Constraining the Pre-atmospheric Parameters of Large Meteoroids: Košice, a Case Study

Abstract

Out of a total around 50,000 meteorites currently known to science, the atmospheric passage was recorded instrumentally in only 25 cases with the potential to derive their atmospheric trajectories and pre-impact heliocentric orbits. Similarly, while observations of meteors generate thousands of new entries per month to existing databases, it is extremely rare they lead to meteorite recovery (http://www.meteoriteorbits.info/). These 25 exceptional cases thus deserve a thorough re-examination by different techniques—not only to ensure that we are able to match the model with the observations, but also to enable the best possible interpretation scenario and facilitate the robust extraction of key characteristics of a meteoroid based on the available data. In this study, we evaluate the dynamic mass of the Košice meteoroid using analysis of drag and mass-loss rate available from the observations. We estimate the dynamic pre-atmospheric meteoroid mass at 1850 kg. The pre-fragmentation size proportions of the Košice meteoroid are estimated based on the statistical distribution of the recovered meteorite fragments. The heliocentric orbit of the Košice meteoroid, derived using numerical integration of the equations of motion, is found to be in close agreement to earlier published results.

Appendices

Appendix 1: List of Symbols

α:

Ballistic coefficient

β:

Mass loss parameter

γ:

Slope between horizon and the trajectory

λ:

Geodetic longitude of the beginning fireball point

μ:

Shape change coefficient

ρ₀ :

Gas density at sea level

ρ _a :

Gas density

ρ _m :

Meteoroid bulk density

ϕ:

Geodetic latitude of the beginning fireball point

a _x , a _y , a _z :

Dimensionless sizes of prefragmented meteoroid

A :

Shape factor of meteoroid

A _e :

Pre-entry shape factor of meteoroid

B ₀ :

Power-law scaling exponent

c _d :

Drag coefficient

c _h :

Heat-transfer coefficient

d :

Shape parameter of meteoroid

F _C (m):

Power-law complementary cumulative distribution function, which is an approximation of meteorite fragment distribution

h :

Height

h ₀ :

Scale height

H* :

Effective destruction enthalpy

L :

Length along atmospheric trajectory

m _i :

Fragment masses

m _L :

Minimum fragment mass limit

m _U :

Power-law cutoff mass

M :

Meteoroid mass

M _e :

Pre-entry meteoroid mass

n :

Number of considered points along the trajectory

N :

Number of fragments

N* :

Piecewise complementary cumulative distribution of meteorite fragments

S :

Head cross section area

S _e :

Pre-entry middle section area of meteoroid

t :

Time

V :

Velocity

V _e :

Pre-entry velocity

ω:

Argument of periapsis

Ω:

Longitude of the ascending node

a :

Semimajor axis

e :

Eccentricity

i :

Inclination

M :

Mean anomaly at epoch

Appendix 2: Dimensionless Quantities

$$ y=h/{h}_0 $$

$$ v=V/{V}_e $$

$$ m=M/{M}_e $$

$$ {A}_e=\frac{S{\uprho}_m^{2/3}}{M^{2/3}} $$

$$ d=1+2\left({a}_x{a}_y+{a}_y{a}_z+{a}_z{a}_x\right){\left({a}_x^2+{a}_y^2+{a}_z^2\right)}^{-1}\vspace*{8pt} $$

$$ \upalpha =0.5{c}_d\frac{\uprho_0{h}_0{S}_e}{M_e \sin \upgamma}\vspace*{8pt} $$

$$ \upbeta =0.5\left(1-\upmu \right)\frac{c_h{V}_e^2}{c_d{H}^{*}}\vspace*{8pt} $$

$$ \upmu ={ \log}_m\frac{S}{S_e}\vspace*{8pt} $$

$$ \Delta =\overline{\mathrm{E}}\mathrm{i}\left(\upbeta \right)-\overline{\mathrm{E}}\mathrm{i}\left(\upbeta {v}^2\right)\vspace*{8pt} $$

$$ {\Delta}_i=\overline{\mathrm{E}}\mathrm{i}\left(\upbeta \right)-\overline{\mathrm{E}}\mathrm{i}\left(\upbeta {v}_i^2\right)=-2 \ln {v}_i+{\displaystyle \sum_{k=1}^{\infty }}\frac{\upbeta^k}{k\cdot k!}\left(1-{v}_i^{2k}\right) $$

$$ {{\left({\Delta}_i\right)}_{\upbeta}}^{\prime }={\displaystyle \sum\nolimits_{k=1}^{\infty }}\frac{\upbeta^{k-1}}{k!}\left(1-{v}_i^{2k}\right) $$

$$ {{\left({\Delta}_i\right)}_{\upbeta}}^{\prime \prime }={\displaystyle \sum\nolimits_{k=2}^{\infty }}\frac{\upbeta^{k-2}}{\left(k-2\right)!\cdot k}\left(1-{v}_i^{2k}\right) $$

Appendix 3: Special Mathematical Function Ēi(x)

The exponential integral Ēi(x), which is defined for real nonzero values of x as:

$$ \overline{\mathrm{E}}\mathrm{i}(x)={\displaystyle \underset{-\infty }{\overset{x}{\int }}}\frac{e^zdz}{z}. $$

The integral has to be understood in terms of the Cauchy principal value, due to the singularity in the integrand at zero.

