Some Problems in Meteor Physics

V. N. Petrov, Leningrad

1. Physical Theories of Meteors

§ 1. The ignition and extinction of meteors occurs in the upper layers of the atmosphere; on average, ignition is observed at an altitude of 115 km, and extinction at an altitude of 80 km. For meteors from various meteor streams these altitudes prove to be different and depend strongly on speed; the greater the speed, the higher the altitude of ignition, and, conversely, the lower the speed, the lower the altitude of ignition.

Determination of the altitudes of ignition and extinction of meteors is carried out on the basis of “corresponding” observations, both visual and photographic, i.e. on the basis of observations from two (or more) points separated from one another by 20–40 km. At present there are about 2,000 separate published determinations of altitudes. In addition, there are 3,500 unpublished altitudes from the Arizona Meteor Expedition, organized by Harvard Observatory (USA) and Cornell College in 1931–1933.

Most altitudes were obtained on the basis of naked-eye observations and are therefore burdened with errors of various kinds, as a result of which deviations from the exact values may reach 15–20%. These errors arise, first, from the physiological properties of the eye, owing to which different observers perceive the moment of ignition and extinction of a meteor differently; second, from the different conditions of visibility of meteors in the region of observation; third, from the presence of a certain “selectivity” in the available material.

For a long time the altitudes of ignition and extinction of telescopic meteors remained unknown. In 1930, independently of one another, E. Epik in Tartu (Estonia) and I. S. Astapovich in Leningrad succeeded in determining them. From 13 directly corresponding observations with comet-seekers from two points, they found that the altitudes of ignition and extinction of telescopic meteors are of the order of 70–90 km.

§ 2. On the basis of the altitudes available in the catalogues of Denning,^3 Lindemann and Dobson^4 in 1922 gave the first serious theory of the physical phenomena in the flight of meteors, which later contributed greatly to the successful development of the study of the stratosphere by the methods of meteor astronomy.

A physical theory of the meteor is a theory of phenomena connected with the motion of a solid body with cosmic velocity in the upper layers of the earth’s atmosphere. In a theory so constructed, such quantities as the dimensions of the meteor, its shape, chemical composition, velocity and inclination of its path, as well as three independent quantities characterizing the upper layer of the atmosphere, must be independent: molecular weight, pressure, temperature and, in part, the direction of displacement of air currents.

Through the principal seven independent quantities one must express the altitudes of ignition and extinction of meteors, the character of the energy radiated by the meteor, its temperature, and its absolute brightness.

The problem as posed in general form is still insoluble. In

...its solution allows a considerable number of simplifications, which make the mathematical side of the problem simpler and amenable to a complete analytic solution. For example: the shape of the meteoric body is taken to be spherical; its dimensions are considered quite insignificant, of the order of several millimeters; the chemical composition is assumed to be similar to the known chemical composition of meteorites.

In solving the problem posed, knowledge of the molecular and chemical composition and structure of the upper layers of the atmosphere is also of particularly great importance. Usually, in this part, geophysical data are used. At the same time, a certain uncertainty may enter into the work, since data on the structure of the upper layers of the atmosphere are still based on indirect observations and investigations.

Lindemann and Dobson, introducing the above-mentioned simplifications with regard to the size, shape, and composition of the meteoric body itself, and taking the chemical composition of those layers of the atmosphere where the flight occurs to be analogous to the composition of the atmosphere at the surface of the Earth, constructed a physical theory of the flight of a meteor in the atmosphere. They investigated the behavior of a meteor of about 1st stellar magnitude, moving with a velocity of 40 km/sec, having an ignition height of 100 km above the Earth, an extinction height of 80 km, and a path length of 60 km. The mean distance of the observer from the trajectory of the meteoric body is taken to be 150 km. The duration of the meteor’s flight is 1.5 sec. The mean dimensions and mass of the meteoric body in this case turn out to be, respectively, 0.6 mm and 6 mg.

A meteoric body, entering the Earth’s atmosphere with cosmic velocity at heights from 100 to 80 km, compresses the air masses encountered along its path and forms in front of itself an “air cushion.”

The process of compressing the air into the cushion takes place very rapidly, so that it is accompanied by a complete absence of heat exchange between the gaseous cushion and the surrounding layers of the Earth’s atmosphere, i.e. the process proceeds according to the equations of the adiabatic state.

Lindemann and Dobson assume that all the kinetic energy of the meteor is expended on the motion of the “air cap” that forms in front of the meteor and on its adiabatic compression.

The meteoric body1 is subjected to impacts by molecules even in the very uppermost layers of the atmosphere, which cause dissociation of its molecules, but these impacts are comparatively few, and their total energy is insufficient for a noticeable luminosity of the meteor. Moreover, the problem of the interaction of the meteoric body with the molecules of the air at those heights at that time was unclear, and Lindemann and Dobson did not succeed in finding its solution on the basis of kinetic theory.

At heights of about 100 km the density of the air already proves to be rather appreciable; the mean free path of the molecules is found to satisfy the case of formation of a gaseous cap, since the molecules no longer have time, after collision with the meteor, simply to scatter in front of it. The deeper the meteor penetrates into the atmosphere, the denser the gaseous cap becomes.

