THE UN I V E R S I T Y OF M I C H I G A N COLLEGE OF ENGINEERING Department of Electrical Engineering Space Physics Research Laboratory Scientific Report No, 1 THE THERMOSPHERE PROBE EXPERIMENT D. R Tatusch G. R. Garignan H. B. Niemani A. F. Nagy ORA Project 07065 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GODDARD SPACE FLIGHT CENTER CONTRACT NO. NAS 5-9113 GREENBELT, MARYLAND administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR March 1965

TABLE OF CONTENTS Page LIST OF FIGURES v 1. DESCRIPTION OF EXPERIMENT1 1.1 Introduction 1.2 The Thermosphere Probe1 1.3 Ejection and Tumble System 2. ASPECT DETERMINATION 2.1 Introduction 2.2 Ejection Kinematics 2.3 Equations of Motion of the Free TP 8 2.4 Tumble Period 13 2.5 Roll Period 13 2.6 Orientation Analysis (Velocity Vector Reference) 14 2.7 Orientation Equations 16 2.8 Orientation Analysis (Earth Normal Reference) 18 2.9 Direct Method of Obtaining a 21 2.10 Computer Analysis 25 2.11 Atmospheric Wind 25 3. DATA REDUCTION 29 3.1 N2 Density Vs. Altitude-Data Analysis 29 3.1a 02, 0 Density Vs. Altitude-Data Analysis 30 3.2 Ambient Neutral Temperature Vs. Altitude-Scale Height Method 34 3.3 Ambient Temperature Vs. Altitude-Velocity Scan Method 36 3.4 Electron Temperature and Density-Data Analysis 37 3.5 Data Processing 43 REFERENCES 45 iii

LIST OF FIGURES Figure Page 1. Assembled TP. 2 2. Ejection nose cone system. 4 3. Ejection kinematics coordinates. 6 4. Coordinates for free TP analysis. 9 5. Coordinates for free TP analysis. 10 6. Coordinates for TP orientation, velocity vector reference. 15 7. Coordinates for TP orientation, earth normal reference. 19 8. Coordinates for direct method of angle of attack analysis. 22 9. Spherical triangles of Fig. 8. 23 10. Computer output for trajectory and angle of attack solutions. 26 11. Horizontal component of wind velocity in the plane of tumble vs. altitude for various Au. 28 12. N2 Pimax Pimin and Pa vs. altitude for Nike-Tomahawk trajectory. 31 13. Temperature effect on Pi vs. G. 38 14. Te determination, NASA 8.19, 504 kilometers; 41 15. Telemetry record from NASA 6.06. 44 v

1. DESCRIPTION OF EXPERIMENT 1.1 INTRODUCTION The thermosphere probe (TP) experiment described herein is the result of a research effort implemented by this laboratory under contract with the NASA Goddard Space Flight Center, Aeronomy and Meteorology Division. The purpose of this effort was to provide an ejectable rocket-borne system capable of making simultaneous direct measurements of gas temperature and density, ion and electron density, and electron temperature in the earth's atmo.sphere in the altitude region between 100 and 350 km, a region within the thermosphere. The primary mission of the experiment is to fill the present measurement gap in this general altitude region which is above the altitude capability of the grenade, falling sphere and pitot-static techniques, and below the altitude of usual satellite measurements. The TP incorporates an omegatron partial pressure gage, a cylindrical electrostatic probe, and a sun-earth aspect measuring system. This complement of instruments provides data for the determination of the previously mentioned desired atmospheric parameters. Subsequent development of the aspect determination system permits an extension of the experiment to determine the horizontal component of atmospheric wind in the plane of tumble of the TP. The ejectable system was chosen for the purpose of removing the TP from the environment of the launch vehicle, similar to the established "Dumbbell" technique,2 and to permit a tumbling motion to be imparted to the package, independent of the launch vehicle. The following report describes the theoretical background and techniques utilized in obtaining the gas temperature and density data, the electron temperature and density data, and atmospheric winds. Only those engineering particulars that bear directly on the actual measurement of the desired quantities required for data analysis are described in this report. A comprehensive engineering report of the system is in preparation. 1.2 THE THERMOSPHERE PROBE The TP is a cylindrical instrument 6 in. in diam, 32 in. long and weighs 40 lb. A photograph of the assembled instrument is shown in Fig. 1. One end of the cylinder contains the omegatron gauge with its circular orifice and breakoff device on the cylindrical axis; the other end of the cylinder contains the earth sensor. The center section contains the sun aspect sensor, 1

..^B~~~p 1'"''''^^~ —"..y s::I r 1 1,~~~~~~~~~~~~~~~~~~~~~~~~~~~~i: Fig. 1. Assembled TP.~~~~~~~~~~~~~~~~~~~::::,1~~~

a small cylindrical electrostatic probe and the telemetry antennae. The outer structure of the probe is made of stainless steel and the assembled instrument is vacuum sealed. The probe system is completely self-contained providing its own power supply, measuring sensors, signal conditioning, and transmission equipment, 1.3 EJECTION AND TUMBLE SYSTEM The ejection nose cone system is shown in Fig. 2. The clamshell-type nose cone halves, which provide the aerodynamic shape of the rocket during powered flight are hinged to the base of the enclosure and are held together against the force of two springs by a magnesium ring which is pyrotechnically fractured to effect opening. The TP rests on a spring-loaded plunger within the enclosure. The plunger is held depressed against the spring force by a latch mechanically linked to a nose cone half so that opening of the nose cone releases the latch, freeing the plunger, which, operating against the compressed spring, ejects the TP from the opened enclosure. A negator motor (constant force spring), with 8 ft of cable is mounted below the plunger. One end of the cable is fastened to the top side of the TP. As the TP leaves the vehicle, it is tumbled in the plane containing the cable hook and the center of gravity. When the TP has tumbled approximately 135~, the cable releases and is reeled back into the vehicleo The ejection system causes the TP to separate from the nose cone at about T fps and the negator or constant force spring imparts the tumble motion with a period of 2.0 sec. The roll period is noncontrolled and is the resultant of the roll period of the rocket at ejection. The opening of the nose cone halves prior to ejection provides an effective despin mechanism assuring a roll rate substantially less than the tumble rate. Thus, with a low roll rate and a moment of inertia ratio of more than 30, the TP can be considered to be tumbling in the plane of the cylindrical axis. 5

