ON THE ROLE OF THE VELOCITY OF A METEOR BODY IN RADAR OBSERVATIONS OF METEORS

E. I. FIALKO

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

Visual observations of meteors yield results that are greatly distorted by selection [1].

The difference between the visible picture and the true one is determined, above all, by a physical factor: depending on the conditions under which meteor bodies encounter the Earth (which determine their geocentric velocity), “meteor bodies give rise to brighter or fainter meteors and produce greater or lesser ionization of the atmosphere” [1].

The important role of the physical factor in the complex set of phenomena associated with the formation of a meteor has been convincingly shown by B. Yu. Levin [1].

Consideration of the role of the physical factor in radar observations of meteors is undoubtedly of considerable interest.

It is known that the visual method has increased sensitivity to fast meteors.

Therefore a natural question arises: is selectivity with respect to the velocity of meteor bodies also inherent in the radar method; if selectivity is present, what is its character—is it the same as in the case of the visual method, or different from it?

To answer these questions, one should consider the dependence of the number of detected meteors on the velocity of the meteor bodies.

However, for an approximate solution of the question one may restrict oneself to a simpler consideration, based on determining the dependence of the power of the echo signal on the velocity of the meteor body. Such an approach makes it possible to replace consideration of the entire set of detected meteors by consideration of only a single meteor. In this case meteor bodies are compared that differ from one another only in the magnitude of their geocentric velocity.

Since the number of meteor bodies increases rapidly as their mass decreases, it seems expedient to consider relatively the weakest meteors, because they constitute the majority of the detected meteors. As is known, weak meteors are detected by radar chiefly in the region of the most intense ionization. It is therefore of interest to determine the de—

dependence of the echo-signal power on the velocity of the meteoric body for the region of the characteristic height.

The dependence of the echo-signal power received by a radar on the velocity of the meteoric body has been analyzed in a number of works [2—4].

However, it has not been possible to obtain a complete picture of the variation of the power of the received radio signal \(\varepsilon\) as a function of the velocity of meteoric bodies \(v\), all other conditions being equal. In particular, it was shown that, for a linear dependence of the ionization probability on the velocity of meteoric bodies, the function \(\varepsilon(v)\) has an extremal character [3]. It remained unclear, however, in what velocity range the extremum lies.

Of special interest, as was indicated above, is the investigation of signals arriving from the region of the most intense ionization (that is, from the region of the characteristic height). The nature of the dependence \(\varepsilon(v)\) in this case is not obvious. Indeed, with increasing velocity (more precisely, in going to a meteoric body of greater velocity but of the same mass), the characteristic height increases. This leads to an increase in the initial radius of the ionized trail, as a result of which the power of the reflected signals should decrease.

At the same time, an increase in velocity is associated with a decrease in the time during which the meteor traverses the first Fresnel zone; if the diffusion coefficient and the length of the first Fresnel zone remained unchanged, the diffusion effect would be weakened and the signal power would increase. However, owing to the increase in the characteristic height, the slant range increases (for an unchanged zenith distance of the radiant), which leads to an increase in the first Fresnel zone; the diffusion coefficient also increases; as a result, the diffusion effect should be manifested more strongly, entailing a decrease in the echo signal.

On the other hand, with increasing velocity the ionization probability increases and, consequently, so does the linear electron density, which causes an increase in the power of the received signal.

Thus, when the velocity increases, some factors tend to increase, and others to decrease, the echo-signal power. Without an appropriate analysis it is difficult to predict which factors will prove predominant. It is not excluded that in one velocity range some factors will prevail, and in another range—others; it is also possible that at certain velocities, under certain conditions, some factors will be compensated by others.

In the present work the dependence \(\varepsilon(v)\) is considered for the case of reception of echo signals arriving from the region of the characteristic height under conditions of normal reflection of radio waves from ionized trails [5].

Even with this formulation of the problem, the analysis is associated with considerable difficulties, since there is still no exact information on the dependence of the ionization probability on the velocity of the meteoric body.

