DISTRIBUTION OF METEOR REFLECTIONS BY DURATION

E. I. Fialko

(Submitted by the scientific seminar of the Faculty of Radio Engineering)

Introduction

Comparison of the analytical and experimental distributions of meteor radio echoes by duration makes it possible to investigate certain characteristics of meteoroid bodies and of the atmosphere and, in particular, the law of distribution of meteoroid bodies by mass.

Knowledge of the analytical distribution of the durations of meteor radio echoes facilitates interpretation of observational results.

In several works, questions related to the distribution of the durations of meteor reflections in the case of a trail of the underdense type have been considered [1–3], which made it possible to extend information on the law of distribution of meteoroid bodies by mass [1, 2, 4]. Only one work [5] is devoted to the distribution of echo-signal durations when trails of the overdense type are recorded. However, these works did not consider the general case—the distribution of the durations of reflections received from the entire echo plane¹.

This problem can be solved in several ways. One of them is as follows: knowing the dependence of the duration of a meteor radio echo on the mass of the meteoroid body and the law of distribution of meteoroid bodies by mass, we find the differential (and then the integral) distribution of reflections by duration for the case of reception from an element of the echo plane [3, 5], and then pass to the distribution characterizing reception from the entire echo plane.

Another method consists in finding the number of detected meteors that give reflections whose durations exceed a specified value (in this case, of course, the distribution of meteoroid bodies by mass is also used).

In earlier papers [3, 5], the first method was implemented; in the present work, the second is.

Obtaining objective data on meteoroid bodies is possible only with a correct interpretation of the results of radar observations of meteors and, in particular, of the distribution of radio echoes by duration.

Often the distribution curve of radio echoes by duration has a kink: as the duration \(\tau\) increases, the decrease in the number of reflections in the region of large \(\tau\) occurs more rapidly than in the region of small \(\tau\) (see Figs. 2 and 7). Naturally the question arises—whether this is only a consequence of the fact that the distribution of meteoroid masses in the region of large masses \(m\) is different from that in the region of small masses, as the authors of [1] suppose; or whether it may prove to be a consequence of the peculiarities of normal reflection of radio waves from meteor trails.

To answer these, as well as many other questions, it is necessary to obtain an analytical distribution of meteor radio echoes by durations.

Here the case is considered of observing a meteor stream under conditions of normal reflection of radio waves from ionized trails [6].

Basic relations

The number of meteoroids with masses in the interval \(m : m+\Delta m\), passing during a time \(\Delta t\) through an area perpendicular to the direction of the stream, can, in the same way as Kaiser does [6, 7], be expressed through the distribution of meteoroids by mass \(p(m)\)

\[ \Delta N_1 = p(m)\,\Delta m\,\Delta s\,\Delta t, \tag{1} \]

where \(\Delta s\) is the surface area of the element.

The function \(p(m)\) is usually represented in the form [6—8]

\[ p(m)=\frac{b}{m^s}, \tag{2} \]

where \(b\) and \(s\) are constants.

The number of meteors giving reflections with durations \(\geq \tau\), detected in the element \(\Delta s\), will be equal to

\[ N_1=\Delta s\,\Delta t \int_m^\infty p(m)\,dm, \tag{3} \]

where \(m\) is the mass of a meteoroid that creates a meteor trail recorded during the time \(\tau\). Let us note that trails detected by chance will not be of interest to us, since the distribution by durations is constructed using reflections whose durations can be measured with a definite accuracy.

Substituting (2) into (3), we obtain

\[ N_1=\frac{b\Delta s\Delta t}{(s-1)m^{s-1}}. \tag{4} \]

Formula (4), which characterizes reception from an elementary area \(\Delta s\), makes it possible to determine the number of meteors recorded in a sector (with aperture \(\Delta\Theta\)) of the echo plane in the interval of slant ranges \(R_1 : R_2\) (see Fig. 1):

\[ N_c=\frac{b\Delta\Theta\Delta t}{s-1}\int_{R_1}^{R_2}\frac{R\,dR}{m^{s-1}}. \tag{5} \]

Sometimes it is more convenient to operate not with the slant ranges \(R\), but with heights above the level of the Earth’s surface \(h\). Let us express \(R\) in terms of \(h\) (Fig. 1):

\[ R=\frac{h}{\cos\Theta \sin\chi}, \tag{6} \]

where \(\chi\) is the zenith distance of the radiant, and \(\Theta\) is the angle between the direction to the meteor and the line of intersection of the echo plane with the plane passing through the zenith \((Z)\), the radiant \((p)\), and the location of the radar

Fig. 1. Relation between the height \(h\) and the slant range \(R\). \(O\) is the location of the radar; \(Z\) is the zenith; \(p\) is the radiant; \(\chi\) is the zenith distance of the radiant; \(\Gamma\) is the horizontal plane; \(B\) is the vertical plane passing through the zenith, the radiant, and the location of the radar; \(\Pi\) is the echo plane (the plane perpendicular to the direction to the radiant); \(M\) is the point of intersection of the meteor trajectory with the echo plane; \(OA\) is the line of intersection of the planes \(\Pi\) and \(B\); \(R=OM\) is the slant range from the radar to the trajectory of the meteor body; \(h=MM'\) is the height of the specularly reflecting center above the ground level; \(G(\Theta)\) is the diagram of the directional action of the antenna in the echo plane; \(\Theta\) is the angle measured in the echo plane from the line \(OA\) to the direction to the meteor; \(\Theta_1\) and \(\Theta_2\) determine the aperture of the antenna pattern in the echo plane; \(OA'\) is the projection of the line \(OA\) onto the horizontal plane; \(h=R\cos\Theta\sin\chi\).

radar \((O)\); the angle \(\Theta\) is measured in the echo plane. We note that formula (6) does not take into account the curvature of the Earth’s surface.

Substituting (6) into (5), we obtain for the height interval \(h_1 \div h_2\):

\[ N_c=\frac{b\Delta\Theta\Delta t}{(\cos\Theta \sin\chi)^2(s-1)} \int_{h_1}^{h_2}\frac{h\,dh}{m^{s-1}} . \tag{7} \]

In order to determine the total number of meteors of the stream registered by the radar, it is necessary to carry out integration over the entire echo plane.

