TH E UNI VE RS ITY OF M I CH I GAN STUDIES IN RADAR CROSS SECTIONS XXVHI THE PHYSICS OF RADIO COMMUNICATION VIA THE MOON M. L. Barasch, H. Brysk, B. A. Harrison, T. B. A. Senior, K. M. Siegel, and H. Weil Prepared under the sponsorship of Rome Air Development Center, ARDC, Griffiss Air Force Base, New York Project No. 4538 Task No. 45653 Contract AF 30(602)-1725 March 1958 University of Michigan Report No. 2673-1-F Approved l Keeve M. Siege~ Project Supervisor The University of Michigan Engineering Research Institute The Radiation Laboratory Ann Arbor, Michigan l~~~~~~~~~~~~~~~~~~~~~ i

THE UNI VE RS ITY OF MI CH I GAN Qualified requestors may obtain copies of this report from the Armed Services Technical Information Agency, Arlington Hall Station, Arlington 12, Virginia. ASTIA Services for Department of Defense Contractors are available through the Field of Interest Register-or a'heed-to-know' certified by the cognizant military agency of their project or contract. iiI

THE UNIVERS ITY OF MICHIGAN 2673-1-F TABLE OF CONTENTS Section Title Page Preface iv | I Introduction I II Reflections from the Moon 4 III Far Field Criteria Applied to the Moon 11 IV The Electromagnetic Constants of the Moon's Surface 15 V The Faraday Effect 23 VI Electromagnetic Scattering by Low-Density Meteor Trails 31 6. 1 Scattering by a Low-Density Uniform Line Distribution of Electrons 31 6.2 The Effect of Ambipolar Diffusion 44 6.3 Additional Refinements 67 6.4 Meteor Velocity Measurements 72 6.5 Meteor Trails as Interference to Communication via the Moon 72 VII Communication Considerations 77 VIII Conclusions 84 References 85 I ~~~~~~~~~~~~~~~~iii-I

THE UNIVE RS ITY OF MI CHIGAN 2673-1-F PREFACE This paper is the twenty-eighth in a series growing out of studies of radar cross| sections at the Engineering Research Institute of The University of Michigan. The| primary aims of this program are: 1. To show that radar cross sections can be determined analytically. 2. A. To obtain means for computing the radiation patterns from antennas by approximate techniques which determine the pattern to the accuracy required in military problems but which do not require the exact solutions. B. To obtain means for computing the radar cross sections of various objects of military interest. (Since 2A and 2B are interrelated by the reciprocity theorem it is necessary to solve only one of these problems. ) 3. To demonstrate that these theoretical cross sections and theoretically determined radiation patterns are in agreement with experimentally determined ones. Intermediate objectives are: 1. A. To compute the exact theoretical cross sections of various simple bodies by solution of the appropriate boundary-value problems arising from electromagnetic theory. B. Compute the exact radiation patterns from infinitesimal sources on the surfaces of simple shapes by the solution of appropriate boundary-value problems arising from electromagnetic theory. (Since 1A and 1B are interrelated by the reciprocity theorem it is necessary to solve only one of these problems. ) 2. To examine the various approximations possible in this problem and to determine the limits of their validity and utility. 3. To find means of combining the simple-body solutions in order to determine the cross sections of composite bodies. iv

THE UNIVERS ITY OF MICHIGAN 2673-1-F 4. To tabulate various formulas and functions necessary to enable such computations to be done quickly for arbitrary objects. 5. To collect, summarize, and evaluate existing experimental data. K. M. Siegel VI

THE UNIVE RS ITY OF MI CHIGAN -- 2673 -1 -F SECTION I INTRODUCTION In order to determine the optimum frequency range for communication by| reflection from the moon, a thorough understanding has to be obtained of the effect on | transmission of the media between the earth and the moon, as well as the effect on reflection of the surface characteristics of the moon. A frequency has to be chosen which will be high enough to get through the ionosphere without significant attenuation or scatter-| ing. A thorough analysis is devoted to meteor scattering, to establish the extent of its effect upon a moon communication system. Even though point to point communication may require that the points be far enough separated that the transmitter's main beam cannot be considered coincident with the receiver's main beam (since we limit this discussion to| the surface of the earth and do not investigate communication with vehicles in outer space), only a small angular difference exists between the axes of symmetry of these main beams.| Nevertheless., since it is not much harder to obtain the bistatic answer than the mono- | static answer., we have determined the Faraday effect., i. e.., the effect of the rotation of | the plane of polarization during propagation through the ionosphere., as a function of the | locations of transmitter and receiver. Practically speaking., for purposes of communica-| tion via the moon it seems wiser to use circular polarization or to choose a high enough | frequency so that this physical effect is not mleasurably significant.....,.. 1 -~~~~~~~bems

TH E UNI VE RS ITY OF M I CH I GAN 2673-1-F A great mvster to all who have studied the moon is the nature of the surface of y the moon. A major requirement on any antenna system'is to understand the properties of the antennas themselves. The moon in this case acts as a parasitic antenna, and it is very important to understand the effect of its surface on reflection of electromagnetic waves before one can design optimally a communication system which has this type of reflection inherent in its operation. In Section II, e'erimental and theoretical support is given for the hypothesis XP that at radar frequencies the moon can be considered to be a smooth reflector. The shape of the echo pulses under this hypothesis is found not to contradict the available experimental evidence. By routine application of the usual antenna far-zo'ne criterion, the earth would be considered to be in the near zone of the moon for all frequencies which would go through the ionosphere. However, for the special case (not previously specifically analyzed) in which the dimensions of the scatterer are extremely large compared with the wavelength, while still very small -compared with the ranges to the transmitter and receiver, a more refined criterion is possible. In Section M, this analysi's is carried out and it is demonstrated that far zone information is available for at least part of the pulse. In Section IV, we determine the dielectric constant and conductivity of the surface of the moon, by using the physical picture given in Section II together with the experimental results of several experimenters (Yaplee, Trexler, Leadabrand., Blevis.. Evans),,

THE UNIVE RS ITY OF MI CHIGAN 2673-1-F The effect of the Faraday rotation on transmission with linear polarization is | derived in Section V. The theory of meteor scattering by under-dense trails is developed in Section VI, with emphasis on its complex character at the higher frequencies;, for the latter, the results differ substantially from previous less sophisticated analyses. While the overdense trails (which are not treated) produce a stronger and longer lasting return than the under-dense trails, their occurrence is sufficiently rare not to affect significantly communication considerations. This Section will of course be of interest to people concerned with meteor scatter communication., and serves to delimit meteor considerations for moon communication. In some of the experimental data, double fades have been observed. That is, if one transmits at one polarization and receives at two orthogonal polarizations, a fade sometimes occurs simultaneously in both receiving antennas. A theory for this type of return has been formulated, but we have left it out of this report because enough time| |has not elapsed to show that the double fade phenomenon is truly reproducible.l In Section VII, we discuss the factors which determine a suitable choice of| frequency and antenna size, and point out that at high frequencies the phase shift as a function of modulation spectrum can be ignored. The optimum pulse width for pulse communication is also considered. In Section VIII we present some conclusions of this investigation, and recommendations concerning communication via the moon.

THE UNIVE RS-ITY OF MI CHIGAN 2673-1 -F SECTION II REFLECTIONS FROM THE MOON The moon has been characterized by astronomers as a'rough" (meaning non - uniformly rough) surface,and this has led many scientists to assum~e that the laws of l scattering from a rough surface are applicable to the moon at all frequencies - radar as well as optical. Because of the assumed roughness, some experimentalists have rejected the idea of using the moon as a means of communication between points on the earth. It is our opinion that the moon is not rough in the radar sense of the word, and that a more accurate description is "quasi-smooth". The meaning to be attached to this terminology will become clear later. One of the reasons for abandoning the roughness concept is provided by the experimental results obtained by Blevis (Ref. 1). With circular polarization, at 488 Mc.. Blevis compared the signals scattered from the moon using two orthogonally polarized receivers and found that a maximum signal in one receiver was generally accompanied by a.- minimum in the other. The existence of maxima and minima is no surprise, and is quite in accord with both rough and smooth scattering theories. Some of the minima, |owever, are below the noise leveluand this fact strongly suggests a smooth scattering surface. To see this., let us consider the radar return from a corner reflector whose sides

TH E UNI VE RS ITY OF MI CH I GAN 2673-1-F magnitudes are of order k/2. When orthogonally polarized receivers are used it is found that the minimum signals always contain at least thirty per cent of the total re-| ceived energy. On the other hand, when a smooth corner reflector is rotated and one of the receivers has the transmitted polarization., the energy contained in a signal at minimum is between 0 and 1 per cent. Which receiver has the minimum depends upon the aspect angle; if the transmitted signal is circularly polarized, the dominant return is in the receiver with the opposite (same) polarization when the aspect is such as to demand| an odd (even) number of reflections. The reason for 99 per cent instead of 100 per cent is because of edge effects. Thus the fact that the minima in the moon's return can be below the noise level shows that the moon has a reflection property in common with a smooth body. In view of the preceding results., it is evident that the deep nulls are best explained by assuming only even or only odd numbers of bounces in the predominant re-| turn. Since there must obviously be a large single bounce return it follows that it is the odd number of bounces that are contributing. Since the contributions from double bounces are negligible,, and since the above-mentioned results hold for a wide range of aspects,| it is probable that single bounces provide the bulk of the received energy. Such a statement does not rule out the existence of double bounce reflections,, but it does imply that| the corresponding energy is not usually returned to a receiver on the earth.| All this is not in disagreement with what the astronomers have determined. By| visual observation it is known that the moon is liberally strewn with mountains and

TH E UNI VE R S ITY OF M I CH I GAN 2673-1-F cratersand in this sense the moon is rough. With this definition of roughness,a 3-sided corner reflector one of whose sides was measured in millions of wavelengths, while the other two sides were only thousands of wavelengths in dimension, would be considered a rough surface, since the smaller sides could be likened to the mountains on the moon.'From the radar point of view, however, the surface would behave as a smooth one. Optical measurements have shown that the light reflectivity of the moon can be accounted for by the assumption that the moon is a rough surface (rough in the sense of| diffusely reflecting) and a directivity factor of 5. 7 has been ascribed to the moon. But a |roughness at optical wavelengths implies only the existence of irregularities whose dimen sions are measured in microns, and obviously a fine sandy surface could provide an appropriate type of roughness. At radar frequencies, however, this surface may well behave as a smooth reflector. In short, we admit the fact that the moon has irregularities whose dimensions are of the order of many microns; we also admit the presence of the large natural features such as mountains and craters. But these facts do not preclude the behavior of the moon as a substantially smooth reflector at radar frequencies. Ob-| viously there will be some return from the mountains and craters at these frequencies| ~~~~~~~~~~~~~~~~~~~ad. superimposed on the smooth sphere return and, in general, rather small compared to it, and it is-for this reason that we have used the word "~quasi-smooth"Z (rather than smooth)| to describe the moon's surface for wavelengths in the meter and centimeter bands.| By using the above theory it is possible to predict the form of a radar pulse|

TH E UNI VE RS ITY OF MI CH IGAN 2673-1-F primarily from the specular point and from the- mountains or craters. If the mountains| are near to the center of the moon, they would be the first features of the moon to give rise to a scattered signal and their contributions would represent the initial portion of | the returned pulse. If the mountains (or surface indentations) are on the limbs of the moon., their contributions would appear as fluctuations superimposed on that portion of| the pulse representing the specular return. In addition,they would give rise to a tail to the pulse, the relative length of the tail depending upon the pulse width. These fluctuations may also be a function of aspect, and hence of time. If the surface of the moon has large areas with comparatively small curvature (which areas could be likened to l parts of an oblate spheroid), the specular region of reflection could move, relative to a fixed point on the surface, as the aspect changes. The phase relations between the individual contributions to the returned pulse would then be a function of time, leading to continuous changes in the fluctuations. Still other types of fluctuations can be expected to-appear. Both the Canadians and the British have observed a fluctuation rate of 9 cycles per second in their moon data at 120 Mc. and 488 Mc. which has been ascribed to the libration of the moon. The| Faraday effect in the ionosphere will also produce fluctuations in the received signal, and fades occuring about every two minutes at 488 Mc. have been attributed to this effect. To prove conclusively that the fades had this origin it would be necessary to measure the whole scattering matrix, rather than merely the signals at two orthogonally polarized re

TH E UNI VE RS ITY OF M I CH IGAN 2673-1-F diurnal variations in ionization and to the presence of stray clouds of high ionization density. Although these fluctuations are important in confirming our views concerning the medium between the earth and the moon, they do not represent all the information which can be obtained from the moon data. Even if we exclude such astronomical considerations as better determinations of the distance from the earth to the moon, additional information is available if it is realized that, for most of the time, the fluctuations are superimposed on the specular return. Consequently by selecting the high-pulse returns and averaging out the fluctuations, it is possible to obtain a measure of the specular return. The theory outlined in Section IV can then be used to determine the permittivity, permeability and conductivity of the relevant portion of the moon's surface. A knowledge of the specular return at two different frequencies is theoretically sufficient to specify the constitutive parameters of the moon's surface, on the assumption that they are frequency independent in the frequency range considered, and by using the results at more than two frequencies it is possible to test the consistency of the theory. An objection to the above work might be based upon a feature of the data obtained using very short pulses. Thus, Trexler found that with a 12 microsecond pulse only fifty per cent of the received energy was returned in the first 50 microseconds (Ref. 2).| Still shorter pulses of 5 microseconds were used by the Royal Radar Establishment Ref.~~~~~~~~~~~~~~~~~~ 3) n yia eunotie yte srpoue nFgr.In site of|

THE UNI VE RS ITY OF MI CH IGAN 2673-1-F 0 10 20 30 40 50 kini. lange FIG. 2-1: AMPLITUDE OF THE MOON ECHO (Ref. 3) the appreciable lengthening of the received pulse, we do not feel that this violates our theory. The theory merely states that, in the strong portion of the return, the signal is due to specular reflection upon which fluctuations are superimposed: and the time at which the strong portion is received is, in general, consistent with specular reflection. | But the theory does not assume that energy cannot be reflected from points other than the specular point; if, for example, there are mountains further back, they will contribute to| |the received signal and their contribution will appear at some time after the specular sig-| nal has been received. As an extreme example of this behavior, consider a sphericallytipped cone having a disc at its rear extending beyond the cone's surface. When viewed aor ner head on, the initial return would be from a specular reflection at the spheric al tip and by concentrating upon this portion of the return it would be possible to deter-|

TH E UNI VE RS ITY OF M I CH I GAN 2673 -1 -F frequencies. This becomes even easier if the radius of the sphere is known. At some time after the specular return a strong signal would be received from the disc at the rear of the coneand thus the complete received signal would be spread out in time by an amount depending upon the length of the cone and the transmitted pulse width. Thus, the fact that significant returns can be obtained from parts of the moon other than the specular point does not prevent us from determining the electromagnetic properties of the moon near its specular point. Naturally the limb of the moon would not contribute as strongly as the disc in the above example, but it could be important enough, at some aspects., to spread out the received energy over a time much greater than that of the transmitted pulse. In practice, the time is limited by the radius of the moon, since it is hard to believe that,, at the-se frequencies, any significant amount of energy could travel sufficiently far around the moon to.get back to earth.

