Influence of Meteor Velocity on Echo Signal Power
Abstract Generated abstract
This paper analyzes how meteor velocity affects the power of radio echo signals reflected from unstable meteor trails near the height of maximum electron line density. Starting from the usual dependence of echo power on the square of electron line density, it argues that a simple scaling with the ionization probability overestimates the velocity effect, because increasing velocity also changes the characteristic height, atmospheric scale height, slant range, and diffusion losses during Fresnel zone formation. Using approximations from prior meteor radar studies, the author derives a dependence of the form ε proportional to v raised to 2(x minus 1.3), where x is the exponent in the assumed ionization probability law β proportional to v raised to x. The paper concludes that quantitative prediction depends chiefly on better determination of this ionization exponent and notes that delayed signal measurements may show additional velocity related attenuation.
Full Text
Effect of Meteor Velocity on Echo-Signal Power
E. I. Fialko
(Presented by the scientific seminar of the Radio Engineering Faculty)
Preliminary Remarks
It is known that the power of a signal reflected from an unstable meteor trail, at the receiver input \(\varepsilon\), is proportional to the square of the linear electron density \(\alpha\)
\[ \varepsilon \sim \alpha^2 . \tag{1} \]
Let us consider how the power of an echo signal arriving from the region of the characteristic height \(h_m\) depends on the velocity of the meteor body.
According to Kaiser [1], the maximum linear electron density \(\alpha_{\max}\) may depend on the meteor velocity because the ionization probability \(\beta\) depends on \(v\). It is usually assumed that
\[ \beta = a \cdot v^x, \tag{2} \]
where \(a\) and \(x\) do not depend on \(v\).
Comparing (1) and (2), one might arrive at the conclusion that, as the velocity increases, the power of the reflected signal increases in proportion to \(v^{2x}\)
\[ \varepsilon \sim v^{2x}. \tag{3} \]
However, this is not correct. In reality, the dependence of \(\varepsilon\) on \(v\) is weaker. First of all, it should be taken into account that, as the velocity increases, \(h_m\) and the scale height of the homogeneous atmosphere \(H\) increase, and moreover [1]
\[ \alpha_{\max} \sim \frac{\beta}{H}\, W, \tag{4} \]
and, as can be shown using the dependences \(h_m(v)\) and \(H(h_m)\) [1],
\[ H = b \cdot v^y, \tag{5} \]
where \(b \simeq 1^{1)}\), \(y \simeq 0.54\). Consequently,
\[ \alpha_{\max} \sim v^{x-y}, \qquad \varepsilon \sim v^{2(x-y)} . \tag{6} \]
\(^{1)}\) We omit the dimensions of the proportionality coefficients.
In addition, a change in \(h_m\) entails a change in the slant range (for a given zenith distance of the radiant \(\chi\), and when the region \(a \simeq a_{\max}\) is irradiated under conditions of normal reflection of radio waves from the meteor trail). As a result, an increase in velocity leads to an increase in the slant range and, consequently, to a decrease in \(\varepsilon\).
A decrease in the echo-signal amplitude due to the fact that the time of passage by the meteor through the first Fresnel zone [1] is finite is also of substantial importance. With increasing velocity, the time required for the formation of the first Fresnel zone decreases, despite some increase in the slant range.
It might seem that, owing to this effect, the decrease in the echo-signal amplitude caused by diffusion would be less sharp than in the case of a lower velocity \(v\). However, the diffusion coefficient increases very rapidly with increasing \(h\), and although the time during which the meteor forms an ionized trail in the region of the first Fresnel zone decreases, the increase of the diffusion coefficient (owing to the increase of \(h_m\)) may lead to a decrease in \(\varepsilon\) (because of the effect of the finite velocity of the meteor).
Thus there naturally arises the question of how, and by how much, the power of the echo signal changes when the meteor velocity changes\(^1\).
Initial relations
Without giving the complete expression for the echo-signal power, we shall take into account only those quantities that depend on \(v\) [1]:
\[ \varepsilon \sim \frac{a^2}{R^3}\cdot \xi . \tag{7} \]
Formula (7) is valid for the case \(a \ll 2.4\cdot 10^{12}\) el/cm, the initial radius of the ionized trail being small in comparison with the wavelength.
The coefficient \(\xi\) takes into account the decrease in the amplitude of the reflected signal owing to the fact that, during the time of formation of the ionized trail in the region of the first Fresnel zone, a substantial broadening of the trail due to diffusion can occur. \(R\) is the slant range from the locator to the section of the trail that normally scatters the radio waves.
If the section of the trail reflecting the radio waves is situated at an altitude \(h \simeq h_m\), then \(a \simeq a_{\max}\).
Using [1], (4), (2), and (5), we represent \(a\) in the form
\[ a = B\cdot \frac{a}{\delta}\cdot v^{x-y}, \tag{8} \]
where
\[ B=\frac{4}{9}\cdot \frac{m}{\mu}\cdot \cos\chi; \]
\(m\) is the mass of the meteor; \(\mu\) is the mass of an atom of the meteoric body. The slant range
\[ R \sim \frac{h_m}{\sin\chi}. \]
Approximating the graphical dependence \(h_m(v)\) [1], we represent it in the form
\(^1\) The deceleration of the meteoric body is not taken into account.
\[ h_m=d\cdot v^z, \tag{9} \]
where
\[ d \simeq 55;\qquad z \simeq 0{,}15. \]
Thus,
\[ R \sim \frac{d}{\sin x}\cdot v^z. \tag{10} \]
To reveal the dependence of the coefficient \(\xi\) on \(v\), we shall use known data [1].
