THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR, MICHIGAN Technical Report THE EFFECT OF THE ATMOSPHERE REFRACTIVE INDEXES ON THE ACCURACY OF DOVAP Joseph Otterman Department of Aeronautical Engineering UMRI Project 2387 DEPARTMENT OF THE ARMY PROJECT NO. 3-17-02-001 METEOROLOGICAL BRANCH, SIGNAL CORPS PROJECT NO. 1052A CONTRACT NO. DA-36-039-SC-64659 FTo MONMOUTH, NEW JERSEY August 1958

TABLE OF CONTENTS Page THE UNIVERSITY OF MICHIGAN PROJECT PERSONNEL iii ABSTRACT iv 1. INTRODUCTION 1 2. NOTATION 2 3. THE BASIC DOVAP EQUATION 4 4. THE EFFECTS OFTHE CHANGING AIR DENSITY 11 5. THE EFFECTS OF THE IONOSPHERE 14 6. CONCLUSIONS 16 7. ACKNOWLEDGMENT 16 APPENDIX A. The Determination of the Distances Transmitter-MissileReceiver for One SM1.O1 Trajectory Point Using Variable Average Velocity of the Radio Waves 17 APPENDIX B. Computation of the Difference Between the Curved Radiation Path and the Actual Distance from the Missile to the Receiver 20 REFERENCES 25 ii

THE UNIVERSITY OF MICHIGAN PROJECT PERSONNEL Both Part-Time and Full-Time Allen, Harold F., Ph.D., Research Engineer Bartman, Frederick L., M.S., Research Engineer Billmeier, William G., Assistant in Research Edman, Marshall W., Assistant in Research Harrison, Lillian M., Secretary Henry, Harold F., Electronic Technician Jew, Howard, M.A., Research Assistant Jones, Leslie M., B.S., Project Supervisor Kakli, G. Murtaza, B.A., Assistant in Research Kakli, M, Sulaiman, M.S., Assistant in Research Lay, Manchiu D. S., M.S., Assistant in Research Liu, Vi-Cheng, Ph.D., Research Engineer Loh, Leslie T., M.S., Research Associate McKenna, Kieth J., Assistant in Research Millard, Wayne A., Assistant in Research Nelson, Wilbur C., M.S.E., Prof. of Aero, Eng. Nichols, Myron H., Ph.D., Prof. of Aero. Eng. Otterman, Jos.eph, Ph.D., Associate Research Engineer Pattinson, Theodore R., Electronic Technician Rock, Allan L., B.S.E., Research Assistant Schumacher, Robert E., B.S., Assistant in Research Taylor, Robert N., Assistant in Research Thayer, Carl A., Assistant in Research Thornton, Charles H., Assistant in Research Titus, Paul A., B.S., Research Associate Wenk, Norman J.,!B^S.E., Research Engineer Wenzel, Elton A., Research Associate Whybra, Melvin G., M.A., Technician Wilkie, Wallace J., M.S.E., Research Engineer Wurster, John R., Assistant in Research Zeeb, Marvin B., Research Technician iii

ABSTRACT This report discusses to what extent the refractive indexes in the atmosphere affect the DOVAP accuracy. The basic DOVAP equation is derived and applied to propagation under the conditions of varying refractive indexes due to air density and ionization. It is suggested that a small improvement in position determination can result if a variable average velocity of radio waves, as a function of altitude, is used. Calculations indicate that the ionosphere propagation will have only a very small effect on the accuracy at altitudes under 90 km. The effect of the curved propagation path, i.e,, the difference between the actual propagation path and the straight-line distance, has been shown to be negligible, contrary to previous estimates of the effect. iv

1. INTRODUCTION DOVAP (derived from Doppler Velocity and Position) is a method for accurately determining the trajectory of a missile. DOVAP provides for transmitting a frequency-stabilized continuous radio-frequency wave from a fixed ground station to the missile, doubling the frequency of this transmitted wave in the missile receiver, and transmitting the doubled frequency from the missile to four fixed ground receivers. The frequency of the missile signal as received at each ground station is heterodyned with twice the frequency of the original transmitted ground signal. This is accomplished by receiving this reference frequency on a separate receiver at each ground station and doubling the frequency by the same method used in the missile receiver. The beat, or doppler, frequency is produced by mixing the radio-frequency output from the two receivers. This frequency is recorded on magnetic tape or on film during the missile flight. Each doppler-frequency cycle represents a change of one half the wavelength of the reference frequency in path length from the transmitter to the missile to the ground receiver. From the path lengths to three or more receivers, the spatial coordinates of the missile at any instant may be computed by solving a group of equations representing the intersection of three or more ellipsoids. The reference frequency of the existing DOVAP system is approximately 37 mc.* Since the retransmitted carrier frequency is twice the received frequency, the missile-borne transmitter operates at about 74 mc. Thus DOVAP is a missile-positioning system in which one-half of the wavelength of the reference frequency is the basic yardstick. This yardstick is affected by the refractive index on the path of propagation, The refractive index in the atmosphere depends on the air density and varies strongly with the degree of ionization. The problem discussed in this report is to what extent the refractive indexes in the atmosphere affect the DOVAP accuracy. Primary consideration is given to the use of DOVAP in the rocket-grenade experiment for upper-atmosphere temperature and winds. *This frequency is assumed in all the computations, except that in Appendix A the actual frequency recorded in the pre-IGY flight SM1.01, 36,938,775 cps, is used. The frequency has been changed to about 38 mec for the IGY series of flights at Ft. Churchill. 1

