C U1 ^~ ~..E ~ ,1 A Studies in Radar Cross -Sections XIX Radar Cross-Section of a Ballistic Missile - II by K. M. Siegel, M. L. Barasch, H. Brysk, J. W. Crispin, T. B. Curtz, and T. A. Kaplan Contract AF 04 (645) -33 January 1956 2428-3-T 2428-3-T - RL-2050 The University of Michigan Engineering Research Institute Willow Run Laboratories 0 Willow Run Airport Yjlp^silanti, Michigan? X, Thisonal documeefense of the United States within the meaning of the Espionage Laws, (Title 8 U>, Sections 793 and 794). Its transmission or the revelation of its contents in an ymrto an unauthorized person is prohibited by law. Pil Ili t4,!4: 11 4 4-:1, I vll A i.A "',04.,,, "Y" " Pr, Al %:z W R-11h, i. 1* g rl 4, t

UN I VERS ITY OF MICHIGAN 2428-3-T STUDIES IN RADAR CROSS-SECTIONS -: I Scattering by a Prolate Spheroid, F.V. Schultz (UMM-42, March 1950), W 33L03-8)-ac-14222. UNCLASSIFIED II The Zeros of the Associated Legendre Functions Pnm(i') of Non-Integral Degree, K. M. Siegel, D.M. Brown, H.E. Hunter, H.A. Alperin, and C.W. Quillen (UMM-82, April 1951), W 33(038)-ac-14222. UNCLASSIFIED III Scattering by a Cone, K. M. Siegel and H. A. Alperin (UMM-87, January 1952), AF 30(602)-9. UNCLASSIFIED IV Comparison Between Theory and Experiment of the Cross-Section of a Cone, K. M. Siegel, H.A. Alperin, J.W. Crispin, H.E. Hunter, R.E. Kleinman, W.C. Orthwein, and C.E. Schensted (UMM-92, February 1953), AF 30(602)-9. UNCLASSIFIED V An Examination of Bistatic Early Warning Radars, K. M. Siegel (UMM-98, August 1952), W-33(038)-ac-14222. SECRET VI Cross-Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies, R.R. Bonkowski, C.R. Lubitz, and C.E. Schensted (UMM-106, October 1953), AF 30(602)-9. SECRET (UNCLASSIFIED when Appendix is removed) VII Summary of Radar Cross-Section Studies Under Project Wizard, K. M. Siegel, J. W. Crispin, and R.E. Kleinman (UMM-108, November 1952), W 33(038)-ac-14222. SECRET VIII Theoretical Cross-Sections as a Function of Separation Angle Between Transmitter and Receiver at Small Wavelengths, K. M. Siegel, H. A. Alperin, R.R. Bonkowski, J. W. Crispin, A.L. Maffett, C.E. Schensted, and I.V. Schensted (UMM-115, October 1953), W 33(038)-ac-14222. UNCLASSIFIED IX Electromagnetic Scattering by an Oblate Spheroid, L. M. Rauch (UMM-116, October 1953), AF 30(602)-9. UNCLASSIFIED X The Radar Cross-Section of a Sphere, H. Weil(2255-20-T, to be published), AF 30(602)-1070. UNCLASSIFIED XI The Numerical Determination of the Radar Cross-Section of a Prolate Spheroid, K. M. Siegel, B.H. Gere, I. Marx, and F.B. Sleator (UMM-126, December 1953), AF 30(602)-9. UNCLASSIFIED XII Summary of Radar Cross-Section Studies Under Project MIRO, K. M. Siegel, M.E. Anderson, R.R. Bonkowski, and W.C. Orthwein (UMM-127, December 1953), AF 30(602)-9. SECRET XIII Description of a Dynamic Measurement Program, K.M. Siegel and J. M. Wolf (UMM-128, July 1954) W 33(038)-ac-14222. CONFIDENTIAL XIV Radar Cross-Section of a Ballistic Missile, K. M. Siegel, M. L. Barasch, J. W. Crispin, I.V. Schensted, W.C. Orthwein, and H. Weil (UMM-134, September 1954), W 33(038)-ac-14222. SECRET XV Radar Cross-Sections of B-47 and B-52 Aircraft, C.E. Schensted, J. W. Crispin, and K. M. Siegel (2260-1-T, August 1954), AF 33(616)-2531. CONFIDENTIAL XVI Microwave Reflection Characteristics of Buildings, H. Weil, R.R. Bonkowski, T.A. Kaplan, and M. Leichter (2255-12-T, May 1955), AF 30(602)-1070. SECRET XVII Complete Scattering Matrices and Circular Polarization Cross-Sections for the B-47 Aircraft at S-Band, A.L. Maffett, M.L. Barasch, W.E. Burdick, R. F. Goodrich, W.C. Orthwein, C.E. Schensted, and K. M. Siegel (2260-6-T, June 1955),AF 33(616)-2531. CONFIDENTIAL XVIII Airborne Passive Measures and Countermeasures (Offense with, and Defense Against, Bomber Decoys (s)), K.M. Siegel, M.L. Barasch, J.W. Crispin, R.F. Goodrich, A.H. Halpin, A,L. Maffett, W.C. Orthwein, C.E. Schensted, and C.J. Titus (2260-29-F, December 1955). AF 33(616)-2531. SECRET XIX Radar Cross-Section of a Ballistic Missile - II, K. M. Siegel, M. L. Barasch, H. Brysk J. W. Crispin, T. B. Curtz, and T. A. Kaplan (2428-3-T, January 1956), AF 04(645)-33. SECRET w --------- ii ~ - ~EZZ Z L~ r 7~p\ CII

UNIVERSITY OF MICHIGAN 2428-3-T PREFACE This paper is the nineteenth in a series of reports growing out of studies of radar cross-sections at the Willow Run Laboratories of 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. To elaborate means for computing cross-sections of various objects of military interest. 3. To demonstrate that these theoretical cross-sections are in agreement with experimentally determined values. Intermediate objectives are: 1. To compute the exact theoretical cross-sections of various simple bodies by solution of the appropriate boundary-value problems arising from electromagnetic theory. 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. 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. Titles of the papers already published or presently in process of publication are listed on the preceding page. The major portion of the studies summarized in this paper were performed under Contract No. AF 04(645)-33, a prime contract of the 111 BeaLZ 7 c^ r?[ ) - ' -='J I1-' I r \ 1

C=- r= F L-I7 U N I VE R S I T RSITY 2z O F 428 -3 -T MI C HIGAN Western Development Division of the Air Research and Development Command (University of Michigan Project 2428); the contract was under the technical management of the Ramo-Wooldridge Corporation. Appendix F was done under Purchase Order No. S-96527 (University of Michigan Project 2355) sponsored by the Radio Division of the Bendix Aircraft Corporation. Other work was done under Purchase Order No. 7540 (University of Michigan Project 2360) under the sponsorship of the Ramo-Wooldridge Corporation. K. M. Siegel iv -I l j11\ =177 mw

VE RS ITY OF M I C H I GAN N,, 2428-3-T U N I TABLE OF CONTENTS Section Title I II III IV V Appendix A B C D E F G H Studies in Radar Cross-Sections Preface List of Figures List of Tables Introduction and Summary The Radar Cross-Section of the Rudolph Missile Reduction of the Detectability of the Rudolph Missile Continuation of the Basic Analysis Involving the Cross-Sections of Ballistic Missiles Conclusions and Recommendations for Future Work The Radar Cross-Section of the R-W Re-entry Bodies (Configurations I - VI) Effect of Perturbation to the Shape of the Re-entry Body on the Cross-Section in the 200 to 3000 Mc Range Camouflage Materials Utilization of Fission Waste Products for Decoy Purposes Decoys Further Analysis of The University of Michigan Drop-Test Data The Cross-Section of the Tanks Ionization and Related Topics References Page ii iii vi xi 1 15 16 18 58 84 85 102 106 161 177 184.w~ V.,. r -7 X- r

-=J =- I1 ' r\ =- 1 UNIVERSITY OF MICHIGAN _______ 2428-3-T LIST OF FIGURES Number Title 2-1 2-2 2-3 2-4 2-5 3-1 3-2 A. 1-1 A. 1-2 A.2-1 A.2-2 A.2-3 A.2-4 A.2-5 A.2-6 A.2-7 A.2-8 Typical Trajectories of an ICBM A Typical Parameterization for the Rudolph Warhead Cross-Section of One of the Parameterizations for the Rudolph Warhead The Terminal Propulsion Unit (Tanks) Cross-Section of the Tanks (Warhead Removed) at 200 and 270 Mc Configuration I with Perturbations Theoretical Nose-On Cross-Sections as a Function of Frequency Configurations I, II, and III Configurations IV, V, and VI Radar Cross-Section of Configuration I Radar Cross-Section of Configuration II Radar Cross-Section of Configuration III Radar Cross-Section of Configuration IV Radar Cross-Section of Configuration V Radar Cross-Section of Configuration VI Comparison of the Cross-Section of Configuration VI with the Cross-Section of a Sphere of Diameter Equal to the Maximum Diameter of Configuration VI R. R. D. E. Experimental Data 5 6 7 8 9 12 13 19 20 26 27 28 29 30 31 32 34 vi L- r=~ i77

_UNIVERSITY OF MICHIGAN _ 2428-3-T LIST OF FIGURES (Cont.) Number Title Page A. 2-9 Experimental Data - Sperry Gyroscope Company 35 A. 3-1 Cross-Sections of Configurations I, II, III, and A at 3000 Me (Method I) 36 A. 3-2 Cross-Sections of Configurations I, II, III, and A at 3000 Mc (Method II) 37 A. 3-3 Cross-Sections of Configurations I, II, III, and A at 1000 Mc (Method I) 38 A. 3-4 Cross-Sections of Configurations I, II, III, and A at 1000 Mc (Method II) 39 A. 3-5 Cross-Sections of Configurations IV, V, VI, and B at 3000 Mc (Method I) 40 A. 3-6 Cross-Sections of Configurations IV, V, VI, and B at 3000 Me (Method II) 41 A. 3-7 Cross-Sections of Configurations IV, V, VI, and B at 1000 Mc (Method I) 42 A. 3-8 Cross-Sections of Configurations IV, V, VI, and B at 1000 Mc (Method II) 43 A. 3-9 Cross-Sections of Configurations I- VI 44 A. 4-1 Carrot and Cone Models 46 A. 4-2 Cross-Section vs. Aspect Angle for Carrot and Cone Data 47 A. 4-3 Largest, Smallest, and Average CrossSections - Carrot and Cone Experimental Data 48 A. 4-4 7-OC Type Models 50 A.4-5 Largest, Smallest, and Average CrossSections - 7-OC Experimental Data 52 - vii XC X[ r\ F

U N I V E R S I T Y OF MIC(HIGAN ______ 2428-3-T LIST OF FIGURES (Cont.) Number Title Page A.4-6 Cross-Section Data at 200 Mc (Vertical Polarization) 54 A. 4-7 Cross-Section Data at 200 Me (Horizontal Polarization) -I 55 A. 4-8 Cross-Section Data at 200 Me (Horizontal Polarization) -II 56 B. 1-1 Configurations A and B 59 B. 1-2 Configurations C and D 59 B. 1-3 Configurations E and F 60 B. 1-4 Configurations G and H 60 B. 1-5 The Rounded Step in Configuration G 61 B.2-1 The R-W Re-entry Designs and Configurations A and B 63 B. 2-2 Cross-Sections of Configurations A - H at 3000 Me 72 B. 2-3 Cross-Sections of Configurations A - H at 1000 Me 73 B. 3-1 Nose-On Cross-Sections of Configurations I - VI and Configurations A - F at 200 Mc 76 B. 3-2 Nose-On Cross-Sections of Configurations I - VI and Configurations A - F at 350 Mc 77 B. 3-3 Nose-On Cross-Sections of Configurations I - VI and Configurations A - F at 500 Mc 78 B. 3-4 Cross-Section of Ogives at 200 Mc 80 lo ^ viii I. '7 7\ r

I ___ UNIVERSITY OF M ICHIGAN 2428-3-T LIST OF FIGURES (Cont.) Number B. 3-5 B.4-1 F.4-1 F.4-2 F.4-3 F.4-4 F.4-5 F.4-6 F.4-7 F.4-8 F.4-9 F.4-10 F.4-11 F.4-12 F.4-13 F.4-14 F.4-15 F.4-16 F.4-17 Title Cross-Section of Configuration I Compared with the Cross-Sections of Configurations G and H at 200 Mc Theoretical Nose-On Cross-Sections as a Function of Frequency for Configurations A, C, E, and G Compared with the Cross-Sections of Configuration I Page 82 83 S(W) S(w) S(w) S(w) S(w) S(w) S(w) S(w) S(w) S(w) S(w) S(w) S(w) S(o) S(W) S(W) S(w) S(W) S(W) S(W) S(w) for for for for for for for for for for for for for for for for for Object Object Object Object Object Object Object Object Object Object Object Object Object Object Object Object Object 59 59 59 59 61, 61 61 61 62 62 62 62 64 64 64 64 65 S-Band S-Band X-Band X-Band S-Band S-Band X-Band X-Band S-Band S-Band X-Band X-Band S-Band S-Band X-Band X-Band S-Band 137 137 138 138 139 139 140 140 141 141 142 142 143 143 144 144 145 ix - IC 1 -3_. L, z ra

I UNIVERSITY OF MICHIGAN 2428-3-T LIST OF FIGURES (Cont.) Number F. 4-18 F.4-19 F. 4-20 G. 1-1 G. 2-1 G. 2-2 G. 2-3 G. 2-4 G. 3-1 G. 4-1 G. 4-2 Title S(w) for Object 65 S-Band S(w) for Object 65 X-Band S(w) for Object 65 X-Band Geometry of "Tanks" Used in Computations A 5/6-Scale Model of the 7-OC Booster Theoretical Cross-Sections of the Tanks Cross-Section of the Tanks at 90 Mc Cross-Section of the Tanks at 270 Mc Cross-Section of the Tanks at 200 Mc (Meth Geometry of the Modified Tanks Used for Cl putational Purposes Cross-Section of the Tanks at 200 Mc (Meth Page 145 146 146 162 163 165 166 167 169 174 176.od 2) om-.od 3) wml I x -- -1\ r7

VERS ITY OF MICHIGAN __ 2428 -3 -T U N I LIST OF TABLES Number A. 2-1 A.4-1 B.3-1 B. 3-2 F. 2-1 F. 2-2 F. 2-3 F. 2-4 F. 2-5 F.2-6 F. 2-7 F. 2-8 F.2-9 F. 2-10 F.4-1 F.4-2 F.4-3 F.4-4 G. 2-1 Title Nose-On Cross-Sections of Configurations I - VI Cross-Section Data for Configurations Shown in Figure A. 4-4 Nose-On Cross-Sections of Configurations A - F Comparison of Dimensions Frequency Distribution Frequency Distribution Frequency Distribution Frequency Distribution Frequency Distribution Frequency Distribution Frequency Distribution Frequency Distribution Frequency Distribution Frequency Distribution -Object -Object - Object - Object - Object - Object - Object - Object - Object - Object 59 59 61 61 62 62 64 64 65 65 (S-Band) (X-Band) (S-Band) (X-Band) (S-Band) (X-Band) (S-Band) (X-Band) (S-Band) (X-Band) Page 24 53 74 79 109 110 111 112 113 114 115 116 117 118 127 129 131 134 163 R(nTo) - S-Band - Swept Back Fins R(nTo) - X-Band - Swept Back Fins R(nTo) - S-Band - Square Fins R(nTo) - X-Band - Square Fins Comparison of Dimensions Between the Tanks and the 5/6-Scale Model of the 7-OC Booster xi 5 LZ X ) - C- r [..

_UNIVERSITY OF MIC1HIGAN 2428-3-T I INTRODUCTION AND SUMMARY This report is a continuation of the radar cross-section study of ballistic missiles begun in Reference 1. In this report the radar crosssections of a new ICBM for several parameterizations have been found for aspect angles out to 60 degrees off-nose, from 200 Mc through Sband. It is concluded that there are several methods, which appear to be feasible from the cross-section point of view, for significantly reducing the radar cross-section of this new warhead; future systems analysis will determine if these changes will unduly disturb the over-all system. A major point established in this report is that fission wasteproducts are not suitable for camouflage purposes. The new ICBM warhead referred to above was designed as the result of studies carried out by leading experts in hydrodynamics, thermodynamics, aerodynamics, and upper-atmosphere physics under the leadership of the Ramo-Wooldridge Corporation. They felt that the third stage of an ICBM as previously designed might vaporize upon re-entry into the atmosphere. As a result, new parameterizations of the third stage of the ICBM were developed by the staff members and consultants of the Ramo-Wooldridge Corporation. The University was requested to determine the radar cross-section of these vehicles. The cross-sections determined and the analyses performed are presented in Appendix A; a brief summary of the results is presented in the body of this report (Sec. II). Enough analysis of the countermeasure problem has been performed to indicate that, if the modifications suggested in this report are made, the radar cross-section of the new ICBM warhead will be significantly reduced. The modifications involve certain changes in the shape of the vehicle while it is out of the atmosphere, without disturbing the vehicle's geometry upon re-entry into the sensible atmosphere. It is assumed that, for effective warning,the search radars must look for the ICBM long before re-entry takes place. 1 c^"I[ ^ ")*=r"

U N I VUE R S ITY OF M I C H I GAN 242 8-3-T Design of perturbations to Rudolph' to perform this function has been carried out; scale drawings of the new shapes and perturbations to them and computations of their cross-sections appear in Appendix B. It is shown there that perturbing the Rudolph shape in the proper manner can result in areduction of the nose-on cross-section by a factor of 105 at S-band. The perturbation method of reducing the detectability of Rudolph requires that the warhead be in approximately a nose-on attitude to the enemy search radar. This in turn requires knowledge of the location of the enemy search radars and means to point the warhead in the proper direction while it is in flight out of the atmosphere. There are three other methods of reducing the detectability of an ICBM which are discussed here: decoys, camouflage, and chaff. Since the aircraft decoy problem has been studied quite thoroughly (Ref. 2), little additional work seemed necessary from the reflectivity point of view to design decoys for the ICBM with the exception of considering certain problems in the decoy field which are essentially different for aircraft in the atmosphere and for missiles out of the atmosphere. As a result, only two efforts other than the perturbationprotuberance method were made in this field. These involved the analysis of the latest proposed camouflage material for aircraft and missiles (App. C), and the possibility of the use of radioactive wastes to serve either as an ionized medium itself or as an inducer of ionization in the atmosphere (App. D). Reference 2 contains a discussion of the possibility of the defense determining the difference between decoys and aircraft. Since the same methods of differentiating decoys from aircraft are available for use in defense against missiles, and since it was felt that these problems warrant further study, these methods are summarized in Appendix E. 'Since no name had been given to the new third stage of the ICBM, the authors have coined the name Rudolph because of the "red nose" upon re-entry. 2 XCX[WLr

UNIVERSITY OF MI (HIGAN 2428-3-T One subject discussed in Appendix E in regard to decoys is the necessity for foreseeing the possible methods available for distinguishing between decoys and actual missiles or airplanes. One proposed method uses the differences in the fine structure of the radar return (scintillation and glint) to make this distinction. Since the auto-correlation functions and power spectra associated with the radar cross-sections of ballistic missiles have not been obtained previously, and since this information might prove important in more sophisticated methods of detection, these quantities were computed using The University of Michigan droptest data. The drop-test experiment is summarized in Reference 3; the experimental cross-section data is presented in Reference 1; and the analysis of this data which yields the power spectra and the autocorrelation functions is presented in Appendix F. One must consider the cross-section and trajectory of the terminal propulsion unit (tanks) as compared with the cross-section and trajectory of the third stage to see if the knowledge of one will give away the location of the other. As a first step in this direction the radar cross-section of the tanks was determined. Also, some analysis was made of different ways of changing the enclosure which previously housed the third stage, in order to reduce the nose-on cross-section of the tanks. Computations for the tanks are given in Appendix G. Additional study has been carried out on the analysis of reflections from ionic trails such as those from meteors and, since it is clear that these trails are worthy of strong consideration by the defensive people in the ICBM field, a summary of thinking on this problem and on other related topics is given in Section IV and Appendix H. Recommendations for future research on the radar-reflection problem for offensive ICBM study appear in Section V. 3 XIL Zam o7 GH^ fr r) ^ _-] c= _[- j r^<

UNIVERSITY OF MICHIGAN 2428-3-T II THE RADAR CROSS-SECTION OF THE RUDOLPH MISSILE The primary purpose in the design of the Rudolph warhead was to make sure that it would not vaporize upon re-entry into the atmosphere. It is expected that any low-drag supersonic missile traveling at the velocity of Rudolph would vaporize away upon re-entry. (Typical trajectories of an ICBM are shown in Figure 2-1. ) The change in configuration was designed to produce a high-drag shape to replace the low-drag shapes previously under investigation (e. g., the 7-OC warhead). As a result, the small conical nose was replaced by a spherical one'. Of course, this aerodynamic difference changed the radar cross-section picture and at small wavelengths increased the cross-sections of the ICBM warhead. The detailed analysis of the cross-section work for the different parameterizations of the Warhead is given in Appendix A. For the sake of clarity and brevity, one parameterization of the six considered in Appendix A is dealt with here; this particular third stage is referred to in Appendix A as Configuration I and is presented in Figure 2-2. In Appendix A the cross-sections of the various parameterizations of the Rudolph warhead are given out to 60 degrees off-nose for wavelengths between 10 cm and 150 cm (i. e., for frequencies between 3000 Me and 200 Mc). For these conditions Configuration I yields the smallest crosssections. The cross-sections of Configuration I are shown in Figure 2-3. The cross-section of the terminal propulsion unit (the "tanks," Fig. 2-4) of Rudolph is discussed in Appendix G. Several different variations of the surface of the enclosure exposed, when the Rudolph warhead has 'The 7-OC warhead has a conical nose of half-angle 15 degrees. An earlier configuration, referred to in Reference 1 as the "needle-nose warhead, " had a conical nose with total-angle equal to 7 degrees. 4

L- r- \\ L _______ UNIVERSITY OF MICHIGAN i28 - 3-T Center of Earh I FIG. 2-1 TYPICAL TRAJECTORIES OF AN ICBM Figure was obtained from Ref. 4) (Information for this 5 I L-.I \ 7~

UNIVERSITY L-II r OF MIC HIGAN _ 2428-3-T Rn = 0.15m Rt = 0.20m R1 = 0.47m R2 = 0.50m FIG. 2-2 A TYPICAL PARAMETERIZATION FOR THE RUDOLPH WARHEAD (CONFIGURATION I) separated, are discussed. These variations are: a. The hole filled with an absorbing material so that only a negligible contribution to the cross-section is received from the interior of the hole. b. The hole lined on the inside with an inverted hemisphere. c. The hole considered empty up to the end of the hat section, so that for the nose-on aspect the return from the hole would be like that from a flat plate. d. The interior of the hole lined as the inverse of the rear portion of the warhead. The first and last of these variations yield the smallest nose-on crosssections. The results (with the warhead removed) are summarized in Figure 2-5. 6 X'7r

L =cDF ~ir7 UNIVERSITY OF MICHIGAN 2428 - 3 - T 04 8 6 4 2 I I. T I I I I i 4 I i i I I I I 4 + I - 103 8 6 4 A 2 102 v) LJU Iu.i LLJ L) z z 0 0 -u u 8 6 4 2 8 6 4 2 0 10 0. -. * 4 + + I. I 1 8 6 4 2 10'1 8 6 4 2 10-2 8 6 4 2 10-3 r — T 1 - I T I l l 1 1 | |200 Mc- | I I I I1 g lO 3000 Mc II km. m.-mm - - - -m~- - I= I= II I I I I I I - 0 20 40 60 80 ASPECT ANGLE / (IN DEGREES OFF NOSE) FIG. 2-3 CROSS-SECTION OF ONE OF THE PARAMETERIZATIONS FOR THE RUDOLPH WARHEAD (CONFIGURATION I) AS A FUNCTION OF ASPECT AND FREQUENCY 7 J 1-\

I ~ prfl -j — (a) Lined with an absorbing material (b) Lined with an (c) Empty inverted hemisphere THE INTERIOR OF THE HAT SECTION (d) Lined with a surface which is the inverse of the rear of the warhead Booster shown in phantom removed at staging \ C7 ttm C/) tj h) 00 I i I! LLFF nFl L=Z - 50" -A FIG. 2 - 4 THE TERMINAL PROPULSION UNIT (TANKS) (The tanks are shown here with a nose cone rather than with one of the Rudolph parameterizations) 0 z --

L[ LDF c= r r I I -7 U N I 104 - 8 6 4 2 -103 8.6 4 2I — 2102 uj 8- ----- I[ 8 ' 6 o 4 z z 2 0 " 10 U) 8 ( 6) 0 6 UZ VERS ITY OF 2428 - 3 - T MICHIGAN - 0 30 60 90 120 150 180 ASPECT ANGLE (IN DEGREES FROM NOSE-ON) FIG. 2-5 CROSS SECTION OF THE TANKS (WARHEAD REMOVED) AT 200 & 270 Mc(See Appendix G for Details) 9 - C( lr

I fc- c- T ET cL — ___ UNIVERSITY OF MIC HIGAN 2428-3-T III REDUCTION OF THE DETECTABILITY OF THE RUDOLPH MISSILE There are four approaches to the problem of reducing the detectability of the Rudolph missile: 1. If it is possible to choose the orientation of the warhead with respect to the radar within certain broad limits (5 to 20 degrees) when it is out of the atmosphere, the best method available seems to be to reduce the crosssection of the Rudolph warhead by the addition of structures which will vaporize readily upon re-entry into the atmosphere, thus re-establishing the desired shape. 2. A large number of false Rudolphs could be produced so that the enemy could not discriminate between the actual warhead and the decoys. 3. Camouflage materials could be applied to reduce the crosssection of the missile. 4. The warhead could be hidden in an almost continuous column of ionized particles or in chaff. 3. 1 PERTURBATION OF THE SHAPE OF THE WARHEAD The discussion of perturbation of the shape of the warhead must be broken up into three cases: a. The assumption that the tanks would not have to be separated from the warhead until after the search phase was ended, b. The assumption that the tanks could be separated from the warhead by a large distance (the tanks could then be blown up), and c. The assumption that the separation would occur, but that the tanks would remain near the warhead. 1 10 AfaL ErJ I

