A fragment cloud model for asteroid breakup and atmospheric energy deposition

A fragment cloud model for asteroid breakup and atmospheric energy deposition ARTICLE IN PRESS JID YICAR [m5G; February 27, 2017;17 2 ] Icarus 0 0 0 (2017) 1–21 Contents lists available at ScienceDire[.]

ARTICLE IN PRESS JID: YICAR [m5G;February 27, 2017;17:2] Icarus 0 (2017) 1–21 Contents lists available at ScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarus A fragment-cloud model for asteroid breakup and atmospheric energy deposition Lorien F Wheeler a,∗, Paul J Register b, Donovan L Mathias c a CSRA, NASA Ames Research Center, MS 258-6, Moffett Field, CA 94035, United States Vanderbilt University, PMB 401807, 2301 Vanderbilt Place, Nashville, TN 37235, United States c NASA Ames Research Center, MS 258-5, Moffett Field, CA 94035, United States b a r t i c l e i n f o Article history: Received December 2016 Revised 13 February 2017 Accepted 17 February 2017 Available online xxx a b s t r a c t As asteroids break up during atmospheric entry, they deposit energy that can be seen in flares of light and, if substantial enough, can produce damaging blast waves Analytic models of asteroid breakup and energy deposition processes are needed in order to assess potential airburst hazards, and to enable inferences about asteroid properties or breakup physics to be made from comparisons with observed meteors This paper presents a fragment-cloud model (FCM) that is able to represent a broad range of breakup behaviors and the resulting variations in energy deposition in ways that make it a useful tool for both applications Sensitivity studies are performed to investigate how variations the model’s fragmentation parameters affect the energy deposition results for asteroids 20–500 m in diameter The model is also used to match observational data from the Chelyabinsk meteor and infer potential asteroid properties and representative modeling parameter ranges Results illustrate how the model’s fragmentation parameters can introduce different energy deposition features, and how much they affect the overall energy deposition rates, magnitudes, and altitudes that would drive ground damage for risk assessment applications © 2017 The Authors Published by Elsevier Inc This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction Throughout its history, Earth has been continually bombarded by asteroids While the vast majority are small and burn up harmlessly in the atmosphere, rare impacts from larger objects have caused notable damage, ranging from the flattened forest in Tunguska (Vasilyev, 1998) to the K-T dinosaur extinction event (Alvarez et al., 1980) More recently, the 20-m asteroid that airburst over Chelyabinsk, Russia in 2013 injured 1500 people and caused $33 M in damage (Popova et al., 2013) This event motivated new assessments of the potential threat posed by midsized asteroids that may not be large enough to cause cratering or global-scale effects, but may still produce significant ground damage To enable assessment of these risks, models of asteroid entry and breakup are needed As a bolide descends toward Earth and breaks up under aerodynamic forces, portions of its kinetic energy are transferred into the atmosphere through drag and thermal ablation This energy pro- ∗ Corresponding author E-mail addresses: lorien.wheeler@nasa.gov (L.F Wheeler), paul.j.register@ vanderbilt.edu (P.J Register), donovan.mathias@nasa.gov (D.L Mathias) duces observable light emissions and, if substantial enough, can generate blast overpressure waves or thermal radiation with the potential to cause significant ground damage However, due to the rarity of sizeable asteroid bursts, limited data regarding pre-entry asteroid properties, and the complex multi-physics processes taking place during atmospheric entry, many key factors of the fragmentation and energy deposition process have remained uncertain The purpose of our work is to develop an analytic approach for modeling asteroid breakup, focused on capturing the atmospheric energy deposition that drives ground damage and observable light emission These energy deposition results are used to evaluate airburst altitudes and severities for probabilistic asteroid impact risk assessments (Mathias et al., 2017), and can also be compared with energy deposition estimates from observed meteor light curves A central challenge in developing such analytic models is including an effective balance of fidelity and efficiency The model needs a sufficient, tractable set of modeling parameters that can reasonably represent key structural properties, combined with enough flexibility in the fragmentation modeling approach to produce a variety of breakup behaviors and energy deposition features At the same time, the model must also remain simple enough to efficiently compute the millions of cases needed for probabilistic risk http://dx.doi.org/10.1016/j.icarus.2017.02.011 0019-1035/© 2017 The Authors Published by Elsevier Inc This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 assessment or explore the many parameter variations needed to match specific features of observed meteor entries We have developed a fragment-cloud model (FCM) that represents the breakup process using a combination of independent fragments and aggregate debris clouds The model has the capability to vary key fragmentation parameters such as the number of fragments produced in each break, the relative mass distributions between fragments and debris clouds, and fragment strengths Combining treatments of both discrete fragments and multiple debris clouds with these variable parameters enables the model to efficiently represent a wide variety of potential energy deposition features, from the broad, smooth flares associated with large bursts to multiple small flares from successive fragmentations This capability provides a means to investigate how asteroids with different structures, material properties, and breakup mechanisms may deposit energy, and how those factors may affect potential ground damage Comparing different parameter variations can help to assess which factors most significantly affect energy deposition, which factors make little difference, and what aspects may warrant further study In addition, adjusting the various modeling parameters to match observational data or high-fidelity simulation results can inform what parameter ranges or assumptions provide the best analytic proxies for the more complex underlying processes Matching observed meteor entries can also potentially provide insight into the object’s pre-entry characteristics, such as its structure, bulk density, porosity, or strength ranges (e.g., Brown et al., 2002; Ceplecha and ReVelle, 2005) This paper gives an overview of the current fragment-cloud model and presents initial sensitivity studies investigating the effects of its various fragmentation parameters on energy deposition The parameter variations assessed include the initial aerodynamic breakup strength, fragment strength scaling, the number of fragments per break, debris cloud mass per break, cloud dispersion rates, and ablation rates We present the resulting effects on energy deposition features and discuss the implications for risk assessment applications and observational inferences We then apply the model to reproduced observational data from the 2013 Chelyabinsk meteor Results demonstrate the model’s ability to estimate the energy deposited during realistic breakup processes, match specific energy deposition features, and provide insight into representative parameter ranges and asteroid properties 1.1 Modeling background A number of analytic asteroid entry models have been previously published, employing a range of simplified approaches for representing the breakup process Many of these models tend to focus on describing specific aspects of the problem—such as strewn fields, crater formation, landed mass, or ablation—but are not applied to energy deposition or airburst assessment (e.g., Passey and Melosh, 1980; Melosh, 1981; Baldwin and Sheaffer, 1971; Borovicˇ ka et al., 2007) The existing breakup models applicable to energy deposition and risk assessment tend to use either a “pancake” approach or a discrete fragmentation approach In the pancake type models, the fragmentation process is treated as a continuous deformation of an aggregate, single-body mass The bolide remains intact until it meets a specified flight condition that triggers its disruption It then begins to spread laterally into a pancake shape, decelerating and ablating as its frontal area grows These models have typically been used to represent catastrophic fragmentation resulting in a single primary flare or burst, and are readily applicable to estimating burst altitudes for risk assessments and damage models Hills and Goda (1993 , 1998) used a pancake approach to model energy deposition and airburst damage for various representative asteroid types Chyba et al (1993) applied a similar pancake approach to estimate energy deposition and burst altitudes for variations of the Tunguska event These models have subsequently been adopted in several asteroid impact risk assessment studies (Stokes et al 2003; Collins et al., 2005; Motiwala et al., 2015) However, the aggregate pancake treatment does not allow for variations that could result from nonuniform asteroid structures and the behavior of large, independent fragments Discrete or progressive fragment approaches, on the other hand, treat the breakup as a successive series of fragmentation events that split the body into individual pieces The pieces increase in strength with each break, and then continue to undergo discrete fragmentations each time their flight conditions exceed their new strength Baldwin and Sheaffer (1971) used this approach to represent swarms of an increasing number of fragmenting pieces in order to account for the effects of fragmentation on mass ablation and landed mass ReVelle (20 05, 20 07) presents several discrete fragmentation treatments in which the fragments either fly side-by-side in formation (collective wake), or are shed into the wake (non-collective wake) Mehta et al (2015) proposed an alternate treatment in which each fragment is assumed to separate enough to fly independently from the other fragments These types of models often assume identical fragments in order to approximate general fragmentation trends and effects Purely pancake or discrete fragmentation models are useful for approximating simple, uniform flares or aggregate fragment