ON MEASURING THE DISTANCE FROM THE EARTH TO THE MOON USING ELECTROMAGNETIC WAVES*)

N. D. Papaleksi

The question of the distance from the Earth to the heavenly bodies and, first of all, to the Moon, has undoubtedly been of lively interest to man since the very dawn of culture. As far as is known, the first attempts to measure the distance to the Moon were made as early as the third century before our era by the Greek astronomer Aristarchus. About 100 years after him, the famous ancient astronomer Hipparchus measured the distance to the Moon with enormous accuracy for that time, obtaining a value close to the true one, which marked a substantial stage in the development of our knowledge of the universe.

As is well known, modern astronomical methods for determining the distance to the Moon are based on measuring its horizontal equatorial parallax, i.e. the angle at which the semimajor axis of the geoid is seen from the Moon. For this purpose a trigonometric method is used, which in principle consists in the simultaneous measurement of the angles at which some definite point on the Moon, for example the summit of a certain mountain, is seen from two points on the Earth sufficiently far apart from one another, the distance between them being precisely known.

Thus, in one of the first accurate determinations of the distance to the Moon, carried out in the middle of the 18th century by the French astronomers Lalande and Lacaille, the former made measurements in Berlin, and the latter—at a distance of 10,000 km, at the Cape of Good Hope. Since the Moon moves around the Earth in an approximately elliptical orbit, whose eccentricity, under the assumption of unperturbed motion, is equal to $\varepsilon = 0.0549$, the distance from the center of the Moon to the center of the Earth does not remain constant, but varies within the limits from 356,000 km to 407,000 km. In astronomy, the distance to the Moon is taken to be a certain mean value $\Delta_{\rm L}$, which, on the basis of numerous observations, is now considered to be, in round numbers, 384,400 km, in accordance with the value

*) A somewhat supplemented report delivered on February 1, 1946, at the colloquium of the Oscillation Laboratory of the P. N. Lebedev Physical Institute of the Academy of Sciences of the USSR.

lunar parallax, equal to \(\pi = 57'2'',7\). The accuracy of determining \(\Delta \xi\) can be estimated at \(1/30000\)*). This gives an uncertainty in the value of the distance to the Moon of the order of \(10\)—\(15\) km.

In view of such great accuracy, attained by astronomers, the question may arise: is there any need for new methods of determining the distance to the Moon, and what interest can they present? It goes without saying that any possibility of further increasing the accuracy of the determination of the distance to the Moon is already of interest for theoretical astronomy. There is also no doubt that a method allowing one quickly to determine, from a single station, the distance to the Moon and, especially, continuously to follow its changes in time would be of special value. The point is that the high astronomical accuracy now achieved is obtained as the result of numerous observations requiring favorable atmospheric conditions and involving measurements from two stations located approximately on the same meridian at a very great distance from one another (most often in different countries), as well as complicated calculations and allowance for numerous corrections. Therefore the development of new accurate and at the same time simpler methods for determining the distance to the Moon from one station is undoubtedly not only of “sporting” interest, but also of great scientific interest.

Such highly promising methods are those based on reflecting electromagnetic waves from the surface of the Moon and measuring the time of their travel to the Moon and back. The idea of the possibility of reflecting light waves from the surface of the Moon arose among astronomers and opticians long ago. However, simple calculations showed that the technical means existing at that time were insufficient not only for producing a reflection from the Moon perceptible on Earth, but even for signaling to the Moon. Even a modern powerful searchlight of billions of candlepower would be visible from Earth on the Moon only as a sixth-magnitude star. Nor is the idea of reflecting radio waves from the Moon a new one. It undoubtedly occurred to many physicists and radio specialists, especially after the appearance of the pulse method, or the so-called “radio-echo method,” first applied by Breit and Tuve in 1925 for measuring the height of atmospheric layers reflecting radio waves. Thus, radio echoes with abnormally large delays (from one or two to 30 sec.), observed in 1928 by the well-known Norwegian investigator of auroras Størmer and the Dutch radio physicist van der Pol, led many radio specialists to suppose that these “cosmic” radio echoes owed their origin to reflection of radio waves from the surface of the Moon. However, elementary calculations—such calculations were made then by

*) According to a kind communication from Prof. M. F. Subbotin, the mean error in determining the distance to the Moon may be estimated at \(\pm 26\) km.

we, Academician L. I. Mandelstam and the author, showed that with the radio-technical means then available, both transmitting and receiving, there could be no question of reflection from the Moon. The idea of the reflection of radio waves from the surface of the Moon was expressed more than once later as well; however, until very recently there were not sufficient technical prerequisites for its realization. These prerequisites have been created in recent years by exceptional progress both in the generation of powerful ultrashort and microwave radio waves and in the technique of concentrating them into narrow directed beams of enormous instantaneous power, which also accounts for the striking successes of radar, which played such an exceptional role in the Second World War.

Our first estimates, made as early as 1943, showed that the contemporary state of pulse radio engineering had made it possible to achieve reflection of radio waves from the Moon, and further calculations confirmed our assumptions that this could be used to create a new precise method for determining the distance to the Moon. Indeed, according to American journals, the accuracy of determination by the radar method under conditions of direct visibility is limited only by the accuracy of timing and, independently of the distance measured, amounts to only tens of meters. This permits one to calculate that the accuracy of measuring the distance to the Moon by the “radio echo” method will exceed, at least by an order of magnitude, the astronomical accuracy, which, as was indicated above, is on the average 20–30 km. In addition, the radio-echo method possesses a number of other advantages, namely: simplicity and speed of measurement, the possibility of measuring directly and, moreover, continuously the distance to the Moon from a definite point on the Earth with an accuracy considerably exceeding the mean astronomical accuracy, and so on. It should be noted that in the radio-echo method the distance is obtained by multiplying the measured delay time by the value of the propagation velocity of radio waves, which is equal to the velocity of light in vacuum, i.e. 299,796 km/sec, and at present is known with an accuracy of up to one hundred-thousandth. Therefore, the development of this method for the purpose of further increasing its accuracy will entail the need for a more precise measurement of the propagation velocity of radio waves.