TH E UNI VE RS ITY OF M I CH I GAN 2673 -1 -F SECTION III FAR FIELD CRITERIA APPLIED TO THE MOON On the assumption that the dimensions of the transmitting/receiving antenna are small compared to the scattering object,, the criterion for the application of far-zone approximations is r 4 a2 >1 where r is the range and 2a is the maximum dimension of the scatterer in the plane perpendicular to the direction of propagation. Equation (1) is based on a maximum spread in phase angle of 7r/4. In the case of the moon we identify a with its radius (1080 miles). We choose the largest value of X (which therefore corresponds to the minimum range) to.3 be 5 x 10- miles and thus obtain 8 r >> 8 x 10 miles. 5 Since the range of the moon is about 2 x 10 milesP we see that, by ordinary antenna theory, the moon must be treated as being in the near zone. Let us now consider a pulse scattered by a sphere of radius a at a distance r from ihe. radar. If Equation (1) is violated, then at least part of the received pulse will be determined b near-zone scattering laws. However an initial part of the pulse of time length t, can only correspond to reflection from the small initially-irradiated portion of the sphere. of the pulse will correspond to ]For- a sufficiently small tl., this part

THE UNI VE RS ITY OF M I CH I GAN 2673-1-F We shall now determine an upper bound for the time tl during which far zone considerations can be applied to the moon. If Equation (1) is replaced by the less stringent condition r, / 4 a - (2) the critical radius for the illuminated zone is 1 r X The depth of this zone on the 2 moon can easily be shown to be d = 2a - d/4a2 -rAl and since A must be much less than 2a, d = rk 8a The corresponding value of tl is tl = r X (3) 4 ac where c is the velocity of light. Extreme values for tI are obtained by taking x = 10 m., giving t1 = 2 sec., and A = 3 cm, giving tj = 0.006 M sec. If the scatterer is a smooth sphere with voltage reflection coefficient R., there| are two cases which can be considered. If the range r satisfies Equation (1), the optics scattering cross section is S J R 12 r a2 (4) 12

TH E UNI VE RS ITY OF M I CH I GAN 2673-1-F This is the leading term in an asymptotic expansion proceeding in inverse powers of (ka)Z/3 where k = 27r/). and,when ka is much greater than unity, Equation (4) gives the cross section to a high degree of accuracy. If r violates Equation (1), let us confine our attention to the "far-zone" portion of the returned pulse. If the dimensions of the corresponding area on the sphere are large compared with X, geometrical optics can| again be used to give the scattering cross section of Equation (4). So much for the initial portion of the return. For application to the problem of reflections from the moon, the transmitted pulse length T can be assumed to exceed tj. Let us measure time t from the instant at which a return is first received. During the interval 0 < t < tj, Equation (4) is applicable. Thereafter, a near-zone contribution |wil1 be superimposed upon the far-zone return., but since the former is of a lower order| |of magnitude,, the overall level of the signal may be expected to remain unchanged. This| will continue until time t = T. When t >', no far-zone contribution is present, and the level of the return will be appreciably reduced, ultimately falling to zero. All this applies to a smooth moon. It will be observed that in this case the magnitude of the reflection coefficient can be deduced from the initial part of the pulse, notwithstanding the fact that the moon as a whole is in the near zone. Moreover, if the moon is "quasi-smooth" in the sense described in Section II, the essential shape of the returned pulse will be unchanged except for the presence of spikes corresponding to |specular reflections from other (discrete) areas of secondary importance.l

T HE UNI VE RS ITY OF MI CH I GAN 2673 -1 -F the wavelength under discussion. There will then be no specular reflection and the level of the return during the time 0 < t < tj will not be significantly -larger than that of the tail. However, there is no moon data at radar frequencies which agrees with rough surface theory.

TH E UNI VE R S ITY OF M I CH I GAN | ~~~~~~~~~~~26 73 -1-F SE CTION IV-I THE ELECTROMAGNETIC CONSTANTS OF THE MOON'S SURFACE In this section we shall attempt to determine the electromagnetic parameters of the moon's surface from measurements of the returned power at different radar frequencies. Since the radius of the moon is very much greater than -the wavelength at all frequencies under consideration, the voltage reflection coefficient R at the specular point is the same as for an infinite slab of the same material as the moon's surface. If a plane electromagnetic wave is incident normally on such a slab, I+ 1+o f i S /I EO0 0f (see for example., Stratton (Ref. 4,.493) ),where i, and S are the permittivity,, permeability,and conductivity respectively of the material. The suffix "o"l denotes the same. quantities for free space and w is the frequency. M. K. S. units are employed and a time dependence e-iL~t has been assumed. The radar cross section of the moon is JR 1 R2 7T a2 (2) whe re a is the mean radius of the moon. JRI may be written in the form

TH E UNI VE RS ITY OF M I CHI GAN 2673-1-F 2 x - +Z 2 Y- i~)Z) J RI...(3) X - 1 +2 ( Y + v/(X+l)f'/2 where | where X = 1l + z X2, Y = (4) 472 2 S2 )1o3 y4 2 2 /4 4r2~ At this stage it is well to review the assumptions which have been made in| carrying out the above analysis. Given the basic assumption that specular reflection itakes place (some of the evidence for this has been discussed in Section II) it follows l that the radius of curvature in la region surrounding the specular point must be large| compared with the wavelength. This in turn justifies the use of a plane slab** to determine an expression for R. and since the distance of the moon from the earth is very | much larger than the maximum separation of a transmitter and receiver on the surface| of the earth, the transmitted field can be treated as a plane wave incident normally on the specular area. In writing down Equation (2) however, it has been assumed that the * For X very smallEquation (3) takes on the form iRI 2 (Y - 1)2 + Z 2 Y. (Y + 1)2 4 (1 + y)2 **An alternative., but entirely equivalent, method for obtaining Equation (2) would have| sphere.|

TH E UNI VE RS ITY OF M I C H I GAN 26 73 -1 -F radius of curvature of this area is equal to the mean radius of the moon. It is obvious that this will not be true'if, for example, the main reflection takes place from a crater, a situation which could arise at particular aspects. Although it seems reasonable to suppose that this will not occur in general, the use of the mean radius a in Equation (2) could be a source of error in comparing different sets of moon data. In seeking to determine Y and Z by fitting Equation (3) to measured values of IR 1 2at different frequencies, it will be assumed that constant values of Y and Z can be used representing averages of Y and Z over the frequency range. This hypothesis is probably a reasonable approximation at the frequencies under consideration and will be put on a firmer basis a posteriori. As the wavelength increases, so does the depth to which the incident field penetrates below the surface of the moon. The effective reflection takes place at a lower level as X increases r n tand if the surface laye s of the moon are ot homogeneous., or are stratified., the assumption of a uniform slab in the derivation of R will not be valid. Indeed the available data does provide some slight evidence of non-uniformity as a function of X (and hence probably of depth), though the accuracy of the experimental values for S'is not sufficient for any definite statement on this point. Let'us now return to Equation (2) in which the scatterin cross section T is given as a function of X. If it is assumed that Y and Z are independent of X, a knowledge of a- for different values of X is sufficient to determine Y and Z (and hence 6 and

TH E UNI VE RS ITY OF MI CH I GAN 2673-1-F 2 W X. = 0. 1 In 0. 0003, ra (ii) X = 0. 6 m T = 0. 03 ffa2 (iii) x = 1. 0 In T = (0. 05-0. 09)w-a2 2 (iv) X = 1. 5 m T = (0. 06 -0. 1 0) 7Ta W X =,2.5 m T = 0. 1 7ra2 The first of these was obtained by Yaplee et al (Ref. 5) at a frequency of 2860 Mc using 2 microsecond pulses. The relatively low'cross section has been attributed by Trexler and Yaplee to modulation loss, and a figure given for this effect is 22 db (private communication from Trexler). The corrected value0f T is then 0. 05 -ff a2 a value more in accord with that obtained by Trexler himself at this frequency. It is this result which will be used in our calculations. The second measurementOf Twas made by Blevis (Ref. 1) at a frequency of 488 Mc using two second pulses. For X = 1. 0 m (300 Mc) the value0f T is due to Trexler (Ref. 2) and has been deduced from his statement that the transmission loss is 258 db when calculated on the basis of isotropic transmitting and receiving antennas. The two values given for a- in (iii) and (iv) are for a moon at perigee and apogee respectively. The fourth result is the average obtained from many measurements made at or near to a frequency of 200 Mc. it has been stated by Trexler (Ref. 2) as a transmission loss of 254 db, although it is pertinent to remark that Trexler himself found an additional loss'of 17 db when using 10A sec pulses at 198 Me. This further reduction in power was attributed to m adulation loss. For X = 2.5 m the value of o is due to Evans (Ref. 6). The result was obtained using 10 H sec pulses and is a refinement of that originally

.THE UNI VE RS ITY OF MI CH IGAN 2673-1-F of. a correlation of all the data available to him, Trexler (Ref. 2) states that for frequencies between 20 Mc and 3000 Mc the average transmission loss increases at a rate of 6 db per octave, passing through the value 258 db at 300 Mc; going back to the radar equation., this implies a constant cross section, The accuracy of the above data'i's difficult to assess. Evans has given a probable error of + 3 db for his result, and an uncertainty of 4 db has been attached to the data quoted by Trexler. It is clearP however, that the magnitude of a- will depend on the manner in which the return is measured., and while most of the results are for the total' returned power, it is the initial peak return (corresponding to the main specular reflection) which is required in the present theory, A good fit of Equation (3) to these experimental results is obtained by choosing Y and Z'in Equation (4) to be Y= 645 Z 0. 1 M (5) From this result we obtain 2.4 0 x 10-5 1. 7 (mhos and A 90 4 (6) S 1. ox 10 mhos /henry A go 2. 1 x lo-11 farads /meter and S 1. 3 x 10 -2 mhos /meter (7)

THE UNI VE RS ITY OF MI CHI GAN 2673-1-F These values of and S are considered to be mean values -in the frequency range examined. By comparison of these values to <- and S.values quoted in-von Hippel (Ref. 8) for various soils, it i's evident that the present results are not opposed to the hypothesis that the surface layers of the moon are mainly composed of dry soil, though certain varieties of crumbly cheese might be possible materials. The curve obtained using Equation (5) is superimposed on the experimental points in Figure 4-1. A further justification for the use of constant values for Y and Z in the curve fitting is now possible. Von Hippel in Ref. 8 reports experimental values, of E and's for wavelengths of.03, 0. 1, and 1. 0 meters, and up. E was found to be quite insensitive to wavelength, while S. varied, very roughly speaking, as X-1. From Equations (3) -1 0 and (4) it is clear that if a frequency dependence of X for S and X for E is assumed, IR 1 2would be independent of frequency. If a frequency dependence somewhat closer to that of actual dry soil samples is assumed and the values of Equation (7) assigned to 1 2 X = 1 m., the slope of the resulting curve of I R computed for. 1 < X < 20 5 m decreases 2 compared to the curve in Figure 4-1.,,but the JR values.remain well within 3 db of those- on the curve. For the purposes of Section VII we require also the phase change on reflection. This can be calculated from Equations (1) and (5), and the curve of arg R as a function of X is given'in Figure 4-2.

