Where did the Chelyabinsk meteorite fall on the map. Fall of a meteorite in Chelyabinsk. Research and study of the Chelyabinsk meteorite

On the map - the approximate trajectory of the fall of the meteorite

Chelyabinsk meteorite- a stone meteoroid that fell on February 15, 2013 near Lake Chebarkul in the Chelyabinsk region. The meteorite fell at 9:20 local time 80 km west of Chelyabinsk. As a result of the fall of the meteorite, 1491 people were injured.

According to experts, the mass of the meteorite was up to 10,000 tons, and the diameter was about 15-17 m. The flight of the meteorite body from the moment it entered the atmosphere lasted 32.5 seconds. During the flight in the atmosphere, the meteorite collapsed into many pieces, and therefore fell to the ground in the form of a meteor shower. At a height of 15-25 meters, the meteorite broke up into several parts as a result of a series of explosions. The fall speed of the car was from 20 to 70 km/s. During the fall, the space object left a bright trace, which was visible even in Kazakhstan and the Samara region.

When the meteorite was destroyed into several parts, shock waves were formed. According to experts, the total amount of energy released during the destruction of the cosmic body amounted to 500 kilotons in TNT equivalent.

Chronicle of the fall of the Chelyabinsk meteorite

At 9:15 local time, the movement of the cosmic body was seen by residents of the Kostanay and Aktobe regions of Kazakhstan. At 9:21 am, a meteor trail was seen in the Orenburg region. The witnesses of the fall of the meteorite were residents of the Sverdlovsk, Tyumen, Kurgan, Samara and Chelyabinsk regions, as well as the Republic of Bashkortostan.

At 9:20 local time, the meteorite fell into Chebarkul Lake, located 1 km from the city of Chebarkul. The fall of parts of the meteorite was observed by fishermen who were fishing near the lake. According to eyewitnesses, about 7 fragments of a cosmic body flew over the lake, one of which fell into the lake, raising a column of water 3-4 meters high. On the satellite map you can see Chebarkul Lake, where the meteorite fell.

As a result of the fall of the meteorite, a blast wave was formed, which, in terms of released energy, exceeded the energy of the atomic bombs dropped on Hiroshima and Nagasaki. Due to the gentle trajectory of the entry of the body into the atmosphere, only part of the released energy reached the settlements.

The consequences of the fall of the Chelyabinsk meteorite

As most of the energy dissipated, the blast mostly shattered windows in buildings in nearby communities. A total of 1,491 people were injured in the meteorite impact, but most of the injuries were cuts and bruises from broken windows. Nevertheless, the number of victims of the Chelyabinsk meteorite has no equal in the world.

The greatest damage from the disaster was suffered by 6 settlements of the Chelyabinsk region: the cities of Yemanzhelinsk, Chelyabinsk, Korkino, Kopeysk, Yuzhnouralsk and the village of Etkul. The shock wave damaged many buildings: the damage from it was estimated from 400 million to 1 billion rubles.

Chelyabinsk zinc plant, the roof of which collapsed from the blast wave of a meteorite

Research and study of the Chelyabinsk meteorite

On February 15, 2013, it was found that fragments of a meteorite fell in the Chebarkul and Zlatoust districts of the Chelyabinsk region. Scientists from URFU have collected meteorite fragments for further study.

Later, the researchers told the press that the meteorite was an ordinary chondrite, which is composed of sulfites, iron, olivine and melting crust.

Early February morning in 2013 unexpectedly became tragic for 1613 residents of Chelyabinsk and its environs. There has never been such a large number of people affected by a fallen meteorite in the history of the Earth's population. During the impact, windows were broken in many buildings, trees were broken and people were injured to varying degrees of severity, as a result of which about 1,613 people were recognized as victims, of which, according to various sources, from 50 to 100 people ended up in hospitals. People who watched the fall of the meteorite that morning were simply shocked by the events taking place. The first versions of what was happening sounded like: a plane crash, a rocket crash and even an alien attack ...

At the moment, the picture of the events of that tragic morning has been fully restored and it is reliably known when and where the meteorite fell in Chelyabinsk.

How it was

At about 9 am on February 15, this “unexpected guest” appeared high in the sky over Chelyabinsk, as a result of which a state of emergency was declared in Chelyabinsk and its surroundings. Previously, the same meteorite was observed by residents of other regions of the Russian Federation, but they were much more fortunate than the residents of Chelyabinsk, because it simply flew past them without causing absolutely no harm. For example, at 7.15 Moscow time or at 9.15 local time, residents of the Aktobe and Kostanay regions of Kazakhstan saw it, and residents of Orenburg observed this amazing phenomenon at 7.21 Moscow time. This meteorite was also clearly visible in Sverdlovsk, Kurgan, Tyumen and their environs, and even 750 km from the place of impact in the village of Prosvet, Volzhsky district, Samara region.

Bright flash

According to the US National Aeronautics and Space Administration (NASA), a meteorite weighing about 10 tons and a diameter of about 17 meters, with a speed of 17 km / s, entered the Earth's atmosphere and after 32 seconds split into many pieces. The destruction of the meteorite was accompanied by a series of explosions, the first of three explosions was the strongest and caused the destruction. It was a bright flash, it lasted about five seconds, and a minute later it came to Earth in the form of a destructive wave. According to scientists, the destruction of the meteorite led to the release of energy, which was approximately equal to 100 to 500 kilotons of TNT. The center of the explosion was not the city of Chelyabinsk itself, but its area, which is located a little to the south and is called Yemanzhelinsk - Yuzhnouralsk.

