ESSAYS ON THE PHYSICS OF THE EARTH’S ATMOSPHERE¹

I. A. Khvostikov, Leningrad

III. The Problem of Atmospheric Ozone

Evidence for the existence of the ozone layer. Oxygen exists in the atmosphere in the form of diatomic molecules $\mathrm{O_2}$, but under the action of various factors—for example, during thunderstorm discharges—dissociation of molecules into atoms may occur:

$\mathrm{O_2 \to O + O}$.

The appearance of atomic oxygen may, in turn, lead to the formation of ozone molecules $\mathrm{O_3}$ by the reaction

$\mathrm{O_2 + O \to O_3}$.

It had long been known that during a thunderstorm the air becomes ozonized; but although as early as the middle of the last century it proved possible, by direct chemical analysis, to demonstrate the presence of ozone in the air, contradictory results were obtained for the ozone content. It was clear only that the ozone content in the air is exceedingly small, which accounts for the great difficulty of quantitative determinations.

Proceeding from the negligible content of ozone, one might have supposed that ozone has no noticeable influence whatever on biological and physical life on the earth. Only comparatively recently did it become clear that this is by no means the case.

The first definite indications of this were obtained in 1913 by Fabry and Buisson¹². They studied the edge of the ultraviolet spectrum of the sun and showed that a considerable part of the sun’s ultraviolet radiation is absorbed by ozone, the total amount of which in the atmosphere must be much greater than had been assumed.

The solar spectrum observed by us breaks off sharply at the wavelength $\lambda = 2900\ \text{Å}$; radiation of shorter wavelength does not reach us. But the sun, considered as a source of radiation, corresponds to a black body with a temperature of about $6000^\circ$. For such a radiator the spectrum should extend far beyond

¹ See Uspekhi fizich. nauk, 19, 49, 1938.

limits \(\lambda = 2900\ \text{\AA}\); if the energy of the solar rays at \(\lambda = 4000\ \text{\AA}\) is taken as 1, then at \(\lambda = 3000\ \text{\AA}\) we should still have an energy equal to 0.61, and at \(\lambda = 2000\ \text{\AA}\), equal to 0.09.

Radiation with a wavelength shorter than \(2000\ \text{\AA}\) is absorbed by atmospheric oxygen, but the interval from 2000 to \(2900\ \text{\AA}\) is absorbed by ozone. This was proved by Fabry and Buisson through careful measurements of the spectrum of the sun at its ultraviolet edge and by studying the absorption curve of ozone under laboratory conditions. The ozone absorption curve is presented in Fig. 3, where the values of the wavelength in angstroms are plotted along the abscissa axis, and along the ordinate axis the absorption coefficient \(\alpha\), determined by the formula

\[ I = I_0 e^{-\alpha d}, \]

where \(d\) is the thickness of the layer. The absorption of ozone is incredibly large—the absorption coefficient exceeds 100 in the middle part of the curve and lies precisely in that region of the spectrum which is absent from the spectrum of the sun. True, absorption by ozone may occur not in the earth’s atmosphere, but in the solar atmosphere itself. But long ago Cornu[^14] drew attention to the fact that the shortening of the solar spectrum is the stronger, the lower the sun stands above the horizon; this clearly indicated that the absorption is determined by the length of the path of the ray through the earth’s atmosphere. Subsequently Fabry and Buisson[^15] were able to prove directly that ozone absorption does not occur in the solar atmosphere. They photographed the spectrum of the solar rays both from the center and from the edge of the solar disk—it turned out that the absorption at the boundary of the spectrum was the same in both cases.

Fig. 3.

Noticeable absorption by ozone begins already at \(3200\ \text{\AA}\), but we are able to observe the sun’s rays down to \(\lambda = 2950\ \text{\AA}\). This means that at the edge of the ozone absorption spectrum, where the magnitude of absorption is already relatively small, the solar rays are only partially absorbed in the earth’s atmosphere. It is precisely this that makes it possible to determine the absolute amount of ozone in the atmosphere, which can be calculated by comparing the curve of the distribution of energy at the edge of the solar spectrum and the ozone absorption curve.

It turned out that the total amount of ozone in the atmosphere is such that, if all of it were collected into a layer at atmospheric pressure, the thickness of the layer would be approximately \(0.3\ \text{cm}\). This amount, as Fabry and Buisson showed, varies depending on the time of year (in spring it is greater than in autumn) and during the day (in the morning it is somewhat greater than in the evening).

The amount of ozone equal to 0.3 cm corresponds to an average ozone content in the air of approximately \(4 \cdot 10^{-7}\) by volume. Rayleigh\(^ {13}\) carried out in 1918 an experiment which, by a purely optical method, was to verify the actual ozone content in the air near the earth’s surface. If one uses the radiation of a mercury quartz lamp, then the intense resonance line

\[ \lambda = 2537 \ \text{\AA} \]

falls precisely at the maximum of ozone absorption. When photographing the spectrum of a mercury lamp from a large distance, the resonance line should be strongly weakened in comparison with other parts of the mercury spectrum. Rayleigh photographed the spectrum of mercury vapor from a distance of about 6 km. If one starts from the average concentration \(4 \cdot 10^{-7}\) by volume, then for a path length of 6 km the thickness of the equivalent ozone layer should be about 2.5 mm. Such an amount of ozone, owing to the enormous absorption coefficient for \(\lambda = 2537 \ \text{\AA}\), should practically completely absorb the resonance line. But Rayleigh was able even from a distance of 6 km to obtain this line with great intensity. This indicated that in the lower layers of the atmosphere the ozone content is much less than the average (according to Rayleigh’s determination—by a factor of 40). But where, then, is the main mass of the ozone located? The answer suggests itself: the greater part of the ozone must be located in the higher layers of the atmosphere.

This assumption subsequently received full confirmation. It proved possible, by studying the edge of the solar spectrum at different positions of the sun above the horizon, to calculate the relative variation of the ozone concentration in the atmosphere with height. The idea of the method was indicated by Fabry and Buisson. The method assumes that the greater part of the ozone is contained in some layer at height \(h\) above the earth’s surface. Depending on the height of the layer, there should be one or another diurnal course of absorption by ozone.

Fig. 4.

Let us imagine an observer located on the earth’s surface at point \(A\) (Fig. 4) and studying the solar ray \(ABC\), which makes an angle \(\beta\) with the vertical \(h\). Let us further imagine that in the atmosphere there is a layer of ozone of some thickness. This layer occupies a position, as yet unknown to us, relative to the earth. Let us take two cases: when the layer is immediately adjacent to the earth’s surface, and when it is located at some height \(h\). If the ray goes vertically (along \(h\)), then its path will be minimal and, moreover, the same in both cases. But if the ray goes not vertically, but at some angle \(\beta\), then the path of the ray through the ozone layer increases, and this increase occurs more rapidly in the lower layer, as is easily seen from the drawing: \(AB > CD\).

Consequently, the higher the ozone layer lies, the more slowly absorption should increase as the sun approaches the horizon. By carrying out a complete geometrical calculation and comparing the results of the computations with the true diurnal course of absorption at the boundary of the spectrum, one can determine the height of the ozone layer. When such a calculation was made, the height of the center of gravity of the ozone layer was found to be of the order of 45 km. Upward and downward from this height, the relative concentration of ozone must decrease rather rapidly.

Doubts were expressed as to whether the method described could give unambiguous results. The well-known astrophysicist Rosseland came to the conclusion that the method can give no unambiguous answer either as to the height of the center of gravity of the ozone layer or as to the vertical distribution of ozone.^16 However, a more rigorous consideration of the question, carried out by the Leningrad theoretician V. A. Ambartsumian,^17 showed the erroneousness of such a conclusion. Ambartsumian gave a theoretical derivation of a formula for determining the vertical distribution of ozone.

The method proposed by Fabry and Buisson proved not to be the only one. Cabannes and Dufay,^18 as well as Götz,^19 proposed two independent methods, likewise geometrical in idea and making it possible to determine the vertical distribution of ozone from observations made at different heights of the sun above the horizon. Especially fruitful proved to be Götz’s method, which in the works of Dobson and Chalonge received broad development. We cannot dwell on this question in greater detail, and refer the reader to the specialized literature.^19, ^20, ^21, ^22

Let us now turn to an account of the results obtained. The total content of ozone, as we have already indicated, corresponds to a layer of 0.3 cm. Table 6 gives the annual course of the ozone content, calculated on the basis of observations at Arosa (Switzerland) in 1926–1929.

TABLE 6

Month	Ozone content	Month	Ozone content
January	0.277	July	0.259
February	0.293	August	0.245
March	0.289	September	0.236
April	0.308	October	0.221
May	0.301	November	0.231
June	0.275	December	0.258

The table shows a clearly expressed maximum in spring and a minimum in autumn. It is very indicative that in the southern hemisphere the opposite situation occurs: a minimum in spring and a maximum in autumn. The corresponding data, relating to a station set up by Dobson in New Zealand (44° south latitude), are given in Table 7.

TABLE 7

Month	Value	Month	Value
January	0.250	July	0.312
February	0.241	August	0.320
March	0.223	September	0.316
April	0.247	October	0.307
May	0.263	November	0.271
June	0.279	December	0.256

A clearly expressed dependence of the ozone content on geographic latitude within the northern hemisphere was also established, as shown by the data of Table 8. These figures

TABLE 8

Latitude	67°	60°	50°	40°	35°	30°	20°	15°
Ozone content in cm	0.375	0.347	0.316	0.296	0.280	0.250	0.218	0.210

refer to April, when the dependence on latitude proves to be the greatest.

As for the height of the center of gravity of the ozone layer, for a number of years all methods seemed to lead to a value on the order of 45–50 km. Thus, Cabannes and Dufay[^18] in 1927 found a height of 45–50 km above Montpellier; MacLennan, Ruedy, and Krotkov[^23], using the Cabannes–Dufay method in 1928, found in Toronto (Canada) a height of 50 km; using the Fabry and Buisson method, in 1927 Lambert, Déjardin, and Chalonge[^24] found a height of 45 km, carrying out observations at the Vallot Observatory on Mont Blanc. Observations by Götz and Dobson[^25] at Arosa, belonging to the period 1927–1929, gave a height of the same order. But the state of affairs began unexpectedly to change in 1931, when Götz made an observation by his new method[^19]. Contrary to all previous data, Götz obtained the center of gravity of ozone at a height of 25 km. Chalonge and Dobson continued Götz’s observations and, likewise by his method, obtained a height of 25 km. These discrepancies with the previous results attracted general attention among geophysicists in 1931–1934, since, besides the great general importance of the ozone layer for the physics of the stratosphere, by 1930 Chapman had succeeded in theoretically explaining the presence of the ozone layer at a height of 45 km from the point of view of the effect on the atmosphere of the Sun’s ultraviolet rays. This theory concerned the most urgent questions of theoretical geophysics—the action of solar rays on the atmosphere—and it was extremely important to determine the degree of agreement between the conclusions of the theory and the observed height.

We shall omit the account of the discussion about the height of the ozone layer, which proceeded very vigorously in the period 1931–1934, since in 1934 Regener26 carried out a direct experiment that unambiguously resolved the question of the height. Regener succeeded in raising a quartz spectrograph to a height of 31 km by attaching it to a hydrogen-filled sounding balloon. The spectrograph had an automatic device that made it possible to photograph spectra of sunlight at different heights. A whole series of spectra was obtained, and Regener’s instrument simultaneously recorded the height of the balloon corresponding to each given spectrum. As the height increases, the boundary of the spectrum shifts toward shorter wavelengths. The total displacement of the boundary at the maximum height is very large; it amounts to several hundred angstroms. By studying the distribution of energy at the edge of spectra taken at different heights, Regener was able to determine the approximate distribution of ozone with height. The supposition that the ozone layer lies lower was fully confirmed. It is enough to say that the spectrum taken at a height of about 30 km showed that below 30 km there is more than 70% of all the ozone. The center of gravity of the layer proved to be at a height of 22 km.

Careful measurements carried out by Dobson, Götz, and Meetham by combined indirect methods (studying the direct and scattered light of the sun at different positions of the sun above the horizon) also fully confirmed this conclusion. In Table 9 are given the results of calculations by Dobson, Götz, and Meetham for the ozone content above Arosa. The distribution of ozone in the lower layers—from 0 to 24 km—is difficult to take into account by the indirect method, and here the question must be resolved by direct measurements during stratosphere flights, which will be discussed below. Table 9 gives only the approximate ozone content in the lower layers; it is impossible to indicate the exact distribution with height. Moreover, it gives the correct value of the total content both in the lower layers and in the higher layers. It is possible that in the future these data will be further refined.

TABLE 9

Layer boundary, km	Mean height, km	% of total amount of ozone
0	3.1	14
8.5–16.2	11.3	15
16.2–23.7	18.8	21
23.7–31.2	26.6	28
31.2–39.2	34.1	18
39.2–50.5	43.3	3

Theory of the formation of the ozone layer. For the formation of ozone molecules, the presence of atomic oxygen is necessary, i.e. dissociation of oxygen molecules into atoms must take place:

\[ \mathrm{O}_2 \to 2\mathrm{O}. \]

Dissociation can occur only under the action of some external agent, since the temperature of the atmosphere at the place where the ozone layer is formed (20–35 km) is certainly too low…

low (−55° C) for thermal dissociation to be able to occur.

Opinions were in complete agreement that the source of the energy of dissociation is the radiation of the sun—this is definitely indicated both by the annual and daily variations in the ozone content, and by the dependence of the amount of ozone on geographical latitude. But, one asks, exactly which radiation of the sun—ultraviolet or corpuscular—causes the dissociation? On this question opinions immediately diverged. Some authors consider the cause of dissociation to be the corpuscular radiation of the sun (French scientists): they proceed from the fact that the corpuscular radiation of the sun, in general, apparently plays a large role in the physical life of the earth’s atmosphere, for example in the occurrence of the aurorae; and, in addition, this is suggested by the character of the dependence of the amount of ozone on latitude (Table 8), namely: an increase with distance from the equator. Indeed, if one assumes, as is usually done, that the corpuscular rays of the sun include charged particles, then, upon entering the earth’s magnetic field, these rays should gather predominantly in the polar regions.

