4% IEEE TRIXSACTIONS os AKTEXXAS AXD PROPIGATIOK, 1-0~. -ip-li, NO. 4, JULY 1969

REFERENCES [ I ] S. A. Schelkunoff, “Theory of ant.enna~ of arbit,rary size and

shape,” Proc. IRE, vol. 29, pp. 193-521, September 1911. [2] -, “Principal and compIement.ary wavw i n antenna^?" Proc.

[3] -, “General theory of symmetrir bicotlical ant.ennas,” J . IRE, vol. 34, pp. 23P-32P, J a n u a ~ 1946.

[4] C. T. Tai, “On t.he t.heory of biconiral antenna*,“ J . ; I p p l . d p p l . Phys., vol. 22, pp. 1330-1332, Sovember 1951.

[5] -L “A study of the E. AI. F. method,” J . . i p p l . Phgs., vol. 20, Phys., vol. 19, pp. 1155-11tj0, Ikernber 1943.

[6] P. D. P. Smit.h, “The conical dipole of wide angle,” J . ;Ippl. pp. 117-i23, July 1949.

[i] C. H. Papas and R. W. P. King, “Input impedance of wide- Phys., vol. 19, pp. 11-23, Januarl- 1948.

angIe conical antennas fed by a coasial line,” Proc. IRE, vol. 3 i , pp. 1269-1271, November 1949.

[8] A., G.. Kandoian, “Three new antenna types and t.heir a p pheat~ons,” Proc. IRE (Waves and Electronsr, rul. 31, pp. TOW- 75W, February 19%.

[9] Haward Radio Res. Lab. St,aff, V e r y High Freqtcmcy Tech-

[lo] E. W . Hobson, Spherical am/ Ell.ipsoidul Harnzortics. Sew niqttes, vol. 1. New Tork: McGraw-Hill, 1947, ch. 4.

[I11 S. A. Schelkunoff, .Applied Mathenzutics .for Engiwxxs and York: Cambridge University Press, 1931, ch. 6, sec. 193.

Scientists, 2nd ed. Princet.on, 5. J. : 1-an Sostrand, 1965, p. 391.

(121 P. K. Karr, “Radiating propert.ies of spherical antennas as a function of the location of the driving force,“ J . Res. .VBS,

[I31 L. Robin and A. Pereira-Fmes, “L’antenne biconique, synlCtri- vol. 46, pp. 422436, May 1951.

que, d’angle quelconque, Ann. Telecomntcn., vol. 8, pp. 382- 390, December 1953.

[14] J. I>. Kraits, --l&vmas. New York: l I d ~ r a ~ - H i l l , 1930.

. ‘ I ’ Relativistic Communications Effects Associated with Moving Space Antennas

Absfrucf-Future space missions may require vehicles traveling at relativistic velocities. Consequently, relativistic effects will have to be considered in the development of future space communication systems. This paper presents a formulation of these relativistic communication effects which are not normally encountered by the designer.

The gain function, power density available at the receiver and apparent frequency of the source can be greatly d e c t e d by the motion of the space craft antenna. In addition, the apparent di- rection of the space craft can be quite dif€erent from its actual direction for both the transmitting and receiving conditions. Formu- lations are presented by which these effects are described for an arbitrary antenna in terms of the antenna characteristics in its rest frame and the actual space craft location, a form that is con- venient for the designer. In addition, numerical examples are shown to illustrate the magnitude of these relativistic effects.

I. ISTROD~-CTIOS UTURE space vehicles ma. be t,raveling at. relativistic velocities [l], and consequently the design of com-

munications systems involving the space ant,ennas on t.hese vehicles will have to consider relativistic effect,s. Probably the most commonly thought of effect is the Doppler shift in frequency. However, i n addition, the radiation pattern of the antenna, the power density avail- able at the receiver, and the apparent direction of the spa.ce

This work was sponsored in part by the U. S. -4ir Force Cam- Manuscript received December 2,1968: revised February 24,1969.

bridge Research Laboratories, Office of Aerospace Research, under Contract F 19(628)-68-C-0126.

University of Rashington, Seatt.le, Wash. 98105. He is now Rith J. F. HoImes was with the Depart.ment of Electrical Engineering.

t.he Department of Electrical and Electronics Engineering, Oregon State Universitr, Corvallis, Oreg. 97331.