4-) U, 0'l C)j b0)?...... U) orl Z10 4

2. ASPECT DETERMINATION 2.1 INTRODUCTION The analysis of the neutral particle-pressure measurement requires that the orientation of the pressure-gauge orifice, -with respect to the velocity vector, be knowno For this purpose, the TP experiment utilizes a sun-earth aspect sensor. The sun aspect sensor views a fanx 360~ wide, the plane of which is oriented perpendicular to the long axis of the TP. As the TP tumbles, and rolls, the fan sweeps out a solid angle of 4Tr steradians, thereby viewing the sun at periodic intervals (every half tumble period). The output of the sun sensor yields the roll position of the TP at the time the sun is viewed o As was described in a previous section of this report, the ejection system for the TP is designed to decrease the roll rate and tumble the instrument in the plane containing the cylindrical axis. Since there are no external torques on the TP after ejection, it will be assumed that the angular momentum vector, L, for the system will remain fixed in inertial space. 2 2 EJECTION KINEMATICS In Fig. 3 the quantities and coordinates of interest are described for the analysis of the equations of motion of the TP during ejection. The Lagrangian for the TP after it has left the plunger is: = l/2mx2 + 1/21Q2 K(x + c cosQ) (2.2.1) where I = maximum moment of Inertia K = constant force spring constant m = mass of TP The equations of motion are mx + K = 0 (2.2o2) and 5

pa e - - K (CONSTANT FORCE SPRING) Fig. 3. Ejection kinematics coordinates. 6

IQ K sin = 0 (2.235) The solutions to 2.2.1 are: -t + V (2.2.4) m and x = - - /2 t2 + Vot + X (2.2.5) The solutions to 2 2.3 are: max max = /2 sQ - cosQmax (2.2.6) where Q = release angle max 2 K, I The release time is: t max d tax 2r7o t = F(k, (22.7) max- where~2 - 1. i Q -11 where P a= sin; k sin 1 - 2 2 p F(k,) = Elliptic Integral of the first Kind. sin n sin - 2 The translational velocity of the TP after being tumbled is given by V = V 2 A tmax KI\ (2.2.8) rrn o' 0 m where V is the initial velocity obtained from the plunger, which is 2 2 2...: Kp(xmax miin)- 2Kx.o (Energy In) (2?. ) m m 7

where Kp = plunger spring constant Xmax = expanded length of plunger spin x i = compressed length of plunger spring x =(x x. ) o max miin In the present configuration, the TP parameters are: m = 1.27 slugs K =100 lb/in I = 1.25 slug ft2 k = 4 lb max 2 Energy In = 61.5 slug ft2/sec2 min.04 slug ft mm n For the solutions we get ~ =,97 ft A2 = 3.10 F(82021,82034t) = 2060 Q =150~ c =.1305 Q = 135 ~ = 1.017 max The solutions are: Vo' 9.8 ft/sec Vin 5 ft/sec tma 1.53 sec W 2 rad/s max max rad/sec Tumble Period = Tmax - 2 sec 2.3 EQUATIONS OF MOTION OF THE FREE TP The TP cylinder is assumed to be a symmetrical top moving in a force free space. Referring to Figs. 4 and 5, the 1',213' axes of a right Cartesian coordinate system are fixed to the TP, and the 1,2,3 axes are fixed in inertial space. The components of angular velocity along the 1, 2',3 axes with respect to the inertial frame of reference in terms of Eulerian angles are: 8

33 WF \Ap — / I -\ ss^ — ur \ /, I Fig. 4. Coordinates for free TP analysis. 9

3. L i -d - 2' \0 I \ Fg C/ \ / III \ /^^"^2/ Fig. 5. Coordinates for free TP analysis. 10

* 0' t= sinQ cosr + ~ cosr 2e 2 ^ sinQ cos* - Q sin* (25.31) WI = c COs i- + 3 The components of the angular momentum for this angular velocities are: LI = TIl (2.3.2) LI = w1I and L2 2 2 2 (2 ) L = Lt + L2 L 1. 2 5 Assuming the angular momentum vector, L, is in the direction of the 3 axis of the inertial coordinate system, the components of L on the 1 Y2q 7) axes are: LI = L sinQ sinf 1 LI = L sinO cost (2.3.4) LV = L cosQ 3 Therefore, from (2,352) and (2.3.4) - _ L sinO sin* 1 I1 L c^ - ~ L sinQ cost (2.9.5) I1 O = L cosQ 3 3 Defining c' as the angular velocity along the axis of maximum moment r of inertia, we have: UCD _ (32 V 2 1/2.6; r 1 2 i 11

W' is the angular velocity along the axis of minimum moment of inertia., 3 The angular velocity is the direction of the 3 axis is the precission angular frequency: "P = = L (2-3-7) IL From (2.3.1) and (2.3.5) d e (I1 -L)sin0 sin = - Ii cost (I i-L)sin0 cos\ = I1Q sin* (253.8) (I3O-L)cos I3* Solving for Q, 9, and r (L is constant in the inertial frame) 0 = 0 L = (2.3.59) I1 cosO ( — - ) 13 I1 Integrating (2..359) yields Q = const. = = k-t + (2.o5l) I1 $ = cosO [L - t + o 3E I Equations (2.3.10) are the linearly time dependent Euler angles of the TP. In section 2.2, the TP parameters were given as: 2 I1 = 12 = 1.25 slug ft 12