TH E UNI VE RS ITY OF MI CH I GAN 2673-1 -F 2 Trexler (NRL) Data.JR 1 2 Yaplee'(NRL) Data 08 06.04 vans Computed Curve (Manchester) Data 02 +3 db Uncertainty data Blevis (DRB) Data Uncertainty due to variation in 4 Moon's range.01 0 2 3 (in meters) FIG. 4 -1: POWER REFLECTION COEFFICIENT JR1 2 AS A'FTTNCTTC)N OF WAVELENGTH

TH E UNIVE RS ITY OF MICH IGAN 2673-1-F Phase (in degrees) 210 200 190 180 0 1 2 3 X (in meters) FIG. 4-2: PHASE OF VOLTAGE REFLECTION COEFFICIENT, R..... ~~~~~~~~~22

TH E UNI VE RS ITY OF M I C H I GAN 2673-1-F SECTION V THE FARADAY EFFECT The Faraday effect is a rotation of the plane of polarization of a plane-polarized wave on propagation through an ionized region in which there is a magnetic field. The rotation is anti-reciprocal, that is., it is doubled rather than cancelled when the path is traversed twice in o posite directions. Since this rotation is manifested as a power.loss., or fade, if linear polarization is used, it must be considered in any discussion of radar reflection from the moon'. The ionosphe re and the geomagnetic field are the sources of the Faraday rotation in this situation. For radar frequencies in. excess of 100 Mc, the expression for the rotation angle on a one-way traversal may be written as 3 e N _S"' H(S —") ds-" ff M2 C2y2 in which e and m are the charge and mass of the electron, c is the velocity of light, and -v the radar frequency. N'is the electron density., and H the magnetic field. The integral is taken along the propagation path. Unfortunately, N is not known very well, so that accurate evaluation of the integral is precluded. As a rough estimate, we might assume (Ref. 6) the value N ds 1013/cm2 for the whole ionosphere. Again, to obtain an order of magnitude -un II Ion fn-r Ir wea n.Q.Qviyma +'kn Y"^^-n a+ -7,a-ni+'k Qn +Iin+ nril-v Oixi xnalr4innI nnmnn-nda-n+ nf T-T

TH E UNI VE RS ITY OF M I CH I GAN 2673-1-F is involved. This we take as 0. 5 gauss, a representative value for the United States. These values yield a rotation of roughly 7r at ~ = 200 Mc. We see then that the Faraday effect is not negligible at lower frequencies. Since it might be desirable to operate in such a manner that the Faraday effect is significant, the bistatic Faraday effect has been analyzed. Such information would be of most value if formulated explicitly in terms of the location of the transmitter and re-| ceiver, and the position of the moon relative to one or both of these. We now proceed to derive such a result. Since the only complexity of the derivation is geometrical, it will not be presented in great detail. | ~As seen from Equation (1) the geometrical dependence of the Faraday effect is l contained in the integral |N(,)H(I) ~ L dl A in which N(Q) is the free electron density, _H(_Q) the geomagnetic field, and L a unit vector along the propagation direction. The integral is taken along the ray path (i. e., from transmitter to moon for the Faraday effect on the transmitted ray, and from moon to receiver for the rotation on the return path). We want to express this integral in convenient coordinates. We designate the transmitter and receiver locations by OT, OT and OR, OR. Here 0 is the usual polar angle (the colatitude), and 0 is longitude (0 < 0 is W. Longitude). The position of the moon is defined by angles )VRO T and (R. T )T is the elevation........ th2agn4ln tOTP fteln f ih rmtetasitrt h on

TH E UNI VE RS ITY OF M I CH I GAN 2673-1 -F YT is measured from the meridian pointing north clockwise to the projection of this line of sight into the tangent plane. R and /R are defined similarly at the receiver. One finds that the vector LT for the path of the transmitted ray is given by + [ sinT sinT sin XT -coseT sin Tc os Xr cOS C/T + SiT cos X sin 1YT] T n Ti XT -T CSTFCos PTO F T co| + [ C~SeT sin XT + sineT Cos ftT Cos T]. According to Mitra (Ref. 9), the main part of the geomagnetic field can bel attributed to a dipole of strength IPI = 8.19 x 1025 cgs., oriented with N. pole at 78. 5 N$ 69~W. Then, where Px = sin 11. 5~ sin 69~ Pz =cos 11.5, we may write H (x,y,z) = x -P +3P x2 3P + i l ___& 1.~~ ~~~ x r2 2'r + similar terms in cyclic order for the other two components. We: want to obtain H(9,) from this. To do this, we first write the position vector S-T(Q) of a point on the transmission path: S~T(Q) =G(x sineTcOSpT+y sinOTsin T+ cos0T)+(-G sin XT+ JjQ2-G2cos2 XT) LT Here G is the radius of the earth, and R, runs from G to D, the mean earth-moon dis-| tance. Now to get HQ), we replace r by ST(J)|, which is Q, and x, y, z by the three components of ST(g). lT25

TH E UNI VE RS ITY OF MI CH I GAN 2673-1-F | H The scalar product T() LT may now be evaluated, yielding 3 ~ G yQ2_G 2cos2 (T+ 1 Clsin?r+C2c2sC rCTsO T+C3 cOs )Tsin 2 q2 - 3G2cos27r 3Gsin XT 2 G2COS2 in which C1 = sin 11.5~ sin OT cos (OT+690)+COS 11.5~ cos OT C2 = sin 11. 5~ cos 0T cos(,0T + 69~) + cos 11.65~ sin 0T l C3 = - sin 11. 5~ sin(OT + 69~). Fina lly, N ( Q) may be obtained from s ome s tanda rd tabl es of atmosphe ric| properties. Such a table would give N(z), where z is altitude above the earth's surface. The replacement z = - G converts this to N(Q). With the expressions given above, the Faraday rotation between the transmitter and moon can be obtained. Of course, a numerical integration would be required. Since the rotation'between the moon and receiver is the same as if the receiver ltransmitted to the moon, the rotation on this second segment of the path can be obtained| simply by changing all subscripts T to R in the expressions given above. However, the angles X(R and CR are obviously redundant, and may be expressed in term of OT, OTV OR' OR' )(T'and T'I Let t =ST() -G(x sin OR os iR + y sin OR sin R + zcos OR). Then if t is the sin OR = O. (xsin0Rc s0R+ Y sin0R R+ ces0R) 26 ve

TH E UNI VE R S ITY OF M I CH I GAN 2673-1 -F Since if the moon is visible at the receiver, it is above the horizon, we know 0 < YR < 7r/2 and thus the sign of cos XR is determined, while Icos XRI is obtained from sin ~R' Further, Cos R Cos t 0(0ROR) = -t (X Cos ORCosOR+yCos0R sin OR ^z sin OR Finally, A A cos;Rsin k =Rt *(0R,~R)-t' (-x sin~R + yCoR). Since cos (R may first be obtained numerically, sin YR and cos rR may be determined this way. All the quantities required to replace T by R to get the Faraday rotation for the return path are now prescribed. However, normal diurnal fluctuations in ionospheric electron density., and those associated with magnetic storms, make it pointless to attempt a numerical evaluation of the Faraday integral. To yield a practically useful result, such evaluation would have to be in terms of site coordinate and time and date (corresponding to the moon's position). But such fluctuations would invalidate the table. For these reasons, we feel the Faraday effect should be circumvented. Receiving two orthogonal polarizations,, or transmitting and receiving circular polarization., are schemes for overcoming power loss due to the Faraday effect. Since it varies inversely with A, operation in the KMc range will eliminate it. | ~In the presence of the Faraday effect., one may ask how the two rotations of the | plane of polarization will disturb or alter the determination of the scattering properties of the moon (or an earth satellite). For a given aspect and a fixed frequency, the full 27

TH E UNI VE R S ITY OF M I C H I GAN 2673-1 -F description of the cross section of a target is specified by the scattering matrix. The scattering matrix of interest is of course that which would be present in the absence of the ionosphere, not that due to the composite system of scatterer plus ionosphere. The complication is that we can only observe it with the effect of the Faraday rotation super-| imposed. For simplicity, the discussion is restricted to back-scattering; no new essential features appear in the general case. Let us define the direction of propagation as the z-axis and let E be the incident field and E the scattered field. The scattering matrix is then described by Es = Si, Ei + S12 El x Snx y s i +2z Ey= S21 Ex + S22 El y x Ey where the Sjk are complex numbers. By reciprocity, S12 = S21. An overall absolute phase is irrelevant. There are thus five independent real numbers that determine the S-matrix: The magnitudes of SH, S22, and S12, and the relative phases between them. Denote the electric polarization vector of the field incident on. the scatterer by (~ cos 0 + y sin 0), and let (x cos a + y sin a) be the polarization vector upon leaving the scatterer of a scattered field whose polarization on arrival at the receiver will coincide with that of the receiver. The field measured at the receiver will be Er s s sne E Ex cos ar + Ey sin a |=E |: S cos 0 cos a + S22 sin 0 sin a + S12 sin (8 + a) Without the ionosphere, 0 and a would be the polarization angles of the transmitter and

THE UNI VE RS ITY OF M I CH I GAN 2673-1-F le. g. for the pairs of values (0 = O. a = 0), (O = r/2, a = 7r/2), and (O = O. a = 7r/2), one obtains (E r/Et) = S11s S22., and S12 respectively., so that if one measures the amplitude of| each of the three returns and the two relative phases the S-matrix is determined.] With the ionosphere in the way, the Faraday effect rotates the polarization vector by an angle A, so that if the radiation is incident with a polarization angle 0 it must have started with an angle 0 = e - r; similarly, to receive the component of the scattered field that leaves the scatterer with polarization angle a one must receive with a polarization angle a = a +'. Thus, if the transmitter has a polarization (x cos 0 + sin ~) and the receiver a polarization (x cos,B + y sin ]), the measured field will be E =Et S[1 cos (0 +A) cos.(~ -) + S22 sin(0 + ) sin(! - ) + S12 sin (0 + f])| (4) E [ S12 sin (0 + A) + (1/2) (Sll - S22) cos (0 + ) + (1/2) (Sn1+S22) cos (0-) cos 2~ + (1/2) (S11+S22)sin( -f)sin 2 This expression contains functions of and 0 which are known for a particular choice of transmitter and receiver polarizations, and the unknown quantities S12, (Sll-S22)j, (SH1 + S22) cos 2 R, and (SH1 + S22) sin 2 }. Four pairs of polarization angle settings are now necessary, supplying seven measurements; these suffice to determine the amplitude and phase of the above unknowns. Since the trigonometric functions are real, there is a redundancy in that the_ phase of the last two unknowns is the same (modulo vT). It is not proper to simply discard one phase measurement on this account; the corresponding l ~~~~~~29l

THE UNI VE RS ITY OF M I CH I GAN 2673-1 -F amplitude is then a quadratic expression which possesses an extraneous root (unless one of the unknowns vanishes). One can, however, use delay line circuitry to superpose the returns without loss of phase before the recording stage in such a way that the resulting linear combinations are the above unknowns; after this is done, only six quantities need be recorded for maximum information about the scattering matrix (discarding thel relative sign of cos 2 and sin Z I is of no importance). These measurements do not, however, completely determine the scattering matrix. While cos 2 ~ (or sin 2 # ) is real, it can be either positive or negative; unless we know which it is, there is an ambiguity of 7r in the phase of (SI, + S22). Because of jthis ambiguity, the measurements yield two scattering matrices to choose between., only one of which is of course correct. (The two are identical for a matrix proportional to the unit matrix such as occurs for the sphere). If we denote the two solutions by superscripts a and b, and fix arbitrarily the overall phase of one with respect to the other, | they are related by a 2b a lb a b S11 = S22, S22 = S, S12 =-s.2 (5) The choice of the correct S-matrix can be made only if either of two additional data are| available: 1. a determination of the Faraday rotation adequate to ascertain the sign of cos 2t (e.g. frequency comparison); or 2 sufficient prior knowledge of the properties of the scatterer to decide which of

TH E UNI VE RS ITY OF M I CH I GAN 2673 -1 -F SECTION VI E-LECTROMAGNETIC SCATTERING BY LOW-DENSITY METEOR TRAILS 6. 1 Scattering y a Low-Density!Jniform Line Distribution of Electrons 6. 1. 1 - Statement of the Problem As a meteoroid approaches the earth, its interaction with the atmosphere leads to the formation of ionization in its wake. This ionized region will scatter electromagnetic energy incident upon it. Although the trail is electrically neutral (i.e.-, there are an equal number of electrons and positive ions), the electrons, because their mass is so much smaller than that of the ions', contribute essentially all of the return. The meteoroid itself is too tiny an object to have a significant effect (the frequency of occur-). rence of meteorites is negligible). Thus, the most elementary model of a meteor trail that suggests itself., for the purpose of a calculation of electromagnetic scattering, is a uniform line distribution of electrons. At radar frequencies., the Compton scattering due to an electron is given by the Thomson cross section e )2 2 d -a d T r 0 A where e and el are the electric polarization vectors of the incident and scattered fields 2/MC2) respectively, r is the classical electron radius (e and dS1 is the solid angle sub0 tended at the detector. — If the transmitter power is P and its gain is G., the power density

