On the Selective Properties of the Radar Method for Observing Meteors

(II. Persistent Trails)

E. I. Fialko

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

Introduction

A number of investigators believe that the radar method of observing meteors possesses selectivity, that is, a different sensitivity to meteors depending on their velocities [1, 2, and others].

It was shown earlier that, in the case of recording ionized meteor trails of the nonpersistent type, selectivity with respect to velocity must indeed occur [3, 4, 5, and others].

However, one cannot confine oneself to considering the recording only of nonpersistent meteor trails, since in measuring meteor velocities persistent trails are used to a considerable extent, and sometimes (especially when operating at relatively long wavelengths and receiving from large distances) the velocities are measured only for those meteor bodies that produce persistent ionized trails.

In this connection it becomes necessary to clarify the selective properties of the radar method in recording meteors (or in measuring their velocities) in the case of persistent meteor trails.

In order for a meteor to be recorded, it is necessary that the duration of reflection of radio waves from the ionized trail exceed some minimum value \(T_{\min}\) (characterizing the response speed of the system [6]). The same also applies when measuring velocity. It is necessary to record pulsations of the echo-signal amplitudes within several Fresnel zones, for which the existence of reflection and its exceeding the corresponding threshold level for a certain time (on the order of tenths of a second) are required.

Strictly speaking, \(T_{\min}\) depends on the measured parameter of the meteor, on the type of measurement (or recording), on the parameters of the locator, etc.; in particular, \(T_{\min}\) may also depend on the velocity of the meteor body \(v\). Restricting ourselves here to an approximate analysis, we shall assume the quantity \(T_{\min}\) to be independent of \(v\).

In addition, we shall assume that reception of echo signals is conducted from the region of the characteristic height [7].

The case of recording signals under conditions of integration, which requires separate consideration, is not discussed in the present work.

The dependence of the number of detected meteors of the underdense type on the parameters of the apparatus, the atmosphere, and the meteoroid bodies was considered by Kaiser [8]; a more detailed treatment was carried out in [9].

Two regimes of reliable detection of underdense meteor trails are possible [9]:

1) A reflection of duration \(T_{\mathrm{sl}} > T_{\min}\) exceeds the threshold level \((\varepsilon_c > \varepsilon_n)\) (Fig. 1).

Fig. 1. Case in which a reflection with duration \(T_{\mathrm{sl}} > T_{\min}\) exceeds the threshold level \(\varepsilon_n\). \(\varepsilon_c\) is the power of the echo signals; \(t\) is time. 1 and 2 are the envelopes of the amplitudes of the echo signals: 1—a reflection reaching the threshold level. 2—a reflection of duration \(T_{\mathrm{sl}} = T_{\min}\).

2) A reflection of duration \(T_{\mathrm{sl}} = T_{\min}\) does not reach the threshold level \((\varepsilon_c < \varepsilon_n)\), and, consequently, the duration of a reflection that has reached the threshold level proves to be \(> T_{\min}\) (Fig. 2).

Fig. 2. Case in which a reflection of duration \(T_{\mathrm{sl}} = T_{\min}\) (2) does not reach the threshold level. 1—a reflection reaching the threshold level \((T_{\mathrm{sl}} > T_{\min})\).

The first regime is possible at large diffusion coefficients, in measurements associated with a very large value of \(T_{\min}\), etc.

The second regime may occur when receiving signals from lateral directions and from large ranges, when using very low-power apparatus, etc.

We shall conventionally call the first regime the case of apparatus of medium sensitivity, and the second—the case of apparatus of low

sensitivity (and we shall assume that highly sensitive equipment detects nonpersistent trails, while equipment of low and medium sensitivity can register only trails of the persistent type).

In order to determine to what extent the selective properties of the radar method of meteor observation are manifested, we shall proceed as follows.

