THE UNIVERSITY OF MICHIGAN 7455-3-Q TECHNICAL REPORT ECOM-01378 -3 APRIL 1966 Missile Plume Radiation Characteristics Third Quarterly Report 1 January to 31 March 1966 Report No. 3 Contract No. DA 28-043 AMC-01378(E) Prepared by M. L. Barasch, J. J. LaRue and C-M Chu BIe University of Michigan Department of Electrical Engineering Radiation Laboratory Ann Arbor For U. S. Army Electronics Research and Development Activity White Sands Missile Range New Mexico

THE UNIVERSITY OF MICHIGAN 7455-3-Q Qualified requestors may obtain copies of this report from the Defense Documentation Center, Cameron Station, Alexandria, Virginia 22314. DDC RELEASE TO OTS NOT AUTHORIZED

THE UNIVERSITY OF MICHIGAN 7455-3-Q TABLE OF CONTENTS Page ABSTRACT iv APPROACH EMPLOYED AND SUMMARY OF PROGRESS 1 1.1 Statement of Problem 1 1. 2 Method of Attack 1 1. 3 Status of Subsidiary Task Areas 3 II SOLUTION OF THE RADIATIVE TRANSFER EQUATION 4 2.1 Formal Solution Neglecting Scattering 4 2. 2 Radiation from Isotropic, Homogeneous Source 5 2. 3 The Finite Cylinder as a Source 6 III SOURCE FUNCTION AND ABSORPTION COEFFICIENT IN THE CONTINUUM 9 3. 1 Bremsstrahlung Power from Electron-Ion Encounters 9 iii

THE UNIVERSITY OF MICHIGAN 7455-3-Q ABSTRACT This is the third quarterly report on Contract DA 28-043 AMC-01378(E), the objective of which is the prediction of intensity of millimeter wave radiation from the exhausts of certain rockets during boost phase. A formal solution for the radiative transfer equation is presented corresponding to a source which includes the features believed essential to the situations of interest here. Some expressions relating to the generation and absorption of power in the continuum are given here. A discussion of the plan of attack and of the progress in the subsidiary areas of research into which the problem has been divided is given.

THE UNIVERSITY OF MICHIGAN 7455-3-Q APPROACH EMPLOYED AND SUMMARY OF PROGRESS 1.1 Statement of Problem This scope of work covers the study of specific missile fuel configurations to determine their theoretical electromagnetic characteristics in the 30 - 300 Gcs frequency range. The radiation characteristics of interest occur during the boost phase for each configuration. The missile fuel configurations to be studied are; a) Pershing booster, b) Nike Hercules booster, c) Nike Zeus booster, d) Nike X booster, e) Honest John, and f) Sergeant. The results required from this study include, but are not limited to, the following: a) Spectral distribution of radiation within the 30 - 300 Gcs region. b) Type of spectra involved and the mechanism causing same. c) Estimate of power levels involved. d) Changes in the characteristics during boost phase due to altitude effects, acceleration effects, velocity effects, etc. e) Comparison of theoretical results with known experimental or field measurements available to the Contractor during the contract period. 1.2 Method of Attack To solve a problem of this type it is necessary, first of all, to obtain the form of the pertinent solution to the radiative transfer equation. This solution will be a function of frequency through the source function and absorption coefficient of the exhaust, and will be a function of aspect in a manner depending on the exhaust geometry. Only when this solution is at hand can further progress be made.

