THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Electrical Engineering Space Physics Research Laboratory Scientific Report No. RS-1 AN INTRODUCTION TO THE THEORY AND DATA REDUCTION METHOD FOR THE PITOT-STATIC TECHNIQUE OF UPPER ATMOSPHERE MEASUREMENT R. W. Simmons ORA Project 05776 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION CONTRACT NO. NAS5-3335 GODDARD SPACE FLIGHT CENTER GREENBELT, MARYLAND administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR March 1964

PREFACE The Department of Electrical Engineering, Space Physics Research Laboratory of The University of Michigan has been for many years actively engaged in various aspects of atmospheric research by rocket-borne techniques. As a result of this work much has been added to our present day knowledge of the earth's upper atmosphere. Because of this ever increasing knowledge and understanding of the upper atmosphere, more and more emphasis is being placed upon accuracy, basic understanding, and speed of processing of experimental data obtained from atmospheric measurements. Thus, the purpose of this paper shall be twofold. First, it will serve to briefly introduce those persons involved in the data-reduction effort of the research program of this laboratory to the terminology and nature of the upper atmosphere as it is known today. Second, it shall develop in detail the theory, equations, conversions, and transformation required in the dataprocessing effort for this research program. Thus, it is hoped that this paper will bridge the gap between the varying technical backgrounds of those persons involved in this most important work and the technical background required, thereby increasing the speed and accuracy of the data processing of any subsequent flights of rocket-borne upper-atmosphere probes. iii

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS ix LIST OF SYMBOLS xi CHAPTER I. THE UPPER ATMOSPHERE AS IT APPEARS TODAY 1 1.1 Basis of our Knowledge of the Upper Atmosphere 1 1.2 The Regions of the Atmosphere 2 1.3 Some Properties of the Atmosphere 5 1.3.1 Sea level 3 1.3.2 Temperature variation with altitude 1.3.3 Variation of atmospheric pressure and density with altitude 6 1.3.4 Electrical properties of the upper atmosphere 6 1.3.5 Winds in the upper atmosphere 8 1.4 Concluding Remarks 9 References for Chapter I 10 II. A COMPARISON AND BRIEF DESCRIPTION OF UPPER ATMOSPHERE NEUTRAL PARTICLE MEASUREMENTS 11 2.1 Classification of Neutral Upper Atmosphere Measurements 11 2.2 Common Neutral Particle Experiments 11 2.3 Description of Common Neutral Particle Experiment 12 2.3.1 The rocket grenade experiment 12 2.3.2 The sodium release experiment 15 2.3.3 The falling sphere experiment 2.4 Purely Aerodynamic Upper Atmospheric Measurements 18 References for Chapter II 19 III. THEORY OF AERODYNAMIC UPPER ATMOSPHERE NEUTRAL PARTICLE MEASUREMENT 21 3.1 Introduction 21 3.2 The Equation of State and Mean Free Path 22 3.2.1 The Maxwell-Boltzmann velocity distribution law 22 3.2.2 Equation of state 32 5.2.3 Mean free path 3.3 The Hydrostatic Law 37 v

TABLE OF CONTENTS (Continued) CHAPTER Page 3.4 Aerodynamic Flow Regions 40 3.4.1 The continuum region 40 3.4.2 Free molecule flow 45 3.4.3 The transition region 57 3.5 Summary of Theoretical Results 58 References for Chapter III 64 IV. THE PITOT-STATIC PROBE 67 4.1 Introduction 67 4.2 The Pitot-Static Probe Configuration 67 4.3 Instrumentation 69 4.3.1 The radioactive ionization-gauge densatron 69 4.3.2 Supporting rocket-borne equipment 73 References for Chapter IV 75 V. TYPICAL DATA OBTAINED FROM THE SPRL PITOT-STATIC PROBE AND PRELIMINARY PROCESSING 77 5.1 Introduction 77 5.2 The Analog Data 77 5.3 Conversion of the Analog Data to "Gauge Pressure" 80 5.53.1 Densatron calibration curves 80 5.3.2 Thermistor calibration curves 82 5.3.3 Measured pressures and temperatures 82 5.534 Gauge pressure 85 5.4 The Trajectory 87 5.5 Example of Raw Flight Data in Final Form 87 VI. DETERMINATION OF AMBIENT ATMOSPHERIC DENSITY FROM IMPACT GAUGE PRESSURE 89 6.1 Introduction 89 6.2 Ambient Density Determination Below 85 km 89 6.3 Ambient Density Determination Above 90 km 93 6.4 Summary of Density Formulae 96 VII. DETERMINATION OF AMBIENT ATMOSPHERIC PRESSURE 99 7.1 Introduction 99 7.2 Directly Measured Ambient Pressure 99 7.3 Ambient Pressure Determination Through the Use of the Hydrostatic Law 101 7.4 Conclusion 103 References for Chapter VII 105 vi

TABLE OF CONTENTS (Concluded) Page CHAPTER VIII. DETERMINATION OF AMBIENT ATMOSPHERIC TEMPERATURE 107 8.1 Introduction 107 8.2 Ambient Temperature Through the Use of the State Equation 107 8.3 Ambient Temperature Through Use of the Hydrostatic Law in Combination with the Equation of State 108 8.4 Ambient Temperature in the Continuum Flow Region From the Rayleigh-Pitot-Tube-Equation 110 8.5 Ambient Temperature in the Free Molecule Flow Region 111 8.6 Ambient Temperature From Direct Integration of Ambient Density 114 8.7 Conclusion 115 APPENDIX: PHYSICAL CONSTANTS AND CONVERSION FACTORS 117 A.1 Introduction 117 A.2 Physical Constants and Conversion Factors 117 A.2.1 Physical constants and formulae 117 A.2.2 Conversion factors 118 A.2.3 Conversion Formulae 119 A.3 Derivation of the Constants A and B 119 A.4 Table and Graph of K(M) vs. M for Air 122 A.5 Table and Graph of F(S) vs. S 127 A.6 Table of g, g/go, and (g/go)(go/R) vs. Altitude for Air from 0 to 100 km 133 vii

LIST OF ILLUSTRATIONS Table Page I.1 Percentage by Volume of Atmospheric Gases at Sea Level. 4 Figure I.1 Ambient temperature vs. altitude U.S. standard atmosphere, 1962. 7 III.1 Velocity space retangular coordinates. 24 111.2 Velocity space polar coordinates. 29 III.3 The ideal gas law model. 33 111.4 The hydrostatic law model. 39 III.5 The Rayleigh pitot-tube equation model. 43 111.6 Free molecule flow across an orifice. 48 IV.1 The pitot-static probe. 68 IV.2 The densatron gauge. 71 V.1 Real time oscillograph data record. 79 ix

LIST OF SYMBOLS A Avogadro's number 6.0247 x 1023 particles per mole f 2 A a dimensional constant equal to 1435.064 ( ft_ kg ] \sec2m3[mm Hg B a dimensional constant equal to 10.257 ft kg K1/2 ec m3 [mm Hg] Cm most probable thermal particle speed given by 2kT m m C aerodynamic coefficient d mean molecular diameter D drag force 2 F(S) a function of the parameter S given by e-S +S w[ (l+erf S) g acceleration due to gravity h distance H altitude k Boltzmann's constant 1.3804 x 10-23 joules/~K 1 K(M) a function of Mach number given by 21 +1) 2+ 2 1 2y 47M2-27+2 I characteristic length m mass M Mach number MW molecular weight N number of particles xi

LIST OF SYMBOLS (Concluded) P pressure R gas constant per unit mass given as k/m S area element, velocity parameter given by - cos a C ma t time T absolute temperature v particle thermal velocity V velocity a angle of attack:(T) one-half of the atmospheric scale parameter m 2kT 7 ratio of specific heats - = 1.4 for air Cv particle mean free path ~ spherical polar angle spherical azimuth angle p mass density T particle mean free time Subscripts a ambient conditions D drag i impact conditions o sea level conditions s condition directly behind a normal shock x,y,z direction along the coordinate axis cal information obtained directly from calibration data gauge actual conditions in a pressure sensor xii

CHAPTER I THE UPPER ATMOSPHERE AS IT APPEARS TODAY 1.1 BASIS OF OUR KNOWLEDGE OF THE UPPER ATMOSPHERE Although research on the upper atmosphere does antedate the Twentieth Century, current knowledge about the upper air comes primarily from work done during the past 50 years. Within this relatively short period of time the atmosphere has revealed itself as a very complex and stimulating object of study, and calls upon an impressive number and variety of techniques in its investigation. Modern high-altitude research is a major field of endeavor claiming the attention of many well-known scientists throughout the world. One of the most challenging features of the upper atmosphere has been, and is, its inaccessability. This challenge has brought about two lines of attack. First, there have been devised a great many indirect* methods of studying the atmosphere and associated phenomena in terms of observations made at the earth's surface. Second, means have been sought to make the upper regions accessible, such as balloon, rocket, and satellite-borne probes. The latter two approaches, in which we are most interested, have met with a great deal of success in recent years.35 Before the advent of the rocket-borne probe, most upper atmosphere measurements were obtained by earth-bound observation of atmospheric-spectra, meteor trails, sound propagation, and by investigation of aurorae and the *Indirect shall refer to measurements made entirely by earth-bound observation. 1

earth's magnetic field. From the above and various theoretical studies, many inferences have been drawn about the structure, composition, and electrical properties of the high atmosphere. These early indirect attacks upon the problem have long been supplemented by the direct approach of making measurements with balloon-borne instruments, and most recently (within the past 20 years) rocket-borne instruments have extended the direct approach much beyond the balloon ceiling of about 30 km to over 1,000 km. 1.2 THE REGIONS OF THE ATMOSPHERE The lowest portion of the atmosphere is referred to as the troposphere. It is in this region that the ordinary meteorological processes occur (rain, snow, hurricanes, tornadoes, monsoons, etc.). One might say it is the "weather sphere." Starting at the ground, the temperature and pressure decreases with height until a minimum temperature is reached at what is called the tropopause.4'5 The height of the tropopause is, of course, variable and ranges from approximately 6 km above the poles of the earth to approximately 18 km above the equator. In the middle latitudes (Central United States) its height is roughly 12 km. Above the troposphere and the tropopause lies the so-called stratosphere, so-called because in the early part of this century this region was thought to be isothermal (constant temperature) and quiet, and hence stratified (separated into distinct layers). In the higher latitudes (over Northern Canada) there does exist an almost isothermal region of 10 or more km in thickness, but it is found that over the equator there is a very sharp temperature inversion at the tropopause. In recent times, the stratosphere has come to be that 2

portion of the atmosphere between 18 and 50 km with a stratopause occurring at approximately 50 km. Continuing beyond the stratosphere there is the upper atmosphere. In the not too distant past this phrase has been used to denote the region of the atmosphere above 30 km (in general, the maximum attainable altitude of balloon-borne probes). It is most common today, however, to denote a middle atmosphere or mesosphere which extends above the stratopause from 50 to approximately 85 km with a mesopause occurring at about 85 km, and an exosphere or more recently named the thermosphere, at the very top where the most significant heating of the atmosphere occurs. Here the atmospheric density is so low that atmospheric particles rise and fall without collision in elliptic paths under the influence of gravity or following parabolic and hyperbolic trajectories, they will escape from the earth entirely. Thus, the thermosphere (exosphere) is accordingly the fringe region between the atmosphere and interplanetary space. The precise location of the upper edge of the thermosphere is of considerable interest and is at present not well-known by virtue of its very nature, but it is believed to extend well above 400 km~ 1.3 SOME PROPERTIES OF THE ATMOSPHERE 1.3.1 Sea Level At the surface of the earth, the following observations of the atmosphere have been made. The average composition of dry air at sea level is given in Table I.7 5

TABLE I PERCENTAGE BY VOLUME OF ATMOSPHERIC GASES AT SEA LEVEL (Dry Air) Molecular Nitrogen (N2) 78.084% Molecular Oxygen (02) 20.9476% Ozone (03) 0-0.000007% Argon (Ar) 0.934% Carbon Dioxide (C02) 0.0314% Molecular Hydrogen (H2) 0.00005% Neon (Ne) 0.001818% Helium (He) 0.000524% Krypton (Kr) 0.000114% Nitrous Oxide (N20) 0.0005% Xenon (Xe) 0.0000087% Methane (CH4) 0.0002% The actual composition at ground level is observed to vary slightly from the above with both position on the surface of the earth and time, and variable amounts of water vapor are always present. The sea level values of pressure, temperature, and density adopted in Ref. 7 are as follows: PO - 1.01325 x 105 nt/rmh 760 mm Hg po - 1.2250 kg/m3 To - 288.150~K The temperature, pressure, and density at the earth's surface are also variable with position and time but the extremes of variation about a mean value are quite limited on an absolute scale. The extreme variation in ground4

level temperature is of the order of 15% of the mean in the middle latitudes and the deviation of atmospheric pressure and density are limited to a very small percent of the mean. 1.3.2 Temperature Variation With Altitude Starting at sea level, temperature and pressure decrease almost, but not quite adiabatically with height through the troposphere until the temperature minimum of between approximately 200 and 230~K is reached at the tropopause, whose exact altitude and temperature value is variable with time and position over the earth. The stratosphere exhibits a nearly constant temperature over an altitude region which is also variable with time and position. Immediately above the stratosphere the atmospheric gases are heated by solar ultraviolet energy absorbed in the atmospheric ozone which is formed from atmospheric molecular oxygen by photochemical processes at the top of a layer sometimes called the ozonosphere which lies between 20 and 50 km. This heating causes a rise in temperature from the cold stratospheric value to a maximum of about 2700K at an altitude of approximately 50 km (the stratopause). Thereafter, the temperature falls again to a low value of the order of 200~K in the neighborhood of 80 km (mesopause). At still higher altitudes in the thermosphere the temperature rises again until it reaches an isothermal value of as low as 800~K at an altitude of approximately 200 km8 during the quiet sun and as high as 1200-1300~K during the maximum solar activity. The exact nature of this heating process is not well known at present. However, several theoretical and experimental studies of the various physicochemical processes 5

of this part of the atmosphere are being carried out with hopes that a proper explanation of the heating process will be determined. Figure I.1 graphically illustrates the various regions and temperature values of the atmosphere. 1.3.3 Variation of Atmospheric Pressure and Density With Altitude Pressures and densities fall off roughly exponentially with altitude from sea level up to an altitude of approximately 110 km. This fact may by expressed mathematically by P = Poe-2(T)gH (I.1) -2(T)gH (. P = poe2(. (I.2) Equations (I.1) and (1.2) are strictly correct if P(T) is considered constant for a small range in altitude where the atmospheric temperature may be considered constant. An approximate value at sea level of 23(T)go is 1/9.6 km for which Eqs. (I.1) and (I.2) shows that all but approximately 35% of the total mass of the atmosphere lies below 9.6 km. Above 110 km the decrease of pressure and density is still exponential; however, the rates of decrease of these parameters is observed to be much less than at the lower altitudeso A good mnemonic is that up to 110 km both pressure and density are divided by 10 for every 10-mile increase in altitude. Also, if pressure is expressed in millimeters of mercury (mm Hg) and density in grams per cubic meter, the numerical values of the two at a given altitude are about the same. 1.3.4 Electrical Properties of the Upper Atmosphere The atmosphere may also be subdivided according to its electrical proper6

