UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F RADC-TRASTIA Document No. STUDIES IN RADAR CROSS SECTIONS XXXVII - ENHANCEMENT OF RADAR CROSS SECTIONS OF WARHEADS AND SATELLITES BY THE PLASMA SHEATH (UNCLASSIFIED REPORT) by D. L.olph and H. Weil December 1959 Report Nqo. 2778 2-F -.on. Contract A~F 30(602)-1853. " Project 5535 -Ta sk- 45773 * ~.,.. ~. % "~. Prepared for ROME AIR DEVELOPMENT CENTER AIR RESEARCH AND DEVELOPMENT COMMAND UNITED STATES AIR FORCE GRIFFISS AIR FORCE BASE, NEW YORK UNCLASSIFIED

THE UNIVERSITY OF MICHIGAN 2778-2-F PATENT NOTICE: When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto. Qualified requestors may obtain copies of this report from the ASTIA Arlington Hall Station, Arlington 12, Virginia. ASTIA Services for the Department of Defense contractors are available through the "Field of Interest Register" on a "need-to-know" certified by the cognizant military agency of their project or contract. ii

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F STUDIES IN RADAR CROSS SECTIONS I "Scattering by a Prolate Spheroid", F.V. Schultz (UMM-42, March 1950), W-33(038)-ac-14222. UNCLASSIFIED. 65 pgs. II "The Zeros of the Associated Legendre Functions P_ (') 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. 20 pgs. III "Scattering by a Cone", K. M. Siegel and H.A. Alperin (UMM-87, January 1952), AF-30(602)-9. UNCLASSIFIED. 56 pgs. IV "Comparison between Theory and Experiment of the Cross Section of a Cone", K.M. Siegel, H.A. Alperin, J. W. Crispin, Jr., H.E. Hunter, R. E. Kleinman, W. C. Orthwein and C.E. Schensted (UMM-92, February 1953), AF-30(602)-9. UNCLASSIFIED. 70 pgs. V "An Examination of Bistatic Early Warning Radars", K. M. Siegel (UMM-98, August 1952), W-33(038)-ac-14222. SECRET. 25 pgs. 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. 63 pgs. VII "Summary of Radar Cross Section Studies under Project Wizard", K.M. Siegel, J.W. Crispin,Jr. and R.E. Kleinman (UMM-108, November 1952), W-33(038)-ac-14222. SECRET. 75 pgs. VIII "Theoretical Cross Section 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, Jr., A.L. Maffett, C.E. Schensted and I.V. Schensted (UMM-115, October 1953), W-33(038)-ac14222. UNCLASSIFIED. 84 pgs. IX "Electromagnetic Scattering by an Oblate Spheroid", L. M. Rauch UMM-116, October 1953), AF-30(602)-9. UNCLASSIFIED. 38 pgs. 111 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F X "Scattering of Electromagnetic Waves by Spheres", H. Weil, M. L. Barasch and T. A. Kaplan (2255-20-T, July 1956), AF-30(602)-1070. UNCLASSIFIED. 104 pgs. 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. 75 pgs. XII "Summary of Radar Cross Section Studies under Project MIRO", K. M. Siegel, M.E. Anderson, R.R. Bonkowski and W.C. Orthwein (UMM127, December 1953), AF-30(602)-9. SECRET. 90 pgs. XIII "Description of a Dynamic Measurement Program", K. M. Siegel and J.M. Wolf (UMM-128, May 1954), W-33(038)-ac-14222. CONFIDENTIAL. 152 pgs. XIV "Radar Cross Section of a Ballistic Missile" K. M. Siegel, M. L. Barasch, J.W. Crispin, Jr., W.C. Orthwein, I.V. Schensted and H. Weil (UMM134, September 1954), W-33(038)-ac-14222. SECRET. 270 pgs. XV "Radar Cross Sections of B-47 and B-52 Aircraft", C.E. Schensted, J.W. Crispin, Jr. and K.M. Siegel (2260-1-T, August 1954), AF-33(616)2531. CONFIDENTIAL. 155 pgs. 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. 148 pgs. 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. 157 pgs. XVIII "Airborne Passive Measures and Countermeasures", K. M. Siegel, M.L. Barasch, J.W. Crispin, Jr., R.F. Goodrich, A.H. Halpin, A. L. Maffett, W.C. Orthwein, C. E. Schensted and C.J. Titus (226029-F, January 1956), AF-33(616)-2531. SECRET. 177 pgs. XIX "Radar Cross Section of a Ballistic Missile II", K. M. Siegel, M.L. Barasch, H. Brysk, J.W. Crispin, Jr., T.B. Curtz and T.A. Kaplan (2428-3-T, January 1956), AF-04(645)-33. SECRET. 189 pgs. ivUNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F XX "Radar Cross Section of Aircraft and Missiles", K. M. Siegel, W. E. Burdick, J.W. Crispin, Jr. and S. Chapman (WR-31-J, March 1956), SECRET. 151 pgs. XXI "Radar Cross Section of a Ballistic Missile III", K.M. Siegel, H. Brysk, J.W. Crispin, Jr. and R.E. Kleinman (2428-19-T, October 1956), AF-04(645)-33. SECRET. 125 pgs. XXII "Elementary Slot Radiators", R. F. Goodrich, A. L. Maffett, N.E. Reitlinger, C.E. Schensted and K. M. Siegel (2472-13-T, November 1956), AF-33(038)-28634, HAC-PO L-265165-F31. UNCLASSIFIED. 100 pgs. XXIII "A Variational Solution to the Problem of Scalar Scattering by a Prolate Spheroid", F.B. Sleator (2591-1-T, March 1957), AF-19(604)-1949, AFCRC-TN-57-586, AD 133631. UNCLASSIFIED. 67 pgs. XXIV "Radar Cross Section of a Ballistic Missile - IV The Problem of Defense", M.L. Barasch, H. Brysk, J.W. Crispin, Jr., B.A. Harrison, T.B.A. Senior, K. M. Siegel, H. Weil and V. H. Weston (2778-1-F, April 1959), AF-30(602)-1853. SECRET. 362 pgs. XXV "Diffraction by an Imperfectly Conducting Wedge", T. B.A. Senior (2591-2-T, October 1957), AF-19(604)-1949, AFCRC-TN-57-591, AD 133746. UNCLASSIFIED. 71 pgs. XXVI "Fock Theory", R. F. Goodrich (2591-3-T, July 1958), AF-19(604)-1949, AFCRC-TN-58-350, AD 160790. UNCLASSIFIED. 73 pgs. XXVII "Calculated Far Field Patterns from Slot Arrays on Conical Shapes", R. E. Doll, R. F. Goodrich, R.E. Kleinman, A. L. Maffett, C.E. Schensted and K. M. Siegel (2713-1-F, February 1958), AF-33(038)28634 and 33(600)-36192; HAC-POs L-265165-F47, 4-500469-FC-47-D and 4-526406-FC-89-3. UNCLASSIFIED. 115 pgs. XXVIII "The Physics of Radio Communication via the Moon", M. L. Barasch, H. Brysk, B.A. Harrison, T. B.A. Senior, K.M. Siegel and H. Weil (2673-1-F, March 1958), AF-30(602)-1725. UNCLASSIFIED. 86 pgs. XXIX "The Determination of Spin, Tumbling Rates and Sizes of Satellites and Missiles", M.L. Barasch, W.E. Burdick, J.W. Crispin, Jr., B.A. Harrison, R.E. Kleinman, R.J. Leite, D.M. Raybin, T.B.A. Senior, K.M. Siegel and H. Weil (2758-1-T, April 1959), AF-33(600)-36793. SECRET. 180 pgs. UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F XXX "The Theory of Scalar Diffraction with Application to the Prolate Spheroid", R. K. Ritt (with Appendix by N.D. Kazarinoff), (2591 -4-T, August 1958), AF-19(604)-1949, AFCRC-TN-58-531, AD 160791. UNCLASSIFIED. 66 pgs. XXXI "Diffraction by an Imperfectly Conducting Half-Plane at Oblique Incidence", T.B.A. Senior (2778-2-T, February 1959), AF-30(602)-1853. UNCLASSIFIED. 35 pgs. XXXII "On the Theory of the Diffraction of a Plane Wave by a Large Perfectly Conducting Circular Cylinder", P. C. Clemmow (2778-3-T, February 1959), AF-30(602)-1853. UNCLASSIFIED. 