THE MODELING OF BOLIDE TERMINAL ...

Earth, M o o n , and Planets (2004) 95: 513-520 D O ! 10.1007/sl 1038-005-9000-7

© Springer 2005

THE MODELING OF BOLIDE TERMINAL EXPLOSIONS G. A. TIRSKIY and D. Yu. Khanukaeva Institute of Mechanics, Moscow State M.V. Lomonosov University, Michurinskiy-1, Moscow, Russia (E-mail: [email protected])

119192,

(Received 15 October 2004; Accepted 13 May 2005)

Abstract. T h e phenomenon of terminal thermal explosions of bolides is considered and mathematically modeled, using the mechanisms of ablation and fragmentation due to mechanical and thermal stresses. The definition and criterion of thermal explosions are given. A n analytical solution is obtained for the model of ablating and mechanically fragmenting meteoroid motion in the atmosphere. Numerical calculations including the terminal stage of the motion are fulfilled for the Tunguska parameters. They demonstrated a very rapid energy loss, corresponding to the terminal flare and full mass loss, explaining the absence meteorites. Keywords: Bolides, fragmentation, meteorites, meteoroids, terminal flares, thermal explosion

1. Introduction The existence of the phenomenon of terminal flares or explosions of bolides is known in Physical Theory of Meteors (Bronshten, 1983; Ceplecha et al., 1998). Professional and amateur observers regularly register flares. For example, according to the latest reports of American Meteor Society more than 20% of meteors demonstrate terminal flashes. Characteristic durations of the flares are of the order of fractions of a second. There can be several flares during meteoroid flight. They usually correspond to intensive (tens of percents) but not total mass loss. There are a number of works, devoted to the discussion of bolide terminal explosions (see i.e. Levin and Bronsten, 1986 and ref. therein). Data of the Prairie and European Networks on fireballs with terminal flares were collected by Sekanina (1983). Thus far, there is no satisfactory physical explanation of the very phenomenon and the absence of meteorites on the surface of the planet after it. The present paper is devoted to physical-mathematical modeling of terminal explosions connected with full mass loss of the body. The suggestion of the paper is that the phenomenon is due to combined action of thermo-mechanical fragmentation and ablation of fragments. We believe that both the preliminary mechanical fragmentation of the meteoroid and the mechanism of thermal stresses play important roles.

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2. The Criterion of a Terminal Thermal Explosion Thermal explosion is a very fast energy transfer to the atmosphere with the conversion of the whole body mass into vapor. Words "very fast" mean in fractions of a second, in a narrow interval of heights (less than 1 km). Thus, there are two conditions, which must be satisfied, in order we could say that a thermal explosion of a bolide took place. The first one is the fast energy transfer to the atmosphere; the second is the total mass loss of the body. The efforts of mathematical modeling of terminal explosions have been done since the 1950s. The first papers (Smith, 1954; Pokrovsky, 1966) considered qualitatively the process of catastrophic fragmentation as an explosion. Quantitative model of such fragmentation was developed by Grigorian (1979). An effect of a very intensive mass loss was achieved by some strong assumptions in (Petrov and Stulov, 1975) and using the idea of porosity in (Liu, 1978). But none of the listed models explained both the rapid energy deposition and full mass loss of the body. Purely gasdynamical considerations were made by Shurshalov (1984) and Kondaurov et al. (1998), but they were not connected with the previous conditions of meteoroid motion. So, the description of the coordinated scenario of meteoroid interaction with the atmosphere including terminal stage remained to be an open problem. The thermal explosion criterion was formulated by Liu (1978) as follows: W * b < l> 0) s sca e where characteristic ballistic time ть ~ тШг- h' ' height, V is the meteoroid velocity, 9 is the trajectory inclination angle to the horizon; characteristic time of full mass loss t m ~ nG jfy- t , G is the drag coefficient, .

(2)

The physical meaning of expression (2) is that the process of energy loss takes very short period of time (less then 1 s), independently of the duration of the body motion. And the same combination of variables as for the first

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case must be much smaller in comparison with the fixed time interval of the order of a second.

