Reliable and Accidental Radio Detection of Persistent Meteor Trails

E. I. Fialko

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

Introduction. Statement of the Problem

Usually, when considering the dependence of the number of recorded meteors on the parameters of a radar, use is made of relations that are valid in the case of reception of signals reflected from nonpersistent ionized trails [1].

At the same time, it may turn out that the system records only trails of the persistent type. The dependence of the observational result (the number) on the parameters of the radar in this case may differ substantially from analogous dependences characteristic of the reception of nonpersistent echoes.

For the correct interpretation of observational results, and also for the reliable calculation of the parameters of the equipment, it is necessary, first of all, to determine what trails the station is capable of recording (only persistent, nonpersistent and persistent, etc.). It is necessary to establish the dependence between the number of registered meteors and the parameters of the radar, the meteor, and the atmosphere.

It should be noted that this dependence may be different not only in the cases of reception of persistent and nonpersistent trails, but also for reliable and accidental detection of meteors of the given type.

To clarify the questions listed, it is necessary to take into account all meteor signals received by the radar, which is associated with great difficulties. First of all, persistent trails can give long-lasting reflections, and as a consequence many of them will have an abnormal character [1].

But even when only normal reflections arriving from the echo plane are taken into account (that is, the plane perpendicular to the direction toward the radiant and containing the point at which the radar is located), an analytical solution of the problem is extremely difficult.

The required dependences can be obtained in explicit form for a particular case of great interest—namely, for reception from the region of characteristic height (that is, the height at which the most intense ionization in the meteor trail occurs) [1].

The problem of the present work is to determine the dependence of the number of detected persistent meteors belonging to a stream—

$K y$, on the parameters of the radar and the characteristics of meteors and of the atmosphere for the case of reception of normally reflected signals arriving from the region of the characteristic height.

Basic relations

Long-lived meteors that give an echo signal exceeding the threshold level are reliably detected by the radar. Ionized trails that exist for a very short time may remain unnoticed by the locator; the detection of such trails is a random event.

For reliable detection of a trail it is necessary that the time during which the echo signals exceed the threshold level be greater than the minimally necessary time $T_{\min}$ [2]; in the particular case $T_{\min}$ is equal to the period of transmission of the probing pulses $T_i$.

Trails of the persistent type (with a linear electron density $\alpha \gg 2.4 \cdot 10^{12}$ electrons/cm [1]) give a sharp rise and a sharp fall of the reflection, as a result of which, for any excess of the signal over the noise level, the duration of meteor registration $(T)$ will be practically the same.

Several cases of detection of a weak meteor are possible. It may turn out that a reflection which has reached the threshold level $\varepsilon_n$ has duration $T < T_{\min}$ (Fig. 1); detection of a meteor giving such a reflection will be random; a reflection with duration $T \approx T_{\min}$

Fig. 1. Cases in which both randomly and reliably detectable trails are registered. $t$—time; $\varepsilon$—power of echo signals; $\varepsilon_n$—threshold-signal level; 1, 2, 3, and 4—envelopes of the amplitudes of echo signals; $T$—duration of the reflection; $T_{\min}$—the minimally necessary time during which echo signals exceed the threshold level, for which detection of the trail becomes reliable. 1—a reflection that has reached the threshold level. 2—a reflection whose duration is equal to $T_{\min}$. Reflection 3 is detected randomly; reflection 4—reliably.

$m_{\min}$—the mass of the meteoric body creating an ionized trail, the duration of reflection from which is equal to $T_{\min}$;

$m_{\min}^{*}$—the mass of the meteoric body creating an ionized trail, the reflection from which reaches the threshold level $\varepsilon_n$.

will exceed the threshold level. But it may turn out that a reflection which has reached the threshold level has duration $> T_{\min}$ (Fig. 2); in this case detection of the meteor will be a reliable event.

In the first case weak meteors are detected accidentally (and strong ones—reliably); in the second, any recorded meteor is detected reliably.

The number of reliably detected meteors passing during the time \(\Delta t\) through an element of the echo plane (whose area is \(\Delta s\)) is equal to

Fig. 2. Case in which reliably detectable meteors are recorded. The notation is the same as in Fig. 1. Any reflection exceeding the threshold level (for example, 3) has duration \(T>T_{\min}\) and is detected reliably.

\[ N_{\mathrm{uv}}=\Delta s\,\Delta t\int_{m_{\min}}^{\infty} p(m)\,dm, \tag{1} \]

where \(m_{\min}\) is the minimum mass of a meteoroid giving a reflection of duration \(T=T_{\min}\) (and a signal exceeding the threshold level \(\varepsilon>\varepsilon_n\), see Fig. 1); \(p(m)\) is the differential law of the mass distribution of meteoroids, usually represented in the form [1]

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

where \(b\) and \(s\) are constants independent of \(m\).

The number of accidentally detected meteors passing through an element of the echo plane \((\Delta s)\), as is not difficult to show, will be equal to:

\[ N_{\mathrm{sl}}=\Delta s\cdot \Delta t\cdot \int_{m^*_{\min}}^{m_{\min}} p(m)q(m)\,dm, \tag{3} \]

where \(m^*_{\min}\) is the minimum mass of a meteoroid corresponding to the threshold signal (Fig. 1); \(q(m)\) is the probability of detecting the ionized trail formed by a meteoroid of mass \(m\).