400 380 360 340 320 300 280 - 260 E 240: 220 200 E 180 0 / 160 140 120 100 / Thermosphere 80 \////:/// Mesopause 60 - \Mesophere 40 20 V/ Stratosphere -,,,,',,',.,,,,,,,,,,, Tropopause 0 I r I Iroppsphere I I I I I I I I I 100 200 400 600 800 1000 1200 1400 Ambient Neutral Particle Temperature ~K Fig. I.1. Ambient temperature vs. altitude U.S. standard atmosphere, 1962. 7

ties. From this point of view, the ionosphere is that portion of the atmosphere in which significant numbers of the constituent particles are charged. The total ionization in the ionosphere varies from altitude to altitude and several well-pronounced maxima are found. The first, or lowest altitude, of these layers which is sometimes found in the neighborhood of 60 km is the Dregion. This region is often associated with strong radio fadeouts. Next in altitude are the E- and F-regions which are the most familiar and most important parts of the ionosphere.9 During the day the F-region contains two layers designated by F1 and F2, which at night coalesce. The E-region diminishes at night and during the day may sometimes exhibit two layers, E1 and E2. From ground-based radio propagation studies and ejectable electrostaticprobe experiments the approximate height of the maximum ionization in the Eregion is placed somewhere between 90 and 120 km, that of F1 appears at about 175 km, with that of F2 at approximately 300 km. Sudden intense clouds of ionization often appear within the E-region and are referred to as sporadic E. Their cause is not well-known, but it is believed that meteors may be one cause of the ionization of the sporadic E. The direct measurement of the properties of the ionosphere is relatively young and is the subject of an intense research effort by many laboratories including the Space Physics Research Laboratory. 1.3.5 Winds in the Upper Atmosphere Many observations by various techniques (long-enduring meteor trains, radar-chaff, sodium vapor trails, etc.) indicate the existence of winds in the upper atmosphere. These observations show that adjacent layers in the high 8

atmosphere often have winds of quite different speeds and directions. Vertical components are also observed in some of the winds. These high-altitude winds sometimes exhibit velocities of several hundred kilometers per hour with average values between 100 and 200 km/hr. Radar studies of meteor trails confirm and supplement visual observations and extend the results to much higher altitudes. Rocket-borne techniques for measuring wind are presently being worked out but results are still fragmentary. One notable characteristic of high-altitude winds is their seeming correspondence to and dependence on the atmospheric temperature profile. It is found that at points of temperature inversion wind shears between adjacent layers are usually present and of a very violent nature. 1.4 CONCLUDING REMARKS It is realized that the foregoing descriptive treatment of the atmosphere has been very general and has offered little in the way of explanation as to the cause of the observed phenomenao It is hoped, however, that it has served to stimulate the interest of the reader in the subject and give some basis for further study. To pursue this study, the interested reader is referred to the references at the end of this chapter. 9

References for Chapter I 1. Sicinski, H. S., Spencer, N. W., and Dow, W. G., "Rocket Measurement of Upper Atmosphere Ambient Temperature and Pressure in the 30 to 70 Kilometer Region," Journal of Applied Physics, 25, 2, 161-168, Feb., 1954. 2. Spencer, No W., Brace, L. H., and Carignan, G. R., "Electron Temperature Evidence for Non Thermal Equil. in the Ionosphere," J.G.R., 67, p. 157, 1962. 3. Horvath, J. J., Simmons, R. W., and Brace, L. H., "Theory and Implementation of the Pitot-Static Technique for Upper Atmospheric Measurements," Office of Research Administration, The University of Michigan, Scientific Report No. NS1, March 1962. 4. Newell, H. E., High Altitude Rocket Research, Academic Press, New York, 19535 5. Ratner, B., Temperature, Pressure, and Relative Humidity Over the United States and Alaska, U.S. Department of Commerce, Washington, D.C., 1945. 6. Humphreys, W. J., Composition of the Atmosphere. International Critical Tables, Vol. I, McGraw-Hill Book Company, New York, 1926. 7. Dubin, M., Sessenwine, N., and Wexler, H., U.S. Standard Atmosphere, 1962, U.S. Government Printing Office, Washington, D.C., Dec., 1962. 8. Results of NASA 6.06 Univ. of Mich., and NASA GSFC. 9. Mitra, S. K., The Upper Atmosphere, 2nd ed. The Asiatic Society, Calcutta, 1952. 10. Malone, T. F., ed., Compendium of Meteorology. American Meteorological Society, Boston, 1951. 11. Murgatroyd, R. J., "Winds and Temperatures Between 20 Km and 100 Km —A Review," Quarterly Journal of the Royal Meteorological Society, 83, No. 358, pp. 417-454, Oct., 1957. 10

CHAPTER II A COMPARISON AND BRIEF DESCRIPTION OF UPPER ATMOSPHERE NEUTTRAL PARTICLE MEASUREMENTS 2.1 CLASSIFICATION OF NEUTRAL UPPER ATMOSPHERE MEASUREMENTS In the investigation of the neutral upper atmosphere the parameters of interest to the experimenter are ambient pressure, temperature, density, and mass motion or wind in the atmosphere. Thus, the study of the upper atmosphere by rocket-borne techniques has brought about many different approaches to the design and implementation of neutral particle measurements. Among the more common theoretical approaches taken in the problem of experiment design are those involving the theory of sound propagation, the theory of gaseous diffusion and dispersion, and basic aerodynamic theory. Thus, experiments which yield measurements of the parameters of the neutral upper atmosphere may be classified according to the theoretical concepts upon which they are based. As will be seen, these experiments involve straightforward applications of well-known theoretical concepts or simple extensions of these wellknown ideas and will fall into the categories mentioned above. 2.2 COMMON NEUTRAL PARTICLE EXPERTMENTS As indicated, several rocket-borne techniques have proved very useful in the direct measurement of the upper atmosphere above balloon altitudes. Among the more successful of these is the rocket-grenade experiment which is based on the theory of sound propagation in a gaseous media such as the atmosphere 11

and is well suited to determining an average temperature profile in the upper atmosphere. A second technique, which has been highly successful and is well adapted to the measurement of winds, is the sodium release experiment which is based on gaseous diffusion and dispersion of a sodium vapor trail left by an ascending rocket under the influence of atmospheric wind. A third experiment which is based on aerodynamic theory involves the measurement of the atmospheric drag on a sphere which is ejected from an ascending rocket and allowed to fall freely back to earth. The measurement of the drag encountered by the falling sphere leads to the determination of atmospheric density. A fourth technique, the pitot-static probe, also based on aerodynamic theory, uses pressure measurements made at suitable points on a rocket surface to obtain the ambient pressure, temperature, and density in the upper stratosphere, mesosphere, and lower thermosphere. A fifth experimental approach based on aerodynamic theory and particularly well suited to determination of ambient pressure, temperature, and density at very high altitudes is the thermosphere probe. This device makes use of a cyclic pressure measurement at a suitable point on a tumbling cylindrical probe ejected from an ascending rocket vehicle. 2.3 DESCRIPTION OF COMMON NEUTRAL PARTICLE EXPERIMENT 2.3.1 The Rocket Grenade Experiment In this experiment, which is based on the speed of propagation of sound 12

through the atmosphere, it is assumed that the ratio of specific heat at constant pressure to that at constant volume, and the average molecular mass or composition of the atmosphere remains essentially unchanged with altitude up to approximately 90 km. Under these conditions it is well-known that the speed of a sound wave traveling through the atmosphere is a function only of 2 the atmospheric temperature. This functional relationship forms the theoretical basis of the rocket-grenade experiment which leads to a determination of an average upper atmosphere temperature profile. The implementation of this experiment requires that grenades be ejected from a flying rocket and exploded at high altitude. Both the times and locations in space of the explosions are accurately measured. The time of arrival of the sound from each grenade at the earth's surface is also measured accurately at each of several listening stations arranged on the ground directly below the rocket. In the ideal case, all grenades would be exploded directly above a listening station, the sound traveling vertically downward to the recording point or listening station. Thus, if A is a listening station on the ground, B a point of explosion vertically above A, and C the position of a second explosion vertically above B, then the sound path from C to A would overlap that of B to A along BA. The transit time of the sound wave from the second explosion minus that of the wave from the first, would then give the transit time from B to C. This transit time could then be used along with the known distance between B and C to calculate the average velocity of sound on the path BC, from which, through the theoretical relation between speed of sound and 15

temperature, and average temperature between B and C can be derived. The practical implementation of the experiment, however, departs from the above idealized situation. The explosion points may be displaced by several degrees away from the vertical over the listening stations. In this real case, the sound wave undergoes continuous refraction (turning) on its way downward toward the recording point, and accordingly follows a curved rather than straight path from the point of explosion to the listening point. In such a case, the various sound paths do not overlap thus making the use of other more limiting assumptions necessary in the derivation of the average temperature profile from this technique. There is, in addition to the above practical limitation, still another effect which must be considered. It is known that appreciable atmospheric winds exist at various altitudes along the sound path. Wind components along the path change the apparent velocity of sound, whereas components of wind perpendicular to the path alter its curvature. Thus, if not corrected for, winds can introduce errors into the determination of air temperatures. Since winds have a directional effect, their influence on the sound path can be separated from that of temperature by measuring not only the times but also angles of arrival of the sound at the various listening stations. With such data and balloon determination of wind and temperature to as high an altitude as possible, the still higher altitude winds can be estimated and taken into account as a correction factor in the determination of temperature by the rocket grenade technique. This experiment has been flown many times by various experimenters with 14

great success in determining average atmospheric temperature profiles to an altitude of approximately 90 km where it becomes very difficult to receive the sound from the grenade burst because of atmospheric attenuation of the sound wave. Also, the experiment has, with some degree of success, given an estimate of atmospheric winds in the upper regions to approximately 90 km. 2.3.2 The Sodium Release Experiment As mentioned earlier, this experiment is used in determining upper atmosphere wind profiles. The implementation of this technique requires that a rocket nose cone be equipped with a canister containing sodium metal and a remotely controlled release valve. As the ascending rocket reaches the altitude at which measurement is to commence, the valve in the canister is opened, thus allowing the sodium to escape into the rare atmosphere where the sodium vaporizes producing a continuous trail of sodium vapor behind the rocket vehicle. This trail is then very accurately photographed from the ground by several precision motion picture cameras from several different locations. Thus, by comparison of the various films a triangulation scheme is used to determine the rate and direction of the dispersion of the sodium vapor trail which yields a direct determination of the upper atmosphere wind which caused the dispersion. A great number of these experiments have been used and have added much to our present-day knowledge olf the upper atmosphere winds to altitudes well in excess of 200 km. This technique, because of its dependence upon ground based photography 1.5

for data acquisition is severely limited for a number of reasons. First, a very accurately surveyed launch area is required for the triangulation procedure used in data analysis. Second, in order to assure data acquisition, very severe limits are placed upon allowable cloud cover. Finally, to assure adequate and proper illumination of the vapor trail, launchings must be scheduled for either sunset or sunrise. 2.3.3 The Falling Sphere Experiment The theoretical basis for this experiment is found in basic aerodynamics; for it is well known that the drag force encountered by a body moving through a fluid media is a direct function of the fluid density, the square of the relative velocity, and the shape of the body. Mathematically this expression would be D = 1/2 p V2 CDS (I.1) where D is the drag force p is the fluid density V is the relative velocity CD is a dimensionless drag coefficient (a function primarily of the body shape) S is the cross-sectional area of the body. In addition to this, Newton's second law of motion states that the sum of the forces acting on a body is equal to the product of the mass of the body and its net acceleration, or: 16

D - mg = m dV (II.2) dt where D is the drag force m is the mass of the body g is the acceleration due to gravity dV is the rate of change of velocity with respect to time or acceleration. dt Now combining Eqs. (II.1) and (II.2) and solving 2m fi+ g P V (II3) VICDS Thus, Eq. (11.3) gives a precise mathematical statement of atmospheric density in terms of known quantities m, g, CD, and S, and quantities to be measured, V, and dV/dt. The implementation of the experiment is carried out by ejecting a spherical body, whose drag coefficients, cross-sectional area, and mass is known, from an ascending rocket and allowed to fall freely back to earth. During the period of free fall the velocity altitude and acceleration of the sphere are carefully determined. The velocity and altitude are determined by groundbased radar tracking and the acceleration is determined by an on-board acceler3meter in the case of the active falling sphere, or by differentiation of the velocity time profile from radar in the case of the passive falling sphere. Through the use of Eq (II.5) and the velocity-acceleration profile, the atmospheric density as a function of altitude may be deduced. It Is easily seen that the falling sphere experiment is very modest in IT7

its requirement of ground support equipment and thus highly portable. This experiment has been used extensively with good results and is presently limited to a maximum altitude of approximately 100 km where, owing to the insensitivity of the device to extremely small drag forces, and inadequate understanding of the very low density drag coefficients, the derived densities are apparently meaningless. 2.4 PURELY AERODYNAMIC UPPER ATMOSPHERIC MEASUREMENTS As with the development of any physical science the need for ever increasing range of measurement and basic accuracy imposes itself on those involved in the science. The pitot-static probe and the thermosphere probe have been developed,3'4 to this end, through the pioneering efforts of the Naval Research Laboratories, NASA, and The University of Michigan. The pitot-static probe and thermosphere probe techniques are based upon a purely aerodynamic approach to the measurement of pressure, temperature, and density in the upper atmosphere, and is the subject of subsequent chapters of this report. In these following chapters a detailed development of the theory, implementation, and subsequent data analysis for the pitotstatic probe will be given. 18

References for Chapter II 1. Newell, H. E., High Altitude Rocket Research, Academic Press, New York, 1953. 2. Liepmann, H. W., and Puckett, A. E., Aerodynamics of a Compressible Fluid, John Wiley and Sons, Inc., New York, 1947. 3. Horvath, J. J., Simmons, R. W., and Brace, L. H., "Theory and Implementation of the Pitot-Static Technique for Upper Atmospheric Measurements," Scientific Report No. NS1, ORA, The University of Michigan, March, 1962. 4. Ainsworth, J. E., Fox, D. F., and LaGow, H. E., "Upper Atmosphere Structure Measurement Made With the Pitot-Static Tube," J.G.R., 66(10), 3191-3211, 1961. 19

CHAPTER III THEORY OF AERODYNAMIC UPPER ATMOSPHERE NEUTRAL PARTICLE MEASUREMENT 3.1 INTRODUCTION In this chapter the theoretical basis for the interpretation of aerodynamic pressure measurements made at suitable points on a rocket-borne probe surface in terms of the ambient properties of the upper atmosphere will be discussed and the governing equations developed. These equations will be developed in detail starting with well-known aerodynamic and gas dynamical concepts, with this development leading to general equations which are readily specialized to the interpretation of a measured quantity in terms of one or more of the desired properties of the atmosphere. In light of the above, certain models of the gaseous media which is known as the upper atmosphere must be presented, and a mathematical analysis of this model made. The results of such analysis leads then to the desired concepts and governing equations. To facilitate the readers' understanding of the next few sections, the following definitions of important terms are presented. 1. Density.-The total mass of the molecules in a unit volume is defined as the density of the gas. Mass is defined as weight divided by the local acc:eleration of gravity. 2. Pressure. —When molecules strike a surface, they rebound; and, by Newton's law, the surface experiences a force equal and opposite to the time 21

rate of change of momentum of the rebounding molecules. Pressure is defined as the force per unit area exerted on a surface immersed in the fluid. Because the molecular bombardment of the surface is considered perfectly elastic we have a first important result: fluid pressure acts normal to a surface. As will be shown in a later section a second important result concerning fluid pressure is: the pressure is proportional to the kinetic energy of the random motion per unit volume. 5. Temperature. -According to the kinetic theory of gases, the absolute temperature is defined to be proportional to the mean kinetic energy of the molecules. It will also be shown to be interpreted in terms of the Equation of State for an ideal gas as P = pRT where P and p are the pressure and density, R is the gas constant per unit mass, and T is the absolute temperature. Thus, for systems in which the density remains constant, such as the addition of heat, that increase in kinetic energy of random motion will increase the temperature and pressure by proportional amountso 532 THE EQUATION OF STATE AND MEAN FREE PATH 3.2.1 The Maxwell-Boltzmann Velocity Distribution Law As has been shown the earth's atmosphere consists of a gaseous mixture of several constituents and exhibits certain measurable properties such as pressure, temperature, density, etc. In order to further understand these properties, a theoretical model has been postulated and several of these 22

known properties have been linked together on the basis of this theoretical model. In order to set down a concrete theory for aerodynamic atmospheric measurements, these concepts must be called upon. The first of these shall be the concept of a molecular velocity distribution function. As a good approximation to the actual gaseous atmosphere let us consider a finite volume of an ideal gas. A finite volume of an ideal gas consists of a huge number of individual molecules of negligible size. These molecules are in random motion and collide with each other and the walls of the containing vessel all the time and at a very high rate. In order to represent mathematically the behavior of this gas we must adopt a molecular point of view where the individual molecule is the important physical quantity. A method of representing this molecular motion is to use a velocity phase space rather than the physical space and use a velocity distribution function which gives the density of the gas molecular velocities through the spectrum of all possible velocities. Consider the velocity space of Fig. III.la where 1v1 = v+vy+vz If at a given time the velocities of all of the molecules could be found, then the behavior of the gas would be known. This leads to the determination of a velocity distribution function f(vx,vy,vz) which gives the density of the molecular velocities through the spectrum. Again let us consider the above velocity space with several velocities of individual molecules shown, as in Fig. III.lb, and further we see that 23

vz \ Velocity vectorof a particle v located at 0 at the instant in view whose components are,/ Vyx V z (a) dv, / I (b). V3 0/ /, / / / Fig. III.1. Velocity space retangular coordinates. 24 dr" / -.Vi

dv = dvdvydvz Now f(vxvyvz)dvxdvydvz equals the fraction of the number of molecules per unit volume whose velocity lies in the range v and v + dv or vxyVx + dvX vy, Vy + dvy v,v + dv Further, if we let n be the number density of the molecules or the number of molecules per unit volume then nf(vx,vyvz)dvxdv ydvz equals the number of molecules per unit volume whose velocity lies in the range v and v+dv. It is easily seen that f(v), the distribution function, represents the probability of the velocity of a molecule lying between v and v+dv. 1,2 To determine f(v) we will follow the approach of Maxwell, which employs the following postulate: f(v) f(vx,vy,vz) = fi(vx)fl(vy)fl(vz) (III.1) where fi(vx) represents the distribution function of the x component of molecular velocity. Physically this postulate implies; (1) the velocity components, vx, vy, and vz, are statistically independent; and (2) the distribution function, f(v), is isotropic (spherically symmetric). Working now with Eq. (III.1) after taking the logarithm of both sides we 25

see in f(v) = in fl(vx) + in fl(vy) + in fi(vz) Taking the derivative of the above equation with respect to vx where v2 = V + Vy + V gives f?(v) av fl(vx) f(v) 6vx fl(vx) but 6v avx v thus! f'(v) _ 1 f(vx) v f(v) vx fi(vx) Introducing the notation 1 f'(v) m(v) = f i1 (v) x) vx fi(vx) we see that O(v) = - 1(v) (EIII 2) Now taking the derivative of Eq. (III.2) with respect to vy or Vz leads to 26