Locations of the fall of fragments

As a result of research conducted by a specially created group, four places were discovered where fragments of the meteorite are supposed to be located. The first two places are in the Chebarkulsky district of the Chelyabinsk region, the third in the Zlatoustovsky district, and the fourth in the Chebarkul lake area. The information that the meteorite is located in the lake was confirmed by the fishermen who were at the crash site. From their stories, members of the search group learned that at the moment the meteorite fell into the lake, a column of water and ice about 3-4 meters high rose from it.

Second largest after Tunguska

As a result of the work carried out in the area of ​​​​Emanzhelinsk and the village of Travniki, about a hundred fragments were found, and about 3 kg of fragments were collected in the lake area. All of them are currently being studied by scientists, according to whom, the meteorite that fell in Chelyabinsk is the second largest after the Tunguska meteorite that fell on the territory of Russia on June 30, 1908.

Full cut video from the event

For some reason, I have not seen serious attempts in the forums today to restore the trajectory of today's Ural car. In the evening I decided to try to do it myself. I came up with this method: we assume the trajectory of a straight line for simplicity, on the pictures from different cities we measure the visible angle α of the trajectory with the horizon. This is the same as the angle between the plane passing through the trajectory and the observer with a horizontal surface. Then the lines of constant α will be direct rays coming out of the "point of incidence", i.e. points of intersection of the trajectory with the ground, assuming that the ground is flat. If you do not assume, they will begin to somehow bend in the distance.

Measurement results:

I have now built some kind of fitting of parameters with a nonlinear least squares method, the results are: the angle of the trajectory to the horizon is 14 °, the azimuth of the trace projection is 280 °, if we count from north to the right. Those. it turned out that he was flying almost to the west, but 10 ° to the north. The coordinates of the "drop point" are 54.8+-0.25, 60.2+-0.9. Those. in latitude to the south of Chebarkul, but in longitude it is very spread out - probably more suitable data are needed. This is very preliminary data, now it's time to sleep and there is no time to check. (UPD3: not very preliminary anymore and α converges with the calculated one everywhere.)

UPD (February 16, 2013 4:47 a.m.): If he didn’t screw up, in equatorial coordinates he arrived approximately from R.a. 21:56 Dec. +6°.

UPD2 (February 16, 2013 13:13): Chelyabinsk and Kamensk-Uralsky had their latitudes mixed up: they were 10° more. Corrected values: inclination of the trajectory to the horizon 13.5°, azimuth 276°, "falling point" 54.72+-0.05, 60.31+-0.09 (errors are estimated from the scatter of data and are probably underestimated). There remains an incomprehensible strong deviation of the calculated value of α (20° in the center, 24° in the south of the city) from the observed value (~34°) for Chelyabinsk. For other points more or less the same. I will sort this out. Probably, it is necessary to take into account data errors more correctly.

UDP3 (16.2.2013 13:39): Made a more correct error model. Previously, there was some kind of heuristic gag instead, from this it was not very correctly taken into account which data should be trusted more, which less. New parameters: inclination of the trajectory to the horizon 15.7°+-3.2°, azimuth 287°+-9°, drop point 55.05+-0.11, 60.00+-0.25. Coordinates can be viewed in maps.google.com by clicking "Maps Labs" on the bottom left and turning on the LatLng Tooltip. All errors at the 2σ level and were calculated from the data scatter. With such a small amount of data, this is not a very accurate estimate of error. Now I will add the calculated α to the table. (UPD3" 14:46: contributed.)

: Measured on the frame that I did for hyperpov , the position of the explosion point. The height in the flat earth approximation is 22.2+-2.0 km, the projection distance to the ground from the "falling point" is 90.7+-8.2 km. If we add the curvature of the earth, the height will be 22.9 + -2.0 km. The main error in measuring the height is associated with the inaccuracy of the azimuth of the trajectory.

The coordinates of the explosion point are 54.84 N, 61.12 E. In longitude, the error is 26 km: in addition to the sources of errors listed above, the main source of error is the inaccuracy of the longitude of the "falling point". In latitude, the error is much smaller, about 5 km. When I determine the absolute azimuths on the photo, the longitude can be measured more accurately. So far, I can only measure relative azimuths.

Here, the errors did not yet take into account the inaccuracy in determining the angular dimensions of the photo - I have not yet verified this by an independent method.

UPD6 (03/22/2013 11:59 AM): Firstly, in UPD5, the angular dimensions of the percent were underestimated by 10, see. Secondly, as a first approximation, I measured the speed of the fireball / meteorite, I don’t know how to do it right now. Here are the measured coordinates of the first 6.67 seconds of the flight in the video from Kamensk-Uralsky (frame numbers 445...644, time 14.848...21.488 sec): http://pastebin.com/x8wh4Mwb . Haven't measured further yet. Here is the processed data: http://pastebin.com/riMkhSFa. -l-- distance to the "falling point" along the trajectory, z-- height, r-- direction from the camera to the car in the camera coordinate system (Cartesian, x right, y up, z forward). The coordinates in the frame are quite accurate, σ~1 pixel spread in both coordinates. IN l And z there is an inaccuracy associated with the parameters of the trajectory. There may be, for example, a multiplicative bias of about 10% (2σ) due to this. Cm. . l(t) lies well on a straight line, even at the beginning in the corner of the frame, the deviation is σ~0.5 km. Here is the chart l(t): http://s017.radikal.ru/i429/1302/17/d73f9782f067.png . Velocity from the slope of the graph v=20.86+-0.03 km/s, plus an error of ~2 km/s due to the inaccuracy of the trajectory parameters.

