ISSN 0038�0946, Solar System Research, 2016, Vol. 50, No. 1, pp. 1–12. © Pleiades Publishing, Inc., 2016.Original Russian Text © V.V. Shuvalov, O.P. Popova, V.V. Svettsov, I.A. Trubetskaya, D.O. Glazachev, 2016, published in Astronomicheskii Vestnik, 2016, Vol. 50, No. 1, pp. 3–14.

1

INTRODUCTION

A fall of a cosmic body into the atmosphere with ahypersonic velocity and extension of the vapor and gascloud (shock plume) forming after the surface impactare accompanied by the appearance of air shock waves(SWs). The destructive effect of SWs is determined bythe pressure jump at their front, the duration of thecompression and rarefaction phases, and the gasvelocity behind the front. Depending on the size of thecosmic body, three scenarios of the impact can be dis�tinguished (Shuvalov and Trubetskaya, 2006). Smallstony meteoroids (<10 m) are as a rule decelerated inthe atmosphere at such a high altitude that the SWsattenuate and reach the surface as a packet of weakacoustic waves. Bodies with a size from 20 to 100 m aredecelerated at heights from several kilometers to 20–30 km. In this case, SWs reach the surface causingconsiderable damage (as, for example, with the fall ofthe Tunguska cosmic body). Bodies with a size largerthan 100 m, in addition to the creation of SWs in theatmosphere, reach the surface on average with a ratherhigh velocity, which causes SWs in the ground andleads to the formation of a shock crater with a diame�ter exceeding the size of the body by several tens oftimes. Propagation of seismic waves in the ground canlead to earthquakes, landslides, and rockfalls at con�siderable distances; an impact at water is accompanied

by the formation of tsunami waves which can propa�gate to great distances without significant attenuation.Note that the boundaries between different scenariosare approximate; they depend on the nature of the fall�ing body and inclination angle of the trajectory. Theaforementioned values of the boundary dimensionsare typical for stony bodies; they are larger for comet�like bodies and smaller for iron bodies.

Since meteoroids with a size less than 100 m inmost cases are disrupted and strongly decelerated inthe Earth’s atmosphere, they transfer their kineticenergy to the surrounding air. In this process, theenergy is released mainly in a small part of the trajec�tory (on the order of the characteristic height of theatmosphere) and, therefore, at great distances fromthe entry point (e.g., on the Earth’s surface), the shockwave looks like a shock wave from an focused explo�sion. For this reason, the process of disruption anddeceleration of a meteoroid in the atmosphere is oftencalled a meteoric explosion and its effect is estimatedusing formulas obtained when performing TNT ornuclear explosions. By tradition, we also use the term“meteoric explosion”, but in quotation marks becausein fact no thermal explosion occurs when the cosmicbody enters the atmosphere; the only mechanism ofthe energy release is the deceleration of the meteoroiditself, as well as of its vapors and fragments. This isaccompanied by the formation of a ballistic shock

Determination of the Height of the “Meteoric Explosion” V. V. Shuvalov, O. P. Popova, V. V. Svettsov, I. A. Trubetskaya, and D. O. Glazachev

Institute of Geosphere Dynamics, Russian Academy of Sciences, Leninskii pr. 38�1, Moscow, 119334, Russia e�mail: shuvalov@idg.chph.ras.ru

Received April 23, 2015

Abstract—When cosmic bodies of asteroidal and cometary origin, with a size from 20 to approximately100 m, enter dense atmospheric layers, they are destroyed with a large probability under the action of aero�dynamic forces and decelerated with the transfer of their energy to the air at heights from 20–30 to severalkilometers. The forming shock wave reaches the Earth’s surface and can cause considerable damage at greatdistances from the entry path similar to the action of a high�altitude explosion. We have performed a numer�ical simulation of the disruption (with allowance for evaporation of fragments) and deceleration of meteor�oids having the aforesaid dimensions and entering the Earth’s atmosphere at different angles and determinedthe height of the equivalent explosion point generating the same shock wave as the fall of a cosmic body withthe given parameters. It turns out that this height does not depend on the velocity of the body and is approx�imately equal to the height at which this velocity is reduced by half. The obtained results were successfullyapproximated by a simple analytical formula allowing one to easily determine the height of an equivalentexplosion depending on the dimensions of the body, its density, and angle of entry into the atmosphere. Acomparison of the obtained results with well�known approximate analytical (pancake) models is presentedand an application of the obtained formula to specific events, in particular, to the fall of the Chelyabinskmeteorite on February 15, 2013, and Tunguska event of 1908, is discussed.

Keywords: asteroid, comet, asteroid�comet hazard, shock wave, meteoric burst, numerical simulation

DOI: 10.1134/S0038094616010056

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wave which is perceived as an explosive one at greatdistances.