U N I VERSITY OF M ICHIGAN 2428-3-T Let us consider (c) first. In that case the defense could fix its search radars on the tanks and predict the trajectory of the warhead to within small bounds. Thus perturbation of the shape of the warhead by itself is not sufficient. Under case (b) the most economical method available for reducing the cross-section of the warhead is to put a thin sheet of tin around Rudolph (Fig. 3-1). It is expected that the tin will vaporize almost immediately upon re-entry, and as a result one will obtain the aerodynamic shape required for re-entry. If properly done, this will reduce the cross-section of the warhead while it is outside the atmosphere by a factor of approximately 105 at S-band in the nose-on region. In Figure 3-2 we have plotted the average cross-section of the Rudolph warhead (Configuration I) within t5 degrees of nose-on as a function of wavelength, with and without the tin structural perturbation. Figure 3-2 also contains the cross-sections of the 7-OC and Needle-Nose warheads as a function of wavelength. In this figure observe that the crosssection of the Rudolph warhead has not only been decreased by a factor of approximately 105 at S-band, but, simultaneously, this cross-section has been reduced by a factor of about 102 at 200 Mc (App. B). Under case (a) the cross-section of the entire missile at the noseon aspect will be comparable to the cross-section of the warhead alone above 200 Mcl. 3.2 THE USE OF DECOYS If additional load capacity is available in the Rudolph missile, this space can be used for decoys having the same cross-section as the Rudolph warhead, and adequate aerodynamic characteristics. For a discussion of the use of decoys see References 2, 5, and 6. This work is summarized 1At lower frequencies the nose-on cross-section of the warhead alone will be less than that of the entire missile because the warhead is in the Rayleigh region. 1 1 ~z~zz:J r- F\ Zr 7

[gCi7 _-' - L-JIr \ -E 1 I UNIVERSITY 2428 OF MICHIGAN -3-T FIG. 3-1 CONFIGURATION I WITH PERTURBATIONS The shaded portions indicate the perturbations which would vaporize upon re-entry. This perturbation is the one which is referred to as Configuration G in Appendix B. 12 D CE I lr

iL C: - 7n UNIVERSITY OF MICHIGAN 2428-3 -T 10 8 6 4 2 1 8 6 4 2 tr) LL.LI 0: e) z z 0 o O V) C) i — 0 u ev 10-1 8 6 4 2 10-2 8 6 4 2 10-3 8 6 4 2 10-4 8 6 4 2 10-5 8 6 4 2 10-6 0 500 1000 1500 2000 2500 3000 FREQUENCY (IN Mc) FIG. 3-2 THEORETICAL NOSE-ON CROSS-SECTIONS AS A FUNCTION OF FREQUENCY 13. - [SC r EII

UN I VE RS ITY OF MI ("HI GAN 242 8-3-T in Appendix E. It is shown in Appendix D that fission waste-products cannot be used to enhance the cross-section of the decoys. 3. 3 THE APPLICATION OF CAMOUFLAGE MATERIALS The University of Michigan's examination of the camouflage field has consisted of a continuous but limited study, under Air Force Contract AF33(602)-1070, over the past 18 months. A portion of this study is summarized in Reference 7. Additional studies were made since the completion of Reference 7. One of these additional studies consisted of an examination of thick and heavy materials (approximately 8 inches thick and 1 1/2 lbs/ft2) for which large reductions in cross-sections can be obtained for operation at frequencies above 500 Mc (Ref. 8). Recently a new camouflage material has been developed by Deutsche Magnesit, AG, Munich, Germany. This new material seems to be the best presently available for aircraft and missile applications. Tests on this camouflage material, carried out at the Rome Air Development Center, are discussed in Appendix C. This material is effective in reducing the back-scattering cross-section over small frequency bands. Unfortunately, its effectiveness may be due not so much to absorbent qualities as to the fact that it scatters the electromagnetic energy into angles other than the back-scattering direction. The above studies are still in process and it is too early to state that there is a camouflage material presently available which is suitable for use on missiles. 3.4 THE USE OF CHAFF OR IONIZED PARTICLES It is clearly not possible to carry enough chaff to cover the space required to camouflage the Rudolph warhead over a significant portion of of its trajectory. The alternative possiblity of using fission wasteproducts to hide the warhead can be discounted on the basis of the results contained in Appendix D. 14 r-IL E> r- - 1I I -\

__ UNIVERSITY OF C-i M 1 ( H I GAN * 2428-3-T IV CONTINUATION OF THE BASIC ANALYSIS INVOLVING THE CROSS-SECTIONS OF BALLISTIC MISSILES 4.1 POWER SPECTRA ANALYSIS One method of discriminating between ballistic missiles and ballisticmissile decoys might be based upon scintillation and glint. Recently, for the Bendix Aviation Corporation, The University of Michigan computed the auto-correlation function and power spectra of the cross-section measurements obtained for one-third scale V-2 models (Ref. 1 and 3). The results of this investigation are given in Appendix F. 4.2 IONIZATION AND SHOCK WAVE CONSIDERATIONS In Appendix H we have summarized the work presented in Reference 1, and information on the analysis performed by the Massachusetts Institute of Technology on the effects of ionization of the atmosphere by the missile. Also included is a discussion of the shape of shock waves formed by a missile as it leaves the atmosphere, as compared to the shape of the shock wave upon re-entry. 4.3 EXPERIMENTAL WORK ON HIGH SPEED PELLETS It has been discovered recently by Dr. T. C. Poulter (of the Poulter Laboratories) that one may obtain experimental velocities in the neighborhood of 15 km/sec. These experiments were performed at Stanford Research Institute. The results are summarized in Appendix H. 15

U N I V E R S I T Y OF M1C( H I G AN 2428-3 -T V CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK The radar cross-sec~tion study of ballistic missiles begun in Reference I has been continued in the present report. In this report the radar cross-section of several parameterizations of the Rudolph warhead have been found from 200 Mc through S-band for aspect angles out to 60 degrees off-nose. It has been concluded here that there are several methods for significantly reducing the radar cross-section of the Rudolph warhead; future systems analysis will determine if these changes will unduly disturb the over-all system. A major point established in this report is that fission waste-products should not be used for camouflage purposes. It is recommended that future work on the radar cross-section problem for the ICBM should include: 1. A thorough analysis of the decoy problem for the ICBM (similar to the thorough analysis given to the decoy problem for aircraft (Ref. 2)), 2. A continuing analysis of the reflection characteristics of ionized media (as associated with problems of propulsion, as well as with ionization of the atmosphere during the re-entry portion of the missile trajectory), 3 A continuation of the perturbation analysis presented in this report. (The University of Michigan expects to do this on an extension of Air Force Contract AF 04(645)-33. ), 4. An experimental check on the cross-section computations performed on the perturbations to Rudolph warhead shapes (It is planned also to include these experimental checks in the extension of Air Force Contract AF 04(645)-33. ), 5. The determination of the bistatic cross-section of the ICBM (due to the advantages to the defense that bistatic operation would yield), and mi~ 1 6 I,-r=-l r

UN I VE RS ITY OF r MIMrCHGAN ____ MI1 ("H1I GA N 2428-3-T 6. Further analysis of camouflage materials, including the considerations of the idea of rotating the polarization of the reflected energy. 17 iLZ'EIz =1

I UNIVERSITY OF MICHIGAN 2428-3-T APPENDIX A THE RADAR CROSS-SECTION OF THE R-W RE-ENTRY BODIES (CONFIGURATIONS I -VI) A. 1 INTRODUCTION This appendix contains the analysis involved, and the results obtained, in the computation of the radar cross-sections of the six RamoWooldridge Corporation re-entry designs. The six configurations are shown in Figures A. 1-1 and A. 1-2.(Throughout this report, they are referred to as Configurations I-VI.) Cross-section computations have been performed for 200, 350, 500, 1000, and 3000 Mc for aspect angles (&) out to 60 degrees off-nose (in the case of 3000 Me, the aspect angle ip ranged from 0 to 180 degrees). Section A. 2 contains the 200 -5 00 Me data and analysis, and Section A. 3 contains a discussion of the 1000 and 3000 Mc considerations. Section A. 4 is devoted to a consideration of experimental data at 200 Mc for shapes whose dimensions are comparable to those of Configurations I-VI. This is done because of the inapplicability of physical optics to the study of the cross-sections of these configurations at 200 Mc. In estimating the cross-sections of the Configurations I-VI at 1000 Mc and at 3000 Mc, the methods of physical optics were used exclusively. At the lower frequencies, due to the inapplicability of physical optics, other methods of approximation were required. 1 For the nose-on aspect an approach based upon the creeping wave method was employed; in the 'At these lower frequencies the ratio of the characteristic dimension of the body to the wavelength is approximately equal to one. When this ratio is much greater than one, the methods of physical optics can be applied, and when the ratio is much less than one, the Rayleigh approximation is valid. 18 _______ 18

-Conf. II -jA Is.) 00 z FIG. Al1-i CONFIGURATIONS I, II, AND III

z dUl 0.45m 1.00 Om 0.94 m --.30 m 0.2m M --- - - -- - - - j C/) ~-q 11,.) >. FIG. A. 1-2 CONFIGURATIONS I IV, V, AND VI

r --- UNIVERSITY OF MICHIGAN 2428-3-T aspect range 0< 4 <60~, cross-section estimates were obtained based upon the results of the study of experimental data (Sec. A. 4), physical optics results, and the data obtained in the study of the nose-on case. The over-all results of any such approximation method should be accurate within a factor of 10. It is of course hoped that the approximations will be better in many cases, but this maximum error is tolerable' because the range of detection varies as the fourth root of the crosssection. In other words, if the radar cross-section is known within a factor of 10, the range performance of the radar system is known within a factor of 1.78. A. 2 CROSS-SECTIONS AT 200 TO 500 Mc A. 2. 1 Cross-Sections for the Nose-On Aspect (O = 0) Each of the Configurations I-VI has a maximum diameter, D, and length, L, such that at 200 Mc, D/\ = 0.67, and 0.93(< L/X< 1.43; at 350 Mc, D/X = 1.17, and 1.63< L/k->2.50; and at 500 Mc, D/X = 1.67, and 2.33< L/X<3.57. Reference to the theoretical sphere curve in Kerr (Ref. 9) and the 10: 1 prolate spheroid curve (Ref. 10), indicates that these values of D/X and L/k are in the resonance region. These ratios are of such magnitude that neither physical optics nor Rayleigh scattering can be considered to be applicable (although at 500 Mc we are approaching values of these ratios for which physical optics could be applied). Thus some other method of approximation must be applied to estimate the nose-on cross-sections of Configurations I-VI for these frequencies. The method applied is similar to the one which was successfully used by Franz and Deppermann (Ref. 11) for the cylinder and is the same as that applied in Reference 1 for the determination of the nose-on crosssection of the 7 -OC warhead.(The 7 -OC calculations yielded results which agreed with experimental data to within a factor of 2. ) Briefly, the method involves considering the back scattering cross-section to be the sum of two contributions. One is due to the radiation scattered from the first Fresnel zone on the body or, in the case of a pointed body, from the tip, and the other is due to the radiation which creeps around 'The accuracy of the results obtainable depends only upon the amount of time and money one wishes to expend in obtaining the results. - 21 - I I.- Li7K

UNIVERSITY OF MICHIGAN 2428-3-T the rear of the body and comes back to interfere coherently with the radiation scattered from the forward part of the body. An estimate of the contribution from the rear was obtained in Reference 1 through the study of sphere cross-sections. To do this the cross-section of a sphere at a peak was written 2 C = A +4A peak g. o. rear j where rpeak is the sphere cross-section at a peak in the resonance region, Ago. is proportional to the field amplitude due to the first Fresnel zone, and Arear is the contribution due to the radiation which 2 2 creeps about the rear of the body. Taking (Ag o ) = mra, where a = the radius of the sphere, and reading values of rpeak off existing sphere graphs such as the one found in Kerr (Ref. 9, p. 453), it was found that in the region of interest (Arear) is given by 0. 033X2 (X/a)1/2 Applying this procedure to Configurations I-VI and considering the contribution from the rear to be approximated by that from a sphere of radius equal to 0.5 m, we obtain for each of the six configurations (Arer) = 0. 128 m2 at 200 Mc, (Arear) = 0.0317 m2 at 350 Mc, and (A ) = 0. 0130 m2 at 500 Mc. rear From the physical optics methods applied for the 1000 and 3000 Mc cases, the contribution from the first Fresnel zone is 0. 0707 m2 for ConfigurationsI and IV, 0.283 m2 for Configurations II and V, and 0. 636 m2 for Configurations III and VI, 22 I i7 -=) r- u-ir-\< ' - 1

UNIVERSITY OF MICHIGAN 2428-3-T for all three frequencies From these computed values "average" cross-sections can now be obtained by merely taking the sum of the two contributions, an upper bound by assuming complete reinforcement, and a lower bound by assuming complete interference between the two contributions; i.e., if -1I is the contribution from the front and ar2 is the contribution from the rear, we have (c1 + %r2) as the "average" value, ( J/~- /- / )2 as the lower bound, and (J/t + /-o)2 as the upper bound. The results appear in Table A.2-1. At 500 Mc the ratios of L/X are approaching values which would permit the application of physical optics. It is of interest to compare the values for 500 Mc given in Table A. 2-1 with those which are obtained through the application of physical optics. The following average values are yielded by the physical optics method for X = 0.6 m: 0. 17 m I = 0.39 m, III = 0.74 m IV = 0.64 m2, cr = 0.84 m2, and oVI = 1.19 m2 It should be noted that these physical optics cross-sections are larger than those given in Table A. 2-1 by factors ranging from one up to eight. A. 2.2 Cross-Sections for the Off-Nose Aspects (0 <, 600~) The "creeping-wave" approach used for the nose-on aspect cannot conveniently be applied for, > 0 since the use of a sphere to determine the contribution from the "rear" for q >0 is a poor geometric approximation to the actual shape. However, estimates of the manner in which the cross-section varies over the interval 0 ~< ' 600 can be obtained through the use of 1. The results obtained for = 0. 1 m and X = 0.3 m (Sec. A.3), 2. The physical optics estimate at 500 Mc, 'This contribution is given by -rRn where Rn equals the radius of the front spherical section of the missile (see Fig. A. 1-1 and A. 1-2). 23

TABLE A. 2-1 ' I NOSE-ON CROSS-SECTIONS OF CONFIGURATIONS I-VI (In Square Meters) 350 Mc 500 Mc CONFIGURATION Lower Av. Upper Lower Av. UpperLower Av. Upper _ Bound Bound Bound Bound Bound 00 I and IV.0083.20.39.0023.084.14 II and V.030.41.79.13.31.50.17.30.42 III and VI.19.76 1.3.38.67.95.47.65.83 o< 0 f1 -j I2 O.-> 0 -c> Z

U N I VERSITY OF M ICHIGAN 2428 -3 -T 3. The geometric optics prolate spheroid (a = L/2, b = D/2) curve in the region 0< (< 60~ (Ref. 10), and 4. The results of the examination of experimental data (Sec. A. 4) which indicated that at 200 Mc o-r() should not vary much in the interval 0 < 4 < 60~ for these configurations, combined with the computed nose-on values given in Table A. 2-1. Crosssection curves are given in Figures A. 2-1 through A. 2-6. Each figure contains, for a given configuration, the cross-section curves at 3000 Mc (X = 0.1 m) and 1000 Mc (X = 0.3 m)(Sec. A. 3), the curves for the prolate spheroid (geometric optics), and points obtained through physical optics at 500 Mc. With these as a guide, estimates of the cross-section curves at 200 Mc, 350 Me, and 500 Mc are made with the cr(0~) point determined from Table A. 2-1. A. 2.3 The Wavelength Dependence in the Region From 200 to 500 Mc Some idea of the manner in which the cross-sections of Configurations I-VI vary as a function of frequency in the region from 200 to 500 Mc is obtained by an examination of Figures A. 2-1 through A. 2-6. However, a better idea may be obtained by analogy with the variations with frequency of the cross-section of the theoretical sphere (Ref. 9) and the theoretical 10:1 prolate spheroid curves (Ref. 10). These shapes display an oscillation of Ca as a function of X in this frequency interval. It is to be expected that the nose-on cross-section of Configurations I-VI will oscillate as a function of X in a somewhat similar manner. Figure A. 2-7 shows the cross-section of a sphere (of diameter equal to the diameter of Configurations I-VI) as a function of 1/(2X), together with the computed "average", lower bound, and upper bound a- at 200, 350, and 500 Mc, for Configuration VI. Examination of this figure yields a fairly good picture of how the nose-on cross-section of Configuration VI varies in the 200 to 500 Mc frequency region, which can be considered to be typical of the variations for all six configurations. Experimental data obtained by the Radar Research and Development Establishment (Ref. 12) and by the Sperry Gyroscope Company (Ref. 13) 25 I I[ 7K

- ui i r U N I VE R S IT Y OF MICHIGAN 2428 - 3 -T cn w LU hz O IU LU io CO C, O e) 0 104 8 6 4 2 103 8 6 4 2 102 8 6 4 2 10 8 6 4 2 1 8 6 4 2 10 1 8 6 4 2 10-2 I All Cross-Sections Are "Average" Cross-Sections Computed Points (200 - 500 Mc): 0 200 Mc, * 350 Mc, & A 500 Mc ("Creeping Wave") - T 500 Mc (Optics) I - 1000 Mc Mc 0 10 20 30 40 50 ASPECT ANGLE 0 (IN DEGREES) 60 FIG. A.2-1 RADAR CROSS-SECTION OF CONFIGURATION I 26. I i 77

WC',r c- L- \\ r-E IN-'j 11\\ '-_ U N I V E R S I T Y OF MICHIGAN 2428 - 3 -T 104 8 6 4 2 103 8 6 4 2 102 8 6 4 T T T m I I A iI i i T 1 I i I I I =AII Cross-Sections Are "Average" Cross-Sections -Computed Points (200 - 500 Mc): - 0 200 Mc, ~ 350 Mc, & A 500 Mc ("Creeping Wave") -v 500 Mc (Optics) I I I I I I I-I I I I I I I I I I I I I1 - LU LLI LU C (I) z z 0 LU,) (I (I) 0 u 2 10 1 8 6 4 2 8 6 A J - - — —tt Prolate Spheriod t - (a=L/2; b=D/2 -- Geometric Optics) I I ) I -p I. 1Conf iurnfien II - 1000 Mc L Configuration II- 200 Mc c %-%Jllllwulullull I i- - i - Ir - I dt r - Configuration II - 3000 Mc - = — =60IGN I -wom 10-1 2 8 6 4 2 4 - - I \Co I Configuration II r 50 Mc & 500 Mc r, I. I I — t 4 + i i I 10-2 8 6 4 1 2 10-3...- I I I 0 10 20 30 40 50 60 ASPECT ANGLE 0 (IN DEGREES) FIG. A.2-2 RADAR CROSS-SECTION OF CONFIGURATION II 27

L- -I 77 UNIVERSITY OF MICHIGAN 2428 - 3 -T 104 8 6 4 I... i I 2,, i 8 6 4 All Cross-Sections Are "Average" Cross-Sections -Computed Points (200 - 500 Mc): - 0200 Mc, U 350 Mc, & A500 Mc ("Creeping Wave") _ 500 Mc (Optics) 102 0 LU ILU LU Qt O z z Z IU 0 U 10 1 tion III - 3000 Mc III - 1000 Mc, 500 Mc, 350 Mc & 200 Mc 10-1 10-2 10-3 0 10 20 30 ASPECT ANGLE 40 50 60 0 (IN DEGREES) FIG. A.2-3 RADAR CROSS-SECTION OF CONFIGURATION III 28 7 I [ r r- E j r-.

LIDF-E i U N I VE R S I T Y OF 2428- 3-T MICHIGAN 104 8 6 4 2 r. i I i i i i I I I i~~ i 8 6 4 2 102 8 6 4 =All Cross-Sections Are "Average" Cross-Sections -Computed Points (200 - 500 Mc): - 0200 Mc, * 350 Mc, & A500 Mc ("Creeping Wave") - v500 Mc (Optics) in LU z z 0 I0 u 2 10 8 6 4 2 8 6 4 2 Prolate Spheriod,(a =L/2; b=D/2 -- Geometric Optics) 1 Jration IV - 1000 Mc uration IV - 500 Mc uration IV - 3000 Mc 10-1 8 6 4 2 10-2 8 6 4 2 10-3 0 10 20 30 40 50 ASPECT ANGLE b (IN DEGREES) 60 FIG. A.2-4 RADAR CROSS-SECTION OF CONFIGURATION IV 29 L- I'r-, I i 77

r - I ir UNIVERSITY OF 2428 - 3 -T MICHIGAN 104 8 6 4 2 8 6 4 2 -All Cross-Sections Are "Average" Cross-Sections:Computed Points (200 - 500 Mc): - 0200 Mc, ~ 350 Mc, & A 500 Mc ("Creeping Wave") - v 500 Mc (Optics) 102 8 6 4 QLL uo LI= ILI) z U n CI u LI O u 0~ u^ 2 10 8 6 4 2 1 8 6 4 2 10-1 8 6 4 2 8 6 4 2 10-3 0 10 20 30 40 50 60 ASPECT ANGLE L (IN DEGREES) FIG. A.2-5 RADAR CROSS-SECTION OF CONFIGURATION V 30 W @ lr r_ %. E i r

L-. ==1 r r I I I -U UNIVE RS ITY OF MICHIGAN 2428 - 3 - T 104 8 6 4 2 103 8 6 4 All Cross-Sections Are "Average" Cross-Sections Computed Points (200 - 500 Mc): o 200 Mc, ~ 350 Mc, & A 500 Mc ("Creeping Wave") v 500 Mc (Optics) 2 102 8 6 4 V) LU ILU LU CO n( Z z 0 u LU::) 0 v u 10 2 8 6 4 2 8 6 4 2 1 10-1 8 6 4 2 10-2 8 6 4 2 10-3 0 10 20 30 40 50 60 ASPECT ANGLE b (IN DEGREES) FIG. A.2-6 RADAR CROSS-SECTION OF CONFIGURATION VI 31 - r --- (,\ -

7 r OF MICHIGAN - 28-3 -T UNIVERSITY 24: 8 --- -- - _ I I I 6 I 1.C 4 2 1.0 I 8I I II it I I I I t 6I I I& 2 100 1/(2 X) FIG. A.2-7 COMPARISON OF THE CROSS-SECTION OF CONFIGURATION VI WITH THE CROSS-SECTION OF A SPHERE OF DIAMETER EQUAL TO THE MAXIMUM DIAMETER OF CONFIGURATION VI (Q=0~ ) 32 ~ [ S =I l -

_UNIVERSITY OF MICH I GAN 2428 -3 -T add additional information relative to the oscillation of Co with changing frequency. The RRDE data is reproduced in Figure A. 2-8 and the Sperry data in Figure A. 2-9. The configurations in these experiments have L/D ratios of from 6: 1 to approximately 10: 1. An examination of the above data, the theoretical sphere data, and the theoretical 10: 1 prolate spheroid data, indicates that Configurations I - VI have a total fluctuation in cross-section (as a function of frequency) of less than a factor of ten in the 200 to 500 Me range. A. 3 CROSS-SECTIONS AT 1000 AND 3000 Mc The cross-sections of Configurations I-VI at 1000 and 3000 Mc were computed by two methods. Method I yields "average cross-sections," and Method II yields cross-sections for which the oscillations due to phase differences are estimated. Since the computations were performed in connection with the perturbations to the six re-entry designs, the computational procedures and equations involved for both sets of calculations appear in Appendix B. The computational procedures followed closely the methods and techniques outlined in Appendix A of Reference 5, and the results obtained are shown in Figures A. 3-1 through A. 3-9. Figures A. 3-1 through A.3-8 cover the aspect interval 0< 4,< 60~ and include the data for two of the perturbations (Configurations A and B) discussed in Appendix B. Figure A. 3-9 contains 3000 Mc data in the aspect range from 60 to 180 degrees. Examination of the data contained in these figures indicates that of the six configurations, Configuration I has the smallest cross-sections at all of the frequencies and aspects considered. A. 4 EXPERIMENTAL DATA AT 200 Mc FOR SHAPES COMPARABLE TO CONFIGURATIONS I-VI A. 4. 1 Introduction An examination of experimental data for configurations having body dimension to wavelength ratios similar to those for Configurations I-VI should give information relative to the magnitudes of the Rudolph crosssections. 33 E [LC [C CDF r —

I Horizontal Polarization Head On Aspect (0 Degrees) Variable 2.97A -j H- j -2X Nose Shape t Nose Insert Back Variable 2.97X -- -] |t2X Nose Shape B A Nose Insert Back Variable -J c c Hemispherical at Each End 1.0 I 1-l li/i c4 Z l< z LU OS u I U LU Iz LU -J C LU LU C, z I U LU Iz LU -J LU C, z I LU LU I z 0 - u LU -- ZD CU LU 0.1 A I I I A I I I I I t I x. I I I I t!M Pi C0 0 -O 0-11 00 I ^>q Iril J 0.1 ( 4 *I11" O-W -. C 1.0o 5.( --- 4.5 6.0 4.5 5.0 5.5 6.0 INSERT LENGTH IN X 0 6.0 7.0 INSERT LENGTH IN X OVERALL LENGTH IN X Z FIG. A.2-8 R.R.D.E. EXPERIMENTAL DATA - EQUIVALENT ECHOING AREA OF FINLESS MISSILE MODEL OF 1.17 A DIAMETER (Figures 5A, 5B, & 6 of Ref.12)

L L IDF I - J lraS 7r - UNIVERSITY OF MICHIGAN 2428 - 3 -T IU 8 I I I I I6 -I II I I II I I 4- I A '4j4 (6 ft 1I nI Ai I i i i n.V 1I II III IIU I I I I-I "I I I I ''I~ C'\/ 4 - I 2 I, 1 n V.. 8 6 4 Experimental Shape: E Vector -. - I | 2a I-I -a=6 2a 2 0 O O Scattering from sphere of radius "a" -: - Experimentally determined 0.01 C I I I I. I I ). I _______ -I- -I-I-I-I I I I I I I.1 8 1.0 2 27ra AT 4 6 8 10 2 4 6 8 100 FIG. A.2-9 EXPERIMENTAL DATA - SPERRY GYROSCOPE COMPANY (From Ref. 13) 35 00, L---. 1 i FEr-7

