Dependence of the Average Hourly Number of Detected Meteors on the Pulse Repetition Frequency of a Radar

E. I. Fialko

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

Introduction

Knowledge of the dependence of the hourly number of detected meteors on the pulse repetition frequency is necessary in interpreting the results of radio observations of meteors, in comparing the number of registrations on different radars, and also in designing radar stations intended for meteor observations.

In the present work, experimental data are compared with simplified theoretical dependences obtained for the reception of reflections arriving from an element of the echo plane lying in the region of the characteristic height; it is assumed that the apparatus has high sensitivity and is capable of detecting underdense ionized trails, i.e., trails with a linear electron density \(\alpha \ll 10^{12}\) electrons/cm \([1]\).

Main Results of the Analysis and Experiment

The number of recorded meteors depends substantially on the operational efficiency of the radar, estimated by the minimum time \(T_{\min}\) during which the reflection must exceed the threshold level corresponding to reliable (certain) detection of meteors.

As was shown earlier, the number of reliably detectable meteors is equal to \([2, 3]^1\):

\[ N_{\text{conf}} = A_1 \cdot e^{-\frac{16\pi^2 D T_{\min}(s-1)}{\lambda^2}}, \tag{1} \]

where \(D\) is the diffusion coefficient; \(s\) is an exponent characterizing the mass distribution of meteoroid bodies \([1]\); \(\lambda\) is the wavelength; \(A_1\) is a coefficient independent of \(T_{\min}\).

\(^1\) In formula (1), the influence of the initial radius of the ionized trail on the number of detectable meteors is not taken into account.

Naturally, we may assume that \(T_{\min}=\dfrac{n}{F_i}\), where \(F_i\) is the pulse repetition frequency, and \(n\) is a constant number depending on the indication conditions, etc.

In Fig. 1 the dependence

\[ \frac{N_{y v}(F_i)}{N_{y v}(F_i=300)} \]

is plotted for \(n=1\) and \(n=2\).

The same figure also shows the curve corresponding to the hourly number, taking into account not only reliably but also accidentally detected meteors, \(N_{\Sigma}(F_i)\) [2, 3] (also referred to the hourly number at \(F_i=300\) pulses/sec:

\[ \frac{N_{\Sigma}(F_i)}{N_{\Sigma}(F_i=300)} \]
).

The total number \(N_{\Sigma}(F_i)\) is found from [2, 3]

Fig. 1. Dependence of the average hourly number of detected meteors \(N\) on the pulse repetition frequency \(F_i\) (wavelength \(\lambda=10\ \mathrm{m}\)). \(N_{y v}\) is the number of reliably detected meteors at \(T_{\min}=\dfrac{n}{F_i}\); the solid curve corresponds to the value \(n=1\), the dashed curve to \(n=2\). \(N_{\Sigma}\) is the total number of meteors detected not only reliably but also accidentally. \(N_{int}\) is the number of meteors recorded in integration mode. Experimental data are marked by points (photographic film speed \(v_{phl}=125\ \mathrm{cm/min}\)).

\[ N_{\Sigma}=N_{y v}\cdot \psi, \tag{2} \]

where

\[ \psi=1+\frac{1}{s-1}\cdot\frac{T_e}{T_{\min}} \left[ (1-s)\frac{T_{\min}}{T_e} -1+e^{\frac{T_{\min}}{T_e}(s-1)} \right], \tag{3} \]

\[ T_e=\frac{\lambda^2}{16\pi^2D} \]

is the reflection duration, measured at the level

\[ \frac{U_{m0}}{e}; \]

\(U_{m0}\) is the greatest (initial) amplitude of the meteor radio echo.

The curves in Fig. 1 were constructed for \(\lambda = 10\ \text{m}\), \(D = 8\ \text{m}^2/\text{s}\), and \(s = 2\). The experimental data are marked by points (at \(F_i = 300\) and \(F_i = 75\ \text{pulses/s}\)).

The experiment was carried out on March 1, 1959, at the TPI-2 station (\(\lambda = 10\ \text{m}\)) [4], using photographic film drawn at a speed \(v_{\text{pl}} = 1.25\ \text{m/min}\).

