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A Numerical Approach to Study Ablation of Large Bolides: Application to Chelyabinsk

A Numerical Approach to Study Ablation of Large Bolides: Application to Chelyabinsk Hindawi Advances in Astronomy Volume 2021, Article ID 8852772, 13 pages https://doi.org/10.1155/2021/8852772 Research Article A Numerical Approach to Study Ablation of Large Bolides: Application to Chelyabinsk 1,2 1,2 3,4,5,6 5† Josep M. Trigo-Rodrı´guez , Joan Dergham, Maria Gritsevich, Esko Lyytinen, 7,8 9 Elizabeth A. Silber and Iwan P. Williams Institut de Ci` encies de l’Espai–CSIC, Campus UAB, Facultat de Ci` encies, Torre C5-parell-2a, 08193 Bellaterra, Barcelona, Catalonia, Spain Institut d’Estudis Espacials de Catalunya (IEEC), Edif. Nexus, c/Gran Capita, 2-4, 08034 Barcelona, Catalonia, Spain Finnish Geospatial Research Institute (FGI), Geodeetinrinne 2, FI-02430 Masala, Finland Department of Physics, University of Helsinki, Gustaf Ha ¨llstro ¨min katu 2a, P.O. Box 64, FI-00014 Helsinki, Finland Finnish Fireball Network, Helsinki, Finland Institute of Physics and Technology, Ural Federal University, Mira str. 19., 620002 Ekaterinburg, Russia Department of Earth Sciences, Western University, London, ON N6A 5B7, Canada ?e Institute for Earth and Space Exploration, Western University, London, ON N6A 3K7, Canada Astronomy Unit, Queen Mary, University of London, Mile End Rd., London E1 4NS, UK Deceased Correspondence should be addressed to Josep M. Trigo-Rodr´ıguez; trigo@ice.csic.es Received 31 August 2020; Revised 13 January 2021; Accepted 2 March 2021; Published 27 March 2021 Academic Editor: Kovacs Tamas Copyright © 2021 Josep M. Trigo-Rodr´ıguez et al. )is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, we investigate the ablation properties of bolides capable of producing meteorites. )e casual dashcam recordings from many locations of the Chelyabinsk superbolide associated with the atmospheric entry of an 18m in diameter near-Earth object (NEO) have provided an excellent opportunity to reconstruct its atmospheric trajectory, deceleration, and heliocentric orbit. In this study, we focus on the study of the ablation properties of the Chelyabinsk bolide on the basis of its deceleration and fragmentation. We explore whether meteoroids exhibitingabruptfragmentation can bestudiedby analyzing segmentsofthe trajectorythatdonotinclude a disruptionepisode. We apply that approach to the lower part of the trajectory of the Chelyabinsk bolide to demonstrate that the obtained parameters are consistent. To do that, we implemented a numerical (Runge–Kutta) method appropriate for deriving the ablation properties of bolides based on observations. )e method was successfully tested with the cases previously published in the literature. Our model yields fits that agree with observations reasonably well. It also produces a good fit to the main observed characteristics of Chelyabinsk superbolide and 2 − 2 provides its averaged ablation coefficient σ �0.034s km . Our study also explores the main implications for impact hazard, concluding that tens of meters in diameter NEOs encountering the Earth in grazing trajectories and exhibiting low geocentric velocities are penetrating deeper into the atmosphere than previously thought and, as such, are capable of producing meteorites and even damage on the ground. year prior, an unexpected impact with an Apollo asteroid 1. Introduction ensued [1]. At 03:20 UTC, a superbolide, also known as the On February 15, 2013, our view about impact hazard was Chelyabinsk superbolide, flew over the Russian territory and seriously challenged. While there was a sense of accom- Kazakhstan. )e possible link between the superbolide and plishment for being able to forecast the close approach of the 2012 DA NEA was discarded by the European Space 2012 DA near-Earth asteroid (NEA) within a distance of Agency (ESA) and the Jet Propulsion Laboratory–National 27700km, eventhough this NEO was discovered only one Aeronautics and Space Administration (JPL-NASA) from 2 Advances in Astronomy (SPMN) pioneered the application of high-sensitivity the reconstruction of the incoming fireball trajectory. )e Chelyabinsk superbolide entered the atmosphere at∼19km/ cameras for detecting fireballs, and it currently maintains an online list of bright events detected over Spain, Portugal, s, and according to the US sensor data (CNEOS fireball list: https://cneos.jpl.nasa.gov/fireballs/), it reached the maxi- Southern France, and Morocco since 1999 [27, 28]. For mum brightness at an altitude of 23.3km with a velocity of example, casual video recordings plus several still photo- 18.6km/s [1, 2], also providing us with valuable samples in graphs of the superbolide in flight allowed us to reconstruct the form of meteorites. the heliocentric orbit of Villalbeto de la Peña meteorite in the )e existence of meteoroid streams capable of producing framework of the SPMN [28]. )e possibility of studying meteorite-dropping bolides is a hot topic in planetary sci- superbolides such as Chelyabinsk is a very attractive mile- stone to be considered. )e software used in this study was ence. Such streams were first proposed by Halliday [3, 4]. )eir existence has important implications because they can developed as part of a master thesis [29] and subsequently tested and validated using several cases discussed by [29], as naturally deliver to the Earth different types of rock-forming materials from potentially hazardous asteroids (PHAs). It is well as events from the 25 video and all-sky CCD stations set up over the Iberian Peninsula by the SPMN. In this context, believed that NEOs in the Earth’s vicinity are undergoing dynamical and collisional evolution on relatively short we have been engaged in studying the dynamic behavior of timescales. We previously identified several NEO complexes meteoroids decelerating in the Earth’s atmosphere [30, 31]. that are producing meteorite-dropping bolides, and we In this study, we study the Chelyabinsk bolide by fol- hypothesize that they could have been produced during close lowing a Runge–Kutta method of meteor investigation approaches to terrestrial planets [5, 6]. Such formation similar to that developed by Bellot Rubio et al. [32]. We aim scenario for this kind of asteroidal complexes is now to test if that specific method is also valid for another mass range, particularly for small asteroids and large meter-sized reinforced with the recent discovery of a complex of NEOs likely associated with the NEA progenitor of the Chelyabinsk meteoroids. We first describe our numerical model and test it against the known meteor events. We compare our code bolide [7]. )e shattered pieces resulting from the disruption of NEOs visiting the inner solar system can spread along the validation results to the results obtained by Bellot Rubio et al. [32] for the same dataset. We then apply our numerical entire parent body orbit on a time-scale of centuries [4, 8, 9]. )is scenario is also consistent with the modern view of model to the Chelyabinsk superbolide in order to study its NEOs being resurfaced as consequence of close approaches dynamical behavior. For the sake of simplicity, our model [10]. Additionally, the meteorites recovered from the considers a constant ablation coefficient and the shape Chelyabinsk have a brecciated nature [11, 12] that is rem- factor, even though these parameters could vary in different iniscent of the complex collisional history and a probable ablation stages [33–35]. rubble pile structure of the asteroid progenitor of the )is study is structured as follows: the data reduction and the theoretical approach pertaining to the Chelyabinsk complex [13]. )e existence of these asteroidal complexes in the near-Earth region has important implications as they bolide are described in the next section. In Section 3, the main implications of this work in the context of fireballs, could be the source of low spatial density meteoroid streams populated by large meteoroids. Such complexes could be the meteorites, and NEO research are discussed. We use the source of the poorly known fireball-producing radiants model to determine the fireball flight parameters, and by [14, 15]. )is could have important implications for the studying the deceleration, we also obtain the ablation co- fraction of sporadic meteoroids producing bright fireballs efficient. Finally, the conclusions of this work are presented and in the physical mechanisms envisioned in the past in Section 4. [16–18]. )e Chelyabinsk event is also of interest because of its 2. Data Reduction, Theoretical Approach, magnitude and energy and due to its relevance to be con- and Observations sidered as a representative example of the most frequent outcome of the impact hazard associated with small aster- )e Chelyabinsk superbolide was an unexpected daylight oids in human timescales. Chelyabinsk also exemplifies the superbolide as many other unpredicted meteorite-drop- importance that fragmentation has for small asteroids, ping bolides in history. Fortunately, numerous casual which can even excavate a crater on the Earth’s surface, video recordings of the bolide trajectory from the ground although rarely [19–23]. Fragmentation is important as it were obtained, given the nowadays common dashcams provides a mechanism in which a significant part of the available in private motor vehicles in Russia. According to kinetic energy associated with small asteroids is released. It the video recordings available, it is possible to study the was certainly a very relevant process for the Tunguska event atmospheric trajectory and deceleration carefully, [24, 25], and in the better-known case of Chelyabinsk, most allowing the reconstruction of the heliocentric orbit in of the kinetic energy was transferred to the internal energy of record time [2, 26]. the air, which is radiated as light [26]. One way to study meteoroids as they enter the Earth’s atmosphere is through video observations of such events. 