Let us suppose that there are several streams with identical spatial densities of meteoroid bodies, with identical laws of distribution of meteoroid bodies by mass, etc.; the streams differ only in the velocities of the meteoroid bodies. Let us determine how the number of meteors \((N)\) recorded by a given radar will vary as a function of velocity \((v)\).

The dependence \(N(v)\) will characterize the selective properties of the radar method of meteor observation.

Equipment of medium sensitivity

In the case when a reflection with duration \(> T_{\min}\) exceeds the threshold signal (Fig. 1), the number of confidently detected meteors of the persistent type per unit time may be represented in the form [9]

\[ N=A_1\cdot \frac{b\cdot \Delta S}{s-1}\left(\frac{\cos\chi}{\mu}\cdot \frac{\lambda^2}{T_{\min}}\cdot \frac{\beta}{HD}\right)^{s-1}, \tag{1} \]

where \(A_1\) is a constant; \(\chi\) is the zenith distance of the radiant; \(\mu\) is the mass of an atom of the meteoroid body; \(\beta\) is the ionization probability; \(H\) is the height of a homogeneous atmosphere; \(D\) is the diffusion coefficient; \(\lambda\) is the wavelength; and \(b\) and \(s\) are constants characterizing the distribution of meteoroid bodies by mass [7, 10],

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

where \(p(m)\Delta m\) determines the number of meteoroid bodies in the mass interval \(m \div m+\Delta m\) passing through a unit area (in the echo plane [7]) per unit time.

Thus, formula (1) determines the number of stream meteors “piercing” an area lying in the echo plane (in the region of the characteristic height), of area \(\Delta S\).

Expressing the coefficient \(b\) in terms of the velocity \((v)\) and the volume density of the meteoroid stream \((\Omega)\),

\[ b=\Omega\cdot v \tag{3} \]

and separating in (1) the factors depending on \(v\), we obtain

\[ N=A_1'\left(\frac{\beta}{HD}\right)^{s-1}\cdot v. \tag{4} \]

Representing the ionization probability \(\beta\) in the form [7]

\[ \beta=a\cdot v^n \tag{5} \]

(where \(a\) and \(n\) are constants independent of \(v\)) and substituting (5) into (4), we obtain

\[ N=A_1'' \frac{v^{n(s-1)+1}}{(HD)^{s-1}} . \tag{6} \]

For the case \(s=2\), (6) takes the form

\[ N=A_1''' \frac{v^{n+1}}{HD}. \tag{7} \]

Taking into account the dependence of the characteristic height on the velocity \(h_m(v)\) [7] and using \(H(h)\) [7] and \(D(h)\) [11], we find \(N(v)\) for a definite value of the exponent \(n\). As is known, there are very contradictory opinions regarding the magnitude of the exponent \(n\) [12].

Relying on the results of theoretical studies [13, 14], we take \(n \simeq 1\).

Figure 3 presents the dependence \(N(v)\), constructed from formula (7) for \(n=1\) (with the ordinate axis showing the ratio

\[ \frac{N(v)}{N_{\max}}, \]

where \(N_{\max}\) corresponds to \(v=30\ \text{km/sec}\)).

Fig. 3. Dependence of the number of meteors recorded by apparatus of “medium” sensitivity on the velocity of meteoroid bodies. \(N\) is the number of recorded meteors having velocity \(v\). \(N_{\max}\) corresponds to \(v \approx 30\ \text{km/sec}\).

Low-sensitivity apparatus

In the case where a reflection that has reached the threshold level has a duration \(>T_{\min}\) (Fig. 2), the number of confidently detected meteors of the persistent type per unit time will be equal to [9]

\[ N=A_2 \frac{b\Delta s}{s-1} \left(\frac{\cos \chi}{\mu}\cdot \frac{\beta}{H}\right)^{s-1} \left(\frac{P_iG^2\lambda^3}{\varepsilon_n R^3}\right)^{2(s-1)}, \tag{8} \]

where \(P_i\) is the radiated power in the pulse; \(G\) is the antenna directivity coefficient; \(\varepsilon_n\) is the threshold-signal power; \(R\) is the slant distance from the locator to the trail along the normal; \(A_2\) is a constant (the remaining notation was defined above).