According to American radio reports of January 26 of this year, on January 10 of this year in North America, with the aid of specially redesigned radar apparatus, successful experiments were carried out for the first time in reflecting radio waves from the surface of the Moon, which gave, for the radio-echo delay time, as was to be expected, about \(2 \tfrac{1}{2}\) sec. Thus, the possibility of obtaining, with the aid of modern technical means, power sufficient for measurements of radio waves reflected from the Moon has been experimentally demonstrated and, thereby, as shown below, the correctness of the calculations we made has been confirmed. In considering the question

about the reflection of radio waves from the Moon, it was quite natural also to analyze the possibility of carrying out, with modern optical means, the reflection of light pulses from the Moon. Before turning to the consideration of each of these cases, we shall derive formulas, common to them, for the magnitude of the power of a pulse of electromagnetic waves reflected from the Moon.

REFLECTION OF A PULSE OF ELECTROMAGNETIC WAVES FROM THE MOON

In order to formulate mathematically the problem of the reflection of electromagnetic waves from the surface of the Moon, it is first of all necessary to adopt a definite idealization of the properties of this surface. From astrophysical data it follows that the surface of the Moon can by no means be regarded as smooth. It is necessary, on the contrary, to assume it to be very uneven and furrowed; apparently, to a considerable extent it consists of porous rocks and is covered with volcanic ash and cosmic dust. Thus one cannot assume that from the surface of the Moon as a whole—apart from individual small areas—regular reflections of either light or micro-radio waves are possible. The surface of the Moon should undoubtedly be regarded more correctly as “rough” in the optical sense. Therefore, as a possible idealization, giving rather an underestimate of the reflected power, one may assume that Lambert’s law is applicable to the surface of the Moon, which we shall regard as a sphere, or, perhaps, even more closely Lommel-Seeliger’s law. Then our problem may be formulated as follows. Let a pulse of electromagnetic waves of arbitrary form, of duration \(\Delta \tau\) and power \(P\), concentrated in a solid angle \(\Omega\), fall on a sphere of radius \(R\) with a rough surface, at a distance \(D \gg R\) from the radiation source and having reflection coefficient \(\sigma\). It is required to find the time dependence of the density of the power flux returning back to the radiation source after reflection from the sphere. It is assumed here that the transmission coefficient (transparency) of the radiation along the path from the source to the sphere is \(\beta\).

Let us first consider the case when the pulse has a rectangular form and its thickness \(\Delta b = c\Delta \tau\) is very small in comparison with \(R\). In the case of the Moon this condition may be regarded as practically fulfilled if the pulse duration \(\Delta \tau\) is of the order of \(10^{-5}\) sec and less. We shall carry out the calculations under the assumption of the applicability both of Lambert’s law and of the Lommel-Seeliger law.

As is seen from Fig. 1, the waves incident on the sphere are successively scattered from different spherical rings, as a result of which the pulse after reflection is stretched out to a length equal to the diameter of the sphere if the solid angle \(\Omega\) is equal to or greater than the angle \(\Omega_0\) under which the sphere is visible, and to a correspondingly smaller length if \(\Omega < \Omega_0\). Since the area of the ring \(\Delta S_\vartheta\), corresponding to the central angle \(\vartheta\)

(Fig. 1), is equal to

\[ \Delta S_{\vartheta}=2\pi R\sin\vartheta\, R\,\Delta\vartheta=2\pi R\,\Delta b, \tag{1} \]

then \(\Delta S_{\vartheta}\) does not depend on \(\vartheta\). The solid angle \(\Delta\Omega_{\vartheta}\), under which the area \(\Delta S_{\vartheta}\) is seen from the radiation source, is equal to

\[ \Delta\Omega_{\vartheta}=\frac{\Delta S_{\vartheta}\cos\vartheta}{D^{2}}. \tag{2} \]

Introducing the concept of the radiation intensity \(I\), corresponding to the luminous intensity in optics, and taking into account that the transmission coefficient is equal to \(\beta\), we obtain for the power flux \(\Delta P_{\vartheta}\) incident on the surface \(\Delta S_{\vartheta}\):

Fig. 1.

\[ \Delta P_{\vartheta}=\beta I\cos\vartheta\,\frac{\Delta S_{\vartheta}}{D^{2}}, \tag{3} \]

and for the power flux scattered from \(\Delta S_{\vartheta}\):

\[ \Delta P'_{\vartheta}=\sigma\Delta P_{\vartheta} =\sigma\beta\cdot I\cos\vartheta\,\frac{\Delta S_{\vartheta}}{D^{2}}. \tag{4} \]

In the case of applicability of Lambert’s law, the brightness \(B'\) of the radiation from the surface \(\Delta S_{\vartheta}\) is related to \(\Delta P'_{\vartheta}\) by the relation:

\[ \pi B'\Delta S_{\vartheta}=\Delta P'_{\vartheta}, \tag{5_1} \]

whereas for the Lommel–Seeliger law we have:

\[ 2\pi B'\cos\vartheta\,\Delta S_{\vartheta}=\Delta P'_{\vartheta}. \tag{5_2} \]

