Study of the Density of the Upper Atmospheric Layers by Meteor Methods

B. Yu. Levin

Until very recently, the study of the upper layers of the atmosphere was possible only with the aid of indirect methods, in which the quantities directly observed and studied are very remote from the fundamental characteristics of the stratosphere that interest us. These are, first of all, density, temperature, and chemical composition. Among the indirect methods for studying the stratosphere is the study of meteors. The processes that occur when a meteoric body moves in the Earth’s atmosphere—its heating, evaporation, deceleration, and luminosity—depend directly on the density and, in part, on the chemical composition of those layers in which these processes take place. In recent years, the successful development of the physical theory of meteors has provided researchers with sufficiently accurate formulae, the application of which to observational data makes it possible to study the structure of the atmosphere up to altitudes of 120–125 km. In the overwhelming majority of cases, meteors appear at altitudes of 100–120 km and disappear at altitudes of 70–90 km. Therefore the most reliable meteor data on the structure of the stratosphere refer to the altitude interval from 70 to 120 km*). However, when slow and at the same time large meteoric bodies enter the Earth’s atmosphere, they penetrate into considerably lower layers—down to altitudes of 20–30 km, and sometimes even lower. Therefore, in prospect, there is every possibility of linking the results of meteor methods with the reliable data accumulated for the lower 30–40 km of our atmosphere.

The study of the stratosphere by meteor methods began 25 years ago. In 1923 Lindemann and Dobson¹, precisely wishing to investigate the high layers of the atmosphere, developed the first physical theory of meteors and applied it to meteor data obtained from visual observations. Subsequently, the Lindemann and Dobson theory was recognized as unsatisfactory; however, much information about the stratosphere obtained at that time with its aid proved to be entirely

*) For photographic observations the upper limit is somewhat lower—about 100 km.

correct. On the basis of the heights of appearance and disappearance of meteors, for altitudes of about 100 km, densities and temperatures were found that considerably exceeded those previously accepted on the basis of the hypothesis of an isothermal stratosphere. In addition, the curve of the distribution of meteor-disappearance heights, constructed on the basis of Denning’s catalogues, revealed a minimum at an altitude of about 60 km. Lindemann and Dobson explained this minimum by the fact that at this altitude there is a layer of elevated temperature (they identified it with the ozone layer). Subsequently the supposition that a layer of elevated temperature exists here was confirmed by Whipple (the elder) and Duckert on the basis of the anomalous propagation of sound waves, by Taylor and Pekeris on the basis of an analysis of the periods of free oscillations of the atmosphere, and partly by Whipple (the younger) on the basis of meteor photographs. However, the original source of this supposition—the minimum on the curve of the distribution of disappearance heights—at present gives rise to great doubts. It is confirmed neither by the observations of the Arizona meteor expedition nor by the English observations included in Porter’s catalogue and processed by the author of the present article. The lower maximum of disappearance heights is apparently explained by the fact that Denning’s catalogues included a disproportionately large number of bright, slow fireballs penetrating deep into the atmosphere.

The theory of Lindemann and Dobson was sufficiently developed mathematically to enable its authors to estimate the density of the atmosphere at the heights of appearance and disappearance of meteors. But, as is now known, the physical picture underlying this theory—namely the assumption that a meteoric body is heated by a cushion of adiabatically compressed air—is incorrect. It was subjected to criticism, and first Sparrow², and then Merris³, proposed new theories based on consideration of collisions of the meteoric body with individual air molecules. These theories were almost not developed mathematically and therefore did not provide new quantitative data on the stratosphere. True, Sparrow, on the basis of his theory, considered that meteor heights could be reconciled with a hydrogen isothermal stratosphere. But this conclusion is based on erroneous data on meteor heights.

In the thirties numerous studies by Öpik⁴ were published, devoted to the physical theory of meteors. In these works, acquaintance with which is made difficult by an extremely heavy style of exposition, alongside interesting results there are also results that are completely erroneous. Unfortunately, up to the present time Öpik remains the only author who has considered the question of the luminosity of meteors.

