Essays on the Physics of the Earth’s Atmosphere

I. A. Khvostikov, Leningrad

I. Structure of the atmosphere. — II. Temperature of the upper layers of the stratosphere. — III. The problem of atmospheric ozone. — IV. Ionization of the atmosphere and the propagation of radio waves. — V. The ionizing action of the Sun’s ultraviolet rays. — VI. The connection between the ionization of the upper layers and other phenomena. Corpuscular theory of ionization. — VII. Polarimetric method for studying the structure of ionized layers.

I. Structure of the Atmosphere

The boundary of the atmosphere. Ever since, in 1645, Pascal performed the experiment with Torricellian vacuum on Mount Puy-de-Dôme and showed that at the summit of the mountain the height of the mercury column was 3 dm \(1 \frac{1}{2}\) lines less than at its foot, it became clear that atmospheric pressure decreases with altitude. Each cubic centimeter of air weighs \(0.0013\) g, and therefore, if the density of the air remained the same at all altitudes, the boundary of the atmosphere would lie at a height of 7.7 km, since atmospheric pressure supports a column of mercury 76 cm high. But the density of the air must rapidly decrease with altitude, because the lower layers are compressed by the weight of the upper layers; hence it follows that the true boundary of the atmosphere lies somewhere much higher.

In the ninth–eleventh centuries the science of the Arabs flourished and then quickly faded. One of the remarkable achievements of Arab science was the method of determining the boundary of the atmosphere indicated by the famous Arab scholar Alhazen. Alhazen’s method is extremely simple, entirely correct, and continues to be widely used even in our day. With its help Alhazen, already in the eleventh century, knowing neither that air has weight (from the time of Aristotle and until Galileo air was considered weightless), nor that atmospheric pressure decreases with altitude (in those days atmospheric pressure was not even suspected), not only correctly concluded that the atmosphere has a finite height, but was also able to determine quite well the order of magnitude of the height of the atmosphere. Alhazen used for this purpose the observation of twilight.

After sunset, night does not come instantaneously, because the upper layers of the air still continue to be illuminated by the sun’s rays. A gradual darkening takes place, and night comes only when the rays of the sun no longer reach even the very upper layers of the atmosphere. Even the ancient Greeks knew that the onset of night always corresponds to one and the same sinking of the sun below the horizon, namely by 18°. Alhazen calculated from this that the height of the atmosphere is approximately 50 km.

True, in his geometrical computations Alhazen made an error: he assumed that the solar ray is reflected from the upper boundary of the atmosphere and precisely for this reason reaches the earth even when the sun is below the horizon. This error was later pointed out by Kepler, who refined Alhazen’s calculations.

Of course, full use of the possibilities provided by twilight observations has become possible only in our time, when the theory of the scattering of light by gases was developed, since twilight is a very complex phenomenon caused by molecular scattering of light. Very precise observations of recent years, chiefly the observations of V. G. Fesenkov, show that the gradual weakening of the brightness of the sky during twilight can be traced up to a moment corresponding to the scattering of light at an altitude of about 200 km. Consequently, the boundary of the atmosphere lies somewhere no lower than 200 km.

It goes without saying that the actual “boundary” of the atmosphere is a conventional concept. Indeed, the density of a gas decreases with height, and if we imagine a point so high that below it is contained 0.999999 of the entire quantity of atmospheric air, and above it only the 0.000001-th part, then, as is easy to see, this 0.000001-th part must occupy a volume no smaller than all the remaining 0.999999 parts, since the density of the gas above our point is on the average a million times less than in the lower layers. In general there is the obvious, yet nevertheless paradoxical, result that at whatever height we draw the conventional boundary of the atmosphere, the gas located above this boundary must occupy a volume no smaller than the entire remaining mass of gas below the “boundary” (within the limits of applicability of the Boyle–Mariotte law).

Therefore it is permissible to speak of the boundary of the atmosphere only in the sense of the possibility of detecting, by some phenomena observed from the earth, the presence of gas at one height or another. In this sense, the height of 200 km obtained from twilight observations is not the greatest. Auroras, the greater part of which have a height of the order of 100 km, are not infrequently observed at heights of 200–300 km, and sometimes they reach heights of 600–800 km. Likewise, observations of the propagation of radio waves indicate occasional reflections of waves from an ionized layer at a height of about 1 thousand km. Finally, the study of the properties of the night sky’s own glow showed that

in the composition of the glow, even at midnight, the presence of polarized scattered sunlight is detected, which indicates scattering of light at an altitude of 2 thousand km. Traces of gas at still greater altitudes have not yet been detected, and for the height of the atmosphere we may, in the sense indicated above, take the value of 2 thousand km.

Composition of the atmosphere. As regards the composition of the air at the surface of the earth, it has been studied very precisely. In addition to nitrogen and oxygen, present in amounts of respectively 78.03 and 20.99% by volume, argon (0.94%), carbon dioxide (0.03%), hydrogen (0.01%), neon (0.0012%), and helium (0.004%) are always present. These figures refer to dry air; the admixture of water vapor is approximately 1%. But the question of the composition of the air in the upper parts of the atmosphere is much more complicated. Considering the atmosphere as a mechanical mixture of various gases and applying to this mixture the laws of ideal gases, one can derive the so-called barometric formula, giving an exponential law for the decrease of pressure in the atmosphere with height

\[ p=p_0 e^{-\frac{Mg}{RT}h}, \tag{1} \]

where \(p\) is the pressure of the gas at height \(h\), \(p_0\) is the pressure at height \(h=0\), \(M\) is the molecular weight of the gas, \(g\) is the acceleration of gravity, \(R\) is the gas constant, and \(T\) is the absolute temperature. The rate at which the pressure decreases depends on the molecular weight; it is greater, the greater the molecular weight. This means that with height the relative content of light gases should increase.

Let us calculate at what height \(h^*\) the pressure of a gas decreases by a factor of \(e\). To do this, we take the logarithm of equation (1):

\[ \lg \frac{p_0}{p}=\frac{Mg}{RT}h. \]

Put \(\lg \frac{p_0}{p}=1\), i.e. \(\frac{p_0}{p}=e\); then

\[ \frac{Mg}{RT}h^*=1 \]

or

\[ h^*=\frac{RT}{Mg}. \]

For oxygen \(M=32\), and we obtain \(h^*=7.1\) km (at \(T=273\) abs.). For nitrogen \(h^*=8.1\) km; for hydrogen \(M=2\) and, consequently, \(h^*=115\) km. These numbers show how strongly the composition of the atmosphere should change at great altitude. At height \(h=115\) km the pressure of hydrogen will fall only by a factor of \(e\), but for oxygen it will decrease by a factor of \(e^{16}\), and for nitrogen by a factor of \(e^{14}\). At an altitude of 100 km hydrogen should have been 95.6%,

nitrogen 3%, helium 1.3% and oxygen 0.11% at a total pressure of 0.007 mm Hg. Beginning at an altitude of 100 km and higher, the atmosphere would have to be almost purely hydrogen.