On the other hand, between \(B'\) and the radiation intensity \(I'\) of the surface \(\Delta S_{\vartheta}\) in the direction \(\vartheta\) to its normal, the following dependence holds:

\[ I'_{\vartheta}=B'\cos\vartheta\,\Delta S_{\vartheta}, \tag{6} \]

whence, on the basis of (4), \((5_1,\,5_2)\), and (6), we obtain:

in the first case:

\[ I'_{\vartheta}=\frac{\sigma\beta}{\pi}\,I\,\frac{\cos^{2}\vartheta}{D^{2}}\,\Delta S_{\vartheta}, \tag{7_1} \]

in the second case:

\[ I'_{\vartheta}=\frac{\sigma\beta}{2\pi}\,I\,\frac{\cos\vartheta}{D^2}\,\Delta S_{\vartheta}. \tag{7_2} \]

Hence at the location of the source of radiation (on the Earth) we have, for the density

\[ p'=\frac{\Delta P'_{\vartheta}}{\Delta S} \]

of the power flux reflected from \(\Delta S_{\vartheta}\):

\[ p'=\frac{\sigma\beta^2}{\pi}\,\frac{I}{D^4}\,\Delta S_{\vartheta}\cos^2\vartheta \quad\text{(Lambert’s law),} \tag{8_1} \]

\[ p'=\frac{\sigma\beta^2}{2\pi}\,\frac{I}{D^4}\,\Delta S_{\vartheta}\cos\vartheta \quad\text{(Lommel–Seeliger law).} \tag{8_2} \]

Substituting into these expressions the values of \(\Delta S_{\vartheta}\) from (1) and taking into account that

\[ \cos\vartheta=\frac{R-x}{R} \]

(Fig. 1), we obtain:

\[ p'=\sigma\beta^2\,\frac{I}{D^4}\,2\Delta b\cdot R \left(1-\frac{x}{R}\right)^2 \quad\text{(Lambert’s law),} \tag{9_1} \]

\[ p'=\sigma\beta^2\,\frac{I}{D^4}\,\Delta b\cdot R \left(1-\frac{x}{R}\right) \quad\text{(Lommel–Seeliger law).} \tag{9_2} \]

If the instant at which the pulse falls on the apex of the sphere (the Moon) is taken as the origin of the time count, then \(x=ct/2\), where \(c\) is the velocity of electromagnetic waves, and we have the following dependence of the density of the reflected power flux on time:

in the case where Lambert’s law is applicable:

\[ p'(t)=\sigma\beta^2\,\frac{I\Delta b}{D^4}\,2R \left(1-\frac{ct}{2R}\right)^2, \tag{10_1} \]

in the case where the Lommel–Seeliger law is applicable:

\[ p'(t)=\sigma\beta^2\,\frac{I\Delta b}{D^4}\,R \left(1-\frac{ct}{2R}\right). \tag{10_2} \]

Here \(t\) must satisfy the condition:

\[ 0\leq t\leq \frac{2R}{c}, \quad \text{if } \Omega \gg \Omega_0 \]

and

\[ 0\leq t\leq \frac{2m}{c}, \quad \text{if } \Omega<\Omega_0, \]

where \(m\) (Fig. 1) is determined from:

\[ m=R-\sqrt{R^2-\frac{\Omega}{\pi}D^2} =\frac{D}{\sqrt{\pi}}\left(\sqrt{\Omega_0}-\sqrt{\Omega_0-\Omega}\right). \tag{11} \]

Using formulas \((9_1,\,9_2)\), one can easily calculate also the case of a pulse whose thickness \(b\) is comparable with \(R\). Indeed, putting \(\Delta b=dx\) and, to simplify the notation,

\[ \sigma\beta^2\,\frac{I}{D^4}\,2R=A, \]

we, obviously, in the case of appli-

of the applicability of Lambert’s law, we obtain:

\[ p' = A \int_{x_1}^{x_2} \left(1-\frac{x}{R}\right)^2 dx, \tag{12} \]

where \(x_1\) and \(x_2\) have different values depending on the positions of the pulse relative to the sphere (Fig. 2), which, for brevity, we shall denote as its phases. In the first phase, which extends from the moment when the front plane of the pulse comes into contact with the sphere until the moment when its rear plane comes into contact,

\[ x_1=0,\qquad x_2=\frac{ct}{2}; \]

in the second phase, which lasts until the moment when the front plane of the pulse passes through the center of the sphere,

\[ x_1=\frac{ct-2b}{2},\qquad x_2=\frac{ct}{2}, \]

and, finally, in the third phase:

\[ x_1=\frac{ct-2b}{2},\qquad x_2=R. \]

Fig. 2.

\(a\) — beginning of the first phase, \(b\) — beginning of the second phase, \(c\) — beginning of the third phase.