UPD7 (26.02.2013 2:14): I measured another video: with it, the azimuth of the trajectory direction is well specified. I re-measured the entire video more precisely, separately the slope before the explosion, separately after, clarified the magnitude of the errors of all slopes. I also wrote and debugged the code for gnuplot, which adjusts the trajectory taking into account the sphericity of the Earth, but I didn’t really pick its results, because to use them you need to write and debug a bunch of new code. Results for a flat Earth (x0, y0 - latitude and longitude of the "point of impact", i.e. the intersection of the continuation of the trajectory with the earth, beta0 - azimuth from east to the left in radians, tana0 - tangent of the angle of the trajectory with the surface):
# Flat Earth, Segment 0 (Pre-Fragmentation) TANA0 = 0.280602 +/- 0.02358 (8.404%) Beta0 = -0.255932 +/- 0.09432 (36.86%) X0 = 55.0351 +/- 0.08824 (0.1603%) Y0 = YU 59.8565 +/ - 0.1833 (0.3062%) # flat Earth, segment 1 (post-fragmentation) tana0 = 0.317638 +/- 0.0115 (3.622%) beta0 = -0.235893 +/- 0.06019 (25.51%) x0 = 54.966 +/- 0.04223 (0.07683 % ) y0 = 60.1681 +/- 0.04489 (0.0746%)
Results with a spherical Earth (ghav, decv -- spherical track direction coordinates in radians, counted in the same way as latitude and longitude latf, lonf):
# spherical Earth, segment 0 (pre-fragmentation) ghav = 2.25177 +/- 0.08172 (3.629%) decv = 0.0818073 +/- 0.04304 (52.61%) latf = 0.960549 +/- 0.001488 (0.1549%) lonf = 1.04 481+/- 0.002962 (0.2835%) # alpha0=15.6769974978532, (latf lonf)=(55.0353931240146 59.8629341269169) # spherical Earth, segment 1 (post-fragmentation) ghav = 2.26859 +/- 0.04871 (2.147%) decv = 0.12456 +/- 0.03287 (26.39 %) latf = 0.959263 +/- 0.0007175 (0.0748%) lonf = 1.05028 +/- 0.0007463 (0.07106%) # alpha0=17.3613472848805, (latf lonf)=(54.9617137981393 60.17 67421945092)
I also measured the fall on the video further, up to 371 frames out of 449. Then it’s somehow not immediately clear which of the wreckage should be monitored. Here are the coordinates inside the frame of the video http://pastebin.com/bcz0qqAF , here are the restored directions in camera coordinates (quite accurate) and the coordinates of the meteorite on its path http://pastebin.com/Ys8rhBVB (there is a systematic error related to the inaccuracy of the trajectory, but it is unlikely that anyone now has less, it seems to me). The biggest explosion is at frame 319 (t=10.64 sec), the first noticeable fragmentation is around t=6.67. After 319 frame l And h in fall.dat are not entirely accurate, because the parameters of the trajectory before the explosion are used everywhere.

In general, this video (from Kamensk-Uralsky) clearly shows the fine details of fragmentation, because the matrix begins to invert the image at high intensities. Even the rays from scattering on the windshield also show these details, albeit a little worse.

Continued in a new post. In general, I was hoping that someone who understands would come and also pick up the data. Alone, a lot of time is wasted, besides, in fact, wasted.

Keywords

annotation scientific article on Earth sciences and related ecological sciences, author of scientific work - Bondarenko Yury Sergeevich, Medvedev Yury Dmitrievich

A technique has been developed to determine trajectory celestial body in the Earth's atmosphere, parameters heliocentric orbit body before it enters the atmosphere, as well as assess the main factors of damage by a shock wave. The technique provides for the study of several scenarios for the development of events due to the passage of an object in the Earth's atmosphere. If the object passed through the atmosphere without colliding with the Earth, the moments of entry and exit of the body from the Earth's atmosphere are determined. An object can collide with the Earth without being destroyed. In this case, the differential equations are integrated until the celestial body reaches the Earth's surface. It was believed that an object burns up in the atmosphere if its radius becomes less than 1 cm. The case was considered separately when the object is destroyed during the movement, and only fragments reach the Earth's surface. The developed technique was implemented in a software-computer complex. One of the advantages of the complex is the ability to save the calculation results in a .kml file, which allows displaying three-dimensional geospatial data in the Google Earth program, as well as on two-dimensional Google maps. In our case, this is the flight path and its projection on the Earth's surface, the places of destruction, explosion and fall of the meteorite, the area of ​​fragments falling and shock wave damage, as well as other useful information. The efficiency of the software and computing system was tested on the motion of the asteroid 2008 TC3 and the Chelyabinsk meteorite. It was shown that the orbits of the 2008 TC3 and Chelyabinsk meteorites before entering the atmosphere turned out to be close to the orbits obtained by other authors, and the parameters air bursts coincide with the original data within their accuracy. The resulting areas of fall of fragments of these meteorites are only a few kilometers from the discovered fragments. The zones of destruction as a result of the action of an air shock wave in the case of the Chelyabinsk meteorite coincide with real data.