One can hardly doubt the applicability of the Boyle–Mariotte and Dalton laws (on the basis of which the barometric formula is derived) to atmospheric gases, and it is all the more surprising that, for the high layers of the atmosphere, a complete discrepancy is obtained between the calculations according to the barometric formula and the actual composition of the air. There is every reason to believe that in the upper layers of the atmosphere hydrogen is completely absent and that the composition of the atmosphere, as below, is essentially nitrogen–oxygen. Thus, for example, the study of the spectra of auroras and of the intrinsic glow of the night sky shows that in the glow of the atmosphere at heights above 100 km there are always lines and bands belonging to the spectra of nitrogen and oxygen, and that the spectrum of hydrogen has never been observed. This circumstance compels one to put forward the hypothesis of the so-called “mixed” atmosphere: under the influence of some entirely unclear causes, the atmosphere is mixed throughout its whole height and has approximately the same composition everywhere. This point of view is now the most widespread; in any case, the opinions of all investigators agree that the atmosphere is completely mixed up to heights of 100–150 km.

The results of the study of air samples taken from great heights during flights into the stratosphere also speak in favor of the theory of the “mixed” atmosphere. Thus, during the flight of the stratospheric balloon SSSR on September 30, 1933, an air sample was brought back from a height of 18.5 km. If the barometric formula were correct, then at this height the content of nitrogen and oxygen should have changed by several percent compared with the composition of the air at the earth’s surface. But analysis of the sample showed that the composition of the air remained the same to an accuracy of 0.1%.

Some of the assumptions that have usually been taken as the basis for theoretical calculations of the composition of the upper layers of the atmosphere may be incorrect. For example, attention has always been paid only to those gases that are found in the composition of the air at the earth’s surface. But the possibility is not excluded that at great height there may exist, in considerable quantity, a gas not detected below. Let us imagine a very light gas that enters into the composition of the air near the earth’s surface in such a small quantity that its presence remains unnoticed. With the small molecular weight of such a gas, its percentage content at great heights, according to formula (1), may become appreciable. It was precisely this kind of hypothesis that Wegener put forward in 1910. At that time the question of the nature of the bright green line, always observed in the spectra of auroras, was very unclear. This line, possessing the greatest intensity in the entire spectrum of auroras, has wavelength \(\lambda = 5577.3\ \text{Å}\), which—

which could not be found in the spectra of any substance. Wegener suggested in this connection that the green line is emitted by some as yet unknown gas, which he named geocoronium. This gas, possessing, according to Wegener, a very small specific weight, is present in negligible quantity in the lower parts of the atmosphere, but in the upper layers its relative content increases greatly owing to its small weight. Wegener believed that above 100 km geocoronium is the chief constituent part of the atmosphere.

This hypothesis has not been justified for many reasons, in particular because the green line mentioned above turned out to belong to the spectrum of atomic oxygen.

An essential difference of the upper layers is the state of dissociation of gas molecules. Below, in the air, we have only molecular oxygen, but at great altitude there is a large amount of oxygen whose molecules have dissociated into atoms. We know well of the presence of atomic oxygen in the upper parts of the atmosphere from the spectra of auroras and of the glow of the night sky. The cause of the dissociation of oxygen molecules is the ultraviolet and corpuscular radiation of the sun. This radiation does not penetrate into the lower layers (it is absorbed by the atmosphere).

Finally, we proceeded from the assumption that the atmosphere is a mechanical mixture of gases chemically not acting upon one another. We are sure of this with respect to the lower layers. But the situation changes at great heights owing to the intervention of an outside agent—the ultraviolet and corpuscular radiation of the sun. This radiation produces various effects: it dissociates molecules, excites atoms and molecules, and, in addition, ionizes them, which extremely complicates the picture of the physical state of the upper layers. As a result of the appearance of atomic oxygen, reactions arise \(O + O_2 \to O_3\), leading to the formation of ozone molecules \(O_3\). The presence of ozone decisively changes the properties of the earth’s atmosphere, chiefly in the optical respect. The action of ozone may be called astonishing, especially if one takes into account its negligibly small content in the atmosphere. By many years of painstaking investigations (which will be discussed in detail in Chapter III) it has been possible to establish that the total amount of ozone in the entire atmosphere is such that, if it were collected in a layer at atmospheric pressure, the thickness of the layer would be only 3 mm; i.e., the mean concentration of ozone is only \(4 \cdot 10^{-7}\). Meanwhile, this amount of ozone, which has very strong absorption in the spectral region from 2900 to 2200 Å, proves sufficient to absorb completely all the energy of the ultraviolet radiation of the sun from 2900 to 2200 Å. As a result, it turns out that the magnitude of the energy of the ultraviolet rays of the sun absorbed by ozone exceeds the energy absorbed by all the rest of the atmosphere. The distribution of energy in the spectrum of radiation of an absolutely black body at a tempe-

ture of 6000°, which the sun approximately is, is such that, if the energy of the sun’s rays for a wavelength of 4000 Å is taken as 1, then at 3000 Å it should amount to 0.61, and at 2000 Å—0.09. Still farther into the ultraviolet part of the spectrum the energy decreases very rapidly, so that, of the ultraviolet portion of the spectrum wholly absorbed by the earth’s atmosphere, beginning at 2900 Å, the greater part of the energy falls precisely within the interval 2200–2900 Å, i.e., to the share of ozone. Hence it is clear what a large role ozone plays in the energy balance of the atmosphere.

As is known, the presence of ozone in the atmosphere has, moreover, great biological significance. The point is that short-wave ultraviolet radiation has an exceedingly strong biological effect, and this effect increases especially near the boundary of ozone transmission.

All these numerous and varied circumstances connected with the presence of ozone in the atmosphere make the question of atmospheric ozone itself one of the central problems of atmospheric physics. Of great theoretical interest are the mechanism of ozone formation and its very uneven distribution with height. These questions have only in the most recent years come close to their solution, and a separate chapter of our article will be devoted to them.

The presence of strong ionization of the upper layers introduces still greater complication into the picture of the physical state of the atmosphere. Both the origin of the ionization itself and its peculiar distribution with height represent a whole complex of fascinating physical problems, far from being completed even at the present day. The presence of ionization entails a series of phenomena affecting the behavior of the earth’s magnetic field and the propagation of radio waves; the presence of ionization is connected with the occurrence of auroras and the glow of the night sky. In the structure of the ionized layers the powerful action of solar, wave and corpuscular radiation upon the earth’s atmosphere is revealed with particular sharpness, and it may be that precisely the study of this structure will make it possible to solve a number of important questions in solar physics. Therefore the problems belonging here are, in the whole study of the earth’s atmosphere, especially important and interesting, and a special place will be allotted to their consideration in our review.

Troposphere and stratosphere. As height above the earth’s surface increases, the temperature of the air decreases on average by 5–6° per 1 km. However, such a fall of temperature occurs only up to a height of 10–12 km, after which the temperature ceases to decrease—the overlying layers are almost isothermal, with a temperature of about −55°, and even a slight increase of temperature is observed. Thus the atmosphere proves to be divid-

divided into two principal layers: the lower one, whose physical processes are closely connected with processes on the Earth’s surface, and the upper one, dependent to a much lesser degree on the physical life of the Earth.