2, NOTATION v velocity of the rocket c velocity of light T(t) number of the wave being emitted by the transmitter at the time t M(t) number of waves propagating on the path transmitter-missile-receiver G(t) number of waves propagating on the direct path transmitter-receiver Rm(t) number of the wave entering the receiver from the missile path Rg(t) number of the wave entering the receiver by the direct propagation from the transmitter S(tl;to) the DOVAP count in the time interval from to to t1 t time to initial time of trajectory computation t! time for which the rocket position is being computed dr incremental distance taken along the path of propagation l(ar) half a wavelength for frequency fl at point r cl(r) velocity of propagation for frequency fl at point r X2(r) wavelength for frequency 2f1 at point r c2(r) velocity of propagation for frequency 2f1 at point r rll transmitter-missile transmission distance Irl-rkl missile-receiver transmission distance tp time of propagation from the transmitter to the receiver rl Cposition of the rocket at t{ ro position of the rocket at to c*(r) velocity of propagation of the path missile-receiver 2

vl(r) twice the wave number at point r for frequency f, v2(r) the wave number at point r for frequency 2f1 fl frequency of the transmitter u1 distance transmitter-missile-receiver at t1 v average of vl and v2 Uo distance transmitter-missile-receiver at to cl(rl) average velocity of propagation for frequency f c2(rl) average velocity of propagation for frequency 2f1 l1(rl) twic e the average wave-number for frequency fl v2(rl) average wave-number for frequency 2fl co velocity of light in vacuum, km/sec p ambient density c(p) velocity of radio waves as a function of the ambient density f frequency x(p) wavelength as a function of the ambient density v(p) wave number as a function of ambient density la ~ refractive index of air T absolute temperature, ~K Hs scale height of the equivalent atmosphere k Boltzmann's constant M mean molecular mass 9g acceleration of gravity r(H1) average refractive index for propagation to altitude H1 H1 altitude of the rocket Ho altitude of the DOVAP array

p(Ho) pressure at the altitude of the DOVAP array p(H1) pressure at altitude H1 n electron density per cubic centimeter e electron charge m electron mass Ce increase in length of RF wavelength in the ionosphere as compared with its length in an un-ionized medium AH/H error in thickness of the layer in DOVAP determination of altitude differences in a uniformly ionized layer AT error in determination of the temperature 5. THE BASIC DOVAP EQUATION In deriving the basic DOVAP equation, a Newtonian inertial frame of reference is assumed. Thus, any possible relativistic correction terms in (v/c)2, where v is the velocity of the rocket and c is the velocity of light, are neglected. Even for a missile traveling with the orbiting velocity such terms are of the order of 10-9. For the purpose of deriving the basic equation, each electromagnetic wave on the path transmitter-missile and on the direct path transmitter-receiver will be counted as two waves. This simplifies the presentation. It is assumed that the rocket does not spino In accordance with the above, each half-wave broadcast by the transmitter is labeled with a consecutive real integer. Each half wave becomes a full wave in the frequency doubling, whether in the rocket or at the receiver, and still retains its identifying numbero The labeling is done on an analog basis, i.e., the idea is extended to fractions of a wave and fractions of a number. Let T(t) be the number of the wave being emitted by the transmitter at the time t; let M(t) be the number of waves propagating on the path transmittermissile-receiver; and let G(t) be the number of waves propagating on the direct path transmitter-receiver. The number of the wave entering the receiver from the missile path, Rm(t), will be 4

