POWER OF THE SIGNAL REFLECTED FROM A METEOR TRAIL UNDER CONDITIONS OF INTENSE DIFFUSION

E. I. Fialko

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

Introduction. Statement of the Problem

Of great interest is the establishment of a quantitative dependence of the power of the echo signal on the parameters of the meteor, the atmosphere, and the radar under conditions of intense diffusion.

The effect of diffusion leads to the fact that, during the time of formation of the ionized trail within the first Fresnel zone, the trail broadens, as a result of which the maximum reflection coefficient (and, consequently, the greatest amplitude of the echo signal) decreases in comparison with the case of absence of diffusion.

Kaiser [1] obtained the ratio \(\zeta\) of the greatest amplitude of the echo signal under diffusion conditions to the steady-state value of the amplitude of the echo signal in the absence of diffusion as a function of

\[ \Delta=\frac{16\pi^2 D R^{1/2}}{V\lambda^{3/2}}, \tag{1} \]

where \(D\) is the ambipolar diffusion coefficient; \(V\) is the velocity of the meteor body; \(R\) is the slant range from the radar to the trail (along the normal); \(\lambda\) is the radar wavelength.

The result of the solution by the method of numerical integration was given by Kaiser in the form of a graph of \(\zeta(\Delta)\) (Fig. 1).

However, Kaiser did not give an analytical dependence \(\zeta(\Delta)\), and therefore did not represent the power of the echo signal as a function of \(V\), \(D\), etc.

Hawkins [2], having investigated the radio echo under conditions of strong diffusion, obtained the approximate dependence

\[ \varepsilon=2.54\cdot 10^{-28}\frac{P_i G^2 \lambda^3 \alpha^2}{R^3}\,\zeta^2, \tag{2} \]

where \(\varepsilon\) is the power of the echo signal; \(P_i\) is the radiated power in the pulse; \(G\) is the antenna directivity coefficient; \(\alpha\) is the linear electron density (in electrons per \(1\ \mathrm{cm}\));

\[ \zeta=\frac{V^2}{\Delta}. \tag{3} \]

As follows from (1), (2), and (3),

\[ \varepsilon \sim \frac{\lambda^{6} V^{2} a^{2}}{D^{2} R^{4}}, \tag{4} \]

whereas in the absence of diffusion, as is known [3],

\[ \varepsilon \sim \frac{\lambda^{3}}{R^{3}} a^{2}. \tag{5} \]

However, the result obtained by Hawkins raises doubts, since precisely under intense diffusion \(\zeta(\Delta)\) according to Hawkins differs somewhat from \(\zeta(\Delta)\) according to Kaiser; Hawkins does not claim, moreover, that the solution is exact in the region of small \(\Delta\).

Fig. 1. Dependence \(\zeta(\Delta)\) according to Kaiser [1].

A more accurate result was obtained by F. I. Peregudov [4]: the power of the signal reflected from a meteor trail at the input of the receiver \(t\) seconds after the meteor body has passed through the middle of the first Fresnel zone is equal to

\[ \varepsilon = \frac{1}{144^{2}\pi^{6}} \left(\frac{e^{2}}{m_e c^{2}}\right)^{2} \cdot \left(\frac{\beta m}{\mu H}\right)^{2} \frac{V^{2}\lambda^{6}P_iG^{2}}{R^{4}D^{2}} \cos^{2}\chi(1-e^{-\Delta})^{2} \cdot e^{-2(ka)^{2}} e^{-\frac{32\pi^{2}Dt}{\lambda^{2}}}, \tag{6} \]

where \(e, m_e\) are the charge and mass of the electron; \(c\) is the speed of light; \(m\) is the mass of the meteor body; \(\mu\) is the mass of an atom of the meteor substance; \(H\) is the height of the homogeneous atmosphere; \(\chi\) is the zenith distance of the meteor radiant; \(\beta\) is the probability of ionization; \(k=\dfrac{2\pi}{\lambda}\); \(a\) is the initial radius of the meteor trail.

Formula (6) was obtained for the case when the reflecting section of the trail is located in the region of the characteristic height, and therefore

\[ a=a_{\max}=\frac{4}{9}\cdot \frac{\beta}{\mu H}\, m\cos\chi; \tag{7} \]

according to Peregudov

\[ \zeta(\Delta)=\sqrt{2}\,\frac{1-e^{-\frac{\Delta}{\sqrt{2}}}}{\Delta}. \tag{8} \]

Although formula (8) was obtained as a result of substantial simplifications, it gives a very good approximation for \(0<\Delta<2\).