The power of the signal received from the region of the characteristic height, in the absence of resonance effects, will be equal to [3, 5]

\[ \varepsilon(t)= A\,\frac{P_i G^2 \lambda^3}{R^3} \left(\frac{\beta m \cos \chi}{\mu H}\right)^2 \xi^2 e^{-2(kr_0)^2} e^{-\frac{32\pi^2Dt}{\lambda^2}}, \tag{1} \]

where

\[ A=\frac{0.7}{81\pi^2} \left(\frac{e^2}{m_e c^2}\right)^2 \approx 7\cdot 10^{-29}; \]

\(e,\ m_e\) are the charge and mass of the electron; \(c\) is the speed of light; \(P_i\) is the power in the radiated pulse; \(G\) is the antenna directivity coefficient; \(\lambda\) is the wavelength; \(R\) is the slant range from the locator to the trail (along the normal); \(\beta\) is the probability of ionization; \(m\) is the mass of the meteoric body; \(\chi\) is the zenith distance of the meteor radiant; \(\mu\) is the mass of the meteoric atom; \(H\) and \(D\) are the height of a homogeneous atmosphere and the diffusion coefficient corresponding to the characteristic height; \(\xi\) is the coefficient characterizing the decrease in the intensity of reflection of radio waves from the ionized trail, caused by the diffusion effect and by the finite time during which the meteor traverses the first Fresnel zone, equal to

\[ \xi=\frac{1-e^{-1.5\Delta^{0.5}}}{1.5\Delta^{0.5}}; \tag{2} \]

\[ \Delta=\frac{16\pi^2 D R^{\frac{1}{2}}}{\upsilon \lambda^{\frac{3}{2}}}; \tag{3} \]

\(\upsilon\) is the velocity of the meteoric body; \(k=\dfrac{2\pi}{\lambda}\) is the wave number; \(r_0\) is the initial radius of the ionized meteor trail; \(t\) is the time elapsed since the meteoric body passed through the middle of the first Fresnel zone, i.e., the point lying at the intersection of the trajectory of the meteoric body with the normal dropped from the position of the locator to the trajectory.

Formula (1) is valid for reflection of radio waves from underdense meteor trails, i.e., for a line density \(\alpha \ll 10^{12}\) electrons/cm.

Passing from power to voltage, we obtain

\[ u(t)=B\,\frac{\xi}{R^{\frac{3}{2}}H}\,e^{-(kr_0)^2}\,e^{-\frac{16\pi^2Dt}{\lambda^2}}, \tag{4} \]

where \(B\) is a coefficient independent of the velocity of the meteoric body.

Formula (4) gives a representation of the change in the envelope of the amplitudes of the received pulses.

The greatest amplitude, as follows from (4), is equal to

\[ u_m=B\,\frac{\xi}{R^{3/2}H}\,e^{-(kr_0)^2}. \tag{5} \]

To determine the dependence \(u_m(\upsilon)\), it is necessary to express each of the factors as a function of \(\upsilon\).

The slant range may be represented in the form

\[ R=\frac{h_m}{\sin\psi}, \tag{6} \]

where \(\psi\) is the elevation angle of the reflecting segment of the meteor trail (in the particular case \(\psi=\chi\)).

Formula (6) does not take into account the curvature of the Earth’s surface. The dependence \(h_m(v)\) was found by Kaiser and Evans [5, 6]; they also obtained the dependence \(H(v)\).

The variation of the diffusion coefficient in the region of the meteor zone \(D(h)\) was found by Greenhow and Neufeld [7]; knowing \(D(h)\) and \(h_m(v)\), it is easy to construct \(D(v)\).

The relation between the initial radius of the ionized trail and the mean free path of the ions \(l_i\) was investigated by Manning [8], who showed that \(r_0 \simeq 14\,l_i\), with \(l_i \simeq \frac{1}{5}l_a\) (where \(l_a\) is the mean free path of neutral atoms).

Using the graph of \(D(v)\) and \(h_m(v)\), we find \(\Delta(v,\psi,\lambda)\) or \(\Delta(v,\chi,\lambda)\), and then \(\xi(v,\chi,\lambda)\).

Next, from \(r_0(v)\) we construct

\[ e^{-\left(\frac{2\pi}{\lambda}\right)^2 r_0^2} \]

as a function of \(v\). We represent the probability of ionization in the form \(\beta \sim v^n\), where \(n\) is a constant [5].

After preliminary constructions we find \(u_m(v)\) for \(\lambda = 4, 6, 8, 10\ \text{m}\) for three values of \(\psi\) (or \(\chi\)): \(20, 45, 70^\circ\).

