Radio Detection of Unstable Meteor Trails in Photographic Registration Mode

E. I. Fialko

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

Introduction

In considering many problems of meteor radar, it is necessary to know the dependence of the number of registered meteors on the parameters of the radar. A number of works [1—6, etc.] are devoted to this question; however, they do not take into account the effect of integration of the signal and noise. At the same time, in most radars used for meteor observations, echo signals are registered by means of photographic film, which is advanced at a relatively low speed. As a result, the integration time, determined by the size of the image of the luminous spot and by the speed of motion of the film, turns out to be many times greater than the pulse repetition period.

For example, with continuous motion of standard photographic film at a speed of several centimeters per minute and good focusing of the electron beam, the accumulation time amounts to several tenths of a second, and at a pulse repetition frequency of several hundred pulses/sec there will be tens of averages. It is quite obvious that under such conditions the integration effect cannot be neglected.

For a rigorous solution of the problem it is necessary to operate with the distribution of instantaneous amplitude values in the presence of signal and interference; one should take into account the nonlinearity of the dependence of the degree of blackening of the photographic film on the brightness of the indicator-screen glow; it is also necessary to take into account the nonuniformity of brightness along the radius of the luminous spot, as well as the finite size of the grains of the photographic-film emulsion; the special features of the operating mode of the cathode-ray indicator must also be taken into account (in particular, the presence of “clipping” of the control voltage), etc., etc.

To obtain an approximate dependence of the number of registered meteors on the radar parameters, however, there is no need to consider questions of scientific photography (which lie far beyond the author’s competence).

We shall assume that the operating mode of the cathode-ray indicator is close to optimal (the level of “clipping” of the control voltage is approximately equal to the mean value of the noise voltage), and that the degree of darkening of the film is a linear function of the control voltage of the cathode-ray tube; in this case, as is known, the intensity of darkening of the photographic film (in the absence of

signal) will be proportional to the square root of the number of averages [7]. Such a simplification substantially facilitates the analytical derivation of the dependence of the number of detected meteors on the parameters of the radar and on the characteristics of the meteors and the atmosphere.

In the present work, the detection of meteor trails of the unstable type [2] is considered for the case of normal reflection of radio waves from ionized columns.

Basic relations

In the absence of integration, the signal would be observed as long as it exceeded the threshold value $\varepsilon_{\text{por}}$ (Fig. 1), determined by the interference level.

The minimum necessary registration time $T_{\min}$ and the value of the threshold level $\varepsilon_{\text{por}}$ determine the minimum value of the initial power of the echo signal $\varepsilon_{m0}$ required for detecting a meteor. Consequently, the number of detected meteors will depend substantially on the responsiveness of the radar, characterized by the parameter $T_{\min}$ [3, 8, and others].

Fig. 1. Signal registration time $T$.
a) Change in the intensity of an unstable echo. $\varepsilon_m$—power of the echo signal; $t$—time; $\varepsilon_{\text{por}}$—threshold level.
b) Schematic representation of the registration of a meteor radio echo on rapidly drawn photographic film. $d_n$—diameter of the image of the light spot.

With integration, however, the parameter $T_{\min}$ may lose its physical meaning, since it may turn out that signals are detected which have not reached the interference level.

The result of integration will depend not only on the initial amplitude of the signal and on the effective value of the noise, but also on the integration time, the repetition frequency, and the character of the envelope of the echo-signal amplitudes.

The integration time $t_i$ is determined by the time during which the photographic film is displaced by a distance equal to the diameter of the image of the light spot $d_n$,

\[ t_i = \frac{d_n}{v_{n l}}, \tag{1} \]

where $v_{n l}$ is the speed of motion of the photographic film.

As a result of accumulation, the level (power) of the interference will increase by \(n_u\) times, where \(n_u\) is the number of accumulations, equal to

\[ n_u=\frac{t_u}{T_i}, \tag{2} \]

and \(T_i\) is the pulse repetition period (naturally, \(n_u\) is the integer part of the ratio \(t_u/T_i\)).

If the signal envelope had a rectangular form (which occurs in the case of stable meteor trails), then the signal power as a result of accumulation would increase by \(n_u^2\) times.