effects, but individually are not able to capture the range of physical processes and energy deposition variations that occur in realistic breakup events (Register et al., 2017) An actual breakup process likely involves a combination of these behaviors, consisting of both larger, intact fragments that can reasonably be treated independently, and smaller debris or dust that is predominantly influenced by the common group aerodynamics Hybrid model concepts like the FCM have been discussed in the literature to some extent (Popova, 2011; Artemieva and Shuvalov, 1996 , 2001), but specific analytic models involving both discrete and aggregate fragmentation components have not been presented Popova et al (2013) showed results of such a model applied to the Chelyabinsk meteor, but did not discuss its implementation or details Broader studies investigating analytic hybrid model behavior, parameter sensitivities, or risk assessment applications are also lacking in the literature High-fidelity hydrocode simulations have also been used to study asteroid entry and breakup These simulations generally consider specific impact events (e.g., Crawford et al., 1995; Boslough and Crawford, 2008; Zahnle and Mac Low, 1994) or investigate a specific aspect of the entry or breakup process (e.g., Robertson and Mathias, 2017; Korycansky et al., 2003; Shuvalov et al., 1999, 2002) However, hydrocodes are not currently used as part of impact risk models or to reproduce specific light curves Even with recent gains in supercomputing power, these simulations remain too computationally expensive for the hundreds (Rumpf et al., 2016) to tens-of-millions (Mathias et al., 2017) of cases needed for statistical assessment of the impact threat For example, the hydrocode simulations of Chelyabinsk-like meteor entries presented in Robertson and Mathias (2017) required around 48 h on 72 supercomputing cores (∼3500 core-hours) for each single case For observational comparisons, current hydrocode applications have not combined all of the relevant three-dimensional physics, object strength, mass ablation, thermal radiation, shock layer chemistry, etc required to computationally create a synthetic light curve Higher fidelity models also require specific definition of the preentry structure and material properties, which are not generally known For these reasons, analytical approximations remain the most efficient modeling constructs for risk assessment and inferring pre-entry characteristics of observed meteors Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 ARTICLE IN PRESS JID: YICAR [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 equations as functions of uniform altitude increments rather than uniform time steps in order to more readily track the final energy deposition as a function of altitude The altitude step can be set as a model input, with 10-m increments used as the default setting to provide sufficient resolution The time-derivatives of velocity, v, flight angle, θ , and mass, m, are given by:   dv = − Cd ρA v2 A/m + g sin θ dt dθ =  v RE + h dm = − g = g0 Fig Diagram of the fragment-cloud model approach In our previous paper, Register et al (2017), we compared several of the existing analytic breakup models using the Chelyabinsk meteor as a test case We then introduced a new model that was able to better reproduce the observed energy deposition features by combining the discrete and pancake approaches That initial combination model generated two independent fragments and a debris cloud with each successive breakup event, using a single radius-split parameter to define the relative sizes of the fragments and remaining cloud mass The fragment-cloud model presented here is a continuation of the combination model work introduced in Register et al (2017) The current model has been extended to include additional fragmentation parameters and the ability to model multiple fragments with variable relative mass distributions between the fragments and the debris clouds This implementation now appears to contain a sufficient mix of parameters to produce a realistic variety of energy deposition features, which can then potentially be tied to relevant physical asteroid properties The focus of the current effort presented here is to begin to evaluate the applicability and effects of the various parameters used in this approach Fragment-cloud model The fragment-cloud model (FCM) is an analytic model for estimating the energy deposited in the atmosphere during asteroid entry and breakup The model integrates the standard single-body meteoric equations of motion and ablation to compute flight trajectories, velocities, and masses of the fragmenting bolide components throughout entry The breakup process is represented as a series of progressive fragmentation events that split the bolide into a number of independent sub-fragments and a debris cloud The model computes the atmospheric energy deposition throughout descent as the total change in kinetic energy of all fragments and clouds as a function of altitude Energy deposition results are expressed as the spatial rate at which energy is deposited in a given altitude increment of atmosphere, i.e., as the amount of energy deposited per unit of altitude change, conventionally given in units of kilotons per kilometer (kT/km) Fig shows a notional diagram of the fragmentation approach and components 2.1 Flight integration The bolide’s flight path angle, velocity, mass, and size are computed by integrating the standard meteor physics equations of motion and ablation (e.g., Opik, 1958) The model integrates the flight dt = + g v  cos θ dt ρA v3 Aσab dt  R 2 E RE + h dh v sin θ (1a) (1b) (1c) (1d) (1e) where Cd is the drag coefficient, A is the instantaneous crosssectional area of the bolide piece, ρ A is the atmospheric density, RE is the radius of the Earth, h is the altitude, σ ab is the ablation parameter, and g is the gravitational acceleration with g0 = 9.81 m/s2 at sea level Here we have used a drag coefficient of 1, so these formulations may differ by factors of from those that use a drag coefficient of 0.5 The initial bolide and all fragments are assumed to be spherical with a constant, uniform density The atmospheric density at each altitude step is computed using a curve fit of the 1976 standard atmosphere tables (U.S Standard Atmosphere, 1976), given by: ρA = −140.2e−0.000187h + 141.4e−0.000186h (2) 2.2 Fragmentation Fragmentation begins when the stagnation pressure exceeds the initial aerodynamic strength parameter of the bolide (Hills and Goda, 1993; Popova et al., 2011; Passey and Melosh 1980) The stagnation pressure is given by: P = ρA v (3) The fragmentation event splits the bolide into a given number of smaller, discrete fragments plus a debris cloud of a given mass fraction The fraction of the mass that goes into the debris cloud, the number of fragments produced per split, and the relative masses of each fragment are all specified as input parameters The fragments produced in each split can either all be the same size, or can be given different fractions of the total fragment mass All components are assumed to begin as spheres with the same density, flight angle, and velocity adopted from the parent fragment at the point of break The debris cloud produced in the break quickly spreads out and slows, based on an adaptation of the Hills and Goda (1993) pancake model In this approach, the cloud is treated as a single body that continuously crumbles and broadens under a common bow shock Each cloud continues descent until it ablates completely or impacts the ground The lateral spread of the cloud’s cross-sectional area is given by a dispersion velocity, vdisp , which is a function of the cloud’s velocity, v, the ratio of the atmospheric density and meteor density, ρ A /ρ m , and a dimensionless dispersion coefficient, Cdisp :  vdisp = v Cdisp ρA /ρm (4) Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 ARTICLE IN PRESS JID: YICAR [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Table Baseline parameter values and variation ranges used for the sensitivity study Initial aerodynamic strength Strength scaling α Cloud mass # Fragments per break Fragment mass splits Ablation coefficients Cloud dispersion coefficient Baseline value Variation range MPa 0.1 50% Even splits 1e-8 kg/J 3.5 0.1–100 MPa 0.05–1 10–90% 2–16 50/50%–90/10% 1e-9–5e-8 kg/J 0.35–7 The dispersion coefficient, Cdisp , is taken to have a baseline value of 3.5 (Hills and Goda, 1993), but can also be varied as an input parameter With each height-step of the flight integration, dh, the radius of the spreading cloud is increased by: dr = vdisp dt =  Cdisp ρA /ρm dh sinθ (5) The fragments produced in each break are assumed to be stronger than the parent fragment, since the break would eliminate some of the larger structural weaknesses (as discussed in Bland and Artemieva, 2006), putting the weaker portions of material into the debris cloud and leaving the stronger portions of the material as the remaining fragments The strength of each child fragment is increased as a function of its reduced size, according a Weibull-like exponential scaling relation (Weibull, 1951 , 1939; Bland and Artemieva, 2006; Artemieva and Shuvalov, 2001; Baldwin and Shaefer 1971):  S f = S0 m0 /m f α (6) where Sf and mf are the post-break strength and mass of the child fragment, S0 and m0 are the pre-break strength and mass of the parent fragment, and α is the strength scaling parameter The stronger child fragments then continue to descend until the stagnation pressure again exceeds their new strength and another break event occurs The fragment strengths continue to increase in this manner with each break unless they reach a maximum strength of 330 MPa, based on laboratory compression tests of meteorite falls (Popova et al., 2013; Popova and Nemtchinov, 2002), at which point they are prevented from further fragmentation and continue descent until they either ablate or impact the ground However, fragments not tend to reach this limit given the fragmentation parameter ranges assumed in the cases presented here Energy deposition modeling sensitivities We performed a series of sensitivity studies to investigate the effects of the model’s various fragmentation parameters on computed energy deposition Results are shown for bolides 20, 50, 100, 20 0, 30 0, and 50 m in diameter All cases are modeled with a baseline density of 2.5 g/cc, entry velocity of 20 km/s, and entry angle of 45° We chose these values to approximate a typical