This remarkable fact of the division of the atmosphere was discovered in 1899 by the French scientist Teisserenc de Bort, who named the lower layer the troposphere, and the upper layer the stratosphere.

Subsequent investigations showed that one must also distinguish, as an independent layer, a third layer—the transition layer from the troposphere to the stratosphere, since this layer is characterized by a peculiar distribution of temperature. Specifically, it turned out that often above the troposphere there is a layer with a temperature inversion: the temperature rises rapidly, then falls again, and only after that begins the slow rise of temperature corresponding to the transition to the stratosphere proper. This intermediate layer received the name substratosphere or (among the English) tropopause. The substratosphere usually has a thickness of from 1 to 4 km.

As for the height at which the boundary of the troposphere passes, many years of observations have shown that it does not remain constant, although its changes are in general quite regular. First, it was established that the height of the troposphere depends on geographic latitude: at the equator the boundary of the troposphere lies higher than at high latitudes. Second, this boundary shifts depending on the season: in winter it descends, in summer it rises. On average, at the pole the stratosphere begins at an altitude of 11 km, and at the equator—at an altitude of 17 km.

The temperature of the stratosphere in the Arctic undergoes very interesting annual changes, as was clarified thanks to the extensive application of the radiosonde method developed by P. A. Molchanov. It turned out that in the Arctic, at an altitude of 10 km, the temperature varies from −61° in January to −46° in July, while at an altitude of 20 km January gives a temperature of −66°, and July −33°. There are also other peculiarities in the temperature curves.

The division of the atmosphere into two clearly expressed layers may be explained theoretically. All geophysicists agree that the cause of the division is the selective absorption of solar heat by water vapor. The quantity of water vapor decreases rapidly with altitude, as is seen from Table 1, where the absolute quantity of water vapor is indicated in grams per cubic meter for different seasons.

The temperature of the air at each given altitude depends on the quantity of solar heat absorbed by water vapor. In the stratosphere the content of water vapor is small; correspondingly, low temperatures of the order of −50–60° prevail there. As the altitude decreases the quantity of water vapor increases, but correspondingly the total density of the air also increases, and with it the heat capacity of a unit volume; therefore a larger quantity of heat, expended on a larger mass of air, still creates the same

TABLE 1

the lowest temperature. It is necessary to take into account here the temperature radiation of the air masses themselves into the surrounding space (radiative equilibrium). But from a certain altitude the increase in the amount of water vapor begins to overtake the increase in the total density, as a result of which the temperature rises—we pass from the stratosphere into the troposphere. The rapid increase in absolute humidity as one approaches the earth’s surface creates a large temperature gradient in the lower layers—in the troposphere.

It can be shown that the annual fluctuations of the level of the stratosphere agree qualitatively with the assumption that the optical thickness of water vapor for this level always has some definite value. In winter, when the humidity is less, the stratosphere must descend lower in order to have the same optical thickness of water vapor. Such a lowering of the stratosphere in winter is in fact observed, as has already been indicated. From this point of view one can also understand the dependence on the latitude of the place. Indeed, at the equator the earth receives more radiant energy than at the pole. Therefore, in comparison with the mean picture, the temperature at the equator will rise, and at the poles will fall. But the higher temperature at the equator causes there a greater humidity, and consequently the lower boundary of the stratosphere must pass at the equator at a greater altitude than at the poles. This is precisely what is observed in reality.

From this point of view the possibility is not excluded of the influence, on the level of the stratosphere, of local fluctuations of humidity near the earth’s surface. Thus, over islands situated in the ocean, the boundary of the stratosphere may pass lower than over large continents.

Very interesting considerations on the “regulating” action of water vapor were expressed by Simpson. Any strengthening of the sol-

nocturnal radiation must be accompanied by an increase in evaporation from the oceans and an intensification of cloudiness. But as a result of this, the absorption of solar heat in the atmosphere increases, and that part of the solar energy which goes to heating the soil and the oceans is reduced. Therefore the increase in temperature near the earth’s surface will be much smaller than would correspond to the initial intensification of solar radiation. Conversely, in the case of a weakening of solar radiation, the content of water vapor in the atmosphere would decrease, which would contribute to a smaller lowering of temperature. Simpson believes that the temperature would change very little if the earth were to receive, for example, as much energy as Venus receives, or as little as Mars receives.

In conclusion to the paragraph on the troposphere and stratosphere, we cannot fail to mention the thermodynamic treatment of the vertical stability of the atmosphere developed by theoretical meteorology. Let us imagine some mass of air situated at the height \(h_1\), and suppose that this mass of air has risen to some greater height \(h_2\). The result of this will be an increase in the volume occupied by the given mass of air, owing to the decrease of pressure with height. As a result of the expansion the gas will cool. If there is no exchange of heat with the surrounding space, i.e. if the process of expansion takes place adiabatically, then it can be calculated that an increase in height by each 100 m will be accompanied by a cooling of \(1^\circ\). Suppose that the actual temperature gradient with height is greater than \(1^\circ\) per 100 m. In this case our rising mass of air, as a result of adiabatic expansion, will be warmer than the surrounding air; consequently, its density will be less than the density of the surrounding space, and it will tend to rise higher and higher. In this case the atmosphere will be in an unstable state. Conversely, if the actual temperature gradient is less than \(1^\circ\) per 100 m, then the rising mass will prove colder, and consequently also heavier, than the surrounding space, and ascent cannot take place. In this case the atmosphere will be in a state of stable equilibrium. Finally, with a gradient of \(1^\circ\) per 100 m the equilibrium will be indifferent.

Since the stratosphere is distinguished by constancy of temperature in the vertical and even by a slight increase of it, we may say that the stratosphere is characterized by a sharp increase in vertical stability.

These exceedingly clear considerations make, however, the question of the composition of the upper layers of the atmosphere still more obscure. Indeed, the state of vertical stability characteristic of the stratosphere would seem to exclude the possibility of the existence of vertical currents, yet we know that the actually observed composition of the upper parts of the atmosphere compels us to suppose the presence

powerful vertical mixing. As a result, the state of affairs in this respect proves to be so unclear that, strictly speaking, no very serious attempts are even made to understand this phenomenon. The question of vertical mixing is an unsolved problem in stratospheric physics.

II. Temperature of the Upper Layers of the Stratosphere

Direct measurement of temperature by means of sounding balloons can be carried out up to an altitude of 30 km, and in rare cases even somewhat higher. Higher layers are inaccessible to direct methods. But in no other question of stratospheric physics do all possible indirect methods find such extensive application. One may count almost a full dozen independent indirect methods that make it possible to draw various conclusions about the temperature of layers up to heights of several hundred kilometers. Here one encounters the use of observations of the propagation of acoustic waves, the application of photometric methods for the study of twilight, analysis of the altitude distribution of auroras, the deciphering of data on the burning of meteors along their path through the atmosphere, and the theoretical treatment of the question from the standpoint of the radiative equilibrium of the stratosphere, as well as other methods. The application of all these methods is a splendid illustration of the possibilities possessed by modern physics for the study even of objects that would seem to be quite inaccessible. And the fact that the results obtained by applying different methods sometimes lead to results that are difficult to reconcile not only does not diminish the value of all this work but, on the contrary, increases the interest of such investigations, since the clash of contradictory results, compelling one to delve more deeply into the subject, leads to the inevitable improvement of the existing picture of the physical state of the atmosphere. Therefore we considered it possible to devote a separate chapter to the comparison and review of all indirect methods for determining temperatures.