However, the envelope of the amplitudes \(U_m\) of echo signals from unstable meteor trails has an exponential form [2] (Fig. 1):

\[ U_m=U_{m0}e^{-\frac{t}{T_{c.l}}}, \tag{3} \]

where \(T_{c.l}=\dfrac{\lambda^2}{16\pi^2D}\) is the reflection duration at the level \(\dfrac{U_{m0}}{e}\);

\(\lambda\) is the wavelength; \(D\) is the diffusion coefficient; \(t\) is the time elapsed since the formation of the trail; \(U_{m0}\) is the initial amplitude of the echo signal (in the case where the “packet” of radio waves covered the trail immediately after its formation).

The effect of signal accumulation may be represented in the form

\[ U_{c\,\mathrm{int}}=\sum_{t=0}^{t_u}U_m. \tag{4} \]

Since irradiation of the meteor trail may begin after a time \(t_1\) following the formation of the ionized column, within the first Fresnel zone (it is quite obvious that \(0\leq t_1\leq T_i\)), expression (4), taking (3) into account, may be written as:

\[ U_{c\,\mathrm{int}}=U_{m0}\left( e^{-\frac{t_1}{T_{c.l}}} +e^{-\frac{t_1+T_i}{T_{c.l}}} +\ldots+ e^{-\frac{t_1+n_uT_i}{T_{c.l}}} \right). \tag{5} \]

After simple transformations, (5) takes the form

\[ U_{c\,\mathrm{int}}=U_{m1}\, \frac{1-e^{-\frac{T_i(n_u+1)}{T_{c.l}}}} {1-e^{-T_i/T_{c.l}}}, \tag{6} \]

where

\[ U_{m1}=U_{m0}e^{-\frac{t_1}{T_{c.l}}}; \tag{7} \]

here \(U_{m1}\) is the amplitude of the first echo signal.

The quantity \(U_{m1}\) may be any value within the limits from \(U_{m0}\) to \(U_{m0}e^{-T_i/T_{c.l}}\).

Let us determine the mean value of the amplitude of the first echo signal \(\overline{U}_{m1}\). For this purpose we shall find the distribution law of the amplitudes \(U_{m1}\).

It is quite obvious that the distribution of the values of \(t_1\) in the interval \(0 \le t_1 \le T_i\) is uniform:

\[ q_1(t_1)=\frac{1}{T_i}. \tag{8} \]

The differential distribution law for the values of \(U_{m1}\) is easily expressed through the probability density \(q_1(t_1)\):

\[ q(U_{m1})=q(t_1)\frac{1}{\left|\dfrac{dU_{m1}}{dt_1}\right|}. \tag{9} \]

After carrying out the necessary transformations and using (7) and (8), we represent (9) in the form

\[ q(U_{m1})=\frac{T_{\mathrm{sl}}}{T_i U_{m1}}. \tag{10} \]

The mean value \(\overline{U}_{m1}\) will be equal to

\[ \overline{U}_{m1}=\int_{U_{m,\min}}^{U_{m,\max}} U_{m1}q(U_{m1})\,dU_{m1}, \tag{11} \]

where, as follows from (7),

\[ U_{m,\min}=U_{mo}e^{-T_i/T_{\mathrm{sl}}}, \quad \text{and} \quad U_{m,\max}=U_{mo}. \]

Substituting expressions (7) and (10) into (11), we obtain

\[ \overline{U}_{m1}=U_{mo}\frac{T_{\mathrm{sl}}}{T_i}\left(1-e^{-T_i/T_{\mathrm{sl}}}\right). \tag{12} \]

Understanding below by \(U_{\mathrm{c\,int}}\) the result of the accumulation of echo signals at the mean value of the amplitude of the first reflection, we represent (6), taking into account (12) and (2), in the form

\[ U_{\mathrm{c\,int}}=U_{mo}\frac{T_{\mathrm{sl}}}{T_i}\left(1-e^{-\frac{t_p+T_i}{T_{\mathrm{sl}}}}\right). \tag{13} \]

The interference power, as already mentioned, as a result of accumulation will become equal to

\[ \varepsilon_{\mathrm{p\,int}}=\varepsilon_p \cdot n_u, \tag{14} \]

where \(\varepsilon_p\) is the interference level in the absence of integration.