stonytype asteroid impact, based on bulk density ranges in Britt et al (2003), velocity distributions in Greenstreet et al (2012), and entry angle assessments in Shoemaker (1962) The modeling parameters investigated in this study are: initial aerodynamic breakup strength, strength scaling parameter, percentage of mass that goes into the debris cloud with each break, number of fragments produced per break, mass distribution among the fragments, ablation parameters, and cloud dispersion rate Each parameter was varied individually while holding the others constant at a set of baseline values Table lists the baseline values and variation ranges used for each parameter The basis for each chosen range is discussed in the respective subsections below Fig FCM energy deposition curves for 20–500 m asteroids with baseline entry and fragmentation parameters: 2.5 g/cc, 20 km/s, 45°, MPa initial strength, 0.1 α , 50% cloud mass, even fragments per break, ablation parameter 1e-8 kg/J, and cloud dispersion coefficient 3.5 Fig shows the FCM energy deposition results for the asteroid sizes assessed, using the baseline entry properties and modeling parameters given above Fig shows an example of how the cloud and fragment components contribute to the total energy deposition for the 100-m baseline case These plots illustrate some of the key energy deposition mechanisms at play in the sensitivity results presented below The cloud components are efficient at depositing energy due to their spreading and deceleration, and contribute most to the overall curves Also, larger initial clouds deposit their energy much more gradually than smaller clouds, and tend to peak lower down, even though they are released higher than the smaller subsequent clouds The fragment components contribute relatively little energy compared to the clouds, but serve to carry mass lower and distribute the altitudes and sizes of the debris clouds released It may be noted that the baseline curves in Fig seem to show higher peak altitudes than those of previous pancake-style models, such as the Tunguska cases presented by Chyba et al (1993), or the Hills and Goda (1993) results adopted by the Stokes et al (2003) risk assessment This is due largely to differences in our baseline parameter assumptions, such as initial density (and associated energy), aerodynamic breakup strengths, cloud mass, and cloud dispersion rates These differences are noted in the relevant parameter discussions below The primary modeling difference is that the faster dissipation of multiple smaller clouds contributes to higher peak altitudes compared to the single large clouds used in pure pancake approaches, as illustrated in Fig In Register et al (2017), we provide a direct comparison of the combination fragment-cloud approach with previous purely pancake and purely discrete fragmentation approaches Additionally, in Mathias et al (2017) we provide a comparison of FCM burst altitudes for the Tunguska-scale cases presented in Chyba et al (1993), using both FCM baseline parameters and the parameters assumed in the Chyba cases When using equivalent input parameters, the FCM approach yields peak altitudes within the range of previously published estimates for Tunguska-scale impacts The following subsections present the ranges of energy deposition curves resulting from each parameter variation Except where noted, the non-varied parameters maintain the baseline values given in Table For each parameter, the effects on energy deposition features—such as the maximum energy deposition rate, the altitude at which the peak occurs, the overall width and shape of the primary peak, emergence of smaller flares, and the amount of energy being deposited at ground level—are discussed and tied to the associated differences in breakup mechanisms The potential Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig Energy deposition contributions of fragment and cloud components for the 100-m diameter baseline asteroid impact case The left-hand figure shows the contributions from all the fragments compared with the contribution from all the clouds, the center figure shows the energy deposition from each individual cloud generation, and the right-hand figure shows the cumulative contribution of each successive cloud generation, building up to the total from all the clouds implications for risk assessment applications or observational light curve comparisons are also noted For observational comparisons, the luminous efficiency must be estimated in order to determine energy deposition from light emission, or vice versa In the discussions below, references to matching observed “light curve” features assume this translation between modeled energy deposition and measured luminous power, recognizing the associated uncertainties For risk assessment applications, estimates of an effective burst energy and burst altitude are needed to assess the ground damage resulting from an airburst blast wave The burst energy and altitude can then be used with yield-scaling relations derived from nuclear test data to obtain damage areas for given overpressure levels (Glasstone and Dolan, 1977) An overpressure threshold of psi has typically been used to define the ground damage area for asteroid impact risk studies (Mathias et al., 2017; Stokes et al., 2003; Motiwala et al., 2015; Hills and Goda, 1993 , 1998; Toon et al., 1997), and we maintain that convention in the damage discussions below In considering the blast damage potential of various energy deposition profiles, note that a lower burst altitude does not always produce more ground damage than a higher burst altitude For a given burst energy, there is an “optimal” burst height that produces the greatest possible ground overpressure levels This altitude is proportional to the cube root of the burst energy, and so increases with asteroid kinetic energy With the baseline density and velocity values used herein, the optimal burst altitudes are around km for the 20 m diameter case, km for the 50 m case, km for the 100 m case, 18 km for the 200 m case, 27 km for the 300 m case, and 46 km for the 500 m case One approach to extracting an effective burst altitude from an energy deposition curve is to use the altitude at which the maximum energy deposition occurs For smooth or relatively narrow profiles where the maximum falls somewhere near the middle of the overall flare (like the baseline cases shown in Fig 2), the peak maximum provides a reasonable proxy for the burst altitude In other cases, however, where the energy deposition profile is very broad or contains substantial sub-flares, the maximum energy deposition point may be overly influenced by minor curve shape features that would not likely influence the ground damage On the other hand, such variations can be advantageous for reproducing the more specific features of observed meteor events and inferring what breakup characteristics could produce those features Parameter ranges that produce variations in detailed features of the energy deposition curves are advantageous for enabling matches to observed meteors, while parameter ranges that vary the overall size and altitude of the primary flare are most relevant to characterizing airburst severity for risk assessment applications 3.1 Cloud mass fraction One of the predominant factors in representing the fragmentation behavior within the FCM is how much of the mass is treated as a debris cloud versus how much is treated as independent fragments The nature of asteroid physical properties and structures is the subject of much debate, with possibilities ranging from cohesive monoliths to loosely bound rubble piles (Britt et al., 2003; Sanchez and Scheeres, 2014) These structural factors could cause large variations in the amount of debris released, with substantial regolith, gravel, or dust liberated during disruption of rubble pile conglomerates, and with coherent or pre-fractured monoliths potentially producing much less dust To explore these variations, we ran cases with cloud masses of 10–90% of the parent mass generated in each break Fig shows the variations in the energy deposition due to different cloud mass fractions For these cases, which use the moderate baseline strength scaling value, lower cloud percentages tend to sharpen the nose of the peak and bring the point of maximum energy deposition toward the top of the flare This is a result of many, closely spaced successive fragmentations following the initial breakup With the baseline two fragments per split, putting little mass into the cloud means that the fragment strengths not increase as quickly, allowing for a faster fragmentation rate Higher cloud percentages, on the other hand, tend to create a more gradual, smoother energy deposition profile with the maximum pushed toward the bottom of the flare This behavior is consistent with the expected tendencies of the purely fragment-based and purely pancake-style models compared in Register et al (2017) The 90% cloud cases are nearly identical to cases run with 100% cloud and provide a comparison of what a pure pancake model would predict given the same baseline parameters The larger initial clouds yield a more gradual energy release as they spread and slow, and also increase the strength gain of the smaller remaining fragments, leading to more widely spaced fragmentations However, it is the large size of the first cloud released, not the smaller subsequent clouds, that pushes the peak down in these high-cloud-mass cases The spacing of the fragmentation events smoothens the release of the subsequent clouds making a more sloped peak, but they still dissipate above the first cloud due to their smaller size, as illustrated in Fig For bolides over 100 m in diameter, very large cloud masses make the energy deposition almost as gradual as the pre-breakup rates of the intact body A notable feature that emerges from the combination of fragment and cloud components produced in each break is the potential for relatively small fragmentation events to cause sharp spikes in energy deposition These spikes are due to the sudden release of small debris clouds that are initially traveling at fragment veloci- Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig Energy deposition sensitivities to cloud mass variations from 10 to 90%, for asteroids 20–500 m in diameter All other parameters maintain the baseline values given in Table ties after penetrating into denser atmosphere, and then spread and slow extremely rapidly as soon as they are released This effect is exaggerated