29 pgs. XXXIII "Exact Near-Field and Far-Field Solution for the Back Scattering of a Pulse from a Perfectly Conducting Sphere", V. H. Weston (2778-4-T, April 1959), AF-30(602)-1853. UNCLASSIFIED. 61 pgs. XXXIV "An Infinite Legendre Integral Transform and Its Inverse", P. C. Clemmow (2778-5-T, March 1959), AF-30(602)-1853. UNCLASSIFIED. 35 pgs. XXXV "On the Scalar Theory of the Diffraction of a Plane Wave by a Large Sphere", P.C. Clemmow (2778-6-T, April 1959), AF-30(602)-1853. UNCLASSIFIED. 39 pgs. XXXVI "Diffraction of a Plane Wave by an Almost Circular Cylinder", P. C. Clemmow and V.H. Weston (2871-3-T, September 1959), AF19(604)-4993. AFCRC-TN-59-955. UNCLASSIFIED. 47 pgs. XXXVII "Enhancement of Radar Cross Sections of Warheads and Satellites by the Plasma Sheath", C. L. Dolph and H. Weil (2778-2-F, December 1959), AF-30(602)-1853. SECRET. XXXVIII "Non-Linear Modeling of Maxwell's Equations", J.E. Belyea, R.D. Low and K.M. Siegel (2871-4-T, December 1959), AF-19(604)-4993. UNCLASSIFIED. vi UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Preface This is the thirty-seventh in a series of reports growing out of the study of radar cross sections at The Radiation Laboratory of The University of Michigan. Titles of the reports already published or presently in process of publication are listed on the preceding pages. When the study was first begun, the primary aim was to show that radar cross sections can be determined theoretically, the results being in good agreement with experiment. It is believed that by and large this aim has been achieved. In continuing this study, the objective is to determine means for computing the radar cross section of objects in a variety of different environments. This has led to an extension of the investigation to include not only the standard boundary-value problems, but also such topics as the emission and propagation of electromagnetic and acoustic waves, and phenomena connected with ionized media. Associated with the theoretical work is an experimental program which embraces (a) measurement of antennas and radar scatterers in order to verify data determined theoretically; (b) investigation of antenna behavior and cross section problems not amenable to theoretical solution; (c) problems associated with the design and development of microwave absorbers; and (d) low and high density ionization phenomena. K. M. Siegel vii UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Table of Contents Page Abstract 1 Introduction 2 1. The Scattering Integral 3 2. Computation of Electron Distribution 7 3. Numerical Results for Electron Density Distribution and Radar Cross Section 23 Appendix - The Reduction of the "Oseen-Like" Linearized Equations of Motion for an Ionized Medium 35 References 42 ix UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F On the Change in Radar Cross Section of a Spherical Satellite Caused by a Plasma Sheath+ by C.L. Dolph, H. Weil Abstract A uniform neutral dilute ionized gas is assumed to be perturbed by a sphere moving through it. The radar return from the disturbed region is obtained by integrating the Compton scattering from the electrons, taking phase into account, but ignoring secondary scattering and attenuation. The electron density distribution for this computation is obtained by integration of the zero'th order velocity distribution function for neutral particles obtained by C. S. Wang Chang as a solution of the Boltzmann transport equation. Numerical results are obtained for the perturbation of the electron distribution by a sphere traveling at 8 km/sec and an altitude of 500 km, and for the radar cross section of this perturbed region when viewed broadside.++ + This portion of the final report consists essentially of a paper presented at the Symposium on the Plasma Sheath; Its Effects on Communication and Detection, sponsored by AFCRC in December 1959. The changes are in Part III which has been expanded here to include more extensive numerical results on radar cross sections. The theoretical work and preliminary computations were supported by USAF Contract AF 30(602)-1853. The final machine computations were carried out as an unsponsored faculty research project MO2-N at the Computation Laboratory of The University of Michigan. UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Introduction A sphere is assumed to move with a constant velocity, V, through a dilute, electrically neutral, ionized gas which, in its unperturbed state, is assumed to have a uniform number distribution of electrons, n. The sphere disturbs the distribution of electrons to a non-uniform one, N, with an excess ahead of the sphere and a deficiency behind it. The simplest estimate of the effect of this non-uniform, but everywhere dilute distribution on the radar return is obtained by summing the scattering by the individual electrons accelerated by the incident field. Electrons only are considered since their return is far greater than that of the much heavier positive ions, or the Rayleigh scattering from non-ionized particles. The incident field on each electron is assumed to be a plane wave; this implies that secondary scattering is ignored. The approach is thus directly analogous to that used to determine the radar return from underdense meteor trails [1]. There are three parts to the paper. The first part consists of a formulation of the expression for the backscattered energy. This expression involves the perturbed density distribution. The condition of neutrality is used in determining this distribution since it forces the electron motion to be governed by the positive ions whose velocities and mass are similar to that of the neutral particles. This will be discussed in the second part of the paper wher'e expressions for the distribution are found. In the third part numerical results for a specific sphere velocity and altitude are presented. UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F 1. The Scattering Integral The radiation field of each electron yields a backscattered power per unit solid angle per electron for unit incident power density given by e [e /(4r e mc2 2 Here e is the charge on the electron, m its mass, eo the permittivity of free space, c the velocity of light and MKSC units are to be used. The incident power is given by PG/ (4wr r2) where P is the total power emitted from the radar antenna, G the antenna gain and r the distance from antenna to electron. The effective collecting area of the antenna is