Rm(t) = T(t) - M(t), (3.01) and the number of the wave entering the receiver by the direct propagation from the transmitter, Rg(t), will be Rg(t) = T(t) - G(t). (3.02) In the interval t1 - to a number of waves Rm(tl) - Rm(to) enter the receiver from the missile direction and a number of waves Rg(tl) - Rg(to) enter the receiver by the direct propagation path. The DOVAP count for this time interval, S(tl;to), represents the difference between those two numbers. S(tl;to) = (Rg(ti) - Rg(to)) - Rm(tl) - Rm(to)} = ([T(tl) - G(tl)] - [T(to) - G(to)]} -= [T(tl) - M(tl)] - [T(to) - M(to)] = [M(tl) - M(to)] -[G(tl) - G(to)] ~ (3.03) We assume now that the frequency of the transmitter and the conditions of propagation on the directpath transmitter-receiver are constant. It follows that G(t) = constant. Then we have S(tl;to) = M(tl) - M(to). (3.04) Equation (35.01) was written on the assumption of no time delay or phase shift between missile receiver and missile transmitter. Equation (3.04) remains valid if a constant phase shift occurs between the missile receiver and the missile transmitter. Assume first that the rocket was at rest with respect to the DOVAP array at the time to and decelerated to rest before t1. Let r1 be the vector from the transmitter to the rocket (see Fig. 1) at tl and let rk be the vector from the transmitter to a receiver. Then we have, from the examination of the physical meaning of M(tl), which is the number of waves in propagation on the path transmitter-missile-receiver, M(t = dr r/ dr M(tl) =(r) c1(r) + 2(r) 5 o r 2(r)

MISSILE AT t, //\ / / / r X I r rK INITIAL POSITION OF MISSILE AT to LAUNCH TRANSMITTER METRO LAUNCH RECEIVER RECEIVER TWIN LAKES RECEIVER DIGGES RECEIVER Fig, 1. Schematic presentation of the DOVAP array at Ft. Churchill.

where dr is incremental distance taken along the path of propagation; where Al(r) and cl(r) are, respectively, half a wavelength and the velocity of propagation for frequency fl at point r; and where %2(r) and c2(r) are, respectively, a wavelength and the velocity of propagation for frequency 2f1 at point r. Let Irll and Irl-rkl be, respectively, the length of the transmitter-missile and the missile-receiver transmission path. (It will be shown later that straight-line propagation can be assumed as a very good approximation.) Then, if average velocities cl(rl) and c2(rl) are defined in the following way II(r) C- ci(r) (3.06) and Jr nkl r rk dr dr -7-rl - - | = A *,(r) (3.061) Eq. (3.05) can be rewritten: M(tl) = 2f1 + Irk- rl ( 7) Since the averaging is done over the inverse of velocity, it might be convenient to use the wave number instead. Let 2~ v(r) = cl(r) (3.08) c?(r) Y2(r) 2f. (3,09) In terms of wave numbers, Eq. (3,05) becomes: pr, rk M(tl) = vl(r) dr + V 2(r) dr. (310) o rl If we define: vl(r) = (r) dr (.11) 7

and 0rk -~(PI) V2(r) dr v2(r) = V2(r)dr (3.12) Irk - rlj then M(tl) = vi(rl)' |ri + 2(rl) |Irk - rl ~ (3.13) Now, let us no longer assume that the rocket is stationary at t1. The argument that led to Eqs. (3.01), (3.02), and (3.03) is unchanged. However, the expression for M(tl) given before in Eq. (3.05) has to be reconsidered, since the path of propagation changes continuously. It will be remembered that M(t1) is the difference between the number of the wave T(tl) being emitted by the transmitter at t1 and the number of the wave Rm(ti) entering the receiver from the missile at tl, Eq. (3.01). M(ti) is thus equal to the number of waves produced by the transmitter from the time wave number Rm(ti) was emitted by the transmitter until time t1. This number is equal to twice the frequency of the transmitter, 2fl, times the time of propagation, tp, of the wave Rm(tl) from the transmitter to the receiver. This wave was received and sent by the missile at time t1 < tl, such that rk dr -(r) = t - t, (514) where rj is the position of the rocket at to and c*(r) is the velocity of propagation on the path missile-receiver for frequency modified by the doppler effect. The time of propagation is therefore given by or r dr rk dr tP J c1(r) + J c(r) (3.15) and the equation for M(ti) is: M(.t1) = 2f1 r{ Tdr /rk dcr_ dy rk drI M(t ).2f, [+ X l j J. (3.16) Comparing Eq. (3.16) with Eq. (5.05), we note that, first of all, the difference lies in the fact that the integration of the inverse of velocity is carried out on a different path: on path 0 - r - rk, instead of 0 - rl - rk. A second difference is that the velocity of propagation in the rn - rk integral is c( r) to denote that the propagation is at a frequency somewhat different from 2f1. The frequency is different from 2fl because of a doppler shift in 8