Let us note that in the case when the mass \(m^*_{\min}\) corresponds to a reflection with duration \(T\geq T_{\min}\) (Fig. 2), \(N_{\mathrm{sl}}=0\), and

\[ N_{\mathrm{uv}}=\Delta s\cdot \Delta t\int_{m^*_{\min}}^{\infty} p(m)\,dm. \tag{4} \]

To find \(N_{uv}\) and \(N_{sl}\), we shall use the known expressions for the echo-signal power \(\varepsilon\) and the reflection duration [1]

\[ \varepsilon=\frac{P_iG^2\lambda^3}{32\pi^4R^3}\,g^2, \tag{5} \]

where \(P_i\) is the radiated power in a 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); \(g\) is the reflection coefficient, equal to

\[ g=\left(\frac{\pi}{2}\right)^{1/2} e^{-1/4} \left(\frac{e^2}{m_ec^2}\right)^{1/4}\alpha^{1/4}, \tag{6} \]

\(e, m_e\) are the charge and mass of the electron; \(c\) is the speed of light; \(e\) is the base of natural logarithms; \(\alpha\) is the linear electron density, equal to

\[ \alpha=\frac{m\beta\cos\chi}{\mu H}\cdot\frac{p}{p_m} \left(1-\frac{1}{3}\frac{p}{p_m}\right)^2; \tag{7} \]

\(\beta\) is the probability of ionization; \(\chi\) is the zenith distance of the radiant; \(\mu\) is the mass of an atom of the meteoric body; \(H\) and \(p\) are the height of the homogeneous atmosphere and the pressure (\(H\), \(p\), and \(\alpha\) correspond to the section of the ionized trail that normally scatters radio waves); \(p_m\) is the pressure at the characteristic height (i.e., the height at which \(\alpha=\alpha_{\max}\)).

For reflections coming from the layer lying in the region of the characteristic height \((p\simeq p_m)\), as follows from (7),

\[ \alpha\simeq\alpha_{\max}=\frac{4}{9}\,\frac{m\beta\cos\chi}{\mu H}, \tag{8} \]

and (5), taking (6) and (8) into account, takes the form

\[ \varepsilon=K_1\sqrt{m}, \tag{9} \]

where

\[ K_1=A_1\frac{P_iG^2\lambda^3}{R^3} \sqrt{\frac{\beta\cos\chi}{\mu H}}, \tag{10} \]

\[ A_1=\frac{1}{96\pi^3}e^{-1/2} \left(\frac{e^2}{m_ec^2}\right)^{1/2} \simeq 1.08\cdot 10^{-10}. \tag{11} \]

A reflection reaching the threshold level \(\varepsilon_n\), as is seen from (9), corresponds to a mass of the meteoric body equal to

\[ m_{\min}^{*}=\left(\frac{\varepsilon_n}{K_1}\right)^2. \tag{12} \]

The duration of reflection from a persistent trail is expressed by the formula [1]

\[ T=1.124\cdot 10^{-12}\frac{\lambda^2}{16\pi^2D}\,\alpha, \tag{13} \]

where \(D\) is the diffusion coefficient.

For signals arriving from the region of the characteristic height, as follows from (8) and (13), we have

\[ T = K_2 \cdot m, \tag{14} \]

where

\[ K_2 = A' \cdot \frac{\beta \cos \chi}{\gamma H D}\,\lambda^2, \tag{15} \]

\[ A' = \frac{1.124}{36\pi}\cdot 10^{-12} = 3.17 \cdot 10^{-15}. \]

The reflection duration \(T = T_{\min}\) corresponds to the mass

\[ m_{\min} = \frac{T_{\min}}{K_2}. \tag{16} \]

Number of meteors recorded under different types of detection

Reliable detection

The number of reliably detected meteors, as follows from (1) and (2), will be equal to

\[ N_{uv} = \frac{b \cdot \Delta s \cdot \Delta t}{s - 1}\cdot \frac{1}{m_{\min}^{\,s-1}}. \tag{17} \]

where \(m_{\min}\) is determined from (16).

Expression (17) is valid in the case when the reflection with duration \(T = T_{\min}\) exceeds the threshold level \(\varepsilon_n\). If, however, the duration of the reflection that has reached the level \(\varepsilon_n\) proves to be greater than \(T_{\min}\), then \(N_{uv}\) is found from (4):

\[ N_{uv} = \frac{b \cdot \Delta s \cdot \Delta t}{s - 1}\cdot \frac{1}{m_{\min}^{* (s-1)}}, \tag{18} \]

where \(m_{\min}^{*}\) is determined by formulas (12), (10), and (11).

Accidental detection

Consider the special case: \(q(m) = \dfrac{T}{T_{\min}}\).

