The University of Tennessee, Knoxville, TN 37996

Stanford University, Stanford, CA

The University of Tennessee, Knoxville, Tennessee 37996

INTRODUCTION

On June 30, 1908, a sudden release of energy in the atmosphere above
the Tunguska region in central Siberia devastated an area of over 2100
square kilometers of forest.^{1} The energy release has been attributed
to a number of sources, but the most widely accepted is the sudden and
complete disruption of a large meteor well before it struck the ground.
The lack of an apparent impact crater indicates that the resulting
debris must have been quite small. Because of the remote location of
the event and the low number of human casualties, its occurrence is not
widely known, either among the general public or in the scientific
community. However, the recent entry of comet Shoemaker-Levy 9 into the
atmosphere of Jupiter has brought attention to the possibility of such a
cataclysm taking place on Earth and, therefore, has refocused interest
on Tunguska, the largest reported event of this type in modern times.

Previous studies of Tunguska have yielded widely differing
interpretations as to the size, composition, velocity, and density of
the body prior to its entry into the atmosphere.^{2,3,4} However, it is
generally agreed that the object disrupted catastrophically at 6 to 10
km above the ground,^{5} depositing approximately 15 Mtons of energy^{6} in a
relatively small altitude band. The atmospheric entry velocity and
entry angle are not well established; experimental reconstruction
of the extent and pattern of treefall suggests a value for the entry
angle, gi between -10 and -60 degrees (with a most probable value
of -30 degrees), while eyewitness accounts report a shallower angle of
-5 to -17 degrees.^{2} Estimates of the atmospheric entry velocity range
from under 15 km/s to over 40 km/s.^{2,3,4} In a comprehensive paper
published in 1983, Sekanina^{3} interpreted the data as indicating
that the body could not have been of cometary origin and
was probably an Apollo-type asteroid with a diameter of 90 to 190
meters.^{3} More recent work by Chyba et al has suggested that the object
was a stony asteroid significantly smaller than that described by
Sekanina (perhaps 60 meters in diameter).^{2} In marked contrast, Turco
et al^{4} concluded that the entry body was of cometary origin with an
effective density of 0.002 gm/cc and a radius of 800 meters.

ATMOSPHERIC ENTRY PHYSICS

The velocity of a body in unpowered, non-lifting atmospheric flight varies according to the following equation:

where V is velocity, C_{d} is the coefficient of drag, A is the cross
sectional area, m is the mass, g is the acceleration due to gravity, rho
the ambient atmospheric density and gamma is the flight path angle (measured
from horizontal with downward being negative). The change in altitude
and flight path angle are given by:

and

where H is the altitude and R is the radius of the Earth plus the
altitude.

Whatever type of object caused the Tunguska event, its passage through the atmosphere imposed enormous pressure and heat transfer loads on the body. The combination of these two effects caused the destruction of the meteor in the altitude band discussed above. Once the internal stresses induced by the pressure load exceeded the yield strength of the material, the body began to fail mechanically. The high pressure in the stagnation region and low pressure on the backside of the body resulted in a compression load, while the confining pressures on the sides were relatively low. As a result, once mechanical failure commenced, the meteor spread laterally resulting in a so-called "pancaking" effect. This increased the frontal area and caused the aerodynamic drag to go up, therefore increasing the deceleration of the body according to Eqn. 1. In Ref. 2 Chyba et al show that the increase in the radius of the body can be approximated by:

where r and rho_{c} are the radius and density of the body.

Aerodynamic heating of the meteor caused a high rate of material ablation resulting in a decreasing body mass. The heating was dominated by radiation from the highly compressed gas behind the bow shock and can be approximated using the Stefan-Boltzmann relation:

where q_{Rad} is the adiabatic radiative heat transfer rate and T_{2} is the
temperature behind the shock wave (which will vary with velocity,
position and shock angle). Because of radiative emission from the gas
as it moves from the shock toward the body, the shock layer is
non-adiabatic and it is necessary to modify the heating as given by the
above equation according to a method developed by Goulard.^{7} For this
case, Goulard's method gives approximately^{8,9}

where q_{R} is the non-adiabatic radiative heat transfer rate and G =
q_{Rad} / ( 4 rho V^{3} ). These equations do not account for the presence
of ablation products in the shock layer and their absorption of some
fraction of the radiative heat input. The significance of this effect
has been analyzed by Gupta^{10} and depends primarily on the ratio of the
freestream mass flow rate to the mass flow rate of the ablation products
(which is simply the heat transfer rate divided by the heat of ablation
of the material). Since experience shows that the vast majority of
the flow energy goes into heating the atmosphere rather than an entry
body,^{8,11} the heating rate was required not to exceed fifteen percent
of the total flow energy (0.075 rho V^{3} (10^{-4}) W/cm^{2}); if the rate as
calculated above was in excess of this amount, it was set equal to the
fifteen percent limit.

