USING THRESHOLD SIGNAL VARIATION TO STUDY THE MASS DISTRIBUTION OF METEOROIDS

E. I. FIALKO

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

Introduction

The mass distribution of meteoroids is characterized by the function \(p(m)\), which makes it possible to determine the number of meteoroids \(\Delta N\) whose masses lie in the interval \(m \div m+\Delta m\) [1]:

\[ \Delta N = p(m)\,\Delta m . \tag{1} \]

Usually the function \(p(m)\) is represented in the form [1]

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

where \(b\) and \(s\) are constants.

Thus, the study of the mass distribution of meteoroids essentially reduces to determining the exponent \(s\).

For this purpose, results are used from visual and photographic observations as well as radar observations of meteors.

There are several methods for determining the exponent \(s\) from radar studies of meteors. One of them is based on the distribution of meteor radio echoes by duration [2, 3, 4]; another, on the dependence of the mean hourly number of recorded meteors on the parameters of the radar [2, 3]. One variant of the latter method is a method for determining \(s\) based on variation of the threshold signal [5].

Earlier [5], the possibility was considered of finding the value \(s\) using highly sensitive equipment that makes it possible to record radio echoes of the unstable type (corresponding to ionized trails with a linear electron density \(\alpha_{\min} \ll 2.4\cdot 10^{12}\) el/cm [3]).

However, the method of threshold-signal variation can also be implemented when using low-sensitivity equipment that makes it possible to detect only stable radio echoes (corresponding to trails with \(\alpha_{\min} \gg 2.4\cdot 10^{12}\) el/cm) or radio echoes of intermediate and stable types (corresponding to \(\alpha_{\min} \approx 2.4\cdot 10^{12}\) el/cm).

Let us find the general expression for the exponent \(s\) under the assumption that the threshold signal is varied in such a way that \(\alpha_{\min}\) remains in one of three regions:

\[ \alpha_{\min} \ll 2.4 \cdot 10^{12}\ \text{electrons/cm}; \]

\[ \alpha_{\min} \gg 2.4 \cdot 10^{12}\ \text{electrons/cm}; \]

\[ \alpha_{\min} \simeq 2.4 \cdot 10^{12}\ \text{electrons/cm}. \]

The exponent \(s\) (general expression)

If the minimum mass of a meteoric body producing a trail recorded by the radar is \(m_{\min}\), then the hourly number of detected meteors \(N\) will be [3]

\[ N \sim \frac{1}{m_{\min}^{s-1}} . \tag{3} \]

The linear electron density in the region of the characteristic height \(\alpha_m\) [3], as is known, is proportional to the mass of the meteoric body

\[ \alpha_m \sim m . \tag{4} \]

The power of the echo signal \(\varepsilon\) is proportional to [3]

\[ \varepsilon \sim \alpha^n, \tag{5} \]

where the exponent \(n\) will be different in the regions of unstable, intermediate, and stable echoes.

Assuming that reception of signals from the region of the characteristic height makes it possible to detect meteors produced by bodies with \(m_{\min}\), we find from (3), (4), and (5):

\[ N \sim \varepsilon^{-\frac{s-1}{n}}, \tag{6} \]

where \(N\) is the number (for example, the mean hourly number) of meteors giving echo signals with power \(\geq \varepsilon\).

Recording the numbers of reflections \(N_1\) and \(N_2\) exceeding, respectively, the threshold levels \(\varepsilon_1\) and \(\varepsilon_2\), we find from (6):

\[ s \simeq 1 + n \frac{\lg \dfrac{N_1}{N_2}}{\lg \dfrac{\varepsilon_2}{\varepsilon_1}} . \tag{7} \]

To obtain a working formula it is necessary to specify the values of the coefficient \(n\).

Determination of the coefficient \(n\)

As is known [3], for \(\alpha \ll 2.4 \cdot 10^{12}\ \text{electrons/cm}\), \(\varepsilon \sim \alpha^2\), and consequently \(n_1 \simeq 2\); for \(\alpha \gg 2.4 \cdot 10^{12}\ \text{electrons/cm}\), \(\varepsilon \sim \alpha^{1/2}\), and consequently \(n_2 \simeq \dfrac{1}{2}\).