For large \(\Delta\), as follows from (8), \(\zeta(\Delta)\simeq \dfrac{\sqrt{2}}{\Delta}\), that is, Hawkins’ result is a special case of formula (8). In this case, as is seen from (6), for \(t\simeq 0\) approximately

\[ \varepsilon \sim \frac{\beta^2\lambda^6 V^2}{D^2R^4}. \tag{9} \]

If we take into account the dependence of \(D(h)\), \(R(h,\chi)\), and \(H(h)\), as well as the dependence of the characteristic height \(h_m\) on the velocity of the meteoric body [4, 13], we obtain

\[ R(h\simeq h_m)\sim V^{0.15}, \]

\[ D(h\simeq h_m)\sim V^{2.3}. \tag{10} \]

Substituting (10) into (9), we find

\[ \varepsilon \sim \beta^2 V^{-3.2}. \tag{11} \]

However, both (9) and (11) require refinement, since in the region of large \(\Delta\) the solution becomes inaccurate.

Thus, in order to refine the dependence of \(\varepsilon\) on \(V\), \(\lambda\), \(R\), and \(D\), it is necessary to refine the analytical expression \(\zeta(\Delta)\). In addition, in a number of cases one cannot restrict oneself to considering the altitude region \(h\simeq h_m\): it is also necessary to investigate another particular case, \(h>h_m\).

For this purpose we shall obtain a general expression for the power of a normally reflected signal as a function of the parameters of the locator, meteor, and atmosphere, refining the dependence \(\zeta(\Delta)\) and not restricting ourselves to the case \(h=h_m\).

Echo-signal power

As is known, the power of the received echo signal is equal to [3, 5]

\[ \varepsilon=\frac{P_iG^2\lambda^2}{64\pi^3R^4}\cdot s_0, \tag{12} \]

where \(s_0\) is the effective scattering surface of the reflecting object (in the present case, of the ionized meteor trail).

In the case where the linear electron density in the trail is \(\alpha \ll 2.4\cdot 10^{12}\) el/cm, the effective scattering surface of a cylindrical trail proves to be equal to [4]

\[ s_0=2\pi\left(\frac{e^2}{m_ec^2}\right)^2 R\lambda\alpha^2 e^{-2(ka)^2}e^{-\frac{32\pi^2Dt}{\lambda^2}}, \tag{13} \]

where \(a\) is the initial radius of the trail, caused by thermal diffusion; \(t\) is the time elapsed since the meteoric body passed through the middle of the first Fresnel zone (that is, the point lying at the intersection of the trajecto-

trajectory of the meteoric body with the perpendicular dropped from the location of the radar to the trajectory).

Formula (13) does not take into account pulsations of the echo-signal amplitude due to Fresnel diffraction.

In addition, (13) does not take into account the change in the amplitude of the reflected signal due to the parabolic shape of the trail caused by the diffusion effect; nor does it take into account the increase in the echo-signal amplitude that may be observed owing to resonance effects in the ionized trail.

In order to find the greatest value of the effective scattering surface in general form, we shall use the known expression for \(s_0\) [5]:

\[ s_0 = 4\pi R^2 \frac{p_2}{p_1}, \tag{14} \]

where \(p_1\) is the power-flux density of the primary field at the target; \(p_2\) is the power-flux density of the secondary field in the region where the radio receiving device is located.

Comparing the effective scattering surfaces of the meteor for unchanged values of \(R\) and \(p_1\), we obtain

\[ \frac{s_{0\mathrm{I}}}{s_{0\mathrm{II}}} = \frac{p_{2\mathrm{I}}}{p_{2\mathrm{II}}}. \tag{15} \]

Passing from the power of the secondary field to the amplitude \(u_m\) of the received signal, we represent (15) in the form

\[ \frac{s_{0\mathrm{I}}}{s_{0\mathrm{II}}} = \left( \frac{u_{m\mathrm{I}}}{u_{m\mathrm{II}}} \right)^2 . \tag{16} \]

Let \(u_{m\mathrm{I}}\) denote the steady-state value of the amplitude of the received signal in the absence of concentration diffusion; the effective scattering surface for this case will be equal to (see (13)):

\[ s_{0\mathrm{I}} = 2\pi \left( \frac{e^2}{m_e c^2} \right)^2 R\lambda \alpha^2 e^{-2(\varkappa a)^2}. \tag{17} \]