Radiative equilibrium of the stratosphere. In principle, the temperature of the stratosphere at any altitude can be calculated purely theoretically. To do this it is necessary to calculate the amount of energy absorbed and, knowing it, to calculate what temperature the gas must have so that the expenditure of energy on thermal radiation into the surrounding space would be equal to the amount of absorbed energy. Of course, the possibility of carrying out such a calculation presupposes the applicability to the radiation of atmospheric gases of the laws of an absolutely black body, which is obviously only approximately true, and therefore a possible source of large errors is hidden here. Nevertheless, it seems very interesting to carry out such a thermodynamic analysis.

Let us consider some element of the volume of air. Let $K_\lambda$ be the measure of absorption of radiant energy by this volume (the ratio of absorp-

value of energy to incident energy) for wavelength \(\lambda\); \(S_\lambda, E_\lambda\), and \(A_\lambda\) are, respectively, the radiation from the Sun, from the Earth, and from the surrounding parts of the atmosphere that reaches the volume under consideration. Finally, \(B_{\lambda,T_0}\) is the energy of black-body radiation at temperature \(T_0\) for wavelength \(\lambda\). In the case of radiative equilibrium the equality

\[ \int_{0}^{\infty} K_\lambda B_{\lambda,T_0}\,d\lambda = \int_{0}^{\infty} K_\lambda (S_\lambda + E_\lambda + A_\lambda)\,d\lambda . \tag{2} \]

must hold.

Absorption \(K_\lambda\) is a very irregular function of wavelength, and therefore the integration can be carried out only numerically. Moreover, the quantity \(A_\lambda\) depends on the temperature of all the other parts of the atmosphere, and the temperature distribution with height has to be found by successive approximations. One cannot neglect the quantity \(A_\lambda\), since it is not small enough for that. Having taken some temperature distribution as the initial one, one must compute the right-hand side of equation (2) for some height; it must be equal to the radiation appearing on the left-hand side of the equality. The temperature \(T_0\) obtained in this way usually does not coincide with the one initially adopted. By changing, through successive approximations, the prescribed temperature curve, one can ultimately obtain a sufficiently good agreement between the initial and calculated temperatures at all heights. Calculations of this kind were carried out mainly by Goen1.

In the practical execution of calculations based on the theory of radiative equilibrium, many additional assumptions have to be introduced. Thus, for example, the atmospheric absorption \(K_\lambda\) at different heights is unknown to us; it depends on the composition and density of the atmosphere, which are unknown to us and which themselves depend on a whole series of conditions, including the very temperature that we seek to determine. Proceeding from the great transparency of atmospheric gases, Goen in his calculations confined himself to taking into account the absorption of energy by water and ozone molecules, neglecting all the remaining constituents of the atmosphere. Therefore the final result of the calculations will be determined by the assumed distribution of ozone and water vapor with height.

A fundamental source of large errors is also connected with the fact that, even when limiting the problem to ozone and water vapor, we must take laboratory data for the quantity \(K_\lambda\). It is true that, for ozone and water vapor, absorption in the ultraviolet, visible, and near-infrared parts of the spectrum is known, but, besides the absence of data for the far-infrared rays, it is known that the magnitude of absorption depends on pressure and temperature. Using for all heights the values of \(K_\lambda\) determined by laboratory methods, we admit a large error.

But, despite all these great difficulties, caused

the general state of the science, the method of radiative equilibrium gives us certain general indications, very important for clarifying the picture of the physical state of the stratosphere and for choosing particular assumptions of one kind or another. The principal result of the theory of radiative equilibrium consists in the fact that at great heights there must prevail temperatures much higher than at the beginning of the stratosphere. If one assumes that the amount of water vapor at great heights is the same as at an altitude of 11 km (there are some data permitting one to consider that this is indeed so), and, with respect to ozone, proceeds from the proven existence of a maximum of ozone concentration at a certain altitude (Ch. III), then in the case of a “mixed” atmosphere the temperature must increase strongly with height. At an altitude of 90 km, or in any case somewhere far above the ozone maximum, the temperature must attain its greatest value, of the order of \(+300^\circ\) C. For still greater heights a decrease in temperature is obtained.

If one proceeds not from a “mixed” atmosphere, but assumes that the mean molecular weight decreases with height in accordance with the barometric formula (1) (Ch. I), then the increase in temperature must occur less rapidly, although even in this case at an altitude of 80–100 km there should be a temperature maximum of the order of \(+100\)—\(200^\circ\).

For altitudes of 40–80 km the theory of radiative equilibrium gives a temperature not lower than minus 10–15°, i.e., also higher than in the lower parts of the stratosphere.

Acoustic method. It has long been known that during powerful explosions, in addition to the principal zone of audibility directly surrounding the point of explosion, there also exists a secondary zone of anomalous audibility, separated from the principal zone by a fairly wide zone of silence (the width of the zone of silence is several tens of kilometers, and sometimes more than 100 km). To explain the zone of anomalous audibility it is necessary to assume that sound waves, in traveling upward, for some reason bend their trajectory to such a degree that part of them returns again to the earth’s surface at some distance from the initial point. A theoretical analysis of the possible trajectories of a sound ray shows that often the trajectories in their upper part enter rather high into the stratosphere—up to 50 km. Since the velocity of propagation of sound waves depends on temperature, it proves possible, on the basis of purely acoustic observations in the zone of anomalous audibility, to draw conclusions about the temperature of the stratosphere up to an altitude of 50 km. In recent years this method has become very widespread, and the results obtained with its aid have proved exceptionally interesting.

As is known, the velocity of sound in a gas \(v\) is expressed by Laplace’s formula:

\[ v=\sqrt{\frac{\gamma_0 R_0 T}{M}}, \tag{3} \]

where \(R_0\) is the gas constant, \(T\) the temperature, \(M\) the molecular weight, and \(\gamma_0\) the ratio of heat capacities. Sound waves, rising upward, pass through layers of different temperature; therefore the velocity of propagation of the ray changes. From the standpoint of the theory of wave processes, a change in velocity corresponds to a change in the refractive index, and we thus have the case of propagation of a sound ray in a medium with a variable refractive index. Therefore the sound ray must undergo refraction analogous to the refraction of light rays in the atmosphere. To calculate the trajectory of a sound ray we may use the equation for the refraction of a light ray in the atmosphere:

\[ (R+H)n\sin \alpha=Rn_0\sin \alpha_0, \tag{4} \]

where \(n\) is the refractive index of the gas at height \(H\) above the earth’s surface, \(\alpha\) is the angle of incidence of the ray on the interface between two layers with different refractive indices (this surface is concentric with the earth’s surface), \(R+H\) is the distance of the point of the trajectory from the center of the earth, and \(n_0\) and \(\alpha_0\) are the corresponding values at the earth’s surface. If the velocity of sound as a function of height is known, then from equation (4) one can determine the zenith distance of the ray \(\alpha\) as a function of height and find the function \(\Delta=f(H)\), where \(\Delta\) is the distance from the sound source, measured along the earth’s surface. From the theory of the refraction of light rays in the atmosphere it is known that the quantity \(\Delta\) is determined by the expression

\[ \Delta=R\int_0^H \frac{\operatorname{tg}\alpha}{R+H}\,dH. \tag{5} \]