The Lindemann–Dobson theory has been set forth in considerable detail in Russian in a number of review articles (by Obolenskii, Khaneskii),[^5][^6] and N. M. Shtude gave some of its development.[^7] The exposition that we give here proceeds in accordance with the corrections introduced into the theory by later investigations, especially the works of Anderson and Radakovich.[^8]

In their work Lindemann and Dobson examine the following main questions: 1) the formation of the gaseous cap in front of the meteoric body, the temperature within it; 2) the process of heating of the meteoric body; the character of the braking—

of its velocity as it moves in the stratosphere; the dependence between the decrease in velocity and the dimensions of the meteor; 3) the temperature of the meteor in the vapor-like state; and 4) the density and temperature of the upper layers of the atmosphere according to meteor observations.

On all these questions Lindemann and Dobson managed to obtain more or less satisfactory answers. Some simplifications and inaccuracies of the calculation were corrected by Radaković, whose work may be regarded as a further development of the work of Lindemann and Dobson.

Radaković determines the density of the air for the case of formation of a gas cap—\(\rho_1\). If we know the air density \((\rho_0)\) and \(L_0\)—the length of the free path of molecules under normal atmospheric conditions, \(V\) and \(v\)—the velocities of motion of the meteor and of the molecules, and \(r\)—the radius of the meteor, then

\[ \rho_1=\frac{\rho_0 L_0\cdot V}{2v\cdot r}. \]

In the gas cap that is formed, part of the kinetic energy of the meteor goes into adiabatic compression of the air masses, while the greater part of it goes into increasing the kinetic energy of motion of the molecules. Considering the conditions of adiabatic compression to correspond to reality (this is the assumption of the whole construction), and proceeding from the equation of state for a definite temperature in the gas cap, a formula of the following form is obtained:

\[ T_1=T\left(\frac{M\cdot v^2}{2R\cdot T}\right)^{\frac{\gamma-1}{\gamma}}, \tag{1} \]

where \(T_1\) is the absolute temperature inside the gas cap, \(T\) is the absolute temperature of those layers of air where the flight of the meteor takes place, \(M\) is the molecular weight of air, \(R\) is the gas constant, and \(\gamma\) is the ratio of heat capacities.

The pressure inside the gas cap is determined by the formula:

\[ p_1=\frac{1}{2}\rho_0 v^3+\frac{v_0^2\cdot \rho_0}{3(\gamma-1)} \left[ \left(\frac{\rho}{\rho_0}\right)^{\frac{\gamma-1}{\gamma}}-1 \right]. \tag{2} \]

After determining the ratio of the amount of energy reaching the meteor to the energy expended by the meteor, which is obtained from the formula:

\[ k'=\frac{v_1+v_2}{V}\cdot \frac{v_1+v_2}{3V}, \]

where \(v_1\) is the velocity of the molecules inside the cap and \(v_2\) is the velocity of the diffusing meteor molecules, the authors obtained formulas for determining the mass and radius of the meteor:

\[ m=\frac{2\int E\,dt}{v^2}; \]

and

\[ r=\left(\frac{3\cdot \int E\,dt}{2\pi\rho_m\cdot v^2}\right)^{\frac{1}{3}}, \]

where \(E\) is the energy emitted by the meteor, \(\rho_m\) is the density of the substance of the meteor body, and \(t\) is the time of flight of the meteor.

Lindemann and Dobson assumed that \(\rho_m\) of meteoric matter corresponds to the density of iron. This made it possible to determine the mass \((m)\) of the meteor as \(6.25\cdot 10^{-3}\) g, and to regard the radius as approximately equal to \(0.6\) mm.

The principal source of the meteor’s luminosity is the kinetic energy of motion of the meteor, expended on heating the gas cap and vaporizing the substance of the meteor itself. Having clarified this process, we can obtain den-

the density of the air at the point of ignition of the meteor—\(\rho_{\mathrm{ign.}}\), and at the point of extinction of the meteor \(\rho_{\mathrm{ext.}}\).

\[ \rho_{\mathrm{ign.}}=\frac{16}{3}\cdot S\cdot T_2\frac{r\rho_m\cos x\cdot g\cdot M}{k\cdot v^2\,RT}; \tag{3a} \]

\[ \rho_{\mathrm{ext.}}=8\cdot l\,\frac{r\rho_m\cos x\cdot g\cdot M}{kv^2\,RT}, \tag{3b} \]

where \(S\) is the heat capacity of the substance of the meteor, \(r\) is the radius of the meteor, \(l\) is the latent heat of evaporation, \(g\) is the acceleration due to gravity, \(M\) is the molecular weight of air, \(x\) is the angle formed by the direction of flight of the meteor with the vertical line.

Using the formulas obtained, Lindemann and Dobson determined the temperature of the gaseous cap, the pressure in it, determined its dimensions, the percentage of light radiation from the envelope and from the meteor body relative to the total radiation of the meteor. According to formula (1), the temperature of the envelope turned out to be equal to \(2000\)—\(3000^\circ\mathrm{K}\). The pressure inside the envelope, obtained from formula (2), is of the order of several hundreds of atmospheres. The main mass of light is emitted by the gaseous envelope, whose dimensions are several tens of times larger than the dimensions of the meteoric body itself.