&'(v) = o Integrating ~(v) = -2 where 5 is a constant. From Eqo (III.2) Bi(vx) = 1 f( = -2. (III.3) Vx f~(vx) Integrating Eq. (III.3) in fl(vx) = - Pvx where a and a are constants. Finally fl(Vx) = e = e e or fl(vx) = A e-v (III.4) where A = e Now from Eq. (IIIol) the form of the distribution function f(v) becomes 3 _P( v2+v 2+v 2) f(v) = A e Vx y (III.5) To normalize Eq. (III.5) we must determine A = e. We know that 27

nf(v)dv = n A3e X dvdvxdv = dn where dn is the fraction of the number of molecules per unit volume whose velocity lies between v and v+dv. Now integrating over the entire range of values of vx, vy, and vz, 00 0~f0 p(V2X +V n (v?+v2+V2) f ffnf(v)dv = fff/ n e d dv ydvd = n -00 -00 we find that A 1/2 Thus, f(v) = e)3/ 2 (TII.6) Equation (III.6) represents the basic form of the Maxwellian velocity distribution function, but because the use of this function often involves rather complicated integrals, it is more convenient to express it in a slightly different form which simplifies the usual integration problem. To facilitate this change we shall define the spherical polar coordinate system shown in Fig. III.2. The volume element in this velocity space is then given by dv = v2 sin Q dQ d~ dv and f(v)dv () 3/ e v2 sin Q dQ dd dv 28

Vz /V sin 9)d> vy Fig. III.2. Velocity space polar coordinates. 29

define -'x 2Jt2 Tr3/2 R~V2 F(v)dv = f sin 2 d f d / e- vdv o o carrying out the integration f(v)dv = 4T ()3/2ae-V2v2dv (I 7) Equation (III.7) now represents the fraction of the molecules per unit volume whose speed lies between v and v+dv. Equations (III06) or (IIo7) may now be used to calculate some average quantities of the gas we are dealing with such, as the mean kinetic energy of the molecules: 1/2 m = 1/2 m2v which is the mathematical expression of mean kinetic energy, where m is the molecular masso Thus, we see that the problem reduces to that of obtaining an expression for 2 (the mean square velocity) in terms of the distribution functions. Recalling that --- 00 0 -s = j vaF(V)dv which upon substitution of Eqo (IIIo7) becomes va = fo vv 4) / e) dv Integrating we obtain s = - i3 25 Thus, the mean kinetic energy of a molecule is given by 30

M1i' mv^ ^ m ^. (III.8) 2 2 2 At this time it is convenient to introduce the concept of absolute temperature and relate this quantity to the mean kinetic energy. This relationship stated mathematically is: 1 2 = 3 kT 2 2 where m is the molecular mass k is Boltzmann's constant or, the gas constant per molecule T is the absolute temperature. This theoretical relationship along with Eq. (III.8) leads to the determination of the constant P. 1 mv2 = Im 3 = 3 kT 2 2 2 2 Thus, 2E kT III.9) Substituting Eq. (III.9) into Eqs. (III.6) and (III.7) we obtain the Maxwell-Boltzmann velocity and speed distribution function in terms of the absolute temperature, T. mv2 f(v) = (m )3/2 kT (III.10) \25TkT/ and 31

my2 F(v) = 4 (2k)3e 2kT2 (III.11) These equations represent the most frequently encountered forms of the Maxwellian distribution law, and have been verified experimentally to very accurately represent the actual velocity distribution for a real gas. 3.2o2 Equation of State The results of the previous derivation may now be utilized to develop an equation of state for an ideal gas. To facilitate this development let the physical modelillustrated by Figo III.3, be considered. To calculate the pressure exerted on the walls of the container we must calculate the force applied to the wall through momentum exchange from the colliding molecules. That is, an expression must be obtained for the momentum transfer to area, S, per unit time. Now, the condition on the molecule at time, t = o, to insure collision with the wall at S in time increment, 5t, is that it have velocity component, vx, such that h < vx 6t or another way of saying this is that at time, t = o, the molecule with velocity component, vx, must be with a volume hS < vx t S Further, the number of molecules, Nvx, in the volume V, of the container having a component of velocity, vx, between vx and vx+dvx is given by 32

z / I / i~Z~,S Vx-x I/ P, N, T Molecule of mass m at time t=O with velocity component Vx which will collide with area S on the positive x wall y Fig. III.35 The ideal gas law model. 33

Nvx = nfl(vx)dvx And the number of molecules hitting surface, S, in time, St, Nvx, S is given by NvxS = vx 5t S nfl(vx)dvx The momentum transferred to S by the individual molecules specified by NVx,S is dMxS = 2m vx vx S 5t nfl(vx)dvx The total momentum transferred to S in 6t is found by integrating the above over the positive range of values of vx. 00 MvS = f 2mvx S St nfl(vx)dvx 0 MvxS = 2nm S St / vx fl(fx)dvx but 00 1 -- Vx f(vx)dv = 2 Therefore, 1 2 MxS = 2n S 5t 2 mvx Recalling that pressure is given by force per unit area and force is given by total exchange of momentum per unit time, we see that the pressure, P, exerted on the walls of the container can be expressed by 34

Mv S P =Mx S 5t or 1 -2 P = 2n - mvx From the previous expression of the absolute temperature and the theory of equipartition of energy we also have that if 1 3 mv2 = kT 2 2 then 2m = 2kT Substituting this in the above expression for P P = nkT Now if we define nm = p the mass density and k- R the gas constant per unit mass. We have as the final form of the equation of state for an ideal gas: P = pRT (III.12) 55

where P is the pressure (force per unit area) p is the mass density of the gas R is the constant per unit mass T is the absolute temperature. Thus, we have a very simple equation which links the three properties of a gas to each other. Experiments with real gases show that this equation accurately predicts the behavior of the gas to within a very minute percent error over a large range of conditions of temperature, pressure, and density. 5.2.3 Mean Free Path Up to now, it has been apparent that molecular collisions must take place, but the details of such collisions have been neglected. Let us now consider the molecular model in which the following assumptions are made: (1) all molecules are spherically symmetric and undeformable; and (2) the force of interaction is in the direction of the line of centers of the two molecules, and its magnitude depends on the distance between the two (i.e., a central force system). Under the conditions of the above assumptions and following the further development of the Kinetic Theory of Gases,1'2 one is lead to the following results: T 1 (III.13) 2 n(ird )v

where T is the mean free time between intermolecular collisions in an ideal gas consisting of spherical elastic molecules n is the number density of the gas d is the mean molecular diameter 18kT v is the mean velocity of the molecules given by,\|jtm and v 1 4 n(td2)v / 2 ntd or -- 2 m = — (III.14 ) k= 2 pjd2 where \ is the mean free path length between intermolecular collisions in an ideal gas consisting of spherical elastic molecules p is the mass density of the gas m is the molecular mass d is the mean molecular diameter v is mean molecular velocity. The above results are obtained under the influence of some rather restrictive assumptions, but are found to be very good first-order approximations to the phenomena observed in real gases. 3.3 THE HYDROSTATIC LAW Considering the atmosphere surrounding the earth and the effects of the 37

gravity field of the earth on the atmospheric gases, a very important physical result is obtained. Namely, that known as the hydrostatic law. Now, except for mass motions, such as winds and convection, the atmospheric pressure, as defined previously, is in equilibrium with the forces of gravity. That is, in an atmosphere at rest the pressure at a specific point is equal to the weight of the gas:.in a vertical column of unit cross-sectional area above that point. Thus, the difference between two altitude levels is due to the weight of the air contained between those levels. To mathematically show the above, consider Fig. III.4 in which a unit cross-sectional cylinder of atmospheric gas between the geometric altitudes H and H+dH above the surface of the earth where the acceleration due to gravity, g, is shown. The weight of gas contained in the incremental volume 1 dH between H1 and H2 is 1 pg dH, acting at the center of the elementary volume. Equating these forces, which is required by equilibrium, we obtain P = (P+ dH + pg dH Cancelling common terms and rearranging -p - p and since we will assume P = P(H) then ~P ^ dP:H dH 38

aP P + dH 3H Altitude H + dH 9 dH egdH Altitude H p P Fig. III.4. The hydrostatic law model.

and knowing that g = g(H) thus, we have P - pg(H) (III.15) dH which is the elementary form of the hydrostatic law, and is valid for all real gases which are under the influence of a central gravity field. 3.4 AERODYNAMIC FLOW REGIONS In the previous sections we have devoted the discussion to the general equation resulting from the consideration of a gas from the molecular point of view. We must now make use of a previously obtained result which becomes important when one considers the measurement of atmospheric parameters by means of a supersonic rocket-borne probe. That is, we must now make use of the mean-free path concept in the determination of aerodynamic flow regions encountered by a supersonic rocket vehicle on a high-altitude mission. In this section, three aerodynamic flow regions will be defined in terms of the mean-free path of the atmospheric particles and pertinent characteristics of each stated. 3.4.1 The Continuum Region The continuum region of the atmosphere is defined as that portion of the atmosphere where the mean-free path of the molecules is much less than a characteristic dimension defined by the dimensions of the instrument used in mak40

ing physical measurements of the atmospheric parameters. For example, the flow region is considered continuous if the mean-free path is less than the orifice diameter of a pressure gauge used to make aerodynamic measurements. It is found that when this condition exists, with. respect to the pressure sensor, there are many more intermolecular collisions than collisions with the walls of the sensor, and thus, to the sensor the gas appears to be a continuous media. Mathematically (III.14) may be used as a first approximation to determine the density for the flow-region to be classified as continuous. m J2 pjtd2 where m is the molecular mass considered constant p is the mass density d is the mean molecular diameter considered constant I is a characteristic dimension of the measuring instrument The above may be written in the following form: m d< p (III.16) J2 iTid2 Equation (111.16) thus gives a lower limit of the magnitude of the density for which continuum flow exists, and further shows that the continuum flow region exists in the portion of the atmosphere directly adjacent to the surface of the earth where the density is the highest. 41

3.4.1.1 The Rayleigh Pitot Tube Equation In the continuum region the gas dynamics of a compressible fluid dictates the theoretical interpretation of aerodynamic pressure measurements made at points on a supersonic vehicle. Let us now consider a rocket-borne probe flying at supersonic speed at zero angle of attack in the continuum region. Further, let this probe assume a configuration similar to that shown in Fig. III.5. The probe is moving forward with x directed velocity Vx which is greater than the local speed of sound and thus is accompanied by the shock front shown in the figure. This implies that the incoming flow Mach M is greater than one. Further, let the undisturbed atmospheric parameters in front of the shock wave be P, Pa, Ta (ambient pressure, ambient density, and ambient temperature). Also, let the pressure directly behind the shock front be Ps and the pressure appearing at the orifice of the impact pressure sensor be Pi. Now, following the development of Ref. 3 we see that EL _ Pip;" Pa S ) P a) Now Pi 1 + +l ^ )' P, 2 where y is ratio of the specific heat of air at constant pressure to that at constant volume 42

M>1 Po Pa Ta Detached Shock Normal Portion Ms< 1 Shock'YP M>1 ~ \\ Impact pressure chamber,j, \^ y Fig. III.5. The Rayleigh pitot tube equation model. 43

Ms is the Mach number directly behind the normal shock position of the shock front and 2 M2 +2 -1 where M is the incoming flow Mach number. Thus, substituting ~s; ~ L 2'V -1 PS 2 y 2 which after some manipulation becomes Pi = R7+l)2M2 PS L4-M2 27+2j We now need PS = 27 M2 - Pa 7+1 K+1l which may be written Ps = 272-7+l Pa 7+1 Combining these results / \/ \ r~ 2 ~22 7 /7-1 Pi _ (P S P 2yM -y+1 (y+l) M Pa \Ps I\Pa J L 7+1 J L4yM2-27y+2 Now, by multiplying and dividing the above by the term 2 (ry+I)2M2 44

we achieve the final desired form Pi = +1 (7) 17 (111.17) Pa \ 2 y 4yM2-27y+2 Where Pi is the measured stagnation pressure or impact pressure Pa is the undisturbed ambient atmospheric pressure M is the incoming flow Mach number y is the ratio of specific heats. Equation (III.17) represents a most useful form of the Rayleigh supersonic pitot-tube equation. In a later chapter it will be shown that this equation allows one to interpret the measured impact pressure in terms of the ambient atmospheric density provided the condition of continuum flow is met. 3.4.2 Free Molecule Flow The free molecule flow region, or FMF as we shall refer to it, is defin-:. iass that region of the atmosphere where the ambient density is sufficiently low so that the fluid media ceases to act as a continuum. Unlike conti'nuum flow, where the important physical phenomena is intermolecular collision, in FMF, the importance is placed upon molecular collision with the walls of an intruding measuring device. In FMF the mean-free path, because of the low density, is sufficiently long compared to the dimensions of the measuring device, that with respect to the sensor, the fluid appears to be a collection of a large number of individual molecules in random mo45

tion. In other words, the sensor sees the molecules of the gas as essentially independent entities obeying the laws of physics as individuals rather than as individual groups of molecules as is the case with continuum flow. In terms of our mean-free path analysis, the condition for the existence of FMF is: m > X = -- > lJ2 pjcd2 which may be written P < m (IIT.18) 2 Qid2 where m is the mean molecular mass p is the mass density d is the mean molecular diameter I is a characteristic dimension of the measuring instrument. Equation (III.18) thus gives an upper limit on the magnitude of the ambient density for which FMF exists, and also shows that FMF can exist only at altitudes above a certain critical altitude where Eq. (III.18) is satisfied. Because of the nature of FMF, it is noted that the aerodynamics of the rocket-borne probes are considerably different than that found in continuum flow. In FMF, the familiar shock wave associated with continuum flow disappears and the term "speed of sound" becomes physically meaningless. Therefore, a theoretical analysis based upon the Kinetic Theory of Gases is again called for. The next section will present such an analysis which will lead 46

to a result analogous to the Rayleigh supersonic pitot-tube equation which will, in a later chapter, be specialized to the interpretation of measured pressures in terms of the desired ambient properties of the atmosphere. 3.4.2.1 The Equations for Gas Flow Equilibrium at the Entrance to a Measuring Chamber in Free Molecule Flow Following the analysis given in Refs. 1,2,6, and 12, we shall direct our attention to a chamber with an opening in one face. The situation is illustrated in Fig. III.6 where Pa is the ambient pressure outside the chamber Ta is the ambient absolute temperature outside the chamber Pi is the measured impact pressure inside the chamber Ti is the measured absolute temperature inside the chamber Vx is the x component of relative velocity between the chamber and the gas vx is the x component of random particle velocity of the ambient gas St is the time increment under consideration na is the number density of the ambient gas ni is the number density of the gas within the chamber. The chamber shown has a velocity relative to the ambient atmosphere, which velocity is assumed to have component Vx normal to the face of the chamber opening. The gas molecules external to the chamber are assumed to have instantaneous thermal velocities v which vary from particle to particle and are a function of time and temperature for a single molecule. The x directed component of v is vx. The absolute temperature and pressure within the chamber 47

