present through the original larger samples. This bias is what likely is causing the difference between our determined Weibull coefficients. It has been suggested that the α for static strength determined from flaw distributions can provide a smaller value then the α for dynamic strength[7], but may explain some of the variation from the Tamdakht stress testing value.
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Figure 6: Plot of likelihoods of densities. Each measurement and error has been been converted into a Gaussian distribution. Then for each source the Gaussians have been summed.
entering the atmosphere and quantify their impact hazard.
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Acknowledgements
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This work was funded by NASAs Planetary Defense Coordination Office (PDCO). We acknowledge the support and discussion from Derek Sears, figures generated by Jessie Dotson, and the curators at the meteorite collections in London and Vienna who allowed access to their collections.
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References
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[1] P. J. Register, D. L. Mathias, L. F. Wheeler, Asteroid fragmentation approaches for modeling atmospheric energy deposition, Icarus 284 (2017) 157 – 166.
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[2] L. F. Wheeler, P. J. Register, D. L. Mathias, A fragment-cloud model for
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[4] D. W. G. Sears, H. Sears, D. R. Ostrowski, K. L. Bryson, J. Dotson, M. B. Syal, D. C. Swift, A meteorite perspective on asteroid hazard mitigation, Planet. Space Sci. 124 (2016) 105–117.
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[5] M. B. Syal, J. M. Owen, P. L. Miller, Deflection by kinetic impact: Sensitivity to asteroid properties, Icarus 269 (2016) 50 – 61.
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[3] O. Popova, J. Borovi˘cka, W. K. Hartmann, P. Spurn´ y, E. Gnos, I. Nemtchinov, J. M. Trigo-Rodr´ıguez, Very low strengths of interplanetary meteoroids and small asteroids, Meteoritics & Planetary Science 46 (2011) 15251550.
[6] E. Asphaug, E. V. Ryan, M. T. Zuber, Asteroid Interiors, in: W. F. Bottke, A. Cellino, P. Paolicchi, R. P. Binzel (Eds.), Asteroids III, University of Arizona Press, 2002, pp. 463–484. [7] K. R. Housen, K. A. Holsapple, Scale Effects in Strength-Dominated Collisions of Rocky Asteroids, Icarus 142 (1999) 21 – 33. [8] B. Baldwin, Y. Sheaffer, Ablation and breakup of large meteoroids during atmospheric entry, Journal of Geophysical Research 76 (1971) 4653–4668.
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[9] D. Cotto-Figueroa, E. Asphaug, L. A. J. Garvie, A. Rai, J. Johnston, L. Borkowski, S. Datta, A. Chattopadhyay, M. A. Morris, Scaledependent measurements of meteorite strength: Implications for asteroid fragmentation, Icarus 277 (2016) 73–77.
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asteroid breakup and atmospheric energy deposition, Icarus 295 (2017) 149 – 169.
[10] C. Steger, An unbiased detector of curvilinear structures, IEEE Transactions on Pattern Analysis and Machine Intelligence 20 (1998) 113–125.
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[11] W. Weibull, A statistical theory of the strength of materials, in: Proceedings of the Royal Swedish Institute for Engineering Research, 151. [12] A. R. Piggot, Fractal relations for the diameter and trace length of discshaped fractures, Journal of Geophysical Research 102 (1997) 18,121– 18,125.
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[14] H. Nagaoka, S. Takasawa, A. M. Nakamura, K. Sangen, Degree of impactor fragmentation under collision with a regolith surfacelaboratory impact experiments of rock projectiles, Meteoritics & Planetary Science 49 (2014) 69–79.
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[13] A. M. Nakamura, P. Michel, M. Setoh, Weibull parameters of Yakuno basalt targets used in documented high-velocity impact experiments, Journal of Geophysical Research 112 (2007).