The greatest value of the amplitude \(u_{m\mathrm{II}}\) in the case where diffusion is present (that is, in the case of a parabolic trail shape), taking into account diffraction in the region of the first Fresnel zone, corresponds to \(s_{0\mathrm{II}}\). From (16) and (17) we obtain

\[ s_{0\mathrm{II}} = 2\pi \left( \frac{e^2}{m_e c^2} \right)^2 R\lambda \alpha^2 e^{-2(\varkappa a)^2}\cdot \zeta^2, \tag{18} \]

where

\[ \zeta = \frac{u_{m\mathrm{II}}}{u_{m\mathrm{I}}}. \]

If, owing to plasma resonance in the meteor trail (which may occur when there is a component of the electric field normal to the axis of the trail, under the conditions \(\alpha \ll 10^{12}\) electrons/cm and \(a \ll \lambda\)), the greatest value of the amplitude of the received signal increases to the value \(u_{m\mathrm{III}}\), then the effective scattering surface \(s_{0\mathrm{III}}\) will be equal to

\[ s_{0\mathrm{III}}=s_{0\mathrm{II}}\cdot\left(\frac{u_{m\mathrm{III}}}{u_{m\mathrm{II}}}\right)^2, \]

and, taking (18) into account,

\[ s_{0\mathrm{III}}=2\pi\left(\frac{e^2}{m_ec^2}\right)^2 R\lambda\alpha^2 e^{-2(\kappa\alpha)^2\zeta^2\eta^2}, \tag{19} \]

where

\[ \eta=\frac{u_{m\mathrm{III}}}{u_{m\mathrm{II}}}. \]

Substituting (19) into (12), we obtain an expression for the greatest value of the power of the received echo signal,

\[ \varepsilon=\frac{1}{32\pi^2}\left(\frac{e^2}{m_ec^2}\right)^2 \frac{P_iG^2\lambda^3}{R^3}\alpha^2 e^{-2(\kappa\alpha)^2\zeta^2\eta^2}. \tag{20} \]

After a time \(t\) has elapsed following the formation of half of the first Fresnel zone, the power \(\varepsilon(t)\), as follows from (12), (13), (16), and (19), will be

\[ \varepsilon(t)=\frac{1}{32\pi^2}\left(\frac{e^2}{m_ec^2}\right)^2 \frac{P_iG^2\lambda^3}{R^3}\alpha^2 e^{-2(\kappa\alpha)^2\zeta^2\eta^2} e^{-\frac{32\pi^2Dt}{\lambda^2}}. \tag{21} \]

As was already mentioned, the dependence of the coefficient \(\zeta\) on \(\Delta\) (1) was obtained by Kaiser [1] by numerical integration and presented graphically (Fig. 1). However, for the solution of a number of problems it is necessary to have an analytical expression \(\zeta(\Delta)\).

Analytical expression for the coefficient \(\zeta(\Delta)\)

As a result of an approximate analysis, F. I. Peregudov [4] found, as mentioned above,

\[ \zeta(\Delta)=\frac{1-e^{-\frac{\Delta}{\sqrt{2}}}}{\Delta}\sqrt{2}. \tag{8} \]

However, in the region of large values of \(\Delta\), formula (8) gives appreciable errors.

To obtain a more accurate analytical expression for \(\zeta(\Delta)\), one should use (8) to determine the structure of a formula that would give a satisfactory approximation to the dependence \(\zeta(\Delta)\) obtained by Kaiser.

Let us represent \(\zeta(\Delta)\) in the form

\[ \zeta(\Delta)=A\,\frac{1-e^{-B\Delta^n}}{B\Delta^n} \tag{22} \]

and find the coefficients \(A\), \(B\), and the exponent \(n\) from the condition that the approximating and approximated curves coincide at \(\Delta=0;\ 1;\ 5\).

According to Kaiser (Fig. 1),

\[ \zeta(\Delta=0)=1.18; \]

\[ \zeta(\Delta=1)=0.62; \tag{23} \]

\[ \zeta(\Delta=5)=0.34. \]

From (22) and (23) we obtain \(A=1.18\) and

\[ \begin{aligned} 0.53\,B &= 1-e^{-B},\\ B\cdot 5^n &\simeq 3.47 . \end{aligned} \tag{24} \]

Solving the first equation graphically with respect to \(B\), we find \(B \approx 1.4\); from the second equation \(n \approx 0.57\). Thus, (22) takes the form

\[ \zeta(\Delta) \approx 0.83\,\frac{1-e^{-1.4\Delta^{0.57}}}{\Delta^{0.57}} . \tag{25} \]

Rounding the values of \(B\) and \(n\) to \(n=0.5\) and \(B=1.5\), we represent (22) in the form

\[ \zeta(\Delta) \approx 1.18\,\frac{1-e^{-1.5\Delta^{0.5}}}{1.5\Delta^{0.5}} . \tag{26} \]

The dependence \(\zeta(\Delta)\), constructed from formula (26), is shown in Fig. 2 (curve 2); the same figure shows the dependence \(\zeta(\Delta)\) according to Kaiser (curve 1). As is seen from Fig. 2, formula (26) gives a very good approximation over a wide range of values of \(\Delta\) \((\Delta=0\div 5)\).

