Resonant Scattering of Radio Waves by Paraboloidal Meteor Trails

E. K. Nemirova

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

Introduction

The radar method of observing meteors has now become widespread and continues to develop rapidly. Along with the improvement and creation of new equipment, and the acquisition and processing of observational material, numerous studies of a theoretical nature are being carried out. In particular, much attention is being devoted to questions of the scattering of radio waves by meteor trails and to the interpretation of observational results with allowance for the regularities of scattering. Research in this direction is very important, since in the radar observation of meteors a whole series of specific phenomena arises, without taking them into account the correct interpretation of observational results is impossible.

One of the features of radar observations of meteors is the possibility of the occurrence of resonant scattering, which causes a dependence of the parameters of the reflected signal on the polarization of the field of the incident wave relative to the axis of the trail.

A number of theoretical works, which have received experimental confirmation, are devoted to questions of resonant scattering. However, these works [1, 2, 3, and others] deal mainly only with the essence and regularities of the phenomenon itself, without applying the conclusions obtained to an analysis of the influence of the polarization effect on observational results.

The task of the present work is to clarify the dependence of the characteristics of the reflected signal in resonant scattering on the parameters of the meteor, the atmosphere, and the equipment. The characteristics of the reflected signal include the amplitude, phase, and form of the diffraction pattern.

Features of Resonant Scattering

A theoretical consideration of the question of the scattering of radio waves by meteor trails [1,2] leads to the conclusion that, when the polarization of the field of the incident wave is perpendicular to the axis of the trail, the phenomenon of plasma resonance may occur. The linear electron density $\alpha$, at which resonance is possible, lies within the limits determined by the inequalities

\[ (kr_0)^2 \ll 11.24 \cdot 10^{-13}\alpha;\ \alpha \ll 2.4 \cdot 10^{12}\ \text{cm}^{-1}, \]

where \(k=\dfrac{2\pi}{\lambda}\); \(\lambda\) is the wavelength; \(r_0\) is the trail radius.

If the conditions necessary for resonance exist, the amplitude and phase of the reflected signal for perpendicular polarization differ from the amplitude and phase for parallel polarization. The ratio of the moduli \(\rho\) and the phase shift \(\psi\) of the scattering coefficients for the two types of field polarization were calculated by Kaiser and Closs for cylindrical trails with a Gaussian distribution of electron density in the radial direction. The dependence of \(\rho\) and \(\psi\) on \(\dfrac{(kr_0)^2}{\alpha}\) is shown in Figs. 1 and 2.

The dependences shown in the figures are valid for an expanding trail having a cylindrical shape. But such a model is approximate. The shape of a real meteor trail is not cylindrical. Since the velocity of the meteor is finite, the parts of the trail formed earlier, under the influence of diffusion, expand more strongly, and the trail acquires a shape close to a paraboloid of revolution.

Fig. 1. Polarization ratio as a function of \((kr_0)^2/\alpha\) for different values of the electron density.

In this case, different sections of the trail will correspond, at the same instant of time, to different resonance frequencies; the values of \(\rho\) and \(\psi\) will not be the same along the trail.

Billem and Brown [3], investigating the question of the influence of this effect on the magnitude of the experimentally observed polarization ratio, calculated the time dependence of the amplitude of the signal reflected from a meteor for the two types of polarization. The calculation was carried out for one particular case. In the calculation the relations used were

\[ E_{\parallel}\sim \int_{-\infty}^{n_0} e^{-\kappa_1\delta(n_0-n)} e^{i\varphi}\,dn, \tag{1} \]

\[ E_{\perp}\sim \int_{-\infty}^{n_0} \rho(n)\, e^{-\kappa_1\delta(n_0-n)} e^{i(\varphi+\psi)}\,dn, \tag{2} \]

where \(n\) is the distance along the trail from the reflecting point in units of \(\sqrt{R\lambda}\); \(k_1 = 16\pi^2D/\lambda^2\) (\(D\) is the diffusion coefficient); \(\delta = \sqrt{R\lambda}/v\) (\(R\) is the range, \(v\) is the meteor velocity); \(\varphi = 2\pi n^2\) is the phase angle of the element \(dn\) relative to the phase at the point \(n=0\); \(\rho, \psi\) are the polarization ratio and the phase of the scattering coefficient for the trail element \(dn\).

Fig. 2. Dependence of the phase of the scattering coefficient \(\Psi\) for perpendicular polarization on the quantity \(\dfrac{(kr_0)^2}{a}\).

The calculation was carried out by numerical integration. As a result, the curves shown in Fig. 3 were obtained.