A vast database on the SW effect on differentobjects was collected after nuclear tests in the 1950s–1960s (Glasstone and Dolan, 1977) and widely used inestimations of effects from cosmic body impacts.However, there is no perfect analogy between a con�centrated explosion (TNT or nuclear) and “meteoricexplosions”; this was demonstrated, e.g., in the simu�lation of the shock wave created in the atmosphere bythe Chelyabinsk meteorite (Popova et al., 2013). Whena cosmic object enters the atmosphere, the area on theEarth’s surface where the destructive effect of theshock wave and other factors is observed is deter�mined, first of all, by the released energy and the shapeof this area depends on details of the energy release inthe process of the body flight and disruption. If anobject enters the atmosphere nonvertically, the shapeof this area differs from the round shape which is typi�cal for a concentrated explosion (Shuvalov et al., 2014;Popova et al., 2013). However, for a rapid approximateestimate of effects caused by the fall of a cosmic body,the explosion analogy is very useful. The questionabout the determination of the height of an equivalent(i.e., producing a similar shock wave) focused explo�sion is still little studied. Determining this height isprecisely the aim of this work (cf. Levin and Bronsh�ten, 1985; Korobeinikov et al., 1991; Grigoryan et al.,2013).

First, we present results of the numerical simula�tion of several impacts of cosmic bodies (including thestages of disruption, deceleration, and propagation ofa shock wave to great distances) and point explosionswith the same energy and determine the relationbetween the deceleration height of the meteoroid andheight of the equivalent explosion. In doing this, theequivalence is determined by the coincidence of max�imum excessive pressures in the epicenter of the explo�sion. Then, we describe a series of calculations per�formed for disruption and deceleration (“explosion”)of comet nuclei and asteroids having different sizesand entering the atmosphere at different angles.Finally, we approximate the numerical resultsobtained by a simple analytical dependence allowingone to easily determine the height of an explosion thatgenerates a shock wave similar to that generated by afall of an asteroid with the given parameters. At the endof the paper, the result obtained is compared withapproximate models proposed earlier and the applica�tion of the obtained formula to specific events is dis�cussed.

CALCULATION TECHNIQUE

The disruption and deceleration of a meteoroid inthe atmosphere and following propagation of theshock wave to long distances were calculated using atwo�step model described in (Shuvalov et al., 2013). Atthe first step, the motion of a meteoroid in the atmo�

sphere was simulated with allowance for its deforma�tion, deceleration, destruction, and evaporation. Thesimulation involved the model, equations, and numer�ical scheme described in (Shuvalov and Artemieva,2002; Shuvalov and Trubetskaya, 2007). The model isapplied when the meteoroid moves at heights whereaerodynamic loads considerably exceed its strength;for this reason, it is supposed that it is already dis�rupted and can be described in the hydrodynamicapproximation. The problem was solved in a coordi�nate system associated with the falling body which wasblown by air whose density varied according to theatmosphere stratification and the flow velocity wasequal to the velocity of the body. The calculations ter�minated when the meteoroid was disrupted and almostcompletely decelerated (its velocity in the coordinatesystem associated with the Earth decreased by fivetimes, i.e., when its further fall had no effect on thedetermined height of the explosion) or reached theEarth’s surface. The distributions of gas�dynamic andthermodynamic parameters in the atmosphere wereused as initial data for the second step of calculations.At that step, the propagation of an air shock wave togreat distances was simulated in a coordinate systemassociated with the Earth’s surface. Both the calcula�tion steps were implemented using the SOVA numeri�cal method (Shuvalov, 1999). The same method wasused for calculating the propagation of the shock wavefrom point explosions.

The calculations involved tables of the equation ofstate and radiation runs of air (Kuznetsov, 1965;Avilova et al., 1970), H chondrite (Kosarev, 1999), andcometary matter (Kosarev et al., 1996) (for the gas�eous phase) and tables obtained using the ANEOSprogram for the equation of state (Thomson and Lau�son, 1972) with initial data (Pierazzo et al., 1997) forgranite. Tables for the equation of state are availableonly for a severely limited set of rocks and we chosegranite as an approximation for a stony body. Thecomet nuclei were considered as consisting of ice forwhich the Tillotson equation of state was used (Tillot�son, 1962). We emphasize that meteoroids of the sizeunder consideration are noticeably deformed and fallinto pieces at heights where aerodynamic loads aresignificantly higher than the plastic limit, which makesit possible to neglect the strength of the body.

CALCULATIONS OF THE EXCESSIVE PRESSURE ON THE EARTH’S SURFACE

FOR IMPACTS OF COSMIC BODIES AND POINT EXPLOSIONS

As examples of falls of cosmic bodies, we considervertical impacts of spherical stony asteroids withdiameters of 40 and 70 m, velocities of 18 km/s, anddensity of 2650 kg/m3 (this is the normal density ofgranite). The energy of such impacts amounts to 1.4 ×1016 and 7.7 × 1016 J (which corresponds to 3.4 and18 Mt TNT). Figure 1 shows the picture of a flow over

