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In this study, we investigate the ablation properties of bolides capable of producing meteorites. The casual dashcam recordings from many locations of the Chelyabinsk superbolide associated with the atmospheric entry of an 18 m 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 exhibiting abrupt fragmentation can be studied by analyzing segments of the trajectory that do not include a disruption episode. 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. The 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 provides its averaged ablation coefficient ^{2} km^{−2}. 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.

On February 15, 2013, our view about impact hazard was seriously challenged. While there was a sense of accomplishment for being able to forecast the close approach of 2012 DA near-Earth asteroid (NEA) within a distance of 27700 km, eventhough this NEO was discovered only one year prior, an unexpected impact with an Apollo asteroid ensued [

The existence of meteoroid streams capable of producing meteorite-dropping bolides is a hot topic in planetary science. Such streams were first proposed by Halliday [

The Chelyabinsk event is also of interest because of its magnitude and energy and due to its relevance to be considered as a representative example of the most frequent outcome of the impact hazard associated with small asteroids in human timescales. Chelyabinsk also exemplifies the importance that fragmentation has for small asteroids, which can even excavate a crater on the Earth’s surface, although rarely [

One way to study meteoroids as they enter the Earth’s atmosphere is through video observations of such events. Consequently, we are developing complementary approaches to study the dynamical behavior of video-recorded bolides in much detail. The SPanish Meteor Network (SPMN) pioneered the application of high-sensitivity cameras for detecting fireballs, and it currently maintains an online list of bright events detected over Spain, Portugal, Southern France, and Morocco since 1999 [

In this study, we study the Chelyabinsk bolide by following a Runge–Kutta method of meteor investigation similar to that developed by Bellot Rubio et al. [

This study is structured as follows: the data reduction and the theoretical approach pertaining to the Chelyabinsk bolide are described in the next section. In

The Chelyabinsk superbolide was an unexpected daylight superbolide as many other unpredicted meteorite-dropping bolides in history. Fortunately, numerous casual video recordings of the bolide trajectory from the ground were obtained, given the nowadays common dashcams available in private motor vehicles in Russia. According to the video recordings available, it is possible to study the atmospheric trajectory and deceleration carefully, allowing the reconstruction of the heliocentric orbit in record time [

There are two main approaches in the study of the dynamic properties of meteors during atmospheric interaction, the quasicontinuous fragmentation (QCF) theory introduced by Novikov et al. [

We remark that the initial dynamic mass estimate or the preatmospheric size can be derived using the methods described in other works [

The fragmentation of meteoroids was studied in detail by various authors [

In practice, a combination of two or more types of fragmentation can be observed in a given meteor event. In fact, it is possible to observe that (a) and (c) fragmentation types described in the preceding paragraph could occur more than once for the same meteor event. The analysis of meteors performed by Jacchia [

The dynamic behavior of a meteoroid as it interacts with the Earth’s atmosphere is described using the drag and mass loss equations. These equations, as presented by Bronshten [_{air} is the air density,

By using equations (

The shape-density coefficient depends on the shape and density of the meteoroid and is expressed as

We must point out that the observational data obtained from the reconstruction of the trajectories of meteors using CCD or video cameras is basically the frame to frame speed of the bolide as a function of the height, requiring another equation to link the time with the altitude:

By substituting equation (

Next, by dividing equation (

Solving this differential equation with the boundary condition of

Now, we combine equations (

In order to obtain the value of

However, by using the concepts introduced above, it is not possible to obtain the values of initial mass (_{o}) and _{o}^{−1/3·}_{o} separately. The remaining expression is the photometric equation:

We define the

Then, the equation to work with is

In this section, we develop a numerical approximation with the aim to describe the meteoroid flight in the atmosphere. Our goal is to obtain the solution that can be used to better understand that physical process. Subsequently, our intention is to develop a numerical approach that can be very valuable in predicting the variation of the parameters along the trajectory segments versus analytical “whole-trajectory smoothing.”

Equation (

The Runge–Kutta method is an iterative technique for the approximation and solution of ordinary differential equations. The method was first developed by Runge [

The Runge–Kutta approximation provides a solution at a determined point of altitude. Application of the Runge–Kutta method requires the initial conditions to be known:

In our case, the initial conditions will be the initial velocity and the altitude of the bolide when ablation starts, “synthetically” written as

We then need to choose a step size (

Once the step size is defined, we define the model coefficients as follows:

For our case, the function to be studied is

Once the coefficients (equation (_{n+1} using the following formula:

For our case (equation (

The result obtained is the solution for the point (_{n+1}, _{n+1}), which becomes the initial condition to find the numerical approximation for the next point. The procedure is repeated until the desired value is reached.