2.1. Single Body ?eory. )ere are two main approaches in Consequently, we are developing complementary ap- the study of the dynamic properties of meteors during at- proaches to study the dynamical behavior of video-recorded mospheric interaction, the quasicontinuous fragmentation bolides in much detail. )e SPanish Meteor Network (QCF) theory introduced by Novikov et al. [37], which was Advances in Astronomy 3 2.3. ?e Single Body ?eory: ?e Drag and Mass Loss later extended by Babadzhanov [38], and the single body theory described by Bronshsten [39]. )ere have been dis- Equations. )e dynamic behavior of a meteoroid as it in- teracts with the Earth’s atmosphere is described using the crepancies in the applicability conditions of both methods: the single body works with basic differential equations, while drag and mass loss equations. )ese equations, as presented the QCF uses semiempirical formulas studying only the by Bronshten [39], are as follows: luminosity produced by the meteor. )e main difference is dv − 1/3 2 (1) that the single body theory obtains smaller dynamical masses � − Kρ m · v , air dt than the QCF method. )us far, neither approach is prev- alent, and the reason why the theories do not converge could dm 2/3 3 (2) � − σKρ m · v . be attributed to the contribution of other key processes such air dt as the fragmentation and deceleration of meteoroids during ablation or to the poorly constrained values of the bulk where K is the shape-density coefficient, ρ is the air air density and/or the luminous efficiency coefficient [40–42]. density, m is the meteoroid mass, v is the instantaneous We remark that the initial dynamic mass estimate or the velocity, and σ is the ablation coefficient. preatmospheric size can be derived using the methods de- By using equations (1) and (2), the parameters to be scribed in other works [43–48]; hence, we leave it out of the identified are K and σ. )e ablation coefficient defines the scope of the present model. We also note that alternative loss of mass for the bolide as it penetrates the atmosphere; models accounting for ablation have been recently devel- the larger the value is, the more the mass will be ablated for a oped; however, further discussion on that topic is beyond the given velocity. )e value of the ablation coefficient depends scope of this study, and the reader is referred to the following on various factors and is expressed as literature [46, 47, 49–52]. As noted in the introduction, high- strength meteoroids from asteroids or planetary bodies σ � , (3) 2ΓQ exhibit a very different behavior than fragile dust aggregates originating from comets [53–57]. 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 and 2.2. ?e Role of Fragmentation. )e fragmentation of me- density of the meteoroid and is expressed as teoroids was studied in detail by various authors [38, 40]. sA K � , After analyzing different photographic observations, Levin (4) 2/3 [40] distinguished four possible types of fragmentation: (a) the decay of the meteoroid into large nonfragmenting debris, where A is the shape factor, s is the cross-section area, and ρ (b) the progressive disintegration of the original meteoroid is the meteoroid bulk density. into fragments that continue to crumble into smaller We must point out that the observational data obtained fragments, (c) the instantaneous ejection of a large number from the reconstruction of the trajectories of meteors using of small particles, which, when affecting the entire mete- CCD or video cameras is basically the frame to frame speed oroid, is called a catastrophic disruption, and finally, (d) the of the bolide as a function of the height, requiring another quasicontinuous fragmentation which consists of the equation to link the time with the altitude: gradual release of a large number of small particles from the dh surface and their subsequent evaporation due to high � − v · cos(z), (5) dt temperatures associated with the shock thermal wave formed around the body. where z is the zenith angle. In practice, a combination of two or more types of By substituting equation (5) into equations (1) and (2), fragmentation can be observed in a given meteor event. In the following expressions are obtained: fact, it is possible to observe that (a) and (c) fragmentation − 1/3 types described in the preceding paragraph could occur dv Kρ m · v air (6) � , more than once for the same meteor event. )e analysis of dh cos(z) meteors performed by Jacchia [58] using Super-Schmidt cameras demonstrated that the single body theory does not 2/3 2 dm σKρ m · v air (7) � . work for cases which suffer abrupt types of fragmentation dh cos(z) along the trajectory. As a direct consequence, meteoroids exhibiting fragmentation of the first (a), second (b), and Next, by dividing equation (2) by equation (1), we obtain third (c) kind should not be studied using this simplified dm single body theory. When a meteoroid suffers an abrupt (8) � mσv. dv fragmentation, the main body loses mass instantaneously, and therefore, the single body equations cannot be applied Solving this differential equation with the boundary since the condition of continuity of mass is not satisfied. condition of m � m , when v � v , the following is obtained: o o )us, the cases with possible abrupt fragmentation epi- 2 2 − (1/2)σ v − v ( o ) (9) sode(s) will not be examined in this work. m � m · e . o 4 Advances in Astronomy Now, we combine equations (9) and (6) to derive data obtained via high-resolution Super-Schmidt cameras [58] can be utilized for a case study. )ese meteor trajectory data are − (1/3) dv 2 2 1 − (1/2)σ v − v ( ) (10) � Kρ 􏼒m · 􏼒e 􏼓􏼓 · v · . used to optimize the procedure to obtain the values of K′ and σ air o dh cos z that fit the best the actual data points. )e procedure presented here produces many synthetic curves. Subsequently, in order to In order to obtain the value of K and σ, we use equation optimize the computing time, a solution to make the equation (10), as it uses substitution for the dependence on the in- converge to the real data is found. stantaneous mass, and thus, we deal with one equation instead of two. Furthermore, there is the initial mass, which is an important parameter to be studied. Equation (10) directly links 3.1.1. ?e Runge–Kutta Method Implementation. )e Run- the deceleration of the meteoroid as a function of different ge–Kutta method is an iterative technique for the approxi- parameters, in particular the zenith angle. )e last term is of mation and solution of ordinary differential equations. )e particular interest as it modulates the full equation. For the ° method was first developed by Runge [60] and Kutta [61]. vertical entry (z � 0 ), the deceleration is maximized, while for z )e Runge–Kutta approximation provides a solution at a close to 90 , the deceleration is minimized. For that reason, determined point of altitude. Application of the Runge–Kutta large meteoroids at grazing angles are able to follow extremely method requires the initial conditions to be known: long trajectories or even escape again into space, such as the Grand Tetons superbolide that spent nearly two minutes . (14) y � f(t, y) ; y ; t o o traveling over several states of the USA and Canada on August 10, 1972 [11, 59]. In our case, the initial conditions will be the initial However, by using the concepts introduced above, it is velocity and the altitude of the bolide when ablation starts, not possible to obtain the values of initial mass (m ) and K. “synthetically” written as )e parameters that can be found are only the expressions of − 1/3· _ dv m K and σ. Consequently, another equation is required (15) � f(h, v) ; v h 􏼁 ; h o o to obtain K and m separately. )e remaining expression is dh the photometric equation: We then need to choose a step size (p) that should be τ dm comparable with the data resolution to allow a better com- I � 􏼠− 􏼡 · v , (11) parison between the model and the observations. )e step size 2 dt defines how many integration steps need to be implemented where τ is the luminous efficiency. )is equation is assuming beforethefinalsolutionisreached.)esmallerthestepsize,the v �constant, and it is often applied to small meteors. )e more accurate the solution will be, noting that this will also luminous efficiency is obtained empirically; thus, any de- increase the computing time. )e step size corresponds to the viations in that value might produce significant changes in characteristic Δh that according to Bellot Rubio et al. [32] can the results. )is equation will be referred to later as a part of be chosen around a few hundred of meters, 100–300m. the mass determination. Now, we focus only on equation Once the step size is defined, we define the model co- (10). efficients as follows: We define the K′ as l � f t , y , 1 n n − 1/3 ′ (12) K � K · m . 1 1 l � f􏼒t + p, y + p l 􏼓, 2 n n 1 2 2 )en, the equation to work with is (16) 1 1 − 1/3 2 2 l � f􏼒t + p, y + p l 􏼓, dv 1 − 1/2σ v − v 3 n n 2 ( o ) ′ (13) � ρ K e · v · , 2 2 􏼒 􏼓 air dh cos z l � f t + p, y + pl . 4 n n 3 with variables K′ and σ. For our case, the function to be studied is − 1/3 3. Results and Discussion 2 2 1 − 1/2σ v − v ( ) ′ (17) f h , v 􏼁 � K ρ(h)􏼒e 􏼓 · v · , n n cos z 3.1. Numerical Approximation. In this section, we develop a numerical approximation with the aim to describe the and the coefficients are calculated as meteoroid flight in the atmosphere. Our goal is to obtain the l � f h , v , 1 n n solution that can be used to better understand that physical process. Subsequently, our intention is to develop a nu- 1 1 merical approach that can be very valuable in predicting the l � f h + p, v + p l , 􏼒 􏼓 2 n n 1 2 2 variation of the parameters along the trajectory segments (18) versus analytical “whole-trajectory smoothing.” 1 1 l � f􏼒h + p, v + p l 􏼓, Equation (13) is the expression used to find the physical 3 n n 2 2 2 parameters. To test our model, we use meteor data pertaining to the meteoroid velocity at different altitudes taken from the l � f h + p, v + pl 􏼁. 4 n n 3 literature. In this regard, the accurate trajectory/velocity meteor Advances in Astronomy 5 Once the coefficients (equation (16)) are calculated, we ξ � v (21) 􏼐 − v ( K, σ)􏼑 , d f compute the solution for the point y using the following n+1 formula: where v is the velocity inferred from the measured data, and v is the computed velocity. (19) y � y + p l + 2l + 2l + l 􏼁. n+1 n 1 2 3 4 Furthermore, we introduce the increment factors for K′ and σ, which are defined as ΔK and Δσ. Following the same For our case (equation (18)), this is procedure as earlier, we compute the error factor for 2 2 (20) ξ � v − v ( K + ΔK, σ) ξ � v − v ( K − ΔK, σ) , v � v + p l + 2l + 2l + l 􏼁. 􏼐 􏼑 􏼐 􏼑 n+1 n 1 2 3 4 d f d f 2 2 ξ � 􏼐v − v ( K, σ + Δσ)􏼑 ξ � 􏼐v − v ( K, σ − Δσ)􏼑 , d f d f )e result obtained is the solution for the point (h , n+1 2 2 v ), which becomes the initial condition to find the nu- n+1 ξ � 􏼐v − v ( K + ΔK, σ + Δσ)􏼑 ξ � 􏼐v − v (K − ΔK, σ − Δσ)􏼑 , d f d f merical approximation for the next point. )e procedure is 2 2 ξ � v − v ( K + ΔK, σ − Δσ) ξ � v − v ( K − ΔK, σ + Δσ) . 