Expressing the probability of detection \(q(m) = \dfrac{T}{T_{\min}}\) through (14) and substituting (2) into (3), we obtain

\[ N_{sl} = \frac{b \cdot \Delta s \cdot \Delta t \cdot K_2}{T_{\min}} \int_{m_{\min}^{*}}^{m_{\min}} \frac{dm}{m^{s-1}}. \tag{19} \]

After integration and simple transformations, we write (19) in the form

\[ N_{\mathrm{acc}} = \frac{b \cdot \Delta S \cdot \Delta t} {(s-2)m_{\min}\cdot (m_{\min}^{*})^{s-2}} \left[ 1-\left(\frac{m_{\min}^{*}}{m_{\min}}\right)^{s-2} \right], \tag{20} \]

where \(m_{\min}\) and \(m_{\min}^{*}\) are determined from (16) and (12).

Expression (20) is valid for \(s \ne 2\).

For \(s=2\), as follows from (19) and (16),

\[ N_{\mathrm{acc}} = \frac{b \cdot \Delta S \cdot \Delta t}{m_{\min}} \ln \frac{m_{\min}}{m_{\min}^{*}} . \tag{21} \]

Total number of registrations

The total number of meteors detected both reliably and accidentally is equal to

\[ N_{\Sigma}=N_{\mathrm{rel}}+N_{\mathrm{acc}} . \tag{22} \]

In the case \(m_{\min}<m_{\min}^{*}\), which occurs if the duration of the reflection that has reached the threshold level proves greater than \(T_{\min}\) (Fig. 2), \(N_{\Sigma}=N_{\mathrm{rel}}\).

Let us write (22) in the form

\[ N_{\Sigma}=N_{\mathrm{rel}}\cdot \Psi , \tag{23} \]

where

\[ \Psi = 1+\psi , \tag{24} \]

and

\[ \psi=\frac{N_{\mathrm{acc}}}{N_{\mathrm{rel}}}. \tag{25} \]

Substituting (17) and (20) into (25) and (24), we obtain

\[ \Psi = 1+\frac{s-1}{s-2} \left[ \left(\frac{m_{\min}}{m_{\min}^{*}}\right)^{s-2}-1 \right]. \tag{26} \]

For \(s=2\), as follows from (24), (17), and (21), we have

\[ \Psi = 1+\ln \frac{m_{\min}}{m_{\min}^{*}} . \tag{27} \]

As is seen from (23), (17), (26), and (27), the total number of registrations may differ substantially from the number of reliably detected meteors.

If the apparatus parameters are such that \(m_{\min}\gg m_{\min}^{*}\), then for \(s>2\)

\[ \Psi \approx \frac{s-1}{s-2} \left(\frac{m_{\min}}{m_{\min}^{*}}\right)^{s-2}, \tag{28} \]

and for \(s=2\)

\[ \Psi \approx \ln \frac{m_{\min}}{m_{\min}^{*}} . \tag{29} \]

In the case \(s < 2\), as follows from (26),

\[ \Psi \approx 1 + \frac{s - 1}{2 - s}. \tag{30} \]

The dependence \(\Psi \left( \dfrac{m_{\min}}{m_{\min}^{*}}; s \right)\), constructed from formulas (26) and (27), is presented in Figs. 3 and 4.

Thus, the deviation of \(N_{\Sigma}\) from \(N_{\mathrm{uv}}\) is the more substantial (all other conditions being the same), the larger the value of the exponent \(s\).

Fig. 3. Relationship between the total number of recorded meteors \(N_{\Sigma}\) and the number of meteors detected with certainty \(N_{\mathrm{uv}}\).
\(m_{\min}\)—mass of a meteoroid corresponding to a reflection duration \(T = T_{\min}\);
\(m_{\min}^{*}\)—mass of a meteoroid corresponding to the threshold echo signal \((\varepsilon = \varepsilon_{n})\);
\(s\)—exponent characterizing the mass distribution of meteoroids.

The figure gives the dependence \(\Psi \left( \dfrac{m_{\min}}{m_{\min}^{*}} \right)\) for \(s = 1 \div 2\).

Knowing the radar parameters \((P_i, G, \lambda, \varepsilon_n\) and \(T_{\min})\), the distance of the meteor from the observation point \((R)\), the meteor velocity \(v\) (which determines the value of the diffusion coefficient \(D\) at the characteristic altitude), and the mass distribution of meteoroids \((s)\), we can estimate:

by what factor the total number of registrations exceeds the number of reliable detections (see (26), (27), (16), (15), (12), (10), and (11)):

\[ \Psi=1+\frac{s-1}{s-2}\left\{\left[3.66\cdot 10^{-6}DT_{\min}\left(\frac{P_iG^2\lambda^2}{\varepsilon_n R^3}\right)^2\right]^{s-2}-1\right\} \]

\[ (\text{for } s\ne 2) \tag{31} \]

\[ \Psi=1+\ln\left[3.66\cdot 10^{-6}DT_{\min}\left(\frac{P_iG^2\lambda^2}{\varepsilon_n R^3}\right)^2\right] \]

\[ (\text{for } s=2). \tag{32} \]

As is clear from the foregoing (in the present paper and in [3]), it is natural to expect that the results of analyzing systems may differ sharply

Fig. 4. Same as in Fig. 3 (for \(s=2.5\div 4\)).

depending on what trails (only persistent ones, or trails of different types) the system records; whether signals are detected only reliably, or whether accidental detection also occurs, etc.