Denoting by \(\varepsilon_{c0}\) the power of the first echo signal when irradiating the trail immediately after its formation and taking into account that \(\varepsilon_{c0}\sim U_{mo}^2\), we obtain, using (13) and (14),

\[ \frac{\varepsilon_{\mathrm{c\,int}}}{\varepsilon_{\mathrm{p\,int}}} = \frac{\varepsilon_{c0}}{\varepsilon_p} \cdot \frac{\psi^2}{n_u}, \tag{15} \]

where

\[ \psi=\frac{T_{c\lambda}}{T_i}\left(1-e^{-\frac{t_u+T_i}{T_{c\lambda}}}\right). \tag{16} \]

A meteor will be detected if

\[ \frac{\varepsilon_{c\,\mathrm{int}}}{\varepsilon_{n\,\mathrm{int}}} \geq k_u, \tag{17} \]

where \(k_u\) is a coefficient depending on the specified probability of detection, etc.

As is seen from (15) and (17), the condition for registering a meteor will be the inequality

\[ \varepsilon_{c0} \geq \varepsilon_n k_u \frac{n_u}{\psi^2}. \tag{18} \]

As is known, in the absence of integration the threshold signal \(\varepsilon_{por}\) can be expressed in terms of the noise level \(\varepsilon_n\) and the signal observability coefficient \(M\) [9].

In the general case the coefficients \(M\) and \(k_u\) may be different. Assuming henceforth \(k_u=M\) and denoting \(k_u\varepsilon_n\) by \(\varepsilon_{por}\), we present the condition for registering a meteor in the integration mode (18) in the form

\[ \varepsilon_{c0} \geq \varepsilon_{por}\cdot \frac{n_u}{\psi^2}. \tag{19} \]

Let us recall that here \(\varepsilon_{c0}\) is the power of the first echo signal (in the case of irradiation of the trail immediately after its formation); \(\varepsilon_{por}\) is the threshold signal in the absence of integration; \(n_u\) is the number of accumulations of the signal (and noise); \(\psi^2\) is a coefficient showing how many times the energy of the accumulated signals exceeds the energy of the first (maximum possible) echo signal.

Number of meteors recorded in the integration mode

As is known, the power of the echo signal \(\varepsilon_{c0}\) depends on a number of parameters of the meteor, the atmosphere, and the radar, as well as on the distance between the meteor trail and the observation point [10].

Substituting the expanded expression for \(\varepsilon_{c0}\) into (19), we obtain

\[ 7\cdot 10^{-29}\, \frac{P_iG^2\lambda^3}{R^3} \left( \frac{\beta m\cos\chi}{\mu H} \right)^2 z^2 e^{-2\left(\frac{2\pi r_0}{\lambda}\right)^2} \geq \varepsilon_{por}\frac{n_u}{\psi^2}, \tag{20} \]

where \(P_i\) is the transmitted power in the pulse; \(G\) is the antenna directivity factor; \(\lambda\) is the wavelength; \(R\) is the slant range from the radar to the trail (along the normal); \(m\) is the mass of the meteoric body; \(\beta\) is the probability of ionization (the ratio of the number of liberated electrons to the number of evaporated atoms of the meteoric body); \(\chi\) is the zenith distance of the radiant; \(\mu\) is the mass of an atom of the meteoric body; \(H\) is the height of the homogeneous atmosphere; \(r_0\) is the initial radius of the ionized trail;

\[ \xi = \frac{1 - e^{-1.5\Delta^{0.5}}}{1.5\Delta^{0.5}}; \qquad [10] \tag{21} \]

\[ \Delta = \frac{16\pi^{2}DR^{1/2}}{\vartheta \lambda^{3/2}} \qquad [2] \tag{22} \]

\(\vartheta\) is the velocity of the meteoric body; the definitions of \(n_u\), \(\Psi\), and \(\varepsilon_{\text{thr}}\) were given above.

Formula (20) is valid for echo signals arriving from the region of the characteristic height; formula (20) also assumes that a meteor of mass \(m\) gives a reflection of duration \(T_{\text{tr}}\) shorter than the integration time \(t_u\).

