ON THE DIURNAL AND SEASONAL VARIATION OF HOURLY METEOR COUNTS

PART II

R. G. LAZAREV

(Presented by the Department of Astronomy of Tomsk State University)

Introduction

In papers [1] and [2], a study was carried out of the diurnal and seasonal variation of hourly meteor counts on the basis of radar observations in Tomsk, and a comparison was made with observations at other stations.

The principal component of both the diurnal and the seasonal variation is the result of kinematic causes—the motion of the Earth around the Sun and its rotation about its axis. In the case of a uniform distribution of true radiants over the celestial sphere, for the diurnal variation we would have a maximum of hourly counts at \(6^{\mathrm h}\) and a minimum at \(18^{\mathrm h}\) local mean time, and for the seasonal variation a maximum in the autumn months and a minimum in the spring. These phenomena, to a first and fairly good approximation, are in fact observed both in the northern and in the southern hemispheres of the Earth [2, 3, 4]. The clearest results are obtained in observations with a nondirectional antenna, irradiating the visible hemisphere more or less uniformly, at wavelengths of \(8\text{–}10\) m, when the hourly counts reach values of 50–100 and higher, and provided that the observations are averaged over a sufficiently long interval of time (from a month to a year or more) [2]^1).

In reality, as is known, the true meteor radiants are distributed nonuniformly over the celestial sphere. There is a concentration toward the plane of the ecliptic [8], which does not substantially change the basic kinematic component mentioned above.

Let us now turn to the distribution of visible radiants along the ecliptic. They are grouped on the ecliptic near the apex, the Sun, and the antisun (in the annual distribution) [8, 9]. This distribution is asymmetric with respect to the apex [1, 2, 10]. The radiants are concentrated toward the antisun to a greater extent than toward the Sun (the phenomenon of asymmetry [1, 2], and also below in the present article).

^1) In paper [5] it is stated that in the southern hemisphere the character of the seasonal, and also of the diurnal, variation is different from that in the northern hemisphere, and that kinematic effects are altogether absent. References are made to the first, small-number observations of A. Weiss, C. Elliott, and C. Keay. However, subsequent observations by the same authors show that this is not so [3, 6, 7].

Let us note that knowledge of the annual-mean diurnal variation is insufficient for characterizing the distribution of meteoric matter along the Earth’s orbit. The seasonal distribution is more complete. In [2] an index of meteor activity, $i(t)$, was obtained, characterizing the distribution of meteor orbits along the Earth’s orbit from month to month. The maximum of the function $i(t)$ occurs in November–December, and the minimum in May–June. During the winter months the maximum in the diurnal monthly-mean histograms shifts to $4^{\mathrm h}$, and in the summer months to $8^{\mathrm h}$ local mean time. In the annual-mean histograms the overall maximum tends to shift from $6^{\mathrm h}$ to $4^{\mathrm h}$. This reflects the predominant influence of the winter months.

Let us also note that all the results refer to the variation in the numbers of meteors of absolute magnitude $3^m \div 7^m$ [2, 6, 7]. For example, for the Tomsk radar station, from the known formula

\[ \alpha_{\min}=\frac{6.2\cdot 10^{13}}{G}\sqrt{\frac{\varepsilon}{P}}\left(\frac{R}{\lambda}\right)^{3/2}, \]

where $G$ is the antenna directivity coefficient,
$P$ is the radiated power,
$\varepsilon$ is the minimum power of the received signal,
$R$ is the mean distance to the reflecting points,
$\lambda$ is the effective wavelength,

we find the minimum linear electron density

\[ \alpha_{\min}=8.75\cdot 10^{10}\ \text{el/cm} \]

and then the maximum absolute magnitude

\[ M=35-2.5\lg \alpha_{\min}=7^m.6. \]

Harmonic analysis of the variation curves

For a more detailed study of the variation in the hourly numbers of meteors, a harmonic analysis of several variation curves was carried out. The annual-mean curves of the diurnal variation of the hourly numbers of meteor echoes were used, based on observations in Tomsk [2], New Zealand (Christchurch) [3], and Canada [11]. In the last case, oblique sounding was used.