University of Washington, Seat,tle, Wash. 98105. 4. Ishimaru-is with the Depart.ment, of Electrical Engineering,

craft ca.n be greatly affected by the motion of the space craft antenna.

The radiation from particular moving sources has been discussed in a number of text books [2>[4], and recently Fujioka et al. [ 5 ] discussed in det.ai1 t,he radiation charac- teristics of a moving point dipole. The purpose of this paper is to present a genera,lized formulat.ion of these relativistic effects in a form that. is appIicabIe t.o any ant.enna whose stationary charact,eristics are known. The results are given in terms of the rest frame gain function for the antenna! the actual location of t,he space craft.. and it.s velocity I a form which should be useful and convenient to a designer.

In the work that follows, \ye will utilize two inertial reference frames designated S and Sf, where it, is asumed that reference frame St is moving with velocity L’ in the f direct.ion with respect to reference frame S , and that they coincide a t t = 0. The problem we are considering is iIIustrat.ed in Figs. 1 and 2 as it, would be viewed from each of these inertial reference frames.

Our approach to finding t.he pox-er density available a t t,he receiving antenna is to develop a relativist,ic t.ransfor- mation for t.he antenna gain function. In this regard the usual definit,ion of antenna gain is inadequat.e when the antenna. is moving. So our first task is to discuss the criteria t,hat an antenna. gain function should have and t,hen t.0 define a relativistic gain function t,hat, satisfies t.hese crit.eria. (Section 11). We will t,hen develop the relat>ivist.ic transformations for t.he gain function (Section 111) and discuss it.s application to moving antenna problems (Section IT).

,$ ) ; OF SOURCE i

/ i i

i 1 ,' -vz

Y '

RECEIVER

Fig. 1. Problem viewed from reference frame S.

X t _ . - v t' X '

-SOURCE

,+-

c vL

Fig. 2. Problem viewed from reference frame S'.

11. RELATIVISTIC ASTESSA GMS F~-SCTIOS When an antenna is moving, the fields that are observed

at a fixed point in reference frame S will vary n-it,h time in a manner t,hat is not, time harmonic even t.hough the source is t.ime harmonic in its rest frame. This is due to the fact t,hat, the energy which makes up the field:: is being released at different points in space as time progreases. I t is t,here- fore convenient. t.0 define a relativist>ic antenna gain function which nleet,s t.he following criteria:

1 ) independent of distance 2) independent. of time 3) experimentally measureable [ at least in theoryj 4) reduces t.0 the usual definitions in the rest frame of

5 ) readily convertible to the Poynting vector at a fixed the ant,enna

point in reference frame S.

Criteria 1 j and 2) preclude the possibilitv of defining the relativistic ant,enna gain funct.ion around a single fixed point in reference frame S . Therefore, n-e will utilize a somewhat artificial but strnightforward approach to t,he problem. If in lieu of a single fixed point in reference frame S we utilize a. set of fixed points, a relritivist.ic antenna gain function which sat,isfies our criteria can be defined.

Let, us first consider an esperimental scheme for measur- ing t.he power density received at some fixed set of a.ngles 4 and 0 with respect to the moving source. Xt, first glance it, would appear that our detector would have to be moving along with the source and therefore would not measure the received poxer density that would be observed in reference frame S. If n.e utilized a single detector, this would indeed be true. Hou-ever, we will specify that the nleasurement,s be made using 3 set of det,ect,ors that are stationary in reference frame S , and we u-ill monit,or the detectors in a. manner that, yields the received power

Fig. 3. Received pom-er density measurement.

density a.t a fixed set of angles.' The measurement scheme is illustrated in Fig. 3. The X represent the array of detec- tors which are strung out in a line (defined by 4, R", and e ) along the direction of motmion of t.he source and separat,ed by a distance d. The 0 represent the retarded posit,ions of t,he source, corresponding t,o each associated det,ector. Kow as the source moves by, the power density being re- ceived by each det,ector is recorded at such a time that it is receiving power radiated by the moving source in a direc- tion defined by 6 and 4. This time, of course, depends on the separation of the detectors, the dist,ance R" that the radiated energy has t,o travel, and the ~ T O U ~ ve1ocit.y. It is given by

nd R* t = - + -

where use has been made of the fact. that. in free space t,he group velocit,y is the speed of light. The paramet,er .n is an int,eger which ident.ifies each det,ect*or. If we now let d get very small, we obtain a continuum of n1easurement.s which gives us the power density received in n direction defined by 0 a.nd 4 and at a. dista,nce R*.