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DETERMINATION OF THE HEIGHT OF THE “METEORIC EXPLOSION” 3

a 40�m asteroid at different heights. At a height ofabout 30 km, the meteoroid begins to be deformed andwave�like perturbations appear on its surface due tothe development of Rayleigh–Taylor and Kelvin–Helmholtz instabilities. An increase in aerodynamicloads flattens the meteoroid; at heights of 20–25 km(of the order of 10 km), as a result of transverse expan�sion, it turns into a pancake�like structure in a qualita�tive agreement with analytical approximate models(Grigoryan, 1979; Chyba et al., 1993). The furtherdevelopment of instabilities causes the fragmentationof the meteoroid; at heights below 15 km, it turns intoa jet consisting of an evaporated matter heated in thebow shock wave of air and fragments of the fallingbody. The velocity of this jet first slightly differs from

the initial meteoroid velocity of 18 km/s. Thus, thedisruption and fragmentation of the body under con�sideration occur before it begins noticeably decelerate.It follows, by the way, that deceleration of such bodiescannot be simulated within equations of the physicaltheory of meteors because they do not describe the gasjet motion. When determining the velocity of thedecelerating object, the motion of the front boundaryof the luminous region (of the meteor) was consid�ered; this region, in turn, was determined as a regionwith a temperature above ~5000 K (i.e., the tempera�ture at which the air becomes nontransparent). Thefragmentation leads to an increase in the evaporatedsurface and, therefore, to an increase in the ablationrate. At a height of about 10 km, meteoroid fragments

0–0.05 0 0.05

0.05

0.10

0.15

R, km

Z,

km

h = 18

0–0.05 0 0.05

0.05

0.10

0.15

R, km

Z,

km

h = 14

0–0.05 0 0.05

0.05

0.10

0.15

R, km

Z,

km

h = 30

0–0.05 0 0.05

0.05

0.10

0.15

R, km

Z,

km

h = 23

Fig. 1. Disruption of an asteroid with a diameter of 40 m in the process of vertical entry to the Earth’s atmosphere. h is the heightof the body’s flight in km. The figures are constructed in a coordinate system associated to the body.

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SHUVALOV et al.

are evaporated completely and the jet turns into a gasjet (of air and vapor) which is decelerated at a height ofabout 6 km.

The shock wave reaches the Earth’s surface with anamplitude corresponding to the excessive pressure Δpof about 20 kPa, reflects from the surface, and inter�acts with the incident wave; this results in the forma�tion of a wave propagating at great distances (of theorder of 10 km) along the surface (implementation ofthe so�called Mach reflection).

The dependence of the excessive pressure on theEarth’s surface at different distances from the impactepicenter, i.e., points at which the vertical trajectoryintersects the Earth’s surface, is shown in Fig. 2.Effects of destruction by a shock wave are usuallyrelated to the excess pressure maximum behind theshock wave front. Results of nuclear tests (Glasstoneand Dolan, 1977) show that masonry walls with athickness of 24–36 cm begin to be destroyed at anexcess pressure of 20 kPa (0.2 bar) and block walls witha thickness of 24–36 cm are completely destroyed atan excess pressure of 35 kPa (0.35 bar). We used thesevalues to estimate the disruptions. As seen from Fig. 2,noticeable damage can be observed after a fall of a40�m diameter asteroid in a zone with dimensions ofthe order of 10 km.

The same Fig. 2 presents calculation results forpoint explosions with an energy equal to the asteroidenergy at different heights h. In the near zone (of theorder of 10 km) where the main damage occurs, thebest coincidence between results of calculations forthe impact of the cosmic body and explosion occurs ath =11–12 km. This height approximately correspondsto the height where the velocity of the disrupted aster�oid decreases by two times. At great distances (morethan 20 km), where the excess pressure amounts to afew percent of the normal atmospheric pressure, itslightly depends on the explosion height.

Figure 3 shows similar dependencies of the excesspressure for the case of the vertical fall of an asteroidwith a diameter of 70 m. In this case, damage (theexcess pressure exceeds 20 kPa) is observed in an areawith a radius of about 20 km. Excess pressuresobtained in calculations of the cosmic body impact atthe surface coincide with a good accuracy with calcu�lations for the excessive pressure from an explosion ofthe same energy at a height of 3 km. This height, likein the preceding case, corresponds to a decrease in thevelocity of the disrupted asteroid by two times.

0.3

0 10 20 30 40

0.4

0.1

0.2

Distance, km

Exc

ess

pres

sure

, ba

r 10

11

12

h = 15

Fig. 2. Dependence of the excess pressure on the Earth’ssurface at different distances from the epicenter of theimpact of an asteroid with a diameter of 40 m, i.e., pointsat which the vertical trajectory intersects the Earth’s sur�face. The thick gray curve corresponds to the excessivepressure obtained in the simulation of the body fall in theatmosphere. The thin black curves show the excessive pres�sure obtained in calculations of point explosions (with anenergy equal to the energy of an asteroid with a diameter of40 m) at different heights h.