=.04 slug ft2 C r = - sin Ii and L _ T = - cos@ 3 I 03 tan 0 - 1 ~ 30 I3 or Q0 88~ Therefore, since the highest roll rate expected was assumed, we conclude that the TP is tumbling in a plane perpendicular to the angular momentum vector. 2.4 TUMBLE PERIOD Two independent methods for the determination of the TP tumble period are available. The period between successive sun pulses can be read more accurately and is used for the tumble-period measurement, with the period between pressure maxima providing confirmation of the results. The tumble period is measured with an error of less than 2 msec (-1 part in 500)Y 2*5 ROLL PERIOD Each cycle of the tumble motion causes the sun sensor to generate an output which is a function of the TP roll position. The roll rate is determined by the following analysis of this information: Let X = roll rate (deg/sec) = roll position of TP k = an integer (0,1,2,..) n = number of 1/2 tumble periods t = time 13

then ^(n+2)-6 k 36 ~ t (n+2 n where t(+2) -t is simply the tumble period> The role rate, in deg/sec, is equal to the number of degrees the TP has apparently rolled in one tumble period (~0Q n+r -Cn), plus the number of complete cycles it has rolled (~k 3600), divided by the tumble period-the time between roll position data inputs, Qn+2 and Qno The plus or minus signs are a consequence of the uncertainty in roll direction. In the TP application, since the tumble period is less than the roll period, k is 0 and the equation becomes ~Qn+2-Qp/tn+2-tn0 Any pair of sensor outputs provide a solution; therefore, successive solutions can be used to prove the assumption k = 0 and also to indicate the correct sign in ~Qn+2, 2.6 ORIENTATION ANALYSIS (Velocity Vector Reference) Figure 6 shows the coordinate system used for the determination of L. It is a right cartesian-coordinate system in which the z axis is pointing at the sun The TP1, and TP2 vectors describe the position of the TP cylinder axis, the direction being that of the normal to the orifice of the pressure gauge. TP1 is the position of the TP at the time it is closest to the velocity vector, i,.e, the time a maximum pressure reading is recorded. TP2 is the position of the TP at the time a sun pulse is received. TP2 is in the x-y plane since the sun sensor is perpendicular to the axis of the TP. The angle Yl is the angle between TP1 and TP2. It is determined by measuring the time difference, At, between a peak pressure reading and a sun pulse. The angle yl is then given by: At rl = 360~ x. At turmble period The angle Y2 is the half angle of the cone, about the z axis, of all possible angular momentum vectors. This is determined by the following analysis: 14

y.iV -TP -X I TP2 (82=9o) I 0I / Half Cone Angle/ /:a I I\/ Sun Fig. 6. Coordinates for TP orientation, velocity vector reference. 15

Let t = time n = number of 1/2 tumble periods = roll rate y7 = cone half angle of L from z axis Q= sun sensor roll position Then, assuming that the roll rate is less than the tumble rate: 272 = Qn+l - ~n + ({(tn+l-tn) Once one knowns ~n and where the sun sensor has rolled to in half a tumble period, Qn+l can only be the received output for one plane of tumble with respect to the sun vector. /2 is the half angle of the cone of possible angular momentum vectors about z, 2.7 ORIENTATION EQUATIONS Referring to Fig. 6 which shows the vectors to be determined, the equations to be solved are the following: Assuming the TP is tumbling in a plane, we get: TP1' =L TP2 - = 0. (2.71L) From our previous definiton of 71: TP1 TP2 = cos 1l 5 (2.7.2) Since the TP1 vector is tangent to the cone of minimum angle of attack, we can say: L * T x = 0. (2-735) The minimum angle of attack, a, is then given by: 16

TP! - V lvi cos a (2.7.4a) lvl or L'.. =V sin a. (2,7.4b) Assuming all vectors, except V, are unit vectors. Using typical spherical coordinates, Q measured from the z axis and, the angle in the x-y plane, measured counterclockwise from the x axis, we can solve the above four equations for the unknown quantity (L' the 0 position of L. QL is by definition equal to y72 The solutions are: 2 2.1/2 A 4-cos i cos( l-22) = I^ +- | (2.7.5) A +1 where A = cos Y7 cos 72 cos 7Yl sin @ cos 7l (2.7.6) cos(O.1-02) i2^ L = - T (2.7-7) 2 B / NrC ~ nB2 - sin(~ - ) = BC ~ 1B-C+1 (2.7.8a) for =2-~L= - 2 L ^2 sin(Ul-4v) = -BC -~ B2 C2+ (2.78b) B'+1 for 02-L = - 2 where 17

2 2 sin (U-i2)+cot 72 sin(- ) = cot2C ~ 2) _ cot 72 cot OV sin ( l-h2) The above equations yield eight solutions for ~L' If this analysis is carried out at two or more times during the flight, with data input from the sun sensor, only one of the eight solutions will yield the same sL each time. This then is the correct rL, and the angular momentum vector is known. 2.8 ORIENTATION ANALYSIS (Earth Normal Reference) In this method, the output from an earth sensing instrument is used to determine the time when the TP is closest to the local earth normal vector, T. The equations for this case are similar to those of the previous method (see Fig. 7). Let TP = position of TP when a sun pulse is recorded TPE = position of TP when closest to N (earth sensor data) N = Local normal to earth's surface L = angular momentum vector 7 = angle between TP2 and TP3 (minimum) (measured)'3 QL= half angle of cone of possible angular momentum vectors (measured) TP2 - = TP3 L = 0 (2.8.1) TP2 TP3 = cos 3 (2.8.2). TP3 = 0 (2.8.3) LV = sina (2.8.4) 18

y T3 N 2 / IL z -/ Sun Fig. 7. Coordinates for TP orinetation, earth normal reference. 19