in these cases because even fragment splits were used, resulting in all child fragments breaking and releasing small clouds at the exact same altitudes The sharp upper nose of the low-cloud-mass cases is also dependent on the choice of the strength scaling parameter Fig shows the same cloud-mass comparisons run with a higher strength scaling parameter of 0.5 For these cases, the break events are fewer, more broadly spaced, and persist to lower altitudes, yielding a more sloped upper peak and more distinctive lower flares penetrating below the point where the largest initial clouds have dissipated In these cases, the amount of cloud mass has more effect on the predominance of small lower spikes and the amount of energy being deposited at ground level than on the main flare As will be discussed in the strength scaling sensitivity section, low cloud fractions and/or high α values result in significant masses of intact fragments continuing to descend after dissipation of the released debris This can be seen in the large hump below the flares, which approximately continues the pre-breakup deposition trend of the initial bolide body At the baseline density and velocity assumed here, the cloud fraction has a strong influence on the altitude at which the maximum energy deposition rate occurs, shifting it by around km for smaller asteroids and by almost 20 km for the 200 m case For the two smaller cases (20 and 50 m), the range of burst altitudes predicted from the differing peak maximums would not cause any 4psi ground damage For the 100 m case, the peak maximum ranges Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig Energy deposition sensitivity to cloud mass variation with high strength scaling α = 0.5, for asteroids 20–500 m in diameter All other parameters maintain the baseline values given in Table from close to the optimal (worst-case) burst height when near the bottom of the flare (potentially causing a 28-km damage radius), to high enough to cause no 4-psi ground damage when near the top of the flare Interestingly, for the 200 m case, there is very little difference in 4-psi blast radius resulting from the minimum and maximum altitudes—only 52 km for the lower altitude compared to 56 km for the higher alt—due to the fact that the range brackets the optimal burst altitude Higher cloud percentages that place the peak at the lower end of the flare provide the most pessimistic (i.e., greater damage) burst assumptions for asteroids in the 100-m range or smaller, while moderate cloud percentages that peak at mid-flare altitudes are the most pessimistic for asteroids a few hundred meters in diameter As such, use of moderate-to-high cloud fractions appears to produce energy deposition profiles that are well suited for characterizing burst altitudes in a risk assessment In these cases, the peak deposition rate falls in the mid-tolow portion of the main flare, making it more representative of the overall flare and generally more pessimistic than peaks at the very top In contrast, the combination of low cloud mass and high α shown in Fig yields curves that would be more difficult to characterize as an airburst at a particular altitude, and not look to be as intuitively representative for smaller asteroid sizes On the other hand, the light curves and associated energy deposition of smaller observed meteor cases are not always dominated by a single, smooth flare, and often exhibit many small, abrupt spikes and jags, such as those seen in the Benešov meteor (Ceplecha and Revelle, 2005) and Košice meteor (Borovicˇ ka et al., 2013) Adjusting the cloud mass to allow the emergence of these Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig Energy deposition sensitivity to fragment number variation, for asteroids 20–500 m in diameter All other parameters maintain the baseline values given in Table features will be important for matching meteor observations, especially when combined with more realistic fragment mass distributions, and may provide insight into the properties of the initial asteroid 3.3 Number of fragments per break Increasing the number of fragments per split introduces more variability in the peak profile Fig shows comparisons of 2–16 fragments per break, for cases with the baseline 50% cloud mass and α of 0.1 Higher fragment numbers push the upper edge of the peak out to a sharper point, introduce small sub-flares, and produce wave-like oscillations The sharpening of the nose is similar to the effect of reducing the amount of cloud mass per break, as more fragments release many more small clouds in the first few breaks The strength of the child fragments also increases more rapidly when divided into smaller pieces, causing slower fragmentation rates and lower-altitude flare oscillations Going from to fragments per break produces the most notable differences, while increasing the fragmentation beyond seems to make relatively less difference, especially for the smaller sizes Again, note that the sharpness of subsequent spikes and the coherent wave-like oscillations in the main peak are magnified in these cases due to even fragment splits releasing identical clouds at the same altitudes The effect of fragment numbers naturally also depends upon the amount of cloud mass released Fig shows the same fragment number variations modeled with 10% and 90% cloud mass for the 100 m case Greater numbers of small clouds produce sharp spikes at the top of the flare, while greater numbers of larger clouds produce the more wave-like oscillations While it is intu- Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig Energy deposition sensitivity to fragment number variation for a 100 m asteroid with low (10%) and high (90%) cloud mass All other parameters have the baseline values given in Table itive to think of the fragment number as a parameter affecting the fragment portion of the process, the way in which it most notably impacts the energy deposition is in how it divides up the released cloud masses For a given cloud mass fraction, the same total cloud mass will be released in each break generation independent of the fragment divisions However, since each fragment subsequently produces its own cloud when it breaks, higher fragment numbers mean many more small clouds rather than fewer larger clouds of the same total mass This is why the cases with more fragments behave more like the low-cloud-mass cases, even though those two factors have opposite effects on the fragment sizes Increasing the fragment count slows fragmentation similarly to increasing α , except with the key difference of also influencing cloud sizes The actual subdivision of the intact fragment masses has little impact on the overall energy deposition rate relative to the cloud contributions In terms of estimating an effective burst altitude for risk assessment applications, use of fragments per break produces the most “well-behaved” energy deposition profiles, i.e., with the maximum deposition altitude most representative of the main bulk of the flare rather than concentrated at the top or subject to small, inconsequential spikes While higher fragment numbers change the detailed features of the profile, the overall magnitude, altitude, and width of the flare, which are the pertinent factors for assessing ground damage, remain similar However, reproducing detailed features from observed light curves will depend significantly on using different numbers of fragments to produce a variety of small flares Furthermore, the number of fragments produced per break could support better inferences about the bolide properties or breakup behavior by enabling the model to also match fallen meteorite masses, sizes, or numbers 3.4 Fragments mass distribution Splitting fragments into even, identical pieces is a convenient simplification, allowing a large number of cases to be evaluated with very fast, memory-efficient computations However, such symmetric fragmentation is obviously unrealistic, and can cause overly magnified spikes in energy deposition by concentrating the fragmentation effects into identical altitudes For cases with two fragments per break, we varied the fragment masses between even 50/50% splits and 90/10% splits Fig compares these variations for cases with the baseline 50% cloud mass and α 0.1 At the moderate baseline cloud percentage and α values, the fragment mass distribution makes only subtle differences, since the larger initial clouds already dominate over the later fragmentations In these cases, larger fragment mass differences push more of the energy release lower, due to larger fragments persisting and continuing to release debris at lower altitudes However, the mass split needs to be highly uneven to make any difference, and even then the differences in peak altitude and magnitude are negligible When α is large enough or cloud mass is low enough for specific fragmentation behaviors to emerge from the dominant initial cloud deposition, then the fragment mass split has more effect Fig shows the fragment mass split variations for the 20-m and 50m sizes, using a high α of 0.5 Using uneven fragment splits reduces the magnitude of sub-peaks but increases the number, as expected The narrow, jagged nature of the sub-peaks, however, tends to persist, with the different fragment sizes tending to separate the energy deposition into more numerous smaller peaks rather than spreading the release into broader sub-peaks This, again, is because debris clouds released from lower fragmentation events slow so rapidly in the denser atmosphere that even small, single breaks manifest as very sharp energy spikes It may also indicate that different dispersion coefficients may be needed for smaller cloud masses While the main flare remains similar for the sizes and baseline parameter assumptions shown here, the introduction and variation in the minor flares will be important to matching specific features of observed meteor profiles For risk assessment applications, even fragment splits provide a comparable representation of the main deposition peak as reasonably uneven split ratios Only very extreme split ratios begin to shift the peak maximum, and such splits may be as unlikely as roughly even splits If an assessment considers the sizes and masses of landed fragments, then variations in fragment mass may be more relevant However, the potential damage radius of a blast wave (either emanating from a ground impact or an airburst) exceeds that of direct impact cratering, and so tends to be the primary concern for risk modeling 3.5 