[LrD r U N I V E R S I T Y 24 UNIVERSITY 4 2 10 8 6 4 2 8 ------------- 10 LU L8 -- -- U~j 6 4 4 - z 2 8 L. ):~ 8 - -- -- - -- III O 4 0 6 U - O 6 = -- - -- - - ta — - - -- - O F [28 - 3 - T MICHIGAN I FIG. A.3-1 I II, 0b (IN DEGREES) CROSS-SECTIONS OF CONFIGURATIONS III, AND A AT 3000 Mc (METHOD I) 36 - [ 1 1 - I \ 1 j~~~~j

Lr 77 U NI V ER SI TY OF 2428 -3 -T M I CHI GA N I c eZ --— t --- I I I - A F (=0.1 M) 2-V U-I LUj LUJ C:E Lf) z z LU L,) 0 U 8 - A A n nliw '1 Li ( I' -+- - 4 I li I I II TU I IT i'i f t i..... Lu I lIII II' I1 iII I I III FII I1 I I II I FT -u iin 5 0 v v 'I tl I t 1 V WI t 1 H 2 I II f 1 H, in 0 3 H, M 1 i i I I II I ly ",I iI I3 Iif IlI U INU I Al 41lItI 0l i i H Hti It i iI v Ii i dI v IV if.I-3t --- —----— t i i 4 4 4 - ----— 4.-i-4 —i I- I I I I'c I V 8 0 41 2 __ _ _ _ V -63L 0 4 - I__ ~ _ _ I__ A _ _ I I__. _ I _ _ _ I _ _ _ I _ _ _ I I_ _ 1 0 20 30 40 50 60 0t (IN DEGREES) FIG. A.3-2 CROSS-SECTIONS OF CONFIGURATIONS If III III, AND A AT 3000 Mc (METHOD II) 37 t ~

r~Fr —IIi r~-i U NIVERSITY OF 2428 - 3 - T MICHIGAN _ 1 1 (n LU tCO Z) z I= u LI) I.) 0 U 0 10 20 30 40 50 601 b (IN DEGREES) FIG. A.3-3 CROSS-SECTIONS OF CONFIGURATIONS I, II, III, AND A AT 1000 Mc (METHOD I) 38 fr^rF F ~r —

UNIVERSITY OF M I CHI 2428- 3-T I ----t GA N I I, I - I-, LU I — WsL LU C 0 LU O,) 0 U 0 10 20 30 40 50 60 0 (IN DEGREES) FIG. A.3-4 CROSS-SECTIONS OF CONFIGURATIONS I, II, III, AND A AT 1000 Mc (METHOD II) 39 I,r-_ 77

L-, r-r r 77 UNIVERSITY 24 10 6 4 2 1 w Iui LI) VI z8 ----- -- UL <t 6 -- --- - --— = ---- L 10 -- -- e 8 6 -i = 0 4 u ua 1 O ~ 6 u O F i28 - 3 -T MICHIGAN 0 10 20 30 40 50 60 0 (IN DEGREES) FIG. A.3-5 CROSS-SECTIONS OF CONFIGURATIONS IV, V, VI, AND B AT 3000 Mc (METHOD I) 40

L-c FELi IU N I VE R SIT Y i OF MI CHIGAN 2428 -3- T LU LUJ LUJ LI) z z 0 LU LI) L/) 0 ix 0 10' 20 30 40 50 60 0t (IN DEGREES) FIG. A.3-6 CROSS-SECTIONS OF CONFIGURATIONS IV, V, VI, AND B AT 3000 Mc (METHOD II) 41 ol I 7 r- 1 I i r

L- I I rx 77 UNIVERSITY 24 O F i28 - 3 -T MICHIGAN I- I 1 LU I — LU LU Ce. D 0 '/) z z 0 u LU U) Cl) 0 w 0 10 20 30 40 50 60 ' (IN DEGREES) FIG. A.3-7 CROSS-SECTIONS OF CONFIGURATIONS IV, V, VI, AND B AT 1000 Mc (METHOD I) 42 IE LDF E 7K I \ L

L- I" r -7 UNIVERSITY U N I V E R S I T Y 2 10 4 10 z ' 82. 61.. = E 1 - - S ^-f *" -,hhF -y^ -r OF MICHIGAN N >428 - 3 -T l 1 14 0 10 20 30 40 50' 60 t (IN DEGREES) FIG. A.3-8 CROSS-SECTIONS OF CONFIGURATIONS IV, V, VI, AND B AT 1000 Mc (METHOD II) 43 XW ll -

LEL r U N I VERSITY OF MICHIGAN 2428 - 3 -T 104 8 6 4 2 103 8 6 4 2 102 8 6 4 2 10 1 8 6 4 2 8 6 4 2 8 6 4 10-1 2 10-2 8 6 4 2 10-3 60 80 100 120 140 160 180 b (IN DEGREES) FIG. A.3-9 CROSS-SECTIONS OF CONFIGURATIONS I - VI AT 3000 Mc IN THE ASPECT RANGE 60~ 0 - < 180~ 44 IELJ r5 i ==, 77

UN I VE RUS ITY OF MI CH I GAN 2428-3-T A. 4. 2 Experimental Data for Carrots and Cones The Microwave Radiation Company, Inc. has performed cross - section measurements for the Cornell Aeronautical Laboratory, Inc. under Contract DA-30-115-ORD-543 between the Cornell Aeronautical Laboratory and the Department of Defense. These experiments were conducted at a frequency of 9375 Mc on models which were 2, 4, 8, and 32 wavelengths long. Two of the models involved were a "Carrot" and a finite cone (Fig. A.4-1). This section contains some of the 2X model data scaled to 200 Mc. This results in full scale configurations which are all 3 meters long with a maximum diameter = 0. 66 m and 0. 75 m. 1 To examine the effect of configuration shape on the model crosssections, these two models are considered as four; i.e., the Carrot with the cone tip in front, the Carrot with the spherical bulb in front, the Cone with the tip in front, and the Cone with its base in front. These four configurations have approximately the same length-to-diameter ratio, but otherwise differ in many respects. The experimental data (Ref. 23) scaled to 200 Mc are displayed graphically in Figure A. 4-2. Average curves can be fitted to this 200 Mc data which serve to predict most of the experimental points to within a factor of three. These curves, together with maximum and minimum curves, are shown in Figure A. 4-3. For horizontal polarization the average curve is within a factor of three of all the experimental data for aspects out to 20 degrees from the front and is always within a factor of eight of the remaining data in the 25 to 60 degree aspect range. For vertical polarization the average curve is within a factor of three of all the data, except for the aspect of 40 degrees, at which the average is within a factor of five of the extremes. 'This may be compared with lengths between 2. 16 and 1. 40 meters and a maximum diameter of 1 meter for the six configurations of interest here. 45 XW E7rK

[ I r- 1 UNIVERSITY OF MICH 2428 - 3 - T 0.66m - -L I I GAN I, J m -1 0I — 0.75m -- _ Carrot 15~ I I 12 a. -1 L A m b II Cone FIG. A.4-1 CARROT AND CONE MODELS (Scaled to 200 Mc size) 46. - F E r

Dr I 7r r I \ U N I VE RS IT Y OF 2428- 3-T MICHIGAN 10 V) Ct I — L) z u 0: Z vO 8 6 4 2 8 6 4 2 1 10-1 8 6 4 2 10-2 — * Carrot (Tip in Front) -o Cone (Tip in Front) *- -* Carrot ("Bulb" in Front) o —o Cone (Base in Front) 10 8 6 4 IV) uj LU LU CI LI) z z 0 LU LI) LI) 0 w (n 1 10-1 0-2 0 10 20 30 40 50 60 ASPECT ANGLE (IN DEGREES) FIG. A.4-2 CROSS-SECTION vs. ASPECT ANGLE FOR CARROT AND CONE DATA (200 Mc) 47 Lr1L L - r -- Flo 7

r orl r L-L Ir 77 U NI V ER SIT Y OF MI CHIGAN 2428- 3-T 10 LUJ III LUE z 0 LLI) (/) 0 0U 8 6 4 2 8 6 4 2 I 10-1 8 6 4 2 10-2 10 LU LILU a (I) z z 0 u LU LI) LI) 0 u 8 6 4 2 8 6 4 1 2 8 6 4 2 10-2 0 10 20 30 40 50 '60 ASPECT ANGLE (IN DEGREES) FIG. A.4-3 LARGEST, SMALLEST, AND AVERAGE CROSS-SECTIONS - CARROT AND CONE EXPERIMENTAL DATA (200 Mc) 48 =:

UNIVERS ITY OF MICHIGAN __ 2428-3-T A. 4. 3 Experimental Data for the 7-OC Static experiments on 7-OC-type missiles were performed by the Microwave Radiation Company, Inc. (MRC), and by the Evans Signal Laboratory (ESL) at the instigation of Project Wizard. These data (scaled to full 7-OC dimensions) are given in Reference 1. Using the 7-OC 75 Mc data appearing in Reference 1 and scaling it to 200 Me yields data for the three configurations shown in Figure A. 4-4. Here we have three configurations for which (at 200 Mc) L/k = 1.54, 1.46, and 1.27; and D/k = 0.346, 0.360, and 0.346.' As in the Carrot and Cone experiments, the number of configurations is increased (over the aspect range of interest) by considering the model rotated through 180 degrees to be a different configuration. Combining the data from the two sources (MRC and ESL) yields ten experimental curves both for horizontal and vertical polarization: 1. 7-OC Model-nose in front (MRC data), 2. 7-OC Model-nose in front (ESL data), 3. 7-OC Model-tail in front (MRC data), 4. 7-OC Model-tail in front (ESL data), 5. 7-OC (Hemispherical Stern)-nose in front (MRC data), 6. 7-OC (Hemispherical Stern)-nose in front (ESL data), 7. 7-OC (Hemispherical Stern)-tail in front (MRC data), 8. 7-OC (Hemispherical Stern)-tail in front (ESL data), 'Compare with the 0.93 < L/X < 1.43 and D/k = 0.67 ratios for the six R-W re-entry designs. 49 XW@Xi7Kr

r-rImr r 77 U NIVERSITY OF MICHIGAN 2428 - 3-T.52 m 2.31 m I=....!-!rAA / - UC lype Model 7 - OC Model with hemispherical stern i'.52m - 1.91 m J I 7- OC Model with flat stern (The Dimensions are obtained by scaling the 75 Mc data from Ref. 1 to 200 Mc) FIG. A.4-4 7-OC TYPE MODELS - 50

UNIVERSITY OF MICHIGAN 2428-3-T 9. 7-OC (Flat Stern)-nose in front (MRC data), and 10. 7-OC (Flat Stern)-tail in front (MRC data). Figure A.4-5 shows the largest, smallest, and average values obtained for the cross-sections of the ten shapes listed above by scaling the 75 Mc-7-OC data to 200 Mc. (Table A.4-1 contains the data from which Fig. A.4-5 was constructed.) A.4.4 Summary of Experimental Data for Carrots, Cones, and the 7-OC By grouping the 7-OC and Carrot and Cone data and considering each of the models both from the front and the back, fourteen experimental cross-section vs. aspect curves are obtained for ten different configurations. In some cases these ten configurations differ markedly (compare Fig. A.4-1 and A.4-4), but the length (L) and maximum diameter (D) of each satisfy the conditions: 1.27< L/k < 2.00 and 0.34 < D/X < 0.50. All of the data for vertical polarization are grouped in Figure A. 4-6, which shows the range of values of Co (largest and smallest) as a function of aspect, together with an average curve. All of the data lie within a factor of five of the average curve. The horizontal polarization data for the five "cone -nosed" configurations are grouped in Figure A.4-7 in a similar manner. In this case the average curve fits the data to within a factor of six, except for the aspect angle range of 10 to 40 degrees. The remaining horizontal polarization data are grouped in Figure A. 4-8. The average curve lies within a factor of five of all the data. The experimental data examined in this section indicates that the cross-section of a configuration does not vary to any appreciable extent with changes in configuration shape if the L/k and D/\ ratios are restricted by the above inequalities. (This is especially true for vertical polarization. ) The configurations involved in these experiments were 51 ~E~L~DF0 Z7

- = I I r- I i r U N I VE R S I T Y OF 2428- 3 -T MICHIGAN (n LLU z-: LU ILU C I)> z z 0 u LU Li) L) 0 Uj 10 8 6 4 2 1 8 6' 4 2 10-1 8 6 4 2 10-2 10 8 6 4 2 1 8 6 4 2 10-1 I 8 6 4 2 10-2 LU C) ILI) n 0 O or) z 0 O u u Vertical Polarization Data I I I I 0 10 20 30 40 150 60 ASPECT ANGLE (IN DEGREES) FIG. A.4-5 LARGEST, SMALLEST, AND AVERAGE CROSS-SECTIONS - 7-OC EXPERIMENTAL DATA (200 Mc) 52 ~o

WC~rr UNIVERSITY OF MICHIGAN 2428-3-T TABLE A.4-1 CROSS-SECTION DATA FOR CONFIGURATIONS SHOWN IN FIGURE A. 4-4 AT 200 Mc (in m2) Configuration Source Pol. Aspect 0~ 100 20~ 30~ 40~ 50~ 60~ 7-OC model-nose in front MRC H.12.07.01.03.31.44.53 7-OC model-nose in front ESL H.17.12.01.07.24.27.42 7-OC model-tail in front MRC H.17. 22.60 2. 1 2.5.93.27 7-OC model-tail in front ESL H.14.34.92 2.2 2.9 1.0.42 7-OC (Hemi. Stern)-nose in front MRC H. 19.18.19.13.04.25.38 7-OC (Hemi. Stern)-nose in front ESL H.28.31.30.22.11.38.49 7-OC (Hemi. Stern)-tail in front MRC H.31.42 1.0 2.2 3.3 1.4.13 7-OC (Hemi. Stern)-tail in front ESL H.32.32.90 3.3 4.2 1.2.13 7-OC (Flat Stern)-nose in front MRC H.36.40.49.77.85.70.21 7-OC (Flat Stern)-tail in front MRC H.62.81 1.0 1.2.77.28.28 7-OC model-nose in front MRC V.13.14.15.08.08.24.34 7-OC model-nose in front ESL V.12.15.17.13.12.24.34 7-OC model-tail in front MRC V.20.21.19.22.21.36.34 7-OC model-tail in front ESL V.14.16.18.19.19.34.29 7-OC (Hemi. Stern)-nose in front MRC V.22.24.27.23.15.25.42 7-OC (Hemi. Stern)-nose in front ESL V.18.22.27.27.21.25.40 7-OC (Hemi. Stern)-tail in front MRC V.42.38.42.56.31.46.56 7-OC (Hemi. Stern)-tail in front ESL V.20.20.22.23.23.42.42 7-OC (Flat Stern)-nose in front MRC V.33.35.33.27.28.36.42 7-OC (Flat Stern)-tail in front MRC V.62.56.46.44.49.56.40 53 I', - ) j r- I i-' 11 \ '

Fr I r I_ 1 I I\ U N I V E R S I TY OF MICHIGAN___ 428- 3 -T 105 8 6 4 I I I I I I I I I I 2 I I I I I I I I I I I -.Limits of Data (Experimental) onl4 Experiments on 10 Different Configurations *-* with a "Non-Cone-Type-Nose" for Which 1.27L L/X - 2.00 and 0.34 < D/X < 0.50 -o ---o Average Values of the Experimental Data 8 6 4 2 103 8 6 4 V) u0 u ct:D 0n O u LU wr U 2 102 8 6 4 2 10 8 6 4 2 Iq j,, ~ r — =E m 0i 4 1 8 6 4 2 -1 10 8 6 4 2 10-2 0 10 20 30 40 50 60 k (IN DEGREES) FIG. A.4-6 CROSS-SECTION DATA AT 200 Mc (VERTICAL POLARIZATION) 54.

I ~ U NIVE R SITY OF M I C HIGAN 9 2428 -3- T 1 05 8 6 4 2 I 04 8 6 4 2 LIu-I a (I) z z 0 LLI) UV) LI) 0 u 8 6 4 2 1 02 8 6 4 2 10 8 6 4 2 I 8 6 4 2 10-1 8 6 4 2 10-.0 10 20 30 40 50 60 0t (IN DEGREES) FIG. A.4-7 CROSS-SECTION DATA AT 200 Mc (HORIZONTAL POLARIZATION) - I 5 5 r_ = ~ L

rIr'mr 77 UNIVE RS ITY OF 2428 - 3 - T M I C HI GA N I 0 5 10 4 1 0 3 8 6 4 2 8 6 4 2 8 6 4 w I — wl V.) z z 0 u C,) 0 ix I 02 10 1 10-1 1 0 0 10 20 30 40 150 60 i/ (IN DEGREES) FIG. A.4-8- CROSS-SECTION DATA AT 200 Mc (HORIZONTAL POLARIZATION)HKE 56 I',

r L L-l UNIVERSITY 24 O F 128 -3 -T M IC HIGAN ___ all pointed at one end only. If the body were pointed at both ends, other data indicate that this relative constancy of the cross-section in the 0 -to 60-degree aspect interval would be expected not to be as pronounced (see Ogive data in App. B). 57 X ' c r\ [r l

UNIVERS ITY OF MICHI GAN 2428-3-T APPENDIX B EFFECT OF PERTURBATION TO THE SHAPE OF THE RE-ENTRY BODY ON THE CROSS-SECTION IN THE 200 TO 3000 Mc RANGE B. 1 INTRODUCTION A part of The University of Michigan Project No. 2428 was the determination of perturbations to the six R-W re-entry designs which might reduce the radar cross-section at 200, 1000, and 3000 Mc for aspect angles out to 60 degrees off-nose. In this perturbation study eight modifications, Configurations A through H, were considered. Configurations A, C, E, and G are modifications of Configurations I-III, and Configurations B, D, F, and H are modifications of Configurations IV-VI. These eight perturbations are shown in Figures B. 1-1 through B. 1-4. Configurations A and B are obtained from the R-W re-entry designs by capping the nose with a cone; Configurations C and D enclose the re-entry shape except the rear hemisphere in an ogive; Configurations E and F are obtained from Configurations I-VI by capping the nose with a cone and filling in and rounding off the step; and Configurations G and H are obtained from the R-W designs by capping both the nose and the tail with cones as well as filling in and rounding off the step. It is intended that these "caps" be made of a material which will vaporize upon re-entry so that the configurations will reduce to the R-W design. Figure B. 1-5 displays the process of rounding off the step in the case of Configuration G. In what follows the computational procedures are outlined and the results obtained are compared with the results for Configuration I. (It will be recalled from Appendix A that of the six R-W re-entry designs, Configuration I yields the smallest cross-section. ) A summary of the nose-on cross-section data is given in Section B. 4. 58__ _ r F-\ Ez-I

FrE 7 UNIVERSITY OF MICHIGAN - 2428 - 3 - T FIG. B.1-1 CONFIGURATIONS A AND B (a=20~ for Conf. A and ax=40~ for Conf. B) B.1-2 CONFIGURATIONS C AI es faired into the hemispherical s 59 ) WWF

rr iScr r r U N I V E R S I T Y OF MICHIGAN 2428 - 3 -T Conf. ca h ~ T | h rm r t I Conf. ( (m) (m ) (m) E 20~ 0.22 0.28 0.20 1.03 F 40~ 0.43 0.07 0.20 1.03 FIG. B.1-3 CONFIGURATIONS E AND F 1-. Conf. a | rm h I (m () (m ) (m) G 20~ 0.28 0.22 1.45 H 40~ 0.07 0.43 1.45 FIG. B.1-4 CONFIGURATIONS G AN[ 60 F-rF, L-LIt I I

C -- r LrI UNIVERSITY 24 O F 128-3 -T M 1 C H I GAN FIG. B.1-5 THE ROUNDED STEP IN CONFIGURATION G B. 2 COMPUTATIONAL PROCEDURES (1000-3000 Mc) The techniques employed in determining the cross-sections at 1000 and 3000 Mc were based upon physical optics. 1 As mentioned in Appendix A, the cross-sections of Configurations I-VI were determined by two methods, one yielding average cross-sections and the other taking into account phase differences. The same approach was used for Configurations A and B. 'The basic theory involved is discussed in detail in Appendix A of Reference 5. 61 L — L I EZ 7r

I UN I VERSITY OF M ICH I GAN 2428-3-T B. 2. 1 Computational Procedures for Configurations I-VI, A and B The formulas used in the computation of the cross-sections of Configurations I-VI, A,and B are listed below by method.' (Since the procedure was applied for aspect angles out to ' = 180 degrees for 3000 Mc, under University of Michigan Project 2360, the required formulas for the additional values of 4 are also included.) The meanings of the various parameters involved in these formulas are defined in Figure B.2-1. B.2. 1. 1 Method I (0 < fr/2) For ' = 00 a- = rRn + r R2 tan a. For 0~ < < an r= R + 8 tan 2 n + t) + tan (an - )] n 8-~fsinO n n For a < 2 — a - R2 + XR2 Fan (2 ). n 8nrsin, n For = — a 2 n 8 [R 3/2 R 3/2cos3/2(a)] 2 9X sin2 an cosan For - - a < < - 2 n 2 X R2 tan2 (an + 9) 8-r sing 'An improvement on the approach used here to estimate the "finite cone" contributions will be discussed in the next report on this contract. 62 I r=-[C \r

ir r UNIVERSITY OF MICHIGAN 2428- 3-T / Ca n O n Rn ot Rt R2 R1 Configuration Configu(m) (m) (m) (m) I 20~ 0.15 17~ 0.20 0.50 0.47 II " 0.30 I I III " 0.45 it i t ~ IV 40" 0.15.. if V " _ 0.30 If If VI f" 0.45. _.... A 200~ 0 B 40" 0.If.. FIG. B.2-1 THE R-W RE-ENTRY DESIGNS (CONFIGURATIONS I - VI) AND CONFIGURATIONS A AND B 63

r —rF00 -r - UNIVERSITY OF M I CH I GAN 2428 -3 -T B.2.1.2 Method II (0~< < rr/2) For ' = 0~ = o- + o2 + 2 / s (' -0 ) For 0 < 4 (a n 1= 1 3 + a3 4 + 2 1/ 1 3 cos(l -1 ~3) + 2 (ro4 cos(1 - q4) + 2 o- cos(q3 -4). Fqr a < - - a n 2 n o-: o-1 + 2 +2/O-O3 cos (4, - 4,); where 2 0' =r R 1 n 2 2 ( = rR tan a 2 2 n k \R2 tan2 (an + 4) 3 8fr sin 4 kRz tan2 (an - 4') 8'rr sin 0 64 [g@ -r

17 U N I V VE R S ITY OF MC HIGAN 2428-3-T I 4 r Rn ( 1 1 k sin a n - + 3z^ 4r Rn / 1 x \ 1 = 4wrr R2 2 tana n /cos4 sin a n IT 4-n=R2 cos (an + P) 3 ksinan IT 4' and 4 41 Rz cos (an - ) X sin a IT 4 1T B. 2. 1.3 Computations For - <.< 'rr In applying Methods I or II in the aspect range out to P = 180 degrees, the formulas listed below are employed. For - 4 -For 2< p< 2 +a = 4 + 2 4 5 004=+cT4 + 0, 6 2 VF X\ R1R2 [tan (' - cat) + tan t 1 an] 4 r sinp tan ' cos L (R -R ) sin; |_\ 2 1 where XR1 tan2 ( - at) (Y = 4 = 8rrsin 65

C7 — r-. 1% F L-7 U NI VE R SITY OF M ICH IGAN____ 2428-3-T 0 X XR/8'rr sin 4' tan 2p and 51 = XR /8wTsirnp tan 2p For 1T= 'rr/2 + 8w FR 32 -R3/2 3/ 9 Xsin2 at Cosai For1-+ a < p< ir2 t 7 +a4 +05 + 6 +2 a7a-4Cos(b-) + 2-~7~047 4 + 2 a-a- Cosq!-) ~w7~u57 5 +2 a7a-06 cs(7 - P6 + 2 a-Y4a -5 cos(04 - 9s) + 2 a4a6co s(4 -+ 2 a-a6co s(~ a-7= R 2 7t 2 - 2 a t wher 66 L-~. C~ L'rt [F'\ _

~cF-FL U NI VE R SIT Y OF M IC H IGAN ____ 2428 -3 -T 2 t t X R tan 2(OP - a~t) T4 = 8irfsin o/' 05 XR /8wTrsinot' tan 2ip T XR /8irsirnp tan 2 O 6 2 4Tr Rt 7 = x =4iTr R 1 4 (Cos IP iT 39 sinG / 4 t 4)5 = 4)4+ r, and 44-r 41(R 2 ~-R1) sin P 6 4 X For 'ir -t a P < Ir +27 4 ~ 2 if0 7 5 ~ 2 a-' G't 7 6 + 2 af 05 +c5 +6 ~6 8 Cos(4 4))+ 2 MICos 7 4 7 5~J Cos 7(q5 7a4~+2 r 6Cos Cos (4) - ) + 2 cr Cos. - Mw 67 w zL NI 1i7K

C — _____UN IVE RSITY OF 2428-3-T M IC HIGAN - iTr - CL -< p < ir (cont.-) + 2 4-Cs( 6 Cos (q5 - 0it) + 2 a' 4 8 + 2 (r I cos(#4 #8) ~2 a'aei Os (OI- 0 Cos (o 0 6 1-~ Cos (o -' # 1) - q 5' + 2 5 ~ 2 Os508 co ~ 2 11 Cos cOS(#" 56 5 ~ 2 a' 011 Cos 0 ~ 2 060a8 Co OS, - #)~+2 0.10. Cos (i 8 56 5 6 58 5 8) - 4511) + 2 1 6 J8 Cos (OI- ) - o 8) where 2 a'.7 = -rRt KR 1 tan2 (q,-(t/ a' 4 8'rrsin P a'5 c6 = 0.1 = X 4- ii 5 1' = Gr" = XR /4 irsin iP 6 2 tan 2'P 2 t an iP XR1 tan 2 ('P + oat) a'.8 = 8ir sin iP 4irfRt /Cos ' 7 K \~incL / '11r 2 I -- I 68 iF