Related Topics scientific papers on Earth sciences and related ecological sciences, author of scientific work - Bondarenko Yury Sergeevich, Medvedev Yury Dmitrievich

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Determination of the trajectory of motion of celestial bodies in the Earth""s atmosphere

The authors have developed and realized the method, allowing to determine the trajectory of motion of celestial bodies in the Earth's atmosphere, to determine the parameters of heliocentric orbit of celestial bodies prior to its entry into the atmosphere, as well as to estimate major factors of damage due to the blast wave . The method researches several scenarios due to the passage of the object in the Earth's atmosphere. In case the object passed through the atmosphere, without colliding with the Earth, the moments of an entrance and exit of a body from the Earth’s atmosphere are determined. The object can collide with the Earth without breakup. In this case, the differential equations are integrated until the celestial body reaches the Earth's surface. It was assumed that the object burns in the atmosphere, if its radius becomes less than 1 cm. The case when object breaks up during the motion and only the fragments reach the Earth's surface was considered separately. The developed method has been implemented in the software package. One of the advantages of the package is the ability to save the results of calculations in the.kml format, allowing to display threedimensional geospatial data in the “Google Earth” as well as two-dimensional data in “Google” maps. In our case these data are the flight trajectory and its projection to the Earth’s surface, the places of meteorite break up and air burst, the impact areas of the fragments, the overpressure areas due the blast wave , as well as other useful information. Using this method the motion of Chelyabinsk and 2008 TC3 meteorites were simulated. It was shown that heliocentric orbital elements of the Chelyabinsk and 2008 TC3 meteorites before entering the Earth 's atmosphere calculated using the developed software are close to the parameters obtained by other authors, the trajectory parameters are in good agreement with the initial data within their accuracy . Estimated impact areas of meteorites fragments are only in a few kilometers from the recovered one. The overpressure areas due to the blast wave in case of “Chelyabinsk” meteorite coincide with the real data.

The text of the scientific work on the topic "Determination of the trajectory of the movement of celestial bodies in the Earth's atmosphere"

UDC 521.35; 523.628.4

Bulletin of SibGAU 2014. No. 4(56). pp. 16-24

DETERMINATION OF THE TRAJECTORY OF MOVEMENT OF CELESTIAL BODIES IN THE EARTH'S ATMOSPHERE

Yu. S. Bondarenko, Yu. D. Medvedev

Institute of Applied Astronomy of the Russian Academy of Sciences Russian Federation, 191187, St. Petersburg, nab. Kutuzova, 10 [email protected]

A technique has been developed that makes it possible to determine the trajectory of a celestial body in the Earth's atmosphere, the parameters of the body's heliocentric orbit prior to its entry into the atmosphere, and also to evaluate the main factors of damage by a shock wave. The technique provides for the study of several options for the development of events due to the passage of an object in the Earth's atmosphere. If the object passed through the atmosphere without colliding with the Earth, the moments of entry and exit of the body from the Earth's atmosphere are determined. An object can collide with the Earth without being destroyed. In this case, the differential equations are integrated until the celestial body reaches the Earth's surface. It was believed that an object burns up in the atmosphere if its radius becomes less than 1 cm. The case was considered separately when the object is destroyed during the movement, and only fragments reach the Earth's surface. The developed technique was implemented in a software-computer complex. One of the advantages of the complex is the ability to save the calculation results in a .kml file, which allows displaying three-dimensional geospatial data in the Google Earth program, as well as on two-dimensional Google maps. In our case, this is the flight path and its projection on the Earth's surface, the places of destruction, explosion and fall of the meteorite, the area of ​​fragments falling and shock wave damage, as well as other useful information. The efficiency of the software and computing system was tested on the motion of the asteroid 2008 TC3 and the Chelyabinsk meteorite. It was shown that the orbits of the 2008 TC3 and Chelyabinsk meteorites before entering the atmosphere turned out to be close to the orbits obtained by other authors, and the parameters of air explosions coincide with the original data within their accuracy. The resulting areas of fall of fragments of these meteorites are only a few kilometers from the discovered fragments. The zones of destruction as a result of the action of an air shock wave in the case of the Chelyabinsk meteorite coincide with real data.

Key words: celestial body, asteroid, meteorite, heliocentric orbit, motion trajectory, Earth's atmosphere, air burst, shock wave, impact area.

Vestnik SibGAU 2014, no. 4(56), P. 16-24

DETERMINATION OF THE TRAJECTORY OF MOTION OF CELESTIAL BODIES

IN THE EARTH'S ATMOSPHERE

Yu. S. Bondarenko, Yu. D. Medvedev

Institute of Applied Astronomy of Russian Academy of Sciences 10, Kutuzova nab., St. Petersburg, 191187, Russian Federation [email protected]

The authors have developed and realized the method, allowing to determine the trajectory of motion of celestial bodies in the Earth's atmosphere, to determine the parameters of heliocentric orbit of celestial bodies prior to its entry into the atmosphere, as well as to estimate major factors of damage due to the blast wave. The method researches several scenarios due to the passage of the object in the Earth's atmosphere. In case the object passed through the atmosphere, without colliding with the Earth, the moments of an entrance and exit of a body from the Earth's atmosphere are determined. The object can collide with the Earth without breakup. In this case, the differential equations are integrated until the celestial body reaches the Earth's surface. It was assumed that the object burns in the atmosphere, if its radius becomes less than 1 cm. The case when object breaks up during the motion and only the fragments reach the Earth's surface was considered separately. The developed method has been implemented in the software package. One of the advantages of the package is the ability to save the results of calculations in the .kml format, allowing to display three-dimensional geospatial data in the "Google Earth" as well as two-dimensional data in "Google" maps. In our case these data are the flight trajectory and its projection to the Earth"s surface, the places of meteorite break up and air burst, the impact areas of the fragments, the overpressure areas due the blast wave, as well as other useful information.

