ANALYSIS OF A FALLING SP-HERE EPERIMENT FOR MEASUREMEiT OF UPPER-ATMOSPiEIRE DENSITY ADID WINDS by J. Otterman, I. J. Sattinger, and D. F. Smith 15 September 1959 2873- 3-T Contract AF 19(604)-5205 The University of Michigan Willow Run Laboratories P. 0. Box 2008 Ann Arbor, Michigan

September, 1959 ANALYSIS OF A FALLING SPHERE EXPERIME!T FOR MEASUREMENT CB U P -ATSP'i ER DEN3I Y At by J. Otterman, I. J. Sattinger, and D. F. Smith 1. General Discussion 1.1 Purpose of Study The research program discussed in this report is being conducted under Contract Number AF 19(604)-5205 for the Geophysics Research Directorate of the Air Force Cambridge Research Center. The purpose of the program is to study methods of conducting an experiment in which a "vp% measurements of air density and winds at high altitudes are carried out by means of a falling sphere ejected from a high-altitude rocket. The measurement of density and winds would be accomplished by measuring the drag acceleration oF the sphere by means of accelerometers mounted on an inertial platform carried in the sphere. The study is intended to establish the range of altitudes and winds over wnich useful data can be obtained and to recommend the equipment and techniques to be used in conducting the experiment. 1.2 Significance of the Experiment to obtain data on The purpose of the described experiment is la -xa raoka xal density and wind structure of the upper atmosphere. Existing methods of high-altitude air density and wind determination have in the past provided useful data,but are limited in attainable accuracy and altitude range. The investigation described in this report indicates triat the proposed experiment promises to yield relatively accurate data about atmospheric density up to aleast 150 km 2.3 and 2.4 and data about horizontal winds up to at least 120 km. (See Section_/.) A variety of methods nave been used by other investigators for obtaining information on air density e CDeslt y h7 f the upper atmospiler~ has been studied by observing meteor trails (Whripple, 1352). A ground based technique of pulsed searchlighting

- 2 -et aci., has been used (Frieland 4 1956). l o ocket techniques have extended the range of measurements and provided a more direct method of attack. One rocket technique which has been successfully used for determining air density consists of the measurement of the drag acceleration of a falling sphere. In the experiment, as being carried out presently (Jones, 195 a,b; et al., Jones B&WXxc 1958), the acceleration of the sphere is measured by a transit is time accelerometer, in which a bobbin/periodically clamped and released. When released, the bobbin is free to travel in any direction until it makes contact with the inside surface of the housing in which it is carried. The time taken for the bobbin to make contact is measured and telemetered to the ground. Since no drag acts on the bobbin, the transit time provides information about However, is obtained the magnitude of the drag acceleration5t/ no information/on the direction of acceleration. Thus, the telemetered signals do not provide direct information of about the trajectory either of the missile or/the sphere after the ejection, and wind determination is not possible. The range of the experiment is 90 km et al., (Jone/, 1958). In spite of the great advances in research techniques, the density of the upper atmosphere has not been sufficiently determined. Thus, at 100 km altitude, reports of different researchers show a variation by a factor of 4 (Nicolet, 1959). Winds observing:XSQUBWff have been determined by/the motions of meteor trails (Whipple, 1952 MMl;Robertson a&ket a4 1953). At an altitude of about 80 km, wind information can be obtained from the movements of noctilucent clouds (Ludlam, 1957). This technique is limited to high geographic latitudes. Rocket techniques for wind measurement consist of observing an artificial sodium cloud 2.