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Scale strength of meteorite to parent body using scaling factor Non-destructive technique to determine scaling factor for strength of meteorites Weibull coefficient within range of previous determined values for meteorites
S0032-0633(17)30411-7
DOI:
10.1016/j.pss.2018.06.018
Reference:
PSS 4569
To appear in:
Planetary and Space Science
Received Date: 3 November 2017 Revised Date:
26 June 2018
Accepted Date: 26 June 2018
Please cite this article as: Bryson, K.L., Ostrowski, D.R., Blasizzo, A., Meteorite flaws and scaling for atmospheric entry, Planetary and Space Science (2018), doi: 10.1016/j.pss.2018.06.018. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Meteorite Flaws and Scaling for Atmospheric Entry
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Kathryn L. Brysona,b,∗, Daniel R. Ostrowskia,b , Aline Blasizzoc a
NASA Ames Research Center Bay Area Environmental Research Institute c Texas Tech University
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Abstract
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Research is being conducted to understand the behavior of asteroids entering the atmosphere in order to help quantify their impact hazard. The strength of the body plays a critical role in determining the outcome of their impact events and is needed for many asteroid mitigation options. Meteorites are the physical material we have here to understand the larger parent body. Our objective is to scale flaw parameters in meteorites to their parent body, and therefore providing a way to scale strength from the smaller meteorites to the larger asteroids.
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The Chelyabinsk meteorite fall early February 15th, 2013 increased the awareness of asteroids impacting Earth. In response there has been an increased focus on the detection of threatening asteroids, deflecting them away from Earth if possible, and the effects of entry and impact should deflection fail. Within the NASA Ames Asteroid Threat Assessment Project (ATAP), multiple analytic models of asteroid breakup have been proposed [1, 2]. Using meteor fall data they have investigated the effects different parameters have on atmospheric breakup. The strength of the body plays a critical role in determining the outcome of their impact events [3, 4] and is needed for many asteroid mitigation options [5]. Meteorites are our only samples here
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1. Introduction
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Keywords: Meteorites, Asteroids, Near-Earth Objects, Weibull, Scaling
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Corresponding author Email address: [email protected] (Kathryn L. Bryson)
Preprint submitted to Planetary and Space Science
June 27, 2018
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2. Experimental
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Meteorites in the Natural History Museums of Vienna and London were examined. In Vienna we examined the ∼1,100 meteorites on public display which includes all meteorite classes, and 35 meteorites with obvious visible flaws were imaged. In London, due to time limitations, we examined their 567 H and 481 L chondrites, and imaged 34 meteorites with obvious visible flaws. The focus in both collections was to examine falls over finds to limit the effects of weathering, so 75% of the meteorites imaged are fall. The flaw patterns in selected individuals were imaged with a Nikon D7100, with an image size of 4000 by 6000 pixels. Flaws include cracks, shock veins, and slickensides, all of which have been shown in stress testing to be points of failure [9]. In addition 6 meteorites on loan from the Antarctic meteorite collection were also imaged.
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2.1. Microscope Imaging Thin sections of selected samples were borrowed from the Natural History Museum, London and from the Antarctic meteorite collection. The selected thin sections matched up to larger imaged meteorites. Photomicrographs were prepared under transmitted light with a Leitz Orthoplan microscope fitted with a 3MP digital camera. Selections of the thin section with images of 2048 by 1536 pixels were stitched together to form a complete image of the thin section.
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on earth to represent their parent bodies. The need is to scale the strength from the meteorite to the larger parent body. Scaling factors for strength vary greatly in terrestrial rocks [6] while a scaling factor of 0.167 has been favored by nature [7]. This factor has been used from some of the earliest meteoroid entry models [8]. A range of scaling factors has been determined previously from light curve fireball data [3], but factors vary greatly between different events and even within a single event. Only limited meteorite stress testing based scaling factors exist [9]. The limited stress testing is based on the quantities of samples needed and the destructive nature of the testing. Our study focuses on quantifying the flaw concentrations in multiple types of meteorites to use to scale to their parent body, therefore providing a scaling factor for meteoroid entry models that is nondestructive.