Using this method the motion of Chelyabinsk and 2008 TC3 meteorites were simulated. It was shown that heliocentric orbital elements of the Chelyabinsk and 2008 TC3 meteorites before entering the Earth's atmosphere calculated using the developed software are close to the parameters obtained by other authors, the trajectory parameters are in good agreement with the initial data within their accuracy The overpressure areas due to the blast wave in case of "Chelyabinsk" meteorite coincide with the real data.

Keywords: celestial body, asteroid, meteorite, heliocentric orbit, trajectory of motion, Earth's atmosphere, air blast, blast wave, impact area.

Introduction. The main disturbing factors in the motion of small bodies in the solar system are the attraction of large planets, which in most cases are considered as material points. However, in the case of a close approach or collision of the object under study with the Earth, it is necessary to take into account such factors as the effect of non-sphericity, the perturbation exerted by the Earth's atmosphere, the mass, composition and shape of the body itself, which presents a certain difficulty for researchers. In this regard, there is a need to develop a technique that allows one to make a fairly accurate estimate of the trajectory of a body when it moves both near and in the Earth's atmosphere.

dynamic model. In the developed dynamic model, if the object under study moves outside the earth's atmosphere, the equations of motion are given in a rectangular heliocentric coordinate system and have the form

where " - gravitational acceleration from the Sun; W2" - perturbing accelerations determined by the attraction of the object under study by the planets; W," - relativistic corrections.

If the body entered the Earth's atmosphere, then there is a transition to the geocentric coordinate system, and the equations of motion change. They add terms that take into account the compression of the Earth and the resistance of the atmosphere. A differential equation is also added that describes the change in the size of an object due to its deceleration in the atmosphere:

7 = W + W2 + W3; I = VI

where W - gravitational acceleration from the Earth, taking into account compression; G2 - gravitational perturbations from the Sun and planets of the Solar system; W, - atmospheric resistance; V is the rate at which the object's size changes.

The perturbing acceleration W, which takes into account the resistance of the atmosphere, is given in the form

W = -1 Cd рУ (

speed; the ratio of the midsection to the mass of the object m characterizes the windage. For convenience, the letter P denotes the pressure exerted by air on the body, and the letter A denotes air resistance.

Assuming that part of the energy arising from atmospheric resistance goes to heating and evaporation of matter from the surface of the body, and the object itself has and retains a spherical shape as a result of evaporation, the rate of change in the body radius will be determined by the following expression:

where y is the amount of energy spent on the sublimation of matter; I is the radius of the object; K is the heat required to vaporize 1 kg of a substance.

Possible development of events. The technique provides for the study of several options for the development of events due to the passage of an object in the Earth's atmosphere. If the object passed through the atmosphere without colliding with the Earth, the moments of entry and exit of the body from the Earth's atmosphere are determined. An object can collide with the Earth without being destroyed. In this case, the differential equations are integrated until the celestial body reaches the Earth's surface. It was believed that an object burns up in the atmosphere if its radius R becomes less than 1 cm. The case was considered separately when the object is destroyed during its movement, and only fragments reach the Earth's surface.

The destruction of the body occurs when the air pressure on the body P reaches the critical value Рmax. The values ​​of critical pressure for various materials of the object under study are presented in Table. 1 . Depending on the given density, the critical pressure values ​​​​are determined from Table. 1 by interpolation.

Table 1

Critical pressure values ​​for various materials

Material Density, kg/m3 Pmax; Pa

Porous rock 1500 105

Hard Rock 3600 10"

Iron 8000 108

where Sp is the coefficient of air resistance; pa - air density; u is the object's velocity vector relative to the Earth's atmosphere; and - vector modulus

Having reached the critical pressure, the body is destroyed, however, for some time the fragments of the body move as a whole, moving away from each other at a speed

bodies at the moment of destruction; p is the density of the body. After destruction, the rate of resizing

object V in the system is taken equal to V. Due to the difference in pressures on the front and back surfaces, the fragmented body, as it were, expands perpendicular to the motion trajectory until the ratio of the current radius to the radius of the body at the moment of destruction R(t)/R reaches the specified limit. Estimates of this value by different authors vary from 2 to 10. In the developed dynamic model, it is considered that an air explosion occurs at the moment when the value of R(t) = 5R, provided that the body has not reached the Earth's surface by this moment. From this moment, it is considered that the fragments begin to move along independent trajectories, and the consequence of their rapid deceleration is a shock wave.

The parameter of the shock wave, which determines its effect on various objects, is the maximum overpressure at the front Apm. On the basis of experimental data for a spherical shock wave, the empirical dependence 1 2

Apm = 0.084 - + 0.27 U- + 0.7 E Fm l l2 l3

where E is the energy of the explosion, measured in kg of TNT equivalent; l - distance from the center of the explosion, m; excess pressure at the front of the shock wave Apm is measured in MPa. This formula is valid for high power explosions: E > 100 kg TNT in the range 0.01< Apm < 1 МПа.