Integrating the Taylor series for function ${e}^{-z}/z$, and extracting the logarithmic singularity, we can derive the following series representation for Ēi(x) for real values of x (see e.g. (Abramovitz and Stegun 1972)):

$$ \overline{\mathrm{E}}\mathrm{i}(x)=c+ \ln x+{\displaystyle \sum\nolimits_{n=1}^{\infty }}\frac{x^n}{n\cdot n!},\kern1.25em x>0, $$

where c is the Euler–Mascheroni constant (also called Euler’s constant). It is defined as the limiting difference between the harmonic series and the natural logarithm:

$$ c=\underset{n\to \infty }{ \lim}\left({\displaystyle \sum\nolimits_{k=1}^n}\frac{1}{k}- \ln n\right)\approx 0.5772. $$

Appendix 4: Relation Between the Shape Factor A and the Shape Parameter d

In our study we use dimensionless shape factor A and dimensionless shape parameter d, defined in Appendix 2. The first one is used in Eq. (7), the second one is derived from Eq. (9). Each of these parameters defines only a subset of objects with the appropriate shape properties. More insights onto the shape of the object may be obtained, if we combine both definitions into the nonlinear system of equations.

The shape factor A can be expressed as follows:

$$ A=k\frac{a_x{a}_y}{{\left({a}_x{a}_y{a}_z\right)}^{2/3}}, $$

where the coefficient k shows how much shape of the considered object differs from the brick-like geometry, for which k = 1, e.g. a spherical object yields k = 1.209.

Since we are looking for the ratio a _x  : a _y  : a _z , and not for the absolute object size, it is convenient to assume that ${a}_z=1$ is the smallest size dimension representing a unit length (i.e. the dimensions are normalized by the actual value a _z ). Then for the remaining dimensions, a _x and a _y , we obtain:

$$ \left\{\begin{array}{l}A=k{\left({a}_x{a}_y\right)}^{1/3},\hfill \\ {}d=1+2\frac{a_x{a}_y+{a}_x+{a}_y}{a_x^2+{a}_y^2+1},\hfill \end{array}\right. $$

$$ \left\{\begin{array}{l}{a}_x{a}_y={A}^3{k}^{-3},\hfill \\ {}d=1+2\frac{A^3{k}^{-3}+{a}_x+{a}_y}{{\left({a}_x+{a}_y\right)}^2-2{A}^3{k}^{-3}+1},\hfill \end{array}\right. $$

$$ \left\{\begin{array}{l}{a}_x{a}_y={A}^3{k}^{-3},\hfill \\ {}{\left({a}_x+{a}_y\right)}^2\left(d-1\right)-2\left({a}_x+{a}_y\right)-2{A}^3{k}^{-3}d+d=1,\hfill \end{array}\right. $$

Here we solve this problem for the brick-like shape of an object (as suggested as a general model for a meteoroid shape, e.g. by Halliday et al. 1989, 1996 based on the investigation of the Innisfree meteorite)

$$ \left\{\begin{array}{l}\left(d-1\right){a}_x^4-2{a}_x^3+\left(\left(d-1\right)-2{A}^3\right){a}_x^2-2{A}^3{a}_x+\left(d-1\right){A}^6=0,\hfill \\ {}{a}_y={A}^3{a}_x^{-1}.\hfill \end{array}\right. $$

If $A=1.5$ and $d=2.8$, then

$$ 0.9{a}_x^4-{a}_x^3+\left(0.9-3.375\right){a}_x^2-3.375{a}_x+10.2515625=0. $$

This equation has two real roots and two complex ones. The real sizes are:

$$ {a}_x=1.69,\;{a}_y=2.0,\kern0.37em \mathrm{or}\kern0.5em {a}_y=2.0,\;{a}_y=1.69. $$

Appendix 5: Results of analyses of cosmogenic radionuclides in the Košice meteorite

After this chapter was sent to the publisher, we discovered another important work with the relevant initial mass estimate by Povinec et al. (2015). The authors have estimated average radius of the meteoroid at 50 cm using both the ⁶⁰Co and ²⁶Al data, and they provide the pre-atmospheric meteoroid mass estimate of 1840 kg, bringing it extremely close to the presented in this chapter calculations.