Once the procedure is defined, we need to validate the code by comparing the results with the previously published data. We use a catalog of very precise photographic trajectories of meteors [

Equation (

We have defined a way to transform the differential equation into an expression that can be iteratively computed. The aim is to find a result for which

Furthermore, we introduce the increment factors for

In principle, we created a 2D matrix of errors. The error parameter can be shown in a table for better visualization of the algorithm (Figure

Basic scheme of the error matrix.

We repeat the same procedure for the next centered value, until we reach the point where the minimum will be centered in the middle of the matrix. Consequently, the minimum error value will correspond to the sought values of

Figure

Normalized velocity vs. altitude fit for the meteor J8945. The result is a good fit to the previously published Figure

We have presented a model capable of obtaining some parameters for meteoroids. 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 et al. [

Comparison of results for the meteoroid J4141. The model cannot produce a well-fitting solution since this meteoroid likely suffered an abrupt disruption. This figure can be compared with that shown in Figure

Despite the difficulties to obtain well-fitting solutions for some events, it is remarkable that our model is able to identify and produce solutions for the events undergoing quasicontinuous fragmentation. A possible way to study meteoroids with abrupt fragmentation is to focus on different segments of the trajectory that do not include a disruption episode. We will apply that approach to the lower part of the trajectory of the Chelyabinsk bolide to demonstrate that the obtained parameters are consistent.

Equation (^{2} km^{−2} and can also be expressed through the dimensionless mass loss parameter [^{2} km^{−2} [

Comparison of meteoroids having the same initial conditions (_{0} = 1 g,

In general, the larger the ablation coefficient is, the faster the body decelerates due to a more rapid mass loss. Consequently, the mass of the body decreases due to ablation; this is described by using the ablation coefficient, and the drag force imposed by the atmosphere has a larger effect. Table

Comparison of the ablation coefficients for various meteor cases from the JVB catalog.

Meteor ID | Jacchia (s^{2}·km^{−2}) | This study (s^{2}·km^{−2}) |
---|---|---|

J6882 | 0.0812 | 0.075 |

J6959 | 0.0331 | 0.0382 |

J7216 | 0.0501 | 0.079 |

J8945 | 0.0354 | 0.036 |

J9030 | 0.0549 | 0.0542 |

J7161 | 0.0354 | 0.0381 |

In this section, the effect of deceleration is examined in more detail. Given that we have the velocity as a function of altitude along the trajectory, we can study deceleration from the behavior of the velocity curve. This is particularly useful as many meteor processing algorithms and detection methods provide velocity values sequentially [_{i}, the code computes an increment of velocity over the increment of distance at the points immediately before and after. This can be expressed as

Considering all the points with known velocity and altitude along the trajectory as input data, equation (

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

The normalized instantaneous mass (_{0}) is the next quantity to be studied. The expression for the normalized instantaneous mass can be derived by rearranging equation (9):

Equation (

We define the normalized mass loss rate as the derivative of the relative mass over the altitude. This value is computed as follows:

Figure

We apply the Runge–Kutta code developed in this work to the famous Chelyabinsk superbolide. On February 15, 2013, it was predicted that the NEA-2012 DA14, discovered a year prior by the Observatory Astronòmic de Mallorca, would approach the Earth within a close minimum distance of only 27700 km. 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. The bolide disintegrated in the proximity of the city of Chelyabinsk [

A casual photograph of the Chelyabinsk bolide taken by Marat Ahmetvaleev. (a) The abrupt fragmentation at a height of 23 km 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.

As the days passed and the orbits were calculated, scientists discarded a possible association between the two NEAs, as they presented very different heliocentric orbits. Thanks to video cameras (dash cams) found in the majority of Russian motor vehicles and surveillance cameras placed on buildings, the initial trajectory of the bolide was reconstructed and the orbit was determined [

After the superbolide sightings, many people uploaded various videos to Internet. Since the geographical location of the recorded videos was known, we reconstructed the bolide trajectory, obtaining the values of velocity as a function of altitude. As shown in Table

Dynamic data of the Chelyabinsk superbolide.

Height (km) | Height (km) | ||
---|---|---|---|

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 |

Figure

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

By studying the dynamic curve, the ablation coefficient can be obtained. The derived value is

It is quite remarkable that this value, derived for the lower trajectory, provides a similar ablation coefficient as that for the fireballs at much higher altitudes, even though Chelyabinsk was the deepest penetrating bolide ever documented, still emitting light even as it reached the troposphere. Notably, the atmospheric density in these lower regions is about four orders of magnitude higher.