the rocket receiver due to the radial velocity of the rocket with respect to the transmitter, and because of the doppler shift for a stationary observer due to the radial velocity of the missile relative to the receiver.* However, the change in velocity caused by this frequency shift is quite insignificant. Thus, we commit an insignificant error if we supplant c2(r) for cd(r) in Eq. (3.16). We obtain M(tl) = 2fl J C( r + c(r) dr (317) o rr and for S(tl;to), if the rocket moves at to, we have S(tl;to) = M(tl) - M(to) = 2f1 Fr 1 dr dr O dr dr =L[ r) + _ C (r) + Jr c(r) r c2(r) (3.18) where rl and r6, the positions of the missile at tj and to, respectively, are defined by rk dr (3519) Ir^ -^ = ti -, (3.19) J oC2(r) and dr r — = to - to. (3.20) Let us compute the order of magnitude of the time interval tl - tV. For a rocket at 120-km altitude this will be 120 km 0.4 10-3sec 300,000 km/sec for a station underneath the rocket. A rocket traveling at the orbiting velocity of about 8 km/sec will move during this time a distance of Irl - ri = 8000. 0.4: 10-3 = 3.2 m. This is not a negligible amount, but smaller than the current accuracy of the DOVAP system. *Actually the spin will also affect the frequency on the path missile-receiver, but ordinarily by an even smaller amount. 9

Thus, for all present-day applications, Eqs. (3.05), (3.10), and (3513) are sufficiently accurate. Strictly speaking, the use of these equations yields the position of the rocket at t{ rather than at t1. The question of the numerical values of the average velocity of light to be used in the equations will be discussed in later chapters. It should be pointed out here that Eqs. (3.06) and(3.061), or (3.11) and (3.12), for computing the average velocity of light, or the average wave number, are based on the knowledge of the propagation path transmitter-missile-receiver. Straightline propagation will be assumed, since it is shown in Appendix B that the effect of bending can be neglected. Knowledge of the position of the rocket is necessary; however, since the velocity of light changes slowly, only an approximate knowledge of the position is required. It should be pointed out further that Eq. (3.07) and Eq. (3.13) are equations in two variables: I rj and Ir2 - rni. Equation (3.13) can be rewritten as follows: = l^rl) lrl l( M(t ) (r)+2 (r) rll+rkrll) + [.(r.)-j.(r.) ( rli-irk-rli) (3.21) Except for ionosphere propagation, i-(rl) is equal to v2(rl); the term in the square brackets will therefore vanish. For ionosphere propagation the term in square brackets can be considered as a small correction term. In either case, the equation can be considered as an equation in one variable: |rli + Irk-rl = ul. If the concept of _-_ vj- + V2 v = o (35.211) is introduced, we have M(tl) V(rl) +,2(rl) ul = v(rl). u1. (3.22) 2 Equations (3.05) to (3.13) and (3.21) to (5.22) were written for ti. A parallel set of equations can be written for to, and in particular M(to) l(r) + 2(r) uo = V(ro) uo (3.25) t = to, then the distance transmitter-missile-receiver ul can be computed from Eq. (3.22), (3.23), and (3.04). u = S(tl;to) + uo. (ro) (5.24) 1(rl) 10

4. THE EFFECTS OF THE CHANGING AIR DENSITY The velocity of light near sea level departs by about 35 parts in 10,000 from the velocity of light in a vacuum. At the present time, the determination of the position of a missile from the DOVAP cycle count is based on the use of a constant velocity of light, which is chosen as an average of the velocities in the vacuum and at the ground. In this chapter, the problem of how the air density affects the numerical value of the average velocity of light to be used in Eq. (3.07), or the average wave number to be used in Eqs.(3.13) and (3.24), is considered. From this discussion it follows that for greater accuracy a variable average velocity of light as a function of altitude of the rocket should be used. A formula for calculating this variable velocity is suggested. It is shown that, for rockets at high altitudes, the current practice can result in errors approximately equivalent to omitting two cycles for each ten thousand cycles counted. The suggested method of calculation assumes that an approximate altitude of the rocket is already known. The velocity of light in vacuum, Co, is thought to be known with an accuracy of about 1 part in 300,000. Its value is given by DuMond and Cohen1 as 299,792.9 (+ 0.8) km/sec, and this value will be used here. For atmospheric propagation, the velocity of radio waves will be a function of the ambient density c(p) c (4.01) 4(P) and so will be the wavelength X(p) = (p) (4..02) and the wave number f L(P) __0 v(p) (p) =. (4 ) The velocity is assumed independent of frequency (except for ionospheric propagation). BRL Report No. 677~ gives the refractive index of air as t = 1.0002726 computed for sea-level pressure and for the temperature of 59~F (150C). This checks closely with a newer experimentally determined value of (-1l)l06 = 288.15 for dry sea-level air at 0~C.3 Since k-l is approximately proportional to the density, we have for the sea-level propagation 11

