On the Minimum Mass of a Meteoroid Producing an Ionized Trail Detectable by Radar

E. I. Fialko

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

Introduction

Determining the dependence of the minimum mass of a meteoroid that creates an ionized trail which can be detected (or used for measurements) by radio means on the parameters of the radar is of great importance in connection with the solution of a number of problems, such as, for example, determining the capabilities of the radar method in the study of faint meteors; choosing the parameters of a radar intended for the study of faint meteors; studying the number of meteors detected or used, for example, for measuring atmospheric parameters; investigating the maximum and mean characteristic height of detected meteors, etc. Questions concerning the minimum mass of detectable meteors have been touched upon in a number of works [1–5 and others]; however, the dependence of the minimum mass of detectable meteors on the response speed of the radar, on the diffusion coefficient, etc., was not taken into account. Nor were questions considered that are connected with the choice of parameters of a radar intended for the study of faint meteors. In the present work a general dependence is given for the minimum mass of a detected (or used) meteor \(m_{\min}\) on the parameters of the radar, the properties of the atmosphere, and certain characteristics of the meteor under normal reflection of a radio wave from meteor trails, and several special cases are considered.

Very faint meteors produced by meteoroids that do not “reach” the boiling point are excluded from consideration.

Basic Relations (Expressions for \(m_{\min}\))

The power of the echo signal \(\varepsilon\), received by the antenna, in the case when the “packet” of the radio wave “covers” the meteor trail at the moment of its formation, is equal to [1, 3, 4]

\[ \varepsilon = 2.52 \cdot 10^{-38} \frac{P_i \gamma^2 G_m^2 S'(z,\psi)}{R^3} \, q^2 \lambda^2 e^{\,2\left(\frac{z-a}{l}\right)^2}, \tag{1} \]

where \(P_i\) is the pulse power of the radar transmitter; \(\lambda\) is the wavelength; \(G_m\) is the antenna directivity factor; \(s(\varphi,\psi)\) is the antenna directivity pattern (in field strength); \(\psi\) and \(\varphi\) are the azimuth and elevation angle of the object (the portion of the ionized meteor trail normally reflecting the radio wave); \(R\) is the slant range from the radar to the normally reflecting portion of the meteor trail; \(\alpha\) is the linear electron density in the portion of the meteor trail normally reflecting radio waves; \(\eta\) is a coefficient allowing for the increase in the amplitude of the reflected signal when resonance effects appear in the ionized trail; \(\xi\) is a coefficient allowing for the decrease in the maximum value of the echo-signal amplitude owing to the fact that, by the moment at which the formation of the trail segment within the first Fresnel zone is completed, the diffusion effect has had time to appear; \(c\) is the initial radius of the meteor trail.

Formula (1) is valid for the case \(\alpha < 2.4 \cdot 10^{12}\) el/cm.

The linear electron density in the meteor trail can be found by considering the process of evaporation of the meteoroid and the ionization produced as a result of collisions of the evaporated meteor atoms with air particles,

\[ \alpha = \frac{d m}{d t} \cdot \frac{\beta}{m_a} \cdot \frac{1}{v}, \tag{2} \]

where \(m\) is the mass of the meteoroid; \(m_a\) is the mass of an atom of the meteoroid; \(\beta\) is a coefficient characterizing the probability that, when a meteor atom evaporates and collides with air particles, one free electron is released; \(v\) is the geocentric velocity of motion of the meteoroid.

The maximum rate of evaporation is equal to [6]

\[ \left(\frac{d m}{d t}\right)_{\max} = 0.73\,\frac{m_0 \cdot v_0 \cdot \cos \chi}{H} \cdot (\mu - 0.1)^{\frac{\mu-0.1}{1.1-\mu}}, \tag{3} \]

where \(m_0\) and \(v_0\) are the initial mass and velocity of the meteoroid; \(H\) is the height of a homogeneous atmosphere (at the characteristic height \(h_m\), to which the most intense evaporation corresponds); \(\mu\) is a coefficient characterizing the relation between the change in mass \(m\) and the change in the area of the midsection \(S\);

\[ \frac{S}{S_0}=\left(\frac{m}{m_0}\right)^\mu; \]

here \(m_0, S_0\) and \(m, S\) are, respectively, the initial and current values of the mass and midsection of the meteoroid. If evaporation changes the shape of the body, then \(\mu\) does not remain constant. In the case of a nonrotating body of arbitrary shape that preserves this shape during evaporation, \(\mu=\frac{2}{3}\) [6].