If observations are conducted over a time \(T=t_2-t_1\), and outside the height interval \(h_{\min}\div h_{\max}\) meteors are practically not detected, while outside the angular interval \(-\Theta_1\div\Theta_2\) the radiation is negligibly small, then the mean hourly number of registered meteors will be equal to

\[ N=\frac{1}{T}\int_{t_1}^{t_2}\frac{b\,dt}{(s-1)\sin^2\chi} \int_{-\Theta_2}^{\Theta_2}\frac{d\Theta}{\cos^2\Theta} \int_{h_{\min}}^{h_{\max}}\frac{h\,dh}{m^{s-1}} . \tag{8} \]

Using (8), one can consider various special cases. Thus, for example, if in the processing only meteors detected in a narrow layer \(\Delta h=h_1-h_2\) are selected, then formula (8) is substantially simplified and takes the form

\[ N=\frac{h\Delta h}{T}\int_{t_1}^{t_2}\frac{b\,dt}{(s-1)\sin^2\chi} \int_{-\Theta_1}^{\Theta_2}\frac{d\Theta}{\cos^2\Theta\,m^{s-1}}, \tag{9} \]

where

\[ h=\frac{h_1+h_2}{2}. \]

In the case of reception from an element of the echo plane, the expression for the mean hourly number is simplified still further, especially if during the observing time \(T\) the stream parameters \(\chi\), \(s\), and \(b\) do not have time to change noticeably:

\[ N=\frac{b\Delta h\Delta\Theta h}{(s-1)\sin^2\chi\cos^2\Theta\,m^{s-1}} . \tag{10} \]

Formulas (8)—(10) determine the number of meteors with masses \(\geq m\).

In order to find the number of reflections whose durations exceed some value \(\tau\), it is necessary to express the mass of the meteoroid \(m\) in terms of the linear electron density \(\alpha\) at the point of normal reflection of radio waves (that is, in the region of the first Fresnel zone) and to use the dependence of the reflection duration on the linear electron density \(\alpha\).

Relation between the mass of a meteoroid and the linear electron density in the trail

As is known, the number of electrons per unit length of the ionized trail (the linear electron density in the trail \(\alpha\)) is equal to [6]

\[ \alpha=m\frac{\beta\cos\chi}{\mu H}\frac{p}{p_m} \left(1-\frac{1}{3}\frac{p}{p_m}\right)^2, \tag{11} \]

where \(m\) is the initial mass of the meteoroid;
\(\beta\) is the probability that, as a result of the evaporation of one atom of the meteoroid, one free electron will be released (the probability of ionization); \(H\) is the height of a homogeneous atmosphere at the point considered; \(\chi\) is the zenith distance of the radiant; \(\mu\) is the mass of an atom of the meteoroid; \(p\) is the atmospheric pressure in the region where the normally reflecting center is located; \(p_m\) is the pressure corresponding to the characteristic height \(h_m\), at which the linear electron density is maximal, equal to

\[ p_m=Qg\frac{\cos\chi}{v^2}\,m^{\frac{1}{3}}, \tag{12} \]

where \(Q\) is a coefficient characterizing the physical and geometrical properties of the meteoric body; \(g\) is the acceleration of gravity; \(v\) is the velocity of the meteoric body.

The mass of a meteoric body forming a trail with electron line density \(\alpha\), as follows from (11), is equal to

\[ m=\alpha\,\frac{\mu H}{\beta\cos\chi\,k_1}, \tag{13} \]

where

\[ k_1=\frac{p}{p_m}\left(1-\frac{1}{3}\frac{p}{p_m}\right)^2 . \tag{14} \]

This form of writing can substantially simplify consideration of the problem in those cases when the characteristic height \(h_m\) is known. If, however, one takes into account the dependence of the pressure \(p_m\) and the coefficient \(k_1\) on the mass \(m\), then, as is not difficult to show, substituting (12) into (11) and solving the equation with respect to \(m\) [7],

\[ m=\left[\frac{1}{3a_1}\left(p+3\sqrt{\frac{\mu H a_1^3}{\beta\cos\chi\,p}\,\alpha}\right)\right]^3, \tag{15} \]

where

\[ a_1=\frac{Qg\cos\chi}{v^2}. \]

In the region of the characteristic height \((p=p_m)\), as follows from (13) or from (15),

\[ m=\frac{9}{4}\,\alpha\,\frac{\mu H}{\beta\cos\chi}. \tag{16} \]

In order to express \(m\) through the duration \(\tau\) of the recorded reflection, one may use the dependence of the time during which the echo signals exceed the threshold level on the parameters of the meteoric body, the atmosphere, and the radar.

As is known, this dependence proves to be different in the cases of underdense and overdense trails.

Echo-signal power and reflection duration

The power of the signal reflected from an underdense meteoric trail \((\alpha \ll 2.4\cdot 10^{12}\ \text{el/cm}\ [6,9])\), \(t\) seconds after the meteoric body has passed the middle of the first Fresnel zone, will be equal to [10, 11]

\[ \varepsilon(t)=A_1\frac{P_iG^2\lambda^3}{R^3}\,\alpha^2 e^{-2\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4Dt)/\xi^2\alpha^2}, \tag{17} \]

where \(P_i\) is the radiated power in the pulse; \(G\) is the antenna gain; \(\lambda\) is the operating wavelength; \(R\) is the slant range from the radar to the meteor trajectory; \(r_0\) is the initial radius of the ionized trail in the region of the first Fresnel zone; \(D\) is the diffusion coefficient (in the region where the part of the trail that normally scatters the radio waves is located); \(\xi\) is a coefficient characterizing the decrease of the initial amplitude of the echo signal caused by the fact that the diffusion effect has time to manifest itself during the formation of the trail within the first Fresnel zone, equal to [11]

\[ \xi=\frac{1-e^{-1.5\Delta^{0.5}}}{1.5\cdot\Delta^{0.5}}; \tag{18} \]

here [6]

\[ \Delta=\frac{16\pi^2 D R^{\frac12}}{v\lambda^{3/2}}; \tag{19} \]

\(\eta\) is a coefficient characterizing the increase in the echo-signal intensity due to plasma resonance in the ionized trail [7, 9]; \(A_1\) is a constant equal to

\[ A_1=\frac{1.4}{32\pi^2}\left(\frac{e^2}{m_e c^2}\right)^2=3.5\cdot10^{-28}\ \mathrm{cm}^2, \]

where \(e\) and \(m_e\) are the charge and mass of the electron; \(c\) is the speed of light.