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Figure 1: Flaws (red tracings) in this slab of Bluff (a) (left) are insensitive to meteorite texture and have no point of origin. Flaws (yellow tracings) in thin section of Bluff (a) (right). The surface area is outlined in red in both images.
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3. Results
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3.1. Images In this study of over a thousand meteorite fragments (mostly hand-sized, some 40 or 50 cm across), we identified six kinds of flaw patterns. Not all fragments examined contained obvious surface or cut face flaws, and were not imaged. As not all fragments were imaged, quantitative statistics are not available from this study.
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2.2. Image Analysis A small initial subsection of meteorites where both a hand sample and thin section were available to image, further image analysis was performed. The digital images were analyzed using the ImageJ software to trace flaw features. In the thin section images flaws were traced using the Ridge Detection plugin [10] with larger flaws traced using the line segment tool (Fig. 1 right). The plugin does detect some noise as features at the level of the wavelength of visible light, but will not be fit in the analysis. In the fragment images, all flaws were traced using the line segment tool (Fig. 1 left). The line segment traced flaws include shock veins and cracks. The density of the flaws in both groups of images was then determined based on the measured surface area in the image.
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3.1.1. Patterns (1) Chondrites usually showed random flaws with no particular sensitivity to meteorite texture. Approximately 80% of these imaged indicated 3
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no point of origin (Fig. 2a), while 20% show a vertex indicating a point of origin for the flaws (Fig. 2b). (2) Approximately 10% of the chondrites imaged, have a distinct and strong network of flaws making an orthogonal (T-intersections) (Fig. 2c) or triple intersections (Y-intersections) (Fig. 2d) structure. The Chelyabinsk meteorite has the many triple intersection of flaws allowing many locations for fragmentation, which explains the very large number of centimeter-sized fragments that showered the Earth. (3) Fine irons with large crystal boundaries show flaws along the crystal boundaries (Fig. 2e). (4) Coarse irons contain flaws along kamacite grain boundaries (Fig. 2f), while other (5) fine irons have random flaws (Fig. 2g), c.f. chondrites. Finally, (6) CM chondrites showed that water-rich meteorites contained flaws around clasts (Fig. 2h).
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Figure 2: The six flaw patterns described in section 3.1.1: a. Pervomaisky (Pattern 1 no point of origin), b. Chandpur (Pattern 1 point of origin), c. Futtehpur (Pattern 2 orthogonal) d. Pacula (Pattern 2 triple-intersections), e. Arispe (Pattern 3), f. Canyon Diablo (Pattern 4), g. Coahuila (Pattern 5), h. Sutters Mill (Pattern 6)
3.2. Density Distributions The density and trace length for the flaws in both a fragment and a thin section of selected sample were measured. For the antarctic meteorites where multiple samples were present in our lab and images more images from different orientations could be taken, 20 images of the hand samples were measured. We assume that flaws follow the Weibull distribution [11], where flaws are assumed to be randomly distributed through the body and 4
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the likelihood of encountering a flaw increases with distance. The strength of the object can then be scaled based on: σl = σs (ns /nl )α
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where σs and σl refers to stress in the small and large object, ns and nl refer to the number of flaws per unit volume of the small and large object, and α is the shape parameter or scaling factor called the Weibull coefficient. The value for α is unclear [6] and a large range in α has been determined from light curve data [3]. Materials with a smaller α are relatively homogeneous, and as α approaches zero the material is more metallic (uniform). The images collected provide a two-dimensional view of the flaws. A relationship exists between the distributions of measured trace length and actual flaw size [12]:
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NL = kL L−1/(2α)+1