If it is assumed that the temperature distribution in the terrestrial atmosphere is isothermal, then, on the basis of the theory developed by these authors, from the altitude of ignition and extinction of meteors we can determine the temperature of the stratosphere by formula (4), since the energy imparted to the gaseous cap depends on the temperature of the atmosphere and is proportional to it:

\[ T=\frac{g(h_{\mathrm{ign.}}-h_{\mathrm{ext.}})}{cR}\cdot M. \tag{4} \]

Proceeding from the same isothermal model of the atmosphere, it is easy to determine the ratio of the pressure at the point of ignition and at the point of extinction of the meteor. It is obtained from the formula:

\[ \frac{P_{\mathrm{ext.}}}{P_{\mathrm{ign.}}} = \frac{\rho_{\mathrm{ext.}}}{\rho_{\mathrm{ign.}}} = e^{-\frac{gM}{RT}(h_{\mathrm{ext.}}-h_{\mathrm{ign.}})}. \tag{5} \]

Proceeding from their theory, the authors investigated the heights of ignition and extinction of the meteors in Denning’s catalogues and, on their basis, by formulas (3a) and (3b) gave values of the densities for the points of ignition and extinction of meteors, and then determined by formula (4) the temperature of the stratosphere at the heights of flight of the meteor.

In order to determine the density of the air, they first assumed that the temperature at these heights is the same as that obtained by aerologists for heights of \(20\)—\(40\) km, i.e. equal to \(220^\circ\mathrm{K}\); in this case, however, the density of the air turned out to be extremely high. In order to arrive at better agreement, the authors assumed that at a height of about \(50\) km there is a temperature inversion, that here the temperature rises to \(300^\circ\mathrm{K}\). This circumstance was then confirmed by a number of other investigations, such as, for example, by the propagation of sound waves, investigations of ozone, and also by theoretical calculations. Taking into account the temperature inversion, the authors made new calculations, which showed that for the points of extinction of meteors the air density obtained from their formulas agrees fairly well with geophysical determinations; for the points of ignition, however, the density of the atmosphere is obtained as considerably greater (by \(10^2\)—\(10^3\) times) than was given by the geophysical determinations.

The resulting discrepancies in the density of the air at the heights of ignition of the meteor could have arisen from two causes: 1) from the fact that the theory itself was erroneous, being based on a number of erroneous assumptions, or 2) from the fact that the value and character of the parameters laid at the basis

SOME PROBLEMS OF METEOR PHYSICS

calculations, does not correspond to reality, and a number of data accepted as typical of the state of the atmosphere at the heights under investigation in fact do not characterize it.

The studies by Soviet astronomers I. S. Astapovich and V. V. Fedynsky, carried out in 1935,⁹ showed that the value of the “meteoric” density obtained by Lindemann and Dobson, 10²–10³ times greater, in fact differs less from the “geophysical” density (being greater than the latter by 10–10² times). The error of these authors consisted in the fact that the heights of meteors were systematically overestimated by them, and the density values obtained in reality should have been assigned to considerably lower layers, and not to those to which Lindemann and Dobson assigned them. A temperature inversion also played a certain role in this overestimate.

§ 3. In 1926 the American scientist Sparrow subjected the theory of Lindemann and Dobson to criticism. He indicated¹⁰ that the principal shortcoming of the Lindemann–Dobson theory consists in the application of the formula for adiabatic compression to the process of formation of the meteor cap. The equations of adiabatic compression would be applicable for a meteor speed small in comparison with the speed of atmospheric molecules. In reality the situation is quite the contrary: the meteor speed is of the order of 30–60 km/sec, whereas the speed of air molecules is only about 0.5 km/sec.

According to Sparrow, the ignition of a meteor occurs not from heating of the shell at the expense of the kinetic energy of the meteor, but at the expense of the numerous impacts of molecules on the meteoric body. The energy released in this case will already be proportional to the square of the body’s velocity, and not to the temperature of the surrounding masses of gas. The rise in gas temperature due to molecular impacts is very considerable and, according to Sparrow, fully explains the character of the ignition and extinction of meteors.

Sparrow points out that the conception of the motion of a meteor in the upper layers of the atmosphere as in a continuous gaseous medium (Lindemann and Dobson) is wholly inconsistent with reality. In fact we have here a more complex process: at heights of 70–160 km, owing to the strong rarefaction of the air, the mean free path of the molecules is large and is often considerably greater than the dimensions of the meteoric body. For example, at a height of 100 km it is equal to 1 cm, while the dimensions of the meteoric body are only fractions of a millimeter. As a result, the molecules do not accumulate around the meteoric body in the form of a continuous gaseous cap, but only strike its surface in large numbers, heating it to a considerable degree.

Sparrow develops his theory of meteor luminosity by considering mutual impacts between the atoms of the meteoric body and the molecules of the air. Collisions between molecules are regarded as not absolutely elastic. The energy obtained in impacts is partly dissipated, being spent on the destruction of the crystalline lattice of the substance of the meteoric body, on the dissociation of molecules and ionization of atoms of the air and of the meteoric body itself, while the greater part of it is transmitted to the atoms of the meteoric body and to the molecules of the air surrounding the meteor to excite luminosity.