3. Meteoroid Mechanical Fragmentation From our point of view at least two mechanisms are responsible for the realization of condition (2). The first one is the mechanical fragmentation due to the aerodynamic load pV2. The ideas of the model, analytical solutions and their consequences are presented in another paper by the authors. Here we cite concisely only its general ideas. Fragmentation starts, when pV2 exceeds the strength limit of the body a*, which is defined according to statistical theory of strength (Weibull, 1939): (3)

where crc is the initial strength of the body, Mc is the initial mass, M* is the total mass at the moment of fragmentation, a is an empirical scale parameter, characterizing the homogeneity of the meteoroid material (the more homogeneous body, the less the value of a). According to relation (3), the strength of the fragment is larger, the smaller its size. While the aerodynamic load is growing, the fragmentation goes on. Following Fadeenko (Fadeenko, 1967) we assume all fragments to be identical, i.e. having one and the same mass equal to Mf = M/N, where N is the number of fragments. If we apply condition (3) to each fragment of mass Mf and strength a and take into account that cr(cr*)_1 = pV2(p^ Vl)~l we obtain: (4)

The subscript star is used in (4) and below for the values of variables at the moment of fragmentation beginning. The values of a, ae and Mc are assumed to be known. Formally, N may be noninteger, but its integer part should be taken when this number is determined. The equations of drag and ablation of fragments conglomerate have the same forms as in the classical physical theory of meteors (Bronshten, 1983) with the total cross section area depending on N:

where 5 is the meteoroid density, the shape factors of all of the fragments ' fk = 1-21 (spheres), so / = 1.21 too.

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G. A. TIRSKIY AND D. YU. KHANUKAEVA

This statement of the problem of progressively fragmenting meteoroid deceleration and ablation allows the following analytical solution: -3g 2

V — V*

м

=

м Л

(5)

у 2

у

-

(6)

" \

where p* = the scale height h, the trajectory slope to the horizon 0; the ablation parameter a, the drag coefficient G are constant. The convenient analytical form of solution (5-6) makes it possible to find all the characteristics of meteoroid fragments at any moment, in particular, the sizes" of fragments. They are important for the evaluation of thermal stresses imposed, which represent the second mechanism involved for the physical explanation of the explosion phenomenon.

4. Thermo-stresses The heat equation in a spherical body of radius R with zero initial temperature and constant surface temperature T w equal to the temperature of the surrounding medium has the solution, which represents the temperature distribution as (see e.g. Koshlyakov et al., 1970 or Tikhonov and Samarskiy, 2004) r(M) = 7 w < f l + I Ttr \

. nnr ( /7Ш\ 2 \ sm—exp( - Ц - ) \ i n=1

n

where % is the temperature conductivity ( x ~ 10 1 — 10 ' 2 cm2 s 1 for iron and stony meteoroids), t is the time, r is the radial coordinate counted from the center of the sphere, so r = R corresponds to the sphere surface. In more accurate statement boundary condition should include thermal radiation from the particle surface, which makes the problem nonlinear and requires numerical solution. Since we use the solution only for estimates of stresses, we remain in the frames of the simplified statement and leave its exact consideration for future works. Temperature gradients inside the body are the reasons of thermo-stresses. The distributions of radial and tensile stresses inside a sphere are (Hopkinson, 1879): °r=^^[T(R,t)-T(r,t)],

(7)

MODELING OF BOLIDES TERMINAL EXPLOSIONS

2pE T{R,t)^-T{r,t)

+

l

-T{r,t)

517

(8)

3(i-0

where ft is the thermal expansion coefficient, E is the Yung module, )

Figure I. The change of energy deposition to the atmosphere with height, (a) single body . model; (b) mechanically fragmenting body model; (c) mechanically and thermally fragmenting body model.

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sonable explanation of bolide flares and the absence of meteorites on the surface of the planet after them. The model assumes independent motion of the fragments, while some time is required for their separation. It puts a restriction to the application of the model for extremely large bodies (with sizes of tens of meters) and small values of a (close to zero). Restriction dealing with body sizes is given in (Svetsov et al. 1995). The separation of fragments was considered in (Khanukaeva, 2002) for various values of a and formulated in terms of the delay before fragments origination. The estimates made for 1 m-sized body gave 24 km shift in altitudes of fragments appearance after the fragmentation condition realization. We neglected this delay in the present work. More accurate consideration should include the stage of interacting fragments motion and may represent further development of the model.

Acknowledgements The work was supported by RFBR Grant N03-01-00-542, Leading Scientific Schools grant N1899.2003.1, Universities of Russia Grant N04.01.020.

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Svetsov, V. V., Nemtchinov, I. V. and Teterev, A. V.: 1995, Icarus 116, 131-153. Tikhonov, A. N. and Samarskiy, A. A. 2004, Mathematical Physics Equations, MSU Press, Moscow, 798 pp. Weibull, W.: 1939, Proc. Roy. Swedish Inst. Eng. Res. 151, 32 pp.