RADIO DETECTION OF PERSISTENT METEOR TRAILS IN INTEGRATION MODE

E. I. Fialko

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

Introduction

Low-sensitivity equipment detects meteor trails only of the persistent type, that is, trails with a linear electron density \(\alpha \gg 10^{12}\) el/cm.

Usually, radars used for meteor observations are capable of recording nonpersistent trails (with \(\alpha \ll 10^{12}\) el/cm); however, with a sharp increase in the noise level, with a decrease in transmitter power, and also in carrying out special experiments connected with varying the threshold signal over wide limits, etc., it may turn out that the radar detects only persistent trails.

The dependence of the number of recorded meteors on the parameters of the radar in such a mode has already been considered earlier [1, 2, and others]; however, the presence of photographic registration was not taken into account. At the same time, for the most part, meteor observations are accompanied by photographic registration. Therefore it becomes necessary to investigate the dependence between the number of detected meteor trails and the parameters of the equipment, the atmosphere, and the meteors, taking into account the integrating properties of the photographic film.

Since persistent trails give long-duration reflections, the analysis may be carried out under the assumption of a large number of integrations of both the signal and the noise, which substantially simplifies consideration of the question, since it makes it possible to assume that the effective value of the noise, with allowance for integration, is proportional to the root mean square of the number of noise averagings \((n_{\sh})\), and that the signal increases in proportion to the number of signal averagings \((n_c)\) [2, 3]:

\[ \varepsilon_{\sh \,\mathrm{int}} = \varepsilon_{\sh} \cdot n_{\sh}, \tag{1} \]

\[ \varepsilon_{c \,\mathrm{int}} = \varepsilon_c \cdot n_c^{2}, \tag{2} \]

where \(\varepsilon_{\sh}\) and \(\varepsilon_c\), \(\varepsilon_{\sh \,\mathrm{int}}\) and \(\varepsilon_{c \,\mathrm{int}}\) are the powers of the signal and the noise at the output of the system in the absence of (and in the presence of) integration.

A meteor will be detected if

\[ \varepsilon_{c \,\mathrm{int}} > k \varepsilon_{\sh \,\mathrm{int}}, \tag{3} \]

where \(k\) is a coefficient of order unity.

As follows from (1)—(3), the condition for detecting a meteor can be represented in the form

\[ \varepsilon_c \geq \varepsilon_{\sh}\cdot k \cdot \frac{n_{\sh}}{n_c^2} \tag{4} \]

in the case when \(n_c<n_{\sh}\), which occurs when the duration of reflection from the meteor trail \(T_{\mathrm{sl}}\) is less than the noise integration time \(t_{\sh}\); \(t_{\sh}\) is the time during which the continuously moving film is displaced by a distance equal to the diameter \(d_n\) of the image, on the film, of the light spot observed on the indicator screen.

If, however, \(t_{\sh}<T_{\mathrm{sl}}\), then \(n_c=n_{\sh}\), and inequality (3) takes the form

\[ \varepsilon_c \geq \varepsilon_{\sh}\frac{k}{n_{\sh}} . \tag{5} \]

Thus, if in the absence of integration the power of the threshold signal is equal to

\[ \varepsilon_n = k\varepsilon_{\sh}, \tag{6} \]

then in the presence of integration

\[ \varepsilon_{n\,\mathrm{int}}=\varepsilon_n\frac{n_{\sh}}{n_c^2} \tag{7} \]

(for \(n_c>n_{\sh}\)), and

\[ \varepsilon_{n\,\mathrm{int}}=\frac{\varepsilon_n}{n_{\sh}} \tag{8} \]

for \(n_c=n_{\sh}\).

The case when the duration of reflection is less than the duration of averaging

The numbers of averagings of the signal \(n_c\) and of the noise \(n_{\sh}\) will be respectively equal to

\[ n_c=T_{\mathrm{sl}}\cdot F_i \tag{9} \]

and

\[ n_{\sh}=t_{\sh}F_i=\frac{d_n}{V_{\mathrm{pl}}}F_i, \tag{10} \]

where \(F_i\) is the pulse repetition frequency; \(V_{\mathrm{pl}}\) is the speed of motion of the film; \(T_{\mathrm{sl}}\) is the duration of reflection from a persistent trail, equal to [1]

\[ T_{\mathrm{sl}}=1.12\frac{10^{-12}}{16\pi^2}\cdot\frac{\lambda^2 a}{D}; \tag{11} \]

\(\lambda\) is the wavelength, \(D\) is the diffusion coefficient; \(a\) is the linear electron density. Formula (11) is valid for normal reflection of radio waves from an ionized trail [1].

