PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Volume 100 　　　　　　　　　　　　　　　　　　　　　　　　　　　1962

APPARATUS FOR DETERMINING THE VELOCITY VECTOR OF A LIGHT WIND

(preliminary considerations)

A. N. KOLESOV

Formulation of the Problem

The problems of studying meteoric matter and related problems require a solution to the question of measuring the velocity vectors of individual meteors (both those belonging to streams and sporadic ones) by radio-engineering means.

The most significant work in this field at the present time is that of Davis [1].

Davis’s method is based, as is known, on determining the magnitude of the velocity of a meteor body from the diffraction pattern and on measuring the time delays between diffraction patterns obtained at three separated points.

Associate Professor E. I. Fialko proposed a new method for determining the velocity vector of a meteor body [2].

When this method is used, there is in principle the possibility of determining the velocity vector of a meteor body in the case when a sufficiently good diffraction pattern is not obtained that would be suitable for determining from it the magnitude of the velocity of the meteor body.

Below, the basic relations for this method are given and its block-diagram solution is presented.

Basic Relations

Let at the point \(O\) (Fig. 1) there be located a bearing-and-ranging installation giving the coordinates of the normally reflecting point of the trail—\(R\), \(\alpha\), \(\beta\), and at the point \(B\) a remote installation receiving and transmitting to point \(O\) signals from the specularly reflecting point of the trail \(3\) (we here neglect the fact that the maximum of the reflected signal corresponds to the end of the first Fresnel zone, and not to its middle). Then, obviously, the radius vector \(R\) \((R, \alpha, \beta)\) determines the plane \(\pi\) (normal to \(R\)), in which the meteor trail lies, and the total range \(D = r_1 + r_2\), measured from the signal received at the remote point, determines an ellipsoid of revolution with foci at the points \(O\) and \(B\). The intersection of this ellipsoid with the plane \(\pi\) gives an ellipse. Tangents to the ellipse drawn from the point of normal reflection \(O_1\) will give the possible positions of the trajectory of the meteor-

body in space. In the general case, to eliminate the resulting ambiguity, one more remote station is needed, not lying on the same straight line as the points \(O\) and \(B\). However, analysis of the reality of the heights of the reflecting points and of the orientation of the velocity vectors for the two possible trajectories may in some cases yield unambiguous data even when only one remote installation is used.

Fig. 1. For the derivation of the basic relations of the method under consideration.

Let us derive the basic relations connecting the measured quantities \((R,\alpha,\beta,D)\) with the direction cosines of the trail trajectory with respect to the chosen coordinate system \(Ox, Oy, Oz\) (Fig. 1). The equation of the plane \(\pi\) has the form:

\[ x\cdot \cos\gamma + y\cdot \cos\delta + z\cdot \cos\varepsilon - R = 0. \tag{1} \]

The equation of the ellipsoid of revolution is written as:

\[ \frac{\left(x-\dfrac{d}{2}\right)^2}{\dfrac{D^2}{4}} + \frac{y^2}{\dfrac{D^2-d^2}{4}} + \frac{z^2}{\dfrac{D^2-d^2}{4}} =1. \tag{2} \]

Here \(\gamma,\delta,\varepsilon\) are the angles of the radius vector \(R\) with the coordinate axes \(Ox, Oy, Oz\), respectively; \(d\) is the length of the baseline \(OB\). The angles \(\gamma,\delta,\varepsilon\) are related to \(\alpha\) and \(\beta\) by the obvious relations:

\[ \cos\gamma=\cos\beta\cdot\cos\alpha, \]

\[ \cos\delta=\cos\beta\cdot\sin\alpha, \]
\[ \cos\varepsilon=\sin\beta. \tag{3} \]

Let us denote the length of the segment \(O_1 3\) \((O_1 3')\) by \(l\), and its projections on the axes \(Ox\), \(Oy\), \(Oz\) by \(l_x\), \(l_y\), \(l_z\), respectively. Then the direction cosines of the trajectory of the meteoric body \((m,n,p)\) with respect to the axes \(Ox\), \(Oy\), \(Oz\) that interest us will be expressed, respectively, as

\[ m=\frac{l_x}{l};\qquad n=\frac{l_y}{l};\qquad p=\frac{l_z}{l}. \tag{4} \]

To determine the magnitude \(l\) and its projections \(l_x, l_y, l_z\), it is necessary to find the coordinates of the points of tangency \(3\) and \(3'\) of the possible trajectories of the meteoric body to the ellipse \(Z\) (Fig. 1). This is easy to do if we pass to the auxiliary coordinate system \(O_1x_1\), \(O_1y_1\), \(O_1z_1\), which is chosen as follows: the origin of the new coordinate system is placed at the point of normal reflection \(O_1\), the axis \(O_1z_1\) coincides in direction with the radius vector \(R\) \((R,\alpha,\beta)\), and the axis \(O_1x_1\) lies in the same plane as the axis \(Ox\) of the old coordinate system.