______ UNIERSIT F-E-iF r 2428 OF MI CH IGAN ~-3-T Itr -o (it-,,p <ir (cont. ) 4 n (OS:[c~ - P iT 08 / na 5 ( = 06= k4 + iT 48 + iT 4-rr (R 2- R 1) sin4'P 04 +x and I t 4'rr (R2 - R ) sin P 08 For 'P = 1 o-=oa. +c, +oa + 2 0.0a' 7 9 1 0 79 Cos (4)' - 4 ) 7 9 ~ 2 T Cos (4)7 - 4 0)) 9 10 os O9 - 10) where 7 t 2 2 =. =irR t an a 09 1 t. 69 77

UNIVERS ITY OF MICHIGAN 2428-3-T P=ir (cont.) 41r3 2R 2 2 10 = 2 (R2 R1 ) 7 kX \innt a 2 4-rR2 r 47rR2l <9 P and 4 >r__ 9 tan a 2 and 0 X tan at The formulas as listed above are used in Method II. To apply Method I it is necessary to delete all of the "cosine terms" from the above formulas (i.e., each factor of cos (Oi - 4j) is a measure of the relative phase and should be replaced by zero). B. 2. 2 Computations for Configurations C, D, E, F, G, and H Configurations C and D, being ogival in the front, were computed using the ogive formulas of Reference 5, which are listed below (b = maximum radius = 0.50 m and a = 1/2 nose angle)': sin2 a ( +c osa c sin a For 0 < i<K - a k2 tan4 a Or = --- —------- 16 ircos6 (1 - tan2 a tan2 )3 lAn extensive discussion of the determinaon of the cross-section of an ogive and a summary of ogive experimental data will be contained in the next report on this contract. I_____70 X-ILZ [C7K F-_ F[-N\ r

UT I VERSITY OF M 1 CHIGAN 2428-3-T For. = — a b 41r tan2 (a/2) iT TV For - -a< p2 h cr = )p (1 - — ). P sinij For Configurations E, F, G, and H, the techniques for C and D were employed in the vicinity of 0 = 0 degrees, and the basic procedure used on Configurations A and B (Method I) were employed in the vicinity of P = 7r/2 - an. The cross-section curve is faired in between the two points determined. B.2. 3 Cross-Section Data at 1000 and 3000 Mc The data for Configurations I - VI, A, and B are given in Figures A. 3-1 -A. 3-9. The cross-sections of Configurations A - H at 1000 and 3000 Me are shown in Figures B.2-2 and B.2-3. In examining Figures B. 2-2 and B. 2-3 it should be recalled that the cross-section of Configuration I is approximately 10-1 m2 throughout the aspect interval. B. 3 CROSS-SECTIONS FOR CONFIGURATIONS A - H IN THE 200 TO 500 Mc RANGE B. 3. 1 Configurations A - F In Appendix A the nose-on cross-sections of Configurations I - VI were computed by a "creeping-wave" method. Applying this technique for Configurations A -F, the results shown in Table B. 3-1 are obtained. The contribution from the rear, o-2, is the same for Configurations A - F as it was for Configurations I - VI, but the nose contribution, 01, 71 - ILZ J I 7K_ 1

L- I v LUNIVERSITY OF MICHIGAN 2428 - 3 - T 10 8 6 4 2 8 6 4 1 LU ILU w O LI) z L1 Lo, O 0 Of: u 2 l-1 10-1 8 6 4 2 10-2 8 6 4 2 8 6 4 2 10 -4 8 6 4 2 8 6 4 2 10-6 0 10 20 30 40 50 60 ' (IN DEGREES) FIG. B.2-2 CROSS-SECTIONS OF CONFIGURATIONS A - H AT 3000 Mc (X=0.1 m) 72 - r — rJ7 77

I 77 IUNIVERSITY OF MICHIGAN 2428 - 3 - T 10 8 6 4 2 1 8 6 4 2 V) 0t uJ ILU1 w (/) z z 0 u (I) (I) 0 Ur u 10'8 6 4 2 10-2 8 6 4 2 10-3 8 6 4 2 10-4 8 6 4 2 10-5 8 6 4 2 10-6 0 10 20 30 40 50 60 b (IN DEGREES) FIG. B.2-3 CROSS-SECTIONS OF CONFIGURATIONS A - H AT 1000 Mc ( = 0.3 m) 73 X i^ I X [r r - r -lo[ I

I - TABLE B.3-1 7Il] i NOSE-ON CROSS-SECTIONS (in Square Meters) OF CONFIGURATIONS A - F __ 200 Mc _350 Mc _ 500 Mc Lower Upper Lower Upper Lower Upper Bound Average Bound Bound Average Bound Bound Average Bound A.11.13.15.026.032.038.011.013. 016 B.047.15. 26. 0086. 039. 069.0029.017. 030 C.017.18.34.0023.048.094.00053.021.042 D.077.53 1.19.034.16.29.019. 077.13 E.11.13.15.026.032.038.011.013.016 F. 047.15.26.0086.039.069.0029.017.030 00! H-4 0 - n, "I0 >e -J mlmmmlmmw

UNIVERSITY OF MICHIGAN 2428 -3 -T is in most cases less for Configurations A- F.1 The determination of o-1 and 0-2 for each configuration yields, as it did in Appendix A for Configurations I -VI, an average cross-section, o-r + cr02, and bound estimates, (/ --- t+/2)2. These three values are given in Table B.3-1. A comparison of these results with those for Configurations I - VI appears in Figures B.3-1 -B. 3-3. Examination of these figures indicates that little can be gained, cross-section-wise, at 200, 350, or 500 Mc, by changing the re-entry shapes to those of Configurations A - F.2 B. 3.2 Configurations G and H Although the cross-sections of Configurations A -F are not significantly less than those of the original Rudolph design, Configurations G and H, due to their ogival shape, can be expected to yield large reductions in cross-section even at 200 Mc. B. 3.2. 1 Experimental Ogive Data at 200 Me Experimental ogive data obtained by Ohio State University, and reported in Reference 15, included measurements made at 3000 Me 'The reader is referred to Section A. 2. 1 for a discussion of this creeping wave method. Its application to Configurations A - F is similar to its use on Configurations I - VI in that the contribution from the rear, 0-2, is approximated by that from a sphere of radius equal to 0.5 m for each of the Configurations A- F. The differences revolve around the determination of the contribution from the nose, o-1. For Configurations I- VI the nose is spherical and thus o-1 in those cases is given by iT Rn2 where Rn is the radius of the spherical nose; for Configurations A- F the nose is pointed and thus the contribution, 0-1, is given by X2 tan4 a 16rr ZThis conclusion is also verified by the experimental data on the Carrot, the Cone, and the 7-OC models discussed in Section A. 4. 75 mi!fr r" Ilr

F[Fl —rv- I T )!= ~ UNIVERSITY OF 2428- 3 -T MICHIGAN 103 8 6 4 2 102 8 6 4 9 10 (n ILUI LU C-,) z z 0 LUj (n) 0 0 2 8 6 4 2 8 6 4 2 1 10-1 8 6 4 2 10-2 8 6 4 2 10-3 8 6 4 2 10-4 FIG. B.3-1 NOSE-ON CROSS-SECTIONS OF CONFIGURATION I - VI AND CONFIGURATIONS A - F AT 200 Mc 76 r__ Ir

r - I ar-r U N I VERSITY OF 2428-3-T MIC HIGAN 103 8 6 4 2 102 8 6 4 2 10 8 6 4,) - LU IU( CO z 0 u 2 1 8 6 4 2 10-1 8 6 4 2 -2 102 8 6 4 2 8 6 4 2 10-4 FIG. B.3-2 NOSE-ON CROSS-SECTIONS OF CONFIGURATION I - VI AND CONFIGURATIONS A - F AT 350 Mc 77 C-=L L_

rlT r —I ) Fr C_- I I I r- 1 UNIVERSITY 24 O F i28 - 3 -T MICHIGAN 103 8 6 4 2 i.. T T i 4 -. * - I I 102 8 6 4 10 gn Iz 0 I,u Lo 0 U 2 8 6 4 2 8 6 4 2 1 T Upper Bound, Average, and Lower Bound = As Computed by the "Creeping Wave" Method > o - ad I. i-o 0 2, -. C- e-: = O c. =:== --- '. = cm 0 -- - U ~L, = O,;~. - e0 i 10-1 8 6 4 2 10-2 8 6 4 2 10-3 8 6 4 2 10-4 FIG. B.3-3 NOSE-ON CROSS-SECTIONS OF CONFIGURATION I - VI AND CONFIGURATIONS A - F AT 500 Mc 78 I',L- r=-, 77

C_F i7 UNIVERSITY 2 - O F 428-3-T MI CHIGAN (X = 10 cm) on two ogives: one of length 10 cm and maximum diameter 2.05 cm, and the other of length 20 cm and maximum diameter 5.18 cm. Upon scaling this data to 200 Mc, one obtains ogives comparable in size and shape to Configuration G. Table B. 3-2 contains a comparison between the dimensions of Configuration I (the unperturbed Rudolph shape), Configuration G, and these two ogives, denoted as "Ogive A" and "Ogive B". The data given in Reference 15 yield values of cross-section for the 10 cm ogive at four values of P(0, 34, 75, and 90 degrees), and for the 20 cm ogive at eight values of P(0, 15, 20, 43, 45, 50, 85, and 90 degrees). Figure B. 3-4 contains these data scaled to 200 Mc. In drawing these curves only the points defined by the angles mentioned above were used. Figure B. 3-4 also contains the theoretical (physical-optics) curves for Ogives A and B. The theoretical curves do not differ from the experimental curves by more than a factor of 10 over most of the interval.,,,,, i., TABLE B. 3-2 COMPARISON OF DIMENSIONS (L = total length; D = maximum diameter; and a = 1/2 nose angle) Configuration L (in m) D (in m) a I 2.14 1.00 G 2.91 1.00 20~ Ogive A 1.50.31 23~ Ogive B 3.00.78 29~ L.9 m 79 )5 1 jZ "

L- r== 7 r- I_ I U NIVE R SITY OF 2428 - 3-T M I C HIGA NI Io23 8 6 4 2 8 6 4 2 10 8 6 4 cui w D (I) z z LLI w (I 2 I 8 6 4 2 10 - 10-2 E e 4 A 10-3 f A f A 10-4 0 20 40 60 80 V' (IN DEGREES) FIG. B.3-4 CROSS-SECTION OF OGIVES AT 200 Mc (L/X =1 AND 2) - - 80

UNIVERSITY OF MICHIGAN ______ 2428-3-T B. 3. 2. 2 Cross-Sections of Configurations G and H at 200 Me In view of the agreement between experimental and theoretical results for ogives shown in Section B. 3.2, physical optics ogive methods were used for Configurations G and H at 200 Mc. 1 The results obtained are shown in Figure B.3-5, where they are compared with the Configuration I data. B. 4 CONCLUSION The perturbation analysis described above has indicated that, of the perturbations considered, the one referred to as Configuration G yields the largest reductions in cross-section in the aspect interval 0~< g < 60~. Capping the nose with a cone without adding any additional protuberances to the Rudolph shape does not decrease the nose-on cross-section of the warhead to any appreciable extent. Capping the nose with a cone and rounding-off the step results in large reductions in the nose-on crosssection at the larger frequencies, but at 200 Mc no appreciable reduction is obtained. If, in addition to capping the nose of the warhead with a cone and rounding-off the step, the rear of the warhead is also capped with a cone (as in Configuration G), then comparatively large reductions in the nose-on cross-section can be obtained at 200 Mc also. This is illustrated in Figure B.4-1 for Configurations A, C, E, and G, compared with Configuration I. The other four perturbations (B, D, F, and H) are not included in this graphic comparison, since they would not enclose Configuration I. It is seen from Figure B.4-1 that the nose-on cross-section of Configuration G is smaller than that of Configuration I by factors of approximately 105 at 3000 Mc, 104 at 1000 Mc, and 102 at 200 Mc. 'The use of these ogive methods at small wavelengths has been shown to be valid by the work of Sletten (Ref. 14) and by the work presented in Reference 40. Since there is some contradictory evidence in recent O. S. U. work, additional discussion on this point will be made in the next report under this contract. r-F-\ cF.7

r -,r -- L-_,Li 7 r UNIVE RS ITY 24 O F 128 - 3 -T MICHIGAN I 102 8 6 4 2 8 6 4 2 10 1 8 6 4 LU IU u nw Z) 0 C) z z 0 IU C,) C,) 0 U 2 10o8 6 4 2 10-2 8 6 4 2 10-3 8 6 4 2 10-4 8 6 4 2 10-5 0 (IN DEGREES) FIG. B.3-5 CROSS-SECTION OF CONFIGURATION I COMPARED WITH THE CROSS-SECTIONS OF CONFIGURATION G AND H AT 200 Mc 82 C^

I, 107 I 1 II 77 UNIVERSITY OF MICHIGAN 2428-3-T 10 8 6 4 2 8 6 4 1 c,. uJ IUJ 0 Z 0O I-. U LIU O U) u) 0 g~ 2 10-1 8 6 4 2 10-2 8 6 4 2 10-3 8 6 4 2 10-4 10 8 8 6 4 2 10-5 8 6 4 2 10-6 0 500 1000 1500 2000 2500 3000 FREQUENCY (IN Mc) FIG. B.4-1 THEORETICAL NOSE-ON CROSS-SECTIONS AS A FUNCTION OF FREQUENCY FOR CONFIGURATIONS A, C, E, AND G COMPARED WITH THE CROSS-SECTIONS OF CONFIGURATION I 83 ^ll - - 77

- II I UNIVERSITY OF MICHIGAN 2428-3-T APPENDIX C CAMOUFLAGE MATERIALS A radar absorbing material has been developed by the Deutsche Magnesit,AG, Munich, Germany. Representatives of this company were invited by the Rome Air Development Center to fabricate their absorbing material at RADC and then apply it to models furnished by RADC. The results are very good when the material is manufactured for use in one small frequency region lying in the S- to K-band region. In fact, it was claimed that there was a 26 db reduction in cross-section at X-band. The German scientists had no mass production methods available for the fabrication of their material. They also had no method available for making broadband materials, but they thought they could do so without too much difficulty. They attempted to do this while at RADC. The results were not good, but this may be because of the insufficient time available. After they had left, work was started at RADC under the direction of J. Vogelman, Director of Electronic Warfare Laboratory, to find out why the obtained reduction existed and to analyze its division into absorption and increased scattering at other than the back-scattering direction. It was found that half the decrease in db was due to absorption and the other half was due to off-angle scattering. Due to the newness of this material, and due to the fact that br)adband techniques have not yet been completed, the only conclusion to be drawn at this time is that the Deutsche Magnesit material may be usable for camouflaging missiles and aircraft. Since RADC is continuing its investigation of this material for the U. S. Air Force, it is felt that its analyses will show how good or bad this material is. At this time the authors do not recommend any further action than that which is presently going on at RADC. It is recommended that people in the ICBM field follow RADC's latest results on this material for possible applications to ICBM. 84 X [rX r[ ^ r- __ F^I - \j r -<_: 1

UNIVERS ITY OF MICHIGAN 2428-3-T APPENDIX D UTILIZATION OF FISSION WASTE PRODUCTS FOR DECOY PURPOSES D. 1 INTRODUCTION Fission waste products are radioactive and therefore produce ionization. It has been conjectured that if the wastes are carried in a decoy, the radar cross -section of the decoy might be sufficiently enhanced to improve the chance of the decoy being confused with the primary vehicle by proximity-fused interceptors. Also, if the decoys were used to draw off enemy defenses to a false location by confusing ground-based radars, one would want to increase their detectability. This idea is considered here for two types of missiles and found impracticable. Another group has reached the same conclusion (Ref. 16) for sowing the wastes from airplanes as a sort of chaff. The first type of missile considered is a hypothetical intercontinental ballistic missile. The following properties, which should be duplicated by an effective decoy, are assumed: a velocity of 10 km/sec, and flight altitudes ranging from ground level to above 200 km. Since in its noseon descent such a missile would not have a large cross-section, it is assumed that something between wT and 10 rr square meters would be a reasonable radar cross-section to simulate. The second type is a V-2, representative of the shorter-range missiles. Experimental data quoted in Reference 17 indicate typical altitudes up to about 90 km and velocities ranging from about 3. 25 km/sec down to about 0. 5 km/sec at different altitudes. The size of the cross-section to be simulated is the same as above. D. 2 PHYSICAL PRINCIPLES INVOLVED D. 2. 1 Critical Density Upon solving the propagation equations in an ionized medium for a given radar frequency, it is found that if the electron density exceeds a I________________________85___ L-i z

UNIVERSITY OF [_..__ FI-ir M ICHIGAN _______ I 2428-3-T critical value the surface bounding this dense region acts as a perfect reflector (Ref. 18). This critical density is given (Ref. 1 ) by Nc = (f2/81) X 106electrons/cm3, where f is the frequency in megacycles. A subcritical-density region surrounding the critical-density contour acts as a diffuse reflector, reducing the back scattering from the critical-density region. (The cross-section of a region which nowhere attains critical density is very much less than that of a geometrically identical criticaldensity region. ) The critical-density region should be at least a few wavelengths thick if the propagation equations are to have meaning. A radius of about one meter should be sufficient, even down to L-band. D. 2.2 The Nature of the Fission Waste Products The radioactivity of the fission waste products consists almost exclusively of P- and V-radiation. Therefore, other types of radiation do not come into consideration. A representative figure for the power released is 100 watts/gm, divided equally between the P3's and T 's. This figure, based on radiochemical analysis, does not take into account the loss of output due to self-absorption in a bulk source. The mean energy of both types of radiation is 0. 7 Mev, the maximum (3-energy about 5 Mev (Ref. 191). D. 2. 3 Range-Energy Relations For I.'s and Y 's For electron energies such that v c (since v/c = 0. 5 corresponds to less than 80 KV, this covers nearly all of the (3-activity present here), the energy loss of a P per g cm-2 of absorber is given by (Ref. 20) k =4 N(Z/A)2 me c2 T3/2 k = 4 N(Z/A)rre- rns c in I-1/ — -1.45, 'The authors wish to express their thanks to J. V. Nehemias for communicating these results to them. 86 7 r - rll- L-i7

UNIVERSITY OF MICHIGAN 2428-3-T where N = Avogadro's number atomic number Z/A = atomic weight, for the stopping element, re = e2/me c2, the classical electron radius, me = electron rest mass, ET = total electron energy, E = rest energy, m c2, and I(Z) = ionization potential of stopping material. Since k varies monotonically by less than 30 per cent from 0 up to 5 Mev, it may be taken as constant at its value for the mean 3 energy of 0. 7 Mev with little error. Using as order of magnitude figures for the fission products Z/A = 0.45 and I(Z) = 8 ev gives k = 1. 6 Mev/g cm2. Using a -3 density p = 5g cm for the wastes yields a loss rate of 8 Mev/cm, or a range of about 1 mm for 0. 7 Mev P's in the source. For ranges in air, Z/A = 0. 5 may be used, and little error is involved in taking I(Z) = 13. 5 ev, correct for atomic oxygen. p must correspond to the altitude considered. The range of the electrons in the waste material being so small, absorption of the electrons in the source itself (self-absorption) will greatly reduce the power output of the source into the air, allowing only those electrons originating near the surface to escape. For electrons of given energy, consider a strip of area A and thickness t equal to the range, and compute what fraction of their energy will escape through the outer side of the strip (averaging over points of origin). This fraction is F=l- (s/t), where s is the mean path of the electron in the strip (since energy loss is proportional to path length). 87 I = c I- Li7K

C7=- r= r ci UNIVERSITY OF 2428-3-T M I C H IGAN __I G A N A t Then, t 2wr F=- J 4r At2 0 cos 1 (a/t) sin do (t - a/cos O) 0 Ada t 0 da t(l - a/t) + a In (a/t)] = -2t2 - t2/2 - t2/4 =-+-. This derivation involves the assumption that the thickness of the strip is small compared to its radius of curvature (which agrees with the actual dimensions considered later), and assumes a straight-line path. If a thicker strip is used, only electrons originating within a distance t from the surface will contribute. Neglecting range straggling, electrons originating at a distance from the surface greater than their range will be absorbed before they reach the surface. The use of a thinner strip is not advisable, since the reduction of surface activity due to the presence of less source more than counterbalances the lowered selfabsorption. 88 - iz EZ 7E [F-N r 7r

UNIVERSITY OF M ICHIGAN 2428-3-T For a 0. 7 Mev electron in air, k = 1. 5 Mev/g cm-2. Near sea level, the air density is about 10-3 g/cm3. Thus the energy loss per unit path length is 1.5 X 10-3 Mev/cm, and the range is about 500 cm or 5 m. As the altitude increases, the air density drops off more and more rapidly (e. g., 10-12 g/cm3 at 170 km) and the range increases proportionately. Gamma-rays can be considered as attenuated as exp(-uix) (at least, for Lx of order unity or less). For the v-energies of interest, the interaction of the gamma-rays with matter is practically all due to the Compton effect (pair production competes prominently only at much higher energies, the photoelectric effect only at very much lower ones). The absorption coefficient, p, for the Compton effect is (Ref. 21) " = 2rr NpAre - In (1 2a) 1 12a) + 3a. + ln (1+2a)- (1+2a)2J' where a = r -energy in units of the electron rest mass. For a 0. 7 Mev -ray in air near sea level, pL = 4X10-5 cm-; hence, the Y-ray loses -63per cent of its energy in about 300 meters. This distance varies with air density (consequently with altitude) in the same manner as the electron range. Thus, the -rays are about 100 times less efficient than the p's in producing ionization. Inasmuch as the available power is evenly divided between I3's and Y's, and that portion of it in the form ofp's is far more effective, consideration is limited to the electrons alone. 1 In view of the high self-absorption of the source for electrons, the power output per gram quoted above (neglecting absorption) is not directly applicable to a bulk source. Instead, the useful quantity is the power output per unit area of the source. There are 50 watts/gm emitted in the form of n's. Using a density p = 5 g/cm3 for the waste products, 'The overly generous assumption that all the radiation is in the form of p's would reduce the source size by less than a factor of two, leaving the conclusions unchanged. I........ 89 S- L-\

U N I V E R S I T Y OF MICHIGAN ______N Z428-3-T as before, this is 250 watts/cm3. From the discussion of self-absorption above, the power released by a layer of thickness t cm, equal to the range, is 1/8 X 250 Xt watts/cm2, and deeper-lying material does not contribute. Since the rate of energy loss is 8 Mev/cm, t = E/(8 Mev/cm) and the output is (250/64) E watts/cm2, where E = electron kinetic energy in Mev. Averaging E over the electron spectrum gives 2. 7 watts/cm2. This is the surface activity to be expected from a thickness equal to at least the range of the most energetic 3's (about 6 mm). A thinner strip will give less surface activity and a thicker one will yield no more. D. 3 COMPUTATIONS D. 3. 1 Introduction This section contains a computation of the source strength required to maintain a critical density of ionization over a volume adequate for decoy purposes. It will be shown that an impractically large source is needed. Since this consideration alone rules out the use of fission wastes, others need not be treated quantitatively. However, these additional limitations are stated here. First, there is the fact that the 3-energy spectrum is continuous. This implies a continuous distribution of ranges, and therefore an ion density distribution which tails off away from the source. Thus, beyond the region of critical density a subcritical density is present which reduces the radar cross-section. Further, the phenomena of recombination and attachment to form negative ions have been ignored. These would work to lower the electron density and require a larger source. On the other hand, if recombination is not sufficiently rapid behind the moving source, an ion trail may be formed. In the latter case, the radar reflection from the ionization produced by the decoy would not resemble a moving missile. The fact that the P's must be relied on to produce the ionization raises a problem as to mounting the source on the decoy. As was indicated in the discussion of self-absorption, one centimeter of solid 1_- ~90 -- urILZ Zj r

UNIVERSITY OF MICHIGAN 2428-3-T material would stop all the P's. This effectively precludes carrying the source inside the decoy. The nature of the fission waste products raises serious handling problems. The high level of radioactivity would make the loading of the decoy unpleasant and result in contamination difficulties at the launching site. The effects of this radioactivity upon the chemical fuels powering the decoy should also be considered. Finally, the large amount of power dissipated in self-absorption would create significant thermal problems. D. 3. 2 Source Strength Requirements for Critical Density To simulate a missile, ionization of at least critical density must be made to fill a volume comparable with the volume of the missile. Hence, at any moment, a critical density region of radius rl (ideally a couple of meters) is required. As the decoy moves, the region that has to be filled changes. If the missile velocity is L cm/sec, a cylinder of radius rl and length L cm must be filled in one second. Since the source is located on the axis of the cylinder, the ion density decreases as the distance from the axis increases. Therefore, critical density must be achieved near the surface of the cylinder (which will result in a higher density being present further inside). For simplicity, assume the source to be a sphere of radius r0 (this shape also minimizes reabsorption, in one part of the source, -of electrons that have escaped the surface at another point). To fool the radar, the ionized layer must be at least a meter thick, requiring that rl - ro > 100 cm. Given a source whose power output per unit surface area is 2. 7 watts/cm2, subject to the conditions just enumerated, the minimum ro required will now be computed. The calculation is not carried out exactly; instead, a lower bound to the answer is obtained in three different ways (which is the most restrictive depends on the altitude and velocity considered). D. 3.2. 1 ICBM (near sea level) To maintain the critical density out to r1, it is necessary to supply, 1 9 1 F-\ Ez-7

zLSD EJ 77 UNIVERSITY OF 2428-3-T MI C HIGAN every second, a cylindrical shell of radius r1 and thickness dr with an amount of energy = 2Trr1 L dr D I(Z) where D = critical density, and I(Z) = ionization potential (13. 5 volts). Let f(E) dE = number of electrons supplied by the source, with energy lying between E and E + dE, k(E) = energy loss of electrons per unit path length (approximately independent of E), ds = path length of electron through the cylindrical shell (ds >~ dr), and E' = minimum energy of electron at source if it is to have a range sufficient to arrive at the cylindrical shell. Then oo ~- k X f(E)dE ds E' c00 = k Ef(E)dE E' co ds <k - E' ds E Ef(E)dE 92 Liz EZErZ 77