The time during which the echo signals exceed the threshold level \(\varepsilon_n\), as follows from (17), is

\[ \tau=\tau_{sl}\ln\left[\sqrt{A_1}\left(\frac{P_iG^2\lambda^3}{\varepsilon_n R^3}\right)^{\frac12}\xi\eta e^{-\left(\frac{2\pi r_0}{\lambda}\right)^2}\right], \tag{20} \]

where \(\tau_{sl}\) is the duration measured at the level \(1/e\) of the initial amplitude, equal to [6, 9]

\[ \tau_{sl}=\frac{\lambda^2}{16\pi^2D}. \tag{21} \]

In the case of reception from the region of the characteristic height, as follows from (20),

\[ \tau=\tau_{sl}\ln(Cm), \tag{22} \]

where

\[ C=\frac{4}{9}\sqrt{A_1}\left(\frac{P_iG^2\lambda^3}{\varepsilon_nR^3}\right)^{\frac12}\xi\eta\frac{\beta\cos\chi}{\mu H}e^{-\left(\frac{2\pi r_0}{\lambda}\right)^2}. \tag{23} \]

The duration of reflection in the case of a persistent trail \((\alpha \gg 2.4\times10^{12}\ \mathrm{el/cm}\ [6, 9])\), as is known, is equal to [6, 9]

\[ \tau=A_3\frac{\alpha^2}{D}, \tag{24} \]

where

\[ A_3=\frac{1.12\cdot10^{-12}}{16\pi^2}=7.1\cdot10^{-15}\ \mathrm{cm}, \]

and, in the case of reception from the region of the characteristic height,

\[ \tau=\frac{4}{9}A_3\frac{\lambda^2\beta\cos\chi}{\mu H D}\,m. \tag{25} \]

Masses of meteoroid bodies producing trails recorded during a time \(\tau\)

An underdense meteor trail that has produced a reflection exceeding the threshold level during a time \(\tau\) corresponds to an electron line density (see (20))

\[ \alpha=\frac{1}{\sqrt{A_1}}\left(\frac{\varepsilon_n R^3}{P_i G^2 \lambda^3}\right)^{\frac12}\frac{1}{\xi\eta} e^{\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4D\tau)} . \tag{26} \]

Substituting (26) into (15) or (13), we obtain

\[ m=\left[\frac{1}{3a_1}\left(p+\frac{3}{A_1^{1/4}} \sqrt{\frac{\mu H a_1^3}{\beta\cos\chi\,p}}\times \right.\right. \]

\[ \left.\left. {}\times\left(\frac{\varepsilon_n R^3}{P_iG^2\lambda^3}\right)^{\frac14} \frac{1}{(\xi\eta)^{1/2}} e^{\frac12\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4D\tau)} \right)\right]^3, \tag{27} \]

or

\[ m=\frac{\mu H}{\sqrt{A_1}\kappa_1\beta\cos\chi} \left(\frac{\varepsilon_n R^3}{P_iG^2\lambda^3}\right)^{\frac12} \frac{1}{\xi\eta} e^{\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4D\tau)} . \tag{28} \]

A persistent trail that gives a reflection of duration \(\tau\) has the electron line density (see (24))

\[ \alpha=\frac{1}{A_3}\frac{D\tau}{\lambda^2}, \tag{29} \]

and the mass of the meteoroid that formed such a trail is equal to (see (15) and (13))

\[ m=\left[\frac{1}{3a_1}\left(p+\frac{3}{\sqrt{A_3}} \sqrt{\frac{\mu H a_1^3}{\beta\cos\chi\,p}\cdot \frac{\sqrt{D\tau}}{\lambda}}\right)\right]^3 \tag{30} \]

or

\[ m=\frac{1}{A_3\kappa_1}\frac{\mu H}{\beta\cos\chi}\frac{D\tau}{\lambda^2}. \tag{31} \]

In the case of reception from the region of the characteristic height, the coefficient \(\kappa_1=\frac{4}{9}\) (see (14)), and formula (28) takes the form:

\[ m=\frac{9}{4\sqrt{A_1}}\frac{\mu H}{\beta\cos\chi} \left(\frac{\varepsilon_n R^3}{P_iG^2\lambda^3}\right)^{\frac12} \frac{1}{\xi\eta} e^{\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4D\tau)} . \tag{32} \]

Expression (32) can also be obtained from (30). Moreover, (32) can be obtained directly from (22):

\[ m=\frac{1}{C}\cdot e^{\frac{16\pi^2D\tau}{\lambda^2}}, \tag{33} \]

where the coefficient \(C\) is determined by relation (23).

In the case when the reflection from a persistent trail comes from the region of the characteristic height, as is evident from (31), (30), or directly from (25),

\[ m=\frac{9}{4A_3}\cdot\frac{\mu H}{\beta\cos\chi}\cdot\frac{D\tau}{\lambda^2}. \tag{34} \]

Number of registered meteors

The mean hourly number of meteors giving a registration duration \(\geq \tau\), detected by a system capable of registering unsta-

stable meteor trails, is determined by formula (8), where \(m\) is found from (27) or (28) and (14).

In the case of using a low-sensitivity system that records only trails of the stable type, the mean hourly number is likewise determined by formula (8), while \(m\) is determined by formulas (30) or (31) and (14).

Let us consider a special case: reception of reliably detectable reflections from an element of the echo plane situated in the region of the characteristic height.

If the system is capable of recording unstable trails, then the mean hourly number will be equal to (see (10) and (32))

\[ N=\left(\frac{4}{9}\sqrt{A_1}\right)^{s-1} \frac{1}{(s-1)} \frac{bh\Delta h\Delta\Theta}{\sin^2\chi\cos^2\Theta} \times \]

\[ \times\left[ \frac{\beta\cos\chi}{\mu H} \left(\frac{P_iG^2\lambda^3}{\varepsilon_n R^3}\right)^{\frac12} \xi\eta e^{-\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4D\tau)} \right]^{s-1}. \tag{35} \]

If the system records only trails of the stable type, then, as follows from (10) and (34),

\[ N=\left(\frac{4}{9}A_3\right)^{s-1} \frac{1}{(s-1)} \frac{bh\Delta h\Delta\Theta}{\sin^2\chi\cos^2\Theta} \left( \frac{\beta\cos\chi}{\mu H}\cdot \frac{\lambda^2}{D\tau} \right)^{s-1}. \tag{36} \]

It should be noted that the dependence of the hourly number on the apparatus parameters may be affected by the features of indication and recording of signals.

The relations presented in the present work (and, in particular, the initial formula (17)) have been obtained without taking into account signal-accumulation effects, which may occur, for example, when photographic recording of meteor radio echoes is used. Radio detection of meteors in the photographic-recording mode has been considered in works [14, 15]. However, in the case when the duration of the reflection is such that the image of the echo signal on the photographic film has an extent considerably exceeding the size of the image of the light spot, the results given above are also acceptable in the photographic-recording mode.

In the work presented here it is assumed that the volume density of electrons in the trail changes with time only as a result of diffusion and, consequently, that the linear electron density \(\alpha\) remains constant. In reality, however, processes take place that alter the linear electron density (recombination of ions and electrons, capture of electrons by neutral particles, etc.). The phenomena of recombination and capture may prove significant only at comparatively low heights and in the region of very large \(\tau\). As a result, the results of investigations carried out without allowance for capture, recombination, etc., may possibly require corrections in the region of low heights and, consequently, at low velocities of meteoric bodies, as well as in the region of very large durations \(\tau\).

