SOME COMPONENTS OF THE ERROR IN VELOCITY MEASUREMENT OF METEORS BY THE DIFFRACTION METHOD

G. V. AVRAAMOVA

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

At present there are three principal methods for measuring the velocity of meteors, based on the phenomenon of reflection of radio waves from an ionized meteor trail. The first, or range–time, method uses the change with time of the slant range to the “head” of the meteor and is applicable only to very large meteors capable of producing non-specular reflection. The second method is based on the diffraction of radio waves by the meteor trail and uses the change with time of the amplitude and phase of the reflected signal.

Observations show that this method is applicable to a much larger number of reflections, since it requires the presence of a reflection only in the immediate vicinity of the point of normal incidence of the radio ray on the meteor trail. The third method has been considered only theoretically. This method uses the change in the frequency of the signal reflected from the moving leading edge of the meteor trail. Like the first, this method is applicable only to very large meteors.

Manning [3] showed that the change in frequency of the signal reflected from the moving leading edge of the meteor trail near point \(A\) (Fig. 1), the point least distant from the radar station, is equal to

\[ F=\frac{2V^2}{\lambda R_0}\,t, \tag{1} \]

where \(V\) is the velocity of the meteor in km/sec,

\(R_0\) is the minimum value of the range to the trail,

\(t\) is the time counted from the moment when the meteor passes point \(A\) (Fig. 1).

Since fixing the instant \(t_0\) at which the meteor passes the middle of the first Fresnel zone with the necessary accuracy presents considerable difficulties, in practical computation of the velocity one usually uses a somewhat different expression, which is obtained from equation (1) after its integration:

\[ V=\frac{\sqrt{m}-\sqrt{n}}{\Delta t}\sqrt{\lambda R_0}. \tag{2} \]

In this expression, \(m\) and \(n\) are the numbers of the periods of oscillation between which the time interval \(\Delta t\) is measured, the beginning of the periods coinciding with the instant \(t_0\).

Exactly the same expressions for the velocity were obtained by McKinley [2], although the basis of his reasoning was the phenomenon of diffraction of radio waves by the forming meteor trail. As the author [2] points out, in the presence of diffraction reflection, equation (1) remains valid only if about ten periods of oscillation near the instant \(t_0\) are excluded. In the literature [1] there are also certain remarks to the effect that the velocity of a meteor found from equation (1) or (2) will differ depending on whether the given reflection is of the diffraction type or of the type of a moving point target; however, no numerical value of the difference in velocities is given anywhere.

Fig. 1.

The diffraction method of measuring meteor velocities uses the time variation of the amplitude of the signal reflected from the forming meteor trail. In this case the envelope of the amplitudes of the reflected signal is characterized by the modulus and phase of a vector whose origin is located at a certain point \(O\) (Fig. 1), while its end slides along the branches of Cornu’s spiral.

It follows from Fig. 1 that the time during which the phase of the amplitude envelope of the reflected signal changes by \(360^\circ\) will be proportional to the lengths of the arcs of Cornu’s spiral \(\smile C_1C_2; \smile C_2C_3\), i.e., to the time taken by the meteor to traverse successive Fresnel zones. Fresnel zones are taken to mean the distance between two such points located on the meteor trail for which the phase of the amplitude envelope of the reflected signal, at the instants when the meteor passes these points, will differ by \(360^\circ\).

Let us now consider the method of measuring velocity based on determining the magnitude of the frequency shift of the signal reflected from a moving point target.

Let us return again to Fig. 1. The change in the phase of the reflected signal when the target traverses some distance \(AA_1\) is characterized by the change in the angle of inclination of the tangent drawn to the spiral at the points corresponding to the beginning and end of the given segment of the path.

Consequently, the portions of the trace from whose ends the reflected signals arrive at the receiver in the same phase will be proportional to the arcs \(AA_1\), \(A_1A_2\), etc.

From Fig. 1 it is seen that \(\overset{\frown}{AA_1}\) is not equal to \(\overset{\frown}{AC_1}\). This leads to the fact that the beat periods of the signal reflected from a moving point target (when a reference voltage is supplied) differ from the periods of the oscillations of the signal envelope that occur in diffraction reflection. Because of this, the velocity calculated from formulas (1) and (2), when diffraction reflection is used, will differ from its true value.

The difference between the true and the calculated velocity decreases as the numbers of the oscillation periods over which the time interval \(\Delta t\) is counted increase. The magnitude of this difference is given in Table 1.

Table 1

Value of \(\delta_1\) in percent

Note. The minus sign means that the true value of the velocity is less than the value determined by formula (2).

