On the Dependence of the Average Hourly Number of Detected Meteors on Certain Radar Parameters (Case of Persistent Radio Echoes)
Abstract Generated abstract
This paper extends a previously developed comparison method for meteor radar systems to the detection of persistent radio echoes from trails with high linear electron density. It derives how the average hourly number of detected meteors depends on pulse power, receiver threshold, pulse shape, receiver resonance characteristic, and the meteoroid mass distribution exponent. Comparing probabilistic, smooth pulse and resonance forms with rectangular approximations under an optimal bandwidth and pulse duration relation, the analysis suggests that smooth forms can yield a higher detection rate, about one and a half times for a typical exponent near 2. The paper concludes that for low-sensitivity meteor radars without stringent coordinate-determination requirements, a resonant amplifier and smooth emitted pulse are preferable to rectangular characteristics.
Full Text
On the Dependence of the Average Hourly Number of Detected Meteors on Certain Radar Parameters
(Case of Persistent Radio Echoes)
E. I. Fialko
(Presented by the scientific seminar of the Radio Engineering Faculty)
Introduction
Earlier, the dependence of the number of detected meteors on the shape of the resonance characteristic of the receiver and on the shape of the envelope of the generated pulse was considered [1]. It was assumed there that the radar can detect trails with a linear electron density \(\alpha < 2.4 \cdot 10^{12}\) el/cm (i.e., trails of the nonpersistent type). In a number of cases, radars used for meteor studies can detect only trails of the persistent type (with \(\alpha > 2.4 \cdot 10^{12}\) el/cm). This occurs when the sensitivity of the radar is low, especially when receiving radio echoes from large distances (for example, with a high position of the radiant of the stream).
The number of meteors detected per unit time in the case \(\alpha > 2.4 \cdot 10^{12}\) el/cm will be equal to [2]
\[ N = A \left( \frac{P_i}{\varepsilon} \right)^{2(s-1)}, \tag{1} \]
where \(P_i\) is the power in the pulse; \(\varepsilon\) is the power of the receiver threshold signal; \(s\) is a constant characterizing the law of distribution of meteoric bodies by mass [2]; \(A\) is a coefficient independent of the signal shape and of the shape of the receiver resonance characteristic.
Using the same method as in [1], we write (1) in the form
\[ N = A(B \cdot D \cdot P_0)^{2(s-1)}, \tag{2} \]
where \(P_0\) is the average power; \(D = 2 \dfrac{T_i}{kt \cdot F}\); \(T_i\) is the pulse repetition period; \(t\) is the absolute room temperature; \(F\) is the noise factor of the receiver (taking external interference into account); \(k\) is Boltzmann’s constant.
\[ B = \frac{\psi^2}{\xi^2 \tau_{\mathrm{eff}} \cdot \Delta f_{\mathrm{sh}}}, \tag{3} \]
\(\tau_{\mathrm{eff}}\) is the effective pulse duration [1]; \(\Delta f_{\mathrm{sh}}\) is the receiver noise bandwidth; \(\psi\) is a coefficient characterizing the variation
of the pulse amplitude due to distortions introduced by the linear part of the receiver; $\xi$ is the minimum signal-to-noise ratio at the detector input that ensures reliable detection of a weak signal against the background of interference.
Comparison method
The ratio of the average hourly numbers of meteors detected by two radars differing only in the shapes of the resonance characteristics of the receivers and in the shapes of the envelopes of the pulses generated by them will depend on the mass distribution of meteor bodies (characterized by the exponent $s$):
\[ \frac{N_1}{N_2}=\left(\frac{B_1}{B_2}\right)^{2(s-1)} . \tag{4} \]
If two radars (with different shapes of the resonance characteristics of the receivers and different pulse shapes) detect the same number of meteors, then the ratio of the average transmitted powers will be equal to (as also in the case $a<2.4\cdot 10^{12}$ el/cm)
\[ \frac{P_{01}}{P_{02}}=\frac{B_2}{B_1} \tag{5} \]
(the indices 1 and 2 in (4) and (5) correspond to the first and second radars).