For each curve an expansion of the following form was obtained:

\[ N(t)=\frac{a_0}{2}+\sum_{k=1}^{3}\left[a_k\cos\frac{k\pi}{12}(t-t_0)+ \right. \]

\[ \left. +b_k\sin\frac{k\pi}{12}(t-t_0)\right], \]

where $t$ is local mean time.

1) Tomsk (Fig. 1a):

\[ N(t)=193.0+88.2\cos\frac{\pi}{12}(t-5.5)-41.7\sin\frac{\pi}{12}(t-5.5)+ \]

\[ +1.4\cos\frac{\pi}{6}(t-5.5)-14.7\sin\frac{\pi}{6}(t-5.5)+ \]

\[ +10.6\cos\frac{\pi}{4}(t-5.5)-2.2\sin\frac{\pi}{4}(t-5.5), \]

\[ 5^{\mathrm h}.5 \leqslant t \leqslant 29^{\mathrm h}.5. \]

The standard deviation is \(\sigma = 9.98\).

2) New Zealand, Christchurch (Fig. 1 б):

\[ N(t)=82.1+47.1\cos\frac{\pi}{12}(t-6.5)-12.4\sin\frac{\pi}{12}(t-6.5)- \]

\[ {}-8.7\cos\frac{\pi}{6}(t-6.5)-7.9\sin\frac{\pi}{6}(t-6.5)- \]

\[ {}-0.6\cos\frac{\pi}{4}(t-6.5)+2.1\sin\frac{\pi}{4}(t-6.5). \]

\[ 6^h.5 \leqslant t \leqslant 30^h.5,\qquad \sigma=6.4. \]

3) Canada (Fig. 1 в):

\[ N(t)=97.9+40.3\cos\frac{\pi}{12}(t-10)-14.8\sin\frac{\pi}{12}(t-10)+ \]

\[ {}+6.0\cos\frac{\pi}{6}(t-10)-5.8\sin\frac{\pi}{6}(t-10)- \]

\[ {}-4.8\cos\frac{\pi}{4}(t-10)-6.7\sin\frac{\pi}{4}(t-10). \]

\[ 10^h \leqslant t \leqslant 34^h,\qquad \sigma=6.4. \]

The observations in Tomsk and Christchurch satisfy the following conditions:

1. A weakly directional antenna is used.
2. The observations continue for no less than a year.
3. The parameters of the apparatus are constant throughout the entire period of observations.
4. The hourly numbers are sufficiently large (\(\overline{N}=213\) for Tomsk, 1957; \(\overline{N}=82\) for Christchurch, 1960–1961).

The minimum of each curve corresponds to the moment \(\sim t_0+12^h\). The variation curves are analogous for both the northern and southern hemispheres of the Earth. It is of interest that, also in the case of scattering forward, the character of the variation curve does not change.

The meaning of the first term of the expansion is evident. The second term, containing \(\cos \frac{\pi}{12}(t-t_0)\), represents, chiefly, the magnitude of the kinematic effect. This is the principal component of the diurnal variation¹. We shall call the term \(a_1\cos \frac{\pi}{12}(t-t_0)\) the “quantity of first order.”

The term \(b_1\sin \frac{\pi}{12}(t-t_0)\) characterizes the asymmetry. The coefficient \(b_1\) is negative for any curve, and in absolute value exceeds the coefficients in all subsequent terms of the expansion. This is the “quantity of second order.”