The power of the echo signal produced by a persistent trail at a distance \(R\) from the locator, as is known, is equal to [1]

\[ \varepsilon_c=A\frac{P_iG^2\lambda^3}{R^3}a^{1/2}, \tag{12} \]

where \(P_i\) is the radiated power in the pulse; \(G\) is the antenna directivity factor; \(A=1.62\cdot10^{-10}\). Substituting into (4) the expres-

tions (12), (9), (10), (11), and (6), we obtain the value of the linear density \(\alpha\) of the detected trail

\[ \alpha > B \left[\frac{\varepsilon_n D^2 R^3 d_n}{P_i G^2 \lambda^7 F_i V_{n\ell}}\right]^{0.4}, \tag{13} \]

where

\[ B=\left[\left(\frac{16\pi^2}{1.12}\right)^2 \cdot \frac{10^{24}}{A}\right]^{0.4} \approx 1.5\cdot 10^{15}. \]

Considering the case of reception from a layer located at the characteristic height [1], we have

\[ \alpha=\frac{4}{9}m\frac{\beta\cos\chi}{\mu H}, \tag{14} \]

where \(\beta\) is the probability of ionization; \(m\) is the mass of the meteoroid; \(\chi\) is the zenith distance of the meteor radiant; \(\mu\) is the mass of an atom of the meteoroid; \(H\) is the height of the homogeneous atmosphere.

The minimum mass of a meteoroid producing a detectable trail, as follows from (13) and (14), is

\[ m_{\min}=\frac{9}{4}B\frac{\mu H}{\beta\cos\chi} \left(\frac{\varepsilon_n D^2 R^3 d_n}{P_i G^2 \lambda^7 F_i V_{n\ell}}\right)^{0.4}. \tag{15} \]

The number of reflections per unit area lying in the echo plane [1], recorded per unit time, will be equal to

\[ N=\int_{m_{\min}}^{\infty} p(m)\,dm; \tag{16} \]

here \(p(m)\) is the differential mass-distribution law for meteoroids, usually represented in the form

\[ p(m)=\frac{b}{m^s}, \tag{17} \]

where \(b\) and \(s\) are coefficients independent of \(m\). Substituting expressions (17) and (15) into (16), we obtain

\[ N=B_1\cdot \frac{b}{s-1} \left\{ \frac{\beta\cos\chi}{\mu H} \left( \frac{P_i G^2 \lambda^7 F_i V_{n\ell}}{\varepsilon_n D^2 R^3 d_n} \right)^{0.4} \right\}^{s-1} \tag{18} \]

where

\[ B_1=\frac{4}{9}B^{1-s}=(2.6\cdot 10^{-16}\ \mathrm{cm})^{s-1}. \]

This dependence, as indicated above, is valid when the condition \(t_{\sh}>T_{\mathrm{sl}}\) is satisfied, that is, when

\[ \frac{\lambda^2 V_{n\ell}}{d_n}<1.4\cdot 10^{-14}\frac{D}{\alpha}. \tag{19} \]

The case when the duration of the reflection is greater than the averaging duration

If \(t_{\sh}<T_{\mathrm{sl}}\), then the numbers of signal and noise averages are identical and, as follows from (5), (8), (10), (12), (14), and (16),

\[ N=B_2\cdot \frac{b}{s-1}\left\{\frac{\beta\cos\chi}{\mu H} \left(\frac{P_i G^2\lambda^3 F_i d_n}{\varepsilon_n R^3 V_{nl}}\right)^2\right\}^{s-1}, \tag{20} \]

where

\[ B_2=\left(\frac{4}{9}A^2\right)^{s-1} =(11.3\cdot 10^{-21}\mathrm{cm})^{s-1}. \]

Formula (20) is valid when the condition

\[ \frac{\lambda^2 V_{nl}}{d_n}\geq 1.4\cdot 10^{14}\frac{D}{a}. \tag{21} \]

is satisfied.