TH E UNI VE RS ITY OF MICH IGAN 2673-1 -F receiver of gain G' for an operating wavelength X is G' V2/47r. Hence, the effective solid angle subtended by the receiver located at a distance r' from the scatterer is G'G 2/41r r'2. Thus, the scattered power received from one electron is 2 \A2 2 2 12 S = PGG' ro (e * e')2/16 r r (2) Consider next a uniform line distribution of electrons, a per unit length. Assume a sufficiently small that the field incident on one electron is essentially unaffected by the presence of the other electrons. The amplitude of the scattered field due to a line segment of length ds will then be a ds times that due to a single electron. Its phase, relative to other segments, will be given by (27r/k) (r + r'). The gains of the transmitter a nd receiver will., in general, vary along the line; this'is most conveniently taken into | account by understanding G and GI to be the maximum gains and using a lumped antennae pattern factor f(s) to denote the reduction in amplitude at the point whose position along| the line is denoted by s. The scattered power due to the line distribution is then 2 k2/16 |) s A a f(s) ei(2r/)(r+r )/rr 2 S = (PGG' ro A! 6 I)er (3) where a. r, r', and (e ~ e') are functions of the position s. We shall take a to be constant, reinterpreting f(s) to include any variation in a over the line. Inasmuch as the altitude of the trail is much larger than its length, the variation in r, r', and (e * e') over| the trail is small and, except in the phase, can be ignored. Letting R and R' represent the distances r and r' measured with respect to a representative point on the trail and (e* e') the value at that point, the scattered power is 32

TH E UNI VE RS ITY OF MI CH I GAN 2673-1 -F S (PGG a2 ro2 A2/167r2 R2 R2)(e *')2 | e ds f(s) ei(2 /k)(r+4r' ) The subsequent discussion concerns itself with the evaluation of the integral I - Jds f(s) ei(27r/l) (r + r') under various conditions. 6. 1. 2 - Integration by Parts It is convenient to rewrite the integral as I = y ds f(s) eiK g(s) where K = (27r/) (R + R') (6a) and g(s) = (r + r')/(R+R'). (6b) For the parameters of interest; K is a very large number., while g is of order unity. | The functions f and g are well-behaved. In any region of s within which g'(s) does not vanish, we can change the variable to| g: g(b)ig I = g(a dg (f/g') eiK (7) g a) For convenience, write G(g) = f/g' (7a) so that g(b) I = g(fa dg G(g) eiKg (8) g a) Integratinlg by parts repeatedly, we obtain 33

TH E UNI VE RS ITY OF MI CH IGAN | ~~~~~~~~~2673-1 -Fi | I = (iK)-1 (-iK)n [G n(g(b) ) eig)- G~n (g (a)) ei~~) (9) n=o If g'(s) vanishes at the point s =sO within a region, we can break up the region into| the interval (s0-h) < s 4E (so+h) about this point, and the two intervals a -, s,c (so-h) and (so+h) < s < b for which the above treatment is still valid: | I= (iK)1 (-iK) -n ZG n(g (so-) eigS-h) -' G n(g (a)) eiKg(a)| n=o + ds f(s) eigs (10) -1 2 K-n t(n) gb) iKg(b) - (n) igS~h)eg(s ~h] n=o nGo or I (iK (-iK) [G) (g (boh) eiKg(b o) G(n) (g (a)i gisgh)| n=o | + J ~~~ds ~)eg.(1 Th (irst l term) is th saea n) the absec of apintsoh at whic g'(s vaih)eigs, whil ca be several). 0-h3

THE UNI VE RS ITY OF MI CH I GAN 2673-1 -F To evaluate the remaining integral, expand g(s) in a Taylor series about the point | S sso:I g(s) = g(so) + 0 g ( So) (12) n! s +h (S -SO) 9 (n)~~~~ (So) 0 dsf(s) eiKg(s): eiKg(so) ~ ds n e o ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ds f sedsf(s)e. (~~~~~~~~~~13) so-h so-h The first step in examining the evaluation of I is to look at the explicit form of g. Take the z-axis along the trail, and the origin at the point on the trail from which R and R' are measured. Then A s= z s R = R(x sin 0 cos + y sin 0 sin0 + z cos 0) -'A'. A A R' = R'(x sin 0' cos 0' + y sin 0' sin O' + z cos 0') (14) and r= R - s _ _ _~ r': R' -S. Evaluating r and r' in terms of the variable s and the constant quantities, r=R~~l-2scos0+s2 1~1/2 2 s cos 0 + sa 21 L R R I~~~~~~~~~~~~ / (15) r =R' [1- 2 s cos O + s2 l/2 R' R12 so that ~ ~' R ~ ~~~~~~2 1/2 1, I~~_.... co 1 +. R 1 2s cos'+ s2' R+R' R R2 R' R'2 (16) 35

THE UNIVE RS ITY OF MI CHIGAN 2673-1 -F Differentiating,, (17) = 3 cs0- C )[1 _ (os0- s _)+ _2s C os' + s ] s R+R R R R..... + sin(19) InR ge, c s R R2 R R' ofth R'dera) 9 3 si 0(Coo co s cos = -2 +sn-Csl )1cos o ( 20 R+R'( R2 R R R 2 -R,2 R' i In general, succes sive derivatives decrease in magnit ude by a factor of the orde r of i /R. to be the origin. It then follows that, at this point Cos 0' =Cos 0 (20) the same (specular condition). The azimuthal angles do not appear in the condition for g'(0) vanishing, hence there i's no requirement that the trail and the two rays be coplanar.| We have then g(O) = 1 (16') g'(0) = 0 (17') f ~) sin 2 0,g"(0): i (18') l | ~~~~~g'"(0) = 3 ( R' -R ) sin2 0 cos 0 (19') l...... 36

TH E UNI VE R S ITY OF M I C H I GAN 2673-1-F 6.1. 3 - Constant Antenna Pattern Factor The special case that first suggests itself is f(s) = 1, a 4 s < b; f(s)= 0, s < a or s > b. This might correspond to a constant uniform line distribution of electrons extending back from the meteoroid, two idealized overlapping sharply defined beams | which have a constant intensity within the beamwidth and zero intensity just beyond it,, and neglect of diffusion of the trail. In this case, we have G(g) = _ (21) g' G'(g) = d 1 = 1 d 1 - _ g" (22) dg g' g' ds g' (g,) The ratio of the second term in the series of Equation (11) to the first is | G'(g) |= g(3 ~~~~g"9 (23) KG(g) K(g')2 Outside the neighborhood of a point where g' = 0, g" is of order R 2 and g' is of order | R-1 so the ratio is of order K-l~-(X/R) which is quite small. Successive terms fall off to the same order. Near a specular point (chosen as s = 0), we use a Taylor series expansion whose successive terms are of order (s/R), which is reasonably small: g'(s) - s g"(0) +...... (24) G'(g) |= kg"(0) = ARR' (25) |KG(g) - 2r (R + R')sz [g"I(0) 2 2r (R + R')s2 sin2 l and a comparable factor for successive terms. We shall accordingly stipulate that 37

THeE UNI VE RS ITY OF MI CH IGAN 2673 -1 -F h >>'csc e. (26) ~27r(R+R') This will permit us to discard all but the first term of the series. (Provisionally, consider that a < -h < h -4 b to streamline the discussion; the other possibilities will be treated later). For the series in the exponential of Equation (13)., the first two terms are3 s2~~~~~ 2 K --- g"(0) =If s R+R' sin 0 (27) K _ gI"(0) =r s3 W2Rsin 2 Cos 0 (28)7l 6 X R2.R,2 snecoe and successive terms are of order (s/R) smaller. If the expression of Equation (28) and the higher order terms are small, the integral reduces to a Fresnel integral; otherwise| it cannot be evaluated analytically. We shall consequently further require that h<< X R2 R'229 hC< iA.......l (29) l 7r (R2 -R'2) sin2 0 cos 0 (If the denominator of Equation (29) is near-vanishing., it may, be necessary to state in l addition the corresponding inequality for the next order term). This inequality is compatible with Equation (26) provided only that X<< R. With h chosen, in accordance with the discussion of the last paragraph, so that only| |the first term of the series in the exponential of Equation (13) is required, the integral | reduces to....... ~~~~~~38

TH E UNIVE RS ITY OF MI CHIGAN 2673 -1 -F / i K ~~~~~~~~~~2~2 s sin 0 I1Al = l ds e iKg(s) = 2e? ds e 2 2XRR' -h ~(30) oXR' i-2 +R =(R+R' ) /csc 0 ee F 2t R'R h sin 0 where F(H) is the Fresnel integral H i (7r/2)x2 F(H) = dxe (30a) From the asymptotic expansions (Ref. 10), -1/2 ilr/4 i-1 l) 3i1i(r/2)H2 F(H) = 2 e - i(rH) e (7r2 H3) e +.... (31) Equation (26) sets a lower limit to H; if we take H as small as 2t/2 the inequality will hold by a factor of about 2.5 and the ratio of terms of Equation (25) will be (2fr)-1*The last term of Equation (31) can then be neglected, being 5 per cent of the total.l Of the two series in Equation (I11) the second is the larger, and we have chosen h l such that only the first term of this series is required. This is I2 = - 1 [ 1 eiKg(h) 1 iKg(-h)] 2 2 e' ( e I2 ~~~~~=..... Ie0 i K g'(h) g -h) iKhg"(0) (32) i XRR' iK (l+h2sin2 0/2RR'),2,X] / cs i1e 2.) e = -Csc 0 e i (7rH) e -r(R +R')h sin20 where the expansion of the exponent has been cut off, as permitted by the choice of h. When I, and I2 are combined, the sum reduces to I =[kRR'/(R+R')]1 scoi(+/2) 39

TH E UNI VE RS ITY OF M I CH I GAN 2673-1 -F because I2 just cancels the contribution of the second term of Equation (31). Since h is| a mathematical construct of no direct physical meaning, its disappearance from the final result is to be expected. The contributions of the end-point series have been neglected; they are of magnitude a and ~~~~~~~hence will (h/2,ff al) and (h/2,xb) respectively vis-% -vis Equation (33) (for H2=Z/) n hnewl -1 contribute less than (2Xr) of I provided Jad and b are both at least twice ho If either end point is closer than this to the specular region, the Fresnel integral on that side should be extended out to it (for this'integral, the second term of Equation (31) will now| have to be retained), eliminating the integration by parts; for the other side, the cancellation will work as before. We see then that the chief contribution to I comes from the neighborhood of the point at which g' = 0. Conceptually, this is to be expected: The phase exponential is a rapidly fluctuating function, and the fluctuations result in contributions from neighboring intervals approximately cancelling; if there is an interval in which the oscillation is temporarily suspended (as indicated by g' = 0), its contribution will remain and will dominate the answer. That is the basis of the principle of stationary phase, and in fact the result could have been obtained by invoking it; this has not been done because the investigation has to include situations in which the conditions for applying the stationary phase evaluation are not satisfied, and the other terms then have to be considered. Hence, for a < -h < h < b and f = I between a and b (and zero outside),l II\2 = ARRI-CSC2 o ~~~~~~(34) I R+R' 4O

THE UNIVE RS ITY OF MICHIGAN 2673-1- F and 2 23 22 | S =PGG' ro csc e ( (35) 2' 16ir RR' (R+R') An equivalent expression is given in Ref. 11 after a heuristic discussion. For backscattering with transmitter and receiver at the same polarization, we have| R = R' 0 = Xr and e - e, so that PGG'a2r0 (36)2l 32r2 R3 which is the Lovell-Clegg formula as quoted in Ref. 12 from a very rough argument. (There are numerical discrepancies from this value in Refs. 13 and 14).l Alternatively, if there is no stationary point in the interval (a, b), only the first series will contribute, and its first term will dominate: I: 1 e - e. iKg(a) l iK g'(b) g'(a) ]J |If the interval (a, b) is far removed from a stationary point, g' will vary slowly so| that, choosing s = 0 to lie within (a, b), we can replace it by g'(0) = _ 0 + cos 0_'. (17") R + R' The magnitude of the two terms in Equation 37 is then the same. The two phases, Kg, |are very large numbers and, consequently, the relative phase of the two terms cannot be| adequately estimated. -This difficulty i's due to the oversimplified model and does not| occur in a more realistic view of the physical situation. Actually, the contribution of the 41

T H E U N I VE'R S I TY OF M I C H I GAN 2673-1-F back end will be reduced by the effect of diffusion. Anticipating this result, let us take I = - iK g'(0) e g() (38) so that (39) 4w2 (cos 0 + cos 0,)2 and PGG' a r o X4AA S PG'(- e. (40) 647r4 R2 R2(cos 0+cos 0')2 In particular, for back-scattering at a single polarization PGG' a2r2 (4 S = o _ (41) 2567r4 R4 cos2 e If there is no stationary point in (a, b),, but there is one near one end of the interval., say a distance p from b, then |g'(b)| C< ]g'(a)~, and furthermore g'(b) = - pg"(0) if we choose the stationary point to be s = 0. Thus Ill: X R 2R' = 12 R R1 (42) 47r2(R + R' p 2sin 4fR Another case to examine is the stationary point within the interval but close to one end,. say a distance p from b, where p ~ h. Again let the stationary point be s = 0. Referring back to Equation (10), we now have the first series and the pertinent part of the integral, i.e. (i K -n o(n) e iKg(-h) _ (n) +~~ d ~ gs' = ( K) t(-iK) G (g(-h)) e (g(a) ) eiKg(a eiKg(s) n=o -h (43)