Another point of view, developed by Chapman, assumes that the dissociation of oxygen is caused by the ultraviolet rays of the sun. It must be said at once that the hypothesis of corpuscular rays, as applied to ozone, has not received any definite development—the matter has been limited to general statements. On the contrary, the hypothesis of ultraviolet rays formed the basis of the quantitative theory of equilibrium of the ozone layer, developed by Chapman and receiving, with each year, more and more new confirmations.

The ultraviolet radiation of the sun plays a dual role with respect to ozone: that of an agent promoting the formation of ozone, and that of an agent destroying the molecules \(O_3\). Radiation of 1300–1850 Å is absorbed by oxygen \(O_2\) and dissociates it into atomic oxygen, thereby creating the prerequisites for the formation of ozone. Since the greater part of the ozone is contained in the layer 20–35 km, we must expect that the indicated radiation penetrates into the earth’s atmosphere at least to a height of 30 km. On the other hand, radiation with wavelengths 2200–3100 Å is absorbed by ozone and causes dissociation of ozone molecules. These two mutually opposite processes must, evidently, balance each other at some definite ozone concentration.

It is not difficult to prove that ozone must exist in the atmosphere in the form of a layer at some height. Indeed, in the upper layers the ozone content must be small, since because of the great intensity of rays with wavelengths 2200–3100 Å a very rapid destruction of ozone molecules occurs. But as this radiation penetrates into the lower layers it is weakened by absorption, and the number of ozone molecules destroyed per unit time rapidly decreases; the ozone content increases. But in still

in the lower layers of the atmosphere, the decrease in the content of atomic oxygen begins to make itself felt, since the radiation producing it, with wavelength \(1300\)—\(1850\,\text{\AA}\), is itself strongly weakened by absorption in the atmosphere. The decrease in the number of oxygen atoms slows the formation of ozone. Thus the ozone layer arises.

The first versions of Chapman’s theory proceeded from an altitude of the ozone layer of \(45\) km, which had been obtained from earlier observations. Subsequently the theory was refined, and its conclusions agreed with the most recent measurements of the vertical distribution of ozone.

The initial process for the formation of ozone is the photodissociation of the oxygen molecule:

\[ \mathrm{O}_2=\mathrm{O}+\mathrm{O}. \tag{a} \]

Oxygen atoms, combining with oxygen molecules, form ozone molecules:

\[ \mathrm{O}+\mathrm{O}_2=\mathrm{O}_3. \tag{b} \]

Ozone molecules, in turn, either dissociate:

\[ \mathrm{O}_3=\mathrm{O}_2+\mathrm{O} \tag{c} \]

or, recombining with oxygen atoms, again lead to the formation of \(\mathrm{O}_2\) molecules:

\[ \mathrm{O}_3+\mathrm{O}=2\mathrm{O}_2, \tag{d} \]

Generally speaking, recombination of oxygen may occur,

\[ \mathrm{O}+\mathrm{O}=\mathrm{O}_2, \tag{e} \]

leading to a decrease in the reserves of \(\mathrm{O}\) atoms and thereby to a slowing of ozone formation. However, as calculations show, the recombination coefficient \((b)\) is very small; according to A. N. Terenin’s formula it is approximately \(10^{-6}\), whereas for reaction \((d)\) the recombination coefficient (according to Harteck) is of the order of \(10^{-3}\). Moreover, it must be borne in mind that in recombination \((b)\) and \((e)\) only one particle is formed (\(\mathrm{O}_3\) or \(\mathrm{O}_2\), respectively), and therefore, for the conservation laws to be fulfilled, a third partner is also required, i.e. these reactions can occur only in triple collisions. Since the probability of triple collisions decreases with decreasing pressure faster than the probability of double collisions, at great altitudes reaction \((e)\) may be neglected in comparison with reaction \((d)\). We may confine ourselves to considering reactions \((a)\), \((b)\), \((c)\), and \((d)\).

The number of ozone molecules formed [reaction \((b)\)] is proportional to the number of oxygen atoms per unit volume \(n_1\) and to the number of oxygen molecules \(n_2\); it may be expressed by the product

\(k_{12}n_1n_2n\), where \(k_{12}\) is a constant characterizing the rate of reaction \((b)\), and the number \(n\) takes into account the number of particles that can participate in reaction \((b)\) as a third partner, since triple collisions are required for this reaction. Similarly, for reaction \((d)\) we may write \(k_{13}n_1n_3\), where \(n_3\) is the number of ozone molecules per unit volume. The loss of ozone is also possible as a result of the reaction

\[ \mathrm{O}_3+\mathrm{O}_3=3\mathrm{O}_2, \tag{f} \]

for which we may write an expression for the number of reactions: \(k_{33}n_3^2\). For the number of dissociation reactions \((a)\) and \((b)\) we may write \(\lambda n_2\) and \(\mu n_3\), where \(\lambda\) and \(\mu\) depend on the energy of the sun’s rays in the corresponding part of the spectrum and on the absorption coefficients of these rays by the molecules \(\mathrm{O}_2\) and \(\mathrm{O}_3\). Then the equilibrium equation for the ozone layer takes the following form:

\[ k_{12}nn_1n_2=\mu n_3+k_{13}n_1n_3+2k_{33}n_3^2, \tag{*} \]

where on the left stands the reaction leading to the formation of ozone molecules, and on the right—the reactions leading to the destruction of these molecules. The last term of equation \((*)\) is supplied with the coefficient 2, since each reaction of this type leads to the disappearance at once of two ozone molecules.

This equation must be supplemented by the equilibrium equation for atomic oxygen:

\[ 2\lambda n_2+\mu n_3=k_{12}nn_1n_2+k_{13}n_1n_3. \tag{**} \]

Detailed calculations show that it is possible to explain completely all the known experimental facts. For example, let us consider the question of the dependence of the amount of ozone on geographical latitude, to which supporters of the corpuscular hypothesis of the origin of ozone have usually referred. The amount of solar energy arriving per unit surface depends on latitude—it increases toward the equator. Table 10 gives the data

TABLE 10

Latitude	0°	10°	20°	30°	40°	50°	60°	70°
Mean annual amount of solar energy	0,3053	0,3011	0,2885	0,2683	0,2412	0,2088	0,1737	0,1445
Equivalent distance in astronomical units	1,00	1,01	1,03	1,07	1,13	1,21	1,32	1,45

for the mean annual amount of solar energy at latitudes from 0 to 70°. This amount may be replaced by a certain fictitious distance from the sun, at which a unit of surfac-

at a given latitude would receive an equivalent amount of solar energy. For latitude \(70^\circ\), the energy decreases by more than a factor of two in comparison with the equator, which, by the inverse-square law for energy with distance, corresponds to an increase in distance by \(45\%\). The corresponding data are given in the last line of Table 10. If we now consider the reactions of dissociation of ozone \((c)\) and molecular oxygen \((a)\), then the number of these reactions must be proportional to the intensity of the light, and hence inversely proportional to the square of the distance, and we may write for the number of reactions the expressions:

\[ \frac{\sigma_3 n_3}{r^2} \quad \text{and} \quad \frac{\sigma_2 n_2}{r^2}, \]

where \(\sigma_3\) and \(\sigma_2\) are certain constants.

Then equations \((*)\) and \((**)\) can be rewritten in the following form:

\[ k_{12}nn_1n_2-\frac{\sigma_3n_3}{r^2}-k_{13}n_1n_3-2k_{33}n_3^2=0, \]

\[ 2\frac{\sigma_2n_2}{r^2}+\frac{\sigma_3n_3}{r^2}-k_{12}nn_1n_2-k_{13}n_1n_3=0. \]

From these two equations we easily find that

\[ \frac{n_3}{n_2}=A\sqrt{nn_2}\cdot r. \tag{10} \]

Since the oxygen content is the same at all latitudes, it follows from equation (10) that the ozone content must be a linear function of \(r\). But \(r\) increases for large latitudes (Table 10), and consequently the increase of ozone with geographic latitude is thereby already qualitatively explained. But here there is not only qualitative, but also good quantitative agreement. Indeed, it can be shown that the data for ozone content at different geographic latitudes (Table 8) fit the following empirical formula:

\[ X=\frac{5}{3}r-\frac{2}{3}; \tag{11} \]

where \(X\) is the ozone content at the latitude for which the equivalent distance is equal to \(r\). Thus the theoretical formula correctly gives the linear dependence on \(r\), and the formulas themselves agree to within the arbitrary constant.

Formula (10), or the corresponding empirical formula (11), also explains other known regularities. In winter we have a decrease in solar energy, which corresponds to an increa-

to \(r\), i.e., to an increase in the ozone content. And indeed, at the end of winter the maximum ozone content is observed (Table 6). Conversely, during the summer the amount of ozone continuously decreases, and in October the amount of ozone proves to be the smallest. In the same way one explains the increase in the amount of ozone toward morning and the decrease toward evening. But the most essential merit of the theory is the possibility of explaining the vertical distribution of ozone. We omit here the lengthy computations relating to the calculation of the ozone content at different heights, and merely point out that in this question complete agreement with the observed facts has been attained.

Of very great interest are the conclusions that can be drawn on the basis of the photochemical theory of the ozone layer concerning the general structure of the atmosphere. It is necessary that radiation with wavelengths \(1300—1850\ \text{Å}\), which dissociates oxygen, penetrate to a height of \(30\ \text{km}\), for it is precisely here that the greater part of the ozone is formed. But molecular oxygen has an enormous absorption coefficient (Fig. 5)—greater than metals for visible light. Hence it follows that above \(35\ \text{km}\) the oxygen atmosphere must be purely atomic. In the ozone layer—\(20—35\ \text{km}\)—we have both atomic and molecular oxygen. Here complete absorption of rays with wavelengths \(1300—1850\ \text{Å}\) takes place, and below \(20\ \text{km}\) oxygen is practically purely molecular.

Fig. 5.

Thus the ozone layer turns out to be, as it were, a transitional layer. Below it the mean molecular weight of the air is 29, above it—24. The sharp decrease of molecular weight above \(35\ \text{km}\) may fully explain the reflection of sound waves, as was indicated in Chapter II.

The presence of atomic oxygen in the upper layers of the atmosphere, directly confirmed by the spectra of the aurorae and by the glow of the night sky, is of great significance for the theory of conducting layers, to which the following chapters of the present article will be devoted.

Direct determination of ozone concentrations in the lower layers of the atmosphere. We have already indicated at the beginning of this chapter that direct determination of the ozone content in the air at the Earth’s surface by chemical methods for a long time led to results not deserving particular confidence. The concentration of ozone at the Earth’s surface is so small that the sensitivity and accuracy of chemical methods prove insufficient. However, it has been possible to solve this problem by purely optical methods. Regener[^13] was the first to photograph

of a resonance line of the mercury spectrum from a great distance and, by this method, as indicated above, was able to determine the approximate ozone content in the air at the earth’s surface. Rayleigh’s first observations gave, for the concentration of ozone in the air at the earth’s surface, a value of the order of \(1—2\cdot 10^{-8}\) by volume. A similar method of measurement was subsequently greatly improved by Buisson\({}^{27}\), Ladenburg and Gett\({}^{28,29}\). To determine the concentration of ozone, the spectrum of a light source rich in ultraviolet energy is photographed. By measuring the change in the relative intensity of lines lying within and outside the region of ozone absorption, when passing from one distance to another, and using the ozone absorption curve measured under laboratory conditions (Fig. 3), the ozone content is determined. By this method quite reliable data are obtained; moreover, by making measurements in the mountains, it is possible to determine the ozone concentration also for higher layers. These latter measurements are of special interest, since they make it possible to detect directly the increase of ozone content with altitude. Numerous measurements of this kind were carried out in the mountains of Switzerland at an altitude of \(2—3.5\) km above sea level, but the greatest altitude was reached on Elbrus, where an expedition of the Academy of Sciences made measurements at an altitude of 4500 m. All the data obtained in this way are given in Table 11, where the

TABLE 11

Measurement site	Provence	Zurich*	Lauterbrunnen	Arosa*	Elbrus	Herknai	Jungfrau	Elbrus
Authors	Buisson	Götz	Chalonge and Götz	Götz, Shaw and Dobson	Balakov, Vafin and Krivun	Götz and Ladenburg	Götz and Chalonge	Balakov, Vafin and Krivun
Altitude above sea level in km	0.3	0.5	0.8	1.8	2.2	2.3	3.5	4.5
Ozone concentration in microns per 1 km	22	10	22	25	28	24	30	42

ozone content is given in microns per kilometer: that is, the thickness of the ozone layer contained in an air layer of 1 km. In other words, this number, divided by \(10^{9}\), i.e., by the number of microns in 1 km, gives the ozone concentration by volume. The scatter of the points is due not only to measurement errors, which here are not very large, but chiefly to the diurnal and annual variation of the ozone content, so

since the measurements were made in different years and at different times of the day and year. The unconditional increase of the ozone concentration with height is noticeable. The measurements of Stoll in Zurich and Götz, Chein and Stoll in Arosa, marked in Table 11 with an asterisk, in contrast to the other measurements, which were made by the photographic method, were made with the aid of so-called photoelectric light counters. Although in this case too the principle of measuring the ozone content remained the same—by measuring the intensity of different portions of the ultraviolet spectrum at different distances from the source—nevertheless, for measuring intensities in this case a chamber was used in which the ultraviolet light falling on a light-sensitive surface caused a discharge in an ionization chamber, and the number of discharges per minute served as a measure of the radiation intensity^30 (a preliminary calibration of the instrument is required).