In 1937–1941 there appeared works by Hoppe⁵, Whipple⁶, and Levin⁷, containing the derivation of the basic equations of the physical theory of meteors—the equation of evaporation and the equation of braking, as well as a number of

resulting from them. In Levin’s work, moreover, a formula is derived for the heating of a meteoric body, making it possible to find the height of appearance of a meteor, i.e. the height at which boiling of the surface layer of the frontal surface begins. The temperature of the frontal surface of a non-rotating stony meteoric body is given by the formula

\[ T=\frac{b\sqrt{H^*}}{2\lambda}\,\frac{v^{5/2}}{\sqrt{\cos z}}\,\rho \]

\[ \left(b=\frac{\lambda}{\delta c}\text{—coefficient of thermal diffusivity, }\lambda\text{—coefficient of thermal conductivity, }H^*\text{—height of a homogeneous atmosphere, }\rho\text{—density of the atmosphere, }v\text{—velocity of the meteor, }z\text{—zenith distance of the radiant}\right). \]

As Levin showed, for meteoric bodies of different velocities the temperature at the moment of appearance is practically the same, and therefore this formula makes it easy to find \(H^*\) for the heights at which meteors appear. The dependence of the height of appearance on velocity contains no constants characterizing the properties of the material of the meteoric body:

\[ \Delta H_1=H_1(v_1)-H_1(v_2)=\frac{5}{2}H^*\ln\frac{v_1}{v_2}. \]

Observational data show that for meteors with \(v=60\text{–}70\ \mathrm{km/sec}\) the height of appearance is \(H_1\approx125\ \mathrm{km}\), and for meteors with \(v=30\ \mathrm{km/sec}\), \(H_1\approx100\ \mathrm{km}\). It follows from this that at an altitude of \(100\text{–}120\ \mathrm{km}\), \(H^*\approx10\text{–}12\ \mathrm{km}\). This value of \(H^*\) corresponds to a mixed atmosphere and can in no way be reconciled with the hypothesis of a hydrogen or helium composition of the stratosphere. It should be emphasized that this result is obtained in all physical theories of meteors, provided only that one starts from correct data on the heights of appearance and their dependence on velocity. Different theories give only small deviations in the numerical values of \(H^*\).

Knowing \(H^*\) and assigning numerical values to the coefficients of thermal conductivity and thermal diffusivity, one can find the density at the height of appearance of meteors of different velocities. The following results are obtained:

The densities cited refer to the height of appearance of stony non-rotating meteoric bodies with a flat frontal surface. For the other limiting case—for a rotating spherical meteoric body—they are 4 times greater.

The determination of \(H^*\) and \(\rho\) from the formulas given above is valuable because it is based on heights of appearance obtained from visual observations and therefore refers to maximum heights.

there, accessible to meteor methods of investigation. In addition, the height of appearance does not depend on the mass of meteoric bodies, whereas all other meteor methods for determining the density of the atmosphere require finding the mass.

Whipple[^8] applied the physical theory developed by him for processing photographic observations of meteors (see below on this) to visual determinations of heights obtained during the Arizona meteor expedition. From the mean heights of appearance, disappearance, and maximum brightness for the Leonids, Perseids, and four groups of meteors of different velocities, he found the atmospheric densities at these heights. The results proved to be in satisfactory agreement with those obtained from meteor photographs.

Before proceeding to the presentation of the results obtained on the basis of meteor photographs, it is necessary to dwell on those data concerning processes in the stratosphere which were obtained from visual determinations of heights. Lindemann and Dobson already noted that, according to Denning’s catalogue data, the principal maximum of the heights of disappearance is located in summer at an altitude of 85 km, and in winter at an altitude of 75 km. This difference could be caused by contraction and expansion of the atmosphere due to temperature changes. In 1936 Epic[^9] found an annual variation in meteor heights from 3½ thousand Arizona observations. The amplitude came out to be \(3.7 \pm 0.7\) km, with a maximum in autumn and a minimum in spring. (Epic analyzed the height of the midpoint of the meteor’s path, and his data refer to an altitude of about 85 km.) In 1940 McIntosh[^10] confirmed the presence of an annual variation on the material of all height catalogues that he was able to obtain (Denning, Herschel, King, Hoffmeister, and others). McIntosh’s amplitude came out substantially greater than Epic’s, and it increases as the heights decrease—from the heights of appearance to the heights of the midpoint of the path and then to the heights of disappearance. However, while Epic, according to his custom, introduces an innumerable number of empirical corrections, which partly improve and partly worsen the material, McIntosh simply averages the observed heights, limiting himself only to excluding meteors of the principal showers. Therefore the results of both investigators do not inspire full confidence. In 1944 Porter, analyzing heights found by him as a result of a thorough reprocessing of English observations, came to the conclusion that changes in heights are accompanied by changes in mean geocentric velocities and can be explained by them. However, the presence of oscillations of the stratosphere associated with temperature changes is confirmed by the densities obtained by Whipple from photographic observations.

