GEOPHYSICS

K. P. STANYUKOVICH and V. A. BRONSHTEN

ON THE VELOCITY AND ENERGY OF THE TUNGUSKA METEORITE

(Presented by Academician V. G. Fesenkov on 8 V 1961)

The study of the region of destruction caused by the fall of the Tunguska meteorite on 30 VI 1908 (^1), and calculations of the parameters of the shock wave formed during its flight through the atmosphere (^2), made it possible to estimate the total energy of destruction at \(\sim 10^{23}\) ergs. The clear picture of the radial fall of the forest from the epicenter indicates the predominant action of the blast wave, while the very character of the fall, in particular the existence of a “zone of indifference” and standing tree trunks (“telegraph poles”), indicates beyond doubt that the explosion of the meteorite occurred in the air (^1). This thereby confirms the assumption of the above-ground character of the explosion, expressed as early as 1925 by A. V. Voznesensky and somewhat later by L. A. Kulik (^3). Possible causes of the explosion will be considered below.

The motion of the meteorite in the atmosphere was considered by V. A. Bronshten (^4) on the basis of the well-known equations of meteor physics (^5, ^6). Solutions were obtained for the range of initial masses \(10^5\)—\(10^7\) tons, initial velocities 11—46 km/sec, and values of the drag coefficient \(c_x/2 = 0.5 \div 2\).

From the values of the final masses and velocities for the entire family of solutions, values of the kinetic energy of the meteorite \(E_k\) were calculated. Comparison of the obtained values of \(E_k\) with the above estimates of the energy of destruction shows that the initial mass of the meteorite in any case exceeded \(10^5\) tons and was apparently within the range \(10^6\)—\(10^7\) tons, which agrees well in order of magnitude with V. G. Fesenkov’s estimate, made on the basis of entirely different considerations (^8).

Regardless of the adopted value of the initial mass, the final velocity and mass of the meteorite must lie within the limits: \(16 < v_k < 30\) km/sec, \(2 \cdot 10^4 < M_k < 7.5 \cdot 10^4\) tons. The physical meaning of the independence of these estimates from \(M_0\) is that, in order to reconcile the value of \(E_k\) with the data on the energy of destruction in the region of the fall (^1,^2), when increasing the estimate of \(M_0\) one must simultaneously increase the adopted value of the drag coefficient \(c_x\), i.e., assume that a larger mass experiences greater resistance in the atmosphere. Ceplecha’s study (^7) of the Příbram meteorite showed that for a large meteoric body \(c_x/2 = 0.43\), and therefore the most probable solution is the variant corresponding to \(M_0 = 10^6\) tons, \(v_0 = 35 \div 43\) km/sec, \(v_k = 30\) km/sec, \(M_k = 2 \cdot 10^4\) tons.

Let us now turn to the physical investigation of the phenomena accompanying the flight of a large meteoric body in the Earth’s atmosphere.

During motion with cosmic velocity, a body of diameter 25—30 m will already at an altitude of 120 km begin to form a shock wave, the temperature at whose front \(T_y^0\) in the ideal case is determined by the relation

\[ \frac{c_v}{\mu} T_y^0 = \frac{v^2}{2}, \tag{1} \]

where \(c_v\) is the heat capacity of air (per mole), and \(\mu\) is its molecular weight (\(\mu = 29\)).

A more exact formula for \(T_y^0\), taking into account the change in the adiabatic exponent \(\gamma=c_p/c_v\), has the form \((^9)\)

\[ T_y^0=T_1\frac{p_2}{p_1}\frac{\rho_1}{\rho_2} =T_1\frac{p_2}{p_1}\frac{\gamma-1}{\gamma+1}, \tag{2} \]

where \(p_{1,2}\) and \(\rho_{1,2}\) are the pressure and density of the air, respectively, before and behind the shock-wave front, and \(T_1\) is the temperature before the front.

In the real case, the temperature at the shock-wave front \(T_y\) will be less than \(T_y^0\) because of energy losses to dissociation and ionization of the gas \((^{10})\). Values of \(T_y^0\) and \(T_y\) for different meteorite velocities are given in Table 1.

Table 1

Thus, the temperature at the shock-wave front of the Tunguska meteorite was \(70\,000 \div 100\,000^\circ\). The transition from \(T_y^0\), determined by formula (1), to the real temperature \(T_y\) can be carried out by the empirical formula

\[ T_y=\eta T_y^0 v^{-0.7}, \tag{3} \]

where \(\eta=2.27\).