It should be noted that, at a low velocity of motion of the trail, short-lived persistent reflections will be integrated in the regime \(t_{sh}>T_{sl}\), and long-lived ones in the regime \(t_{sh}<T_{sl}\).

Analysis of the results

As can be seen from (18) and (20), the dependence of the number of meteor registrations on the parameters of the locator (and also on the parameters of the meteor and of the atmosphere) may be different for long-time \((t_{sh}>T_{sl})\) and short-time \((t_{sh}<T_{sl})\) averaging.

In the regime of long-time averaging, \(N\) depends only weakly on the repetition frequency,

\[ N\sim F_i^{0.4(s-1)}, \tag{22} \]

but depends strongly on the wavelength,

\[ N\sim \lambda^{2.8(s-1)}. \tag{23} \]

In the case of short-time averaging \((t_{sh}<T_{sl})\), the number depends much more sharply (than in the case \(t_{sh}>T_{sl}\)) on the repetition frequency,

\[ N\sim F_i^{2(s-1)} \tag{24} \]

and on the wavelength,

\[ N\sim \lambda^{6(s-1)}. \tag{25} \]

Let us note that the dependence \(N(\lambda,\varepsilon_n,P_i,G,R)\) in the regime \(t_{sh}<T_{sl}\) has the same character as in the absence of integration, but with high system efficiency [2].

It is interesting to note that for \(s\simeq 2\), as follows from (23), the number of registered meteors is approximately proportional to the cube of the wavelength \((N\sim \lambda^3)\); practically the same result also occurs in the detection of nonpersistent trails (without allowance for integration) [4].

It is usually assumed that, in the regime of detecting persistent trails, the number depends sharply on the transmitted power, and therefore the requirement on the stability of the transmitter power becomes considerably more stringent than in the registration of nonpersistent trails [5, 6]. This conclusion was drawn on the basis of an analysis of the dependence \(N(P_i)\sim P_i^2\), obtained without taking into account the averaging effect that occurs under photoindication.

As can be seen from (18),

\[ N\sim P_i^{0.4(s-1)}, \tag{26} \]

which is closer to

\[ N\sim P_i^{0.5(s-1)}, \tag{27} \]

valid in the observation of nonpersistent trails (in the absence of averaging), than to

\[ N\sim P_i^{2(s-1)}, \tag{28} \]

that occurs in the detection of persistent trails (also without averaging).

But, as has already been noted, when condition (21) is fulfilled, relations (22), (23), and (26) lose their force, and the dependence of \(N\) on \(\lambda\), \(F_i\), \(P_i\), etc., becomes very sharp. The natural question arises as to which regime ((19) or (21)) obtains under real conditions, which of these regimes is more advantageous, and what determines the transition from one regime to the other.

It is easy to see that one of the factors determining the integration regime is the speed of motion of the photographic film.

Dependence of the number of recorded meteors on the photographic-film pull speed \(V_{n.l}\)

As long as the film speed remains small \((t_{\mathrm{sh}}>T_{s.l}\) and, consequently,

\[ V_{n.l}<V^{*}_{n.l}=\frac{d_n}{T_{s.l}}), \]

as follows from (18),

\[ N\sim V_{n.l}^{0.4(s-1)} . \tag{29} \]

As soon as the speed of motion of the film becomes greater than \(V^{*}_{n.l}\), a further increase of \(V_{n.l}\) leads to a decrease in the number of registrations (20):

\[ N\sim V_{n.l}^{-2(s-1)} . \tag{30} \]