Possibly, there exist oscillations of the stratosphere associated with the diurnal course of temperature. According to the data of the Arizona expedition[^9], the mean heights of the midpoint of the path decrease during the night by \(300 \pm 90\) m per hour. It must, however, be noted that in detecting

of this nocturnal effect, Öpik introduced a correction for the diurnal change in the mean velocity of meteors, which has a maximum in the morning. At the same time, the same course of heights was obtained by Hey and Stewart^11 from radar observations without introducing any corrections. The introduction of corrections into radar observations may lead to the complete disappearance of the diurnal course of heights.

Owing to the inaccuracy in plotting on a map the visible path of a meteor, the heights obtained from visual observations may be erroneous by many kilometers and even tens of kilometers. In addition to observational errors, the obtained heights were also affected by shortcomings in the methods of reduction that had been used until quite recently. Methods of reduction that do not introduce additional errors began to be applied only 10–15 years ago, and priority in this matter belongs to Soviet meteor researchers and, first of all, to V. V. Fedynsky.

Still less accurate are determinations of the velocity of meteors—they are distorted not only by errors in determining the true path of the meteor in the atmosphere, but also by errors in estimating its duration. Only a few years ago it was possible to become convinced that all meteoric bodies (perhaps with the exception of an imperceptibly small fraction) are members of the solar system and, consequently, their heliocentric velocity does not exceed 42 km/sec. This means that the velocities with which meteoric bodies enter the Earth’s atmosphere lie within the limits from 11 to 72 km/sec. Establishing the absence of velocities exceeding 72 km/sec contributed to the recognition of the erroneousness of those enormous heights of meteor appearance—150–200 and even 250 km—which had been obtained as a result of errors in visual observations. The fastest meteoric bodies, flying toward the Earth, appear at a height of 120–125 km, while the slowest, overtaking bodies appear at a height of 80–90 km (and perhaps even lower). The height of disappearance depends not only on the velocity, but also on the mass of the meteoric body, and for large slow bodies it may be arbitrarily small.

Because of the large random errors that burden determinations of meteor velocities based on visual observations, the following two recommendations must be made:

1) in all studies of processes occurring during the motion of meteors in the atmosphere, the basis should be meteors of streams—those streams whose heliocentric velocities are known either from the orbit of the comet that produced them, or from photographic determinations;

2) when using visual observations of the remaining meteors, they should be assigned a velocity computed from the assumption that their heliocentric velocity is close to parabolic. Considerable changes in the dimensions of the orbit correspond to only small changes in heliocentric velocity, and therefore the velocities in the atmosphere, compu-

changed in this way will on average be much more accurate than the observed velocities.

However, in visual observations, even in the best case of meteors of streams, we may know the velocities of entry into the atmosphere, but we know nothing about the deceleration of the meteor during its flight. At the same time, deceleration can be found from photographic observations. Moreover, photographs make it possible to obtain the curve of the change of brightness along the path. Thus, photographic observations give not only more accurate, but also more complete information about the meteor than visual observations.

To obtain accurate heights and the direction of the meteor’s path, it is necessary to photograph it simultaneously from two stations. To determine velocity and deceleration, it is necessary that, at least at one station, an obturator rotate in front of the camera lens, interrupting the exposure 10–20 times per second. In this case the trace of the meteor on the plate is broken into separate dashes, the measurement of which precisely makes it possible to find the law of motion.

For the first time such a method was applied by Elkin at the end of the last and the beginning of the present century. At the beginning of the 1920s, successful photographs were obtained by Lindemann and Dobson. In 1932, photography with an obturator was organized in Moscow. The work was carried out not by professional astronomers, but by amateurs—members of the MOVAGO Observers’ Collective under the direction of V. V. Fedynsky. In the very first year, on August 14, 1932, successful photographs of a bright meteor were obtained. They were reduced by Fedynsky and Stanyukovich, and the results were published in 1935.^12 The determination of heights was carried out with the aid of a new, simple and convenient method developed by them. The absence at that time of a satisfactory physical theory of meteors led to the fact that conclusions about the structure of the stratosphere were made on the basis of extremely simplified assumptions. Nevertheless, the temperature distribution found in the stratosphere is in qualitative agreement with modern data.