It is therefore necessary to obtain criteria that make it possible to determine the conditions for detecting persistent trails.

Conditions for reliable and accidental detection of persistent trails

The dependence of the signal power on the linear electron density (in a persistent trail) can be obtained from (5) and (6):

\[ \varepsilon = A_2 \frac{P_i G^2 \lambda^3}{R^3}\,\alpha^{1/2}, \tag{33} \]

where

\[ A_2=\frac{1}{64\pi^3 e^{1/2}} \left(\frac{e^2}{m_e c^2}\right)^{1/2} \simeq 1.6\cdot 10^{-10}. \]

A trail of the persistent type will correspond to the threshold signal \(\varepsilon_n\) if

\[ \alpha(\varepsilon_n)\gtrsim n\cdot 2.4\cdot 10^{12}\ \text{el/cm}. \tag{34} \]

For \(\alpha \gtrsim 10^{13}\) el/cm, the ionized trail possesses the characteristic features of persistent trails [1]. Putting \(n \simeq 4 \div 5\), we obtain

\[ \frac{P_iG^2}{\varepsilon_n} \left(\frac{\lambda}{R}\right)^3 \lesssim 2\cdot 10^3. \tag{35} \]

When condition (35) is fulfilled, the system registers neither trails of the nonpersistent type \((\alpha \ll 2.4\cdot 10^{12})\), nor trails of the intermediate type \((\alpha \simeq 10^{12})\), but can register persistent trails \((\alpha \gg 2.4\cdot 10^{12})\).

Detection is reliable if the signal at the threshold level \((\varepsilon=\varepsilon_n)\) has a duration \(T \geq T_{\min}\), that is, if \(m^*_{\min} \geq m_{\min}\). This condition, as follows from (12) and (16), can be represented in the form

\[ \left(\frac{\varepsilon_n}{K_1}\right)^2 \geq \frac{T_{\min}}{K_2}. \tag{36} \]

Substituting \(K_1\) (10) and \(K_2\) (15) into (36), we obtain

\[ \frac{P_iG^2}{\varepsilon_n} \left(\frac{\lambda}{R}\right)^3 \leq \frac{520\lambda}{\sqrt{D T_{\min}}}. \tag{37} \]

When condition (37) is fulfilled, the detection is reliable.

Accidental detection of meteor trails occurs if the reflections that have reached the threshold level \((\varepsilon=\varepsilon_n)\) have a duration \(T<T_{\min}\), that is, if \(m^*_{\min}<m_{\min}\). Using (12) and (16), we obtain

\[ \left(\frac{\varepsilon_n}{K}\right)^2 < \frac{T_{\min}}{K_2}, \tag{38} \]

and, after substituting expressions (10) and (15) into (38), inequality (38) takes the form

\[ \frac{P_iG^2}{\varepsilon_n} \left(\frac{\lambda}{R}\right)^3 \geq \frac{520\lambda}{\sqrt{D T_{\min}}}. \tag{39} \]

Thus, the radar records trails of the persistent type, and the detection is either accidental or reliable in character if conditions (35) and (39) are satisfied, that is, if

\[ 2 \cdot 10^3 \gtrless \frac{P_i G^2}{e_n}\left(\frac{\lambda}{R}\right)^3 > \frac{520\lambda}{\sqrt{D T_{\min}}}. \tag{40} \]

It is quite obvious that condition (40) has physical meaning if

\[ \frac{520\lambda}{\sqrt{D T_{\min}}} > 2 \cdot 10^3, \tag{41} \]

which can be represented in the form

\[ T_{\min} > 7 \cdot 10^{-2}\frac{\lambda^2}{D}. \tag{42} \]

Usually, for meteor observations the wavelength range used is \(\lambda \simeq 4 \div 10\ \text{m}\); the diffusion coefficient is \(D \simeq 20\ \text{m}^2/\text{s}\) (for meteor velocities of about \(60\ \text{km/s}\) and lower); consequently, the right-hand side of inequality (42) practically always exceeds the value \(0.056\ \text{s}\).

In other words, accidental detection of persistent trails at \(\lambda \geq 4\ \text{m}\) and \(v \leq 60\ \text{km/s}\) is possible only when \(T_{\min} > 0.056\ \text{s}\). Systems whose response speed is characterized by \(T_{\min} \leq 0.056\ \text{s}\) operate in the mode of reliable detection of persistent trails (if, of course, condition (35) is satisfied).

It is quite obvious that high-speed radars capable of recording trails with a duration exceeding the pulse-repetition period will operate in the mode of reliable pulse detection, since the repetition frequencies of known radars are tens and hundreds of pulses per second, and only in P-2M radars (modernization by K. V. Kostylev) is the effective repetition frequency \(F = 25\) pulses/s.

In practice, however, for the registration of a meteor it is necessary that the duration of the reflection \(T\) exceed several sending periods (\(T_{\min}=n_1 \cdot T_i\)). Therefore, accidental detection of persistent trails is practically possible; in the case of high-speed radars it does not occur.

In the wavelength range \(\lambda < 4\ \text{m}\), it is natural to expect that the majority of recorded meteors will be detected accidentally.

Having clarified the characteristic features of high- and low-sensitivity radars, let us turn to an analysis of the dependence of the number of detected meteors on the radar parameters.