The addend \(b_2\sin \frac{\pi}{6}(t-t_0)\) also constitutes a certain (small) fraction of the asymmetric part of the radiants. This also includes part of the signals observed in the daytime. In [11] it is noted—

Fig. 1. Annual-mean curves of the diurnal variation of hourly numbers of meteor echoes:
\(N\)—hourly number, \(t\)—local mean time

a) Tomsk, July–December 1957.
b) Christchurch, New Zealand, Feb. 1960–Jan. 1961.
c) Canada, Jan.–Dec. 1953. Forward-scatter case.

an increase in signals in the daytime. It also mentions the presence of a semidiurnal component, which is considered doubtful. However, as a result of analyzing a series of curves, the value of the semidiurnal component is found, as is evident, quite definitely.

All other factors have a significantly smaller influence on the distribution of visible radiants. These include the dispersion of meteor masses and velocities and other physical characteristics, the different sensitivity of the apparatus (and, in general, the instrumental conditions of observation), the geographical location of the station, and, finally, fluctuations in the number of meteors caused by various random causes. All these quantities must be represented by terms of higher orders of smallness.

Schematically, the result obtained may be represented in the form of the equality

\[ N(t)=N_1(t)+N_2(t)+N_3(t)+\varepsilon(t), \]

where \(N(t)\) is the curve of diurnal variation,
\(N_1(t)\) is the magnitude of the kinematic effect.

\[ N_1(t)=\frac{a_0}{2}+a_1\cos\frac{\pi}{12}(t-t_0)+a_2\cos\frac{\pi}{6}(t-t_0). \]

Let us note that \(N_1(t)\) also contains the symmetric component of the solar and antisolar sources\(^1\).

\(N_2(t)\) is the magnitude of the asymmetry.

\[ N_2(t)=b_1\sin\frac{\pi}{12}(t-t_0)+b_2\sin\frac{\pi}{6}(t-t_0), \]

\(N_3 t\) is the influence of physical factors and instrumental conditions.

\(\varepsilon(t)\) denotes other deviations.

The directions for further investigation should consist in determining the structure of \(N_3(t)\) and \(\varepsilon(t)\), i.e., quantities of “higher orders of smallness.”

Index of Meteor Activity (i.m.a.)

The concept of the index of meteor activity \(i(t)\) was introduced in papers [1, 2], where its physical justification was given. We shall now define the index \(i(t)\) purely formally.

Suppose we have a histogram of the diurnal variation of hourly meteor counts (Fig. 1), obtained, generally speaking, by any method (visually, photographically, or with the aid of radar). Let \(t\) be the local time, and \(N\) the hourly number of meteors. Arrange the histogram so that the minimum\(^2\) of the hourly counts falls at its center (usually it corresponds to \(t=18^h\)). Denote by \(N_i\) the hourly number of meteors for the \(i\)-th hour (or the number of meteors from the \(i\)-th to the \(i+1\)-st hour), where the index \(i\) takes the values lying to the right of the minimum (corresponding to the values \(t=18^h,19^h,\ldots,5^h\)).

Similarly, \(N_k\) is the hourly number of meteors for the points lying to the left of the minimum (the index \(k\) takes the values corresponding to \(t=17^h,16^h,\ldots,6^h\)).

Let \(N_{\mathrm{right}}=\sum N_i\) be the sum of all \(N_i\), and \(N_{\mathrm{left}}=\sum N_k\) be the sum of all \(N_k\).

\(^1\) The terms “solar,” “apex,” and “antisolar sources” (Sun, apex and antisun sources) are used in paper [6]. We shall follow this terminology.
\(^2\) Or the principal minimum, if there are several.

Then the index of meteor activity (m. a.) can be introduced in one of the following ways:

\[ \text{1) } i(t)=N_{\mathrm{pr}}/N_{\mathrm{lev}}, \]

\[ \text{2) } i'(t)=\lg i(t)=\lg N_{\mathrm{pr}}/N_{\mathrm{lev}}. \]

In article [2] the first definition was used. In the present work we shall use definition 2) as the best one (see Appendix 2).