􏼐 􏼑 􏼐 􏼑 repeated until the desired value is reached. d f d f (22) In principle, we created a 2D matrix of errors. )e error 3.1.2. Validating the Code Using the Jacchia Catalog. parameter can be shown in a table for better visualization of Once the procedure is defined, we need to validate the the algorithm (Figure 1). Once all the error values are code by comparing the results with the previously computed, the search for the minimum value is performed, published data. We use a catalog of very precise pho- and the result is considered the new centered value for the tographic trajectories of meteors [58], hereafter referred next computation iteration. to as the JVB catalog, obtained using high spatial reso- We repeat the same procedure for the next centered value, lution Super-Schmidt cameras. Jacchia’ [58] study syn- until we reach the point where the minimum will be centered in thesizes the physical inferred parameters for 413 meteors the middle of the matrix. Consequently, the minimum error between − 5 and +2, 5 stellar magnitude ranges obtained value will correspond to the sought values of K′ and σ. If very during a multistation meteor network operated in the small increments of K′ and σ are used, the solution will be more fifties and sixties in New Mexico, USA. )e data provide accurate (at thecost of increasedcomputing time).If wesetlarge the meteor velocity and magnitude as a function of the increments of K′ and σ, the solution will be reached faster, but at altitude, the derived preatmospheric velocity, the de- the expense of the resolution of the solution. In order to deal celeration, and some additional information for the with this conundrum, the code is optimized such that it is set to observed meteors. All JVB catalog events were named work with large increments of K′ and σ at the beginning. Once using a number. In this project, we have used the same the ‘first approximation’ solution is found, the code switches to Jacchia and Whipple [58] numbering, but included a J at smaller increments until the optimal resolution is reached. the beginning for the computational reasons. For ex- Figure 2 shows an example of the autofitting procedure. ample, we later discuss meteor J8945 (listed in JVB as We have chosen the J8945 meteor for the comparison with 8945). published data [32]. Other cases were studied and compared Equation (13) also requires the knowledge of air density. in Dergham [29]. By comparing the velocity vs. altitude We adopted a general model widely used in meteor studies graphs, it is evident that the fitting curves are very similar to [45], the United States (US) standard atmosphere [62]. )e the observed data. US standard atmosphere was originally developed in 1958 by the US Committee on Extension to the Standard Atmo- sphere and improved in 1976. It is a series of tables which 3.1.4. Abrupt Disruption Cases. We have presented a model approximate the values for atmospheric temperature, den- capable of obtaining some parameters for meteoroids. sity, pressure, and other properties over a wide range of altitudes. However, as previously mentioned, not all meteoroids can be studied using this particular model because if they undergo fragmentation, the results might be skewed. Bellot Rubio 3.1.3. Autofitting Procedure. We have defined a way to et al. [32] also mentioned that there are a significant number transform the differential equation into an expression that of cases in the JVB catalog that cannot be fitted, likely as a can be iteratively computed. )e aim is to find a result for consequence of undergoing abrupt disruptions. Figure 3 which K′ and σ produce the closest curve fitting to the shows the results of the velocity curve using our model observational data points. In order to find the best for the case J4141. )e results are also compared with that matching values, we introduce the following autofitting obtained by Bellot Rubio et al. [32]. procedure. We start by choosing two random values of K′ Despite the difficulties to obtain well-fitting solutions for and σ set as initial approximation. )e meteor data to be some events, it is remarkable that our model is able to investigated include the velocity of a meteor at specified identify and produce solutions for the events undergoing points of altitude; we compare this velocity with the ve- quasicontinuous fragmentation. A possible way to study locity simulated by our code at the same altitude points. )e meteoroids with abrupt fragmentation is to focus on dif- error factor is then calculated as follows: ferent segments of the trajectory that do not include a 6 Advances in Astronomy 2 2 2 ξ = (v – v (K + ∆K, σ – ∆σ)) ξ = (v – v (K + ∆K, σ)) ξ = (v – v (K + ∆K, σ + ∆σ)) +K d f d f d f 2 2 ξ = (v – v (K, σ)) ξ = (v – v (K, σ + ∆σ)) ξ = (v – v (K, σ – ∆σ)) d f d f d f 2 2 2 ξ = (v – v (K – ∆K, σ – ∆σ)) ξ = (v – v (K – ∆K, σ)) ξ = (v – v (K – ∆K, σ + ∆σ)) d f d f d f +σ Figure 1: Basic scheme of the error matrix. 3.2. ?e Ablation Coefficient. Equation (13) has several 1.00 unknowns. )e ability of a meteoroid to produce light can be linked to its mass loss or the ablation coefficient σ [34, 42]. In principle, the ablation coefficient reflects how rapidly the 0.99 meteoroid loses its mass as it interacts with the atmosphere. Low values of the ablation coefficient indicate that the object is losing a lesser amount of mass compared to that having a 0.98 higher value of the ablation coefficient. )e ablation coef- 2 − 2 ficient is normally represented in units of s km and can also be expressed through the dimensionless mass loss pa- 0.97 rameter [44, 63] used through the range of meteor physics studies. )e value of the ablation coefficient depends of many factors, such as chemical composition, grain size, 0.96 density, porosity, and body shape, among others. In general, the values of the ablation coefficient range between 0.01 and 2 − 2 86 88 90 92 94 96 98 100 102 0.3s km [25]. In order to exemplify this, we applied Altitude (km) different ablation coefficients to a 1g meteoroid with the preatmospheric velocity of 25km/s and which starts to This study Bellot Rubio et al. (2002) decelerate at the altitude of 100km. )e results are shown in Figure 4. Figure 2: Normalized velocity vs. altitude fit for the meteor J8945. In general, the larger the ablation coefficient is, the faster )e result is a good fit to the previously published Figure 1 of Bellot the body decelerates due to a more rapid mass loss. Con- Rubio et al. [32]. sequently, the mass of the body decreases due to ablation; this is described by using the ablation coefficient, and the 17.0 drag force imposed by the atmosphere has a larger effect. Table 1 shows the comparison of our results to several events 16.8 described in the JVB catalog. 16.6 3.3. Deceleration, Normalized Instantaneous Mass, and Mass Loss Rate. In this section, the effect of deceleration is ex- 16.4 amined in more detail. Given that we have the velocity as a function of altitude along the trajectory, we can study de- 16.2 celeration from the behavior of the velocity curve. )is is particularly useful as many meteor processing algorithms and detection methods provide velocity values sequentially 16.0 [15, 64]. At a certain point h , the code computes an in- crement of velocity over the increment of distance at the 15.8 77 78 79 80 81 82 83 84 points immediately before and after. )is can be expressed as Altitude (km) dv Δv v − v h2 h1 Figure 3: Comparison of results for the meteoroid J4141. )e h􏼁 � � . (23) dh Δh h − h model cannot produce a well-fitting solution since this meteoroid 2 1 likely suffered an abrupt disruption. )is figure can be compared Considering all the points with known velocity and with that shown in Figure 1 of Bellot Rubio et al. [32]. altitude along the trajectory as input data, equation (23) can be applied directly. Figure 5(a) shows the velocity curve as a disruption episode. We will apply that approach to the lower function of altitude for the J8945 meteoroid. part of the trajectory of the Chelyabinsk bolide to demon- )e normalized instantaneous mass (m/m ) is the next strate that the obtained parameters are consistent. quantity to be studied. )e expression for the normalized v/v Velocity (km/s) 0 Advances in Astronomy 7 26 predicted that the NEA-2012 DA14, discovered a year prior by the Observatory Astronomic ` de Mallorca, would ap- proach the Earth within a close minimum distance of only 27700km. However, while all the attention was focused on anticipating that encounter, another NEA unexpectedly entered the atmosphere over central Asia on February 15, 2013 at 03:20:33 UTC. )e bolide disintegrated in the proximity of the city of Chelyabinsk [1]. )e Chelyabinsk bolide reached the brightness magnitude of − 28, which is brighter than the moon (Figure 6). As the days passed and the orbits were calculated, sci- entists discarded a possible association between the two NEAs, as they presented very different heliocentric orbits. )anks to video cameras (dash cams) found in the majority 16 of Russian motor vehicles and surveillance cameras placed 70 75 80 85 90 95 100 on buildings, the initial trajectory of the bolide was Altitude (km) reconstructed and the orbit was determined [2]. σ = 0.01 After the superbolide sightings, many people uploaded σ = 0.03 various videos to Internet. Since the geographical location of σ = 0.05 the recorded videos was known, we reconstructed the bolide Figure 4: Comparison of meteoroids having the same initial trajectory, obtaining the values of velocity as a function of conditions (m �1 g, v �25km/s) but different ablation altitude. As shown in Table 2, the data were obtained from the coefficients. analysis of our video compilation, the bolide velocity along the terminal part of its trajectory just after the massive fragmen- tation event taking place at the altitude of 26km. )e step size Table 1: Comparison of the ablation coefficients for various meteor was determined by the video frame rate, corresponding to cases from the JVB catalog. differences in height of 200–150m. Table 2 shows these data. 2 − 2 2 − 2 Figure 7(a) depicts the dynamical data after the dis- Meteor ID Jacchia (s ·km ) )is study (s ·km ) ruption had occurred at an altitude of 26km, and the fit is J6882 0.0812 0.075 obtained by our model. )e graph shows a quite uniform J6959 0.0331 0.0382 behavior of the Chelyabinsk superbolide after the main J7216 0.0501 0.079 fragmentation event. J8945 0.0354 0.036 J9030 0.0549 0.0542 By studying the dynamic curve, the ablation coefficient J7161 0.0354 0.0381 can be obtained. )e derived value is 2 − 2 σ � 0.034 s · km . (26) instantaneous mass can be derived by rearranging equation (9): It is quite remarkable that this value, derived for the m 2 2 lower trajectory, provides a similar ablation coefficient as − (1/2)σ v − v ( ) � e . (24) that for the fireballs at much higher altitudes, even though Chelyabinsk was the deepest penetrating bolide ever Equation (24) expresses the normalized instantaneous documented, still emitting light even as it reached the mass as a function of velocity, whereas the values of velocity troposphere. Notably, the atmospheric density in these lower are the functions of altitude. Figure 5(b) shows the behavior regions is about four orders of magnitude