A Numerical Approach to Study Ablation of Large Bolides ...

Research ArticleA Numerical Approach to Study Ablation of Large BolidesApplication to Chelyabinsk

Josep M Trigo-Rodrıguez 12 Joan Dergham12 Maria Gritsevich3456 Esko Lyytinen5dagger

Elizabeth A Silber 78 and Iwan P Williams9

1Institut de Ciencies de lrsquoEspaindashCSIC Campus UAB Facultat de Ciencies Torre C5-parell-2a 08193 Bellaterra BarcelonaCatalonia Spain2Institut drsquoEstudis Espacials de Catalunya (IEEC) Edif Nexus cGran Capita 2-4 08034 Barcelona Catalonia Spain3Finnish Geospatial Research Institute (FGI) Geodeetinrinne 2 FI-02430 Masala Finland4Department of Physics University of Helsinki Gustaf Hallstromin katu 2a PO Box 64 FI-00014 Helsinki Finland5Finnish Fireball Network Helsinki Finland6Institute of Physics and Technology Ural Federal University Mira str 19 620002 Ekaterinburg Russia7Department of Earth Sciences Western University London ON N6A 5B7 Canada8e Institute for Earth and Space Exploration Western University London ON N6A 3K7 Canada9Astronomy Unit Queen Mary University of London Mile End Rd London E1 4NS UKdaggerDeceased

Correspondence should be addressed to Josep M Trigo-Rodrıguez trigoicecsices

Received 31 August 2020 Revised 13 January 2021 Accepted 2 March 2021 Published 27 March 2021

Academic Editor Kovacs Tamas

Copyright copy 2021 Josep M Trigo-Rodrıguez et al )is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

In this study we investigate the ablation properties of bolides capable of producing meteorites)e casual dashcam recordings frommanylocations of the Chelyabinsk superbolide associated with the atmospheric entry of an 18m in diameter near-Earth object (NEO) haveprovided an excellent opportunity to reconstruct its atmospheric trajectory deceleration and heliocentric orbit In this study we focus onthe study of the ablation properties of the Chelyabinsk bolide on the basis of its deceleration and fragmentation We explore whethermeteoroids exhibiting abrupt fragmentation can be studied by analyzing segments of the trajectory that do not include a disruption episodeWe apply that approach to the lower part of the trajectory of the Chelyabinsk bolide to demonstrate that the obtained parameters areconsistent To do that we implemented a numerical (RungendashKutta) method appropriate for deriving the ablation properties of bolidesbased on observations )e method was successfully tested with the cases previously published in the literature Our model yields fits thatagree with observations reasonably well It also produces a good fit to the main observed characteristics of Chelyabinsk superbolide andprovides its averaged ablation coefficient σ 0034 s2 kmminus 2 Our study also explores the main implications for impact hazard concludingthat tens ofmeters in diameter NEOs encountering the Earth in grazing trajectories and exhibiting low geocentric velocities are penetratingdeeper into the atmosphere than previously thought and as such are capable of producing meteorites and even damage on the ground

1 Introduction

On February 15 2013 our view about impact hazard wasseriously challenged While there was a sense of accom-plishment for being able to forecast the close approach of2012 DA near-Earth asteroid (NEA) within a distance of27700 km eventhough this NEO was discovered only one

year prior an unexpected impact with an Apollo asteroidensued [1] At 0320 UTC a superbolide also known as theChelyabinsk superbolide flew over the Russian territory andKazakhstan )e possible link between the superbolide andthe 2012 DA NEA was discarded by the European SpaceAgency (ESA) and the Jet Propulsion LaboratoryndashNationalAeronautics and Space Administration (JPL-NASA) from

HindawiAdvances in AstronomyVolume 2021 Article ID 8852772 13 pageshttpsdoiorg10115520218852772

the reconstruction of the incoming fireball trajectory )eChelyabinsk superbolide entered the atmosphere at sim19 kms and according to the US sensor data (CNEOS fireball listhttpscneosjplnasagovfireballs) it reached the maxi-mum brightness at an altitude of 233 km with a velocity of186 kms [1 2] also providing us with valuable samples inthe form of meteorites

)e existence of meteoroid streams capable of producingmeteorite-dropping bolides is a hot topic in planetary sci-ence Such streams were first proposed by Halliday [3 4])eir existence has important implications because they cannaturally deliver to the Earth different types of rock-formingmaterials from potentially hazardous asteroids (PHAs) It isbelieved that NEOs in the Earthrsquos vicinity are undergoingdynamical and collisional evolution on relatively shorttimescales We previously identified several NEO complexesthat are producing meteorite-dropping bolides and wehypothesize that they could have been produced during closeapproaches to terrestrial planets [5 6] Such formationscenario for this kind of asteroidal complexes is nowreinforced with the recent discovery of a complex of NEOslikely associated with the NEA progenitor of the Chelyabinskbolide [7])e shattered pieces resulting from the disruptionof NEOs visiting the inner solar system can spread along theentire parent body orbit on a time-scale of centuries [4 8 9])is scenario is also consistent with the modern view ofNEOs being resurfaced as consequence of close approaches[10] Additionally the meteorites recovered from theChelyabinsk have a brecciated nature [11 12] that is rem-iniscent of the complex collisional history and a probablerubble pile structure of the asteroid progenitor of thecomplex [13] )e existence of these asteroidal complexes inthe near-Earth region has important implications as theycould be the source of low spatial density meteoroid streamspopulated by large meteoroids Such complexes could be thesource of the poorly known fireball-producing radiants[14 15] )is could have important implications for thefraction of sporadic meteoroids producing bright fireballsand in the physical mechanisms envisioned in the past[16ndash18]

)e Chelyabinsk event is also of interest because of itsmagnitude and energy and due to its relevance to be con-sidered as a representative example of the most frequentoutcome of the impact hazard associated with small aster-oids in human timescales Chelyabinsk also exemplifies theimportance that fragmentation has for small asteroidswhich can even excavate a crater on the Earthrsquos surfacealthough rarely [19ndash23] Fragmentation is important as itprovides a mechanism in which a significant part of thekinetic energy associated with small asteroids is released Itwas certainly a very relevant process for the Tunguska event[24 25] and in the better-known case of Chelyabinsk mostof the kinetic energy was transferred to the internal energy ofthe air which is radiated as light [26]

One way to study meteoroids as they enter the Earthrsquosatmosphere is through video observations of such eventsConsequently we are developing complementary ap-proaches to study the dynamical behavior of video-recordedbolides in much detail )e SPanish Meteor Network

(SPMN) pioneered the application of high-sensitivitycameras for detecting fireballs and it currently maintains anonline list of bright events detected over Spain PortugalSouthern France and Morocco since 1999 [27 28] Forexample casual video recordings plus several still photo-graphs of the superbolide in flight allowed us to reconstructthe heliocentric orbit of Villalbeto de la Pentildeameteorite in theframework of the SPMN [28] )e possibility of studyingsuperbolides such as Chelyabinsk is a very attractive mile-stone to be considered )e software used in this study wasdeveloped as part of a master thesis [29] and subsequentlytested and validated using several cases discussed by [29] aswell as events from the 25 video and all-sky CCD stations setup over the Iberian Peninsula by the SPMN In this contextwe have been engaged in studying the dynamic behavior ofmeteoroids decelerating in the Earthrsquos atmosphere [30 31]

In this study we study the Chelyabinsk bolide by fol-lowing a RungendashKutta method of meteor investigationsimilar to that developed by Bellot Rubio et al [32] We aimto test if that specific method is also valid for another massrange particularly for small asteroids and large meter-sizedmeteoroids We first describe our numerical model and testit against the known meteor events We compare our codevalidation results to the results obtained by Bellot Rubio et al[32] for the same dataset We then apply our numericalmodel to the Chelyabinsk superbolide in order to study itsdynamical behavior For the sake of simplicity our modelconsiders a constant ablation coefficient and the shapefactor even though these parameters could vary in differentablation stages [33ndash35]

)is study is structured as follows the data reductionand the theoretical approach pertaining to the Chelyabinskbolide are described in the next section In Section 3 themain implications of this work in the context of fireballsmeteorites and NEO research are discussed We use themodel to determine the fireball flight parameters and bystudying the deceleration we also obtain the ablation co-efficient Finally the conclusions of this work are presentedin Section 4

2 Data Reduction Theoretical Approachand Observations

)e Chelyabinsk superbolide was an unexpected daylightsuperbolide as many other unpredicted meteorite-drop-ping bolides in history Fortunately numerous casualvideo recordings of the bolide trajectory from the groundwere obtained given the nowadays common dashcamsavailable in private motor vehicles in Russia According tothe video recordings available it is possible to study theatmospheric trajectory and deceleration carefullyallowing the reconstruction of the heliocentric orbit inrecord time [2 26]

21 Single Body eory )ere are two main approaches inthe study of the dynamic properties of meteors during at-mospheric interaction the quasicontinuous fragmentation(QCF) theory introduced by Novikov et al [37] which was

