Applying the formula for mass distribution of fragments of disrupted body to meteorite showers

Abstract

Formula for the cumulative mass distribution of fragments of disrupted body, obtained as function of fragment mass, mass fraction of the largest fragment(s), number of largest fragments, and power index, is used to describe mass distribution of recovered meteorites for eight meteorite showers.

1. Introduction

To model independent motion and ablation of meteoroid fragments it is necessary to know their mass distribution. In this regard, an analogy can be drawn with impact experiments on high-velocity collisions, which were performed to simulate asteroid destruction. In many experiments [1, 2, and others] it was noted that cumulative mass distribution curve is well described by a power law. This is usually represented as cumulative number of fragments with masses larger than or equal to m is proportional to the power of m. This correlation between cumulative number of fragments and mass gives a linear plot in log–log coordinates with power index slope. It was also noted that whole mass distribution curve usually cannot be well represented by a single exponent in the power law and is divided into two or three segments with a steeper slope for larger fragments. 

Here, the power law is used not for cumulative mass distribution, but for distribution of probability density. In this case, cumulative distribution is not a linear function of mass in log–log coordinates and enables to adequately describe results of impact experiments by single curve, i.e. using single exponent. We compare the proposed cumulative distribution with mass distributions of meteorites when a large number was collected.

Earlier, the power law distribution in discrete form was used for the grain mass distribution in studies of small meteoroids [3, 4, and others]. Different approaches were suggested to approximate mass distribution of found meteorites [5–7, and others].

2. Probability and cumulative distributions

We assume the power law for probability density function n_m:

n_m = Dm^-β-1,   n_m = –dN_m/dm.          (1)         

Here m is fragment mass, N_m is cumulative number of fragments with masses larger than or equal to m; exponent β is constant. Normalizing coefficient D is found using the equation of conservation of the total mass of fragments M (mass of the meteoroid just before breakup, mass of the target in experiments). Then probability density n_m has a form (β < 1)

n_m = M(1-β)/(m_l^1-β)m^-β-1 .                 (2)

Here m_l is mass of the largest fragment. Cumulative number of fragments N_m is found by integration of the second equation (1)

N_m = (1-β)/(βl^1-β){(m/M)^-β-l^-β}+c.      (3)      

Here c is the number of largest fragments, l = m_l/M is mass fraction of the largest fragment. Probability density n_m (2) can be used to model meteoroid fragmentation; the total mass and energy deposition can be found by integration over all fragment initial masses.

3. Meteorite distributions

We test formula (3) by comparing with mass distributions of recovered meteorites for eight meteorite showers: Tsarev, Sikhote-Alin, Mbale, Bassikounou, Almahata Sitta, Košice, Sutter’s Mill, and Chelyabinsk. Comparisons show that formula (3) satisfactory describes meteorite mass distribution in cases of uniform change of fragment masses without gaps. The example of such distribution is shown in Figure 1. Asteroid 2008 TC₃ entered the Earth’s atmosphere on October 7, 2008 and fragmented over the Nubian Desert in Northern Sudan; more than 662 meteorites were recovered, named Almahata Sitta [8]. We used data from tables [8] to construct the cumulative mass distribution of meteorites.

Figure 1: Almahata Sitta meteorites. Violet dots: data from catalog [8], red and green lines: formula (3) at β = 0.67 and 0.6; 662 fragments, M = 10.55 kg, m_l = 0.379 kg.

Figure 2: Sutter’s Mill meteorites. Violet dots: data from table [9], red line: formula (3) at β = 0.5; 77 fragments, M = 942.7 g, m_l= 204.6 g.

In cases where the largest fragment(s) is several times larger than the next one, formula (3) satisfactory describes meteorite mass distribution starting from the second one. Figure 2 shows the cumulative mass distribution of Sutter’s Mill (fall on April 22, 2012) meteorites, where mass of the largest fragment is almost 5 times that of the second. To construct the meteorites distribution we used data from table [9].

4. Summary and Conclusions

Formula (3) adequately describes cumulative distribution of recovered meteorites for considered meteorite showes. Difference between analytical and empirical distributions at very small masses is natural and should be, because, unlike laboratory experiments, it is problematic to find most small particles. Preliminary estimate of the most probable range of exponent β for meteorite distributions is from 0.5 to 0.7; further research is needed to determine more accurately the range of possible β values that could be used in probability density distribution for modelling meteoroid fragmentation.

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.

References

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