Separating in (8) the factors that depend on \(v\), and taking into account that \(R\sim h_m\), we obtain:

\[ N = A_2' \frac{v^{n(s-1)+1}}{H^{s-1} h_m^{6(s-1)}} . \tag{9} \]

For \(s = 2\)

\[ N = A_2' \frac{v^{n+1}}{H h_m^6} . \tag{10} \]

Taking, as in the preceding case, \(n = 1\) and allowing for \(h_m(v)\) [7] and \(H(h)\) [7], we construct \(N(v)\) (see Fig. 4), in which along the ordinate axis the ratio

\[ \frac{N(v)}{N_{\max}}, \]

is plotted, where \(N_{\max}\) corresponds to \(v = 70\ \text{km/sec}\).

Fig. 4. Dependence of the number of meteors \(N\), registered by “low-sensitivity” equipment, on the velocity of meteor bodies \(v\). \(N_{\max}\) corresponds to \(v = 70\ \text{km/sec}\).

Discussion of the obtained dependences \(N(v)\)

As is evident from consideration of the curves shown in Figs. 3 and 4, in the registration of persistent meteor trails the selective properties of the radiolocation method become apparent; moreover, the character of the selectivity is essentially determined by the parameters of the equipment, as well as by certain characteristics of the stream.

Equipment with a relatively large transmitted power and a short wavelength proves to be most sensitive to persistent meteors with velocities \(v \simeq 30\ \text{km/sec}\).

Low-power equipment, receiving from large ranges, possesses increased sensitivity to fast meteors (within the velocity range \(v \simeq 20 \div 70\ \text{km/sec}\) the sensitivity increases monotonically as \(v\) increases).

Thus, in order to determine the true distribution of meteor bodies by velocities, it is necessary to know on what equipment the measurements of meteor velocities were made.

In the case of very low-sensitivity equipment registering persistent trails (as also in the case of very high-sensitivity equipment registering nonpersistent trails [4, 5]), the true distribution of meteor bodies by velocities will have (in comparison with the apparent distribution) a considerably greater rise in the region of small velocities.

In the case of a system of “medium sensitivity,” substantial corrections must be introduced in the region of fast meteors.

It should be noted that the fraction of meteors of cosmic origin remains small even after corrections are introduced in the case of a system of “medium” sensitivity (we note that these conclusions were obtained under the assumption of a linear dependence of the ionization probability \(\beta\) on the velocity of the meteoroid).

Thus, in order to introduce corrections into the measurement results, it is necessary to have criteria that would make it possible to determine to which class the apparatus belongs—of “medium” or of “low” sensitivity.

Criteria for Apparatus of “Medium” and “Low” Sensitivity

A radar registers only trails of the persistent type (that is, trails with linear electron density \(\alpha \gg 10^{12}\ \mathrm{el}/\mathrm{cm}\)), if the condition [9, 15] is satisfied:

\[ \frac{P_i G^2}{\varepsilon_n}\left(\frac{\lambda}{R}\right)^3 \gtrless 2\cdot 10^3 . \tag{11} \]

Reflection from a persistent trail of duration \(\ge T_{\min}\) exceeds the threshold level \(\varepsilon_n\), if [9, 15]

\[ 520\,\frac{\lambda}{\sqrt{D T_{\min}}} < \frac{P_i G^2}{\varepsilon_n}\left(\frac{\lambda}{R}\right)^3 \lessgtr 2\cdot 10^3, \tag{12} \]

and does not reach the threshold level if conditions (11) and (13) [9, 15] are simultaneously fulfilled:

\[ \frac{P_i G^2}{\varepsilon_n}\left(\frac{\lambda}{R}\right)^3 < \frac{520\lambda}{\sqrt{D T_{\min}}}. \tag{13} \]

Thus, a meteor radar of “medium” sensitivity corresponds to condition (12), and a radar of “low” sensitivity to conditions (11) and (13).