As follows from the table, in a rough determination of velocity, or when large values of \(m\) and \(n\) are used, the correction \(\delta_1\), caused by the nonlinear dependence of the change in the oscillation frequency of the reflected-signal amplitude envelope on time, may be disregarded.

However, when pulse techniques are used, where the working periods are the first oscillation periods, the correction must be taken into account.

When meteor velocities are measured in the regime of undamped oscillations, data processing is further complicated by the fact that the initial phase of the beats of the output signal, \(\varphi_0\), and consequently also the numbers \(m\) and \(n\), will depend on the phase shift \(\psi\) between the voltage of the “ground wave” \(\dot U_{\text{g.w.}}\) and the signal voltage \(\dot U_c\) at the moment when the meteor passes through the middle of the first Fresnel zone.

To determine the initial phase of the beats, it is necessary to know the moment \(t_0\), which, as was said above, is in practice very difficult to do. Let us try to establish how the phase \(\varphi_0\) can be determined without knowing the time \(t_0\), and how \(\varphi_0\) affects the accuracy of velocity measurement.

If the voltage of the “ground wave” is approximately an order of magnitude higher than the maximum value of the signal, then the diffraction pattern has—

has a form symmetric with respect to the instant \(t_0\) (Fig. 2), so that the phase \(\varphi_0\) can be found quite easily.

If the amplitudes of the signal voltage and of the “ground wave” have approximately the same magnitude, then the diffraction pattern is asymmetric, and the initial phase of the beats depends both on the phase shift between the vectors \(\overline{U_c}\) and \(\overline{U_{gw}}\) and on the ratio of their moduli \(|\overline{U_c}|/|\overline{U_{gw}}|\). To determine the angle \(\varphi_0\) for some reflection, it is necessary to compare the form of the diffraction pattern obtained on the indicator screen with a series of diffraction patterns constructed in advance for specified values of the ratio \(|\overline{U_c}|/|\overline{U_{gw}}|\) and for different phase shifts between \(\overline{U_c}\) and \(\overline{U_{gw}}\). The initial phase of the ground-wave voltage is determined by the angle of inclination of the vector \(\overline{O_1S}\) to the horizontal axis. The form of the diffraction patterns and the corresponding angles \(\varphi_0\) can be determined by an analytical or graphical method. When the analytical method is used, great difficulties arise, since the equation of the Cornu spiral is expressed only through Fresnel integrals. Therefore only the graphical method is of practical interest. Its essence is as follows.

Fig. 2.

1. A spiral is constructed for the given value of \(|\overline{U_c}|\).

2. A circle is constructed for the given value of \(\overline{U_{gw}}\) \((r=|\overline{U_{gw}}|;\) the origin of coordinates of the circle is displaced from the point \(O_1\) by the amounts \(r\sin\psi\) and \(r\cos\psi\) along the \(X\) and \(Y\) axes, respectively).

3. The segments \(a_1b_1\), \(a_2b_2\), \(a'_1b'_1\), etc., are found.

4. The diffraction curve is constructed; its ordinates are the segments found, and its abscissae are the lengths of the arcs of the spiral, measured from the point \(A\) to the points \(a_1\), \(a_2\), etc., respectively.

5. \(\varphi_0\) is found as the angle through which the vector \(\overline{O_1A_1}\) must be rotated so that its end falls on the line of zero phases (point \(F\)).

The line of zero phases is an arc of a circle of radius \(r\), if \(|\overline{U_c}|\simeq|\overline{U_{gw}}|\); if \(|\overline{U_c}|\ll|\overline{U_{gw}}|\), then this line may be regarded as a straight line perpendicular to the vector \(\overline{U_{gw}}\).

1. Similar constructions are performed for different \(|\overline{U_c}|:|\overline{U_{gw}}|\) and \(\psi\).

When meteor velocities are measured by the pulsed-diffraction method, the initial phase of the beats (as follows from Fig. 1) is constant and is equal to

\[ \frac{\pi}{2}. \]

The magnitude of the error \(\delta_2\), caused by the inaccuracy in determining the numbers \(m\) and \(n\), is given in Tables 2–5 for several values of \(\psi\).

It must be said that, because of the error in determining \(m\) and \(n\) caused by not knowing the exact value of \(\varphi_0\), we shall introduce an additional error \(\delta_3\), which is due to the fact that instead of \(\delta_1\varphi\), corresponding to \(m_\varphi\) and \(n_\varphi\), we take \(\delta_{10}\), corresponding to \(m_0\) and \(n_0\),

Table 2

Value of \(\delta_2\) in percent for \(\varphi_0=\pi/4\)

Table 3

Value of \(\delta_2\) in percent for \(\varphi_0=-\dfrac{\pi}{2}\)

\[ \left(m_{\varphi}=m_0+\frac{\varphi_0}{2\pi};\qquad n_{\varphi}=n_0+\frac{\varphi_0}{2\pi}, \right. \]
and \(m_0\) and \(n_0\) are the number of periods of oscillations for the case when the initial phase is zero).