The author assumes that as a result of the entire sum of impacts the temperature of the meteoric body rises to no less than 2000–2500° K (the melting temperature of stony rocks). The high temperature causes considerable evaporation of the substance, which leads to the “ignition” of the meteor, and it becomes visible.

Sparrow, proceeding from a definite character of the distribution of gases in the atmosphere and from a definite composition of it, attempted to calculate the theoretical heights of ignition and extinction of meteors. He found that, owing to the increase in the energy of individual impacts of molecules, which increases in proportion to the square of the meteor’s speed, faster meteors should ignite at a more considerable height. This conclusion of Sparrow is confirmed directly by observations.¹¹ For example, H. Newton (H. A. Newton), C. P. Olivier, G. Nissl, W. F. Denning, many years ago, and V. A. Maltsev¹² in 1930 showed that a difference in the velocities of meteors of 30–40 km gives a difference between the heights of ignition of meteors of

40–50 km.

In conclusion, Sparrow finds that there is no increase in temperature in the upper layers of the atmosphere. He points out that the data at his disposal, obtained from observations of meteors, agree well with the actual facts in the case where it is assumed that the isothermal atmosphere at those altitudes has a temperature of \(-54^\circ\) C. But Sparrow came to this conclusion while assuming that in the upper layers of the atmosphere only hydrogen and partly helium are present; therefore his conclusions cannot be considered correct.

§ 4. A newer point of view in meteor physics was the theory of meteors proposed by the English geophysicist Maris¹³; however, from the mathematical side it is developed in less detail than the theories of Lindemann–Dobson and Sparrow.

At the time Maris created his theory of meteors, the heights of telescopic meteors were not yet known. There were only isolated considerations by a number of authors who regarded the heights of occurrence and disappearance of these meteors as greater than those of normal meteors. It was indicated that some meteors are visible at altitudes of about 500 km, and according to Denning—even at an altitude of 2,000 km.

At so great an altitude a gaseous cap cannot form, since the density of the atmosphere there is small and the mean free path of molecules is so large that molecules, after colliding with the meteoric body, manage to fly out of the sphere that characterizes the formation of the cap.

At such altitudes the luminosity of meteors can occur solely through collisions of the meteoric body with molecules. The kinetic energy of motion of the molecules and of the meteoric body itself, being transformed into heat, heats its substance. But the bombarding molecules cause not only heating of the body—which, as experiments carried out under laboratory conditions with canal rays have shown, must be very considerable—but also strong dissociation of the substance.

During the flight of a meteor, molecules of air collide with it and rebound from it with high speeds. Despite the great rarefaction of the atmosphere in the upper layers, in addition to direct impacts on the meteoric body there are also mutual collisions of the rebounding molecules with one another. The ratio of the number of direct impacts of molecules to the number of mutual collisions is determined approximately by the following formula:

\[ \beta=\frac{d^{2}}{\gamma\cdot l_{1}^{2}}, \tag{6} \]

where \(d\) is the magnitude of the meteor’s cross section.

At great altitudes \(\beta\) is small, i.e., there are few mutual molecular collisions and no gas envelope is formed; but at small altitudes \(\beta\) becomes, owing to the large number of collisions, very significant, and conditions favorable for the formation of a cap arise.

Maris studies an average meteor entering the atmosphere with a velocity of 40 km/sec and flying 60 km in the atmosphere; the mass of the meteoric body is taken to be \(6.25\cdot10^{-5}\) g, its diameter 0.1 mm, and its composition iron. Maris considers the collisions between the molecules of the air and the atoms of the meteoric body to be elastic. In collision, the mechanical energy of motion is transformed into light energy, and the energy released is then determined from the relation:

\[ E=\frac{MV^{2}}{2}=\rho\cdot\frac{dV^{3}}{2}, \tag{7} \]

where \(M\) is the mass of air carried along by the meteor \((M=dv)\), \(\rho\) is the density of the air at this altitude, and \(d\) is the magnitude of the cross-sectional area. If into formula (7) we substitute the value of the air density \((\rho)\), the magnitude of the meteor’s cross section \((d)\), and the speed of motion of a typical meteor, then \(E\) is obtained equal to:

SOME PROBLEMS OF METEOR PHYSICS

\[ E=\frac{\rho dV^{3}}{2}=2\cdot 10^{8}\ \text{erg/sec}. \]

A meteor expends its kinetic energy while moving in the atmosphere: 1) on radiation, 2) on the transfer of energy to air molecules, 3) on the dissociation of air molecules and the ionization of atoms.

Marris points out that 97% of the entire kinetic energy of the meteor is given to the surrounding air space as a result of collisions with air molecules and through the evaporation of atoms of matter from the meteoric body.

The principal share of the light energy is radiated by the excited air molecules surrounding the meteor, and only a quite insignificant share of the light radiation is emitted by the meteoric body itself.