Thus, in the case \(b \le R\), we have:

\[ \begin{aligned} p'(t) &= \frac{AR}{3}\left[1-\left(1-\frac{ct}{2R}\right)^3\right], && \text{for } 0 \le t < \frac{2b}{c},\\ p'(t) &= \frac{AR}{3}\left[\left(1-\frac{ct-2b}{2R}\right)^3-\left(1-\frac{ct}{2R}\right)^3\right], && \text{for } \frac{2b}{c} \le t \le \frac{2R}{c},\\ p'(t) &= \frac{AR}{3}\left(1-\frac{ct-2b}{2R}\right)^3, && \text{for } \frac{2R}{c} \le t \le \frac{2(R+b)}{c}, \end{aligned} \tag{13_1} \]

and in the case \(b>R\),

\[ \begin{aligned} p'(t) &= \frac{AR}{3}\left[1-\left(1-\frac{ct}{2R}\right)^3\right], && \text{for } 0 \le t \le \frac{2R}{c},\\ p'(t) &= \frac{AR}{3}, && \text{for } \frac{2R}{c} \le t \le \frac{2b}{c},\\ p'(t) &= \frac{AR}{3}\left[1-\frac{ct-2b}{2R}\right]^3, && \text{for } \frac{2b}{c} \le t \le \frac{2(b+R)}{c}. \end{aligned} \tag{13_2} \]

Formulas (13) show how, after reflection, the shape and intensity of a pulse of thickness \(b\) change, if \(\Omega \ge \Omega_0\). It is easy to see that in the case \(\Omega < \Omega_0\) we shall have, for the reflected pulse, the following formulas:

in the case \(b \le m\)

\[ \begin{aligned} p'(t)&=\frac{AR}{3}\left[1-\left(1-\frac{ct}{2R}\right)^3\right], &&\text{if } 0\le t\le \frac{2b}{c},\\ p'(t)&=\frac{AR}{3}\left[\left(1-\frac{ct-2b}{2R}\right)^3-\left(1-\frac{ct}{2R}\right)^3\right], &&\text{if } \frac{2b}{c}\le t\le \frac{2m}{c},\\ p'(t)&=\frac{AR}{3}\left[\left(1-\frac{ct-2b}{2R}\right)^3-\left(1-\frac{m}{R}\right)^3\right], &&\text{if } \frac{2m}{c}\le t\le \frac{2(m+b)}{c}; \end{aligned} \tag{14} \]

in the case \(b>m\)

\[ \begin{aligned} p'(t)&=\frac{AR}{3}\left[1-\left(1-\frac{ct}{2R}\right)^3\right], &&\text{if } 0\le t\le \frac{2m}{c},\\ p'(t)&=\frac{AR}{3}\left[1-\left(1-\frac{m}{R}\right)^3\right], &&\text{if } \frac{2m}{c}\le t\le \frac{2b}{c},\\ p'(t)&=\frac{AR}{3}\left[\left(1-\frac{ct-2b}{2R}\right)^3-\left(1-\frac{m}{R}\right)^3\right], &&\text{if } \frac{2b}{c}\le t\le \frac{2(b+m)}{c}. \end{aligned} \tag{15} \]

Fig. 3. Shape of pulses of equal total energy after reflection.
I. The pulse thickness is small in comparison with \(R\). II. The pulse thickness is equal to \(0.1R\). III. The pulse thickness is equal to \(0.2R\). IV. The pulse thickness is equal to \(R\).

In Figs. (3 and 4) is shown the change, after reflection, in the shape of rectangular pulses of different thickness (duration)\(^6\) under the condition that the total energy of the pulse incident on the sphere is the same, i.e.

\[ I_t=\frac{Ib}{c}=\mathrm{const} \]

or, equivalently, \(Ab=\mathrm{const}\). As is seen from Fig. 3, the shorter the pulse, the more sharply (steeply) the front of the reflected pulse rises and the greater is its maximum density (“illumination”) at the place of emission on the earth. For \(b \ll R\) (\(b=3.5\ \text{km}\) and shorter in the case of reflection from the Moon) the pulse shape is given by formulas (10), and the maximum density of the reflec-

Fig. 4. Change in the shape of a pulse of very small thickness after reflection:
a) in the case of reflection according to Lambert’s law.
b) in the case of reflection according to Lommel–Seeliger’s law.

reflected power will be expressed as:

\[ p'_m=\sigma\beta^2\frac{lb}{D^4}\,2R =\frac{\sigma\beta^2}{\pi}\frac{W}{D^2}\frac{2c}{R}\frac{\Omega_0}{\Omega} \quad\text{(Lambert’s law)} \tag{16_1} \]

or

\[ p'_m=\frac{\sigma\beta^2}{\pi}\frac{W}{D^2}\frac{c}{R}\frac{\Omega_0}{\Omega} \quad\text{(Lommel–Seeliger law).} \tag{16_2} \]

It is interesting to note that the quantity \(W'=\int p'(t)\,dt\), i.e. the density of the entire energy of the pulse reflected from the sphere in the direction opposite to its incidence, is related to the entire radiated energy of the pulse \(W\) by the relation:

in the case of Lambert’s law:

\[ W'=\frac{4}{3\pi}\sigma\beta^2\frac{W}{D^2}\frac{\Omega_0}{\Omega}, \tag{17_1} \]

in the case of the Lommel–Seeliger law:

\[ W'=\frac{\sigma\beta^2}{\pi}\frac{W}{D^2}\frac{\Omega_0}{\Omega}. \tag{17_2} \]

Formulas \((16_1, 16_2, 17_1\) and \(17_2)\) may be used as the basis for estimating the power of a pulse of electromagnetic waves required so that it can be detected on Earth after reflection from the Moon. For this purpose one may use both radio waves (meter, decimeter range and shorter) and visible light. In view of the features inherent both in the conditions for producing powerful pulses of each type of electromagnetic radiation and in the conditions for receiving them after reflection, it is advisable to consider separately the cases of the application of radio waves and of visible light.