1sea level = 1 + 2 726 10-4 288 (4.04) T where T is the temperature in ~K. Thus, if the DOVAP array is substantially at sea level and if the initial position of the missile at to is very close to the ground (this was the case in the IGY rocket firings in Fort Churchill, Canada), the number of waves M(to) can be determined from [see Eq. (3.23)] M(to) = v(ro) ~ uo = 2 u uo = - 2.726 10 (4.05) The calculations of the average velocity of light (i.e., the calculations of the average wave number) for different altitudes of the rocket are not complicated. The calculations are simplest for points at altitudes higher than about 40 km. For such trajectory points, more than 99% of the atmosphere lies under the altitude of the missile; and for the purpose of our calculation, it can be assumed that the missile is "outside the atmosphere." The calculations will be developed first for such points. A well-known notion of an equivalent atmosphere, of uniform (sea-level) density, is thought to simplify the presentation. The thickness of such equivalent atmosphere is referred to as scale height (at sea level). The propagation up to one-scale-height altitude is assumed to be taking place with the sea-level velocity; and the propagation above the scale height and up to the missile is assumed to be taking place with the vacuum velocity of light. The height of the equivalent atmosphere (equal scale height) is given by Hs = T = 2.926' 10-2 T km, (4.06) Mg where k is Boltzmann's constant, T is the absolute temperature in OK, M is the mean molecular mass, and, g is the acceleration due to gravity. For the rocket altitude of H1, we average the refractive index in the following way: propagation in vacuum (i.e., refractive index equal to unity) over the altitude H1 - H., and propagation at the ground density for an altitude Hs. -(H1) = l(Hl-Hs) + 4sea level Hs 1 + (>sea level - 1) H.(4.07) HI H1 From this equation and from Eq. (4.04), it follows that j>(H1) = 1 + 2.726. i0-4 — 8 - (4.08) 12

and when the value of the scale height Hs is introduced from Eq. (4.06), we have (H1) 1 + 2.726 10-4 288 2.926 10-T = 1 + 2.726 104 8 427 T H1 H1 (4.09) where H1 is the altitude in km. It can readily be seen that the value of the average coefficient of refraction is independent of the temperature. The formula is written in this way to bring out the fact that 8.427 1kn is'the scale height for the temperature of 288~K, for which the value of (,sea level' 1) is 2.726 ~ 10-4. Equation (4.09) applies for the trajectory points outside the atmosphere, i.e., at altitudes greater than several scale heights, with the transmitter and receiver located at sea level. For points lower than 40 km and when the DOVAP array is located at altitude Ho different from sea level, the average refractive index will be given by the formula: H HL(p)d'H IH (1+ [sea level -1 d ( Ho fHO \ Psea level ~(H1) = -- =' —----.. HI - Ho Hi - Ho - H H H1-HI HI-Ho (#sea level -1) fH1 pdH (#ssea level 1l) fH1 dp (~sea level -1) fH~ Hi-H o (HI-Ho) Psea level (Hi-Ho)Psea level' g (sea level- 1) [p(Ho) - p(H1)] p(Ho) - p(Hi) 2.72 4 288 + (H-Ho) Psea level * (Hi-Ho) Pealevel g T (4.10) and since sea levPsea level M (4.1) Psea level =..- (4.11) kT we have (H1) = p(Ho) - (H1) kT 2.726 288 ~l. - ~ --- ~ 2.726 ~ 10'4 (H1-Ho) Psea level g T p(Ho) - p(H1) 288 8.427 p(Io)- p(H1) = 1 +.- level Ts'2.726 10 -- = 1 + 2.726 *10'4 H- -- -----. (Hi-Ho)Psea H. 11H Psea level (4.12) 13

In the simpler case of Eq. (4.09), Ho is zero, p(Ho) is taken to be the sealevel pressure, and p(H ) is taken to be vanishingly small. The average refractive indexes for the average wave numbers to be used in Eq. (3.24)] are functions of altitude. The error which results from using the current approach is calculated for one trajectory point of Aerobee SMl.O1 in Appendix A. 5. THE EFFECTS OF THE IONOSPHERE The problem of errors caused by the ionosphere in the DOVAP determination of a rocket trajectory is discussed by deBey and Hoffleit in BRL Report No, 677, already mentioned.5 They discuss the effect of change of wavelength, i.e., effective change of the yardstick, and the effect of bending, i.e., the difference in length between the curved-path propagation and straight-line transmitterreceiver distance. Calculations were carried out which indicated that the BRL figures regarding the magnitude of the bending effect were apparently too large by several orders of magnitude. The actual differences between the curved path and the straight-line transmitter-receiver distance appear to be negligible compared with the change of wavelength effect (see Appendix B). Accordingly, this second effect is not considered further in the body of this report. The index of refraction,)p.., neglecting the effects of the earth's magnetic field and electron collisions, is given by the formula:. = (1 - ne2/imf2)1/2 (1 - 80.5 106 n/f2) /2 (5.01) where n is the electron density per cubic centimeter, e the electron charge, m the electron mass, and f the radio frequency. Thus, the RF wavelength becomes longer in the ionosphere by a fraction e as compared with its length in an un-ionized medium: 1 1 ~e =~~~~~ — ~~1 =1.502) p. - -1 (1 - 80.5 106 n/f2)1/2 (502) and for << 1 E r 40.25' 106 n/f2. (5.03) For the transmitter-missile path we have 14