As a result of the processes of dissociation and ionization of air atoms, ionized (gaseous) trails are formed, and, as a result of the evaporated matter of the meteor, dust trails are formed. Ordinarily, atoms of chemical elements in the ionized state and molecules of chemical compounds in the dissociated state exist for small fractions of a second. But Marris points out that, in the presence of vapors of metals evaporated from the meteoric body, in the regions of flight of “high” meteors they may exist for several minutes.

At altitudes of about 80 km and below, the density of the air becomes such that formula (6) loses its meaning. Here, a dense gaseous cap already forms around the meteor, preventing further direct impacts of molecules against the meteoric body. At altitudes of about 80 km the meteor will already give off a more considerable amount of energy; by formula (7) we obtain it as equal to:

\[ E=5\cdot 10^{11}\ \text{erg/sec}. \]

In his work Marris criticizes the theories of Lindemann—Dobson and Sparrow.

Marris quite correctly points out and gives a number of arguments in favor of the fact that in the gaseous cap of a meteor, where there is no thermodynamic equilibrium, the equation of adiabatic compression introduced by Lindemann and Dobson is inapplicable. But Sparrow’s equation as well, in deriving which he considered the impacts between molecules to be elastic, also proves not to correspond to reality, since with such a kind of collision too little energy is released in comparison with the amount needed for the excitation of masses of gas and the dissociation of molecules of the matter of the meteoric body.

According to Marris, the main mass of radiation arising owing to the collision of air molecules with the meteoric body is emitted in the extreme ultraviolet part of the spectrum and partly in the region of X-rays. The visible region of the spectrum accounts for approximately only 8–10% of all the energy.

Marris’s views are likewise not free of many shortcomings. Let us cite a number of remarks concerning this theory. It is hardly permissible to transfer laboratory experiments on the collision of atoms with molecules to the conditions prevailing during a meteor’s flight in the upper layers of the atmosphere. Further, the energy obtained from the collision of the meteor with air molecules, by virtue of scattering, must be smaller than the value assumed by Marris.

The latest determinations of the heights of telescopic meteors, carried out in 1930 by E. Öpik^1 in Estonia, by I. S. Astapovich in the USSR^2, by the Arizona expedition of the Harvard Observatory in 1931–1933, and in 1935 by V. A. Bronshten^14, indicate that the heights of telescopic meteors are considerably lower than Marris assumed. We see the flight of telescopic meteors at an altitude of 70–90 km, and their ignition has never been noted above 100 km. As a result, that part of Marris’s work which treats the cause of the ignition of telescopic meteors at great altitudes loses its significance.

§ 5. A substantial role in the development of meteor physics was played by the work of the Estonian astronomer E. Öpik15, published in 1933.

The intensity of the meteor radiation, the character of the glow of the meteor and of the meteor cap are considered by the author as consequences of the process of collisions of the atoms of the meteor and the atoms of the atmosphere. E. Öpik investigates the problem both by the methods of the Bohr model and with the aid of quantum mechanics.

The conclusions at which the author arrives are the following:

1. The kinetic energy of atoms at cosmic velocities with which meteors fly is very considerable and quite sufficient for ionization. For atoms of hydrogen, nitrogen, and iron the following data are given (the energy is expressed in volts).

The energy obtained is quite sufficient to cause secondary, tertiary, etc. ionizations of atoms (ionization potentials, for example, of potassium, total 6.1 V, iron—7.8 V, hydrogen—13.5 V, oxygen—13.6 V, nitrogen 14.5 V, and of the molecules H₂—16 V, O₂—13 V, N₂—16 V, etc.16).

1. The work analyzes the flight of a meteor in a nitrogen atmosphere; the solution obtained for the problem can also be transferred to the case of an oxygen or mixed oxygen–nitrogen atmosphere (since their ionization and radiation potentials are almost identical).

2. In contrast to Lindemann and Dobson, Öpik finds that a noticeable gaseous cap is possessed in practice not by all meteors, but only by exceptionally large fireballs and meteorites. Moreover, the greater the cosmic velocity of the meteor, the greater will be the gaseous envelope around it. The dependence of the radius of the meteor on the velocity at which a gaseous cap can form is illustrated by the following table:

Ordinary meteors must possess a very small cap, inside which, despite its small dimensions, there is a high pressure. The cap does not admit the oncoming air molecules to the meteor and radiates considerably more light than the body itself. The larger the meteoric body, the larger the cap that forms around it.

1. The radiation emitted by a meteor consists of two parts: the radiation of ionized atoms and temperature radiation. From the excited and multiply ionized atoms of the gaseous cap, very short-wave radiation will be emitted. Owing to the heating of the meteoric body, radiation will be emitted with a maximum in the red part of the visible spectrum.

2. During flight, products of evaporation will fly off from the surface of the meteoric body; these (if there are many of them) will give rise to the meteor tail. The larger the dimensions of the meteor, the more products of evaporation there will be, and the more significant and bright the meteor’s tail will be1).

1) One should distinguish the meteor trail (Meteorspur, meteor streak) from the meteor tail (Meteorschweif, meteor train): the first is formed after the meteor’s flight in the surrounding air, the second follows the meteor, closing its envelope.