The direct impact of excess pressure at the front of the shock wave leads to partial or complete destruction of buildings, structures and other objects. Depending on the magnitude of the excess pressure, various destruction zones are distinguished, the values ​​of which are presented in Table. 2. The lesion on flat terrain is conditionally limited to a radius with an overpressure of 10 kPa (0.1 kgf/cm).

The energy of an air explosion is determined by the amount of energy released during deceleration of a collapsing body, according to the formula

E = l-tiT, 2

where m is the mass of the body at the moment of destruction; n is the fraction of energy released almost instantaneously during deceleration of small fragments. Thus, knowing the energy and height of the explosion, the dimensions of the destruction zones are found.

table 2

Destruction under the influence of a shock wave

Destruction zones Apm, kPa

Glass strength threshold 1

10% glass broken 2

Minor damage to buildings 5

Partial destruction 10

Medium destruction 20

Strong Destruction 30

Complete destruction 50

destruction of the object into fragments. To estimate the area of ​​impact, the developed method jointly integrates the movement of 4 fragments, which scatter in opposite directions in a plane perpendicular to the velocity vector of the body at the moment of destruction um with velocities V = -\[p~1rot. These

directions are shown in fig. 1. In this case, the velocity vectors of each of the four fragments u, uE, and are given by the formulas

Tl Yu - - Tl Yu X°T

uW = uT + V-; uN \u003d uT + V--r

Suppose that during the movement of the body in the Earth's atmosphere at some point in time T,

uE = uT - VuW ; uS = uT - VuN,

where rä = uT x ¥T ; ¥T - body position vector at the moment of destruction. The fragment radius is taken equal to Rf = RT/n , where n is the number of fragments; RT - radius

object at the time of destruction. The coordinates of the fall fragments indicated in fig. 1 points W, E, N and S are calculated taking into account the parameters of precession and nutation of the Earth's axis, and the area of ​​incidence is approximated by an ellipse passing through these points.

The developed technique was implemented in a software-computer complex. One of the advantages of the complex is the ability to save the calculation results in a .kml file, which allows you to display three-dimensional geospatial data in the Google Earth program

And also on two-dimensional Google maps. In our case, this is the flight path and its projection on the Earth's surface, the places of destruction, explosion and fall of the meteorite, the area of ​​fragments falling and shock wave damage, as well as other useful information. The efficiency of the software and computing complex was tested on the motion of the asteroid 2008 TC3 and the Chelyabinsk meteorite.

Asteroid 2008 TC3. Asteroid 2008 TC3 was discovered on the morning of October 6, 2008 at Mount Lemmon Observatory. Operational calculations of the preliminary orbit showed that this asteroid should collide with the Earth in the next 24 hours. It was the first celestial body discovered before entering the Earth's atmosphere. Its diameter was estimated in the range from 2 to 5 m. On October 7, the meteorite was destroyed when it fell in the atmosphere over the desert territory of Sudan at an altitude of 37 km with coordinates of 20.8 ° N. sh. and 32.2° E. d.

More than 600 asteroid fragments with a total mass of 10.7 kg were later found.

At the first stage, using the method of determining orbits based on the enumeration of orbital planes , the elements of the heliocentric orbit were obtained (Table 3), which represent 589 positional observations of the asteroid 2008 TC3 with a root-mean-square error c = 2.0"" for epoch 2454746.5 JD (7 October 2008). These elements define the so-called nominal orbit, i.e., satisfying the conditions of the least squares method. For comparison in table. Figure 3 also shows orbital elements obtained by the Jet Propulsion Laboratory (JPL).

Further, using the obtained elements of the orbit, the motion of the asteroid 2008 TC3 was simulated until the moment of its collision with the Earth. In the adopted model, the equations of motion take into account gravitational perturbations from all the major planets, the Moon and Pluto. The coordinates of the perturbing planets were calculated from the numerical ephemeris EPM. Numerical integration of the equations of motion was performed by the 4th order Runge-Kutta method with automatic selection of the step according to the velocity. The air density was calculated from the US 1976 Standard Atmosphere Tables, in which the atmosphere is divided into seven consecutive layers with a linear dependence of temperature on height. The Earth's surface was approximated by an ellipsoid of revolution. Assuming that the object was spherical, the drag coefficient

air Cn was taken equal to 2 . The amount of energy spent on the sublimation of matter y was taken equal to 10-3 for the main body, and 10-2 for fragments. It was also believed that 600 cal/g is needed to evaporate 1 kg of the substance of the asteroid 2008 TC3.

The results of simulation of the motion of the asteroid 2008 TC3 in the Earth's atmosphere are shown in Figs. 2, which shows a satellite image of the area, on which the black line shows the trajectory of the meteorite, obtained from the elements of the nominal orbit, and the white line shows its projection onto the Earth's surface. The places of the beginning of the destruction and explosion of the meteorite are designated by the letters A and B, respectively, and their parameters in comparison with satellite data are given in Table. 4. The numbers mark the places of the discovered fragments of the meteorite, and their masses and coordinates are given in Table. 5.