Distribution of radio echoes by duration

If during the observation time \(N(\tau_{\min})\) reflections with durations \(>\tau_{\min}\) have been recorded, and the number of reflections with durations \(>\tau\) is equal to \(N(\tau)\), then the normalized integral distribution law of radio echoes by duration may be represented in the form

\[ P(\tau)=\frac{N(\tau)}{N(\tau_{\min})}. \tag{35} \]

Substituting into (35) the expressions for the hour numbers (8), corresponding to \(m_{\min}(\tau_{\min})\) and \(m(\tau)\), we obtain

\[ P(\tau)= \frac{ \displaystyle \int_{t_1}^{t_2}\frac{b\,dt}{(s-1)\sin^2\chi} \int_{-\Theta_1}^{\Theta_2}\frac{d\Theta}{\cos^2\Theta} \int_{h_{\min}}^{h_{\max}}\frac{h\,dh}{m^{s-1}} }{ \displaystyle \int_{t_1}^{t_2}\frac{b\,dt}{(s-1)\sin^2\chi} \int_{-\Theta_1}^{\Theta_2}\frac{d\Theta}{\cos^2\Theta} \int_{h_{\min}}^{h_{\max}}\frac{h\,dh}{m_{\min}^{s-1}} }. \tag{36} \]

In the case of reception from a comparatively narrow sector over a short interval of time (during which \(b\), \(\chi\), and \(s\) are approximately constant), expression (36) is simplified to

\[ P(\tau)= \frac{ \displaystyle \int_{h_{\min}}^{h_{\max}}\frac{h\,dh}{m^{s-1}} }{ \displaystyle \int_{h_{\min}}^{h_{\max}}\frac{h\,dh}{m_{\min}^{s-1}} }. \]

The form of the function \(m(\tau)\) is determined by the type of ionized trail (see (27) and (30)).

\(P(\tau)\) in the case of persistent trails

The distribution of persistent radio echoes by duration has the form (see (36) and (30))

\[ P(\tau)= \frac{ \displaystyle \int_{t_1}^{t_2}\frac{b\,dt}{(s-1)\sin^2\chi} \int_{-\Theta_1}^{\Theta_2}\frac{d\Theta}{\cos^2\Theta} \int_{h_{\min}}^{h_{\max}} \left[ \frac{1}{3a_1} \left( p+ \frac{3}{\sqrt{A_3}} \sqrt{ \frac{\mu H a_1^3}{\beta\cos\chi\,p} \cdot \frac{\sqrt{D\tau}}{\lambda} } \right) \right]^{-3(s-1)} h\,dh }{ \displaystyle \int_{t_1}^{t_2}\frac{b\,dt}{(s-1)\sin^2\chi} \int_{-\Theta_1}^{\Theta_2}\frac{d\Theta}{\cos^2\Theta} \int_{h_{\min}}^{h_{\max}} \left[ \frac{1}{3a_1} \left( p+ \frac{3}{\sqrt{A_3}} \sqrt{ \frac{\mu H a_1^3}{\beta\cos\chi\,p} \cdot \frac{\sqrt{D\tau_{\min}}}{\lambda} } \right) \right]^{-3(s-1)} h\,dh }; \tag{37} \]

Formula (37) characterizes reception from the echo plane.

Analysis and construction of the function \(P(\tau)\) in the general case are made difficult not only by the need to use graphical or numerical integration, but also because some quantities entering into (37) are unknown or are known only very approximately (\(b\), \(\beta\), \(a_1\), etc.).

Let us consider some special cases.

1) Distribution of very long reflections.

Suppose that the condition is satisfied

\[ p \ll \frac{3^{2/3}}{A_{3}^{1/3}}\,a_{1}\left(\frac{\mu H \cdot D\tau}{\beta \cos \chi}\right)^{1/3}\frac{1}{\lambda^{2/3}} . \tag{38} \]

This can occur either in the case of reception from high layers, where the pressure \(p\) is small (in comparison with the pressure at the characteristic height), and the diffusion coefficient is large, or in the case of selection of very long echoes.

When condition (38) is fulfilled, the number of reflections with durations \(\ge \tau\) can be represented in the form

\[ N(\tau)=\frac{B_{1}}{\tau^{\frac{3}{2}(s-1)}} , \tag{39} \]

where

\[ \begin{aligned} B_{1}={}& \left(\frac{A_{3}}{a_{1}}\right)^{\frac{3}{2}(s-1)} \frac{\lambda^{3(s-1)}}{T} \left(\frac{\beta}{\mu}\right)^{\frac{3}{2}(s-1)} \bigl(\tg \Theta_{2}+{} \\ &{}+\tg \Theta_{1}\bigr) \int_{t_{1}}^{t_{2}} \frac{b(\cos \chi)^{\frac{3}{2}(s-1)}}{(s-1)\sin^{2}\chi}\, \int_{h_{\min}}^{h_{\max}} \left(\frac{p}{DH}\right)^{\frac{3}{2}(s-1)} h\,dh . \end{aligned} \tag{40} \]

and the distribution of durations (37) can be represented in explicit and, moreover, very simple form

\[ P(\tau)=\left(\frac{\tau_{\min}}{\tau}\right)^{\frac{3}{2}(s-1)} . \tag{41} \]

2) Distribution of durations of radio echoes arriving from a thin layer lying in the region of the characteristic height.

The number of reflections with durations \(\ge \tau\), received from a thin layer lying in the region of the characteristic height \(h_m\), as is not difficult to see by substituting (34) into (8) or (9), will be equal to

\[ N(\tau)=\frac{B_{2}}{\tau^{s-1}}, \tag{42} \]

where

\[ B_{2}= \left(\frac{4}{9}A_{3}\right)^{s-1} \frac{1}{T} \left(\frac{\beta\lambda^{2}}{\mu}\right)^{s-1} \frac{h_m\Delta h}{(DH)^{s-1}} (\tg \Theta_{2}+\tg \Theta_{1}) \int_{t_{1}}^{t_{2}} \frac{b(\cos \chi)^{s-1}}{(s-1)\sin^{2}\chi}. \tag{43} \]

The normalized integral distribution of radio echoes by durations, as follows from (35) and (42), in this case has the form

\[ P(\tau)=\left(\frac{\tau_{\min}}{\tau}\right)^{s-1}. \tag{44} \]

Formulas (42) and (44) were obtained earlier for reception from an element of the echo plane located at height \(h_m\) [3,16]. It should be clarified that in the present case the characteristic height is understood to be \(h_m\). As is known, the characteristic height corresponds to the most intense evaporation of the meteoric body, and, consequently, to the greatest linear electron density [6]. But the value \(h_m\) depends

on the mass of the meteoroid (see (12)). The duration of the reflection in the region of the characteristic height also depends on \(m\) (see (25)).

Thus, strictly speaking, when passing from some reflection durations to others (under the condition of reception from the region of the characteristic height), the quantity \(h_m\) does not remain constant.

But when the height changes, the diffusion coefficient \(D\) changes.