TH E UNI VE RS ITY OF M I CH I GAN 2673-1-F The evaluation of the sum and the portion of the integral for negative s proceeds as in the original case (see Equations 30-33 and associated discussion). The integral from 0 to p is proportional to a Fresnel integral, as in Equation (30), but it cannot now in | general be replaced by the first term of the asymptotic expansion. We thus have 1/212 =1 (RR' ) e csc e + 2 F 2 R p sin ) (44) 2-+R 2 RR' If p is sufficiently large, this can be simplified by means of Equation (31). One more case remains to be mentioned for the sake of completeness, a simpler one the conditions for which have not been met by past experimental parameters. It occurs |when the irradiated portion of the trail contains the specular point and is short enough so| that the higher terms in the phase can be neglected throughout it (i. e. the expression in Equation (28) is small for a S s 4< b, and so are the successive terms). We then perform the integration for the whole region at once, obtaining (45) I 2XIP' 1/2 ( R['12RR / |= A eiF csc 0 F 2 \a I [ + 2 b sin0 2R+R'/ ~2 RR'AI Depending on the magnitude of the argument, it may be possible to approximate one or both of the Fresnel integrals by means of Equation (31). Surveying the results of this section, we find that for f =1 the return can vary from| the specular value given by Equation (35) (or actually somewhat higher at the maximum of the diffraction pattern given by Equation (44)) to the non-specular value given by Equation (40) wit a ontiuousspetrumof iterediae vauesindiatedby quatons(44), (42),L - ~~~43

TH E UNI VE RS ITY OF MI CH I GAN 2673-1-F (37) and (38). The extremum values differ by the order of A/R, with a numerical factor that tends to worsen the gap (at a few meters wavelength, where the meteor trail observations have been made, the ratio of returns might be 106). The specular returns have a X3 dependence, the non-specular a X dependence, with an irregular transition in the intermediate region. The experimental results on the wavelength dependence of meteor trail scattering can be accounted for in terms of these results. Comparing simultaneous returns at 4 and 8 meters from the same trail, Lovell and Clegg (Ref. 13) report the A3 dependence char-| acteristic of specular scattering. They state that the sensitivity of their equipment was such that a meteor trail would not be detected at 4 m. unless its aspect were essentially specular. The intensity of the return increases rapidly with the wavelength, so that its detectability improves. The non-specular return varies as X4, increasing faster than the specular return. Since there are obviously far more non-specular than specular trails (except possibly for some special conditions of shower observations) the wavelength dependence at longer wavelengths can be expected to become X4. This is corroborated by Eckersley's observation (Ref. 15) of a X4 increase in total returns for wavelengths between 16 and 32 meters. 6.2 The Effect of Ambipolar Diffusion. 6. 2.1 - Local Attenuation A l ine distribution of ionization will not maintain itself as a line but will undergo arebipolar diffusion. As a result of this diffusion, electrons change position | L ~~~~~~44

TH E UNIVE RS ITY OF MI CH IGAN 2673-1-F relative to each other, and hence the relative phase of their contribution to the return also changes. Herlofson (Ref. 16) proposed taking the attenuation due to these phase differences into account by integrating the amplitude of the electron returns over a plane perpendicular to the trail and taking the ratio of this value to that obtained for a line distribution of the same number of electrons. His calculation contains several oversimplifications: he implicitly assumes the meteoroid to have an infinite velocity (taking the diffusion to be identical all along the trail), and he uses a two-dimensional geometryl which implies that the transmitter and receiver are lines of infinite extent parallel to the axis of the trail. His idea can be used for a more nearly correct treatment: consider a short element of length of the trail, and carry out the phase integral over a slice perpendicular to this element of length, for a point transmitter and a separate point receiver; then use the result as a factor in f(s) for the integral I. To start with, we must determine the distribution function, due to diffusion. For the moment, we shall assume that the ionization is formed at rest along the line (the consequences of this approximation will be discussed later). For each electron freed at a point at time t = O., there is a probability density (Ref. 17) _ r2 C =(47r Dt)-3/ e (46) ifor the electron to be at time t at a point r distant from the point of ionization, where | D is the diffusion coefficient. Actually, as the meteoroid moves with velocity v, there will be a succession of such electrons originating at different points at different times.

THE UNIVE RS ITY OF MI CHIGAN 2673-1-F meteoroid, the origin being the meteoroid's position (the trail extends over positive values of z). If an electron originated at a time t in the past, it must have then been at the point (z = vt, p= 0). Hence, the electron density due to it at the point (z,p ) is 3/2 _(z-vt) +l C (47r Dt)-3/ 4Dt'(7 We have stipulated before that in traversing a distance dz vdt the meteoroid produces a free electrons. To find the electron density due to the meteoroid, we must integrate over the path covered, vdt, for all past time: 2 2 OD~~~~ (z -vt) +p -3/2 N a v dt (47 Dt) e 4Dte (48) 0 1/2 2~2 -1/2 - (v/2D) [ (z2+ -Z (Cl v/4 7r D) (z + p 2) e upon application of Reference 18, for instance. In reality, the trail originated at a finite time in the past, T. The error incurred in taking the upper limit infinite is 00~~~~~ (-vt)2 + P2 -32 | N = a v 7 dt (47r Dt) e 4Dt T r [(z2+~2)/4DT]'1/~ < 271/2 -1/2- (vz/2D)_(v2T/4D) -U < (2/ ) (a v/4rD) (z2 + / 2) | du e u 0/ < (a v/47rD)(z2 +? 2)-1/2 - (v/2D)[ (vT/2) - z] < (a v/40 (z + Pe (49) 1/2 so that for the approximation to be valid it suffices that vT/2 > (z2 + p2) At a later stage of the calculation, it will be shown that contributions to f(s) from values of? > Pc 46

TH E: UNIVE RS ITY OF MI CH IGAN 2673-1-F can be neglected for the situations of interest, where )o2cnbe kept below Xr/7r and l | xr'1/ r. Hence., wi thin the li mi ta tions of the sub sequ ent d iscu s sion., E quati on (4 8) c an be| used if the trail extends as far back again behind the furthest point of interest as the. latter is behind.the meteoroid. We now seek to determine the attenuation factor f( z) e dT pdp N ed d dp N. (50) 0 0 ~0 The denominator is simply a, the electron density per unit length of trail, as expected. For the phase, for fixed z we can express the range to a point in terms of the range| to the axis and the distance off the axis, i.e. r(p) = r(0) - p (51) where p = jO(~cosT +Ysinf ) (51a) Temporarily taking the origin at the point on the axis, r (0) and r' (0) are given by Equation (14), and we find |.2()= r2 (0) + p2-2r(0) psin8(cos0coskP +sin sinT ) (52) r'2(p) = r'2(0)+ p2-2r'(0) p sin0' (cos~' cos Vt + sin' sin ~ ) so that (dropping the zeros for convenience) Kg = (27r/) { r [1 - 2(P/r) sin O cos (,' - ) +,2/r2 ] 2 1/2 ~(53) r' [1 -2(p/r') sin0' cos(p -0') + p2/r ]. 47

THE UNIVE RS ITY OF MI CHIGAN 2673-1 -F Break up the o - integration in Equation (50) into two portions: 0 - p < Pc and PC < /p < co. Take Pc small enough so that the first two terms in the binomial expansion of the radicals suffice for p o Pc2 i. e n Kg= (27r/ {r+r' sinOcos f-0)+sin@'cos OP -P1 (54) p With a little trigonometric manipulation, the coefficient of p can be rewritten in the | form sin 0 cos (, -0) + sin o' cos ( W -p')= ~ cos ('' -, ) (55) where }:= ssin e + sin 0' + 2 sin o sin 0' cos (0 - p') (55a) and tan ~: (sin o sin ~ + sin O' sin p')/(sin o cos 0 + sin 0' cos 0'). (55b) Instead of using the azimuthal angles, Y can be expressed in terms of the scattering| angle/ - cos: = r'rr'l = cos O cos 0' +sin 0 sin O' cos(p-0') (56) namely 2: 4 cos2 (f/2) - (cos O + cos 0')2 (57) The angular integration can now be carried out, just as in Herlofson's case:| f 2iKg i(2-r/),)(r+r') 2 -i(27r/k), )o cos (''- ) d+ e:e dI e 0; (58) i(27r/M(r+r') e 27r Jo (27r l / k) Thus, the attenuation factor can be written as L ~~~~~~48

TH E UNI VE R S ITY OF M I CH I GAN 2673-1-F vz/2D OD -1/2 +/2D) (Z 2 +.p 2) 1/2 f(z) (v/2D) e p dp (z 2 +)02) e Jo (2ir 6Yp /;k) 0 (59) 2?r OD 1/2 vz/2D 2 2 -(v/ 2D) (Z2+))2 (v/47rD) e d4f d p (z +,p e e 0 PC i(27r/;k)(r+r 27r OD -1 (Z2 21/2 + (v/4?rD) evz/2D e- dy p dp (z2+p2) /2e-(v/ 2D) +P e iKg(s,,p 0 PC can be chosen sufficiently large, subject to the earl' ition the Provided that PC ler cond $ last two terms can be dropped. Before deciding this point,, we must know the first term. If we apply to the first term the coordinate change 2 + p2)1/2 t (2ir/)L) X (z (60) it can be evaluated as a Laplace transform (Ref. 19) vz/2D -1/2 (v/2D)(Z2+.p2) 1/2 f(z) (v/ 2D) e P dp (z2+p2) e- Jo (27r iK A) 0 vz/2D co -V>,Lt/47rD Y 1/2 = (v;k/ 47rD a) e dt e J0 49 2 W2Z2/ A5 27r x z /)L (61) -1/2 -(27r/X) Jz (vX/47D'') 2 + 1 1/2 + (vz/2D) = (v)L/47rDJ) (0,/471) 2f)2 + 1 e In all cases of interest, the expression simplifies because (vX,/4-rDX)>,> 1 (taking typical

TH E UNI VE R S ITY OF M I CH I GAN.2673-1-F even for a very unfavorable combination of parameters, this. lower limiting value will not exceed 4 cm.) Hence, upon expanding the exponent 2 2 -4-r it Dz / vx Az) e (62) An upper bound on the last two terms of Equation (59) can be obtained by replacing the integrands by their absolute value: 27r co _1P 2 vz/2D -(v/2D)(z4p2)1/ 6 Az) (v/470 e d tf p d p (Z2+p 2) e 0 PC 1/2 2 2 (63) e -(v/2D) (z + P) -ZI In order for these terms to be negligible,, it is necessary that the exponent of Equation (63) be large and that it be larger than that of Equation (62). Solving the resulting in-.equalities for Pe and recapitulating the previous condition, we have -;kr 2 > > (4&/v2) + (4Dz/v) (64) c 2 Ar (4?r I'Dz/VX)2 1 V>j2 >> + (27r D/ (65) c Even for the least favorable situation, the first term on the right side of Equation (64) is -4 2 less than 10 m; the second term taking z as. large as 20 km, is no more than some 2 100 m Even at the minimum range., Equation (64) can be satisfied provided only that X be more than 3 mm. In Equation (65), the second term in the bracket is small compare with the first (see comment below Equation 61). The remaining inequality becomes

T H E U N I VE R S I TY OF M I CH I GAN 2673-1-F For the typical values given above, with the most unfavorable range and trail length, this reduces to X >> 0. 5m; for the unfavorable set of parameters it is X >> 2m. Actually, the restriction implied by Equation (66) can be further limited. A large value of z can have such a strong attenuation fac tor connected with it as to be of no interest. A very safe criterion for the maximum relevant z is a value for which the attenuation exceeds the amplitude reduction from specular to non-specular scattering by, say, a factor of 10. Thi's reduction (taking Equations 36 and 41 with cos 0 1 for simplicity) is -1 1/2 (Snon-s/Sdl = (2r) ()L/r) (67) Thus- the maximum z we need consider will be given by -4r2 2f 2 Dz/V)L2 1/2 e (2 010 Ck/ r) (68) which yields 2 Z (4/47 D ln 207r - (1/2) ln X + (1/2) ln r (69) As X decreases$ Zmax decreases faster, so that the sense of the inequality in Equation (66) gets reversed, i.e. if Equation (69) is substituted into Equation (66) there results 44 7r 2 r ln 2ft - (1/2) ln X + (1/2) ln r -2 (70) which reduces to > 4C.4C (42 /6) r (701) Thus for any z of conceivable interest, f(z) will be given by Equation (62) provided only that (4(2/6) r >-,v )L >> 47r a( D/V (71)

TH E. UNI VE R S ITY OF M I CH I GAN 2673-1-F As a special case of Equation (62)., if the point on the axis happens to be the specular point., Z( simplifies and 2 2 2 -167r (Dz vX) cos (P/2) Az) e (621) This expression was obtained by Eshleman (Ref. 11) by a slight extension of Herlofson's calculation (Ref. 16). Herlofson's own result applies to the further specialization of normal incidence with specular scattering: 2 -167r Dz/vx A Z) e (6211) 6. 2. 2 - Scattering by the Trail Equation (62) specifies the amplitude attenuation due to ambipolar diffusion in a plane perpendicular to the trail axis (within the wavelength limitations specified). The effect of this attenuation upon the return from the whole visible trail remains to be examined. If we assume that the specular point z zo lies within the visible. region for sharplydefined overlapping beam patterns., and if we measure s from the specular point the pattern factor becomes now -47r 2 e 2Dz/vx -47r2CY2Dzo/VA! -4v2 2Ds/VA! As) e e e (72) Inste ad of Equation (30), there occurs 2 2 h i?r R+R1 _47r2 jr2Ds/ 0 iK -47r 2 ),2 Dzo/0 _F 2 s sin 0 I, e e ds e X RR' e -h (73)