2 Advances in Astronomy

later extended by Babadzhanov [38] and the single bodytheory described by Bronshsten [39] )ere have been dis-crepancies in the applicability conditions of both methodsthe single body works with basic differential equations whilethe QCF uses semiempirical formulas studying only theluminosity produced by the meteor )e main difference isthat the single body theory obtains smaller dynamical massesthan the QCF method )us far neither approach is prev-alent and the reason why the theories do not converge couldbe attributed to the contribution of other key processes suchas the fragmentation and deceleration of meteoroids duringablation or to the poorly constrained values of the bulkdensity andor the luminous efficiency coefficient [40ndash42]

We remark that the initial dynamic mass estimate or thepreatmospheric size can be derived using the methods de-scribed in other works [43ndash48] hence we leave it out of thescope of the present model We also note that alternativemodels accounting for ablation have been recently devel-oped however further discussion on that topic is beyond thescope of this study and the reader is referred to the followingliterature [46 47 49ndash52] As noted in the introduction high-strength meteoroids from asteroids or planetary bodiesexhibit a very different behavior than fragile dust aggregatesoriginating from comets [53ndash57]

22 e Role of Fragmentation )e fragmentation of me-teoroids was studied in detail by various authors [38 40]After analyzing different photographic observations Levin[40] distinguished four possible types of fragmentation (a)the decay of themeteoroid into large nonfragmenting debris(b) the progressive disintegration of the original meteoroidinto fragments that continue to crumble into smallerfragments (c) the instantaneous ejection of a large numberof small particles which when affecting the entire mete-oroid is called a catastrophic disruption and finally (d) thequasicontinuous fragmentation which consists of thegradual release of a large number of small particles from thesurface and their subsequent evaporation due to hightemperatures associated with the shock thermal waveformed around the body

In practice a combination of two or more types offragmentation can be observed in a given meteor event Infact it is possible to observe that (a) and (c) fragmentationtypes described in the preceding paragraph could occurmore than once for the same meteor event )e analysis ofmeteors performed by Jacchia [58] using Super-Schmidtcameras demonstrated that the single body theory does notwork for cases which suffer abrupt types of fragmentationalong the trajectory As a direct consequence meteoroidsexhibiting fragmentation of the first (a) second (b) andthird (c) kind should not be studied using this simplifiedsingle body theory When a meteoroid suffers an abruptfragmentation the main body loses mass instantaneouslyand therefore the single body equations cannot be appliedsince the condition of continuity of mass is not satisfied)us the cases with possible abrupt fragmentation epi-sode(s) will not be examined in this work

23 e Single Body eory e Drag and Mass LossEquations )e dynamic behavior of a meteoroid as it in-teracts with the Earthrsquos atmosphere is described using thedrag and mass loss equations )ese equations as presentedby Bronshten [39] are as follows

dv

dt minus Kρairm

minus 13middot v

2 (1)

dm

dt minus σKρairm

23middot v

3 (2)

where K is the shape-density coefficient ρair is the airdensity m is the meteoroid mass v is the instantaneousvelocity and σ is the ablation coefficient

By using equations (1) and (2) the parameters to beidentified are K and σ )e ablation coefficient defines theloss of mass for the bolide as it penetrates the atmospherethe larger the value is the more the mass will be ablated for agiven velocity )e value of the ablation coefficient dependson various factors and is expressed as

σ Λ

2ΓQ (3)

where Λ is the heat transfer coefficient Γ is the drag coef-ficient and Q is the heat of ablation

)e shape-density coefficient depends on the shape anddensity of the meteoroid and is expressed as

K sA

ρ23 (4)

where A is the shape factor s is the cross-section area and ρis the meteoroid bulk density

We must point out that the observational data obtainedfrom the reconstruction of the trajectories of meteors usingCCD or video cameras is basically the frame to frame speedof the bolide as a function of the height requiring anotherequation to link the time with the altitude

dh

dt minus v middot cos(z) (5)

where z is the zenith angleBy substituting equation (5) into equations (1) and (2)

the following expressions are obtained

dv

dh

Kρairmminus 13

middot v

cos(z) (6)

dm

dhσKρairm

23middot v

2

cos(z) (7)

Next by dividing equation (2) by equation (1) we obtain

dm

dv mσv (8)

Solving this differential equation with the boundarycondition of m mo when v vo the following is obtained

m mo middot eminus (12)σ vo

2minus v2( ) (9)

Advances in Astronomy 3

Now we combine equations (9) and (6) to derive

dv

dh Kρair mo middot e

minus (12)σ vo2minus v2( )1113874 11138751113874 1113875

minus (13)

middot v middot1

cos z (10)

In order to obtain the value of K and σ we use equation(10) as it uses substitution for the dependence on the in-stantaneous mass and thus we deal with one equation insteadof two Furthermore there is the initial mass which is animportant parameter to be studied Equation (10) directly linksthe deceleration of the meteoroid as a function of differentparameters in particular the zenith angle )e last term is ofparticular interest as it modulates the full equation For thevertical entry (z 0deg) the deceleration is maximized while for zclose to 90deg the deceleration is minimized For that reasonlarge meteoroids at grazing angles are able to follow extremelylong trajectories or even escape again into space such as theGrand Tetons superbolide that spent nearly two minutestraveling over several states of the USA and Canada on August10 1972 [11 59]

However by using the concepts introduced above it isnot possible to obtain the values of initial mass (mo) and K)e parameters that can be found are only the expressions ofmo

minus 13middotK and σ Consequently another equation is requiredto obtain K and mo separately )e remaining expression isthe photometric equation

I τ2

minusdm

dt1113888 1113889 middot v

2 (11)

where τ is the luminous efficiency)is equation is assumingv constant and it is often applied to small meteors )eluminous efficiency is obtained empirically thus any de-viations in that value might produce significant changes inthe results )is equation will be referred to later as a part ofthe mass determination Now we focus only on equation(10)

We define the Kprime as

Kprime K middot mminus 13o (12)

)en the equation to work with is

dv

dh ρairKprime e

minus 12σ vo2minus v2( )1113874 1113875

minus 13middot v middot

1cos z

(13)

with variables Kprime and σ

3 Results and Discussion

31 Numerical Approximation In this section we develop anumerical approximation with the aim to describe themeteoroid flight in the atmosphere Our goal is to obtain thesolution that can be used to better understand that physicalprocess Subsequently our intention is to develop a nu-merical approach that can be very valuable in predicting thevariation of the parameters along the trajectory segmentsversus analytical ldquowhole-trajectory smoothingrdquo

Equation (13) is the expression used to find the physicalparameters To test ourmodel we usemeteor data pertaining tothe meteoroid velocity at different altitudes taken from theliterature In this regard the accurate trajectoryvelocity meteor

data obtained via high-resolution Super-Schmidt cameras [58]can be utilized for a case study)ese meteor trajectory data areused to optimize the procedure to obtain the values of Kprime and σthat fit the best the actual data points )e procedure presentedhere produces many synthetic curves Subsequently in order tooptimize the computing time a solution to make the equationconverge to the real data is found

311 e RungendashKutta Method Implementation )e Run-gendashKutta method is an iterative technique for the approxi-mation and solution of ordinary differential equations )emethod was first developed by Runge [60] and Kutta [61]

)e RungendashKutta approximation provides a solution at adetermined point of altitude Application of the RungendashKuttamethod requires the initial conditions to be known

yprime f(t y)middot

yo to (14)

In our case the initial conditions will be the initialvelocity and the altitude of the bolide when ablation startsldquosyntheticallyrdquo written as

_dv

dh f(h v) v ho( 1113857 ho

(15)

We then need to choose a step size (p) that should becomparable with the data resolution to allow a better com-parison between the model and the observations )e step sizedefines how many integration steps need to be implementedbefore the final solution is reached)e smaller the step size themore accurate the solution will be noting that this will alsoincrease the computing time )e step size corresponds to thecharacteristic Δh that according to Bellot Rubio et al [32] canbe chosen around a few hundred of meters 100ndash300m

Once the step size is defined we define the model co-efficients as follows

l1 f tn yn( 1113857

l2 f tn +12

p yn + p12l11113874 1113875

l3 f tn +12

p yn + p12l21113874 1113875

l4 f tn + p yn + pl3( 1113857

(16)

For our case the function to be studied is

f hn vn( 1113857 Kprimeρ(h) eminus 12σ vo

2minus v2( )1113874 1113875minus 13

middot v middot1

cos z (17)

and the coefficients are calculated as

l1 f hn vn( 1113857

l2 f hn +12

p vn + p12l11113874 1113875

l3 f hn +12

p vn + p12l21113874 1113875

l4 f hn + p vn + pl3( 1113857

(18)


Once the coefficients (equation (16)) are calculated wecompute the solution for the point yn+1 using the followingformula

yn+1 yn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (19)

For our case (equation (18)) this is

vn+1 vn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (20)

)e result obtained is the solution for the point (hn+1vn+1) which becomes the initial condition to find the nu-merical approximation for the next point )e procedure isrepeated until the desired value is reached