However, the optical methods for determining ozone concentrations are so cumbersome that, unfortunately, it is impossible even to attempt to apply them on a stratosphere balloon or an airplane for the investigation of higher layers. Thus the possibility of direct analysis of the air for ozone content is limited to heights of 4–5 km; it may be considered that the measurements on Elbrus (4.5 km) are the limit in terms of the height attained. Meanwhile, it is scarcely desirable to leave the investigation of the ozone content in higher layers to indirect methods, since precisely for the lower part of the ozone layer they, by their very nature, give less definite results. In this respect special importance should attach to a new method developed very recently at the Institute of Physics of the Academy of Sciences of the USSR. This method was developed by M. A. Konstantinova-Shlezinger from an idea proposed by Academician S. I. Vavilov. The great simplicity of the method makes it possible to use it on any stratospheric-balloon flights, and its enormous sensitivity permits one to hope that, up to the ceiling of modern stratosphere balloons (15–20 km), the ozone content will be determined by direct measurements. The essence of this original method comes down to applying the principle of fluorescence analysis and consists in the following. A reaction is sought in which, owing to interaction with ozone, a fluorescing substance would be obtained, and moreover in a concentration strictly proportional to the amount of ozone entering into the reaction. The intensity of the fluorescence in this case is a direct measure of the amount of ozone, and the limit of sensitivity of the method depends wholly on how small luminescence intensities can still be measured. For these purposes the so-called quenching method, developed by Academician S. I. Vavilov, is used, in which an eye adapted to darkness is used to measure intensity. The quenching method is the most sensitive of all existing photometric methods^31. As the reaction, the oxidation by ozone of dihydroacridine into fluorescing acridine was chosen^32. In its final form the Konstantinova-Shlezinger method possesses such high sensitivity that, for determining the cont—

To determine ozone content it is sufficient to pass only 2–4 liters of air through an alcoholic solution of dihydroacridine. Since at present there exists a good method that makes it possible, during stratosonde flights, to take air samples from the ceiling in quantities of several liters, the problem of direct determinations of ozone concentrations up to an altitude of 20 km may now be considered fully solved. Already in 1936, during substratosonde flights, air samples were taken for ozone analysis at an altitude of 9.5 km, and in the autumn of 1937 a stratosonde delivered air samples from an altitude of 16 km. A strong increase of ozone concentrations with altitude has been established. There is no doubt that in the very near future the exact curve of the vertical distribution of ozone up to an altitude of 25 km will become known to us; owing to the exceptional role played by ozone in many processes occurring in the stratosphere, this curve will make it possible to refine considerably the existing theories and the general picture of the structure of the stratosphere.

IV. Ionization of the Atmosphere and the Propagation of Radio Waves

The hypothesis of a conducting layer. The history of the discovery of strong ionization in the upper layers of the atmosphere is especially rich in bold hypotheses that later, in a remarkable way, received full confirmation. The first such hypothesis was put forward as early as 1878 by Balfour Stewart[^33] in connection with the observed periodic daily variations of the Earth’s magnetic field.

According to this hypothesis of Stewart’s, which was at once quite energetically supported by A. Schuster[^34], the cause of the indicated daily variations might be a certain phenomenon that subsequently received the expressive name “dynamo effect.” Somewhere in the atmosphere, apparently at a great height, there must exist a gas possessing strong electrical conductivity. When the conducting gas moves relative to the Earth (as a result of periodic daily heating and cooling, the action of tidal forces, etc.), a phenomenon arises corresponding to the motion of a conductor in the Earth’s magnetic field. According to the laws of electromagnetic induction, an electric current thereby arises in the conducting layer, accompanied by a magnetic field superposed on the Earth’s magnetic field. Schuster’s calculations showed that from this point of view it is possible to explain a large share of the daily variations, if one assumes sufficient ionization of the upper layers of the atmosphere.

In view of the impossibility of directly verifying this hypothesis, it remained unused for many years. However, at the very beginning of the twentieth century it became necessary to return to this question again, and for an entirely different reason.

As is known, Hertz’s famous experiments with electromagnetic

waves became the starting point for the development of radio engineering. Hertz’s own experiments had only fundamental, and not technical, significance, since the reception of electromagnetic oscillations was accomplished by him only at a very short distance. The well-known inventor Marconi, analyzing Hertz’s experiments from the point of view of increasing the reception range, came to the conclusion that the reason for the small radius of action of Hertzian waves was their very small wavelength. Marconi correctly reasoned that such rays, in their properties, are close to light rays and, being incapable of undergoing any significant diffraction, propagate practically in straight lines. Marconi decided to try rays with a much greater wavelength—on the order of kilometers—hoping that, owing to diffraction, they would be able to bend around the earth’s surface and make transmission over large distances possible. Marconi’s attempt was crowned with brilliant success, although, as we shall now see, not at all for the reasons that Marconi himself supposed.

In December 1901 Marconi transmitted a radio signal across the Atlantic Ocean at a large wavelength. It turned out that, indeed, rays with a very great wavelength are capable of bending around terrestrial obstacles much better than short-wave rays. However, when Rayleigh^35 applied the rigorous theory of wave diffraction to the calculation of the range of Marconi’s radio signals, it immediately became clear that diffraction could in no way explain so great a range as that achieved by Marconi. Diffraction can account for an increase in range by tens of kilometers, but not by thousands of kilometers, as proved to be the case in reality.

But the facts continued, it seemed, to speak for themselves. Transmission over great distances succeeded well on long waves of the order of 10–20 km, but did not succeed on waves of hundreds of meters, which, on the contrary, served well at short distances. The greatest theoreticians (including Rayleigh, Sommerfeld, Poincaré, Nicholson, and many others) tried to modify and refine the theory of diffraction, striving to achieve satisfactory agreement with the experimental data, but all attempts proved unsuccessful. Meanwhile, as early as 1902, immediately after Marconi’s first experiments, a hypothesis had been advanced that pointed to the solution of the problem that had arisen in an entirely different direction. This hypothesis, whose appearance and initial development are associated with the names of Kennelly and Heaviside, was confirmed only in the 1920s.

Kennelly^36 and Heaviside^37 pointed to the idea, already expressed by Stewart, of the existence of conductivity in the upper layers of the atmosphere. The presence of a conducting layer can have a decisive effect on the propagation of radio waves, for the following reasons. From the standpoint of the theory of electromagnetic oscillations, the conductivity of a substance, while causing the absorption of electromagnetic waves by the substance, at the same time increases its reflecting power. Thus a layer of gas at great altitude, if it possesses sufficient electrical conductivity, can reflect a radio signal.

Since the ocean surface must possess analogous properties, waves turn out to be enclosed between two reflecting surfaces and along such a “corridor” can reach very distant points.

The views expressed by Kennelly and Heaviside were considerably refined in 1912 by Eccles[^38]. Kennelly and Heaviside’s reasoning about the “reflection” of radio waves is schematic. Eccles pointed out that we have every reason to suppose that the upper layers of the atmosphere are characterized by a gradual increase in the ionization of the air. An increase in ionization should be accompanied, according to Eccles’s calculations, by an increase in the phase velocity of wave propagation. But according to wave theory, an increase in the velocity of wave propagation corresponds to a decrease in the refractive index. Thus an electromagnetic wave directed upward enters a layer with a gradually decreasing refractive index, and therefore it must deviate more and more from the vertical. At some altitude the direction of the ray may become parallel to the Earth’s surface; from there the wave will proceed along the second branch of the trajectory, completely symmetrical to the first and with its end resting on the Earth. This theory of ionic refraction, developed by Eccles, shows that we should properly speak not of the “reflection” of radio waves, but rather of “total internal reflection.”

Eccles’s enormous achievement, in addition to developing the theory of ionic refraction, was to indicate that the cause of the appearance of ionization in the upper layers of the atmosphere is the ionizing action of solar radiation. This ionizing action of solar radiation also figures in modern theories as the principal cause, and Eccles may be regarded as the founder of the theory of atmospheric ionization. But this will be discussed in the following chapters; for the moment it is essential for us only that Eccles, having correctly guessed the nature of ionization, could conclude that the magnitude of ionization must decrease downward owing to the absorption of solar radiation in the atmosphere. Precisely in this case there can occur a gradual deflection of the ray, leading to its “total internal reflection.”

The theory of ionic refraction was subsequently refined in the works of Zenneck[^39] and van der Pol[^40], but of special importance was the investigation of Larmor[^41], who in 1924 showed that especially good agreement with experimental data is obtained if one assumes the presence in the ionized layers of a large number of free electrons. Let us examine these questions somewhat more closely.

Propagation of a radio signal in an ionized gaseous medium. For simplicity we shall have in mind a plane wave and shall confine ourselves to considering the basic facts, neglecting details. Let us consider the influence of free electrons on the velocity of the wave, assuming at first that the electrons do not undergo collisions with the surrounding particles. If a variable electric force \(E\), changing with the course of time, acts on an electron

periodically according to the law \(E = E_0 \sin \omega t\), where \(\omega\) is the frequency, then the electron acquires a variable acceleration whose magnitude is equal to

\[ \frac{dv}{dt}=-\frac{e}{m}E_0\sin\omega t. \tag{12} \]

Here \(m\) is the mass, \(e\) the charge of the electron, and \(v\) the velocity. The variable component of the electron velocity is equal to

\[ v=\frac{e}{m\omega}E_0\cos\omega t. \tag{13} \]

If the number of free electrons in \(1\ \mathrm{cm}^3\) is \(N\), then the result of the accelerating action of the electric force \(E\) will be the appearance of a current whose density is equal to

\[ j=-eNv=-\frac{Ne^2}{m\omega}E_0\cos\omega t. \tag{14} \]

The minus sign indicates that the direction of the current is opposite to the direction of the electric force. Let us consider a plane capacitor of thickness \(1\ \mathrm{cm}\), whose plates, each having an area of \(1\ \mathrm{cm}^2\), are arranged perpendicular to the electric force \(E\). In vacuum the capacitance of the capacitor is equal to \(\frac{1}{4\pi}\). Under the action of the alternating field \(E\) a capacitive current arises in the capacitor

\[ j'=\frac{E_0\omega}{4\pi}\cos\omega t. \tag{15} \]

If this takes place not in vacuum, but in the ionized gaseous medium under consideration, with \(N\) free electrons in \(1\ \mathrm{cm}^3\), then the current formed by the electrons (formula 14) will add to the capacitive current (15), and we obtain the total current

\[ I=j+j'=\frac{E_0\omega}{4\pi}\left[1-\frac{4\pi Ne^2}{m\omega^2}\right]\cos\omega t. \tag{16} \]

Thus the presence of free electrons changes the current by a factor \(\varepsilon\), where

\[ \varepsilon=1-\frac{4\pi Ne^2}{m\omega^2}. \tag{17} \]

Formally one may picture the matter as follows: the appearance of free electrons changes the dielectric constant by a factor \(\varepsilon\). Using the relation according to which the dielectric constant is equal to the square of the refractive index, we may say that the presence of \(N\) free electrons in \(1\ \mathrm{cm}^3\) leads to a change of the refractive index \(n\) by a factor \(\sqrt{\varepsilon}\). For vacuum

$n=1$, and consequently, for our ionized gaseous medium

\[ n=\sqrt{1-\frac{4\pi Ne^2}{m\omega^2}}\simeq 1-\frac{2\pi Ne^2}{m\omega^2} \tag{18} \]

in view of the smallness of the second term. The refractive index proves to be less than unity, and it decreases with increasing ionization, i.e., with increasing $N$. It is precisely for this reason that, in an atmosphere with gradually increasing ionization of the upper layers, “total internal reflection” of a radio signal may occur, as was indicated above.

If one knows the distribution of ionization with height and uses the refraction equation (4), which was discussed in Chapter II, then one can calculate the trajectory of the ray in exactly the same way as this was done for sound waves (Chapter II).

As for the influence of collisions of free electrons with the surrounding particles, which we have so far neglected, it must manifest itself in the loss of part of the velocity acquired by the electrons under the action of the electric field $E$, and consequently must somewhat reduce the influence of the free electrons. Calculations show that in this case, instead of formula (18), the following expression is obtained for the refractive index:

\[ n=1-\frac{2\pi Ne^2}{m(\omega^2+\alpha^2)}. \tag{19} \]

Here $\alpha=k\eta$, where $k$ is the number of collisions per second, and $\eta$ is the fraction of the average loss of velocity in collisions.

Determination of the ion concentration in the reflecting layer. The formulas obtained above make it possible, by calculation, to determine the ion concentration at the height at which the ray turns back toward the earth. Such a calculation is of interest in the sense that it will be possible to discuss the data obtained from the point of view of the probability of the existence of the corresponding conditions in the upper layers of the atmosphere.

Fig. 6. (diagram labels: “reflecting layer,” $A$, $\alpha_0$, $\alpha$, $H$, $R$.)

For the calculation we shall use the refraction equation for a light ray in the atmosphere, which we examined in Chapter II (formula 4). For convenience we rewrite this equation once more:

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

Here $n$ is the refractive index of the gas at height $H$ above the earth (Fig. 6), $\alpha$ is the angle of incidence of the ray on the interface of two

of layers with different refractive indices, located at height \(H\) (this surface is concentric with the earth’s surface); \(n_0\) and \(\alpha_0\) are the corresponding values at the earth’s surface; \(R\) is the radius of the terrestrial sphere. We shall consider a ray leaving radio station \(A\) tangentially to the earth’s surface (Fig. 6); this gives \(\sin \alpha_0 = 1\). Moreover, in view of the absence of appreciable ionization of the air near the earth’s surface, one may put \(n_0 = 1\). Finally, \(\sin \alpha\) should also be set equal to 1, because what interests us is precisely the height at which the ray travels parallel to the earth’s surface (the highest point of the ray trajectory). Then equation (4) is simplified and takes the following form:

\[ (R+H)n = R, \tag{20} \]

where \(n\) is now the refractive index of the layer at the height at which the ray turns back. Taking for \(n\) its value from formula (18), we obtain the following equality:

\[ \frac{R}{R+H} = 1 - \frac{2\pi Ne^2}{m\omega^2}. \]

Expressing the frequency \(\omega\) in terms of the wavelength \(\lambda\): \(\omega = \frac{2\pi c}{\lambda}\), where \(c\) is the speed of light, we find:

\[ N = \frac{2\pi mc^2}{e^2\lambda^2}\cdot \frac{H}{R}. \tag{21} \]

Substituting numerical values for the mass and charge of the electron, as well as for the speed of light and the earth’s radius, we obtain

\[ N \simeq 4\cdot 10^4 \frac{H}{\lambda^2}. \tag{21′} \]

Fulfillment of this condition is necessary for the ray to become parallel to the earth’s surface, but it is not sufficient for the return of the ray to the earth, since, propagating farther in a straight line, it will, owing to the curvature of the earth, pass beyond the limits of the atmosphere. For the ray to return to the earth, the radius of curvature of its trajectory in its upper part must be less than \(R+H\). This leads to the requirement of a sufficiently large ionization gradient:

\[ \frac{dN}{dH} > \frac{2\pi c^2}{e^2\lambda^2 R} \simeq \frac{4\cdot 10^4}{\lambda^2}. \tag{22} \]

If in formula (21) we put \(\lambda = 100\ \text{m}\), and assume the “reflection” to occur at height \(H = 100\ \text{km}\), then \(N\) must be no less than \(4\cdot 10^3\) electrons in \(1\ \text{cm}^3\); for \(\lambda = 10\ \text{m}\) we obtain \(N = 4\cdot 10^5\), i.e. the “reflecting power” depends sharply on the wavelength.