In the region of intense diffusion (\(\Delta\) large)

\[ \zeta \sim \frac{1}{\Delta^{0.5}}, \]

whereas according to Hawkins [3] and Peregudov [4]

\[ \zeta \sim \frac{1}{\Delta}. \]

It should be noted that here the case of diffusion so intense that the size of the reflecting portion of the trail becomes considerably smaller than the length of the first Fresnel zone [6] is not considered.

Fig. 2. Dependence \(\zeta^2(\Delta)\):
\(1\)—\(\zeta^2(\Delta)\) according to Kaiser;
\(2\)—\(\zeta^2(\Delta)=\left(1.18\,\dfrac{1-e^{-1.5\Delta^{0.5}}}{1.5\Delta^{0.5}}\right)^2\).

Dependence of the echo-signal power on the radar parameters and on the linear electron density in the meteor trail

Substituting (26) into (21), we obtain

\[ \varepsilon(t)=A_1\,\frac{P_tG^2\lambda^3}{R^3}\,a^2 \left(\frac{1-e^{-1.5\Delta^{0.5}}}{1.5\Delta^{0.5}}\right)^2 e^{-2(ka)^2}e^{-\frac{32\pi^2Dt}{\lambda^2}}\cdot\eta^2, \tag{27} \]

where

\[ A_1=\frac{1.18^2}{32\pi^2} \left(\frac{e^2}{m_ec^2}\right)^2 =3.5\cdot 10^{-28}. \]

For weak diffusion, \(\zeta \approx 1\), and \(\varepsilon \sim \dfrac{\lambda^3}{R^3}\alpha^2\), which coincides with (5).

For strong diffusion (\(\Delta\) is large),

\[ e^{-1.5\Delta^{0.5}} \ll 1 \quad \text{and} \quad \varepsilon(t) \sim \frac{1}{\Delta}. \]

And since

\[ \Delta = \frac{16\pi^2 D R^{1/2}}{V\lambda^{3/2}}, \tag{1} \]

then

\[ \varepsilon(t)=A'_2\,\frac{P_iG^2\lambda^{4.5}V}{D R^{3.5}}\, \alpha^2 e^{-2(\varkappa a)^2} e^{-\frac{32\pi^2Dt}{\lambda^2}}\eta^2, \tag{28} \]

where

\[ A'_2=\frac{1.18^2}{32\pi^4} \left(\frac{e^2}{m_ec^2}\right)^2 \frac{1}{1.5^2\cdot 16} \approx 0.98\cdot 10^{-30}, \]

that is,

\[ \varepsilon \sim \frac{\lambda^{4.5}V}{D R^{3.5}}\cdot \alpha^2. \tag{29} \]

In the further consideration of the question we shall restrict ourselves to the case of absence of resonance (more precisely, to the case of horizontal polarization of the radio waves relative to the trail) and shall take \(\eta=1\).

Comparing (29) with (5), we see that the dependence of the echo-signal power on \(\lambda\) and \(R\) in the case of intense diffusion becomes sharper than in the absence of diffusion; at the same time, this dependence is less strong than follows from (4) and (9).

Moreover, as is seen from comparison of (29) with (4) and (9), the echo-signal power is proportional to the velocity (and not to the square of the velocity) and inversely proportional to the diffusion coefficient (and not to the square of the diffusion coefficient).

This conclusion is valid when considering the power of an echo signal obtained from a portion of the trail with a given linear electron density; such a formulation of the question has physical meaning in a number of cases, including comparison of observational results at different wavelengths.