To construct histograms of the diurnal variation, monthly mean values of the hourly numbers are often taken. Thus the index \(\lg i(t)\) can be calculated for each month. Plotting the values of \(\lg i\) from month to month, we obtain its dependence on \(t\) (where \(t=\) I, II,...XII month). In Fig. 2 the dependence of \(\lg i\) on \(t\) is given for the case of a symmetrical distribution of visible radiants relative to the apex. The curve is symmetrical with respect to the point \(t=\mathrm{IX}\). To construct the index \(\lg i\) in this case we used the theoretical curves given in [9, 12].

Fig. 2. Index of meteor activity in the case of a symmetrical distribution of visible radiants.

Asymmetry of the variation curve

In Fig. 3 a curve of \(\lg i\) is constructed from the data of radar observations in Tomsk (a detailed analysis of the index \(i(t)\) is given in [2]). The positive branches of the curve considerably exceed its negative part. The maxima of \(\lg i\) are observed in November–December. The minima in May–June are much smaller in absolute value. This is a consequence of the asymmetric distribution of visible radiants relative to the apex. The mean annual number of meteors having radiants near the antisolar point is greater than the number of meteors whose visible radiants are grouped near the Sun. In other words, the mean-

the annual power of the antisolar source is greater than that of the solar one¹. This phenomenon was called an asymmetry in [1, 2]. Let us investigate the asymmetry for the southern hemisphere of the Earth, using the concept of the meteor-activity index and the data of [6].

Fig. 3. Meteor-activity index from observations in Tomsk, 1957–1960.

For brevity, we shall denote the integral power of the solar, apex, and antisolar sources, respectively, by \(F_1\), \(F_2\), and \(F_3\).

Meteor observations in Antarctica at Mawson Station

C. S. Nilsson and A. A. Weiss published in [6] the results of radar observations of meteors at Mawson Station (\(68^\circ\) S, \(67^\circ\) E). For the observations a system of half-wave dipoles was used, installed in a single row above a metal mesh in a direction from east to west. The axes of the dipoles were oriented in the same direction.

Registration of radio echoes was carried out over several months in 1957–1960, so that, in total, observations were obtained for all 12 months of the year. The authors constructed curves of the diurnal variation for each month. In addition, theoretical curves of the diurnal variation were calculated, likewise for each month. The hypothesis of three sources (Sun, apex, antisun) was used, and it was assumed that the solar and antisolar sources were identical in power (\(F_1 = F_3\)). The authors point out a noticeable difference between the theoretical curves obtained and the observed ones. The observed minimum during August–December is shifted from \(17^h\) to \(20^h\) (the shift is even larger), whereas the theoretical minimum is shifted in the opposite direction. Nilsson and Weiss explain this discrepancy by the influence of ionospheric interference or by the general unreliability of the observational data. Let us note that in the first half of the year as well, the shift of the observed minimum is opposite to that calculated theoretically, which the authors do not mention².

This displacement of the observed minimum is in fact easy to explain if it is assumed that the sources \(F_1\) and \(F_3\) are not equal in strength, the strength of the antisolar source being greater than that of the solar one, \(F_3 > F_1\), i.e., that an asymmetry is present. The asymmetry is indicated, first of all, by the course of the change in the index of meteor activity. The index \(\lg i(t)\) for the observed and theoretical curves of the diurnal variation for Mawson station is shown in Figs. 4a and 4b. The curve in Fig. 4a is entirely analogous to the index \(\lg i(t)\) for Tomsk (Fig. 3) and indicates the phenomenon of asymmetry for the southern hemisphere¹, of course with a shift by half a year. At the same time, the curve in Fig. 4b has an entirely different character, showing that the hypothesis of symmetry of the sources \(F_1\) and \(F_3\) is unacceptable. The theoretical curves [6] give only a first approximation for the diurnal variation, a quantity of the first order. Taking the asymmetry effect into account makes it possible to obtain a second approximation².