-3 -(Edwards, 1956), chaff (Smith, 1958) and propagation of sound from grenades at high altitudes (Stroud dt al.,, 1958). Of the tnree above metnods, only the sodium method extends somewnat above 100 km. At the present the wind information up to 100 km is rather incomplete, and above 100 km very scanty. The atmosbotch pheric motions above 100 km are of great alrrent scientific interest,/ %neoretically, since their theory hinges upon the characteristics of the atmosphere as a wnole, and more practically, since tney affect radio communication and radio astronomy. Hines (1959) in his kfUMZ article on motions in the ionosphere lists almost 70 references, practically all recent, dealing with various aspects of this problem. The opening sentence of his conclusions reads "Perhaps the most immediate conclusion that can be drawn from all these remarks is that a very great deal has yet to be learned about motions in the ionosphere" While the results sought in this experiment are of basic scientific interest, to the they -have a direct application to the flights of ballistic missiles and/study of effects of explosions at high altitudes. 3

8 Sept 1959 The University of Michigan * WILLOW RUN LABORATORIES /h3 Description of the Experiment /,"3./ Method of Data Analysis The falling sphere method for the determination of winds and density in the upper atmosphere consists of measuring thle drag acceleration of a sphere IJones and Bartman, 1956) ejected from a high altitude rocket. Thile drag acceleration vector a is related to the drag force vector D by the equation D = ima 1.1 where m is the mass accelerated by the drag force. The drag force is a function of the air density ACJS | c D. = — > -1.2 where A is the sphere cross-section area, Cd the coefficient of drag and c the the sphere's velocity vector relative to/ambient air. The coefficient of generally varies drag Cd/depends on the Mach number and Reynolds number, but rather slowly in most of the range of interest. Thus, if m and A are known, and a is measured, the density,, can be determined. Data on winds can also be determined from the same measurements, by making use of the fact that the drag acceleration and the velocity of the sphere relative to the ambient air are colinear and in opposite directions. It is shown in Section 2.2 that a solution of the equations representing conditions existing at any instant during the flight can be performed to determine horizontal winds if the assumption is made of zero vertical winds. This is the same assumption that is currently being used in the data xeduction of the rocket grenade experiment for upper atmosphere temperature and windso(Stroud et al., 1958). L/

The University of Michigan ~ WILLOW RUN LABORATORIES The vertical winds can be determined if the data reduction on both the upleg and downleg of the sphere trajectory for a given altitude is carried out simultaneously. If the density and wind conditions are assumed constant in the time interval between the upleg and downleg (this time interval will be of the order of 10 minutes))an overdeterminate solution is obtained and can be solved for the best fit of the data for the density and the wind components. Alternatively, a determinate solution can be obtained for a given altitude by assuming that the horizontal winds do not remain constant. l,3.Experimental Procedure In order to measure drag acceleration, an inertial platform, similar to those used for the guidance of aEi ~fIfIballistic missiles, is mounted in the sphere. The inertial platform contains three accelerometers measuring acceleration along, three mutually-perpendicular axes, which are or rotated at a constant rate maintained fixed/with respect to inertial space. The experiment thus consists of shooting a rocket carrying an inflatable sphere inside which is located an inertial platform. At an altitude of approximately 50 km this instrumentation system is ejected from the rocket and the folded container is inflated to form a sphere with the inertial platform at its center, the total system weighing in the neighborhood of 120 lbs. The sphere continues on a trajectory which is, except for drag, a freefall trajectory. Its peak should be of the order of 250 km. of thrust and drag accelerations are The accelerometer data/a telemetered to the ground throughout the they are possibly flight, whereM J converted/by means of an electronic computer into data cy)