In the case where an optimal relation is maintained between the bandwidth of the linear part of the receiver $\Delta f$ and the duration of the input pulse $\tau$ ($\Delta f_{\mathrm{opt}}\tau=c$), and there is also a linear dependence between the noise bandwidth $\Delta f_{\mathrm{sh}}$ and the passband $\Delta f$, as well as between $\tau$ and the effective pulse duration $\tau_{\mathrm{ef}}$ ($\tau_{\mathrm{ef}}=a\tau$; $\Delta f_{\mathrm{sh}}=b\cdot \Delta f$), expression (2) for the coefficient $B$ (as also in the case of reception of nonpersistent radio echoes) takes the form
\[ B=\frac{\psi^2}{\xi^2 a b c}. \tag{6} \]
Case of probabilistic and rectangular approximations
In the case of the probabilistic approximation we have [1] $a_1\simeq 0.63$; $b_1=1.065$; $c_1\simeq 0.75$; $\psi_1=0.707$, and, consequently,
\[ B_1\simeq \frac{1}{\xi_1^2}. \tag{7} \]
In the case of the rectangular approximation [1] $a_2=1$; $b_2=1$; $c_2\simeq 1.37$; $\psi_2\simeq 1.062$ and, consequently,
\[ B_2\simeq \frac{0.825}{\xi_2^2}. \tag{8} \]
Substituting (7) and (8) into (4), we obtain
\[ \frac{N_1}{N_2}\simeq\left(1.21\cdot \frac{\xi_2^2}{\xi_1^2}\right)^{2(s-1)} . \tag{9} \]
For a rigorous solution of the problem, a careful investigation is necessary of the observability of weak rectangular and probabilistic signals,
(passed through filters with optimal passbands) against the background of noise on the indicator screen (or on photographic film).
Restricting ourselves to an approximate treatment, we may take [1]
\[ \xi_1 \simeq \xi_2 . \]
The coefficient \(s\) varies from 1 to 4 [3], with \(\bar{s} \simeq 2\).
Figure 1 presents the dependence
\[ \frac{N_1}{N_2}(s), \]
constructed under the assumption
\[ \xi_1=\xi_2 . \]
Thus, in the case of the probabilistic approximation and \(s \simeq 2\), there is an almost one-and-a-half-fold gain in \(N\) compared with the case of the rectangular approximation. It should be noted that the distribution of meteors with respect to the exponent \(s\) has a mode at \(s \simeq 2\); cases of large \(s\) are very rare, and therefore a large (two- to threefold) gain (as follows from Fig. 1) would be very rare.
Fig. 1. Comparison of the probabilistic and rectangular approximations; \(N_1\) is the mean hourly number of meteors detected by a system with a smooth envelope shape of the emitted pulse and a smooth shape of the resonance characteristic; \(N_2\) is the same for a system with a rectangular pulse envelope and a rectangular shape of the receiver resonance characteristic; \(S\) is the exponent characterizing the distribution of meteoroid bodies by mass.
Conclusions
-
The method presented earlier [1] for comparing different forms of the resonance characteristics of receiving devices and the forms of the envelopes of generated pulses (from the point of view of the number of detected meteors) has been applied to the case of persistent radio echoes.
-
Consideration of the limiting cases (rectangular and probabilistic approximations) shows that the difference between pulse shapes and resonance characteristics for \(a > 2.4 \cdot 10^{12}\ \mathrm{el/cm}\) is more significant than for \(a < 2.4 \cdot 10^{12}\ \mathrm{el/cm}\) (from the point of view of the number of detected meteors).
-
In the case of the optimal relation between pulse duration and passband, a somewhat better result is given by a system with bell-shaped forms of the pulse and of the receiver resonance characteristic.
-
If, for meteor observations, a radar with low sensitivity is used and no special requirements are imposed on it (precise determination of coordinates, etc.), then a resonant amplifier should be used (and not a band amplifier with a resonance characteristic close to rectangular) and a smooth (not rectangular) pulse.
LITERATURE
-
Fialko E. I., On the influence of certain radar parameters on the average hourly number of detected meteors, Izv. TPI, vol. 86, 27, 1958.
-
Kaiser T. R. Theory of the meteor height distribution obtained from radio-echo observations, Mon. Not. R. A. S., 1954, v. 114, No. 1, 39.
-
Fialko E. I., Some results of the study of the distribution of meteoroid bodies by mass, Astr. Circular, No. 195, 22, 1958.