Fig. 4a. Index of meteor activity for Mawson station (Antarctica), obtained from the observed curves of the diurnal variation, 1957–1960.
Fig. 4b. Index of meteor activity for Mawson station, obtained from theoretical curves.

We recalculated the strength of the source \(F_3\) under the assumption that the strengths \(F_1\) and \(F_2\) remain the same as those adopted in [6]. The strength of the sources \(F_1\) and \(F_2\) is still specified by the integral [6]

\[ F(z)=\int_0^{\pi} D(\varepsilon)\sin \varepsilon\, d\varepsilon \int_0^{2\pi} f(A)\, dA, \]

where \(z\) is the zenith distance of the center of the source,
\(F\) is the strength of the source,
\(\varepsilon\) is the elongation of the radiant from the center of the source,
\(D(\varepsilon)\) is the density of radiants,

\(\gamma\) — zenith distance of the radiant,

\(f(\gamma)\) — a function of the instrumental parameters,

\(A\) — the angle at the center of the source between the direction to the zenith and the given radiant \(R\) (Fig. 5).

To compute the strength of the anti-solar source we solved the problem inverse to the one considered by Nilsson and Weiss. The observed curves of the diurnal variation were taken as the initial data. The strength

Fig. 5.

of the anti-solar source \(F_3(z)\) was computed as the difference between the observed values of the hourly numbers and the sum \(F_1(z) + F_2(z)\) for the Sun and the apex.

The curves for all three sources are given in Fig. 6; \(F_1(z)\) and \(F_2(z)\) are taken from [6], and \(F_3(z)\) is the corrected anti-solar source. The curve \(F_3(z)\) gives the best agreement with the observations.

Fig. 6. Strengths of meteor sources as a function of zenith distance \(z\). \(F_1(z)\) — solar, \(F_2(z)\) — apex,
\(F_3(z)\) — anti-solar source.

The power of the antisolar source exceeds, at its maximum, the power of the solar source by

\[ \frac{6.9}{5.0}=1.38 \]

times. The curve \(F_3(z)\) has a steeper inclination to the \(z\)-axis, i.e., the antisolar source is denser than the solar one (there is reason to believe that it practically ceases to act already at \(z=90^\circ\), i.e., the curve is directed still more steeply, as shown by the dashed line).

Thus, the presence of asymmetry is confirmed by observations in the southern hemisphere of the Earth as well (which, incidentally, is quite natural).

Let us note that, according to observations in Tomsk and Christchurch, the apex source is more powerful than for the Mawson station (\(F_2>F_1\) and even \(F_2>F_3\)). It is possible that here the dependence of the diurnal variation on geographic latitude is simply being manifested.

Observations of meteor counts in Kharkov in 1958–1960

Radar observations of meteors in Kharkov were carried out at the wavelength \(\lambda=8\) m. The following apparatus was used [13].

1. From December 1957 to August 1958, a five-element “wave-channel” antenna was used for transmitting signals, and a half-wave vibrator for reception.

2. From September 1958 to March 1960, transmission was with an eight-element antenna, and reception on a half-wave vibrator.

3. From April to December 1960, reception and transmission were carried out using a five-element “wave-channel” antenna.

Fig. 7. Meteor activity index for Kharkov (1958–1960).

The index \(\lg i(t)\) (Fig. 7) was compiled for Kharkov with allowance for the orientation of the transmitting antenna [14]. Although the reception and transmission conditions changed with time, and a weakly directional antenna was used only in part, the index \(\lg i(t)\) for Kharkov is in general agreement with the Tomsk index (Figs. 3 and 7). The table of asymmetry based on the Kharkov observations is given in Appendix 1.

Observations at Christchurch (New Zealand). 43°.6 S lat.,
February 1960—January 1961

The index $\lg i$ for Christchurch was obtained from the data published in [15]. The maximum value of $\lg i$ is reached in the month of July (Fig. 8).