Dependence of the number of reliably detected meteors of the persistent type on the radar parameters

As already indicated, two modes of operation are possible that provide reliable detection. In the first case, all recorded meteors are detected reliably; this occurs when the reflection that has reached the threshold level has a duration \(T > T_{\min}\left(m^*_{\min} > m_{\min},\ \text{Fig. 2}\right)\). In the second case, both accidentally and reliably detected trails are recorded, and the reflection that has reached the threshold level has a duration \(T < T_{\min}\left(m^*_{\min} < m_{\min},\ \text{Fig. 1}\right)\). Consider two cases:

1. Reliable detection (the case \(m_{\min}^{*} > m_{\min}\)).

Such a regime occurs when condition (37) is satisfied. In this case, naturally, \(N_{\Sigma}=N_{\mathrm{uv}}\).

Substituting into (18) the expression for \(m_{\min}^{*}\) (12), taking into account (10) and (11), we obtain:

\[ N_{\mathrm{uv}}=B_{1}\left(\frac{P_iG^2\lambda^3}{R^3\varepsilon_n}\right)^{2(s-1)}, \tag{43} \]

where

\[ B_{1}=(1.08\cdot 10^{-10})^{2(s-1)}\cdot \frac{b\cdot \Delta s \Delta t}{s-1} \left(\frac{\beta\cos\chi}{\mu H}\right)^{s-1}. \tag{44} \]

In the wavelength range \(\lambda \simeq 3\div 10\,m\), the threshold signal \(\varepsilon_n\sim \lambda^{2.3}\) [4], and the dependence \(N_{\mathrm{uv}}(\lambda)\) may be represented in the form

\[ N_{\mathrm{uv}}\sim \lambda^{1.4(s-1)}. \tag{45} \]

1. The case \(m_{\min}^{*}<m_{\min}\).

This regime occurs when condition (39) is satisfied. The number of reliably detected meteors is determined by formula (17).

Substituting into (17) the expanded expression for \(m_{\min}\) (see (16) and (15)), we obtain

\[ N_{\mathrm{uv}}=B_{2}\cdot \frac{\lambda^{2(s-1)}}{T_{\min}^{s-1}}, \tag{46} \]

where

\[ B_{2}=(3.17\cdot 10^{-15})^{s-1}\cdot \frac{b\cdot \Delta s\cdot \Delta t}{s-1} \left(\frac{\beta\cos\chi}{\mu HD}\right)^{s-1}. \tag{47} \]

As is evident from (46) and (47), in the latter case the number of reliably detected persistent-type meteors depends neither on the transmitter power, nor on the antenna-system directivity coefficient, nor on the slant range, which at first glance might seem unnatural.

However, the independence of \(N_{\mathrm{uv}}\) from \(P_i\), \(\varepsilon_n\), \(G\), and \(R\) is quite justified: the condition \(m_{\min}^{*}<m_{\min}\) means that the mass \(m_{\min}\) of a meteoroid producing a reflection of duration \(T_{\min}\) is greater than the mass \(m_{\min}^{*}\) corresponding to the threshold signal (Fig. 1), and since the value of the mass \(m_{\min}\) is determined not by amplitude but by time relations, it is natural that in this case there is no explicit dependence of \(N_{\mathrm{uv}}\) on \(P_i\), \(\varepsilon_n\), \(G\), and \(R\).

This does not mean, however, that such a result will occur for arbitrary values of \(P_i\), etc. On the contrary, the result obtained is valid only when condition (39) is fulfilled, in which the power \(P_i\), the directivity coefficient \(G\), etc., do appear.

Thus, if the radar records only reliably detected trails, \(N_{\mathrm{uv}}\) depends substantially on the radar parameters \(P_i\), \(G\), \(\varepsilon_n\), \(\lambda\) (and on the distribution of meteoroids by mass); moreover, in the wavelength range usually used for radar observations of meteors, at \(s\simeq 2\), \(N_{\mathrm{uv}}\sim \lambda^{1.4}\). If, however, the radar records both reliably and accidentally detected trails, then \(N_{\mathrm{uv}}\) depends on the radar parameters \(\lambda\) and \(T_{\min}\) (and on the exponent \(s\), characterizing the distribution of meteoroids by mass); at \(s\simeq 2\), \(N_{\mathrm{uv}}\sim \lambda^{2}\).

Dependence of the total number of recorded persistent-type meteors on the radar parameters

In the case \(m_{\min}^{*}>m_{\min}\), all recorded meteors are detected reliably, i.e., \(N_{\Sigma}=N_{\mathrm{uv}}\).

If, however, \(m_{\min}^{*}<m_{\min}\), then both reliable and accidental detection occur, and \(N_\Sigma=N_{\mathrm{uv}}\cdot \Psi\) (see (23)—(25)).