The tail that is formed, not being in thermodynamic equilibrium, rapidly scatters and mixes with the gases of the upper layers of the atmosphere. The broad meteor trails visible fairly often for a long time after the passage of a meteoric body are explained by the author either by the recognition of the meteor tail (dust trail), or by the ionization of air atoms by the short-wave radiation of the meteor. Trails of the first type must have in their spectrum both metal lines and nonmetal lines; spectra of trails of the second kind must consist exclusively of gas lines.

On the basis of Öpik’s theory it is possible to attempt an estimate of the mass of meteors. In 1922, even before the publication of the cited work, Öpik, considering that a meteor radiates like α-Persei, estimated the mass of a Perseid of the 2nd stellar magnitude at 0.3 mg[^17]. In doing so he assumed that all the kinetic energy is converted into light radiation and that the radiation of the meteor is analogous to the radiation of an absolutely black body with a temperature of \(6000^\circ\) K. If one determines the mass of a Perseid of the 2nd stellar magnitude under all of Öpik’s assumptions, but using the figures of 1933, then one obtains a mass equal to 12 mg[^18].

This value of the mass of a Perseid meteor agrees fairly well with the determination of meteor masses made by V. V. Fedynsky (Moscow) by an entirely different route.

In Öpik’s works of 1937[^19] and 1938 the processes of atomic and atom-molecular collisions during the flight of a meteor are analyzed in greater detail. By calculation the author finds the intensity values of a number of lines of meteor spectra; the results of the calculation coincide with the values obtained indirectly from observations.

In these works the author dwells in more detail on the elucidation of a number of physical conditions of the flight of a meteoric body in the Earth’s atmosphere, namely: changes in the velocity of motion, decrease in the dimensions of the meteoric body, increase of the temperature and pressure inside the gas cap, etc.

Investigating an iron meteor with density \(\rho = 7.8\) and molecular weight \(\mu = 53\), Öpik arrived at a number of formulae on the basis of which the following table was constructed.

TABLE 1

\(\Delta R\)—change in size, \(\Delta T\)—change in temperature, \(m\)—brightness of the meteor

| Velocity | \multicolumn{3}{c}{\(R = 1\) cm} | \multicolumn{3}{c}{\(R = 0.1\) cm} | \multicolumn{3}{c}{\(R = 0.02\) cm} |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | \(\Delta R\) | \(\Delta T\) | \(m\) | \(\Delta R\) | \(\Delta T\) | \(m\) | \(\Delta R\) | \(\Delta T\) | \(m\) |
| \(W = 16\) km/sec | 0.035 | 24000° | -1 | 0.011 | 800° | 7 | 0.005 | 70° | 13 |
| \(W = 40\) ” ” | 0.054 | 92000 | -5 | 0.017 | 3100 | 5 | 0.008 | 270 | 10 |
| \(W = 90\) ” ” | 0.081 | 320000 | -9 | 0.026 | 10000 | 2 | 0.012 | 870 | 7 |

Öpik’s investigations of “stony” meteors lead qualitatively to the same results.

II. COMPOSITION AND STRUCTURE OF THE STRATOSPHERE ACCORDING TO METEORS. MOTION OF METEORS IN THE STRATOSPHERE

§ 6. A number of authors have tried, on the basis of one or another meteoric material, to make certain conclusions about the density, temperature, and composition of layers of the atmosphere. We shall dwell on works carried out mainly in our Union, and shall speak only briefly about the works of foreign scientists.

As early as 1933 the Moscow astronomer A. B. Severnyi^20 analyzed the distribution of meteors by their luminosity, or so-called “absolute brightness,” at various heights, and applied his results to the clarification of a number of data on the structure of the stratosphere.

A. B. Severnyi, basing himself on the catalogue of 108 heights of meteors of the Perseid stream, compiled by Ph. Broch, calculated, by the formula of I. S. Astapovich,^21 their absolute brightness in international candles. It has the form:

\[ \lg J = 2 \lg R - 0.4m + 0.327, \tag{8} \]

where \(J\) is the “absolute brightness” of the meteor in international candles, \(R\) is the distance to the meteor, \(m\) is its apparent stellar magnitude (\(+0.327\) is a constant of zero point).

Having investigated the distribution of the absolute brightness of meteors with height, Severnyi found that it increases with increasing height. The observed increase in the absolute brightness of meteors with height is obtained as a consequence of the difference in the masses of meteoric bodies and of the physical conditions in the upper layers of the atmosphere. Assuming that all the visible radiation of a meteor is produced at the expense of the gas cap, the author, by formula (1), determined by a method of successive approximations the temperature of the undisturbed atmosphere.

Severnyi finds that the theoretical data best satisfy the observations if one assumes the predominance, at great heights (from 100 to 150–200 km), of helium. The temperature of this layer of the stratosphere should be about 400–500°K. Taking the obtained temperature values of the upper layers of the atmosphere and considering that the laws of ideal gases are applicable to them, and also allowing for the presence of a change in molecular weight in the direction of its decrease with height, the author succeeded in composing the density equation:

\[ \frac{1}{\rho}\frac{d\rho_0}{dh} + \frac{1}{R'}\frac{dR'}{dh} + \frac{1}{T}\frac{dT}{dh} + \bar{g} = 0, \tag{9} \]

where \(\bar{g}\) is the ratio of the force of gravity at height \(H\) to the acceleration of gravity at the surface of the Earth, \(T\) is the temperature of the undisturbed atmosphere, \(R'\) is a constant equal to \(\frac{R}{M_0}\), where \(R\) is the gas constant, and \(M_0\) is the molecular weight, \(\rho_0\) is the density of the atmosphere.