If, further, it is assumed that the conductivity of the upper layers is due not to free electrons but to ions, then in formula (21) for \(m\) one must take the mass of the ion. For ions of atomic oxygen \(m\) is almost 30,000 times greater than for the electron. In this case, for

$\lambda = 100\ m$ would require a concentration on the order of $10^8$ ions in $1\ \mathrm{cm}^3$ in order for “reflection” to be able to occur. For $\lambda = 10\ m$ we obtain $N = 10^{10}$, i.e. an enormous concentration of ions.

We shall return to these important questions later.

Justification of the hypothesis of the conducting layer. Although the Kennelly—Heaviside hypothesis explained the propagation of radio waves quite plausibly, nevertheless the very existence of a conducting layer in the upper parts of the atmosphere remained unproved for more than 20 years. But in the 1920s a number of remarkable facts were discovered, for the most part completely unexpected, which made the presence of a conducting layer quite evident. The successes achieved here were connected with the development of the technique of radio transmission on short waves.

We have already said that, since the time of Marconi, long-distance radio communication had been carried out only on very long (many-kilometer) waves. If waves of several hundred meters were used (usually not below $300\ m$), then only over short distances: over large distances they failed to work. But the development of vacuum technology made it possible to construct highly perfected radio tubes, with whose help it was very convenient to generate short waves—below $200\ m$. Since technical and state radio stations considered the use of such short waves inexpedient, it turned out that radio transmission on short waves at first came into wide use only among amateurs. But, unexpectedly for everyone, it became clear that at a wavelength of about $200\ m$ it was possible to establish communication over distances considerably exceeding the usual range of long-wave stations, although the communication was rather unstable. Still more unexpected was the discovery that, when passing to still shorter wavelengths, one can not only achieve a very great range, but the communication becomes more stable. Soon, when waves below $50\ m$ were reached, an astonishing fact was discovered, the comprehensive study of which led to unconditional proof of the existence of a conducting layer “reflecting” radio waves. The so-called “zone of silence” was discovered.

In the preceding paragraph we showed that, for reflection of a signal with wavelength $\lambda$, the concentration of free electrons must be not less than $N = 4 \cdot 10^4 \dfrac{H}{\lambda^2}$ (formulas 21 and 21′). As $\lambda$ decreases, an ever larger and larger concentration of electrons is required. Suppose that in the conducting layer we actually have a concentration $N'$ of electrons per $1\ \mathrm{cm}^3$. To this number $N'$ there corresponds a certain wavelength $\lambda'$; if the wavelength of a radio signal $\lambda$ is less than $\lambda'$, then this signal will not be able to be reflected. The question arises: what will happen to such a signal? It will pass through the layer and will not return to the earth. Consequently, as the wavelength of the radio signal decreases, we may reach such a value that reflection of the wave will no longer take place. Let us consider this important question in somewhat greater detail.

In deriving formula (21) we considered a ray emerging from radio station \(A\) tangentially to the earth’s surface (Fig. 6). But what relations will hold for a ray emerging upward at some angle to the horizon (Fig. 7)? In this case, in the refraction equation (formula 4) we cannot put \(\sin \alpha_0 = 1\), since \(\alpha_0\) is no longer equal to \(90^\circ\). Consequently, instead of formula (20) we must write

\[ (R+H)n = R \sin \alpha_0. \tag{22a} \]

It is clear that in this case, for the ray to return downward, it must undergo a much greater deviation, and the electron concentration required for this must also be greater. For a given concentration we may encounter the case that a ray traveling more gently undergoes reflection, while a ray of the same wavelength but emerging at a larger angle to the horizon cannot be reflected and, having passed through the conducting layer, will go off into outer space. Thus there arises a zone of silence, shown in Fig. 8. The boundary of the zone of silence is determined by the critical angle \(\alpha_k\), and on the other side—by the range of action of the wave propagating along the surface of the earth.

Fig. 7.

It should be noted that the range of action of the surface wave is somewhat increased owing to the presence of a certain conductivity of the surface of the terrestrial sphere.

Fig. 8.

The magnitude of the critical angle increases as the wavelength decreases, i.e. in this case the extent of the zone of silence correspondingly increases. At a sufficiently small wavelength the zone of silence may encompass the entire terrestrial sphere—the ray, at any angle of emission, penetrates through the conducting layer and goes off into outer space.

All the indicated circumstances are observed experimentally. Zones of silence appear when the wavelength is less than \(50\) m; for wavelengths of the order of \(8\)—\(10\) m the zone of silence encompasses the entire terrestrial sphere.

These facts make the hypothesis of the conducting layer quite reliable. We do not speak here of a whole series of other, less существен-

... facts, also indicating the presence of a conducting layer.

It is clear that the phenomena listed may also serve for studying the physical structure of the conducting layer. For example, using formula (21) or (21′) and taking the value for the limiting wavelength for which reflection still exists, we can determine the concentration of free electrons or ions. For the study of the structure of the atmosphere this is a very important quantity. But, in addition, we can also determine the height of the reflecting layer, and such a measurement can be made even in several ways.

The most widespread method at the present time is based on measuring the delay time of the “sky ray” (i.e. the ray reflected from the conducting layer) relative to the “ground” ray (i.e. the ray arriving in a straight line directly above the surface of the earth). For this purpose, at two points separated from one another by a distance of only a few kilometers, two horizontal antennas are installed. To reduce direct coupling between them to a minimum, their axes are arranged along one straight line. From the first station they begin to transmit radio signals of very short duration (ten-thousandths of a second and less), following one another at equal intervals of time. The second station receives each signal twice: through the ground ray and the sky ray. If the distance between the stations is equal to \(r\), and the height of the reflecting layer is \(H\), then the first signal arrives in the time \(t_1 = \dfrac{r}{c}\) sec. (\(c\) is the speed of propagation of electromagnetic waves in a medium whose dielectric constant is equal to 1), and the second signal in the time

\[ t_2=\frac{2}{c}\sqrt{\left(\frac{r}{2}\right)^2+H^2}\ \text{sec.} \]

At the second station the delay time \(t_2 - t_1\) is measured directly (oscillographically), from which the height \(H\) is calculated.

In this way one can determine both the height of the conducting layers and the concentration of ions or free electrons existing at that height. The results of the measurements are essentially reduced to the following. First, there is not one conducting layer, but several—this result is of enormous importance for the physics of the atmosphere. Two layers are most clearly expressed: one at a height of about 220 km with very strong ionization (it is called the layer \(F_1\)), and the second at a height of about 100 km with weaker ionization (this is the Kennelly–Heaviside layer; otherwise it is called the layer \(E\)). In addition, the latest radio-technical data indicate the presence of a weakly ionized region at a height of approximately 50 km (layer \(D\)).

The height of these layers is given approximately, since it is subject to considerable changes with the passage of time. Among these

changes are found to be both periodic and regular, and sudden. There is undoubtedly a diurnal variation: during the day the height is lower than at night, while the concentration of charges in the daytime is approximately 10 times greater. There is also an annual variation and, apparently, an 11-year cycle corresponding to the 11-year period of solar activity. All these facts directly indicate the great role played by solar radiation in producing conductivity. In addition to periodic changes, irregular, sudden changes of height are often recorded, associated with certain special phenomena.

In addition to the three layers indicated, reflection is undoubtedly observed from a strongly conducting layer at a height of about 300 km (the layer \(F_2\)), and sometimes also from a layer at a height above 1000 km, which gives a very large delay of the celestial ray. Calculation of the charge concentration shows that in these layers we are dealing with 100% ionization, which is easily explained if one takes into account the colossal power of solar radiation, including ultraviolet radiation, capable of ionizing gases. The situation is different with the layers \(E\) and \(F_1\). In them the ionization is undoubtedly not complete, but a more precise determination of the charge concentration encounters fundamental difficulties. We have already said that, in calculating the concentration of charges by formula (21), one has to choose between electrons and ions. Owing to the large mass of the ion in comparison with the mass of the electron, the concentration of ions required for reflection of a radio wave must be tens of thousands of times greater than the corresponding concentration of electrons. Thus, for example, if one assumes that the conductivity is due to free electrons, then for the layer \(E\) we obtain a concentration of \(1.5 \cdot 10^5\) electrons/cm\(^3\) (at noon at the equator), and for the layer \(F_1\)—\(3 \cdot 10^5\). If, however, one starts from the ionic nature of the conductivity, the corresponding figures will be of the order of \(0.5 \cdot 10^{10}\) and \(1 \cdot 10^{10}\) ions/cm\(^3\). The difference is too great for one to be indifferent to the choice of one of these hypotheses. Unfortunately, it must be stated that this question remains unresolved for the time being. Although radio engineers more often incline toward free electrons, a number of important facts from other fields apparently point to ionic conductivity. This question will be considered more closely below.

We can now briefly summarize that observations of the propagation of radio waves prove the presence in the atmosphere of several ionized layers. Why are there several layers? Why are they located precisely at these heights and not at others? How does ionization arise in general? Theory must give an answer to these questions. The task of theoretical interpretation became especially attractive when it became clear that a whole series of other geophysical problems is connected with the phenomenon of ionization; moreover, in this phenomenon the most essential features characterizing the general effect of solar radiation on the atmosphere are most sharply revealed. Since this effect is the principal factor determining the course and character of physical phenomena in the atmosphere, the study

nature of ionization proves to be the key to understanding a number of fundamental questions of the whole of atmospheric physics. In the following chapters the existing theories will be considered.

V. The Ionizing Action of the Sun’s Ultraviolet Rays

General remarks on the theory of ionization. It appears unquestionable that the ionization of the upper layers of the atmosphere is due, in its existence, chiefly to solar radiation. However, it is by no means clear which particular radiation of the sun is to be meant here: ultraviolet radiation or corpuscular rays. To a certain extent the observed phenomena can be explained from both points of view, but there is no direct evidence that would make it possible to choose unequivocally, or at least, if both types of radiation act jointly, to separate the regions of their influence. Moreover, in attempts to carry out a quantitative calculation, both theories often encounter insurmountable difficulties. But work on improving and refining the theory has in recent years been so intense that, despite the impossibility of an unambiguous explanation, very rich material has been obtained, on the basis of which we can picture many processes occurring in the atmosphere much more clearly and deeply than was possible even 5–10 years ago.

In attempts to clarify the mechanism by which ionization arises, one must immediately take into account the connection which undoubtedly exists between ionization and other geophysical problems. Thus, for example, the diurnal variations of the magnetic field depend on the behavior of the ionized layers; on the other hand, magnetic storms are associated with increased activity of the sun, which, as direct observations show, also affects the structure of the ionized layers. But with this same increased activity of the sun are associated periods of especially intense and frequent auroras. The very explanation of magnetic storms and powerful auroras proves to be connected with an analysis of the motion of streams of charged particles in the upper parts of the atmosphere. Thus a certain special line is created in the development of the theory of ionization, which, in addition to satisfactorily explaining the basic properties of the ionized layers, also requires an explanation of the regularities observed in adjacent fields of phenomena.

We shall first consider the foundations and, so to speak, the prerequisites of the theory: how in general the ionization processes could proceed, and to what extent quantitative agreement can be achieved in explaining the occurrence of a number of layers at different heights; and we shall begin with the theory of ultraviolet radiation.

Action of different regions of the ultraviolet spectrum. From laboratory experiments the ionization potentials of the gases that make up the atmosphere are well known. Thus, for example,

for the ionization of atomic oxygen, an energy of not less than 13.6 eV is required; for molecular oxygen \(O_2\), not less than 16.1 eV. Hence, bearing in mind photoionization, one can calculate the energy of a light quantum capable of producing ionization, and consequently also its frequency and wavelength. The calculation shows that for the photoionization of oxygen atoms, quanta with a wavelength of no more than \(910\ \text{\AA}\) are required, and for oxygen molecules, no more than \(770\ \text{\AA}\). The results of such calculations, carried out for the principal constituent parts of air, are given in Table 12.

TABLE 12

Gas	O	\(O_2\)	\(O_3\)	\(H_2\)	\(N_2\)	N	He
Dissociation
Energy in V	—	6.50	4.25	—	12.0	—	—
Wavelength (in \(\text{\AA}\)) not greater than	—	1850	2900	—	1030	—	—
Ionization
Energy	13.6	16.1	—	16.1	16.9	14.1	25.3
Wavelength in \(\text{\AA}\)	910	770	—	770	730	875	490

In addition to O, \(O_2\), \(O_3\), \(H_2\), \(N_2\), and He, atomic nitrogen N is included in the table, since in recent years it has been shown that photodissociation of nitrogen molecules may occur, the dissociation potential of \(N_2\) into atoms being equal to 12 V (A. N. Terenin). The region of the spectrum active in the sense of photodissociation of nitrogen is bounded by wavelengths from 1030 to \(950\ \text{\AA}\). Calculations show that in layers located above 220 km, complete photodissociation of nitrogen may occur\(^{46}\).