To construct the trajectory of a sound ray near the points of departure and reception of the sound, one uses the temperatures \(T\) determined by means of balloon soundings or, in the absence of such data, the mean temperatures at various heights. Having obtained two segments of the trajectory—its beginning and its end—one interpolates the trajectory at intermediate points in such a way that the observed time of passage of the sound coincides with that calculated by the formula:

\[ t=2\int_0^{H_0}\frac{dH}{v\cos\alpha}, \tag{6} \]

where \(H_0\) is the maximum height of the trajectory. Usually agreement is obtained only after several trials made for different assumptions about the course of temperature with height in the stratosphere.

The results of the calculations prove to be as follows. From the very fact of the return of sound waves it follows that, on their upward path, sound rays deviate more and more from the vertical direction. At a certain height the trajectory becomes horizontal, and then deviates back downward. The deviation of the ray from the verti-

rays, i.e., from the perpendicular to the boundary between layers with different refractive indices, shows that the refractive index must decrease with altitude, and consequently the speed of sound must increase. But according to Laplace’s formula (formula 3), the cause of the increase in the speed of sound may be either an increase in temperature \(T\), or a decrease in molecular weight \(M\), or an increase in the ratio of heat capacities \(\gamma_0\). The last assumption is rejected at once; as for a decrease in the molecular weight \(M\), from the point of view of the law of decrease of density according to the barometric formula (1) it may occur. However, more precise calculations by formula (6) showed that, starting from the assumption of a decrease in molecular weight, it is impossible even approximately to obtain a curve of travel times agreeing with the observations. It turned out that the required degree of decrease in molecular weight is such that, in order to obtain any satisfactory agreement, the hydrogen content already at an altitude of 40 km would have to amount to more than 25%. The known facts about the composition of the atmosphere rule out such a possibility; therefore it was decided that a change in molecular weight cannot be the cause of the high speeds of sound in the stratosphere. Thus there remained only the possibility of an explanation by means of the assumption of an increase in temperature.

Exact calculations give the following values for the speed of sound at various altitudes (Table 2).

TABLE 2

According to Laplace’s formula, on the assumption that only \(T\) changes, the ratio of the speeds is proportional to the ratio of the square roots of the absolute temperatures. On this basis the following values of temperature as a function of altitude are obtained (Table 3).

TABLE 3

Thus it is found that the temperature, which on the basis of direct measurements remains equal to −55—−50° up to an altitude of 30 km, after 30 km begins to rise rapidly. A sharply expressed temperature inversion must take place.

Many attempts were made to explain this rise in temperature at an altitude of 30—50 km. In the period 1927—1932,

a number of data were obtained on the distribution of ozone with height, which made it possible to suppose that the cause of the rise in temperature lay in the absorption of solar rays by ozone. Indeed, ozone is strongly absorbed both in the ultraviolet and visible parts of the spectrum, and also in the infrared part. At the same time, on the basis of purely spectroscopic observations, it was considered established that ozone is concentrated in a layer at an altitude precisely of 40–60 km. It was therefore concluded that the rise in temperature has a plausible explanation in the action of ozone, and that, consequently, the temperature inversion may be regarded as the principal cause determining the peculiarities of the propagation of sound waves in the atmosphere.

The hypothesis of a rise of temperature in the upper layers of the atmosphere is also in accord with the general conclusions of the theory of radiative equilibrium set forth in the preceding paragraph.

However, the last three years (1934–1937) have brought us a substantial change in views on the distribution of ozone with height. Quite unexpectedly for everyone, data were obtained which proved unconditionally that the ozone layer in reality lies 20 km lower than had previously been supposed. As for the question of atmospheric ozone itself, the next chapter will be specially devoted to it. But at present we are particularly interested in those changes which, because of this, must occur in views on the temperature interpretation of acoustic experiments.

Ozone has its maximum concentration at an altitude of about 25 km, i.e., below the place where the assumed temperature inversion begins. At an altitude of 50 km, where the rise in temperature is supposed to continue, the concentration of ozone is already very small. This casts doubt on the hypothesis of an ozone origin of the high temperature. In this connection, experiments carried out during the 2nd International Polar Year in America in 1932–1933 deserve special attention.

If the temperature inversion in the upper layers is created by the absorption of solar rays, then during the polar night this inversion should not exist. In that case, anomalous zones of audibility should not have been observed either. However, a special expedition which carried out observations at four different points in the Arctic found that a zone of anomalous audibility exists also during the polar night.

The most recent data from theoretical consideration of the ozone problem make it possible to give an entirely different interpretation of the origin of zones of anomalous audibility. In the very latest period theories have appeared which treat the question of the formation of the ozone layer and arrive at the conclusion that above the ozone layer oxygen must be in a state of complete dissociation. From this point of view, the ozone layer as it were divides the oxygen atmosphere into two parts: atomic (the upper layers) and molecular (the lower layers). In the intermediate layer, at the boundary of the atomic and molecular atmospheres (at an altitude of 20–30 km), reactions of ozone formation occur.

\(\mathrm{O} + \mathrm{O}_2 \to \mathrm{O}_3\). In layers lying above 35 km, all oxygen must already be atomic. The dissociating agent is the ultraviolet radiation of the sun.

The appearance of atomic oxygen sharply changes the molecular weight of the air from 29 to 24 (with complete dissociation of oxygen). Such a decrease in molecular weight gives, according to Laplace’s formula, an increase in the speed of sound by \(\sqrt{\frac{29}{24}}\) times, i.e. by 10%. The figures given in Table 2 show that such an increase in speed is quite sufficient for resolving the whole problem of the sound zones of anomalous audibility.

It is possible that precisely this interpretation corresponds most closely to reality, but since the indicated theory of the stratification of the oxygen atmosphere into two parts has not yet received any fairly broad recognition and has not even been subjected to general discussion (it was put forward quite recently), for the time being one must refrain from any final conclusions.

Observation of meteors. The developed technique of meteor observations makes it possible to determine a whole series of quantities characterizing a meteor: the speed and direction of its flight, the surface temperature, the height of appearance and disappearance, the length of the visible path, the dimensions and density of the meteor, and so on. On the other hand, since the very phenomenon of the burning of meteors is caused by its heating through friction against the air and depends on the density of the air at those heights, it is possible, in principle, theoretically speaking, to derive a relation between the observed quantities and the density of the atmosphere. But the density of the air depends on temperature, and thus we have at our disposal yet another indirect method that makes it possible to judge the temperature of the upper layers of the atmosphere. A number of investigators have developed this problem (Lindemann and Dobson \(^{2}\), Sparrow \(^{3}\), and others) and found its solution, which, although it inevitably encounters a number of difficulties, nevertheless can serve as a basis for further improvement of this interesting method.