Successful photography with an obturator continued in the Observers’ Collective in subsequent years as well. It was even possible to obtain a triple photograph including the spectrogram of a meteor. However, the meteor of 1932 remained the only one whose reduction was brought to publication. Likewise, up to the present time the numerous photographs obtained before the war at the Stalinabad Observatory, with the aid of a special meteor patrol consisting of two dozen cameras, have not been published.

Meanwhile, beginning in 1936, at the Harvard Observatory, under Whipple’s direction, systematic photography of meteors began with the aid of two (only two!) cameras located at the ends of a base 38 km long. The results of the reduction of photographs of the first 6 meteors, including from the standpoint of densities in the stratosphere, were published in 1938,^6 and in 1943 there was published

a special article was published, devoted to the structure of the stratosphere and based on the processing of observations of 18 meteors^8. As has already been noted above, Whipple also brings into consideration the results of the visual observations of the Arizona expedition. The observational data refer to the altitude interval from 45 to 120 km, but the majority are concentrated in the interval 60–100 km.

The theory used by Whipple makes it possible to find the density of the atmosphere at four points of the meteor’s path—at the beginning, at the end, at the point of maximum brilliance, and at some point near the middle of the path for which the deceleration has been found (on long paths there may be several such points).

The density is determined most simply and most reliably at the point for which the deceleration is known. From the equation of deceleration

\[ m\frac{dv}{dt}=-C_1 S\rho v^2 \]

(\(S\) is the cross-sectional area of the meteoric body, \(C_1\) is a coefficient of proportionality) the mass of the meteoric body \(m\) is found. Since the kinetic energy of the meteoric body is hundreds of times greater than the energy required for its evaporation, and it is expended mainly in collisions of evaporated molecules of the meteoric body with air molecules, the assumption is made that the luminous intensity of the meteor \(I\) is proportional to the loss of energy per unit time

\[ I=-\frac12 \frac{dm}{dt}v^2\tau \]

(\(\tau\) is a coefficient of proportionality). Substituting from the equation of evaporation

\[ \frac{dm}{dt}=-C_2\rho v^3, \]

we obtain

\[ I=C_3\tau m^{2/3}\rho v^5. \]

(Here it is assumed that, while decreasing, the meteoric body does not change its shape, but remains similar to itself, i.e. \(S=Am^{2/3}\).) The luminous intensity depends sharply on velocity, as \(v^5\). For bright meteors the dependence is still stronger, since, according to Öpik, for them

\[ \tau=\tau_0 v. \]

Substituting the mass from the equation of deceleration, we find

\[ \rho=Kv^{-10/3}\left(-\frac{dv}{dt}\right)^{2/3}I^{1/3}. \]

In this formula (as also in all other formulas for density) the luminous intensity enters to the power \(1/3\), which increases the accuracy of the results.

Density at the point of maximum brightness is found by logarithmic differentiation of the formula for luminous intensity. The resulting derivatives $\dfrac{dm}{dt}$ and $\dfrac{dv}{dt}$ are substituted from the equations of evaporation and deceleration, while the term with $\dfrac{d\rho}{dt}$ is transformed as follows:

\[ \frac{1}{\rho}\frac{d\rho}{dt} = - v \frac{1}{\rho}\frac{d\rho}{dH}\cos z = \frac{v\cos z}{H^*}. \]

Hence it is clear that, in this method, it is necessary to know $H^*$ in advance, i.e., to make in advance certain assumptions about the structure of the atmosphere.

Since the brightness curve of individual meteors may differ appreciably from the theoretical curve, the densities at the point of maximum brightness show a greater dispersion than the densities found from deceleration.

The density at the beginning of the path is determined for that point at which the meteor’s brightness has reached a measurable value, but at the same time it may still be assumed that its mass is equal to the initial mass. The latter is found from the formula

\[ E = \int I\,dt = \tau \frac{m_0 v_0^2}{2}, \]

in which the deceleration of the meteor is neglected. Substituting from this the mass into the formula for luminous intensity, we find

\[ \rho = K_1 I E^{-2/3} v_0^{-4}. \]

Despite possible systematic errors, the use of this formula is valuable in that it makes it possible to advance to the maximum heights accessible to the photographic method. The use of Levin’s formula for the temperature at the moment of appearance in photographic observations is impossible, since in the overwhelming majority of cases the latter give not the true height of appearance, but the height at which the meteor’s brightness became sufficient to produce a trail on the plate.