The solutions are: 2 2 1/2 A' + cos 73 / cos(3-^) =2 12 (2.8.5) A? + 1 where A' =cos Y3 cosQL cos y3 COS9L sinQ3 = c 73 (2.8.6) cos(y - y) 3 2 ~2 - T = ~ /2 (2.8.7) where, for 2 - T= /2 BtC + J C2 + 1 sin(l3D - VN) = 2- (2.8.8) B' + 1 and for ~2 - nL = - /2 sin(3 - =) -BtC' ~ 2 -B 2 _ C (2.8.9) B' + 1 where 2 2 B,= sin (3 - 2) + cot QL sin(0 - i2)cos(5 -g2) cotQLcotQN C = sin (3 - 5 ) As for the previous case (Sec. 2*7), the equations yield eight solutions for rL' An analysis at several points along the trajectory eliminates the ambiguity. 20

2.9 DIRECT METHOD OF OBTAINING a Another method of finding a directly is discussed below. This method gives a more understandable physical picture of the TP's motion and yields ac directly for each time sun sensor data are available. Figure 8 shows the plane of tumble (shaded plane) and its relationship to the measured quantities, y1 and Y2. The cone half angle AD is equal to (2/2)-72. It is the cone to which all possible tumble planes must be tangent. EB is the angle 71, the angle measured between a sun-pulse and a peak-pressure measurement (at B). The circle about the sun vector of radius AB describes the locus of all possible positions of the TP when a peak-pressure measurement was received. The criterion for a solution is that the tumble plane must be tangent to the cone AD and must be tangent to a cone about the velocity vector (cone half angle BC) at a point on the cricle of radius AB (Q1 of our previous analysis). The angle BC is by definition the minimum angle of attack a, to be solved for. The problem then is to solve the spherical triangles for the angle BC = c in terms of the known quantities AB = aQ, BD = (g/2)-vl, AC = O@ AD = (Er/2)-72 (see Fig. 9). cos AC = cos BC cos AB + sin BC sin AB cos(90~-i) or cos AC = cos BC cos AB - sin BC sin AB sin i (2o9 also cos AB = cos AD cos BD + sin AD sin BD cos 90~ or cos A.B = cos AD cos BD-. (2.9.2) Now, from the sine law: sin AB sin AD sin 2 sin or sin A = sin A.D (2.9.5) sin ( 21

\ 1 X I/ 01 B Fig. 8. Coordinates for direct method of angle of attack analysis. 22

A D B Fig. 9. Spherical triangles of Fig. 8. 23

Substituting Eqs. (2.9.2) and (2.9.3) into Eq. (2.9.1), we get: cos AC = cos BC cos AD cos iD - sin BC sin AD. (2.9.4) Rearranging cos AC - cos BC cos AD cos BD =- s/l-cos BC sin AD. Squaring both sides 2 — 2 — 2 2 — cos AC + cos BC cos AD cos BD - 2 cos AC cos BC cos AD cos BD 2- 2- 2= sin AD - cos BC sin AD Rearranging 2- 2- 2- 2 —cos BC[cos AD cos BD + sin AD] - 2 cos BC(cos AC cos AD cos ED) 2 — 2+ (cos AC-sin AD) = 0 Let = cos AD cos BD + sin D = cos( - y) cos - - + sin2( - Y2) m = cos AC cos AD cos BD = cos QV cos (- ) cos ) 2- 2- 2 n = cos AC - sin AD = cos Q - sin ( - 2- - cos BC - 2m cos BC + n = 0 and m + jm2-en cos BC =.. 2. (2.9.5) This analysis was carried out with the assumption that the position of the TP at minimum angle of attack was describable in the top hemisphere of the sun coordinate system shown in Fig. 8. If the TP is positioned on the bottom hemisphere [AB > (</2)] then the angle BD is [ (/2)+Yl)] 24

For AB < 7, <' 2 2 2 2 2 = sin 72 sin yl + cos 72 m = cos QV sin 72 sin yl 2 2 n = cos QV - co 72 and for AB > - 72 < 2 2 2 2 2 ~ = sin 72 sin 1 + cos 72 m = - cos ~V sin 72 sin y1 2 2 n = cos Q- cos 7 V C0572 where m ~ |m2 -n cos a= -.M...- (2.9.6) ~ 2.10 COMPUTER ANALYSIS The orientation analysis described in Sections 2.5 and 2.6 have been programmed for digital computer solutions. The program instructs the computer to produce a theoretical trajectory, matched to radar data, with velocity components in both earth fixed and sun fixed coordinates. This trajectory is then used to obtain the eight solutions for each of two analysis points. The two groups of solutions are compared, and the correct angular momentum. vector is selected and used to determine the minimum angle of attack versus flight time. Other parameters obtained from the results of the solution are shown in Fig. 10. 2.11 ATMOSPHERIC WIND The use of the two orientation analyses described in Sections 2.4 through 2.6 permits a determination of the component of horizontal atmospheric wind in the plane of tumble of the TP. Since the velocity vector reference method determines the motion and, hence, angle of attack of the TP, with respect to the atmosphere and the sun, and the earth reference method determines the 25

