RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 7, ES6004, doi:10.2205/2005ES000193, 2005
[2] Jones et al., [2002] proposed a hypothesis according to
which the fall of a large meteorite (such as the
Chicxulub meteorite 10-16 km in diameter, which fell
65 Myr ago at a velocity of about 11 km s
-1 and produced a
crater 180-300 km in size) can activate convection in the
lower mantle and provoke the formation of plumes in the
D'' layer. An attempt to estimate the order of magnitude
of elastic stresses arising at the time moments of such
events is made in the given paper.
[3] The main uncertainty involved in the estimation of
elastic stresses is related to the question of which part
of the kinetic energy of a meteorite is converted into
heat and which part, into the energy of seismic
vibrations. It is natural to divide the fall process into
two phases: (1) the phase of inelastic impact
(accompanied by the fracture of material in a region
comparable in size with the crater) and (2) the phase of
elastic interaction of the resulting fragments with the
mantle.
[4] As follows from the law of conservation of momentum, the
average velocity of fragments in the first phase is
determined by the relation
 | (1) |
where ( m1, r1 ) and ( m2, r2 ) are the mass and
average radius of the meteorite and the crater,
respectively, and
v1 is the velocity of the meteorite
fall. Accordingly, the ration of the total kinetic energy
of fragments to the initial energy of the meteorite is
 | (2) |
[5] The phase of elastic interaction of fragments with the
mantle outside the crater can be described by the elastic
energy balance
 | (3) |
where
 | (4) |
is the volume density of elastic energy,
l and
m are the
Lamé coefficients,
u is the displacement vector,
eik is the stress tensor, and
 | (5) |
is the
k th component of the Poynting vector in a
Cartesian coordinate system (summation over repeated
indices is assumed in (4) and (5)). Neglecting the depth
dependence of the elastic moduli, we may assume that the
elastic energy in mantle is concentrated in a spherical
layer centered at the impact point and bounded by spheres
of radii
R1 and
R2; the thickness of the layer
R2-R1 is determined by (i) the finite time of the
interaction of fragments with the lower boundary of the
crater and (ii) the diffusion of the wavefront associated
with the dispersion of seismic wave velocities. In the
case of body waves, the velocity dispersion is only due
to the inelasticity of the medium (the velocities depend
on the period of vibrations and are independent of the
wavelength). Given the quality factor values
characteristic of the Earth's mantle ( Qm
102-103 ),
the velocity dispersion is
dVp/Vp
dVs/Vs
10-2,
and the value of the wavefront dispersion due to the velocity
dispersion is of the order of
dR
R1dV(p,s)/V(p,s)
10-2R1. It is
easy to show that this effect is negligible compared to
the first effect. To estimate the finite time of the
interaction of fragments with the inner boundary of the
crater
dt, note that elastic strains at the lower
boundary of the crater assume have ultimate (failure)
values typical of rocks:
e max
10-3
10-4.
In accordance with the law of conservation of momentum, we
have for the interaction process of fragments with the
inner boundary of the crater:
 | (6) |
where the average interaction force is of the order of
 | (7) |
Hence, we have
dt
m1 v1/(me max r22) and
 | (8) |
[6] In accordance with the equation of the energy balance in
the second phase of the process, the stresses in the
wavefront region
efr are determined by the relation
 | (9) |
where
 | (10) |
is the volume of the spherical layer in which the energy
of elastic vibrations is concentrated. Accordingly, the
maximum stresses in the mantle at the distance
R1 from
the center of the crater can be estimated as
 | (11) |
mr3/21 r2-1/2 R-11 (v1 e max/(2pVp, s))1/2.
Setting
R1 = r2 in this formula and equating the result
to the known value of elastic stresses at the lower
boundary of the crater
me max, we obtain
 | (12) |
which yields
r2
r1 (v1/ (2pVp, s e max))1/3.
With characteristic values of ultimate tangential
stresses of rocks of the order of
10-3-10-4 and
v1
pVp, s
10 km s-1,
this formula yields the estimate
r2
(10-25)r1 relating the sizes of a meteorite and its
crater and agreeing well with data on the Chicxulub
meteorite. To obtain numerical estimates of stresses that
developed in the D'' layer at the time moment of the
Chicxulub meteorite fall, one can use either formula (11)
giving
r2/r1
(10-25) or the aforementioned geological data
according to which
r2/r1
(180-300) km/(10-16) km~ 10-30.
Given
e max
10-3-10-4 and
m
1011 dyn cm-2,
we obtain in both cases
mefr
106-107 dyn cm-2 =
0.1-1 MPa.
[7] This value is about an order of magnitude smaller
than the tectonic stresses developing in the
crust and upper mantle before an earthquake.

Citation: Molodensky, S. M. (2005), On the estimation of elastic stresses in the mantle at the time moment of a large meteorite fall, Russ. J. Earth Sci., 7, ES6004, doi:10.2205/2005ES000193.
Copyright 2005 by the Russian Journal of Earth Sciences
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