TH E UNI VE RS ITY OF MI CHIGAN 2673-1 -F We next make the changes of variable =~~~~~~~~~ U = 2 f(R+R')/2)LRR']' s sin 0 (73a) 1/2 H = 2 [(R+R')/2kRR'] h sin O (73b) 1/2 2T A = [(R + R')I/2X RR' ] v 2 sin 0/2 2D (73c) so that Equation (73) is rewritten as H iK -42 ~2Dzo/VX2 1/2 )2 U/A Ix = e e sc DXRR/2(R+R) cscO ) du e e. e -H i r/)2-A (73') In this form, the various parameters that affect the integral are lumped into the two combinations H and A; it will be convenient below to refer to values of H and A rather than run through various combinations of the larger set of parameters. The integrand can be put into a standard form by resort to the complex variable w = u + i/rA. (74) I Then, H i(7r/2)u2-u/A i/(27rA2) H+i/7rA i(r/2)w2.du e =e dw e -H +i/MrA (75) i/(27rA2) i (7/A)A i(7r/2)w = e dw e + dw e 0 0

THE UNIVERS ITY OF MICHIGAN 2673-1 -F The last integrals are Fresnel integrals of complex argument, in terms of which| Equation (73') becomes (76) i K+(1/2rA2 _4X2 62 DZo/VX2 [R'(76) I I= e e )AR 2(R+R ) CSCO [F(H+i/7rA)+F(H-i/7rA)]. The introduction of f(s) from Equation (72) also changes Equation (32) to iK -4X 7'2 Dzo/Vo2 1/2 i(Tr/2)H2 I2 = e e [ RR'/2(R+R')7 csc 0 (2i/TH) e cosh (H/A). (77) The expansion of the Fresnel integral given in Equation (31) is valid for complex argument, U., provided that -7r/4 < arg t/ < 3r/4 (78) as can be verified, for instance, from the confluent hypergeometric function representation of the Fresnel integral (Ref. 20). Equation (78) is satisfied for both Fresnel integrals of Equation (76) provided that r A H > 1. (79) In that case, Equation (31) yields 1/2 ir/4 F(H + i/7rA) + F(H - i/7rA) = 2 e (80) i(7T/2)H2_i/27rA2 -1 -(2i/7r) e [H2+(7rA) [H cosh (H/A)+(i/7rA)sinh (H/A)] if terms beyond the second in Equation (31) are neglected. Unlike the situation in the absence. of diffusion, addi-ngEquation (77)1to Eqaioni (76;) doesq -not nowqi u cncel- the5

TH E UNIVE RS ITY OF MI CH I GAN 2673-1-F second term in the expansion of the Fresnel integral. The condition that the remainder be at least a factor (2,r) smaller than the first term of Equation (80) is expressed by i (7r/2)2 -1?2.1/2 - | | 2A e 1[+(7rAH)2] [sinh (H/A)+i(7rAH) cosh (H/A)] -, 2 /(2r) (81) | We must re-examine the requirements imposed by the sequence of integrations by l parts and its truncation. We now have G'(g) = d f = 1 d f = f (82) dg g' g' ds g' (g') so that Equation (25) is replaced by G'(g) | = Ig" + g' (-f'/f)} = 1 + (H/A) (83) KG(g) K(g')2 7rH and we now stipulate that (TrI~) [ 1 + (H/A)] ~ (2fr)-1. (84) Equations (81) and (84), each set a lower bound on A, expressed as a function of H.| Inasmuch as this bound increases with H in the first case and decreases with H in the second (in both instances monotonically),I. the minimum acceptable value of A is obtained| by requiring the equality to hold in each case and solving simultaneously. The result is A 1, H 2. 8 With these values, the neglect of the higher order terms in the expansion of the Fresnel integrals in Equation (80) is justified. Furthermore, the phase (27rA2)can be omitted. The combination of Equations (76) and (77) then leads to 1 ~~~~~~55

TH E UNI VE RS ITY OF M I CH I GAN 2673-1 -F i [K+ (r/4)] -167 (DZo/VAF)cos (,/2) 1/2 I=e e [ RR'/(R + R1 csc O (85) which is just the stationary-phase value. The discussion so far has left out of consideration the contributions of the end-point| series. Because of the exponential factor introduced by consideration of diffusion, the front-end contribution ("head echo") is now enhanced relative to the specular value and may become competitive with it in magnitude. It yields (86) iKg(a) _472rz 2z VZo/V2 -4vr22DaI/VX2I I'= i [XRR'/27r(R+R') a sin2 0] e e e iKg(a) -167?(Dzo/VA2)CoS2(p3/2) 1/2 -1/2 1 L/A = -i e e csce 2 (rL) e where L -2 [ (R+R)/2XRR'1 a sin. (86a) The validity of Equation (86) is limited by the fact that it represents the first term in a sequence of integrations by parts. Corresponding to Equation (84), there is therefore the requirement that -1 -1 (7rL2) [ 1 + (L/A)] 4 (2r) (87) for Equation (86) to be a good approximation. The relative magnitude of the head echo as against the specular return is L/A I'/II 2 1/2 rL)-1 L (88) The minimum value of this expression for a given A occurs for L - A. Hence, if the ratio is to be less than (27r)-1 for any L at all, A must be at least 21/2 e. If it is to 56

g ~TH E UNIVE RS ITY OF MI CH IGA 2673 -1 -F remain that small over a significant range of L, a larger A is required-. Obviously, the| |larger the L that has to be considered, the greater A must be before the head echo can| |be neglected. At the other end, Equation (86) alone suffices provided that the ratio of l Equation (88) is greater than 2,ff and Equation (87) is satisfied. It should be noted that, if| |these conditions are met, Equation (86) can be used for A < 1, since for smaller A the| specular return will be less than the value given by Equation (85). If Equation (87) is satisfied and the ratio of Equation (88) has an intermediate value, we need the sum of l Equations (85) and (86):l -167r2(Dzo/VAF)CoS2(p/2) 1 Itot =e [MR'(R+R') csc 0 (89) [i [K+(1/2rTA )+(1r/4)] iKg(a) 212(r)- LA Equation (89) holds up to. terms smaller by a factor (2,ff)-1 with no other restriction on A| than Equation (87); the additional constraints due to the limitations on Equation (85) can| |be consistently relaxed because of the contribution of the head-echo term. The realm of| validity of Equations (85), (86), and (89) in terms of values of A and L is displayedl graphically in Figure 6-1.| Let u's next consider the case where the front end of the visible trail i's located at the specular point (if this front end is defined by the meteoroid, and not by a beam cut-off,| zo is zero). Ins tead of Equation (76) the re now occursl T. i C K+(1 /2,rA2)] -167r2(Dzo/vA2)COS 2(B/ 2 )YW 1P4DIl 1/2 Al. 01,,A1

THE UNIVE RS ITY OF MI CHIGAN 2673-1-F A 10 6 6 i + 8 | E Lo Eq. 85 Eqo 86 Eq. 98 2 Eq. 89 8Eq..~~~~86.6 Eq. 103 ~4 Eq. 103 —.2 l l l | 1~~105\| l l l l l l l -4 -3 -2 -1 0 1 2 3 4 5 6 7 FIG. 6-1- REALMS OF VALIDITY OF EQUATIONS| 58.

TH E UNI VE RS ITY OF M I CH I GAN 2673-1-F and instead of Equation (77) (91) 2(p M -16w(Dzo/vx)Cos /2) 1/2 i(r/2)1? -H/A I2 e e XIM 2 (R+R Csc0 4/70 e e We can now always choose H sufficiently large that the first Fresnel integral ca'be replaced by the first term of its asymptotic expansion and that the contribution of Equation.(91) can be neglected (For small A, the negative exponential predominates strongly enough to insure this despite the near-cancella'tion of the two terms'n Equation 90). Thus, we have (92) i -l67r2(Dzo/vX2)cos 2(P/2) i7r/4 I = e e XRR1/2(R+RI)3 csco _1P e -F(i/7rA For A sufficiently large, the argument of the Fresnel integral in Equation (92) is small, and the Fresnel integral is approximately equal to its argument. In order for the higher terms in the power series to be'negligible to order (2,r)-l we must have A >.67. Then (93) i (K+(l 27rA2 -167r2(Pzo/vA)cos 20/2) 1/2 ir/4 I = e e XM I/ 2 (R+R csco _1P e -i/7rA If A > 2 3/2 the contribution of F(i/7rA) can in fact be completely neglected, and the 2 phase factor (2?rA can be ignored, leaving i[K+(7r/4)) -167r2(Dzo/vAF)Cos 2(P/ 2) 1/2 I= (1/2) e e;kRRI/(R+Rl) csCV (94) which, except for the negative exponential factor, is the value obtained without consideration of diffusion.

THE UNIVE RS ITY OF MI CHIGAN 2673-1 -F At the other end, for small A the asymptotic expansion holds for the Fresnel| integral in Equation (92), namely _~p2 i~r/4 -i/2fA2 -i/27rAl F(i/,rA) = 2 e -A e - iwrA3 e +.... (95) If the third term is a factor (2~r)-l smaller than the second, i.e. if A C 2-0/27r-1 Equation (92) can be reduced to iK -16~2(Dzo/vX2)cos2(p/2) I = A e e XRR'/2(R+R' csc 0 (96) This is the value that would be obtained by pulling the phase factor out of the integral. | It thus represents a situation in which the negative exponential dominates the behavior | and overrides the phase. So far, the discussion has covered the front of the visible trail being well ahead of| the specular point (i. e. Equation (87) being satisfied) or being right at it. Next, we cover the intermediate situation. This involves adding to Equation (92) K -167r2(Dz~/vX2)cos2(13/Z)/ (/2 0 i(vr/2)u2 -u/A I' = e e A 2(R+R csco du e e (97) | [K+(l /2rA2)] -1672(Dzo/v)~) CoS:(,/2) l. = e e )[RR'/2(R+R')l csco [F(i/OrA) -F(-L+i/7rA to obtain i(K+(1/27MA)3 -167 (Dzo/ X)o2,/2) 1/2 r iv/4 I = e e >RR'/2(R+R'e csc8 2-1/2e -F(-L+i/erA) (98) or alternatively, since F is an odd function, i +(1/27rA2)] -16X2(nZo/Vk2)eos2(f~/2) l2 _/ ~/ | I = e e tA~~~~[RR'/2(R+R')] csc0 [2-1/e +F(L-i/rA)].* 60

THE UNIVERS ITY OF MICHIGAN 2673-1 -F A n exp re s sion re se mbling thi s i s appa rently u se d by Lo ewenthal (Re f. 2 1), though thel approximations he made in obtaining it are not sufficiently justified, and multiple typographical errors obscure the results. For L sufficiently large, or A sufficiently small, F can be replaced by its asymptotic| expansion. As the criterion for the usefulness of this expansion we impose the requirement that the magnitude of the third term be no more than (27r)-1 of the second, whence L2 + (rA)-2 ~ 2. (100) If 7rAL ~ 1, Equation (99) then yields -1 67r(Dzo/vX2)cos 2(P/ 2) H I =e oARR'/(R+R cscO i[K+(1/27rA2)+(a'/4)] i[K+(7r/2)L2] -1 L/A A(l te -lie 2 1/2 (L-i/7rA e J (101) l which looks just like Equation (89) except that the phase factor of the second term has been expanded; and (L-i/,rA) appears instead of L in the denominator. If 7rAL > 27r, as it must be whenever Equation (87) is satisfied, this latter difference can be neglected. Thus, Equations (89) and (101) are mutually consistent. The specular term cannot dominate the head-echo term by a factor 27r for any value of the parameters for which Equation (87) does not hold. On the other hand, the special case does occur for which the head-echo term dominates by a factor 27r, i. e. [L2+(z.A)_ Z1-/2 L/A 23/2 2 When this happens,..... 61..........~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

THE UNIVERS IT Y OF MICHIGAN 2673-1 -F i[K+(r/Z)L2] -16r2(Dzo/Vk2)cos2(p/2)' 1/ -1 L/A I =i e e [XRR'/2(R+R')] csc0 [T(-L+i/rA)] e (103) If 7rAL < 1 and Equation (100) holds, the asymptotic. expansion can be applied to thel Fresnel integral of Equation (98). When this is done, Equation (103) emerges again. Pushing on further, if |rAL, (2)-1, (104) Equation (103) simplifies to | i[K+(7/2)L2] -16'r(Dzo/v2cs(/)1 L/Al I = e 2 (R+Re csc0 A e. (105) In Figure 6-1, the realm of validity'in terms of values of A and L of the successively| more general Equations (105), (103), (101), and (98) is displayed graphically. |The extension to the front of. the visible trail falling somewhat short of the specularl point is now very simple. Equation (98) applies with L negative. If Equation (100) is satisfied, this reduces to Equation (103). If in addition |rA ILI 4 (2,r)-!, (106) it reduces further to Equation (105). On the other hand, if (7rn2)-! [1 - (L/A)| < (2r)-1, (107) Equation (86) can be used. The realm of validity of the various equations for L negative is also depicted in Figure 6-1. For very narrow beams including the specular point, there would occur the generalizationl |of Equation (45)l. he z~6 i i i i L ii iU