higher. of normalized mass expressed as a function of altitude for the )e normalized mass evolution for that lower part of J8945 event. the trajectory is shown in Figure 7(b), and the mass loss rate We define the normalized mass loss rate as the derivative evolution is shown in Figure 7(c). In particular, it is cer- of the relative mass over the altitude. )is value is computed tainly encouraging that the model predicts quite well the as follows: ablation behavior of the Chelyabinsk bolide in the lower part of its atmospheric trajectory. )e light curve of the m/m Δm/m m/m − m/m o o oh2 oh1 􏼢 􏼣 � � . (25) Chelyabinsk event was normalized using the US govern- h Δh h − h 2 1 ment sensor data at a peak of brightness value of 13 − 1 2.7 we10 Ws·r , corresponding to an absolute astro- Figure 5(c) shows the direct application of this approach nomical magnitude of − 28 [1]. According to the NASA JPL for the J8945 event. Chelyabinsk final report [65], the maximum brightness was achieved at an altitude of 23.3 km. )is is consistent with our results as the maximum inferred value of mass loss rate 3.4. Application to the Chelyabinsk Superbolide. We apply the from our dynamic data that occur at an altitude of∼23.5 km Runge–Kutta code developed in this work to the famous (Figure 7(c)). Chelyabinsk superbolide. On February 15, 2013, it was Velocity (km/s) 8 Advances in Astronomy 1.00 0.0 0.75 –0.1 –0.2 0.50 –0.3 0.25 88 90 92 94 96 98 100 102 88 90 92 94 96 98 100 102 Altitude (km) Altitude (km) (a) (b) 0.08 0.06 0.04 0.02 88 90 92 94 96 98 100 102 Altitude (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. (a) (b) Figure 6: A casual photograph of the Chelyabinsk bolide taken by Marat Ahmetvaleev. (a) )e abrupt fragmentation at a height of 23km is clearly distinguishable by the main flare. (b) Pictures of the dust trail by the same photographer reveal multiple paths produced by the fragments. Both pictures are courtesy of the author. It is well known that the maximum brightness is impactor that produced an airburst over Siberia in 1908 achieved shortly after the meteoroid catastrophically breaks when it entered the atmosphere at a velocity of ∼30km/s apart due to the fragmented/pulverized material being ex- [66]. )e Chelyabinsk superbolide had the preatmospheric posed to the heat generated by the resulting shock wave. velocity of 19km/s, and its trajectory was consistent with a An interesting conclusion that can be obtained directly grazing geometry (high zenith angle). from these results is the importance of the atmosphere. As To exemplify this, we have plotted the entry of the mentioned before, the faster the meteoroid, the more rapid Chelyabinsk superbolide for different initial velocities the ablation process. )us, the atmosphere could effectively (Figure 8). Of course, disrupting NEOs can produce frag- shield the Earth from very fast impacts since such objects are ments tens of meters in size. If these fragments encounter preferentially and more quickly ablated. However, the less our planet under certain geometric conditions, they might favorable cases are very large objects (especially if their become a significant source of damage on the ground and preatmospheric velocity is low), such as the Tunguska casualties. )us, identifying the existence of asteroidal Acceleration (km/s ) Relative mass loss rate Mass (m/m ) 0 Advances in Astronomy 9 Table 2: Dynamic data of the Chelyabinsk superbolide. Height (km) V (km/s) Height (km) v (km/s) 18.98 14.04 15.66 9.73 18.78 13.86 15.53 9.46 18.58 13.68 15.39 9.20 18.38 13.49 15.26 8.94 18.18 13.29 15.13 8.68 17.99 13.09 15.01 8.42 17.80 12.88 14.89 8.17 17.62 12.66 14.77 7.92 17.44 12.44 14.66 7.67 17.26 12.22 14.55 7.43 17.08 11.99 14.45 7.19 16.91 11.75 14.34 6.96 16.74 11.51 14.24 6.74 16.58 11.27 14.15 6.52 16.42 11.02 14.06 6.31 16.26 10.76 13.97 6.11 16.10 10.51 13.88 5.92 15.95 10.25 13.80 5.73 18.0 0.96 13.5 0.64 9.0 0.32 4.5 0.00 12 14 16 18 20 22 24 26 12 14 16 18 20 22 24 26 Altitude (km) Altitude (km) (a) (b) 0.13 0.09 0.04 0.00 12 14 16 18 20 22 24 26 Altitude (km) (c) Figure 7: Velocity evolution (a), mass evolution (b), and mass loss rate (c) as a function of altitude for the Chelyabinsk superbolide. complexes in the near-Earth environment is crucial for moving at higher velocity could have undergone a much better assessment of impact hazard. faster ablation process, culminating in an explosive airburst, If the Chelyabinsk superbolide entered at a higher ve- similar to that occurring over Tunguska. Such conclusions locity, it would have slowed down faster due to a more rapid imply that the efficiency of the Earth’s atmosphere to shield mass loss. According to our model, the maximum brightness us from dangerous tens of meters in size asteroids that of a meteoroid occurs when the mass loss reaches the peak. strongly depend on the relative velocity of the encounter Consequently, the model predicts that a Chelyabinsk-like with our planet. asteroid moving at a lower velocity could have more de- It is important to remark that new improvements in structive potential on the Earth’s surface (Figure 8). On the detection of fireballs from space could provide an additional other hand, a projectile of the same rocky composition and progress in studying the luminous efficiency of bolides [34]. Acceleration (km/s ) Relative mass loss rate Mass (m/m ) 0 10 Advances in Astronomy 30 probably for low-velocity projectiles under favorable geo- metric circumstances, crater excavation is still possible. A good example of this was the so-called Carancas event: an impact crater excavated in the Peru’s ´ Altiplane by a one meter-sized chondritic meteorite [71]. Being a quite unusual impact event, it is also likely that a significant number of these events are rarely studied because they happen in re- mote locations and remain unnoticed. It is obvious that 15 Chelyabinsk and other well-recorded events (e.g., by fireball networks) provide an opportunity to understand in the required detail the behavior of meter-sized meteoroids and their capacity to directly cause human injuries and even casualties. 4. Conclusions 12 14 16 18 20 22 24 Altitude (km) We have developed a numerical model employing the Runge–Kutta method to predict the dynamic behavior of v = 17.1 km/s meteoroids penetrating the Earth’s atmosphere based on the v = 23.0 km/s meteor physics equations [39]. To test the numerical model, v = 27.0 km/s we successfully applied it to several meteor events described Figure 8: Simulation of the Chelyabinsk superbolide with different in the scientific literature. Having validated the numerical initial velocities (v �17.1, 23, and 27km/s). model, we studied the deceleration behavior of the Che- lyabinsk superbolide in the lower part of its atmospheric trajectory, just at the region following the main fragmen- In fact, several detections of the Chelyabinsk bolide from tation event where such an approach is applicable. )is space have been reported [67, 68]. Future studies of common scheme represents a novel way to study complicated me- events detected from both the ground and space could constrain the role of the observing geometry in signal loss teoric events by examining only the portion of the trajectory during which the object does not undergo abrupt and possible biases in the subsequent determination of velocity, radiated energy, and determination of the orbital fragmentation. Our study of the deceleration profile of the Chelyabinsk elements. superbolide has allowed us to reach the following conclusions: 3.5. Implications for Impact Hazard. Considering the de- (a) Our numerical model, successfully applied to the structive nature of large extraterrestrial objects (e.g., [24]), it lower part of the fireball trajectory, predicts well the is crucial to identify the existence of asteroidal complexes in main observed characteristics of the Chelyabinsk the near-Earth environment. Indeed, the Chelyabinsk scale superbolide. )is is quite remarkable as the lower or larger extraterrestrial bodies (10s of meters in size) are of trajectory studied here has a similar ablation be- special interest to the planetary defense community because haviour compared to fireballs at higher altitudes. It depending on their orbital geometry and impact angle, these should be noted that the Chelyabinsk event is the objects could pose significant hazard for humans and in- deepest penetrating bolide ever documented, bor- frastructure on the ground. )us, studying the near-Earth derline emitting light as it reached the troposphere. environment along with the well-documented events such as )us, our approach offers a promising venue in the Chelyabinsk superbolide can shed more light on the studying complex meteoric events in a streamlined dynamic processes, as well as fundamental properties of and simplified manner. these objects. (b) )e best fit to the deceleration pattern provides an From the study of meteorite falls and the relative absence 2 − 2 average ablation coefficient σ �0.034s ·km , which of small impact craters [19], we know that meter-sized as- is in the range of those derived in the scientific teroids are efficiently fragmented when they penetrate into literature the stratosphere at hypervelocity. )e loading pressure in front of the body produces the rock fracture when it (c) )e ablation coefficient is considered constant overcomes its tensile strength. As a natural consequence, within each studied trajectory interval. )is sim- meteorite falls often deliver tens to hundreds of stones, just plistic approach is probably one of the reasons why after this type of break-ups [69, 70]. We have previously this model is not applicable to the entire trajectory of described such behavior when discussing the evolution of the meteoroids suffering significant fragmentation Chelyabinsk. 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Research at the Ural Federal University was Cambridge, UK, 2004. supported by the Russian Foundation for Basic Research (18- [14] A. Terentjeva, “Fireball streams,” in Asteroids, Comets Meteors 08-00074 and 19-05-00028). During the peer-review of this III, C.I. Lagerkvist, H. Rickman and B. A. Lindblad, Eds., manuscript, the authors lost a mastermind, dear friend, and p. 579, Uppsala Astronomical Observatory, Uppsala, Sweden, the co-author Esko Lyytinen. )e authors dedicate this common effort in memorial to his enormous science figure [15] I. Halliday, A. A. Griffin, and A. T. 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Jenniskens, Meteor Showers and ?eir Parent Comets, Cambridge University Press, Cambridge, UK, 2006. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Astronomy Hindawi Publishing Corporation

A Numerical Approach to Study Ablation of Large Bolides: Application to Chelyabinsk