Dependence of the echo-signal power on the parameters of the radar and on the mass and velocity of the meteoric body

Expressing the linear electron density in the meteor trail through the mass of the meteoric body [1],

\[ \alpha=\frac{\beta m\cos\chi}{\mu H}\cdot \frac{p}{p_m} \left(1-\frac{1}{3}\frac{p}{p_m}\right)^2, \tag{30} \]

where \(p\) is the atmospheric pressure at the point with linear electron density \(\alpha\); \(p_m\) is the pressure at the characteristic height (that is, at the height corresponding to the maximum of \(\alpha\)), equal to

\[ p_m=\frac{Qm^{1/2}}{V^2}\cos\chi; \]

\[ Q=\frac{2gl}{\Lambda A}; \tag{31} \]

\(l\)—the energy required for heating, melting, and evaporation of 1 g of meteoric substance; \(g\)—acceleration due to gravity; \(\Lambda\)—heat-transfer coefficient; \(A\)—the shape coefficient of the meteoric body, depending on the shape and density of the meteoric body.

The remaining notation entering into (30) and (31) has been explained above (see (6) and (1)).

Substituting (30) into (27), we obtain

\[ \varepsilon(t)=3.5\cdot 10^{-28}\, \frac{P_iG^2\lambda^3}{R^3} \left(\frac{\beta m\cos\chi}{\mu H}\right)^2 \left(\frac{p}{p_m}\right)^2 \left(1-\frac{1}{3}\frac{p}{p_m}\right)^4 \cdot \left(\frac{1-e^{-1.5\Delta^{0.5}}}{1.5\Delta^{0.5}}\right)^2 e^{-2(\kappa a)^2}e^{-\frac{32\pi^2Dt}{\lambda^2}} \tag{32} \]

(as was stipulated above, we take \(\eta=1\)).

For weak diffusion \((\Delta \ll 1)\), formula (32) takes the form

\[ \varepsilon(t)=3.5\cdot 10^{-28}\, \frac{P_iG^2\lambda^3}{R^3} \left(\frac{\beta m\cos\chi}{\mu H}\right)^2 \left(\frac{p}{p_m}\right)^2 \left(1-\frac{1}{3}\frac{p}{p_m}\right)^4 e^{-2(\kappa a)^2}e^{-\frac{32\pi^2Dt}{\lambda^2}}. \tag{33} \]

In the case of reception of reflections from the region of characteristic height \(h \approx h_m\):

\[ \varepsilon(t)\approx A_2\, \frac{P_iG^2\lambda^3}{R^3} \left(\frac{\beta m\cos\chi}{\mu H}\right)^2 \cdot e^{-2(\kappa a)^2}\cdot e^{-\frac{32\pi^2Dt}{\lambda^2}}, \tag{34} \]

where

\[ A_2=A_1\left(\frac{2}{3}\right)^4=0.69\cdot 10^{-28}, \]

that is (for \(\kappa a\ll 1\)):

\[ \varepsilon\sim \frac{\lambda^3\beta^2}{R^3H^2}\,m^2. \tag{35} \]

In the case of reception from the region of heights \(h>h_m\), with \(p\ll p_m\), formula (32) takes the form

\[ \varepsilon(t)=3.5\cdot 10^{-28}\, \frac{P_iG^2\lambda^3}{R^3} \left(\frac{\beta m\cos\chi}{\mu H}\right)^2 \left(\frac{p}{p_m}\right)^2 \cdot e^{-2(\kappa a)^2}\cdot e^{-\frac{32\pi^2Dt}{\lambda^2}}, \tag{36} \]

and, taking (31) into account,

\[ \varepsilon(t)\approx 3.5\cdot 10^{-28}\, \frac{P_iG^2\lambda^3}{R^3} \left(\frac{\beta}{\mu H}\right)^2 \frac{V^4m^{4/3}}{Q^2}\, p^2 e^{-2(\kappa a)^2}\cdot e^{-\frac{32\pi^2Dt}{\lambda^2}}, \]

that is (for \(\kappa a\ll 1\))

\[ \varepsilon=\frac{\lambda^3\beta^2V^4}{R^3H^2}\,m^{4/3}p^2. \tag{37} \]

For intense diffusion, \(e^{-1.5\Delta^{0.5}}\ll 1\), and expression (32), taking (1) into account, assumes the form

\[ \varepsilon(t)=A_3\, \frac{P_iG^2\lambda^{4.5}V}{R^{3.5}D} \left(\frac{\beta m\cos\chi}{\mu H}\right)^2 \left(\frac{p}{p_m}\right)^2 \left(1-\frac{1}{3}\frac{p}{p_m}\right)^4 e^{-2(\kappa a)^2}e^{-\frac{32\pi^2Dt}{\lambda^2}}, \tag{38} \]

where

\[ A_3=\frac{A_1}{1.5^2 16\pi^2}=0.98\cdot 10^{-30}. \]