MOMENTUM VECTOR CETERMINATION TPE-L-141.CC --- - GA 1 Al 36.400 THETA1 100.OCC -- - LX LY LZ PHIL THETAL PHI1-PHI2 PHI1-PHIV -.73152 -.68039.04408 133.456 37.300 30.390 37.347.26239 -.93260.2478' -107.325 37.3CC -30.390 37.347 " IDnc34 — 99357.05253 -132.194 37.3CC -30.390 12.478 -.81658 -.52640.23685 108.587 37.300 30.390 12.478.79540.39315 -.46127 -96.368 142.70C 30.390 -12.478.12662.98888.07799 22.851 142.700 -30.390 -12.478.37488.91945.11866 -2.C19 142.7CC -30.390- -37.347L —.67206 ___.30624 -.67420 -121.238 142.700 30.390 -37.347 TIME 335.500 GAtMA1 57.300 THETA1 80.000 LX LY LZ PHIL THETAL PHI1-PHI2 PHII-PHIV 37 5.41T~ —-^< 72^ —-— 1 —.IT'864 -2.071 142.700 51.095 19.536 - -.27627.92099 -.27467 -284.261 142.7CO -51.095 19.536 -.13343.98766 -.08201 -307.966 142.70C -51.095 -4.170.59C18.80448.06701 -25,777 142.7CC 51.095 -4.170 -.51955 -.84914 -.09498 -197.437 37.3CC 51.095 4.170 rou.19248 -.97051.14513 -119.627 37.300 -51.095 4.170.0046" —- -.-9987- -.015 6 2 -143.332 37.300 -51.095 -19.536 -.70023 -.71385.C0999 -221.143 37.300 51.095 -19.536 THE'CORRECT' MCMENTUM VECTCR IS.37488.91945.11866 TIME ALTITUDE RANGE PCSITICN INERTIAL VELCCITY VELOCITY WRT EARTH FIXEC VEL AZIMUTH ELEVATION ALPHA CEGREES FEET METERS FEET METERS FEET DEGREES DEGREES S~F~C"' " —F FEET LATITUDE TOTAL TOTAL TOTAL TOTAL X INERTIAL INERTiAL DEGREES METERS METERS LONGITUDE HORIZONTAL HORIZCNTAL F-ORIZCNTAL HORIZONTAL Y WRT EARTH WRT EARTH RADIANS GEOPCT METER VERTICAL VERTICAL VERTICAL VERTICAL Z COSINE 393 214390 383189 37.08 5372 1638 5046 1538 667 112.32 -66.31 11.850 65346 116766 -74.57 2158 658 1123 342 -761 136.86 -77.14.207 4682 -- -4920 -1500 -4920 -1500 -4944.9777

motion with respect to the earth and sun, any differences between the two methods larger than the expected error must be due to atmospheric motion. Therefore, the quantity determined is an apparent difference in minimum angle of attack, A0o Although true wind velocity determination depends upon ao, the minimum angle of attack, and other data input magnitudes, a theoretical plot of the horizontal wind velocity component in the plane of tumble versus altitude for a typical Nike-Tomahawk trajectory is given in Fig. 11. 27

400'''''' 300 >200 4j4 U 100 A = 5 \~~~~~~~~30 Ac=20 100 200 300 Altitude (Km.) Fig. 11. Horizontal component of wind velocity in the plane of tumble vs. altitude for various Ac. 28

3, DATA REDUCTI ON 3.1 N2 DENSITY VS. ALTITUDE-DATA. ANALYSIS The pressure relationship across the orifice of a pressure gauge mounted within a moving, rotating body in a free-molecular-flow region in a planetary atmosphere, is given by the thermal transpiration equation as modified by drift velocity considerationso From Ref. 3 Pi T/o f(s) (5.11) o where 2 f(s) = e- + rs(l+erfs) s = V cos 3/u u = J2kT/m P = pressure T = temperature V = vehicle velocity = angle between the normal to the pressure-gauge orifice and the velocity vector u = most probable thermal velocity for molecules of mass m i = subscript denoting quantities inside the pressure gauge o = subscript denoting quantities outside the pressure gauge For the TP experiment, the pressure gauge is tumbling in a plane which is at an angle ac (minimum angle) from the velocity vector, Considering the maximum change in pressure during one tumble period: Pimax - Pimin = PT/To Lf(s)-f(-s)] (3.1.2) since 29

f(+s) - f(-s) = s s[erf(s)-erf(-s)] = 2 s. We get: Pi -Pi = 2P10cs \T1/T = AP (5.135) max min i From the ideal gas law: P = pRT 0 0o we get: APi = J2 Rpo TT s Which, for ambient N2 density, is: APi 2 R T. T (.) substituting u = 2kT/m, and rearranging yields: APi ~p APi (see Ref. 4) (5315) o = C ui V cos a In terms of number density: kTiAnN2i ni m = -. nN2o mN2 l-UiVcosc or AnN2i UN2i N2i N2i nN2 = 2 Vics. ^ (3.1_6) n2~ 2 ~7 Vcos6 Figure 12 gives typical values for N2Pi max Pimin' and Po for a typical NikeTomahawk trajectory. 3.1a 02' 0 DENSITY VSo ALTITUDE-DATA ANALYSIS Omagatron 02 measurement-total recombination inside gauge assumed for o ->- n -3o 50

5 10 1 6 P _ —.1 7 10 P ~~E~ ~ALTITUDE (KM.) 101 10 130 170 210 250 290 330 31.