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Hindawi Advances in Astronomy Volume 2021, Article ID 8852772, 13 pages https://doi.org/10.1155/2021/8852772 Research Article A Numerical Approach to Study Ablation of Large Bolides: Application to Chelyabinsk 1,2 1,2 3,4,5,6 5† Josep M. Trigo-Rodrı´guez , Joan Dergham, Maria Gritsevich, Esko Lyytinen, 7,8 9 Elizabeth A. Silber and Iwan P. Williams Institut de Ci` encies de l’Espai–CSIC, Campus UAB, Facultat de Ci` encies, Torre C5-parell-2a, 08193 Bellaterra, Barcelona, Catalonia, Spain Institut d’Estudis Espacials de Catalunya (IEEC), Edif. Nexus, c/Gran Capita, 2-4, 08034 Barcelona, Catalonia, Spain Finnish Geospatial Research Institute (FGI), Geodeetinrinne 2, FI-02430 Masala, Finland Department of Physics, University of Helsinki, Gustaf Ha ¨llstro ¨min katu 2a, P.O. Box 64, FI-00014 Helsinki, Finland Finnish Fireball Network, Helsinki, Finland Institute of Physics and Technology, Ural Federal University, Mira str. 19., 620002 Ekaterinburg, Russia Department of Earth Sciences, Western University, London, ON N6A 5B7, Canada ?e Institute for Earth and Space Exploration, Western University, London, ON N6A 3K7, Canada Astronomy Unit, Queen Mary, University of London, Mile End Rd., London E1 4NS, UK Deceased Correspondence should be addressed to Josep M. Trigo-Rodr´ıguez; trigo@ice.csic.es Received 31 August 2020; Revised 13 January 2021; Accepted 2 March 2021; Published 27 March 2021 Academic Editor: Kovacs Tamas Copyright © 2021 Josep M. Trigo-Rodr´ıguez et al. )is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, we investigate the ablation properties of bolides capable of producing meteorites. )e casual dashcam recordings from many locations of the Chelyabinsk superbolide associated with the atmospheric entry of an 18m in diameter near-Earth object (NEO) have provided an excellent opportunity to reconstruct its atmospheric trajectory, deceleration, and heliocentric orbit. In this study, we focus on the study of the ablation properties of the Chelyabinsk bolide on the basis of its deceleration and fragmentation. We explore whether meteoroids exhibitingabruptfragmentation can bestudiedby analyzing segmentsofthe trajectorythatdonotinclude a disruptionepisode. We apply that approach to the lower part of the trajectory of the Chelyabinsk bolide to demonstrate that the obtained parameters are consistent. To do that, we implemented a numerical (Runge–Kutta) method appropriate for deriving the ablation properties of bolides based on observations. )e method was successfully tested with the cases previously published in the literature. Our model yields fits that agree with observations reasonably well. It also produces a good fit to the main observed characteristics of Chelyabinsk superbolide and 2 − 2 provides its averaged ablation coefficient σ �0.034s km . Our study also explores the main implications for impact hazard, concluding that tens of meters in diameter NEOs encountering the Earth in grazing trajectories and exhibiting low geocentric velocities are penetrating deeper into the atmosphere than previously thought and, as such, are capable of producing meteorites and even damage on the ground. year prior, an unexpected impact with an Apollo asteroid 1. Introduction ensued [1]. At 03:20 UTC, a superbolide, also known as the On February 15, 2013, our view about impact hazard was Chelyabinsk superbolide, flew over the Russian territory and seriously challenged. While there was a sense of accom- Kazakhstan. )e possible link between the superbolide and plishment for being able to forecast the close approach of the 2012 DA NEA was discarded by the European Space 2012 DA near-Earth asteroid (NEA) within a distance of Agency (ESA) and the Jet Propulsion Laboratory–National 27700km, eventhough this NEO was discovered only one Aeronautics and Space Administration (JPL-NASA) from 2 Advances in Astronomy (SPMN) pioneered the application of high-sensitivity the reconstruction of the incoming fireball trajectory. )e Chelyabinsk superbolide entered the atmosphere at∼19km/ cameras for detecting fireballs, and it currently maintains an online list of bright events detected over Spain, Portugal, s, and according to the US sensor data (CNEOS fireball list: https://cneos.jpl.nasa.gov/fireballs/), it reached the maxi- Southern France, and Morocco since 1999 [27, 28]. For mum brightness at an altitude of 23.3km with a velocity of example, casual video recordings plus several still photo- 18.6km/s [1, 2], also providing us with valuable samples in graphs of the superbolide in flight allowed us to reconstruct the form of meteorites. the heliocentric orbit of Villalbeto de la Peña meteorite in the )e existence of meteoroid streams capable of producing framework of the SPMN [28]. )e possibility of studying meteorite-dropping bolides is a hot topic in planetary sci- superbolides such as Chelyabinsk is a very attractive mile- stone to be considered. )e software used in this study was ence. Such streams were first proposed by Halliday [3, 4]. )eir existence has important implications because they can developed as part of a master thesis [29] and subsequently tested and validated using several cases discussed by [29], as naturally deliver to the Earth different types of rock-forming materials from potentially hazardous asteroids (PHAs). It is well as events from the 25 video and all-sky CCD stations set up over the Iberian Peninsula by the SPMN. In this context, believed that NEOs in the Earth’s vicinity are undergoing dynamical and collisional evolution on relatively short we have been engaged in studying the dynamic behavior of timescales. We previously identified several NEO complexes meteoroids decelerating in the Earth’s atmosphere [30, 31]. that are producing meteorite-dropping bolides, and we In this study, we study the Chelyabinsk bolide by fol- hypothesize that they could have been produced during close lowing a Runge–Kutta method of meteor investigation approaches to terrestrial planets [5, 6]. Such formation similar to that developed by Bellot Rubio et al. [32]. We aim scenario for this kind of asteroidal complexes is now to test if that specific method is also valid for another mass range, particularly for small asteroids and large meter-sized reinforced with the recent discovery of a complex of NEOs likely associated with the NEA progenitor of the Chelyabinsk meteoroids. We first describe our numerical model and test it against the known meteor events. We compare our code bolide [7]. )e shattered pieces resulting from the disruption of NEOs visiting the inner solar system can spread along the validation results to the results obtained by Bellot Rubio et al. [32] for the same dataset. We then apply our numerical entire parent body orbit on a time-scale of centuries [4, 8, 9]. )is scenario is also consistent with the modern view of model to the Chelyabinsk superbolide in order to study its NEOs being resurfaced as consequence of close approaches dynamical behavior. For the sake of simplicity, our model [10]. Additionally, the meteorites recovered from the considers a constant ablation coefficient and the shape Chelyabinsk have a brecciated nature [11, 12] that is rem- factor, even though these parameters could vary in different iniscent of the complex collisional history and a probable ablation stages [33–35]. rubble pile structure of the asteroid progenitor of the )is study is structured as follows: the data reduction and the theoretical approach pertaining to the Chelyabinsk complex [13]. )e existence of these asteroidal complexes in the near-Earth region has important implications as they bolide are described in the next section. In Section 3, the main implications of this work in the context of fireballs, could be the source of low spatial density meteoroid streams populated by large meteoroids. Such complexes could be the meteorites, and NEO research are discussed. We use the source of the poorly known fireball-producing radiants model to determine the fireball flight parameters, and by [14, 15]. )is could have important implications for the studying the deceleration, we also obtain the ablation co- fraction of sporadic meteoroids producing bright fireballs efficient. Finally, the conclusions of this work are presented and in the physical mechanisms envisioned in the past in Section 4. [16–18]. )e Chelyabinsk event is also of interest because of its 2. Data Reduction, Theoretical Approach, magnitude and energy and due to its relevance to be con- and Observations sidered as a representative example of the most frequent outcome of the impact hazard associated with small aster- )e Chelyabinsk superbolide was an unexpected daylight oids in human timescales. Chelyabinsk also exemplifies the superbolide as many other unpredicted meteorite-drop- importance that fragmentation has for small asteroids, ping bolides in history. Fortunately, numerous casual which can even excavate a crater on the Earth’s surface, video recordings of the bolide trajectory from the ground although rarely [19–23]. Fragmentation is important as it were obtained, given the nowadays common dashcams provides a mechanism in which a significant part of the available in private motor vehicles in Russia. According to kinetic energy associated with small asteroids is released. It the video recordings available, it is possible to study the was certainly a very relevant process for the Tunguska event atmospheric trajectory and deceleration carefully, [24, 25], and in the better-known case of Chelyabinsk, most allowing the reconstruction of the heliocentric orbit in of the kinetic energy was transferred to the internal energy of record time [2, 26]. the air, which is radiated as light [26]. One way to study meteoroids as they enter the Earth’s atmosphere is through video observations of such events. 