on the velocity and position of the rocket and, subsequent to tiae ejection, of the sphere. The computation involves integration of rocket and sphere measured acceleration due to thrust and drag, and computed acceleration due to gravity, and conversion from an inertial system of coordinates to an earth system of coordinates. Seven coordinate systems have been studied; however, no choice has yet been made on which system to use. Values of the components of acceleration, velocity, and position can then be used to-determine density and winds as a function of altitude. The basic equations are derived in Section 2.1. If radar or optical tracking systems are available at the site of the experiment, they may be used to supplement the accelerometer data in determining sphere velocity and position as a function of time. They cannot, however, provide data of sufficient accuracy to determine drag acceleration, since the drag is so small at the altitudes of greatest interest that the deviation of the sphere from a trajectory without drag would be too small to detect from the ground. 1.4 Threshold Altitudes for Density and Wind Determination The analysis given in Section 2.3 provides an indication of the greatest altitude at which satisfactory data can be obtained on air density. In this analysis a 12-ft. (3.66 m.) diameter, 120 lb (5 4.4 kg.) sphere is assumed to fall vertically through the atmosphere. The vertical velocity which it must have as it falls through a given altitude in order to produce a drag acceleration of lxlO g is determined. (This acceleration value is 5 times the assumed accelerometer threshold error of 2xlO-5g.) This velocity can then be used to compute the required peak of the trajectory.

Assuming the Model A atmosphere of Kallmant(l958), the analysis shows that the density-measuring threshold occurs at an altitude of about 166 km if the sphere has a velocity of 1500 m/sec, which is equivalent to a trajectory peak of 166 / 120 = 286 km. Since the accelerometer error is the dominant one at high altitudes in the determination of density, the density can thus be determined within about 20-. Assuming the 1956 ARDC standard atmosphere (Minzner and Ripley, 1956) the density threshold occurs at about 147 km. The analysis givei n Section? provides an indication of the greatest altitude at which satisfactory data can be obtained on winds. It is assumed arbitrarily that the wind threshold will occur when the error in the horizontal wind component is 10 m/sec. The crucial factor in determining the altitudee limit to wind measurement is the instrument error caused by the accelerometers. As assumed previously, for small accelerations the accelerometer error is approximately 2x10 5g. For the Model A atmosphere, the wind measurement thres-8 hold then will occur at about 120 km where the density is 6.9x10- kg/m3. For the 1956 A;DC standard atmosphere the wind threshold will occur at about 116 km. values used for the tle The/weight of the sphere and the accuracy capabilities of/inertial package representp a realistic estimate of capabilities of present day systems. The investigation thus indicates that the proposed experiment promises to yield relatively accurate data about atmospheric density up to at least 150 km, and data about winds up to at least 120 km.

The University of Michigan ~ WILLOW RUN LABORATORIES A Equipment Considerations In this section, tile major features of the instrumentation Hystem to be used in the experiment are discussed. Although final decisions have not yet been reached regarding the selection of components and operating methods, certain assumptions have been made in this discussion to permit the presentation of at least a first order design of the system. Further theoretical and experimental analysis will be necessary to arrive at the most suitable design. /iA Inertial Platform System The inertial platform system consists of a 4-gimbal stabilized platform, a group of electronic circuits for controlling the platform and operating the gyros and accelerometers, and the necessary power supply. Accuracy of the accelerometers and the gyros of the inertial platform determine the obtainable accuracy of the experiment. By taking all possible precautions before take-off to eliminate bias errors, it is believed that accelerometer errq can be held to values as low as 2x10-g. ii Gyro the experiment, which will continue for about aaBBa1w rift durinB twenty minutes, is expected to be about.002 radians. The over-all dimensions and weight of the inertial platform equipment influence the size of the rocket required to send the sphere into the upper atmosphere. In addition, size and weight affect the area-to-mass ratio of the falling sphere and hence the sensitivity of the experiment. The weight the of the inertial platform system, including the batteries and/transmitter, is of the order of 80 lbs. The dimensions of the platform are approximately 10 inches in diameter and 15 inches long.?