Rice. 1. Determining the area of ​​falling fragments

IPA 330.7502 234.4474 194.1011 2.5416 0.311995 0.658783

CXR 330.7541 234.4490 194.1011 2.5422 0.312065 0.658707

Table 4

Parameters of the places where the destruction and explosion of the asteroid 2008 TSZ began

IPA parameter Satellite data (KABA/KHR, 2008)

Destruction Explosion

Altitude, km 36.9 35.2 37

Time, IT 02:45:51 02:45:51 02:45:45

Latitude, ° with. sh. 20.72 20.71 20.8

Longitude, ° in. 32.15 32.19 32.2

Table 5

Parameters of found fragments of asteroid 2008 TZ

Parameter 1 2 3 4 5 6 7

Weight, g 4.412 78.201 65.733 141.842 378.710 259.860 303.690

Latitude, ° with. sh. 20.77 20.74 20.74 20.70 20.68 20.70 20.70

Longitude, ° in. 32.29 32.33 32.36 32.49 32.50 32.50 32.52

Rice. 2. Simulation results of the motion of the 2008 TC3 meteorite in the Earth's atmosphere

From Table. Figure 5 shows that the masses of the detected fragments do not exceed a kilogram, therefore, after the meteorite explosion, the motion of fragments with masses in the range from 100 to 700 g was simulated. to files. The figure shows the probable regions of impact of fragments of various masses, obtained from the nominal orbit and its two variations. The letters A and B denote the regions where the fragments with the smallest and largest masses fell out, respectively. On fig. 2 shows a good agreement between the results of the assessment of the impact areas with the found fragments, and small deviations can be explained, for example, by the effect of wind. Table data. 4 also indicate a good agreement between the simulation results and satellite data.

Meteorite "Chelyabinsk". On the morning of February 15, 2013, a bright flash was observed in the sky over Chelyabinsk, which was caused by a relatively small asteroid approximately 17-20 m in diameter, which entered the Earth's atmosphere at high speed and at a small angle. At that moment, a huge amount of energy was released, and the body itself collapsed into many parts of different sizes, which fell to the ground. Since this event took place over a populous city, it differs from similar events in the number of eyewitness accounts. It was recorded by a large number of video recorders and video cameras. In addition, meteorological satellites

MyeoBa! 9 and MeleoBa1 10 were able to photograph the condensation trail from the passage of a meteorite in the Earth's atmosphere, and a fragment of a meteorite about a meter in size and weighing approximately 600 kg was raised from the bottom of Lake Chebarkul.

To model the movement of the meteorite, the most accurate data to date were used as the initial parameters, which were obtained by equipment installed on geostationary satellites operating in the interests of the US Department of Defense and the US Department of Energy. This equipment makes it possible to track nuclear air explosions, as well as to measure the luminosity curves of fireballs burning up in the atmosphere. According to these data, the moment of maximum brightness occurred on February 15, 2013 at 03:20:33 GMT at an altitude of 23.3 km with coordinates of 54.8° N. sh. and 61.1° E. e. The speed of the object at the moment of maximum brightness was 18.6 km / s, and the released energy was 440 Kg in TNT equivalent.

The trajectory azimuth and inclination, obtained by Colombian astronomers from numerous records from video recorders and surveillance cameras, were taken to be 285 ± 2° and 15.8 ± 0.3°, respectively. The found remains of a meteorite indicate that it was an ordinary chondrite with a density of approximately 3.6 g / cm3. The diameter of the object before entering the atmosphere was taken to be 18 m.

These parameters were used to calculate the elements of the object's heliocentric orbit prior to its entry into the atmosphere at epoch 2456336.5 AR (February 13, 2013). These elements, in comparison with the results of other authors, are presented in Table. 6 in the first line.

Table 6

Comparison of the parameters of the resulting heliocentric orbit

IPA 0.70 0.56 100.90 326.46 4.27 1.60

7u1^a 0.71 0.48 97.98 326.47 4.31 1.37

1Аu 3423 0.77 0.5 109.7 326.41 3.6 1.55

INASAN 0.74 0.58 108.3 326.44 4.93 1.76

KhNU 0.65 0.65 97.2 326.42 12.06 1.83

Rice. 3. Heliocentric orbit of the Chelyabinsk meteorite

Rice. 4. Simulation results of the motion of the Chelyabinsk meteorite in the Earth's atmosphere

Rice. 5. Areas of falling fragments of the meteorite "Chelyabinsk"

On fig. 3 shows the heliocentric orbit of the Chelyabinsk meteorite in the plane of the ecliptic according to the calculated elements, obtained using the NLBU software complex. As can be seen from fig. 3, the asteroid's orbit reaches the orbit of Venus at perihelion and the asteroid belt at aphelion. A numerical calculation of evolution shows that the asteroid could move along this orbit for thousands of years, repeatedly crossing the Earth's orbit. It is likely that this asteroid was formed as a result of collisional processes in the main belt. Being at the perihelion of its orbit approximately two and a half months before the collision, it approached the Earth from the side of the Sun, which prevented its early detection by observatories that constantly monitor the small bodies of the solar system.

Table 7

Parameters of the beginning of the destruction and explosion of the meteorite "Chelyabinsk"

Parameter Destruction Explosion

Height, km 27.7 24.5

Time, IT 03:20:32 03:20:33

Latitude, ° with. sh. 54.78 54.81

Longitude, ° in. d. 61.20 61.04

The black line in Fig. 4 shows the trajectory of the fall, white is the projection of the trajectory, the places of destruction

and explosion at points L and B, respectively, the area where the fragments fell, as well as the nearest settlements superimposed on a satellite image of the area.

