Dependence of the Average Hourly Number of Detected Meteors on the Wavelength of a Radar Station
Abstract Generated abstract
This paper analyzes how the average hourly number of meteors detected by a radar depends on the radar wavelength when receiver threshold power and pulsed operation are taken into account. Starting from Kaiser’s relation for normal reflection from meteor trails, it incorporates the wavelength dependence of receiver noise, especially external cosmic noise in the meter range, and then applies Bullough’s correction for meteors missed because their echo duration is shorter than the pulse repetition period. The analysis suggests that if threshold power is held constant, the echo rate varies approximately as wavelength to the 3/2 power for sporadic meteors, but when threshold noise increases with wavelength the dependence may become very weak. For low pulse-repetition frequencies, however, missed short-lived echoes restore an approximately quadratic dependence, with an estimated exponent near 2.1 under typical parameter values.
Full Text
Dependence of the Average Hourly Number of Detected Meteors on the Wavelength of a Radar Station
E. I. Fialko
(Presented by the scientific seminar of the Radio Engineering Faculty)
As is known, the number of meteors detected by a radar station depends substantially on its parameters and, in particular, on the wavelength. On the basis of experimental data, McKinley [5] concluded that the number of detected meteors varies in proportion to the square of the wavelength.
Kaiser [6], considering the case of normal reflection of radio waves from a meteor trail, found that, approximately,
\[ N \sim \left(\lambda^3\right)^{\frac{S-1}{2}}, \]
where \(N\) is the average hourly number of meteors detected with the aid of a radar,
\(\lambda\) is the wavelength,
\(S\) is a constant characterizing the distribution law of meteor masses.
For \(S \simeq 2\),
\[ N \sim \lambda^{\frac{3}{2}}. \]
It should be noted, however, that these results are valid for an unchanged threshold signal.
In reality, a change in wavelength, as a rule, leads to a change in the magnitude of the threshold signal; the question arises as to the dependence of \(N\) on \(\lambda\) in this case. For example, how will the number of detected meteors change with an increase in the wavelength if, at the same time, the pulse duration and the receiver passband remain unchanged?
Basic Relations
The average number of meteors detected by a radar station per unit time \((N)\) can be found from an approximate
formula obtained by Kaiser [6] for the case of normal reflection of radio waves from meteor trails:
\[ N=A\cdot\left(\frac{\lambda^3}{\varepsilon}\right)^{\frac{S-1}{2}}, \tag{1} \]
where \(\varepsilon\) is the threshold-signal power,
\(A\) is a coefficient independent of the wavelength \(\lambda\).
The threshold-signal power of the receiver is equal to
\[ \varepsilon=k\cdot T_0\cdot \Delta f\cdot F\cdot M, \tag{2} \]
where
\(k\) is Boltzmann’s constant;
\(T_0\) is room temperature;
\(\Delta f\) is the effective frequency band (noise passband) of the receiver;
\(M\) is a coefficient characterizing the observability of a weak signal;
\(F\) is the receiver noise factor.
As is known [1],
\[ F=\frac{T_a}{T_0}-1+F_{\text{own}}. \tag{3} \]
where
\(F_{\text{own}}\) is the noise factor due to the receiver’s internal noise;
\(T_a\) is the effective antenna temperature.
With increasing wavelength, \(F_{\text{own}}\) decreases, while \(T_a\) increases [3].
In the meter-wave range the level of external interference (in particular, of cosmic origin) considerably exceeds the level of the receiver’s internal noise. The noise factor may be represented in the form
\[ F=B\cdot\lambda^l, \tag{4} \]
where \(B\) and \(l\) are constants (formula (4) is approximate). In the range \(\lambda=3\text{--}9\ \mathrm{m}\) [2, 3, 4]
\[ 2<l<3. \]
From (1)—(4) we have:
\[ N=A_1\cdot\lambda^{\frac{(3-l)(S-1)}{2}}, \tag{5} \]
where \(A_1\) is a coefficient independent of \(\lambda\). Since for sporadic meteors \(S\simeq 2\) [6, 7], then
\[ N=A_1\cdot\lambda^r, \]
where
\[ 0<r<0.5, \]
which indicates a weak dependence of the average hourly number of meteors on the radar wavelength. This conclusion was obtained in prev-
of the assumption that all meteors that can produce reflections exceeding the threshold signal will be detected.
In reality, however, some such meteors will not be detected, since the duration of the radio-wave reflection from weak meteor trails may prove to be very short—less than the pulse repetition period.