Fig. 8. Index of meteor activity for Christchurch (New Zealand), Feb. 1960—Jan. 1961.

Latitudinal variation of m. a.

The latitudinal variation of m. a. is characterized by the following table:

Here $i_{\min}$ is the mean value of $\min i$ over the corresponding years of observations, and $i_{\max}$ is the mean value of $\max i$.

It is seen that $i_{\max}$ and $\Delta i = i_{\max} - i_{\min}$ decrease from the equator toward the pole (although theoretically one might have expected the opposite). The ratio

$$ K=\frac{|\lg i_{\max}|}{|\lg i_{\min}|}>1, $$

indicates an asymmetric distribution of radiants (for a symmetric distribution $K = 1$). For the Northern Hemisphere $K = 2.4 \div 3.2$, and for the Southern Hemisphere $K = 4.1 \div 7.6$. It is possible that this difference is caused by daytime streams, especially numerous in June.

Conclusion

Thus, from the data for four stations (two in the Northern Hemisphere of the Earth—Tomsk, Kharkov—and two in the Southern—Christchurch, Mawson), one may conclude that m. a. is a stable characteristic, despite considerable differences in observing conditions. To reveal finer features, it is necessary to use identical (same-type) equipment and to keep its parameters constant throughout the entire observing period. The registration of meteor counts is best carried out with a nondirectional antenna.

In [13, 14] a broad belt of orbits of meteoric bodies is discussed, which the Earth crosses twice a year. The boundaries of the belt, especially taking into account the results obtained at other stations, prove to be still more diffuse than indicated in [13, 14]. M. a. characte-

...sets a more “stable”¹ feature in the distribution of meteoric bodies—the ratio of the number of “external” particles crossing the Earth’s orbit “from outside inward” to the number of “internal” particles flying in the opposite direction. This ratio shows that the number of “external” particles is greater than that of “internal” ones, which is natural, since meteoric particles arrive mainly from outside—from the zone of Jupiter and the asteroids.

APPENDIX 1

Formal Definition of Asymmetry

Let there be an annual histogram of the diurnal variation of hourly meteor counts.

Let \(t\) be local time, and \(N\) the hourly meteor count. Arrange the histogram so that the minimum of the hourly counts falls at its middle (for example, let the minimum be at \(t = 18^h\)).

Denote by \(N_i\) the hourly meteor counts for values of \(t\) lying to the right of the minimum \((t = 18^h, 19^h, \ldots, 5^h)\), and by \(N_k\) the hourly counts for values to the left of the minimum \((t = 17^h, 16^h, \ldots, 6^h)\).

Then the asymmetry will be defined as the difference \(N_{\mathrm{as}} = N_i - N_k\) (or \(\Delta N_{\mathrm{as}} = N_i - N_k\)), where \(N_i\) and \(N_k\) are the corresponding values of \(N\), located symmetrically to the right and left of the minimum.

Tables of Asymmetry of Variation Curves

On p. 15 are given tabular values of asymmetry from observations at Tomsk (USSR), Kharkov (USSR), Christchurch (New Zealand), Canada (forward scatter), and Mawson (Antarctica).

It is evident that, for different stations with rather different observing conditions, the asymmetry has approximately the same character, and its maximum falls at approximately the same hours \((24^h—2^h)\) of local mean time.

The physical meaning of the asymmetry is considered in [2]. Asymmetry means that the right-hand part of the curve is higher and steeper than the left-hand part. The right-hand part corresponds to the activity of the antisolar point, and the left-hand part to the solar point. The asymmetry is, as a rule, positive and greatest in magnitude at the moment of upper culmination of the antisolar source (\(\sim 1^h\) local mean time).

Asymmetry in Forward Scatter

It should be noted that an asymmetric distribution of radio echoes is also observed in forward scattering of signals. This is indicated by the results of an analysis of Canadian observations (Tables 5 and 6).