I E I \ X ~I I77 U N I V E R S ITY OF 2428-3-T MIC HIGAN oo 0 <kd, Ef(E)dE. For different orientations of the electron path, ds is proportional to the path length traveled in order to reach the cylindrical shell (minimum range), which in turn is proportional to the minimum required energy E'. Hence, ds/E' is independent of direction and can be replaced by dr/E1, where E1 is the minimum energy for radial motion. If the dimensions of the source are small compared with rl, E1 = kr1. Hence, E <kds E' dr - k1 -k E1 00 0 Ef (E)dE 00 0 Ef(E)dE co dr J Ef(E)dE r1 0 oO and 0 Ef(E)dE > Trr 2 L D I(Z). I 93 riLZE

I _ UNIVERSITY OF MIC HIGAN ____ 2428-3-T co The energy supplied, Ef(E)dE, comes from a sphere of radius ro. In one second, this source supplies 2. 7 joules per cm2 of area or a total of 2. 7X 4rr 2 joules = 2. 12 X 1020 r0 ev. For minimum energy input for the ICBM (L = 106 cm), 2. 12 X 1020 r02 ev = 2Trr12 lcm X 6. 9 X1011 X 13. 5 ev r02 r2 = 0.28 rl r0 = 0.53 r. Sinice rl - r 1 O00cm for simulation, the minimum ro = 110 cm, so the diameter of the sphere has to be more than two meters. Thus at low altitudes, the fission products ionization source is at best no more effective as a decoy than a metallic object of comparable size would be. Neglect of the source dimensions is not essential. A / - - 94 ) — Zr7

E_77 UNIVERSITY OF MICHIGAN 2428-3-T The path difference of an electron arriving at a point on the cylinder from different places on the source can be taken into account. Assuming isotropy of emission direction from the source, the correction factor is (l)av. rl From the geometry, r2 = r2 + r2 1- 0 max singdG r 0 fmax singdO 0 - 2r 0r cos 9 and 9 = cos1 r0 0 1 max r1 r Let x = --, cos 9 = u, then rI r rl 1+x2 -Zxu x 1 + x2 - 2x u du \r av - (1 - X) 3x 1 du x [ (+x)3/2 _ (1 -) 3/2. 1 For x = 2, 0.7. av The minimum diameter of the sphere is changed to 1 1/2 meters. The conclusions are unaltered. 95 I c 1 7r t 1E - r

U N I VE RS ITY OF MICHIGAN 2428-3-T D. 3. 2. 2 ICBM (higher altitudes) At low air densities where the range of most of the source electrons greatly exceeds r1, the replacement of ds/E by ds/E' represents a gross underestimate of the source strength required. Instead, 1/E is left inside the integral and a mean value of ds, or better ds/dr, is used: ds s 1 dr r = cos 0 r 1 Electrons will leave the cylinder at angle 0 ranging from zero to a maximum value such that s is the electron range. An overestimate is obtained by choosing smax = Emax/k and assuming an equal number of electrons leaving at all values of 9 in the range so defined (i. e., 04 0< 0max = cos-1 (krl/Emax) ). Then Omax dO $ cose /ds) 0 1 ay< Imax = ln (tan 0max + sec 0max) dr rv max JdO 0 96 ~~Izz F-\I77

I ---r-. 77 _______ UNIVERSITY OF M C H I GAN 2428-3-T 1 in 0max 1 + sin 0max cos 0max = os-1 -1 1 + 1 + krl ) v \ max/ Emax/ kr1 Emax At low densities, kr1 << Emax, and av 2 Emax kr1 2 Emax krl o00,6 = k f(E)dE ds < -k drin 1T E' 00 f(E)dE E' < -k dr In IT 2 Emax krl 2 Emax krl oo 0 0 f(E)dE E f(E)dE, 2 1T k - dr In E where E is the mean energy. 00 Jo E f(E)dE > w2 rl L D I(z) E 2Emax k Inkr krl 97 [l f E ^^^ l' ' ^\\ II E I 77

I'C_- r- r [7 I UNIVERSITY OF M1 CHI 2428-3-T;AN __ and, at the minimum, 2 1.89 X 10-8 L r1 r0 6. 25 P In P r1 The condition kr << Emax means 1.6 X 106 pr << 5 X 106 or- >r. P 6.25 1 ri 1 Thus, In Irn - In, and pr1 p 6.25 p r 2 1.89X10-8 L 0_ = — r1 P In 1 P Near sea level, p - 10-3 gm/cm3 and a reasonable r1 is 200 cm, so that krl/Emax. 0. 06. At higher altitudes, the inequality is even better satisfied. For the minimum ro, impose again the condition that rl - rO = 100 cm (minimum ionization layer for simulation), r2 1. 89 X 10-8 L r0 + 100 p In For the ICBM at different altitudes, the source size is given below: Altitude in km Air Density in gm/cm3 (Ref. 22) Diameter 10 4.21X 10-4 0.5 m 20 9.34 X 105 1.2 m 50 1.16 X 10-6 26 m 100 8.60 1010 21 km 200 1.71 X 10-13 7. 5X104 km.. 98 [- fZC 'i '

__ UNIVERSITY OF 2428-3-T ri7K M I C H 1 G AN I Thus, while this lower bound on the source is somewhat less stringent than the previous one at low altitudes, it imposes far greater (in fact, quite unattainable) limitations at the higher altitudes. It would appear at first sight that the large value of ro obtained at the higher altitudes conflicts with the omission of source dimension considerations from the derivation. This is not correct. Indeed, including these considerations results in changing the maximum of s to (Emax/k) + ro. This simply makes Omax = cos-l r = cosEmax + ro k kr1 Emax (+ kr0 \ \ maxJ Since kr0 < kr << Emax, this correction factor can be neglected. D. 3.2. 3 The V-2 The V-2 has a lower velocity than an ICBM. Furthermore, its maximum altitude is only about 150 km, which is considerably less than that of an ICBM. These differences reduce somewhat the power requirements. Applying the calculation of Section D. 2, Altitude in km 20 32 48 63 88 Velocity in km/sec 0.995 1.49 1.32 1.08 3.25 (Ref. 17) Density in gm/cm3 9.34X10-5 1.37 X 105 1.48 106 2.42 107 5.65 X10 (Ref. 22) Diameter.16 m 1.1 m 3.8m 13 m 1.1 km 99 c-L I \ -

C= rUN I VE RS ITY I OF CHIGAN ___OF MI CHIGAN 2428-3-T As the altitude increases, Omax reaches a cutoff value due to the size of the cylinder, namely Omax = tan'- (L/2r1). L/2 Then (ds) dr av 1- ln(tan Omax Omax + sec Omax) 1 tan- 1 L 2r1 L In + 1 + Zr1 For (L/2)2 >> ri, this leads to a maximum value: ds \drI av 2 L -ln - Tr r1 Then 2. 12 X 1020 r02 w2 r 1 L D I (z) E kln,I r1 4.02 X 1012 rl L pln -L r1 100 LIZ

_ _ UNIVERS I TY OFM I_____ UN IVE RS ITY OF M H I AN MI1C HIG A N ______ r 2428-3-T r02in L rl 1.9 X 10-8 L p Set r1 = r0 + 100 cm (minimum): r0 In r l + 100 r + 100 1.9 X10-8 L p Results obtained in this way are: Altitude in km 32 48 63 Diameter in m 1 1/2 7 40 At an altitude of 88 km, this approach becomes meaningless because r > L. D. 4 CONCLUSIONS The proposal to utilize fission waste products in decoys to simulate the radar cross-section of conventional missiles or the ICBM has been examined. From source strength (and size) considerations alone, the idea has been shown to be impractical. Even near sea level, a lower limit on the power required leads to a source of size comparable to that of a metallic object giving the same cross-section. At higher altitudes (over most of the range for the V-2 and practically all of the range for the ICBM), the source required is unattainably large. Other difficulties, mentioned but not treated quantitatively, raise further doubts as to the feasibility of the proposal. 101 7 I "' 1 77

I'Z77 U N I VE R S I TY 24 O F 28-3-T M I C H I GA__A APPENDIX E DECOYS Before considering possible defenses against decoys from the radar cross-section point of view, we will summarize the work which has been done for the aircraft case. One first computes the radar cross-section of the aircraft. Then one investigates means of simulating its radar cross-section and flight characteristics with a smaller body. In Reference 5, the radar crosssections of the B-47 and B-52 were computed from 65 Me through X-band. It was found that an object of approximately 1/100th the volume of the aircraft could be used to reproduce the radar cross-section the object could also reproduce the flight trajectory for one flight. In Reference 2 the possible defensive methods outlined below and analyses for the design of decoys to simulate the reflection characteristics of the aircraft were presented. In Reference 6, the scattering matrix of the B-47 was determined at S-band. The latter was useful in discussing polarization methods of distinguishing between decoys and aircraft. If one assumes that the defense has linearly polarized monostatic radars of fixed frequency (L-band or higher), then the radar cross-section of the aircraft can be simulated through the use of corner reflectors in decoys. However, the enemy may then use frequency comparison methods to determine whether the vehicle is an aircraft or a decoy. For example, if the enemy's main radar is at S-band and its frequency comparator radar is at X-band, one would expect the cross-section of the aircraft to remain approximately the same for the two radars while the crosssection of the decoy would increase by a factor of 10 from S-band to X-band, due to the fact that the cross-section of a corner reflector depends on wavelength approximately as 1/X2. On the other hand, it has been our experience (Ref. 23) that the radar cross-sections of aircraft such as the B-47 and the F-86 are only slowly varying functions of wavelength as soon as the wavelength becomes small with respect to the major dimensions of the aircraft. 102 L- Z Lr7

-=3 I I \\ 7 -_______ UNIVERSITY OF M C H G AN 2428-3-T This detection scheme can be countered as follows. Corner reflectors with appropriately curved faces will simulate the aircraft at S-band but cannot be recognized by the above technique. This is so because curvature of the faces of a corner reflector greatly modifies the behavior of its cross-section as a function of X. A feeling for this can be gotten by observing that the angle error formulas for corner reflectors are wavelength-dependent, and thus changes in angle which are insignificant at one wavelength can cause severe reduction in the radar cross-section at smaller wavelengths. It has been shown that a corner reflector with spherical faces can be designed that eliminates the 1/X2 wavelength dependence. The beauty of the spherical face idea is that after one matches the cross-section at one frequency it remains essentially a constant for all higher frequencies. Another means of "unmasking" corner reflectors in decoys is the use of a transmitter which is circularly polarized (e. g., to the right), together with two receiving antennas, one of which is right-circularly polarized and the other of which is left-circularly polarized. When the corner reflectors are so oriented that single or triple reflections dominate, the left-circular polarization receiver will get 10 to 20 times as much power as the right polarized receiver. When double reflections dominate, the reverse situation applies. In any practical case, one of these orientations will occur. Thus, decoys using corner reflectors could easily be differentiated from aircraft (for which the power distribution between the two receivers varies between 3:2 and 2:3). This method of detection can be countered by the use of dielectric material on one of the faces of the corner reflector (Ref. 24). The procedure consists of properly coating one of the faces with a dielectric, so that the ray picture remains essentially unchanged, but the polarization is rotated in such a manner that the return is now distributed between the two receivers in roughly the ratio expected from aircraft. Another means of defeating this method of detection is by the use of holes of appropriate size in one of the cornerreflector faces or the use of one-half-wavelength protuberances. These last two approaches are not as efficient as the Ohio State University technique (Ref. 24) because they cause a significant degradation in the energy returned. I103

U N I VE RS I TY OF MIC HIGAN __I A 2428-3-T Another possible means of discriminating between decoys and aircraft would be the use of bistatic radars (Ref. 25 and 26). Bistatic radars are presently being used in the McGill Fence and the DEW line. Discrimination is based, in this case, on having one receiver near the transmitter and one or more receivers at a distance from the transmitter. In the case of aircraft (assumed to fly between the remote receiver and the transmitter), the radar cross-section at the remote site is often much larger than, and in almost all cases at least comparable to, the radar cross-section at. the receiver which is near the transmitter. In the case of decoys, the cross-section at the transmitter site is always much larger than the cross-section at the distant receiver, due to the property of the corner reflector of reradiating energy into the quadrant from which it came. Some tentative schemes (obviously not fully adequate) have been discovered to counter this method of detection (Ref. 2). Two other methods for the detection of decoys have been brought forward; one involves scintillation and glint and the other broadside discrimination (it is assumed in the discussion that the frequency of detection is above 1000 Mc). The power spectra for one-third-scale V-2 type ballistic missiles are contained in this report. The authors have no reason to believe that the power spectra which would be obtained for the Rudolph missile would be significantly different from these. In fact, if one analyzes the return from aircraft like the B-52 and B-47 (Ref. 27), he finds the shapes of the power spectra are rather similar to those for the missiles. This, of course, does not mean that special equipment cannot be built to detect spectral differences. However, it would be extremely difficult to do this without a thorough knowledge of the structural behavior of the enemy's missile and the enemy's decoy. The broadside discrimination method is more applicable to the aircraft problem than the missile problem, since in the search phase the Rudolph missile would be out of the sensible atmosphere, and as a result Rudolph never would have to "put its broadside forward." In fact, one of the best methods of reducing the cross-section of the Rudolph missile would be to guarantee that its nose is always pointing in the general direction of the enemy's search radars, with the temporary alteration of the metallic skir4 described in this report applied, so that the cross-section of the Rudolph warhead in the nose-on direction would be significantly reduced. 104 r — Z- r

UN I VE RS ITY OF MI C"HIGAN__ 2428-3-T In the case of low frequency detection (e. g., the 60 to 100 Mc range), the decoys must carry barrage jammers if they are to duplicate the low frequency cross-sections of the Rudolph missile. However, the background due to cosmic noise, meteors, and stray ionization sharply limits the feasibility of detection by search radars in that range. It has been said that if American radars were operated at 60 Mc, it might be possible for the "Commissar of Jamming of the USSR" to push a button and create so much radiation that he could jam out all our receivers in this frequency range. Due to the great range of propagation at these low frequencies, it is clear that the above statement has an aura of truth about it. The bistatic method for detecting decoys is not presently being countered. However, the data-gathering and data-processing equipment necessary for the use of this method would tax our national economy. Although it has been assumed that the United States has more computing equipment and data-processing equipment available than the Russians, it is felt that even the United States could not afford in the next few years to use a great many remote receivers and the associated data-processing equipment in order to discriminate between decoys and aircraft. Thus, it seems reasonable to assume that the USSR also will not be able to afford such defensive procedures. It appears, therefore, that decoys could substantially increase the "probability of kill" of the ICBM. Of course, until one is sure that the enemy can defend against such missiles, the necessity for a decoy effort would not seem to be established. However, if one waits until the knowledge is obtained that a defense is possible, it might easily be too late to design and build the necessary number of decoys. Decoys, of course, can be used for both the long-range and shortrange efforts; they can be used against search radars and against local defense radars. It may be possible to store decoys in the tanks section (if it is not far removed from the warhead) which can be used against either set of radars. It is possible to make decoys which are aerodynamically more highly performing than the Rudolph warhead (they might even catch up with and pass the warhead in flight after re-entry into the atmosphere). 105 riLZ" L

I U N I VE RS ITY OF M I C H I GAN 2428- 3-T APPENDIX F FURTHER ANALYSIS OF THE UNIVERSITY OF MICHIGAN DROP-TEST DATA (POWER SPECTRA FOR EXPERIMENTAL DATA) F. 1 INTRODUCTION AND SUMMARY As stated in Section IV and in Appendix E, one method the defense might attempt to employ to discriminate between decoys and aircraft is based upon possible differences in the fine structure of the radar return. An investigation of this possibility involves a thorough study of the fine structure of the radar return from aircraft, decoys, and missiles. Others have expressed the belief that power spectra associated with the radar return from these vehicles would be of significance in this detection problem. Power spectra for manned aircraft have previously been studied (Ref. 27); to the best knowledge of the authors, the power spectra presented here are the first that have been computed for the radar return from missiles. The experimental data involved in the present analysis are the measurements made during The University of Michigan's drop tests reported in References 1 and 3. These computations were sponsored by the Bendix Aviation Corporation under Purchase Order No. S-96527. The one-third-scale V-2-type models involved in the experiments are referred to here as Objects 59, 61, 62, 64, and 65. The first three models had swept-back fins and the last two had rectangular fins. Three separate items were determined under this study: a frequency distribution of the radar cross-sections of each model at each wavelength; the autocorrelation function for each run; and the power spectrum for each run. (The procedures employed and the data obtained appear in Sec. F. 2-F. 4.) The study brings into focus two basic questions: first, whether or not a fine structure analysis for the missile-decoy problem can be expected to provide significant information; and second, the question of ____ ____ ___ ____ ____ __ o6 L- r-~ 77

UNIVERSITY OF M I CH I GAN - 2428-3-T whether or not the power spectra are the correct quantities to consider in this analysis. This study deals with finite samples. The classical theory is based upon a sample of infinite size. A certain amount of arbitrariness is involved when the theory is modified for application to a finite sample. Alternative definitions of the power spectrum and autocorrelation function have been given in the literature (Refs. 28 and 31). It is pointed out that these definitions are definitely not equivalent (Sec. F. 5). The power spectra obtained by the procedure outlined in this appendix are displayed graphically in Section F.4. Even though the main contribution to the radar cross-sections of V-2-type missiles usually comes from the fins, the variation in the shapes of the curves found for Objects 59, 61, and 62 (swept-back fins) appears to be as great as the variation between these and Objects 64 and 65 (rectangular fins)1. It is possible that a study of a larger sample of missile runs by the approach used here might begin to display a pattern of differences between power spectra. An attempt to obtain greater discrimination by lengthening the time of observation for a given run would have no bearing on the practical defense problem, since the times available are even less than the intervals of observation used in the drop tests. The practical problem is further restricted, if a pulse-modulated radar is used, since the accessible frequency interval is limited; for the drop-test data, the upper limit was approximately 400 cps. Some other approach to the question of how the infinite sample theory can be modified for application to a finite sample might conceivably yield spectra which are more sensitive to differences between radar returns for different objects. If one is convinced that the fine structure of the target return is the most important information available for discriminating between aircraft 'In fact, there appears to be little difference between the spectra found here for missiles and those obtained for a B-47 aircraft (Ref. 27). 107

UNIVERSITY OF M ICHIGAN 2428-3-T and decoys, it seems clear that much further analysis is necessary before one can hope to instrument this concept into a feasible discriminator between decoys and aircraft. Such an investigation should include an analysis of the mathematical functional representation of the fine structure to yield the maximum amount of information. After the above method of representation is established and a method of presentation is fixed, one should determine (probably experimentally) whether the differences in the fine structure between missile and decoy are greater than the differences between missile and missile. The latter differences are expected because of differences in the conditions under which the observations are made or unanticipated alterations in enemy missiles (as to structure or structural behavior). If this investigation is to be used to analyze vehicles which are out of the atmosphere for almost all of their trajectories, one should bear in mind the possibility that transient effects (which disappear rapidly in flights through only the sensible atmosphere) might not be damped out. It is clear that, for economic reasons, vehicles cannot be fired out of the atmosphere to analyze transient effects which, by their very nature, may not be reproducible from air frame to air frame. In short, one should first complete a theoretical investigation to determine the best method of collecting fine structure data, assembling it, representing it, and then displaying it. When this is done, one should see whether, after considering the physics of the ICBM problem, the observables are really significant and physically meaningful in the optimal theory. F. 2 THE FREQUENCY DISTRIBUTIONS The frequency distributions obtained for each of the ten runs appear in Tables F. 2. 1 - F. 2. 10. Each value of cross-section was computed by averaging the power received over consecutive groups of six pulses each. The aspect angles for the measurements contained in these tables are such that the aspect angle never exceeds 15 degrees off-nose and in most cases is between 5 and 10 degrees. 108 C I,[-r

mI 77 -UNIVERSITY OF MICHIGAN 2428-3-T TABLE F.2-1 FREQUENCY DISTRIBUTION -OBJECT 59(S-band) - C- Interval (m2) 0.00 - 0.01 No. of values in interval 208 107 45 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 20 12 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 109 L-z 8 16 26 6 448 rEZ2 77

I H r ---- I, ~ - UNIVERSITY OF MICHIGAN __ 2428 -3 -T TABLE F.2-2 FREQUENCY DISTRIBUTION -OBJECT 59(X-band) II I i i I I I I I I I II I I o- Interval (m2) 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - 0.10 0. 10 - 0.11 0.11 - 0.12 0.12 - 0. 13 0.13 - 0.14 0.14 - 0.15 0.15 - 0.16 0.16 - 0.17 No. in of values interval 45 50 55 54 48 28 28 16 19 19 16 10 5 5 5 9 1 o- Interval (m2) 0.17 - 0.18 0.18 - 0.19 0.19 - 0.20 0.20 - 0.21 0.21 - 0.22 0.22 - 0.23 0.23 - 0.24 0.24 - 0.25 0.25 - 0.26 0.26 - 0.27 0.27 - 0.28 0.29 - 0.30 0.31 - 0.32 0.59 - 0.60 0.64 - 0.65 No. of values in interval 4 5 5 5 4 3 4 3 1 1 4 3 1 4 3 463. 110 @IZlr I i7K

UNIVERSITY OF 2428-3-T air MCH IGAN MI ("HIGAN I TABLE F.2-3 FREQUENCY DISTRIBUTION -OBJECT 61 (S-band) a Interval (m2) 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - 0.10 0.10 - 0. 11 0.11 - 0.12 0.12 - 0. 13 0.13 - 0.14 No. in of values interval 40 44 27 16 8 13 14 22 17 6 5 4 3 2 221 111 -5CI- LZlZ

I lS I < r UNIVERSITY OF MICHIGAN 2428-3 -T TABLE F.2-4 FREQUENCY DISTRIBUTION - OBJECT 61 (X-band) I _^^___^_^It - IIII I I I I I II III o- Interval (m2) 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0. 05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - 0.10 0.10 - 0.11 0.11 - 0.12 0.12 - 0.13 0.13 - 0.14 0.14 - 0.15 0.15 - 0.16 0.16 - 0.17 0.17 - 0.18 0.18 - 0.19 0.19 - 0.20 No. in of values interval 49 47 17 22 12 12 5 4 3 10 9 6 8 4 5 3 1 3 2 1 223 112 rL — - 3[I \ I

I UNIVERSITY OF MI(CHIGAN 2428-3-T TABLE F.2-5 FREQUENCY DISTRIBUTION — OBJECT 62(S-band),, r Interval (m2) 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - 0.10 0.10 - 0.11 0.11 - 0.12 0. 12 - 0. 13 0.13 - 0. 14 0.14 - 0.15 0. 16 - 0. 17 0.17 - 0.18 No. of values in interval 123 107 117 62 55 48 46 21 14 12 15 3 6 4 4 2 2 641. 113 5-LZ I \

I UNIVERSITY OF MI(CHIGAN 2428 -3-T TABLE F.2-6 FREQUENCY DISTRIBUTION -OBJECT 62(X-band) ~ I~I III I I I I I III I o- Interval (m2) 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - 0.10 0. 10 - 0.11 0.11 - 0.12 0.12 - 0.13 0.13 - 0.14 0. 14 - 0.15 No. in of values interval 152 115 80 74 43 35 25 13 15 19 12 8 10 9 6 C( Interval (m2) 0.15 - 0.16 0.16 - 0.17 0.17 - 0.18 0. 18 - 0.19 0.19 - 0.20 0.20 - 0.21 0.21 - 0.22 0.22 - 0.23 0.25 - 0.26 0.27 - 0.28 0.35 - 0.36 0.36 - 0.37 0.39 - 0.40 No. of values in interval 3 6 2 3 2 5 3 2 1 1 1 1 3 649 =- - I ^\ L

UIVERSTY 7N1 r OF MICHIGAN 2428-3-T TABLE F.2-7 FREQUENCY DISTRIBUTION -OBJECT 64(S-band) r Interval (m2) No. of values in interval a- Interval (m2) No. of values in interval 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.01 0.02 0.03 0. 04 0. 05 0.06 0.07 0. 08 0.09 0.10 0.11 0.12 0. 13 0.14 0.15 0.16 0.17 0. 18 0.19 0.20 0. 21 0.22 0.23 0.24 0.25 0.26 0.27 49 129 147 123 65 52 51 59 61 73 60 28 33 19 15 21 14 18 18 6 11 9 7 8 7 6 10 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 6 4 3 6 2 2 2 3 3 3 5 4 2 1 5 6 6 3 4 1 7 4 7 3 1 1192 Mw 115 r-IZEIZJ EZ F 77

- - UNIVERSITY OF MICHIGAN 2428-3-T TABLE F. 2-8 FREQUENCY DISTRIBUTION -OBJECT 64(X-band) I a- Interval (m2) 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0. 04 - 0. 05 0.05 - 0.06 0. 06 - 0. 07 0.07 - 0.08 0.08 - 0.09 0.09 - 0.10 0. 10 - 0. 11 0.11 - 0.12 0.12 - 0.13 0.13 - 0.14 0.14 - 0.15 0.15 - 0. 16 0.16 - 0.17 0. 17 - 0.18 0.18 - 0. 19 0.19 - 0.20 0.20 - 0.21 0.21 - 0.22 0.22 - 0.23 0. 23 - 0.24 0.24 - 0.25 0.25 - 0.26 0.26 - 0.27 0.27 - 0.28 0.28 - 0.29 0.29 - 0.30 0.30 - 0.31 0.31 - 0.32 0.32 - 0.33 0.33 - 0.34 0.34 - 0.35 0. 35 - 0. 36 0.36 - 0.37 0.37 - 0.38 0.38 - 0.39 0.39 - 0.40 0.40 - 0.41 0.41 - 0.42 0.42 - 0.43 0.43 - 0.44 No. of values in interval 56 61 54 60 52 39 46 39 33 30 41 26 26 21 33 27 33 35 26 25 34 27 30 34 21 16 15 10 11 13 13 12 14 6 18 14 6 10 8 6 5 5 6 3 o- Interval (m2) 0.44 - 0. 45 0.45 - 0.46 0.46 - 0.47 0.47 - 0.48 0.48 - 0.49 0.49 - 0.50 0.50 - 0.51 0.51 - 0.52 0.52 - 0.53 0.53 - 0.54 0.54 - 0.55 0.55 - 0.56 0.57 - 0.58 0.59 - 0.60 0.60 - 0.61 0.61 - 0.62 0.63 - 0.64 0.65 - 0.66 0.66 - 0.67 0.67 - 0.68 0.69 - 0.70 0.70 - 0.71 0.74 - 0.75 0.75 - 0.76 0.80 - 0.81 0.81 - 0.82 0.83 - 0.84 0.84 - 0.85 0.85 - 0.86 0.87 - 0.88 0.88 - 0.89 0.89 - 0.90 0.94 - 0.95 1.00 - 1.01 1. 04 - 1. 05 1.10 - 1.11 1.12 - 1.13 1.13 - 1.14 1.16 - 1. 17 1.27 - 1.28 1.28 - 1.29 1.30 - 1.31 1.36 - 1.37 No. of values in interval 6 4 4 4 4 6 4 1 2 2 5 1 4 1 1 1 3 1 2 1 1 1 1 1 2 2 2 1 1 1 3 1 1 1 3 2 1 3 3 1 1 1 1 1192 7 r 116 [Im- IWar =L