The calculations showed that at heights of 90–100 km there is a sharp change in density.

As early as 1926 Sparrrow^10 also discovered an anomalous change in density at these heights, proceeding from other considerations.

Severnyi’s conclusions, as subsequent investigations have shown, are not final: 1) the increase in brightness with height could have occurred because of heights overestimated owing to observational errors; 2) Broch’s catalogue is not free from the indicated errors and was used because of the absence of other suitable data; 3) the decrease of mean molecular weight with height may testify not to the presence of helium at great heights, but to the presence of strong dissociation of such gases as nitrogen and oxygen.

In 1935 K. P. Staniukovich published a note,^22 in which, relying on the material of V. A. Mal’tsev, who gave the height of combustion of meteors as a function of their geocentric velocities for 484 bolides of the Niessl-Hoffmeister catalogue, he attempted to determine the character of the pressure distribution in the upper layers of the atmosphere.

He proceeds from the basic propositions of Sparrrow’s theory and assumes that: 1) the composition of the upper layers of the atmosphere is chemically homogeneous, 2) the mass and dimensions of the meteoric bodies are equal, and 3) their form is spherical.

If one investigates two meteors with initial geocentric velocities \(V_0'\) and \(V_0''\), whose ignition points are located at heights \(H_1'\) and \(H_1''\), and which undergo, from the beginning of the boundary of the atmosphere to heights \(H_1'\) and \(H_1''\), collisions with gas molecules of total mass \(m'\) and \(m''\), then, under the assumption

constancy of the meteor’s velocity \(V\) (which, in a first approximation, corresponds to reality), the following relation is obtained:

\[ V_0' \cdot m' = V_0'' m'', \]

which makes it possible to determine directly the pressure in the upper layers of the atmosphere.

At an altitude of 320 km above the earth, Stanyukovich considers the density of the atmosphere to be zero. The pressure at other, lower, nearby altitudes \(H_i'\) and \(H_i''\) is determined by the formula:

\[ m'' - m' = \frac{\triangle V}{k} \cdot \mu \left( \frac{1}{V'} - \frac{1}{V''} \right). \]

The data obtained by K. P. Stanyukovich, based on heterogeneous material, however, represent the character of the density distribution very poorly. At present they can be regarded only as an illustration of the method, interesting in its originality.

In 1938 there appeared a work by B. Yu. Levin (Moscow), in which the author, on the basis of the latest investigations\(^9\), which give a lowering of the visual heights of meteors by 15–25%, statistically considers the distribution of the number of meteors by height as a function of the parallax error.

B. Yu. Levin\(^ {23}\) points to the presence of several special layers in the atmosphere, differing from one another both in temperature and in pressure, namely: a layer at an altitude of 110–115 km, where the heights of ignition of meteors and fireballs are concentrated; the already well-known layer at an altitude of 80 km, where the heights of extinction of ordinary meteors are concentrated; layers at altitudes of 49, then 36 and 27 km, where the heights of extinction of bright fireballs are found.

The presence of changes in the density gradient at altitudes of 30–35, 80 and 105–115 km above the earth has also been detected in recent years by acoustic, electrophysical and other methods (from aurorae, etc.).

§ 7. Of special value for clarifying the physical conditions during the flight of a meteor (the temperature and pressure in the gaseous envelope, the evaporation of matter from the surface of the meteoric body, the nature of ionization and excitation of the surrounding air, etc.) is the study of meteor spectra.

The study of spectra already obtained, and the work of photographing new meteor spectra, really began only about 7–8 years ago (before 1931 only 8 meteor spectra were known, obtained mostly by chance, with the exception of 3 spectra obtained by the Russian astronomer S. N. Blazhko in Moscow in 1904 and 1907). But from 1931, when a special program for photographing spectra was worked out in the United States at the Harvard Observatory, the number of spectrograms obtained increased considerably. By the autumn of 1938, 47 photographs of meteor spectra had been obtained throughout the world. Especially many meteor spectra were taken in 1932–1934 (28 spectrograms).

The study of the spectrograms obtained, carried out by the American astronomer P. Millman\(^ {24}\), indicated that the meteoric body consists of iron, silicon, magnesium, calcium, aluminum, chromium, and manganese, i.e., has a composition very close to that of stony meteorites (some meteors have a metallic character).

From the intensities of individual spectral lines this investigator succeeded in determining the so-called effective temperature of the gaseous envelope; it was assumed that the masses of gas in the envelope are in thermodynamic equilibrium. Since in fact equilibrium is absent, the temperature obtained in this calculation Millman calls “effective.” The “effective” temperature of the meteoric gaseous envelope lies within the limits from 1700 to 3400° K. Despite the fact that during the period of a sharp flash of the meteor the brightness of the spectral lines increases by tens of times, the temperature of the meteor remains unchanged, and in some meteors it even proves to be lower by 100–300°, which

rich in meteor streams (Perseids, Leonids, and Orionids), as well as sporadic ones.