2.1. Single Body ?eory. )ere are two main approaches in Consequently, we are developing complementary ap- the study of the dynamic properties of meteors during at- proaches to study the dynamical behavior of video-recorded mospheric interaction, the quasicontinuous fragmentation bolides in much detail. )e SPanish Meteor Network (QCF) theory introduced by Novikov et al. [37], which was Advances in Astronomy 3 2.3. ?e Single Body ?eory: ?e Drag and Mass Loss later extended by Babadzhanov [38], and the single body theory described by Bronshsten [39]. )ere have been dis- Equations. )e dynamic behavior of a meteoroid as it in- teracts with the Earth’s atmosphere is described using the crepancies in the applicability conditions of both methods: the single body works with basic differential equations, while drag and mass loss equations. )ese equations, as presented the QCF uses semiempirical formulas studying only the by Bronshten [39], are as follows: luminosity produced by the meteor. )e main difference is dv − 1/3 2 (1) that the single body theory obtains smaller dynamical masses � − Kρ m · v , air dt than the QCF method. )us far, neither approach is prev- alent, and the reason why the theories do not converge could dm 2/3 3 (2) � − σKρ m · v . be attributed to the contribution of other key processes such air dt as the fragmentation and deceleration of meteoroids during ablation or to the poorly constrained values of the bulk where K is the shape-density coefficient, ρ is the air air density and/or the luminous efficiency coefficient [40–42]. density, m is the meteoroid mass, v is the instantaneous We remark that the initial dynamic mass estimate or the velocity, and σ is the ablation coefficient. preatmospheric size can be derived using the methods de- By using equations (1) and (2), the parameters to be scribed in other works [43–48]; hence, we leave it out of the identified are K and σ. )e ablation coefficient defines the scope of the present model. We also note that alternative loss of mass for the bolide as it penetrates the atmosphere; models accounting for ablation have been recently devel- the larger the value is, the more the mass will be ablated for a oped; however, further discussion on that topic is beyond the given velocity. )e value of the ablation coefficient depends scope of this study, and the reader is referred to the following on various factors and is expressed as literature [46, 47, 49–52]. As noted in the introduction, high- strength meteoroids from asteroids or planetary bodies σ � , (3) 2ΓQ exhibit a very different behavior than fragile dust aggregates originating from comets [53–57]. 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 and 2.2. ?e Role of Fragmentation. )e fragmentation of me- density of the meteoroid and is expressed as teoroids was studied in detail by various authors [38, 40]. sA K � , After analyzing different photographic observations, Levin (4) 2/3 [40] distinguished four possible types of fragmentation: (a) the decay of the meteoroid into large nonfragmenting debris, where A is the shape factor, s is the cross-section area, and ρ (b) the progressive disintegration of the original meteoroid is the meteoroid bulk density. into fragments that continue to crumble into smaller We must point out that the observational data obtained fragments, (c) the instantaneous ejection of a large number from the reconstruction of the trajectories of meteors using of small particles, which, when affecting the entire mete- CCD or video cameras is basically the frame to frame speed oroid, is called a catastrophic disruption, and finally, (d) the of the bolide as a function of the height, requiring another quasicontinuous fragmentation which consists of the equation to link the time with the altitude: gradual release of a large number of small particles from the dh surface and their subsequent evaporation due to high � − v · cos(z), (5) dt temperatures associated with the shock thermal wave formed around the body. where z is the zenith angle. In practice, a combination of two or more types of By substituting equation (5) into equations (1) and (2), fragmentation can be observed in a given meteor event. In the following expressions are obtained: fact, it is possible to observe that (a) and (c) fragmentation − 1/3 types described in the preceding paragraph could occur dv Kρ m · v air (6) � , more than once for the same meteor event. )e analysis of dh cos(z) meteors performed by Jacchia [58] using Super-Schmidt cameras demonstrated that the single body theory does not 2/3 2 dm σKρ m · v air (7) � . work for cases which suffer abrupt types of fragmentation dh cos(z) along the trajectory. As a direct consequence, meteoroids exhibiting fragmentation of the first (a), second (b), and Next, by dividing equation (2) by equation (1), we obtain third (c) kind should not be studied using this simplified dm single body theory. When a meteoroid suffers an abrupt (8) � mσv. dv fragmentation, the main body loses mass instantaneously, and therefore, the single body equations cannot be applied Solving this differential equation with the boundary since the condition of continuity of mass is not satisfied. condition of m � m , when v � v , the following is obtained: o o )us, the cases with possible abrupt fragmentation epi- 2 2 − (1/2)σ v − v ( o ) (9) sode(s) will not be examined in this work. m � m · e . o 4 Advances in Astronomy Now, we combine equations (9) and (6) to derive data obtained via high-resolution Super-Schmidt cameras [58] can be utilized for a case study. )ese meteor trajectory data are − (1/3) dv 2 2 1 − (1/2)σ v − v ( ) (10) � Kρ 􏼒m · 􏼒e 􏼓􏼓 · v · . used to optimize the procedure to obtain the values of K′ and σ air o dh cos z that fit the best the actual data points. )e procedure presented here produces many synthetic curves. Subsequently, in order to In order to obtain the value of K and σ, we use equation optimize the computing time, a solution to make the equation (10), as it uses substitution for the dependence on the in- converge to the real data is found. stantaneous mass, and thus, we deal with one equation instead of two. Furthermore, there is the initial mass, which is an important parameter to be studied. Equation (10) directly links 3.1.1. ?e Runge–Kutta Method Implementation. )e Run- the deceleration of the meteoroid as a function of different ge–Kutta method is an iterative technique for the approxi- parameters, in particular the zenith angle. )e last term is of mation and solution of ordinary differential equations. )e particular interest as it modulates the full equation. For the ° method was first developed by Runge [60] and Kutta [61]. vertical entry (z � 0 ), the deceleration is maximized, while for z )e Runge–Kutta approximation provides a solution at a close to 90 , the deceleration is minimized. For that reason, determined point of altitude. Application of the Runge–Kutta large meteoroids at grazing angles are able to follow extremely method requires the initial conditions to be known: long trajectories or even escape again into space, such as the Grand Tetons superbolide that spent nearly two minutes . (14) y � f(t, y) ; y ; t o o traveling over several states of the USA and Canada on August 10, 1972 [11, 59]. In our case, the initial conditions will be the initial However, by using the concepts introduced above, it is velocity and the altitude of the bolide when ablation starts, not possible to obtain the values of initial mass (m ) and K. “synthetically” written as )e parameters that can be found are only the expressions of − 1/3· _ dv m K and σ. Consequently, another equation is required (15) � f(h, v) ; v h 􏼁 ; h o o to obtain K and m separately. )e remaining expression is dh the photometric equation: We then need to choose a step size (p) that should be τ dm comparable with the data resolution to allow a better com- I � 􏼠− 􏼡 · v , (11) parison between the model and the observations. )e step size 2 dt defines how many integration steps need to be implemented where τ is the luminous efficiency. )is equation is assuming beforethefinalsolutionisreached.)esmallerthestepsize,the v �constant, and it is often applied to small meteors. )e more accurate the solution will be, noting that this will also luminous efficiency is obtained empirically; thus, any de- increase the computing time. )e step size corresponds to the viations in that value might produce significant changes in characteristic Δh that according to Bellot Rubio et al. [32] can the results. )is equation will be referred to later as a part of be chosen around a few hundred of meters, 100–300m. the mass determination. Now, we focus only on equation Once the step size is defined, we define the model co- (10). efficients as follows: We define the K′ as l � f t , y , 1 n n − 1/3 ′ (12) K � K · m . 1 1 l � f􏼒t + p, y + p l 􏼓, 2 n n 1 2 2 )en, the equation to work with is (16) 1 1 − 1/3 2 2 l � f􏼒t + p, y + p l 􏼓, dv 1 − 1/2σ v − v 3 n n 2 ( o ) ′ (13) � ρ K e · v · , 2 2 􏼒 􏼓 air dh cos z l � f t + p, y + pl . 4 n n 3 with variables K′ and σ. For our case, the function to be studied is − 1/3 3. Results and Discussion 2 2 1 − 1/2σ v − v ( ) ′ (17) f h , v 􏼁 � K ρ(h)􏼒e 􏼓 · v · , n n cos z 3.1. Numerical Approximation. In this section, we develop a numerical approximation with the aim to describe the and the coefficients are calculated as meteoroid flight in the atmosphere. Our goal is to obtain the l � f h , v , 1 n n solution that can be used to better understand that physical process. Subsequently, our intention is to develop a nu- 1 1 merical approach that can be very valuable in predicting the l � f h + p, v + p l , 􏼒 􏼓 2 n n 1 2 2 variation of the parameters along the trajectory segments (18) versus analytical “whole-trajectory smoothing.” 1 1 l � f􏼒h + p, v + p l 􏼓, Equation (13) is the expression used to find the physical 3 n n 2 2 2 parameters. To test our model, we use meteor data pertaining to the meteoroid velocity at different altitudes taken from the l � f h + p, v + pl 􏼁. 4 n n 3 literature. In this regard, the accurate trajectory/velocity meteor Advances in Astronomy 5 Once the coefficients (equation (16)) are calculated, we ξ � v (21) 􏼐 − v ( K, σ)􏼑 , d f compute the solution for the point y using the following n+1 formula: where v is the velocity inferred from the measured data, and v is the computed velocity. (19) y � y + p l + 2l + 2l + l 􏼁. n+1 n 1 2 3 4 Furthermore, we introduce the increment factors for K′ and σ, which are defined as ΔK and Δσ. Following the same For our case (equation (18)), this is procedure as earlier, we compute the error factor for 2 2 (20) ξ � v − v ( K + ΔK, σ) ξ � v − v ( K − ΔK, σ) , v � v + p l + 2l + 2l + l 􏼁. 􏼐 􏼑 􏼐 􏼑 n+1 n 1 2 3 4 d f d f 2 2 ξ � 􏼐v − v ( K, σ + Δσ)􏼑 ξ � 􏼐v − v ( K, σ − Δσ)􏼑 , d f d f )e result obtained is the solution for the point (h , n+1 2 2 v ), which becomes the initial condition to find the nu- n+1 ξ � 􏼐v − v ( K + ΔK, σ + Δσ)􏼑 ξ � 􏼐v − v (K − ΔK, σ − Δσ)􏼑 , d f d f merical approximation for the next point. )e procedure is 2 2 ξ � v − v ( K + ΔK, σ − Δσ) ξ � v − v ( K − ΔK, σ + Δσ) . 