E ~ 40.25' 106 n/372' 1012 = 2.94 n ~ 10-8 (5.04) and for the missile-receiver path E C 40.25 * 106 n/742 * 1012 = 0.74 n * 108. (5.05) The actual error in missile position obviously depends on the geometry of the missile DOVAP system. The magnitude of the error can be readily computed for a nearly vertical trajectory if it is borne in mind that at a high altitude the angle between the transmitter-missile path and the missile-receiver path is small. A horizontally stratified ionosphere is assumed. Under these conditions, the DOVAP determination of altitude differences in an uniformly ionized layer will be approximately in error by a fraction: AH 2.94 +0.74 0- -8 H = 2.94 + 7 o10 =.84 n 18 108 n (5.06) H 2 The determination of the layer thickness will be too low, and therefore the determination of the velocity of sound will be too high by the same fraction 1.84' 10-8 n, where n is the average electron density per cubic centimeter in the layer between two successive grenade explosions. The temperature will be too high by the fraction 2 * 1.84 * 10-8 n = 3.68 * 10-8 n or (in the 80-90-km region) by about 205' 3.68 ~ 10-8 n = 7.54' 10-6 n OK. The electron-density data at altitudes under 100 km seemed somewhat contraversial. Kuiper6 quotes Piggott (1950) to the effect that the noon values of n at 90 km are about 1.5' 104/cm3. On the other hand, Berning7 reported electron densities as high as 7 104/cm3 at 60 km. From conversation with Berning (October, 1956), it transpired that he considered his electron-density data not reliable in the D-region. In view of this fact, the density of 2' 104/cm3 will be used as the basis for calculating the temperature error. This figure was given by J. C. Seddon (private correspondence) as the outside value for noon-time electron density at 90 km at White Sands Proving Ground, New Mexico. Assuming that this density prevails throughout the layer between two successive grenade explosions, the determination of the thickness of the layer will be in error by AH = 1.84 - 10 2 04 =.68 10-4 =0.037, (5.07) H and the temperature will be in error by: AT = 7.54 ~ 10-6. 2 * 104 = 0.158Q. (5.08) 15

This is only about 1/35 of the average probable error for the temperature data (+ 5~) of the 1950-1953 rocket-grenade experiments carried out at White Sands8 and 1/3 of the probable error (+ 0.4~) for the most accurate experiment of the series (24 April 1956). The ionosphere error, as calculated.above, is expected to occur only close to 90 km, where the accuracy of the experiment might be lower than at lower altitudes. Thus, the conclusion might be drawn that the ionosphere does not greatly influence the accuracy of the experiment. It should be pointed out that, at high latitudes, transient ionospheric abnormalities are very pronounced.9 Electron-density profiles are not available, however, at the present but might become available during the International Geophysical Year. At altitudes higher than those of interest in the rocket-grenade experiment, the ionospheric effect can be much more pronounced. Assuming an electron density of 3 * 10 electrons/cc to exist in a sporadic E-layer,10 the DOVAP determination of the thickness of a layer will be in error by AH = 1.84 10-8. 3 106 = 5.5 ~ 10-2 = 5.5% (5.09) 6. CONCLUSIONS The effect of the atmospheric refractive indexes on the accuracy of the DOVAP system has been discussed. It has been shown that a small improvement in position determination can result if a variable average velocity of the radio waves, as a function of altitude, is used. This is due to effect of air density. The calculations indicate that the ionosphere propagation will have only an extremely small effect at the altitudes of the rocket-grenade experiment. This conclusion, however, should be further checked when more information about the ionosphere at high latitudes is available. The effect of the curved propagation path, i.e., the difference between the actual propagation path and the straight-line distance, has been shown to be negligible,, contrary to previous estimates of the effect. 7. ACKNOWLEDGMENT The comments of Frederick L. W, Bartman and Hal F. Schulte, Jr., which resulted in significant improvements in the plesentaLion of the subject matter, are gratefully acknowledged. 16