These data show that the ionization of atomic oxygen must be of great importance. Indeed, as Table 12 shows, only rays with wavelengths shorter than \(910\ \text{\AA}\) possess an ionizing action in general. But the energy of the solar rays must decrease rapidly with decreasing wavelength, if the solar radiation is assumed to be thermal for \(T = 6600^\circ\). The wide wavelength interval from 910 to \(770\ \text{\AA}\), especially rich in energy, is absorbed by oxygen and nitrogen in the atomic state; in the case of nitrogen the active region is \(875\)—\(840\ \text{\AA}\). But because atomic nitrogen is found only in very high parts of the atmosphere—above 220 km—where the gas density is very small,

atomic nitrogen can absorb only 8% of the radiation with wavelengths 875–840 Å. The remaining 92% pass through and remain for atomic oxygen.^46 Thus the interval 910–770 Å goes practically entirely into the ionization of atomic oxygen. O₂ and H₂ account for the narrow region 770–730 Å, which is moreover partly absorbed by atomic oxygen. Very little energy remains for helium, since in the corresponding region of the spectrum all the other gases also absorb.

As for the absolute amount of ionized gas, it depends on the absorption coefficient of each gas for radiation of the corresponding wavelength and on the partial pressure of these gases.

The absorption coefficient and the recombination coefficient. The theory of photoelectric ionization, developed by Kramers,^42 Milne^43 and Gaunt,^44 makes it possible to calculate the photoelectric atomic absorption coefficient \(\gamma\). According to Milne’s theory, for a gas whose ionization potential corresponds to the frequency \(\nu_0\), the coefficient of absorption of radiation of frequency \(\nu\) is approximately:

\[ \gamma=\frac{16\pi^2}{3\sqrt{3}}\cdot\frac{Z^2e^6\nu_0}{ch\nu^3}. \tag{23} \]

Here \(Z\) is the atomic number of the element, \(e\) the charge of the electron, \(c\) the speed of light, and \(h\) Planck’s constant. If \(\gamma\) is calculated for the limiting frequency \(\nu=\nu_0\), using the atomic number \(Z\) also for molecules, the following values are obtained (absorption per 1 particle):

Gas	O	O₂	N₂	H₂
\(\gamma=\)	\(2.5\cdot10^{-16}\)	\(2\cdot10^{-16}\)	\(1.2\cdot10^{-16}\)	\(3\cdot10^{-18}\)

It should be noted that Gaunt,^44 using wave mechanics to calculate the absorption coefficient, obtained an expression somewhat different from formula (23). In essence, the difference is due mainly to allowance for screening by the outer electrons, which leads to a certain decrease in \(\gamma\). In the case of atomic oxygen this may give, as detailed calculations show, a decrease of \(\gamma\) by a factor of 2.5; this is equivalent to the fact that the height of the corresponding ionization will also be somewhat lower, since to obtain the same amount of absorbed energy a density 2.5 times greater will be required. It is therefore not difficult to estimate the resulting uncertainty in the estimate of the height of the ionized layer. According to the barometric formula (1), for air with mean molecular weight 29, an increase in density by a factor of 2.5 will take place when the height of the layer is decreased by 7.5 km (Chapter I). Consequently, if one speaks of an ionized-

layers \(E\) and \(F_1\) (Chapter IV), situated at altitudes of the order of 100 and 220 km, then the possible relative error in determining the height, due to a possible discrepancy between the calculated value and the actual one, will be approximately equal to 7.5 and 3.0%, respectively.

The ionization effect produced by the absorbed solar energy will, to some extent, be compensated by recombination of ions, which must therefore be taken into account in calculating the distribution of ionization with height. If we assume that we are dealing chiefly with the recombination of ions with electrons, then for the recombination coefficient \(\alpha\) we may use the expression given by Milne’s theory:

\[ \alpha=\frac{32\pi}{3}\left(\frac{2\pi m_e}{3k}\right)^{\frac12} \frac{Z^2 e^6}{c^2 m_e^2}\cdot \frac{1}{hT^{\frac12}}, \tag{24} \]

where \(k\) is Boltzmann’s constant, \(m_e\) is the mass of the electron, and \(T\) is the absolute temperature of the gas. For atomic oxygen (\(Z=8\)) and for the temperature \(T=400^\circ\mathrm{K}\) we obtain \(\alpha=3.2\cdot10^{-10}\). It should be noted that a change in temperature has very little effect on the value of \(\alpha\): for \(T=300^\circ\mathrm{K}\), \(\alpha=3.7\cdot10^{-10}\).

The calculation of the concentration of ions at a given height amounts to determining the equilibrium state, when the reactions of formation and recombination of ions occur in equal numbers. If we pose the problem in such a way that radiation of temperature \(T_1\) (\(T_1=6000^\circ\)) passes through an atmosphere at temperature \(T\), and the density of the atmosphere decreases with height according to an exponential law, then at the height at which the total number of molecules in \(1\ \mathrm{cm}^3\) is equal to \(n\), the equilibrium state is established, as Chapman’s calculations\({}^{45}\) show, at the following number of ions per \(1\ \mathrm{cm}^3\):

\[ n_e=\left(\frac{Kn}{kT}\right)^{\frac12} l^{-\frac12\gamma H n}, \tag{25} \]

where

\[ H=\frac{RT}{Mg}, \]

and \(K\) is determined by Saha’s formula:

\[ K=\beta\frac{(2\pi m_e)^{\frac32}k^{\frac52}}{h^3}\, T_1 T^{\frac32}\,l^{-\frac{h\nu_0}{kT_1}}. \tag{26} \]

\(\beta\) is the solid angle under which the earth is seen from the sun, \(M\) is the molecular weight, and \(g\) is the acceleration due to gravity.

Maximum concentration of ions. With the help of the relations indicated above it is possible to determine the concentration of ions in the layer where it reaches its maximum value. It is only necessary to bear in mind that, since these relations were derived for the equilibrium state, when the number of ions being formed and recombining is the same, the concentration values obtained will give only a certain limiting value. In fact, in order to establish the equilibrium state, the action of the ionizing radiation must continue for a sufficiently long time. But in reality, owing to the motion of the sun across the celestial vault, the angle at which the rays penetrate into the atmosphere is changing all the time, and consequently the quantity of energy absorbed in each given layer is also continuously changing. We must expect that what actually takes place is a non-steady process, which should be described by the following equation:

\[ \frac{dn}{dt}=-\alpha n^2+Y, \tag{27} \]

where \(n\) is the number of ion pairs, and \(Y\) is the number of ion pairs arising in \(1\ \mathrm{cm}^3\) in 1 sec. owing to photoionization. Our previous relations referred to the case when \(\frac{dn}{dt}=0\). Kryuchkov\({}^{46}\) in Leningrad made an attempt to calculate the ionic concentrations starting from the general case (27). But in order to obtain a general idea of what maximum ionization can be produced by the solar rays, one may start from the stationary process, assuming a sufficiently long action of radiation whose intensity has the greatest of all possible values. This, evidently, will occur at the equator at noon with the sun at the zenith (the vertical passage of the rays through the atmosphere). The results of calculations for the indicated conditions are given in Table 13.

TABLE 13

Gas	O	O\(_2\)	N\(_2\)	H\(_2\)	N
\((n_e)_{\max}\) in \(1\ \mathrm{cm}^3\)	\(4\cdot10^6\)	\(6\cdot10^5\)	\(2.5\cdot10^5\)	\(10^6\)	\(7\cdot10^5\)

The numbers \(n_e\) give the number of ion pairs—ion + electron, i.e. they give at the same time the concentration of free electrons (the first 4 figures are given according to Chapman’s calculations\({}^{45}\), and the last figure according to Kryuchkov’s calculations\({}^{46}\)).

Of chief interest is the comparison of the obtained values

...causes with the data provided by observations of the propagation of radio waves. As was indicated in Chapter V, the concentration of free electrons in layer \(E\), according to radiometric analysis, is \(1.5 \cdot 10^5\), in layer \(F_1\)—\(3 \cdot 10^5\), and in layer \(F_2\)—\(10^6\). Consequently, it may be asserted with confidence that photoionization by the ultraviolet rays of the Sun is capable of producing the concentration of ions actually observed, although not all gases can participate in the phenomenon to the same degree.

Height of the ionized layers. Using the formulas for the absorption of solar radiation in the atmosphere, one can calculate the height of the ionized layers. Omitting the calculations themselves, we shall give the final results.

The principal result reduces to the fact that none of the gases appearing in Table 13 can give heights of maximum ionization of the order of 100 km, where layer \(E\) is located: all the heights turn out to be much greater. Thus, for molecular nitrogen the maximum ionization is obtained at a height of 170–180 km, for atomic nitrogen—at a height of the order of 220 km, for atomic oxygen—170 km. If one assumes that, within heights up to 170 km, all oxygen is in the molecular state, then the layer of ionized molecular oxygen has a maximum at a height of 130 km. However, the role of molecular oxygen appears rather unclear, since from the point of view of the theory of the ozone layer (Chapter III) all oxygen above 35 km should be in the atomic state. A number of other facts also speak in favor of a significant content of atomic oxygen in the upper layers.

Consequently, we may consider that the upper ionized layers (layers \(F_1\) and \(F_2\)) can undoubtedly be explained by the action of the Sun’s ultraviolet radiation; but to explain the Kennelly–Heaviside layer at a height of 100 km, it is necessary to assume the action of some other agent; such an ionizing agent may be corpuscular radiation from the Sun. True, attempts are being made to explain the lower layer as well by the action of ultraviolet rays: for this it is assumed that at the corresponding heights there is some gas whose absorption lies in the spectral region corresponding to a “window” in the absorption diagram of the other gases, i.e., in a region not absorbed by the other gases—in this case the corresponding radiation can penetrate sufficiently deeply into the atmosphere. To satisfy this last condition, the absorption coefficient of the assumed gas itself must also be sufficiently small. There are indications that neon may satisfy all the requirements mentioned, but they are of very recent date and have not yet been subjected to any broad discussion.

However, the question of the sufficiency of the ionization produced by ultraviolet rays begins to appear in an entirely different light if one looks at it from the point of view of a number of other geophysical phenomena connected with ionization. We shall now turn to the consideration of these questions.

VI. The Connection of the Ionization of the Upper Layers with Other Phenomena. Corpuscular Theory of Ionization

Diurnal Variations of the Earth’s Magnetic Field.

These variations reach a magnitude of 0.0005 CGS, i.e., amount to 0.1% of the strength of the terrestrial field. The diurnal changes are greater by day than by night, and greater in summer than in winter. Plotting on a map the results of simultaneous measurements made at different points of the globe, we obtain a complete picture of the phenomenon, the analysis of which leads to quite definite results.

The magnetic field, known over the entire surface of a sphere, can, using Gauss’s theorems, be decomposed into three parts: two parts dependent on a potential and giving the magnetic action of currents or permanent magnets situated inside or outside the sphere, and a third, having no potential, which may be regarded as the result of the action of vertical currents penetrating the surface of the sphere along the normal. Calculations have shown that for the Earth’s magnetic field this third part, which has no potential, is entirely absent, and that the constant component of the field has a source that may be regarded as wholly concentrated within the terrestrial globe; conversely, for the variable component of the field, a source localized above the Earth’s surface is obtained. Since the supposition of the existence of permanent magnets outside the Earth’s surface cannot be made, the matter is evidently the presence of a system of electric currents in the atmosphere. We saw at the beginning of Chapter IV that the question was posed in just this way by Balfour Stewart as early as the 1870s; but at that time the assumption that the upper layers of the atmosphere were conductive was an interesting but scarcely probable hypothesis. The latest data on the propagation of radio waves (Chapter IV) have proved the presence of conducting layers, and the corresponding theories have obtained a solid basis for their development. It may be said that it was precisely as a result of the successful study of the ionosphere by means of radio waves that the intensive development of the corresponding chapters of the doctrine of terrestrial magnetism, observed during the last decade, became possible.

Electric currents in the atmosphere can only be the result of some movements of ionized masses of air. Since, from the observed pattern of variations of the magnetic field, one can in a purely formal way construct the corresponding pattern of currents in the atmosphere, it is evident that the study of phenomena connected with the variation of the magnetic field can provide material for judging the properties of the ionized masses (layers) of the atmosphere. It is in this connection that this question must be considered here.

As for the mechanism of the influence of the conductivity of the upper layers on the magnetic field, three principal possible cases are indicated here. The first of them is the so-called dynamo effect, which was discussed at the beginning of Chapter IV. As Chapman’s calculations^47 have shown, the strength of the current induced in the conducting layer can be

can be roughly determined as the product \(\sigma D u Z\), where \(\sigma\) is the mean specific electrical conductivity, \(D\) is the thickness of the layer, \(u\) is the horizontal velocity of the air, and \(Z\) is the vertical component of the earth’s constant magnetic field. The variations may also be due to changes in \(u\) and changes in \(\sigma\), which in fact do occur. In some way it is necessary to separate the contribution of these quantities individually.

The second phenomenon, which has been given the name “magneto-gravitational drift,” is as follows. At the equator the earth’s magnetic field is directed horizontally (to the north), and if the conducting layer for some reason begins to move in the vertical direction, then, owing to the action of the magnetic field on the moving charge, a force will appear which imparts to the charge an acceleration perpendicular to the direction of the magnetic force and to the direction of motion. A descending layer of charged particles will “drift” eastward or westward (at the equator), depending on the sign of the charge. A horizontal electric current is formed in the atmosphere, which can occur only in very high layers, where the mean free path is sufficiently large.

Finally, the third theory proceeds from the fact that the upper conducting layers of the atmosphere must be diamagnetic. The intensity of magnetization \(i\) is determined by the relation:

\[ i = -\frac{nkT}{H}, \tag{28} \]

where \(n\) is the number of charges in \(1\ \mathrm{cm}^3\), \(T\) is the absolute temperature, \(k\) is Boltzmann’s constant, and \(H\) is the intensity of the earth’s magnetic field. Since the concentration of charges \(n\) is subject to diurnal variations, the additional force caused by the magnetization of the conducting layers must have a diurnal variation. This theory can satisfactorily explain a number of the principal phenomena.