The transition to the new coordinate system is effected by translating the origin of coordinates to the point \(O_1\) \((x_{0_1}=R\cdot\cos\gamma;\; y_{0_1}=R\cos\delta,\; z_{0_1}=R\cdot\cos\varepsilon)\) and rotating the axes as given by Table 1.

The equation of the plane (1) in the new coordinate system, obviously, is written as \(z_1=0\). Taking this into account, from (2) we obtain the equation of the ellipse \(Z\) in the new coordinate system in the form:

Table 1

\[ Ax_1^2+By_1^2+Cx_1+F=0, \tag{5} \]

where

\[ A=\frac{D^2-d^2\cdot\sin^2\gamma}{4};\qquad B=\frac{D^2}{4}; \]

\[ C=-\frac{d}{4}\,(D^2-d^2+2Rd\cos\gamma)\sin\gamma; \tag{5a} \]

\[ F=\frac{1}{16}\,[4R^2D^2-(D^2-d^2+2Rd\cos\gamma)^2]. \]

The equation of the tangent to the ellipse will obviously have the form:

\[ (2Ax_{1k}+C)(X-x_{1k})+2By_{1k}(Y-y_{1k})=0. \tag{6} \]

Here \(x_{1k}, y_{1k}\) are the coordinates of the points of tangency in the new coordinate system; \(X, Y\) are the current coordinates of the tangent.

Putting \(X=Y=0\) in (6) (the tangents pass through the origin of coordinates), we obtain, together with (5), a system of equations for determining the coordinates of the points of tangency:

\[ 2Ax_{1k}^{2}+2By_{1k}^{2}+Cx_{1k}=0, \]

\[ Ax_{1k}^{2}+By_{1k}^{2}+Cx_{1k}+F=0. \]

Hence we obtain

\[ x_{1k}=-\frac{2F}{C}; \qquad y_{1k}=\pm \sqrt{\frac{C^{2}F-4AF^{2}}{BC^{2}}}. \tag{7} \]

Obviously,

\[ l=\sqrt{x_{1k}^{2}+y_{1k}^{2}} =\frac{1}{C}\sqrt{\frac{4BF^{2}+C^{2}F-4AF^{2}}{B}}. \]

Finally we obtain:

\[ l=\frac{R\sqrt{4R^{2}D^{2}-(D^{2}-d^{2}+2Rd\cos\gamma)^{2}}} {D^{2}-d^{2}+2Rd\cos\gamma}. \tag{8} \]

Returning to the old coordinate system, we obtain expressions for the projections of \(l\):

\[ l_x=x_{1k}\sin\gamma; \qquad l_y=-\frac{\cos\gamma\cdot\cos\delta}{\sin\gamma}x_{1k} +\frac{\cos\varepsilon}{\sin\gamma}y_{1k}; \]

\[ l_z=-\frac{\cos\gamma\cdot\cos\varepsilon}{\sin\gamma}x_{1k} -\frac{\cos\delta}{\sin\gamma}y_{1k}. \]

Substituting the expressions obtained into (4) and taking account of (7) and (5a), we obtain expressions for the direction cosines:

\[ m=\frac{\sqrt{4R^{2}D^{2}-(D^{2}-d^{2}+2Rd\cos\gamma)^{2}}}{2Rd}; \]

\[ n_{1,2}=-\frac{\cos\gamma\cdot\cos\delta}{\sin^{2}\gamma}\,m +\frac{\cos\varepsilon}{\sin^{2}\gamma} \left(\pm\sqrt{\sin^{2}\gamma-m^{2}}\right); \tag{9} \]

\[ p_{1,2}=-\frac{\cos\gamma\cdot\cos\varepsilon}{\sin^{2}\gamma}\,m -\frac{\cos\delta}{\sin^{2}\gamma} \left(\pm\sqrt{\sin^{2}\gamma-m^{2}}\right). \]

Here \(\cos\gamma\), \(\cos\delta\), and \(\cos\varepsilon\) are determined through \(\alpha\) and \(\beta\) by relations (3).

The transition from the direction cosines of the trail, given by relations (9), to any of the astronomical coordinate systems presents no difficulty. Of course, for this it is necessary to record the time of appearance of the reflection from the meteor trail.

As for determining the magnitude of the velocity of the meteoric body, it can be determined if, in addition, one measures the time shift \(T\) between the diffraction patterns obtained at the principal and remote points (analogously to Davis’s measurements [1]). Indeed, during the time \(T\), obviously, there is formed a section of the trail of length \(l\), determined by (8). Hence we obtain:

\[ |v|=\frac{l}{T}, \tag{10} \]

where \(|v|\) is the magnitude of the velocity of the meteoric body. Naturally, with high-quality diffraction patterns, the velocity magnitude can be obtained in the same way as Davis did.