OBJECTIVES AND METHODOLOGY

The objective of this work was to determine the type(s) and size(s)
of the objects which could have been responsible for the Tunguska event.
The equations of motion described above were numerically integrated for
a wide range of initial bolides and entry conditions, while the mass of
the body was decreased according to the ablation rate. For asteroids,
entry velocities from 12.5 to 20 km/s were considered (recent work
indicates that this range should include the great majority of
Earth-crossing asteroids), while for comets, entries were examined only
at 20 km/s, an approximate lower limit for cometary entry speeds.^{12}
The atmospheric density was calculated as a function of altitude based
on curve fits of the 1976 U.S. Standard Atmosphere.^{13} At each time
step in the trajectory, the temperature behind the shock was calculated
as a function of position and shock angle by solving for the equilibrium
species concentrations. (The shock angle was assumed to be 70 degrees
at the body's edge.) This was used to determine the local heat transfer
rate which was corrected using the technique of Goulard. Gupta's
corrections for ablating surfaces were applied by curve fitting his
plots of heat transfer ratio (ablating to non-ablating surface) vs mass
flow ratio. The body was assumed to have a C_{d} of 1.2 for the nominal
cases, although other values were considered. (The value of 1.7 used by
Chyba et al^{2} is too high for irregularly shaped bodies with rounded
edges such as asteroids.) Chyba's pancake model of mechanical
deformation was adopted, and the increasing frontal surface area was
used for both drag and heat transfer calculations. The physical
characteristics of the various bolide materials are listed in Table I.

RESULTS AND CONCLUSIONS

Figure 1 shows a representative heat transfer pulse calculated as
described above for a carbonaceous meteor compared to the pulse
calculated for the same body using the method of Ref. 2 (which assumed
an adiabatic shock layer at a uniform temperature of 25000 degrees
Kelvin). By accounting for the non-adiabatic nature of the shock layer
and the reduction of surface heating due to radiation absorption by
ablation products, the current method yields much lower heating rates
(and less ablative mass loss) than previous studies.^{2,3}

Figure 2 shows the bolide velocity as a function of time for two representative carbonaceous bodies. The airburst altitude (defined here as the altitude at which a meteor has lost fifty percent of its initial kinetic energy) is plotted in Figure 3 for carbonaceous chondrites as a function of the initial body radius over a range of atmospheric entry angles. (This definition of airburst altitude yields a value which is very close to that at which the rate of energy release is a maximum.) Figure 4 shows a similar calculation for a stony asteroids. The atmospheric entry velocity in these figures is assumed to be 15 km/s. The airburst altitude was found to be only a weak function of the atmospheric entry velocity. (Since the altitude indicated in these figures is with reference to sea level, it should be realized that the average elevation in the devastated area was 500 to 1000 meters.) From these plots, it is apparent that a body of either stony or carbonaceous composition could have broken apart and deposited its energy at the appropriate altitude.

Previous studies have suggested that carbonaceous bodies would have
released their energy too high in the atmosphere to have been
responsible for Tunguska while comets either would have entirely ablated
at altitudes in excess of 20 km or required extremely high initial
masses.^{2,3} However, using the present model, a given bolide will
airburst at a significantly lower altitude than predicted in Ref. 2 and
3. This is caused by both the lower value used for the drag
coefficient and the less severe heating environment (see Fig.1). As a
result, carbonaceous chondrites, which previously had been considered
unlikely causes of the Tunguska event,2 become the most likely
candidates. (It should be noted that Reference 14 gives slightly
higher values for the density and heat of ablation of carbonaceous
material; the use of these values would further serve to lower the
airburst altitudes for carbonaceous bodies.) This finding is reinforced
by the fact that carbonaceous bodies are the most common type of meteors
to enter the Earth's atmosphere. Moreover, the present model predicts
that a cometary body with a radius of 75 meters, an entry velocity of 20
km/s, and an entry angle of -45 degrees would airburst at 9.2 km.
Therefore, unlike previous studies^{2,3} which have excluded comets as the
cause of the Tunguska event, our model indicates that such an object can
not be eliminated on the basis of its terminal altitude alone.

The implication of this study is that meteors do not have to be as large or dense or have as steep an atmospheric entry angle as previously believed either to reach the ground or to cause significant damage due to an airburst.

REFERENCES

1. Gallant, R.A., Sky and Telescope, 38-43, June 1994.

2. Chyba, C.F., Thomas, P.J., and Zahnle, K.J., Nature, 361, 40-44,
1993.

3. Sekanina, Z., The Astronomical J., 88, 1382-1414, 1983.

4. Turco, R.P., Toon, O.B., Park, C., Whitten, R.C., Pollack, J.B., and
Noerdlinger, P., Icarus, 50, 1-52, 1982.

5. Ben-Menahem, A., Phys. Earth and Planet. Inter., 11, 11-35, 1975.

6. Hughes, D.W., Nature, 259, 626-627, 1976.

7. Goulard, R., AIAA J., 2, 494-502, 1964.

8. Page, W.A., Compton, D.L., Borucki, W.J., Ciffone, D.L., and Cooper,
D.M., AIAA Paper 68-784, 1968.

9. Tauber, M.E., J. Spacecraft and Rockets, 6, 1103-1109, 1969.

10. Gupta, R.N., Lee, K.P., Moss, J.N., and Sutton, K., J. Spacecraft
and Rockets, 29, 173-181, 1994.

11. Tauber, M.E. and Sutton, K., J. Spacecraft and Rockets, 28,
40-42, 1991.

12. Chyba, C.F., Icarus, 92, 217-233, 1991.

13. U.S. Standard Atmosphere, 1976, NOAA ST 76-1562, October 1976.

14. Baldwin, B. and Scheaffer, Y., J. Geophys. Res., 76, pp. 4653-4668,
1971.

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MATERIAL DENSITY (GM/CC) HEAT OF ABLATION (MJ/KG)
YIELD STRENGTH (ATM)

CARBONACEOUS 2.2 5.0
10.0

STONE 3.5 8.0
100.0

COMET 1.0 2.5
1.0

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