APPLICATION OF RADIO WAVES

To estimate the required magnitude of the radiation power, it is necessary to start from the minimum value of the density \(p'\) of the radiation reflected from the Moon that can be reliably registered on Earth. Since

\[ p'=\frac{cE^2}{4\pi}, \tag{18} \]

where \(E\) is the electric-field strength of the wave, then, having specified the field strength necessary for reception, from (18) we obtain the required value of \(p'\). In doing so, it should be taken into account that, owing to depolarization of the radiation upon reflection from a rough surface, in the case of ordinary dipole reception we must evidently put \(E^2=kE_1^2\), where \(E_1\) is the field strength necessary for reception and \(k=2\) in the case of complete depolarization. Taking this into account, we obtain, proceeding from (18), the following expression

for \(E\) (\(k = 2\)):

\[ \left[E_1 \frac{\mu\mathrm{V}}{m}\right]^2 = 60\pi \cdot 10^6 p' \left[\frac{\mathrm{W}}{m^2}\right]. \tag{19} \]

As is known, for reliable reception of short pulses on meter waves it is necessary to have a field strength of the order of a microvolt per meter, whereas for waves in the decimeter range it must be an order of magnitude greater. If for the first case we set \(E_1 = 2\ \mu\mathrm{V}/\mathrm{meter}\), and for the second \(E_1 = 20\ \mu\mathrm{V}/\mathrm{meter}\), then for the quantity \(p'\) we obtain, respectively, the following values:

\[ p' = 2.1 \cdot 10^{-14}\ \frac{\mathrm{W}}{m^2} \quad \text{and} \quad p' = 2.1 \cdot 10^{-12}\ \frac{\mathrm{W}}{m^2}. \]

In our calculations we shall proceed from these quantities.

From formula (16), which we may rewrite in the form

\[ W = \frac{\pi}{\sigma \beta^2}\, D^2 \frac{R}{2c}\frac{\Omega}{\Omega_0}\, p'_m , \tag{20} \]

it is evident that \(W\) depends substantially on \(\sigma \beta^2\). The value of \(\sigma\) for the reflection of light from the Moon may, on the basis of astrometric observations, be taken as equal to 0.07. If one takes into account that a considerable part of the surface of the Moon is covered by a volcanic ash layer and cosmic dust, then it must be supposed that for microwaves of the order of decimeters, and still more for meter waves, the value of \(\sigma\) will be greater. We shall take \(\sigma = 0.1\). As for the value of \(\beta\), it depends strongly on the wavelength. According to Bouguer’s formula\(^2\) \(I = I_0 u^\mu\), where \(\mu\) is the mass of the atmosphere traversed by the rays and referred to the mass traversed in the vertical direction of the ray, and \(u\) is the coefficient of transparency. The dependence of \(\mu\) on the zenith angle \(\zeta\) may be judged from the following table\(^2\):

Table 1

The quantity \(u\), in turn, depends on the wavelength of light \(\lambda\) in the following way\(^2\):

Table 2

We shall also note that, from observations of the brightness of stars for \(\mu\), an average value of \(0.83\) is obtained. All these data allow us to consider that we are not making an error in the direction of exaggeration if we assume that for microradiowaves \(\beta^2=u^2\mu\) is not less than \(0.8\) for \(\zeta \leq 40^\circ\). Thus, taking \(\dfrac{1}{\beta_0^2}=15\), we are hardly making an excessively optimistic estimate. Substituting now this value of \(\dfrac{1}{\beta_0^2}\) into (20) and taking into account that for the Moon, in round numbers, \(D^2=15\cdot 10^{18}\,m^2,\ R=1738\ km\), we obtain:

\[ W=43\,\frac{\Omega}{\Omega_0}\quad \text{kilojoules for waves of the decimeter range,} \]

\[ W=0.43\,\frac{\Omega}{\Omega_0}\quad \text{kilojoules for meter waves} \]

for a pulse duration \(\Delta \tau=10^{-5}\) sec; this gives for the power \(P\), respectively,

\[ P=4.3\cdot 10^6\,\frac{\Omega}{\Omega_0}\ kwt \quad \text{and} \quad P=4.3\cdot 10^4\,\frac{\Omega}{\Omega_0}\ kwt, \]

i.e., quantities which for radio waves are obviously unrealizable at the presently attainable values of \(\dfrac{\Omega}{\Omega_0}\). Thus, the problem of producing a reflection of radio waves from the Moon that could be measured with certainty on the Earth would be insoluble if we did not now possess means that make it possible to concentrate the microradiowave radiation incident on the Earth by thousands of times. If the coefficient of concentration of power in reception is denoted by \(g\), then formula (20) takes the form:

\[ p'_m = g\,\frac{\zeta\beta^2}{\pi}\, \frac{W}{D^2}\, \frac{2c}{R}\, \frac{\Omega_0}{\Omega}. \tag{21} \]

The coefficient of concentration of power in reception, as, of course, also in transmission, can be made the greater the shorter the wavelength. If we put \(g=5000\), which is comparatively not difficult to realize for waves of the decimeter range, then we obtain for the required value of the radiation energy:

for decimeter waves

\[ W=8.6\,\frac{\Omega}{\Omega_0}\quad \text{joules,} \]

whereas for meter waves (with \(g=500\)) we have

\[ W=0.86\,\frac{\Omega}{\Omega_0}\quad \text{joules.} \]

This gives, for pulse durations of respectively \(10^{-5}\) sec and \(10^{-4}\) sec, for the power \(P\), respectively:

\[ P=860\,\frac{\Omega}{\Omega_0}\quad \text{kilowatts} \quad \text{and} \quad P=8.6\,\frac{\Omega}{\Omega_0}\quad \text{kilowatts,} \]

and at experimentally attainable values of $\dfrac{\Omega}{\Omega_0}$ this is entirely feasible.