Block Diagram of the Installation

From what has been considered above, the required composition of the block diagram of the installation for determining the velocity vector of a weak meteor becomes clear.

At the remote station I there must be a receiving device 1 (Fig. 2) for receiving signals reflected from the trail, and a transmitter 2 with amplitude-pulse modulation for retransmitting the received signals to the main station (the location of the bearing-and-ranging installation).

Fig. 2. Block diagram of an installation for determining the velocity vector of a meteoric body.

At the main station there is the bearing-and-ranging installation II itself, which includes transmitter 3, irradiating the meteor trail; receiving-indicating range-finder unit 4; and receiving-indicating goniometric unit 5. As already mentioned above, the bearing-and-ranging installation measures the range \(R\) and the angular coordinates \(\alpha\) and \(\beta\) of the normally reflecting point of the trail.

At the main station there is also an attachment III to the bearing-and-ranging installation. It includes receiving device 6 for receiving the signals retransmitted from the remote station; discriminator 7 for separating the signals from interference; indicator device 8 for indicating the diffraction patterns formed both by the signals received by the main station and by the signals arriving from the remote station; and photosynchronization unit 9, which ensures synchronous operation of the photographic attachments recording the range \(R\), the total range \(D\), the angular coordinates \(\alpha\) and \(\beta\), and the diffraction patterns.

Figure 2 also shows the principal communication lines between the individual parts of the block diagram, explaining their interaction.

The lines “Ж” show the synchronization circuits for the operation of the entire main station as a whole. The lines “а” show the paths of the signals received from the remote station. These signals enter range-finder unit 4 in order to obtain a mark of the total range on the markers.

range, to discriminator 7 for obtaining signal-illumination pulses separated from interference and for triggering the sweep unit of the diffraction-pattern indicator 8, and to the deflecting plates of one of the tubes of the diffraction-pattern indicator. Line “6” shows the path of the discriminated signal. Line “3” shows the path of the photosynchronization signal arriving from the diffraction-pattern indicator 8 to the photosynchronization unit 9. Lines “8” show the path of the photosynchronization and illumination signals generated by unit 9. Along lines “2” and “d” from the range-finder section 4 there arrive at the diffraction-pattern indicator 8 the trigger pulses for the sweep unit of indicator 8, the illumination pulses of the signals received by the main station, and the signals themselves. Along line “e,” from indicator 8 to the bearing-range installation there arrive the illumination pulses of the signals received from the remote station.

The sweep unit of indicator 8 is triggered either by pulses from the main station or by pulses from the remote station, depending on where the signal appears earlier. The sweep voltage for the markers of the diffraction patterns of both the main and the remote stations is common. Therefore, relative to the beginning of the sweep, the diffraction patterns on the markers will be located differently, depending on whether the meteoric body is moving from the direction of the main station or from the direction of the remote station. This provides additional information that facilitates processing of the measurement results. An approximate appearance of the marker screens for different directions of motion of the meteoric body is shown in Fig. 3.

During the existence of the sweep in indicator 8, the illuminations of all signals registered at the main station are fed to the cathode-ray tubes, and photoregistration of the signals is carried out by photographic attachments whose shutters are open continuously. After the end of the sweep in indicator 8, access of the illuminations to the markers is terminated, the frames in the photographic attachments are changed automatically, and the installation is again ready to receive the next reflection from a meteor.

Fig. 3. I—screen of the diffraction-pattern indicator obtained from the main station. II—screen of the diffraction-pattern indicator obtained from the remote station. Case “a” corresponds to motion of the meteoric body from the direction of the main station; case “b” corresponds to motion of the meteoric body from the direction of the remote station.

The content of the general block diagram in the presence of a second remote station now becomes obvious.

Conclusion

1. The method considered makes it possible to determine the velocity vector of a weak meteor (even of those that do not give a clearly expressed diffraction pattern).

2. In connection with implementation of the method considered, it is necessary to examine the following questions: a) determination of the required accuracies of measurement of all quantities necessary for determining the velocity vector of the meteoric body; b) determination of the regions of space in which, with the method considered and with the given measurement accuracies, determination of the velocity vector of the meteoric body is possible with an admissible error; c) development of a method for processing

measurement results (which also include questions of eliminating the ambiguity that arises when there is only one off-site station).

In the future, the results of considering the questions posed in item 2 of the conclusion, as well as the results of the experiment, will be published.

References

1. G. Davies, J. C. Gill. Monthly Notices of the Royal Astr. Society, vol. 116, No. 1, pp. 105–113, 1956.

2. E. I. Fialko. On the question of determining the velocities and radiants of sporadic meteors, Bulletin of the Commission on Comets and Meteors of the Astronomical Council of the Academy of Sciences of the USSR (in press), 1958.