In the case of reception from the region \(h \simeq h_m\):

\[ \varepsilon(t) \approx 0.2 \cdot 10^{-30} \frac{P_i G^2 \lambda^{4.5} V}{R^{3.5}D} \left(\frac{\beta m \cos \chi}{\mu H}\right)^2 e^{-2(\kappa a)^2} e^{-\frac{32\pi^2Dt}{\lambda^2}}, \tag{39} \]

that is, for \(\kappa a \ll 1\):

\[ \varepsilon \sim \frac{\lambda^{4.5} V \beta^2}{R^{3.5}H^2D}\,m^2 . \tag{40} \]

For reception from the region \(h > h_m\) (with \(p \ll p_m\)),

\[ \varepsilon(t)=98\cdot10^{-32} \left(\frac{e^2}{m_e c^2}\right)^2 \frac{P_iG^2\lambda^{4.5}V}{R^{3.5}D} \left(\frac{\beta}{\mu H}\right)^2 \frac{m^{1/3}V^4}{Q^2} p^2 e^{-2(\kappa a)^2} e^{-\frac{32\pi^2Dt}{\lambda^2}}, \tag{41} \]

that is, for \(\kappa a \gg 1\):

\[ \varepsilon \sim \frac{\lambda^{4.5}V^5\beta^2}{R^{3.5}H^2D}\, m^{1/3}p^2 . \tag{42} \]

Formulas (35), (37), (40), and (42) give an idea of the nature of the dependence of the echo-signal power on certain parameters of the meteor, the atmosphere, and the radar when the normally reflecting segment is located in the region of the characteristic height and in the region situated above \(h_m\), both under conditions of weak diffusion and under conditions of intense diffusion (when the inequality \(2(\kappa a)^2 \ll 1\) is satisfied).

Comparing (35) with (37) and (40) with (42), one may note that reflections from high layers \((h>h_m,\ p\ll p_m)\) are characterized by a sharper dependence of the signal power on the velocity of the meteoric body and by a weaker dependence of the power on the mass of the meteoric body than reflections arriving from the region of the characteristic height.

From a comparison of (35) with (40) and (37) with (42), it follows that an increase in diffusion leads to a certain strengthening of the dependence of the echo-signal power on the wavelength, the slant range to the meteor trail, and the velocity of the meteoric body.

Let us consider in greater detail the dependence \(\varepsilon(V)\).

Dependence of the echo-signal power on the velocity of the meteoric body

Let us first consider reception of normally reflected signals from a region lying above the characteristic height \((h>h_m;\ p\ll p_m)\). This case is of practical interest, in particular, in the study of radio echoes arriving from large distances.

In the case of weak and strong diffusion we have, respectively (see (37) and (42)),

\[ \varepsilon \sim \beta^2 V^4 \tag{43} \]

and

\[ \varepsilon \sim \beta^2 V^5 . \tag{44} \]

The ionization probability may be represented in the form [1]

\[ \beta=dV^n, \tag{45} \]

where \(d\) and \(n\) are constants.

The numerical value of the exponent \(n\) has not yet been established at present. Theoretical works give approximately \(n \simeq 1\) [7, 8, 9]. Attempts to determine \(n\) from experimental data lead to pro-

contradictory results; the experimentally found values of \(n\) lie in the range \(n = 0 \div 5.6\) [10, 11].

Thus, (43) and (44), taking (45) into account, can be represented in the form

\[ \varepsilon \sim V^{2(n+2)} \tag{46} \]

and

\[ \varepsilon \sim V^{2(n+2.5)}, \tag{47} \]

and although the exact value of \(n\) is unknown, it is quite obvious that the dependence of the echo-signal power on the velocity of the meteoric body is extremely strong both in the case of weak diffusion and in the case of intense diffusion.

This confirms the proposition, advanced by B. Yu. Levin, concerning the important role of the physical factor [12]. The result obtained is also clear from physical considerations: with increasing velocity, \(a_m\) increases and \(a(h)\) is shifted toward greater heights [13]; therefore, for \(h > h_m\), an increase in \(V\) must lead to a sharp increase in \(a\), and consequently also in \(\varepsilon\).

Let us proceed to consider the reception of signals from a layer in the region of the characteristic height. To determine the dependence \(\varepsilon(V)\), it should be taken into account that \(R\), \(H\), and \(D\) are functions of height, while the characteristic height is a function of the velocity of the meteoric body.