APPENDIX A THE DETERMINATION OF THE DISTANCES TRANSMITTER-MISSILERECEIVER FOR ONE SM1.O0 TRAJECTORY POINT USING VARIABLE AVERAGE VELOCITY OF THE RADIO WAVES NOTATION T(H ) ground wavelength, m o jT(61.2) average refractive index for 61.2-km altitude ((61.2) average wavelength at 61.2-km altitude, m -BRL average wavelength recommended by BRL, m S(tl;to) spin-corrected cycle count from wire break up to the time t of the last explosion. U(ti) distance transmitter-missile-receiver at the time of the last explosion u(to) distance transmitter-missile-receiver at the wire-break time Au(tl) differences between the variable-velocity method and the current method In this Appendix the numerical differences between the computation by the current method and the computation by the suggested method of variable average velocity of the radio waves (as a function of altitude) will be demonstrated for a single trajectory point of the SMl.01 rocket flight, namely, the explosion of the last grenade. The comparison shows that the differences in the distance transmitter-missile-receiver for various receiving stations are equivalent to between three and five cycles. At the time of Aerobee SMl.01 firing, on November 12, 1957, the temperature at the ground level at Fort Churchill, Manitoba, Canada, was about -25~C. The recorded frequency of the DOVAP transmitter was 36,938,775 cps. The explosion of the last grenade occurred at about 61.2 km above msl. The refractive index for propagation near the ground, in accordance with Eq. (4.04), will be = isea, level = 1 + 2.726 * 104 288/T - 1 + 2.726' 10'4 288/248 = 1.0003165 (A.01) 17

and the ground wavelength ( ).0001 — 6 29792.9 4.056686 m. (A.02) * 0) 1.0003165 2 36938775 The average refractive index for the altitude of 61.2 km will be, in accordance with (4.09), 1(61.2) = 1 + 2.726 * 10-4 8.42 = 100005754, (A'726 0- 61.2 1.00003754' (A.03 and the average wavelength 299,792.9 X(6l.2) = ----- 29.729 - 4.057818 m. (A.04) N2(62) / 1.00003754. 2 * 36938775 The average wavelength recommended by BRL for SM1.01 data reduction is.BARL = 4.05736248 m. (A.05) The distances p.(to) for wire-break time and the spin-corrected cycle counts from wire-break time up to time t1 of the last explosion are tabulated in meters here for the four receiving stations. Launch Digges Twin Lakes Metro u(to) 320.51 28,300.34 13,149.42 14,769.52 S(tl;to) 31,391.34 24,139.68 27,722.12 27,886.57 Computation of the distance u(tl) by the current method is carried out according to the formula u(tl) = XBRL S(tl;to) + u(to) (A.06) and the results are tabulated in meters below. Launch Digges Twin Lakes Metro u(to) 320.51 28,300.34 1,149.42 14,769.52 XBRL-S (t; to) 127,366.05 97,945.43 112,478.69 115,145.92 u(tl) 127,686.56 126,243.77 125,628.11 127,915,44 18

Computation of the. distances u(tj) by the method of variable light velocity is carried out according to the formula [see Eq.(3.24)] u(tl) = X(H1) Fs(ti;to) + ( = \(H1) S(t;to) + u(t) t(H1)) S(tlto) +- ~ u(to) L J(Ho) (A.07) and the results are tabulated in meters below. Launch Digges TWin Lakes Metro U (612) 320.60 28,308.24 13,153.09 14,773.64 o(Ho) )(6l.2)-S(tl;to) 127,380.34 97,954.43 112,491.32 113,158.63 u(t1) 127,700.94 126,262.67 125,644.41 127,932.27 The differences Au(tl) between the variable-velocity method and the current method are (in meters): Launch Digges Twin Lakes Metro - 14.38 + 18.90 + 16.30 + 16.83 In general, comparison of Eqs. (A.06) and (A.07) will give for Au(tl) Au(ti) = [O(H1)-\BRL]S(tl;to) H ).(H (to) (A.08) T(H19) 19

APPENDIX B COMPUTATION OF THE DIFFERENCE BETWEEN THE CURVED RADIATION PATH AND THE ACTUAL DISTANCE FROM THE MISSILE TO THE RECEIVER NOTATION M missile R receiver MR actual radiation path missile-receiverMO line through M, parallel to the portion of the path with the largest zenith angle Pi B1 ~largest zenith angle 32 smallest zenith angle BP zenith angle of the missile as seen from the receiver RO line tangent to the actual radiation path at R I curved path RO + OM H vertical elevation of the missile over the receiver h vertical elevation of point T over the receiver AP angle P2 - PI d distance OT C13 angle p1 - (C2 angle - p2, =l-AZ smallest index of refraction throughout the path This computation presents a simple formula for estimating the effect of the curved radiation path, The assumptions are such that the formula provides an outside limit rather than the correct numerical value. 20