During t,he above discussion w e have been careful t.0 distinguish between the power radiated and the pon-er received because, when the source i s moving v-it.h respect to the receiver, these two quant,ities are not the same. This is beca,use the energy radiat,ed per unit solid angle in a direction defined by 0 and 4 by a moving source during a differehal time interval will not. necessarily be received during a like int.erval of time [Z, p. 3021.

We a.re now in a position tjo define t.he re1at.ivistic antenna gain function. It will first, be described in words a.nd then by a mathema.t.ica1 formulation:

time average power density received* G, =

spacial average power radiat,ed* per unit solid angle

lvhere the asterisk indicates that t,he power measuren1ent.s are done ni th a suita.bly scanned set of detect,ors as n-as previously described. In mathemntical notation this becomes

1' c

(-hrR*?I I N ' G, =

point intxoduced by Lewis and Presmal1 [7] . This technique is similar to the concept of retarded reference

486 IEEE TRINSACTIOXS ON AMTEXNAS AKD PROPAGATION, KLT 1969

where N is the Poynting vector and is evaluat.ed at the appropriate time a.nd then time averaged.

111. RELATIVISTIC TRASSFORXATIOX FOR THE GAIX FIXCTIOK

In this section Re niu derive the relativistic transfornu- tion for the antenna gain function. First we trill consider how the Poynting vector for the radiation field in free space transforms betn-een inertial reference frames. The transformation equations can be obtained by considering how certain elements of the stress-energy-momentum tensor transform between inertial reference frames. The stress-energy-momentum tensor is given by [3, p. 3851

r N1' -

C

N 3 - c

where T' is the Maxwell stress tensor, t,7' is the electro- magnetic energy density, and N1', hT2', and N 3 ' are the three components of the Poynting vector. Using the transformation properties of a doubly contrsvariant tensor [SI IT is given by

G = S" = Cy~,&S'". (3)

Now expanding (3) we find that

where

and TS' is the three-three component of the Maxn-ell stress tensor. Kow for the radiation field, E and B are orthogonal to E* and t.o ea.ch ot.her [2, p. 3001. and t,hese relationships are independent of the velocity. Con- sequent,ly,

I N' i = c C ' ( 3

N3' = CL?' cos 8'. (6)

Using this in (4) it becomes

NOK in lieu of trying to find T33' in terms of L7', rre d l consider the transformation for N3' and then use this addit,iona,l equation to eliminate T33' from (7). The

transfornlat.ion is given by

l"\'om eliminating TB' between (7) and (8) and solving for C , we obtain

where use has been made of t,he ray-angle transformations [3, p. 3641. The electromagnetic energy density C?' would be known as a funct,ion of 8' and @'. However, these angles can be transformed t o t.he "unprime" frame by again using the ray-angle t,ransformations. Equation (9) then yields l7 as a function of the angles measured in the "unprime" frame. The transformations for the Poynting vector then are given by

I N' I

Son- using (1) and (10) we conclude that

where use has been made of the fact that the total radiated power is invariant [S, p. 921 G,' is the gain function in the rest frame of the source, and

Equation (11) is the relativistic transformation for the antenna gain function.

IV. APPLICATION The transformations derived in the previous section

will now be applied to the problem of a moving antenna in free space. It. appears that a systems designer considering a moving antenna would be interested in four things: the power density available at the receiving antenna, the appa.rent frequency of the received signal, and which diredon to point the receiving a.ntenna for both the receiving and transmitting conditions. These four pa.ram- eters will of course be a function of time and can be ob- tained in a simple manner from the solution to the problem in the rest fra.me of t.he antenna. It is the purpose of this section to formulate these four parameters in terms of the present location of the source ( T and 8,) and the gain function for the ant,enna in its rest frame.