Initial aerodynamic strength parameter In any breakup model, one of the most fundamental factors is the condition under which breakup is assumed to begin This parameter is generally referred to as the “strength” of the asteroid, but is really a proxy for the pressure at which the body is assumed to undergo breakup, and does not represent a specific physical strength of the material (such as compressive, tensile, or shear strength) As such, it remains a fairly poorly constrained modeling parameter more than a measurable property of the asteroid This is particularly true for larger bolides, which may range from loose Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS 10 [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig Energy deposition sensitivity to fragment mass distributions with two fragments per break, for asteroids 20–500 m in diameter All other parameters maintain the baseline values given in Table rubble piles to monoliths, and for which there is little-to-no direct observational data for comparison For this reason, the following cases compare a very large range of breakup strength, from 0.1 MPa to 100 MPa An upper limit of 10 MPa is likely a more reasonable bound for most stony type asteroids (Popova et al., 2011), but the range was extended up to 100 MPa for this study to include the potential for very strong coherent objects and to see at what point the impact shifts from depositing most of its energy in an airburst to carrying most of it directly to ground The higher strengths are also included to provide comparison with the values used in previously published models For example, Chyba et al (1993) used a breakup condition equivalent to approximately 24 MPa of stagnation pressure for their stone type Tunguska cases Hills and Goda (1993) used 200 MPa for their iron cases and 50 MPa for their hard stone cases, which were also subsequently used in the Stokes et al (2003) airburst risk assessment These significantly higher strength assumptions are one of the primary differences between the peak altitudes in our baseline set of cases and those of the previous pancake models Fig 10 shows the variation of energy deposition with breakup strength, for cases run with the baseline 50% cloud mass and 0.1 α The top of the energy deposition peak moves down in altitude with increasing breakup strength, as expected, but otherwise it maintains the same slope and form Rather than shifting the whole peak down, however, cases with strengths up to ∼10 MPa or so all tend to converge to the same peak maximum near the bottom of the curves and all dissipate at around the same altitude and rate, relatively insensitively to where breakup began Even when disrup- Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS L.F Wheeler et al / Icarus 000 (2017) 1–21 [m5G;February 27, 2017;17:2] 11 Fig Energy deposition sensitivity to fragment mass distributions with two fragments per break, for asteroids 20–50 m in diameter modeled using a higher strength scaling α of 0.5 All other parameters maintain the baseline values given in Table Sizes over 100 m are large enough to subsume the sub-peaks and not show significant variation for these cases tion begins very early on, the debris does not spread and slow significantly until it reaches thicker atmosphere The initial strength only begins to notably affect the peak maximum if it is so high that breakup doesn’t begin until near the altitude at which the debris clouds undergo significant dissipation These trends hold similarly over a moderate range of cloud masses and strength scaling values This behavior is consistent with results of high-fidelity hydrocode simulations of asteroid breakup using a range of material strength models (Robertson and Mathias, 2017) Those simulations indicated that weaker asteroids tend to have peak energy deposition around the same altitude regardless of initial strength, while stronger asteroid materials tended to burst more suddenly, close to the altitude at which the pressure first exceeded their strength If only the peak energy deposition point is used to characterize burst altitude, then the blast risk would be relatively insensitive to initial strength over a large range of values However, if the width of the energy deposition peak can be correlated to the ground overpressures produced, rather than relying solely on analogs to static nuclear point sources, then initial strength becomes a more important factor In terms of relating breakup behaviors to observed meteor light curves, broad, blunt peaks such as those produced by our lowerstrength cases may indicate early release of significant amounts of debris acting in an aggregate cloud-like fashion While higher initial breakup in thinner atmosphere does spread out the subsequent discrete fragmentations somewhat for the weaker cases, the discrete fragmentation still only occurs within the first 10 or so km after initial disruption and ceases well above where the predominant cloud energy deposition dissipates 3.5 Strength scaling parameter The fragment strength scaling parameter, α , used to increase the aerodynamic breakup strengths of child fragments relative to their decreased size (Eq 6), serves to control the rate at which successive breakup events are triggered Published estimates of α -coefficient values range widely from 0.03 to 0.57 (Yoshinaka et al., 2008; Cotto-Figueroa et al., 2016) Popova and Nemtchinov (2002) and Svetsov et al (1995) summarize ranges of published measurements and modeling conventions, and estimate values of 0.1–0.5 for stony bodies Large uncertainties remain, however, regarding how well measurements of small terrestrial rocks or fallen meteorite samples may represent pre-entry asteroid materials, inhomogeneous rubble pile conglomerations, or larger size ranges To account for these uncertainties, we extended the estimated range for stony bodies by a factor of two and considered values from 0.05 to for this study Fig 11 compares the energy deposition curves for strength scaling parameter values from 0.05 to Increasing α slows the fragmentation rate, causing larger fragments to persist longer and carry energy deposition lower This steepens the upper slope of the profile, sharpens the peak toward the lower end, and introduces sub-flares below the main peak Although increasing α has a strong effect on the qualitative shape of the peak, the altitude of the peak maximum varies only slightly for cases with moderate cloud masses, due to the dominant initial clouds already pushing the maximum deposition point toward the base of the peak Only very high α values increase the peak magnitude or introduce significant sub-flares These high strength gains produce cases that only undergo partial breakup and have relatively large pieces striking the ground intact For example, the 20-m and 50-m cases only break into a total of 16 fragments with an α of 1, as compared to the thousands (40 0 and 80 0, respectively) produced with the baseline α of 0.1 The persistence of these large fragments results in the residual humps seen below the main flare for the smaller bursts with α over 0.5 For twofragment cases with the baseline cloud mass and fragment splits, such high α values not appear to be consistent with bursts in which the asteroid undergoes thorough breakup; more moderate values in the 0.1–0.3 range seem to better represent such cases If moderate-to-high cloud fractions are assumed, then the sensitivity of a risk model to the strength scaling parameter would depend largely on whether the width of the peak or amount of energy impacting the ground are considered For cases with less than 30% cloud mass, however, α has a stronger effect on peak altitudes Fig 12 shows the same α variations for cases with 10% and 30% cloud mass Less cloud mass means that the maximum energy deposition point falls at the top edge of peak, which is shifted down by increasing α There is also less initial cloud mass to dissipate before smaller sub-flares can emerge below the main flare Likewise, cases with more fragments per split will compound the effect that α has on fragmentation rate One distinction to note is that α can have a counterintuitive effect on peak altitude for cases modeled with moderate cloud masses, due to the subtle differences in the shape of the peak’s nose For example, in the 100-m 50% cloud case, increasing α from 0.1 to 0.3 moves the peak maximum up ∼2.5 km in altitude, rather than down, because it happens to push the top of the nose out further than the bottom In reality these subtle differences in Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS 12 [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig 10 Energy deposition sensitivity to initial aerodynamic breakup strength, for asteroids 20–500 m in diameter All other parameters maintain the baseline values given in Table nose shape are unlikely to have significant effect on the resulting ground damage However, it is a factor to be considered when extracting effective burst altitudes from variably shaped energy deposition curves, or when attempting to represent physical strengths with these modeling parameters Across the cases considered, α values of 0.1–0.3 seem to provide the most representative energy deposition profiles for approximating burst altitudes 3.6 Ablation rates Although not strictly a fragmentation parameter, the effect of the assumed ablation rate on energy deposition may vary depending upon the breakup mechanics, such as the number and size of fragments and clouds produced The FCM allows different con- stant ablation coefficient values, σ ab , to be set for the fragment and cloud components, with a baseline value of 1e-8 kg/J adopted from Hills and Goda (1993) In reality, the ablation rate should vary with fragment size, shape, speed, and altitude throughout entry (Baldwin and Sheaffer, 1971) However, appropriate values for those rates and how much they may vary throughout entry remain uncertain (Johnston et al., 2015) Most simulation studies and theoretical formulations of asteroid ablation tend to assume spherical, independent fragments and may not be accurate for aggregate debris clouds, fragments flying in close proximity, irregular shapes, or very porous surfaces Alternately, most observationally derived ablation estimates are for small, high-altitude meteors, and their applicability to larger, deeper-penetrating asteroids is uncertain For Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS L.F Wheeler et al / Icarus 000 (2017) 1–21 [m5G;February 27, 2017;17:2] 13 Fig 11 Energy