G X2/ (4wr) so that the scattered power per electron received by the radar for large r is pG2 x2 a e e 16 r2 r4 We now assume the radar is well out of the ionized region of interest so that r is always large. Then the net power received from this region is e G e2ikr Nd where dv is a volume element. For simplicity a beam width wide enough to be essentially constant over the disturbance is assumed, and the slowly varying factor r2 replaced by the range Roto the sphere and removed UNCLASS IFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F from under the integral sign. Finally N is referred to the constant n by writing N = (N-n) + n. The integral of n e2ikr vanishes except for contributions at the "edges" of the region of integration. Of course the distribution n extends beyond the beamwidth of the radar and thus we know these "edges" are not physically significant. They may be neglected with the result that the desired quantity, the net power received due to the disturbance of the density distribution is given by 2 2 T PG 2ikr SD e e (N-n) dv 16ir2 R4 The integration is to be extended over the region of interest. In general this will include the entire region over which N-n differs appreciably from zero. However, one might also be interested in considering separately the effects of the region ahead of the sphere and the region behind it. If these were to act as independent scatterers the average returned power (averaged over all relative phases) would be the sum of two expressions SD1 and SD2 corresponding to SD with the integration in SD1 over the region ahead of the sphere and the region in SD over the region behind the sphere. To put the integral in S in a form suitable for computation it is convenient to refer to Figure 1. The density N must be symmetric about z so that it is convenient to use cylindrical coordinates p, //, z in the integration. Furthermore one can simplify the integral by using in the expression for r(p) UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F z h P Figure 1. 2 2 2 r(p) =r; r- r +p -2f. O O the fact that - << 1, so that rO r =r [1+ P _ 2p sinO cos (P -)2 0 2 r ~r -p sinO cos (p- b)+ 1 PP2 sin20 cos2(0 - 0 2 r 8r O o In turn ro is approximated by 2 z 1 z2 2 r R + — - 2z cos~ -cos +. 0 2R0 8 R UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F We shall be interested in 600 < e < 1200 and hence will neglect the last term as well as those of higher order in z/Ro. Then 2 00 00 2 2 2 2r 2 2(e ePG2 | [ 2ikr(p,'/, z) S = 2 4PO | d42 d dz p dp N(p, z)-n ] e o 0 -co 0 00 00 XA PG 2 ( 4R4 4 3 2R 3 p2 The term - is neglected in the phase since it will appreciably affect the 2 r phase only if 3 =900 and then only where p exceeds.1 \fi. For such large p's the amplitude N-n is negligible. This is not quite true for corresponding values of z. In this consideration we have assumed k < 20 m. Note that in this integral sinO (z)=R sinO/ r(z). UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F 2. Computation of the Electron Distribution The steady-state problem of a point charge moving through a fully ionized medium of sufficiently low density has been treated by Kraus and Watson[2]. Their work was extended to the case where a constant magnetic field is present by Greifinger [ 3]. A good deal of insight into the physical meaning of the above theories which started from the linearized Landau-Vaslov equation was provided by the report of Pappert [4]. In this report Pappert deduced the results of Kraus and Watson from the random phase approximation of Bohm and Pines[ 5] and also demonstrated the equivalence of these methods to the linearization of the equations of motion and continuity for the ions and electrons under the assumption that an isothermal state exists.+ The problem of an object of finite size has been approached by using the expression obtained in 1950 by C. S. Wang Chang [6]. Chang obtained the zero'th order velocity distribution function for a sphere of radius R at rest in a neutral gas with a streaming-velocity V under the assumption that the sphere was sufficiently small and the gas sufficiently dilute that the collisions between the main stream particles and those reflected from the sphere could be neglected. The distribution function so determined satisfies: +An alternate and simpler derivation of a more general version of the uncoupled microscopic equations of Kraus, Watson and Pappert has been worked out by C. L. Dolph and is given in an Appendix. UNCLASSIFIEED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F (A) The collision-free Boltzmann transport equation in the absence of external forces; (B) A boundary condition on the surface of the sphere that implies that the sphere neither absorbs nor emits gas particles by itself so that all particles that hit the sphere are re-emitted; (C) The property that it reduces to the Maxwell-Boltzmann distribution around the streaming velocity at infinity independent of angle. In addition, this distribution allows for arbitrary amounts of diffuse or specular reflection at the spherical surface. Since the region of interest here involves velocities of the sphere much greater than that of the ions and at the same time much smaller than that of the electrons and altitudes where the mean free paths are large compared to the expected dimensions of the disturbed area, Chang's distribution may be used to provide an order of magnitude estimate of the electron distribution around the sphere when it is assumed that the charge on the sphere is so small that it can be ignored so that the ions will (to all intents and purposes) behave as neutral particles as far as their interaction with the sphere is concerned. The strong coulomb forces should provide electrical neutrality which will then force the electrons to assume a distribution identical in form to Chang's in which only the mass and velocity parameters can be different. This use of electrical neutrality, while appropriate in ionospheric physics,is less exact than the assumptions usually used in physics of confined plasmas where [71 it is more customary to assume electrical UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F neutrality except for consideration of Poisson's equation while here we ignore any deviations from neutrality in this equation as well. While it would be more exact to assume the Chang distribution as the first term in a perturbation procedure for the Landau-Vaslov equations, it is unlikely that such a refined analysis would affect the radar cross section results. Other approaches such as that used by Bernstein and Rabinowitz [9] in their discussion of spherical probes would seem to encounter even more intractable analytical difficulties were the Chang distribution to be used in place of the mono-energetic one used by them. C n Figure 2. +Actually the indicated procedure for the second approximation is presently under investigation under another contract for a different purpose. Preliminary analysis seems to indicate the existence of oscillatory solutions for the electron density with frequencies of the order of those characteristic for plasmas modified by an increment dependent upon electron temperature and form factors appropriate to the geometry. See Reference 8. 9 UNCLASSIFIED