From our estimate of the magnitude of the radiation power required in order to receive radio waves reflected from the Moon, it is clear that the main task here consists not only in obtaining pulses of sufficient power and in using receivers of sufficient sensitivity, but also in the possibility, on the one hand, of concentrating the radiated power within a sufficiently small solid angle and, on the other hand, of collecting, from as large an area as possible, the power incident on the Earth after reflection. These tasks are at present technically quite solvable both by means of systems of coherent dipole radiators (at meter wavelengths) and by means of mirrors in the microwave decimeter and centimeter ranges. This is fully confirmed by the first successful experiments mentioned above on producing the reflection of radio waves from the Moon, carried out in January of this year in North America. As can be seen from the published data $^{3,4,5}$, meter waves were used in these experiments ($f = 112$ Mc, i.e. $\lambda = 2.68$ m), and as the transmitting and receiving antennas there served plane systems of in-phase dipole antennas $(2 \times 32)$ with reflectors (Fig. 5), placed on masts 30 m high. The pulse duration varied in different experi-

Fig. 5. Transmitting and receiving plane antenna system.

from 0.1 to 0.5 sec, while the interval between pulses varied from 3 to 5 sec. Such a long pulse duration made it possible to use an exceptionally selective receiving device with fourfold frequency conversion (with four intermediate frequencies) and to narrow the passband to 57 cycles. Thanks to these measures it was possible to reduce the noise level extraordinarily and bring the receiver sensitivity to 0.01 μV. It should be noted that such a narrow receiver passband required, on the one hand, very great stability of the radio-wave frequency, which was achieved by piezoquartz stabilization of the initial transmitter frequency (516.2 kc), which was then brought up to its final value by successive multiplication of the frequency.

Fig. 6. Oscillogram obtained during the experiments of 22/I 1946. The ratio of the signal strength to the “noise background” was 20 db, and the change in the frequency of the signal wave due to the Doppler effect was 227 cycles.

to its final value. On the other hand, when tuning the receiver it was necessary to take into account the circumstance that the pulse frequency upon reflection from the Moon (in view of the relative motion of the Earth and the Moon) changes according to the Doppler law. Under the conditions of the experiment described, carried out at moonrise, the frequency of the pulse wave increased upon reflection from the Moon by 200–300 cycles. Here it should be especially emphasized that such a narrow-band receiver, tuned to a frequency different from the frequency of the emitted wave, does not respond to signals reflected from stationary and slowly moving objects and, consequently, produces the necessary discrimination.

It is of interest to compare the experimental results obtained in the experiments just described with those that could have been expected on the basis of the calculations given above. According to the experimental data ³˒⁵, the emitted peak power of the pulses was

about 4 kW; the energy-concentration coefficient at reception \(g=200\)—\(250\); the aperture angle of the radiation cone is about \(15^\circ\), which corresponds to \(\dfrac{\Omega}{\Omega_0}=900\). Assuming further the pulse duration \(\Delta\tau=\) \(0.25\) sec, we may take the energy \(W\) radiated during one pulse to be equal to 1 kilojoule. Since \(R \ll b\), from formula \((13_2)\) we obtain for \(p'_m\):

\[ p'_m=\frac{AR}{3}=g\,\frac{2\pi^2}{3\pi}\,\frac{\Omega_0}{\Omega}\,\frac{W}{D^2}\,\frac{c}{b}. \]

Substituting the numerical values of \(g,\ \dfrac{\Omega_0}{\Omega},\ W,\ D^2,\ c\) and \(b\), we have:

\[ p'_m=1.5\sigma\beta^2\,10^{-15}\ \frac{\mathrm{W}}{\mathrm{m}^2}. \]

Since the experiment was carried out at moonrise, i.e. the radio waves passed through the entire thickness of the atmosphere along the earth, \(\beta\) must have been considerably smaller than at small zenith angles. If \(\sigma\beta^2\) is taken equal to \(1/30\), then we have:

\[ p'_m=5\cdot 10^{-17}\ \mathrm{W}/\mathrm{m}^2. \]

On the other hand, with a receiver sensitivity of \(0.01\,\mu\mathrm{V}\) the signal reception strength exceeded the “noise” intensity by \(20\) db, i.e. the voltage from the signal at the receiver terminals was about \(0.1\,\mu\mathrm{V}\). Since, further, at \(\lambda=2.68\ \mathrm{m}\) and with half-wave antennas the effective height of the antenna was approximately \(\lambda/\pi=1.2\ \mathrm{m}\), the field strength was of the order of \(0.1\) microvolt/m, whence by formula (19) we obtain for \(p''_m\):

\[ p''_m=8\cdot 10^{-17}\ \mathrm{W}/\mathrm{m}^2. \]

As we see, the agreement of the experimental data with the results of the calculations is quite satisfactory, especially if one takes into account that the purpose of the experiments was to obtain only qualitative results. It should be noted that in subsequent experiments it is intended to proceed to decimeter waves (\(f\) from 500 to 1000 Mc/s), for the concentration of which the use of parabolic reflectors with an aperture diameter of 12—15 m is being designed.