Finding from (20) the value of the minimum mass of the meteoric body, \(m_{\min}\), that produces a recorded trail, we express the number of meteors of the stream detected per unit time in the form [2]

\[ N = \frac{b\Delta s}{(s-1)m_{\min}^{s-1}} . \tag{23} \]

Formula (23) determines the number of meteoric bodies with masses \(m \geq m_{\min}\) that have flown through an area (of area \(\Delta s\)) lying in the echo plane [2] in the region of the characteristic height; the constants \(b\) and \(s\) characterize the distribution of meteoric bodies by mass [2]:

\[ q_m(m)=\frac{b}{m^s}. \tag{24} \]

Substituting into (23) the expression for \(m_{\min}\) found from (20), we obtain

\[ N=A\left[ \left(\frac{\lambda^3}{\varepsilon_{\text{thr}}}\right)^{1/2} \frac{\Psi}{\sqrt{n_u}}\, \xi e^{-\left(\frac{2\pi r_0}{\lambda}\right)^2} \right]^{s-1}, \tag{25} \]

where

\[ A=\left(7\cdot10^{-29}\right)^{\frac{s-1}{2}} \frac{b\Delta s}{s-1} \left(\frac{P_iG^2}{R^3}\right)^{\frac{s-1}{2}} \left(\frac{\beta\cos\chi}{\mu H}\right)^{s-1}. \tag{26} \]

As was to be expected, \(N\) does not depend on the radar parameter \(T_{\min}\).

Let us recall that formula (25) (which takes into account the accumulation of signals) is valid for the case of receiving reflections from the region of the characteristic height under conditions of normal reflection of radio waves from unstable meteor trails of the stream.

Let us also recall that (25) loses its validity when the integration time \(t_u\) approaches the value of the transmission period \(T_i\) or becomes small in comparison with the lifetime of the trail \(T_{\text{tr}}\).

As is seen from (25) and (26), the dependence of the number of detected meteors on \(P_i\), \(G\), and \(R\) when photographic registration is used turns out to be the same as in the absence of integration of the signal (and interference); however, the dependence of \(N\) on \(\lambda\) and \(F_i\) turns out to be different.

Dependence of the Number of Detected Meteors on the Pulse Repetition Frequency

As follows from (25), (2), and (16), in the case when the integration time considerably exceeds the value of the pulse transmission period \((t_i \gg 1/F_i\) and, consequently, \(n_i \gg 1)\),

\[ N = A_1 \cdot F_i^{\frac{s-1}{2}}, \tag{27} \]

where \(A_1\) is a coefficient independent of the pulse repetition frequency.

Thus, the dependence \(N(F_i)\) has the same character at different wavelengths \(\lambda\); in the case where there is no integration, however, the character of the dependence \(N(F_i)\) is determined by the value of \(\lambda\) [3].

Figure 2 gives a comparison of the theoretical dependence (27) for the case \(s=2\) \((N \sim \sqrt{F_i})\) with the results of an experiment carried out by F. I. Peregudov at \(\lambda = 4\ \mathrm{m}\) from January 26 to February 5, 1959, in the photographic registration mode for meteors (film-drawing speed \(v_{\mathrm{pl}} \simeq 2\ \mathrm{cm/min}\)) [11] (Table 1).

As is seen from Fig. 2, in the repetition-frequency range \(100 \div 300\ \mathrm{imp/sec}\) the experimental points deviate only very slightly from the theoretical curve (see \(F_i = 100,\ 150,\ 200,\) and \(300\ \mathrm{imp/sec}\)).

Fig. 2. Variation of the hourly number of detected meteors as a function of the repetition frequency \(F_i\). The solid curve corresponds to the integration mode \((N \sim \sqrt{F_i})\). The dashed curve corresponds to the absence of accumulation effects

\[ \left( N \sim e^{-\frac{16\pi^2 D T_{\min}}{\lambda^2}};\quad T_{\min}=2T_i \right). \]

The points are the experimental result.

However, at the repetition frequency \(F_i = 50\ \mathrm{imp/sec}\) the number proved to be almost 3 times smaller than follows from the analysis.