Unit cross sectional area Pa T, P a, n., ma ma Pta,/E Td21 v, st % Pi Ti, ni Impact pessure chamber Vx y Fig. III.6. Free molecule flow across an orifice. 48

will be Ti and Pi, respectively, and those of the ambient atmosphere will be Ta and Pa. Further assumptions required by this analysis are that the molecules both inside and outside the chamber obey the previously derived MaxwellBoltzmann velocity distribution law and that the particles in the chamber are in thermodynamic equilibrium with the walls of the chamber. Also, Pa, Ta, Pi, Ti, and Vx must be assumed independent of time for a period time St where St is an increment of time long enough to assure that an equilibrium state exists and short enough so that Pa, Ta, Pi, Ti, and Vx may be considered time independent. Now, for equilibrium the number of particles entering the chamber through the orifice must be equal to the number leaving. We shall first consider striking the surface area of the orifice per unit time, assuming that there is no relative velocity between the chamber and the atmosphere. In this case Vx is zero. Consider the imaginary cylinder vxSt long of unit cross-sectional area shown in the previous figure. Any particle within this cylinder that has a velocity with an x cormporn.nt of at least vx will reach the orifice in the time increment bt, provided it does not suffer an intermolecular collision before entering the chamber. Because we are considering FMF we will assume that the number of such collisions is negligible. Now, let us lift the restriction on Vx and assume that a relative drift velocity V does exist whose x directed component is Vxo Thus, it is necessary to change the length of the imaginary cylinlder from vTxt to (vx+Vx)bt 49

as shown in the previous figure in order to insure that a molecule whose x directed component of thermal velocity vx will enter the chamber. Either vx or Vx may be negative, as long as their sum is positive, otherwise no molecules will enter the chamber volume. Now, let dNa be defined as the number of external molecules with x component of thermal velocity in the range between vx and vx+dvx that strike the surface area of the orifice in time increment St. The number dNa may be determined by multiplying the volume of the imaginary cylinder by dna, which is the number of external molecules per unit volume whose x directed thermal velocity is between vx and vx+dvx. This may be stated mathematically: dNa = (vx+Vx)l St dna. Now, Vx may vary from large negative values to large positive values so, in order to find the total number Na of particles entering the chamber is 5t, it is necessary to integrate the above expression over all possible values of (vx+Vx), recalling that dna is a function only of vx. Thus: (vx+Vx) = ~ Na = (vx+Vx)bt dna (vx+Vx) = 0 recalling that dna = nafl(vx)dvx which may be written in terms of the most probable x directed thermal velocity 50

which we shall call cma which is given by 2kTa Cma = m where cma is the most probable thermal velocity of the ambient atmospheric molecules k is Boltzmann's constant Ta is the absolute ambient temperature m is the mean molecular mass. Introducing the expression for cma into Eq. (III.4) with appropriate value of A inserted, we find 2 f(Vx ) Q ka ) 1/2e ycma Similarly dvx = (X ) /d (X ) m Y ^mac Thus Vx na -Ce ma~ d vx dna = ee d X V mal Introducing the above equation into the expression for Na with the appropriate change of limits, one finds v x Na = (vx+Vx) Bt e- a) d( -) IVx = aVx Cma Ca Cma 51

which can also be written as 5t n C N a ma -L-e- a m/a d ~ ) Na = x cmaj vx cma + 5 t n Vx V x cma Let us introduce the notation Vx vx S = - and s = - cma Cma Then ot na c2 N tnaa cm se sds+ t na e ds a -_-S - _S which may be separated as follows 00 0 t n a ca -s ta x -S2 Na = e ds + e ds vrt'- -S St -S 5t na Vx 2 f -s2d + - eas 2 ^rt1 Carrying out the integration with the aid of Refs. 9 and 10 we see that Wt na aVx beseps rt na Vx[ 2 esds ~00 _Is 2 52 t nacm se 2 t nacm - 52

But r 2 2_ / d - eref r f Thus, the above integral becomes 51t n, Vx /- St nV 6t na V- X e e ds =- 5 erf S'~ err S Now, finally 5t na VX[ 2 e e2ds 6= t na Vx Therefore, our expression for Na becomes after some manipulation N a t nala IV J xS2 ~ V Na = a e + /;x erf S +. a< i - a "P Recall that VxX oIn t r,. -- Equa.tion (III.19) represonts. the rlilber of ambilent atmospheric molecules entering the chamber orifice in time inrrement 51t. per irnit area of orifice opening. Trw-), wne m1_ist consider the nimrb=nr of parti-les leaving Wthe chamber through the orifice in time St per unit area of orifice. Let this number be Ni- This number may be calculated in a msanner similar to that! uiseid to calculate Na, re553

membering that Vx is equal to zero and that we must use values of n, dn, and Cm for the interior of the chamber. Thus, vx = Ni = 6t v dni vx = 0 where ni - (v dni = e d Substituting into the above integral gives Vx v 2 = 00 t ni Cm Cmi V - Ni X e mi d ocmi Cmi After integration we find. t ni Cmi Ni = (III.20) Recalling now that equilibrium exists when the number of particles entering the chamber is equal to the number leaving. That is to say Ni = Na or st na cma -r 2 I)] t ni cmi Rearranging nicmi = Fe2 + S (l+erf S. (III.21) naCma 5e-S

In terms of pressure, we know from the equation of state that Pi = ni k Ti and P = n k T a a k Ta Now 2kT Cm m Therefore, T = m cm 2k Using this expression in the equation of state P, n. micm i p = _ i n^am,,. we will now assume that mi = m, Wit:! tl i s assimrnption the ratio 1a na inma - 55

or niCmi CmaPi naa mmiPa Substituting this expression into Eq. (III.21) and rearranging, we find Pi Cmi -s2 - = L — e + S (l+erf S Pa Cma But Cmi Ti Ta Cma T So Pi= T S + S2 (l+erf S) Pa T a Let us now define F(S) = [e-s +S (l+erf S) Finally i = TiaF(S) (III.22) where Pi is a measured impact pressure Pa is the ambient atmospheric pressure Ti is the measured absolute temperature interior to the chamber Ta is the absolute ambient atmospheric temperature 56

F(S) = [e-S + S 4 (l+erf S) Vx Vx Cma 2kT / m Equation (111.22) now gives the desired relationship between the measured impact pressure and the ambient atmospheric pressure. This expression will be specialized in a later chapter to allow the interpretation of measured impact pressure in terms of the ambient atmospheric density. It is interesting to note the behavior of F(S), (see Appendix) and the fact thatfor Sequal to zero, (corresponding to Vx equal to zero) the value of F(S) is unity, that is Pi- = (III.23) Pa Ta when there is no relative drift velocity between the chamber orifice and the gas (Vx = 0). 3.4.3 The Transition Region As a logical conclusion, in light of the previous two sections, there must exist a flow region intermediate to continuum and free molecule flow regions. This is the region known as transition. In this region, the flow exhibits properties of both continuum and free molecule flow, and because of the rather wide separation between the theoretical approaches used in analyzing the two flow regions, it does not lend itself readily to theoretical analysis. Considerable effort along the line has been given by several individuals. One very promising effort is given in Refo 11. 57

Again using the mean-free path approach, the transition region is that region where m,- m -J2 pjtd or P - 7 4 * (III.24).2 7td I Here we see that the mean-free path is approximately equal to the characteristic dimension, so that both intermolecular and molecular-wall collision become important. Let us then conclude that, to date, no satisfactory theory exists by which one may interpret measured aerodynamics pressures in terms of ambient atmospheric properties in the transition region. 3.5 SUMMARY OF THEORETICAL RESULTS By way of summary, let us simply restate the important final equations developed in the previous sections of this chapter and the appropriate assumptions and restriction on the use of each. Maxwell-Boltzmann Velocity Distribution Law f(v) = (m /2 e (III.10) where

m is the mean molecular mass k is the Boltzmann constant or gas constant per molecule v is the random thermal velocity Assumptions: (1) The velocity components of vx, Vy and vz, of v are statistically independento (2) The distribution function f(v) is isotropic. Validity~ Several direct experimental verifications of the law for several common gases and vapors have been made and may be found in the literature, On the basis of this experiment we shall consider the law valid for the atmospheric gas and thus the atmosphere. Th-. Equation of' State fur an Ideal Gas P = RT (i T.l) whe re P is Yen gas pre- suire.x t.td normal to a surface.mmesed i.n the fluid p is the gas mass d.'rs:ity R.is t'he gas constant per unit mass T is the absolute temperature of the gaso Assumptions: (I) Ideal gas 59

Validity: Even though the law applies strictly to an ideal gas, it has been shown experimentally to accurately predict the properties of a real gas at the value pressures and densities encountered in the atmosphereo It does not hold for extremely high densities and pressure approaching those of the critical values for the gas. This is because the extremely strong intermolecular forces are not accounted for in the ideal gas analysis, Thus, because a real gas approaches the behavior of an ideal gas at and below the values of pressure and density on the surface of the earth, we shall consider the law valid for the properties of the earth's atmosphereo The Hydrostatic Law dP_ = P - Pag5) dH where dPa is the rate of change of ambient atmospheric pressure with altitude dH H is the altitude in the atmosphere above the surface of the earth Pa is the ambient atmospheric pressure Pa is the ambient atmosphere gas mass density g is the local acceleration due to gravity Assumptions: (1) No mass motion (2) Equilibrium between atmospheric pressure and the forces of gravity. 60

Validity: It is known that over a large altitude interval, very violent mass motion of the atmosphere occurs. However, if the altitude increment dH is chosen arbitrarily small, then the mass motion occurring within the increment may be neglected and again only equilibrium consideredd The Rayleigh Supersonic Pitot-Tube Equation Pi 12 F(\ +()2M 12 y Pa (2) 42 L47M2-27+22 ( 7) where Pi is the measured impact pressure Pa is the ambient atmospheric pressure M is the incoming flow Mach number 7 is the ratio of specific heatso Assumptions: (1) M > (2) i Continuum flow (3) Normal shock exists in front of measuring sensor orifice (4) Zero angle of attacko Validity: In all upper atmosphere rocket-borne probes the first th'ree conditions above are met for at least the low-altitude portion of the flight Condition four, however, is rarely satisfied. Hlowever~ wind-tunnel tests of 6.1

devices similar to the one to be considered show that negligible error is introduced by angles of attack up to at least 10~. Gas Flow Equilibrium at an Orifice in Free Molecule Flow i = T F(S) (II1.22) Pa MNTa where Pi is a measured impact pressure Pa is the ambient atmospheric pressure Ti is the measured absolute temperature interior to the measuring chamber F(S) = [eS2 +4 S(l+erf S)] S is the ratio of the x directed component of relative drift velocity to the most probable thermal velocity. Assumptions: (1) Free molecule flow (2) Thermodynamic equilibrium between particles within the measuring chamber and the chamber walls. Validity: Provided that a high enough altitude is reached by the rocket-borne probe, the conditions for FMF will be met. Thermodynamics equilibrium can be assured if at least three collisions with the interior of the wall take place before the processes of measurement occur. For the third assumption, it is realized that there is some disassociation of atmospheric oxygen mole62

cules above the altitude where transition -to FMF occurs for the probe to be considered. It is therefore, entirely possible that there may be some recombination within the chamber, where the gas is not subject to the strong radiation fields which originally caused disassociation. and because the probability of recombination within the chamber is much greater than that outsideo The magnitude of this effect can be a source of error in any measurement made; however, from past experience it is believed that this error is negligible when compared to other errors which may be presento In view of this then, we shall consider Eqo (ITLL22) valid for all measurements made by the device in question in the free molecule flow regiono 65

References for Chapter I-I 1. Patterson, G. N., Molec^ullar Flow of Gases, JbohnW Wileyo and Sons, 1956. 2. Liui, V. C., Class Notes for Topics in Gas Dyrnamic;:, The U.nirversity of Michigan, College of Engineering, DeptO of Aero. and Astro. Eng., Ann Arbor, Spring, 1962. 3. Shapiro, A. H., Dynamics and Thermodynamics of Compressible Fluid Flow, 1, The Ronald Press, New York, 19530 4. Laurmann, J. A., "Low Density Characteristics of an Aerobee-Hi PitotStatic Probe,"' Univ. of Calif., InstJitolte of Engineering Research, Tech. Report, HE-150-i56, May, 1950. i Kuethe, A M., and Schetzer, D Foundations of Aerodynamics, John Wiley and Sons, New fork, 1950. 6. Schultz, F. V., Spencer, No W., and Reifman, A., "Atmospheric Pressure ar. Tempers r.ire MIeasurernents Between the Altitudes o-f 40 and 110 Kilomet.rs"." TUpJer Atmosphere Report No_ 2, TUnivo of Micho Research Institute, Ann Ar'bcor, Jt.ly, 1948. 7 Perkins, E. W., an(d. JTorgerisen, Lo H., "Comparison of Experimental and Theoretical Normal.-Force Distr:ibutji.r (In' Luding Reynolds Number Effects) on an Ogive-Cylinder Body at Mach Number o98," NA( Technical Note 7,', a4 s jinrg,ton, D.C. May, 1956. Aibnsworth, j.E., Fox, )D F., and LaGow, IH. E., "Upper Atmosphere Strucur,-e rMeas!rements Macde With the PitottS'ta'tic Tube," JG.R. 66(10), f59 321 I521, 1'96hl 64 64

References (Concluded) 9. Hodgman, C. D., Ed., Standard Mathematical Tables, Tenth Edition, Chemical Rubber Publishing Co., Cleveland, Ohio, 1955. 10. Churchill, R. V., Operational Mathematics, Second Edition, McGraw-Hill Publishing Co., New York, 1958. 11. Lui, V. C., "On Pitot Pressure in an Almost-Free-Molecule Flow-A Physical Theory for Rarefied-Gas Flows," Journal of the Aero/Space Sciences, 25, No. 1, December, 1958. 12. Hsu, Y. S., "The Thermal Transpiration Equation; Tables of F(S) vs. S," Scientific Report No. MS-2, ORA, Univ. of Mich., June, 1963o

CHAPTER IV THE PITOT-STATIC PROBE 4o INTRODUCTION Following Refso 1 and 2, the pitot-static technique is an aerodynamic approach to the measurement of pressure, temperature, and density in the upper atmosphere through a simple implementation of the basic theories presented in Chapter IIIo "Pitot-static" implies the two primary measurements which are inherent to the method: (1) the pitot or impact pressure; and (2) the static or ambient pressure. The pitot-static probe design presented here is extended to include a spin pressure measurement which, after further study and development, may lead to a determination of upper atmosphere winds. The pitot-static technique has been extensively studied and developed by the scientists and engineers of the Space Physics Research Laboratory (SPRL) of The University of Michigano It is this design and instrumentation which will be discussed in this chaptero 4.2 THE PITOT-STATIC PROBE CONFIGURATION The configuration of the SPRL pitot-static probe is shown in Figo IV lo As can be seen, the probe is a cylindrical tube, 3,5 inK in diameter, approximately 37 ino long, with a hemispherical nose tip and a cylinder-conecylinder afterbodyo The figure shows the single orifice. of the impact and spin pressure chambers and the many orifices of the manifold ambient pressure chamber. 67

IMPACT PRESSURE AMBIENT PRESSURE CHAMBER CHAMBlaER JDENSATRON POLARIZATION CYL WIND PRESSURE —, COLLECTOR-..".. TRANSMITTER ASSY ECHAMBER K E L-F VACUUM SEAL:: SOURCE PLATE. //' /.". ASPECT SENSOR DENSATRON:.INTERNAL SUPPORT MAGNETOMETER.EM ERL THIOKOL'APACHE' MEMBERS P ROCKET MOTOR l:ip':..:.: j jj, -:.::::: ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........ AMPLIFIER SECTION CC\ IRRADIATED SURFACE AMBIENT CHAMBER ORIFICES (10), PITOT STATIC PROBE SPACE PHYSICS RESEARC LABRTY UNIVEftsirrr CC MCHIGAN OUADRALOOP ANTENNAE (2) DOVAP TRANSPONDER'REMOVABLE SKIN, TODOVA ANTENNA 4 E (4):sz~Pli~,y;~Fig. T.p t i r Fig. IV-1. The pitot-static probe.

