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Modeling of Energy Release at the Final Stage of the Meteoroid Movement

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Modeling of Energy Release at the Final Stage of the Meteoroid Movement Open Astron. 2018; 27: 290–293 Research Article Lidia A. Egorova* and Valery V. Lokhin Modeling of energy release at the final stage of the meteoroid movement https://doi.org/10.1515/astro-2018-0035 Received Feb 06, 2018; accepted Aug 16, 2018 Abstract: The paper continues to build upon the author’s previous research on fireballs fragmentation. A model of the sudden explosive destruction of the cosmic body at the height of the maximum flash is used. After the fragmentation, the kinetic energy of the moving particles of a meteoroid passes into the thermal energy of the gas volume in which their motion takes place. The temperature of a gas cloud calculated analytically using energy conservation law and equations of physical theory of meteors. The mass distribution of fragments was taken from the literature. The high temperature of the gas in a cloud allows us to talk about the phenomenon of a "thermal explosion". Keywords: meteoroid fragmentation; fireball motion in atmosphere; physical theory of meteors 1 Introduction brightest point was proposed by Kruchynenko & Warm- bein (2001). Recent researches contain simulation of the entry and Simulation of meteoroid interaction with the Earth at- destruction of meteoric bodies in the atmosphere to eval- mosphere continues to be in a high interest since the uate the released energy. Shuvalov et al. (2013) introduced Chelyabinsk event in 2013. The Chelyabinsk phenomena the concept of “air giant bolide (GB) or air “meteor ex- shows that even a body 10 m radius can cause catastrophic plosion” (when the products of a completely disintegrated events. It occured in a densely populated area so the nu- and evaporated meteoroid are decelerated in the atmo- merous observations was made as mentioned by Popova et sphere and do not reach the Earth’s surface but the shock al. (2013). These data permit scientists to verify their mod- wave and thermal radiation produce noticeable destruc- els. tion and fires)” for the Chelyabinsk event. Fragmented The phenomena of a similar size take place quite of- body came under full vaporization according to model ten and at the same time they are practically impossible with air jet flow formation. Numerical simulations was in to detect outside the atmosphere (see Popova & Nemchi- a good agreements with observation data. nov 2008; Shuvalov et al. 2013). Therefore, the study of this The above works are very useful for practice, but have class of meteoroid motion and destruction are of particular not clarified the source of released thermal energy after interest to researchers. crushing. Thus we proposed a model of transferring the ki- Ground and space observations of meteoric bodies netic energy into thermal energy. record their brightness along the trajectory. The results are usually represented as a graph - the luminosity curve. For meteoroids larger than 10 cm a sharp increase of emission 1.1 Catastrophic Fragmentation of the is often recorded. This increase often interpreted as body fragmentation or even more as terminal explosion.The Meteoroid theory of terminal explosion based on the proposal that There are plenty of models for meteoroid fragmentation. the maximum of deceleration should be achieved at the The review can be found in Popova (2004); Popova & Nemchinov (2008). We concern with a class of phenom- ena when the energy of a fragmented meteoroid reaches the Earth surface. We assumed that the body breaks up Corresponding Author: Lidia A. Egorova: Institute of Mechan- into separate fragments of different sizes moving indepen- ics of Lomonosov Moscow State University, 119192, Michurinskiy, 1, dently. The explosive nature of the observed effects sug- Moscow, Russia; Email: [email protected] Valery V. Lokhin: Institute of Mechanics of Lomonosov Moscow gests that the aerodynamic drag is several times greater State University, 119192, Michurinskiy, 1, Moscow, Russia than the strength of the body or fragments of the pre- Open Access. © 2018 L. A. Egorova and V. V. Lokhin, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License L. A. Egorova and V. V. Lokhin, Modeling of energy release at the final stage of the meteoroid movement Ë 291 destroyed body. The high velocity of the meteoroid and the The solution of equation (5) could be found