Taking (2) and (3) into account, the maximum linear electron density \(\alpha_m\) is equal to (assuming \(v \simeq v_0\)):

\[ \alpha_m = 0.73\,(\mu-0.1)^{\frac{\mu-0.1}{1.1-\mu}} \cdot \frac{m_0 \cdot \beta \cdot \cos \chi}{H \cdot m_a}, \]

and for the case

\[ \mu=\frac{2}{3} \qquad \alpha_m = 0.47\,\frac{m_0 \cdot \beta \cdot \cos \chi}{H \cdot m_a}. \tag{4} \]

For \(h \ne h_m\) [3]

\[ a=\frac{m_0\cdot \cos \chi}{m_a\cdot H}\cdot \frac{p}{p_m}\cdot \left(1-\frac{1}{3}\cdot \frac{p}{p_m}\right)^2\cdot \beta, \]

where \(p\) is the atmospheric pressure at the location of the portion of the meteor trail that normally reflects radio waves; \(p_m\) is the pressure at the characteristic height \(h_m\).

Fig. 1. Case in which the linear electron density in the portion that normally reflects radio waves is maximal. \(L\) is the location of the radar. \(O-l\) is the meteor trail.

Obviously, the minimum mass of a detectable meteoroid (for a given meteor velocity) will correspond to the case of a favorable position of the trail relative to the radar, namely: the case of reflection of radio waves from a portion of the trail with nearly maximum electron density (Fig. 1), equal to

\[ a_{\min}=\frac{4}{9}\cdot m_0\cdot \frac{\beta\cdot \cos \chi}{m_a\cdot H}. \tag{5} \]

Expressions (4) and (5), obtained from [6] and [3], differ quite insignificantly.

The power of the received signal in the case \(h=h_m\), as follows from (1) and (4) or (5), will be equal to

\[ \varepsilon =0.55\cdot 10^{-28}\, \frac{P_i\lambda^3\cdot G_m^2\cdot s^1(\varphi,\psi)}{R^3}\, \gamma^2\cdot \xi\cdot m_0^2\, \frac{\beta^2\cdot \cos^2\chi}{m_a^2\cdot H^2}\, e^{-2\left(\frac{2\pi a}{\lambda}\right)^2}. \]

The minimum mass of a meteoroid \(m_{0\min}\) producing a detectable (or usable for measurements) trail will correspond to the minimum signal power \(\varepsilon_{\min}\):

\[ m_{\min} =1.35\cdot 10^{14} \left(\frac{\varepsilon_{\min}}{P_i}\right)^{\frac12} \cdot \frac{1}{\lambda^{3/2}\,G_m\cdot s^2(\varphi,\psi)} \cdot \frac{R^{3/2}\cdot H\cdot m_a}{\beta\cdot \cos \chi\cdot \gamma_1\cdot \sqrt{\xi}} \cdot e^{\left(\frac{2\pi a}{\lambda}\right)^2}. \tag{6} \]

The power \(\varepsilon_{\min}\) is determined by the power of the threshold signal \(\varepsilon_{\mathrm{por}}\) and by the minimum time \(T_{\min}\) during which the reflection must exceed the level \(K_n\cdot \varepsilon_{\mathrm{por}}\), in order to ensure the required accuracy of measurements or to detect the meteor; the magnitude of the coefficient \(K_n\) is also determined by the required accuracy of the measurements.

If \(U_{m0}\) is the initial amplitude of the reflection, then \(t\) seconds after the formation of the trail the amplitude of the echo signal will be equal to [1]

\[ U_m=U_{m0}\cdot e^{-\frac{t}{T_{\mathrm{sl}}}}, \]

Note. In formula (6) and below, the index “0” in the expression for the mass is omitted. Instead of \(m_{0\min}\), the notation \(m_{\min}\) is used.

i.e.,

\[ T_{c.l}=\frac{\lambda^2}{16\cdot \pi^2\cdot D}; \tag{7} \]

\(T_{c.l}\) is the duration of reflection from the trail (at the level \(\frac{U_{mo}}{e}\));

\(D\) is the diffusion coefficient.