10

0 10 20 30

100

0.1

1

Distance, km

Exc

ess

pres

sure

, ba

rFig. 3. Dependence of the excess pressure on the Earth’ssurface at different distances from the epicenter of theimpact of an asteroid with a diameter of 70 m, i.e., pointsat which the vertical trajectory intersects the Earth’s sur�face. The thick gray curve corresponds to the excessivepressure obtained in the simulation of the body fall. Thethin black curve shows the excess pressure obtained in thecalculation of a point explosion with an energy equal to theenergy of an asteroid with a diameter of 70 m at a heighth = 3 km.

SOLAR SYSTEM RESEARCH Vol. 50 No. 1 2016

DETERMINATION OF THE HEIGHT OF THE “METEORIC EXPLOSION” 5

DETERMINATION OF THE DECELERATION HEIGHT OF COMET NUCLEI

AND DIFFERENT SIZED ASTEROIDS ENTERING THE ATMOSPHERE

AT DIFFERENT ANGLES

It was shown in the preceding section that effects ofthe action of a shock wave from a falling cosmic bodycan be simulated within a certain accuracy by use ofexplosions, the energy of which is determined by theenergy of the falling body and the height is determinedby the height at which the body is noticeably deceler�ated and loses its energy; more exactly, it is determinedby the height at which the velocity of the body ishalved.

Based on this conclusion, we performed a series ofcalculations of disruption and deceleration in theatmosphere for comet nuclei and asteroids of differentsizes falling at different angles and determined for theconsidered cases the heights at which the velocity ishalved (i.e., the effective heights of explosions). Thedensity of the comet nuclei was supposed to be equalto 1000 kg/m3; the density of asteroids, 2650 kg/m3.

Even first calculations showed that the resultsslightly depend on the velocity of cosmic bodiesaccording to the approximate model (Grigoryan,1979; Grigoryan et al., 2013) because an increase inthe transverse cross section of the body and the effi�ciency of deceleration depend for the most part onlyon the covered distance (see Eq. (7) below). For thisreason, we considered below only the fall of cometnuclei with a velocity of 50 km/s and of asteroids witha velocity of 18 km/s. Some results are presented inTables 1 and 2.

Tables 1 and 2 are not very convenient for the prac�tical use if we want to obtain the explosion height foran impact of a cosmic body with given parameters. Itis more convenient to have an interpolation relation�ship. To understand what such a dependence can looklike, let us consider simple estimates. It is reasonableto assume that the cosmic body is noticeably deceler�ated when it contacts an air mass comparable with themass of the meteoroid itself (per unit of the transversecross section area), i.e.,

(1)

where ρa and ρm are the densities of the air and mete�oroid, D is the meteoroid diameter, K is a dimension�less coefficient, dx is the increment of distance alongthe trajectory, and the integration is performed fromthe deceleration point to infinity. If the angle of thetrajectory inclination to the horizontal equals α, thendx = dh/sinα, where h is the height. For simplicity, weassume that the atmosphere is isothermal with a char�acteristic height H. Then, one can obtain from (1) anexpression for the deceleration height ht

(2)

Here, ρ0 is the sea level atmospheric density. It followsfrom this estimate that it is reasonable to approximatethe solution using the dimensionless variable

Such a variable always appears

when solving equations of the physical theory of mete�

∞

ρ = ρ∫ ,a m

s

dx K D

⎛ ⎞ρ= − α⎜ ⎟ρ⎝ ⎠0

ln ln sin .t mh DKH H

⎛ ⎞ρϑ = α⎜ ⎟ρ⎝ ⎠1

0

sin .mDH

Table 1. Effective heights of explosions (in km) for impacts of asteroids having a diameter D and falling at an angle α; theunfilled cells correspond to variants in which the most part of the initial energy is released at the instant of the impact at theEarth's surface and the effective height over the surface equals zero

D, m α = 90 α = 60 α = 45 α = 30 α = 15 α = 10 α = 5

20 16 18 19 23 29 32 35

40 10 13 14.5 16 21 25 29

70 3 6 8 13 19 23 26

100 0.2 8 13 16 21

200 1 6 12 18

500 0 7.5

Table 2. Effective heights of explosions (in km) for impacts of comets having a diameter D and falling at an angle α

D, m α = 90 α = 60 α = 45 α = 30 α = 15 α = 10 α = 5

40 15 18 21 25 31 32 36

100 6 8 14 18 24 28

200 5.3 10 18 21

500 6 12.5

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SHUVALOV et al.

ors (see, e.g., Hills and Goda, 1993; Svettsov et al.,1995). However, this estimate does not take intoaccount the transverse expansion of the meteoroid(and an increase in the area of its transverse cross sec�tion) after the disruption.

The velocity of the transverse expansion of a falling

body u is determined by the formula

(Grigoryan 1979; Hills and Goda, 1993), and varia�tion in the transverse size r is described by the equation

(3)

where V is the meteoroid velocity and C is a constanton the order of unity. The condition of decelerationcan be obtained by assuming that the meteoroidincreases its size r two fold during the time of descentto the height H; our calculations show that a strongfragmentation occurs at approximately such increasein size. From this condition, we obtain

(4)

i.e., the solution depends on the dimensionless vari�

able Certainly, these estimates

are rather rough but they suggest the form of the

ρ=

ρ,a

m

u CV

ρ=

ρ

( ),a

m

tdr CVdt

⎛ ⎞ρ= − α⎜ ⎟ρ⎝ ⎠0

2 ln sin ,t mh DCH H

⎛ ⎞ρϑ = α⎜ ⎟ρ⎝ ⎠2

0

sin .mDH

approximation to be found. Since it is not clear before�hand which of the dimensionless variables, ϑ1 or ϑ2, isbetter, we tried approximations with respect to the

variable with different values of β.