Fig. 3. Change in the amplitude of the reflected signal with time for two types of polarization:
— parallel polarization,
– – – perpendicular polarization.

The authors came to the conclusion that the finite-velocity effect increases the observed polarization ratio. This conclusion, however,

... has only a qualitative character and was made on the basis of consideration of a single particular case.

A more detailed consideration of the question of the degree to which the resonance effect manifests itself under different observational conditions gives a number of new results in comparison with those available in the published literature.

Length of the Resonating Section of the Trail

The influence of the resonance effect on the characteristics of the signal reflected from a meteor depends to a considerable extent on the conditions under which the observations were made. At the same time, one of the principal factors is the nonconstancy of the radius and of the electron density along the trail.

Fig. 4. Geometry of reflection of a radio wave from a meteor trail.

The nonconstancy of the radius along the length of the trail may be caused by the following reasons (apart from various kinds of irregular phenomena in the atmosphere):

1. Non-simultaneous formation of individual sections of the meteor trail, owing to the finite velocity of the meteoroid.

2. Nonconstancy of the diffusion coefficient along the length of the trail because of the different heights of the trail sections.

3. Nonconstancy of the initial radius.

In the general case, the radius of an element of the trail \(dl\) (Fig. 4) is equal to

\[ r_0(l)=\sqrt{[r_n(l)]^2+4D(l)\cdot\frac{l_0-l}{v}}, \tag{3} \]

where \(l_0\) is the instantaneous position of the head part of the trail.

It has been shown theoretically [4] and experimentally [5] that the degree of ionization depends on height, as shown in Fig. 5.

If \(r_0\) and \(\alpha\) are nonconstant along the trail, then at a given instant of time not the entire trail will be in a state of resonance, but only some section adjacent to its beginning. The dependences of \(\rho\) and \(\psi\) on \(l\) are illustrated in Fig. 4. The length of the resonating section \(l_{\mathrm{res}}\) may conventionally be defined as the length of that part of the trail for which \(\psi\) differs appreciably from zero, since the variation of \(\psi\) along the trail has a greater influence than the variation of \(\rho\).

In the general case

\[ l_{\mathrm{res}}=l_0-l_{\mathrm{gr}}, \]

where \(l_{\mathrm{gr}}\) is the distance from the reflecting point to that section of the trail where \(\psi\) becomes close to zero. The quantity \(l_{\mathrm{gr}}\) depends on the form of the function \(\psi(l)\), which is determined by the law of variation of \(r_0\) and \(\alpha\) along the trail. As is seen from Fig. 2, \(\psi\) may be considered close to zero when \((kr_0)^2/\alpha=10^{-12}\ \mathrm{cm}\). Taking this into account, \(l_{\mathrm{gr}}\) is the root of the equation

\[ k^2\left[r_n^2(l)+4D(l)\frac{l_0-l}{v}\right]-\alpha(l)\cdot10^{-12}=0. \tag{4} \]

In the simplest case, i.e., for \(r_H=0\) and \(\alpha=\mathrm{const}\),

\[ 4k^2D\frac{l_0-l_{2p}}{v}=a\cdot 10^{-12} \]

and

\[ l_{\mathrm{res}_0}=l_0-l_{2p}\simeq 0.65\cdot 10^{-13}\frac{\alpha\lambda^2}{\pi^2D}\,v. \tag{5} \]

The nonconstancy of the initial radius and of the diffusion coefficient leads to a shortening of \(l_{\mathrm{res}}\) in comparison with \(l_{\mathrm{res}_0}\). Owing to the variation of \(a\) with height, when a meteor is detected at a height greater than the characteristic one, \(l_{\mathrm{res}}<l_{\mathrm{res}_0}\); at heights less than \(h_m\), \(l_{\mathrm{res}}>l_{\mathrm{res}_0}\).

Fig. 5. Variation of \(\alpha\) with height.
— — — — theoretical variation,
——— experimental result.

The absolute value of \(l_{\mathrm{res}}\) cannot completely determine the degree to which the resonance effect is manifested. The influence of resonance on the parameters of the reflected signal depends on the relation between \(l_{\mathrm{res}}\) and the Fresnel zones. The sizes of the Fresnel zones are determined by the quantity \(\sqrt{R\lambda}\).

Calculations show that the length of the resonating section of the trail in most cases is comparable with the length of the first Fresnel zone, and under some conditions is much smaller than it.

Some characteristics of the signal under resonance

When analyzing the influence of the resonance effect on the signal reflected from a meteor, it is useful to consider such signal characteristics as the maximum value of the amplitude, the character of the change of amplitude with time, the character of the change of phase with time, the time shift of the maximum amplitude relative to the time at which the reflecting point is passed by the meteor, the periods of amplitude pulsations, and the depth of the pulsations.