Such a discrepancy is quite understandable.

At \(F_i = 50\ \mathrm{imp/sec}\) and \(\lambda = 4\ \mathrm{m}\), the reflection duration \(T_{cd}\) and the repetition period \(T_i\) are of the same order, and signal averaging is practically absent. It is therefore natural to expect that the result of the expe-

…of the experiment will prove characteristic of the case in which there is no signal integration.

Table 1

Dependence of the mean hourly number of registered meteors \(N\) on the repetition frequency \(F_i\) at \(\lambda = 4\) m

If, for example, it is assumed that for reliable detection of a meteor it is necessary that at least 2 echo signals exceed the threshold level \(\varepsilon_{por}\) (that is, that \(T_{\min}=2T_i\)), then the dependence \(N(F_i)\) may be represented in the form [3]

\[ N(F_i)\sim e^{-\frac{16\pi^2DT_{\min}}{\lambda^2}} = e^{-\frac{32\pi^2D}{\lambda^2F_i}} . \tag{28} \]

The ratio

\[ \frac{N(F_i)}{N(F_i=300)} \]

is shown in Fig. 2 by the dashed curve, constructed from formula (28) for \(\lambda=4\) m and \(D=8\ \text{m}^2/\text{sec}\). As we see, the experimental point at \(F_i=50\) pulses/sec lies between the theoretical curves, but, as was to be expected, closer to the curve corresponding to the absence of integration. The fact that the experimental result at \(F_i=50\) pulses/sec exceeded the number predicted by formula (28) is also quite natural, since some meteors may be recorded “by chance,” that is, when the possible duration of the signals above the threshold level proves to be less than \(T_{\min}\) (which is not taken into account by formula (28)).

Thus, it may be considered established that in the integration mode (and, in particular, in the photographic-registration mode) the hourly number of detected meteors is proportional to the square root of the signal repetition frequency (at any wavelengths). In the case where there is no signal integration, a decrease in the repetition frequency in the region of small \(F_i\) leads to a sharper decrease in the hourly number than in the presence of integration.

Dependence of the hourly number of detected meteors on wavelength

As follows from (25), (26), (16), and (2),

\[ N = B\left[ \frac{\lambda^{3.5}}{\varepsilon_{por}^{1/2}} \left( 1-e^{-\frac{t_u+1/F_i}{T_{c\lambda}}} \right) \xi e^{-\left(\frac{2\pi r_0}{\lambda}\right)^2} \right]^{s-1}, \tag{29} \]

where

\[ B = A\left( \sqrt{\frac{F_i}{t_u}}\cdot \frac{1}{16\pi^2D} \right)^{s-1} \]

is a coefficient independent of \(\lambda\); the value of the coefficient \(A\) is given by formula (26).

In the range \(\lambda\simeq 4\div 10\) m, \(\varepsilon_{por}\sim \lambda^{2.3}\) [12]; in the range \(\lambda\simeq 1\div 4\) m, however, one may approximately take \(\varepsilon_{por}\approx \text{const}\).

In connection with this, \(N(\lambda)\) may be represented in the form

\[ N \sim \left[ \lambda^l \left( 1 - e^{-\frac{t_u+1/F_i}{T_{c\lambda}}} \right) \xi e^{-\left(\frac{2\pi r_0}{\lambda}\right)^2} \right]^{s-1}, \tag{30} \]

where \(l=2.35\) for \(\lambda \simeq 4 \div 10\ \mathrm{m}\) and \(l=3.5\) for \(\lambda \simeq 1 \div 4\ \mathrm{m}\).

Let us consider several special cases.

In the region of relatively large values of \(\lambda\) \((\lambda \simeq 8 \div 10\ \mathrm{m})\), for a large integration time, as follows from (30), (2), (21), and (22),

\[ N \sim \lambda^{2.35}, \tag{31} \]

that is, as the wavelength decreases, the number falls somewhat more sharply than in proportion to the square of the wavelength. In the case of no integration, with a highly responsive system (\(T_{\min}\) small), as was shown earlier [3], \(N \sim \lambda^{0.35}\).

In the region of small \(\lambda\), as is seen from (30), the dependence \(N(\lambda)\) proves to be much stronger than would follow from (31).