This particular configuration was chosen for two reasons: first, it meets the requirements of good supersonic aerodynami c design of a rocketborne probe; and second, this configuration readily lends itself to theoretical analysis and, in particular, to the theories presented in Chapter III 4 3 INSTRUMENTATION The raw data from the experiment consists of the continuous measurement of pressure in the three chambers by electronic pressure sensors which are connected directly to the three chambers. These pressure sensors convert the pressure information received from the chambers into their electrical analogo This analog or electric voltage, as is the exact case, is then transferred by internal cable to the instrumentation section whereit is further acted upon and combined with other analog information and finally transmitted via UHF FM radio to receivers and data recorders located on the groundo In this section we will discuss in detail the pressure sensors and associated rocket-borne support equipment. 4o3 1 The Radioactive Ionization-Gauge Densatron The pressure sensors called densatrons utilized with the SPRL pitotstatic probe combine a radioactive ionization chamber with a multirange electrometer amplifier to form an instrument capable of detecting with good resolution the pressures found in the region of the earth's atmosphere from 30 to 120 kmo As with other ionization gauges, neutral gas ionization results in a current which may be measured with a suitable ammeter, and through laboratory cal69

ibration, interpreted in terms of gauge pressure. In view of this, it is to be noted that in an ionization gauge the important factor in deriving a current to be measured is the number of gas molecules present in the gauge, or the gas density, rather than gas pressure. However, if the temperature is known, the gas density is proportional to the pressure. So, in effect, all ionization gauges are density sensing units which are calibrated in terms of pre s sure o The actual configuration of the densatron ionization chamber or gauge is shown in Fig. IV.2. The figure shows the important features of this ionization chamber. In operation, the neutral gas enters the chamber through the gas admittance port in the top seal plateo Upon entering the chamber it is bombarded by primary beta particles emitted by the radioactive source material (tritium) which is contained on the inside surface of the top seal plate. This process causes the neutral gas to become ionized. The ions and electrons are then acted upon by the electric field existing in the chamber between the polarization cylinder which is maintained at a positive potential with respect to ground and the central collector wire which is negative. Under the influence of this field the positive ions move to the collector where they are collected to form the ion current which is then measured and converted to an output voltage from the electrometer amplifier. Now, because the densatron is a density sensor which is calibrated in terms of pressure at a constant gauge temperature, we must determine what effect a change in gauge temperature will have on the pressure measurement. This is easily done through use of the equation of state. Here we assume 70

Source Plate Active area Base plate Terminal Polarization term ina Fig. IV.2. The densatron gauge. 71

that during the calibration the gauge temperature and gas temperature are identical and constant at a value Tcal* In this case Pcal = pR Tca1 where Pcal is the pressure corresponding to p and Tca, p is the gas density Tcal is the absolute gauge temperature during calibration R is the gas constant per unit mass. Now, let us maintain the density p in the gauge constant (which will give the same densatron output voltage as in the previous case) but vary the gauge temperature, and thus the gas temperature, to a value T. We now have gauge Pgauge = pR Tgauge where Pgauge is the pressure corresponding to p and Tgauge p is the gas density Tgauge is the new absolute gauge temperature R is the gas constant per unit mass. Now, since p remained constant, we have Pgauge = Pcal R Tgauge R Tcal which gives Pgauge = cal Tcal ( ) 72

where Pgauge is the real gas pressure for a given densatron output voltage at any gauge temperature Tcalo Tgauge is the absolute gauge temperature during actual operation as measured by the glass bead thermistor in the densatron gauge unito Pcal is the pressure obtained during calibration at gauge temperature Tcal, and is the pressure indicated by the gauge calibration for a given output voltageo Tcal is the absolute gauge temperature during calibrationo Equation (IV.1) must be used whenever the gauge temperature during operation is different from that at which gauge calibration took place 4.3.2 Supporting Rocket-Borne Equipment In addition to the various pressure measurements, the pitot-static payload includes a four-channel FM/FM telemetry system, a DOVAP (Doppler-Velocity-And-Position) transponder for velocity and altitude (trajectory) determination, and an optical aspect system for angle of attack determinationo This equipment is housed along with the battery power supply in the lower cylindrical portion of the probe called the instrumentation sectiono The FM/FM telemetry system consists of a UHF FM radio transmitter and associated electronics which serve as the data transmission link for the analog data from the probe during flight to the groundo The DOVAP transponder enables a supporting agency to determine the precise altitude and velocity versus time history of the probe during flighto The data from the optical aspect sensor allows the determination of the 73

probe in-flight angle of attack~ This information, as will be seen later, is of prime importance in subsequent data reduction of primary probe measurements. 74

References for Chapter IV 1. Ainsworth, J. E., Fox, D. F., and LaGow, H. E., "Upper Atmosphere Struc ture Measurement Made With the Pitot-Static Tube," J.GoRo 66(10), 3191-3211, 1961. 2. Horvath, J. J., Simmons, R. W., and Brace, L. H., "Theory and Implementation of the Pitot-Static Technique for Upper Atmospheric Measurements," Scientific Report No. NS1, ORA, The University of Michigan, March, 1962o 75

CHAPTER V TYPICAL DATA OBTAINED FROM THE SPRL PITOT-STATIC PROBE AND PRELIMINARY PROCESSING 5.1 INTRODUCTION As stated earlier, all primary (pressure) and secondary (gauge temperature, etc.) data from the pitot-static probe are transmitted via radio from the probe in flight to ground based receiverso The analog data is then transferred from the receiver on to magnetic tape for storage and to a "real time" oscillograph paper record for immediate use. For the present, we shall be primarily concerned with the "real time" oscillograph record~ This chapter will explain in detail, the process by which the pressure and gauge temperature information is obtained from the "real time" record and combined with the appropriate trajectory information to give final raw data, which may then be interpreted in terms of the three desired atmospheric parameters 5.2 THE ANALOG DATA The analog data from a typical SPRL pitot-static probe flight is received at SPRL in the form of a "real time' paper oscillograph record, This record is nothing more than a graph of analog voltages versus time, where the voltage is given as the vertical axis and time as the horizontalo The time axis increments are spaced OoOOl sec apart with the actual time given by a binary code in days, hours, minutes, and seconds with respect to the solar observa

tory at Greenwich, England, or Greenwich Mean Time (GMT). The voltages presented on the vertical axis in general lie somewhere on an arbitrary linear scale whose end points are zero volts and five volts respectivelyo A portion of the flight record, Fig. V.1 of NASA 14.21 SPRL pitot-static probe, is given for reference. On this sample of record is shown the 0-5 volt vertical scale, and the horizontal binary time coded scale, in addition, this sample shows 30 actual commutated data segments which are repeated cyclically throughout the entire flight. Each data segment represents one analog data pointo That is, each data segment gives one voltage output which occurred at the time given on the horizontal axis. An example of this is shown for segment no. 28. With the aid of the inflight voltage calibrate points, which are usually located in the first few segments of each cycle, and the various oscillograph reading devices available in the laboratory, each data point voltage is read and recorded along with time (GMT) at which each occurred. This procedure usually involves the reading of approximately 7,500 data point voltages, 7,500 segment numbers, and 250 reference times for each pitot-static probe flight. From the 250 reference times, the time of each individual data segment is computed and recorded. Thus, for each pitot-static flight approximately 22,500 bits of information are obtained, 7,500 voltage points, 7,500 times, and 7,500 commutator segment numbers. For proper handling of this information, it must be recorded as ordered triples of numbers, a segment number, an output voltage, and the time of the data segment. Having completed the reading and recording of the flight data as described above, the flight information is subjected to a sorting procedure. 78

i~~~~~~~~~~ I'm.. - 4.- - ft *~ i 1:............ _...:.. l _.. ^...,,.., ^.,. _ _ ^ ^^ X f~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f -44 10.............: Ie.c i I I _ _ _ _ _ _ 110) 1 1;''.? * ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~................................................................. f r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f (D9 ps,; * * ^ - ^ _ _ 0. _ ~ r H L......,....:. *S~ ~ ~~~ * -.. 1 <~- *m (0 4 > 11 ft' r * (1). 7 (. 1 (Oja!?> o 4 -- -' —----------- -----------------— 1 -: I a 0) 4 1 1 ^~'.';} ":0i-: C+ 7 ^ ^.^^^.^^^.,^.,,,...._.,^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..... O-q~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i o';1*'0: t,;: 1.'r:; ^^ i lii^^^^^ ^d I j|lt F-1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~(D aaBtI ov " - -------— 4-0 —— i —-- - -'CD I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-:-:: 42.~~~~~~~~~~~~~~~~~~~~~~~~r r4f N' 1ii i ift-^. o I I: ft::. ^~~~~~~~~~~~~~~~~~ 1tli^^^^ 51. ^:1j 1 l:lll;, E?.^ i ^^^^^ 1:~~::.. t.:: ^ i?.i i ^ *', ^ I1 ~:'~~1 f fts ft. >:r ^^ ~ ~ ~ ~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1,.:11 -:' a-^'"';*-;**'-...* *;:**.:.~':l71:~ i:.:: *:.. *'; t..*" * *. *.:^:^ ^ ^-:..: *-' **"....** ** **~*...".. * i ** * **. r. * *..-:..'.'. *':: * -.~~~~~~~~~~~~~~~~~5_ ~~::: i

The sorting is carried out according to segment number and time sequence. That is, the flight information is sorted according to experiment by segment number in increasing time sequence. This sorting process has now eliminated the segment or bit number from the triplet and replaced it with a title or experiment name. The primary experiment titles are Impact Densatron Output, Ambient Densatron Output, Impact Densatron Range, Ambient Densatron Range, etc. Some secondary experiment titles are Impact Densatron Gauge Thermistor Output, Impact Densatron Amplifier Thermistor Output, etc. Having completed the sorting process, the preliminary data "conditioning" is complete and the actual data reduction process may proceed. 5.3 CONVERSION OF THE ANALOG DATA TO "GAUGE PRESSURE" In this section, we shall determine the process by which the previously obtained analog voltages are converted into actual "gauge pressures" through the use of the densatron pressure calibration curves obtained in the laboratory preceding each flight. 5.3.1 Densatron Calibration Curves Each densatron to be flown must, prior to flight, be placed on a laboratory vacuum system where a pressure-voltage characteristic is determined~ This procedure is known as "calibration." During the calibration, the pressure in each densatron unit is varied over a range that best simulates the pressure range that will be experienced by the unit during flight. While this procedure is being carried out a particular pressure level in the pressure range is set in each densatron unit and is carefully measured by means of 80

a precision pressure standard~ This pressure is then recorded along with the individual densatron output voltages and densatron gauge temperatures. Upon completing the measurement of this pressure-voltage-temperature point, the pressure in the densatron units is changed to a new value and the above measurements are again made and recordedo This procedure is continued unitl the entire pressure range of the densatron units has been coveredo In order to assure proper pressure sensitivity throughout its entire pressure measuring region, the densatron is so designed that this region is divided into five separate ranges with each successive range six times as sensitive as the precedingO To facilitate the interpretation of the voltage data in terms of pressure, the above recorded pressure-voltage information is presented in the form of five separate graphs of Calibration Pressure versus Densatron Output Voltage at constant calibration temperature Each graph then represents the voltage-pressure characteristic of a particular densatron over one of its five rangeso In addition to the actual graph of Calibration Pressure versus Densatron Output Voltage, there will also appear on each curve a constant calibration temperature and Densatron Amplifier Calibration Voltageo The use of these numbers will be demonstrated later It must be remembered that each densatron has one set of five pressure calibration curveso Thus, for each pitot-static probe in which there are three densatrons, there must be 15 such separate calibration curves, one for each range of each densatrono It is now a simple matter to use the calibration curves to determine the 81

measured pressure corresponding to each analog point tabulated in the manner described in the last section. 5.3.2 Thermistor Calibration Curves A thermistor is simply a semiconductor resistance element that exhibits a negative thermal coefficient of resistance. The thermistor may, therefore9 when employed in a proper circuit, be used to determine an analog temperatureo This analog is again electrical voltageo Similarly, a temperature-voltage calibration of the thermistor circuit must be made. Such a calibration results in a graph of temperature versus thermistor voltage outputo In the SPRL pitot-static porbe, all thermistors are uniform and therefore require only one such calibration curve. With the thermistor calibration curve, the inflight thermistor output voltages may easily be converted to temperature. 55.33 Measured Pressures and Temperatures Assuming that all of the necessary calibration curves are available along with the recorded densatron flight data mentioned in Section 5.2 we may now proceed with the preliminary data processingo Our goal is to convert the tabulated data to pressures and temperatures. To accomplish this, we must first make use of the Amplifier Calibration Voltage. This voltage gives a direct measure of the performance of the Densatron Amplifier during flight, and allows us to correct any error in the Densatron Output Voltage caused by amplifier drift. This correction will be illustrated below 82

Assume for example that Impact Densatron Output ) = 1.22 volts Voltage and Impact Densatron Amplifier 2.54 volts Calibration Voltage Now we shall assume for the sake of illustration that Impact Densatron Amplifier Calibration Voltage given on the Range 1 Impact Densatron Pressure Calibration curve is equal to 2.56 volts. Comparing this with flight data we see a difference of 0.020 voltse This means that a correction of O0020 volts must be made to the output voltage given above. Or simply Amplifier Drif Densatron Impact e 1.22 volts + 0o020 volts Densatron Output Voltage This correction must be made to all densatron output voltages as long as the inflight Densatron Amplifier Calibration Voltage remains different from that given on the individual densatron calibration curves. 83

As a general rule for making this correction to the analog data let us state the following: Let DACVc be a Densatron Amplifier Calibration Voltage obtained during pressure calibration and DACVf be the Densatron Amplifier Calibration Voltage obtained from flight data. Further, take the absolute magnitude of the difference of DACVc and DACVf which is IDACVc - DACVf = X Now is X = 0 then a correction to the Densatron Output Voltage is required, and may be applied as follows: If DACVc > DACVf then add X to all Densatron Output Voltages to obtain the corrected values, that is Amplifier Drift Densatron Corrected Densatron = Output Voltage \Output Voltage 84

If DACVC < DACVf then subtract X from Densatron Output Voltages to obtain the correct values5 that is Amplifier Drift Corrected Densatron) = ensatron ) -X u t pu Output Voltage \Output Voltage/ At this time, the analog data for each pressure experiment has been recorded in time sequence along with the appropriate thermistor output datao The amplifier drift correction may now be applied to the Densatron Output Voltage as previously describedo Having completed the above correction for each of the densatrons we mary now make use of the densatron and thermistor calibration curves to find the pressure and temperature data corresponding to each output voltageo Thi s is easily done by simply looking up the pressure or temperature corresponding to the given voltage In the case of the densatron, be sure to use the Amp.ifier Drift Corrected Value in obtaining the pressureo This procedure must be carried out through the entire measurement range of each densatron pressure experimento This data now represents the measured pressures and temperatures with respect to timeo 5.5.34 Gauge Pressure To complete the preliminary data processing, it is now necessary to util85

ize Eq. (IVol) and the measured gauge temperature along with the constant calibration temperature to compute the actual pressures (Pgauge) experienced by the densatrons during flight. Recalling Eq. (IVol) we see, guage Pguage = PCal Tal (IV.l) where Pgauge is the actual densatron pressure Pal is the pressure obtained for a given densatron output from the densatron calibration curve Tgauge is the absolute temperature existing in the densatron gauge during flight as measured by the densatron thermistor Tcal is the absolute temperature of the densatron gauge and gas temperature during calibration. Equation (IV.l) may be used directly to compute the actual gauge pressures from the previously obtained measured pressure as follows: For each measured densatron pressure and densatron thermistor temperature, compute the ratio Tguage/Tcal using the value of Tcal given on the particular densatron calibration curve, multiply the measured pressure by this ratio, thus obtaining the desired value of actual densatron gauge pressure. Such a procedure must be completed for each of the flight densatron experiments (Impact, Ambient, and Spin). This completes all but one final step in processing and organizing the raw flight data. 86

5.4 THE TRAJECTORY The trajectory, which is given as the position and velocity of the pitotstatic probe in space versus time, is received by SPRL from an outside agency. It is received in the form of punched IBM cards and a set of tabulated information printed from the IBM cardso The information contained on the cards is a time history of total probe velocity, Cartesian total velocity components, probe range, and the Cartesian range componentso A computer program has been written which uses the information on the punched IBM cards as input data whose output consists of a tabulation of probe total velocity and altitude at the exact time from launch of each of the flight data points. As a final step in the organization of the raw data, the above trajectory information (total velocity, V, and altitude, H) is included in the tabulation given below. 5.5 EXAMPLE OF RAW FLIGHT DATA IN FINAL FORM The data presented in this section is actual flight data from SPRL NASA 14.21 flown from Wallops Island, Virginia on 7 Dec. 1963. It is given as illustration of the final form of raw flight data. 87