Millman and Hoffleit, from the photographs obtained, calculated the mean height of the meteor’s flight \((H)\); the distance separating the path of the meteoric body from the camera, corrected for the curvature of the terrestrial sphere \((a)\); the observed velocity of the meteor’s flight relative to the Earth’s surface, reduced to the zenith \((k)\); the geocentric velocity of the meteor, corrected for the acceleration produced by the action on the meteor’s motion of the Earth’s gravitational field, but not corrected for the effect of diurnal aberration; and, finally, the heliocentric velocity of the meteoric body which it would have had while moving around the Sun at a distance equal to the distance separating our Earth from it.

All these data are summarized in Table 2.

TABLE 2

It turns out that the heliocentric velocity for meteors of the Leonid stream is equal to 31 km/sec; the calculations give 41.5 km/sec. This difference between the calculated and observed heliocentric velocities of meteors of one and the same meteor stream indicates that a velocity of 10–11 km/sec is lost by the meteor during its motion in the atmosphere down to the mean height of the meteor’s ignition.

The corresponding value of the heliocentric velocity, not corrected for the influence of the atmosphere, for meteors of the Perseid and Orionid streams is obtained as, respectively, 29 km/sec and 19 km/sec. If these numbers are again compared with the calculated heliocentric velocities, it again turns out that meteors from the Perseid stream lose, during flight in the atmosphere, a velocity of 10 km/sec, while meteors from the Orionid stream lose as much as 14–15 km/sec.

From the velocigrams obtained it was possible to investigate the character of the deceleration of meteors (see p. 463).

The presence of deceleration was noted for all 14 observed meteors.

Interesting information on deceleration during flight in the atmosphere has also been obtained by a number of authors for fireballs. For example, A. Wegener (A. Wegener), in 1927, on the basis of the Niessl-Hoffmeister catalogue of fireballs, found that the mean loss of velocity of fireballs is equal to \(24\% \pm 5\%\). With this determination of the magnitude of the loss of velocity of fireballs, Wegener assumed that all heliocentric velocities of their motion are on average identical, and that the relative percentage of velocity loss for all fireballs has the same magnitude.

From numerous observations of bright fireballs moving at a small angle to the plane of the horizon (i.e., whose flight altitude changes little) and having a long flight trajectory in the atmosphere, Nissl, and then Hoffmeister by his method, established that they have enormous losses of velocity.

Hoffmeister, on the basis of the study of two bright fireballs observed on September 3, 1916, and August 17, 1922, found that the resistance of the air to the motion of fireballs was approximately proportional to the 8th power of the velocity.

In our Union, the Moscow astronomer I. S. Astapovich has been engaged with this question for a number of years.^25

The denser the layers into which a fireball descends, the more significantly its braking is obtained. Nissl, after careful processing of all the material known up to 1917, found that on the average the velocity of fireballs decreases by from 30 to 70% of its initial value; the decrease in velocity is the more considerable, the lower the fireball descends into the atmosphere.

LITERATURE

1. E. K. Öpik, Telescopic Observations of Meteors at the Tartu Observatory, Publ. O. Astr. Observ. Tartu, XXVIII, No. 2, 1930.

2. I. S. Astapovich, Corresponding observations of telescopic meteors in 1930, Bull. of the Collective of Observers (BKN) of VAGO, No. 17, 1932; and also, On the nature of telescopic meteors, Astron. Zhurn., 12, 60—100, 1935.

3. W. F. Denning, Monthly Notices of the R. A. S., No. 57, 1897; No. 72, 1912; No. 76, 1916.

4. F. A. Lindemann and G. M. Dobson, Theory of Meteors, and the Density and Temperature of the outer Atmosphere etc., Proc. Roy. Astr. Soc., London, A, 102, 717, 1922, and also, Note on the Temperature of the Air etc., Proc. Roy. Astr. Soc., London, A 103, 721, 1923.

5a. I. S. Astapovich, On certain meteor methods of investigating the stratosphere, Astron. Zhurn., 16, 23—41, 1939.

1. Obolensky, Lindemann–Dobson’s theory of meteors, Meteoro. Vestnik, No. 8, August 1927.

2. Khanevsky, On the structure of the stratosphere, Zhurn. Geofiziki, No. 3, 1933.

3. N. M. Shtaude, Lindemann–Dobson’s theory of meteors and certain consequences following from it, Trudy VKIS, Publishing House of the Academy of Sciences, L., pp. 481—487, 1935.

4. Rudakowic, Meteorolog. Z., 43, 441, 1926 and 44, 326, 1927.

1. At present, in meteor astronomy, the following are distinguished: a) meteoric body, i.e. a solid body of cosmic origin; b) meteor—a phenomenon produced by the motion of a meteoric body in the atmosphere of some celestial body (planet, etc.); and c) meteorite—a meteoric body that has reached the surface of a planet. ↩