deposition sensitivity to fragment strength scaling parameter, α , for asteroids 20–500 m in diameter All other parameters maintain the baseline values given in Table these reasons, ablation parameters remain poorly constrained for simplified analytic models Some models neglect ablation entirely, arguing that it does not make a significant difference for meteors large enough to pose any threat (Collins et al., 2005) In contrast, Svetsov (1996) proposed that radiant fireball heating could have caused full ablation of all material in the Tunguska event Borovicˇ ka et al (2007) fit ablation coefficient values on the order of 2–3 × 10−8 kg/J (0.02– 0.03 s2 /km2 ) for their cometary dustball model of Draconid meteors To assess the potential impact that uncertain ablation rates may have on energy deposition, we varied the ablation coefficient over a large range, from 1e-9 to 5e-8 kg/J Reducing the ablation coefficient values further than 1e-9 kg/J (including neglecting ablation completely) made negligible difference to the energy deposition results for all case variations examined, so those results are omitted from the range shown Coefficient values larger than 1e-7 kg/J not produce reasonable-looking results for sizable asteroids in our model We applied the ablation parameter variations first to just the fragments, then to just the clouds, and finally to all components together Fig 13 compares the results for the 100-m diameter case, with the baseline 50% cloud mass and also with lower cloud mass fractions of 30% and 10% to allow the fragment ablation effects to appear as predominantly as possible The same trends hold equivalently for all the other sizes considered in this study For moderate cloud percentages, variation of the fragment ablation rate makes some difference in the energy deposition rates before breakup begins, but makes practically no difference to the Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS 14 [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig 12 Energy deposition sensitivity to fragment strength scaling parameter, α , for 50, 10 0, and 30 m asteroids modeled with lower cloud masses of 10% (left) and 30% (right) All other parameters maintain the baseline values given in Table main peak At the very highest rate applied, the subpeaks in the 10% cloud case are slightly reduced due to more of the fragment mass having ablated pre-breakup, leaving less cloud mass Variation of the cloud ablation rate shifts the overall altitude and width of the flare similarly for the different cloud percentages Cloud ablation rates within a factor of of the basline value have only small effect on the peak maximum, and the sharp sub-flares from small debris released in the 10% cloud case remain generally similar The predominance and magnitude of the smaller flare details that would be of interest to observational studies appear to be influenced by the ablation rate of the parent fragment more than the ablation rate of the cloud that produces the flare This is because the energy deposition of small clouds is driven predominantly by their change in kinetic energy, which occurs rapidly enough that there is little time for ablation Only the large clouds released at the beginning of the breakup disperse slowly enough for ablation to notably affect their energy deposition Again, however, the range of values applied here represents significant uncertainty and may not be approprate for aggregate debris clouds that undergo significant, rapid spreading and slowing The results included here are intended to be investigative rather than representative of likely ranges Further simulation studies and comparisons with meteor observations are needed to better bound appropriate ranges for the ways in which this model represents the fragmenting material Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS L.F Wheeler et al / Icarus 000 (2017) 1–21 [m5G;February 27, 2017;17:2] 15 Fig 13 Energy deposition sensitivity to ablation parameter variations for the 100-m asteroid case, modeled with 50% (top), 30% (middle), and 10% (bottom) cloud mass The plots in the leftmost column show variation of fragment ablation, the middle column shows variation of cloud ablation, and the rightmost column shows variation of both together All other parameters maintain the baseline values given in Table 3.7 Cloud dispersion velocity The lateral spread rate with which a cloud’s effective cross sectional area increases affects how quickly it slows due to drag forces and how much area is exposed to thermal ablation The dispersion velocity in Eq (4) and the baseline Cdisp coefficient value of 3.5 are adopted from the pancake model of Hills and Goda (1993) In that model, the asteroid mass is treated as a single continuously fragmenting body, which spreads due to shear forces under a common bow shock The dispersion velocity formula was derived based on the drag work required to spread the material to twice the original radius Passey and Melosh (1980) also arrive at an equivalent formula based on the bow shock interactions between two spheres flying in proximity and separating until they are a certain number of meteoroid radii apart Assuming identical radii and an ultimate separation distance of twice the meteoroid radius in that formulation gives a coefficient value of 3, very similar to the Hills & Goda value Their analysis of strewn fields, however, gives separation distances of 0.02–1.52 radii, leading to analogous dispersion coefficient values between ∼0.03 and 2.3 times the radius ratio of the two fragments (with 1.5 as their nominal value for even pieces, estimated to be accurate to within a factor of two) Artemieva and Shuvalov (1996) performed a similar analysis of fragment repulsion and found a coefficient value of 0.2 for symmetrical fragments Chyba et al (1993) used a spread rate formula based on pressure differentials acting on the sidewalls of a cylindrical object, which may yield slower dispersion than the Hills & Goda formula adopted in the FCM Recent high-fidelity hydrocode simulations of asteroid breakup using different material strength models (Robertson and Mathias, 2017) indicate that disrupted material does not tend to spread laterally very much, potentially resulting in peak energy deposition altitudes a few kilometers lower than the faster-spreading analytic models However, those simulations assumed uniform bolide material and did not explicitly include a treatment of dust or non-homogenous rubble Since the FCM applies the dispersion velocity to multiple debris clouds of many different sizes, rather than to the separation of discrete fragments or to one large mass representing the entire asteroid body, it is uncertain what spread rates will yield the best energy deposition estimates With the model’s current baseline assumptions, large debris clouds spread and deposit energy gradu- Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS 16 [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig 14 Energy deposition sensitivity to variations in cloud dispersion velocity coefficient, Cdisp , for asteroids 20–500 m in diameter All other parameters maintain the baseline values given in Table ally and smoothly, while smaller debris clouds released in later fragmentations spread and slow extremely rapidly, causing sharp spikes in energy deposition We varied the cloud velocity dispersion coefficient, Cdisp in Eq (4), from 0.35 to (one tenth to twice the baseline value) to assess its effects on energy deposition Fig 14 shows the energy deposition comparisons for cases with the baseline 50% cloud mass, and Fig 15 compares the variations for the 100-m case run with 20% and 80% cloud masses The primary effect of the spread rate for these cases is in changing the width of the primary peak Increasing the spread rate raises and narrows the width of the main peak by making the large, initial debris clouds slow more rapidly Conversely, decreasing the spread rate lowers and broadens the overall width of the peak Spikes from the small clouds released with later fragmentations remain relatively unaffected by the range of dispersion coefficients considered here These small clouds slow so abruptly that varying the spread rate by the range shown here has little effect on their energy deposition contributions In these ways, the dispersion rate has similar effects as the cloud ablation rate For observational comparisons, the dispersion rate provides a means of broadening the peak width, independently of the fragmentation rates and masses that may be needed to match more specific flares and features Since this is currently a highly uncertain parameter, however, initial observational comparisons will likely be more useful for refining realistic parameter bounds than making direct inferences about the object For burst altitudes taken from the maximum energy deposition point, dispersion coefficients within a factor of of the baseline Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS L.F Wheeler et al / Icarus 000 (2017) 1–21 [m5G;February 27, 2017;17:2] 17 Fig 15 Energy deposition sensitivity to variations in cloud dispersion velocity coefficient, Cdisp , for the 100-m diameter case run with 20% and 80% cloud masses All other parameters maintain the baseline values given in Table value shift the altitude up or down by under km Reducing the coefficient by an order of magnitude, however, can lower the altitude by as much as 5–7 km For smaller diameters or cases with low cloud mass fractions, reducing the spread rate coefficient by a factor of 10 can also affect the profile shape enough to shift the maximum deposition point from the top nose of the peak to the bottom, resulting in more significant peak altitude differences For example, the 100-m, 20% cloud mass case shown in Fig 15 maintains the same peak altitude at 34 km for dispersion coefficients greater than 0.6, but drops the peak altitude to 18 km when the coefficient is 0.5 or less In these ways, the burst altitude can range from being insensitive over reasonably large variations in dispersion coefficient, to being extremely sensitive to small variations Chelyabinsk meteor comparisons The Chelyabinsk meteor, which airburst over Chelyabinsk, Russia in 2013, is one of the most well-observed sizable asteroid strikes to date and provides a valuable comparison case for model development and asteroid studies We modeled the Chelyabinsk meteor to validate the FCM approach, see what ranges of parameters produced reasonable energy deposition estimates compared to a real