in Stulov exponential growth of the air density as the altitude de- et al. (1995) and is rather complicated. So the simplifica- creases make this proposal reasonable. Under those condi- tions are following relying on the nature of its constituent tions we assume the distribution of fragments in size simi- constants. lar to the distribution of fragments of a body suddenly de- We propose CH = 0.01 as we consider rather small par- stroyed by an explosion. There are a number of papers de- ticles of the meteoroid, CD = 1 for the spherical particles at scribing that distributions of fragments. For instance see altitude less than 50 km, Q = 1.6747·104 m2/c2 as suggested Fujiwara (1986) and Wohletz & Brown (1995). We take into in Stulov et al. (1995). Thus we estimate the parameter of account the distribution from Nemchinov et al. (1999) ablation κ = CH << 1. Also we assume that particles ve- 6CD Q locity at the end of their pass is a small quantity compared dNm k −2 = Cm 3 , k = 1.2. (1) V2 to body velocity before the fragmentation 2 << 1. Under dm V0 these assumptions the velocity Here m is a mass of fragment and Nm is a number of fragments having mass m. C and k are constants. (︂ )︂ (︁ 2)︁ z V = V0 exp −A 1 + κV0 , (6) In Egorova & Lokhin (2016) for this type of fragmen- r0 tation we calculated the light curve analytically from the 3 ρ A = − C . luminosity relation and the equations of the physical the- 8 D δ ory of meteors. The obtained analytical solution for lumi- nosity along the trajectory has a close fit with the Marshall Islands fireball light curve. This fragmentation model is 1.3 Temperature of a Gas Cloud used in present paper. We assumed the destruction of the meteoroid into many fragments. The part of kinetic energy of the moving parti- 1.2 Solution of Equations of the Physical cles passes into the thermal energy of the gas volume in Theory of Meteors which their motions take place. Let us estimate the loss of the kinetic energy E of the whole ensemble of particles To find the parameters it is necessary to solve the equa- with a size distribution (1), which has converted to ther- tions of the physical theory of meteors for particle mal energy during deceleration (during the traversed path z) after fragmentation. Let us introduce the initial relative dV 1 2 M = − SC ρV (2) radius r0 = r0/R, mass m(r0) = m0/M and relative velocity dt 2 D V(r0, z) = V/V0. Then we integrate kinetic energy of par- dm 1 3 ticle over all radiuses and subtract from the initial kinetic Q = − SCH ρV (3) dt 2 energy where V, m, S, Q – velocity, mass, midsection area of par- 2 ticle and enthalpy of mass loss due to aerodynamic forces V0 E = M0 (7) and heat transfer, CD and CH are the drag and heat transfer 2 1 coefficients (constants), ρ is atmosphere density. Z " 2 !# 2 d V (r0, z) Let us divide equation (3) by (2) and transfer terms − M0V0 N (r0) m(r0) dr0 dr0 2 with variables m and V in different directions and solve 0 2 the equation. We assume that the body remains spherical V0 = M0 with radius r and moves vertically. It gives for the radius 2 8 1 9 " 2 !# Z 2A(1+κV )z [︂ C (︁ )︁]︂ < 2 (︁ −9/5 )︁ d 3 − 0 = H 2 2 · 1 − r − 1 r e Rr0 dr . r = r0 exp − V0 − V (4) 3 0 dr 0 0 6CD Q : 0 ; 0 where V , r are the initial velocity and radius. 0 0 Denoting If entered variable height (z) instead of time the mo- (︁ )︁ z mentum equation could be written as B = 2A 1 + κV2 0 R dV 3 ρC dz = − D , (5) V 8 δ r we calculate 1 ∞ where r is expressed by (4). Here δ is the meteoroid density, Z Z 6 −1/5 − B −B 6/5 −6/5 −y z is the vertical coordinate measured from the altitude of r e r0 dr = e − B y e dy = (8) 5 0 0 fragmentation. 0 B 292 Ë L. A. Egorova and V. V. Lokhin, Modeling of energy release at the final stage of the meteoroid movement 0 B 1 (︂4)︂ Z e−B − 5Be−B + 5B6/5 Γ − y−1/5e−y dy . @ 5 A 0 Here Γ is gamma function. Estimations shows that B << 1, then the integrand can be expanded in a series and then it is easy to calculate B Z 5 (︁ )︁ y−1/5e−y dy = B4/5 + O B4/5 , (9) 4 0 where 8 ∞ 9 2 Z V0 < −B 5/6 −1/5 −y = E = M0 1 − e + (B) y e dy (10) 2 : ; B (︂ )︂ E 6/5 4 2 = 1 − 6B + 5B Γ + O(B) (11) V0 5 M0 2 The volume of the gas heated by the particles in first (a) approximation is a cylinder of length L (for a typical case of a fireball flight we put it equal to 1 km) and radius R*. The radius of the cylinder is equal to the speed of prop- agation of the perturbations (we set it equal to the speed of sound) multiplied by the time t.
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