Coefficient \(\xi(v)\)
In [1] a graphical dependence is given for the maximum amplitude of the echo signal as a function of
\[ \Delta=16\pi^2\cdot D\cdot R^{\frac12}\,(v\cdot \lambda^{\frac32})^{-1}, \tag{11} \]
where \(D\) is the diffusion coefficient; \(\lambda\) is the wavelength.
Let us denote by \(\xi_1\) the coefficient characterizing the decrease in amplitude. Then the coefficient characterizing the decrease in power, \(\xi\), will be equal to
\[ \xi=\xi_1^2. \tag{12} \]
Approximating \(\xi_1(\Delta)\) and taking (12) into account, we obtain
\[ \xi(\Delta)=a_1\cdot \Delta^{x_1}, \tag{13} \]
where
\[ a_1 \simeq 0{,}4,\qquad x_1 \simeq 0{,}75. \]
The dependence of the diffusion coefficient on height was determined experimentally [2]. An approximation of the dependence \(D(h)\) gives
\[ D=a_2\cdot h^{x_2}, \tag{14} \]
where
\[ a_2 \simeq 6{,}3\cdot 10^{-27};\qquad x_2 \simeq 15{,}7. \]
Substituting into formula (14) the expression for \(h\) in the form (9), we obtain
\[ D=a_3\cdot v^{x_3}, \tag{15} \]
where
\[ a_3 \simeq 12{,}6,\qquad x_3 \simeq 2{,}36. \]
Taking (15), (10), and (9) into account, expression (11), after simple transformations, takes the form
\[ \Delta=a_4 v^{x_4}, \tag{16} \]
where
\[ x_4 \simeq 1{,}4;\qquad a_4 \simeq \frac{1{,}5\cdot 10^4}{\lambda^{3/2}}(\sin x)^{-\frac12}. \]
Substituting (16) into (13), we find
\[ \xi=a_5\cdot v^{x_5}, \tag{17} \]
where
\[ x_5 \simeq 1{,}07;\qquad a_5 \simeq 2{,}7\cdot 10^{-4}\cdot \lambda^{1{,}14}\cdot (\sin x)^{0{,}38}. \]
Let us note that \(\xi \sim \dfrac{1}{v}\), i.e., with increasing velocity the effect of diffusion weakens the echo signal.
Substituting into (7) the expressions found for \(x(v)\), \(R(v)\), and \(\xi(v)\), i.e. (8), (10), and (17), we obtain
\[ \varepsilon \sim a_0 \cdot v^n, \tag{18} \]
where \(n = 2(x - 1.3)\), and \(a_0\) is a coefficient independent of \(v\).
To determine the quantitative dependence \(\varepsilon(v)\), it is necessary to know the numerical value of the exponent \(x\).
At present there are still no reliable data on the magnitude of \(x\) [3]. However, as may be concluded from theoretical works [4 and others], the value of the exponent \(x\) is close to 1.
Thus, as follows from (18), an increase in the meteor velocity leads to a change in the power of the echo signal corresponding to the region \(h_m\); however, this change is not as large as would follow from (3).
Further consideration of \(\varepsilon(v)\) is connected with the necessity of studying the probability of ionization as a function of the velocity of the meteoric body, \(\beta(v)\), and also with allowance for the initial radius of the meteor trail.
Some remarks
In a number of cases it is of interest to determine the magnitude of the signal power after some time \((T)\) has elapsed since the formation of the trail, \(\varepsilon(T)\). This occurs, for example, in measurements whose performance requires a time \(T \gg T_{\min}\) (and, consequently, the signal must exceed a certain level during this time).
The results obtained above are valid for the case of highly operative systems (\(T_{\min}\) is very small) [5].
In the case of low-operative systems (\(T_{\min}\) is relatively large), with increasing velocity the signal power \(\varepsilon(T_{\min})\) decreases. This becomes clear if one takes into account [6] that
\[ \varepsilon(T_{\min}) \sim \frac{x^2}{R^3} \cdot \xi e^{-\frac{4\pi}{\lambda^2}\left(\rho_{\mathrm{н}}^2 + 4DT_{\min}\right)}, \tag{19} \]
where \(\rho_{\mathrm{н}}\) is the initial radius of the meteor trail.
A detailed consideration of (19) is not part of the task of the present work.
Conclusions
-
The velocity of a meteor has a substantial effect on the power of the echo signal. In the case of reception of signals normally reflected from a region of the trail with an electron line density close to the maximum, the echo-signal power is
\[ \varepsilon \sim v^{2(x-1.3)}; \]
the exponent \(x\) determines the dependence of the probability of ionization on the velocity of the meteoric body, \(\beta \sim v^x\). -
Determination of the quantitative dependence of \(\varepsilon\) on \(v\) is connected with refinement of the value of the exponent \(x\).
References
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Kaiser T. R., Radio echo studies of meteor ionization, Phil. Mag., v. 2, No. 8, 495, 1953.
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Greenhow J. S., Neufeld E. L., The diffusion of ionized meteor trails in the upper atmosphere, Journ. Atm. Terr. Phys., 6, No. 2–3, 133, 1955.
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Hawkins J. S., Meteor ionization and its dependence on velocity, Astr. J., v. 124, No. 1, 311, 1956.
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Sida D. W., Atomic collisions in meteor trails, “Meteors,” Ed. by T. R. Kaiser, 26, 1955.
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Fialko E. I., On the influence of wavelength on the effectiveness of the radar method of meteor investigation, Proceedings of the Siberian Physico-Technical Institute, issue 37, 219, 1959.
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Greenhow J. S. Characteristics of radio echoes from meteor trails, Proc. Phys. Soc., B., v. 65, 1952.