􏼐 􏼑 􏼐 􏼑 repeated until the desired value is reached. d f d f (22) In principle, we created a 2D matrix of errors. )e error 3.1.2. Validating the Code Using the Jacchia Catalog. parameter can be shown in a table for better visualization of Once the procedure is defined, we need to validate the the algorithm (Figure 1). Once all the error values are code by comparing the results with the previously computed, the search for the minimum value is performed, published data. We use a catalog of very precise pho- and the result is considered the new centered value for the tographic trajectories of meteors [58], hereafter referred next computation iteration. to as the JVB catalog, obtained using high spatial reso- We repeat the same procedure for the next centered value, lution Super-Schmidt cameras. Jacchia’ [58] study syn- until we reach the point where the minimum will be centered in thesizes the physical inferred parameters for 413 meteors the middle of the matrix. Consequently, the minimum error between − 5 and +2, 5 stellar magnitude ranges obtained value will correspond to the sought values of K′ and σ. If very during a multistation meteor network operated in the small increments of K′ and σ are used, the solution will be more fifties and sixties in New Mexico, USA. )e data provide accurate (at thecost of increasedcomputing time).If wesetlarge the meteor velocity and magnitude as a function of the increments of K′ and σ, the solution will be reached faster, but at altitude, the derived preatmospheric velocity, the de- the expense of the resolution of the solution. In order to deal celeration, and some additional information for the with this conundrum, the code is optimized such that it is set to observed meteors. All JVB catalog events were named work with large increments of K′ and σ at the beginning. Once using a number. In this project, we have used the same the ‘first approximation’ solution is found, the code switches to Jacchia and Whipple [58] numbering, but included a J at smaller increments until the optimal resolution is reached. the beginning for the computational reasons. For ex- Figure 2 shows an example of the autofitting procedure. ample, we later discuss meteor J8945 (listed in JVB as We have chosen the J8945 meteor for the comparison with 8945). published data [32]. Other cases were studied and compared Equation (13) also requires the knowledge of air density. in Dergham [29]. By comparing the velocity vs. altitude We adopted a general model widely used in meteor studies graphs, it is evident that the fitting curves are very similar to [45], the United States (US) standard atmosphere [62]. )e the observed data. US standard atmosphere was originally developed in 1958 by the US Committee on Extension to the Standard Atmo- sphere and improved in 1976. It is a series of tables which 3.1.4. Abrupt Disruption Cases. We have presented a model approximate the values for atmospheric temperature, den- capable of obtaining some parameters for meteoroids. sity, pressure, and other properties over a wide range of altitudes. However, as previously mentioned, not all meteoroids can be studied using this particular model because if they undergo fragmentation, the results might be skewed. Bellot Rubio 3.1.3. Autofitting Procedure. We have defined a way to et al. [32] also mentioned that there are a significant number transform the differential equation into an expression that of cases in the JVB catalog that cannot be fitted, likely as a can be iteratively computed. )e aim is to find a result for consequence of undergoing abrupt disruptions. Figure 3 which K′ and σ produce the closest curve fitting to the shows the results of the velocity curve using our model observational data points. In order to find the best for the case J4141. )e results are also compared with that matching values, we introduce the following autofitting obtained by Bellot Rubio et al. [32]. procedure. We start by choosing two random values of K′ Despite the difficulties to obtain well-fitting solutions for and σ set as initial approximation. )e meteor data to be some events, it is remarkable that our model is able to investigated include the velocity of a meteor at specified identify and produce solutions for the events undergoing points of altitude; we compare this velocity with the ve- quasicontinuous fragmentation. A possible way to study locity simulated by our code at the same altitude points. )e meteoroids with abrupt fragmentation is to focus on dif- error factor is then calculated as follows: ferent segments of the trajectory that do not include a 6 Advances in Astronomy 2 2 2 ξ = (v – v (K + ∆K, σ – ∆σ)) ξ = (v – v (K + ∆K, σ)) ξ = (v – v (K + ∆K, σ + ∆σ)) +K d f d f d f 2 2 ξ = (v – v (K, σ)) ξ = (v – v (K, σ + ∆σ)) ξ = (v – v (K, σ – ∆σ)) d f d f d f 2 2 2 ξ = (v – v (K – ∆K, σ – ∆σ)) ξ = (v – v (K – ∆K, σ)) ξ = (v – v (K – ∆K, σ + ∆σ)) d f d f d f +σ Figure 1: Basic scheme of the error matrix. 3.2. ?e Ablation Coefficient. Equation (13) has several 1.00 unknowns. )e ability of a meteoroid to produce light can be linked to its mass loss or the ablation coefficient σ [34, 42]. In principle, the ablation coefficient reflects how rapidly the 0.99 meteoroid loses its mass as it interacts with the atmosphere. Low values of the ablation coefficient indicate that the object is losing a lesser amount of mass compared to that having a 0.98 higher value of the ablation coefficient. )e ablation coef- 2 − 2 ficient is normally represented in units of s km and can also be expressed through the dimensionless mass loss pa- 0.97 rameter [44, 63] used through the range of meteor physics studies. )e value of the ablation coefficient depends of many factors, such as chemical composition, grain size, 0.96 density, porosity, and body shape, among others. In general, the values of the ablation coefficient range between 0.01 and 2 − 2 86 88 90 92 94 96 98 100 102 0.3s km [25]. In order to exemplify this, we applied Altitude (km) different ablation coefficients to a 1g meteoroid with the preatmospheric velocity of 25km/s and which starts to This study Bellot Rubio et al. (2002) decelerate at the altitude of 100km. )e results are shown in Figure 4. Figure 2: Normalized velocity vs. altitude fit for the meteor J8945. In general, the larger the ablation coefficient is, the faster )e result is a good fit to the previously published Figure 1 of Bellot the body decelerates due to a more rapid mass loss. Con- Rubio et al. [32]. sequently, the mass of the body decreases due to ablation; this is described by using the ablation coefficient, and the 17.0 drag force imposed by the atmosphere has a larger effect. Table 1 shows the comparison of our results to several events 16.8 described in the JVB catalog. 16.6 3.3. Deceleration, Normalized Instantaneous Mass, and Mass Loss Rate. In this section, the effect of deceleration is ex- 16.4 amined in more detail. Given that we have the velocity as a function of altitude along the trajectory, we can study de- 16.2 celeration from the behavior of the velocity curve. )is is particularly useful as many meteor processing algorithms and detection methods provide velocity values sequentially 16.0 [15, 64]. At a certain point h , the code computes an in- crement of velocity over the increment of distance at the 15.8 77 78 79 80 81 82 83 84 points immediately before and after. )is can be expressed as Altitude (km) dv Δv v − v h2 h1 Figure 3: Comparison of results for the meteoroid J4141. )e h􏼁 � � . (23) dh Δh h − h model cannot produce a well-fitting solution since this meteoroid 2 1 likely suffered an abrupt disruption. )is figure can be compared Considering all the points with known velocity and with that shown in Figure 1 of Bellot Rubio et al. [32]. altitude along the trajectory as input data, equation (23) can be applied directly. Figure 5(a) shows the velocity curve as a disruption episode. We will apply that approach to the lower function of altitude for the J8945 meteoroid. part of the trajectory of the Chelyabinsk bolide to demon- )e normalized instantaneous mass (m/m ) is the next strate that the obtained parameters are consistent. quantity to be studied. )e expression for the normalized v/v Velocity (km/s) 0 Advances in Astronomy 7 26 predicted that the NEA-2012 DA14, discovered a year prior by the Observatory Astronomic ` de Mallorca, would ap- proach the Earth within a close minimum distance of only 27700km. However, while all the attention was focused on anticipating that encounter, another NEA unexpectedly entered the atmosphere over central Asia on February 15, 2013 at 03:20:33 UTC. )e bolide disintegrated in the proximity of the city of Chelyabinsk [1]. )e Chelyabinsk bolide reached the brightness magnitude of − 28, which is brighter than the moon (Figure 6). As the days passed and the orbits were calculated, sci- entists discarded a possible association between the two NEAs, as they presented very different heliocentric orbits. )anks to video cameras (dash cams) found in the majority 16 of Russian motor vehicles and surveillance cameras placed 70 75 80 85 90 95 100 on buildings, the initial trajectory of the bolide was Altitude (km) reconstructed and the orbit was determined [2]. σ = 0.01 After the superbolide sightings, many people uploaded σ = 0.03 various videos to Internet. Since the geographical location of σ = 0.05 the recorded videos was known, we reconstructed the bolide Figure 4: Comparison of meteoroids having the same initial trajectory, obtaining the values of velocity as a function of conditions (m �1 g, v �25km/s) but different ablation altitude. As shown in Table 2, the data were obtained from the coefficients. analysis of our video compilation, the bolide velocity along the terminal part of its trajectory just after the massive fragmen- tation event taking place at the altitude of 26km. )e step size Table 1: Comparison of the ablation coefficients for various meteor was determined by the video frame rate, corresponding to cases from the JVB catalog. differences in height of 200–150m. Table 2 shows these data. 2 − 2 2 − 2 Figure 7(a) depicts the dynamical data after the dis- Meteor ID Jacchia (s ·km ) )is study (s ·km ) ruption had occurred at an altitude of 26km, and the fit is J6882 0.0812 0.075 obtained by our model. )e graph shows a quite uniform J6959 0.0331 0.0382 behavior of the Chelyabinsk superbolide after the main J7216 0.0501 0.079 fragmentation event. J8945 0.0354 0.036 J9030 0.0549 0.0542 By studying the dynamic curve, the ablation coefficient J7161 0.0354 0.0381 can be obtained. )e derived value is 2 − 2 σ � 0.034 s · km . (26) instantaneous mass can be derived by rearranging equation (9): It is quite remarkable that this value, derived for the m 2 2 lower trajectory, provides a similar ablation coefficient as − (1/2)σ v − v ( ) � e . (24) that for the fireballs at much higher altitudes, even though Chelyabinsk was the deepest penetrating bolide ever Equation (24) expresses the normalized instantaneous documented, still emitting light even as it reached the mass as a function of velocity, whereas the values of velocity troposphere. Notably, the atmospheric density in these lower are the functions of altitude. Figure 5(b) shows the behavior regions is about four orders of magnitude higher. of normalized mass expressed as a function of altitude for the )e normalized mass evolution for that lower part of J8945 event. the trajectory is shown in Figure 7(b), and the mass loss rate We