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where NL is the number per unit area of 2D flaw traces greater than length, L, and kL is a constant. Therefore, the slope of a log-log plot of trace length versus flaw density is proportional to α. This technique has been shown by others such as Housen and Holsapple, 1999, to provide a value for α for static strength that does vary slightly from a scaling factor determined for dynamic strength. However the non-destructive nature of this technique can still help to limit ranges and determine trends. 3.2.1. Density Plots A small number of meteorites were examined with this technique as the first attempt to try this on meteorites. Figure 3 plots the flaw lengths and densities measured of both the thin section (blue) and the hand sample (red) of four meteorites from the collections of meteorites imaged in Vienna and London. In these plots, it shows that the flaw distribution is not a power law over the entire range of sizes displayed, this flattening has been observed in other terrestrial samples [7] but is also due to the limitations in counts at the smaller end and limitations of the sample size at the larger end for both the thin section and hand samples independently. A power law can still be fit to the data and an α determined (Table 1). The error shaded is based on the uncertainty of where the lower limit in the thin section measurements is part of the power law fit. From the fit, the lower limit of the fit was varied by ±9 × 10−5 − 1.5 × 10−3 cm depending on the ridge detection plugins distribution of flaw sizes. The line fit is based on the median lower limit
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α = 0.186 ± 0.013 100
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value and includes all measured traces above that value. The error shaded in the figure allows for both kL to vary in the fit and held constant, as we are not examining kL properties of the meteorites at this time as they are not a constraint for our current meteoroid entry models. The line is fit to all the data presented from the beginning of the error shaded area to the largest flaws. In the case of a fully cracked stone described by Housen and Holsapple (1999) with an α of 0.167 equation 2 becomes:
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Figure 3: Distribution of flaw trace length for meteorites measured in thin and hand sections. The black line is based on the relationship between trace density and length, with a slope providing α. The grey shaded area displays the error on α. The blue points are the measurements from the thin section, while the red points are from the hand sample. The dotted line represents the fully cracked stone [7].
NL = 0.64L−2
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The dotted lines indicate the fully cracked stone [7], indicating that these samples were not fully cracked. In comparison figure 4 contains plots of five antarctic meteorites. Similarly to the plots in figure 3, the flaw distribution is not a power law over the 6
DOM 10004
α = 0.172 ± 0.032
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Figure 4: Distribution of flaw trace length for Antarctic meteorites measured in thin and hand sections. The black line is based on the relationship between trace density and length, with a slope providing α. The grey shaded area displays the error on α. The blue points are the measurements from the thin section, while the red points are from the hand sample. The dotted line represents the fully cracked stone [7].
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Table 1: Calculated Weibull coefficients
Bluff (a) Pervomaisky Futtehpur Tamdakht ALHA77294 DOM 10004 GRA 06116 MIL 07036 WSG 95300 LAR 06286
L5 L5 L6 H5 H5 L5 H5 H5 H3.3 H6
0.013 0.021 0.021 0.025 0.032 0.026 0.017 0.018 0.016 0.032
0.186 0.216 0.205 0.188 0.172 0.174 0.189 0.201 0.196 0.154
No. of Hand Samples 1 1 1 1 9 5 5 5 5 4
Flaw Pattern
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Pattern Pattern Pattern Pattern Pattern Pattern Pattern Pattern Pattern Pattern
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1 1 2 1 1 1 1 1 1 1
no origin no origin orthogonal no origin no origin no origin no origin no origin no origin no origin
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4. Discussion
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Figure 5 compares the varying Weibull coefficients of terrestrial samples to meteorites by meteorite group. The ordinary chondrites examined have similar coefficients. Both the random and orthogonal flaw patterns were measured in the ordinary chondrites, and no correlation is observed to flaw pattern. The α values are slightly higher then the commonly used 0.167, which is granite, and the fully cracked stone value possibly favored by nature [7]. This provides a range for the Weibull coefficient for entry modeling of asteroids that match to ordinary chondrites. In addition, the terrestrial samples may not provide representative values. The Yakuno basalt [13] and pyrophyllite [14], though helpful for high-velocity impact testing, do not match well to the values observed in meteorites. As previously mentioned very few Weibull coefficients have been determined for meteorites and their parent bodies. In 2011, Popova et al determined that there was a large range of α based on their estimated bulk strength from fireball data at multiple breakups of nine cases of meteoroids. From that data, using a modified version of equation 1 comparing masses
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entire range, but a power law can still be fit. Multiple fragments of antarctic meteorites allow for multiple hand sample traces causing the scatter seen in the hand sample traces (red). In comparison to the fully cracked stone (dotted line), all of the antarctic samples appear to not be fully cracked.