312 Validating the Code Using the Jacchia CatalogOnce the procedure is defined we need to validate thecode by comparing the results with the previouslypublished data We use a catalog of very precise pho-tographic trajectories of meteors [58] hereafter referredto as the JVB catalog obtained using high spatial reso-lution Super-Schmidt cameras Jacchiarsquo [58] study syn-thesizes the physical inferred parameters for 413 meteorsbetween minus 5 and +2 5 stellar magnitude ranges obtainedduring a multistation meteor network operated in thefifties and sixties in New Mexico USA )e data providethe meteor velocity and magnitude as a function of thealtitude the derived preatmospheric velocity the de-celeration and some additional information for theobserved meteors All JVB catalog events were namedusing a number In this project we have used the sameJacchia and Whipple [58] numbering but included a J atthe beginning for the computational reasons For ex-ample we later discuss meteor J8945 (listed in JVB as8945)

Equation (13) also requires the knowledge of air densityWe adopted a general model widely used in meteor studies[45] the United States (US) standard atmosphere [62] )eUS standard atmosphere was originally developed in 1958 bythe US Committee on Extension to the Standard Atmo-sphere and improved in 1976 It is a series of tables whichapproximate the values for atmospheric temperature den-sity pressure and other properties over a wide range ofaltitudes

313 Autofitting Procedure We have defined a way totransform the differential equation into an expression thatcan be iteratively computed )e aim is to find a result forwhich Kprime and σ produce the closest curve fitting to theobservational data points In order to find the bestmatching values we introduce the following autofittingprocedure We start by choosing two random values of Kprimeand σ set as initial approximation )e meteor data to beinvestigated include the velocity of a meteor at specifiedpoints of altitude we compare this velocity with the ve-locity simulated by our code at the same altitude points)eerror factor is then calculated as follows

ξ vd minus vf( K σ)1113872 11138732 (21)

where vd is the velocity inferred from the measured data andvf is the computed velocity

Furthermore we introduce the increment factors for Kprimeand σ which are defined as ΔK and Δσ Following the sameprocedure as earlier we compute the error factor for

ξ vd minus vf( K + ΔK σ)1113872 11138732ξ vd minus vf( K minus ΔK σ)1113872 1113873

2

ξ vd minus vf( K σ + Δσ)1113872 11138732ξ vd minus vf( K σ minus Δσ)1113872 1113873

2

ξ vd minus vf( K + ΔK σ + Δσ)1113872 11138732ξ vd minus vf(K minus ΔK σ minus Δσ)1113872 1113873

2

ξ vd minus vf( K + ΔK σ minus Δσ)1113872 11138732ξ vd minus vf( K minus ΔK σ + Δσ)1113872 1113873

2

(22)

In principle we created a 2D matrix of errors )e errorparameter can be shown in a table for better visualization ofthe algorithm (Figure 1) Once all the error values arecomputed the search for the minimum value is performedand the result is considered the new centered value for thenext computation iteration

We repeat the same procedure for the next centered valueuntil we reach the point where the minimumwill be centered inthe middle of the matrix Consequently the minimum errorvalue will correspond to the sought values of Kprime and σ If verysmall increments ofKprime and σ are used the solution will be moreaccurate (at the cost of increased computing time) If we set largeincrements ofKprime and σ the solutionwill be reached faster but atthe expense of the resolution of the solution In order to dealwith this conundrum the code is optimized such that it is set towork with large increments of Kprime and σ at the beginning Oncethe lsquofirst approximationrsquo solution is found the code switches tosmaller increments until the optimal resolution is reached

Figure 2 shows an example of the autofitting procedureWe have chosen the J8945 meteor for the comparison withpublished data [32] Other cases were studied and comparedin Dergham [29] By comparing the velocity vs altitudegraphs it is evident that the fitting curves are very similar tothe observed data

314 Abrupt Disruption Cases We have presented a modelcapable of obtaining some parameters for meteoroidsHowever as previouslymentioned not all meteoroids can bestudied using this particular model because if they undergofragmentation the results might be skewed Bellot Rubioet al [32] also mentioned that there are a significant numberof cases in the JVB catalog that cannot be fitted likely as aconsequence of undergoing abrupt disruptions Figure 3shows the results of the velocity curve using our modelfor the case J4141 )e results are also compared with thatobtained by Bellot Rubio et al [32]

Despite the difficulties to obtain well-fitting solutions forsome events it is remarkable that our model is able toidentify and produce solutions for the events undergoingquasicontinuous fragmentation A possible way to studymeteoroids with abrupt fragmentation is to focus on dif-ferent segments of the trajectory that do not include a


disruption episode We will apply that approach to the lowerpart of the trajectory of the Chelyabinsk bolide to demon-strate that the obtained parameters are consistent

32 e Ablation Coefficient Equation (13) has severalunknowns)e ability of a meteoroid to produce light can belinked to its mass loss or the ablation coefficient σ [34 42] Inprinciple the ablation coefficient reflects how rapidly themeteoroid loses its mass as it interacts with the atmosphereLow values of the ablation coefficient indicate that the objectis losing a lesser amount of mass compared to that having ahigher value of the ablation coefficient )e ablation coef-ficient is normally represented in units of s2 kmminus 2 and canalso be expressed through the dimensionless mass loss pa-rameter [44 63] used through the range of meteor physicsstudies )e value of the ablation coefficient depends ofmany factors such as chemical composition grain sizedensity porosity and body shape among others In generalthe values of the ablation coefficient range between 001 and03 s2 kmminus 2 [25] In order to exemplify this we applieddifferent ablation coefficients to a 1 g meteoroid with thepreatmospheric velocity of 25 kms and which starts todecelerate at the altitude of 100 km )e results are shown inFigure 4

In general the larger the ablation coefficient is the fasterthe body decelerates due to a more rapid mass loss Con-sequently the mass of the body decreases due to ablationthis is described by using the ablation coefficient and thedrag force imposed by the atmosphere has a larger effectTable 1 shows the comparison of our results to several eventsdescribed in the JVB catalog

33 Deceleration Normalized Instantaneous Mass and MassLoss Rate In this section the effect of deceleration is ex-amined in more detail Given that we have the velocity as afunction of altitude along the trajectory we can study de-celeration from the behavior of the velocity curve )is isparticularly useful as many meteor processing algorithmsand detection methods provide velocity values sequentially[15 64] At a certain point hi the code computes an in-crement of velocity over the increment of distance at thepoints immediately before and after)is can be expressed as

dv

dhhi( 1113857 ΔvΔh

vh2 minus vh1

h2 minus h1 (23)

Considering all the points with known velocity andaltitude along the trajectory as input data equation (23) canbe applied directly Figure 5(a) shows the velocity curve as afunction of altitude for the J8945 meteoroid

)e normalized instantaneous mass (mm0) is the nextquantity to be studied )e expression for the normalized

ξ = (vd ndash vf (K + ∆K σ ndash ∆σ))2

ξ = (vd ndash vf (K ndash ∆K σ ndash ∆σ))2

ξ = (vd ndash vf (K σ ndash ∆σ))2

ξ = (vd ndash vf (K + ∆K σ + ∆σ))2

ξ = (vd ndash vf (K ndash ∆K σ + ∆σ))2

ξ = (vd ndash vf (K σ + ∆σ))2

ξ = (vd ndash vf (K + ∆K σ))2

ξ = (vd ndash vf (K ndash ∆K σ))2

ξ = (vd ndash vf (K σ))2

+σ

+K

Figure 1 Basic scheme of the error matrix

100

099

098

097

096

vv 0

This studyBellot Rubio et al (2002)

1021009290 94 988886 96Altitude (km)

Figure 2 Normalized velocity vs altitude fit for the meteor J8945)e result is a good fit to the previously published Figure 1 of BellotRubio et al [32]

77 78 79 80 81 82 83 84Altitude (km)

170

168

166

164

162

160

158

Velo

city

(km

s)

Figure 3 Comparison of results for the meteoroid J4141 )emodel cannot produce a well-fitting solution since this meteoroidlikely suffered an abrupt disruption )is figure can be comparedwith that shown in Figure 1 of Bellot Rubio et al [32]


instantaneous mass can be derived by rearranging equation(9)

m

mo

eminus (12)σ vo

2minus v2( ) (24)

Equation (24) expresses the normalized instantaneousmass as a function of velocity whereas the values of velocityare the functions of altitude Figure 5(b) shows the behaviorof normalizedmass expressed as a function of altitude for theJ8945 event

We define the normalized mass loss rate as the derivativeof the relative mass over the altitude )is value is computedas follows

dmmo

h1113890 1113891

Δmmo

Δh

mmoh2 minus mmoh1

h2 minus h1 (25)

Figure 5(c) shows the direct application of this approachfor the J8945 event

34Application to theChelyabinskSuperbolide We apply theRungendashKutta code developed in this work to the famousChelyabinsk superbolide On February 15 2013 it was

predicted that the NEA-2012 DA14 discovered a year priorby the Observatory Astronomic de Mallorca would ap-proach the Earth within a close minimum distance of only27700 km However while all the attention was focused onanticipating that encounter another NEA unexpectedlyentered the atmosphere over central Asia on February 152013 at 032033 UTC )e bolide disintegrated in theproximity of the city of Chelyabinsk [1] )e Chelyabinskbolide reached the brightness magnitude of minus 28 which isbrighter than the moon (Figure 6)