However, integration somewhat weakens the dependence of \(N\) on \(\lambda\) in comparison with the case of a low-response system (\(T_{\min}\) large), in which there are no effects of signal accumulation (see (28) and [3]).

In Fig. 3 the dependence is plotted

\[ \frac{N(\lambda)}{N(\lambda_1)} = \left\{ \left(\frac{\lambda}{\lambda_1}\right)^l \frac{ 1-e^{-\frac{16\pi^2 D d_n}{v_n \lambda^2}} }{ 1-e^{-\frac{16\pi^2 D d_n}{v_n \lambda_1^2}} } \cdot \frac{ \xi(\lambda)\, e^{\left(\frac{2\pi r_0}{\lambda_1}\right)^2} }{ \xi(\lambda_1)\, e^{\left(\frac{2\pi r_0}{\lambda}\right)^2} } \right\}^{s-1}, \tag{32} \]

obtained from (30) under the assumption that \(t_u \gg \frac{1}{F_i}\); for the range \(\lambda \simeq 4 \div 10\ \mathrm{m}\), \(l=2.35\), \(\lambda_1=10\ \mathrm{m}\); for the range \(\lambda \simeq 1 \div 4\ \mathrm{m}\), \(l=3.5\), \(\lambda_1=4\ \mathrm{m}\).

In the calculations the following values of the parameters entering formula (32) were adopted: exponent \(s=2\); zenith distance of the radiant \(\chi \simeq 45^\circ\); velocity of the meteoric body \(v=45\ \mathrm{km/sec}\), the corresponding diffusion coefficient \(D=8\ \mathrm{m^2/sec}\) [2, 13], and initial radius of the ionized trail \(r_0=0.17\ \mathrm{m}\) [14]; speed of motion of the photographic film \(v_{\rm pl}=0.5\ \mathrm{mm/sec}\); diameter of the spot image on the film \(d_n=0.1\ \mathrm{mm}\).

The experimental data are indicated in Fig. 3 by points (see also Table 2).

The experiment was carried out on February 28, 1959, at the TPI-2 stations \((\lambda=10\ \mathrm{m})\), M-3 \((\lambda \simeq 4\ \mathrm{m})\), and “Most-2” \((\lambda=1.5\ \mathrm{m})\), with a pulse repetition frequency \(F_i=300\ \mathrm{pulses/sec}\) and a photographic-film pulling speed \(v_{\rm pl} \simeq 3\ \mathrm{cm/min}\).

As is seen from Fig. 3, the agreement between the analytical and experimental data is quite satisfactory. Thus, in the presence of photographic registration, under real conditions, a decrease in wavelength in the region \(\lambda \simeq 4 \div 10\ \mathrm{m}\) leads to a sharp decrease in the hourly number (in going from \(\lambda=10\ \mathrm{m}\) to \(\lambda=4\ \mathrm{m}\), the hourly number decreases by more than an order of magnitude).

At wavelengths \(\lambda\) of about \(\lambda=1\ \mathrm{m}\), the hourly number is two orders of magnitude smaller than at \(\lambda=10\ \mathrm{m}\).

It should be noted that there is a difference in the character of the dependence \(N(\lambda)\) in the presence and in the absence of signal integration: in the first case there is practically no point of inflection (Fig. 3), and the decrease in number in the region \(\lambda \simeq 8 \div 10\ \mathrm{m}\) occurs very sharply.

As is seen from (29), (30), and (32), the character of the dependence \(N(\lambda)\) is to some extent determined by the film transport speed \(v_{\text{pl}}\).

It is not difficult to show that there exists an optimal film speed equal to

\[ v_{\text{pl opt}} = 12.8 \pi^{2} d_n \frac{D}{\lambda^{2}}, \tag{33} \]

corresponding to the optimal integration time

Fig. 3. Change in the hourly number \(N\) as a function of the wavelength \(\lambda\).
The solid curve corresponds to a reflector motion speed \(V_{\text{pl}} = 0.5\ \text{mm/sec}\) for all \(\lambda\).
The dashed curve corresponds to the optimal film-transport speed \(V_{\text{pl opt}}\), which is a function of wavelength. Points are experimental data.