UNCLASSIFIED THE UNIVERS ITY OF MICHIGAN 2778-2-F.SS =O V17 ( < V Sin~ _ Cos O1 = 1 r2 Figure 3. The zero'th order velocity distribution obtained by Chang is expressed in terms of the following variables. V' = the velocity of the main stream R = the radius of the sphere r = the point under consideration, or the point at which the velocity is being calculated n - the outward normal to the sphere which passes through the A point; r = r n if the center of the sphere is taken as the origin of the coordinate system C' = the velocity vector of gas particles a - = the fraction of molecules that is diffusely reflected from the sphere 10 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F 1- a = the fraction of molecules that is specularly reflected from the sphere V = V' / 2kt/ m, non-dimensional streaming velocity C =- C'/ 2kt/ m, non-dimensional velocity vector n' -= the normal to the sphere at the point from which the molecules arriving at r with velocity C originated. From Figure 2 it is found that A r r C I(r )2 R2 C = C+ r (1 ) - RC2 r2C2 r RC A nC - unit vector in the direction of C n - the number of particles present in the unperturbed state OD 2 2 A -t erfc (x) - e dt =1 - erf (x) A S - S(x, y, z, n a discontinuous function which is zero if the particles of velocity C at point r(y, x, z) come from the A sphere and which is one otherwise. That is, S = 0 if nc points away from the sphere and lies in the solid angle subtended by the sphere at the point under consideration. The distribution can be written as f(a) += n Se + a f(V[n') erfc (V. (l-S)eC 1 [1C - V - 2n-en' ) + (1 - a) (1-S) e The function S which is described above can be represented analytically as 11 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F S - (1 - sign(nC)+ A(1+sign (.C + sign i(1-R2 ) C 2 - -2 C and it has the properties that for r approaching infinity, S approaches 1, while for r approaching the sphere R, S (1 - sign nC). Furthermore, if 0 is the angle between r and C then, as can be seen from Figure 3, r 1/2 S = O if l1cos0 )(1 r2 ) =cos 1 S = 1 if cos 0 < cos 0 The Density Distribution for Ions and Electrons in the Neighborhood of the Sphere for the Case of Diffuse Reflection The necessary calculations are considerably simplified if it is assumed that only diffuse reflection occurs so that a may be set equal to unity. Fortunately this appears to be a good approximation to the physical situation [6], [10o, [1j1] 12 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F We shall therefore calculate Sf(1) d3 C and take up the two terms separately, treating first the contribution -(C - V)2 = S e dV'1= 3/2 SSeICV)d from the main stream. To evaluate I1 we determine S in rectangular C space C,, C C. For convenience set n in the direction of Cz. R2 1/2 Cz - z then C > 0 and S =- [i - sign Cz =0. Cz > 0 Also, for all C for which R2 1/2 C> r2 C2 + C 2 + C 2 x y ) S = 1. Since both sides of the first inequality are positive it may be squared and solved for C Z. The result is 13 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN' 2778-2-F r2 1/2 21 /2 C >( -1) (C 2+C2) R2 x y so that if CZ ) C, S =0. Similarly if CZ < C, S =1. 0 J Thus we must evaluate 3/2' dC dCY dCZe Y The C, integral is C;Zo (2 CZo-VZ eC dC dt= [erf (C - V )+1] -00 -00 Hence n n C +(CyVY )2 l2 2~ r e erf(C-V)dCx dC -00 -00 Introduce polar coordinates as follows: i1 2 2 C= P cosp V- = (V - V cos - X z~ C-psinp Vy=V -V ZsinV. Then 14 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F /2 2 o I UL I + ~ e~v-vz2~ dppe erf(P s v) s e2p V _c cos(p- ) A 0 000 I1 2 ~r R~ dp2pe erf(p -V)I(2 V ) where Io( ) is the Bessel function of imaginary argument and zero order, and V =V.n =V coso V2 V2 =V2sin2.0 z Hence X1. r 2 2 02 Z { Vi+ -V sin 2os( n In the limiting case, r = R Il = 2-V1+2e _ f =V2s2.dppeP _ sin2_1 1I~~~~~~~15 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F To carry out the integration, integrate once by parts to obtain 00 00 0. 2 2 2 with u =V sin. The integral on the right is given in the Bateman Manuscript Project series [12] leading to the result u2 1 2 2 I - + - ue I1 ( ). 2 2 1/2 2 Since 2 I (z) = -Z) sinh z 1/2 7rz one gets 2 1 u I - e 2 and hence for r R, I1 =- 1+ erf (-V cosO) t - n erfc (Vcose ). 1 2 2 o This checks the direct integration of I1 for r set equal to R in advance, since in this case, S = 1 only if Cx.0. When r - 0 co, erf (p - -1-V) eerf(oo) =1 16 UNCLASSIFIED

UNCLASSIFIE D THE UNIVERSITY OF MICHIGAN 2778-2-F and one obtains n 1+1 =n as one should. For the two cases of 00 = 0 and 00 = ir the quantity I1 can be evaluated exactly if spherical coordinates are used. If one uses r as axis r = r(O, O, 1) V = V(sino cos 0, sino sinso0, coso ) C = C(sinO cost, sinO sinp, cos0) and 00 2x 01 = n 3/2 C2 dc d dp 5 sinO 0 0 0 exp -(C2+V2)+2CV [cos cos0e +sinO sin0o(cosp-po)]} dO. By straightforward integration of this form of I1 when 0 = O0 one finds ( it -V2sin20 1 I1(0) =n erfc V+e cosO 1 erfc(-Vcoso 1 ) 11(0)~ 2 17 2 NCLASSIFIE n -"'2 R2