In order that, over the time \(t \geq T_{\min}\), the amplitude of the echo signal exceed the level \(U_m=\sqrt{K_n}\cdot U_{\mathrm{thr}}\), the condition must be fulfilled

\[ U_{mo}\geq U_{\mathrm{thr}}\cdot \sqrt{K_n}\cdot e^{\frac{T_{\min}}{T_{c.l}}}, \quad \text{and since } \varepsilon \sim U^2,\text{ then} \]

\[ \varepsilon_{\min}=\varepsilon_{\mathrm{thr}}\cdot K_n\cdot e^{2\frac{T_{\min}}{T_{c.l}}}. \tag{8} \]

Substituting (8) into (6), we obtain

\[ m_{\min}=1.35\cdot 10^{14} \sqrt{\frac{\varepsilon_{\mathrm{thr}}}{P_i}}\cdot \frac{1}{\lambda^{3/2}G_m s^2(\varphi,\psi)} \cdot \frac{R^{3/2}\cdot H\cdot m_a}{\beta\cdot \cos\chi\cdot \eta\sqrt{\xi}} \cdot \sqrt{K_n}\cdot e^{\left(\frac{2\pi a}{\lambda}\right)^2+\frac{T_{\min}}{T_{c.l}}}, \]

or, taking (7) into account,

\[ m_{\min}=1.35\cdot 10^{14} \sqrt{\frac{\varepsilon_{\mathrm{thr}}}{P_i}}\cdot \frac{1}{\lambda^{3/2}G_m s^2(\varphi,\psi)} \cdot \frac{R^{3/2}\cdot H\cdot m_a}{\beta\cdot \cos\chi\cdot \eta\sqrt{\xi}} \cdot e^{\left(\frac{2\pi}{\lambda}\right)^2\left(a^2+4DT_{\min}\right)}. \tag{9} \]

The magnitude of the minimum mass \(m_{\min}\) at \(h=h_m\) is determined by the parameters of the radar \((P_i,\lambda,G_m,\eta_{\mathrm{thr}},E_{\min})\), the parameters of the meteor \((\chi,m_a,v,\) since \(\beta,\xi,D,a\) depend on \(v)\), the parameters of the atmosphere \((H,D)\), the mutual position of the meteor trail and the radar \((R,s(\varphi,\psi),\eta)\), and the required accuracy of the measurements \((K_n)\).

The relation between the quantities entering into (9) is, in a number of cases, very complex. For example, \(D,a,H\) depend on the characteristic height \(h_m\), which is determined by the quantities \(v,\chi,m\), etc.

Thus, for given parameters of the radar and of the meteor, the quantity \(m_{\min}\) depends on the position of the meteor relative to the radar. Formula (9) in principle makes it possible to find \(m_{\min}\) for the general case.

From (9) one can also find the smallest value of the minimum mass \(m_{\min}^{*}\) for given \(v\) and \(\chi\). \(m_{\min}^{*}\) will correspond to a definite direction \((\varphi,\psi)\).

To find \(m_{\min}^{*}\), we express the slant range through the characteristic height \(h_m\), the zenith distance of the radiant \(\chi\), and the angle \(\Theta\), measured in the echo plane from the line of intersection of this plane \((LL_1)\) with the normal plane passing through the radar location point, the zenith, and the radiant (Fig. 2):

\[ R=\frac{h_m}{\sin\chi\cdot \cos\Theta}. \]

Thus, in the case when the direction of maximum radiation lies in the intersection of the echo plane with the plane containing-

of the point of the radiant and the zenith and the point at which the radar is located, \(m_{\min}\) can be represented in the form

\[ m_{\min} = 1.35\cdot 10^{14}\sqrt{K_n}\cdot \sqrt{\frac{\varepsilon_{\mathrm{thr}}}{P_i}}\cdot \frac{1}{\lambda^{3/2}\cdot G_m}\cdot \frac{h_m^{3/2}\cdot H\cdot m_a}{\beta\cdot \eta\cdot V^{\xi}} \cdot \frac{1}{s(\Theta)\cdot \cos\chi\cdot \sin\chi\cdot \cos\Theta} \cdot e^{\left(\frac{2\pi}{\lambda}\right)^2(\alpha^2+4D)T_{\min}}, \tag{10} \]

where \(s(\Theta)\) is the line of intersection of the volumetric directional pattern with the echo plane.

