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[.]

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deposition

a CSRA, NASA Ames Research Center, MS 258-6, Moffett Field, CA 94035, United States

b 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

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

1 Introduction

Throughout its history, Earth has been continually bombarded

by asteroids While the vast majority are small and burn up harm-

lessly in the atmosphere, rare impacts from larger objects have

caused notable damage, ranging from the ﬂattened forest in Tun-

guska ( 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 ( Popovaetal.,2013) This event motivated new assess-

ments 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 aerody-

namic 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 tak- ing 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 ( Mathiasetal.,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 flexi- bility 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 effi- ciently 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

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2 L.F Wheeler et al / Icarus 0 0 0 (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 rep-

resents the breakup process using a combination of independent

fragments and aggregate debris clouds The model has the capabil-

ity 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 de-

bris 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 capa-

bility provides a means to investigate how asteroids with different

structures, material properties, and breakup mechanisms may de-

posit 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 fur-

ther 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 an-

alytic proxies for the more complex underlying processes Match-

ing 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., Brownetal.,2002;

CeplechaandReVelle,2005)

This paper gives an overview of the current fragment-cloud

model and presents initial sensitivity studies investigating the ef-

fects of its various fragmentation parameters on energy deposition

The parameter variations assessed include the initial aerodynamic

breakup strength, fragment strength scaling, the number of frag-

ments per break, debris cloud mass per break, cloud dispersion

rates, and ablation rates We present the resulting effects on en-

ergy deposition features and discuss the implications for risk as-

sessment applications and observational inferences We then ap-

ply the model to reproduced observational data from the 2013

Chelyabinsk meteor Results demonstrate the model’s ability to es-

timate 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 previ-

ously 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 ap-

plied to energy deposition or airburst assessment (e.g., Passeyand

Melosh,1980;Melosh,1981;BaldwinandSheaffer,1971;Boroviˇcka

etal.,2007) The existing breakup models applicable to energy de-

position and risk assessment tend to use either a “pancake” ap-

proach 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 lat-

erally 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 HillsandGoda(1993,1998)

used a pancake approach to model energy deposition and air-

burst damage for various representative asteroid types Chybaetal

(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; Collinsetal., 2005; Motiwalaetal., 2015) However, the aggregate pancake treatment does not allow for variations that could result from non- uniform 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 BaldwinandSheaffer (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(2005,2007) 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 al- ternate 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 in- ﬂuenced by the common group aerodynamics Hybrid model con- cepts like the FCM have been discussed in the literature to some extent ( Popova, 2011; Artemievaand 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 con- sider speciﬁc impact events (e.g., Crawfordet al., 1995;Bosloughand Crawford, 2008; Zahnle and Mac Low, 1994) or investigate

a speciﬁc aspect of the entry or breakup process (e.g., RobertsonandMathias,2017; Korycanskyetal.,2003; Shuvalovetal.,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 ( Mathiasetal.,2017) of cases needed for statistical assessment of the impact threat For example, the hydrocode simulations of Chelyabinsk-like meteor entries presented

in Robertson andMathias (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 pre- entry structure and material properties, which are not generally known For these reasons, analytical approximations remain the most efficient modeling constructs for risk assessment and infer- ring pre-entry characteristics of observed meteors

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Fig 1 Diagram of the fragment-cloud model approach

In our previous paper, Registeretal.(2017), we compared sev-

eral 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 po-

tentially be tied to relevant physical asteroid properties The focus

of the current effort presented here is to begin to evaluate the ap-

plicability and effects of the various parameters used in this ap-

proach

2 Fragment-cloud model

The fragment-cloud model (FCM) is an analytic model for es-

timating 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 tra-

jectories, velocities, and masses of the fragmenting bolide com-

ponents 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 through-

out 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 en-

ergy deposited per unit of altitude change, conventionally given in

units of kilotons per kilometer (kT/km) Fig.1shows a notional di-

agram of the fragmentation approach and components

2.1 Flight integration

The bolide’s ﬂight path angle, velocity, mass, and size are com-

puted by integrating the standard meteor physics equations of mo-

tion and ablation (e.g., Opik,1958) The model integrates the ﬂight

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.StandardAtmosphere,1976), given by:

ρA=−140.2e−0.0 0 0187h+141.4e−0.0 0 0186h (2)

2.2 Fragmentation

Fragmentation begins when the stagnation pressure exceeds the initial aerodynamic strength parameter of the bolide ( Hills andGoda, 1993; Popova et al., 2011; Passey and Melosh 1980) The stagnation pressure is given by:

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 com- pletely or impacts the ground The lateral spread of the cloud’s cross-sectional area is given by a dispersion velocity, v disp, which

is a function of the cloud’s velocity, v, the ratio of the atmospheric density and meteor density, ρA /ρm, and a dimensionless dispersion coeﬃcient, C disp:

vdisp = v

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# Fragments per break 2 2–16

Fragment mass splits Even splits 50/50%–90/10%

Ablation coeﬃcients 1e-8 kg/J 1e-9–5e-8 kg/J

Cloud dispersion coeﬃcient 3.5 0.35–7

The dispersion coeﬃcient, C disp, is taken to have a baseline

value of 3.5 ( HillsandGoda,1993), but can also be varied as an in-

put parameter With each height-step of the ﬂight integration, dh,

the radius of the spreading cloud is increased by:

d r= vdisp d t=

C dispρA /ρm

d h

The fragments produced in each break are assumed to be

stronger than the parent fragment, since the break would elim-

inate some of the larger structural weaknesses (as discussed in

BlandandArtemieva,2006), putting the weaker portions of mate-

rial 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;

BlandandArtemieva,2006; ArtemievaandShuvalov,2001;

Bald-winandShaefer1971):

S f= S0

m0/ m fα

(6)

where S f and m f are the post-break strength and mass of the

child fragment, S 0 and m 0 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 me-

teorite falls ( Popovaet al., 2013;Popova andNemtchinov, 2002),

at which point they are prevented from further fragmentation and

continue descent until they either ablate or impact the ground

However, fragments do not tend to reach this limit given the frag-

mentation parameter ranges assumed in the cases presented here

3 Energy deposition modeling sensitivities

We performed a series of sensitivity studies to investigate the

effects of the model’s various fragmentation parameters on com-

puted energy deposition Results are shown for bolides 20, 50, 100,

20 0, 30 0, and 50 0 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 stony-

type asteroid impact, based on bulk density ranges in Britt etal

(2003), velocity distributions in Greenstreet etal.(2012), and en-

try angle assessments in Shoemaker(1962)

The modeling parameters investigated in this study are: ini-

tial aerodynamic breakup strength, strength scaling parameter, per-

centage 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 con-

stant at a set of baseline values Table1 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 2 FCM energy deposition curves for 20–500 m asteroids with baseline en-

try and fragmentation parameters: 2.5 g/cc, 20 km/s, 45 °, 1 MPa initial strength, 0.1

α, 50% cloud mass, 2 even fragments per break, ablation parameter 1e-8 kg/J, and cloud dispersion coeﬃcient 3.5

Fig.2shows the FCM energy deposition results for the asteroid sizes assessed, using the baseline entry properties and modeling parameters given above Fig.3shows 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 deposit- ing 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.2seem to show higher peak altitudes than those of previous pancake-style models, such as the Tunguska cases presented by Chybaetal.(1993),

or the HillsandGoda (1993) results adopted by the Stokeset 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 3 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

etal.(2017) we provide a comparison of FCM burst altitudes for the Tunguska-scale cases presented in Chybaet 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 Table1 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

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Fig 3 Energy deposition contributions of fragment and cloud components for the 100-m diameter baseline asteroid impact case The left-hand ﬁgure shows the contribu-

tions 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 emis-

sion, 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, recog-

nizing 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 alti-

tude can then be used with yield-scaling relations derived from

nuclear test data to obtain damage areas for given overpressure

levels ( Glasstone andDolan, 1977) An overpressure threshold of

4 psi has typically been used to deﬁne the ground damage area

for asteroid impact risk studies ( Mathiasetal.,2017;Stokesetal.,

2003; Motiwalaetal., 2015;HillsandGoda, 1993,1998;Toonet

al.,1997), and we maintain that convention in the damage discus-

sions below

In considering the blast damage potential of various energy de-

position 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 pro-

duces the greatest possible ground overpressure levels This alti-

tude 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 2 km for the 20 m diameter case, 5 km for the 50 m case,