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F A similar discussion for 0 = r leads to the value 0 2 R2 2r2 r2 e For r =R both of these reduce to our previously obtained approximation since the additional term is zero there. Likewise in the limit as r — oo, we obtain the free stream density as we should. For either 0 = 0 or 0 =r the above reduces to lirm i1 j - [erfc (V)+ erfc (-V)] =n r -+ co which is correct. To evaluate the term containing the effect of the diffusely reflected particles, it is necessary to consider the integral A 22 ( - (V n') -C2'2 7.3/2 d C e - vF(V. n') erfc (V n') (l-S) e Introducing r as the polar axis again so that' - [(cos) r/R+r/R cos (sin cos - (1 [(-cos0) r/R+r/R cos20 - (1 - R/r2)] sin sink, (r/R) [(-rcos0)/R+r/R cos20 -(1 -R2/r2)]cos0 leads to 18 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F' -Vr V n' = Vr cos0o [cos Ceos20 -(1-R2/r2)] [cos cosOo+sinO sinSOcos(o -po)] so that I2 is quite intractable without approximation. However, as mentioned earlier (1-S) = 1 if and only if 0~ 0 ~ 01 arcos (1 R2 )1 ~1 r~2 and in this range 2 R2 coso - \]cos 0 -(1.2 is a very slowly varying function which has extremes at 0 = 0, and 0 = 0 with values (1 - R/ r) and\1 respectively. We will therefore r2 replace the complicated expression for (V. n') in exp -(Vn') and erfc(V.n') by its first term cos0 V. =n Vr ~- R The integral 12 is therefore evaluated as 2~r~ V22 Go 2 27r 1 - r 2o A n ~ -C2 2 2IO n/ -C2 C2 dC do sin dO e r 0 - 0 s0 - l'(Vi.n')erfc (VrcosoO )| 19 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGA N 2778-2-F where the full expression for (V, n') is used where it appears explicitly. Straightforward integration then yields the following expression for I2A 22 V 2r 2 ~z-: 0 1 cos 0 0 -R n coso] e -j 0 2 2os2 ~20 SS f e rfc cos (Vnn')sinO dO do 47r R o 0 0 2 2 n= i 1 2-e R2 R o n Vr 1 Re R'0 - Ffferfc(-V cos0 )1 2 r2 r2 R 2 3 where 2w 0 13 = (Von') sin0 dO do 0 0 =2w cosO - o- 2cosO 0010 so R 0 0 r2'j 0) 0~~~~ ~20 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F difference between I2A and I2 is bounded above by Vr )2 ( Vr R2 ( o 2 2 R _( ) (c) (VrC2 cosj 1 0 ~~R 0 2 R -- e r -e Vr 2 2 2 2 _ -(- (cos1 + ] )2 R rr 2R2 1 ~o 2 -- -— cos e0

I

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F 3. Numerical Results for Electron Density Distribution and Radar Cross Sections The formulas of Part 2 were applied to a typical [ 71 case of interest for which V = 5. This corresponds, for example, to a satellite altitude of 500 km and speed V' = 8 km/sec. Curves of constant relative density N/n are presented in Figures 4 and 5. They clearly show the build-up of density ahead of the sphere and the'hole" developed in the rear. The ion deficiency in the rear extends to about 50 sphere radii. More detailed data for N on the sphere and along the positive z axis (behind the sphere) is given in Table I and Table II. Table I was computed on a desk calculator using the exact formulas for R=r or 0 0=0, pp 16, 17. Table IIwas computedby numerical integration on an IBM 704 of the integral on page 15. The integration steps were not small enough to get a good check on the exact results for p = 0, 1 < z < 2.5. Radar cross sections (4wr times differential cross sections) were computed from the formula 2 4Ce n2 ( 1 ) e2ikr dv according to the development in Part 1. For electrons 4ir -e 10 m2. A value of n = 1012 electrons/m3 is used for the electron density at 500 km altitude and the resulting values of C are given in Table II for a sphere of im radius. Three aspect angles e (see Figure 1) and three wavelengths, X = 15, 6.3, and 1 23 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Table I Density Ratio = Density = N Free Stream Density n On Sphere, r = R Behind Sphere at 0~ 0 N/n z N/n 00 9.0 x 10-131 9 0 x 10 50 1.0923 x 10-12 2 1.6718 x 10-3 100 1.9443 x 10-12 3 5.8621 x 10-2 150 4. 9876 x 10-12 4 2.0296 x 10-1 200 1.7947 x 10-11 5 3.6045 x 10-1 250 8.7137 x 10-11 6 4. 9237 x 10-1 300 5.4587 x 10-10 7 5.9422 x 10-1 350 4.1834 x 109 8 6. 7133 x 10-1 400 3.6932 x 10-8 8. 5 7.0258 x 10-1 450 3.5343 x 10-7 9 7.2990 x 10-1 500 3.4398 x 10-6 9.5 7.5384 x 10-1 550 3.1943 x 10-5 10 7. 7490 x 10-1 600 2.6707 x 10-4 20 9. 3824 x 10-1 65 1.9583 x 103 30 9.7206 x 10-1 700 1. 1018 x 10-2 40 9. 8419 x 10-1 750 5.0194 x 10-2 50 9.8985 x 10-1 800 1. 7613 x 10-1 850 4. 7471 x 10-1 860 5.6145 x 10-1 870 6.5761 x 10-1 880 7.6306 x 10-1 890 8 7739 x 10-1 90~ 1.0 24 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2 -F Table II Table of N (p, z) used in Numerical Integration 0O -0.5 -1.0 -1.5 -2*0 -2.5 -3.0 -3.5 -4.0 -4.5 -5.0 RHO 0. 9*862 3.102 2.042 1*62:3 1,413 1*294 1.220 117t1 1136 0o100 8.939 3.085 2.038 1.621 1.413 1.294 1,220 1*170 1.136 0,200 7*978 3,037 2.025 1.616 1*410 1*2'93 1.219 1.170 1.136 0.300 7.027 2*961 2'.004 1*608 1.407 1291 12.18 116-9 1l135 0O40f0 6.133 2*861 1.976 1.597 1*402 1o2.88 12'l216 1,168 1135 0.500 5*323 2.746 19-42 1.5:84 1.396 1*285 1.215 1&167 1*134 0*600 46.13 2.619 1.e903 1*568 13-88 1,281 1,212 l.166 11.33 07000 4,007 2.488 l1859 1550 380 1.380 1.276 1.210 1,164 1l132 0.800 3E499 2.357 I1814 1*531 1.370 1.271 1*207 1,162 1.131 0O900 4*273 3,077 2.2:30 1*7.67 1.510 10360 1.266 1,203 1.e160 1,129 100O0 1*000T 3*161 2*730 2110 1.719 1.48-9 1.3'4"9 1.2160 1.20.0 1l158 1,128 1e100 1e290 2*565 2.446 1.e998 l&672 1.467 1,338 l,2153 1.196 1.156 le.126 1 2 O0 1.223 2.239 2*214 1.895 1.6.26 14.444 1.32i6 1.244.6 1.1:9:2 1. i153 1.125 1.300 le180 1*914 2.024 1,802 1.581 1.422 1,314 1.239 l1188 1 *150 1.123 1.400 1.14-9 le723 1.868 1.718 1.538 1*400 13'01 1*232 1.183 1.l147 1l121 1.500 1.127 1.582 1,740 1,642 1,498 1.37'8 1.289 1*22'5 l1.179 1.144 1.119 1&600 1*109 1.475 1.634 1.575 16460 1.357 1.277 1.217 1I174 1,14:1 1117 1.700 1.095 1.371 1l546 1.515 1*425 1*336 1*264 1*210 1.169 l138 e1,l15 1.800 1.084 1.327 1l473 1.462 1.392 1,316 1.252 1*203 1,164 1.135 1l112 1.900 1*074 1*276 1*411 1l415 13561 1,297 1.24i 1.1995 1,159 1,132 1.110 2C000 1*066 1l211 1.359 1.374 1i333 1.279 1.229 1.188 1.155 1*128 1.108 2.100 1.060 1,202 1.315 1.337 1.307 1,252 1*218 1.180 1.150 1.125 1,105 2.200 1.054 1.174 1.278 1,304 1,284 1.246 1.208 1*173 1.145 1,122 1.103 2.300 1.028 1.151 1.246 1,275 1*262 1l231 1*197 e1166 1.140 1.I18 14101 2.400 1.045 14126 1.219 1*249 1*242 1.217 1,187 1*160 1.135 1.115 109-8 2,500 1*041 1,116 1.195 1,227 1.224 1.203 1.178 1.153 1.131 1,112 1.096 2.600 1.038 1.103 1.175 1.206 1.207 1.191 1.169 1.146 1.126 1.108 1.094 2.700 1*035 1*091 1.157 1.188 1.191 1,179 1.160 1.140 1,122 1,105 16091 2*800 1.032 1.081 1.142 1l172 1.177 1,168 1*152 1.134 T1117 1.102 1o089 2.900 1*030 1*073 1.128 1*157 1*165 1.158 1*144 1.128 1.13' 1.099 1,086 3,000' 1.028 1,065 1.116 1,144 1*153 1*148 1,137 1.12.3 1.109 1*096 1.084 3,100 1*026 1,059 1.106 1.133 1.143 1.140 1*130 1*118 1,105 1.093 1.082 34200 1.024 1.053 1.096 1.122 1.132 1, 131 1*123 1*112 1l101 1.090 lO80 3.300 1,023 1.049 1.088 1.113 1.124 1*123 1 117 1*108 1,097 1.087 1.077 3.400 1.021 1*044 1.080 1.104 1.115 1.116 1.111 1*103 1,093 1,084 1*075 3.500 1,020 1*040 1,074 1.097 1.108 1.109 1*106 1.098 1.090 1,081 1.073 3.600 1,019 1.037 1.068 1.089 1.100 16103 1,100.1.094 1.086 1.078 1O071 36700 1.018 1*034 1.063 1*083 1l094 16097 1*095 1.090 1-#083 1,076 l1069 3*800 1o017 1*031 1.058 1.077 1*088 16092 1,090 1,086 1.080 1l073 l1067 3,900 1,016 1o029 1.054 1.072 1*082 1.087 1.086 1.082 1*077 1*071 1.065 4*000 1l015 1.026 1.049 1.067 1*077 1*082 1*082 1*079 1.074 1*068 1,063 4*100 1*014 1.024 1.046 1,062 1.073 1.077 1*078 16075 1.071 1,066 1.061 4.200 1,014 1.022 1.043 1,058 1*068 1C073 1.074 1.072 1.068 1,064 1.059 4*300 1.013 1l021 1.040 1.055 1.064 1*069 1070' 1.069 1,066 1.062 1o057 4.400 1*012 1.019 1.037 1.051 1.060 1.065 1*067 1.066 1.063 1o060 1.056 4,500 1.012 1l018 1.035 1*048 1.057 1*062 1.064 1.063 1.061 le058 1l054 4,600 1.011 1.017 1.032 1.045 1.054 1*059 1.061 1.060 1,059 1.056 1.052 4.700 1.011 1.016 1.030 1.042 1.051 1.056 1.058 1.058 1.056 1.054 1.051 4,800 1.010 1.015 1.028 1.040 1.048 1.053 l1055 1.056 1,054 1.052 l1049 4.900 1.010 1.014 1.027 1.037 1.045 1.050 1.053 1.053 1,052 1.050 l1048 5.000 1I009 1.013 1.025 1,035 1.043 1,048 1.050 1.051 1.050 1.048 1.046 25 UNCLASSIFIED