What, then, can the radio-echo method give for the study of the Moon? As is evident from formulas (13) and clearly from Fig. 3, the accuracy of determining the distance to the Moon depends substantially on the pulse thickness, since it determines the steepness of the reflected pulse. If we assume that the moment of return of the reflected pulse can be determined with an accuracy of tenths of its width, then this means that with a pulse duration of \(10^{-5}\) sec, i.e. with a thickness of 3 km, the distance to the Moon can be determined with an accuracy of at least on the order of one kilometer. And this is already by an order of magnitude

exceeds the average astronomical accuracy. If one takes into account that the radio-echo method gives such accuracy with a single observation and that, moreover, it makes it possible to follow continuously the variation of the distance to the Moon with time, then its value for lunar theory appears beyond doubt. Further, as is evident from formulas \((10_1, 10_2)\), the intensity and form of the reflected pulse depend essentially on the coefficient of reflection of radio waves, which may differ for one or another annular zone of the Moon’s surface. Thus, from the form of the curve of the reflected pulse it will be possible to draw a conclusion both as to the applicability of one or another law of reflection from the Moon’s surface and as to the character of the distribution of the mean reflection coefficients over annular zones of its surface. Moreover, by carrying out measurements at different wavelengths, it will be possible to obtain additional data both on the character of the reflection and on the structure of the Moon’s surface. At present, of course, it is still difficult to say to what extent the radio-echo method will justify itself in practice, and also what possibilities it still conceals. In any case, there is no doubt as to the necessity of the earliest possible development of this method.

THE CASE OF LIGHT WAVES

In contrast to the case of radio waves considered above, the problem of producing reflection of light rays from the Moon has one essential peculiarity. Whereas for radio waves, in estimating the power density of the radiation incident on the Earth from the Moon, we proceeded only from the sensitivity of the receiving apparatus, in the case of light waves we must also take into account the illumination which is produced on the Earth either by the Moon at night or by the Sun (by day or in the evening). Moreover, two different methods of detecting light waves are possible here in principle: a subjective one—with the aid of vision, and an objective one, for example with the aid of photoelements.

Let us consider the question of the minimum density of the luminous flux \(p'_m\) required for the reliable detection on the Earth, with the aid of photoelements, of light reflected from the Moon, on the assumption that extraneous light is absent. One can arrive at an idea of the real values of \(p'_m\) sufficient for reliable registration in various ways. According to Strongin (Lehrbuch der Experim. Physik, vol. XXVI, p. 914, 1937), the astronomer Smith at the Mount Wilson Observatory detected, with the aid of a telescope of diameter \(150\ \text{cm}\) and a photoelement, stars of the 14th magnitude. A star of the 14th magnitude produces on the Earth an illumination of \(1.3\cdot 10^{-12}\) lux, or, in other words, for registration it is sufficient to have

\[ p'_m = 2\cdot 10^{-15}\ \frac{\mathrm{W}}{\mathrm{m}^2}. \]

On the other hand, the sensitivity of modern photoelements (see, for example, \({}^{6}\)) reaches \(2\cdot 10^{-11}\) lm or \(3\cdot 10^{-14}\ \frac{\mathrm{W}}{\mathrm{m}^2}\), which, for the working area of the photoele-

of an element of \(3\ \mathrm{cm}^{2}\) corresponds to a luminous-flux density \(p_{0}=10^{-10}\ \dfrac{\mathrm{W}}{\mathrm{m}^{2}}\).

If we denote by \(g\) the concentration factor of the light arriving after reflection, then for \(g=5\cdot 10^{4}\) we arrive at the same value \(p'_m=2\cdot 10^{-15}\ \dfrac{\mathrm{W}}{\mathrm{m}^{2}}\) as above. To estimate the total power of the light pulse sent from the Earth, necessary in order to obtain on the Earth a noticeable reflection from the Moon, we shall proceed from the value \(p_{0}=10^{-10}\ \dfrac{\mathrm{W}}{\mathrm{m}^{2}}\). Substituting into formula (16), which now takes the form:

\[ W=\frac{\pi}{\varphi^{3}}\frac{D^{2}R}{c}\frac{\Omega}{\Omega_{0}}\frac{10^{-10}}{g}, \tag{16_3} \]

the numerical values of \(D\), \(R\), and \(c\), and also for \(\dfrac{1}{\varphi^{3}}\), which for light may be taken equal to 35, we have:

\[ W=9.6\cdot 10^{3}\cdot \frac{1}{g}\frac{\Omega}{\Omega_{0}}\ \text{kilojoules}. \]

Under various assumptions concerning \(g\) and \(\dfrac{\Omega}{\Omega_{0}}\), this gives the following table for \(W\):

Table 3

Values of \(W\) in joules

Hence, assuming the pulse duration \(\Delta\tau=10^{-4}\ \mathrm{sec}\), we obtain for \(P\) Table 4.

Table 4

\(P\) in kilowatts

We arrive at quantities of the same orders for \(W\), proceeding from formula \((17_2)\) and assuming that, for reliable detection of a light echo from the Moon, it is sufficient to have a density \(W'_m\) of all the energy of the pulse arriving back at the Earth after reflection equal to 200 photons per square meter. Since for visible light \(h\nu = 4 \cdot 10^{-19}\) joule, \(200h\nu = 8 \cdot 10^{-17}\ \text{joule}/m^2\). Substituting this value into \((17_2)\), we obtain for \(W\) at \(\dfrac{\Omega}{\Omega_0} = 1\):

\[ W = 13.2 \cdot 10^2\ \text{joules} \]

and

\[ P = 13200\ \text{kilowatts}. \]