If we take into account that \(D \sim \dfrac{1}{p}\) [6], \(p_m \sim m^{1/3}\), and \(\tau \sim \dfrac{m}{D}\) (see (12) and (25)), then we obtain

\[ \tau \sim m^{4/3}. \tag{45} \]

As is not difficult to show, proceeding from (9) and (45), in the case of reception from the region of the characteristic height, we have (taking into account the dependence \(D(h_m)\))

\[ N(\tau) \sim \tau^{-\frac{3}{4}(s-1)} \tag{46} \]

and

\[ P(\tau)=\left(\frac{\tau_{\min}}{\tau}\right)^{\frac{3}{4}(s-1)}. \tag{47} \]

An analogous result in the form of a differential distribution law was obtained by Kaiser, who considered reception from the height corresponding to the greatest reflection duration (for a given meteoroid mass) [1, 6].

Fig. 2. Integral distribution of meteors by radio-echo duration (region of persistent trails). The points are the result of an experiment (2.00–3.00 hours local Tomsk time, August 12, 1957—the Perseid stream). \(N\) is the number of reflections with duration \(\geq \tau\) sec.

As is seen from comparison of (41), (44), and (47), for one and the same law of distribution of meteoroid bodies by mass (that is, for one and the same value of the index \(s\)), in the region of very great durations the decrease of \(P(\tau)\) with increasing \(\tau\) should occur more sharply,

than in the region of small \(\tau\) (normal reflections from stable trails are meant).

Experiment confirms this assumption. As an example, Fig. 2 gives the integral distribution of radio echoes by duration during the Perseid shower epoch (from 2:00 to 3:00 hours, local Tomsk time, August 12, 1957). In the region of durations \(\tau \approx 10\) sec there is a break in the dependence \(\lg N(\lg \tau)\), and the function \(N(\tau)\) in the region \(\tau > 10\) sec decreases with increasing \(\tau\) more rapidly than in the region \(\tau < 10\) sec.

\(P(\tau)\) in the case of unstable trails

The normalized integral distribution law of unstable radio echoes by duration has the form (see (36)):

\[ \begin{aligned} P(\tau)= &\frac{ \displaystyle \int_{t_1}^{t_2} \frac{b\,dt}{(s-1)\sin^2\chi} \int_{-\Theta_1}^{\Theta_2}\frac{d\Theta}{\cos^2\Theta} \int_{h_{\min}}^{h_{\max}} \left[ \frac{1}{3a_1}\left(p+\frac{3}{A_1^{1/4}}\right) \right. }{ \displaystyle \int_{t_1}^{t_2} \frac{b\,dt}{(s-1)\sin^2\chi} \int_{-\Theta_1}^{\Theta_2}\frac{d\Theta}{\cos^2\Theta} \int_{h_{\min}}^{h_{\max}} \left[ \frac{1}{3a_1}\left(p+\frac{3}{A_1^{1/4}}\right) \right. } \\[0.5em] &\quad \frac{ \displaystyle \left. \cdot \sqrt{\frac{\mu H a_1^3}{\beta \cos\chi\,p}} \left(\frac{\varepsilon_n R^3}{P_iG^2\lambda^3}\right)^{1/4} \frac{1}{(\xi\eta)^{1/2}} e^{\frac12\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4D\tau)} \right]^{-3(s-1)} h\,dh }{ \displaystyle \left. \cdot \sqrt{\frac{\mu H a_1^3}{\beta \cos\chi\,p}} \left(\frac{\varepsilon_n R^3}{P_iG^2\lambda^3}\right)^{1/4} \frac{1}{(\xi\eta)^{1/2}} e^{\frac12\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4D\tau_{\min})} \right]^{-3(s-1)} h\,dh }\, . \end{aligned} \tag{48} \]

Formula (48) characterizes reception from the plane of the echo. The analysis and construction of the function (48) present still greater difficulties than the analysis of formula (37) in the case of stable trails.

Let us consider several special cases

1) Distribution of reflections received from a narrow layer under conditions of intense diffusion.

Let us suppose that the condition is satisfied

\[ p \ll \frac{3^{2/3}}{A_1^{1/6}} \left(\frac{\mu H}{\beta\cos\chi}\right)^{1/3} a_1 \left(\frac{\varepsilon_n R^3}{P_iG^2\lambda^3}\right)^{1/6} \frac{1}{(\xi\eta)^{1/3}} e^{\frac13\left(\frac{2\pi}{\lambda}\right)^2(r_0^2+4D\tau)} . \tag{49} \]

This may occur in cases of reception from a layer in which the pressure is much lower than at the characteristic height; with very intense diffusion, but with a comparatively large duration of the recorded reflection \(\tau\) (in this case in the region of the first Fresnel zone \(\alpha \ll 2.4\cdot 10^{12}\) el/cm).

The number of reflections with durations \(\geq \tau\), received during one hour from a narrow layer for which condition (49) is fulfilled, may be represented in the form

\[ N(\tau)=B_3 e^{-6\left(\frac{2\pi}{\lambda}\right)^2(s-1)D\tau}, \tag{50} \]

where

\[ \begin{aligned} B_3={}& \left(\frac{A_1}{a_1^2}\right)^{\frac{3}{4}(s-1)} \left(\frac{P_iG^2\lambda^3}{\varepsilon_n}\right)^{\frac{3}{4}(s-1)} \left(\frac{\beta\rho}{\mu H}\right)^{\frac{3}{2}(s-1)} e^{-\frac{3}{2}(s-1)\left(\frac{2\pi r_0}{\lambda}\right)^2} \frac{1}{T} \\ &\cdot \int_{-\Theta_1}^{\Theta_2}(\cos\Theta)^{\frac{3}{4}(s-1)+2}\,d\Theta\cdot h^{\frac{1}{4}(3s-7)}\,\Delta h \\ &\cdot \int_{t_1}^{t_2}\frac{b}{s-1}(\sin\chi)^{\frac{3}{4}(s-1)-2}(\cos\chi)^{\frac{3}{2}(s-1)}\,dt, \end{aligned} \tag{51} \]

and the duration distribution (48) is reduced to the simple form

\[ P(\tau)=e^{-6\left(\frac{2\pi}{\lambda}\right)^2(s-1)D(\tau-\tau_{\min})}. \tag{52} \]

Thus, the character of the distribution \(P(\tau)\) in this case is determined not only by the type of distribution of meteoroids by mass (that is, not only by the value of the exponent \(s\)), but also by the atmospheric parameter \(D\) and by the equipment parameter \(\lambda\).

2) Distribution of the durations of radio echoes received from a thin layer lying in the region of the characteristic height.