As the days passed and the orbits were calculated sci-entists discarded a possible association between the twoNEAs as they presented very different heliocentric orbits)anks to video cameras (dash cams) found in the majorityof Russian motor vehicles and surveillance cameras placedon buildings the initial trajectory of the bolide wasreconstructed and the orbit was determined [2]

After the superbolide sightings many people uploadedvarious videos to Internet Since the geographical location ofthe recorded videos was known we reconstructed the bolidetrajectory obtaining the values of velocity as a function ofaltitude As shown in Table 2 the data were obtained from theanalysis of our video compilation the bolide velocity along theterminal part of its trajectory just after the massive fragmen-tation event taking place at the altitude of 26 km )e step sizewas determined by the video frame rate corresponding todifferences in height of 200ndash150m Table 2 shows these data

Figure 7(a) depicts the dynamical data after the dis-ruption had occurred at an altitude of 26 km and the fit isobtained by our model )e graph shows a quite uniformbehavior of the Chelyabinsk superbolide after the mainfragmentation event

By studying the dynamic curve the ablation coefficientcan be obtained )e derived value is

σ 0034 s2 middot kmminus 2 (26)

It is quite remarkable that this value derived for thelower trajectory provides a similar ablation coefficient asthat for the fireballs at much higher altitudes even thoughChelyabinsk was the deepest penetrating bolide everdocumented still emitting light even as it reached thetroposphere Notably the atmospheric density in these lowerregions is about four orders of magnitude higher

)e normalized mass evolution for that lower part ofthe trajectory is shown in Figure 7(b) and the mass loss rateevolution is shown in Figure 7(c) In particular it is cer-tainly encouraging that the model predicts quite well theablation behavior of the Chelyabinsk bolide in the lowerpart of its atmospheric trajectory )e light curve of theChelyabinsk event was normalized using the US govern-ment sensor data at a peak of brightness value of27 we1013W smiddotrminus 1 corresponding to an absolute astro-nomical magnitude of minus 28 [1] According to the NASA JPLChelyabinsk final report [65] the maximum brightness wasachieved at an altitude of 233 km )is is consistent withour results as the maximum inferred value of mass loss ratefrom our dynamic data that occur at an altitude of sim235 km(Figure 7(c))

85 9080 100957570Altitude (km)

16

18

20

22

24

26

Velo

city

(km

s)

σ = 001σ = 003σ = 005

Figure 4 Comparison of meteoroids having the same initialconditions (m0 1 g v0 25 kms) but different ablationcoefficients

Table 1 Comparison of the ablation coefficients for various meteorcases from the JVB catalog

Meteor ID Jacchia (s2middotkmminus 2) )is study (s2middotkmminus 2)J6882 00812 0075J6959 00331 00382J7216 00501 0079J8945 00354 0036J9030 00549 00542J7161 00354 00381


It is well known that the maximum brightness isachieved shortly after the meteoroid catastrophically breaksapart due to the fragmentedpulverized material being ex-posed to the heat generated by the resulting shock wave

An interesting conclusion that can be obtained directlyfrom these results is the importance of the atmosphere Asmentioned before the faster the meteoroid the more rapidthe ablation process )us the atmosphere could effectivelyshield the Earth from very fast impacts since such objects arepreferentially and more quickly ablated However the lessfavorable cases are very large objects (especially if theirpreatmospheric velocity is low) such as the Tunguska

impactor that produced an airburst over Siberia in 1908when it entered the atmosphere at a velocity of sim30 kms[66] )e Chelyabinsk superbolide had the preatmosphericvelocity of 19 kms and its trajectory was consistent with agrazing geometry (high zenith angle)

To exemplify this we have plotted the entry of theChelyabinsk superbolide for different initial velocities(Figure 8) Of course disrupting NEOs can produce frag-ments tens of meters in size If these fragments encounterour planet under certain geometric conditions they mightbecome a significant source of damage on the ground andcasualties )us identifying the existence of asteroidal

(a) (b)

Figure 6 A casual photograph of the Chelyabinsk bolide taken by Marat Ahmetvaleev (a) )e abrupt fragmentation at a height of 23 km isclearly distinguishable by the main flare (b) Pictures of the dust trail by the same photographer reveal multiple paths produced by thefragments Both pictures are courtesy of the author

00

ndash01

ndash02

ndash03Acce

lera

tion

(km

s2 )

88 90 92 94 96 98 100 102Altitude (km)

(a)

Mas

s (m

m0)

100

075

050

025

88 90 92 94 96 98 100 102Altitude (km)

(b)

Relat

ive m

ass l

oss r

ate 008

006

004

002

88 90 92 94 96 98 100 102Altitude (km)

(c)

Figure 5 Curves of the acceleration (a) mass evolution (b) and relative mass loss rate (c) as a function of altitude for the meteoroid J8945


complexes in the near-Earth environment is crucial forbetter assessment of impact hazard

If the Chelyabinsk superbolide entered at a higher ve-locity it would have slowed down faster due to a more rapidmass loss According to our model the maximum brightnessof a meteoroid occurs when the mass loss reaches the peakConsequently the model predicts that a Chelyabinsk-likeasteroid moving at a lower velocity could have more de-structive potential on the Earthrsquos surface (Figure 8) On theother hand a projectile of the same rocky composition and

moving at higher velocity could have undergone a muchfaster ablation process culminating in an explosive airburstsimilar to that occurring over Tunguska Such conclusionsimply that the efficiency of the Earthrsquos atmosphere to shieldus from dangerous tens of meters in size asteroids thatstrongly depend on the relative velocity of the encounterwith our planet

It is important to remark that new improvements indetection of fireballs from space could provide an additionalprogress in studying the luminous efficiency of bolides [34]

Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)

Figure 7 Velocity evolution (a) mass evolution (b) and mass loss rate (c) as a function of altitude for the Chelyabinsk superbolide

Table 2 Dynamic data of the Chelyabinsk superbolide

Height (km) V (kms) Height (km) v (kms)1898 1404 1566 9731878 1386 1553 9461858 1368 1539 9201838 1349 1526 8941818 1329 1513 8681799 1309 1501 8421780 1288 1489 8171762 1266 1477 7921744 1244 1466 7671726 1222 1455 7431708 1199 1445 7191691 1175 1434 6961674 1151 1424 6741658 1127 1415 6521642 1102 1406 6311626 1076 1397 6111610 1051 1388 5921595 1025 1380 573


In fact several detections of the Chelyabinsk bolide fromspace have been reported [67 68] Future studies of commonevents detected from both the ground and space couldconstrain the role of the observing geometry in signal lossand possible biases in the subsequent determination ofvelocity radiated energy and determination of the orbitalelements

35 Implications for Impact Hazard Considering the de-structive nature of large extraterrestrial objects (eg [24]) itis crucial to identify the existence of asteroidal complexes inthe near-Earth environment Indeed the Chelyabinsk scaleor larger extraterrestrial bodies (10s of meters in size) are ofspecial interest to the planetary defense community becausedepending on their orbital geometry and impact angle theseobjects could pose significant hazard for humans and in-frastructure on the ground )us studying the near-Earthenvironment along with the well-documented events such asthe Chelyabinsk superbolide can shed more light on thedynamic processes as well as fundamental properties ofthese objects

From the study of meteorite falls and the relative absenceof small impact craters [19] we know that meter-sized as-teroids are efficiently fragmented when they penetrate intothe stratosphere at hypervelocity )e loading pressure infront of the body produces the rock fracture when itovercomes its tensile strength As a natural consequencemeteorite falls often deliver tens to hundreds of stones justafter this type of break-ups [69 70] We have previouslydescribed such behavior when discussing the evolution ofChelyabinsk Additionally there is a consensual view thatthe terrestrial atmosphere behaves as an efficient shield formeter to 10s of meters in diameter projectiles Despite ofthis we still need to increase our statistics as more rarely and

probably for low-velocity projectiles under favorable geo-metric circumstances crater excavation is still possible Agood example of this was the so-called Carancas event animpact crater excavated in the Perursquos Altiplane by a onemeter-sized chondritic meteorite [71] Being a quite unusualimpact event it is also likely that a significant number ofthese events are rarely studied because they happen in re-mote locations and remain unnoticed It is obvious thatChelyabinsk and other well-recorded events (eg by fireballnetworks) provide an opportunity to understand in therequired detail the behavior of meter-sized meteoroids andtheir capacity to directly cause human injuries and evencasualties

4 Conclusions

We have developed a numerical model employing theRungendashKutta method to predict the dynamic behavior ofmeteoroids penetrating the Earthrsquos atmosphere based on themeteor physics equations [39] To test the numerical modelwe successfully applied it to several meteor events describedin the scientific literature Having validated the numericalmodel we studied the deceleration behavior of the Che-lyabinsk superbolide in the lower part of its atmospherictrajectory just at the region following the main fragmen-tation event where such an approach is applicable )isscheme represents a novel way to study complicated me-teoric events by examining only the portion of the trajectoryduring which the object does not undergo abruptfragmentation