The analysis showed that the best results are obtained

at β = 2/3. The dimensionless dependence

at β = 2/3 and Н = 7.5 km is shown in Fig. 4. It can beapproximated by a simple function, linear with respectto the logarithm of ϑ:

(5)

DISCUSSION

Results of individual calculations can differ (anddiffer) from the approximate dependence by 2–3 km.Such a spread seems to be natural because even resultsof several calculations with the same initial data cannoticeably differ (by the same 2–3 km) from eachother (Shuvalov and Trubetskaya, 2006). The fact isthat the process of deformation and fragmentation ofa quasi�liquid meteoroid is accompanied by the devel�opment of Rayleigh–Taylor and Kelvin–Helmholtzhydrodynamic instabilities on the body surface(Svetsov et al., 1995). The development of instabilitiesis of random character; therefore, deformation, frag�mentation, and deceleration processes develop differ�ently in different calculations with the same initialdata, which just leads to different deceleration heights.We also note that the calculations did not take intoaccount the possible influence of internal frictionwhich can decrease the deceleration height by severalkilometers due to the fact that the friction impedes thedevelopment of instabilities (Ivanov et al., 1997; Shu�valov and Trubetskaya, 2010).

The model we use and, correspondingly, formula (5)can be applied with certainty only for sufficiently large(more than 10–30 m) stony and cometary bodieswhose interaction with the atmosphere can bedescribed in the quasi�liquid approximation. Largebodies have lesser strength and penetrate into denseratmospheric layers where aerodynamic loads arehigher. For bodies with a lesser size (10–20 m andless), their strength and features of the disruption pro�cess play a significant role. If such bodies disintegrateinto a few fragments, then, during the time betweenindividual instants of disintegration, the fragments canmove apart to distances comparable with their ownsize and, at the next stage, fly and disintegrate in factindependently of one another. In this case, a consider�able part of the fragments, being decelerated, falls tothe ground in the form of meteorites. If the body dis�integrates into a large number of fragments which have

β⎛ ⎞ρϑ = α⎜ ⎟ρ⎝ ⎠0

sinmDH

= ϑ( )thf

H

⎛ ⎞⎛ ⎞ρ= − α +⎜ ⎟⎜ ⎟⎜ ⎟ρ⎝ ⎠⎝ ⎠

2 3

0

1.3 ln sin 1.t mh DH H

4

–40

–3 –2 –1 0 1

3

6

5

2

1

+

+

+

+

+ +

+ +

++

+

+

++

Dim

ensi

onle

ss h

eigh

t, h

t/H

lnϑ

Fig. 4. Dimensionless effective height of explosion ht/H asa function of the dimensionless variable ϑ. The blackpoints show calculation results for asteroids; crosses, forcomets. The gray line is the calculation result approxi�

mated by the linear function = − ϑ +1.3 ln 1.th

H

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DETERMINATION OF THE HEIGHT OF THE “METEORIC EXPLOSION” 7

no time to separate and continue to fly together, thenthis cloud of vapors and fragments after the disruptioncan be deformed as a fluid and decelerated as a whole(Grigoryan, 1979). Conditions in the hot region leadto a more intense evaporation of fragments as if theywould fly separately and to a lesser fraction of the massfalling out as meteorites. Both the scenarios of disrup�tion (formation of independently flying fragments andformation of a cloud of fragments and vapors movingas a whole) were observed during the fall of the Chely�abinsk meteorite (Popova et al., 2013). Since thedeformation and fragmentation of a meteoroid areconsidered in our calculations under the assumptionthat it is already disrupted (i.e., in the hydrodynamicapproximation), without regard to its strength and fea�tures of the disintegration process which are importantfor relatively small bodies (less than 10–30 m), ourmodel does not allow one to determine the effectiveheight of the maximum energy release during the fallof such small bodies. From the viewpoint of the aster�oid damage, this is not very important because bodieswith a size of 10 m and less do not lead to considerabledestruction (Shuvalov et al., 2013), although cancause light damage.