V noTo 0: ni Ti For equilibrium flow at orifice: U02 i f = 020 no f0 (so2) + 1/2 noU0f(So) (51-7) where U = most probable thermal speed Substituting the temperature and mass dependent expression for U02i and dividing the right hand part of the equation - n o2i n2 f(So2) + 1/2 no f(S0) (3.1.8) Ti Mo Ti 0 2 _jr Vcosac An En= --— Vo [n02 + 1/2no ] 02 U 020 0 02i 02. or oAn02 U2 - f (An ) (5.1. 9) no + 1/2no = A c~2i 0i 0o 0 2TV cos Assuming diffusion equilibrium, the hydrostatic equation for each species can be written: dno dT0 mult. by 1/2 kT - + k -- n = /2Mo2 gno dh dh 2 dno2 dT and kT + k n2 -2 gn02 add 52

df(An) dTo kT0 dt + k f(An) = _-2 g(1n n ) (351-10) dh dh 22 02 or ~l/) +n = -k df(An) dT 1/4n + n = + o f(An)] -.g(An) o~ Mo2g dh dh (3.1.11) Using equation for g(AN) and f(AN) (3.l.9) and (3.1.11) we get: no? -2g(An) - f(An) (53*1.12) n0 - 4 [f(An) + g(An)] and no 4[g(An) + f(tn)] (3 n02 2g(An) + f(An) also n g(An) = _.... f(An) n ((5.1.14) n 2 --- + 4 no02 0 -f(An) 0.5 -.900f(An) 1.0 -.833f(An) 2.0 -.750f(An) ~00 -.500of(An) Therefore, the ambient number densities of 02 and 0 are separable and determinable from the 02 density measurement within a gauge open to the atmosphere through a knife edge orifice. The validity of the assumption that total recombination of 0 into O2 occurs within the gauge is questionable. However, measurements of O during any typical flight would yield the data required to determine what fraction has recombined into 02, and similar equations can easily be derived. 33

3.2 AMBIENT NEUTRAL TEMPERATURE VS. ALTITUDE-SCALE HEIGHT METHOD The determination of ambient gas temperature from pressure measurements in a moving and tumbling pressure gauge can be accomplished by two independent methodso One method is the determination of a scale height for the ambient gas; the other uses the "velocity scan" technique 3 which determines the relationship between the vehicle velocity, a known parameter, and the most probable thermal velocity of the ambient particles, a quantity proportional to the square root of the temperature. For the first method, we assume an atmosphere at equilibrium suc.h that the hydrostatic equation holds: dP pg9 (5.2.1) dh Also, we assume the ideal gas law is valid: P - pRT ( 32o2) Differentiating Eq. (352.2) with respect to altitude, we getP. RT dp pR n.2o5) dh dh dh Substituting Eq, (3.2.3) into Eq. (3.2.1); RT - + pg + pR.d ~0 dh dh or RT +p ( R ) = 0~ (3 2o4) dh dh For the TP experiment, the expression for ambient density was derived previously: APi P - ---- (5.2.5) Ui j V cos Now dp 1 AP i Pi dV an APi dh Ui V coss dh V dh dh (.2.6) 54

Substituting Eqso (3.2.5) and (352.6) into Eq. (53,24) and cancelling common terms T+ d= T (.+ daT) ---- (3.2.7) R dh dAPi APi dV du dh V d + tan a APi - dh, V dh dh This expression relates the ambient temperature to the basie pressure measurement and trajectory information. To change the equation to a form more suitable for data reduction, we multiply the numerator and denominator by Vz = dh/dt, T _ g+ dT4 IP_ Vz__.' (3.2.8) R d APi APi dV. du dt V dt + tan a Ai T Equation (3.2.8) allows one to reduce much of the data9 in terms of flight time, from the original telemetry records, eliminating trajectory information requirements until final analysis. Returning to Eqs. (3.2.1) and (35.22), we see that Eq. (3.2.1) can Te expressed as: h P ~ P z pgdh. (5o29) Where P1 is the ambient pressure at altitude hl and P2 is the ambient pressure at altitude h2. From Eq. (352o2), we can express ambient temperature as: T = P/pR Therefore, h 2 pgdh + P2 Tz {3.2.10) p iR P2 can be determined by T2 = P2/P2R Where T2 is obtained from the data using Eq (35.2.8) 55

Equations (3.2.8) and (3.2.10) are both valid for the assumption of Eqs. (3.2.1) and (3.2.2). Both equations have been used for data reduction and excellent agreement in the results is obtained. Another technique for temperature determination which is independent of Eq. (3.2.1) is discussed in the following section 353 AMBIENT TEMPERATURE VS. ALTITUDE-VELOCITY SCAN METHOD The velocity scan method for determining ambient temiperature has been derived previously and is reported in Ref. 3. For this t;he thermal transpiration equation is used. | a- i/To f(s) (35.51) Since the TP is tumbling i-n a plane whose angle from t.he velocity vector is a, the angle of attack, P, for any given pressure reading is: cos c = cos a cos G where Q is the rotation angle in degrees in the plane of turifble. 0 is zero when P = a, and a peak pressure reading is obtained. For any given tumble period, it can be assumed that the ambient temperature is constant, therefore, the ratio of (3.-31) at Pi, to (53.3.) at P2 yields: (PiB1//PBiP) - Isf )'S C2) i: 2) where f(s) _ e + j s(l+erfs) Si = V cos a cos Qi/U0 uo - TkT0/m As is known: lim f(s) = 2 s. S- 00 Therefore, for high S (s > 2), the linearity of f(s) causesG Pi/Pi cos 1/co s I 2 36