Fig. 2. On expressing slant range in terms of height.
\(M\) is the “normally reflecting point” on the meteor trail.
\(L\) is the radar, \(Z\) is the zenith, \(LP\) is the direction toward the radiant.
1 is the horizontal plane. 2 is the echo plane (normal to the direction \(LP\)). 3 is the vertical plane passing through the radar, the pole, and the zenith. \(R\) is the slant range. \(h_m\) is the characteristic height. \(\chi\) is the zenith distance of the radiant. \(\psi\) is the angle of position of the meteor. \(LL_1\) is the line of intersection of planes 2 and 3. The direction of maximum radiation is made coincident with \(LL_1\).

For given \(v\) and \(\chi\), the minimum mass will be smallest at the maximum value of the product \(s(\Theta)\cdot \cos\Theta\).

It is quite obvious that this will occur at \(\Theta=0\), with \([s(\Theta)\cdot \cos\Theta]_{\max}=1\). Physically this corresponds to the case in which the direction toward the normally reflecting segment of the meteor trail, in which \(\alpha=\alpha_m\), coincides with the direction of maximum radiation.

Thus, for a given velocity \(v\) and zenith distance of the radiant \(\chi\), the smallest value of the minimum mass is equal to

\[ m_{\min}^{*} = 2.7\cdot 10^{14}\sqrt{K_n}\cdot \sqrt{\frac{\varepsilon_{\mathrm{thr}}}{P_i}}\cdot \frac{1}{\lambda^{3/2}\cdot G_m}\cdot \frac{h_m^{3/2}\cdot H\cdot m_a}{\eta\cdot \eta_1 V^{\xi}\cdot \sin 2\chi} \cdot e^{\left(\frac{2\pi}{\lambda}\right)^2(\alpha^2+4D)T_{\min}}. \tag{11} \]

Formula (11) makes it possible to examine the dependence of the minimum-mass value \(m_{\min}^{*}\) on the radar parameters (for given meteor parameters ...

the meteor) and choose the optimal radar parameters that ensure the smallest value of \(m_{\min}^{*}\). By analyzing [11], one can also consider the variation of \(m_{\min}^{*}\) as a function of the meteor parameters for given radar parameters and determine the limiting capabilities both of the given type of radar and of a radar possessing optimal parameters.

Limiting capabilities of the radar \((m_{\min \min})\)

To determine the minimum mass of a meteoroid producing a meteor trail that can be used in a given type of measurements performed by means of the given radar, it is necessary to construct the dependence \(m_{\min}^{*}(\chi, \vartheta)\) for a given value of \(T_{\min}\) and, having determined \(\chi_{\mathrm{opt}}\) and \(\vartheta_{\mathrm{opt}}\), to find the corresponding value of \(m_{\min}^{*}\). A general solution of this problem at present does not yet seem possible.

In the case of detection of a meteor, however, the problem is simplified; in this case it is expedient to use a system with very high operating speed (\(T_{\min}\) of the order of the transmission period).

The value of the limiting small mass of a meteoroid that can be detected (from its ionized trail) corresponds to the case when the trail is “covered” by a wave packet at the moment of completion of the formation of the first Fresnel zone, and the reflected signal then exceeds the threshold-signal level for a very short time \(T_{\min}=\Delta t\); in the case of using “marked” pulses, \(\Delta t\) is the time during which the whole “marked” pulse is emitted.

It can be shown that in this case (in particular, at comparatively large \(\lambda\)) the smallest value of the minimum mass corresponds to \(\chi \simeq 45^\circ\) and \(\vartheta_{\mathrm{opt}}\) (moreover, at such \(\lambda\), \(\vartheta_{\mathrm{opt}}\) lies in the region of large velocities).