THE UNIVERSITY OF MICHIGAN 2673-1-F i K+(1 /27rA 2) -1672(Dzo /VX)CoS2(p/2) I e e XRR 2 (R+R CSCO F(L+i/7rA)+F(L'-i/7rA -08) where Li 2 (R+R')/2;kRRI b sin 0 (108a) and the F's may or may not be expandable. Wherever in the foregoing the exact Fresnel integral of complex argument has to be resorted to numerical values for it can be obtained from Reference 22. In that table, the function computed'is 00 (z) 27r -1/2 eZ2 e dt (109) z In terms of this function, the Fresnel integral is given by i(7r/2)U2 F(U) = 2-1P e'7r/4 1 - e 7r/ 2) e-i7/4 U (110) For completely non-specular scattering, we parallel closely the treatment for f 1 above, except that the negative exponential reduces the contribution from the back end of the trail relative to that from the front, thus eliminating the first of the two terms in Equation (37) and with it the phase difficulty. Instead of Equation (38)., there now occurs 2 -ef 2 2 iK(a) -47r Dz(a)/vx I i X/27r (cosO + cosO e e The validity of this expression is limited by the requirement G11 (112) 2 KG(g) K(g 27r As the first term of Equation (112) cannot threaten the inequality for non-specular

THE UNIVERS ITY OF MI CHIGAN 2673-1-F (Kg') (-f [ ~X/2r(coso + coso 4] [ 2D/vX!] < (2r) l (113) or A> 4xr2?2D/v (cos e +cos e') (114) If we set D =4 m2/sec, v = 40 km/sec, and ignore the angular factors (both in 2 and in the denominator), this requires A > 1. 6 cm. The fact that Equation (1 11) holds down to quite small wavelengths is not surprising.l Away from the specular point, the oscillations of the phase factor become more violent and| are able to override a stronger damping factor in dominating the characteristics of the scattering. Having obtained results for the scattering by the trail in various positions, we can now| relate these results to map out the intensity variation as the trail moves across the field of view. It has been assumed in the literature until now that, if the specular point is in the field of view, this variation is given by an exponential decay beyond the specular point (Equation 85). As we have seen, this is true only in the large A (i. e. large wavelength limit. As A decreases, the variation becomes less strong than the exponential, and in fact reaches a constant in the limit of small A. To illustrate this behavior, Figure 6-2 exhibits the scattering intensity as a function of L for various A's; for convenience, the received power is expressed in units of the no-diffusion specular return (Equation 35). For purposes of quick orientation, Table I lists the A values corresponding to some representative situations. In part a), specular backscattering is considered; in part b),..... 64......

TH E UNI VE RS ITY OF MI CH I GAN iS/So 2673-1-F 1. 0_.8 —.6- A=5.4-.2_.A=2 ~ 08 A*,.06A=.4 e 04 A-'-.02 —. ~~~~~~~~A =.6.01~008 - ~006~004- ~ A=.i..001-1 ~~02 L4 FIG.6-2 REATIVE SCATTERING INTENSITY

...THE UNIVE RS ITY OF MICHIGAN 2673-1 -F Table I Values of A as a Function of Range (R) and Wavelength () (assuming v = 40 km/sec, D = 4 m2/sec) a) Specular Backscattering R(km) 10 8 6 4 2, 1 90 13 9. 6 6.2 3.4 1.2.42 200 8.9 6.4 4.2 2.3.80.28 300 7.3 5.2 3.4 1.9.65.23 600 5.2 3.7 2.4 1.3.46.16 1000 4.0 2.9 1.9 1.0.36.13 b) Specular "Forward" Scattering (for a horizontal trail, at an altitude of 90 km, coplanar with the transmitter and receiver) m) R(km 10 8 6 4 Z 1 90 13 9.6 6.2 3.4 1.2.42 200 19 14 9.0 4.9 1.7..6 300 23 16 11 5.7 2.0.72 600 26 19 12 6.7 2.4.84 1000 24 17 11 6.1 2.2.76. Transmitter-receiver separation( / ) along great-circle path corresponding to "forward" scattering ranges: R(km) 90 200 300 600 1000 [ (km) 0 350 530 1200 2000... -6.......

TH E UNI VE RS ITY OF MI CH I GAN 2673-1-F receiver. Typical values, of the parameters v = 40 km/sec and D = 4 m2/sec have been assumed for a trail altitude of 90 km. The reader can easily find A from Equation (73c) for other geometries, different wavelengths and ranges, and alternative values of the velocity and the diffusion constant. 6. 3 Additional Refinements 6. 3. 1 Initial Stage of Trail Formation In the ambipolar diffusion treatment above,'there is implicit the assumption that the ionization is formed at rest at the site of the impact. Actually, the high velocity of the meteoroid'Implies that the particles will start with a large kinetic energy, and hence the ionization will initially spread out much more rapidly. This first stage of dissemination (until the ionization is slowed down to thermal velocities) is very brief, so that effectively the ambipolar diffusion starts from an instantaneous distributed source instead of a point source. The longitudinal component of the initial penetration (along the axis of the trail) -primarily just displaces the head of the trail relative to the position of the meteoroid. The transverse component has been studied b Opik (Ref.. 23), who first pointed out this phenoy menon., He obtains a Gaussian distribution, and quotes the results of numerical calculations of its width for different meteor velocities and compositions; the width is inversely proportional to the atmospheric density, and hence is highly altitude-dependent. To account for this effect 1 e the instantaneous point source expr ssed by Equation (47) 1- A - f,.2.1 -.1 P.1- -1 AA A I A

THE UNIVE RS ITY OF MICHIGAN 2673-1 -F and then integrating with respect to P and 11 I over the normalized distribution function )2 (-ra2) -1 e - (pl/a (116) Thus instead of the C of Equation (47) there will occur OD -(PI/a) 2 27r - Z-vt)2 2+,P!2-2pp'cos V1/4M 2 -1 I +Y C' = (4rDt)_3/26ra PUP e d e 0 0 (117) The integral is similar to Equation (58) 27r PP'cos T 2Dt d4i e 27r Io pp 1/2Dt) (118) 0 The remaining pI integral is Weber's'integral (Ref. 24) 00 12 1/ 2) - P a + (1/4Dt p'dp e Io ( pp I/ 2 Dt) 0 (p/2Dt)2 /4 [(I /a2)+(1/4Dt I 1 + I e (I 1 9) 2 4Dt a2 so that 2 2 2 _(z -vt) /4Dt - P (4Dt4-a CI 7r (47rDt)_'/2 (4Dt+a2)-' e e (120) With C' replacing C, the computation of f(z) proceeds along the same lines as before. Subject to the additional condition c>> a (121) which can be easily satisfied for Opik's values of a, the analysis i's essentially identical. ('122) The attenuation factor is now Go 00 2-r 00 00 7r

THE UNIVERS ITY OF MICHIGAN 2673-1 -F It is convenient to defer the t integration. The q integration in the numerator has already been performed (Equation 58), and the subsequent/p integral is again Weber's integral (Ref. 24) | -p/(4Dt+a2) _2 a2 (4Dt+a2) /A Jpdp e Jo(27r arp /k) = (1/2)(4Dt+a2) e (123) 0 The p integration in the denominator is trivial: pdp e -P/(4Dt+a2)= (1/2)(4Dt+a2) There remains co _r2,2 a2/X2 f -1/2 -(z2/4Dt)-(v2t/4D) [ I + (4?r ID/Vx)2] e dt t e f(z) ~ (1z5) 00 -(z2/4Dt) -(v2t/4D) fdt t-~/ e 0 These integrals are given by Reference 25: Go -(z /4Dt)-(v2t/4D) [1 + (47rX D/v)2| dt t-l/2 e (126) 0 o.~~~~~~~~~~~~~~~~~~~~~~~~~~~~1/2 ~0 1/2 -1-1/2 -(vz/2D) [1+(47r-D/v02]k = (4,rD) v 1+(4,r D/vA) e ao, -(z2/4Dt)-(v2t/4D) 1/2 -vz/2D dt t- /2e = (47rD) v-1 e (127) 0 Hence -r22a2/k2 -(vz/2D) { [1+(47r D/vX)2] 1 1 f(z) = e e (128) -i2 y2 a2/2 _402 42 Dzl2 VX -e e 69...'

THE UNI VE RS ITY OF MI CHIGAN 2673-1-F |here, in the last step, the integral has been expanded justaswasdoneingoing from Equation (61) to Equation (62). As a result of the rapid initial spread, there is thus a uniform reduction of the return| all along the trail, compared with the ambipolar diffusion result. Alternatively, if we use the ambipolar diffusion treatment we must re-interpret a to be an effective electron density,, related to the -true electron density a by ao a:~~< ao a/e2 (129)l The exponent is a function of the wavelength, orientation, altitude, and composition of the meteoroid, and can be large for parameters of interest. 6.3.2 - Antenna Patterns The effect of different antenna patterns can be visualized rather simply. Apart from the case in which the front end of the visible trail is near the specular point, the scattering amplitude'is in general a sum of two bLerms - one of which represents the contri-| bultion of the specular region (if it lies within the overlap of the beams) and the other that of| the front end of the visible trail. Provided that the lumped pattern factor does not have any discontinuities (or near-discontinuities), except for the cut-off at the ends, the only modifi|cation that is necessary is to multiply each of these two terms by the value of the patternl factor at the relevant point (specular point or front end). When the front end is near the specular point, a variable antennae pattern factor still presents no problem as long as the variation is small over the distance between the two points (i. e. as long as a Taylor expan-.....70....

THE UN'IVE RS ITY OF MICHIGAN 2673-1 -F of integrals). In effect, it would take antenna patterns fluctuating at a rate comparable with the phase oscillations to change the picture qualitatively. For any less pathological case, |the adjustment of the results to take into account variation in the antenna patterns is very | simple - provided that the overlapping patterns are adequately mapped and there is a means of fully locating the trail within them. The question of the significance of the sharp cut-off is a trifle more subtle. If the | meteoroid is within the beams, it provides the cut-off and there is no problem. If the an-ntennae pattern has an abrupt edge, this edge takes over as the cut-off for the visible trail as the meteoroid crosses it. If there is an area of decline before the edge of the beams is reached, this decline will be reflected in the contribution of the front end as the meteoroid approaches the edge, and the edge will still take over as cut-off as the meteoroid crosses it. On the other hand, if the pattern tapers off gradually and there is no sharp boundary I (e. g. a Gaussian), the meteoroid remains indefinitely in sight, providing a continuously attenuated front-end contribution. Of course, if the meteoroid burns out during the observation, the cut-off is quite unambiguous. We see then that the gradualness with which the.pattern terminates has no effect as long as the meteoroid is -squarely in sight,, but can re|duce the head-echo contribution after the meteoroid has passed into the boundary region. l lFor small A (small wavelengths), this has no importance because diffusion does not allow| the return to persist long behind the meteoroid. For large A (large wavelengths), it leaves the specular return unchanged, but affects the behavior of a non-specular return subsequent

TH E UNI VE RS ITY OF MI CH I GAN 2673 -1 -F 6.4 Meteor Velocily Measurements The predominant technique for measuring the velocity of radio meteors is based upon the distance between maxima of the diffraction pattern, on an amplitude-time record, Since only those returns that show the initial rise of Figure 6-2 are used, the orientation of the trail is essentiallyspecular. The measurements are made for backscattering (For "forward" scattering, the sin 0 factor in L would introduce some ambiguity - 0 i's not completely determined by the scattering angle.) In interpreting the data, the Fresnel integrals of real argument (corresponding to * = 00) have been used. It is of interest, therefore, to examine the effect of taking a finite * upon the validity of the results (i. e. to determine to how small a wavelength they can be trusted). For this purpose., Table II lists the locations of the successive maxima in the diffraction pattern for various A's. it turns out that the location of all the maxima but the first i's very little affected by the value of A, and that the error incurred in using the values obtained for the Fresnel integrals of real arguemnt is always less than one per cent. Thus, the technique'is theoretically applicable to small wavelengths without modification. The onl limitation is the practical one that the oscillations damp out with decreasing wavelength, and ultimately become indiscernible. 6.5 Meteor Trails as Interference to Radio Communication via the Moon In Section IV, the data on the return from the moon at different frequencies havp. hP.P.-n enrrP.1atPd_ Tn thp, nrpspmt spntinn_ tbp. rpturn to bp. P.,xnPrtPO frnm a mptpnr

THE UNI VE RS ITY OF MICH IGAN 2673-1-F Table I| Location of Maxima of Diffraction Pattern A Li L L3 L4 L5 L6 00 1.23 2.33 3.07 3.67 4.18 4.64 5 1.13 2.31 3.06 3.66 4.17 4.63 2 1.02 2.31 3.06 3.66 4.17 4063 1.5 0.97 2.31 3.06 3.66 4.17 4.63 1.0 0.88 2.30 3.06 3.66 4.17 4.63 0.9 0.86 2.30 3.06 3.66 4.17 4.63 0.8 0.82 2.30 3.06 3.66 4.17 4.63 0.6 0.74 2.30 3.05 ll No maxima beyond this point. 73..