FLIGHT DATA GAUGE PRESSURE FLIGHT NO. NASA 14.21 UA DATE 12/27/65.. DENSATRON Impact DATE OF FLIGHT 7 Dec. 1963 RECORDER_____ SERIAL NO. Unit No. 18 Source No. 7 G.M.T. OF LAUNCH 1:4 z SHEET OF____ DENSATRON SEGMENTS 4, 8, 1, 18, 22, 26, 0 LAUNCH SITE Wallops Is., Va. RANGE SEGMENTS 15?NT THERMISTOR SEGMENTS( GAGE, 24 _______________________________________________________________________(AM PLIFIER) 20 ~MATnsTIME FROM DENSATRON RANGE DENSATRON AMPLIFIERON AMPLIFIER A F MEASURED DENSATRONAI THERMISTOR AMPLIFIER CORRECTION DRIFT COR ESUREC AUE TEMR TAUGE T ALTITUDE LAUNCH OUTPUT OUTPUT OUTPUT CALIBRATION FACTOR DENSATRON. PRESSURE VELOCITY H(KM) (SEC.) VOLTS VOLTS VOLTS VOLTAGE VOLTS OUTPUT TGAUE K) PEAL( MM HG) T K)E VT/EC) __________ _________ __________ __________ ~~~~AL( MM HG) TOAL ~GAUGE (MMH VP/SC F57.QO2.00 1.04 4.56 0.490 0.03 2.03 4.08 0 285.973 3.97 0 4.367 3 57.62 57.065 2.02____ ___________2.05 4.05 3.94 4.365 57.69 57.155 2.05____ ____________2.08 4.01 5.90 4.362 57.78 57.205 2.08 _____ 2.11 53.97 3______________.86 __4.360 57.86 57.261 2.10 ______ 2.13 3.94 3.83 4.355 57.93 57.317 2.14 ______ 2.17 3.89 35.78 4.357 58.00 57. 75 2.18 __________ 2.21 5.83 3.75 4.55 58.07 57.428 2.18 1.0. 4.57 0.490 2.21 3.83 285.973 3.73 4.55 8.14 57.484 2.21 ___________2.24 3.79 __3.69 __4.551 58.21 57.552 2.24 ____ ____.... 2.27 3.7,5 3..6.5 4.549 58.50 57.622 2.28 _____2.31 3.69 5 4.547 58.58 57.678 2.30 2.33 5.67 3.57 __ 4.346 58.45 co 57.734 2.534 2.37 3.61 3.51 4.344 58.52 co 57.789 2.36 2.39 3.58 __3.48 4.348 _8.559 57.844 2.38 1.04 4.56 0.490 0.03 2.41 3.55 __ 285.973 3.45 __ 4.341 8.66 57-900 2.42..2.45 3.50 3.41 __ 4.340 58.73 57.970 2.46 2.49.44 3.3 4.358 8.82 58.040 2.49 2.52 3.4o 3. 31 __ 4.3356 58.90 8.096 2.52 2.55 3-37 3.28 __ 4.334 58.97 58.152 2.54 2.57 3-34 3.2 4.555 5.04 58.208 2.56 ____ 2.59 3-31 3.22 __ 4.331 59.11 58.264 2.58 1.04 4.54 0.490 0.03 2.61 3.29 285.973 3.20 __ 4.329 9.18 58.320 2.60 2.63 3.26 3-17 4.527 59.2 58.588 2.62 2.65 3.24 3.1 __ 4.525 59.4 58.458 2.64 2.67 3.21 5.12 4.322 59.42 58.514 2.68.2.71 3.16.07 4.521 9.49 58.570 2.68 2.71 3.16 5.07 _ 4.519 59.56 58.627 2.72 2.75 3.11 5.05 4.318 59.63 58.682 2.74 1.04 4.56 0.490 0.05 2.77 3.08 285.973 3.00 4.316 59.70 58.736 2.76 __________2.79 3.06 2.98 4.315 59.77 58.805 2.79 2.82 5.02 __ 2.94 4.313 58.874 2.85 2.86 2.97 ___ 2.89 __ 4.511 5, 58.930 2.84 2.87 2.95 2.87 4.310 60.00 58.984 2.88 _____2.91 2.90 2.82 4.3o860.07 59.040 2.0 1.04 4.56 0.490 0.03 2.95 2.88 0 2.85.973 2.80 0 4.307 60.14 __......

CHAPTER VI DETERMINATION OF AMBIENT ATMOSPHERIC DENSITY FROM IMPACT GAUGE PRESSURE 601 INTRODUCTION In this chapter, the technique by which the impact gauge pressure is interpreted in terms of the ambient density of the atmosphere will be developed, The technique will be based on the theory presented in Chapter III. As mentioned earlier, the aerodynamics of the probe vehicle depend on the, flow regiono The flow regions being defined by the dimensions of the probe relative to the mean-free path of the atmospheric particleso Because the meanfree path is a function of the density, and the density a function of altitude, the flow regions may be specified as ranges in altitude for any given probe. In light of the above, the derivation of ambient density will be carried out using continuum theory at altitudes below approximately 85 km (continuum flow region for the present SPRL pitot-static probe), and free molecular flow theory at altitudes above approximately 90 kmo The transition region for the present probe exists between 85 and 90 kmo The system of units to be used in this, and following chapters will be,a modified MKS system. 602 AMBIENT DENSITY DETERMINATION BELOW 85 KM In the continuum region, the equations employed to deduce the ambient density from impact gauge pressure are the Rayl eigh Supersonic Pitot-Tube Equa89

tion, Eq. (III.17) and the equation of state, Eq. (III.12) which are 1 Pa 2 2) L47M2-27+2 (III.17) and Pa = PaRTa (III.12) where the symbols have the meanings assigned in Chapter III. Equation (III.12) can be rearranged into the form = RT Paa To obtain an expression for the ambient density, the above equation is multiplied with Eqo (IIIl17) with the following result 1 7Pi = () M2RT (7+l M Pa 2 2 L4yM2-27+2 Recalling from gas dynamics that a2 = RTa where a is the speed of sound in the atmosphere at temperature Ta y is the ratio of specific heats for air R is the gas constant per. unit mass Ta is the ambient atmospheric'temperature 90

and that the definition of Mach number is M- V = V2 a2 yRTa where V is the total velocity of the vehicle to which the Mach number M refers M is the Mach number a is the local speed of soundo Substituting these equations into the expression for Pi/Pa 1 P - = +1 2 V fRTa [( )l M_ (7Pa \2 y RTa 47M2 -27+2] which after simplification becomes 1 Pi 27' V2 7+ r(Y+l)2M2 ](y21) Pa Solving for the ambient density, Pa we have Pi Pa = 1 l.+l) (7+l)2M2 V2 (+) [ 27 4yM M-2+2 Let us define the function K(M) = +l-) y+l)2M2- (7-1) 27 y,/47M2-27+2j K(M) is found to be a very weak function of Mach number (see Appendix) which 91

converges very rapidly to its asymptotic value of 1 K(i) -' + I-1 Because of this weak dependence upon M and its rapid convergence, it can be approximated to within less than 1% error over a small range of Mach number by a known constant Let this constant be K(M1) where Ml is the approximate Mach number of the probe based on its knowrn velocity and altitude and the standard speed of sound at that altitude taken from a standard reference atmosphere. Using the value of K(M1) and the proper units conversion factor for our modified MKS system we see that A Pi (VI1 ) =a K(M) (VI ol) where Pa is the ambient atmospheric density (kg/m ) for altitudes less than 85 km / ft2 kg A* is a units conversion factor equal to 14355.064 3[mmHg]sec \M3 [mmHg sec2 Pi is the impact gauge pressure (mmHg) V is the probe total velocity (ft/sec) K(M1)** is the approximate probe Mach numbera Equation (VIol) represents a very simple and very close approximation to *For derivation of the constant A, see the Appendixo **For tabulated values of K(M) versus M, see the Appendix. 92

the ambient atmospheric density in terms of the impact gauge pressure for the continuum flow regiono The degree of approximation in Eq. (VI.1) may be made closer through an itterative technique. This, however, is usually not necessaryo The results using the first approximation are to within 1% when M1 is properly chosen and at the angles of attack experienced in flighto 6.3 AMBIENT DENSITY DETERMINATION ABOVE 90 KM In the free molecule flow region, the interpretation of impact gauge pressure is modified due to the change in flow conditions, related to the change in mean-free path of the ambient atmospheric particles relative to the probe dimensions~ In this region the governing equation is Eqo (IIIo22) which is Pi = T F(S) (IIIo22) -?a -N Taa where F(S) = eS + S^ [l+erf S] A careful study of the behavior of F(S) (see Appendix for plot of F(S) versus S) shows that F(S) becomes linear in S for S sufficiently large. That is with negligible error F(S) = 2 S S > 1.5 It is easily shown that the condition that S be greater than one implies that the Mach number M of the probe must satisfy the following condition 93

22 M > S - 1.80 for 7 = 1.4 This condition is easily satisfied by the SPRL pitot-static probe throughout its entire measurement region from 20 to 120 km. Under this minor restriction, then, Eq. (III.22) may be rewritten Pia Pa Recalling that Vx f 2RTa The above statement becomes Pi _ 2 t Vx Ti Pa Ta a 2R Making use of the equation of state, Eq. (III.12) in the above, we have that Pa = PaRTa and pL 2 RVx E Pa.2R Now, solving for the ambient atmospheric density, Pa, and simplifying Pa Pi942RTi Vx 94

We also know that Vx = V cos a where V is the total velocity of the probe a is the probe angle of attack. Making this substitution, the expression for Pa becomes Pi Pa a 2jrR T i V cos a Let us now define B - A where A is the same constant given in Eq. (VI.2) with this, we have finally B Pi Pa = /Ti V cos aC where p is the ambient atmosphere density (kg/m3) for altitudes greater than 90 km B* is a dimensional constant equal to 10.257 ft kg _ based on the sec m3[mm Hg] assumed value of molecular weight for air of 28.72 which is valid for the atmosphere between 90 and 120 km *The constant B is derived in the Appendixo 95

Pi is the impact gauge pressure (mm Hg) Ti is the measured impact densatron gauge absolute temperature (~K) V is the probe total velocity (ft/sec) a is the probe angle of attack (degrees). Through the use of Eq. (VI.2) we are now in the position to interpret the impact gauge pressure in terms of the ambient atmospheric density at altitudes above 90 km for the present SPRL pitot-static probe. This equation, it must be remembered, is an approximation dependent not only upon the simplifying assumptions made in its derivation, but on the mean molecular mass of the atmospheric particle. The error involved in the assumption that F(S) is linear in S for M > 1.5 introduces negligible error in the final density expression. The error introduced by assuming a constant molecular weight for the entire FMF measurement (90-120 km) may be further minimized by assuming a variable composition. This, however, based on our present knowledge of the composition of the Upper Atmosphere, is not necessary, for the error introduced by the assumption is of the order of 0.53% at the end points of the region. 6.4 SUMMARY OF DENSITY FORMULAE To summarize the results of the two previous sections, we will simply restate the final expressions and the applicable flow region. 96

Continuum Flow P 1435.o64 Pi (v l Pa K(M1)V2 v where p, a is the ambient atmospheric density (kg/m3) Pi is the impact gauge pressure (mm Hg) V is the probe total velocity (ft/sec) K(M1) is the value of K(M) at M1 (see the Appendix) M1 is the approximate probe Mach number. Equation (VI.1) is applicable for altitudes below 85 km. Free Molecule Flow P - 10257 Pi (VI.2) P JTi V cos where Pa is the ambient atmospheric density (kg/m3) Pi is the impact gauge pressure (mm Hg) Ti is the impact densatron gauge temperature (~K) V is the probe total velocity (ft/sec) a is the probe angle of attack (degrees) Equation (VI.2) is applicable for altitudes between 90 and 120 km. 97

CHAPTER VII DETERMINATION OF AMBIENT ATMOSPHERIC PRESSURE 7.1 INTRODUCTION The determination of the ambient atmospheric pressure from the raw data of the pitot-static probe is considerably easier than either of the other two desired parameters. The reason for this is, in accordance with Refso 1 and 2, that the SPRL pitot-static probe is designed such that the ambient densatron gauge pressure is very close to the ambient atmospheric pressure in the continuum flow region (below approximately 85 km). In addition to this direct measurement of the ambient atmospheric pressure, the Hydrostatic Law may be used in a straightforward manner to deduce the ambient atmospheric pressure from the previously determined ambient density. 7.2 DIRECTLY MEASURED AMBIENT PRESSURE In the continuum-flow region, the ambient pressure is measured directly at the manifold ambient chamber located as shown in Fig. IV.1. Wind tunnel investigations by Laurmann1 show that the surface pressure at this location on the cylindrical body of the probe is within less than 1% of ambient, and that the manifold chamber minimizes deviations from the ambient pressure caused by bow-shock perturbations, boundary layer interference due to the expansion at the flare, and nonaxial flow about the probe. Recent flight testdata from two SPRL pitot-static probes indicates, contrary to wind tunnel information, that surface pressure at the ambient chamber is approximately 40 99

lower than actual ambient pressure. Expressed mathematically, the above statement becomes Pa - o104 (VIIol) Pgauge where Pgauge is the ambient densatron gauge pressure Pa is the ambient atmospheric pressure. Equation (VII.1) is valid in the continuum flow region only. As discussed in Chapter III the pressure measured in the ambient chamber in the free molecule flow region will differ in magnitude from the actual ambient pressure due to the thermal transpiration effect as given by Eqo (IIIo25) P Tgaue gaue = gauge (III.23) Pa q Ta where Pgauge is the ambient densatron gauge pressure Pa is the ambient atmospheric pressure Tgauge is the ambient densatron gauge temperature Ta is the ambient atmospheric temperature. Thus, according to Eq. (III.23), the ambient pressure cannot be found by using the manifold chamber pressure until the ambient atmospheric temperature is known. This problem is circumvented by the consideration of the next section. 100

7.3 AMBIENT PRESSURE DETERMINATION THROUGH THE USE OF THE HYDROSTATIC LAW An alternate approach, which is independent of the flow region, to the determination of the ambient atmospheric pressure utilizes the previously determined ambient density and the Hydrostatic Law as given by Eq. (II1.15) dP = - Pag(H) (II 5) dH Equation (III.15) may be integrated with respect to H as follows H1 Pa]H f Pag dH + Pa]Hef (VII2) Href where H1 is an arbitrary altitude at which the value of Pa is desired Href is an arbitrary reference altitude greater than H1o The integral indicated by Eq. (VII.2) must be evaluated numerically. To illustrate the use of this technique in determining the ambient pressure profile, let us rewrite Eqo (VII.2) in a slightly different but equivalent formo PaHj PaHref + Pal + Aa2 +'oe + APaj or J a =Hj P + APa (VII53) = Hj Pa,Href /- a,n n=l where PaHj is the ambient atmospheric pressure at an altitude Hj PaHf is the ambient atmospheric pressure at an arbitrary high reference altitude. 101

Hn APan = pg dH H(nl) H(n-l) > Hn,Ho = Href To evaluate the integral in Eq. (VIio53), we first plot, using linear scales, the ambient atmospheric density in kilograms per cubic meter versus altitude in kilometersc On this plot subdivide the altitude range into one-half kilometer increments and integrate graphically, using the value of g corresponding to the altitude in question. The value of APa would then be a n given as Pan Sn(SF) D (V-4) where Hn AP n / pg dH (mm Hg) JH(n-l) Sn is the total area under the density profile as determined by graphical integration between H(n-l) and Hn SF is the syale factor of the plot of ambient density versus altitude (kg/m2 ina2) g is the acceleration due to gravity at an altitude midway between H(n-1) and H(n). (m/sec2) go is the acceleration due to gravity at the surface of the eartho (m/sec2) D is a units conversion factor equal to 0.07356 (m Hg]m \ kg 102

The value of Pa H r or reference pressure may be determined in either of two ways, the latter being preferred. First, PaHref may be assumed to be the value of Pa obtained from a Standard Reference Atmosphere corresponding to the altitude Href Second, Pa H may be computed by substituting r ef the previously determined density at Href and an assumed value of atmospheric temperature obtained from the Standard Atmosphere corresponding to the altitude Href into the equation of stateo Since the actual integration process described here takes place starting at a high altitude (Href) and proceeds to a low altitude, the error incurred because of an improper choice of PaHf is minimized. It can be shown that the choice of Pa Hrf is relatively unimportant and Eqo (VIIo3) will rapidly converge to within less than 1% of the true ambient pressure when the difference between the altitude in question and the altitude Href is equal to approximately 15 km. A poor choice of Pa Hf is readily apparent by its complete incompatability with the resulting pressure profile, and another value whice gives more rapid convergence should be selected. 7.4 CONCLUSION As is readily seen, the ambient atmospheric pressure is measured directly at the manifold chamber in the continuum flow region, and is found through the use of the Hydrostatic Law in both the free molecule and continuum flow regionso This redundency of measurement is inherent in the multiple sensing system employed by the pitot-static probe. It is important to maintain a rather high degree of measurement redundency in the design of upper atmosphere experiments for several reasonso Two of which are: (1) 103

if self-consistency exists among the data resulting from the various methods of determining an atmospheric parameter, increased confidence can be had in the measured values of these parameters; and (2) redundency also helps maintain minimum data loss in the event of failure or improper operation of independent parts of the overall system. 104