/, 2 Inflatable Sphere An inflatable mylar sphere of 12-ft. diameter will be used in the experiment. packed It will be/ i into a section of the missile with all of the experimental equipment inside the folded sphere. Details of ejection mechanism and equipment attachment remain to be worked out. However, the techniques have been used various successfully by/tl C groupsX (Jones, 1952; Kenletand Patterson, 1959, Arthur D. Little inc., 1958). Particular attention must be given to the problem of limiting the Iates angular of angular velocity and/acceleration of the sphere throughout the flight so that they do not exceed the capabilities of the inertial platform control system. sphere Adequate roll control of the missile, and avoidance of/ejection transients are approaches which can be used to avoid this difficulty. The inflated sphere will have an ingternal pressure of the order of 2 inches of water. It will therefore begin to collapse at a point in the trajectory where the total pressure exceeds this value. This will not occur until the sphere altitude is below the region of interest, and actually under altitudes which will provide a check point for accuracy of density determination. In view of the high cost of the inertial platform, it would be very desirable to avoid destruction of the platform at impact. This could be accomplished if a parachute could be incorporated into the assembly which would open automatically at some prescribed altitude. Inertial platforms are reasonably rugged devices some of which can withstand accelerations at high as 20g without damage anJ accelerations up to 50g with only minor damage. The possibility of successfully recovering the platform by the use of a parachute therefore deserves further consideration.

The University of Michigan a WILLOW RUN LABORATORIES i.'.3 Telemetering Equipment The form and content of the signal output of the accelerometers affects the method of nuttioning and telemetering the signal and hence the accuracy of the data received 0I B0&Mf = on the ground. The method of i1ftp transmission assumed in this discussion is based on the use of an accelerometer which produces an output signal consisting of a precision current which periodically reverses direction at a frequency of several hundred cycles per second. The difference in the time duration of current in each direction is amndication of the acceleration. The instants at which zero crossover occurs can be transmitted to the ground by ans of a 30 kc signal channel with a 30 db signal-to-noise ratio. The received signal can be converted to a measurement of acceleration by referring it to a 1 mes clock pilse system. It is believed that this technique can also be adapted to an acceerometer having a non-pulsating d-c output. Thus, the telemetry system must provide for three 30 kc signal channels with a 30 db signal-to-noise ratio. This can be provided by means of a conventional FM/FM telemetry system. A turnstile antenna mounted inside the sphere can provide a non-ditectional pattern of radiation, so that ground reception will not be dependent on the attitude of the sphere. 1^5.4 Auxiliary Equipnent The sphere must carry a primary source of power for the inertial platforn and communication equipament. Nickel-cadmium batteries, which may be conservatively assumed to have a rating of 10 watt-hrs per lb., can be used for

The University of Michigan * WILLOW RUN LABORATORIES this purpose. Additional equipment may also be required to convert this d-c power to a form compatible with the equipment using it. Power dissipation capabilities of the sphere should also be investigated in relation to the power produced by the enclosed electronic equipment. Coolant bottles can be used, if necessary, to absorb the excess heat. o.6.5 Missile Characteristics No detailed attention has yet been given to the selection of a missile for the specific application discussed in this r*port. The primary requirements which must be met include the following: 1. The payload will weigh approximately 120 lbs. 2. The payload will have a diameter of approximately 12 inches. 3. The payload must be carried to an altitude of 50 km. and ejected into a trajectory reaching an altitude of ibout-250 km. 4. Peak acceleration during powered flight should preferably not exceed 15g. 5 An antenna must be provided to transmit drag acceleration data to the ground..'6 aGround Based Equipmant As in the case of the missile characteristics, no detailed attention has been given to the selection of the ground-based equipment required for this experiment. The functions which must be performed by the ground-based equipment include the following: 1. Handling and launching of the missile. //

The University of Michigan * WILLOW RUN LABORATORIES 2. Checkout of the iltrumentation equipment, and in particular, alignment of the inertial platform system. 3. Receipt and conversion of telemetering signals to proper form for entry into data-processing system. 4. Digital computation. (This function might be performed at a centralized computing facility, if more convenient. In this case, magnetic tape recording equipment would be required at the site of the experiment). 5. Radar or optical tracking, if desired, to provide supplementary trajectory information. 16.; Cost of the Bperiment The cost of the inertial stable platform package with accelerometers ranges from $80,000 to $200,000 per unit in small batches. It is suggested thatan initial series of experiments should provide for three to five rocket shots. an A very rough cost estimate for such/initial series puts the cost at 1.2 to 2.5.million dollars. /2.