Conclusions

1. Radar observations, when registering both persistent and nonpersistent trails, possess different sensitivity to meteor trails produced by bodies with different geocentric velocities.

2. The nature of the selectivity may be different depending on the properties of the apparatus.

It may turn out that the velocity selectivity has a monotonically increasing character, as, for example, in the case of very low-sensitivity (low-power) apparatus. But the selectivity may also have an extremal character, with an “emphasis” on slow persistent meteors in the region \(v \simeq 30\ \mathrm{km/sec}\) (we note that in the case of registering nonpersistent trails, fast meteors with velocities \(\simeq 50\ \mathrm{km/sec}\) are “emphasized” [1, 16]).

1. The selectivity of the radar method may lead to a substantial distortion of the picture of the distribution of meteoroids by velocity.

2. In passing from the apparent distribution of meteoroids by velocities to the true distribution, it is necessary to know the parameters (and, above all, the sensitivity) of the apparatus.

1. The fraction of meteors of cosmic origin (more precisely, meteors with geocentric velocities \(v > 72\) km/sec) is insignificant.

2. For a thorough study of the selective properties of the radio-location method, it is necessary to refine the dependence of the ionization probability on the velocity of the meteoroid and to consider the reception of reflections from the entire echo plane.

REFERENCES

1. E. J. Öpik. Meteors and the upper atmosphere, Irish Astronomical Journ., v. 3, N 6, p. 166, 1955.

2. Yu. A. Loshilov. Dissipation of the meteor trail and questions of the selectivity of radio-location observations, Report at the jubilee scientific conference of the Radio-Engineering Faculty of the Tomsk Polytechnic Institute, 1958.

3. E. I. Fialko. The power of the signal reflected from a meteor trail under conditions of intense diffusion, Izvestiya TPI, vol. 100, p. 40, 1962.

4. E. I. Fialko. On the question of the role of the velocity of the meteoroid in radio-location observations of meteors, Izvestiya TPI, vol. 114, 1962 (in press).

5. E. I. Fialko. Toward an analysis of the dependence of the hourly number of registered meteors on the parameters of meteoroids, the atmosphere, and the apparatus. Bulletin of the Commission on Comets and Meteors of the Academy of Sciences of the USSR, No. 5, p. 9, 1961.

6. E. I. Fialko. On the efficiency of a meteor radar, Izvestiya vyzov MVO (section Radio Engineering), vol. 3, No. 5, p. 529, 1960.

7. T. R. Kaiser. Radio-echo studies of meteor ionization, J. Adv. Phys., v. 2, N. 8, p. 495, 1953.

8. T. R. Kaiser. Theory of the meteor height distribution obtained from radio-echo observations, M.N.R.A.S., 114, N 1, 39, 1954.

9. E. I. Fialko. Certain and random radio detection of stable meteor trails, Izvestiya TPI, vol. 100, p. 84, 1962.

10. 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.

11. J. S. Greenhow, E. L. Neufeld. The diffusion of ionized meteor trails in the upper atmosphere, Journ. Atm. Terr. Phys., v. 6, N 2—3, 133, 1955.

12. G. S. Hawkins. Meteor ionization and its dependence on the velocity, Astroph. J., 124, N 1, 311, 1956.

13. H. S. Massey, D. W. Sida. Collison processes in meteor trails, Phil. Mag., v. 46, N 373, 190, 1956.

14. Yu. A. Loshilov. Application of the theory of atomic collisions to ionization processes in meteor trails. Izv. Astronom. observ. of Odessa University, vol. 5, issue 1, p. 39, 1959.

15. E. I. Fialko. On the conditions for registering ionized meteor trails of various types, Izvestiya TPI, vol. 100, p. 139, 1962.

16. E. I. Fialko. On the probability of meteor ionization, Radio Engineering and Electronics, vol. 4, issue 7, p. 1206, 1959.