Fig. 1 also shows the approximate dependence of the average hourly number on the repetition frequency in the case where averaging effects are present [5],

\[ \frac{N(F_i)}{N(F_i = 300)} = \sqrt{\frac{F_i}{300}}, \tag{4} \]

constructed for \(s = 2\).

As can be seen from Fig. 1, at a high film-drawing speed the accumulation effects are practically absent, and the average hourly number of detected meteors is determined by the operational speed of the radar, \(T_{\min} = \frac{1}{F_i} \div \frac{2}{F_i}\).

However, photographic recording of meteor radio echoes is more often carried out with slowly drawn film (at a speed on the order of several \(\text{cm/min}\)).

In such a recording mode, signal-accumulation effects cannot be neglected.

Fig. 2 presents the dependence (4), corresponding to the integration mode (solid curve), and the ratio

\[ \frac{N_{\text{uv}}(F_i)}{N_{\text{uv}}(F_i = 300)}, \]

constructed from formula (1) for \(\lambda = 4\ \text{m}\), \(D = 8\ \text{m}^2/\text{s}\), \(s = 2\), and \(n = 2\) (dashed curve).

Fig. 2. Dependence \(N(F_i)\) for \(\lambda = 4\ \text{m}\). The dashed curve corresponds to the number of reliably detected meteors at \(T_{\min} = \frac{2}{F_i}\); the solid curve corresponds to the number of registrations in the integration mode. The experimental data are marked by points (photographic-film motion speed \(v_{\text{pl}} = 2\ \text{cm/min}\)).

Fig. 2 also marks the results of an experiment carried out from 26/I to 2/II-1959 on the M-3 radar (\(\lambda = 4\ \text{m}\)) [6].

As can be seen from Fig. 2, in the repetition-frequency range \(F_i = 100 \div 300\) pulses/sec the experimental data agree well with the analytical dependence \(N(F_i)\) corresponding to the presence of integration. At low repetition frequencies (\(F_i = 50\) pulses/sec), the repetition period \(T_i\) and the duration of the reflection \(T_{\mathrm{sl}}\) prove to be approximately the same, and integration is practically absent; the experimental data correspond to the dependence \(N(F_i)\) characteristic of the case in which accumulation effects are absent.

Conclusions

From consideration of the dependences \(N(F_i)\) and of the experimental results presented in Figs. 1 and 2, a number of conclusions may be drawn.

1. In the absence of accumulation effects, changing the repetition frequency in the range \(F_i \ge 100\) pulses/sec causes an insignificant increase in the hourly number of detected meteors \(N\) at comparatively long wavelengths (\(\lambda\) about \(10\) m); as the wavelength is shortened, the change in \(N\) with variation of \(F_i\) becomes sharper.

In the range \(F_i < 100\) pulses/sec, however, a decrease in the repetition frequency leads to a sharp decrease in the hourly number; the decrease in \(N\) is the greater, the shorter the operating wavelength.

1. In the presence of signal-accumulation effects, the dependence \(N(F_i)\) is (approximately) a parabola, and for \(s = 2\), \(N \sim \sqrt{F_i}\).

2. In photographic recording using slowly moving film, as the repetition frequency is decreased the integration effects weaken, and the dependence \(N(F_i)/N(F_i = 300)\) approaches more and more closely the curve \(N(F_i)\) characteristic of the case in which integration is absent.

3. When the repetition frequency is decreased, the hourly number of detected meteors falls; at large wavelengths (about \(\lambda = 10\) m), integration makes this fall sharper, while at small wavelengths it slows it down.

References

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 radar method of meteor investigation. Izvestiya Sibirskogo fiziko-tekhnicheskogo instituta, issue 37, p. 229, 1959.

3. E. I. Fialko, Confident and random detection of unstable meteor trails. Radiotekhnika, 16, No. 6, p. 24, 1961.

4. E. I. Fialko. Radar observations of meteors at wavelength \(\lambda = 10\) m. Astr. tsirkulyar, No. 175, p. 21, 1956.

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

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