Let us consider the results which these theories give with respect to the properties of the conducting layers. To estimate the possible velocity and horizontal displacement of air masses in the upper layers of the atmosphere, the influence was taken into account of tidal forces acting from the sun and the moon, as well as the influence of nonuniform heating by the sun’s rays. These latter causes of a thermal character had to be set aside, since they would give variations not coinciding in phase with the observed ones. If one restricts oneself to consideration of tides of the simplest character, then from the observed variations of atmospheric pressure it is possible to calculate that there exists a velocity of \(1\ \mathrm{km}\) per hour at the equator. If one starts from an electron concentration of the order of \(10^5\)—\(10^6\) in \(1\ \mathrm{cm}^3\), which radiometric investigations give for the ionized layers, then the corresponding electrical conductivity proves to be wholly insufficient to explain the observed magnitude of the variations of the magnetic field. With the indicated concentration of free electrons, a wind having a velocity of several thousand kilometers per second would be required.

In this connection one must recall that the determination of the concentration of charges in the conducting layer by the radio-wave method (Chapter IV) is associated with a fundamental ambiguity: one has to choose between free electrons and ions, and in the latter case the concentration obtained is tens of thousands of times greater. If it is assumed that the conductivity of the upper layers is due not to free electrons but to ions, then by the radio-wave method concentrations (formula 21, Chapter IV) of the order of \(10^9\)—\(10^{10}\) ions/\(1\ \mathrm{cm}^3\) are obtained; in this case, to explain the magnetic variations, velocities of several tens of meters per hour are sufficient. The existence of such air currents could easily be explained.

With respect to other theories of magnetic variations the situation proves analogous: for a quantitative explanation of the observed magnitude of the diurnal variations of the magnetic field, the concentrations of charged particles given by the radio-wave method under the assumption that the conductivity is due to free electrons turn out to be wholly insufficient; conversely, quantitative agreement can be achieved if one starts from the ionic nature of the conductivity.

This result is of great importance for the theory of conducting layers. If one starts from the assumption of free electrons, toward which investigators of the ionosphere by the radio-wave method are inclined, then the computed electron concentrations lie within the limits of what can be produced by the ionizing action of the Sun’s ultraviolet radiation (Table 13, Chapter V). In the opposite case, one has to deal with exceptionally large and difficult-to-explain ion concentrations. The question of which of these two alternatives corresponds more closely to reality is, properly speaking, an open one. It is not impossible that, in resolving this question, the method of polarization analysis of twilight, which will be discussed in the following chapter, may prove useful.

Magnetic storms and aurorae. Up to now we have spoken only of periodic changes of the magnetic field, and only of diurnal ones. Nonperiodic changes are also of great interest. We shall speak of the most sudden and strong disturbances, which have received the name magnetic storms. The chief interest from the point of view of the theory of ionized layers in the atmosphere is the fact that strong magnetic storms are usually accompanied by aurorae and, in addition, are characterized by disruption of radio communication on short waves.

Of very great importance for the theory of the phenomena we are considering is the connection of magnetic storms with solar activity. It is known that the number of sunspots changes from year to year, and that there exists a well-expressed period of the order of 11 years. It turned out that the frequency and intensity of magnetic storms also have the same period, with the years of maximum sunspots account—

and the greatest number of magnetic storms is also observed. A more detailed study has shown that the development of magnetic storms is influenced chiefly by spots located near the central meridian of the Sun.

In addition, it has been established that for not very strong magnetic storms there exists a 27-day period; this too is an indication of a connection with solar activity, since 27 days is precisely the period of the Sun’s rotation.

Thus it may be asserted with certainty that increased activity of the Sun, finding its outward expression in the number of spots, prominences, faculae, etc., affects physical processes in the Earth’s atmosphere: it changes the state of the ionized layers (disruption of short-wave radio communication), causes magnetic storms, and excites intense aurorae. This directly indicates that the cause of the phenomena mentioned is solar radiation—the only question being what kind: ultraviolet or corpuscular.

Hulburt and Meris \(^{48,49}\) developed a theory according to which the exciting agent is the ultraviolet radiation of the Sun. The basic propositions of this theory are as follows. At an altitude of more than 300 km the rarefaction of the gases is such that the mean free path becomes extraordinarily large: \(10^{16}\) molecules contained in a column with a base of \(1 \text{ cm}^2\) above 300 km undergo, in 1 sec., only \(10^{14}\) collisions, which corresponds to a mean free path of several tens of kilometers. Therefore the molecules execute upward and downward motion resembling “jumps,” if they receive from below an upward impulse of sufficient strength. The authors believe that, of the \(10^{14}\) collisions, \(10^7\) are impacts of the second kind with excited molecules and atoms, as a result of which a velocity of the order of \(10^6 \text{ cm/sec}\) is acquired; therefore particles that have received a velocity in the upward direction can fly to an altitude of several tens of thousands of kilometers in 3–5 hours. Under the action of the Sun’s ultraviolet radiation, the particles have time to become ionized within 3–5 hours, and therefore on the return path they proceed along magnetic lines of force, returning to the lower layers already chiefly in the polar regions. In doing so they release their energy in the form of aurorae and magnetic disturbances; the energy carried by the particles into the polar regions amounts to \(5 \cdot 10^{13}\) erg/sec, which approximately corresponds to the energy of not very intense aurorae. In addition, it is assumed that during periods of increased activity the Sun emits, over the course of several tens of minutes, intense ultraviolet radiation, which increases the ionization of the upper layers of the atmosphere and causes heating. As a result, air currents are formed in the upper layers, leading to the occurrence of magnetic storms and intense aurorae. This theory explains the geographical distribution of aurorae in agreement with observations: it is known that the greatest frequency of auroral appearances falls at a magnetic latitude of approximately 67°; this latitude is also given by the theoretical

calculation as a zone where the majority of trajectories along which highly ejected ionized particles return converge. In addition, the theory explains certain regularities in magnetic phenomena. A more detailed exposition of this theory, given by Halbert himself, may be found in a review recently published in Uspekhi fiz. nauk[^58]. Unfortunately, in this review the author did not at all dwell on those difficulties encountered by this theory. Meanwhile Chapman[^50] gave a very serious critique of the theory, one that concerns the very essence of the matter. Chapman pointed out that, according to calculations, the magnetic lines of force that intersect the earth in the polar regions rise at low latitudes to a height of 30,000–50,000 km above the surface of the earth. Therefore, in order to reach the polar regions and produce aurorae there, the ions must rise at low latitudes to a height of up to 50,000 km. The gas density at such heights is vanishingly small; for a “jump” to such a great height, a particle, being in the layer 300–400 km/sec, must acquire a velocity of 10 km/sec in the upward direction. The supposition that such velocities are acquired owing to collisions of the second kind with excited atoms and molecules is doubtful. But even if one admits the existence of all the assumed conditions, it still proves impossible to explain the observed facts.

Observations show (Chapter II) that a large part of aurorae occurs at a height of the order of 100 km. To penetrate into such low layers, the exciting particles must possess very great velocities, which in no way can be ascribed to the hypothetical particles of Meris and Halbert: calculation shows that the particles cannot enter a layer below 200 km above the earth’s surface. Equally great difficulties are encountered in explaining magnetic storms.

All these circumstances are the reason why we must inevitably make an attempt to explain the phenomena in question from the point of view of corpuscular rays of the sun. We now turn to a consideration of the corpuscular theory.

Corpuscular theory. The hypothesis that magnetic phenomena and aurorae are connected with the action of corpuscular rays of the sun has a long history. This hypothesis was first put forward as early as 1881 by Goldstein. Fifteen years later Birkeland performed his famous experiment, which proved that a magnetic pole causes a stream of electrons to converge to a single point. This experiment, as is known, enabled Størmer to substantiate the well-known theory of aurorae. According to Størmer’s theory[^7], aurorae are excited by streams of charged particles arriving from the sun, which are deflected by the earth’s magnetic field and therefore gather in the polar regions. Størmer calculated the corresponding trajectories of charged particles and obtained good agreement with the observed geographical distribution of aurorae. The motion of powerful streams of charged pa-

particles must create an additional magnetic field, and, consequently, the basic fact—namely that intense auroras are accompanied by magnetic storms—finds a natural explanation.

As Schuster later pointed out[^51], Størmer’s theory encounters difficulties which at one time seemed insurmountable. Lindemann[^52], developing Schuster’s ideas, calculated that a ray composed of charges of the same sign must have so great a density that, owing to electrostatic repulsion, it could exist with such a density only at a distance of no more than 1–3 million km from the Sun, whereas in reality the ray must traverse a distance to the Earth of about 150 million km. At lower densities, however, the actual energy of magnetic storms cannot be explained. Moreover, the influx of charged particles must almost instantaneously impart to the Earth so large a charge that the further approach of beams of charged particles to the Earth becomes impossible because of electrostatic repulsion. Charged particles can reach the Earth in sufficiently large numbers only during the first few seconds, whereas the duration of auroras and magnetic storms corresponds to much longer intervals of time.

This difficulty of the corpuscular theory, however, proved possible to overcome. Chapman and Ferraro[^53] put forward the supposition that the stream of solar corpuscles is, as a whole, neutral, although ionized: it consists of equal numbers of positively and negatively charged particles. When such a stream of corpuscles approaches the Earth, the Earth’s magnetic field, acting in opposite directions on the positive and negative particles, as it were “splits” the initial beam into two beams, which go around the Earth on opposite sides. Such a neutral, but ionized, beam must possess far greater stability, and thus the difficulty indicated by Schuster is avoided.

The Chapman–Ferraro theory, which is the most recent (1931) version of the corpuscular theory, explains a number of known facts fairly well, and we shall dwell on it somewhat more fully. This theory is intended to explain magnetic storms; although, in view of the connection of the latter with the phenomenon of auroras, its conclusions are also essential for interpreting the nature of auroras.

It is assumed that the duration of a magnetic storm (a day or more) is approximately equal to the time during which the Earth crosses the stream of corpuscles; the cross-section of the beam at the Earth’s orbit must in this case be about \(10^6\) km, which corresponds to angular dimensions on the solar surface of the order of 5–15°. In magnetic storms two principal phases usually stand out sharply: the initial phase, characterized by a rapid (within minutes) increase in the horizontal component, and the subsequent, so-called main phase, when there occurs a slow (over days), but nonuniform, decrease of the field strength, until after some time it becomes even less than normal.

...and only then, with the passage of time, is it equalized. The currents created by the flux of charged particles that have approached the terrestrial globe, and the diamagnetic effects of this flux, create an additional magnetic field, which accounts for the first phase of a magnetic storm. Chapman and Ferraro propose the following explanation for the main phase.

Charged particles, as they approach the Earth, must be decelerated. Indeed, the energy of the additional magnetic field of the first phase can appear only at the expense of a decrease in the kinetic energy of the corpuscular stream. It is clear that the greatest deceleration will occur at point \(A\) (Fig. 9), the point closest to the terrestrial globe. The edges of the stream, \(B\) and \(C\), will experience less deceleration and, as a result, will overtake the central part \(A\). The consequence of this will be the formation of a hollow region, as shown in Fig. 9. According to the calculations of Chapman and Ferraro, at the moment of the onset of a storm of average intensity, point \(A\) is at a distance of four Earth radii from the surface of the Earth. But the action of the Earth’s magnetic field on the frontal part of the stream leads to its “polarization,” so that negative charges are concentrated on the surface \(AB\), and positive charges on the surface \(AC\). Such bending of the corpuscles by the Earth’s magnetic field promotes the formation of a cavity near \(A\), since here the greatest change occurs not only in the magnitude of the velocity, but also in its direction. But owing to the separation of charges, forces of electrostatic attraction arise, which tend to close the oppositely charged “flanges” \(B\) and \(C\). As a result, the terrestrial globe becomes surrounded by a ring current, which continues for several days, gradually dying out and producing the main phase of the storm—the progressive diminution of the disturbed magnetic field of the Earth.

Fig. 9.

Omitting the details, one may say that the Chapman–Ferraro theory explains well the basic facts characterizing the course of magnetic storms. Unfortunately, it has one point whose interpretation encounters extraordinary difficulties. The densities of the corpuscular stream that follow from the Chapman–Ferraro theory prove to be extremely large: of the order of \(10^8\) charged particles per \(1\ \mathrm{cm}^3\). How large this density is may be judged by comparison with the density of the solar corona: the latter is a quantity 10 times smaller, while one must still take into account the influence of the purely geometrical expansion of the corpuscular stream on its path from the Sun to the Earth.

The mechanism of the eruption of corpuscular streams

the sun. When it is said that the cause of various phenomena in the earth’s atmosphere may be the corpuscular radiation of the sun, it is natural to raise the question of the very possibility of the existence of corpuscular solar rays. Can the ejection by the sun of streams of corpuscles occur at all? Of course, at the very earliest stages in the development of Størmer’s corpuscular theory, such a question could not be posed, since our knowledge of the physical processes on the sun was far too limited. But the brilliant successes of astrophysics over the last 10–15 years now make it possible to speak with considerable confidence about many fundamental features characterizing the picture of the physical structure of the sun—of course, in its surface layers. It has become possible to pose the question indicated above as well, and it has become clear that not only are there grounds for considering the occurrence of corpuscular eruptions of the sun possible, but it is even possible to calculate quantitatively certain properties of such corpuscular streams. We have in mind Milne’s theory, proposed in 1926.

In connection with the problems that interest us, we shall formulate the following three questions: 1) how corpuscular streams can arise in general, 2) what the velocity of propagation of the streams must be. This question arises in connection with the fact that, according to observations, magnetic storms (and the intense aurorae accompanying them) begin 20–30 hours after the passage of large spots across the central meridian of the sun, 3) what is the penetrating power of solar corpuscular radiation. This question arises from the standpoint of the possibility of corpuscles penetrating into the earth’s atmosphere down to an altitude of 100 km, where the greater part of the aurorae is concentrated.