THE UNIVERSITY OF MICHIGAN 7455-3-Q To serve as inputs to this solution, the source function and absorption coefficient within the exhaust are required. In general, these will contain contributions from both continuous-spectra processes and line processes. The lines may be regarded as merely superposed on the continuum at the correct frequency, since they are not coherent with it. The continuous and discrete spectrum mechanisms have therefore been treated separately. The question of altitude dependence of the radiation should be answered by combination of the formal solution to the radiative transfer equation and the expressions corresponding to radiation and absorption mechanisms. That is one, possible source of altitude variation is aspect dependence of the formal solution. Another is altitude sensitivity of the solution through the source function and absorption coefficient, which do involve atomic and molecular densities, electron densities, and temperatures within the exhaust. These are, in principle, functions of ambient atmospheric conditions against which the exhaust expands. It is true, however, that the altitude region of interest is one over which the ambient conditions do not vary sharply. It has not been possible to consider inhomogeneity of the exhaust, nor to obtain or generate information on the variation of exhaust composition with altitude due to changes in chemical reaction rates. Neither has the exact shape of the exhaust been determined. It is our decision that the interests of the sponsor have been best served by attempting to formulate a radiation model which is not based on such dubious and conjectural information. If the radiation model thus furnished agrees sufficiently well with experimental evidence available to the sponsor, time will not have been wasted in constructing elaborate theory on insecure foundations, but will have been spent on trying to do a respectable engineering type of approximation which includes those features of the problem expected to be most significant. If, on the other hand, satisfactory agreement with theory is not obtained, the interests of the sponsor would be best served by supporting a small pilot study to

THE UNIVERSITY OF MICHIGAN 7455-3-Q decide which of the features omitted here accounts for the discrepancies, and whether theoretical apparatus of sufficient power exists to incorporate them into a radiation model. A related question is whether experimental determination of some quantities would be necessary and possible, in order to permit numerical application of an enhanced theory. Answering these questions would aid the sponsor in determining the optimum course for obtaining a closely-fitted theoretical model, and understanding which features of each vehicle cause it to radiate in its characteristic manner. 1.3 Status of Subsidiary Task Areas a) Solution of the Radiative Transfer Equation This has been completed, subject to certain simplifying assumptions believed to be reasonable for us. The solution is ready for use, and is presented in Chapter II of this report. b) Continuum Radiation The expressions for the source function arising from Bremsstrahlung of electrons on ions, and for the free-free absorption coefficient corresponding to the process inverse to this Bremsstrahlung have been obtained from the literature, and the criteria for their validity here checked. They appear in Chapter III of this report. c) Line Radiation For CO, there is no difficulty in writing down source function and absorption coefficient pertaining to the rotational spectra. This has been given in a previous report. The H20 molecule, which seems to present theoretical difficulties, is being given renewed attention, but no results for it have been obtained yet. At this date, we have reason to question some of the statements in the literature regarding the H20 line at 183 Gcs. Thus, some of the fundamental theoretical work on it appearing in previous papers must be re-investigated before using it as a basis for radiation model predictions.

THE UNIVERSITY OF MICHIGAN 7455-3-Q SOLUTION OF THE RADIATIVE TRANSFER EQUATION 2.1 Formal Solution Neglecting Scattering In the absence of scattering, which is not expected to introduce significant modifications in the specific intensity outside the source for situations of interest to us, the transport equation satisfied by the specific intensity, i, takes the form i2+a Vi(ri(r, )=S(r,2). (2.1) Here a is the effective absorption coefficient and S the source or production function, i.e. the spectral density of power per unit volume produced within the source. All quantities are understood to be evaluated at the same frequency, and the frequency dependence of i is to be obtained. The frequency dependence of ac and S will depend on the mechanisms operating within the source to generate and absorb radiation. Now let us consider a bounded volume of radiating and absorbing systems as shown in Fig. 2-1. FIGURE 2-1