The situation is illustrated in Fig. 4. If we assume that in reference frame S t,he receiver is located at ( -ZO, -yo, - zo ) , then r and 0, are given by

HOLUES A X D ISHIU.4RD: REL4TIYISTIC EFFECTS A S D XOVING SPACE ASTESSAS 487

ACTUAL SOURCE -, I O , O . V t )

~ A N T E N N A , R E F E R E N C E FRAME s ( -X0! -Yo, - z o '

Fig. 4. Moving antenna problem.

which of course depend on the velocity and the time in reference frame S. Son- transforming the location of the receiver to reference frame S', we find that

T' = [X? + ya + r2(zo + ~ t ) q 1 / 2 ( 1 3

We n-odd now like t,o find R*, OPR, and OPT in t,ernls of r and e,. Using Fig. 4 and (13)-(16) we find that they are given by

R* = m[dl - (u2/c2) sin2 0, - (z!:cj cos e,] (17)

-COS e, + (c/c) dl - ( u p / c ~ ) sin2 e8 41 - (o~/c?) sin 28, - ( c / c ) cos e, 1

(18)

ePT = COS-' 1 . (19)

OPR = ?r - cosu1 [

[ ( u / c ) 4 1 - (vz / /c") sin? e, + cos e, ( u j c ) cos e, + dl-- (Y?:'?) sin2 e,

Now using (1) and (11) u-e find that the received power density is given by

where

e = COS^ -cos e, + ( U I C ) 4 1 - (L<Z,;'C~) sin2 e, di 1 ( V ~ / C ? ) sin2 e, - (c!:c) cos e,

~~ ~ ~~

and W 8 is the total power radiated by the source. The last parameter that, we must consider is the a . p

parent frequency of t.he received signal. It is given byv the usual formula for t,he Doppler frequency [3, p. 3M] which is

w 1 w0 r[i - ( v / c ) COS e] '

_ - - (22)

Equations (17) - (22) give t.he power demit- available at the receiving ant,enna, the point.ing angle for the re- ceiving antenna, the pointing angle for the transmit,ting antenna, and the apparent frequency of t.he s0urc.e in t,erms of the rest frame gain function for the t,ransmit,ting antenna and it.s present location. Consequently, antenna problems do not have to be solved again when the antenna is moving in order to det,ermine t.ho relativistic effects.

0 -/4 */ 2 3-/4 T

B,,RAOIANS

Fig. 5. Ratio of received pox-er densitiw for an isotropic radiator.

>- 2.0 z 0 W 3

W LT U

0 1.5

2 '1.0 N

8-0.7 \ /-----

8 5 , RADIANS

Fig. 6. Kormalized frequency.

~~

E-

W

0 n/4 J/ 2 3=/4 a s , RADIANS

Fig. 7. Receiver p o i n h g error.

Figs. $7 illustrat,e the magnitude of t.he relativistic effects caused by t.he relat,ive motion of t,he t,ransmitter a.nd receiver. The ratio of the relativistically correct received power density to the received power density calculated by assuming the tmnsnlitter t.o be stationary at each point, along its tmjector?: is given by

1N( - - I N \stationary G' (a - Oy,9i

[dl 2 ($/?) sin? e, - (c;c) cos e,]? [l - (v2//c') sin? e,]' . (23)

Equation (23) is plotted (in decibels) versus 8, in Fig. 5 for t.he case of an isotropic transmitting antenna. The normalized frequency (22) and the receiver pointing error sPR& are plotted versus 8, in Figs. 6 and 7. -4 plot for the transmitter point,ing error (0, - OPT) would be identical t.0 Fig. 7 if the horizontal scale were reversed, and so it has not been included.

In any pract.ica1 problem the t.raject.org of the antenna

488 IEEE TRASSACTIONS ON ASTEXNAS .*X’D PROPAGATION, VOL. AP-17, NO. 4, ~ L Y 1969

probably \i-ould not be a stra,ight line, and the a,ntenna [3] J. 1). Jarkson, Clossi~:n/ Elert,~od~tra,,l~cs. New York: Wile).,

would probably be programmed to point at the apparent [-I] L. D. Landau alld E. 51. Lifshitz, The clnssical ~ h e o r y of Fields. receiver or transmit.ter. This can be taken into account PerganlO1l, 1~J62.

by a rotation of coordinates in reference frame S.

1962.