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instead number of flaws:
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Figure 5: Weibull values plotted by meteorite group in comparison to terrestrial rocks. The ordinary chondrites have similar values for α yet offset from terrestrial samples.
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(4)
you can fit a value for α for each case and not just the range for all (Table 2). None of these fireballs match to the currently analyzed meteorites from this study. Due to the limited number of falls observed, figure 6 presents the values for α and error converted into a Gaussian distribution presented in dark blue. You can see that it covers a large range of values likely because of the varying meteorite types (Table 2), but also the limited number of fireballs observed. In comparison the two published values for α from meteorite stress testing by Cotto-Figueroa et al [9] are plotted similarly in light blue. Also in figure 6 are the same commonly used values based on terrestrial rocks from figure 5. These are in comparison to our calculated values of α based on flaw image analysis in red. Similar to figure 5 our values do overlap with the nature favored 0.167, but do vary out from there. From stress testing the value of α for Allende overlaps with our range, but the value for their sample of Tamdakht does not. The cutting of samples during preparation for
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Table 2: Calculated values for α based on Popova et al’s [3] estimated bulk strength values for fireball breakups
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Meteorite Type Almahata Sitta Ureilite-an Bunburra Rockhole Eucrite Jesenice L6 Lost City H5 Mor´avka H5 Tagish Lake C2-ung Villalbeto de la Pe˜ na L6 Innisfree L5 Peekskill H6
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Based on the meteorites examined in our study, six flaw patterns have been observed. The majority of the meteorites displayed no particular sensitivity to meteorite texture. The ordinary chondrites analyzed had either this flaw behavior or the orthogonal pattern, and all but one had a Weibull coefficients higher than the commonly used 0.167. Within this small sampling, there was not a correlation of flaw pattern to α, but this study will continue to examine a larger selection of meteorites to see if there is a correlation between flaw pattern and α. The H and L ordinary chondrites examined here have an overlapping range of α, and this study will also examine more meteorite classes. Varying meteorite types and flaw patterns may explain the variations in α determined from fireball data [3]. If no correlation is found a reasonable range may be determined from these results. Values of α will be used in models created by the Asteroid Threat Assessment Project (ATAP) to try to determine the behavior of asteroids
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Cotto-Figueroa et al.’s stress testing did break several cubes biasing their test cubes and providing a more heterogeneous flaw distribution than was present through the original larger samples. This bias is what likely is causing the difference between our determined Weibull coefficients. It has been suggested that the α for static strength determined from flaw distributions can provide a smaller value then the α for dynamic strength[7], but may explain some of the variation from the Tamdakht stress testing value.
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Figure 6: Plot of likelihoods of densities. Each measurement and error has been been converted into a Gaussian distribution. Then for each source the Gaussians have been summed.
entering the atmosphere and quantify their impact hazard.
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Acknowledgements
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This work was funded by NASAs Planetary Defense Coordination Office (PDCO). We acknowledge the support and discussion from Derek Sears, figures generated by Jessie Dotson, and the curators at the meteorite collections in London and Vienna who allowed access to their collections.
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[14] H. Nagaoka, S. Takasawa, A. M. Nakamura, K. Sangen, Degree of impactor fragmentation under collision with a regolith surfacelaboratory impact experiments of rock projectiles, Meteoritics & Planetary Science 49 (2014) 69–79.
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[13] A. M. Nakamura, P. Michel, M. Setoh, Weibull parameters of Yakuno basalt targets used in documented high-velocity impact experiments, Journal of Geophysical Research 112 (2007).
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Scale strength of meteorite to parent body using scaling factor Non-destructive technique to determine scaling factor for strength of meteorites Weibull coefficient within range of previous determined values for meteorites