The parameters listed describe the signal rather fully and are used in measuring certain characteristics of the meteor and of the atmosphere (meteor velocity, diffusion coefficient, etc.). The signal parameters under resonance must be compared with the correspond-

...the same parameters in the absence of resonance (all other conditions being equal).

Constructing diffraction patterns (for amplitude and phase) in the presence and absence of resonance makes it possible to obtain all the signal characteristics listed above.

The computational method, based on the use of relations (1) and (2), is set forth in [6]. The calculation was carried out for a number of cases characterized by a definite set of observing conditions.

Case 1

Calculation conditions: 1) initial radius \(r_\mu=0\); 2) \(\alpha\) and \(D\) are constant. In this case the diffraction pattern is completely determined by the quantities

\[ l_{\mathrm{res}}/\sqrt{R\lambda}\quad \text{and}\quad B=\frac{8\pi^2 D\sqrt{R\lambda}}{\lambda^2 v} =\frac{1}{\sqrt{2}}\cdot \frac{t_1}{\tau}, \]

where \(t_1\) is the time taken by the meteor to traverse the first Fresnel zone, and \(\tau\) is the amplitude damping constant in the absence of resonance [1, 2]. The smaller \(B\), the closer the trail shape is to cylindrical.

For the first case, three families of diffraction patterns (18 curves) were calculated for the values \(B=0.08\), \(B=0.15\), and \(B=0.226\).

The calculations made it possible to establish that, under the influence of resonance, the diffraction pattern changes markedly. This change is expressed in an increase in the signal amplitude, a time delay of the diffraction pattern, and a change in the periods and depth of the pulsations of amplitude and phase. A quantitative analysis of these changes showed that the principal quantity determining the degree to which the resonance effect is manifested is the ratio \(l_{\mathrm{res}}/\sqrt{R\lambda}\); the influence of the parameter \(B\) is weak.

The dependence of the ratio of maximum amplitudes with resonance and without resonance, obtained as a result of the calculations, is shown in Fig. 6¹. The results presented differ substantially from those of Kaiser and Closs [2], which do not take into account the difference between the trail shape and a cylindrical one. According to [2], \(\rho_{1\max}\) is constant; in accordance with Fig. 6, \(\rho_{1\max}\) depends strongly on \(l_{\mathrm{res}}/\sqrt{R\lambda}\), approaching a maximum at \(l_{\mathrm{res}}/\sqrt{R\lambda}\simeq 1\); when \(l_{\mathrm{res}}/\sqrt{R\lambda}\) is decreased or increased in comparison—

Fig. 6. Polarization ratio for maximum amplitudes \(\rho_{1\max}\) as a function of the length of the resonating segment. \(\cdot\)—\(B=0.08\), \(\times\)—0.15, \(\odot\)—0.226.

Fig. 7. Maximum value of the phase shift at resonance, \(\Psi_{\max}\), and the phase at the moment of maximum amplitude, \(\Psi_m\), as functions of \(l_{\mathrm{res}}/\sqrt{R\lambda}\).

Fig. 8. Delay of the signal maximum at resonance, \(\Delta t\), relative to the first pulsation period of the ideal diffraction pattern, \(T_1\).

Fig. 9. Ratio of the first pulsation period at resonance, \(T'_1\), to the first pulsation period of the ideal diffraction pattern, \(T_1\).

\(\times\)—\(B=0.08\), \(\odot\)—\(B=0.15\), \(\cdot\)—\(B=0.226\).

Fig. 10. Ratio of the period \(T_{2}^{1}\) to the second period of the ideal diffraction pattern.
\(\times\)—\(B=0.08\), \(\odot\)—\(B=0.15\), \(\cdot\)—\(B=0.226\).

Fig. 11. Depth of pulsations of the diffraction pattern at resonance. \(\Delta u_{1}\)—difference between the second maximum and the first minimum at resonance; \(\Delta u_{1}\)—the same (for the ideal case).
\(\times\)—\(B=0.08\), \(\odot\)—\(B=0.15\), \(\cdot\)—\(B=0.226\).

Fig. 12. Polarization ratio taking into account the initial radius
\((B=0.08,\ \lambda=10\ \text{m})\)
——— \(r_{\mathrm{n}}=0\); 1—20 cm; 2—28 cm; 3—40 cm; 4—51 cm.