Our study of the deceleration profile of the Chelyabinsksuperbolide has allowed us to reach the followingconclusions

(a) Our numerical model successfully applied to thelower part of the fireball trajectory predicts well themain observed characteristics of the Chelyabinsksuperbolide )is is quite remarkable as the lowertrajectory studied here has a similar ablation be-haviour compared to fireballs at higher altitudes Itshould be noted that the Chelyabinsk event is thedeepest penetrating bolide ever documented bor-derline emitting light as it reached the troposphere)us our approach offers a promising venue instudying complex meteoric events in a streamlinedand simplified manner

(b) )e best fit to the deceleration pattern provides anaverage ablation coefficient σ 0034 s2middotkmminus 2 whichis in the range of those derived in the scientificliterature

(c) )e ablation coefficient is considered constantwithin each studied trajectory interval )is sim-plistic approach is probably one of the reasons whythis model is not applicable to the entire trajectory ofthe meteoroids suffering significant fragmentationand catastrophic disruptions In any case for thecases studied here the ablation coefficients obtainedin our work are consistent with those reported inscientific literature

30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)

v = 171kmsv = 230kmsv = 270kms

Figure 8 Simulation of the Chelyabinsk superbolide with differentinitial velocities (v 171 23 and 27 kms)


(d) A comparison of the fireball main parametersinferred from both the ground and space couldconstrain the role of the observing geometry in signalloss and biases in the determination of the velocityradiated energy and the orbital elements

(e) NEOs disrupting in the near-Earth region canproduce fragments tens of meters in size which ifencountering our planet under the right geometriccircumstances might be a significant source ofhazard for humans and infrastructure on the groundConsequently we suggest that identifying the exis-tence of asteroidal complexes in the near-Earthenvironment is crucial for a better assessment ofimpact hazard

(f ) Finally we should not underestimate the hazardouspotential of small asteroids as our model indicatesthat the ability of the Earthrsquos atmosphere to shield usfrom such objects strongly depends on the relativevelocity of the encounter with our planet

Data Availability

)e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was funded by the research project (PGC2018-097374-B-I00 PI JMT-R) funded by FEDERMinisteriode Ciencia e InnovacionndashAgencia Estatal de InvestigacionMG acknowledges support from the Academy of Finland(325806) Research at the Ural Federal University wassupported by the Russian Foundation for Basic Research (18-08-00074 and 19-05-00028) During the peer-review of thismanuscript the authors lost a mastermind dear friend andthe co-author Esko Lyytinen )e authors dedicate thiscommon effort in memorial to his enormous science figureand also to his friendship insight and understanding sharedover the years )e authors thank Dr Oleksandr Girin andtwo anonymous reviewers for their valuable and con-structive comments )e authors also thank MaratAhmetvaleev for providing them with the amazing picturesof the Chelyabinsk bolide and its dust trail

References

[1] P G Brown J D Assink L Astiz et al ldquoA 500-kilotonairburst over Chelyabinsk and an enhanced hazard from smallimpactorsrdquo Nature vol 503 no 7475 pp 238ndash241 2013

[2] J Borovicka P Spurny P Brown et al ldquo)e trajectorystructure and origin of the Chelyabinsk asteroidal impactorrdquoNature vol 503 no 7475 pp 235ndash237 2013

[3] I Halliday ldquoDetection of a meteorite stream observations of asecond meteorite fall from the orbit of the Innisfree chon-driterdquo Icarus vol 69 no 3 pp 550ndash556 1987

[4] T J Jopek G B Valsecchi and C Froeschle ldquoAsteroidmeteoroid streamsrdquo in Asteroids III W F Bottke JrA Cellino P Paolicchi and R P Binzel Eds pp 645ndash652University of Arizona Press Tucson AZ USA 2002

[5] J M Trigo-Rodrıguez E Lyytinen D C Jones et al ldquoAs-teroid 2002NY40 as source of meteorite-dropping bolidesrdquoMonthly Notices of the Royal Astronomical Society vol 382pp 1933ndash1939 2007

[6] J M Madiedo J M Trigo-Rodrıguez I P WilliamsJ L Ortiz and J Cabrera ldquo)e Northern χ-Orionid mete-oroid stream and possible association with the potentiallyhazardous asteroid 2008XM1rdquo Monthly Notices of the RoyalAstronomical Society vol 431 no 3 pp 2464ndash2470 2013

[7] C de la Fuente Marcos and R de la Fuente Marcos ldquo)eChelyabinsk superbolide a fragment of asteroid 2011 EO40rdquoMonthly Notices of the Royal Astronomical Society Lettersvol 436 no 1 pp L15ndashL19 2013

[8] T J Jopek G B Valsecchi and C Froeschle ldquoPossiblemeteoroid streams associated with (69230) Hermes and 2002SY50rdquo Earth Moon and Planets vol 95 pp 5ndash10 2004

[9] J Toth P Veres and L Kornos ldquoTidal disruption of NEAs-acase of Prıbram meteoriterdquo Monthly Notices of the RoyalAstronomical Society vol 415 no 2 pp 1527ndash1533 2011

[10] J Toth P Veres and L Kornos ldquoTidal disruption ofNEAs - a caseof Prıbram meteoriterdquoMonthly Notices of the Royal AstronomicalSociety vol 415 no 2 pp 1527ndash1533 2011

[11] A Bischoff M Horstmann C Vollmer U Heitmann andS Decker ldquoChelyabinskmdashnot only another ordinary LL5chondrite but a spectacular chondrite brecciardquo Meteoriticsand Planetary Science vol 48 2013

[12] T Kohout M Gritsevich V I Grokhovsky et al ldquoMiner-alogy reflectance spectra and physical properties of theChelyabinsk LL5 chondrite - insight into shock-inducedchanges in asteroid regolithsrdquo Icarus vol 228 pp 78ndash852014

[13] E Asphaug ldquoInterior structures for asteroids and cometarynucleirdquo in Mitigation of Hazardous Comets and AsteroidsM J S Belton T H Morgan N Samarasinha andD K Yeomans Eds p 66 Cambridge University PressCambridge UK 2004

[14] A Terentjeva ldquoFireball streamsrdquo inAsteroids Comets MeteorsIII CI Lagerkvist H Rickman and B A Lindblad Edsp 579 Uppsala Astronomical Observatory Uppsala Sweden1990

[15] I Halliday A A Griffin and A T Blackwell ldquoDetailed datafor 259 fireballs from the Canadian camera network andinferences concerning the influx of large meteoroidsrdquo Me-teoritics amp Planetary Science vol 31 no 2 pp 185ndash217 1996

[16] Z Ceplecha J Borovicka W G Elford et al ldquoMeteorPhenomena and bodiesrdquo Space Science Reviews vol 84 no 34 pp 327ndash471 1998

[17] I P ReVelle ldquo)e evolution of meteoroid streamsrdquo in Me-teors in the Earthrsquos Atmosphere E Murad and I P WilliamsEds pp 13ndash32 Cambridge University Press Cambridge UK2002

[18] M di Martino and A Cellino ldquoPhysical properties of cometsand asteroids inferred from fireball observationsrdquo in Miti-gation of Hazardous Comets and Asteroids M J S BeltonT H Morgan N Samarasinha and D K Yeomans Edsp 153 Cambridge University Press Cambridge UK 2004

[19] P A Bland and N A Artemieva ldquoEfficient disruption of smallasteroids by Earthrsquos atmosphererdquo Nature vol 424 no 6946pp 288ndash291 2003


[20] P Brown D O ReVelle E A Silber et al ldquoAnalysis of acrater-forming meteorite impact in Perurdquo Journal of Geo-physical Research vol 113 no E9 p E09007 2008

[21] M I Edwards V P Stulov and L I Turchak ldquoConsequencesof collisions of natural cosmic bodies with the Earthrsquos at-mosphere and surfacerdquo Cosmic Research vol 50 no 1pp 56ndash64 2012

[22] M I Gritsevich V P Stulov and L I Turchak ldquoFormation oflarge craters on the Earth as a result of impacts of naturalcosmic bodiesrdquo Doklady Physics vol 58 no 1 pp 37ndash392013 httpsdoiorg101134S1028335813010059

[23] L I Turchak and M I Gritsevich ldquoMeteoroids interactionwith the Earth atmosphererdquo Journal of eoretical and Ap-plied Mechanics vol 44 no 4 pp 15ndash28 2014

[24] M B E Boslough and D A Crawford ldquoLow-altitude air-bursts and the impact threatrdquo International Journal of ImpactEngineering vol 35 no 12 pp 1441ndash1448 2008

[25] E A Silber M Boslough W K Hocking M Gritsevich andR W Whitaker ldquoPhysics of meteor generated shock waves inthe Earthrsquos atmospheremdasha reviewrdquo Advances in Space Re-search vol 62 no 3 pp 489ndash532 2018