Bullough showed [7] that the number of actually detected meteors \(N_0\) is equal to
\[ N_0 = N \cdot \xi . \tag{6} \]
Here \(N\) is the number of meteors whose reflections could exceed the threshold signal under continuous irradiation of the meteor trail;
\[ \xi = p^{1-S} + \frac{1}{\ln p \cdot (s-1)} \left[ 1 - p^{1-S} \left\{ 1 + (s-1)\ln p \right\} \right], \tag{7} \]
where
\[ p = e^{\frac{T_i}{T_{\mathrm{c}\lambda}}}, \tag{8} \]
\(T_i\) is the pulse repetition period;
\(T_{\mathrm{c}\lambda}\) is the reflection duration, measured at the level \(1/e\) of the maximum amplitude of the reflected signal [8].
For linear electron density
\[ \alpha < 2.4 \cdot 10^{12}\ \text{el}/\text{cm}, \]
the reflection duration is [8]
\[ T_{\mathrm{c}\lambda} = \frac{\lambda^2}{16 \cdot \pi^2 \cdot D}, \tag{9} \]
where \(D\) is the diffusion coefficient for the altitude at which the reflecting portion of the meteor trail is located.
Let us consider the special case when the pulse repetition period considerably exceeds the lifetime of the trail:
\[ T_i \gg T_{\mathrm{c}\lambda}. \]
Since \(S \simeq 2\), it is not difficult to show that formula (7) takes the form
\[ \xi \simeq \frac{1}{\ln p \cdot (S-1)} \]
and, taking (8) and (9) into account,
\[ \xi \simeq \frac{T_{\mathrm{c}\lambda}}{T_i(S-1)} = A_2 \lambda^2 . \tag{10} \]
where
\[ A_2=\frac{1}{T_i\cdot(S-1)\cdot 16\cdot \pi^2\cdot D} \]
Thus, the average hourly number of actually detected meteors from (5), (6), and (10) will be equal to
\[ N_0 \simeq A_3\lambda^K, \]
where
\[ A_3=A_1\cdot A_2, \]
\[ K=2+\frac{(3-l)(S-1)}{2}. \]
For sporadic meteors \(S\simeq 2\); according to Shklovsky [4] \(l\simeq 2.8\), and in this case \(K\simeq 2.1\).
Thus, in the case where meteor observations are carried out with radars having a low pulse-repetition frequency, the average hourly number of detected meteors is proportional (approximately) to the square of the wavelength.
As the pulse-transmission frequency is increased, \(K\) decreases.
It should be noted that with increasing wavelength the minimum value of the mass of the detected meteors decreases somewhat; the radiation pattern of reflection of radio waves from the meteor trail becomes less sharp [5]; the number of “abnormal” reflections increases, etc.
Consideration of these questions, as well as discussion of recommendations for choosing the wavelength of a radar intended for meteor observations, lies beyond the scope of the present article.*)
Conclusions
- In the case where, when the wavelength is changed, the threshold-signal power is kept unchanged, the average hourly number of detected meteors is proportional to
\[ N\sim \lambda^{\frac{3(S-1)}{2}} \]
and for \(S\simeq 2\)
\[ N\sim \lambda^{\frac{3}{2}}. \]
- But the power of the threshold signal depends on the wavelength. In the case where all meteors capable of giving a reflection exceeding the threshold signal were detected, the average hourly number
\[ N\sim \lambda^{\frac{(3-l)(S-1)}{2}} \]
*) Here the case of the presence of atmospheric interference is not considered, and the operational efficiency of the system is not taken into account.
and for \(l \simeq 2.8,\ S \simeq 2\)
\[ N \sim \lambda^{0.1}. \]
- Since, in pulsed operation, some of the short-lived meteors remain undetected, although the signals reflected from them would exceed the threshold signal, the average hourly number of meteors proves to depend on the ratio of the duration of existence of the trail to the period of transmission of the signals.
In particular, at a very low pulse-repetition frequency and for \(l \simeq 2.8,\ S \simeq 2\)
\[ N \sim \lambda^{2.1}. \]
References
- Siforov V. I. Radio-Receiving Devices, Voenizdat, 1954.
- Kolosov A. A. Resonant Systems and Resonant Amplifiers, Svyazizdat, 1949.
- Sivers A. P. Radar Receivers, Svyazizdat, 1953.
- Shklovskii I. S. On the Nature of Radio Emission from the Galaxy, Astronomicheskii zhurnal, 1952, vol. 29, no. 4.
- McKinley D. W. R. Variation of Meteor Echo Rates with Radar System Parameters, Can. Journ. Phys., 1951, v. 29, No. 5, pp. 403—426.
- Kaiser T. R. Theory of the Meteor Height Distribution Obtained from Radio-Echo Observations, Mon. Not. Roy. Astr. Soc., 1954, v. 114, No. 1, pp. 39—62.
- Bullough K. Radio Echo Observations of the Day-and Night-time Streams. 1951 October to 1953 December. Jodrell Bank Annals, 1954, Ser. 1.1, No. 3, pp. 68—97.
- Kaiser T. R., Closs R. L. Theory of Radio Reflections from Meteor Trails. Phil. Mag., Ser. 7, 1952, v. 43, No. 336, pp. 1—32.