Fig. 13. Maximum value of the phase shift for different initial radii
\((B=0.08,\ \lambda=10\ \text{m})\)
— — — \(r_n=0;\ 1—20\ \text{cm};\ 2—28\ \text{cm};\ 3—40\ \text{cm};\ 4—51\ \text{cm}.\)

Fig. 14. Ratio of the time delay of the diffraction pattern \(\Delta t\) to the first pulsation period \(T_1\) for different \(r_n\) \((B=0.08)\).
— — — \(r_n=0;\ 1—20\ \text{cm};\ 2—28\ \text{cm};\ 3—40\ \text{cm};\ 4—51\ \text{cm}.\)

Fig. 15. Relative depth of pulsations, taking \(r_n\) into account
\((B=0.226,\ \lambda=5\ \text{m})\). — — — \(r_n=0,\)
\(1—17\ \text{cm};\ 2—25\ \text{cm};\ 3—30\ \text{cm};\ 4—35\ \text{cm};\)
\(5—39\ \text{cm}.\)

toward unity, there is a decrease in the polarization ratio. The phase of the scattering coefficient \(\psi\) at the Kaiser and Closs resonance changes with time, as shown in Fig. 2. When the amplitude reaches its maximum value, \(\psi_m \simeq 45^\circ\). The values of \(\psi_m\), taking into account the deviation of the trail shape from cylindrical as a function of the length of the resonating section, are given in Fig. 7. The curve for the maximum value of the phase shift is also given there (according to Kaiser and Closs, \(\psi_{\max}=180^\circ\)).

In addition to the amplitude and phase, resonance also changes the temporal relationships of the diffraction pattern. This is expressed in a delay of the diffraction pattern and in a change in the periods of the pulsations. Calculations for case 1 showed that the moment of the amplitude maximum at resonance occurs later; the time shift is determined by the ratio \(l_{\mathrm{rez}}/\sqrt{R\lambda}\) (Fig. 8).

The periods of the pulsations of the diffraction pattern remain unchanged for small lengths of the resonating section; as \(l_{\mathrm{rez}}\) increases, the periods begin to decrease, and this decrease is appreciable mainly for the first two periods (Figs. 9 and 10). As the periods decrease, the pulsations are smoothed out. The maximum decrease in the periods for pulsations that are still visible is 15–20%. The smoothing of the pulsations occurs in the region of large lengths of the resonating section. The behavior of the pulsation depth for different \(l_{\mathrm{rez}}\) is illustrated in Fig. 11.

The regularities obtained are valid for small heights and large wavelengths, when the magnitude of the initial radius can be neglected.

Calculation of the diffraction patterns with allowance for \(r_n\) showed that, owing to the influence of the initial radius, the maximum value \(\rho_{1\max}\) decreases, while the polarization ratio reaches its maximum value at larger values of \(l_{\mathrm{rez}}/\sqrt{R\lambda}\). As \(r_n\) increases, a sharp decrease in \(\rho_{1\max}\) is observed at small \(l_{\mathrm{rez}}/\sqrt{R\lambda}\), and a rise in the region of large lengths of the resonating section (Fig. 12).

The influence of \(r_n\) causes a decrease in the phase shift (Fig. 13), in the time lag (Fig. 14), and weakens the effect of resonance on the pulsation periods. The depth of the amplitude pulsations decreases with increasing \(r_n\) in the region of small values of \(l_{\mathrm{rez}}/\sqrt{R\lambda}\); the smoothing of the pulsations occurs at larger lengths of the resonating section in comparison with the case \(r_n=0\) (Fig. 15).

Thus, plasma resonance in trails of the unstable type has a noticeable influence on such characteristics of the reflected signal as the maximum value of the amplitude, the character of the change in amplitude, the magnitude of the phase shift and the character of the change in phase with time, and the form of the diffraction pattern.

REFERENCES

1. N. Herlofson, Plasma resonance in ionospheric irregularities, Arkiv för Fysik, 3, 247, 1951;

2. T. R. Kaiser, R. L. Closs. Theory of radio reflections from meteor trails. Phil. Mag., ser. 7, 43, N 336, 1, 1952.

3. E. R. Billam, I. C. Browne. Characteristics of radio echoes from meteor trails, IV. Polarization effects, Proc. Phys. Soc., Sec. B, 69, Part 1, N 433, 98, 1956.

4. T. R. Kaiser. Radio echo studies of meteor ionisation, Phil. Mag., 2, N 8, 495, 1953.

5. J. S. Greenhow, E. L. Neufeld, The variation of ionization along a meteor trail, Mon. Not. RAS, 117, No. 4, 1957.

6. E. K. Nemirova. On the role of resonance effects in measuring meteor velocities, Astr. journal, vol. XXXVI, issue 3, 1959.

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

1. Some scatter of the points is explained by errors in the calculation, which was carried out by a graph-analytic method. ↩