As was mentioned above, the actual magnitude of the meteor velocity is equal to

\[ V_d=\frac{\sqrt{m_{\varphi}}-\sqrt{n_{\varphi}}}{\Delta t_d}\sqrt{\lambda R_0}, \]

where \(\Delta t_d\) is the time between \(m_{\varphi}\) and \(n_{\varphi}\) periods of oscillations in the case of reflection of radio waves from a moving target.

The magnitude of the velocity measured from the diffraction pattern is determined as follows:

\[ V_u=\frac{\sqrt{m_0}-\sqrt{n_0}}{\Delta t_{\varphi}}\sqrt{\lambda R_0}, \tag{3} \]

where \(\Delta t_{\varphi}\) is the time between the \(m_{\varphi}\)-th and \(n_{\varphi}\)-th periods of beats of the amplitude of the output signal under diffraction reflection of radio waves from a meteor trail.

Table 4

Value of \(\delta_2\), in percent, for \(\varphi_0=\pi\)

Table 5

Value of \(\delta_2\), in percent, for \(\varphi_1=\dfrac{3\pi}{2}\)

If in expression (3) the numerator is multiplied by

\[ \frac{\sqrt{m_{\varphi}}-\sqrt{n_{\varphi}}}{\sqrt{m_0}-\sqrt{n_0}}, \]

and the denominator by \(\Delta t_d/\Delta t_{\varphi}\), then we obtain the true value of the velocity

\[ V_d=\frac{\sqrt{m_{\varphi}}-\sqrt{n_{\varphi}}}{\Delta t_d}\sqrt{\lambda R_0}. \]

Let us denote:

\[ \delta_1=\frac{\Delta t_{\phi}-\Delta t_d}{\Delta t_{\phi}}, \qquad \delta_2=\frac{\left(\sqrt{m_0}-\sqrt{n_0}\right)-\left(\sqrt{m_{\phi}}-\sqrt{n_{\phi}}\right)}{\sqrt{m_0}-\sqrt{n_0}}, \]

then

\[ V_d=V_u\frac{1-\delta_2}{1-\delta_1}. \tag{4} \]

Taking into account the correction \(\delta_3\), formula (4) assumes the form

\[ V_d=V_u\frac{1-\delta_2}{(1-\delta_1)(1-\delta_3)}, \tag{5} \]

where

\[ \delta_3=\frac{\delta_{10}-\delta_{1\phi}}{\delta_{10}}. \tag{6} \]

In practical calculations the correction \(\delta_3\) may be disregarded, since its magnitude is approximately an order of magnitude smaller than the magnitudes of \(\delta_1\) and \(\delta_2\).

Conclusions

1. The diffraction method of measuring meteor velocities gives a certain error caused by the inaccuracy in determining the phase \(\varphi_0\) and by the nonlinear dependence of the oscillation frequency of the envelope of the reflected-signal amplitude on time, the first component of the error being considerably greater than the second.

2. As the numbers of the oscillations by which the velocity is determined increase, both components of the error decrease.

3. When the pulse method is used, the initial phase

\[ \varphi_0=-\frac{\pi}{2}-\mathrm{const}, \]

and consequently the error \(\delta_2\) for the given \(m\) and \(n\) is constant (Table 3).

1. When the continuous-wave method is used, the magnitude of \(\delta_2\) depends chiefly on the initial phase \(\varphi_0\), the accuracy of whose determination depends, in turn, on the ratio of the amplitudes \(\overline{U}_c\) and \(\overline{U}_{zv}\).

Calculations show that, in order to determine the velocity with an accuracy of \(\simeq 1\%\), the accuracy in determining the phase \(\varphi_0\) must be approximately \(45^\circ\). For this it is necessary that the voltage \(\left|\overline{U}_{zv}\right|\) be an order of magnitude greater than the maximum signal voltage.

1. The correction must be taken into account when deciding the question of the existence of hyperbolic velocities.

References

1. Lovell B. and Clegg D., Radio Astronomy, Foreign Literature Publishing House, 1953.

2. McKinley D., “Determination of meteor velocities by radio observations,” Astrophysical Journal, vol. 113, no. 2, 1951.

3. Manning L., “Theory of radio detection of meteors,” Journal of Applied Physics, vol. 11, no. 8, 1948.

4. Tables of Fresnel Integrals. Publishing House of the Academy of Sciences of the USSR, 1953.