The radiation energy of the shock wave is

\[ E_{iy}=\sigma S_iT_y^4, \tag{4} \]

where \(\sigma\) is the Stefan–Boltzmann constant, and \(S_i\) is the radiating area of the shock wave. It may be assumed that \(S_i=\beta S_m\), where \(S_m\) is the area of the meteorite, which in first approximation we take to be spherical; \(\beta=5\div10\). Since

\[ S_m=4\pi\left(\frac{3}{4\pi}\frac{M}{\delta}\right)^{2/3} =4\pi\left(\frac{3}{2\pi}\frac{E_m}{\delta v^2}\right)^{2/3}, \tag{5} \]

then, substituting (1), (3), and (5) into (4), we obtain

\[ E_i=\beta\sigma\frac{\pi}{4}\left(\frac{3}{2\pi}\right)^{2/3} \left(\frac{\mu\eta}{c_v}\right)^4 \left(\frac{E_m}{\delta}\right)^{2/3} v^{3.9}. \tag{6} \]

Here \(E_m\) is the total energy of the flying meteorite, and \(\delta\) is its density. For a given energy \(E_m\), the radiation energy increases directly in proportion to the fourth power of the velocity.

In this connection it is necessary to note the erroneous nature of A. V. Zolotov’s calculations \((^{11})\), who took the color temperature of the bolide (upper limit \(6000^\circ\)) for the temperature of the shock wave and tried from this to calculate the velocity of the flying body by formula (1). It should be borne in mind that the radiation maximum at \(T_y=70\,000^\circ\) lies in the ultraviolet part of the spectrum. For such radiation, air is practically opaque. However, ahead of the shock-wave front there arises a heated zone with a radiating area much larger than the radiating area of the shock wave. Re-emission occurs in this case, and the temperature of the outer zone will be lower than \(T_y\), while its radiation will shift into the visible part of the spectrum. Part of this radiation is perceived by the eye as the yellow color of the bolide. Thus, it is obvious that A. V. Zolotov’s attempt to determine the temperature of the shock wave from the color of the bolide is untenable, and calculating the velocity of the meteoric body from this temperature is meaningless.

The use by A. V. Zolotov of the formula relating the luminous energy of the explosion \(E_c\) and the luminous impulse \(I_c\) is likewise unfounded:

\[ E_c=\frac{I_c\cdot 4\pi R^2}{e^{-\mu(R-r)}} ; \tag{7} \]

where \(R\) is the distance from the explosion site, \(r\) is the radius of the luminous region, and \(\mu\) is the coefficient of light absorption in the atmosphere. The latter was taken to be \(0.033\ \mathrm{km}^{-1}\), which corresponds to an unusually high transparency coefficient \(p=0.93\), quite uncharacteristic of taiga regions. If, however, one takes the more realistic, though still high, value \(p=0.80\), then we obtain \(\mu=0.1\ \mathrm{km}^{-1}\), and all of A. V. Zolotov’s estimates change by several orders of magnitude.

Let us conclude by considering the probable nature of the explosion of the Tunguska meteorite. The general equation for the thermal balance of a meteorite has the form [\(^{10}\)]

\[ \left(\Lambda \frac{\rho v^3}{2}+W_{\mathrm{изл}}\right)S\,dt = E_{\mathrm{нагр}} + \sigma\left(T^4-T_a^4\right)S_m\,dt + QmNS_m\,dt, \tag{8} \]

where \(W_{\mathrm{изл}}\) is the flux density of radiation from the shock wave; \(E_{\mathrm{нагр}}\) is the part of the energy going into heating the body; \(T\) and \(T_a\) are the temperatures of the meteorite and of the atmosphere; \(S\) is the midsection area; \(S_m\) is the surface of the body; \(Q\) is the heat of evaporation; \(N\) is the number of evaporating molecules \((\mathrm{cm}^{-2}\cdot\mathrm{sec}^{-1})\); \(m\) is the mass of a molecule.

Analysis of equation (8) shows that the second term on the left-hand side, due to radiation, is much larger than the first, due to heat transfer in flow. The expenditure of heat on evaporation (the third term on the right-hand side) rapidly becomes much larger than the expenditure of heat on radiation from the surface of the meteoric body (the second term), and therefore it is sufficient to consider the supply of heat due to radiation from the shock-wave front and the expenditure of heat on evaporation. Their dependence on altitude is shown for an iron meteorite in Fig. 1.

Fig. 1. Energy balance during the motion of an iron meteoric body:
1 — total energy input (iron, \(v_0=60\ \mathrm{km/sec}\), \(i=72^\circ\), \(r_0=10^2\ \mathrm{cm}\));
2 — energy input from the shock wave due to radiation;
3 — energy input due to flow around the body;
4 — energy expenditure on evaporation.

As can be seen from Fig. 1, at a certain altitude \(h_{\mathrm{равн}}=18\ \mathrm{km}\) the supply and expenditure of heat become equal, heating ceases, after which the body begins to cool and, while simultaneously braking, reaches the surface of the Earth. A similar picture will hold for a stony meteorite.