To determine the exponent \(n_3\), corresponding to the intermediate region \(\left(\alpha \simeq 2.4 \cdot 10^{12}\ \text{el/cm}\right)\), we shall use the dependence of the echo-signal amplitude on \(\alpha\) for the region \(10^{12} < \alpha < 5 \cdot 10^{12}\ \text{el/cm}\) [3]. Passing from voltage to power \(\varepsilon\), we represent \(\lg \varepsilon\) as a function of \(\lg \alpha\) (Fig. 1).

Fig. 1. Toward the approximation of the dependence of the echo-signal power \(\varepsilon\) on the linear electron density \(\alpha\) in the ionized trail (for radio echoes of intermediate type). The points are the theoretical result [3]; \(AB\) is the approximating line.

Approximating, as before, the echo-signal power by the parabola \(\varepsilon \sim \alpha^n\), we find from Fig. 1 \(n_3 \simeq \dfrac{2}{3}\).

Working formulas

Thus, formula (7) takes the form.

1. When the threshold signal is varied in the region of unstable echoes
   \((\alpha_{\min 1} \ll 2.4 \cdot 10^{12}\ \text{el/cm},\ \alpha_{\min 2} \ll 2.4 \cdot 10^{12}\ \text{el/cm})\):

\[ s \simeq 1 + 2 \frac{\lg \dfrac{N_1}{N_2}}{\lg \dfrac{\varepsilon_2}{\varepsilon_1}} . \tag{8} \]

Equation (8) coincides with the formula obtained earlier [5].

1. When the threshold signal is varied in the region of stable echoes
   \((\alpha_{\min 1} \gg 2.4 \cdot 10^{12}\ \text{el/cm};\ \alpha_{\min 2} \gg 2.4 \cdot 10^{12}\ \text{el/cm})\):

\[ s \simeq 1 + \frac{1}{2} \cdot \frac{\lg \dfrac{N_1}{N_2}}{\lg \dfrac{\varepsilon_2}{\varepsilon_1}} . \tag{9} \]

1. When the threshold signal is varied in the region of intermediate echoes (\(\alpha_{\min 1}\) and \(\alpha_{\min 2}\) of the order of \(2.4 \cdot 10^{12}\) electrons/cm):

\[ s \approx 1 + \frac{2}{3}\cdot \frac{\lg \dfrac{N_1}{N_2}} {\lg \dfrac{\varepsilon_2}{\varepsilon_1}} . \tag{10} \]

Remarks

Thus, in determining the exponent \(s\), it is necessary to know the regions to which the threshold signals \(\varepsilon_1\) and \(\varepsilon_2\) correspond—stable, intermediate, or unstable echoes. Otherwise the calculated value of \(s\) may differ from the true one by several times.

Changing the threshold signal \(\varepsilon\) within very wide limits may lead to a situation in which \(\varepsilon_1\) corresponds, for example, to unstable echoes, while \(\varepsilon_2\) corresponds to stable ones. In this case formulas (7)—(10) cease to be valid.

Consideration of the general case—the transition from one region to another (for example, from the region \(\alpha_{\min} \ll 2.4 \cdot 10^{12}\) electrons/cm to the region \(\alpha_{\min} > 2.4 \cdot 10^{12}\) electrons/cm)—is a separate problem and is not given here.

Nor is an estimate of the errors inherent in the variants considered given here.

Conclusions

1. The determination of the exponent \(s\), characterizing the mass distribution of meteoroids by the method of threshold-signal variation, must be carried out with allowance for the peculiarities of radio-wave reflection from trails of different density.

2. The transition from the region of unstable to the region of intermediate and stable meteor radio echoes may lead to gross errors in determining \(s\) (if the necessary corrections are not introduced).

REFERENCES

1. B. Yu. Levin, Physical Theory of Meteors and Meteoritic Matter in the Solar System, Publishing House of the Academy of Sciences of the USSR, 1956.

2. I. C. Browne, K. Bullough, S. Evans and T. R. Kaiser, Characteristics of radioechoes from meteor trails. II. The distribution of meteor magnitudes and masses, Proc. Phys. Soc. (B), 1956, v. 69, part I., No. 433, B., pp. 83—97.

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

4. E. I. Fialko, Distribution of stable meteor radio echoes by duration. (The article is included in this collection.)

5. E. I. Fialko, On the study of the mass distribution of meteoroids, A. Zh., vol. 34, issue 3, pp. 419—423, 1957.