UNCLASSIFIED THE UNIVE RSITY OF MICHIGAN 2778-2-F Table II (continued) Table of N(p,z) used in Numerical Integration''"-0 0. 05 1.. 0 15 2. 0 2 5 3, 30 35 4e 0 4.5 RHO 0. 0, 01 0, 001 0.02-5 0.018 0.6059 0*125 O0-204 0 2:84 0Q100 0.001 0.001 0e030 0.020 0*064 06130 0.209 01289 0.200 0*000. 0.001 0,051 0.028 0*078 0l..1,46 0*224 0.302 0.300 0.000 0.001 0O011 0.045 0*102 0.173 0.249 0.324 0*400 0.000 0O002 0O024 0.073 06.138 0. 211 0.284 0.354 0.500 0.000 0.009 0.050 0.'115 0.a187 0.259 0.327 0.391 0.600 0o001 0.030' 0*099 0.176 0*250 03-.17 0.378 0.434 0*700 0.009 0.085 0,177 0,257 0*324 0*382.0.43:4 0a48:2 0.800 0.068 0.192 0,285 0.35.4 0*408 0.454. 0.495.0.53-3 0.900 0.032 0.2:50 0.355 0.417 0.462 0,4-98 05.29 0.558 0.585 1.000 1.000 0.514 0.528 0.543 0*557 0,572 0*587 0.603 0.620 0.637 1.100 16290 0.8'92 0.772 0.715 0.688 0.676 0.672 0,674 0.680 0.688 1.200 1.223 0*984 0.912 0*8.42 0.79:5 0e766 0.749 07*T39 Q. 7'35 O0.736 1*300 1*180 0e998 0.971 0.922 0.875 0.839 0.814 0.797 0.786 0.780 1*400 1*150 1.000 0.991 0,964 0*928 0*894 0.866 0.845 0.83'0 0.820 1e500 l1.127 1l000 0.997 0.985 0*961 0*933 0.907 0*885 0886-8 0.855 1.600 1O110 le000 0.999 0O994 0.979 0.959 0.937 0.917 0.899 0.e88:5 1.700 1*096 1.000 1.000 00,997 0990 0.976 0.959 0.9:41 0.924 019'10 1;800 1.084 l000 1.000 0.998 0.995 0.986 0.973 0*959 0.944 09:30 1.900 1.075 1*000 1.000 16000 0*997 0.992 0*983 0*972 0e959 O.*947 2,000 1.067 1*000 1.000 1*000 0*999 0*996 0.990 0*969 0.971 0.6960 2.100 1.060'1.000 1.000 1.000 0O999 0.998 0.994 0.9'87 0.979 0.970 2.200 1*055 1*000 1.000 1l000 1*000 0.999 0*996 0*99'2 0.986 0.978 2.300 1.029 1*000 1000 1O000 1.000 0.978 0.9-98 0O995 0*990 0.984 2.400 1t045 1*000 1.000 1.000 1.000 1,000 0,999 0,9996 0.993 0.988 2.500 1.042 1.000 1.000 1.000 1.000 1.000 0*999 0*970 0.995 0*992 2.600 1*038 1*000 1.000 1,000 1.000 1*000 0.999 0.9-99 0.997 0.994 2*700 1.035 1.000 1.000 1.000 1.000 1.000 1.00 0.999 0.998 0*996 2.800 l1033 1.000 1.000 10000 1.000 1.000 1.0:00 0.999 0.999 09'997 2*900 1*031 1*000 1000 1.000 1*000 1l000 1.000 1l000 0.999 0.9,98 3.000 1.028 1.000 1.000 1O000 1000 1*000 1,000 1.000 0.999 0.999 3.100 1.027 1l000 1*000 1,000 1.000 1.000 1,000 1,000 1.000 0,999 3.200 1,025 1*000 1l000 1000 1.000 l.000 1.000 1O000 1000 0.999 3.300 1e024-1,000 1000 1,000 1O000 10:00 1,000 1O000 1.000 0.999 3,400 1,022 10000 1.000 1,000 1.000 1.000 1O000 1.'000 1,000 l1000 3,500 1,021 1,000 1O000 1*000 1,000 1l000 1l000 1000 1000 le00 3,600 1.020 1,000 1,000 1,000 1,000 1.000 1,0O00 1000 1000 1,000 3.700 1,019 1,000 1,000 1,000 1.000 1.000 1.000 1 l000 1.000 1,000 3*800 1.018 1*000 1.000 1.000 1O000 1*000 1.000 1.000 1.000 1.000 3.900 1*017 1l000 1.000 l1000 1,000 1,000 1.000 1.000 1l000 1,000 4.000 1016 1.000 1.000 1,000 1000 1000 1000 100 00 1000 1.000 4*100 1O015 1e000 1.000 140l00 1.000 1,000 1l000 140:00 1.000 1000 4,200 1*014 1,000 1l000 1.000 1l000 1000 1*000 1l000 1000 1,000 4*300 1,014 1.000 1,000 1.000 1l000 1,6000 1l000 1.000 1.000 1.000 4,600 l1012 1,000 1.000 1,000 1,000 1.000 1.000 1*000 l.000 1,000 4,700 1.011 1O000 1000 1.000 l.000 1.0O0 l.000 1.000 1.000 1O000 4e800 1*011 1.000 1*000 1.000 1.000 1.000 1,000 1.000 1.000 1,000 4*900 1*010 1*000 1.000 1o000 1.000 1.000 1.000 1.000 1.O00 1o000 5.000 1O 10 1.000 1.000 l.000 1.000 l.000 1.000 l.000 1,000 leQ00 26 UNCLASSIFIED,

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Table II (continued) Table of N (p, z) used in Numerical Integration 50 5*5 6.0 6.5 7*0 7.5 8.0 8.5 9.0 RHO O. 0 36.1 0e.431 0Oa493 0a547 0.'595 0 6386 0 672*703 0.7730 o0100 0.e 365 0 434 0.6496 0. 549 0 597 0 637 0 673 0*704 0e7'32 0 200 0 3776 0 443 0*503 0*556 0.601 0 a.6 O..-:67'6 0 707 07 735 0,3 00 0.395 0 45-9 0 515 0.566 0 610 0 648 0 684 0 711 0 73 O0400 0 4'20 0 479 0 532 0 579 0 621 0 657 0 692 077 0 742 0,500 0.045 0*504 0.553 30596 0*6.34 0.668 0.700. 00*725 07 a 49 0 600 0.486 0,534 0.5577 0*616 0.651 0.682 08709 0:734 0.76 0.700 0*526 0.567 0.604 06*638 06699 0.697 0a. 722 0.145 0'76:5 o*800 0*5:68 0.601 0*633- 0.662 0,688 0 7:13 0 T36 0 76 0 775 0.900 0612 0.638 0.66'3 0.687 0.709 0.-731 0750 0.7O8 0785 7.000 0. 65-6 0.675 0.694 0O713 01731 0a 749 0a766 0*781 0796 100 0.699 0.711 0.*725 0~739 Oa753 0&767 0.-781 0.795 0m.,807 l 200 0*740 0.74.6 0T755 0*765 0*775 0.786 0.797 0:809 O0.88 1 300 0.778 0.780 0.a78:q4 0~790:0.797 0805 0.8::14 0822 0.8311 400 0 8 -14 0~ 811 0811 0 814 8 08182 0823 0 829 0* 836 0.1 -8 16500 0.846 0.840 0,837 0.836 0.838 0*84;1 0.845 0*850 0:855 1,600 0.873 00*866 0860 0.858 0.857 0.858 0.860 0.863 0:.867 1.700 0*898 0*888 0.882 0.877 0&875 08a.874 084 0876 0819 1 800 0.918 0,908 0.900 0.895 0.891 0l8888 0.8'88 0.888 0-889 1,900 093:5 0*925 0 o917 0*910 0.906 08902 0O900 0*899 0.8'89 2.000 0.950 0.940 0.932 09:25 0O919 0.915 0.912 09'10 0.909 2 100 0&961 0.952 0.944 0.937 0.931 092;6 0 - 923 0920 09"19 2.200 0.970 09 "62 0e954 0.9'47 0s.941 0s9'36 0*932 0.929- 0.9O 2,s00 0.977 0.970 0*963 0.957 0*951 06.946 0*941 0.938 0.935 2.,-o00 0098`3 0.977 0*971 0.964 00959 01954 0*948 0.*946 0 943 2 500 0.987 0.9-82 0.977 0,971 0.966 0O961 0.956 0.-953 0. 949 2,600 0*991 0.986 0.981 0.976 0,972 0.967 0.963 0.959 0.956 2.700 0.993 0.990 0*985 0.980 0.977 00972 0.968 0.964 0*961 2 800 0.995 0.992 0O989 0.985 0&981 0.97'7 0a973 0969 0 a966 2.900 0.996 0,994 0*991 0*988 0&984 0.981 0i977 0a974 0.971 3.000 01997 01995 0*9'93 0.990 04987 0 98:4 0.981 0,978 0.975 3.100 00998 0.9906.0995 0.992 0.989 0,987 0#984 0.981 09'78 3,200 0.999 0.997 0.996 0.994 0O992 0198:9 0*987 0.984 0.981 3.300 0e999 0499:8 0.997 0.995 0.993 0991 9 09986 0.984 3, 4 00 0&999 0.999 0.998 0,9'96 0s99:5 0a993 0.991 0O988 0.986 3. 500 1O000 0O999 0.998 0,997 0.99;6 0e994 0&992 0O990 00988 3*600 1.000 0.999 0a0999 0,998 0.997 0O995 0,994 099:92 0&990 3,700 1*000 0.999 0.999 0.998 0.997 0.996 0.995 099'3 0.9"91 3.800 18000 0*999 0.999 0.999 0.998 09-97 6.996 0O994 0.993 3,900 1000 1.000 0.999 0.999 0,998 0.997 09996 0.995 099'4 4. 000 1.000 1l000 00999 0.999 0.999 00998 0.997 0*996 01995 4o100 1.000 11000 1.000 0 09-99 0.999 0.998 0*998 0.997 04995 4, 200 1000 1.000 1l000 0,999 0.999 09-99 0*998 0.997 0e996 4,300 1.000 1.000 1.000 1*000 0.999 0*999 0&998 0.998 09"97 4,L00 1. 1000 1.000 1.000 1.000 0.999 0.999 0.999 0.998 0.997 4.500 1e000 1.000 1.000 1.000 le000 00999 0.999 0.998 08998 4.600 1.000 1,000 1.000 l1000 1.000 00999 00999 0.999 0998 4.700 1.000 1.000 1o000 1.000 1.000 01999 0.999 0,999 0*999 4.80100 0 11000 1.000 1.000 1.0000 1000 1.000 0.999 0.999 0.999 490,0 1.000 1.000 1.000 1.000 1.O00 1.000 0.999 0.999 0.999 5.000 1.000 1.000 1.000 1.000 1.O0 1.000 1.000 1 0000 0.999 27 UNCLASSIFIED