Without presenting here the intermediate graphs, we give the ratio

\[ \frac{u_m(v)}{u_{m\max}(v)}, \]

where \(u_{m\max}\) is the greatest value of the amplitude in the given wave range (in our case for \(\lambda = 10\ \text{m}\)).

Since the value of the exponent \(n\), characterizing \(\beta(v)\), has not yet been refined, we take as a basis the data of theoretical works [9, 10] (\(n \simeq 1\)) and consider \(n\) within the limits \(0 \leq n \leq 2\).

The results of the calculations are presented in Figs. 1 and 2 for \(\chi = 45^\circ\), \(\lambda = 4; 6; 8; 10\ \text{m}\) and \(n = 0; 1; 2\).

Fig. 1. Dependence of the intensity of a meteor radio echo on the velocity of the meteor body,

\[ \frac{u_m}{u_{m\max}}(v). \]

\(u_m\)—the greatest amplitude of the radio echo for the given \(\lambda, v\); \(u_{m\max}\)—the same for \(\lambda = 10\ \text{m}\); \(v\)—velocity of the meteor body in km/sec; \(n = 0\); \(\chi = 45^\circ\).

As can be seen from Figs. 1 and 2, the change in the amplitude of the echo signal as a function of velocity has an extremal character for \(n \simeq 1 \div 2\), that is

there is an optimal velocity \(v_{\mathrm{opt}}\), at which, for given \(n\) and \(\lambda\), the highest sensitivity of the locator corresponds (to weak meteors). The optimal velocities increase with increasing \(n\) and \(\lambda\), but remain within the range of registered geocentric velocities of meteor bodies.

Fig. 2. \(\dfrac{u_m}{u_{m\max}}\,(v)\). \(n=1\)—solid curves; \(n=2\)—dashed; (the remaining data are the same as in Fig. 1).

A change in the angle \(\varphi\) of the trail (or \(\chi\) of the radiant) within the limits \(20\text{–}70^\circ\) has only a slight effect on the character of the dependence \(u_m(v)\), as is illustrated by Figs. 2 and 3, in which the function \(\dfrac{u_m}{u_{m\max}}(v,\chi)\) is presented for \(\chi=20, 45, 75^\circ\) at \(n=1\).

Fig. 3. \(\dfrac{u_m}{u_{m\max}}\). \(n=1\); \(\chi=20^\circ\)—solid curves; \(n=1\); \(\chi=70^\circ\)—dashed; (the remaining data are the same as in Fig. 1).

The value of the optimal velocity for a given \(n\) changes only slightly within the range \(\chi=20\text{–}70^\circ\) (Fig. 4).

The character of the variation of \(v_{\mathrm{opt}}(\lambda)\) depends on the value of the exponent \(n\) (Fig. 4).

From the preceding discussion it follows that the radar method has reduced sensitivity to slow and very fast meteors and increased sensitivity to meteors with medium and high velocities. Thus, for \(n \simeq 1\), in the wavelength range most often used for meteor observations \((\lambda = 4 \div 10\ \text{m})\), \(v_{\mathrm{opt}} \simeq 35 \div 45\ \text{km/sec}\); for \(n \simeq 1.5\), \(v_{\mathrm{opt}} \simeq 45 \div 55\ \text{km/sec}\) (Figs. 2, 3, 4).

However, for \(n \simeq 1\) the selectivity of the radar method is weakly expressed, which is not inconsistent with the result obtained earlier [4].

As \(n\) increases, the sensitivity with respect to fast meteors, as should be expected, increases. We note that even for \(n=2\) at \(\lambda = 4\ \text{m}\) there is appreciable selectivity with respect to slow and fast meteors.

Fig. 4. Dependence of the optimal velocity \(v_{\mathrm{opt}}\) on the wavelength \(\lambda\).
\(v_{\mathrm{opt}}\)—in \(\text{km/sec}\); \(\lambda\)—in \(\text{m}\). Solid curves are plotted for \(\chi = 45^\circ\) \((n=0.5;\ 1;\ 1.5;\ 2)\). The dashed curve corresponds to \(n=1\) and \(\chi = 70^\circ\); the dash-dotted curve to \(n=1\) and \(\chi = 20^\circ\).