event, and demonstrate how the various fragmentation parameters can replicate specific observed features We compared FCM results with an energy deposition curve that Brown et al (2013) derived from the Chelyabinsk light curve, and varied the model’s fragmentation parameters to best match the observational data Based on estimates from Popova et al (2013), we initially assumed the bolide to be 19.8 m in diameter, with a density of 3.3 g/cc, entering the atmosphere at a velocity of 19.16 km/s and an angle of 18.3° from horizontal The initial strength parameter was set between 1.3 and 1.6 MPa to align with the beginning of main burst at around 40 km, although minor disruption and debris shedding began earlier For these comparisons, we note that there is inherent uncertainty both in the pre-entry asteroid properties assumed and in any observationally derived energy deposition estimates Estimating energy deposition magnitudes from observed light emissions, or vice versa, requires simplified assumptions about the luminous efficiency in the measured passband (Ceplecha and ReVelle, 2005; ReVelle and Ceplecha 2001; Brown et al., 2013) This uncertainty can affect the magnitudes of the energy deposition estimates, but the general curve features (such as the presence, altitudes, and relative sizes of various flares) remain reasonably comparable For this exercise, we applied the same fragmentation parameters to all successive breaks, rather than allowing them to vary from break to break The preliminary version of the combination model presented in Register et al (2017) was also previously used to reproduce the Chelyabinsk energy deposition curve by stochastically varying fragment/cloud split ratios and strength scaling parameters with each break While applying different parameters to each break can enable more explicit matching of the curve shape, we used fixed parameters here in order to capture more generally representative properties and trends The FCM was able to produce multiple good matches using a range of fragment numbers and different combinations of the other parameters Use of 2–6 fragments per break provided sufficient variability and control to produce good matches for both the primary and secondary flare, without requiring excessive manipulation of too many specific fragment masses Fig 16 shows sample matches obtained using different numbers of fragments per break, assuming the initial asteroid properties cited from Popova et al (2013) Fig 17 shows improved matches obtained by reducing the asteroid density or radius compared to the initial assumptions Each plot includes the model results both at the full 10-m altitude-step resolution and averaged to a 1-km altitude resolution The kilometer-averaged results are closer to the resolution of the observational data and may be more representative of how energy deposition details could blend in an observed light curve Table lists the modeling parameters used to obtain each match Across all variations considered, the best matches were obtained using large cloud mass percentages of 70–88%, moderately high strength scaling α values of 0.3–0.4, and fairly narrow fragment mass distributions in which the sibling fragments were all of different but similar sizes For example, fragments were generally between 45–55% of the parent mass for 2-fragment splits, 20–30% for 4-fragment splits, or 10–20% for 6-fragment splits The main peak was not very sensitive to the fragment mass distributions, but the appearance of the sub-peak(s) below changed significantly with even small shifts in the mass splits, either merging with the main peak or splitting into a many more small flares Specific fragment mass divisions enable much better matching of detailed light curve features, and could be matched with surviving meteorite falls, but are not necessary for a blast damage model Cases modeled with the initial mass assumption all tended to over-estimate the peak magnitude of the main flare by around 40 kT/km or 50% (123 vs 82 kT/km observed) across broad variations of the fragmentation and ablation parameters Given the uncertainties in the initial bolide properties cited in Popova et al (2013), and uncertainties in converting luminous power output to energy deposition (Brown et al., 2013), this is not unreasonable The overestimation may also be due in part to the modeling sim- Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 ARTICLE IN PRESS JID: YICAR 18 [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 Fig 16 Comparisons of FCM energy deposition results with the Chelyabinsk meteor energy deposition curve (Brown et al., 2013), assuming an initial diameter of 19.8 m and density of 3.3 g/cc The left-hand Fig (Case A) shows a match obtained using fragments per break, and the right-hand figure (Case B) shows a match obtained using fragments per break Fig 17 Comparisons of FCM energy deposition results with the Chelyabinsk meteor energy deposition curve (Brown et al., 2013), using different bulk densities and/or diameters The left-hand figure (Case C) used a density of 2.5 g/cc with the original 19.8 diameter, the middle figure (Case D) used a diameter of 18 m with the original 3.3 g/cc density, and the right-hand figure (Case E) used a density of 2.3 g/cc and a diameter of 20 m Table Asteroid properties and modeling parameters used to generate the Chelyabinsk energy deposition matches shown in Figs 15 and 16 Diameter (m) Density (g/cc) Strength (MPa) α coefficient Cloud mass percentage # Fragments per break Fragment masses Fragment ablation (kg/J) Cloud ablation (kg/J) Cloud dispersion coeff Case A Case B Case C Case D Case E 19.8 3.3 1.5 0.36 86% 28/26/24/22% 1e-8 5e-9 3.5 19.8 3.3 1.6 0.315 85% 20/18/17/16/15/14% 1e-8 5e-9 3.5 19.8 2.5 1.4 0.38 84% 28/26/24/22% 1e-8 5e-9 18 3.3 1.4 0.385 84% 28/26/24/22% 1e-8 7e-9 20 2.3 1.3 0.38 85% 27/26/24/23% 1e-8 5e-9 1.5 plification of summing the energy deposition from all components as a function of altitude without accounting for potential differences in arrival times, whereas the observational data focuses on the output of the primary fireball as a function of time The consistent overestimation of the peak may also indicate that the initial bolide was less massive than the estimates adopted from Popova et al (2013) Decreasing the initial mass from 1.3e7 kg to 0.9e7–1.0e7 kg, either by reducing the density to 2.2–2.6 g/cc or by reducing the diameter to around 18 m, provided much better agreement with the observed peak energy deposition estimates, as shown in Fig 17 Given that the 3.3 g/cc density estimate was derived from the surviving meteorite falls, and a high percentage of asteroids are thought to be rubble piles with notable macroporosity (Britt et al., 2003), it is likely that the bulk density of the initial body was lower than the meteorite-derived estimate Using a bulk density of 2.2–2.6 g/cc with the original diameter estimate of 19.8 m gives a macro-porosity of 21–33% compared to the meteoritic density of 3.3 g/cc This porosity is within estimated macroporosity ranges for fractured bodies (15–25%) or rubble piles (30– 70%) (Britt et al., 2003) For the cases run with lower initial mass, reducing the cloud dispersion and ablation parameters from their baseline values also helped to better replicate the width, shape, and slope of the observed flares Since the spread rate depends upon the air-to-mass density ratio (Eq (4)), reducing the material density increases the spread rate (creating narrower peaks) unless the dispersion coefficient is reduced For most cases considered, cloud velocity dispersion coefficients of 1.5–2.5 (vs the baseline of 3.5) and cloud ab- Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS L.F Wheeler et al / Icarus 000 (2017) 1–21 lation parameters of 4e-9 to 8e-9 (vs the 1e-8 baseline) provided a slight broadening of the main flare that better matched the observed profile Ablation and dispersion rates for aggregate clouds of debris are not well constrained and would benefit from additional simulation studies or experimental validation Like the fragment mass splits, these variations improved observational matches, but did not impact the main energy deposition enough to be significant for a risk assessment In addition to matching the energy deposition profile, the amount of fallen fragment mass persisting to the ground was also considered The fits shown in Fig 17 yielded total fallen fragment masses between 50 0–650 kg, compared to the 40 0–60 0 kg estimated in Popova et al (2013) Additional variations of the cloud and fragment masses were also able to produce similarly good energy deposition fits, but yielded much higher fallen masses Restricting the cases to the relatively few fragments per break used here resulted in too many large fragments striking the ground, although the maximum sizes were below the largest >570 kg piece that was recovered from Lake Chebarkul (Popova et al., 2013) Most of the smaller meteorite finds are assumed to be carried within the debris clouds in this model, but higher fragment counts with broader mass distributions should provide a more realistic range of meteorites sizes The larger final fragment masses and omission of the higheraltitude debris shedding above the main flare are limitations of assuming a uniform material for the initial body and applying fixed fragmentation parameters to every break These assumptions make it difficult to represent larger coherent chunks that are able to persist longer and break lower, as was the case with the Chelyabinsk meteor Instead, multiple moderately large fragments of similar sizes needed to persist and burst below the main flare in order to replicate the secondary flare Similarly, the assumptions of a uniform body and fixed fragmentation parameters not allow for small fragments, debris, or regolith to be shed before the catastrophic fragmentation of the main burst begins For this reason, the cases presented here are unable to capture the small hump seen above the main flare in the observational data For risk assessment applications, these sorts of low energy, high altitude features are insignificant and can be neglected However, capturing such features is relevant to matching data from observed meteors and making inferences about pre-entry structures and properties To enable more realistic representation of varied asteroid structures and their breakup characteristics, the model is currently being extended to incorporate initial distributions of rubble pile boulderds, different material and bulk densities for porous