But if we imagine that we are dealing with the nucleus of a small comet, as I. S. Astapovich and Whipple once supposed, and assume that this body, like all comet nuclei, is a conglomerate of methane-ammonia ices also containing stone blocks and dust, then the picture of the phenomena will be different. Namely, for an icy block with \(r=10^3\ \mathrm{cm}\), \(v=60\ \mathrm{km/sec}\), \(i=72^\circ\), at an altitude of 50 km the energy going into evaporation is an order of magnitude less than the energy received by the body from the shock wave. As a result, the body is strongly heated in depth and evaporates ever faster, i.e., the boundary of the evaporated layer moves ever faster toward the center. In a comparatively short time (\(\sim 0.2\ \mathrm{sec}\)) a significant mass of material evaporates (about 30%). If the process proceeds sufficiently rapidly, then the evaporated particles,

flying apart, can create a strong spherical shock wave, and the phenomenon will have the character of an extended explosion.

At a velocity \(v = 30\) km/sec the power of the process \((2 \cdot 10^{13}\) erg/g·sec) is comparable with the explosion of gunpowder \((10^{13}\) erg/g·sec). Such a phenomenon, studied in greater detail by K. P. Stanyukovich and V. P. Shalimov \({}^{12}\), may be called a “thermal explosion.”

We have considered above a mechanism that can lead to an explosion when a single body enters the atmosphere. However, there is also another possible point of view on the structure of cometary nuclei. Thus, V. G. Fesenkov regards the nucleus of a comet as a dense swarm of comparatively small bodies.

Analysis of the directions of tree fall clearly indicates the presence not of one, but of several centers of fall, to which L. A. Kulik had already drawn attention. Taking this into account, V. G. Fesenkov proposed \({}^{13}\) yet another possible mechanism of destruction under the action of the entry of a cometary nucleus, which the Tunguska meteorite undoubtedly was.

When a sufficiently dense swarm of bodies enters the Earth’s atmosphere with cosmic velocity, the swarm may be surrounded by a common shock wave. However, as it penetrates into the lower layers of the atmosphere, owing to differences in mass and, hence, in deceleration, the density of the swarm must decrease, and its transverse dimensions and “length” increase. As a result, each individual body, or group of bodies close in mass, will possess individual shock waves. In this case the process of destruction of the swarm bodies will end even before they reach the Earth’s surface (because of the increase of surface area per unit mass), while the shock waves, on reaching the Earth, will cause all the observed destruction, and the radiation of the powerful shock wave will produce radiant burns on a number of objects and, in particular, on trees.

In conclusion the authors express their gratitude to Acad. V. G. Fesenkov for useful discussions.

Committee on Meteorites
Academy of Sciences of the USSR

Received
5 V 1961

CITED LITERATURE

\({}^{1}\) K. P. Florenskii, Yu. M. Emel’yanov et al., Meteoritika, 19 (1960).
\({}^{2}\) M. A. Tsikulin, Approximate estimate of the parameters of the Tunguska meteorite of 1908 from the pattern of destruction of the forest massif, Narodnokhoziaistvennoe ispol’zovanie vzryva, vol. 6, Publ. Siberian Branch, Academy of Sciences of the USSR, 1959.
\({}^{3}\) E. L. Krinov, The Tunguska Meteorite, Publ. Academy of Sciences of the USSR, 1949.
\({}^{4}\) V. A. Bronshten, Meteoritika, 20 (1961).
\({}^{5}\) B. Yu. Levin, Physical Theory of Meteors and Meteoritic Matter in the Solar System, Publ. Academy of Sciences of the USSR, 1956.
\({}^{6}\) V. G. Fesenkov, Meteoritika, 12, 72 (1955).
\({}^{7}\) Z. Ceplecha, Bull. Astr. Inst. of Czechoslovakia, 11, No. 1 (1959);
\({}^{3}\) Tsikulin, Meteoritika, 20 (1961).
\({}^{8}\) V. G. Fesenkov, Meteoritika, 6 (1949).
\({}^{9}\) K. P. Stanyukovich, Unsteady Motions of a Continuous Medium, 1955, p. 233.
\({}^{10}\) V. V. Selivanov, I. Ya. Shlyapintokh, ZhFKh, 32, No. 3, 670 (1958).
\({}^{11}\) A. V. Zolotov, DAN, 136, No. 1 (1961).
\({}^{12}\) K. P. Stanyukovich, V. P. Shalimov, Meteoritika, 20 (1961).
\({}^{13}\) V. G. Fesenkov, On the cometary nature of the Tunguska meteorite, Meteoritika, 22 (in press).