Using the experimentally found dependences \(h_m(V)\) [11], \(H(h_m)\) [11], and \(D(h)\) [14], and assuming \(R \simeq \dfrac{h}{\sin \chi}\), it is possible, by means of a number of approximations, to find the following approximate dependences [15]:

\[ h_m(V) \sim V^{0.15}; \]

\[ H(V) \sim V^{0.54}; \tag{48} \]

\[ D(V) \sim V^{2.36}. \]

Taking (45) and (48) into account, we obtain for the case of weak diffusion (see (35))

\[ \varepsilon \sim \frac{\beta^2}{R^3 H^2} \sim V^{2(n-0.75)} \tag{49} \]

and for the case of intense diffusion (see (40))

\[ \varepsilon \sim \frac{\beta^2 V}{R^{3.5} H^2 D} \sim V^{2(n-1.5)}. \tag{50} \]

Thus, in the case of reception of normally reflected echoes from the layer adjacent to \(h_m\), the power of the echo signal depends on the velocity of the meteoric body (but this dependence is considerably weaker than in the case \(h > h_m\)). It is not yet possible to draw more definite conclusions at present because of the lack of reliable data on the magnitude of the exponent \(n\). But if one is guided by the data obtained theoretically [7, 8, 9] (\(n \simeq 1\)), then the dependence \(\varepsilon(V)\) in the case of weak diffusion will have the form (49):

\[ \varepsilon \sim \sqrt{V}, \tag{51} \]

and in the case of intense diffusion (see (50))

\[ \varepsilon \sim \frac{1}{\sqrt{V}}. \tag{52} \]

From (51) and (52) there follows an interesting conclusion concerning the selectivity of radar observations. Let us compare meteoric bodies with different velocities. The transition to meteoric bodies with higher velocities in the region of small \(V\) leads to an increase in the power of the echo signal, since in this case the diffusion coefficient is comparatively small (slow meteors evaporate at relatively low altitudes), and formula (51) is valid.

As the velocity increases, the characteristic altitude increases, which is associated with an increase in the diffusion coefficient; when the diffusion becomes very intense, a further increase in the velocity of the meteoric body leads to a decrease in the echo-signal power (52). Consequently, there exists an optimum value of the velocity \(V_{\mathrm{opt}}\), to which corresponds \(\varepsilon=\varepsilon_{\max}\) (for given \(m\) and \(\lambda\)). Thus, there is selectivity of radar observations with respect to the velocity of meteoric bodies, a point already noted by Yu. A. Loshchilov, Hoffmeister, and others.

However, if the dependences (51) and (52) are valid, the role of such selectivity is small, since the dependence \(\varepsilon(V)\) is weak, while the dependence of the number of detected meteors \(N(V)\) will be still weaker, since, roughly speaking,

\[ N \sim \frac{1}{\sqrt{\varepsilon}} . \]

Some remarks

Above, we considered the dependence of the greatest value of the echo-signal power on certain parameters of the locator, the meteor, and the atmosphere under normal reflection of radio waves from an ionized meteor trail, on the assumption that the initial radius of the trail \(a\) is small in comparison with the wavelength, and it may be assumed that

\[ 2\left(\frac{2\pi a}{\lambda}\right)^2 \ll 1 . \]

This condition is fulfilled in the region of low altitudes; in the region of high altitudes, however, it is fulfilled only for comparatively large wavelengths.

For small wavelengths, the dependence of \(\varepsilon\) on \(\lambda\) and \(V\) in the region of large \(V\) will apparently be sharper than follows from the formula obtained above (if the value of the exponent \(n \simeq 1\)).

In some cases it is necessary to know what level the power reaches after the time \(t=T_{\min}\) following the formation of the trail (more precisely, after the passage of the meteoric body through the middle of the first Fresnel zone). The time \(T_{\min}\), necessary for recording the meteor (detecting it or measuring some parameter), characterizes the response speed of the system. In the case of high-speed systems operating at comparatively long wavelengths (for example, at \(\lambda \simeq 10\) m), under conditions of weak diffusion

\[ \frac{32\pi^2 D T_{\min}}{\lambda^2} \ll 1 \quad \left(u^{-2}(ka)^2 \ll 1\right); \]

in this case the conclusions obtained above are valid.

In the case of a low-speed system and a comparatively short wavelength under conditions of intense diffusion, the quantity \(\varepsilon(T_{\min})\) will depend substantially on the factor

\[ e^{-\frac{32\pi^2 D T_{\min}}{\lambda^2}} \]

and the character of the dependence of \(\varepsilon(T_{\min})\) on \(V\) will be different from that in the preceding case. A detailed consideration of this question is not given here.