Thus, the limiting value of the minimum mass of a meteoroid producing a detectable trail will be equal to:

\[ m_{\min \min} = 2{,}7 \cdot 10^{14} \cdot m_a \cdot \sqrt{\frac{\xi_{\mathrm{por}}}{P_i}} \cdot \frac{1}{\lambda^{3/2} \cdot G_m} \cdot \left[ \frac{h_m^{3/2}\cdot H}{\beta \cdot \sqrt{\xi}} \cdot e^{\left(\frac{2\pi}{\lambda}\right)^2\left(a^2+4DT_{\min}\right)} \right]_{(\vartheta_{\mathrm{opt}};\,\chi_{\mathrm{opt}})} \cdot \frac{1}{\eta}. \tag{12} \]

The coefficient \(\eta\) characterizes the increase in the intensity of the reflected signal in the case of the manifestation of resonance effects in the meteor trail.

As is known, resonance appears when there is a component of the electric field \(E\) normal to the axis of the trail in the case \(a<2{,}4\cdot10^{12}\) and \(a\ll\lambda\); at the same time, for very small \(a\) the resonance effect is manifested very weakly. In practice, resonance occurs for [7] \(10^{10}<a<10^{12}\); in this case [8] \(\eta\simeq1\div4\). However, if the sensitivity of the apparatus is high and the mass \(m_{\min}\) corresponds to \(a<10^{10}\), then the reflection intensity for normal polarization will be practically the same as for parallel polarization, i.e. \(\eta\simeq1\).

For an approximate determination of \(m_{\min \min}\), one may take \(\vartheta \simeq 50\div60\ \mathrm{km/sec}\); \(h_m\simeq100\ \mathrm{km}\); \(H\sim8\ \mathrm{km}\); \(\sqrt{\xi}\simeq0.5\); \(\beta\simeq1\). Assuming that \(m_{\min \min}\) will correspond to \(a<10^{10}\), we take \(\eta=1\).

The molecular weight of an atom of the meteoric body (iron) is \(m_a \simeq 10^{-22}\)....

Substituting these quantities into (16), we obtain:

\[ m_{\min\min} \simeq 1.35 \cdot 10^9 \sqrt{\frac{\varepsilon_{\mathrm{por}}}{P_i}}\cdot \frac{1}{\lambda^{3.2}\cdot G_m}\cdot e^{\left(\frac{2\pi}{\lambda}\right)^2\left(a^2+4DT_{\min}\right)} . \]

In the case

\[ \left(\frac{2\pi}{\lambda}\right)^2 \left(a^2+4DT_{\min}\right)\ll 1 \qquad m_{\min\min}\simeq (1\div 2)\cdot 10^9 \sqrt{\frac{\varepsilon_{\mathrm{por}}}{P_i}}\cdot \frac{1}{\lambda^{3.2}\cdot G_m}; \tag{13} \]

(here \(\lambda\) is measured in centimeters, \(m\) in grams).

Let us note that the coefficient \(\xi\) depends on \(v,\lambda,R,D\), and so on, and, naturally, its value will not remain constant when \(v\) and \(\lambda\) are varied. The influence of the time taken by the meteor to traverse the first Fresnel zone on the value of \(m_{\min}\) is not considered here\(^{1)}\).

Thus, the value of the limiting minimum mass of a detectable meteoric body, \(m_{\min\min}\), depends substantially on the parameters of the radar used for the observations\(^{2)}\).

Dependence of the Minimum Mass on Radar Parameters

As follows from (11), the quantity \(m_{\min}\) depends on a number of radar parameters: the wavelength \(\lambda\), the transmitter power \(P_i\), the value of the threshold signal \(\varepsilon_{\mathrm{por}}\), the antenna directivity factor \(G_m\), and also on the speed of response of the system, characterized by the time \(T_{\min}\) minimally necessary for detecting a meteor or for carrying out a particular measurement,

\[ m_{\min}^{*}\sim \sqrt{\frac{\varepsilon_{\mathrm{por}}}{P_i}}\cdot \frac{1}{\lambda^{3.2}\cdot G_m\sqrt{\xi}}\cdot e^{\left(\frac{2\pi}{\lambda}\right)^2\left(a^2+4D\right)T_{\min}} . \tag{14} \]

Strictly speaking, it is also necessary to take into account the dependence of \(h_m\) on \(m_{\min}^{*}\) [10], and consequently also on the radar parameters (let us recall that \(D\) depends on \(h_m\)); \(H(m_{\min}^{*})\) [10], etc.