The calculations are carried out for the missile-receiver propagation path. The axes are vertical and horizontal lines at the receiver. Let M represent the missile (see Fig. 2), R the receiver; and let the curved path MR be the actual radiation path. Let MO be a line through M parallel to the portion of the path with the largest zenith angle pi (in the region of smallest refractive index, i.e., the greatest electron density), and let RO (zenith angle (2) be tangent to the actual radiation path at R. The angle ( is the zenith angle of the missile as seen from the receiver. It can be seen (the sum of the lengths of two sides of a triangle is greater than the length of the third side; see Fig. 3) that RO + OM = I is greater than the actual curved path RM. Let OT be perpendicular to RM and let H and h be the vertical elevations of the missile and point T, respectively, over the receiver. From the geometry in Fig. 2, and because AP = P1 - P2 is a small angle, we have (we designate distance TO as d) 2 = RO + OM = RT2 + O + +TO - J(h/cos 3)2 + d2 + (H-h/cos p)2 d2 (B.01) i. 4 + 1/2 [d cos t+ - 1 + 1/2 ( d cos cos ph cos H -h H Hd2 cos p + cos p 2(H-h)h = H + d2 cos23 ((B.02) cos p l 2(H-h)h d = TO _ CO2h _, (H-h) (B.0) cos 3 cos p e1 + a2 AP (B.04) al(H-h) r (A - a2) (H-h) X a2h (B.05) a H-h Ap (B. 6) H h (H-h) d.- cos A (B.07) cosBP H 21

MISSILE H /R__"2 RECE IVER Fig. 2. Geometry of ionospheric propagation. MIS: RO tOM is / / Qa = RO+OM is greater than the curved line RM RECEIVER Fig. 5. Comparison of distance I and the actual path. 22

+ d2 cos2} H + (H-h)h()2 (B.8)2 cos X 4+2(H-h)h =co+ 2H2 (B08) In the following, pi = 1 - A[L designates the smallest index of refraction throughout the path. sin 2 - _ A sin P1 cos AP - cos P sin AP sin p- cos $1-A sin p1 sin P1 sin Pi (B.09) AP _ A. tan P1 _ AB tan P (B.10) H. _ 4- (H-h)h (A2)2 inP e H cos + 2H2 t 1 (B.ll) The expression (H-h)h/H2 takes a maximum value of 1/4 for h = H/2. Assuming that this maximum exists, we have = oHs [1 +A L —2 tan2' I Cos 1+ tanp]. (B.12) The difference in path length Al between the curved path I and the straight-line distance is A = 8cs (AL) tan2A. (B,15) 8 cos p Expressed in percentage, Al/I is - < 12.5 (A^)2 tan2k % (B.14) The numerical example given on pages 116 and 117 of BRL Report No. 677 by deBey and Hoffleit discusses the ionospheric refraction in the following case: H 100 km A. =.00348 tan =- 0.20, cos p = 0.98 The above reference estimates the difference between the curved radiation path and the distance from the missile to the receiver to be approximately 30 m, whereas actually 23

A I 105. 3.482 10"6 0.2 8 0.98 = o.6 102 m = 0.6 cm The formulas (B.i35)) and (B.14) are applicable to the effect of curved path due to changing atmospheric density, with the difference that A1. = -.-l, where L represents the largest index of refraction throughout the path. The numerical example given on pages 114 and 115 of BRL Report No. 677 discusses the refraction due to the atmospheric density in the following case: H - 140 km AU = 0.00027 tan B = 1 cos P = 0.707 The reference estimates the effect of the curved path to be 4 m, whereas actually AA I.1.4 105' 2.72. 10-8 8 0.707 = 1.8 10o = 0.18 cm 24

REFERENCES 1o Do Ho Menzel, Fundamental Formulas of Physics, Prentice Hall, Inc. New York, 1955, pc 149. 2, Lo Go deBey and E, Do Hoffleit, DOVAP - Instrumentation and an Analysis of Operational Results, Ballistic Research Laboratories Report 677, 1948. 3. Lo Essen and K. Do Froome, "Dialectric Constant and Refractive Index of Air and Its Principal Constituents at 24,000 Mc/s," Nature, 167, 512 (March, 1951)o 4. Ro Mo Goody, The Physics of the Stratosphere, Cambridge University Press, 1954, ppo 21-227 5. deBey and Hoffleit, pp. 116-117. 6. P. Kuiper, The Earth as a Planets The University of Chicago Press, 1954, p. 598. 7. Wo Wo Berning, "The Determination of Charge Density in the Ionosphere by Radio Doppler Techniques," Rocket Exploration of the Upper Atmosphere, Pergamon Press, 1954, p. 265. 80 W. G. Stroud, W, Nordberg, and J. Ro Walsh, "Atmospheric Temperatures and Winds Between 30 and 80 km," Jo Geophys. Res., 61, 1, 45 (March, 1956)o 9. Co Go Little, Wo Mo Rayton, and Ro Bo Roof, "Review of Ionosphere Effects at VHF and UHF," Proceedings IRE, 44, 8, 1006-1008 (August, 1956). 10. So Lo Seaton in the Compendium of Meteorology, edited by To Fo Malon.e American Meteorological Society, Boston, 1951, p. 337o 25