Unfortunately, at present it still cannot be asserted with certainty that the true picture is the one corresponding to \(n \simeq 1\); however, there are also no weighty grounds for asserting an increased sensitivity of the locator to very fast meteors (as, for example, for \(n>2\)). We note that some investigators believe that the radar has reduced sensitivity to fast meteors [11, 12].

In all probability, outside the zone \(v = 30\text{--}60\ \text{km/sec}\) the sensitivity of the radar method decreases appreciably (which corresponds to the exponent \(n \simeq 1 \div 1.5\)).

From what has been said, it is clear how important it is to obtain objective data on the character of the dependence \(\beta(v)\).

Above, the dependence of the initial signal intensity on the velocity of the meteoroid was considered; that is, it was assumed that the locator has a very high operational speed, allowing the trail to be recorded immediately after its formation.

If, however, after the formation of the trail a time \(T_{\min}\), necessary for the detection of meteors, must elapse, then one should consider not the initial intensity of the echo signal corresponding to \(t=0\) (see formula (4)), but the intensity of reflection at the instant \(t = T_{\min}\). That is, in formula (4) it is necessary to take into account one more factor,

\[ e^{-\frac{16\pi^2 D T_{\min}}{\lambda^2}} . \]

Consideration of a low-operational-speed system is not given here.

In addition, \(\varepsilon_{\max}\) will correspond not to the characteristic height \(h_m\), but to some optimal height: under conditions of intense diffusion \(h_{\mathrm{opt}} < h_m\), and with increased recombination \(h_{\mathrm{opt}} > h_m\). Allowance for the difference between \(h_{\mathrm{opt}}\) and \(h_m\) and the introduction of the associated corrections into \(v_{\mathrm{opt}}(n,\lambda)\) and into \(u_m(v,\lambda,n)\) will be carried out additionally.

It should also be noted that the dependence \(\varepsilon(v)\) for regions lying both above and below \(h_m\) will be different from the dependence \(\varepsilon(v)\),

obtained for reception from the region of the characteristic height. Therefore the notion of the selectivity of the radar method set forth above may require corrections.

Conclusions

1. For a value of the exponent \(n \simeq 1 \div 2\), the dependence of the echo-signal intensity on the velocity of the meteoric body has an extremal character. In this case the optimum velocity \(v_{\mathrm{opt}}\) lies in the range of velocities usually recorded.

2. With a linear dependence of the ionization probability on velocity \((n \simeq 1)\), the radar method is most sensitive to meteors with intermediate velocities. However, the selectivity in this case is weakly expressed.

3. For \(n \simeq 1.5\), the radar method has increased sensitivity to fast meteors (with velocities \(v \simeq 50 \div 60\ \mathrm{km/sec}\)). For relatively slow meteors the sensitivity decreases noticeably.

4. To clarify the role of the physical factor it is necessary, first of all, to determine the character of the dependence \(\beta(v)\) and to consider the change in the number of recorded meteors as a function of velocity.

References

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

2. Peregudov F. I. On the influence of meteor velocities on the hourly number under conditions of radio observations, A. Zh., vol. 35, No. 6, p. 888, 1958.

3. Fialko E. I. Power of the signal reflected from a meteor trail under conditions of intense diffusion, Izvestiya TPI, vol. 109, p. 40, 1962.

4. Fialko E. I. An approximate estimate of the probability of meteor ionization, A. Zh., 1959, 36, issue 3, 491.

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

6. Evans E. Scale heights and pressures in the upper atmosphere from radio-echo observation of meteors, M. N. R. A. S., v. 114, No. 1, p. 63, 1954.

7. Greenhow J. S., Neufeld E. L. The diffusion of ionized meteor trails in the upper atmosphere, Journ. Atm. Terr. Phys., v. 6, No. 2–3, p. 133, 1955.

8. Manning L. A. The initial radius of meteoric ionization trails, Journ. Geoph. Res., v. 63, No. 1, p. 181, 1958.

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

10. Lomikov Yu. A. On determining the probability of ionization in meteor trails by the method of radio-visual observations, Izvestiya astronomicheskoi observatorii, 5, issue 1, 35, 1959.

11. Hoffmeister C. Jonization durch Meteore, Sterne, v. 33, No. 3–4, s. 62, 1957.

12. Öpik E. J. Meteors and upper atmosphere, Irish Astronomical Journ., v. 3, No. 6, p. 166, 1955.