On the Cedar Rapids—Ottawa path, the minimum of the variation curve is shifted to \(20^h—22^h\). However, the character of the curve and the magnitude of the asymmetry remained, in general, the same as in other cases. The single negative value of the asymmetry is explained by a certain increase in the frequency of occurrence of signals in the daytime compared with nighttime (this increase in the number of radio echoes is noted in [11]). The data for another path (Greenwood—Ottawa) are entirely “classical.” Thus, the asymmetry in forward scatter follows in general the same law as in the return-son—

… in the recording. Even the orientation of the path evidently does not have a significant effect.

Asymmetry Tables

1. Tomsk (USSR)

2. Kharkov (USSR)

3. Christchurch (New Zealand)

4. Mawson (Antarctica)

5. Canada (Cedar Rapids (Iowa)—Ottawa track)

6. Canada (Greenwood—Ottawa routes)

APPENDIX 2

On the determination of the index of meteor activity

Let \(N_{\text{right}}\) be the sum of all hourly numbers to the right of the minimum, and \(N_{\text{left}}\) the sum of the hourly numbers to the left of the minimum. As an index of meteor activity one could take one of the following quantities:

\[ \begin{aligned} 1)\quad & i_1(t)=N_{\text{right}}-N_{\text{left}},\\ 2)\quad & i_2(t)=N_{\text{right}}/N_{\text{left}},\\ 3)\quad & i_3(t)=\lg i_2(t)=\lg N_{\text{right}}/N_{\text{left}}. \end{aligned} \]

However, the values of \(i_1(t)\) are substantially distorted when the sensitivity of the equipment changes from month to month. This quantity does not make it possible to compare observations from different stations. The ratio

\[ \frac{N_{\text{right}}}{N_{\text{left}}} \]

is free of these shortcomings. In [2] this quantity was adopted as the index of meteor activity. However, the best form of the index is \(i_3(t)\), since in the case of a distribution of visible radiants symmetric with respect to the apex, a symmetric annual curve is obtained. In the present work, the index

\[ i_3(t)=\lg \frac{N_{\text{right}}}{N_{\text{left}}} \]

is introduced as the definitive one.

Observations in various regions of the globe with identical equipment and under unchanging conditions may, in the future, make it possible to determine the variation of \(i_3(t)\) from year to year. Then this index would play the same role in the study of meteor activity as, for example, Wolf numbers do in the study of solar activity.

REFERENCES

1. R. G. Lazarev. On the diurnal and seasonal variation of hourly meteor counts. Proceedings of the Second Siberian Conference on Mathematics and Mechanics, Tomsk, p. 144, 1962.

2. R. G. Lazarev. On the diurnal and seasonal variation of hourly meteor counts. Part I. Izvestiya TIRiETa, 3, p. 86, 1964.

3. C. Ellyett and C. S. L. Keay. Journal of Geoph. Res., 66, p. 2590, 1961.

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9. G. S. Hawkins. Month. Not. Royal Astron. Soc., 116, p. 92, 1956.

10. J. G. Davies. Adv. Electr. and Electr. Phys., 9, p. 95, 1957.

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12. B. Lovell. Meteor Astronomy, p. 143, 1958.

13. B. L. Kashcheev and V. N. Lebedinets. Radar Studies of Meteor Phenomena, Moscow, p. 45, 1961.

14. B. L. Kashcheev, V. N. Lebedinets, D. N. Lukyashko. IGY, Information Bulletin No. 2, Kiev, p. 32, 1960.

15. C. Ellyett and C. S. L. Keay. Month. Not. Royal Astron. Soc., 125, No. 4, p. 325, 1963.

1. Precisely because the m. d. a. is a relative quantity (see Appendix 2). ↩↩↩↩

2. One may also say that the first approximation is the kinematic effects under the assumption that the radiants are distributed uniformly. The second approximation is the hypothesis of three sources. Taking into account the asymmetry of the solar and antisolar sources is already a third approximation. ↩↩