Let us denote by \(\Delta h\) the difference between the heights of appearance and disappearance of the meteor, \(L\) the length of the visible path, \(d\) the density of the meteor, \(S\) its heat capacity, \(l\) the latent heat of evaporation of the meteor, \(v\) the velocity of the meteor, \(\varphi\) the angle of the meteor’s path with the vertical, \(r\) and \(r_0\) the radius of the meteor at its appearance and disappearance, \(T_2\) the surface temperature of the meteor, and \(k\) the effective factor of heating of the meteor (the fraction of energy going into heating). These are all quantities characterizing the meteor itself. In addition, it is also necessary to take into account the cap of compressed air formed in front of the meteor owing to its rapid motion (the average velocity of meteors is 40 km/sec); the presence of this cap affects the character of the phenomenon. It proved necessary to introduce into the consideration the velocity of the gas molecules forming the cap and having a component of motion directed toward the meteor; let us denote this velocity by \(V_1\), and the velocity

molecules of the gas having a component of motion away from the meteor—by \(V_2\). Finally, it is also necessary to take into account the properties of the gas at this height. Let us denote by \(T_0\) the temperature, and by \(M_0\) the molecular weight of air. Then, according to the calculations of Lindemann and Dobson\(^2\), for the air densities \(\rho_a\) and \(\rho_b\) at the height of the appearance and disappearance of the meteor, the following formulas are obtained:

\[ \rho_a = \frac{16}{3}\,\frac{T_2 r d S \cos \varphi}{k v^2}\,\frac{g M_0}{R T_0}, \tag{7} \]

\[ \rho_b = \frac{24 r_0}{V_1 - V_2}\,\frac{l \Delta h}{v L}\,\frac{d g M_0}{R T_0}. \tag{8} \]

Here \(g\) is the acceleration of gravity, \(R\) is the gas constant; the meaning of the remaining letters has been explained above.

The formulas obtained are cumbersome, but this is determined by the complexity of the phenomenon itself. At the basis of the derivation of these formulas lie certain assumptions that can hardly be fulfilled exactly. It is assumed that the heating of the air by the meteor takes place adiabatically, that the luminosity of the meteor appears only at the moment when its violent evaporation begins, and so on. But the conclusions obtained from these formulas agree very well with the results of applying other methods. Proceeding from the hypothesis of a mixed atmosphere, the authors obtain, for layers at a height of \(60\) km and more, temperatures of the order of from \(+25\) to \(+75^\circ\). When the calculation is carried out in a somewhat different way, still larger values are obtained for the temperature—of the order of \(+250^\circ\).

A purely theoretical consideration of the temperature problem from the point of view of radiative equilibrium leads, as we saw above, also to high temperatures: the maximum temperature, occurring at a height of about \(90\) km, may reach \(+300^\circ\). Thus, if not an exact quantitative agreement—which, in general, should hardly be expected in investigations of this kind—is obtained, then in any case there is complete agreement of the conclusions speaking in favor of high temperatures in the stratosphere.

In recent years meteor observations have been widely developed in our country, in the USSR, where much has also been done for the further improvement of the theory.

Width of the green line. The green line with wavelength \(5577.3\) Å is always present in the spectra of aurorae and of the night-sky glow. As early as 1922, Babcock succeeded in applying an interference method to the study of the structure of this line, by means of which he established the absence of fine structure in the green line, determined the exact value of the wavelength, and measured the width of the line. For the wavelength he obtained the value

\[ 5577.350 \pm 0.005 \text{ Å}, \]

and for the width \(0.035\) Å. Knowing the width of the line, Babcock attempted to determine the temperature of the upper layers of the strato-

of the sphere, proceeding from the assumption that the width is due to the Doppler—Fizeau effect. In that case, if the radiation belongs to a gas at temperature \(T\) and having molecular weight \(M\), the Doppler width of the line \(\Delta\), as Fabry and Buisson first calculated, is determined by the following expression:

\[ \Delta = 0.82 \cdot 10^{-6}\lambda \sqrt{\frac{T}{M}} . \tag{9} \]

Strictly speaking, at that time Babcock was interested not in the question of temperature, but in that of molecular weight. Until 1925 the nature of the green line remained unclear; it was not known to which gas it belonged. Using the line width, Babcock wanted to determine the molecular weight \(M\), taking the temperature to be approximately known. In accordance with the views widely held at that time, Babcock assumed that the temperature of the stratosphere, even in its very upper layers, was approximately the same as at the beginning of the stratosphere, i.e. \(-55^\circ\) (the stratosphere was considered isothermal). In this case formula (9) gave, for the carrier of the green line, \(M = 3.8\). This came closest to the atomic weight of helium. For hydrogen we would have had to obtain a width of \(0.07\) Å, which is twice the observed value. Conversely, for heavier gases the width would have had to be smaller than the observed one. True, it would have been quite possible to reconcile the data by assuming a higher temperature of the upper layers of the atmosphere, but at that time there were no grounds whatever for such an assumption.

In 1925 McLennan[^5] published the results of his brilliant experiments, which resolved the question of the origin of the green line. The conditions for the appearance of the green line proved to be very unusual, which also explained the fact that previously no one had ever succeeded in observing the emission of oxygen with such a wavelength. The green line appears in an electrical discharge in an oxygen atmosphere, but its intensity turns out to be large only if there is a considerable admixture of an inert gas. In a mixture of argon \(+\) \(1/10\) part by pressure of oxygen, the brightness of the green line turns out to be 100 times greater than in pure oxygen.

The attribution of the green line to atomic oxygen makes it possible to reconsider the conclusions about the temperature of the stratosphere drawn on the basis of formula (9) for the Doppler width. Taking \(M = 16\), for \(\Delta = 0.35\) we obtain a temperature of the order of \(+600^\circ\) for those very high layers where auroras and the glow of the night sky arise. Thus we again arrive at the same high temperatures that are obtained both by the meteor method and by the theory of radiative equilibrium.

Of course, the Doppler effect may not be the only cause determining the width of the green line. McLennan measured the width of the line obtained under laboratory conditions: it proved to be \(0.030\) Å. Consequently, in the stratosphere the temperature

nevertheless higher than the temperature of a gas discharge under MacLennan’s conditions. Taking this latter as equal to \(+50^\circ\) (room temperature \(+\) the unavoidable heating of the tube, since the exposures in MacLennan’s experiments lasted no less than 1 hour), we obtain that the temperature of the upper layers of the stratosphere must be no lower than \(+200^\circ\).