bodies, structural pre-fracturing, or outer regolith layers Conclusions & future work The fragment-cloud model presented here provides an efficient and flexible tool for modeling the atmospheric energy deposition of fragmenting asteroids The combination of discrete fragments and debris clouds allows the model to represent the varied energy deposition characteristics that could result from different asteroid structures, material properties, or unpredictable aerodynamic interactions These varied energy deposition results could enable risk assessment studies to better evaluate the range of potential impact damage for asteroids with uncertain characteristics The model can also reproduce specific energy deposition features from observed light curves, which can help to further refine modeling assumptions, parameter ranges, and knowledge of asteroid properties The model was able to produce excellent matches to energy deposition estimates from the Chelyabinsk meteor light curve These matches demonstrated the ability of the current parameters to produce realistic energy deposition features, and enable inferences about asteroid properties and breakup behaviors In this case, we [m5G;February 27, 2017;17:2] 19 found that the lower initial energy needed to match the peak magnitude indicates a macro-porosity range of ∼21–33% and bulk density of 2.2–2.6 g/cc compared to the found meteorite densities of 3.3 g/cc These findings are in good agreement with likely ranges for stony type asteroids with fractured or rubble pile structures Furthermore, the best fits were achieved with large amounts of debris released into clouds, moderately high fragment strength scaling, reduced cloud dispersion and ablation rates, and fairly small differences in fragment mass distributions Sensitivity studies investigated the effects that variations in the model’s fragmentation parameters have on the resulting energy deposition profiles Overall, the factor that most influences the nature of the energy deposition is the distribution of the cloud masses released at various altitudes Inclusion of discrete fragments provides a means of varying the numbers, sizes, altitudes, and rates of debris clouds released, but it is the resulting cloud components more than the fragments themselves that define both the primary and detailed features of the flare For the nominal set of asteroid and entry properties considered in this study, the energy deposition comparisons showed the following primary trends Assuming a moderate cloud mass fraction (30–80%) produces fairly smooth, moderate-width flares that peak in the mid-to-low portion of the overall flare Small cloud masses below 10–20%, on the other hand, push the peak deposition to a sharper peak at the very top of the flare, while putting 90% of the mass into debris clouds produce more of a gradually sloping profile than a distinct flare For risk applications, cloud masses 50–70% appear to produce energy deposition profiles for which peak altitude appears most representative of the overall flare Adjusting the number and masses of the fragments introduces sub-flares, spikes, and other shape variations that can reproduce observed light curve features and guide inferences about the bolide properties or breakup behavior However, for the range of cases considered here, these detailed features not affect the overall energy deposition in ways that would make a significant difference in blast wave generation, and are therefore not a crucial consideration for risk assessment Use of two even fragments appears to produce energy deposition profiles that are most representative of the overall blast-relevant flare behavior This is an advantageous simplification because use of identical fragments is computationally efficient and enables fast analysis of the millions of cases needed for a probabilistic risk assessment The initial aerodynamic breakup strength and fragment strength scaling primarily affect the width of the energy deposition profile by shifting the altitude or slope of the upper edge of the flare For breakups with moderate to high debris cloud masses (above ∼30%), the peak energy deposition point remains fairly insensitive to these variations in the upper curves because the largest initial debris clouds still tend to dissipate most of their energy in the lower portion of the flare, regardless of how high up they are released Only cases with very little debris released peak at the top of the flare and make the associated burst altitudes sensitive to shifts in the upper curve Risk models that base burst altitudes on the maximum energy deposition point may therefore be fairly insensitive to initial strengths up to around 10 MPa, which is the upper end of current estimates for typical stony type asteroids (Popova et al., 2011) Strength scaling exponents between 0.1 and 0.3 seem to produce the most representative range of peaks for using maximum energy deposition as a proxy for burst altitude Risk models would need to develop relations for estimating effects of flare width on ground overpressures, rather than using point-source estimates, in order to account for the range of differences due to strength However, the shape variations produced by these strength parameters provide additional ways to match specific flare shapes and features Higher values of α , in particular, are Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 JID: YICAR ARTICLE IN PRESS 20 [m5G;February 27, 2017;17:2] L.F Wheeler et al / Icarus 000 (2017) 1–21 needed to produce secondary peaks below the main flare, such as that seen in the Chelyabinsk meteor light curve The width of the peak is also influenced somewhat by variations in the ablation and lateral dispersion rates assumed for the cloud components Appropriate ranges for these parameters remain uncertain, and future studies would benefit from additional observational comparisons, high-fidelity simulations, or experimental testing The main flare is insensitive to even large variations in the ablation rate of the fragment components However, development of variable ablation rates may provide more accurate energy deposition estimates for fragments that persist below the main burst, and provide better matching to observed light curve data More accurate ablation rates may in turn enable better inferences of initial size and density from the pre-breakup energy deposition, and better matching of fallen fragment masses The analysis presented here focused on systematically assessing sensitivity to individual parameters, using one set of representative asteroid and entry properties Allowing for combinations of multiple parameter variations with ranges of fragment numbers and mass distributions produces much more diverse energy deposition features that can be matched to a variety of observed meteor light curves Likewise, different trends may emerge for other asteroid densities or entry properties Planned future work will extend these sensitivity studies to quantify and prioritize factors that drive ground damage estimates over a broader range of asteroid densities, entries, and structural types The capabilities and initial findings from the FCM also enable and motivate several other future studies and model developments to advance the state-of-the-art for asteroid risk assessment Because the FCM is able to produce a range of more realistic energy deposition profiles efficiently enough for risk modeling applications, it could help to bridge the gap between high-fidelity simulations, observations of small non-damaging meteors, and damage estimates for larger asteroids Comparisons with high-fidelity simulations of asteroid breakup dynamics and material properties will be performed to better bound uncertain parameter ranges, such as strength and spread rate High-fidelity blast propagation simulations could also take FCM energy deposition results as inputs to investigate the effects of peak width and shape on the resulting ground damage footprints as compared to point-source blast estimates Development of simulation-anchored blast propagation relations for varied energy deposition profiles could significantly improve the accuracy of blast damage estimates compared to the current standard static point-source approximations The fragment-cloud model is also being extended to represent different initial asteroid structures, such as loosely bound regolith layers or rubble piles with an initial distribution of boulder sizes and strengths within a remaining mass of smaller particulates Inclusion of these initial structural variations enables the model to provide more representative matches to a variety of observed light curves, such as those from the Chelyabinsk, Benešov, and Košice meteors For example, including high-altitude regolith blow-off can capture the small upper flare seen in the Chelyabinsk light curve, and initial rubble boulders with a range of independent strengths and sizes allows larger fragments to persists lower down to better represent multiple main bursts Together, these advancements will improve the fidelity and range of impact consequences for asteroid risk assessments and can help to 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asteroids Meteorit Planet Sci 46 (10), 1525–1550 Register, P.J., Mathias, D.L., Wheeler, L.F., 2017 Asteroid fragmentation approaches for modeling atmospheric energy deposition Icarus 284C (2017), 157–166 doi:10.1016/j.icarus.2016.11.020 ReVelle, D.O., Ceplecha, Z., 2001 Bolide physical theory with application to PN and EN fireballs In: Proceedings of the Meteoroids 2001 Conference, 6–10 Aug 2001 (ESA SP-495, Nov 2001) ReVelle, D.O., 2005 Recent advances in bolide entry modeling: a bolide potpourri Earth, Moon, Planets 95, 441–476 Please cite this article as: L.F Wheeler et al., A fragment-cloud model for asteroid breakup and atmospheric energy deposition, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2017.02.011 ... estimating burst altitudes for risk assessments and damage models Hills and Goda (1993 , 1998) used a pancake approach to model energy deposition and airburst damage for various representative asteroid. .. and their applicability to larger, deeper-penetrating asteroids is uncertain For Please cite this article as: L.F Wheeler et al., A fragment- cloud model for asteroid breakup and atmospheric energy. .. additional ways to match specific flare shapes and features Higher values of α , in particular, are Please cite this article as: L.F Wheeler et al., A fragment- cloud model for asteroid breakup and atmospheric