The normalized mass evolution for that lower part of the trajectory is shown in Figure ^{13} W s·r^{−1}, corresponding to an absolute astronomical magnitude of −28 [

It is well known that the maximum brightness is achieved shortly after the meteoroid catastrophically breaks apart due to the fragmented/pulverized material being exposed to the heat generated by the resulting shock wave.

An interesting conclusion that can be obtained directly from these results is the importance of the atmosphere. As mentioned before, the faster the meteoroid, the more rapid the ablation process. Thus, the atmosphere could effectively shield the Earth from very fast impacts since such objects are preferentially and more quickly ablated. However, the less favorable cases are very large objects (especially if their preatmospheric velocity is low), such as the Tunguska impactor that produced an airburst over Siberia in 1908 when it entered the atmosphere at a velocity of ∼30 km/s [

To exemplify this, we have plotted the entry of the Chelyabinsk superbolide for different initial velocities (Figure

Simulation of the Chelyabinsk superbolide with different initial velocities (

If the Chelyabinsk superbolide entered at a higher velocity, it would have slowed down faster due to a more rapid mass loss. According to our model, the maximum brightness of a meteoroid occurs when the mass loss reaches the peak. Consequently, the model predicts that a Chelyabinsk-like asteroid moving at a lower velocity could have more destructive potential on the Earth’s surface (Figure

It is important to remark that new improvements in detection of fireballs from space could provide an additional progress in studying the luminous efficiency of bolides [

Considering the destructive nature of large extraterrestrial objects (e.g., [

From the study of meteorite falls and the relative absence of small impact craters [

We have developed a numerical model employing the Runge–Kutta method to predict the dynamic behavior of meteoroids penetrating the Earth’s atmosphere based on the meteor physics equations [

Our study of the deceleration profile of the Chelyabinsk superbolide has allowed us to reach the following conclusions:

Our numerical model, successfully applied to the lower part of the fireball trajectory, predicts well the main observed characteristics of the Chelyabinsk superbolide. This is quite remarkable as the lower trajectory studied here has a similar ablation behaviour compared to fireballs at higher altitudes. It should be noted that the Chelyabinsk event is the deepest penetrating bolide ever documented, borderline emitting light as it reached the troposphere. Thus, our approach offers a promising venue in studying complex meteoric events in a streamlined and simplified manner.

The best fit to the deceleration pattern provides an average ablation coefficient ^{2}·km^{−2}, which is in the range of those derived in the scientific literature

The ablation coefficient is considered constant within each studied trajectory interval. This simplistic approach is probably one of the reasons why this model is not applicable to the entire trajectory of the meteoroids suffering significant fragmentation and catastrophic disruptions. In any case, for the cases studied here, the ablation coefficients obtained in our work are consistent with those reported in scientific literature.

A comparison of the fireball main parameters inferred from both the ground and space could constrain the role of the observing geometry in signal loss and biases in the determination of the velocity, radiated energy, and the orbital elements

NEOs disrupting in the near-Earth region can produce fragments tens of meters in size, which, if encountering our planet under the right geometric circumstances, might be a significant source of hazard for humans and infrastructure on the ground. Consequently, we suggest that identifying the existence of asteroidal complexes in the near-Earth environment is crucial for a better assessment of impact hazard.

Finally, we should not underestimate the hazardous potential of small asteroids as our model indicates that the ability of the Earth’s atmosphere to shield us from such objects strongly depends on the relative velocity of the encounter with our planet

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This research was funded by the research project (PGC2018-097374-B-I00, P.I. J.M.T-R), funded by FEDER/Ministerio de Ciencia e Innovación–Agencia Estatal de Investigación. MG acknowledges support from the Academy of Finland (325806). Research at the Ural Federal University was supported by the Russian Foundation for Basic Research (18-08-00074 and 19-05-00028). During the peer-review of this manuscript, the authors lost a mastermind, dear friend, and the co-author Esko Lyytinen. The authors dedicate this common effort in memorial to his enormous science figure and also to his friendship, insight, and understanding shared over the years. The authors thank Dr. Oleksandr Girin and two anonymous reviewers for their valuable and constructive comments. The authors also thank Marat Ahmetvaleev for providing them with the amazing pictures of the Chelyabinsk bolide and its dust trail.

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