Substituting into (23) the expression for \(N_{\mathrm{uv}}\) (46) and \(\Psi\) (31) and (32), we obtain:

1) for \(s\ne 2\)

\[ N_\Sigma = B_2\frac{\lambda^{2(s-1)}}{T_{\min}^{(s-1)}} \left\{ 1+\frac{s-1}{s-2} \left[ \left( 3.66\cdot 10^{-6}DT_{\min} \left[ \frac{P_iG^2\lambda^2}{\varepsilon_n R^3} \right]^2 \right)^{s-2} -1 \right] \right\} \tag{48} \]

\[ \left(B_2\right)\text{—see (47);} \]

2) for \(s=2\)

\[ N_\Sigma = B_2\frac{\lambda^2}{T_{\min}} \left\{ 1+\ln \left( 3.66\cdot 10^{-6}DT_{\min} \left[ \frac{P_iG^2\lambda^2}{\varepsilon_n R^3} \right]^2 \right) \right\}. \tag{49} \]

As we see, the dependence of \(N_\Sigma\) on the parameters of the radar is quite complex.

Let us consider the particular case \(m_{\min}^{*}\ll m_{\min}\), which occurs, as follows from (12), (16), (10), and (15), when the condition

\[ \frac{P_iG^2}{\varepsilon_n} \left( \frac{\lambda}{R} \right)^3 \gg \frac{520\lambda}{\sqrt{DT_{\min}}}. \tag{50} \]

is fulfilled.

If the exponent \(s>2\), then the formula determining the coefficient \(\Psi\) ((26) and (31)) is simplified (see (28)), and expression (48) takes the form

\[ N_\Sigma \simeq B_3 \left[ \frac{P_iG^2}{\varepsilon_n R^3} \right]^{2(s-2)} \frac{\lambda^{6s-10}}{T_{\min}}, \tag{51} \]

where

\[ B_3 = [3.17\cdot 10^{-15}]^{s-1}\cdot [3.66\cdot 10^{-6}]^{s-2} \frac{b\cdot \Delta S\cdot \Delta t}{s-2} \left( \frac{\beta\cos\chi}{\mu H} \right)^{s-1} \cdot \frac{1}{D}. \tag{52} \]

(see (47), (48)).

For \(s=2\) the expression for the coefficient \(\Psi\) ((27) and (32)) is also simplified (29), and formula (49) takes the form

\[ N_\Sigma = B_2\frac{\lambda^2}{T_{\min}} \ln \left( 3.66\cdot 10^{-6}DT_{\min} \left[ \frac{P_iG^2\lambda^2}{\varepsilon_n R^3} \right]^2 \right) \tag{53} \]

\[ B_2\text{—see (47).} \]

For \(s<2\) the coefficient \(\Psi\) ((26) and (31)) is transformed into (30), and the total number of recorded meteors (48) becomes equal to

\[ N_\Sigma \simeq B_2\cdot \frac{\lambda^{2(s-1)}}{T_{\min}^{\,s-1}} \left( 1+\frac{s-1}{2-s} \right). \tag{54} \]

As is seen from (51) and (54), in the case \(s>2\), \(N_\Sigma\) depends on the radar parameters \(P_i\), \(G\), \(\varepsilon_n\), \(\lambda\), and \(T_{\min}\), whereas for \(s<2\) it depends only on \(\lambda\) and \(T_{\min}\).

The dependence \(N_\Sigma(\lambda)\) in the range of wavelengths usually used in meteor radar (\(\lambda\simeq 4\div 10\ \mathrm{m}\); \(\varepsilon_n\sim \lambda^{2.3}\)) may be represented in the form:

1) \(s>2\) (see (51)):

\[ N_{\Sigma}=B_4\cdot \frac{\lambda^{1.4(s-0.7)}}{T_{\min}} . \tag{55} \]

where

\[ B_4=B_3\left[\frac{P_iG^2}{a_1R^3}\right]^{2(s-2)} \tag{56} \]

\[ \left(B_3\text{—see }[52];\qquad a_1=\frac{\varepsilon_n}{\lambda^{2.3}}\right). \]

2) \(s=2\) (see (53)):

\[ N_{\Sigma}=B_2\cdot \frac{\lambda^2}{T_{\min}}\left\{2\ln\left(\frac{\lambda^2}{\varepsilon_n}\right)+a_2\right\}, \tag{57} \]

where

\[ a_2=\ln\left(3.66\cdot 10^{-6}DT_{\min}\left[\frac{P_iG^2}{R^3}\right]^2\right) \tag{58} \]

\[ (B_2\text{—see (47)}). \]
For \(\lambda \simeq 4 \div 10\) m,

\[ N_{\Sigma}=B_2\frac{\lambda^2}{T_{\min}}\{a_3-0.6\ln\lambda\}, \tag{59} \]

where

\[ a_3=\ln\left(3.66\cdot 10^{-6}DT_{\min}\left[\frac{P_iG^2}{a_1R^3}\right]^2\right). \tag{60} \]

Let us recall that formulas (51)—(60) are valid for the case \(m_{\min}^{*}\ll m_{\min}\).

3) \(s<2\) (see (54)):

\[ N_{\Sigma}=B_5\frac{\lambda^{2(s-1)}}{T_{\min}^{s-1}}, \tag{61} \]

where

\[ B_5=\left(1+\frac{s-1}{2-s}\right)B_2 . \]

The results obtained for \(\lambda \simeq 4 \div 10\) m can be represented as follows.