References for Chapter VII 1. Laurmann, J. A., "Low Density Characteristics of an Aerobee-Hi PitotStatic Probe," Univ. of Calif., Institute of Engineering Research, Tech. Report, HE-150-156, May, 1950O 2. Perkins, E. W., and L. H. Jorgensen, "Comparison of Experimental and Theoretical Normal-Force Distribution (Including Reynolds Number Effects) on an Ogive-Cylinder Body at Mach Number 1.98," NACA Technical Note 3716, Washington, D.C., May, 1956. 105

CHAPTER VIII DETERMINATION OF AMBIENT ATMOSPHERIC TEMPERATURE 8.1 INTRODUCTION Ambient temperature in the upper atmosphere is perhaps the most significant of the three parameters yet discussed. It is this parameter which is most closely related to the important physiochemical reactions which result in the heating and cooling of the upper atmosphereo These heating and cooling processes then determine the basic meteorology of the upper atmosphere. Thus, a need for a knowledge of the ambient temperature is readily apparent. In addition to being the more basic and important of the parameters of the atmosphere it is also the most difficult to obtain. At present, there is no simple method by which the ambient temperature may be measured directly with the degree of accuracy required. Thus, the experimenter must rely on techniques for interpreting directly measured quantities in terms of the ambient temperature. 8.2 AMBIENT TEMPERATURE THROUGH THE USE OF THE STATE EQUATION A first method for deducing the ambient atmospheric temperature in terms of the ambient density and pressure is a straightforward substituting of the known quantities into the equation of state which may be stated Ta - a (VIII.1) a - P PaR 107

where Ta is the ambient atmospheric temperature at an altitude H(~K) Pa is the ambient atmospheric pressure at the altitude H Pa is the ambient [mm Hg] atmospheric density at the altitude Ho (kg/m3) R is the gas constant per unit mass whose value is 2o153 [mm Hg]m kg OK based on a molecular weight for air of 2809644. This equation may easily be set up in tabular form and the required calculation carried out for each altitude at which a value of ambient density and pressure exists. Equation (VIIIol) represents an extremely simple method for determing the ambient temperature from the primary data obtained from the pitot-static probe. Because the primary data from the pitot-static is continuous with respect to altitude, a continuous temperature profile is readily obtainedo 803 AMBIENT TEMPERATURE THROUGH USE OF THE HYDROSTATIC LAW IN COMBINATION WITH THE EQUATION OF STATE Still a second technique which is based on the so called pressure scale height may be derived from the Hydrostatic Law as given by Eq. (III.15) dP = - Pag(H) (III.15);dH and Eq. (III12) P a PaTRa ~ ( III.12) Equations (IIIol5) and (I1II.12) may be combined as follows 108

dPa = g dH Pa RTa which, upon integration between the limits of Pal and Pa2, and H1 and H2 respectively, ln a2 - in Pal RT (H-Ha Here it should be noted that g, R, and Ta have been considered constant with respect to the integrand dH on the right hand side of the equationo This assumption is valid as long as the altitude increment H2-H1 is kept very small. The above expression may now be solved for Ta, and after some manipulation the expression for Ta becomes go g AH Ta = - - (VIIi 2) R go ln P_( P2 where Ta is the ambient atmospheric temperature (~K) g is the acceleration due to gravity at altitude H(m/sec2) go is the acceleration due to gravity at the surface of the earth (m/sec2) go/R is the ratio of the acceleration due to gravity at the earth's surface to the gas constant per unit mass whose value is 345154 (~K/km) based on a molecular weight of 2839644 for air AH is the small altitude increment (H2-H1) under consideration (km) P1 is the ambient atmospheric pressure at altitude H1 (mm Hg) P2 is the ambient atmospheric pressure at altitude H2 (mm Hg)o 109

Equation (VIII 2) is readily adapted to tabular computation using the values of g/go given in the Appendix. The value of AH should be the orderof one kilometer to avoid the possibility of "too much averaging" in the final resulting temperature~ This calculation should be carried out using all ambient pressure data obtained, thus yielding a continuous temperature profile for the entire measurements region of the p tot-static probe, 8.4 AMBIENT TEMPERATURE IN THE CONTTINUrM FLOW REGION FROM THE RAYLEIGHPITOT-TUBE EQUATION Still another technique for deducing the ambient atmospheric temperature in the continuum flow region from the primary pressure data of the pitot probe may be derived from the Rayleigh Supersonic Pitot-Tube Equation, Eq. (I'ol?) 1 L- (~7+1.> I (7+l)2M21 Y V- 2 L4yM2-2+2j which may be put in the form of Eqo (VIol) A P4 Pa " e ~ (VI 1) Pa =K(M1)V-2 Recall the equation of state Pa PaRTa (IIo12) or a RTa This expression may be substituted into Eq~ (VIol) to give 110

P A __i RTa K(M )V2 which may now be solved for the ambient temperature as Ta K(M1)V2 Pa (VIL.) A R Pi where Ta is the ambient atmospheric temperature (cK) K(M1) is the value of K(M) at M1 dimensionless M1 is the approximate probe Mach number V is the probe total velocity Pi is the impact densatron gauge pressure (mm Hg) Pa is the ambient atmospheric pressure (mm Hg) A is a unit conversion factor equal to 1435.064 ft2kg1 - - ) [mm Hg]sec/ R is the gas constant per unit mass equal to 2.1535 [mm Hg m kg OK I Equation (VIIIe3) may be used on a point by point basis in the continuum flow region to obtain a continuous ambient temperature profile over the entire flow region. It is important to remember that Eqo (VIII.3) is not valid in the free molecule flow region. 8.5 AMBIENT TEMPERATURE IN THE FREE MOLECULE FLOW REGION An expression for the ambient atmospheric temperature in the free molecule flow region may be developed in a manner similar to that for Eqo (VIIIo53) For this development we make use of Eq, (VIo2) 111

B Pi (VI 2) Pa = -B^ (v1.2) Ti V cos a and the equation of state Pa = PaRTa o (IIl2) Combining Eq. (VI.2) and (III12) and solving for Ta we have Ta 4 -i V cos c _ (VIIIo4) Ta = i c (viii.4) BR Pi where Ta is the ambient atmospheric temperature (~K) Ti is the measured impact densatron gauge temperature (~K) V isthe probe total velocity (ft/sec) (X is the probe angle of attack (degrees) Pa is the ambient atmospheric pressure (mm Hg) Pi is the impact densatron gauge pressure (mm Hg) B is a units conversion factor equal to 10o257 ft kg K~, sec m3[mm Hg] based on the assumed value of molecular weight for air of 28o72 for altitudes between 90 and 120 km R is the gas constant per unit mass equal to 2o172 [mm Hg]m kg OK based on molecular weight of 28o720 Equation (VIII.4) thus gives the ambient temperature in terms of the primary pressure measurements made by the pitot-static probe in the free molecule flow region. Another interesting approach to the determination of ambient temperature 112

in the free molecule flow region which has not yet been employed with the pitot-static probe data is based on the previously mentioned thermal transpiration effecto This technique requires an accurate determination of ambient gauge pressure and gauge temperatureo To illustrate, recall Eqo (III.23) Pgauge Tgauge (III23) Pa Ta where Pgauge is the ambient gauge pressure in free molecule flow Pa is the ambient atmospheric pressure Tgauge is the ambient gauge temperature Ta is the ambient atmospheric temperatureo Writing Eqo (III.23) in terms of ambient temperature, we have Ta = Tgauge a ) (vIro5).Pgaugey Equation (VIII.5) shows that if the three parameters, Tgauge Pa' and Pgauge can be accurately determined in the free molecule flow region, then the ambient temperature may easily be deducedo This method shows much promise for the near future, for it is now becoming possible, with increased sensitivity of the densatron, to obtain an accurate measure of the ambient gauge pressure in free molecule flowo The gauge temperature is directly measured, and the ambient atmospheric pressure is easily obtained from the impact pressure measurement, 113

8.6 AMBIENT TEMPERATURE FROM DIRECT INTEGRATION OF AMBIENT DENSITY In addition to the method of determining developed above, one may develop an expression for atmospheric temperature at altitude "H" in terms of the temperature at a reference altitude "Href" and the ambient density profile. This development again makes use of the theory presented in Chapter III. To begin, we make use of Eq. (III.15) which is the Hydrostatic Law, dP = - pag(H) (II1.15) dH and the equation of state Pa = PaRT (111.12) Equation (III.12) may be differentiated to yield dPa Rd(PaT) which may be substituted into Eqo (III.15) and written as d(pTa) = g dH (vIII.6) R Equation (VIII.6) may be integrated between the limits H and Href giving the desired result H TaH TaHref PaHref 1 Pag dH (VIII.7) Pa,H aiR tref Href > H 114

where TaH is the ambient atmospheric temperature at altitude H (OK) TaHref is the ambient atmospheric temperature at altitude Href PaHref is the ambient atmospheric density at altitude Href PaH is the ambient atmospheric density at altitude H R is the gas constant per unit mass Href is an arbitrary high reference altitude H is a general altitude g is the acceleration due to gravity. In the above, Eq. (VIII.7), Href is specified to be greater than H for the following reason: If the computation of temperature is begun at a high altitude and continued downward, the error incurred in the integration becomes progressively more independent of the error in Ta,Href and of the error in the density profile in the vicinity of PaHref, but becomes almost entirely dependent upon the error in the density profile in the region just prior to Pa H' 8.7 CONCLUSION An important conclusion to be drawn at this time is based on the fact that even though we have present four formulae for determining the ambient atmospheric temperature, there is in reality only two basic approaches usedo Careful study of Eqs. (VIII.1), (VIII.3) and (VIIIo4) show that these are actually the equation of state written in a form which utilizes the pitotstatic probe data directlyo Equations (VIII.2) and (VIII.7) on the other hand are simply forms of the Hydrostatic Law. There are of course several 115

other expressions of ambient temperature in terms of the primary pitotstatic data but they must all be based either on the equation of state or the Hydrostatic Law. 116

APPENDIX PHYSICAL CONSTANTS AND CONVERSION FACTORS A l INTRODUCTION To facilitate the data reduction procedures presented in the body of this report, the following sections containing necessary constants, conversion factors, and tabulated data are presented. In these sections, the following information will be found: (1) Often used physical constants and units conversion factors; (2) Derivation of the constants A and B found in Chapter VI; (3) Tabulation and plot of the function K(M) found in Chapter VI; (4) Tabulation and plot of the function F(S) also found in Chapter VI; Tabulation of the values of the ratio g/go for an altitude range of 0 to 100 km. A.2 PHYSICAL CONSTANTS AND CONVERSION FACTORS A.2.1 Physical Constants and Formulae The following is a list of the most often used physical constants: (1) Avogadro's Number (number of molecules in one mole) A = 6.0247 x 10-23 molecules mole (2) Boltzmann's Constant k = 1.3804 x 10-23 joules/~K 117

(3) Density of Mercury at 273515~K PHg = 13,595.04 k_ (4) Acceleration due to Gravity at Sea Level 908066 m sec (5) Molecular Weight of Air at Sea Level MWo = 28.9644 x 10-3 kg mole (6) Gas Constant per Unit Mass of Air at Sea Level R = 287.128 joule kg ~K R = 2.153 m3[mm Hg] kg OK (7) The Ratio go/R for Air at Sea Level go/R = 34.154 x 10-3 K m A.2.2 Conversion Factors The following important, often used, conversions are presented for reference o 1 nt = 1 kgm sec2 1 joule - 1 nt m 1kg m2 sec2 1 kgFORCE 9.8066 kgs 118

1 mm Hg = 13.595gFORCE m2 1 FORCE = 0.07356 [mm Hg] m2 A.2.3 Conversion Formulae In general, the physical constants presented in A.2.1 are valid for use in the data reduction formulae in the altitude range of 0 to 90 km. In the event, that it is necessary to reduce data for altitudes above 90 km, the following formulae may be used to recompute the values of the necessary constants taking into account the change in molecular weight of air which occurs above this altitude. R kA joule = 8.3165 joule MW kg ~K (MW) kg OK R - x 0.00750 m3[mm Hg - 0.06237 m3[mm Hg] RMW xO.O75Okg OK (MW) kg oK go/R -= g(Mwi = 1.1792 (MW) K kA m m A.3 DERIVATION OF THE CONSTANTS A AND B As stated in Chapter VI, the constants A and B are units conversion factors. To determine the value for each of these constants, it is necessary to make a dimensional analysis of the formulae in which they are found. The Constant A This constant is found in Eq. (VI.1) Pa = KM)V kg * (VI.1) K(M1)V2 m3 119

If Pi is given in [mm Hg] and V2 in (ft/sec)2 then: kg - A[mm Hg]sec2 m3 ft2 Thus, the units of A are: 2 kg ft 1 m3 sec2 [mm Hg] or ft2 kgFORCE m kg A = 10 764 x 13595 ORC x 9.8066 k m2 m [mm Hg] sec2 kgFORCE Carrying out the operations kg ft2 1 A = 1435o0 4 m3 sec2 [mm Hg] The Constant B Similarly, this constant is found in Eqo (VIo2) Pa B P1 kg (VI.2 \ Ti V cos a m3 where B is given by B _A ft kg __K 42irR sec m3 [mm Hg] or A- -A _/MW B - ^\12T kA 3.2808 1435 064 J7o2283 x 352808 120

B = 605214fM t kg___ K sec m3 [mm Hg] which is the general equation for B. If the molecular weight of air in the altitude range 90 to 120 km is taken as MW = 28.72 x 10-3 kg mole Then B becomes B - 10.257 f k J W sec m3 [mm Hg] 121