THE UNIVERSITY OF MICHIGAN 7455-3-Q Since we expect eventually to specialize to a cylinder, all figures will relate to a cylindrical system. The specific intensity in the direction ~2 at any exterior point P ( ) will be obtained by integrating Eq. (2.1) at fixed 2. In Fig. 2-1, 0 denotes an arbitrary origin. As is well known, for any direction Q in which the intensity is appreciable, the extension along - ~2 from P must intersect the source. The points of intersection are denoted by A and B; their separation by d. If one measures distance along the direction 2 from point A and designates it x.2' then (2.1) becomes d- i(x QJ )+a(x Q )i(x Q )=S(x. (2. 2) This is the first-order equation with the obvious boundary condition that no radiation be incident on the cylinder from outside, or i (O,Q)=0. (2.3) The standard procedure for solution of such a problem can be shown to yield i(x~2, Q)=e J0S(xtQ i) e dxr (2.4) Since S and a vanish for x~ 2> AB, (2.4) may be reduced, for exterior points, to 2 dx QY (X r2) Q A i(r, 2)=f S(x (2)e d' (2.5) 2.Since 2 Radiation from Isotropic, Homogeneous Source, to For the isotropic and homogeneous source, which we shall deal with, (2. 5) 2. 2 Radiation from Isotropic, Homogeneous, Souirce

THE UNIVERSITY OF MICHIGAN 7455-3-Q takes the form (d -a(d x' -cd i(,) = S e dx' = S/a(1-e ) (2.6) This solution exhibits correct limiting forms, since for the'thin source', we have ad <<1, and i(r,) =Sd (1 — ad2), (2.7) while for the thick source, ad >> 1 and i ( r, ~) S/a, as for a black body. (2.8) For more exact computations geometry enters the problem. In (2.6) the quantity dQ is to be evaluated explicitly. It should be noted that up to this point, no specialization to a cylindrical source has actually been made, and all work is completely general. 2.3 The Finite Cylinder as a Source Let us consider a cylinder of length L and radius a, as shown in Figs. 2-2. e, 0 D _ FIG. 2-2a: Side View FIG. 2-2b: Top View

THE UNIVERSITY OF MICHIGAN 7455-3-Q If the purpose of the investigation is to obtain i at point P a distance D from the axis of the cylinder and at axial distance h from one end, and if we define Q2 by the angles 0 and 0 in a set of local spherical coordinates centered at P, it is evident that i = o 0 => 0 sin -D or 0 < 0 or 0 > 7-. (2. 9)'Ii o D 2' (I 9 Exact calculation of 01 and 02 depends on the'end configuration' of the cylinder. In practical cases, these configurations are not well-defined. However since we are interested in far-field applications, it is satisfactory to write -1 L-h -1 h 01 tan D Lh 2 = tan -D (2.10) For any ~Q defined by 0 and 0 such that L is not trivally zero, geometrical considerations yield the result 2_ 2 2 d=2 2 a -D sin0 csc0 (2.11) Thus we have -2 acsc a -D sin 0) (2.12) ip( =1 (2.12) and may write an expression for the spectral density of power intercepted per unit area by a receiver oriented normal to the axial distance D (Fig. 2-2b), Power1 oS a2 _2 sin 2 Power S sin2 -2acsce a -D sin 0 Asin OdO cosOd0 ( 1 -e Area a 2 o0 (2.1p)

THE UNIVERSITY OF MICHIGAN 7455-3-Q For an angular receiver oriented toward the source, then, one may write Power -. S sin-2 acscO a -D sin2 0 — foi osinodo df o( 1 -e Effective Area ac 02 o -0 (2.14) Numerical evaluation of these integrals should not be difficult.