The density at the end of the path is found from the following considerations: from the total luminous energy, by the method described above, the initial mass is found. Next, deceleration is neglected and the total mass of the atmosphere along the entire path above the point of disappearance is found. To pass to the density at the end of the path, it is again necessary to assume some value of $H^*$.

The coefficients of proportionality entering the deceleration equation and the evaporation equation are regarded by Whipple as practically constant, and he finds their numerical value on the basis of theoretical and experimental data on the motion of bodies with supersonic velocities (Taylor and Maccoll, 1933–1934). However, these data refer to motion in a fairly dense gaseous medium, when shock waves are formed. Meanwhile, in ordinary

of meteors the phenomenon takes place at those heights where the mean free path of the molecules \(\lambda\) is comparable with the transverse dimension of the meteor and where, consequently, there occurs a transition from individual collisions of the meteor body with separate air molecules to the formation of a shock wave. Bright meteors appearing on photographs have a transverse dimension of the order of \(1\ \mathrm{cm}\). At the same time, at an air density \(\rho = 10^{-10}\ \mathrm{g/cm^3}\) \((H \simeq 120\text{–}125\ \mathrm{km})\), \(\lambda = 100\ \mathrm{cm}\); at \(\rho = 10^{-9}\) \((H \simeq 110\ \mathrm{km})\), \(\lambda = 10\ \mathrm{cm}\); at \(\rho = 10^{-8}\) \((H \simeq 95\ \mathrm{km})\), \(\lambda = 1\ \mathrm{cm}\); and at \(\rho = 10^{-7}\) \((H \simeq 80\ \mathrm{km})\), \(\lambda = 0.1\ \mathrm{cm}\). Although there is as yet no shock wave, the meteoric body is enveloped by a progressively densifying cloud of evaporated molecules, and the possibility of describing the phenomenon by those simplified formulas used by Whipple requires careful verification. For the present, the only justification of Whipple’s method can be that with its aid he obtains satisfactory results—satisfactory in the sense that the densities obtained by all four methods do not show large systematic differences.

The most reliable density values, relating to the point with known deceleration, revealed a correlation between density deviations and the near-surface temperature \(\left(\frac{2}{3}\right.\) of the mean temperature for the preceding day \(+\frac{1}{3}\) of the mean temperature for the following day). An increase of temperature by \(1^\circ\mathrm{C}\) corresponds to an upward displacement of the layers under consideration by \(0.15\ \mathrm{km}\). Taking into account small systematic corrections (differences between the densities obtained by different methods) led to the following most reliable density values:

These values refer to a near-surface temperature of \(+28^\circ\mathrm{C}\) \((301^\circ\mathrm{K})\). The scatter of the points is shown in Fig. 1. The two lower averaged points practically lie on the density curve corresponding to the temperature-distribution curve shown in Fig. 2, which was taken as the basis for the calculations of \(H^*\) (in those cases where \(H^*\) had to be known in advance). Above \(78\ \mathrm{km}\) a linear course of \(\lg \rho\) is obtained, corresponding to \(T = 250^\circ\mathrm{K}\). It is expressed by the formula

\[ \lg \rho = -2.45 - 0.0592H\ (\mathrm{km}). \]

Above \(100\ \mathrm{km}\) the density data are unreliable.

The density curve found from the meteor data corresponds to the following course of temperature: it increases gently with height and at about \(62\ \mathrm{km}\) reaches a maximum of \(375^\circ\mathrm{K}\). Above \(68\ \mathrm{km}\) there is a rapid decrease, and at about \(78\ \mathrm{km}\) a temperature of \(250^\circ\mathrm{K}\) is reached.

Only for the point with known deceleration do the residual deviations of density show a strong positive correlation with the near-

with the ground temperature. The remaining data give an unclear picture (Fig. 3), and for the beginning of the path even a negative correlation is obtained. A weighted solution over all points gives a rise of \(0.19 \pm 0.04\) km for an increase in temperature of \(1^\circ\). This gives, for Boston, an annual amplitude of heights of \(5.3 \pm 0.1\) km, which agrees well

Fig. 1.

with the Atkinson value \(3.7 \pm 0.7\) km for Flagstaff (Arizona).