However, since \(h_m\) and \(H\) change only slightly when \(m_{\min}^{*}\) changes, an analysis of the influence of these relations is not given here.

The dependence of \(m_{\min}^{*}\) on the transmitter power and the antenna directivity factor is quite obvious:

\[ m_{\min}^{*}\sim \frac{1}{G_m\sqrt{P_i}} . \]

Let us consider the dependence \(m_{\min}^{*}(\lambda)\):

\[ m_{\min}^{*}\sim \frac{\sqrt{\varepsilon_{\mathrm{por}}}}{\lambda^{3.2}\sqrt{\xi}}\cdot e^{\left(\frac{2\pi}{\lambda}\right)^2\left(a^2+4D\right)T_{\min}} . \tag{15} \]

Analysis of formula (15), taking into account the dependence of the noise level on wavelength, establishes the existence of an optimum wavelength \(\lambda_{\mathrm{opt}}\), depending on the speed of response of the system (\(T_{\min}\)) and on the meteor velocity.

In the case of highly responsive systems, \(\lambda_{\mathrm{opt}}\simeq 12\ \mathrm{m}\). Therefore, for investigations of meteors in the region of small meteoric masses, one should use-

\(^{1)}\) The influence of the time taken by the meteor to traverse the first Fresnel zone on the value of \(m_{\min}\) was considered by F. I. Peregudov [9] after the present paper had been written.

\(^{2)}\) The dependence of \(\eta\) on \(z\), \(v\), etc., is not considered here.

to use the range \(\lambda>12\ \mathrm{m}\), since for \(\lambda>12\ \mathrm{m}\) the minimum mass \(m^*_{\min}\) increases and, in addition, the dimensions of the system increase. It is known that in the meteor wave range \((\lambda \simeq 4 \div 12\ \mathrm{m})\) the power of the threshold signal, determined mainly by the level of cosmic noise, can be represented in the form \(\varepsilon \sim \lambda^l\), where [11, 12] \(l \simeq 2.3 \div 2.8\); for \(\lambda>12\ \mathrm{m}\), \(l\) increases owing to the manifestation of atmospheric interference.

The coefficient \(\xi\) does not remain constant as the wavelength changes. With increasing \(\lambda\), \(\xi\) increases [3]; a detailed analysis of \(m_{\min}(\xi)\) will be made separately. Thus,

\[ m^*_{\min} \sim \lambda^{\frac{l-3}{2}} \cdot e^{\left(\frac{2\pi}{\lambda}\right)^2\left(a^2 + 4DT_{\min}\right)} \cdot \frac{1}{\sqrt{\xi}} . \tag{16} \]

As is seen from (16), increasing the wavelength leads to a decrease in the minimum mass. It can be shown that in the case of high-\(Q\) systems \(m^*_{\min}\) changes somewhat more sharply than inversely proportional to the wavelength (in the meteor-wave range). In the case of low-\(Q\) systems, however, decreasing \(\lambda\) leads to a sharp increase of \(m^*_{\min}\); this increase is the more substantial, the greater the velocity of the meteors being investigated.

Formula (16) is valid for the case \(G_m=\mathrm{const}\) when \(\lambda\) is varied (which occurs, for example, when comparing systems with identical antenna types and, in particular, when using a half-wave vibrator). As follows from (16), in the case of low-\(Q\) systems, comparatively large wavelengths \((\lambda \simeq 10 \div 12\ \mathrm{m})\) should be used to investigate faint meteors. In the case of high-\(Q\) systems, the “long-wave” part of the meteor range should also be used, \(\lambda = 8 \div 12\ \mathrm{m}\). In choosing the wavelength, the need to use a highly directional antenna should be taken into account. Therefore, if for some reason the dimensions of the antenna prove to be limited, it is more expedient to use shorter waves (for example, \(\lambda \simeq 4\ \mathrm{m}\)).