7 71.0 6 cl.9999 5 ~.9990.9975 C.950 < 03 "~~~~~~~~~C z~~ m - ~~~~~~~~~~~~~~~~est = Cnst', I I'796 t\ V):iol~~~~1' 9 8 7 6 1 3 2 2 3i s \ 4 10\ 3 TS 9 o~ Free Stream Density 00l,, N ~O Fiur 4 ELD CTRS Cr3 C~~~~3 Ln CDC N D MLO o r Upstream Downstream Density Cnt Free Stream Density Figure 4. EQUI-DENSITY CONTOURS

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F. meter. Only the X = 15m and X = 6.3 meter returns are large enough to be of interest. For these wavelengths, the major part of the region in which the density gradients are large all contribute in phase except for the 1800 phasing effect due to the change in sign of N-n ahead and behind the sphere. For these wavelengths the factors which express the asymmetry of the problem about 0 = 900 do not play an important role so that the data for e = 520 can be used for ~ = 1280 and the C's for ~ = 750 are essentially those for ~ = 1050. Table III r in cm2 Region X = 15m X =6.3mX =lm 52~ 1.6.09. z > 0 (behind) 75~ 9.3.5 18 x 10-6 90~ 12. 1.3 53 x 10-6 52 18..95 ---- z < 0 (ahead) 750 13..58 31 x 10-6 900 12..49 180 x 10-6 520 32. 1.5 ---- -oo < z < oo 750 19. 1.7 97 x 10-6 (entire cloud) 900.04.21 74 x 10-6 -ODo e z co( 520 20. 1.0 ---- (entire cloud with 750 23. 1.1 49 x 10-6 power contributions 900 24. 1.8 230 x 10-6 from z > 0 and z 0 added (i.e., average result for random relative phase) 29 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F tO co I 0 co 4a)t4-. ~30 S 10 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F It is of interest to compare these results with a comparable cross section estimate obtained by Davis [13] in a much cruder but far simpler fashion. Davis neglected the density build-up ahead of the sphere and assumed the electrons were completely swept out of a cylindrical column one sphere diameter wide. For this assumption his computations led to a length estimate of 10 sphere diameters behind the sphere. The radar return from such a cavity is that of a column of electrons embedded in a vacuum and of electron number density given by the unperturbed number density. This number of electrons per unit volume was then referred to an equivalent line density and the problem replaced by that of coherent scattering by a line source. It is not clear in [13] but Davis apparently used a wavelength of 15m. To scale Davis' numerical result of C- =.1 cm2 for a.25 m radius sphere in a medium of n = 1012/m2 to the present 1 m radius sphere problem his equivalent line density (2 x 109/cm) is scaled by (100/25)2 = 16 and the fact that -r is proportional to line density squared leads to a scale factor of 256 or C = 25.6 cm2. An instructive insight into the behavior of the perturbation at various distances ahead or behind the sphere is furnished by a plot of the contribution to the volume integral of the various axial stations; i. e., a plot of S(z) = S p dp ( - 1 ) Jo [ 2kp sinO] 31 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F vs z. A typical graph (for the X 6.3m, G = 900 case) is furnished in Figure 6. The somewhat odd transition in S(z) as z crosses the origin reflects the fact that N within the sphere is zero. To complete the volume integration this function 2 is to be multiplied by exp [2ik ( 2R - z cosO)] and integrated. It is clear that as k increases the rate of oscillation of the exponential will increase and the net contributions from both regions z > 0 and z ( 0, will rapidly decrease. 32 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Sl(z) 0.9 0.8 0. 7 Sl(z) vs. z where 0.6 Sl(z) pdp N -1) (2kpsin9) 0.5 0.4 0.3 0.2 0.1 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -0.1-0.2 l, Figure 6. -0. 33 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Acknowledgements The authors would like to thank Professor K. M. Siegel for suggesting the physical bases which enabled the postulate of a neutral gas to be used for determining the ion and electron distributions. The authors would also like to acknowledge helpful discussions with Professor G.E. Uhlenbeck and Professor P.C. Clemmow. Finally, the authors wish to acknowledge the help of Larry B. Evans in coding the formulas for computation and to express their appreciation to The University of Michigan Computing Center, directed by Professor R. C. F. Bartels, for the computing time necessary to complete this work. 34 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Appendix The Reduction of the "Oseen-Like" Linearized Equations of Motion for an Ionized Medium by C. L. Dolph The problem of the flow in which a sphere is immersed has always been a difficult one in fluid mechanics. Thus, for the case of a sphere with cavitation in an ideal fluid, [14] the flow has been solved only approximately by use of sources leading to a potential 0 = U(r cos 0 + a2/ r2) and F 2 2 a2 2 in a stream function // =U r sin ) - 2 sin (/2), in a spherical coordinate system a, 0, 0 with origin at the sphere center. In addition, the fluid is removed bodily from the cavity. Similarly for viscous flows, there are only the classical solutions of Stokes and Oseen. It is the philosophy of the latter which is of interest here for it is a linearization about the free stream velocity. For an ionized medium the problem of the sphere has been solved only for point sources (Kraus and Watson [2] and P. Greifinger E3] )on the basis of linearized theory starting from the transport equation, and for spherical probes (Bernstein and Rabinowitz L9] ) by use of orbital analysis and assumed mono-energetic distributions. The starting point for our remarks is the interesting report by R. Pappert [4] who examines the Kraus and Watson solution and