UNIVERSITY OF MIC HIGAN 2428 -3 -T TABLE F.2-9 FREQUENCY DISTRIBUTION - OBJECT 65(S-band) or Interval No. of values Co Interval No. of values (m2) in interval (m ) in interval 0.00 - 0.01 78 0.20 - 0.21 7 0.01 - 0.02 44 0.21 - 0.22 5 0.02 - 0.03 63 0.22 - 0.23 3 0.03 - 0.04 74 0.23 - 0.24 6 0.04 - 0.05 81 0.24 - 0.25 7 0.05 - 0.06 65 0.25 - 0.26 2 0.06 - 0.07 75 0.26 - 0.27 5 0.07 - 0.08 61 0.27 - 0.28 4 0.08 - 0.09 44 0.28 -0.29 6 0.09 - 0.10 35 0.29 - 0.30 5 0.10 - 0.11 31 0.30 - 0.31 4 0.11 - 0.12 38 0.31 - 0.32 5 0.12 - 0. 13 21 0.32 - 0.33 8 0.13 - 0.14 31 0.33 - 0.34 5 0.14 - 0.15 15 0.34 -0.35 1 0.15 - 0.16 9 0.35 - 0.36 2 0.16 - 0.17 7 0.36 - 0.37 3 0.17 - 0. 18 14 0.37 - 0.38 3 0.18 - 0.19 7 0.38 - 0.39 2 0.19 - 0.20 4880 117 5_ILZ \ Fi

I r= IlL r - - _ UN I VE RS ITY 2 O F 428 -3 -T MICHIGAN I TABLE F.2-10 FREQUENCY DISTRIBUTION -OBJECT 65 (X-band) or Interval (m2) 0.00 - 0.01 0.01 - 0.02 0.02 - 0.03 0.03 - 0.04 0.04 - 0.05 0.05 - 0.06 0.06 - 0.07 0.07 - 0.08 0.08 - 0.09 0.09 - 0.10 0.10 - 0.11 0.11 - 0.12 0.12 - 0.13 0.13 - 0. 14 0.14 - 0. 15 0.15 - 0.16 0.16 - 0.17 0.17 - 0. 18 0.18 - 0. 19 0.19 - 0.20 0.20 - 0.21 0.21 - 0.22 0.22 - 0.23 0.23 - 0.24 0.24 - 0.25 0.25 - 0.26 0.26 - 0.27 0.27 - 0.28 0.28 - 0.29 No. of values in interval 32 49 33 33 43 51 33 38 36 31 31 33 25 27 24 20 17 23 16 19 22 16 20 15 23 17 12 13 10 (' Interval (m2) 0.29 - 0.30 0.30 - 0.31 0.31 - 0.32 0.32 - 0.33 0.33 - 0.34 0.34 - 0.35 0.35 - 0.36 0.36 - 0.37 0.37 - 0.38 0.38 - 0.39 0.39 - 0.40 0.40 - 0.41 0.41 - 0.42 0.42 - 0.43 0.43 - 0.44 0.44 - 0.45 0.45 - 0.46 0.46 - 0.47 0.47 - 0.48 0.48 - 0.49 0.49 - 0.50 0.50 - 0.51 0.52 - 0.53 0.53 - 0.54 0.55 - 0.56 0.60 - 0.61 0.62 - 0.63 0.67 - 0.68 No. of values in interval 13 9 8 5 6 10 12 7 3 9 5 1 5 9 6 1 2 4 3 7 2 1 1 2 2 1 1 1 898 118 m r cDFEr r I I, r

GEES [^ Lr I _UNIVERSITY OF MICHIGAN 2428-3-T F. 3 THE DEFINITIONS OF THE AUTOCORRELATION FUNCTION,,i R(T), AND THE POWER SPECTRUM, S(W), USED IN THE COMPUTATIONS In this section, a brief summary of pertinent facts concerning the usual R(T) and S(w), the definitions of R(r) and S(w) used in the computations, the relationship between R(T) and S(w), and one aspect of the significance of the definitions are presented. When a real random function y(t) is known for all times t from 0 to oo, the autocorrelation function R (r) and power spectrum, S(.(w), are defined as T R (r) = lim -T y(t) y(t+r) dt T T co 0 (F. 3-1) and ZIA(w, T) 12 S (W) = lim -2 T T -)o- * T (F. 3-2) where T A(w, T) = 14 f y(t)e-t dt Note tat Note that (F. 3-3) Se0(W) ~- 0 and RO(Tr) < RO(0), o TR (r) cosw r d r. 0 (F. 3-4) (F. 3-5) 2 S(w) = - 7r The last equation is known as the Wiener-Khinchine Theorem. -- 119 1 -I 11r I -:Ij F r

- UNIVERSITY U N I VE R S I TY i7K OF MICHIGAN 2428-3-T For a random function y(nTo), n = 0, 1,..., N (that is, for a finite set of data), the autocorrelation function, R(r, T), and the power spectrum, S(w), will be defined here by 1 R(rs, T) = N N-s L y(nTo) y(nTo+ TS), 0 < Ts < NTO n=0 = 0, T < 0 and T5 > NTO (F. 3-6) where Ts = sTo, s = 0, 1,..., N, and S(w) = - F(w, T), (F. 3-7) where T F(, T) =NrZ77 N y(nT) -iwnTo y(nTo)e (F. 3-8) Note that the summation in R(-r, T) is divided by N, the total number of values for y(nTo), rather than by N- s, the number of values of y(nTo) x y(nTo + Ts) in the summation. The reason for this is to preserve the relation 2To fJi i S(Wo) =- { R(sTo, T) cos wsTo - R(0, T) S =O (F. 3-9) which is the analogue of the Wiener-Khinchine relation. A proof of Equation F. 3-9 is the following: Using the definition of F(a, T) given in Equation F. 3-8, 120 I =7K

U N I VE RS ITY OF M I C H I GAN 2428-3-T r F(w, T) 2 2 7r N N n = O 2 7r N m=0 n - s=n-N N y(nTO) y(mTo)eiw(n-m)To y(nT) y (n-s)To e To 0 r lq z7 y(nTo) y[(n-s)Toe sTo y = s Y( NTO) Y [(n-S) TO] e iS To - I S=-N T 2 0 27r N+s n = y2(nTO) + 2 N s= 1 N y(nTo) y [(n-s)To coswsTo. n = s From the definition of R(rs, T) R(sTo, T) = n=s y(nTo) y[(n-s)To], where T = NTO. 121 r rF m L~tr7.

r? 1 ma 1=-,1 I '\ L-I M1( M 1 C H 1 I UNIVERSITY OF 2428-3-T CAN ___ Thus 2 To S(w ) F(w, T) =-{R (o, T)+2 T 7 N j R(sTo, T) cos o sTo s=l J from which Equation F. 3-9 immediately follows. One aspect of the significance from the following considerations. f(t) 4 x x x x I I I I x I I I I x x x I of the definition of F(w, T) may be seen R(Tr, T) and S(w) were computed according to the above equations, using for y(nTo) the averaged values of cross-section described in Section F. 2; that is, each - is the average over six successive values of oa with no overlap in neighboring r's. Consider the Fourier analysis of an arbitrary function f(t) which passes through a given set of points fnm defined at the times tnm, m = 0, 1,..., M, n = 0, 1,..., N, and f(t) = 0 for t<too and t > tN, M+l (The fnm correspond to the C values, M+ 1 of these being used to obtain a value of. ) The Fourier I I Set of Discrete Data M = 5 m too to1 A 06- t = t a o6 lo transform of f(t) is given by tN, M+1 F(w to o to f(t)e dt = - N n= tn, m+l -iwt f(t)e dt, tnm the latter expression being valid provided tn, M+I = tn+l, (this insures that the whole integration range be covered). 122 i- I \\

UNIVERSIT OF MICHIGAN 1 2428- 3-T if tn, m+1 tn.,m =-Aand t0 = 0 then tnm = n(M+1) A + MA and tn. m+ 1 = n(M+ 1) A + (m+ 1) A tn. m+1I tnm - iwd f (t) e dt =eiwtnmf 0 dt' f(t' + tnm)el wtI if -WAK< 1,9 N F (w) - p e -iw [n(M+ 1) A +mA&]T. e f~nm - where tn, m+1I f = I nm =A/ tnm If fur the r wiMA& << 1,I f(t) dt ToJZ~ If n= fne -iwnT0 Ip 123 oll Lr- - - F-\ 77

VERSITY OF 2428-3-T -- LI~I U N I MI (HIGAN __ where To = (M+1) A and -f 1 n M+ MI+ 1 p fnm n The formal similarity between this expression for F(&) and f(w, T) as defined in Equation F. 3-8 is apparent. If now f(t) is slowly varying over time intervals of length A (i. e., f(t) is band limited with wmaximum<< 1/A), then fm- fnm-fm+ll that is, may be replaced by fnm, in which case Tn is merely the average of the M+1 values: fnO, fnl..., fnMo Thus, the definition of F(w, T) given above and used in the computations gives the power spectrum, for w << 1/To (To is the interval over which the data are averaged), of a function which is zero outside the observation interval and which passes through all the points of the data before averaging. It is assumed that the function is band limited with maximum frequency << 1/A (A is the pulse repetition period of the radar used in obtaining the original data). In this case, A = 1/409. 75 sec. and To = 6A (M = 5). In actual practice, the following procedure was used: R(r) = R(sTO) = N-s -(nTo) C ((n+s) To), where values of s used were s = 0(5)150(25)1250 s = 0(5)200(25)1250 for S-band, for X-band. These are the values shown on the accompanying list. 124 - I,7 r= F- F-\ r 77

(=- [= -= j:- -' 1-< L-ir [ _UNIVERSITY OF MICH 2428-3-T I GAN Then S(w) for w A< Zr was approximated as 2To S(o) = ~ 7T N as R(sTo) cos(wsTo), where ao = 5/2 as = 5 s 90 f 5 < s < 145 for < s < 195 for s = 150 s = 200 as = 15 for S-band for X-band for S-band for X-band ' 150 for S-band;50 for X-band ' 175 < s < 12 = 25 125 < s < 12 This procedure was used to obtain S(o) for w/27r = 0(. 1)1. 0 cycles per second; for larger values of w, the method must be altered, inasmuch as taking s at intervals of 25 does not yield sufficient accuracy. In the interest of economy of time and money, it was decided to return to the basic definition of the power spectrum, namely 2 2 S(a)-T F(I, T) Numerically, this means 25T 2 S(=) = N -f (5nTo) cos(5nTo)j + l 5n( 5nTo) sin(5nwTo) This is the formula used to obtain S(w) for w/2r = 1. 1, 1.2, 1.5, 2.0, 3. 0, 4. 0, 5. 5, 6. 0, 6. 5, 7. 5 cycles per second. To get S(w) for higher values of w would require taking more than every fifth value of - which was not considered warranted. 125 r I, I - ir

UNIVERSITY OF MICHIGAN 2428-3-T F. 4 COMPUTATIONAL RESULTS FOR R(') AND S(W) The results of the computations of the autocorrelation functions and spectra for the various objects are given in Tables F. 4-1 - F.4-4 and Figures F. 4-1 - F.4-20. There are two curves for each object and frequency, one of which shows all of the computed values and is faired in at /21Tr = 1.1 cycles per second, the other of which shows the region <227r in more detail. The spectra, as expected, all show a large peak at o = 0. They then drop off rapidly and then oscillate more or less randomly. For only three of the ten cases did the ratio of the second largest peak to that at w = 0 exceed 1/10: for Object 59 at S-band this ratio is-0.2; for Object 61 at S-band it is -0. 3; and for Object 61 at X-band, there are two large subsidiary peaks with ratios to the zero-frequency peak of-0. 5 and 0. 2. The observation times for Objects 59 and 61 were considerably less than for the others. (The sample lengths were approximately 7, 3, and 3 seconds for Object 59 at S-band, Object 61 at S-band, and Object 61 at X-band, respectively. For the other objects, the sample length was >10 seconds.) Before interpreting the peaks in terms of physical characteristics of the missile, one should take into account the fact that the zerofrequency signal from the missile will, for short observation times, contribute appreciable subsidiary peaks to the spectrum (in addition to the one at w = 0). When this is done, it is found that the results are not inconsistent with the assumption that the spectra are due merely to a zerofrequency signal with superimposed noise characteristics inherent to the small sample. If the missile generates any other frequencies, these are not likely to be detected from the power spectra. F. 5 FURTHER ANALYSIS OF R(T) AND S(W) The finite nature of the sample involved in these computations of the autocorrelation function and power spectrum for the radar cross-section data obtained in the drop tests necessitates a more detailed study, which is presented in this section. 126 S = r

U N I VERS I TY OF j 2428-3-T TABLE F.4-1 LT rCHGAN M I C H I GA N R(n To) - S-BAND - SWEPT BACK FINS n Object 59 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145.0008453 0008353 0008000 0007538 0006946 0006308 0005674 0005087 0004469 0003924 0003453 0003045 0002761 0002554 0002460 0002395 0002451 0002522 0002656 0002725 0002775 0002768 0002732 0002676 0002540 0002395 0002230 0002076 0001864 0001721 Object 61.0029390 0023537 0017303 0011459 0007092 0005899 0005798 0006064 0007060 0007693 0008243 0009202 0010578 0012257 0014454 0013674 0011573 0009491 0006780 0005560 0005202 0004491 0003977 0003463 0004005 0005413 0006821 0008294 0009101 0009339 Object 62.0023321 0018821 0013379 0009538 0007803 0007903 0008497 0009763 0010370 0010078 0009771 0010260 0010491 0010660 0010257 0009476 0008434 0007511 0007332 0007779 0008361 0008304 0007481 0007005 0007566 0008530 0009498 0010403 0010378 0009784 127 I'-E —, Lr- Fi F

I"'C:= r--n r i7 UNIVERSITY OF MICHIGAN 2428 -3 -T TABLE F.4-1 (Cont.) R(n T0) - S-BAND - SWEPT BACK FINS n Object 59 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800.0001623 0001308 0001248 0001098 0001533 0001949 0001547 0000730 0000348 0000442 0000435 0000129 0000000 Object 61.0007954 0005069 0002642 0006688 0000477 0000812 0003202 0000193 0000615 Object 62.0008917 0006795 0007798 0006138 0006063 0007759 0007138 0005265 0007893 0005356 0003522 0006118 0003889 0003649 0003116 0002660 0002799 0002389 0003582 0003334 0002809 0002497 0001704 0002867 0000575 0002067 0000082 128 ~ 3 I"1 r-IL LZ ZZ F-\

I'E_- r= —, r- r- - U N I VE RS ITY OF MIC HIGAN 428 -3 -T TABLE F. 4-2 R(n To) - X-BAND - SWEPT BACK FINS il 1 l I I II I I! I I n 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 Object 59.0141270 0071525 0070421 0059138 0053559 0044026 0054613 0043404 0040270 0045957 0041832 0033812 0034853 0033959 0032140 0034870 0036184 0036834 0044616 0055883 0056242 0056562 0049853 0041834 0039607 0036294 0037868 0037045 0036235 0033121 0029456 0026168 0024110 0024711 Object 61.0046145 0041448 0034787 0028063 0021480 0015557 0010679 0006991 0004715 0003385 0002416 0001878 0001489 0001520 0002208 0003448 0004371 0005240 0005950 0007081 0008855 0011552 0014348 0016783 0018905 0020430 0022077 0022308 0020588 0017566 0014172 0009584 0005271 0002434 Object 62.0050741 0025862 0020497 0017938 0021512 0025621 0021173 0018398 0018803 0023149 0022695 0016563 0018283 0018065 0019548 0022300 0017712 0021593 0017755 0017029 0016308 0018463 0017689 0017517 0015197 0015692 0019122 0018591 0017432 0017085 0014481 0014074 0012283 0010523 7 r 129 CX i7K

- I i r 7 UNIVERSITY OF 2428-3-T M C HIGAN __A TABLE F.4-2 (Cont. ) R(n To) - X-BAND - SWEPT BACK FINS n 170 175 180 185 190 195 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 Object 59.0026162 0026965 0027853 0029782 0032698 0035881.0039490 0034259 0022680 0021086 0022799 0023015 0013603 0012445 0009518 0007890 0002006 Object 61.0001081 0000774 0000973 0001878 0003357 0003873.0003661 0003475 0002882 0001199 0000566 0001348 0001063 Object 62.0010752 0012779 0013834 0011650 0013454 0010824.0011133 0011588 0009446 0007613 0006469 0008158 0006938 0005678 0004964 0003596 0004875 0004372 0004175 0004257 0004977 0003584 0005147 0003615 0002293 0002339 0001387 0000443 0000183 0000331 0000000... 130..- I\ Z EZ

SC_ L7 UN IVERSITY OF M1CHI 2428-3-T TABLE F.4-3 R(n To) - S-BAND - SQUARE FINS C(AN _ n 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 Object 64.0213705 0200862 0178543 0156342 0138965 0125703 0115042 0106352 0098932 0092760 0088143 0085914 0087777 0091778 0095362 0096523 0095611 0093766 0091029 0087031 0082757 0078319 0073477 0068420 0064605 0063129 0063519 0064878 0067283 0071034 Object 65.0135879 0102389 0066842 0061679 0063365 0068854 0075865 0079173 0077494 0064617 0057250 0060818 0065239 0066173 0067007 0067176 0061667 0054871 0055849 0063276 0073871 0073213 0058965 0049297 0047688 0051110 0058077 0059460 0058439 0057026 131 53LZ fi J -= E~u 11~\ '= 7T

I= — rl —, r W-r UNIVERS ITY 2 O F 428 -3 -T M ICHIGAN __ n 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 TABLE F.4-3 (Cont.) R(n To) - S-BAND - SQUARE FINS Object 64.0075680 0089139 0077239 0060214 0063744 0066082 0045265 0061719 0070511 0056870 0045454 0045930 0041060 0040726 0048994 0048728 0043212 0062451 0067761 0050572 0062977 0072518 0056768 0039070 0035771 0032301 0027729 0026783 0051205 0035928 Object 65.0052668 0053268 0059709 0045687 0037299 0042200 0044469 0036720 0036091 0037898 0040035 0028989 0029714 0032119 0039226 0029245 0025457 0028174 0038098 0022803 0026753 0038686 0028555 001,8952 0017655 0015259 0009849 0007866 0007180 0008692 I I 1 32 rzz. I' I I rz\zF-.

C=- I' lK\ 77 UNIVERSITY OF MI C H I GAN 2428 -3 -T TABLE F.4-3 (Cont.) R(n To) - S-BAND - SQUARE FINS n 900 925 950 975 1000 1025 1050 1075 1100 1125 1150 1175 1200 1225 Object 64.0019334 0036836 0027423 0013044 0020053 0023112 0016797 0007424 0005445 0009259 0002440 0002134 0006488 0006437 133 Object 65.0004677 0001206 ~ILZ~ 7

UNIVERSITY OF 2428-3-T i7 MI C HIGAN ____ n 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 TABLE F.4-4 R(n To) - X-BAND - SQUARE FINS Object 64.0731492 0496405 0415901 0418286 0359315 0301876 0310012 0311995 0350713 0377331 0422809 0432972 0370850 0359330 0362422 0348520 0299206 0289381 0330233 0332032 0329484 0400315 0364368 0347348 0358902 0335385 0290651 0272524 0280468 0301092 Object 65.0401553 0269065 0257876 0257512 0245474 0238165 0244880 0242412 0226309 0255553 0256857 0253318 0232201 0252210 0235390 0251941 0224900 0217394 0238443 0240687 0243688 0241638 0232737 0216312 0237124 0218293 0205318 0213441 0226513 0235193 134 EZ7 ji EZZJ EZT

ICc I S:fj r — OF M I C H I GAN __ 428 -3 -T UNIVERSITY TABLE F.4-4 (Cont.) R(n To) - X-BAND - SQUARE FINS n 150 155 160 165 170 175 180 185 190 195 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 Object 64.0291169 0333597 0355673 0326777 0333055 0351747 0319484 0296904 0284441 0282039.0279145 0300524 0251444 0272144 0260056 0256764 0250086 0246935 0261566 0237105 0274815 0204867 0229171 0226876 0204659 0185845 0181684 0173128 Object 65.0237621 0222940 0209540 0225274 0221028 0212266 0205320 0199072 0206678 0225460.0220419 0190359 0181058 0174472 0159126 0159699 0152185 0145686 0139706 0112744 0114447 0109566 0098070 0098986 0084603 0081030 0063576 0052469 135 I,-~ILZ~ r- 1 JF-\ C7 7

_ I '-7 - I I r\ - UNIVERSITY OF M1(CH1 2428-3-T TABLE F.4-4 (Cont. ) R(n To) - X-BAND - SQUARE FINS GAN n 650 675 700 725 750 775 800 825 850 875 900 925 950 975 1000 1025 1050 1075 1100 1125 1150 1175 1200 1225 Object 64.0151293 0148216 0137398 0133801 0088473 0089325 0077180 0069846 0080214 0073055 0052689 0053285 0040715 0039828 0035806 0026802 0031196 0031564 0024238 0020768 0019607 0014503 0011415 0003088 Object 65.004985 1 0041182 0037598 0032464 0021562 0017225 0013484 0008427 0007767 0008462 0003787 0002023 136 I z- L 7 ZL

L-. Lr- r-_ - UNIVE RS ITY OF MICH I 2428-3-T GA N t-t I th 0 La), f= w/27r(cps) -FIG. F.4-1 S (W) FOR OBJECT 59 S- BAND u LE x - f = W /2 7r (cps) (w) FOR OBJECT 59 S- BAND FIG. F.4 -2 S 137 CJ r~\ i7

rrr-~I r~-ir U NI V ER SIT Y OF MI CHIGAN 2428- 3-T 4A16.0 E 2. 0 x U) 8.C f = w /2 7z (cps) FIG. FA4- 3 5 (W) FOR OBJECT 59 X -BAND - + i i i 4 + 4 4 4 + - - i i 4 4 4 - In C) a) E 0 x U,) 20.0 - -1 16.0 12.0 8.C 4.0 _ _ _ _ _ _ 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 f =w /2 7r (cps) 7. F.4 -4 5 (w) FOR OBJECT 59 X -BAND 138

I ---I rUNIVERSITY OF MICHIGAN 2428-3-T 20.0 - 16.0 Q_.IE 12.0 0 x - n 8.0 - f= w/27r (cps) FIG. F.4-5 S (W) FOR OBJECT 61 S-BAND aQu LI) E 0 O x 3 en V) 20.0 - 16.0 -- 12.0 - - 8.0 4.0 - l -1 n I i,1r nr I I I.U 2.U 3.U 4.0 5.0 6.0 7.0 8.0 f= w/2 r(cps) F.4-6 S (o)FOR OBJECT 61 S-BAND 139 - Il- I

F-rrjrI-i I UNIVERSITY OF MICHIGAN 2428-3-T V) C) - E o x -s O I I A I I 25.0 20.0 15.0 25.0 --------------------------------- --- 0.0 --- —------------------------------------------- 0.2.4.6.8 1.0 1.2 1.4 1.6 f= w/2rr(cps) FIG. F.4-7 S (w) FOR OBJECT 61 X- BAND?5.0 0.0 5.0 0.0 5.0 ------------------------ 2 2 u aE 0 x u) U 1.0 2.0 3.U 4.0 5.U O.U f = w /27 (cps) FIG. F.4-8 S (w) FOR OBJECT 61 X- BAND 140 /.U b.U L-. I - r -. -

r m0 77 UNI VE RS ITY OF MICHIGAN 2428-3-T 40.0 - 32.0 _ u' 24.0 E 24.0 a -- -- -- --- - xcn 16.0 ----- - f= o/27r(cps) FIG. F.4-9 S () FOR OBJECT 62 S-BAND I i I I I I i I I A... An n Q. u i () a) -< E 0 x O V) 32.0 24.0 - - 16.0 8.0 - - - - ------------------------ u0 1.0 2. 3.0 4.0 5.0 6.0 7.0 8.0 f= w/2r (cps) F.4-10 S (w) FOR OBJECT 62 S-BAND 141 3 I 1 r i - I- i

— UN IVE RS ITY OF MICHIGAN p 2428- 3-T 10.01 0~ Da1) 0 C,, f = w / 27r (cps) FIG. FA4- 11 S (to) FOR OBJECT 62 X- BAND i i i i i i i i i i i i i i i i I - - ine.0i - i — i + i I i I i f i- ii i t I i kn8.( a) E6.1 0 1< a C/) 4.( 0 - _ 0 -0 0 -I -L. ~ - - I I t I - I I — II -_ -. 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 f = w /2 7r (cps) FIG. F.4 -12 S (w) FOR OBJECT 62 X -BAND 142 ________ I,7 7 r- r

- Lj I Ir U NI VE RS ITY OF MI CH IGAN 2428-3-T m 40.0 0. u 1) E 30.0 CO x.2 -3 Xo 20.0 f = /27r(cps) FIG. F.4 -13 S (W) FOR OBJECT 64 S - BAND )I - 4 - 1 - 1 14 l l l l - 4 - l I I I I 5U.0 - 4 - 1 i- i i 4 4 4 4 4 - I-4. 4 $ - - I - - a) a) E 0 CO x t/) 40. 30. 0 — C- - - --------- _ _I__- -] I D -- --------- ----— ____ C - - -.., __. ~~ 20.( I U. 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 f = c /2 7r (cps) FIG. F.4 -14 S (L ) FOR OBJECT 64 S - BAND 143 7 r- II I i r

