A. A. Dmitriev

DISTRIBUTION OF RADIATION INTENSITY MEASURED BY A WIDE-ANGLE RECEIVER

(Presented by Academician I. V. Obreimov, 11 VII 1962)

Measurement of the brightness distribution along a given almucantar or along some other observing path is usually carried out with apparatus having a wide angle of view. As a result of observations it is often necessary to obtain the distribution of the intensity referred to a sufficiently small solid angle. This proves possible when the angular distribution \(K(a)\) of the sensitivity of the radiant-energy receiver is known.

Let us consider, for example, the measured brightness field \(\widetilde{I}(\psi)\) as a function of the azimuth \(\psi\), using a receiver that cuts out a section of the path from \(\psi-\omega\) to \(\psi+\omega\).

The observed smoothed intensity \(\widetilde{I}(\psi)\) is related to the true \(I(\varphi)\) by the integral equation

\[ \widetilde{I}(\psi)=\frac{1}{2\omega}\int_{\psi-\omega}^{\psi+\omega}K(\psi-\varphi)\,I(\varphi)\,d\varphi . \tag{1} \]

The Fourier series expansions of the known functions—the measured intensity and the sensitivity—have the form

\[ \widetilde{I}(\psi)=\frac{A_0}{2}+\sum_{n=1}^{\infty} A_n \cos n\psi; \tag{2} \]

\[ K(\psi-\varphi)=\frac{a_0}{2}+\sum_{n=1}^{\infty} a_n \cos n\frac{\pi}{\omega}(\psi-\varphi). \tag{3} \]

We shall seek the unknown intensity in the form of the expansion

\[ I(\varphi)=\frac{\alpha_0}{2}+\sum_{n=1}^{\infty}\alpha_n\cos n\varphi+\sum_{n=1}^{\infty}\beta_m\sin m\varphi . \tag{4} \]

Substitution of (2), (3), (4) into (1) makes it possible to find

\[ \alpha_n= \frac{2\omega A_n}{ \displaystyle \sum_{m=0}^{\infty} a_m \left[ \frac{\sin (m\pi/\omega+n)\omega}{m\pi/\omega+n} + \frac{\sin (m\pi/\omega-n)\omega}{m\pi/\omega-n} \right] }; \tag{5} \]

\[ \beta_n= \frac{2\omega B_n}{ \displaystyle \sum_{m=0}^{\infty} a_m \left[ \frac{\sin (m\pi/\omega+n)\omega}{m\pi/\omega+n} + \frac{\sin (m\pi/\omega-n)\omega}{m\pi/\omega-n} \right] }, \tag{6} \]

where for \(m=0\) one should take \(a_0/2\), and for \(n=0\), \(A_0/2\).

Figure 1 gives the distribution \(\widetilde{I}\) (curve 1) for \(K=1\) and \(\omega=70^\circ\), and also (curve 2) the corresponding intensity reconstructed

by formulas (5) and (6):

\[ \tilde I=\frac{1}{2}I^*\left(1+\frac{\sin 2\omega}{2\omega}\cos 2\psi\right); \tag{7} \]

\[ I=\frac{1}{2}I^*(1+\cos 2\psi). \tag{8} \]

In the case of an open registration path, when \(\tilde I_{\bullet}=\tilde I(x)\), a particularly simple case is that of a U-shaped sensitivity characteristic \(K=b\) for \(|x-\xi|\leq a\), when instead of (1) we have

Fig. 1

\[ \tilde I(x)=\frac{b}{2a}\int_{x-a}^{x+a} I(\xi)\,d\xi . \tag{9} \]

Differentiating (9) with respect to \(x\), it is easy to obtain the formula

\[ I(x+a)=I(x-a)+\frac{2a}{b}\tilde I'(x). \tag{10} \]

Expression (10) makes it possible to find the intensity values over an interval \(2a\) from some initial value of it and from the measured smoothed values.

In conclusion I express my gratitude to K. Ya. Kondrat’ev, who took an active part in discussing the formulation of the present problem.

Peoples’ Friendship University
named after Patrice Lumumba

Received
16 VI 1962