Thus, the greatest number of detected meteors corresponds to the speed \(V^{*}_{n.l}=d_n/T_{s.l}\), and the dependence \(N(V_{n.l})\), as follows from (29) and (30), can be represented in the form

\[ \frac{N(V)}{N_{\max}}=\left(\frac{V_{n.l}}{V^{*}_{n.l}}\right)^{0.4(s-1)} \quad \text{for } V_{n.l}<V^{*}_{n.l}; \tag{31} \]

\[ \frac{N(V)}{N_{\max}}=\left(\frac{V_{n.l}}{V^{*}_{n.l}}\right)^{-2(s-1)} \quad \text{for } V_{n.l}>V^{*}_{n.l}, \tag{32} \]

where \(V^{*}_{n.l}\) (see (19) or (21)) is equal to

\[ V^{*}_{n.l}=1.4\cdot 10^{14}\frac{d_nD}{a\lambda^2}. \tag{33} \]

If one takes \(d_n=0.1\ \mathrm{mm}\), \(D=8\ \mathrm{m^2/sec}\) (which corresponds to a meteor speed \(V\simeq45\ \mathrm{km/sec}\) [1, 7]), and \(a\simeq10^{13}\ \mathrm{el/cm}\), then

\[ V^{*}_{n.l}\simeq \frac{100}{\lambda^2} \]

(here \(\lambda\) is in m, \(V^{*}\) in mm/sec). For \(\lambda=10\ \mathrm{m}\), \(V^{*}_{n.l}\simeq1\ \mathrm{mm/sec}=6\ \mathrm{cm/min}\).

Figure 1 presents the dependence \(\frac{N(V)}{N_{\max}}\), constructed from formulas (31) and (32) for \(s=2\).

As can be seen from Fig. 1, increasing \(V_{n.l}\) in the region \(V_{n.l}>V^{*}_{n.l}\) leads to a sharp decrease in the number of registrations. Thus, for example, at \(V_{n.l}=2V^{*}_{n.l}\) the number of detected meteors decreases by a factor of 4.

In this connection, and also for economical use of film, it is advisable to use a film speed \(V_{n.l}\simeq 0.5\,V^*_{n.l}\); in this case the loss in number will amount to 20–30% of \(N_{\max}\).

When \(V_{n.l}<V^*_{n.l}\) is chosen, the dependence of the number on the apparatus parameters, determined by formula (18), will be weaker than in the absence of averaging.

It should be noted that in the averaging mode the dependence of the number of registrations on the film speed in the detection of persistent trails proves to be sharper than in the detection of nonpersistent trails [8].

Conclusions

The averaging mode can substantially change the dependence of the number of detected meteors on the parameters of the radar.

If the averaging time is greater than the duration of the reflections, then (for \(s\simeq 2\)) the number of recorded persistent meteor trails is proportional (approximately) to the cube of the wavelength and to the square root of the pulse repetition frequency.

The dependence of the number of registrations on the speed of motion of the photographic film is nonmonotonic. It is advisable to choose a film-drawing speed two times smaller than \(V^*_{n.l}\), corresponding to the maximum number of registrations.

Fig. 1. Dependence of the number of registrations \(N\) on the speed of motion of the film \(V_{n.l}\) for \(s=2\). For \(V<V^*_{n.l}\)

\[ \frac{N(V_{n.l})}{N_{\max}}=\left(\frac{V_{n.l}}{V^*_{n.l}}\right)^{0.4}; \]

for \(V>V^*_{n.l}\)

\[ \frac{N(V_{n.l})}{N_{\max}}=\left(\frac{V^*_{n.l}}{V_{n.l}}\right)^2 . \]

References

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2. E. I. Fialko. Confident and random radio detection of persistent meteor trails, Izvestiya TPI, vol. 100, 84, 1961.

3. G. S. Zubarev. On the question of the real sensitivity of a pulse receiver under conditions of a brightness mark and photographic recording of signals, Candidate dissertation, 1955.

4. E. I. Fialko. On the dependence of the hourly number of detected meteors on the wavelength of the radar, Radiotekhnika, vol. 15, No. 12, 10, 1960.

5. F. I. Peregudov. Some problems of meteor radiolocation, Candidate dissertation, Tomsk Polytechnic Institute, 1959.

6. E. I. Fialko. On the accuracy of measuring the parameters of a radar intended for meteor observations, Radiotekhnika, vol. 15, No. 6, 67, 1960.

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

8. E. I. Fialko. Radio detection of nonpersistent meteor trails in the photographic-recording mode, Izvestiya TPI, vol. 100, 100, 1962.