UNI VERS ITY OF MI CHIGAN 2428- 3-T A[T[1]111 20. 'A a) E x (/, 16.1 12.~ 8.i 4.i 0 -0 _ _ 0 2.4.6.8 1.0 1.2 1.4 1.6 f j w/2 7z(Cps) FIG. F.4 -15 S() FOR OBJECT 64 X -BAND 20.4 I 'A 0) a) E 0 x V) 16.0 - 12.0 - 8.0 -- - _ --- ----- 4.0 -- ni I- - -I I I- - -- - u01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 f = w /2 7r (cps) IF.4 -16 S (W) FOR OBJECT 64 X -BAND 1 4 4_ _ _ _ _ _ _ _ _ _ _ _ _ _

L- L DF- E~ir U NI VERS ITY OF 2428- 3-T M I C HIGA N --- i m i i i i i i i i i i i i i i U) U1) U) E 0 x LO In, 40.0 32.C0 -_ _ 24.0 - 16.0 - _ - 8.0 - _ 0.2.4.6.8 1.0 1.2 1.4 1.6 f = co2 r (cps) FIG. F.4 - 17 S(w FOR OBJECT 65 S - BAND 40.0 -__ 32.0 24.0 ____ 16.0 _ 8.0 -_ _o — =0- - - - - - - - v I.u 2.U 3.0 4.U 5.0 6.0 7.0 8.0 f = W/ 2 7(Cps) F.4 -18 S (W) FOR OBJECT 65 S -BAND 145 __ _ _ __ _ _ _ __ _ _ _ E 7 F- ri F 77

I ii7F L I L UNIVERSITY OF MICHIGAN___ 2428-3-T U) u LE CN 0 x 3 v, nO Du a) E CM 0 x v) 10.0 -- 8.0 6.0 4.0 2.0 -- 0.2.4.6.8 1.0 1.2 1.4 1.6 f = w /2 r (cps) FIG. F.4 - 19 S (W) FOR OBJECT 65 X-BAND 10.0 8.0 6.0 - --. 4.0 -- 2.0 -- 2.1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 f = w/27r (cps) FIG. F.4-20 S (w) FOR OBJECT 65 X - BAND 146 II I,-r=- r- F-\ r

= r-! Il-'J I \ UN I VERSITY OF 2428-3-T MICHIAN i7 MICHIGAN F. 5. 1 Definition of the Autocorrelation Function and Power Spectrum for a Continuous Sample on a Finite Interval - In defining the spectral density (power spectrum) and autocorrelation function when the data is given for only a finite interval (2T), these quantities must approach, continuously, the corresponding definitions given in Equations F. 3-1 and F. 3-2 as T becomes larger. Definitions for the continuum corresponding to those of Section F. 3 are |Fi(w,T) 12 S1(w, T) =I( T (F. 5-1) 00 1 R1(TrT) = T y(t,T) y(t+r,T) dt, -00 (F. 5-2) where T F1(w, T) = - S -T y(t)T) = y(t, T) = i L O y(t)eit dt It| < T tl > T (F. 5-3) These functions have the properties Sl(w, T) > 0, R(Tr, T) = R1(-, T), R1(0, T) > R1 (r, T), and they satisfy the Wiener-Khinchine relation: 00 S1(,T) = 2- Ri(-r,T) cos-rd-r. 0 (F. 5-4) 147 )C p =7 J - L^ Uu -\ '=

EJ LCLerxL-I U NI V ER SIT Y OF MI CH IGAN 2428- 3-T The proof of Equation F. 5-4: By definition, S 1(w, T) = 1 27rT S dt jdt' - 00 -00 y(t, T) y(t'1, T) elO)i(tt ) d~r y(t + -,T) e dty(t, T) 27rT Jd -00 Joo 1 -00i R 1(-r, T) (F. 5-5) From IE tion F.I 1~quation F. 5-2, it follows that Rl(-'-r, T) = Rl(-r, T), so that Equa5-5 implies Equation F. 5-4. Inverting, we have R 1(T, T) = 1 00 -i - Jdwe -1W S1(W.,T) - 00 (F. 5-6) Thus, c5 R I(rT) K< dw Sl(w, T) or R 1Tr, T) ~R1I(O, T) 148 ~~I,-Z[~ (F. 5-7) LE= EZZ zzi 77

- - Ifr I EZJ~ 7N \ 27 U N I VE R S I TTY 2428 - OF MICHIGAN -3-T Since the limiting value, R(T), is the time average of the product y(t) y(t+r), another reasonable definition, R2(T, T), would be 1 00 RZ(, T) = 2T - I y(t, T) y(t+r,T) dt; -00 (F. 5-8) that is, the integral 00 { y(t, T) y(t+r, T) dt -00 is divided by the interval used IR2(T, T) I can be greater than in the figure, R2(0, T) = to/T, 0 <to < T. T-r7 I T y(t) y(t+r) dt, T > 0 in averaging y(t) y(t+r). Note that RZ(O, T). For example, if y(t) is as shown whereas R2(2T-to, T) = 1 > R2(0, T) for y(t) 1 1 -T -T+to T-to T One could also define a spectral density as the transform of R2(r, T): SZ(w, T) -00 oo0 R2(-, T) cos wrd. (F. 5-9) 149. r I \ rEZ= 77

U N I VE R S I TY OF MICHIGAN 2428-3-T That S2(w, T) is not non-negative may be seen by the example y(t) = 1. Then R2(rT) = 1, and S2(w, T) = (2/7rw) sin 2oT. Note that 2T T} r S1(w T) = - ( 2T) RZ(rT) cosw-rdT 0 i. e., S(wJ, co) is the first order Cesaro sum of SZ(w, oo), so that when the latter exists it is equal to Sl(w, oo). Before comparing the relative merits of these various definitions, the question as to what is desirable in the approximate quantities must be discussed. Clearly the S(w, T) and R(r, T) which minimize IS(w, T)-S(w)| and I R(r, T)-R(r)I, for all o and r, respectively, would be the most desirable. On the other hand, the usual types of criteria, wherein a single parameter such as L1 = S((, T)-S(w) dw 0 or 0 L2 = JS(,, T)-S(0)] dow is minimized would not necessarily yield the most desirable approximate function. For example, if y(t) consists of a random (noise) component and a signal made up of a finite number of sine waves, and the purpose is to detect this signal and determine the frequencies, one would choose the S(w, T) which gave the "sharpest" peaks at (or near) those frequencies, provided it did not give sharp peaks at other frequencies. It can be shown that, in general, neither L1 nor L2 are good comparison parameters for a,n analysis of this kind. 150 * [gvi7K

_________ UNIVERSITY OF M I(CHI AN 1 A N 2428-3-T Criteria involving S(w) and R(T), the true spectrum and correlation function, cannot be applied directly to problems such as the data analysis for the Bendix Aviation Corporation, since the limitation to a finite sample excludes the possibility of knowing S(w) and R(Tr). However, it is conceivable that some process of guessing the statistics and finding optimum definitions for R(T, T) and S(w, T) could lead one to better definitions (Ref. 29)1. This approach, which seems worthy of investigation, will not be considered further here. 'In Reference 29, the finite time average, T T + Z(t) dt, 0 where Z(t) is some functional of y(t) (both y(t) and Z(t) are assumed to be stationary) is generalized to T g(t) Z(t)dt=M(T); the quantity [<M(T)2 - <M(T)>2] /<M(T) >2-=L 0 is then minimized (<>denotes an ensemble average). This leads to an integral equation for the "optimum" g(t), having the autocorrelation function <Z(t) Z(t+T-)> as its kernel. One can apply this to the autocorrelation function <y(t) y(t+r)> by taking Z(t) = y(t) y(t+-r) and, in addition, modifying the limits of integration to T-r 0 However, the process of deriving the spectrum from the optimum autocorrelation function through the Wiener-Khinchine relation does not, in general, give an optimum definition for the spectrum. Further, the analysis of Reference 29 is of no help in trying to minimize L directly for the spectrum, i. e., taking T Z(t) = f K(t) y(t)e dt. -T 151 I >L ELi r- -\\Ezr 7

U N I VE R S I TY OF MIC H IGAN ___ A 2428-3-T Other properties which might be considered desirable are those described by Equation F. 3-4. These will not be assumed here (if they were, then the pair R2(-, T) and S(w, T) would immediately be ruled out). Instead, y(t) = A cos wo t + N(t) will be investigated, where the function N(t) is assumed to have some standard noise properties. S(w, T) will be required to consist of a continuous (noise) component plus a peak' located near = wo(the purpose of the peak is to allow one to say whether or not the signal, Acos w t, is present and, further, to determine the value of WO). This will be done for the two functions S1(w, T) and SZ(w, T); it will be seen that the nature of the signal peak, and in fact the whole function, is essentially different for these two definitions. Some remarks about the comparison of R1(T, T) and RZ(T, T) will also be made. For y(t) of the form y(t) = ro(t) + rI(t) (F.5-10) (o-(t) is considered to be signal, t7(t) to be noise), 00 R1(, T) = J U(t, T) U(t+r, T) f[a(t) + 1(t)] [f-(t+r) + n(t+r)] R -T) = 2T -00 where 1 t|l < T U(t,T) = 0, |t| >T 'It is tempting to consider the peaked component an approximation to the Dirac 6-function. Care should be taken in this respect, however, since there the manner in which the approximation approaches the 6-function is important. For example, the function 6(w, T) = 2/7r TcoswZT2 satisfies the integral property f 6{(t,- w') f(w') do' = f(w) — 00O in the limit T-~oo for most well-behaved functions f(x), but this 6(w, T) would not be satisfactory for our purposes. 1352 A veJm:J: i^ ~r

C_r UNVRIYOF MICHIGAN ______ U N I V E R S I T Y 2428-3-T Thus, T- -r R (Tr, T) = Rca (r(, T) + RnO (Tf T) + 1 1 1 ZTjT where R p(T, T) = 1 (t) P(t+ir)dt. Taking the ensemble average., [oit) rJ(t+Tr) + o-(t+'r) ri(t) ]dt.P (F. 5-1 1) KR1(-r T)> = R 1'(-, T)+ R 1 (-rT) +< T -T p= a-, Ti[-(t) + -t r dt 0~~r-2T (F. 5 -1 2) i where 77(t) is assumed to be a stationary random process; this assump tion also leads to <R n (r.,T)>= - Ir R 'n (-r), 2T 0~-r~ZT (F. 5-13) where R 'a (1r = <17(t n 0(t+i-)> (F. 5-14) Noting that R 2 (r, T) = ZT R(rT (F. 5-15) 1 53 ~ILZ El" 7

EZ -F IU N I VE R SIT Y OF MIC~iJGJAN _____ 2428- 3-T and p 2 Sk(W.,T) = 2T 0 R p(-r, T) cos w-r d-r then 1s 'w T)>=S(w, T) + S "(w, T)> + q> <Sw 1 1 wrT = so,(w),T) + < S 7(w,,T) > +<Kn > 1 1 ' 7rwT 2T To -T 2T 0 (F. 5 -16) and <S?(w, T)>? = S2a'(Ed, T) + < SI(w. T)> + -<7 IT 0 - T d t o-t) + alt+ -)]0 (F. 5-17) Clearly 2T KS 1(w,),T) > = j. 0 T} R'7 (r) c os w -r d Tr (F. 5-18) and 2T KS (w, T)> = F R'(-r) cos w -dT- = KS "(wT)> 1 2ir jI1 wT 0 2T 0 (F. 5-19) 1 154 5rILZ LZ E~ fl F-. -F-\ EZ

C= r Lr UNIVERS ITY OF MICHIGAN __ 2428-3-T Assuming <n> = 0, the autocorrelation functions and spectra are simply sums of the contributions due to signal alone and noise alone. Furthermore, from Equation F. 5-19 it follows that for T ~ the correlation time of the noise, <S1,T)> <S2 ''T)> Consider th e correlation function and spectrum due to signal alone. For o-(t) = Acos ot (F. 5-20) simple integration leads to -R (IT) = A2 Tr COSw - 1 sinao(T -O)2T, R (r, T) = A - os -r + -, 0 <r< 2T 2 c [ + 2T-r ~o 0 (F. 5-21) (F. 5-22) a' A in(- 2o)T sin (w+wo)T S (W,T) = A Li2T+( -+2 sin(- wo)T sin(w+wo)T] (W-wo) (w+wo) J * (F. 5-23) Calculating SO(a, T) from the definition, (a, A2 sin2(w-wo )T sin2(w+w,)T S2,T) --- -+ 7+-o ^ ^TT "^Q CL+a~O A2 + I(, T), Br<oO (F. 5-24) 155 I I rC C 7

U N I VERSITY OF M1 IC H I GAN 2428-3-T ] where 2T cos wT sin Oo(2T-Tr) I(w, T) = d - 2T — 0 It is easy to show that I(wT) = cos2T {i[2(w+wo)T] - Si[2(w-(o)T] ' = 2 sinZwT 2 [2( - o) T]1 3 (F. 5-25) where x sin t - Si(x) = sint dt t t 0 x P(x) = -os t dt = Inx - Ci(x) 0 Ci(x) = - cost dt and t x V = 1.781072, Euler's constant. 1 Consider first the spectra for the case o = 0. Then a' 2A2 S (o, T) = 1 7r sin2 wT o2 T (F. 5-26) and (Wo = 0) ST) = 2A2 sin 2 wT S (wT) = 2 7T 156 - " i7K -^ - - U r 1 1

E_3 I C UNIVERSITY OF 2428-3-T r-7 M I C H IG AN (since I(w, T)/wo = (sin ZwT)/w when wo = 0). The interesting feature here is that, while for T-*ooboth functions satisfy, to within a normalization constant, the property o00 6(w) f(w) do = f(0) -0o (for most well-behaved f(w)), they differ in a number of other essential properties. Some of these are advantageous (from the viewpoint of signal detection and resolution) for one, some for the other. For example, the fact that S2(w, T) is twice as high as S1(w, T) at w = 0, and half as wide (i. e., the first zero of S2(w, T) occurs at half that of S(w, T)), works in favor of S'(w, T). On the other hand, the property ST(w, T)TO, u = constant / 0, which is not exhibited by S (w, T), tends to favor S'(w, T). 2' 1 Another property which favors Sl(w, T) is that the ratio of the height of the peak at w = 0 to that of the neighboring peak (which occurs at a = 1- 3r/2T) is almost three times the corresponding ratio for S{(w, T): S (0, T) /S_ T) 24 S(, - 5r/2 -= 9 T)-2.8. Sa( 0 T)/S ',T) 2 2 2Z (F. 5-27) Consider wo > 0. Then, S.T A2T sin(-wo )T S(W, T) =-27r Lwo)2T2 + A(w. T S( ' A2 T rsin2(c-oo)T 7r LZ(- wo)T 2(wT)j i (F. 5-28) (F. 5-29) where the functions sin(w- wo)T sin(w+wo)T sin (w+wo)T (w- wo)T (w+wo)T (+ wo)Z T2 9 (F. 5-30) * 157 I X 1X

r [ i7 UN I VERSITY OF M IC H I 2428-3-T sin2(w+Wo)T I(w, T) A (w, T) = ~ + 2 2(w+wo)T 8 wo0T ' ( A N (F. 5-31) tend to distort the peaks; these peaks are represented by the terms (A2/2rT) [sin2( - o)T] /(W -a )2 and (A2/27r) [sin2(w -o)T] (w-w(o) ( i (S~ in S(w, T) and SZ(w, T), respectively, which are simply 1/4 the functions for wo = 0 (Eq. F. 5-26), but centered around wo. Clearly there is not much distortion in S (w, T) for woT >>r; for example, Ai(w,T) = O(w-1T-1 ) at w = wo. To estimate the distortion in S(w, T), examine the asymptotic expressions (Ref. 30) 7 Cos X Si(x) - - 2 x sin x iP(x) ln 7 x - x — x valid for x > 27r. Considering again the value at w = wo, A (w,T) [(ln4ywoT)sin2woT+icos2uT] + 4woT ~. L J -t(^o0 (F. 5-32) which goes to zero as woT-'o; the logarithmic term, however, tends to cause more distortion in the case of S'(w, T). Thus the two functions, S1 (, T) and S2(w, T), which differ in several essential respects, both show peaks at the signal frequency, plus other maxima which are less than 1/7 the height of the signal peak. For the purpose of the Bendix contract, wherein the only a priori information was that there was a large zero-frequency component, with the object 158 [\- I\1

___ UN I VE RS ITY OF MI C HIGAN _____ 2428-3-T being to discover any other components if they existed, the choice uetween the various definitions was a matter of indifference. Before concluding this section, one further remark concerning the two definitions of the autocorrelation functions remains to be made. R1(r, T) is identically zero for T-2T and is near zero for i-r2T, whereas R2(r, T) will not generally approach zero as r-r-2T (-r <2T). Therefore, R2(-r, T) could be considered, in general, a better approximation to the true R(-r). However, as r->-2T, the length of sample used in computing either R(-r, T) goes to zero, so that the results for - -2T will be statistically unreliable; hence, essentially nothing is lost in this respect by using R 1 (rT, T). F. 5. 2 The Autocorrelation Function and Power Spectrum for a Finite Sample The drop-test data, being the result of measurement with a pulsed radar (prr -400 sec-1), yielded a finite number of values of the crosssection spaced at intervals A 1/400 sec. The problem of defining the autocorrelation function and power spectrum in terms of such data is essentially the same as in the case of a continuous sample of finite length with the exception that, in the discrete case, one must make an assumption as to band limitation. This corresponds to the fact that cos(w1 t) and cos(02 t) will look precisely the same if sampled at times tn = nA(n = 0, 1,..., N) if Zw = w1 + 2s7r/A, s any integer. The limitation imposed by the discrete observations is not restricted, a priori, to low frequencies. It is necessary only that two frequencies differing by 2s7r/A be indistinguishable (no matter how large N may be). However, physical considerations lead to the assumption that the function being sampled is essentially band-limited by w<27r/A -400 cps (that is, the amplitudes of the higher frequency components are much less than the amplitudes of those within this band). The actual computation was not based on the original data, but on a reduced set obtained from the original by averaging, with no overlap, successive sets of six points. Hence, the data used was at intervals of To = 6A. One may ask whether it is necessary to assume a band limitation of c<27r/To_ 66 cps. It was shown in Section F. 3 that, in terms of the definitions used, this was not 159 I ir7K

LCL tr\ -L U_ UNIVERSITY OF M ICHIGAN 2428-3-T the case - one need assume only the band limitation imposed by the spacing of the original data, provided that the spectrum is computed for w << 27r/To. Analogously to the case of a continuous sample, there are, as alternatives, the definitions of Section F. 3: 1's R'(s',T) N-s+l N-s A y(nTO) y(nTo + s) n=0 and 2T 1 S'(w) = - T R'(sTo,T)cososTo -- R'(O T 7 2 s=0 For the case y(t) = 1, the summations are easily carried out, yielding T() O (N+i) sin2 (N+l)wTo 2 sin2 WTo 2 S'(w) = 7T + cos NwTo Hence, the remarks of Section F. 5. 1 concerning the various definitions apply here as well. comparison of the 160 r ra\ EL -H

U NIVERSITY OF MICHIGAN 2428-3-T APPENDIX G THE CROSS-SECTION OF THE TANKS G. 1 INTRODUCTION The cross-section of the terminal propulsion unit (tanks) at 200 Mc was determined by essentially three different methods of approach: 1. Using previously obtained theoretical and experimental data on the 7-OC booster (Ref. 1), 2. Using physical optics methods on the breakdown of the tanks shown in Figure G. 1-1 (a drawing of the tanks appears in Fig. 2-41), and 3. Using physical optics methods on a modification of the breakdown used in Method 2. In the consideration of the problem by Methods 2 and 3, the crosssection was determined for-the tanks with the warhead attached, with the warhead removed, and with both the warhead and hat section removed. G. 2 METHOD 1 -SCALED 7-OC BOOSTER DATA This section contains an estimate of the radar cross-section of the terminal propulsion unit (tanks) at 200 Mc based upon the results obtained previously on the 7-OC booster. Experimental and theoretical data on the cross-section of the 7-OC booster appear in Reference 1. If the 7-OC booster is scaled down by a factor of 5/6 (Fig. G. 2-1), the resulting model closely resembles that of the tanks. The dimension of the 5/6-scale 7-OC booster and the tanks are compared in Table G. 2-1. 'The drawing of the tanks shown in Figure 2-4 shows the warhead in the shape of a cone rather than as one of the Rudolph parameterizations. Thus, the Method 2 approach, in computing the cross-section with the warhead attached, assumes the warhead to be in the shape of the cone shown on Figure 2-4. 1 161 I'- N L, C_77 = D - I = C_-< ^

r I r Ir UNIVERSITY OF 2428 - 3 -T M I C H I G A N 120" - - FRONT SECTION CENTER SECTION L20" 1 - 173" - a >v. b; - d ia - (Ogive: P=100") 50" t — 50' 1 REAR SECTION a - cylinder: 103" long; dia. = 10" (sphere cap) b - cylinder: 50" long; dia. = 18" c - sphere: dia. = 15" d - sphere: dia. = 20" FIG. G.1-1 GEOMETRY OF "TANKS" USED IN COMPUTATIONS 1 62 L-, II\ -H

I r VE RS I TY OF 2428-3-T MICHIGAN MI( HIGAN I UNI 100 56 -- 56 ~1 211 163 — ~ -I- ' n rl - 4- av JV/ 49 (all dimensions are in inches) FIG. G.2-1 A 5/6 -SCALE MODEL OF THE 7-OC BOOSTER TABLE G.2-1 COMPARISON OF DIMENSIONS BETWEEN THE TANKS AND THE 5/6-SCALE MODEL OF THE 7-OC BOOSTER Tanks 7-OC Booster Maximum diameter 120 in. 120 in. Front diameter 48 in. 45 in. Rear diameter 50 in. 56 in. Length-truncated cone 204 in. 211 in. Length-cylinder section 497 in. 507 in. Length-rear sloped section 45 in. 49 in. Length-tail cylinder 128 in. 163 in. Nose angle (truncated cone) 10~ 10~, I i i,,.... i i i 163 5)ILZX..

UNIVERSITY OF MIC HIGAN 2428-3-T Application of modeling theory to the data for the 7-OC booster given in Reference 1 yields cross-section data on the "5/6-scale 7-OC booster" (and thus on the tanks) at 90, 270, 720, and 1200 Mc. Figure G. 2-2 shows the theoretical cross-section (average o-) of the "5/6-scale 7-OC booster" at 270, 720, and 1200 Me, as scaled from the data of Reference 1. The theoretical curves given in Figure G. 2-2 for the cross-section, combined with the experimental data shown in Figures G. 2-3 and G. 2-4', yield a reasonably good estimate of the cross-section pattern to be expected for the tanks at 200 Mc in the interval 0 < P <90~. G. 3 METHOD 2 - PHYSICAL OPTICS CROSS-SECTION OF THE TANKS G.3. 1 Cross-Sections The physical optics method was applied to the determination of the cross-sections of the tanks using the formulas given in Section G. 3. 2 which were evolved from the breakdown of the tanks by the methods of Appendix A of Reference 5. The cross-sections were determined at 200 Mc for the tanks with: 1. The nose-cone attached; 2. The nose-cone removed and the resulting hoie assumed to be: a. Filled with an absorbing material so that no return is received from the interior of the hole, b. Lined on the inside with an inverted hemisphere, and c. Empty up to the end of the hat section, so that from the nose-on aspect the return would be like that from a flat plate; and 3. Both the nose-cone and hat section removed. 'Figures G. 2-2 - G. 2-4 were taken directly from Reference 1, with the appropriate scale factors introduced. 164 [ IL ^7i7 rJ^ - j r^\< _^ 1

E- rcFE ir r.^:1 1 -<_- IIir UNIVERSITY OF 2428 - 3 -T MICHIGAN N,-, o LO E Cs CO 0 20 40 60 80 100 120 140 ASPECT ANGLE IN DEGREES FROM NOSE-ON FIG. G.2-2 THEORETICAL CROSS-SECTIONS OF THE TANKS (Based upon a 5/6-scale model of the 7-OC Booster) 1 65 Mm 00 I -- I i FWEr7

r - i rirFImr i " 7r U N I V E R S ITY OF 2428 - 3 -T MICHIGAN I Innn1 f I i 4 i 4! i E b L) 04 %O -C" 6 E Horizontal Polarization (ESL) --- Vertical Polarization (ESL) 100 8 4 21.|I: -,=-,~,i- -i IsI ' I A, nn A11:In f1n n 11 n 1 1 1 Q Mr U zu 4U OU ou I UU I Lz 14U I U I OU ASPECT ANGLE IN DEGREES FROM NOSE-ON FIG. G.2-3 CROSS-SECTION OF THE TANKS (NOSE-CONE REMOVED) AT 90 Mc (Based upon a 5/6-Scale Model of the 7-OC Booster) 166 rirF r L- r- \ r

Irn I uu - I h i F I i i i ff i Ui I i 4 -4 4- 1 -S i M11 LJD~ [Tfill Hir -Ji C4 E b CN 10 Cl) 4 F 10 6?I AV 1.0 4 Ii Theoretical __ -f - - Vertical Polarization (ESL) I IHorizontal Polarization (ESL) 2 0.1 I 00 1) z C-1 Ve *.t~m C/) 0 — 0 zT [Ml 5111i =r(r L I u 20 30V 40 50 60 70 80 90 100 FIG. G.2-4 CROSS-SECTION OF THE TANKS (NOSE-CONE REMOVED) AT 270 Mc (Based upon a 5/6-scale model of the 7-OC Booster) -j

UN I VERSITY OF M 1 (HIGAN 2428-3-T The results obtained are shown in Figure G. 3-1. G. 3.2 Computational Procedures For the Tanks In determining the cross-section, using the breakdown shown in Figure G. 1-1, the primary contributions to the cross-section were assumed to come from: 1. The tip of the nose-cone (o.1), 2. The junction of the nose-cone and the hat section (a3 and -4), 3. The junction of the extended hat section and the central cylindrical portion (r-2 and o-5), 4. The junction of the cylindrical portion and the rear conical surface (o-6 and 0-8), and 5. The engine breakdown back of the rear cone (0-7). At certain aspects (namely those which are normal to the conical or cylindrical surfaces), other formulations were involved. The formulas involved are listed below: o = o- +1T R2 tan2 (10~) +-R1 [tana - tan(10~)]2 for = 0~ 5 -= -., for 0~< <10~ i=1 6 or = o- + o., for 10 ~<p< a i = 3 7 - =- + l-., for a<v<90~ - a i=4 - 168 El LZJ EZ