THE UNIVERSITY OF MICHIGAN 7455-3-Q III SOURCE FUNCTION AND ABSORPTION COEFFICIENT IN THE CONTINUUM 3.1 Bremsstrahlung Power from Electron-Ion Encounters a) Contribution of the Faster Electrons to Source Function Since different expressions are required for the faster and slower electrons, the resulting source functions being additive, we list the expressions separately. For the faster electrons, Equations (5. 7) and (5. 8) of a previous Laboratory report + are to be employed. The result, after minor corrections of notation, is that the power per unit volume in frequency interval dw (or spectral density of source function) generated by the mechanism is given by 16 6 4ir m 3/2 Pd = n n1- e ( ) dwI2 (3.1) ei 3 45 27rkT 2 m c where I2 = PkT 1/ in 2 2 e -Ei(- 2 )(3.2) In (3.1), the symbols e and m are the charge and mass of the electron. T is temperature, c and k are the physical constants usually denoted by these letters. In (3. 2) /l is mc, the rest energy of the electrons, e is the Naperi an base, 6 is 4kT/K, where K is the photon energy hv or-1w.Ei(-x) is the exponential integral, a tabulated function. c = y kT/p K, where y =-ic/X, in which X is the Debye length 2 1/2 (kT/47rn e ). n and n. are the volume density of electrons and ions respece e 1 tively. Single ionization, rather than multiple, has been assumed. At the temperatures we expect, this assumption is valid. +Barasch, M.L. (August 1960), "Studies in Radar Cross Sections XLII: Microwave Bremsstrahlung From a Cool Plasma," The University of Michigan Radiation Laboratory Report 2764-3-T, AD 245070. UNCLASSIFIED.

THE UNIVERSITY OF MICHIGAN 7455-3-Q b) Contribution of the Slower Electrons to Source Function This will be obtained from Eq. (3. 28) of Report 2764-3-T+. This reads 16 nine dw m 3/2 Pdw 4 ( I (3.3) Pdw3 45 4(27rkT 3 m c in which 2kTL' -p2/2pkT 13 3 PdPe( e2 K i- K )K 2 [2E xK) + 2(), (3.4) In I3 the functions K0 and K1 are'modified Hankel functions' e denotes the DeBroglie wavelength, e =-i/mv. Since P = mvoc, e is a function of the variable of integration rather than constant. X is still the Debye distance, and all symbols previously defined retain their original significance. The symbol F' used in the argument of the K function is an abbreviation for that long radical which is written out in detail in the first place it appears inside the brackets. I3 will require numerical evaluation, since no closed analytical form has been obtained for it. c) Free-free Absorption by Faster Electrons in the Fields of Ions It follows from the Report 2764-3-T (op. cit.) that this contribution is given formally by OD P dP -P2 /s2 FF( = n n 16 2 2) c 3 2 2 2 F(w P1 P ) ~~FF ei ~ 3 o 3' 112 0 s (3.5)

THE UNIVERSITY OF MICHIGAN 7455-3-Q The new symbols here are r, the classical electron radius = e /, and s = (2kT)1/ while as usual a denotes the fine-structure constant. Now for the faster electrons the Born approximation is to be employed, and eventually one is able to write for this contribution, 16 2 2 c 3 4p a(FF() = n ni( arr )(-) -I2 (3. 6) where the integral I2 has been evaluated previously (Eq. 3. 2). d) Free-free Absorption by Slower Electrons in the Fields of Ions To obtain this contribution to the continuum absorption, one starts with (3. 5). However, in this case, numerical evaluation is required with the special valid form F E )2+(MWE)2K )K 1 F(C)2+2( 2]K2' 2 ( )K0 1 2 LAPC) 1(i )-K( (3.7) Numerical evaluation is required. 11

UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA- R&D (Security clesification of title, body of abstract andc indexing annotation must be entered when the overall report is classified) OSRIj4NATIN G A TIU T Coprate a hor2.a REPORT SECURITY C LASSIFICATION the umverslW oI lyvilcgan 1aaiation Laboratory UNCLASSIFIED epartment of Electrical Engineering 2b GROUP 3 REPORT TITLE MISSILE PLUME RADIATION CHARACTERISTICS 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) QUARTERLY REPORT 1 January through 31 March 1966 5. AUTHOR(S) (Last name, first name, inftill) Barasch, Murray L., Chu, Chiao-Min and LaRue, John J. 6. RIPO RT QATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFG 30 April 1966 8a. CONTRACT OR GRANT NO. E41. ORIGINATOR'S REPORT NUMBER(S) DA 28-043 AMC-01378(E) 7455-3-Q b. pROJECT NO: C. Sb. OTHER RIPORT NQ(O') (Any other nwmbera that may be assigned this report) d. ECOM 01378-3 10. A VA IL ABILITY/.!.MITATI'ON' NOTIES Qualified requestors may obtain copies of this report from DDC. DDC release to OTS not authorized. 1. SUPP- EMENTARY NOTES II. SPONSORING MI..ITARY ACTIVITY U.S.Army Electronics Research and Development Activity, White Sands Missile Range, New Mex. 13. ABSTRACT This is the third quarterly report on Contract DA 28-043 AMC-01378(E), the objective of which is the prediction of intensity of millimeter wave radiation from the exhausts of certain rockets during boost phase. A formal solution for the radiative transfer equation is presented corresponding to a source which includes the features believed essential to the situations of interest here. Some expressions relating to the generation and absorption of power in the continuum are given here. A discussion of the plan of attack and of the progress in the subsidiary areas of research into which the problem has been divided is given. DfD FORMN 1473 UNCLASSIFIED Security Classification

