Comparative study of models of meteoroid disruption into a cloud of fragments

Abstract

Models of meteoroid disruption into a cloud of fragments are considered: the two-parameter model, which takes into account changes in the cloud shape and density, and simple models without accounting these effects. Models are used to reproduce the energy deposition of the Chelyabinsk superbolide. Influence of ablation model is studied; for simple fragmentation models, optimal coefficient is proposed depending on heat transfer coefficient.

When meteoroid breaks up into a large number of fragments, at the first stage they move with a common shock wave, before dispersing enough distance to move independently. To simulate   meteoroid disruption at this stage, models of a cloud of fragments moving as a single body were proposed and used [1–7, and others]. Such a cloud is flattened by pressure forces: it is compressed in a flight direction, and spreads out in a lateral direction. Fragment cloud models differ in equations for the rate of cloud lateral expansion. Comparison of models [3, 4] was made [7] for energy deposition modeling for the Chelyabinsk and Tunguska events. Here we consider two-parameter model [6], which takes into account changes in the fragment cloud shape and density, and simple models, for example [1, 4], without accounting these effects. We also consider the above models with limited midsection radius growth. We compare models as applied to reproducing the energy deposition of the Chelyabinsk superbolide. For correct comparison, we use them with the same ablation and drag coefficient models.

2. Fragmentation models

Spherical shape of meteoroid is assumed before breakup starts, then it continues its flight as cloud of fragments and vapor, moving as a single body, and sphere is transformed into spheroid with ratio of axes k. In model [6], in addition to flattening parameter k, parameter γ is introduced, characterizing decrease of fragmented meteoroid density δ due to increase of spacing between fragments: δ = δ_e/γ³ (γ ≤ 3), where δ_e is initial meteoroid density. Equation for the rate of increase of the fragment cloud midsection radius R_S in two-parameter model [6] has a form:

 dR_s/dt = (γ³/k)^1/2(ρ/δ_e)^1/2V,   k = 4πδ_eR_S³/(3Mγ³),  γ = (γ_m - 1)( ρ^1/2 - ρ_f^1/2)/(ρ_m^1/2 - ρ_f^1/2)      (1)

Here t is time, ρ is atmospheric density, V and M are meteoroid velocity and mass; subscripts f and m correspond to values at heights of fragmentation start h_f and maximum bolide brightness h_m. Parameter γ_m is adjusted to match the observed height h_m.

In simple models equation for the midsection radius R_S is

dR_s/dt = c(ρ/δ_e)^1/2V,   c = const                   (2)

Here c = 1 and c = (7/2)^1/2 correspond to models [1, 5] and [4]. Simple models, in contrast to the two-parameter model, do not take into account decrease of density of disrupted meteoroid and change of its shape. Solution of equation (2) shows that midsection radius is determined only by initial parameters, ablation does not affect it, so fragmentation problem is separated from ablation and motion problem. For two-parameter model these problems are coupled.

3. Calculation results

We used various fragment cloud models to simulate interaction of the Chelyabinsk meteoroid with the atmosphere, solving ablation and motion equations together with equations (1), or (2), by Runge-Kutta method. Data [9] were used as initial parameters. Unknown entry mass was determined to match the observed maximum energy deposition [10]. For the drag coefficient we used: C_D = 1.78 – 0.85/k. For the radiative heat transfer coefficient C_H we used approximate formula depending on V, ρ, R_S and k [6, 8]. Uncertainty parameter ψ is introduced in С_Н formula to account for effects of precursor, absorption by vapor layer, and other uncertain factors. Effect of C_H coefficient on meteoroid mass loss, midsection radius, energy deposition, and entry mass estimate is studied by varying parameter ψ. Constant C_H values are also used.

Figure 1: Energy deposition per unit height for two-parameter model (a) and models [1] (b) and [4] (c); black curve is observational data [10]; h_f  =  45 km.

Figure 1 shows that two-parameter model and simple model [1] enable satisfactory simulation of the observational energy deposition curve down to 27 km, respectively,  at ψ = 1 (and С_Н = 0.1), and at ψ = 0.3 (and С_Н = 0.03). They give estimates of the Chelyabinsk meteoroid entry mass, close to estimates [9, 11]. Model [4] does not reproduce observational data. We found coefficient c in equation (2) for simple fragmentation models, which provides satisfactory agreement with observational data, depending on heat transfer coefficient. Two-parameter model gives much smaller increase of midsection radius than simple models. Limiting the midsection radius growth does not improve any models.

Acknowledgements

This work was performed according to the plan of Institute of Mechanics of Lomonosov Moscow State University and was partially funded by Russian Foundation for Basic Research, grant 18-01-00740.

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