U NIVERSITY OF 2428 - 3 - T i -C MICHIGAN M I C HI GAN I_____ 1 06 8 6 4 2 8 6 4 2 104 8 6 4 C) w ILU gI 0 z I — z 0O u gO 2 8 6 4 2 102 8 6 4 2 10 8 6 4 2 1 8 6 4 2 101 b (IN DEGREES) FIG. G.3-1 CROSS-SECTION OF THE TANKS AT 200 Mc (METHOD 2) 169 rXW I l7

C_- r=ii7KL UN IVE RSITY OF MI (HIGAN ____ 2428 -3 -T 9X sin aQCosGa 7 a'= az'., folI i=4 + 7 z0 1 for 0P = 9Q00 - ct 0 0 r 90 -a <iP <80 8if [(R 2)3/2 _ (R 1)3/2]2 9 X in2 (100) Cos (100) + 0-6+ ~0 7-7 for 0= 800 7 a' = 7c-., for 8 00 <KiP< 900 2 a- = x +o a,+oa7 for = 900 7 0-= 27-.,for 900 < k<9goo +QC i ____ 8i [R23/2 - _)3/f2] 9 X sin2 a, Cosa +a4 +05 + -7,for = 9 00 ~ ci 7 a- = a'.2 2 i =_4__ 7 0- = 27. 2 i= for 90 0 ~ci.< 0t<18O0 - a for 1800 -ci. <iP < 170 0 - 170 ~7ILZ LIZJ rzr LZZ 7

C_'L~-J- Z7 U NI VE R SIT Y OF MJICHI-IGAN 2428 -3 -T 8-I o- =(F i =6.,for 170 80 a- =7M(rdt) + T(R2 ) tan2C aowhere N. = 1. 5 nr a- = tan ' RI =0. 61 i R 2= 1.52 i r = 0. 23 i rt O. 63 r and 7, for 0 = 18 00 (24/29) (2a.) 1 20S(2a)J 1.2.,for 1 00< K<a X2 tan4 a I 16w F 1 ~ cosl -cos(2 L') + c \R tan 2(10 0- b) 2 87r sin o - 8Rsi Itan 2 (a. - 0 - I a'4 8ijinp (Itan (ai + l - Itan (i00 + 2 171 F-\ 7

I- I rX7 u N I VE R S ITY r --- O F:28 -3 -T M I C H I G A N 24 KR2- T I + O)[ - Itan1 } a'5 = 8i sing 1tan(1 + 6 XR2 s an2 6 8-rI sin4' ), for 10~< < a XR2 ( i2 8~ r | sin l Itan -l - |tan( - a)lJ, for a< ~ < 180~ - a (p / 2 or - + a) 2 2 IT 2 2] ~' 2 (.51) + (.37), for a <~ip<90~ 7 2 L = [(5 2 ( 25] = — [(.51) + (.37) + (.25), for 90~ 0< < 180~ = (.37) + (.25), for P = 180~ 8 8r Isinl an (a + } The formulas above were applied for the tanks with the nose-cone attached. In considering the cases of the tanks with the nose-cone removed, the expressions for a-., i = 1, 3, and 4 had to be changed and, at the aspect P = 90 - a, the normal incidence term deleted. The new forms of o-3 and o-4 are 3 4 ao' =8 I tan2 (10~ - O), 3 8-rrsine XR1 2 o-' = sin tan2 (10~ + ), 4 8rr sin~ 0~< < 10~, p / 80~; 172 rIT- ji III rIZ

W(CEl U N I V E R S I T Y O F MIC HIGAN _ 2428-3-T and s is replaced by: 1. The wire loop formula for the case cited in (2a) of Section G.3. 1, 2. The wire loop and sphere formulas for the case cited in (2b), and 3. The wire loop and circular flat plate formulas for the case cited in (2c). In determining the cross-section for the tanks with both warhead and hat section removed, the approach outlined above for the case cited in (Zc) was employed, with the value of R1 changed to 0. 88. G. 4 METHOD 3 - CROSS-SECTION OF MODIFIED TANKS In performing the analysis and computations for the Method 2 approach, it was observed that the base of the nose-cone shown on the drawing furnished to us of the terminal propulsion unit was larger in diameter than the maximum diameter of the Rudolph warhead. Thus, the tanks were modified slightly (the hat section was lengthened), and the computations were repeated. The warhead used in these computations is Configuration IV. Configuration IV was chosen because it most closely resembled the nose-cone shown on the "tanks" drawing. One further modification was made in the breakdown, and that involved the breakdown of the "engine configuration. " (This latter modification had no effect upon the crosssection curves obtained. ) The new breakdown, as it pertained to the front portion of the tanks, is shown in Figure G.4-1. The computations were performed for: 1. The tanks with the warhead (Configuration IV) attached, 173 CILZX1IF

r~rcr~ LI_- r=x-~ -'K - N UNIVERS I TY OF MICHIGAN 2428 - 3 -T 10~ R-W All Dimensions Shown Are in Meters Central Section (Cylinder: Radius = 1.52, Length = 12.6) (See Figure G.1-1 for Breakdown of Aft Portion of Tank) FIG. G.4-1 GEOMETRY OF THE MODIFIED TANKS USED FOR COMPUTATIONAL PURPOSES - 174 -- M, Ir

___ UN I VE RS UITY OF 2428-3-T iT7 MIrCHIGAN MI1 ("H IGANI 2. The tanks with the warhead removed, the resulting "hole" being the mirror image of the rear of Configuration IV, and 3. The tanks with warhead and hat section removed. The lengthening of the hat section led to a slightly larger value for the cross-sections for Cases 1 and 2 at P = 80~. Otherwise, the only noticeable changes from the results of the Method 2 approach are in the cross-sections for V = 00. For Case 1, the a- of Section G. 3. 2 was replaced by the nose contribution of Configuration IV (App. A); and for Case 2, a-, was replaced by the wire loop formula used in Method 2 plus a contribution from the inside of the hole (estimated to be equivalent to that obtained from a sphere of radius 0. 2 m, the radius of the rear hemisphere of Configuration VI). Case 3 was identical to the Case 3 of Method 2. The results obtained are shown in Figure G. 4 -2. - 175 EZ =LI r"'< =

I i I r-1 — U N I V E R S I T Y OF MICHIGAN ____ 2428 - 3 -T 105 8 6 4 I t 2 104 8 6 4 2 103 8 6 4 V) ix LUi LLI LU LUJ 0 L) z z 0 LUJ L/) L/) 0 u 2 102 8 6 4 2 10 8 6 4 - 1 Cross-Section of Tanks with Warhead (Conf. IV) Cross-Section of Tanks without Warhead Cross-Section of Tanks without Warhead or 3 Hat Section -- 1 2..._.,. 1 2 8 6 4 2 10-1 8 6 4 2 10-Z 0 20 40 60 80 100 120 140 160 180 b (IN DEGREES) FIG. G.4-2 CROSS-SECTION OF THE TANKS AT 200 Mc (METHOD 3) 176 LL F 1ir

UNIVERSITY OF MICHIGAN 2428-3-T APPENDIX H IONIZATION AND RELATED TOPICS H. 1 INTRODUCTION The ionization which appears about a high-speed missile as it descends through the atmosphere could be a contributor to the radar reflections from the missile. An analysis of this problem was begun in Reference 1 and the results of that analysis are summarized below. Other related investigations carried out since the publication of Reference 1 are also discussed. H. 2 SUMMARY OF IONIZATION STUDIES REPORTED IN REFERENCE 1 This section contains a summary of the work on ionization produced by a ballistic missile performed at The University of Michigan and reported in Reference 1. This study was necessitated by the fact that, if missiles produced sufficient ionization, their radar cross-section might be significantly enhanced. Further, if they formed persistent ion trails, they might be confused with meteors in radar observation. The work was based on the fact that, to have perfect reflection from an ion cloud, the electron density must exceed a critical value given by f2 Nc = - X 106 electrons/cm3, 81 where f is the observing radar frequency, measured in Mc. Operation at 25 Mc was assumed; in practice a higher frequency would certainly be used, and thus a greater electron density would be required. The ionization was assumed to be caused by secondary collisions, in which atmospheric particles struck by the missile rebound elastically and ionization occurs in their collisions with other atmospheric particles. The primary collisions were treated by two opposite approximations: rigid 1 177 HiL) zlr ~7\~77

UNIVERS ITY OF MIC1 HIGAN 2428-3-T binding, where the missile atoms may be assigned infinite mass; and no binding, in which they are taken as free. Ionization in the primary collision is energetically forbidden, and no other mechanisms were clearly establis hed as physically significant. The ion distribution about the missile nose is shown to resemble a hyperboloid. Supracritical density is assumed within such a contour, and the effect of the subcritical density outside, which would reduce the radar cross-section, is neglected. At other points in the computation, approximations favorable to ion production are made. A missile velocity of 10 km/sec at an altitude of 200 km was assumed. Maintenance of a contour large enough to enhance the missile's nose-on cross-section by iTm2 was shown to require a probability> 1 for ionization in a sufficiently energetic secondary collision. It was therefore concluded that the nose-on cross-section of a 10 km/sec missile at that altitude would not be enhanced by ionization. It was further shown that the broadside cross-section would not be enhanced, since few electrons would be produced along the sides of the missile. Also, the electron density produced is too small to form a significant trail behind the missile. Since the relative importance of the possible mechanisms for ionization is affected by altitude and speed, the fact that meteors are observed by radar reflection does not constitute an objection to the conclusions reached in Reference 1. Meteor observations by radar usually take place for meteor altitudes lower than 120 km (Ref. 32) and most meteor speeds greatly exceed the missile speeds considered. (Reference 33 contains a summary of the theoretical work done on the meteor problem. ) H. 3 SHOCK WAVES H. 3. 1 Reflections from Shock Waves Reflections from shock waves were also considered in Reference 1 where it was concluded that, for almost all missile cross-section problems of interest, the effects of shock fronts may be neglected. 178 EZC 77

I 1 -UN I VERSITY OF MIC HI GAN 2428-3-T H. 3. 2 Shape of Shock Waves During Ascent and Descent of an ICBM It has been conjectured that the shape of the shock wave associated with an ICBM might differ on the ascending and descending portions of its trajectory. Using trajectory data given (Ref. 34) for Atlas, this section will demonstrate the validity of the conjecture. The problem is simple for the initial portion of the ascent. Once the transonic region is traversed, a normal, attached, acute-angled shock will be present. On the basis of some photographs made at the Aberdeen ballistic range, and quoted by von Karman (Ref. 35), it is estimated that the normal pattern is established by about Mach 2. This pattern should persist until the missile leaves the continuum regime of aerodynamics and enters the slip-flow regime. The shock will then become diffuse, gradually reaching a state in which its existence is questionable. Using the Atlas trajectory data, the time history of the shock during the missile's ascent is determined below. First, consider the 5500-nautical-mile trajectory data given on page II-114 of Reference 34. After about 60 seconds, the transonic region has been traversed. A reasonable criterion for the lower limit of the slipflow regime is (Ref. 17) M X/I = 0. 02, where M = free-stream Mach number, X = mean free path, and I is a characteristic length of the missile. The velocity of sound and mean free path at higher altitudes are taken from a Rocket Panel paper (Ref. 22). Computation then shows this missile to enter the slip-flow regime at an altitude of about 70 km, about 160 seconds after launching. Thus, on the way up, the missile is subsonic for the first 40 seconds, transonic for the next 20 or so, and finally exhibits supersonic flow with a narrow shock for about the next 100 seconds. Starting at about that time, the shock gradually becomes diffuse and disappears. We consider next the ascent of the midwing-configuration glide missile (p. 11-119, Ref. 34) for 5500-nautical-mile range. Here the transonic region is passed about 60 seconds after launching, and this configuration never attains sufficient altitude to leave the continuum regime of aerodynamics. Thus, it retains its narrow conical shock. I179 FsiiLZ ~ i7

U N I V E R S I T Y OF MICHIGAN ___ 11 A N 2428-3-T The powered-flight trajectory for maximum range is given on page II-125 of Reference 34. This missile's speed exceeds Mach 2 after 60 seconds, and it leaves the continuum regime about 140 seconds after launching, at an altitude of about 80 km. Finally, consider (p. 11-127, Ref. 34) the trajectory for the 1000 -nautical-mile range. Here, Mach 2 is passed after about 62 seconds, and the missile remains always in the continuum regime. Now consider the descent portions of these trajectories. For the glide missile, and the 1000-nautical-mile trajectory, unless a blunt re-entry shape gives rise to a detached shock, the shock configuration of the ascent is retained. This, however, is not true in the other two cases. For those cases, consider the possibility that a gas cap forms before the missile re-enters the continuum regime where a shock can exist. If this should happen, the shock would form around the rounded contour of the gas cap and therefore be rounded or possibly detached. The criterion for the existence of a gas cap, formed by trapped air molecules, is (Ref. 36) rv where X is the mean free path and V the thermal velocity of air constituents, v the missile velocity, and r the missile nose radius if it were spherical. Since the authors have not located the assumptions underlying the analyses of Herlofson (Ref. 39), they have not followed his work here, but have used the less recent Reference 36. Here r will be taken as 3 inches; for an order of magnitude calculation this should suffice. For the 5500-nautical-mile powered-flight trajectory, use of the Rocket Panel data and the trajectories given by Convair indicate that the gas cap begins to form at about 110 km. The shock, however, doesn't begin to form until the missile has descended to about 60 km. Therefore, since the gas cap exists prior to the shock formation, a rounded or detached shock is to be expected. 180 LI LI EZ rEZL 77

UN I VERSITY OF MIC HIGAN 2428-3-T Finally, examine the powered-flight trajectory for maximum range. The Convair trajectories do not present the portion of the flight immediately preceding re-entry, but the curves given are sufficiently similar to those for the 5500-nautical-mile range that the conclusions drawn for the latter retain qualitative validity. H. 4 WORK PERFORMED AT OTHER ESTABLISHMENTS H. 4. 1 Lincoln Laboratory of the Massachusetts Institute of Technology Reference 1 contains a list of references containing additional material on ionization and related studies. Included here is the work of the Lincoln Laboratory of the Massachusetts Institute of Technology. Their work on meteors and the ICBM is reported in Reference 37. Some of the conclusions reached to date are: 1. For the megawatt radars being contemplated for ICBM, one may expect meteor echoes at the rate of several thousand per hour. These meteor trail echoes will have a Doppler character during the trail formation, and will persist as relatively stationary target echoes for several seconds after formation. Unless the radar is specifically designed to reject meteor echoes, these can be a serious clutter and false-alarm problem. 2. On the basis of the available meteor-rate data it appears that the number of meteors seen by a radar should not increase with antenna gain but will increase with transmitter power. Hence, to get longerrange radars without increasing meteor background, one has more to gain from higher gain antennas than higher power transmitters. 3. The need for further experiments is indicated, particularly with higher-powered radars and shorter wavelengths than have been used until now. Such data are needed to establish the extent to which the meteor background is cut down by using higher frequencies and to determine the distribution of meteors corresponding to the smaller ion density trails. 181 I -LZ r -c=f^ " F-\ "T

UN I VE RS I TY OF MIH ("H AI __AN_ 2428-3-T 4. The probability of a meteor hitting an offensive missile is negligible. Other work performed at the Lincoln Laboratory on the ICBM problem is reported in Reference 38. H. 4. 2 Poulter Laboratories of the Stanford Research Institute Information relative to experimental work on "Extreme Velocity Pellets" performed at the Poulter Laboratories of the Stanford Research Institute has been received in personal correspondence from Dr. Poulter to one of the authors. Dr. Poulter states that much work remains to be done to develop the controls for velocity, pellet shape, etc., but that they believe the solutions of these problems are rather straight-forward and that the methods which they have employed offer a possibility of obtaining any fragment velocity that will be required in missile studies. By using a Mach stem detonation along the axis of a comparatively small-angle concave conical detonation front (apparently one of the most effective means of collecting and concentrating energy into a very small space) and using 1. 5 pounds of explosive, the Poulter Laboratories have consistently been able to photograph with the framing camera 0. 2-gram pellets having velocities of approximately 12 mm/p. sec. In one case, the pellet was traveling in a lucite tube in which the air pressure was reduced to 50 microns. The luminous pellet, at the time the pictures were taken, was traveling at the tip of the jet. From the scale in the pictures and a camera speed of one frame per microsecond, a velocity of 13.2 km/sec was calculated. In a second case the pellet was traveling in an atmosphere of helium at a pressure of one atmosphere. The luminous conical shock pulse surrounding this pellet was clearly visible in the pictures and was seen to travel 230 mm in 17p. sec, giving a velocity of 13. 5 km/sec. One picture was taken prior to firing the charge for purposes of reference, the next one was taken only a few microseconds after the detonation front struck the rear of the metal liner and before the central 182 LZ- 7I [1

I"' r--, r Z77 UNIVERSITY OF 2428-3-T M ICHIGAN ____ portion of the center of the liner had moved perceptibly. An approximate velocity can therefore be determined for the pellet in terms of the velocity of the central portion of the liner and the distance that the center of the plate has traveled. The most conservative value that it is possible to obtain from this picture is in excess of 60 km/sec. Thus, the Poulter Laboratories feel that this offers a feasible method of obtaining extremely high-velocity pellets in the range of from 10 to more than 50 km/sec. 183 W^ILZEIZJ EZ\

UNIVERSITY OF MICHIGAN _______ 2428-3 -T REFERENCES Number 1 K. M. Siegel, M. L. Barasch, J. W. Crispin, I. V. Schensted, W. C. Orthwein, and H. Weil, "Studies in Radar Cross-Sections XIV -Radar Cross-Section of a Ballistic Missile", The University of Michigan, Engineering Research Institute, Report No. UMM-134 (September 1954). SECRET 2 K. M. Siegel, M. L. Barasch, J. W. Crispin, R. F. Goodrich, A. H. Halpin, A. L. Maffett, W. C. Orthwein, C. E. Schensted, and C. J. Titus, "Studies in Radar CrossSections XVIII -Airborne Passive Measures and Countermeasures", The University of Michigan, Engineering Research Institute, Report No. 2260-29-F (January 1956). SECRET | 3 K. M. Siegei and J. M. Wolf, "Studies in Radar CrossSections XIII - Description of a Dynamic Measurement Program", The University of Michigan, Engineering Research Institute, Report No. UMM-128 (July 1954). CONFIDENTIAL 4 W. B. Graham, "A PreliminaryDiscussion of Intercontinental Ballistic Missile Defense", The Rand Corporation, Report RM-1388 (January 1955). SECRET, RESTRICTED DATA 5 C. E. Schensted, J. W. Crispin, and K. M. Siegel, "Studies in Radar Cross-Sections XV -Radar CrossSections of B-47 and B-52 Aircraft", The University of Michigan, Engineering Research Institute, Report No. 2260-1-T (August 1954). CONFIDENTIAL 6 A. L. Maffett, M. L. Barasch, W. E. Burdick, R. F. Goodrich, W. C. Orthwein, C. E. Schensted and K. M. Siegel, "Studies in Radar Cross-Sections XVII -Complete Scattering Matrices and Circular Polarization Cross - Sections for the B-47 Aircraft at S-Band"l, The University of Michigan, Engineering Research Institute, Report No. 2260-6-T (June 1955). CONFIDENTIAL I 184 -\D -m I r

UNIVERS I TY OF MI CHIGAN 2428 -3 -T REFERENCES (Cont.) Number 7 H. Well, R. R. Bonkowski, T. A. Kaplan, and M. Leichter, "Studies in Radar Cross-Sections XVI - Microwave Reflection Characteristics of Buildings" The University of Michigan, Engineering Research Institute, Report No. 2255-12-T (May 1955). SECRE F 8 W. H. Emerson, A. G. Sands, and M. V. McDowell, "Development of Broadband Absorbing Materials For Frequencies as Low as 500 Me", Naval Research Laboratory, Report No. 300 (May 1954). UNCLASSIFIED 9 D. E. Kerr, Propagation of Short Radio Waves, McGrawHill (1951). 10 K. M. Siegel, B. H. Gere, I. Marx, and F. B. Sleator, "Studies in Radar Cross-Sections XI - The Numerical Determination of the Radar Cross-Section of a Prolate Spheroid", The University of Michigan, Engineering Research Institute, Report No. UMM-126 (December 1953). UNCLASSIFIED 11 W. Franz and K. Deppermann, "Theory of Diffraction by a Cylinder as Affected by the Surface Wave", Annalen der Physik, 10, 361 (1952). 12 "The Equivalent Echoing Area of a Rocket Missile Near Head-on Aspect", Radar Research and Development Establishment, RRDE Report 370 (August 1952). CONFIDENTIAL 13 "Radar System Analysis - Comparative Performance Study of Pulse, F-M, and Doppler C-W Techniques for Ground Based Long-Range Search and MTI Radar Systems", Sperry Gyroscope Company, Sperry Report No. 5223-1109 (June 1948). SECRET 14 C. J. Sletten, "Electromagnetic Scattering Fr6m Wedges and Cones", Air Force Cambridge Research Center, Report CRC-E-5090 (July 1952). UNCLASSIFIED 185 ILZ\ rL

_UNIVERSITY OF MICHIGAN 2428 -3 -T REFERENCES (Cont.) Number 15 "Determination of Echoing Area Characteristics of Various Objects", The Ohio State University Research Foundation, Report 302-7 (October 1947). CONFIDENTIAL 16 "Fifth Quarterly Progress Report on Development of Chaff Material - 10 July to 10 October 1955", The University of Michigan, Engineering Research Institute, Report No. 2273-9-P. SECRET 17 K. M. Siegel, "Boundaries of Fluid Mechanics", Journal of Aeronautical Sciences, 17, 191 (1950). 18 J. Stratton, Electromagnetic Theory, McGraw-Hill (1941). 19 J. V. Nehemias, "Use of Computers in Determining TimeDependence of Fission Product Distribution", Nuclear Energy Conference in Geneva (1955). 20 D. H. Menzel, Fundamental Formulas of Physics, PrenticeHall (1955). 21 C. M. Davisson and R. D. Evans, "Gamma Ray Absorption Coefficients", Reviews of Modern Physics, 24, 79 (1952). 22 "Pressures, Densities, and Temperatures in the Upper Atmosphere", The Rocket Panel, Physical Review, 88, 1027 (1952). 23 K. M. Siegel, "Radar Cross-Section of Aircraft", to be presented 1 March 1956 at the Naval Research Laboratory at a symposium on Radar Detection Theory. CONFIDENTIAL 24 "Quarterly Progress Report - Polarization Dependence of Radar Echoes, Contract AF-30(635)-2811, Period Ending 31 March 1955", The Ohio State University Research Foundation, Antenna Laboratory, Report No. 612-2 (16 March 1955). UNCLASSIFIED 186 -- L' | -- --

U N I V E R S I T Y OF MICHGAN ___ H I A N 2428-3-T REFERENCES (Cont.) Number 25 K. M. Siegel, "Studies in Radar Cross-Sections V -An Examination of Bistatic Early Warning Radars", The University of Michigan, Engineering Research Institute, Report No. UMM-98 (19 August 1952). SECRET 26 E. A. Sloane, E. S. Candidus, J. Salerno, M. Skolnik, "A Bistatic CW Radar", Massachusetts Institute of Technology, Lincoln Laboratory, Technical Report No. 82 (6 June 1955). SECRET 27 B. L. Lewis, "Radar Characteristics of Jet Exhaust Gases", Tracking Branch, Radar Division, Naval Research Laboratory, Report 4438 (2 December 1954). SECRET 28 J. L. Lawson and G. E. Uhlenbeck, Threshold Signals, McGraw-Hill (1950). 29 W. B. Davenport, Jr., R. A. Johnson, and D. Middleton, "Statistical Errors in Measurements on Random Time Functions", Journal of Applied Physics, Vol. 23, No. 4, 377 (April 1952). 30 E. Jahnke and F. Emde, Tables of Functions, Dover (1945). 31 James, Nichols, and Phillips, Theory of Servomechanisms, Radiation Laboratory Series, Volume 25, McGraw-Hill Book Company, Inc. 32 "Thumper Project: The Meteor Problem (In connection with High Altitude, High-Velocity, Anti-Missile Research)", General Electric Company, Report No. TR-55038 (April 1947). CONFIDENTIAL 33 T. R. Kaiser, "Radio Echo Studies of Meteor Ionization", Advances in Physics, Vol. 2, No. 8, 495 (October 1953). - 187 - L z I r I 11

UNIVERS ITY OF M I CHIGAN 2428 -3 -T REFERENCES (Cont.) Number 34 "Summary Report ZM-7-011, Project Atlas", Consolidated Vultee Aircraft Corporation (January 1952). SECRET 35 T. von Karman, "Supersonic Aerodynamics - Principles and Applications", Journal of Aeronautical Sciences, 14, 373 (1947). 36 F. A. Lindemann and G. M. B. Dobson, "A Theory of Meteors and the Density and Temperature of the Outer Atmosphere to Which it Leads", Proceedings of the Royal Society, 102, 411 (1923). 37 D. L. Falkoff, "Meteors and Intercontinental Ballistic Missiles", Massachusetts Institute of Technology, Lincoln Laboratory, Group Report No. 23.10 (4 February 1954). SECRET 38 D. L. Falkoff, E. C. Lerner, "Intercontinental Ballistic Missiles", Massachusetts Institute of Technology, Lincoln Laboratory, Group Report No. 23.2 (28 December 1953). SECRET 39 N. Herlofson, Physical Society Report on Progress in Physics, 11, 44 (1948). 40 K. M. Siegel, H. A. Alperin, R. R. Bonkowski, J. W. Crispin, A. L. Maffett, C. E. Schensted, and I. V. Schensted, "Bistatic Radar Cross-Sections of Surfaces of Revolution", Journal of Applied Physics, 26, 297 (1955). 188 1 7- Ez EZ 77 I F-\ EZ-I

FrC-hm r-1 U N I VE RS I TY OF 2144 -3-T MIC HIGAN ____ DISTRIBUTION To be distributed with the terms of in accordance the contract. - — l ~ 189 LI ~I77

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