deduces 35 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F it from the Bohm-Pines Theory of the so-called random phase approximation in which the interaction terms in the Fourier expansion of the Coloumb forces are neglected. (Bohm and Pines [53 ). In addition to this he deduces the equations of Bohm and Pines from the linearized equations of motion and continuity for ions and electrons and poisson's equation. As in Oseen's treatment, these equations are linearized about the particle in a reference frame moving with it at a constant velocity. In addition, an isothermal equation of state is assumed. In view of the similarity of these equations to those used by Oseen in his treatment of the flow past a sphere as well as the difficulties associated with such flows in general, it seems worthwhile to attempt a generalization of Oseen's method for this problem. In this Appendix we successfully complete the first step in this program - namely, the reduction of the system of nine scalar equations for the ion and electron velocities, and their respective densities and the static potential by the introduction of two scalar potentials. (Unlike Oseen's case, two scalar potentials must be used since the ions and electrons satisfy separate equations of motion and of continuity because internal static fields are allowed. ) Our starting point is the system of Pappert generalized to include an interaction term proportional to the relative velocity, a scalar viscosity and a gravitational potential, which may be written as follows if the xl-direction is taken as the direction of rectilinear motion of the sphere. 36 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F 3vk+ kT+ ap+ e a 2 G av0 vkt+ e + + + _ (v -Vi)+ a - ~ ax1 m+Po axk m+ xk c e mp axaxao axk (1) avk- kT ap e a e a vk G V0 + - (Ve-Vi) + aG ax1 m Poaxk m axk c e m ax~ax) axk (2) F? avk + 1_ ap -+ v =0 (3) j ~k ~ 37 aVk- 1 apaax i: aXk 1 ax a o ax1 j=i k Po V =-47re (p+-p) (5) In these equations the electron flow velocity = -vc i + v + the ion flow velocity =-vo i+ v the ambient electron density = the ambient ion density =po the electron density = po + P the ion density = po + P 37 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F the electron mass = m the ion mass = m the electrostatic potential = the unit vector in the xl direction = i. This system of equations is readily deduced from those given by Spitzer (Physics of Fully Ionized Gases) when an isothermal equation of state is assumed. (Note that equations (1) and (2) must be added to each other to give the full equation of motion. ) To reduce this system let us make the assumption that 2 + + a "v vk = (6) axlaxk This implies that the flow is irrotational. Inserting this into equation (3) with the plus superscript, and into equation (4) with the minus superscript, we find 3 + +j a _ + ap a v2 ++ =. (k= xXk)2 p0 ax1 ax LP Thus these equations will be satisfied if we set + P, 2 + P v V r. (7) 38 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Inserting these values into equation (5) we find that 2 VePo [2 + 2 - vO Thus this equation will be satisfied if we set 0=47r - v + arbitrary harmonic function. (8) We ignore this arbitrariness in the sequel, although its inclusion would cause no difficulties. Inserting (6), (7) and (8) into equations (1) and (2) yields: a3~- - _kT a 2 + 4re a [] v0 2 + + v x x axk m v0 axk m- axk 24- 2 4 + 2 C aXlaxk aXlaXk m+pO ax ax ax. aXk axk ax1axk kxx mPo ok k =_a a kT+- 2 + 4re ax a2 + + 2 3 + axk l C L ai t+ ai7 + mcpO a 33v 39 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F Thus these equations will be satisfied if?+ satisfies 32+ 4+re P 2 ______ kT- 2 + 4 + =0 o 2 + + L x m m- + th ese equations reduce tos 2 32 kkT+22 2 4 p__ e1 FVe2 a+ 4e PO L7 7v v 2 + + am x ax m me- r 2re2 24 kT 2 2 a -6i4e + — o V2 - -mm m- ax M M we see that the general system as well as the reduced system may be uncoupled easily. For example, for the reduced system, we have 2 2 kT 27a2 2,r e po3+ 2 2 41re po _T 2 2 2 IIPo 2 2 a 0 2 + 0 2 + M. a~x M M. a3xi m 4 7r e24e 2 p e2P 16 7e4 2 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F This may be simplified to become kT 2 2 a2I kT 2 2 22 (- 44 ae ppok (+ 2 m + 0 2 m m ax1i A similar process may be applied to -. We hope to be able to accomplish the next step in the Oseen formulation which consists in finding simple singular solutions of these equations. This matter is now under investigation.+ The reduced equations are identical with those treated by Pappert. Even if viscosity, and interaction are allowed the full system admits a partial separation in any coordinate system in which there is no coupling with the xl-direction if it is assumed that the functional form of the dependence on the other coordinates is the same for both the ions and the electrons. This assumption still permits different behavior for the ions and electrons since the separation constants can be different for them. + It now appears that this process can be carried through (see a forthcoming paper by C. L. Dolph, R. F. Goodrich). 41 UNCLASSIFIED

UNCLASSIFIED THE UNIVERSITY OF MICHIGAN 2778-2-F UNIVERSITY OF MICHIGAN 3 9015 02082 8144 References 1. H. Brysk, Journ. Geophys. Res., 63, pp. 693-716, 1958. 2. L. Kraus and K. M. Watson, Physics of Fluids, 1, pg. 480, 1958. 3. P. Greifinger, Rand Report R339, Paper No. 19, June 1959. 4. R. Pappert, Convair Report ZPH-026, December 1958. 5. D. Pines and R. Bohm, Phys. Rev., 85, pg. 338, 1952. 6. C.S. Wang Chang, University of Michigan, Engineering Research Institute Report CM-654, December 1950. 7. L. Spitzer, Jr., "Physics of Fully Ionized Gases", Interscience, New York, 1956. 8. C. L. Dolph and R.K. Osborn, University of Michigan Radiation Laboratory Memo No. 2764-547-M, 1959. 9. I. B. Bernstein and I. N. Rabinowitz, Physics of Fluids, 2, pg. 112, 1959. 10. Staedler and Zurich, NACA TN 2423, July 1951. 11. F.C. Hurlbut, Rand Report R339, Paper 21, June 1959. 12. A. Erdelyi, et al., Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953. Eq. 27, pg. 92. 13. A. Davis, Proceedings of the Conference on Interaction of Satellites with the Ionosphere, Naval Research Laboratory, pp. 33, 34, March 28, 1958. 14. W. Bauer, Ann. der Phys. 82, pg. 1014, 1927. 15. M. L. Barasch, H. Brysk, J. W. Crispin, Jr., B. A. Harrison, T. B. A. Senior, K. M. Siegel, H. Weil and V. H. Weston, "Studies in Radar Cross Sections XXIV - Radar Cross Section of a Ballistic Missile - IV: The Problem of Defense", (2778-1-F, April 1959), AF30(602)-1853. SECRET. 42 UNCLASSIFIED