2. Detailed Discussion the ^ ist In/!i sections of this report, tne most important results of/investigation into methods of conducting a falling-sphere experiment to determine highaltitude density and winds have been discussed in a general manner. The detailed analysis of available equipment and techniques and tile derivation of mathematical relationships on which the previous discussion was based are presented this section. inn The basic equations which describe the motion of a falling sphere subsequent to ejection and inflation are described in Section. An explanation of the method of calculating density and winds from these basic equations is Included. The primary item of instrumentation for obtaining the data from which density and winds are to be computed is the inertial navigation equipment. The factor of major importance in the selection of this equipment concerns the accuracy acceleration determined with which/data can be/! si^;S however, other factors, such as size, weight, cost, and adaptability to environment must also receive due consideration. All of the above factors are discussed in Section. The potential value of the experiment depends on the altitude range over which useful information can be obtained. A mathematical analysis is presented in Section 2 which results in the computation of threshold of maximum altitude at which air density can be determined. A corresponding analysis for wind threshold is presented in Section,. /3

-2 -2. 1 BA^SIC EQUI ATmIONS A set of equations can be written describing the forces and motion of the falling sphere subsequent to the ejection and inflation. From the equations, density and winds can be calculated, as explained hereunder., rag and thrust; ae; an; az East, North, and Up components of/!II 3 S tne acceleration in/earth-coordinate system. v; v; v East, North, and Up components of sphere velocit-y relative to the earth-coordinate system. ce; c; cz East, North, and Up components of sphere velocity tlae relative to the ambient air in/earth-coordinate system. we; wn; wz East, North, and Up component of wind. The basic drag equation is: 2 2 2 CL A 2 2 2 (e t a a2 1a CenA (Ce + cn + c) 2.1 I ~ 2m where Cd is the coefficient of drag, e the density, A the cross-section of the sphere and m its mass.+ he drag acceleration and the velocity of the \ sphere relative to the ambient air are colinear and in the pposite <f tfe aerodynamic forces caused bty l e spinning of t:.e sphere are neglected,. direction/iiWaresuits in the following equations: an nC -- 2.3 z //

(3) By combination of Equations (1), (2), and (3) 2m _az^ 2.2 Cd A c2 [1 + (ae/a)2 + (an/az)2 The wind component equations are as follows: a W -Ve Ce Ve - Cz 2.5 a Wn V - n n - n - c 2.6 z W = v - c z z z 2.7 2.2 through 2.7, In Equati ons/n ve, vn, vz and ae, an, az are known trajectory data. The equations contain seven unknowns; ce, Cn, Cz, we, wn, wz, and e,.Jhus)the solution is impossible solely on the ulieg some or solely on the downleg without/additional assumption. It is proposed to use the assumption of zero vertical winds, the same assumption that is reduction currently being used in the data/!WlSW r of the rocket grenade experiment (Stroudp for upper atmosphere temperature and winds EQB6SQ9BX 1958). w 0 2.8 The errors introduced by this assumption both in density and wind determination are discussed later, If the data reduction on both the upleg and doWnleg is carried out simultaneously for the same altitude, the vertical winds can be determined on the basis of certain assumptions. This, of course, is possible only at altitudes higher than the ejection of the sphere, / ~