According to calculations, 474 kt of TNT energy was released at the moment of the explosion. In this case, the radius of the destruction zone with an excess pressure at the front of a shock wave of 1 kPa turns out to be equal to 127 km and 51 km for 2 kPa. Such pressure values ​​correspond to the glass strength threshold (see Table 2). The destruction zones are shown in fig. 4 white circles.

After the explosion of the meteorite, the motion of 20 groups of fragments with sizes ranging from 1.8 to 0.4 m was simulated. 5 asterisk marks the place where the largest fragment of a meteorite fell, about a meter in size and weighing 654 kg, found in Lake Chebarkul. Numbers 1, 2, and 3 designate the obtained probable areas of fall of the fragments, located in the immediate vicinity of the found fragment, and their parameters are presented in Table. 8.

Table 8

Fragment Drop Area Parameters

Parameter 1 2 3

Fragment size, m 0.7 0.6 0.6

Fragment weight, kg 646 517 420

Latitude of the center of the region, ° N sh. 54.94 54.93 54.93

Longitude of the center of the region, ° E 60.31 60.33 60.35

Area size, m 1270x354 1216x346 1166x336

Conclusion. The results obtained in the work show that the developed method allows one to calculate the trajectory of a celestial body in the Earth's atmosphere, the parameters of the heliocentric orbit of the body before it enters the atmosphere, to evaluate the area where fragments fall and the main factors of damage. It was shown that the orbits of the 2008 TC3 and Chelyabinsk meteorites before entering the atmosphere turned out to be close to the orbits obtained by other authors, and the parameters of air explosions coincide with the initial data within their accuracy. The resulting areas of fall of fragments of these meteorites are only a few kilometers from the discovered fragments. The zones of destruction as a result of the action of an air shock wave in the case of the Chelyabinsk meteorite coincide with real data, according to which about 7320 buildings were damaged. In some buildings, windows were broken, in others the frames were completely knocked out of the windows. In the Etkulsky district, which became the epicenter of the explosion, 865 windows in residential buildings and 1.1 thousand windows in other buildings were damaged.

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On the map - the approximate trajectory of the fall of the meteorite

Chelyabinsk meteorite- a stone meteoroid that fell on February 15, 2013 near Lake Chebarkul in the Chelyabinsk region. The meteorite fell at 9:20 local time 80 km west of Chelyabinsk. As a result of the fall of the meteorite, 1491 people were injured.

According to experts, the mass of the meteorite was up to 10,000 tons, and the diameter was about 15-17 m. The flight of the meteorite body from the moment it entered the atmosphere lasted 32.5 seconds. During the flight in the atmosphere, the meteorite collapsed into many pieces, and therefore fell to the ground in the form of a meteor shower. At a height of 15-25 meters, the meteorite broke up into several parts as a result of a series of explosions. The fall speed of the car was from 20 to 70 km/s. During the fall, the space object left a bright trace, which was visible even in Kazakhstan and the Samara region.

When the meteorite was destroyed into several parts, shock waves were formed. According to experts, the total amount of energy released during the destruction of the cosmic body amounted to 500 kilotons in TNT equivalent.

Chronicle of the fall of the Chelyabinsk meteorite

At 9:15 local time, the movement of the cosmic body was seen by residents of the Kostanay and Aktobe regions of Kazakhstan. At 9:21 am, a meteor trail was seen in the Orenburg region. The witnesses of the fall of the meteorite were residents of the Sverdlovsk, Tyumen, Kurgan, Samara and Chelyabinsk regions, as well as the Republic of Bashkortostan.

At 9:20 local time, the meteorite fell into Chebarkul Lake, located 1 km from the city of Chebarkul. The fall of parts of the meteorite was observed by fishermen who were fishing near the lake. According to eyewitnesses, about 7 fragments of a cosmic body flew over the lake, one of which fell into the lake, raising a column of water 3-4 meters high. On the satellite map you can see Chebarkul Lake, where the meteorite fell.

As a result of the fall of the meteorite, a blast wave was formed, which, in terms of released energy, exceeded the energy of the atomic bombs dropped on Hiroshima and Nagasaki. Due to the gentle trajectory of the entry of the body into the atmosphere, only part of the released energy reached the settlements.

The consequences of the fall of the Chelyabinsk meteorite

As most of the energy dissipated, the blast mostly shattered windows in buildings in nearby communities. A total of 1,491 people were injured in the meteorite impact, but most of the injuries were cuts and bruises from broken windows. Nevertheless, the number of victims of the Chelyabinsk meteorite has no equal in the world.

The greatest damage from the disaster was suffered by 6 settlements of the Chelyabinsk region: the cities of Yemanzhelinsk, Chelyabinsk, Korkino, Kopeysk, Yuzhnouralsk and the village of Etkul. The shock wave damaged many buildings: the damage from it was estimated from 400 million to 1 billion rubles.

Chelyabinsk zinc plant, the roof of which collapsed from the blast wave of a meteorite

Research and study of the Chelyabinsk meteorite

On February 15, 2013, it was found that fragments of a meteorite fell in the Chebarkul and Zlatoust districts of the Chelyabinsk region. Scientists from URFU have collected meteorite fragments for further study.

Later, the researchers told the press that the meteorite was an ordinary chondrite, which is composed of sulfites, iron, olivine and melting crust.