The helium method. Helium is contained in the air at the surface of the earth in an amount of \(0.04\%\) by volume. The principal source of helium is considered to be radioactive processes, as a result of which helium is liberated from radioactive rocks located in the earth’s crust. If we regard the atmosphere as mixed and therefore as having the same composition at all heights, then one can calculate the total amount of helium contained in the atmosphere. Jeffreys\(^6\) showed that a layer of radioactive rocks 100 m thick could have released this amount of helium in 160 million years. But the time of existence of the earth’s crust is in fact greater, and, in addition, the thickness of the radioactive layer of the earth’s crust considerably exceeds 100 m; consequently, the reserves of helium in the atmosphere ought greatly to exceed its actual amount. On this basis one may conclude that gas is continuously leaking from the atmosphere into world space. But in order to overcome the force of the earth’s gravity and break away from the earth’s atmosphere, a molecule must have a considerable velocity. If we denote the vertical component of the velocity of a gas molecule by \(v\), its mass by \(m\), and the acceleration due to gravity by \(g\), then, in order to fly to a height \(h\) above the surface of the earth, the molecule must have a velocity not less than a certain value found from the condition

\[ \frac{1}{2} m V_h^2 = mgh . \]

At high velocities a molecule can fly to such a great distance that the decrease in the force of the earth’s gravity already begins to make itself felt and must be taken into account. Calculation shows that at a velocity above 11 km/sec a molecule can leave the field of the earth’s gravity.

According to Maxwell’s law of distribution of the velocities of gas molecules, at any temperature some number of molecules will have a velocity greater than 11 km/sec. It is only necessary that this number be sufficiently large to produce the observed leakage of helium from the atmosphere; that is, the helium atmosphere must have a sufficiently high temperature. In connection with this, the exceptionally ingenious idea was put forward that, knowing the magnitude of the presumed leakage of helium, we can calculate the temperature of the upper layers of the atmosphere.

Detailed calculations show that, at a temperature of the upper layers of the atmosphere corresponding to a mean molecular velocity of 2.3 km/sec, about \(10^9\) years would be required for the complete disappea-

of the atmosphere’s disappearance. If, however, one takes the temperature corresponding to an average velocity of 2.6 km/sec, \(10^6\) years are required, and at a velocity of 2.9 km/sec—only \(10^3\) years. Thus the possible values prove to be confined within very narrow limits. Taking \(10^6\) years as the time for the disappearance of the helium atmosphere, we obtain an average molecular velocity of 2.6 km/sec. This average velocity corresponds to a temperature of \(+700^\circ\). Thus here, too, a high temperature is obtained for the upper parts of the atmosphere.

Of course, the foundations of this method contain much more that is hypothetical than do other methods, but its conclusions should hardly be neglected. If we analyze the basic assumptions, then, despite their unreliability, most of them can hardly be in any significant disagreement with the facts. Indeed, as regards the content of helium in the atmosphere, its total amount may be considered known to us with an error scarcely exceeding 10%. The hypothesis of a well-mixed atmosphere has been experimentally proved up to an altitude of 20 km (air samples taken during stratosphere-balloon flights), and in this layer 90% of all atmospheric gas is contained. It could happen that in the upper layers the helium content increases greatly, but this is contradicted by all known facts (the spectra of the aurorae and of the night-sky glow). Since the absolute amount of gas in the rarefied upper layers is negligible, their influence on the total amount of gas is small. Even if the relative content of helium in the layers above 20 km increased by a factor of 10, the order of magnitude for the total amount of helium would nevertheless remain the same.

On the other hand, the dependence of the time of disappearance of the helium atmosphere on the mean molecular velocity, and hence on the temperature, proves, as we saw above, to be exceptionally sharp. A change in velocity by 12% (from 2.3 to 2.6 km/sec) gives a change in the time of disappearance of helium by a factor of 1000; a change in the mean velocity by 12% corresponds to a change in temperature by 25%. But we may consider that the geological periods of existence of the earth’s crust are known to us with such accuracy that a thousandfold error in the estimate of these periods is certainly excluded. Consequently, an estimate of temperatures by this method may contain an error, due to the inaccuracy of geological periods, certainly smaller than 25%. The content in the earth’s crust of radioactive rocks and the amount of helium produced in radioactive decay may be considered sufficiently well known to us.

Thus the agreement of the conclusions of the helium theory with the conclusions of other methods concerning the presence of high temperatures in the upper parts of the earth’s atmosphere should hardly be regarded as accidental.

Aurorae illuminated by the sun. On the evening of September 8, 1926, Størmer \(^{7}\) observed an exceptional aurora of unusually great intensity. Størmer and his collaborators succeeded in obtaining a series of good photographs of the aurora simul-

simultaneously from two stations, which made it possible to calculate the height and position of individual rays of the aurora. It turned out that many rays lie at an unusual height—from 300 to 500 km. When Størmer processed all the data, it appeared that the greater part of the rays lay in a region still illuminated by the sun’s rays. Størmer’s results are presented graphically in Fig. 1, where a section through the center of the earth in the direction toward the sun is given; the tangent determines the boundary between the illuminated and unilluminated part of the atmosphere. The dotted lines mark levels of height, measured from the surface of the earth. Along the lower line, corresponding to the surface of the earth, the distance is indicated from the point at which the sun’s rays touch the earth’s surface. The position of each computed point of an auroral ray with respect to the earth’s shadow is marked by a small black circle. On each ray two points were computed, determining its direction. The general appearance of this aurora is such as though the rays were suspended in the illuminated part of the atmosphere. Below the shadow lie only 2 rays.

Fig. 1.

Fig. 2.

A similar appearance of an aurora situated at an extraordinarily great height, whose increased brightness is connected with the illumination of this part of the atmosphere by direct solar rays, was subsequently noted by Størmer several times. Sometimes the phenomenon proceeded in such a way that, besides rays situated in the illuminated part of the atmosphere, there were many rays lying in the earth’s shadow. One such case is presented in Fig. 2.

Here the rays are quite clearly divided into 2 groups, separated from one another by a dark space extending for several hundred kilometers. It is perfectly clear that these 2 groups of rays have essentially different origins. One group of auroral rays is evidently connected with the action of the sun’s rays: the lower boundary of this group is cut off, with remarkable precision and over an enormous extent (more than 1000 km), by the earth’s shadow. The other group of rays lies much farther away and, on the contrary, is entirely below the shadow.

For the question of the temperature of the stratosphere, the most essential circumstance is that these 2 groups of rays lie, on the average, at quite different heights, as is clearly seen from the drawing: the greater part of the illuminated rays, with their lower ends, lies above 200–300 km, while their upper ends reach 800, and sometimes even 1000 km; the second group, however, as is usual and is observed for normal auroras, begins mainly at a height of 100 km and does not extend beyond 400 km.

Several years ago Angenheister8 pointed out that the observed difference in the height of auroras in the illuminated and darkened parts of the atmosphere can be used to determine the temperature of the upper layers of the atmosphere. Angenheister’s method, apart from its novelty, deserves attention also in that it contains the original idea of an atmosphere pulsating over the course of a day, rejecting the previously used model of an atmosphere in static equilibrium.

Angenheister’s basic assumption is that the cause of the change in the heights of auroras is the redistribution of air masses under the influence of thermal expansion. Angenheister considers that the lower boundary of auroras, averaging 80 km for the unilluminated part of the atmosphere and 200 km for the illuminated part, corresponds to the complete absorption of the rays that produce the aurora by the overlying masses of air. Therefore the optical thickness of the atmospheric layer lying above 80 km “at night” is the same as the optical thickness of the layer lying above 200 km “by day.” In other words, the pressure which “at night” pertains to a height of 80 km, “by day” corresponds to a height of 200 km.