TH E UNI VE R S ITY OF M I C H I GAN 2673 -1 -F competition of the two processes. As the most convenient basis for comparison, Table DI lists the value of a (obtained with Table Ia and consequently implying a choice of typical parameters) at which vertical backscattering by a meteor trail will give a return equal to that from the moon, and also the value of a (using a typical value of a from Opik for an 0 altitude of 90 km), for a number of representative frequencies. Various lines of argument 12 show that the underdense-trail formalism is valid up to a line density somewhat below,10 electrons/cm, and that- above that value the return continues to rise but at an increasingly reduced rate. It is thus obvious that meteor returns far in excess of the moon returns are possible, and that at the least some form of signal limitation or range gating in the receiver is desirable as protection against them, unless a frequency above 500 Mc i's used. The exponential due to Opik's correction has the effect of sharply suppressing meteor signals at higher frequencies. The other question of interest is how often an interruption of radio communication via the moon can be expected because of meteor trail interference. For this purpose, we refer to a Stanford study of meteor rates (Ref. 26) carried out at 23 Mc with a fan-shaped beam. They quote a maximum count of 5000 meteors per hour for equipment which (on the basis of the Lovell-Clegg formula,.which is adequate for this low frequency) can detect trails down 8 to an electron density of 10 cm. They also found that the count varied linearly with the transmitter current. This implies that the number of meteors varies inversely as the cutoff value of electron density. Neglecting any effect of a smaller beamwidth'in reducing the

THE UNIVERS ITY OF MICHIGAN 2673-1-F Table HI Meteor Trails which Scatter as Strongly as the Moon wavelength 10 m 3 m 1.5 m 1 m 75 cm 60 cm freq. (Mc) 30 100 200 300 400 500 78 8 99 9 a 2x107 1.4xl 08 4xl 08 1. lxl9 2x109 4x1 09 7 8 9...11 * 0' 15 la~ 2x107 2.3x10 4x10 1.3x10 1 lx0l 3x10 0~~~~231 maximum number 20 000 2000 140 4 < 0 per hour 20 These values correspond to overdense trails, so that the use of the equations above to obtain them is not adequate. Actually, a higher electron density is necessary to yield the desired signal strength. 75

THE UNIVE RS ITY OF MICHIGAN 2673-1-F We conclude that meteor trails need not be considered for any moon communication system operating significantly above 500 Mc..... 76...l....

TH E UNI VE RS ITY OF MI CHIGAN 2673-1 -F SECTION VII COMMUNICATION CONSIDERATIONS The previous chapters have given detailed consideration to the conditions which | affect communication between points on the earth's surface by using the moon as a par as itic an tenna. The c ons ide rations help us to dete rmine the p aram ete rs which oc cur| in the design of a communication system. Al though th ere i s va ri ation i n the p ul se -to -p uls e r etu rn f rom the m oon, the p ul se s | exhibit a relatively sharp rise in the leading edge. As discussed in Section H., the shape| of the pulse behind the leading edge can depend uon the effect of returns from portions| of the moon other than the specular point. The sharp rise in the leading edge of the| pulse is clearly seen when sufficient power is transmitted to bring the return above the| noise, or when pulse integration or superposition is used. The primarily specular nature of the return implies that only a small part of the | lunar surface contributes most of the detected energy. Estimates of the size of the main reflecting area have ranlged from such gross descriptions as less than one-tird of the surface to less than the first five miles of depth. However, the specular portion of the return., which is evident in the leading edge of the pulse., need come from an area only| a few multiples of wavelength in diameter, and the size of this area will depend upon the characteristics of the illuminated7portion of the moon surface. If the specular region is in a large crater,, the main part of the pulse may come from a very small area,

THE UNIVE RS ITY OF M I CHIGAN 2673-1 -F A communication system which takes advantage of the characteristics of the lunar echo need not use the entire return pulse. It is not necessary to wait until all of the Epulse has been detected, since a major part of the received energy (the specular return)| is contained in a small initial portion of the pulse. Such a communication system could, for example, transmit in two alternate frequenc ies. The receiver would detect only the| leading edge of a pulse transmitted in one frequency and then switch to detection of the subsequent pulse transmitted in the second frequency. Los se s due to dens e me teo r trails s houl d b e an inf re quent oc currenc e. Los se s due | to double fade s have b ee n ob se rv ed, but th e f req uency an d du ration of s uch fade s ha s | not yet been established by experimenters. It may be desirable to repeat a message when| necessary to overcome -loss of information through. meteor scattering and double fades.| The frequency of such occurrences will depend on geography and time of year and will | have to be determined experimentally. At high frequencies, they are not expected to be serious. The range of useful frequencies for moon communication is bounded on one side by| liitations imposed by ionospheric attenuation and meteor scattering and on the other | side by attenuation due to losses from atmospheric oxygen and water vapor. Theoretical values for oxygen and water vapor attenuation, as given by Kerr (Ref. 27), show that for wavelengths shorterthan3cm the loss due to these factors islargest. Thus apeak frequency bound is about 10,000 megacycles. Using a value of maximum electron density, 6 N,~~~~~~~~~~~~~~~~~~~ of5x1

THE UNIVE RS ITY' OF MI CH IGAN r ~~~~~~~~2673-1-F| critical frequency, a value of 20 megacycles is obtained. However, a frequency larger than the critical | frequency should be chosen as a low frequency limit because a more conservative esti- | mate is necessitated by uncertainties in' the value of N. Thirty megacycles appears to | b e a reasonable lower limit in order to avoid ionospheric absorption, | A further consideration involved in choice of an appropriate frequency range is that| of the beam width achievable with the antennas available. The moon subtends an are of | approximately 0. 5 degrees from the earth If it were necessary to illuminate the whole| moon., the hig frequency cutoff with the 60-foot antenna would be 2300 Mc., and 6900 Mc with the 20-foot antenna. However,, as indicated earlier., a very much narrower beam | would be sufficient for the observed specular reflection. The approximate relation be- | tween beam width in degrees at the half -power points,, wavelength, and antenna diameter| is given by the well-known formula (Ref. 28 for example),l @= 70.| For antenna diameters of 20 feet and 60 feet, the narrowest obtainable beam widths for| the shortest feasible wavelength, 3 cm, are 0. 3 degrees and 0. 1 degrees., respectively. Us e of 1the narrow be am will, of course., c ons erve pow er since it is not nec es sary to | illuminate the whole moon to obtain a usable echo. Because of the desirability of a| narrow beam., a larger diameter antenna is preferable to a smaller diameter antenna |

TH E UNIVE RS ITY OF MI CH IGAN 2673-1-F shorter wavelength is preferable to a longer wavelength. Moreover, as shown in Section IV, the radar cross section of the moon drops off very little as wavelength| decreases, so there is no drawback on this score to going down to a wavelength of 3 cm for communication. Finally, we note that the Faraday rotation for, a wavelength of 3 cm| would be considerably less than one degree It appears from all this that 3 cm is the best wavelength to use in moon communication. Consideration should be given to automatic methods of tracking the moon. This is | especially needed when small beam widths are employed. A computer operating with a| predetermined lunar orbit and angular tracking rates adjusted for a particular site may be desirable. The limited times when the moon is visible simultaneously from both receiver and transmitter must also be computed (Ref. 29). It is not feasible to achieve partial communication security by narrowing the beam so that the signal reflected from the moon will irradiate only part of the hemisphere of the earth facing the moon. The angle of the transmitted beam which will, upon specular reflection from the moon., completely cover the hemisphere of the earth facing the moon is given by ( dal 0~r2 where d is the radius- of the earth, a is'the radius of the moon, and r is the distance from -the earth to the moon. This beam angle is approximately three seconds of arc., l To obtain such a beamwidth, the ratio of antenna diameter to wavelength would have to be of the order of 105, which is impractical. Even if such a beam width were achievable, 80

TH E UNI VE RS ITY OF M I CH I GAN 2673-1-F The receiver noise and the desired information rate place limits on the useful band width for a communication system., Receiver noise power N is given by the following| relationship (Ref. 30):l \ ~~~~~~~~N k TB(NF -1) g lwhere k is Boltzmann's constant, T is 290 degrees Kelvin., B is the bandwidth, NF is the |noise figure of the reciever., and g is receiver gain.| An upper bound for the maximum usable band width is that the, receiver output noise power be equal to the power of the minimum desired signal at the output of the receiver. The desired signal in this case is moon echo. This minimum signal power is obtained from the range equation: (4xr r| where: Pt is taken to be 250 kilowatts for purposes of illustration,l G for the 60-foot antenna is taken to be 2. 6:x 106, X is 3 cm., or is 4. 5 x loll m2., andl r is 2.5 x 1034m4. The cross section of the moon at 3 cm is computed using a value for JRl=. 047 from l Section IV.,(Fig. 4-1). The minimum signal power obtained with these values is approxi-| mately 1. 4 x 1 0watts.l The band width can now be computed for the representative case in which Smin isL 1. 4x 10-l watts.- We equnateq g t inodroderiethbawdthIn this L

TH E UNI VE R S ITY OF MI CHI GAN 2673-1-F calculation, a nois e figure of 1 1 is taken to be representative of receivers working in the 101 000 megacycle range. The allowable band width is then computed to be 350 Mc. For a reliable moon communication system a signal-to-noise ratio of 40 db would be a more likely value. This choice would result in a decrease in the band width to 35 Kc in the above illustration. Since the effective pulse length is approximately the reciprocal of the band width, the shortest pulse length permitted by these considerations would be 29 Pt sec. If the sun is in the radar beam, solar noise, in addition to the receiver noise is superposed on the signal. For a quiescent sun, this is mainly thermal noise. (Pawsey and Bracewell. Ref 31). The band width is then min B = k F -1) To+ a Ts where: = sun temperature 0000 K for X=3 cm) TS a = a factor which depends on the radar beam pattern and positions of the sun and moon in the beam. For the narrow beam recommended in this report., a should be so small that B is essentially unaffected by solar radiation. For a very wide beam receiver this would not be true, and solar radiation could become of the same order of magnitude as the receiver noise. The required information rate will depend upon the use to which the moon communication system is put. For example, the message rate for a 16-channel military teletype

TH E UNI VE RS ITY OF MI CH I GAN 2673-1-F pulse length for all 16 channels transmitting simultaneously would be 1300 microseconds. Meteor communication systems acquire information at relatively slow teletype rates., and store the information for transmission at about 100 times the acquisition rate when conditions for transmission are right.'Ihis transmission rate may be 10, 000 bits/see. The maximum allowable pulse length for such a. system would be 100 microseconds. The 29Tnicrosecond pulse length of the example of the previous paragraph would be obviously more than adequate for either type of teletype transmission. For a final design, therefore, less power than the 250 kilowatts of this example would be adequate for this type of communication. However, the signal-to-noise ratio could be raised to 45 db if 100 microsecond pulses were us ed rather than 29 microsecond pulses. In a simi ar fashion, the values of parameters for other types of communication, and other information rates, can be worked out. In the course of using frequency Shift transmission., frequencies over a band width of say 100 Kc may be considered. It is of interest to note that no appreciable variation in phase of the frequency components of the received signal will be obtained over such a band. The chief source of such changes would be in the variation of phase with frequency on'. reflection. For bands of 100 Kc., Figure 4-2 shows the change'in phase shift over the band is negligible.

THE UNI VE RS ITY OF MI CH IGAN 2673 -1 -F SE C TION VII[ CONCLUSIONS Due consideration should be given to the design of a frequency-modulated pulsed communication system with approximately the following characteristics —, frequency 10 000 Mc transmitted power 250 Kw band width Kc pulse length 30 Asec. The information rate could be as large as that attained during transmission bursts with meteor scatter systems, but now, of course., uninterrupted communication is possible for as long as the moon is visible to both the transmitter and -the receiver. The theory developed in Sections H and IV shows the feasibility of predictin9 possible constituents of the moon's surface. For this purpose more accurate dataon the power reflected from the moon.should be obtained, with particular emphasis on the initial part of the returned pulse., This will require echoes of the clarity obtained, for example, by the Royal Radar Establishment (Ref. 3). Finally, it is worth remarking that more refined radar techniques might well allow a'limited form of geological survey of parts of the moon surface.

TH EUNI E R ITYOF I CH IGAN 1.. H. Balevis (private communication).mn,'RaarEches ro 2. J. H. Trexler.,"Lunar Radio Echoe", Proceedipgs of the Institute of Radio Epginershl. ol. 46, 286(1958). e hsca ocey(BVo,6, 0(15) J. A. Staton ipe, Electromjic Theil ad~~liain, JonWlyrNwYr Enines ol 6 9327(95) S.KMta h pprAmsPh ical Society (B), Vol. 70, 1105 Be1112 1957).

B. van de P lasma Resonance in Iratonospheric Irreglarities".0 Ariv fai 16. H, erlofson.,o o h r R d o W ve, M ~ aw E il Bo k C m an, I c Fys1]ir, 3p,2473(1951). New he Mahe at cs of Difu io.C ar nd n re s, ox or,. 7( 95 ) DF. ~inter "R Nouveleletos Tables- d'ntegrales efin Editin of 1867 1 8.. B e r e s d eor r c t e G.. S e c h r t a n d C m p a n, N e Y o r j T a b e 9 8 E q n 1 6 ( 9 3 9 ) 19.. van dr Po and H.. Bremmerj 0 tiooldSifnal. acls., Cambrdge Uniersity Press.y, Cambridge10 (19.50). J.L. Pawse Oberhettinger and TricomiAsro, Higher rd aanscendental o