The number of reflections with durations \(\geq \tau\), received per unit time from a thin layer located in the region of the characteristic height \(h_m\), as can be shown by substituting (32) into (8) or (9), will be equal to

\[ N(\tau)=B_4 e^{-\left(\frac{4\pi}{\lambda}\right)^2(s-1)D\tau}, \tag{53} \]

where

\[ \begin{aligned} B_4={}& \left(\frac{4}{9}A_1^{\frac{1}{2}}\right)^{s-1} \left(\frac{\beta}{\mu H}\right)^{s-1} \left(\frac{P_iG^2\lambda^3}{\varepsilon_n}\right)^{\frac{s-1}{2}} e^{-\left(\frac{2\pi r_0}{\lambda}\right)^2(s-1)} \frac{1}{T} \\ &\cdot \int_{t_1}^{t_2}\frac{b}{s-1} (\sin\chi)^{\frac{3}{2}(s-1)-2} (\cos\chi)^{s-1}\,dt \int_{-\Theta_1}^{\Theta_2} (\cos\Theta)^{\frac{3}{2}(s-1)-2}\,d\Theta . \end{aligned} \tag{54} \]

The normalized integral law of the distribution of radio echoes by duration, as follows from (35) and (53), in this case has the form

\[ P(\tau)=e^{-\left(\frac{4\pi}{\lambda}\right)^2(s-1)D(\tau-\tau_{\min})}. \tag{55} \]

As is seen from (55), the integral distribution (as also in the preceding case (52)) is described by an exponential (Fig. 3). The form of the exponent is determined not only by the parameter \(s\), but also by the intensity of diffusion and by the wavelength of the radar.

The change of \(P(\tau)\) as \(\tau\) increases, in the case of large \(\tau\) and \(D\), occurs more sharply than in the case of reception from the region \(h \approx h_m\) (see (52) and (55)).

For convenience of terminology, in what follows we shall understand by short-duration reflections—radio echoes from unstable trails

($\tau$—of the order of hundredths–tenths of a second); reflections of medium duration and long-duration reflections we shall assign to radio echoes from stable trails. By reflections of medium duration we shall understand radio echoes with $\tau$ of the order of seconds, and by long-duration ones—of the order of tens of seconds (for $\lambda \approx 8 \div 10$ m). We note that the “boundary” between short-lived, medium, and long-duration reflections

Fig. 3. Integral distribution of meteors by radio-echo durations (region of unstable trails). Points are the result of an experiment (from 1 h 10 min to 5 h 20 min local Tomsk time on December 16, 1958—the epoch of the Geminid stream). $N$ is the number of meteors producing reflections of duration $> \tau_{\rm sec}$.

is determined by the quantities $D$ and $\lambda$. Very long-duration reflections, existing for many minutes and not obeying Kaiser’s theory, are not considered here.

Methods for determining the exponent $s$

From the distribution of meteor radio echoes by duration one can determine the exponent characterizing the distribution of meteor bodies by mass. The determination of $s$ from $P(\tau)$ can be carried out in several ways.

Approximating the experimental distribution of reflections of medium duration (belonging to stable trails) $P(\tau)$ by an inverse-power function, we obtain, using (44),

\[ s = 1 + \frac{\lg \dfrac{N_1}{N_2}}{\lg \dfrac{\tau_2}{\tau_1}}, \tag{56} \]

or, using (47),

\[ s = 1 + \frac{4}{3}\,\frac{\lg \dfrac{N_1}{N_2}}{\lg \dfrac{\tau_2}{\tau_1}}, \tag{57} \]

where \(N_1\) (and \(N_2\)) is the number of reflections with durations \(\tau \ge \tau_1\) (or \(\ge \tau_2\)).

Figure 2 presents the dependence of \(N\) on \(\tau\) on a logarithmic scale. The points mark the results of observations carried out in Tomsk during the Perseid stream on August 12, 1957 (one one-hour interval from 2:00 to 3:00 local time is shown); the observations were carried out at the TPI-2 station \((\lambda = 10\ \text{m})\) [4,17].

Determination of the exponent \(s\) by formula (56) gives \(s \approx 1.43\), and by formula (57), \(s \approx 1.57\); the value \(s \approx 1.57\) agrees better with the known data [6,1].

In the region of long-duration reflections (also corresponding to stable trails), as is seen from (41),

\[ s \approx 1 + \frac{2}{3}\, \frac{\lg \dfrac{N_1}{N_2}} {\lg \dfrac{\tau_2}{\tau_1}} . \tag{58} \]

Approximating the distribution of reflections by duration in the interval \(\tau = 10 \div 60\ \text{s}\) (Fig. 2) also by an inverse-power dependence, we find, using (58), \(s \approx 1.52\). This result differs from \(s \approx 1.57\) by only \(\approx 3\%\).

If, in the region of long-duration reflections, the exponent \(s\) were determined in the same way as in the region of medium durations (that is, by formula (57)), then for \(\tau > 10\ \text{s}\) one would obtain \(s \approx 2.06\). This is precisely the error made by the authors of [1], determining \(s\) in the region of medium and large \(\tau\) in the same way. Thus, for example, for the Geminid stream of 1954 the values \(s \approx 1.7\) and \(s \approx 3.7\) [1] were obtained.

In the region of short-duration reflections, caused by unstable trails, the distribution \(P(\tau)\) is exponential in character. If the reflections arriving from the region of characteristic height are selected, then, as follows from (53),

\[ s = 1 + \frac{\lambda^2}{16\pi^2 D}\, \frac{\ln \dfrac{N_1}{N_2}}{\tau_2 - \tau_1}. \tag{59} \]

Thus, in order to find the exponent \(s\) from the distribution \(P(\tau)\) in the region of short-duration reflections, it is necessary to know the operating wavelength and the diffusion coefficient.

If reception is carried out from the entire echo plane, then the result obtained from (59) (as well as from (57) and (58)) becomes approximate.

Figure 3 presents the distribution of the durations of short-time echoes in the interval \(\tau = 0.1 \div 0.5\ \text{s}\). The observations were carried out in Tomsk on December 16, 1958, from 1 h 10 min to 5 h 22 min local time. The TPI-2 station was used \((\lambda = 10\ \text{m})\) [4,17] with an increased film-advance speed \(v_{\text{pl}} = 125\ \text{cm/min}\) (instead of the normally used mode for photographic recording of echo signals at

\[ v_{\text{pl}} = 3\ \frac{\text{cm}}{\text{min}} \]
).

Such a high film-advance speed made it possible to measure the duration of reflections with an accuracy of up to several hundredths of a second.