The Chelyabinsk meteoroid, the size of which isestimated at 18–20 m, and the height of maximumenergy release is about 30 km (Brown et al., 2013;Popova et al., 2013; Grigoryan et al., 2013) can serveas an example of such a relatively small body. By for�mula (5), the effective energy release height for a bodyof about 20 m amounts to 23.8 km at an angle of entryof 19° and meteoroid density of 3200 kg/m3. Theobservation data for smaller bodies with a size of about10 m are very limited; one can mention fireballs ofFebruary 1, 1994; January 14, 1999; and October 8,2009, whose dimensions are estimated at 5–12 m at aninitial energy of ~10–70 kt TNT (Popova and Nem�chinov, 2005; Silber et al., 2011). The heat releasemaximum for the fireball of February 1, 1994 enteringthe atmosphere at an angle of about 45° was at theheight of 21 km (McCord et al., 1995). There are nodata about the angle of entry for two other fireballs.The height of the energy release maximum for the fire�ball of January 14, 1999, was 35 km. For the fireball ofOctober 8, 2009, the energy release height is estimatedindirectly by an infrasonic signal, and the authors (Sil�ber et al., 2011) believe that the main energy releaseoccurred at heights of 30–50 and 15–20 km. The esti�mate for the effective explosion height for stony bodieswith a diameter of 5–10 m by formula (5) lies withinthe range of 21–41 km for angles of entry of 90°–10°and within the range of 24–28 km for an angle of 45°.There are more comprehensive data (including themeasured territory and collected meteoritic matter)for several bodies with a size of 3–4 m (Sutter’s Mill,Almahata Sitta, and Tagish Lake (Jenniskens et al.,2012)), which are beyond the validity limits of ourmodel. Comparison with these data makes it possibleto estimate the extrapolation error of formula (5) for

bodies with a size less than 20–30 m. It turns out thatthe error in the determination of the effective heightfor 10–20 m bodies can reach 10–15 km. It should beemphasized that the deceleration height (the height ofthe main energy release) for small bodies stronglydepends on the meteoroid strength and structure (sys�tem of fractures and inhomogeneities) which cannotbe determined before the fall. For this reason, it is dif�ficult to predict the “explosion” height for such smallbodies with a good accuracy, although there exist somespecial models relying on statistical results.

The effective energy release height can be also rap�idly estimated by the widely known Earth ImpactEffects Program (Collins et al., 2005) posted on theInternet. For stony bodies, it is assumed that the dis�ruption begins at loads of 0.2–0.6 MPa (depending onthe meteoroid density), which does not contradict esti�mates for the strength upon the first disruption in theatmosphere according to observation data (0.1–1 MPa(Popova et al., 2011)). The effective energy releaseheight is determined using the pancake model describ�ing the transverse spreading of a fragmented meteor�oid (Chyba et al., 1993). It somewhat differs from themodels (Grigoryan, 1979; Hills and Goda, 1993).

In Fig. 5, effective heights of the meteoric explo�sion by formula (5) are compared with effective decel�eration heights calculated by the program (Collinset al., 2005) for body diameters from 20 to 100 m anddifferent angles of entry into the atmosphere. Theseeffective heights also correspond to a drop in the bodyvelocity approximately by a factor of two. We chose anasteroid density of 3300 kg/m3 (the typical density ofordinary chondrites) and comet nucleus density of1000 kg/m3. Since the calculation results obtained forthe explosion height by use of the program depend on thevelocity (although very weakly, plus or minus 1 km), weput the asteroid velocity to be equal to 20 km/s; thecomet velocity, to 50 km/s. The comparison of for�mula (5) with estimates by Collins et al. (2005) showsthat they can differ by 5–7 km; the largest difference isobserved for small angles of entry (5°–15°).

Since we believe that the model (Chyba et al.,1993) is less justified than the pancake model (Grigo�ryan, 1979; Hills and Goda, 1993), we compared for�mula (5) also with this pancake model for which oneobtains the explosion height. If the equation of thebody motion is written as

(6)

where m is the body mass, t is time, and CD is the resis�tance factor, then one can obtain from Eqs. (3) and (6)the following expression for the effective explosion

= − ρ π +

2 2( 2 ) ,D adVm C V D utdt

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SHUVALOV et al.

height ht of a spherical meteoroid (when its velocitydrops by a factor of two) in an isothermal atmosphere:

(7)

It is written in an implicit form because ρa =ρ0exp(–ht/H).The numerical solution of (7) different sizes D andangles of entry α is presented in Fig. 6, where it is com�pared with formula (5). We set CD = 0.5 and C = 0.7. Itis seen that the model (Grigoryan, 1979; Hills andGoda, 1993) agrees much better with formula (5) thanthe model (Chyba et al., 1993) used in the program.The largest difference was 2.5 km.

In addition, the Earth Impact Effects Program(Collins et al., 2005) considerably underestimates theexcess pressure at the surface. In the zone of destruc�tion from the Chelyabinsk fireball (at distances of upto 40 km from the trajectory projection), the excesspressure is estimated at 2000–4000 Pa (Brown et al.,2013; Popova et al., 2013) while the approximationproposed by Collins et al. (2005) predicts a pressure ofonly 100–200 Pa. Such low pressures would hardlycause noticeable damage in populated localities andwould not cause the observed seismic effect with a

⎛ ⎛ ⎞ρ ρ+⎜ ⎜ ⎟⎜α ρ α ρ⎝ ⎠⎝⎞⎛ ⎞ρ+ =⎟⎜ ⎟ ⎟ρ α⎝ ⎠ ⎠

3 2

2

3 162 sin 3 sin

8 ln 2.sin

a aD

m m

a

m

C H CHD D

CH

D

magnitude of 3.7–4 (Tauzin et al., 2013; Brown et al.,2013).