Table 2

Experimental data

\[ t_{i\,\text{opt}} = 1.25\,T_{\text{с.л.}}. \tag{34} \]

Substituting (34) into (30), or (33) into (32), we obtain

\[ \frac{N(\lambda)}{N(\lambda_1)} = \left\{ \left(\frac{\lambda}{\lambda_1}\right)^l \frac{\xi(\lambda)}{\xi(\lambda_1)} \frac{ e^{\left(\frac{2\pi r_0}{\lambda_1}\right)^2} }{ e^{\left(\frac{2\pi r_0}{\lambda}\right)^2} } \right\}^{s-1}. \tag{35} \]

where \(l=2.35\) and \(\lambda=10\ m\) for \(\lambda \simeq 4 \div 10\ m\), and \(l=3.5\), \(\lambda=4\ m\) for \(\lambda \simeq 1 \div 4\ m\).

The dependence \(N(\lambda, v_{n l\,opt})\), constructed from formula (35) for \(\chi=45^\circ\), \(v=45\ km/sec\), \(s=2\), \(D=8\ m^2/sec\), \(r_0=0.17\ m\), is shown in Fig. 3 by the dashed curve. As is seen from a comparison of the solid and dashed curves, the deviation from \(v_{n l\,opt}\) has little effect on the character of the dependence \(N(\lambda)\).

Conclusions

1. The signal-integration mode changes the character of the dependence of the hourly number of detected meteors \(N\) on wavelength and pulse repetition frequency in comparison with the mode in which accumulation effects are absent.

2. In the integration mode the hourly number \(N\) is proportional to the square root of the pulse repetition frequency practically at any wavelengths used in meteor radiolocation, if the distribution of meteor bodies by mass is characterized by an index \(s \simeq 2\).

At low repetition frequencies and small \(\lambda\), when the reflection duration and the repetition period are of the same order, signal accumulation is practically absent, and the decrease in the hourly number as the repetition frequency \(F_i\) is reduced occurs more sharply than according to the law \(\sqrt{F_i}\).

1. In the integration mode, the decrease in the hourly number as the wavelength is reduced occurs very sharply: in going from \(\lambda=10\ m\) to \(\lambda=4\ m\), the hourly number decreases by more than an order of magnitude.

Literature

1. D. W. R. McKinley. Variation of meteor echo rates with radar system parameters, Can. Journ. Phys., 1951, v. 29, No. 5, 403.

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

3. E. I. Fialko. On the influence of wavelength on the effectiveness of the radio-location method of studying meteors, Proceedings of the Siberian Physico-Technical Institute, issue 37, p. 229, 1959.

4. F. I. Peregudov. On the influence of meteor velocities on the hourly number under conditions of radio detection, Astr. Zhurn., 1958, 35, issue 6, 888.

5. E. I. Fialko. Certain and random detection of unstable meteor trails, Radiotekhnika, 1961, vol. 16, No. 6, 25.

6. E. I. Fialko. Certain and random radio detection of stable meteor trails, Izvestiya TPI, 1961, vol. 100, 84.

7. G. S. Zubarev. On the question of the real sensitivity of a pulse receiver under conditions of brightness measurement and photographic recording of signals, dissertation for the degree of Candidate of Technical Sciences, 1955.

8. E. I. Fialko. On the efficiency of a meteor radar, Izvestiya VUZov MVO SSSR (Radio Engineering section), 1960, vol. 3, issue 5, 690.

9. A. Ya. Breitbart. Brief Fundamentals of Radiolocation, Sovradio, 1951 (ed.).

10. E. I. Fialko. Power of a signal reflected from a meteor trail under conditions of intense diffusion, Izvestiya TPI, vol. 100, 40, 1962.

11. F. I. Peregudov. Some questions of meteor radiolocation, Candidate’s dissertation, Tomsk Polytechnic Institute, 1959.

12. H. V. Cottony, J. R. Johler, Cosmic radio noise intensities in the VHF band, Proc. I. R. E., 1952, 40, No. 9, 1053.

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

14. L. A. Manning. The initial radius of meteoric ionization trails, Journ. Geoph. Res., 1958, 33, No. 1, 181.