Milne’s theory, as one of the possible cases, considers the ejection of calcium ions CaII, whose presence in large quantity in the solar chromosphere is indicated by spectral-analysis data. Radiation from the inner parts of the sun, possessing a continuous spectrum, is partially absorbed by CaII ions, which results in the appearance of the corresponding Fraunhofer absorption line. In Fig. 10 an approximate contour of the absorption line is schematically represented: the maximum of absorption falls at a certain wavelength \(\lambda_0\), the width of the line is \(\lambda_0-\lambda_2=\Delta\lambda\); we denote the intensities of sunlight for the wavelengths \(\lambda_0\), \(\lambda_1\), and \(\lambda_2\) respectively by \(I_0\), \(I_1\), and \(I_2\), where, obviously, \(I_0<I_1<I_2\). Suppose that a CaII ion, for some reason (for example, owing to random collisions or eruptions on the surface of the sun), has acquired a positive velocity relative to the sun (ascending motion). Owing to the Doppler–Fizeau effect, the frequencies of the sunlight overtaking it from behind

by the CaII ion, will be perceived by the latter as somewhat reduced, and the proper frequency of the ion’s maximum absorption, corresponding to the wavelength $\lambda_0$ in the solar spectrum, will correspond to some wavelength $\lambda_1$ from the violet wing of the absorption line. Milne takes radiation pressure into account: since for $\lambda_1$ the energy $I_1$ in the solar spectrum is greater than the energy $I_0$ for $\lambda_0$, the result of the indicated influence of the Doppler–Fizeau effect will be an increase in the radiation pressure. Owing to this latter, the ion’s velocity increases still further, which leads to a further increase of the radiation pressure, and the process continues automatically until the ion “emerges” beyond the contours of the line $\lambda_2$ and begins to absorb at frequencies corresponding to the continuous spectrum near the absorption line. However, as the distance from the Sun increases, the acceleration decreases, since the radiation pressure diminishes in proportion to the square of the distance, and it is clear that, in the end, the atom will acquire a certain limiting velocity $V_\infty$, corresponding to an infinite distance from the Sun.

Thus arises the corpuscular radiation of the Sun, which, naturally, should occur chiefly in regions of increased solar activity (spots, prominences, faculae), since in these places one may expect the presence of ascending local streams of solar matter and an increased intensity of radiation.

As for the magnitude of the limiting velocity $V_\infty$, Milne’s calculations give $V_\infty = 1630$ km/sec, and a velocity of the same order is obtained also for other atoms (H, Fe, He, etc.) located in the chromosphere. The distance from the Sun to the Earth will be traversed in approximately 27 hours, which is in surprisingly good agreement with observations (question 2). Matters are somewhat worse with the penetrating power: at a velocity of 1600 km/sec, a CaII atom can penetrate only through a layer of air 1.5 mm thick at normal pressure. If, however, one speaks of the possibility of penetrating into the atmospheric layer situated at an altitude of 100–120 km above the Earth’s surface (the altitude of most auroras, the altitude of the Kennelly–Heaviside layer), then this corresponds to a layer of the order of 3–5 mm under normal conditions. This difficulty can be overcome by assuming that there is some initial velocity of the corpuscles’ departure from the Sun. In general we must acknowledge that Milne’s theory gives results very encouraging for the corpuscular hypothesis.

It must also be noted that, although radiation pressure can exert the described action only on atoms and ions, and not on electrons, nevertheless the latter are carried along together with the positive ions owing to electrostatic attraction. This is necessary in order to obtain the neutral ionized beam considered in the Chapman–Ferraro theory.

Corpuscular eclipse. Concluding remarks. In 1932 Chapman pointed out a very original method by means of which one can resolve the question of the nature

of ionizing radiation for different conducting layers. In Chapter IV we saw that, from observations of the propagation of radio waves, it is possible to determine both the height of the conducting layers and the concentration of ions at the height where the “reflection” of the radio wave occurs.

Chapman proposed making such measurements during total solar eclipses. During an eclipse there should be a decrease in ionization. If the ionization is caused by the Sun’s ultraviolet rays, then the observed decrease in ionization should coincide in time with the moment of the visible eclipse. If, on the other hand, corpuscular rays are responsible for the ionization, then the moment of decrease in ionization will not coincide with the visible eclipse, since the velocity of propagation of corpuscular rays is less than the velocity of light. Calculation shows that such a “corpuscular eclipse” should occur earlier than the visible one (and not later, as might at first sight seem).

Such observations were carried out during several eclipses, including that of August 31, 1932, in the USA, and that of June 19, 1936, in the USSR. It turned out that for the layers \(E\) and \(F_1\) the decrease in ionization coincides with the moment of eclipse, but for the layer \(F_2\) this decrease in ionization takes place 2 hours before the eclipse. These results may be considered to some extent unexpected. On the one hand, it had seemed beyond doubt that precisely the uppermost layer—the layer \(F_2\)—should have arisen under the action of ultraviolet radiation, since the intensity of solar radiation at great heights is so large that it can undoubtedly produce complete ionization of this layer. But the data of solar eclipses indicate a corpuscular origin of the layer \(F_2\). On the other hand, as theoretical calculation shows (Chapter V), it is difficult to explain the penetration of ionizing ultraviolet radiation to a height of 100 km: it should be completely absorbed at great heights. Meanwhile it turns out that the layer \(E\) is associated with ultraviolet radiation.

All these circumstances make the question of the mechanism of the origin of ionization still unresolved. Both theories of ionization set forth above allow us to form a general idea of how the processes of ionization might proceed, but they do not give us the possibility of constructing an exact and complete picture of the phenomenon. However, there is no reason to doubt that these theories will form the basis for the complete solution of the problem of ionization, which will undoubtedly be found with the help of further systematic investigations.

VII. Polarimetric method for studying the structure of ionized layers

General considerations. As has already been mentioned more than once, observations during twilight can serve for the study of certain properties of the stratosphere. The theory developed by V. G. Fesenkov* makes it possible, from photometric measurements of the decreasing

to calculate changes in the brightness of the sky during twilight, under the assumption of one or another molecular weight of the air at great heights, and the vertical distribution of atmospheric density. In Chapter II we pointed out that, knowing the density and molecular weight, it is possible to determine the temperature of the air up to a height of 100 km. The actual determination of density from twilight observations can be made up to heights of 170–200 km.

Within this range of heights there exists, as we know, the ionized layer \(E\)—the Kennelly–Heaviside layer; however, this layer is not revealed in the twilight curves. Since photometric analysis of twilight gives directly the variation of air density with height, one may think that the change in density accompanying ionization is so small that it cannot be detected by photometric observations. The photometric method proves insufficiently sensitive.

In 1936 the author indicated that the situation might become substantially different if one measured not brightness but the polarization of twilight light. Indeed, twilight light is caused by the scattering of the Sun’s rays by gas molecules, and if the presence of ionization, having little effect on density, does not exert a substantial influence on the intensity of the scattered light, then it may noticeably affect the polarization, since the latter depends on the anisotropy of the scattering particles, which may change greatly in the presence of ionization.

Observations made in the summer of 1936 and then continued in the summer of 1937 showed that the assumptions mentioned above are justified: the curve giving the change in polarization of the scattered light of the sky during twilight always has several maxima and minima, showing that at certain heights the properties of the particles producing the scattering of light change substantially. The values of the heights obtained show that an especially strong change in polarization is associated with the region where the Kennelly–Heaviside layer is located, although besides this latter the polarization curves indicate the existence of a number of other layers at different heights.

Thus, the indicated measurements of polarization can serve as a good method for studying the structure of ionized layers. It is very important that this method is completely independent of other methods and, moreover, evidently makes it possible to determine the height of ionized layers most accurately. If further developed, the method will make it possible not only to check the data obtained from observations of radio-wave propagation, but also to considerably refine and expand the range of known facts. Let us consider in somewhat greater detail several questions relating to this.

Laws of the scattering of light by gases. As is known, the phenomenon of the scattering of light by gas molecules is well described by Rayleigh’s theory. The greatest success of this theory was the explanation of the blue color of the sky, and the results obtained on this question—

...the results were characterized not only by qualitative but also by good quantitative agreement: the value of Avogadro’s number obtained from such observations differs little from the values obtained by other methods.

One of the principal results of Rayleigh’s theory is the sharp dependence of the coefficient of scattering of light on the wavelength and on the direction of the light vector. The numerical value of the coefficient is determined by the following formulas:

\[ \begin{aligned} K_{\perp}&=\frac{\pi^{2}(n^{2}-1)^{2}}{2\lambda^{4}N},\\ K_{\parallel}&=\frac{\pi^{2}(n^{2}-1)^{2}}{2\lambda^{4}N}\cos^{2}\theta. \end{aligned} \tag{29} \]

Here \(n\) is the refractive index of the scattering gas, \(\lambda\) is the wavelength of the light, \(N\) is Avogadro’s number, and \(\theta\) is the angle of scattering, i.e. the angle between the direction of the incident ray and that of the scattered ray. Let us imagine the plane drawn through these two rays—this is the so-called plane of vision. \(K_{\perp}\) is the coefficient of scattering of light for a light beam in which the light oscillations take place perpendicular to the plane of vision, while \(K_{\parallel}\) is for light oscillations parallel to the plane of vision. The total intensity of the scattered light in this case, if in the incident ray the two components of the light vector are equal to one another (i.e. if the incident ray is unpolarized), is determined by the sum

\[ K=K_{\perp}+K_{\parallel} =\frac{\pi^{2}(n^{2}-1)^{2}}{2\lambda^{4}N}(1+\cos^{2}\theta). \tag{30} \]

These relations show that the coefficient of scattering varies inversely as the fourth power of the wavelength of the light, which, as is known, determines the blue color of the sky (the color of the sky is determined by the product of three curves: the curve of the distribution of energy in the spectrum of the solar rays, the curve of the dependence of \(K\) on \(\lambda\), and the curve of the spectral sensitivity of the human eye). Formula (30) shows that the intensity of scattering has its greatest value along the incident beam of rays (\(\theta=0\) or \(\pi\)); in directions perpendicular to the incident ray the intensity is smallest \(\left(\theta=\frac{\pi}{2}\right)\). From the point of view of the theory of polarization, it is very important that the dependence of the coefficient of scattering on the angle of scattering exists only for oscillations taking place parallel to the plane of vision, as follows from formula (29). At \(\theta=\frac{\pi}{2}\) the parallel component will be entirely absent:

\[ K_{\parallel}=0. \]

This means that the scattered light must be polarized. In lateral observation, when \(\theta=\frac{\pi}{2}\), the polarization

must be complete; for \(\theta \ne \frac{\pi}{2}\) the polarization is partial, and only when observed along the incident ray (\(\theta=0\) or \(\pi\)) should the scattered light be unpolarized (if the incident ray itself is unpolarized). If we denote the intensity of the unpolarized incident light by \(I\), and the intensity of the scattered light for light vibrations in the two directions under consideration by \(I_{\perp}\) and \(I_{\parallel}\), then, according to the definition of the scattering coefficient, we may write that \(I_{\perp}=K_{\perp}I\) and \(I_{\parallel}=K_{\parallel}I\). As is known, the measure of the degree of polarization \(p\) is the ratio

\[ p=\frac{I_{\perp}-I_{\parallel}}{I_{\perp}+I_{\parallel}}. \]

Using formula (29), we may write:

\[ p=\frac{1-\cos^{2}\theta}{1+\cos^{2}\theta}. \tag{31} \]

This formula gives the dependence of the degree of polarization of the scattered light on the scattering angle. Graphically this dependence is presented in Fig. 11 (curve I).

Fig. 11

Refinement of the theory of polarization. It follows from formula (31) that if one observes perpendicular to the incident rays \(\left(\theta=\frac{\pi}{2}\right)\), then \(p\) should be equal to 1. However, under real conditions of the earth’s atmosphere it has never been possible to observe 100% polarization, although the general dependence on the scattering angle is fully confirmed: the polarization is greatest at \(\theta=90^\circ\) and decreases as the angle \(\theta\) increases or decreases. Usually the polarization of the sky under an angle of \(90^\circ\) to the direction of the solar rays does not exceed 75–80% (it depends on meteorological conditions); in especially rare cases one can observe polarization of the light of the daytime sky up to 85%, but polarization greater than 85% has never been observed. The question naturally arises whether this is connected with the presence in the lower layers of the atmosphere of impurities (dust, the finest droplets of water, etc.), and also with the depolarizing action of secondary scattering. From this point of view, laboratory measurements of polarization in the scattering of light by gases are of great interest. The experiments of Rayleigh\(^ {53}\), Cabannes\(^ {54}\), Raman and Rao\(^ {55}\), and others showed that although under laboratory conditions greater polarization is observed than in the atmosphere, even it is less than 100%. For air, polarization of no more than 92% was observed.

On the basis of the fluctuation theory of scattering it was possible to show\(^ {54}\).

that, taking into account the anisotropy of molecules, one can explain the incomplete polarization of light scattered at an angle of \(90^\circ\). In this connection the so-called optical coefficient of anisotropy of molecules \(a\) is introduced, which is defined as the ratio \(a=\dfrac{B}{A}\), where \(A\) is the component of the light vector of the scattered light in the direction perpendicular to the plane of vision, and \(B\) is the component parallel to the plane of vision. It is not difficult to see that the coefficient of anisotropy \(a\) is related to the degree of polarization \(p\) by the relation \(p=\dfrac{1-a}{1+a}\). If the scattering angle \(\theta\) is not equal to \(90^\circ\), then, as Tikhanovskii \(^{56}\) showed, this relation takes the form

\[ p=\frac{1-a}{1+a+2\,\operatorname{ctg}^{2}\theta}. \tag{32} \]

Taking for air the maximum observed \(p=0.92\), we obtain for air molecules (i.e. nitrogen and oxygen molecules) \(a=0.04\). Substituting this value into formula (32) for various \(\theta\), we obtain the dependence of the degree of polarization of scattered light on the scattering angle, shown in Fig. 11 (curve II).