The case of iron cosmic bodies which possesshigher strength requires special consideration. Suchcosmic bodies occur much more rarely than stonyones. Their falls amount to about 3–5% of all observedfalls of cosmic bodies but it is remnants of iron mete�oroids that are found in a considerable number of largeremaining impact craters. Even small (on the order ofa meter) iron bodies penetrate into the Earth’s atmo�sphere deeper than stony bodies and, all the more,comet�like ones; for this reason, they can be ratherdangerous.

A good example is the fall of the Sikhote Alin ironasteroid with a mass from 200 to 500 t (size of 4–5 m)(Nemchinova and Popova, 1997). According to thesynthesis of the eyewitness evidence, the fragmenta�tion occurred in a few stages at heights of 58, 34, 16,and 6 km (Divari, 1959). Individual fragments with amass of up to 1700 kg, after the deceleration, fell onthe surface with velocities of several kilometers persecond (Svettsov, 1998). The main crater field of theSikhote Alin fall (the area of possible damages) haddimensions of 0.3 × 0.6 km. The effects of the fall ofsuch bodies can be predicted only under certainassumptions about the strength which can vary withinrather wide limits. Strong bodies can reach the Earth’ssurface almost undisrupted and can fractionize duringflight into individual large fragments which diverge at

30

200

40 60 80 100

20

10

Asteroid, 20 km/s

Exp

losi

on h

eigh

t, k

m

Body diameter, m

30

200

40 60 80 100

20

10

Comet, 50 km/s

Exp

losi

on h

eigh

t, k

m

Body diameter, m

40

50

Fig. 5. Comparison of formula (5) with results of calculations according to the technique by (Collins et al., 2005). The figureshows dependencies of the effective explosion height on the body diameter for different angles of the entry into the atmosphere.The solid curves were constructed by formula (5); dashed curves, according to (Collins et al., 2005). Each curve was constructedfor a definite angle of entry. These angles are equal to 90°, 60°, 45°, 30°, and 15° and correspond to curves in the sequence frombottom to top. The left figure contains dependencies for a body with a density of 3300 kg/m3 and velocity of 20 km/s; the rightfigure, for a body with a density of 1000 kg/m3 and velocity of 50 km/s. The characteristic height of the atmosphere is H = 8 km.

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DETERMINATION OF THE HEIGHT OF THE “METEORIC EXPLOSION” 9

a certain distance and then can continue to fractionizeinto smaller fragments. For such cases, approximatemodels were proposed (see, e.g., Svetsov et al., 1995);however, they yield a large spread of results and theanalogy with an focused explosion becomes rougherbecause the deceleration occurs on a longer part of thetrajectory.

ESTIMATES FOR THE TUNGUSKA EVENT

Using formula (5), one can make some estimatesfor the Tunguska event of 1908 when the release ofenergy by a cosmic body in the atmosphere caused aforest fall by the shock wave over an area of about 2000km2, a seismic wave, a forest fire, and a series of otherphenomena (see, e.g., Vasil’ev, 2004). Using the regionof the forest fall and seismic effect, one can estimatethe range of the explosion heights and energies corre�sponding to this event. In (Svettsov, 2007), numericalsimulation of focused explosions in the atmospherewas performed and their energies and heights that bestagree with features of the forest fall in the Tunguskaevent were determined. The features are as follows:radius of the zone of standing trees near the explosionepicenter, maximum distance from the epicenter witha strong forest fall, and average radius correspondingto light injures in trees at the periphery. According tocalculations of (Svettsov, 2007), the explosion energyin the Tunguska event amounted to from ~6.5 to ~19 Mtat a height of ~6.5 to 10 km; for each value of energy,

the interval of heights at which a satisfactory agree�ment in the forest fall was reached was determined.For example, for the explosion energy of 10 Mt, agood agreement was found for an explosion heightfrom 6.8 to 8.2 km. The less the energy, the lower theexplosion. Figure 7 shows the region of possibleheights and energies that yield a good agreement withthe forest fall; this region is contained inside a closedcontour.

Using formula (5), we can construct curves ofexplosion heights as functions of the body energy fordifferent velocities and angles of entry and determineat which body parameters these dependencies fallwithin the region of possible heights and energies. Indoing this, we assume that the angles of entry cannotexceed 45°. The fact is that the trajectory of the Tun�guska cosmic body, according to evidences of eyewit�nesses from Preobrazhenka village (Vasil’ev et al.,1981), went somewhere over this village situated at347 km from the epicenter. Inhabitants of Preo�brazhenka could observe near the zenith either a flightof a fireball or a plume, i.e., particles ejected upwardalong the trace of the cosmic body in the atmosphere.In both cases, the inclination angle of the trajectory tothe horizontal could not exceed 45° because the fire�ball glow begins only at heights of about 100 km andthe visible part of a plume ejected upward does notreach values corresponding to the distance form thePreobrazhenka village if the inclination angle of thetrajectory to the horizontal exceeds 45°.