If all recorded meteors are detected reliably, then

\[ N_{\Sigma}=N_{\mathrm{uv}}\sim \lambda^{1.4(s-1)} . \]

If both reliably and accidentally detected meteors are recorded, then

\[ N_{\mathrm{uv}}\sim \frac{\lambda^{2(s-1)}}{T_{\min}^{s-1}} . \]

In the particular case \(m_{\min}^{*}\ll m_{\min}\):

\[ N_{\Sigma}\sim \frac{\lambda^{1.4(s-0.57)}}{T_{\min}}\qquad \text{for } s>2; \]

\[ N_{\Sigma}\sim \frac{\lambda^2}{T_{\min}}(a_3-0.6\ln\lambda)\qquad \text{for } s=2; \]

\[ N_{\Sigma}\sim \frac{\lambda^{2(s-1)}}{T_{\min}^{s-1}}\qquad \text{for } s<2 . \]

The analysis carried out makes it possible to draw a number of conclusions concerning the detection of persistent trails.

The frequency dependence of the hourly number of recorded meteors of the persistent type is substantially determined by the value of the exponent \(s\), which characterizes the law of distribution of meteoroid masses.

In the case where the system provides “non-missing” detection of persistent trails, the number of registrations at small \(s\) depends relatively weakly on the wavelength. For example, for \(s<2\), \(N_\Sigma=N_{\mathrm{uv}}\sim\lambda^n\), where \(n<1.4\). At the same time, for large \(s\) the dependence \(N_\Sigma(\lambda)\) may prove strong. For example, for \(s\simeq3.5\), \(n\simeq3.5\).

In the case where the system is capable of registering trails detected accidentally, the dependence of the number of confidently detected meteors on wavelength proves somewhat sharper than in the preceding case; for \(s\simeq2\), \(N_{\mathrm{uv}}\sim\lambda^2\).

If the mass of the meteoroid that corresponds to reflection at the threshold level is much smaller than the mass giving a reflection whose duration is \(T=T_{\min}\), then for \(s>2\) detection will be mainly accidental in character. If, however, \(s<2\), then a significant (and for \(s<1.5\), the greater) part of the meteors is detected confidently (Fig. 2).

The dependence of the number of detected meteors \(N_\Sigma\) on wavelength proves somewhat stronger than in the case of non-missing detection.

If detection has both a confident and an accidental character, then the dependence \(N_\Sigma(\lambda)\) differs little from the dependence \(N_{\mathrm{uv}}(\lambda)\) (and in the region \(s<2\) they have the same character). If \(N_\Sigma(\lambda)\) is represented in the form \(N_\Sigma(\lambda)\sim\lambda^n\), then for \(s<2\), \(n<2\); for \(s=2\), \(n\simeq2\); for \(s\simeq3.5\), \(n\simeq4\). Thus, at large values of the exponent \(s\), the dependence \(N_\Sigma(\lambda)\) can be quite strong.

In the case where the system is capable of registering accidentally detected trails, both \(N_\Sigma\) and \(N_{\mathrm{uv}}\) depend substantially on the operating speed of the system: \(N_{\mathrm{uv}}\) and \(N_\Sigma\) (for \(s<2\)) \(\sim \dfrac{1}{T_{\min}^{\,s-1}}\); for \(s>2\), \(N_\Sigma\sim\dfrac{1}{T_{\min}}\).

As is evident from the relations given, the character of the function \(N_\Sigma(\lambda)\) depends only weakly on the operating speed of the system (\(T_{\min}\)).

In turn, the change in \(N_\Sigma\) due to an increase or decrease in the operating speed of the system is practically unrelated to the wavelength.

Comparison of \(N_\Sigma(\lambda)\) in the registration of persistent and nonpersistent trails

As is evident from a comparison of the results of the analysis carried out in the present work and in [3], in the registration of nonpersistent and persistent trails the dependences \(N_\Sigma(\lambda, T_{\min})\) have a completely different character.

First of all, in the case of a highly sensitive system capable of registering nonpersistent trails, the character of the dependence \(N_\Sigma(\lambda)\) is substantially determined by the operating speed of the system: at small \(T_{\min}\), \(N_\Sigma\) changes little in passing from one wave to another (within the range \(\lambda\simeq4\div10\) m); at large \(T_{\min}\), a change in \(\lambda\) leads to a very sharp change in \(N_\Sigma\).

In the case of a low-sensitivity system, however, the dependence of \(N_\Sigma\) on \(\lambda\) and \(T_{\min}\) proves much weaker than in the case of highly sensitive systems at small \(\lambda\) and at large \(T_{\min}\), but stronger than at large \(\lambda\) and small \(T_{\min}\).

Conclusions

In the case of a low-sensitivity system that detects only trails of the persistent type, the dependence of the number of recorded meteors on the parameters of the apparatus is determined by the operating mode of the locator and by the nature of the mass distribution of meteoroid bodies.

1) First mode—continuous detection of meteor echoes that have exceeded the threshold level.

1. The number of detected meteors \((N_{\Sigma}=N_{\mathrm{conf}})\) depends very sharply on the radar parameters \(P_i, G, \varepsilon_n, \lambda\), and also on the slant range \(R\); this dependence is the sharper, the larger the parameter \(s\) characterizing the mass distribution of meteoroid bodies.