Conclusions

1. A refined expression has been obtained for the power of the echo signal under conditions of normal reflection of radio waves from an ionized meteor trail (27).

2. Under conditions of intense diffusion, the echo-signal power

\[ \varepsilon \sim \frac{1}{\Delta}, \]

and not

\[ \frac{1}{\Delta^2}, \]

as follows from Hawkins’s work [3].

1. The power of echo signals received from the region of characteristic height under conditions of intense diffusion is proportional to

\[ \varepsilon \sim \frac{\lambda^{4.5} V}{R^{3.5}D}\,\beta^2; \]

the dependence of \(\varepsilon\) on \(\lambda\) and \(R\) proves to be sharper than in weak diffusion, when the relation holds:

\[ \varepsilon \sim \frac{\lambda^3}{R^3}\,\beta^2; \]

however, the dependence \(\varepsilon(\lambda, R, D, V)\) is weaker than in Hawkins,

\[ \varepsilon \sim \frac{\lambda^6 V^2}{R^4 D^2}. \]

1. The power of echo signals arriving from the region located above \(h_m\) (for \(p \ll p_m\)) depends strongly on the velocity of the meteor body:

\[ \varepsilon \sim \beta^2 V^4 \sim V^{2(n+2)} \text{—for weak diffusion} \]

and

\[ \varepsilon \sim \beta^2 V^5 \sim V^{2(n+2.5)} \text{—for strong diffusion.} \]

1. The dependence of the power of the echo signal received from the region \(h \simeq h_m\) on the velocity of the meteor body proves to be weaker than in the case \(h > h_m\):

\[ \varepsilon \sim V^{2(n-0.75)} \text{—for weak diffusion} \]

and

\[ \varepsilon \sim V^{2(n-1.5)} \text{—for strong diffusion.} \]

1. The assumptions regarding the selectivity of radar observations of meteors with respect to the velocity of meteor bodies are confirmed.

Thus, if the ionization probability is proportional to the velocity to the first power (\(n \simeq 1\)), then

\[ \varepsilon \sim V\sqrt{V} \text{—for weak diffusion (i.e., in the case of slow meteors)} \]

and

\[ \varepsilon \sim \frac{1}{\sqrt{V}} \text{—for intense diffusion (i.e., in the case of fast meteors).} \]

However, for \(n \simeq 1\) the selectivity with respect to velocity is weak.

References

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

2. Hawkins G. S. Radar echoes from meteor trails under conditions of severe diffusion, Proc. I. R. E., No. 9, 1192, 1956.

3. Lovell B. and Clegg J., Radio Astronomy, Foreign Literature Publishing House, 1953.

4. Peregudov F. I. On the influence of meteor velocities on the hourly number under conditions of radio observations, A. Zh., 35, 888, 1958.

5. Bogomolov A. F. Fundamentals of Radiolocation, Sov. Radio, 1954.

6. Flood W. A. Meteor echoes at ultra-high frequencies, Journ. Geophys. Res., 62, No. 1, 79, 1957.

7. Massey H. S., Sida D. W., Collision processes in meteor trails, Phil. Mag., 46, No. 373, 190, 1955.

8. Sida D. W. Atomic collisions in meteor trails, “Meteors,” Ed. by T. R. Kaiser, p. 26, 1955.

9. Loshchilov Yu. A. Application of the theory of atomic collisions to ionization processes in meteor trails, Izv. Astron. Observ. of Odessa University, 5, issue 1, 39, 1959.

10. Evans S. Scale heights and pressures in the upper atmosphere from radio-echo observations of meteors, M. N. R. A. S., 114, No. 1, 63, 1957.

11. Hawkins G. S. Meteor ionization and its dependence on velocity, Astr. J., 124, No. 1, 311, 1956.

12. 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.

13. Fialko E. I. On the question of the dependence of the linear electron density in a meteor trail on the velocity and mass of the meteoroid, Proceedings of the Tomsk Polytechnic Institute, vol. 105, 250, 1960.

14. Greenhow J. S., Neufeld E. L., The diffusion of ionized meteor trails in the upper atmosphere, Journ. Atm. Terr. Phys., 6, Nos. 2–3, 133, 1955.

15. Fialko E. I. The influence of meteor velocity on the echo-signal power, Proceedings of the Tomsk Polytechnic Institute, vol. 105, 260, 1960.