-4 -If the density and wind conditions are assumed constant in the time interval between the upleg and downleg (th-is time interval will be of the order of Eqs. 10 minutes) we have at each altitude point 12 equations (each of the /tB_ w twice) with 10 unknowns; Ce, cn Cz, we, Wn, wz, and ce, cn; cz, * * * where Ce, Cn, cz refer to upleg and Ce, Cn, and cz to downleg only. This set of equations is overdeterminate, and can be solved for t-ie best fit in we wn, wz and 0. Alternately, it can be assumed tnat the horizontal winds * we, wn do not remain constant. Then two additional unknowns, i.e., we and w appear and we have a system of twelve equations with twelve unknowns. a. _ N INERTIAL NAVIGATION EQUIPE2iT In order to compute trie information required for air density and wind determination, it is necessary to obtain altitude the components of sphere acceleration, velocity, and/t3 X XXMMOW with respect to a known coordinate system. The primary source of this information is an inertial platform system mounted in the sphere. It would be possible to obtain from the accelerometers of this system either the (and a gravWty- eomputer) components of acceleration or, if integrating accelerometers/are- used, the components of velocity. From the accelerometer outputs, all data on acceleration, velocity, and position could be computed in tie sphere and transmitted to the ground. Alternatively, only the acceleration data or the velocity data could be transmitted, from wnich a ground-based computer could compute the remaining information. The alternative of computing all information within tie sphere requires that the inertial navigation system contain accurate computing equipment but minimizes the accuracy required of the telemetering system. The alternative

-5 -of transmitting only acceleration data to the ground requires a telemetering system handling only three channels of information but requires high accuracy (in the neighborhood of 0.1 percent of the transmitted quantity) in the telemetering operation. The latter alternative is considered preferable, because of the importance of minimizing size and complexity of the airborne equipment. The following discussion of inertial navigation equipment is based on the assumed use of this latter alternative. In order to determine which inertial navigation system would be most suitable for the purpose, a number of systems have been investigated. The following manufacturers have been contacted during this investigation: Arma Division, American Bosch Arma Corporation Reeves Instrument Corporation Ford Instrument Company Litton Industries Honeywell Corporation Autonetics Division, North American Aviation Nortronics Division, Northrop Corporation AC Spark Plug Division, General Motors Corporation Kearfott Company, Inc. The following paragraphs summarize the information -r obtained on the characteristics of inertial navigation systems which are important to the application under consideration. t Accuracy Accuracy of the accelerometers and the gyros of the inertial platform determine the obtainable accuracy of the experiment. Accelerometer accuracies /7

- 6 - of 0.1% of the measured acceleration are sufficient to permit computation of good velocity and position data. No difficulty is anticipated in meeting this requirement with any of the systems investigated. In addition, threshold acceleration errors should be kept as low as possible in order to maximize the altitude at which useable drag data can be obtained. Accelerometer errors are in tie form of bias (zero uncertainty), threshold, and scalefactor and nonlinearity errors. 3By taking all possible precautions before take-off, bias errors can be minimized. Moreover, it is believed tiat the bias error can be eliminated through an appropriate correction of after-flight data, as the following consideration shows. An analysis of the trajectory indicates that close to the peak, the drag in tne z direction should be, under ideal conditions, approximately as given in Figure /. TDie time origin in this Iigure: the corresponds to/peak, as determined by vt = 0.,, At f trMeasured ~\ i o > 10 A Acceleration Zero { I... Bias > *;f I L<.t C D, ( Sec '. True I \ ^-^Acceleration Figure 1 Alternatively, (this peak time can be determined alternately by minimum-slope point of the drag data.) However, in Ik case Lozero bias exists, the curve will be lifted as indicated in the figure. up or down by the amount of the bias / Thus, the bias can be determined, and observing subsequently eliminated by/the z accelerometer readings at the trajectory peak. /B

It is thought that the accelerometer data close to the threshold can contain spurious readings due to the sphere spin. If the center of rotation of the outer body around the stable platform does not correspond to the center of gravity of the outer body, the stable platform will be subject to sinusoidal accelerations at the spin frequency. These accelerations can seriously mask the drag accelerations at the threshold, and care must be exercised in balancing the sphere and minimizing the spin. Even so, it is thought that these sinusoidal accelerations will be present to same degree, but it is believed that their relatively steady pattern can be discerned and largely filtered out by data inspection. These two considerations, of moving the zero bias and spin effects, present disadvantages at on-line data reduction by computers during flight at altitudes of density threshold and wind threshold determination. /9