It should also be noted that, for the investigation of meteors in the region of small \(m\), systems with \(\lambda>12\ \mathrm{m}\) may be used, but they can be employed only during times of a reduced level of interference (atmospheric).

On the capabilities of the radar method for the investigation of faint meteors

Let us estimate the minimum mass of a meteoroid that produces an ionized trail which can be recorded by the radio method.

In the case of a wavelength close to the optimum (and \(T_{\min}\) sufficiently small), formula (13) may be used:

\[ m_{\min\min} \simeq (1 \div 2)\cdot 10^9 \sqrt{ \frac{\varepsilon_{\mathrm{th}}}{P_i} \cdot \frac{1}{G_m \lambda^{3/2}} }. \]

As calculation shows, for real values of the antenna directivity factor \((G \simeq 1 \div 100)\), pulse duration \((\tau \simeq 10 \div 100\ \mu\mathrm{sec})\), and pulse power \((P_i \simeq 100 \div 200\ \mathrm{kW})\), with \(\lambda\) close to \(\lambda_{\mathrm{opt}}\), radars can detect meteor trails with \(a_m \simeq 10^8 \div 10^{11}\ \mathrm{el}/\mathrm{cm}\), which at \(v \simeq 50 \div 60\ \mathrm{km/sec}\) corresponds to meteoroid masses \(m_{\min\min} \simeq 10^{-8} \div 10^{-5}\ \mathrm{g}\) and magnitudes \(M \simeq 8 \div 15\).

stellar magnitudes. The calculations of \(m_{\min\min}\) carried out above were performed under the assumption that, during the motion of the meteoric body, its intense evaporation takes place.

It is known that small bodies are decelerated without “reaching” the boiling point [6].

As is seen from the graph given in [6], the critical size (radius) of a meteoric body lies within the limits \(r = 10^{-4} \div 10^{-3}\) cm at

\(v \simeq 60\) km/sec. For iron particles (density \(\rho \simeq 7.6\ \dfrac{g}{cm^3}\)) this will correspond to the critical value of the mass

\[ m_{\mathrm{cr}}=\pi r^3 \rho=\pi\cdot 7.6\cdot(10^{-4}\div 10^{-3})^3 \simeq 2.5\cdot(10^{-8}\div 10^{-11})\ g . \]

From comparison of the values of \(m_{\min\min}\) obtained above with \(m_{\mathrm{cr}}\), it is seen that the smallest meteoric bodies detected by radar undergo intense evaporation.

It should be borne in mind that the values calculated above, \(m_{\min\min}\simeq 10^{-8}\div 10^{-5}\) g, correspond to the most favorable case (irradiation of the trail occurs at the moment when the formation of the 1st Fresnel zone is completed, within which the most intense ionization occurs; the direction of maximum radiation coincides with the direction toward the portion of the trail containing the 1st Fresnel zone). Under real conditions, “covering” of the trail at the moment of its formation will occur very rarely, and consequently trails produced by meteoric bodies with such small masses will be detected very rarely.

Comparison of systems

It is sometimes necessary to compare two systems from the point of view of their capabilities in detecting (or in being used for various measurements of) meteors.

As follows from (11), the ratio of the minimum masses of meteoric bodies accessible to two radars will be equal to

\[ \frac{m_{\min 1}^{*}}{m_{\min 2}^{*}} = \sqrt{\frac{K_{m1}}{K_{m2}}}\cdot \sqrt{\frac{\varepsilon_{\mathrm{por}1}}{\varepsilon_{\mathrm{por}2}}}\cdot \sqrt{\frac{P_{i2}}{P_{i1}}}\cdot \left(\frac{\lambda_2}{\lambda_1}\right)^{3/2}\cdot \frac{G_{m2}}{G_{m1}} \left(\frac{h_{m1}}{h_{m2}}\right)^3 \cdot \frac{H_1}{H_2} \]
\[ {}\cdot \sqrt{\frac{\xi_2}{\xi_1}}\cdot \frac{\beta_2}{\beta_1}\cdot e^{4\pi^2 \left[ \frac{a_1^2\cdot 4D_1T_{\min 1}}{\lambda_1^2} - \frac{a_2^2\cdot 4D_2T_{\min 2}}{\lambda_2^2} \right]}, \tag{17} \]