9 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 max-

imum 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 de-

position point may be overly inﬂuenced by minor curve shape fea-

tures 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 infer-

ring what breakup characteristics could produce those features Pa-

rameter 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 charac-

terizing 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 co- hesive monoliths to loosely bound rubble piles ( Brittetal.,2003;SanchezandScheeres,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.4shows 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 do 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 tenden- cies 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 spac- ing 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.3 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-

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Fig 4 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 1

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 de-

pendent on the choice of the strength scaling parameter Fig

5 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 αvalues 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 5 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 pre- dicted from the differing peak maximums would not cause any 4- psi ground damage For the 100 m case, the peak maximum ranges

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Fig 5 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 1

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 lit-

tle difference in 4-psi blast radius resulting from the minimum

and maximum altitudes—only 52 km for the lower altitude com-

pared 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 pes-

simistic (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-to- low 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.5yields curves that would be more diﬃcult to char- acterize as an airburst at a particular altitude, and do 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 domi- nated by a single, smooth ﬂare, and often do exhibit many small, abrupt spikes and jags, such as those seen in the Benešov meteor ( CeplechaandRevelle, 2005) and Košice meteor ( Boroviˇckaetal.,

2013) Adjusting the cloud mass to allow the emergence of these

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Fig 6 Energy deposition sensitivity to fragment number variation, for asteroids 20–500 m in diameter All other parameters maintain the baseline values given in Table 1

features will be important for matching meteor observations, es-

pecially when combined with more realistic fragment mass distri-

butions, 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 6 shows comparisons of 2–16

fragments per break, for cases with the baseline 50% cloud mass

and αof 0.1 Higher fragment numbers push the upper edge of the

peak out to a sharper point, introduce small sub-ﬂares, and pro-

duce 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 2 to 4 fragments per break produces the most notable differences, while increasing the fragmentation beyond 4 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 7 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-

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Fig 7 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 1

itive to think of the fragment number as a parameter affecting the

fragment portion of the process, the way in which it most no-

tably 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 inde-

pendent of the fragment divisions However, since each fragment

subsequently produces its own cloud when it breaks, higher frag-

ment 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 as-

sessment applications, use of 2 fragments per break produces the

most “well-behaved” energy deposition proﬁles, i.e., with the max-

imum deposition altitude most representative of the main bulk of

the ﬂare rather than concentrated at the top or subject to small, in-

consequential 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 in-

ferences about the bolide properties or breakup behavior by en-

abling 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 evalu-

ated 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.8compares 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 spe- ciﬁc fragmentation behaviors to emerge from the dominant initial cloud deposition, then the fragment mass split has more effect Fig

9shows the fragment mass split variations for the 20-m and 50-

m sizes, using a high α of 0.5 Using uneven fragment splits re- duces 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

Trang 10

Fig 8 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 1

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 as-

teroids ( Popovaet al., 2011), but the range was extended up to

100 MPa for this study to include the potential for very strong co-

herent objects and to see at what point the impact shifts from de-

positing most of its energy in an airburst to carrying most of it di-

rectly to ground The higher strengths are also included to provide

comparison with the values used in previously published models

For example, Chybaetal.(1993)used a breakup condition equiva-

lent to approximately 24 MPa of stagnation pressure for their stone

type Tunguska cases HillsandGoda(1993) used 200 MPa for their

iron cases and 50 MPa for their hard stone cases, which were also subsequently used in the Stokes etal.(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.10shows 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 ∼10MPa 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-

Tiêu đề	A fragment cloud model for asteroid breakup and atmospheric energy deposition
Tác giả	Lorien F. Wheeler, Paul J. Register, Donovan L. Mathias
Trường học	NASA Ames Research Center; Vanderbilt University
Chuyên ngành	Planetary science
Thể loại	Journal article
Năm xuất bản	2017
Thành phố	Moffett Field, CA

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Số trang	21
Dung lượng	8,68 MB