[26] D A Kring M Boslough and ldquo Chelyabinsk ldquoChelyabinskPortrait of an asteroid airburstrdquo Physics Today vol 67 no 9p 32 2014

[27] J M Trigo-Rodrıguez A J Castro-Tirado J Llorca et al ldquo)edevelopment of the Spanish Fireball Network using a new all-sky CCD systemrdquo Earth Moon Planets vol 95 pp 375ndash3872004

[28] J M Trigo-Rodrıguez J Llorca A J Castro-TiradoJ A Docobo and J Fabregat ldquo)e Spanish fireball networkrdquoAstronomy amp Geophysics vol 47 no 6 pp 26ndash6 2006

[29] J Dergham and J M Trigo-Rodrıguez ldquoA numerical methodto investigate the Ablation coefficient of the ChelyabinskSuperboliderdquo in Proceedings of the Astronomical ConferenceHeld at AM University T J Jopek F J M RietmeijerJ Watanabe and I P Williams Eds AM University PressPoznan Poland pp 11ndash17 August 2013

[30] J M Trigo-Rodrıguez J M Madiedo I P Williams et alldquo)e 2011 October Draconids outburst - I Orbital elementsmeteoroid fluxes and 21PGiacobini-Zinner delivered mass toEarthrdquo Monthly Notices of the Royal Astronomical Societyvol 433 no 1 pp 560ndash570 2013

[31] M Moreno-Ibantildeez J M Trigo-Rodrıguez J M Madiedoet al ldquoMulti-instrumental observations of the 2014 Ursidmeteor outburstrdquo Monthly Notices of the Royal AstronomicalSociety vol 468 no 2 pp 2206ndash2213 2017

[32] L R Bellot-Rubio M J Martınez Gonzalez L Ruiz Herreraet al ldquoModeling the photometric and dynamical behavior ofSuper-Schmidt meteors in the Earthrsquos atmosphererdquo Astron-omy amp Astrophysics vol 389 no 2 pp 680ndash691 2002

[33] M I Gritsevich T S Guslov D M Kozhemyakina andA D Novikov ldquoOn change in the mass loss parameter alongthe trajectory of a meteor bodyrdquo Solar System Researchvol 45 no 4 p 336 2011

[34] A Bouquet D Baratoux J Vaubaillon et al ldquoSimulation ofthe capabilities of an orbiter for monitoring the entry ofinterplanetary matter into the terrestrial atmosphererdquo Plan-etary and Space Science vol 103 pp 238ndash249 2014

[35] E K Sansom M Gritsevich H A R Devillepoix et alldquoDetermining fireball fates using the α-β criterionrdquo e As-trophysical Journal vol 885 no 2 p 7 Article ID 115 2019

[36] E Lyytinen ldquoAmerican Meteor Society webpagerdquo 2013httpwwwamsmeteorsorg201302large-daytime-fireball-hits-russia

[37] G G Novikov V N Lebedinet and A V Blokhin ldquoFor-mation of meteor tails during the quasi-continuous frag-mentation of meteor bodiesrdquo Tr Inst Ehksp MeteorolGoskomgidrometa No 18119 Atmospheric Optics MoscowRussia 1986pp 75ndash82 in Russian

[38] P B Babadzhanov ldquoFragmentation and densities of mete-oroidsrdquo Astronomy amp Astrophysics vol 384 no 1pp 317ndash321 2002

[39] V A Bronshten Physics of Meteoric Phenomena ReidelPublishing Co Dordrecht Netherlands 1983

[40] B Levin Physikalische eorie der Meteore und die meteor-itische Substanz im Sonnensystem Akademie-Verlag BerlinGermany 1961

[41] M I Gritsevich ldquoValidity of the photometric formula forestimating the mass of a fireball projectilerdquo Doklady Physicsvol 53 no 2 pp 97ndash102 2008a

[42] M Gritsevich and D Koschny ldquoConstraining the luminousefficiency of meteorsrdquo Icarus vol 212 no 2 pp 877ndash8842011

[43] M I Gritsevich and V P Stulov ldquoExtra-atmospheric massesof the Canadian network bolidesrdquo Solar System Researchvol 40 no 6 pp 477ndash484 2006

[44] M I Gritsevich ldquoDetermination of parameters of meteorbodies based on flight observational datardquo Advances in SpaceResearch vol 44 no 3 pp 323ndash334 2009

[45] E Lyytinen and M Gritsevich ldquoImplications of the atmo-spheric density profile in the processing of fireball observa-tionsrdquo Planetary and Space Science vol 120 pp 35ndash42 2016

[46] M M M Meier K C Welten M E I Riebe et al ldquoParkForest (L5) and the asteroidal source of shocked L chon-dritesrdquo Meteoritics amp Planetary Science vol 52 no 8pp 1561ndash1576 2017

[47] M I Gritsevich V Dmitriev V Vinnikov et al ldquoCon-straining the pre-atmospheric parameters of large meteoroidsKosice a case studyrdquo in Assessment andMitigation of AsteroidImpact Hazards Astrophysics and Space Science ProceedingsJ Trigo-Rodrıguez M Gritsevich and H Palme Eds vol 46p 153 Springer Cham Switzerland 2017

[48] J Moilanen M Gritsevich and E Lyytinen ldquoDeterminationof strewn fields for meteorite fallsrdquo Monthly Notices of theRoyal Astronomical Society 2021

[49] C O Johnston E C Stern and L F Wheeler ldquoRadiativeheating of large meteoroids during atmospheric entryrdquo Icarusvol 309 pp 25ndash44 2018

[50] C O Johnston E C Stern and J Borovicka ldquoSimulating theBenesov bolide flowfield and spectrum at altitudes of 47 and57 kmrdquo Icarus vol 354 Article ID 114037 2021

[51] F Bariselli A Frezzotti A Hubin and T E MaginldquoAerothermodynamic modelling of meteor entry flowsrdquoMonthly Notices of the Royal Astronomical Society vol 492no 2 pp 2308ndash2325 2020

[52] B Dias A Turchi E C Stern and T E Magin ldquoA model formeteoroid ablation including melting and vaporizationrdquo Ic-arus vol 345 p 113710 2020

[53] J M Trigo-Rodrıguez and J Llorca ldquo)e strength of cometarymeteoroids clues to the structure and evolution of cometsrdquoMonthly Notices of the Royal Astronomical Society vol 372no 2 pp 655ndash660 2006

[54] J M Trigo-Rodrıguez and J Llorca ldquoErratum the strength ofcometary meteoroids clues to the structure and evolution ofcometsrdquo Monthly Notices of the Royal Astronomical Societyvol 375 no 1 p 415 2007

[55] J M Trigo-Rodrıguez and J Blum ldquoTensile strength as anindicator of the degree of primitiveness of undifferentiated


bodiesrdquo Planetary and Space Science vol 57 no 2pp 243ndash249 2009

[56] M Moreno-Ibantildeez M Gritsevich and J M Trigo-RodrıguezldquoNew methodology to determine the terminal height of afireballrdquo Icarus vol 250 pp 544ndash552 2015

[57] J M Trigo-Rodrıguez ldquo)e flux of meteoroids over timemeteor emission spectroscopy and the delivery of volatilesand chondritic materials to Earthrdquo in Hypersonic MeteoroidEntry Physics IOP Series in Plasma Physics G ColonnaM Capitelli and A Laricchiuta Eds Institute of PhysicsPublishing 2019

[58] L G Jacchia and F L Whipple ldquoPrecision orbits of 413photographic meteorsrdquo Smithsonian Contributions to As-trophysics vol 4 no 4 pp 97ndash129 1961

[59] L G Jacchia ldquoA meteorite that missed the Earthrdquo Sky andTelescope vol 48 p 4 1972

[60] C D T Runge ldquoUber die numerische Auflosung von Dif-ferentialgleichungenrdquoMathematische Annalen vol 46 no 2pp 167ndash178 1895

[61] M Kutta ldquoBeitrag zur naherungsweisen Integration totalerdifferential gleichungenrdquo Zeitschrift fur Mathematik undPhysik vol 46 pp 435ndash453 1901

[62] US Government Printing Office US Standard AtmosphereUS Government Printing Office Washington DC USA1976

[63] M Moreno-Ibantildeez M Gritsevich J M Trigo-Rodrıguez andE A Silber ldquoPhysically based alternative to the PE criterionfor meteoroidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 494 no 1 pp 316ndash324 2020

[64] M I Gritsevich ldquo)e Pribram Lost City Innisfree andNeuschwanstein Falls an analysis of the atmospheric tra-jectoriesrdquo Solar System Research vol 42 no 5 pp 372ndash3902008b

[65] D Yeomans and P Chodas ldquoCNEOS JPL Homepagerdquo 2013httpscneosjplnasagovnewsfireball_130301html

[66] Z Sekanina ldquo)e Tunguska eventmdashno cometary signature inevidencerdquo e Astronomical Journal vol 88 pp 1382ndash14131983