where the indices 1 and 2 correspond to the 1st and 2nd radars. Relation (17) was obtained under the assumption that both radars detect meteors with identical \(\gamma\), and that in both cases reflections occur from portions located at characteristic heights (i.e. \(a\simeq a_m\)). In the general case, finding the exact value of

\[ \frac{m_{\min 1}^{*}}{m_{\min 2}^{*}} \]

is very difficult. This is explained, in particular, by the dependence of \(D\) and \(H\) on the characteristic height \(h_m\), which, in turn, depends on \(m_{\min}^{*}\) [10]. In some special cases, finding

\[ \frac{m_{\min 1}^{*}}{m_{\min 2}^{*}} \]

is simplified. As an example, let us compare 2 systems possessing identical wavelengths and identical operational efficiencies.

This case occurs in the transition from operation using a half-wave vibrator to the use of a directional antenna, when changing the magnitude of the threshold signal, when adjusting the transmitter power, etc. (where the changes in \(P_i\), \(\varepsilon\), and \(G\) do not occur over very wide limits, so that changes in both \(h_m\) and \(D\) and \(a\) may be neglected). In this case, the ratio of the minimum masses is determined, in a first approximation, only by the parameters of the radars \((\varepsilon_{\text{th}}, P_i, G_m)\):

\[ \frac{m^*_{\min 1}}{m^*_{\min 2}} \simeq \sqrt{ \frac{\varepsilon_{\text{th}1}}{\varepsilon_{\text{th}2}} \cdot \frac{P_{i2}}{P_{i1}} \cdot \frac{G_{m2}}{G_{m1}} }. \]

Conclusions

1. The minimum mass of a meteoroid creating a meteor that can be registered by means of radar depends both on the parameters of the radar and on the parameters of the meteor and certain characteristics of the atmosphere.

2. Of very substantial importance is the response speed of the system, characterized by the minimum time \(T_{\min}\) during which the amplitudes of the reflected signals must exceed a certain level (exceeding the threshold signal by a definite number of times or equal to the threshold signal). The value \(T_{\min}\) is determined by the type of measurement (the measured parameter), the method of measurement, and the required accuracy of measurement.

3. The values of the limiting minimum masses, depending on the parameters of the apparatus, lie within the limits \(m_{\min\min}\simeq 10^{-5}\div 10^{-8}\) g, which corresponds to \(a_{m\min}\simeq 10^{11}\div 10^{8}\) el/cm, and, for \(v\simeq 50\div 60\) km/sec, to meteor brightnesses \(M_{\min}\simeq 8—15\) stellar magnitudes. In confident detection, and still more so in measurements, the minimum masses substantially exceed \(m_{\min\min}\).

LITERATURE

1. T. R. Kaiser, MNRAS, 1954, v. 114, N 1, 39.
2. T. R. Kaiser, MNRAS, 1954, v. 114, N 1, 52.
3. T. R. Kaiser, I. Adv. Phys., 1953, v. 2, N 8, 495.
4. S. Evans, MNRAS, 1954, v. 114, N 1, 63.
5. Fialko E. I., Astr. zhurn., 1957, 34, issue 3, 421.
6. Levin B. Yu., Physical Theory of Meteors and Meteoritic Matter in the Solar System, Publishing House of the Academy of Sciences of the USSR, 1956.
7. Nemirova E. K., Bull. of the Commission on Comets and Meteors of the Astronomical Council of the Academy of Sciences of the USSR, No. 2, 46, 1958.
8. E. R. Billam, J. C. Browne, “Meteors,” Ed. by T. R. Kaiser, 1955, 73.
9. Peresudov F. I., Astr. zhurn., 35, issue 6, 888, 1958.
10. Fialko E. I., Astr. zhurn., 35, issue 6, 881, 1958.
11. H. V. Cottony, J. R. Johler, Proc. I.R.E., 1953, v. 40; N 9; 1053.
12. Shklovskii I. S., Astr. zhurn., vol. 29, issue 4, 418, 1952.