Since on December 16 the Geminid stream was still active, it is reasonable to take the mean meteor velocity as \(v \simeq 34 \div 35\ \text{km/sec}\) [2] and the diffusion coefficient \(D\) in the region \(h_m\) as \(\simeq 4.2\ \text{m}^2/\text{sec}\) [6, 18]. For \(D \simeq 4.2\ \text{m}^2/\text{sec}\) and \(\lambda = 10\ \text{m}\), formula (59) takes the form

\[ s \simeq 1 + 0.346\, \frac{\lg \dfrac{N_1}{N_2}}{\tau_2 - \tau_1}. \tag{60} \]

Approximating the dependence \(\lg N(\tau)\) by a straight line, we find \(s \simeq 1.94\). This result apparently indicates the predominance of the sporadic background during the observation period, since in the first phase of the stream (December 9–12) \(s \simeq 1.62 \pm 0.04\) [1].

Some results of the study of the distribution of meteoroid bodies by mass

To study the distribution of meteoroid bodies by mass, distributions of the durations of persistent radio echoes (of medium duration) were used.

The exponent \(s\) was found from formula (57). The value of \(s\), as was to be expected, does not remain constant in time (Fig. 4). In addi-

Fig. 4. Variation of the exponent \(s\) as a function of time. \(t\) is Tomsk decretal time. The points are the result of the experiment (from 13:00 on August 12 to 11:00 on August 13, 1958).

tion to determining the mean value of \(s\) for various streams and for the sporadic background, it is of great interest to determine the statistical properties of the exponent \(s\) when considering the entire aggregate of meteors.

In this connection it is useful to find the limits of variation, the mean and modal values of the quantity \(s\), and also to construct the distribution law of the index \(s\).

For this purpose, having found from \(P(\tau)\) the mean value of the index \(s\) for each hour, one can determine the number of meteors \(N\) to which a given value of \(s\) corresponds, and represent \(N\) as a function of \(s\).

In Fig. 5 the differential distribution of the index \(s\) has been constructed from the results of observations carried out in Tomsk at the TPI-2 station from August 1957 through May 1958 [4]. During this time 174,380 meteors were recorded, including 28,540 meteors with duration \(\tau > 1\) sec.

The value of the index \(s\) varied within the limits from 1.25 to 4; the mean value \(\bar{s} \approx 2.08\); the dispersion of \(s\) in the histogram (Fig. 5) is relatively small. This evidently indicates the quantitative predominance of sporadic meteors over shower meteors [4].

Fig. 5. Differential distribution of meteors by the index \(s\). \(N\) is the number of meteoric bodies with index \(s\) in the interval

\[ s-\frac{\Delta s}{2}<s<s+\frac{\Delta s}{2}. \]

\(N_{\max}\) corresponds to the interval \(2<s<2.25\).

On the possibility of measuring the diffusion coefficient

Constructing the distribution of short-lived radio echoes by duration makes it possible to establish a relation between the index \(s\) and the diffusion coefficient \(D\) (see (59)).

The index \(s\) can be found from the distribution of durations of persistent radio echoes (see (57)). If, simultaneously with the recording of short-lived reflections, reflections from persistent trails are also recorded, then, by constructing \(P(\tau)\) in the range of radio echoes of mean duration, we find \(s\), and then from \(P(\tau)\) in the range of short-lived reflections we find \(D\) (see (59)).

Comparison of the distributions of the masses of meteoric bodies and of radio-echo durations

The character of the distribution of radio echoes by duration may differ substantially from the character of the distribution of meteoric bodies by mass.

If, as is usually assumed, the integral law of distribution of meteoric bodies by mass is represented by an inverse power function

\[ P_m(m)=\left(\frac{m_{\min}}{m}\right)^{s-1}, \tag{61} \]

then the integral law of distribution of radio echoes by duration \(P(\tau)\) is determined in the general case by function (36), and in the case of obser-

... of stable and unstable trails, respectively, by formulas (37) and (48).

In the region of medium durations of meteor radio echoes (corresponding to stable trails), the distribution of durations approximately repeats the mass distribution (see (61) and (44)). However, in the region of long-duration reflections, the decrease in the number of echo signals as \(\tau\) increases occurs more rapidly than the decrease in the number of meteoroid bodies as the mass increases; nevertheless, the distribution of durations also obeys an inverse-power law (see (61) and (41)).

In the region of short durations (corresponding to unstable trails), the increase in the number of short-lived reflections as \(\tau\) decreases occurs more slowly than the increase in the number of meteoroid bodies as the mass \(m\) decreases; in this case, the distribution of the durations of short-lived reflections obeys an exponential law (see (61) and (55)).

The distributions \(P_m(m)\) and \(P(\tau)\) are illustrated by Figure 6.

Fig. 6. Deviation of the distribution \(P(\tau)\) from the distribution \(P_m(m)\) (illustration). \(P\)—integral distribution law; \(m\)—mass of the meteoroid body; \(\tau\)—duration of the meteor radio echo; I—region of unstable trails (short-lived echoes). II and III—region of stable trails (II—reflections of medium duration; III—long-duration reflections).

The deviations of \(P(\tau)\) from \(P_m(m)\) are explained by the peculiarities of meteor detection under conditions of normal scattering of radio waves by ionized trails.

It should be noted that, strictly speaking, in the region of medium durations as well, \(P(\tau)\) differs from \(P_m(m)\) (see (61) and (47)).

Thus, when interpreting experimental results, it is necessary to bear in mind that the difference in the character of the distribution of meteor radio echoes by duration in different ranges of \(\tau\) may by no means...

be associated with a change in the exponent \(s\) (that is, with a change in the law of distribution of meteoroid bodies by mass) in the corresponding regions of \(m\).

Therefore, when determining the exponent \(s\) from the distribution \(P(\tau)\), caution must be exercised: the application of one and the same computational formula to the region of small and large durations may lead to an incorrect conclusion about the character of \(P_m(m)\).

This, of course, does not mean that meteoroid bodies producing radio echoes of duration of hundredths of a second and meteoroid bodies forming trails that exist for tens of seconds obey the same distribution law \(P_m(m)\). It is quite possible that in the region of large \(m\) the character of the function \(P_m(m)\) will be different than in the region of small \(m\), but this can be established provided that the distribution \(P(\tau)\) is used correctly, taking into account the peculiarities of the reflection of radio waves from stable and unstable meteor trails.

Some remarks

A substantial work by Brown, Bullough, Evans, and Kaiser [1] is devoted to the distribution of meteoroid bodies by mass. In this work a difference is noted in the character of the distribution of meteor reflections by duration in the region of small and large \(\tau\) (in the case of stable radio echoes). As a result, the authors [1] in a number of cases determine two values of the exponent \(s\), characterizing the inverse-power law of distribution of meteoroid bodies by mass; thus, for example, for the Geminids of 1954, \(s=1.7\pm0.2\) and \(s=3.7\pm0.3\). That is, the authors come to the conclusion that there is a substantial difference in the character of the distribution of meteoroid bodies by mass in the mass regions corresponding to reflection durations \(\tau=1\div5\) sec and \(\tau=5\div10\) sec (see Fig. 7).