However, if the 90~ point (Q = 90~), for examrple, and the peak pressure point (Q = 0~) are chosen for the pressure reading. Pia /Pi9. = f ) o (5.55) From this ratio, an S can be determined. The ambient temperature is then given by: ITITo os ( C3.4) T~ 2k s 2 where V, s, U, m and k are known quantities. It will briefly be noted here that the errors involved in reducing data from Eq. (35.54) above become quite large for high vehicle velocities, since all the temperature information is contained at points on thie pressure curve near Q = 900, where S is small (Fig, 13)o Also, the inherent inaccuracy of a linear amplifier at low outputs, compared to full scale outputs, causes errors in P90o that appear to approach and even exceed 100% especially when background pressure effects are also present. A thorough error analysis of this technique has been initiated and will be reported separately when completed. Therefore, the data presented in a later section was reduced usiig the previous method (scale height). In either case, assuming the gauge ras a linear pressure-current characteristic, a systematic calibration error does not cause an error in the computed ambient temperature, since only ratios of the measured pressures are involved in the expressions used. 3.4 ELECTRON TEMPERATURE AND DENSITY-DATA ANALYSIS The equations for the current collected by a stationary cylindrical probe immersed in a plasma were derived by Mott-Smith, and Langmrir (1926)o An extension of this work to moving cylindrical probes was carried os:t recently by Kanal (1964)o. The thermal velocity of the electrons is very large in comparison with typical spacecraft velocities, thus for electron current calculations the probe can in effect be considered stationary. The Debye length corresponding to typical F-region conditions is of' the order of 1 cm. The dimension of the sheath which surrounds the collector is of the order of the Debye length and since the radius of the collector used in these experiments is only 0.027 cm, a large sheath radius to probe radius ratio results. The retarded and accelerated current equations under these conditions are given by (5341o) and (3.4.2), respectivelyo 57

""' f''' I I 1 I I I I I I I I I' 5.0 4.0 3.0 V = 2000 m/sec. c(j)-x-^ yO~: T= 500 K; S= 3.69 cos > (i) (T: T=700'K; S= 3.12 cosE 1.0 (: T=900 OK; S= 2.75 cose 130 120 110 100 90 80 70 60 50 40 30 20 10 0 -10 130 120 110 100 90 80 70 60 50 40 30 20 10 0 -10 -20 0 (tumble angle: degrees. ) Fig. 13. Temperature effect on Pi vs. Q.

= Qie 1/2 Neq A exp(Vo) (3.4.1) \2~me/ 1/2 I Ia = -(- ) NeA V 1/2 + exp(Vo)erfc(Vo )) \2^eme Lf (3.4.2) where: k = Boltzmann's constant Te = temperature me = mass N = number density q = magnitude of electronic charge A = collector area V = qVpp kTe V = potential difference between the probe and the ambient plasma PP V + VR V = applied voltage ap VR = potential of the Thermosphere Probe with respect to the plasma erfc (x) = complimentary error function x 12 / exp[- 2 ] dp 1/2 o Let Eq. (3.4.1) be rewritten in the following manner: IkTe 2NeAexp T qVRV I = ) NqAexpI -ap exp I- kTe4 \i2me /kTe ke Takingthe natural logarithm of Eqo (35.43) one obtains: 39

[ ~'~e -Vap q R n(I ) i - n. (e Nea. 4 + (4.4) r e +A ---- kT (3~4~4) LK2eI kie kTe The sawtooth voltage applied to the probe Ls made to varyt f-rom about -0.5V to approximately +5-OV at a rate of 4-5 c/s. This rate is sufficiently high to justify the assumption that the anmbient parameters (density and temperature) remain constant during one sweep. Thus, assuming that the equilibrium potential of the TP as -well as the atmospheric parameters remainr constant during a sweep, differentiation of Eq. (354.4) with respect to 7ap yieldso dLil n(lr)] q (5o 5) d[Vap kTe Te Therefore if S n(Ir) is plotted versus Vap a straight line results if the velocity distribution of the electrons is Maxwelliano The slope of this curve directly yields the electron temperature Te. Such a curve obtained from typical. experimental results is shown in Fig~ 14. The retarded current equation [Eqo(5.4L) ] shows an exponeLntial relation between the collected current and the voltage, whereas the equat:ion for accelerated current [Eq. (3.4.2)] shows a strong departure from such behavior. Therefore, ideally the plot of the electron current a a function of the applied voltage on a semilog paper will result in a straight line for positive applied voltages, up to the point where the cylinder reaches the plasma potential. The applied potential corresponding to this bFreakpoint is then equal in magnitude and opposite in sign to the equilibrium reference (TP) potential. Such a curve, constructed from typical experimental points was shown in Fig. 14. The establishment of this breakpoint therefore, yields the equilibrium potential of the TP body. The electron current collected by a probe at the plasma potential, referred to as the random, current, is given by Eq. (3.4,6). kTe \ 1/2 ler - ) NeqA (35.46) Once the equilibrium potential (VIF) is determined by this "breakpoinrt method" the value of the random electron current can easily by found. Using the electron temperatures derived from the slope of retarded characteristics, the ambient electron densities are calc'ul ated from. the random, electron current Equation (304o2) for the accelerated current contains two unnownso the ambient electron density and the reference potentialO The fact that these unknowns and the other parameters in Eqo (534.2) can be considered constant during one sweep, as discussed above, leads to another technique 40

Current (amps) 0' 0, 0, 0 (O, 01 tzJCD C-F CrD 0 - CD \ H _0 DO D I \ (D (o 1 0 0 (C,,,,, |,,, D,,,,.I co 0 (tD) r./ on_ 0'1. I I. 1 a I I I I I II l I I i Ii I I