- 6b - -5 Threshold errors of accelerometers as low as 2l O g are claimed by more than one manufacturer, and this value is used in the analysis of errors. Scale-factor and nonlinearity errors can be neglected, insofar as drag and wind measurements are concerned. Gyro accuracies are generally given as the sum of several components, some of which can be eliminated by a preflignt check or by acceleration compensation. The eliminated 4 random error which cannot be/!lnXZ" tends to be in the neighborhood of 5x10 radians per hour or less. Without acceleration compensation, gyro drift during the experiment, which will be about 20 minutes long, is expected to be about.002 radians. It is this value for gyro drift that is used in Sections 2.3 and 2.4 on density and wind measurement errors. 2.2.2 Size and Weight The over-all dimensions and weight of the inertial navigation equipment influence the size of the rocket required to send the sphere into the upper atmosphere. In addition, size and weight affect the area-to-mass ratio of the falling sphere and hence the sensitivity of the experiment. The weight of the inertial platform system, including the batteries and transmitter, can be of the order of 80 lbs. This estimate is based on information concerning two of the lightest platform systems of those investigated. One of the smallest platforms is 9.5 inches in diameter and 14.5 inches long. Smaller

- 7 - systems are under development which would further reduce this space requirement. ^ _j Spin Capability Tne Jaertial platforms investigated were in all cases four-gimballed systems. Although the four-gimballed system is heavier and costlier, it avoids the possibility of gimbal-lock. In addition to avoiding gimbal lock, the system must be able to withstand continuous tumfbling, of the sphere without serious degradation of system accuracy. Allowable spin velocities ranged from 3 to 7 radians per second up to 20, and perhaps 30, radians per second. Infoimation on angular acceleration capabilities were obtained for one of the systems. Allowable accelerations ranged from 8 radians/sec.2 in pitch to 35 radians/sec.2 in azimuth. It is expected that it will be possible to maintain spin rates within the magnitudes mentioned, provided sufficient care is taken during the rocket ascent and sphere ejection periods. 9 Signal Output The form and content of the signal output of the accelerometers will affect the method of conditioning and telemetering the signal and hence the accuracy of the data received by the ground-based computer. Certain mechanizations of the signal conversion and transmission process appear to be capable of achieving the required accuracy. One method, which has been completely developed and is in current use, is based on the use of vibrating-string accelerometers. The output of this type of accelerometer consists of two a-c voltages, whose difference in frequency is proportional to the acceleration. It is possible to determine the acceleration during a short sampling period by a method which counts the integral and

- 8 - -8 -fractional number of cycles of each string to a high accuracy and determines the difference in these numbers for the two strings. Tssentially, this system is capable of obtaining high resolution by virtue of the fact that it can determine small fractions of a cycle, that is, fractions of a single increment of velocity change. It has the advantage that a conventional FM4/FM telemetering system of moderate accuracy can be used. Another method is based on the use of an accelerometer which produces an output signal consisting of a precision current which periodically reverses direction at a frequency of several hundred cycles per second. The difference in the time duration of current in each direction is an indication of the acceleration. The instants at which zero crossover occurs can be transmitted to the ground by means of a 30 kc signal channel with a 30 db signal-to-noise ratio. The received signal can be converted to a measurement of acceleration by referring it to a high-frequency clock pulse system. j3 d )( ^ock Resistance The resistance of the system to shock and vibration will affect the possibility of salvaging and reusing the equipment after each flight. For most systems investigated, available information consisted of eot -a es of allowable accelerations. Generally speaking, these systems w-i- all withstand at least 20g, and will sustain minor damage at 30g to 5gg. < X Cost The cost of an inertial platform system, complete witl platfoim electronics, ranges from $80,000 to $200,000 per uLit.