Fig. 7. Integral distribution of radio echoes by duration (Geminids, 1954) [1]. \(N\) is the number of meteors that gave reflections of duration \(\ge \tau\).

Of course, over a large interval of \(m\) the law \(P_m(m)\) does not remain unchanged. However, within comparatively small limits of \(m\) the exponent \(s\) changes not so substantially as follows from the work [1].

As has already been noted, in the region of echoes of medium duration (the case of stable trails) \(P(\tau)\), approximately repeating \(P_m(m)\), is an inverse-power law. At smaller durations (in the region of unstable trails) \(P(\tau)\) is an exponential; at larger durations it is also a hyperbola, but a steeper one than in the region of medium durations.

If we assume that in [1] the exponent \(s\approx1.7\) for the interval \(\tau\approx1\div5\) seconds was found correctly, and recalculate the exponent

\(s\) for \(\tau \simeq 5 \div 10\) sec (which was found by the authors in the same way as for the interval \(\tau \simeq 1 \div 5\) sec) by formula (58), then instead of the value \(s = 3.7\) we obtain \(s = 2.34\); that is, within the limits of masses \(m\) corresponding to durations \(\tau \simeq 1 \div 10\) sec, the exponent \(s\) changes from \(s = 1.7\) to \(s = 2.34\), i.e., by approximately 30%, and not by a factor of 2.2, as was obtained by Browne, Bullough, Evans, and Kaiser.

In conclusion, let us recall that very long-lived trails are not considered here.

Conclusions

Analytical distributions of radio echoes by durations \(P(\tau)\) have been obtained both for the case of stable and for unstable meteor trails. In the first case, \(P(\tau)\) is (approximately) an inverse power function; in the second, an exponential.

The integral law of the distribution of durations of radio echoes of the stable type is determined, in the first approximation, only by the exponent \(s\), which characterizes the distribution of meteoric bodies by mass.

In the case of unstable trails, however, \(P(\tau)\) is characterized not only by the value of the exponent \(s\), but also by the diffusion coefficient \(D\) and the wavelength \(\lambda\).

The different character of \(P(\tau)\) in the region of short, medium, and long-duration reflections and the presence of a break in the \(P(\tau)\) curve are a natural manifestation of the features of normal scattering of radio waves by meteor trails; this, of course, does not refute the possibility of a difference in the distribution of meteoric bodies by mass in the region of small and large \(m\), nor is a violation of the conditions of normal scattering of radio waves by long-lived meteor trails (under the influence of vortex currents in the meteor zone) ruled out. But it is possible that in a number of cases the character of \(P(\tau)\) in the region of large \(\tau\) is determined not by turbulence and not by a change in the exponent \(s\), but by the nature of normal scattering of radio waves by ionized trails.

When interpreting the results of observations, it is necessary to keep in mind the change in the character of \(P(\tau)\) in different regions of \(\tau\): a) because of the nonconstancy of the exponent \(s\); b) as a consequence of the different character of the scattering of radio waves by stable and unstable trails; c) because of violation of the conditions of normal scattering of radio waves (in particular, as a consequence of ionospheric vortices).

A simple method is given for finding the exponent \(s\) in the case of stable and unstable trails.

The possibility is shown of admitting a serious error in estimating the value of the exponent \(s\) without taking into account the difference in the distribution of reflections of medium and long duration (in the case of stable trails).

The distribution of the exponent \(s\) has been found from observations of meteors at \(\lambda = 10\) m; \(1.25 < s < 4\), \(s \simeq 2\).

A simple method is given for determining the diffusion coefficient from the distribution of the durations of reflections from unstable and stable trails.

LITERATURE

1. I. C. Browne, K. Bullough, S. Evans, T. R. Kaiser. Characteristics of radio echoes from meteor trails. II. The distribution of meteor magnitudes and masses, Proc. Phys. Soc., B, 69, 83, 1956.

2. A. C. B. Lovell. Meteor astronomy, Oxford, 1954. B. Lovell, Meteor Astronomy, translated under the editorship of B. Yu. Levin, State Publishing House of Physico-Mathematical Literature, Moscow, 1958.

3. E. I. Fialko. Distribution of meteor radio echoes by duration (1. Reflections from stable trails), Astronomical Journal, 36, no. 5, 867, 1959.

1. E. I. Fialko. Some results of a study of meteor bodies by mass. Astr. Circ., No. 195, 22, 1958.

2. E. I. Fialko. Distribution of meteor radio echoes by duration (II. Reflections from unstable trails). Astronomical Journal, 37, issue 3, 526, 1960.

3. T. R. Kaiser. Radio-echo studies of meteor ionization, J. Adv. Phys., 2, No. 8, 495, 1953.

4. T. R. Kaiser. Theory of the meteor height distribution obtained from radio-echo observations. M. N. R. A. S., 114, No. 1, 39, 1954.

5. B. Yu. Levin. Physical Theory of Meteors and Meteoric Matter in the Solar System, Publishing House of the Academy of Sciences of the USSR, 1956.

6. T. R. Kaiser, R. L. Closs. Theory of radio reflections from meteor trails, Phil. Mag., Ser. 7, 43, No. 336, 1, 1952.

7. F. I. Peregudov. On the influence of meteor velocities on the hourly number under conditions of radio observations. Astr. Journal, 35, issue 6, 888, 1958.

8. E. I. Fialko. The power of the signal reflected from a meteor trail under conditions of intense diffusion, Proceedings of TPI, 100, 40, 1961.

9. E. I. Fialko. On the influence of wavelength on the effectiveness of the radar method for studying meteors. Proceedings of the Siberian Physico-Technical Institute, issue 37, 229, 1959.

10. F. I. Peregudov. Dependence of the registration time of meteor reflections on the parameters of a radar station. Astr. Journal, 37, issue 3, 530, 1960.

11. E. I. Fialko. Radio detection of unstable meteor trails in the photoregistration mode, Proceedings of TPI, 100, 100, 1961.

12. E. I. Fialko. Radio detection of stable meteor trails in the integration mode, Proceedings of TPI, 100, 111, 1961.

13. E. I. Fialko. Distribution of stable meteor radio echoes by duration, Proceedings of the Tomsk Polytechnic Institute, 105, 50, 1960.

14. E. I. Fialko, F. I. Peregudov, E. K. Nemirova, G. S. Zubarev, I. D. Zolotarev, L. A. Pokrovsky. Apparatus of the Tomsk Polytechnic Institute for radar observations of meteors. Results of investigations under the MGT program. Ionosphere, Collection No. 8, 65, 1961.

15. J. S. Greenhow. E. L. Neufeld. The diffusion of ionized meteor trails in the upper atmosphere. Journ. Atm. Terr. Phys., 6, No. 2—3, 133, 1955.

1. The echo plane is the plane perpendicular to the direction toward the meteor radiant and passing through the point of observation [6]. ↩