UNCLASSIFIED Security Classification 14. IGLINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT PASSIVE RADIATION EHE ROCKET PLUMES BREMSSTRAHLUNG PROPELLANTS INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements of the contractor, subcontractor, grantee, Department of De- such as: fense activity or other organization (corporate author) issuing (1) "Qualified requesters may obtain copies of this the report. report from DDC." 2a. REPORT SECURITY CLASSIFICATION: Enter the over2a..REPORT SECU.1TY CLASSIFICATION: Enter the over- (2) "Foreign announcement and dissemination of this all security classification of the report. Indicate whether "Restricted Data" is included. Marking is toeport by DDC is not authorized." ance with appropriate security regulations. (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC 2b. GROUP: Automatic downgrading is specified in DoD Di- users shall request through rective 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional markings -have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of this ized. report directly from PDC. Other qualified users 3. REPORT TITLE: Enter the complete report title in all shall request through capital letters. Titles in all cases should be unclassified.,. If a meaningful title cannot be selected without classific.tion, show title classification in all capitals in parenthesis (5) "All distribution of this report is contlolled. Qualimmediately following the title. ified DDC users shall request through 4. DESCRIPTIVE NOTES: If appropriate, enter the type of., report, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of Technical Give the inclusive dates when a specific reporting period is Services, Department of Commerce, for sale to the public, indicovered. cate this fact and enter the price, if known. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on 1L SUPPLEMENTARY NOTES: Use for additional explanaor in the report. Entet last name, first name, middle initial. tory notes If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (pay6. REPORT DATE;: Enter the date of the report as day, ing for) the research and development. Include address. month, year; or month, year. If more than one date appears on the report, use date of publication. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though 7a. TOTAL NUMBER OF PAGES: The total page count it may also appear elsewhere inthe body of the technical reshould follow normal pagination procedures, i.e., enter the port. If additional space is required, a continuation sheet shall number of pages containing information. be attached. 7b. NUMBER OF REFERENCES: Enter the total number of | It is highly desirable that the abstract of classified reports references cited in the report. be unclassified. Each paragraph of the abstract shall end with 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the inthe applicable number of the contract or grant under which formation in the paragraph, represented as (TS), (S), (C). or (U). the report was written. There is no limitation on the length of the abstract. How8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from 150 to 225 words. military department identification, such as project number, 14. KEY WORDS: Key words are technically meaningful terms subproject number, system numbers, task number, etc. or short phrases that characterize a report and may be used as 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- index entries for cataloging the report. Key words must be cial report number by which the document will be identified selected so that no security classification is required. Identiand controlled by the originating activity. This number must fiers, such as equipment model designation, trade name, military be unique to this report. project code name, geographic location, may be used as key 9b. OTHER REPORT NU NjB ER(S): If the report has been words but will be followed by an indication of technical conassigned any other report.inmbers (either by the originator text. The assignment of links, rules, and weights is optional. or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those UNCLASSIFIED Security Classification

UNIVERSITY OF MICHIGAN 3 901111111111111115 02227 1335 3 9015 02227 1335