30

200

40 60 80 100

20

10

Asteroid

Exp

losi

on h

eigh

t, k

m

Body diameter, m

30

200

40 60 80 100

20

10

Comet

Exp

losi

on h

eigh

t, k

m

Body diameter, m

40

Fig. 6. Comparison of formula (5) with the model (Grigoryan, 1979; Hills and Goda, 1993). The figure shows dependencies ofthe effective explosion height on the body diameter for different angles of the entry into the atmosphere. The solid curves wereconstructed by formula (5); dashed curves, according to Eq. (7). Each curve was constructed for a definite angle of entry. Theseangles are equal to 90°, 60°, 45°, 30°, and 15° and correspond to curves in the sequence from bottom to top. The left figure con�tains dependencies for a body with a density of 3300 kg/m3; the right figure, for a body with a density of 1000 kg/m3. The char�acteristic height of the atmosphere is H = 8 km.

10

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SHUVALOV et al.

Figure 7a presents curves of the effective explosionheight as a function of the body energy by formula (5)for a body with a density of 3300 kg/m3. It is seen fromthis figure that, at large velocities, a stony body thatmight cause the Tunguska event turns out to have a rel�atively small size, decelerates too high, and the explo�sion height does not fall within the region of possiblevalues. For a body of cometary origin, restrictionsupon its velocity are even stronger. We obtain that thelimit velocity above which curves (5) pass beyond theregion of possible values for all angles of entry is equalto 33 km/s at ρ = 3300 kg/m3, 26 km/s at ρ =2000 kg/m3, 18 km/s at ρ = 1000 kg/m3, and 14 km/sat ρ = 600 kg/m3. The higher the velocity, the narrowerthe range of admissible angles. Indeed, as is seen fromFig. 7, at ρ = 3300 kg/m3 and V = 30 km/s, the anglesof entry can be not less than 41°; at ρ = 1000 kg/m3

and V = 15 km/s, not less than 38°. Even at a lowvelocity of 12 km/s and high body density ρ =3300 kg/m3, the limit angle of entry is equal to 23°.

Thus, our estimates exclude bodies of cometaryorigin entering the atmosphere with a high velocityand very acute angles of entry. Since the velocities ofmeteoric showers observed on the Earth are high,more than 18 km/s, we can exclude the origin of theTunguska cosmic body as a result of disintegration ofcomets creating these showers. But we cannot exclude

short�period comets whose velocities of impacts at theEarth can be lower than 15 km/s (Jeffers et al., 2001).It is evident that if the angles of entry exceeded 20°, itwas impossible to see a fireball flight from Preo�brazhenka and the inhabitants observed a glowingplume. We suppose that modeling the plume with anestimate of its visibility at different angles and dimen�sions of the body could even more narrow down thepossible range of parameters of the Tunguska event.

MAIN CONCLUSIONS

The calculations performed showed that the explo�sive analogy can be used as a fast estimate of effectscaused by the fall of cosmic bodies on the Earth. Theeffective height of the “meteoric explosion” can beestimated by formula (5) obtained based on the com�parison of direct numerical calculations of cosmicbody impacts and focused explosions. The formula isapplicable for the case of sufficiently large (more than10–30 m) stony and cometary bodies with densitiesfrom ~1000 to ~4000 kg/m3 and entry angles of up to 5°.However, to obtain more reliable results, especially forsmall inclination angles of the trajectory, it is necessaryto solve the complete problem of the impact, i.e., toperform a numerical simulation based on physico�mathematical models.

7

66

7 8 9 10 11 12 13 14 15 16 17 18 19 20

9

8

11

10

25 23

4145

30

V = 12V = 15

V = 20

V = 30

30

45

45

45

Body energy, Mt

Exp

losi

on h

eigh

t, k

m(а)

7

66

7 8 9 10 11 12 13 14 15 16 17 18 19 20

9

8

11

10

38

45

32

V = 12

V = 1545

Body energy, MtE

xplo

sion

hei

ght,

km

(b)

Fig. 7. Determination of possible parameters of the Tunguska cosmic body. The closed contour shown by the solid thick curveoutlines the region of possible values of energies and heights of an explosion equivalent to the Tunguska event. The curves inter�secting this region are dependencies obtained by formula (5) using the relation of the body size to the body energy and velocityfor different velocities from 12 to 30 km/s and angles of entry from 45° to 23°. The dependencies are shown for a body with adensity of (a) 3300 and (b) 1000 kg/m3. The Tunguska event is associated with curve segments falling within the region of possiblevalues.

SOLAR SYSTEM RESEARCH Vol. 50 No. 1 2016

DETERMINATION OF THE HEIGHT OF THE “METEORIC EXPLOSION” 11

ACKNOWLEDGMENTS

This work was supported by the Russian Founda�tion for Basic Research, project no. 13_05_00309_a.

We would like to thank anonymous reviewers foruseful remarks.

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Translated by A. Nikol’skii