Returning to the question of polarization during twilight, we refer the reader to curve III in Fig. 11. This curve represents the results of one of the observations made at the zenith. When the sun is at the horizon we have \(\theta=90^\circ\) (zenith observations), and as the sun sinks below the horizon the scattering angle \(\theta\) increases. If the properties of the higher layers of the atmosphere with respect to light scattering remained unchanged, then curve III would have had to run parallel to the theoretical curve II, which is approximately what is observed at the beginning, for angles \(\theta\) close to \(90^\circ\).

But at angles of the order of \(95—97^\circ\) (the sun’s depression below the horizon by \(5—7^\circ\)) the actual degree of polarization sharply decreases and then, remaining at this lower level, undergoes several successive increases and decreases. This directly indicates a sharp increase in the optical coefficient of anisotropy of molecules occurring in the corresponding high layers of the atmosphere. It is, of course, very interesting to determine the height of the layers with increased anisotropy of the molecules.

Determination of the height of the scattering layer. Let us first calculate the height of the shadow of the sun’s rays when observing at the zenith. Let \(ABCD\) (Fig. 12) be the surface of the terrestrial sphere, point \(O\) the center of the earth, the dotted straight line the horizon, \(BE\) the direction of the sun’s rays, the angle \(\varphi\) the depth of the sun’s depression below the horizon, \(CE\) the sought height \(H\) of the shadow of the sun’s rays, \(OC\) the radius of the terrestrial sphere, which we shall denote by \(R\), and \(OB\) the radius drawn to the point of tangency of the sun’s rays. Obviously,

that the angle \(BOE\) is also equal to \(\varphi\). From triangle \(BOE\) we have \(OB=OE\cos\varphi\), or \(R=(R+H)\cos\varphi\), whence we obtain that

\[ R(1-\cos\varphi)=H\cos\varphi, \]

or

\[ H=\frac{1-\cos\varphi}{\cos\varphi}\,R=R(\sec\varphi-1). \tag{33} \]

At any given moment a layer of air from this height upward, to the boundary of the atmosphere, is illuminated. But not all parts of the illuminated layer take equal part in the scattering of light. The total intensity of the scattered light \(I_H\) will, under the conditions considered, be determined by the integral:

\[ I_H=BK\left\{1+\cos^2\left(\frac{\pi}{2}+\varphi\right)\right\} \int_H^\infty I_s\rho\,dH, \tag{34} \]

where \(I_s\) is the intensity of the solar rays illuminating a gas layer of thickness \(dH\), \(\rho\) is the density of the gas in the given layer \(dH\), \(K\) is the coefficient of scattering of light under certain specified conditions, the bracket

\[ \left\{1+\cos^2\left(\frac{\pi}{2}+\varphi\right)\right\} \]

determines the dependence of the intensity of light scattering on direction, and \(B\) is a proportionality factor. The quantities under the integral, \(I_s\) and \(\rho\), are functions of height: the intensity of the solar rays \(I_s\) is different for different heights, since the closer to the earth’s surface the rays pass, the more they are weakened owing to absorption—that is, it is a function that increases rapidly with height; \(\rho\) is the density, and it decreases with height according to a logarithmic law. Thus the integral expression is the product of the monotonically decreasing function \(\rho\) and the monotonically increasing function \(I_s\). At small heights the increase of \(I_s\) occurs more rapidly than the decrease of \(\rho\)—on the whole, the product increases. But at great heights, where absorption is already small, the increase of \(I_s\) slows and lags behind the decrease of \(\rho\). As a result, the value of the integrand, at first rapidly increasing, after reaching its greatest value begins to decrease rapidly. To find the height \(H_m\) at which the integrand has a maximum, we set the derivative with respect to height equal to zero:

Fig. 12.

\[ I_s(H)\frac{d\rho(H)}{dH}+\rho(H)\frac{dI_s(H)}{dH}=0. \tag{35} \]

If this differential equation is solved and numerical data are substituted both for the absorption of light in the atmosphere and for determining the density of air with height, the following result is obtained:

\[ H_m = H + 20, \tag{36} \]

if the height is measured in kilometers. This means that the most effective, at all depressions of the sun, are the rays bending around the earth at a height of the order of 20 km. It is very significant that, as calculations show, the maximum is very sharply expressed. Rays passing closer than 20 km from the earth’s surface (counting the height at the point where the solar rays touch the earth’s surface) undergo very great absorption and take almost no part in the phenomenon; very high rays, on the other hand, enter very rarefied layers of air and likewise exert little influence. The greater part of twilight at any given moment is produced by a comparatively thin layer of gas, of the order of 15–25 km.

For illustration of this result we give a table of the percentages of twilight brightness (Table 14) produced by different layers of air when the sun is depressed below the horizon by \(7^\circ\) (according to the calculations of Fesenkov\(^9\)).

TABLE 14

Height of layer in km	% of the total brightness of twilight
from 25 to 35	0.20
35 to 45	6.8
45 to 55	33.4
85 to 95	3.0

The small thickness of the effective scattering layer not only considerably simplifies the calculations, since instead of the cumbersome computations of the integral (34) for each given height one may approximately use formula (36), but also has an important physical meaning from the point of view of the possibilities possessed by the twilight method. Indeed, if in the atmosphere there exists even a comparatively thin (on the scale of the atmosphere) layer 15–25 km thick, whose molecules possess a different anisotropy than in the rest of the atmosphere, then one may hope to detect it from observations of the polarization of twilight.

Thus we have the possibility of at once associating every segment of the experimental polarization curve (curve III in Fig. 11), which reveals any peculiarities in the behavior of the optical coefficient of anisotropy of the molecules, with a definite height of the scattering layer in the atmosphere. We may, therefore, directly construct a curve giving the value of the anisotropy coefficient as a function of height in the atmosphere.

Results of observations. Curve III in Fig. 11 is one of the curves observed in the summer of 1936 in the Caucasus\(^57\). It must be said that the curves obtained from one twilight to another are not completely identical, although the most essential features are always repeated: there is always a steep drop of polarization at the beginning of the curves and there are always several maxima and minima after it. For

to obtain complete data it is necessary to carry out systematic observations at several points over a long period of time and to process the material statistically, as is always done in the case of atmospheric observations. From this point of view the observations already carried out in the summer of 1936 and, chiefly, in the summer of 1937 should be regarded only as the beginning of work on the use of the polarimetric method. However, some preliminary results can already be discussed even now.

In Fig. 13 the mean curve of polarization of the evening twilight is presented, derived from 10 curves measured by the observers E. A. Aksenova and M. D. Glushchenko in July–August 1937 in the Crimea (curve I). For comparison the theoretical curve II is given, calculated by formula (32) under the assumption of a constant optical coefficient of anisotropy of the molecules over the entire height. Along the abscissa axis is plotted the height of the scattering layer in km, and along the ordinate axis—the degree of polarization of the scattered light in percent. These curves show that if at a height of 40 km the curves practically coincide, then from a height of 50 km there occurs a sharp decrease in the observed degree of polarization, which indicates a sharp increase in the optical coefficient of anisotropy of the molecules at these heights. It may be worth recalling that radiometric observations indicate the presence of an ionized region at a height of approximately 50 km (layer D, Chapter IV). This increase in anisotropy apparently ceases at a height of about 75 km, but after 80 km it appears with still greater strength. In the layer 85–115 km the magnitude of the anisotropy becomes especially large. Apparently this portion of the polarization curve conveys details of the structure of the Kennelly–Heaviside layer. In the layers 120–185 km the anisotropy, although it undergoes certain changes, nevertheless in general remains on average at a certain level, higher than in the layer 85–115 km. After the height of 185 km one can speak of a new noticeable increase in anisotropy, which apparently is connected with the layer.

Fig. 13.

We repeat that curve I in Fig. 13 is apparently only one of the possible types of polarization twilight curves, and that when observing at another time and under other conditions curves of a somewhat different form will be obtained (in further observations it is necessary to pay special attention to excluding local meteorological influences, for which it is desirable to carry out

…make observations simultaneously at two points separated from one another by a distance of 5–15 km). But one can say definitely that curve I in Fig. 13 shows how much the polarimetric method can yield for the study of the structure of ionized layers. In order to understand this question more clearly, let us consider a possible mechanism by which gas ionization affects the polarization of light scattered by this gas.

Possible causes of the depolarizing action of ionization. What determines the optical coefficient of anisotropy of a molecule? If \(a=0\), this indicates complete isotropy of the optical properties of the molecule: the amplitude of the light vibrations does not depend on direction. Conversely, when \(a>0\) we have a molecule for which some direction is preferred. If we confine ourselves to the crudest model of the phenomenon and consider, from the standpoint of optical anisotropy, an ion of a diatomic molecule, then one may assume that such an ion possesses complete anisotropy of optical properties; in other words, it may be likened to a linear oscillator. In general one may suppose that the presence of anisotropy must be connected with the presence of an electric dipole moment. Of course, this connection is very complex, and our assumption that an ion of a diatomic molecule can be likened to a linear oscillator gives only a crude first approximation, which nevertheless allows us to understand the questions that interest us.

Let us now imagine an aggregate of a large number of randomly oriented linear oscillators, illuminated by solar rays (unpolarized light) and scattering this light. It may be calculated that, when observed at an angle of \(90^\circ\), the degree of polarization \(p\) of the scattered light must be 33%. Using the relation \(p=\frac{1-a}{1+a}\), one may obtain that the gas under consideration, which is an aggregate of randomly oriented linear oscillators, possesses the optical coefficient of anisotropy \(a=0.5\).

The scattered light of the daytime sky has a polarization giving \(a<0.1\), but at twilight we have a sharp decrease in the degree of polarization (Fig. 13). There can hardly be any doubt that this decrease is connected with ionization of the high layers of the atmosphere, since it follows from the considerations set forth above that the presence of ionization must necessarily be accompanied by depolarization. If we turn to the layers at a height of 130–150 km, then the degree of polarization of the light scattered by these layers is of the order of 35%. Using formula (32), one may calculate that this corresponds to an optical coefficient of anisotropy of the molecule \(a=0.45\). Consequently, we may assume that at these heights there exists ionization of a very large magnitude, perhaps approaching complete ionization.

In this connection one should recall the ambiguity that occurs when determining, by the radio-wave method, the concentration

ions in the reflecting layer (Chapter IV). If it is assumed that the conductivity is due to free electrons, then in layers \(E\) and \(F_1\) we obtain ion concentrations of \(1.5\cdot 10^5\) and \(3\cdot 10^5\) per \(1\ \mathrm{cm}^3\). If, however, it is assumed that the conductivity is due to the presence of molecular and atomic positive and negative ions, then the concentrations obtained are much greater: of the order of \(5\cdot 10^9\) and \(10^{10}\) ions per \(1\ \mathrm{cm}^3\). We have seen that it is not possible to make an unambiguous choice between these two alternatives: while the data obtained by the radio-wave method speak rather in favor of free electrons, all existing theories of variations of the earth’s magnetic field, without exception, require ion concentrations corresponding to the hypothesis of ionic conductivity (Chapter VI).

If the considerations set forth above concerning the mechanism of the depolarizing action of ionization correspond at least approximately to reality, then it may be asserted that the data of polarimetric analysis of twilight speak, unquestionably, in favor of the hypothesis of ionic conductivity. Indeed, if in the layers \(130\)—\(150\ \mathrm{km}\) we have an optical coefficient of anisotropy of the order of \(0.45\), then this forces one to assume a very large ion concentration. At these altitudes the density of the atmosphere is such that the number of particles in \(1\ \mathrm{cm}^3\) must be of the order of \(10^{10}\)—\(10^{11}\) (if one proceeds from the assumption of an atmosphere mixed up to these altitudes). If the ion concentration at these altitudes is a quantity of the order of \(10^9\)—\(10^{10}\) ions per \(1\ \mathrm{cm}^3\), as follows from the hypothesis of ionic conductivity, then the observed depolarization of scattered light can be explained quite satisfactorily. Conversely, at ion concentrations of the order of \(10^5\), corresponding to the hypothesis of electronic conductivity, the percentage content of ions proves too small—approximately 1 ion and 1 free electron per \(10^5\)—\(10^6\) molecules.

Thus, polarimetric analysis of twilight can be of great benefit in the study of the ionosphere. The curve in Fig. 13 shows that polarimetric observations apparently give even details of the structure of the ionized layers. If in the future it proves possible to connect the observed depolarization of scattered light unambiguously and accurately with the magnitude of the ionization of the scattering gas, then from polarimetric twilight curves it will be possible to calculate directly the distribution of ionization with altitude. Of course, very great difficulties lie on the path to this: even the assumption that the ion of a diatomic molecule is similar to a linear oscillator is only a rough approximation, and the question of the properties of atomic ions appears still more obscure. The role of free electrons must also be taken into account, as well as the influence of the electric field created by the ions on neutral molecules. In the final theory of the polarization of twilight light, the possible influence of meteoric dust, formed when meteors burn up at altitudes of \(80\)—\(120\ \mathrm{km}\), must also be taken into account.

Concluding remarks. The most recent data of rese—

...of the ionosphere with the aid of radio waves indicates the presence of a “fine structure” of the ionized layers; over the extent of a single ionized layer we have a number of maxima and minima of ionization. Apparently the polarimetric observations of twilight also speak to this: the curve in Fig. 13 has very many maxima and minima. Of course, in this latter case the possibility is not excluded that such a sinuous form of the curve is partly connected with observational errors, and this question will be resolved by further systematic observations. But if one raises the question of the possible causes of the “fine structure” of the ionized layers, it may be explained by the presence of intense emission lines in the far-ultraviolet spectrum of the sun \((\lambda < 2000\ \text{Å})\). In this sense one may say that further study of the “fine structure” will allow us to draw one or another conclusion about the properties of the sun’s radiation in so distant a region of the ultraviolet spectrum that direct observation of these rays, owing to their complete absorption in the atmosphere, proves impossible. From this point of view, the investigation of the ionosphere is significant not only for the science of the earth’s atmosphere, but also for astrophysical theories devoted to the question of the structure of the sun.

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