define the normalized mass loss rate as the derivative evolution is shown in Figure 7(c). In particular, it is cer- of the relative mass over the altitude. )is value is computed tainly encouraging that the model predicts quite well the as follows: ablation behavior of the Chelyabinsk bolide in the lower part of its atmospheric trajectory. )e light curve of the m/m Δm/m m/m − m/m o o oh2 oh1 􏼢 􏼣 � � . (25) Chelyabinsk event was normalized using the US govern- h Δh h − h 2 1 ment sensor data at a peak of brightness value of 13 − 1 2.7 we10 Ws·r , corresponding to an absolute astro- Figure 5(c) shows the direct application of this approach nomical magnitude of − 28 [1]. According to the NASA JPL for the J8945 event. Chelyabinsk final report [65], the maximum brightness was achieved at an altitude of 23.3 km. )is is consistent with our results as the maximum inferred value of mass loss rate 3.4. Application to the Chelyabinsk Superbolide. We apply the from our dynamic data that occur at an altitude of∼23.5 km Runge–Kutta code developed in this work to the famous (Figure 7(c)). Chelyabinsk superbolide. On February 15, 2013, it was Velocity (km/s) 8 Advances in Astronomy 1.00 0.0 0.75 –0.1 –0.2 0.50 –0.3 0.25 88 90 92 94 96 98 100 102 88 90 92 94 96 98 100 102 Altitude (km) Altitude (km) (a) (b) 0.08 0.06 0.04 0.02 88 90 92 94 96 98 100 102 Altitude (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. (a) (b) Figure 6: A casual photograph of the Chelyabinsk bolide taken by Marat Ahmetvaleev. (a) )e abrupt fragmentation at a height of 23km is clearly distinguishable by the main flare. (b) Pictures of the dust trail by the same photographer reveal multiple paths produced by the fragments. Both pictures are courtesy of the author. It is well known that the maximum brightness is impactor that produced an airburst over Siberia in 1908 achieved shortly after the meteoroid catastrophically breaks when it entered the atmosphere at a velocity of ∼30km/s apart due to the fragmented/pulverized material being ex- [66]. )e Chelyabinsk superbolide had the preatmospheric posed to the heat generated by the resulting shock wave. velocity of 19km/s, and its trajectory was consistent with a An interesting conclusion that can be obtained directly grazing geometry (high zenith angle). from these results is the importance of the atmosphere. As To exemplify this, we have plotted the entry of the mentioned before, the faster the meteoroid, the more rapid Chelyabinsk superbolide for different initial velocities the ablation process. )us, the atmosphere could effectively (Figure 8). Of course, disrupting NEOs can produce frag- shield the Earth from very fast impacts since such objects are ments tens of meters in size. If these fragments encounter preferentially and more quickly ablated. However, the less our planet under certain geometric conditions, they might favorable cases are very large objects (especially if their become a significant source of damage on the ground and preatmospheric velocity is low), such as the Tunguska casualties. )us, identifying the existence of asteroidal Acceleration (km/s ) Relative mass loss rate Mass (m/m ) 0 Advances in Astronomy 9 Table 2: Dynamic data of the Chelyabinsk superbolide. Height (km) V (km/s) Height (km) v (km/s) 18.98 14.04 15.66 9.73 18.78 13.86 15.53 9.46 18.58 13.68 15.39 9.20 18.38 13.49 15.26 8.94 18.18 13.29 15.13 8.68 17.99 13.09 15.01 8.42 17.80 12.88 14.89 8.17 17.62 12.66 14.77 7.92 17.44 12.44 14.66 7.67 17.26 12.22 14.55 7.43 17.08 11.99 14.45 7.19 16.91 11.75 14.34 6.96 16.74 11.51 14.24 6.74 16.58 11.27 14.15 6.52 16.42 11.02 14.06 6.31 16.26 10.76 13.97 6.11 16.10 10.51 13.88 5.92 15.95 10.25 13.80 5.73 18.0 0.96 13.5 0.64 9.0 0.32 4.5 0.00 12 14 16 18 20 22 24 26 12 14 16 18 20 22 24 26 Altitude (km) Altitude (km) (a) (b) 0.13 0.09 0.04 0.00 12 14 16 18 20 22 24 26 Altitude (km) (c) Figure 7: Velocity evolution (a), mass evolution (b), and mass loss rate (c) as a function of altitude for the Chelyabinsk superbolide. complexes in the near-Earth environment is crucial for moving at higher velocity could have undergone a much better assessment of impact hazard. faster ablation process, culminating in an explosive airburst, If the Chelyabinsk superbolide entered at a higher ve- similar to that occurring over Tunguska. Such conclusions locity, it would have slowed down faster due to a more rapid imply that the efficiency of the Earth’s atmosphere to shield mass loss. According to our model, the maximum brightness us from dangerous tens of meters in size asteroids that of a meteoroid occurs when the mass loss reaches the peak. strongly depend on the relative velocity of the encounter Consequently, the model predicts that a Chelyabinsk-like with our planet. asteroid moving at a lower velocity could have more de- It is important to remark that new improvements in structive potential on the Earth’s surface (Figure 8). On the detection of fireballs from space could provide an additional other hand, a projectile of the same rocky composition and progress in studying the luminous efficiency of bolides [34]. Acceleration (km/s ) Relative mass loss rate Mass (m/m ) 0 10 Advances in Astronomy 30 probably for low-velocity projectiles under favorable geo- metric circumstances, crater excavation is still possible. A good example of this was the so-called Carancas event: an impact crater excavated in the Peru’s ´ Altiplane by a one meter-sized chondritic meteorite [71]. Being a quite unusual impact event, it is also likely that a significant number of these events are rarely studied because they happen in re- mote locations and remain unnoticed. It is obvious that 15 Chelyabinsk and other well-recorded events (e.g., by fireball networks) provide an opportunity to understand in the required detail the behavior of meter-sized meteoroids and their capacity to directly cause human injuries and even casualties. 4. Conclusions 12 14 16 18 20 22 24 Altitude (km) We have developed a numerical model employing the Runge–Kutta method to predict the dynamic behavior of v = 17.1 km/s meteoroids penetrating the Earth’s atmosphere based on the v = 23.0 km/s meteor physics equations [39]. To test the numerical model, v = 27.0 km/s we successfully applied it to several meteor events described Figure 8: Simulation of the Chelyabinsk superbolide with different in the scientific literature. Having validated the numerical initial velocities (v �17.1, 23, and 27km/s). model, we studied the deceleration behavior of the Che- lyabinsk superbolide in the lower part of its atmospheric trajectory, just at the region following the main fragmen- In fact, several detections of the Chelyabinsk bolide from tation event where such an approach is applicable. )is space have been reported [67, 68]. Future studies of common scheme represents a novel way to study complicated me- events detected from both the ground and space could constrain the role of the observing geometry in signal loss teoric events by examining only the portion of the trajectory during which the object does not undergo abrupt and possible biases in the subsequent determination of velocity, radiated energy, and determination of the orbital fragmentation. Our study of the deceleration profile of the Chelyabinsk elements. superbolide has allowed us to reach the following conclusions: 3.5. Implications for Impact Hazard. Considering the de- (a) Our numerical model, successfully applied to the structive nature of large extraterrestrial objects (e.g., [24]), it lower part of the fireball trajectory, predicts well the is crucial to identify the existence of asteroidal complexes in main observed characteristics of the Chelyabinsk the near-Earth environment. Indeed, the Chelyabinsk scale superbolide. )is is quite remarkable as the lower or larger extraterrestrial bodies (10s of meters in size) are of trajectory studied here has a similar ablation be- special interest to the planetary defense community because haviour compared to fireballs at higher altitudes. It depending on their orbital geometry and impact angle, these should be noted that the Chelyabinsk event is the objects could pose significant hazard for humans and in- deepest penetrating bolide ever documented, bor- frastructure on the ground. )us, studying the near-Earth derline emitting light as it reached the troposphere. environment along with the well-documented events such as )us, our approach offers a promising venue in the Chelyabinsk superbolide can shed more light on the studying complex meteoric events in a streamlined dynamic processes, as well as fundamental properties of and simplified manner. these objects. (b) )e best fit to the deceleration pattern provides an From the study of meteorite falls and the relative absence 2 − 2 average ablation coefficient σ �0.034s ·km , which of small impact craters [19], we know that meter-sized as- is in the range of those derived in the scientific teroids are efficiently fragmented when they penetrate into literature the stratosphere at hypervelocity. )e loading pressure in front of the body produces the rock fracture when it (c) )e ablation coefficient is considered constant overcomes its tensile strength. As a natural consequence, within each studied trajectory interval. )is sim- meteorite falls often deliver tens to hundreds of stones, just plistic approach is probably one of the reasons why after this type of break-ups [69, 70]. We have previously this model is not applicable to the entire trajectory of described such behavior when discussing the evolution of the meteoroids suffering significant fragmentation Chelyabinsk. 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Decker, “Chelyabinsk—not only another ordinary LL5 Conflicts of Interest chondrite, but a spectacular chondrite breccia,” Meteoritics and Planetary Science, vol. 48, 2013. )e authors declare that they have no conflicts of interest. [12] T. DeMeo, M. Gritsevich, V. I. Grokhovsky et al., “Miner- alogy, reflectance spectra, and physical properties of the Chelyabinsk LL5 chondrite - insight into shock-induced Acknowledgments changes in asteroid regoliths,” Icarus, vol. 228, pp. 78–85, )is research was funded by the research project (PGC2018- [13] E. Asphaug, “Interior structures for asteroids and cometary 097374-B-I00, P.I. J.M.T-R), funded by FEDER/Ministerio nuclei,” in Mitigation of Hazardous Comets and Asteroids, de Ciencia e Innovacion–Agencia ´ Estatal de Investigacion. ´ M. J. S. Belton, T. H. Morgan, N. Samarasinha, and MG acknowledges support from the Academy of Finland D. K. Yeomans, Eds., p. 66, Cambridge University Press, (325806). 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Advances in AstronomyHindawi Publishing Corporation

Published: Mar 27, 2021

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