[67] S D Miller W C Straka III A S Bachmeier T J SchmitP T Partain and Y J Noh ldquoEarth-viewing satellite per-spectives on the Chelyabinsk meteor eventrdquo Proceedings of theNational Academy of Sciences vol 110 no 45 pp 18092ndash18097 2013

[68] S R Proud ldquoReconstructing the orbit of the Chelyabinskmeteor using satellite observationsrdquo Geophysical ResearchLetters vol 40 no 1 pp 3351ndash3355 2013

[69] M Gritsevich V Vinnikov T Kohout et al ldquoA compre-hensive study of distribution laws for the fragments of Kosicemeteoriterdquo Meteoritics amp Planetary Science vol 49 no 3pp 328ndash345 2014

[70] V Vinnikov M Gritsevich and L Turchak ldquoShape esti-mation for Kosice Almahata Sitta and bassikounou mete-oroidsrdquo Proceedings of the International Astronomical Unionvol 10 no S306 pp 394ndash396 2015

[71] G Tancredi J Ishitsuka P H Schultz et al ldquoA meteoritecrater on Earth formed on September 15 2007 the Carancashypervelocity impactrdquo Meteoritics amp Planetary Sciencevol 44 no 12 pp 1967ndash1984 2009

[72] P Jenniskens Meteor Showers and eir Parent CometsCambridge University Press Cambridge UK 2006


the reconstruction of the incoming fireball trajectory )eChelyabinsk superbolide entered the atmosphere at sim19 kms and according to the US sensor data (CNEOS fireball listhttpscneosjplnasagovfireballs) it reached the maxi-mum brightness at an altitude of 233 km with a velocity of186 kms [1 2] also providing us with valuable samples inthe form of meteorites

)e existence of meteoroid streams capable of producingmeteorite-dropping bolides is a hot topic in planetary sci-ence Such streams were first proposed by Halliday [3 4])eir existence has important implications because they cannaturally deliver to the Earth different types of rock-formingmaterials from potentially hazardous asteroids (PHAs) It isbelieved that NEOs in the Earthrsquos vicinity are undergoingdynamical and collisional evolution on relatively shorttimescales We previously identified several NEO complexesthat are producing meteorite-dropping bolides and wehypothesize that they could have been produced during closeapproaches to terrestrial planets [5 6] Such formationscenario for this kind of asteroidal complexes is nowreinforced with the recent discovery of a complex of NEOslikely associated with the NEA progenitor of the Chelyabinskbolide [7])e shattered pieces resulting from the disruptionof NEOs visiting the inner solar system can spread along theentire parent body orbit on a time-scale of centuries [4 8 9])is scenario is also consistent with the modern view ofNEOs being resurfaced as consequence of close approaches[10] Additionally the meteorites recovered from theChelyabinsk have a brecciated nature [11 12] that is rem-iniscent of the complex collisional history and a probablerubble pile structure of the asteroid progenitor of thecomplex [13] )e existence of these asteroidal complexes inthe near-Earth region has important implications as theycould be the source of low spatial density meteoroid streamspopulated by large meteoroids Such complexes could be thesource of the poorly known fireball-producing radiants[14 15] )is could have important implications for thefraction of sporadic meteoroids producing bright fireballsand in the physical mechanisms envisioned in the past[16ndash18]

)e Chelyabinsk event is also of interest because of itsmagnitude and energy and due to its relevance to be con-sidered as a representative example of the most frequentoutcome of the impact hazard associated with small aster-oids in human timescales Chelyabinsk also exemplifies theimportance that fragmentation has for small asteroidswhich can even excavate a crater on the Earthrsquos surfacealthough rarely [19ndash23] Fragmentation is important as itprovides a mechanism in which a significant part of thekinetic energy associated with small asteroids is released Itwas certainly a very relevant process for the Tunguska event[24 25] and in the better-known case of Chelyabinsk mostof the kinetic energy was transferred to the internal energy ofthe air which is radiated as light [26]

One way to study meteoroids as they enter the Earthrsquosatmosphere is through video observations of such eventsConsequently we are developing complementary ap-proaches to study the dynamical behavior of video-recordedbolides in much detail )e SPanish Meteor Network

(SPMN) pioneered the application of high-sensitivitycameras for detecting fireballs and it currently maintains anonline list of bright events detected over Spain PortugalSouthern France and Morocco since 1999 [27 28] Forexample casual video recordings plus several still photo-graphs of the superbolide in flight allowed us to reconstructthe heliocentric orbit of Villalbeto de la Pentildeameteorite in theframework of the SPMN [28] )e possibility of studyingsuperbolides such as Chelyabinsk is a very attractive mile-stone to be considered )e software used in this study wasdeveloped as part of a master thesis [29] and subsequentlytested and validated using several cases discussed by [29] aswell as events from the 25 video and all-sky CCD stations setup over the Iberian Peninsula by the SPMN In this contextwe have been engaged in studying the dynamic behavior ofmeteoroids decelerating in the Earthrsquos atmosphere [30 31]

In this study we study the Chelyabinsk bolide by fol-lowing a RungendashKutta method of meteor investigationsimilar to that developed by Bellot Rubio et al [32] We aimto test if that specific method is also valid for another massrange particularly for small asteroids and large meter-sizedmeteoroids We first describe our numerical model and testit against the known meteor events We compare our codevalidation results to the results obtained by Bellot Rubio et al[32] for the same dataset We then apply our numericalmodel to the Chelyabinsk superbolide in order to study itsdynamical behavior For the sake of simplicity our modelconsiders a constant ablation coefficient and the shapefactor even though these parameters could vary in differentablation stages [33ndash35]

)is study is structured as follows the data reductionand the theoretical approach pertaining to the Chelyabinskbolide are described in the next section In Section 3 themain implications of this work in the context of fireballsmeteorites and NEO research are discussed We use themodel to determine the fireball flight parameters and bystudying the deceleration we also obtain the ablation co-efficient Finally the conclusions of this work are presentedin Section 4

2 Data Reduction Theoretical Approachand Observations

)e Chelyabinsk superbolide was an unexpected daylightsuperbolide as many other unpredicted meteorite-drop-ping bolides in history Fortunately numerous casualvideo recordings of the bolide trajectory from the groundwere obtained given the nowadays common dashcamsavailable in private motor vehicles in Russia According tothe video recordings available it is possible to study theatmospheric trajectory and deceleration carefullyallowing the reconstruction of the heliocentric orbit inrecord time [2 26]

21 Single Body eory )ere are two main approaches inthe study of the dynamic properties of meteors during at-mospheric interaction the quasicontinuous fragmentation(QCF) theory introduced by Novikov et al [37] which was







dv

dt minus Kρairm

minus 13middot v

2 (1)

dm

dt minus σKρairm

23middot v

3 (2)



σ Λ

2ΓQ (3)



K sA

ρ23 (4)



dh




dv

dh

Kρairmminus 13

middot v

cos(z) (6)

dm

dhσKρairm

23middot v

2

cos(z) (7)


dm

dv mσv (8)



2minus v2( ) (9)



dv


minus (12)σ vo2minus v2( )1113874 11138751113874 1113875

minus (13)

middot v middot1

cos z (10)




I τ2

minusdm

dt1113888 1113889 middot v

2 (11)





dv

dh ρairKprime e

minus 12σ vo2minus v2( )1113874 1113875


1cos z

(13)








yprime f(t y)middot

yo to (14)


_dv

dh f(h v) v ho( 1113857 ho

(15)



l1 f tn yn( 1113857

l2 f tn +12

p yn + p12l11113874 1113875

l3 f tn +12

p yn + p12l21113874 1113875

l4 f tn + p yn + pl3( 1113857

(16)



2minus v2( )1113874 1113875minus 13

middot v middot1

cos z (17)


l1 f hn vn( 1113857

l2 f hn +12

p vn + p12l11113874 1113875

l3 f hn +12

p vn + p12l21113874 1113875

l4 f hn + p vn + pl3( 1113857

(18)



yn+1 yn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (19)


vn+1 vn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (20)





ξ vd minus vf( K σ)1113872 11138732 (21)




2


2


2


2

(22)











dv


vh2 minus vh1

h2 minus h1 (23)












+σ

+K


100

099

098

097

096

vv 0


1021009290 94 988886 96Altitude (km)


77 78 79 80 81 82 83 84Altitude (km)

170

168

166

164

162

160

158

Velo

city

(km

s)




m

mo

eminus (12)σ vo

2minus v2( ) (24)



dmmo

h1113890 1113891

Δmmo

Δh

mmoh2 minus mmoh1

h2 minus h1 (25)











85 9080 100957570Altitude (km)

16

18

20

22

24

26

Velo

city

(km

s)

σ = 001σ = 003σ = 005









(a) (b)


00

ndash01

ndash02

ndash03Acce

lera

tion

(km

s2 )

88 90 92 94 96 98 100 102Altitude (km)

(a)

Mas

s (m

m0)

100

075

050

025

88 90 92 94 96 98 100 102Altitude (km)

(b)

Relat

ive m

ass l

oss r

ate 008

006

004

002

88 90 92 94 96 98 100 102Altitude (km)

(c)







Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)









4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References


















































































dv

dt minus Kρairm

minus 13middot v

2 (1)

dm

dt minus σKρairm

23middot v

3 (2)



σ Λ

2ΓQ (3)



K sA

ρ23 (4)



dh




dv

dh

Kρairmminus 13

middot v

cos(z) (6)

dm

dhσKρairm

23middot v

2

cos(z) (7)


dm

dv mσv (8)



2minus v2( ) (9)



dv


minus (12)σ vo2minus v2( )1113874 11138751113874 1113875

minus (13)

middot v middot1

cos z (10)




I τ2

minusdm

dt1113888 1113889 middot v

2 (11)





dv

dh ρairKprime e

minus 12σ vo2minus v2( )1113874 1113875


1cos z

(13)








yprime f(t y)middot

yo to (14)


_dv

dh f(h v) v ho( 1113857 ho

(15)



l1 f tn yn( 1113857

l2 f tn +12

p yn + p12l11113874 1113875

l3 f tn +12

p yn + p12l21113874 1113875

l4 f tn + p yn + pl3( 1113857

(16)



2minus v2( )1113874 1113875minus 13

middot v middot1

cos z (17)


l1 f hn vn( 1113857

l2 f hn +12

p vn + p12l11113874 1113875

l3 f hn +12

p vn + p12l21113874 1113875

l4 f hn + p vn + pl3( 1113857

(18)



yn+1 yn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (19)


vn+1 vn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (20)





ξ vd minus vf( K σ)1113872 11138732 (21)




2


2


2


2

(22)











dv


vh2 minus vh1

h2 minus h1 (23)












+σ

+K


100

099

098

097

096

vv 0


1021009290 94 988886 96Altitude (km)


77 78 79 80 81 82 83 84Altitude (km)

170

168

166

164

162

160

158

Velo

city

(km

s)




m

mo

eminus (12)σ vo

2minus v2( ) (24)



dmmo

h1113890 1113891

Δmmo

Δh

mmoh2 minus mmoh1

h2 minus h1 (25)











85 9080 100957570Altitude (km)

16

18

20

22

24

26

Velo

city

(km

s)

σ = 001σ = 003σ = 005









(a) (b)


00

ndash01

ndash02

ndash03Acce

lera

tion

(km

s2 )

88 90 92 94 96 98 100 102Altitude (km)

(a)

Mas

s (m

m0)

100

075

050

025

88 90 92 94 96 98 100 102Altitude (km)

(b)

Relat

ive m

ass l

oss r

ate 008

006

004

002

88 90 92 94 96 98 100 102Altitude (km)

(c)







Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)









4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References














































































dv


minus (12)σ vo2minus v2( )1113874 11138751113874 1113875

minus (13)

middot v middot1

cos z (10)




I τ2

minusdm

dt1113888 1113889 middot v

2 (11)





dv

dh ρairKprime e

minus 12σ vo2minus v2( )1113874 1113875


1cos z

(13)








yprime f(t y)middot

yo to (14)


_dv

dh f(h v) v ho( 1113857 ho

(15)



l1 f tn yn( 1113857

l2 f tn +12

p yn + p12l11113874 1113875

l3 f tn +12

p yn + p12l21113874 1113875

l4 f tn + p yn + pl3( 1113857

(16)



2minus v2( )1113874 1113875minus 13

middot v middot1

cos z (17)


l1 f hn vn( 1113857

l2 f hn +12

p vn + p12l11113874 1113875

l3 f hn +12

p vn + p12l21113874 1113875

l4 f hn + p vn + pl3( 1113857

(18)



yn+1 yn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (19)


vn+1 vn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (20)





ξ vd minus vf( K σ)1113872 11138732 (21)




2


2


2


2

(22)











dv


vh2 minus vh1

h2 minus h1 (23)












+σ

+K


100

099

098

097

096

vv 0


1021009290 94 988886 96Altitude (km)


77 78 79 80 81 82 83 84Altitude (km)

170

168

166

164

162

160

158

Velo

city

(km

s)




m

mo

eminus (12)σ vo

2minus v2( ) (24)



dmmo

h1113890 1113891

Δmmo

Δh

mmoh2 minus mmoh1

h2 minus h1 (25)











85 9080 100957570Altitude (km)

16

18

20

22

24

26

Velo

city

(km

s)

σ = 001σ = 003σ = 005









(a) (b)


00

ndash01

ndash02

ndash03Acce

lera

tion

(km

s2 )

88 90 92 94 96 98 100 102Altitude (km)

(a)

Mas

s (m

m0)

100

075

050

025

88 90 92 94 96 98 100 102Altitude (km)

(b)

Relat

ive m

ass l

oss r

ate 008

006

004

002

88 90 92 94 96 98 100 102Altitude (km)

(c)







Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)









4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References














































































yn+1 yn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (19)


vn+1 vn +16

p l1 + 2l2 + 2l3 + l4( 1113857 (20)





ξ vd minus vf( K σ)1113872 11138732 (21)




2


2


2


2

(22)











dv


vh2 minus vh1

h2 minus h1 (23)












+σ

+K


100

099

098

097

096

vv 0


1021009290 94 988886 96Altitude (km)


77 78 79 80 81 82 83 84Altitude (km)

170

168

166

164

162

160

158

Velo

city

(km

s)




m

mo

eminus (12)σ vo

2minus v2( ) (24)



dmmo

h1113890 1113891

Δmmo

Δh

mmoh2 minus mmoh1

h2 minus h1 (25)











85 9080 100957570Altitude (km)

16

18

20

22

24

26

Velo

city

(km

s)

σ = 001σ = 003σ = 005









(a) (b)


00

ndash01

ndash02

ndash03Acce

lera

tion

(km

s2 )

88 90 92 94 96 98 100 102Altitude (km)

(a)

Mas

s (m

m0)

100

075

050

025

88 90 92 94 96 98 100 102Altitude (km)

(b)

Relat

ive m

ass l

oss r

ate 008

006

004

002

88 90 92 94 96 98 100 102Altitude (km)

(c)







Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)









4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References

















































































dv


vh2 minus vh1

h2 minus h1 (23)












+σ

+K


100

099

098

097

096

vv 0


1021009290 94 988886 96Altitude (km)


77 78 79 80 81 82 83 84Altitude (km)

170

168

166

164

162

160

158

Velo

city

(km

s)




m

mo

eminus (12)σ vo

2minus v2( ) (24)



dmmo

h1113890 1113891

Δmmo

Δh

mmoh2 minus mmoh1

h2 minus h1 (25)











85 9080 100957570Altitude (km)

16

18

20

22

24

26

Velo

city

(km

s)

σ = 001σ = 003σ = 005









(a) (b)


00

ndash01

ndash02

ndash03Acce

lera

tion

(km

s2 )

88 90 92 94 96 98 100 102Altitude (km)

(a)

Mas

s (m

m0)

100

075

050

025

88 90 92 94 96 98 100 102Altitude (km)

(b)

Relat

ive m

ass l

oss r

ate 008

006

004

002

88 90 92 94 96 98 100 102Altitude (km)

(c)







Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)









4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References














































































m

mo

eminus (12)σ vo

2minus v2( ) (24)



dmmo

h1113890 1113891

Δmmo

Δh

mmoh2 minus mmoh1

h2 minus h1 (25)











85 9080 100957570Altitude (km)

16

18

20

22

24

26

Velo

city

(km

s)

σ = 001σ = 003σ = 005









(a) (b)


00

ndash01

ndash02

ndash03Acce

lera

tion

(km

s2 )

88 90 92 94 96 98 100 102Altitude (km)

(a)

Mas

s (m

m0)

100

075

050

025

88 90 92 94 96 98 100 102Altitude (km)

(b)

Relat

ive m

ass l

oss r

ate 008

006

004

002

88 90 92 94 96 98 100 102Altitude (km)

(c)







Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)









4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References

















































































(a) (b)


00

ndash01

ndash02

ndash03Acce

lera

tion

(km

s2 )

88 90 92 94 96 98 100 102Altitude (km)

(a)

Mas

s (m

m0)

100

075

050

025

88 90 92 94 96 98 100 102Altitude (km)

(b)

Relat

ive m

ass l

oss r

ate 008

006

004

002

88 90 92 94 96 98 100 102Altitude (km)

(c)







Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)









4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References

















































































Acce

lera

tion

(km

s2 )

12 14 16 18 20 22 24 26Altitude (km)

180

135

90

45

(a)

Mas

s (m

m0)

12 14 16 18 20 22 24 26Altitude (km)

096

064

032

000

(b)

Relat

ive m

ass l

oss r

ate

12 14 16 18 20 22 24 26Altitude (km)

013

009

004

000

(c)









4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References

















































































4 Conclusions






30

25

20

15

10

5

Velo

city

(km

s)

14 16 18 20 22 2412Altitude (km)







Data Availability




Acknowledgments


References
















































































Data Availability




Acknowledgments


References













































































A Numerical Approach to Study Ablation of Large Bolides ...

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Transcript of A Numerical Approach to Study Ablation of Large Bolides ...