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A method for determining the residual meteoritical mass in the Barringer Meteor Crater

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Summary

This paper displays some historical background of the Barringer Meteor Crater. Also shown is a description of methods for determining not only the mass of meteoritical material remaining in the crater, but also depth to the residual material.

TheHarding (1954) andRegan (1967) gravity surveys are presented, together with the electrical potential survey ofJakosky (1932). A method is outlined such that this information can be processed by various numerical techniques to include Lagrangian interpolation and a least squares analysis, then mapped to the grid points of a partial polar grid which overlays the crater. Residual meteoritical mass and depth to the mass can be found by solving a polynomial expression, which connects electrical potential with depth, simultaneously with Newton's Law of Gravitation which connects mass and depth. A solution of this problem can be found by using the Newton-Raphson iterative procedure at each grid point. Total mass can be found by numerical integration over the partial polar grid.

A method for experimentally verifying the analysis is displayed. Suggestions for future activity in this area conclude the paper.

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Abbreviations

A :

ann×n symmetric matrix, used in a least squares scheme

A −1 :

the inverse of the matrixA

a i :

coefficients in a polynomial representation connecting electrical potential and depth

ā i :

value of the coefficients inP(s)

\(\vec a\) :

ann-dimensional column vector ofa i 's written thus (a 1,a 2,...,a n )T

C :

a constant defined as follows,C=F/K m 1

F :

gravitational attractive force between massesm 1 andm 2, when the centers of gravity of these masses are separated by a distances

f(m 2):

a function ofm 2, used in the Newton-Raphson iterative scheme

G(r) :

value of gravitational attraction as a function of the polar radiusr

i :

index of a product or summation

j :

index of a product or summation

K :

a gravitational constant, used in Newton's Law of Gravitation equation

k :

an upper limit in a summation process

k :

an iteration index

m 1,m 2 :

expressions for masses, used in Newton's gravitational equation

m:

mean mass of meteorities, entering the earth's atmosphere, that formed The Barringer Meteor Crater

P(s) :

annth degree polynomial representation of electrical potential as a function of depth

P(s):

an averagenth degree polynomial representation of electrical potential. This function ofs is located at a partial polar grid point

r :

radial extending from the pole in a two dimensional polar coordinate system

r 1 :

a fixed value of the polar radiusr

S :

sum of the squares of residuals in a least squares procedure

s :

depth to the residual meteoritical material

s 0,s 1 :

fixed values of the depths

\(\vec x\) :

ann-dimensional column vector ofx i 's, written thus (x 1,x 2,...,x n )T)

α:

a fixed angle in a polar coordinate system (lower limit of θ in a partial polar coordinate grid system)

β:

a fixed angle in a polar coordinate system (upper limit of θ in a partial polar coordinate grid system)

ε:

a prescribed positive number

ε i :

theith residual in a least squares procedure

θ:

variable angle in a polar coordinate system

km:

kilometers

\(\frac{{\overrightarrow {\partial S} }}{{\partial a}}\) :

ann-dimensional column vector of partial derivatives ofS with respect toa i , written thus,\(\left( {\frac{{\partial S}}{{\partial a_1 }},\frac{{\partial S}}{{\partial a_2 }}, \ldots ,\frac{{\partial S}}{{\partial a_n }}} \right)^T \cdot \)

Δr :

increment in the variable polar radiusr

Δθ:

increment in the variable polar angle θ

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Crowson, H.L. A method for determining the residual meteoritical mass in the Barringer Meteor Crater. PAGEOPH 85, 38–68 (1971). https://doi.org/10.1007/BF00875398

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  • DOI: https://doi.org/10.1007/BF00875398

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