As is known, with a condensed powerful spark discharge the energy released in the spark is measured in thousands and tens of thousands of joules, and the power may reach hundreds of thousands of kilowatts \(^{8,9,10}\). Approximately the same powers are obtained also in pulsed scattering of wires. Even if one assumes that the light output in this case amounts to only 3–4% of the total power, then even at \(10^5\) kW we shall have 3–4 thousand kW of light energy. Thus, as is seen from the numerical data given above, with concentrations feasible for light both of the radiated power into a very narrow beam \(\left(\dfrac{\Omega}{\Omega_0} \leq 0.1\right)\) and of the reflected power from as large an area as possible \((3–4\ m^2)\), one can indeed obtain a power of light waves reflected from the Moon sufficient for receiving them by means of photoelements. However, here, in contrast to the case of applying radio waves, where the sensitivity of reception is limited mainly only by the intrinsic “noise” of the radio apparatus, the problem of receiving light waves reflected from the Moon is complicated by the fact that the reflected light pulse must be separated against the background of sunlight reflected from the Moon. Since the illumination produced by the Moon on the Earth is, at full moon, 0.2 lux, i.e. \(3 \cdot 10^{-4}\ \text{watt}/m^2\), then, for example, at

\[ p'_m = 10^{-14}\ \frac{\text{watt}}{m^2} \]

the reflected light of the pulse will amount to only about \(3 \cdot 10^{-11}\) of the constant lunar light, and it would seem that reliable detection of it must be associated with enormous difficulties or even simply be impossible. However, in reality the situation is not so hopeless. The point is that, first, the experiment can be performed during a total lunar eclipse, as was pointed out during the discussion by N. Ya. Boguslavskaya, when the illumination from the Moon weakens by tens of thousands of times and even more*). On the other hand, in view of the comparative constancy of lunar light, its effect can be

*) I express my gratitude to A. V. Markov for communicating that, according to his measurements, the illumination from the Moon during the total lunar eclipse of February 8, 1925, fell, in comparison with the illumination outside the eclipse, by \(4.5 \cdot 10^5\) times. This result was obtained after subtracting from the brightnesses measured on the Moon the brightness of the background of the night sky.

considerably compensated for, so that the illumination from the reflected pulse can be compared not with the full illumination from the Moon, but only with its fluctuations. It must also be borne in mind that the light energy of the pulse must be compared with the energy of the lunar light falling not during a second, but only during the duration of the pulse. In other words, the comparison must be made with that energy which will fall on the photocell if an obturator is placed before it, opening only for the time of arrival of the reflected pulse. It should be noted that the effective application of the obturator method is substantially facilitated by the circumstance that the distance to the Moon can be calculated in advance with an accuracy of several tens of kilometers. If we assume that the illumination from the Moon during a total lunar eclipse is weakened by \(10^5\) times, and further, that the action of the remaining constant background can be compensated for by \(10^3\)—\(10^4\) times and that the pulse duration is \(10^{-4}\) sec, while the opening time of the obturator is \(10^{-3}\) sec, then we obtain for the light flux from the Moon, instead of \(3\cdot 10^{-4}\) watt/\(m^2\), only \(3\cdot 10^{-15}\) watt/\(m^2\), i.e., a quantity smaller than the expected light flux of the pulse reflected from the Moon.

A very effective means for distinguishing a light pulse reflected from the Moon against the background of lunar light also appears to be the use, instead of “white,” of “colored” pulses containing only a few separate spectral lines, as, for example, occurs in a condensed spark discharge in mercury vapor. In this case the spectral decomposition of the reflected pulse can substantially help to separate it from the continuous light background. In addition, the use of electrical resonance may be of great assistance in distinguishing the light pulse reflected from the Moon against the background of the direct current produced by the illumination of the Moon—for isolating the alternating current obtained, for example, when sending \(N\) times per second pulses equally spaced from one another. Since the travel time of the pulse to the Moon and back lasts about \(2.5\) sec, we must have an alternation of intervals of transmission and reception whose duration must be equal to this travel time. There is reason to think that the use of the means indicated above in one combination or another will make it possible to solve the problem of receiving light waves reflected from the Moon, at least during a total lunar eclipse.

In conclusion, I shall permit myself to make the following remark. As is well known to everyone from reports published on the radio and in print, the explosion of an atomic bomb is accompanied by a flash of light of gigantic intensity, many times exceeding the strength of sunlight. Therefore it would seem extremely interesting to carry out observations of the Moon during explosions of atomic bombs, which will probably be produced for scientific purposes. Of course, for these observations the appropriate sites and a suitable time must be chosen.

References

1. Science News Letter, 2 February, 1946.
2. P. N. Tverskoi, Course of Geophysics, 1936, pp. 477–478.
3. John H. De Witt, Radiocraft, pp. 464, 501, April, 1946.
4. Tom Gootée, Radio News, p. 25, April, 1946.
5. Jack Mofenson, Electronics, p. 43, April, 1946.
6. N. S. Khlebnikov, “Electronic Multipliers,” UFN XXIV, issue 3, p. 370, 1940.
7. L. A. Kubetskii, Izv. AN SSSR, Physical Series VIII, No. 6, pp. 357–365, 1944.
8. A. Babushkin, “Spectroscopy of Powerful Discharges,” Izv. AN SSSR, Physical Series IX, No. 3, 1945.
9. K. S. Vul’fson, “On Pulse Discharge in Inert Gases,” Izv. AN SSSR, Physical Series IX, p. 239, 1945.
10. I. S. Stekol’nikov, “A Super-Powerful Generator of Pulsed Currents,” Elektrichestvo, No. 3, p. 81, 1946.