A 4 TABLE AND GRAPH OF K(M) VS. M FOR AIR (pps. 125-126) 7 = 1.4 GRAPH: K(M) vso M 7 = 1)4 122

M K () M K(M) m K(M) M K(M) M K M) 1.50 1.C8358 2.05 1.00274 2.60.97013 3.15.95364 3.70.94412 1.51 1.08116 2.06, 1.00189 2.61.96973 3.16.95342 3.71.94399 1.52 1.07879 2.07 1.00105 2.62.96933 3.17.95320 3.72.94386 1.53 1.07648 2.08 1.00022 2.63.96894 3.18.95299 3.73.94372 1.54 1.07422 2.09.99941 2.64.96856 3.19.95277 3.74.94359 1.55 1.07201 2.10.99860 2.65.96818 3.20.95256 3.75.94346 1.56 1.06985 2.11.99781 2.66.96780 3.21.95235 3.76.94334 1.57 1.06773 2.12.99704 2.67.96743 3.22.95214 3.77.94321 1.58 1.06566 2.13.99627 2.68.96706 3.23.95194 3.78.94308 1.59 1.06364 2.14.99551 2.69.96669 3.24.95174 3.79.94296 1.60 1.06166 2.15.99477 2.70.96633 3.25.95153 3.80.94283 1.61 1.05972 2.16.99404 2.71.96598 3.26.95133 3.81.94271 1.62 1.05782 2.17.99332 2.72.96563 3.27.95114 3.82.94259 1.63 1.05596 2.18.99261 2.73.96528 3.28.95094 3.83.94246 1.64 1.05413 2.19.99191 2.74.96494 3.29.95075 3.84.94234 1.65 1.05235 2.20.99122 2.75.96460 3.30.95055 3.85.94222 1.66 1.05060 2.21.99054 2.76.96426 3.31.95036 3.86.94211 1.67 1.04889 2.22.98987 2.77.96393 3.32.95017 3.87.94199 1.68 1.04721 2.23.98920 2.78.96360 3.33.94999 3.88.94187 1.69 1.04557 2.24.98855 2.79.96328 3.34.94980 3.89.94176 1.70 1.04395 2.25.98791 2.80.96295 3.35.94962 3.90.94164 1.71 1.04237 2.26.98728 2.81.96264 3.36.94944 3.91.94153 1.72 1.04C82 2.27.98665 2.82.96232 3.37.94926 3.92.94141 1.73 1.03930 2.28.98604 2.83.96201 3.38.94908 3.93.94130 1.74 i.03781 2.29.98543 2.84.96171 3.39.94890 3.94.94119 1.75 1.03635 2.30.98483 2.85.96140 3.40.94872 3.95.94108 so 1.76 1.03491 2.31.98424 2.86.96110 3.41.94855 3.96.94097 1.77 1.03350 2.32.98366 2.87.96080 3.42.94838 3.97.94086 1.78 1.03212 2.33.98309 2.88.96051 3.43.94821 3.98.94075 1.79 1.03077 2.34.98252 2.89.96022 3.44.94804 3.99.94065 1.80 1.02943 2.35.98196 2.90.95993 3.45.94787 4.00.94054 1.81 1.02813 2.36.98141 2.91.95965 3.46.94771 4.01.94043 1.82 1.02684 2.37.98087 2.92.95937 3.47.94754 4.02.94033 1.83 1.02558 2.38.98033 2.93.95909 3.48.94738 4.03.94023 1.84 6 1.02435 2.39.97980 2.94.95881 3.49.94722 4.04.94012 1.85 1.02313 2.40.97928 2.95.95854 3.50.94706 4.05.94002 1.86 1.02194 2.41.97876 2.96.95827 3.51.94690 4.06.93992 1.87 1.02077 2.42.97826 2.97.95800 3.52.94674 4.07.93982 1.88 1.01~61 2.43.97775 2.98.95774 3.53.94658 4.08.93972 1.89 1.01848 2.44.97726 2.99.95748 3.54.94643 4.09.93962 1.90 1.01737 2.45.97677 3.00.95722 3.55.94627 4.10.93952 1.91 1.01627 2.46.97629 3.01.95696 3.56.946124.11.93942 1.92 1.01520 2.47.97581 3.02.95671 3.57.94597 4.12.93932 1.93 1.01414 2.48.97534 3.03.95646 3.58.94582 4.13.93923 1.94 1.01310 2.49.97487 3.04.95621 3.59.945674.14.93913 1.95 1.01208 2.50.97442 3.05.95597 3.60.945534.15.93904 1.96 1.01108 2.51.97396 3.06.95572 3.61.94538 4.16.93894 1.97 i.01C09 2.52.97352 3.07.95548 3.62.945244.1.93885 1.98 1.00912 2.53.97307 3.08.95524 3.63.945094.18.93876 1.99 1.00816 2.54.97264 3.09.95501 3.64.94495 4.19.93866 2.00 1.00722 2.55.97221 3.10.95477 3.65.944814.20.93857 2.01 1.00630 2.56.97178 3.11.95454 3.66.944674.21.93848 2.02 1.00539 2.57.97136 3.12.95431 3.67.944534.22.93839 2.03 1.CC4492.58.97095 3.13.95409 3.68.944394.23.93830 2.04 1.CC361 2.59.97053 3.14.95386 3.69.944264.24.93821

M K(M)M K(M) M K(M M K(M) MK(M) 4.25.93812 4.80.93410 5.35.931265.90.929196.45.92763 4.26.93804 4.81.93404 5.36.931225.91.929166.46.92760 4.27.93795 4.82.93398 5.37.931185.92.929126.47.92758 4.28.93786 4.83.93392 5.38.93113 5.93.929096.48.92755 4.29.93778 4.84.93386 5.39.931095.94.929066.49.92753 4.30.93769 4.85.93380 5.40.93105 5.95.929036.50.92751 4.31.93761 4.86.93374 5.41.931005.96.929006.51.92748 4.32.93752 4.87.93368 5.42.930965.97.928976.52.92746 4.33.93744 4.88.93362 5.43.93092 5.98.928946.53.92743 4.34.93736 4.89.93357 5.44.93088 5.99.928906.54.92741 4.35.93728 4.90.93351 5.45.93084 6.C0.92887 6.55.92739 4.36.93719 4.91.93345 5.46.93080 6.01.928846.56.92736 4.37.93711 4.92.93340 5.47.93076 6.02.928816.57.92734 4.38.93703 4.93.93334 5.48.93071 6.03.928786.58.92732 4.39.93695 4.94.93328 5.49.93067 6.04.92875 6.59.92729 4.40.93687 4.95.93323 5.50.93063 6.05.928726.60.92727 4.41.93679 4.96.93317 5.51.93059 6.06.928696.61.92725 4.42.93672 4.97.93312 5.52.93055 6.07.928666.62.92722 4.43.93664 4.98.93306 5.53.93051 6.08.928636.63.92720 4.44.93656 4.99.93301 5.54.93048 6.09.928606.64.92718 4.45.93648 5.00.93296 5.55.93044 6.10.92857 6.65.92715 4.46.93641 5.01.93290 5.56.93040 6.11.92854 6.66.92713 4.47.93633 5.02.93285 5.57.93036 6.12.92851 6.67.92711 4.48.93626 5.03.93280 5.58.930326.13.928486.68.92709 4.49.93618 5.04.93274 5.59.93028 6.14.928466.69.92707 4.5C.93611 5.05.93269 5.60.93024 6.15.92843 6.70.92704 4.51.93603 5.06.93264 5.61.93021 6.16.92840 6.71.92702 p- 4.52.93596 5.07.93259 5.62.93017 6.17.928376.72.92700 4.53.93589 5.08.93254 5.63.93013 6.18.92834 6.73.92698 4.54.93582 5.09.93249 5.64.93009 6.19.92831 6.74.92696 4.55.93574 5.10.93244 5.65.93006 6.20.92829 6.75.92693 4.56.93567 5.11.93239 5.66.93002 6.21.92826 6.76.92691 4.57.93560 5.12.93234 5.67.92998 6.22.92823 6.77.92689 4.58.93553 5.13.93229 5.68.92995 6.23.92820 6.78.92687 4.59.93546 5.14.93224 5.69.92991 6.24.92818 6.79.92685 4.60.93539 5.15.93219 5.70.92987 6.25.92815 6.80.92683 4.61.93532 5.16.93214 5.71.92984 6.26.92812 6.81.92681 4.62.93526 5.17.93209 5.72.92980 6.27.92809 6.82.92679 4.63.93519 5.18.93204 5.73.92977 6.28.92807 6.83.92676 4.64.93512 5.19.93199 5.74.92973 6.29.92804 6.84.92674 4.65.93505 5.20.93195 5.75.92970 6.30.92801 6.85.92672 4.66.93499 5.21.93190 5.76.92966 6.31.92799 6.86.92670 4.b7.93492 5.22.93185 5.77.92963 6.32.92796 6.87.92668 4.68.93485 5.23.93180 5.78.92959 6.33.92793 6.88.92666 4.69.93479 5.24.93176 5.79.92956 6.34.92791 6.89.92664 4.70.33472 5.25.93171 5.80.92952 6.35.92788 6.90.92662 4.71.93466 5.26.93167 5.81.92949 6.36.92786 6.91.92660 4.72.93460 5.27.93162 5.82.92945 6.37.92783 6.92.92658 4.73.93453 5.28.93157 5.83.92942 6.38.92780 6.93.92656 4.74.93447 5.29.93153 5.84.92939 6.39.92778 6.94.92654 4.75.93441 5.30.93148 5.85.92935 6.40.92775 6.95.92652 4.76.93434 5.31.93144 5.86.92932 6.41.92773 6.96.92650 4.77.93428 5.32.93139 5.87.92929 6.42.92770 6.97.92648 4.78.93422 5.33.93135 5.88.92925 6.43.92768 6.98.92646 4.79.93416 5.34.93131 5.89.92922 6.44.92765 6.99.92644

M K(M) M K(M)! K(M) K(M) M K(M) 7.00.92642 7.21.92603 7.42.92568 7.62.92537 7.82.92508 7.01.92640 7.22.92602 7.43.92566 7.63.92535 7.83.92506 7.02.92638 7.23.92600 7.44.92565 7.64.92534 7.84.92505 7.03.92637 7.24.92598 7.45.92563 7.65.92532 7.85.92504 7.04.92635 7.25.92596 7.46.92561 7.66.92531 7.86.92502 7.05.92633 7.26.92595 7.47.92560 7.67.92529 7.87.92501 7.06.92631 7.27.92593 7.48.92558 7.68.92528 7.88.92500 7.07.92629 7.28.92591 7.49.92557 7.69.92526 7.89.92498 7.08.92627 7.29.92590 7.50.92555 7.70.92525 7.90.92497 7.09.92625 7.30.92588 7.51.92554 7.71.92523 7.91.92496 7.10.92623 7.31.92586 7.52.92552 7.72.92522 7.92.92494 7.11.92622 7.32.92584 7.53.92550 7.73.92521 7.93.92493 7.12.92620 7.33.92583 71.54.92549 7.74.92519 7.94.92492 7.13.92618 7.34.92581 7.55.92547 7.75.92518 7.95.92490 7.14.92616 7.35.92579 7.56.92546 7.76.92516 7.96.92489 7.15.92614 7.36.92578 7.57.92544 7.77.92515 7.97.92488 7.16.92612 7.37.92576 7.58.92543 7.78.92513 7.98.92486 7.17.92611 7.38.92574 7.59.92541 7.79.92512 7.99.92485 7.18.92609 7.39.92573 7.60.92540 7.80.92511 8.00.92484 7.19.92607 7.40.92571 7.61.92538 7.81.92509 8.01.92482 7.20.92605 7.41.92569'C1

1.01 7 = 1. F(M) VS M / _ 4

A.5 TABLE AND GRAPH OF F(S) VS. S (pps. 128-132) GRAPH: F(S) vs. S F(S) vs. -S 127

S F(S) -3.00.00001 -2.90.00001 -2.80.00002 -2.70.00004 -2.60.00007 -2.50.00013 -2.40.00022 -2.30.00038 -2.20.00064 -2.10.00107 -2.00.00173 -1.90.00277 -1.80.00436 -1.70.00674 -1.60.01023 -1.50.01529 -1.40.02246 -1.30.03247 -1.20.C4618 -1.10.C6646 -1.00.08908 -.90.12089 -.80.16161 -.70.21288 -.60.27640 -.50.35387 -.40.44690 -. 30.55695 -.20.68525 -.10.83274 00 1.00000.10 1.18722.20 1.39421.30 1.62039.40 1.86482.50 2.12627.60 2.40328.70 2.69424.80 2.99745.90 3.31121 1.00 3.63388 1.10 3.96392 1.20 4.29994 1.30 4.64071 1.40 4.98518 1.50 5.33249 1.60 5.68191 1.70 6.03290 1.80 6.38500 1.90 6.73789 2.00 7.09133 2.10 7.44515 2.20 7.79920 2.30 8.15342 2.40 8.50774 128

S F(S) 2.50 8.86213 2.60 9.21655 2.70 9.57100 2.80 9.92546 2.90 10.27993 3.00 10.63441 3.10 10.98888 3.20 11.34336 3.30 11.69784 3.40 12.05232 3.50 12.40680 3.60 12.76128 3.70 13.11576 3.80 13.47024 3.90 13.82472 4.00 14.17920 4.10 14.53368 4.20 14.88816 4.30 15.24264 4.40 15.59712 4.50 15.95160 4.60 16.30608 4.70 16.66056 4.80 17.01504 4.90 17.36952 5.00 17.72400 5.10 18.07848 5.20 18.43296 5.30 18.78744 5.40 19.14192 5.50 19.49640 5.60 19.85088 5.70 20.20536 5.80 20.55984 5.90 20.91432 6.00 21.26880 6.10 21.62328 6.20 21.97776 6.30 22.33224 6.40 22.68672 6.50 23.04120 6.60 23.39568 6.70 23.75016 6.80 24.10464 6.90 24.45912 7.00 24.81360 7.10 25.16808 7.20 25.52256 7.30 25.87704 7.40 26.23152 7.50 26.5860C 7.60 26.94048 7.70 27.29496 7.80 27.64944 7.90 28.00392 129

S F(S) 8.00 28.35840 8.10 28.71288 8.20 29.06736 8.30 29.42184 8.40 29.77632 8.50 30.13080 8.60 30.48528 8.70 30.83976 8.80 31.19424 8.90 31.54872 9.00 31.90320 **** ALL INPUT DATA HAVE BEEN PROCESSED. AT LOC 26650 130

12 10 f(s) = Es-+ v/''( 1 + erf s) 9 8 7 6 4 3 2 0 -3 -2 -I 0 1 2 3 S F(S) vs. S 151

f(s) S2+ /rFS ( +erf(s)) -4 10 -6 l0 10 10 -9 -10 08 _69 _ -II -5 -4 -3 -2 -I. --- S F(S) vs. -S 132

A.6 TABLE OF g, g/go, AND (g/go)(go/R) VS. ALTITUDE FOR AIR FROM 0 TO 100 km (pp. 134-135) 133

AL G/GC (G/GO)(GO/K) 0 9.8066 1.000 34.1540 1 9.3035.9997 34.1434 2 9.8005.9994 34.1328 3 9.7974.9991 34.1220 4 9.7943.9987 34.1112 5 9.7912.9984 34.1005 6 9.7882.9981 34.0899 7 9.7851.9978 34.0791 U 9.7820.9975 34.0683 9 9.7789.9972 34.0577 10 9. 7759.9969 34.0471 11 9.7728.9966 34.0363 12 9.7697.9962 34.0255 13 9.7666.9959 34.0149 14 9.7636.9956 34. 042 15 9.7605.9953 33.9936 1l 9.7575.9950 33.9830 17 9.7544.9947 33.9722 18 9.7513.9944 33.9614 19 9.7482.9940 33.9503 20 9.7452.9937 33.9402 21 9.7421.9934 33.9295 22 9.7391.9931 33.9189 23 9.7360.9928 33. 9083 24 9.7330.9925 33.8977 25 9.7299.9922 33.8870 26 9.7269.9919 33.87.64 21 9.7238.9916 33.8658 28 9.7208.9913 33.8552 29 9.7177.9909 33.8446 30 9.7147.9906 33.6339 31 9.7116.9903 33.8233 32 9.7086.9900 33.8127 33 9.7056.9897 33.8022 34 9.7026.9894 33.7918 35 9.6995.9891 33.7812 36 9.6965.9888 33. 705 37 9.6934.9885 33.7599 38 9.6904.9882 33. 7493 39 9.6874.9878 33.7389 40 9.6844.9875 33.7284 41 9.6813.9872 33.7178 42 9.6783.9869 33.7072 43 9.6753.9866 33. 6967 44 9.6723.9863 33.6863 45 9.6693.9860 33.6758 46 9.6663.9857 33.6654 47 9.6632.9854 33.6547 48 9.6602.9851 33.6441 49 9.6557.9846 33.6285 50 9.6512.9842 53.6123 51 9.6497.9840 33.6076 52 9.6482.9838 33.602 3 53 9.6451.9835 33.5911 54 9.6421.9832 33.5811 154

ALT G G/GO (G/GO) (GO/) 55 9.6391.9829 33.5706 56 9.6361.9826 33.5602 57 9.6331.9823 33.5497 58 9.6301.9820 33.5393 59 9.6271.9817 33.5288 60 9.6241.9814 33.5184 61 9.6211.9811 33.5079 62 9.6181.9808 33.4975 63 9.6151.9805 33.4871 64 9.6121.9802 33.4766 65 9.6091.9799 33.4662 66 9.6061.9796 33.4557 67 9.6031.9793 33.4454 68 9.6002.9790 33.4352 69 9.5972.9786 33.4247 70 9.5942.9783 33.4143 71 9.5912.9780 33.4038 72 9.5882.9777 33.3934 73 9.5852.9774 33.3831 74 9.5823.9771 33.3728 75 9.5793.9768 33.3624 76 9.5763.9765 33.3519 77 9.5733.9762 33.3415 78 9.5703.9759 33.3310 79 9.5673.9756 33.3207 80 9.5644.9753 33.3105 81 9.5614.9750 33.30C2 82 9.5585.9747 33.2899 83 9.5555.9744 33.2795 84 9.5525.9741 33.2690 85 9.5495.9738 33.2588 86 9.5466.9735 33.2485 87 9.5436.9732 33.2382 88 9.5407.9729 33.2279 89 9.5377.9726 33.2177 90 9.5348.9723 33.2074 91 9.5318.9720 33.1970 92 9.5288.9717 33.1867 93 9.5259.9714 33.1763 94 9.5229.9711 33.1659 95 9.5199.9708 33.1557 96 9.5170.9705 33. 1454 97 9.5140.9702 33.1351 98 9.5111.9699 33.1248 99 9.5081.9696 33.1146 1CO.5052.9693 33.1043 **** E sRRO1 RETUJRN T1 SYSTEM 155

UNIVERSITY OF MICHIGAN I 01111 0tlM ilE 3 9015 03525 0516