(Method II) for the determination of the electron density and the TP potential. The method employed is simple~ by noting the collected current for two different applied voltage values (selected to ensure accelerating conditions), two equations with two urknowns are obtained, which then are solved. A computer program has been written to obtain such solutions from the actual volt-ampere characteristics. Six to eight points from each curve are typically read into the computer and substituted into Eqo (35.47) for the number density Ne (354.7) k-) XqA Vo / + exp(Vo) erfc(VoL2 Ner = 09e72qA V 1/ 2 gme LL This equation is obtained from simple rearrangement of Eqo (3.4.2). Since Ne is assumed constant during one sweep, a value for the TP equilibrim potential, VR, is obtained corresponding to each pair of data pointsO In this program a value of VR is first calculated for all the possible combinations of pairs and from these an average value of VR is obtained~ This average value of VR is then used to calculate the ambient electron density from each of the data points with the aid of Eqo (534.7)~ The resulting density values are then also averaged. These averaging processes minimize the reading errors. Another method, which is a variation of the one previously described (No. II), can also be used to obtain electron density and TP potential informationo This approach is again based on solving the accelerated electron current equation for VR and Neo Considering two points on the "accelerated' portion of the volt-ampere curves, the ratio of currents is 2 1/2 Ial 2- -7 (Vol) + exp (Vol) erfc (Vol) (l) Ia2 2 1/2 F (o 2) Ia2 1-p (Vo2) 2 + exp (Vo2) erfc o(V) 2) (5.4.8) where V = — ( 4- n v 12 on kT (Vapn D VD)2 e If the two points under consideration correspond to the maximum applied voltage, Vapm, and that applied voltage which brings the cylinder to the plasma potential, IVRI, respectively, one can write: am F (v ) (5. 49) er 42

Equation (3.4.9) provides a functional relationship between Ier and VR with Vapm as a parameter, thus: Ier = G (VR) (3.4.10) The experimental volt-ampere curve can be expressed as Ik= H(k) (3 4..11) When Vk = -VR, the measured current corresponds to the random current Ier. Thus if Eq. (3.4.10) is plotted on the same scale as the experimental curve, the two curves will intersect at one point only. At this intersection point the applied voltage is equal in magnitude and opposite in sign to V. and the current is the random current Ier from which the electron density can be computed. The electron temperature measurement technique described here has been used successfully for a number of years for both rocket and satellite applications.810 This wealth of experience provides great confidence in the results. The use of the electron current characteristics for ionospheric electron density measurements, as discussed here, is a more recent approach, l12 nevertheless the results so far indicate that if the experimental parameters are properly selected (e.g., current detector sensitivity, voltage sweep rate, etc.) the accuracy of the density and equilibrium potential measurements is in the order of 5-10% 355 DATA PROCESSING The theory and analysis methods described in this report have been used in analyzing data obtained from four successful TP payloads launched from Wallops Island, Virginia. All four used Sparrowbee launch vehicles which attained peak altitudes of approximately 300 km. The resulting geophysical data is presented in Refo 12. A section of a telemetry record, obtained during the flight of NASA 6.06, is shown in Fig. 15. To date, the data have been reduced to engineering terms from records similar to this. The voltage deflections are read with data reducers, such as the Gerber GDDRS-3B. Much of the analysis to reduce these data into atmospheric parameters has been programed for computer solutions. In the near future, a data processing system will be available for use in processing the magnetic tapes of the telemetered data. Such a system in conjunction with a computer, also to be available, will consi drably decrease the time and human effort inolved in obtaining the desired geophysical parameters. 453

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REFERENCES 1. Niemann, H. Bo, and Kennedy, B Co "OOmegatron Mass Spectrometer for Parial Pressure Measurements in the Upper Atmosphere," (to be submitted to Rev. Sci, Inst.), 1965. 2. Carignan, G. R, and Brace, Lo H., "The Dumbbell Electrostatic Ionosphere Probe: Engineering Aspects," Univo of Mich., ORA Report 03599-6-S, Ann Arbor, November 1961 o 35 Schultz, F. V,, Spencer, No. W., and Reifman, A., "Atmospheric Pressure and Temperature Measurement Between the Altitude of 40 and 110 Kilometers," Upper-Air Research Program, Report No. 2, Univ. of Micho Res. Inst. Report, Ann Arbor, July 1948 4. Horowitz, R, and LaGow, Ho Eo, "Upper Air Pressure and Density Measurements from 90 to 220 Kilometers With the Viking 7 Rocket " Journal of Geophysics Research, 62:57-78 (1957). 5. Nagy, A. F., Spence, No Wo, Niemann, H. Bo, and Carignan, Go Ro, "Measurements of Atmospheric Pressure, Temperature, and Density at Very High Altitudes," Univ. of Mich., ORA. Report 02804-7-F, August 1961. 6. Mott-Smith, Ho M. and Langmuir, I. "The Theory of Collectors in Gaseous Discharges," Phys. Revo, 28, 727-763, 1926. 7. Kanal, Mo, "Theory of Current Collection of Moving Cylindrical Probes/ J. Applo Phys., 35, 1697-1703, L964, 8. Spencer, N. W,O Brace, Lo Ho and Carignan, G, R., "Electron Temperature Evidence for Nonthermal E uilibrium in the Ionosphere," J. Geophys. Res., 67, 157-175, 1962. 9. Nagy, Ao F., Brace, Lo H,, Carignan, Go Ro and Kanal M., "Direct Measurements Bearing on the Extent of Thermal Nonequilibrium in the Ionosphere," J. Geophys. Reso, 68, 6401-6412, 1963o 10. Brace, Lo H., Spencer, No W., and Carignan, Go R,, "Ionosphere Electron Temperature Measurements and Their Implications," J. GeophysO Res, 68 5397-5412, 19653 11. Nagy, A. F. and Faruqi, A. Zo, "lonospheric Electron Density and Body Potential Measurements by a Cylindrical Probe," UnivJ of Micho. ORA Report 05671-4-S, September 19614 45

REFERENCES (Concluded) 12. Spencer, N. W., Brace, Lo H., Carignan, G. R., Taeusch, D. R. and Niemann, H. B., "Electron and Molecular Nitrogen Temperature and Density in the Thermosphere," (submitted for publication, Jo Geophyso Res.). 46~

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