Although the probable errors of a single determination of \(\lg \rho\), after allowance for systematic differences, become sufficiently small and amount to from \(\pm 0.1\) to \(\pm 0.2\) (for the different methods), nevertheless the interpretation of the results is not reliable and unambiguous. The density values for points with known deceleration, which are the most reliable, fit excellently the solution assuming an isothermal stratosphere plus a dependence on the near-ground temperature. The temperature for heights of 60–110 km is found to be \(256^\circ\) K, and the annual amplitude is 10 km.

This result compels Whipple to write the following: “The data speak in favor of the existence of a high-temperature zone at about 60 km or a little higher, but they do not permit this question to be settled firmly.” In addition, Whipple notes that “the observations give no indication of the existence of a narrow layer with low temperature at an altitude of about 82 km.”

Some additional information on these questions can be extracted from an analysis of the curves of the distribution of brightness along the path of a meteor.

Fig. 2.

Legend:

— adopted curve
V — Vegard
-- Degenes (curve B)
··· Degenes (curve E)

Axes:

Temperature in degrees abs.
\(H\), km

Fig. 3.

Legend:

● braking
○ end of trajectory
× maximum brightness
+ meteor ignition

Axes:

\(\Delta \log \rho\)
\(\Delta T (^\circ r \times 10^6)\)

If deceleration is neglected, then in an isothermal atmosphere all light curves must be exactly the same. Observations indicate, however, that with increasing velocity the maximum shifts toward the end of the path. It turns out that this shift can be explained by the presence of a hot layer at an altitude of about 60 km and, consequently, is an argument in favor of its existence. As for the narrow temperature minimum near 82 km, the data on the shift of the brightness maximum can be better explained on the assumption that it is absent.

It would be surprising if meteor data alone made it possible to construct a complete picture of the structure of the stratosphere at altitudes of 70–120 km—accurate observations are too few, and the physical theory by means of which they are processed is still too insufficiently developed. In both respects, further progress is undoubtedly possible. Yet already now the meteor data on stratospheric density given above are probably no less accurate than the data obtained for these altitudes by various other methods. They must therefore be taken into account when bringing together all our present knowledge of air densities in the stratosphere.

REFERENCES CITED

1. F. A. Lindemann and G. M. B. Dobson, Proc. Roy. Soc., London, A102, 411, 1923; A103, 339, 1923; G. M. B. Dobson, Quart. J. Roy. Met. Soc., 49, No. 207, 1923; F. A. Lindemann, Nature, 118, 195, 1926 (see also N. Obolensky, Meteorological Bulletin, 37, 173, 1927; N. M. Shtaude, Proceedings of the All-Union Conference on the Study of the Stratosphere, p. 481, 1935).
2. C. M. Sparrow, Astrophys. Journ., 63, 90, 1926.
3. N. V. Maris, Terr. Magn. Atm. Electr., 31, 303, 1929 (see also A. B. Severny, Astr. Journ., 11, 613, 1934).
4. E. Öpik, Acta et Comment. Univ. Tartu, A26, 1, 1933, Harv. Repr., No. 100; Publ. Obs. Astr. Tartu, 29, No. 5, 1937.
5. J. Hoppe, Astr. Nachr., 262, 169, 1937.
6. F. L. Whipple, Proc. Amer. Phil. Soc., 79, 499, 1938; Harv. Repr., No. 156.
7. B. Yu. Levin, DAN, 25, 372, 1939; Astr. Journ., 17, 12, 1940; 18, 331, 1941.
8. F. L. Whipple, Rev. Mod. Phys., 15, 246, 1943; Harv Repr. Ser. II, No. 6.
9. E. Öpik, Proc. Nat. Acad. Sci., 22, 525, 1936; Harv. Repr. No. 127; Harv. Ann., 105, 549, 1938 (see also I. S. Astapovich, World Studies, 26, 217, 1937).
10. R. A. McIntosh, Monthly Not. R. A. S., 100, 510, 1940.
11. J. S. Hey and G. S. Stewart, Proc. Phys. Soc., 59, 858, 1947.
12. V. V. Fedynsky and K. P. Stanyukovich, Astr. Journ., 12, 440, 1935.