Starting from certain generally accepted assumptions, Angenheister calculates the pressure at a height of 80 km at night. Considering the atmosphere to be mixed, he assumes a certain definite law of temperature change with height at night. Then one obtains a certain curve giving the nighttime distribution of density with height. On the unknown curve of the daytime density distribution we thus have one point, giving the magnitude of the pressure at a height of 200 km. The assumption that the upper boundaries of the “nighttime” and “daytime” aurorae (about 1000 km by day and 400 km at night) also correspond to the same pressure gives us a second point on the daytime curve. Drawing through the 2 points obtained a curve in its upper part and matching the daytime curve with the nighttime one below

43 km, Angenheister obtains the temperature distribution with height by day. The results of Angenheister’s calculations are given in Tables 4 and 5.

TABLE 4

Temperature at night (assumption)

TABLE 5

Temperature by day (conclusion)

Thus, the daytime temperatures of the upper layers also turn out to be very high.

From the very course of the calculations it is clear that Angenheister’s method rests on the results of determining temperature by other methods; in particular, as is evident from Table 4, Angenheister adopts the hypothesis of the existence of a temperature inversion above 35 km. But the conclusions from the data of the acoustic method, on the basis of which the conclusion was made about a temperature inversion at an altitude of 30—50 km, themselves require reconsideration in connection with the newest theories of the ozone layer (see the paragraph on the acoustic method). Consequently, Angenheister’s method is not entirely independent; it is, as it were, an extrapolator of other methods to great heights. As for Angenheister’s calculations themselves, they in fact give an approximate upper limit for daytime temperatures, since it is assumed that the pulsation of the atmosphere is caused only by heating. In reality there may also be other causes of this grandiose pulsation, and in that case the heating must be less than assumed.

The twilight method. At the very beginning of the article we already mentioned that the moment of the end of twilight (the onset of night) can serve for determining the upper boundary of the atmosphere. As

as the work of the last two decades has shown, chiefly the studies of Acad. V. G. Fesenkov, observations in twilight can also serve for studying the structure of the atmosphere throughout its extent, up to heights of 200 km. If one carries out photometry of the brightness of the sky in twilight and represents the results graphically, plotting along the abscissa axis the depth of the sun’s immersion below the horizon in degrees, and along the ordinate axis the logarithm of the brightness of the sky, one obtains a curve which is called the twilight curve. Analysis of this twilight curve makes it possible to calculate the distribution of density in the atmosphere as a function of height.

Let us imagine that at some moment of twilight, when the boundary of the solar rays passes at the height \(h_1\), the brightness of the sky has turned out to be equal to \(I_1\), while for a later moment, corresponding to the height \(h_2\), the brightness is equal to \(I_2\). The brightness \(I_2\) is less than \(I_1\), since during this time the layer of air enclosed within the height limits from \(h_1\) to \(h_2\) has ceased to be illuminated. Thus the decrease in the brightness of the sky \(I_1 - I_2\) is a measure of the intensity of the light scattered by the layer \(h_2 - h_1\). But the intensity of scattering depends on the density of the gas, and thereby the loss of brightness \(I_1 - I_2\) can serve as a measure of the density of the atmosphere in the layer \(h_2 - h_1\).

The scattering of light by the atmosphere corresponds basically to molecular scattering, described by Rayleigh’s theory. According to this theory, the scattering power of a gas, other conditions being equal, is proportional to the factor

\[ \frac{(n^2 - 1)^2}{N}, \]

where \(N\) is the number of molecules per unit volume, and \(n\) is the refractive index of the gas. But the refractive index of a gas is connected with its density \(\rho\) by the known relation \(n^2 - 1 = A\rho\), where \(A\) is a certain constant. The density of a gas is proportional to the product of the molecular weight \(M\) by the number \(N\), and consequently, in the final analysis, the scattering power of the atmosphere is determined by the quantity \(M^2N\), or, what is the same, \(M\rho\). Having every reason to regard the atmosphere as mixed, we obtain the possibility, by extracting from the twilight curve data on the change with height of the reflecting capacity of the atmosphere, of constructing directly the density distribution curve. Knowing the density and using the known dependence of density on temperature, we can calculate the temperature of the atmosphere at different heights.

The practical implementation of this method of calculation encounters, however, great mathematical difficulties, since the phenomenon of twilight is in reality very complex. The brightness of twilight, measured by a photometer, is composed of the brightness of all elements of the atmosphere illuminated by the sun along the line of sight. But the brightness of each element depends in fact not only on its scattering power, but also on the brightness of the solar ray falling upon it. This latter, however, depends on absorption,

experienced by the ray on its path through the atmosphere; moreover, the magnitude of the absorption depends on the height at which the ray passes above the Earth’s surface. Thus the illumination of the various elements of the atmosphere turns out to be by no means the same. In addition, the scattered ray itself undergoes absorption on its way to the observer. Finally, it must be taken into account that each element of the atmosphere is illuminated not only by the direct rays of the Sun, but also by light scattered by the rest of the atmosphere; and this secondary scattering amounts to tens of percent.

The complete theory of this complex phenomenon, developed by Acad. V. G. Fesenkov,^9 makes it possible to carry the calculations through to the end and to obtain the desired curve for the distribution of density and temperature with height.^10,11

The results obtained by this method are as follows. The temperature inversion at heights of 30–50 km, supposedly detected by acoustic experiments, is not confirmed: up to a height of 55 km there continues an approximately isothermal stratosphere with a temperature of minus 55–60°. But for higher layers an increase in temperature is obtained, reaching its maximum somewhere at a height of about 70 km. Here the temperature is approximately \(+60^\circ\), after which the temperature begins to fall. In this part there is fairly good qualitative agreement with the conclusions of the theory of radiative equilibrium.

For layers above 100 km, the twilight method gives results that can be reconciled with the hypothesis of high temperatures in the upper part of the atmosphere. Here, however, the twilight data are already rather uncertain. And although, generally speaking, the twilight curve can be traced up to a height of 200 km, for determining temperatures it can be used with confidence only up to 100 km.

Distribution of intensity in the band spectrum of nitrogen. In the spectra of auroras and of the glow of the night sky several systems of bands of the molecular spectrum of nitrogen are observed. The band spectrum of nitrogen has been well studied also under laboratory conditions. The quantum theory of diatomic molecules makes it possible to calculate the distribution of energy in the band spectrum of nitrogen, and it has turned out that this distribution should change when the temperature changes. This conclusion of the theory was confirmed experimentally, and Vegard attempted in this way to determine the temperature of those layers of air in which auroras arise. For a height of 100–125 km he obtained a temperature of the order of \(-30^\circ\). This value, although it differs from the results of other methods, which usually give higher temperatures, nevertheless also indicates that, in comparison with the lower layers of the stratosphere, where the temperature is \(-50\)–\(60^\circ\), the upper layers are distinguished by a higher temperature.

(To be continued)

1. The literature will be indicated at the end of the article. ↩