For example, for \(s=2\),

\[ N_{\Sigma}\sim \frac{P_i^2G^4\lambda^6}{R^6\varepsilon_n^2}. \]

1. In the wavelength range usually used for meteor radar \((\lambda \simeq 4\div 10\,\text{m})\), the number of recorded echoes is \(\sim \lambda^{1.4(s-1)}\).

2. A change in the duty cycle of the system (provided that \(T_{\min}\) remains small) does not affect the number of detected meteors.

3. If the radar parameters are unstable and \(G\) and \(R\) vary (owing to a change in the zenith distance of the stream radiant \(\chi\)), \(N_{\Sigma}\) changes very sharply—and the more sharply, the larger \(s\).

4. The requirements on the stability of the radar parameters \(P_i, \varepsilon_n, \lambda\), and \(G\) are very stringent.

2) Second mode—only those signals are registered whose duration exceeds some minimum value \(T_{\min}\) (although the system could also detect signals with \(T<T_{\min}\) that exceeded the threshold level).

1. The number of detected meteors \((N_{\mathrm{conf}})\) depends on the wavelength \(\lambda\) and on the duty cycle of the system \((T_{\min})\), and this dependence is the stronger, the larger \(s\); for \(s=2\),

\[ N_{\mathrm{conf}}\sim \frac{\lambda^2}{T_{\min}}. \]

1. \(N_{\mathrm{conf}}\) does not depend on \(P_i, \varepsilon_n, G, R\).

2. Instability of the parameters \(P_i\) and \(\varepsilon_n\), and changes in \(G\) and \(R\) (for example, owing to a change in the stream’s \(\chi\)) do not change the number \(N_{\mathrm{conf}}\) (provided that the condition \(m_{\min}^{*}<m_{\min}\) is not violated). Instability of \(\lambda\) and \(T_{\min}\) leads to a change in \(N_{\mathrm{conf}}\).

3) Third mode—all echoes that have exceeded the threshold level are registered, with long-duration echoes being recorded reliably (without omissions), and short-duration ones accidentally (that is, they may be missed).

1. The total number of recorded meteors \(N_{\Sigma}\), detected both reliably and accidentally, depends on the locator parameters \(P_i, \varepsilon_n, G, \lambda, T_{\min}\), and also on \(R\). This dependence is very complex and is determined by the nature of the law of the mass distribution of meteoroid bodies.

2. The relative number of accidentally detected meteors is substantially determined by the meteor-stream parameter \(s\): the larger \(s\), the more appreciable the fraction of accidentally detected meteors.

3. If the relative number of accidentally detected meteors is significant (which occurs when \(m_{\min}^{*}\ll m_{\min}\)), the character of the dependence of \(N_{\Sigma}\) on the system parameters is qualitatively different for different values of \(s\):

a) for \(s>2\), \(N_{\Sigma}\) depends very sharply on \(P_i, G, \varepsilon_n, \lambda, T_{\min}\), and also on \(R\);

b) for $s=2$ the dependence of $N_{\Sigma}$ on $\lambda$ and $T_{\min}$ is strong, and on $P_i$, $G$, $\varepsilon_n$ and $R$—weaker than in the case $s>2$;

c) for $s<2$, $N_{\Sigma}$ depends on $\lambda$ and $T_{\min}$ (but more weakly than in the case $s=2$) and is practically independent of $P_i$, $G$, $\varepsilon_n$ and $R$.

1. Thus, a change in the apparatus parameters $P_i$, $G$, $\varepsilon_n$, as well as a change in $R$ owing to the variability of the zenith distance of the stream radiant $\chi$, causes a sharp change in the numbers when observing streams with large $s$, but practically does not lead to a change in $N_{\Sigma}$ when observing streams with small $s$. Instability of $\lambda$ and $T_{\min}$ affects $N_{\Sigma}$ both for large and for small $s$.

2. The requirements on the stability of the apparatus parameters increase as $s$ increases; moreover, the requirements on the stability of $\lambda$ and $T_{\min}$ remain stringent for all values of $s$.

The requirements on the stability of $P_i$, $\varepsilon_n$, $G$ are high in the case of streams with $s>2$, weaker for $s \simeq 2$, and low when observing streams with $s<2$ (but, naturally, under the condition that the inequality $m^*_{\min}<m_{\min}$ is not violated).

4) On corrections.

1. In radiolocation of persistent trails, it is necessary to introduce corrections because of the instability of the transmitter power and of the threshold-signal power: a) in non-stop detection of meteors, and b) in recording reflections that exceeded the threshold level, when observing streams (and sporadic meteors) with $s \geq 2$.

2. The introduction of corrections caused by the instability of $P_i$ and $\varepsilon_n$ is insignificant in the case of recording reflections with duration $T>T_{\min}$, and also when recording all reflections that exceeded the threshold level, if $s<2$.

Let us recall that all the conclusions formulated are valid only for relatively low-sensitivity systems that record trails of the persistent type.

LITERATURE

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

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

3. E. I. Fialko. Reliable and accidental detection of non-persistent meteor trails, Radiotekhnika, vol. 16, No. 6, p. 25, 1961.

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