[ 5 ] H. Fujioka, T. Shiozaxa. and X. Kumagai, “Electromaguet.ic radiat.ion from all elertrir dipole moving with relativistic ve- locity,” Proc. Electrotr. u d (‘our~~rutc. Engrs. Japa.n, vol. 49,

[l] -4. I-. Balakrihman, Space Co?nmmicntions. New York: Mc- [6] D. F. Lan-den, . l t t Itdrod!rrtio,l Lo Tetrsor Ca.lctclus and Relativity.

London: l\let.hnen, 1967: pp. 7-13. 81-89.

[Z] ‘Ar. K. H. Panofsky and 31. Phillips, Classical Electricity and electromaglletir ware?; from moving so~~rces,” Radio Sei., vol. 1, [i] R. 11. LeKis and IT. Pre4s~nm~, “Radiat.ion and propagation of

REFEREXCES Juue 1966.

GranW-Hill, 1963, p. 29.

Xing~refistt!. Reading. Ilass.: Addison-Wesley, 1955. pp. 1029-1040, September 1986.

Radiation from a Circularly Polarized Antenna Through the Ionized Wake of a

Mars-Entry Capsule

Abstract-The effects of the ionized wake :of a Mars-entq capsule on radiation from a circularly polarized antenna operating at 400 and 2295 MHz are studied. The circularly polarized antenna is represented by a turnstile antenna h/4 above a ground plane, while the ionized wake is approximated by a cylindrically stratilied plasma consisting of N-plasma regions. Integral expressions for the fields are obtained for the antenna located in the wake and are evaluated using saddle-point integration to yield the radiation patterns. Computed patterns for the two simplest configurations N = 1 and N = 2 are presented. The radiation patterns for both near- and far-wake electron-density proliles develop a conical null region whose extent is proportional to the peak electron density in the wake. For the near-wake electron-density profile, sharp peaks which are attributable to leaky-wave radiation, appear within the null region of the patterns. The effect of the conical null region in the patterns is to prolong blackout time for communi- cation cone angles that lie within the null region. There are no serious depolarization effects in the nonnull region and satisfactory communications can be carried out.

I. I>-TRODKCTIOS t -RISG the ent.ry of a blunt-body capsule into the Martian atmosphere a plasma will surround and trail

the capsule. The antenna for a mission of this type is located in t,he aft portion of the capsule and propagat.ion takes place through the ionized wake. The capsule diam- eter i r l presently conceived designs for Mars-entry capsules is approsimately l i 5 em, and communications links with frequencies of 400 and 2295 MHz are planned. Blackout.

This research mas sponsored by IiASA under Cont,ract NAS ’7-100. Manuscript received Sovember 29, 1968; revied 3Iarch 13, 1969.

stit.nte of Technology, Pasadena, Calif. R. Woo is with the Jet Propulsion Laboratory, California In-

and is R Consultant. to the Jet. Propulsion Laboratory. A. Ishinlaru is with the University of Washington, Seatt.le, Wash.?

will occur at both frequencies [l]. The purpose of this paper is to invest.igate t,he effects of the ionized wake on comnlunications prior t.0 and aft.er blackout. The antenna is circularly polarized and will be represented by a. turn- stile antenna A:4 above a ground plane.

The plasma properties of the ionized wake must be knon-n before t,he esectu of the ionized wake can be con- sidered. Deternlination of these propert.ies is a formidable problem by itaself, because it involves a detailed knowledge of aerodynamic flow fields and flow rates which, i n turm are dependent on the environment. in which the capsule is moving. Severtheless, a reasonable representation of the ionized wake can be made.

Since the plasma associated uith Martian entry has a low temperature of -800-1300” I< [Z] and is weakly ionized, the magnetoionic theory description of the plasma is assumed. No static magnetic field is present in the vicinity of Mars and the plasma is also isotropic [3]. Furthermore, the electron-neutral collision frequency is at least. three orders of magnitude lower t,han the electron plasma frequency and collision losses can be neglecOed [l]. Since ohe electron-density profiles in the wake a.re axisym- metric, t,he ionized m-a,ke will be represented by a cylin- drically stratified plasma. In the near-wake region, electron density increases at. first and then decreases in the radial direct,ion.,uith the maximum occurring near the edge of the wake phile in the far-wake region electron density is maximum on axis and decreases in the radial direction [4]. Both types of elect,ron-density profiles will therefore be approximat,ed.

In this paper the fields are solved for a horizontal electric dipole located in a cylindrically stratified plasma con-