2.3 DENSITY DTERTMINATION 2.3.1 Threshold Altitude Assuming a particular sphere size and mass falling through a given altitudedensity distribution, the drag Equation 2.1 can be used to determine the necessary sphere velocity to give a minimum measurdble drag acceleration at a given altitude. The object is to find the maximum altitude at which the necessary velocity is reaeanably low. Neglecting winds, and assuming a vertically falling sphere, equation 2.1 with velocity as the independent variable is 2 2azm Cv A c K2.9 Table I is a tabulation of velocity vs. altitude using a 12 ft. (3.66m) diameter, 120 pound (54.4 kg) sphere and assuming the minimum determinable drag acceleration az to be lxlO' g. It is believed that the value of az used represents an acceleration that can be determined with an error of about 20% and velocities considered. The.euivalent free fall distance tabulated in Table I is the distance the sphere would have to fall to obtain thle required velocity (assuming that the only force acting on the sphere is gravity). The equation for the free fall distance is v2 h 2g 2.10 where h is the free fall distance and g is the average gravitational acceleration at the altitudes considered. Two altitude-density distributions are used for comparison in Table I; the ARDC model atmosphere (Minzner and Bi Ripley, 1956) and the Model A atmosphere of Kallmann(1959). According to Kallmann the Model A atmosphere c23

- 10 - a recent is a density curve which represents/mY smooth fit of rocket and satellite data. Assuming the Model A atmosphere, Table I shows that the density measuring threshold occurs at an altitude of t 166 km if the sphere has a velocity of 1500 m/sec, which is equivalent to a trajectory peak of 166 t 120 = 2 5 km. 2.3.2 Density Error Analysis To consider errors in density measurement, notice that equation (4) a, gives density as a function of Cd, cz, azfnd an. On a probabilistic basis, the percent error in density measurement is of the form 100 L K cza 7( V ae~tL an )2an 2 2 where (x) designates the expected rrors in x. BY use of equation (4) this becomes 10 7 2 C\2 2t 3i2 2 2 2ea a ar where 1B: (a/as)2 (an/a)2/ (within tee properorder of magnitude) For errors near the density threshold, the nominal values of the variablesA may be taken as:

Cd= 2 cZ= 1500 m/sec az l-4g- 10-3 m/sec2 ae= 2x10-4 m/sec2 (high value) an= 2x10'4 m/sec2 (high value) It is unlikely that ce and cn will be larger than 300 m/sec, even if a strong wind blows in tne direction opposite to ve or vi. From equations 2.2 and 2.3 the above values of ae and an have been detemnined by the relations ae= az ce/cz and an = az Cn/Cz From Equation 2.7 ^(cZ): (Zr)- e) 2.13 The state of the art of inertial guidance systems indicates that velocities can be determined to within 0.5 m/sec, thus ~$)= 0.5 m/sec. Assuming that the vertical winds, wz, encountered will not be larger than 10 m/sec, the use of the assumption c = vz implies 6(cz) ' 10 m/sec. The accelerometer errors arise from two sources, the platform misalignment, and the accelerometer itself. For a small error.)in platform alignment the error in the z accelerometer is 2 an2) 1 m az) az * (ae + an. 0 ) 2.14 The expected platform misalignment during the short flight duration of the experiment will be of the order of (- 0.002 radians. Inserting the proper values into equation 2.14 gives an acceleration error due to platform misalignment of (az) 2xl' 9 t 4 x 10'7 4 ~ - x 10-7 m/sec2

- 12 -which is small compared with the accelerometer instrument errors,as discussed in Section/^ ihese errors are in the form of threshold and lin2X earity errors. The threshold is about/10-5g and the linearity error is on the order of 10 a. Thus, for the z accelerometer i(a) 2x10'5g 10- a WS2xl0-4 m/eec (for small accelerations) By applying equations similar to/J the errors in the n and e accelerometers f for small accelerations, a. l.+-. A" (ar): (<ae )2xlO 4 m/sec2 -Bi.Q'-.......J _-f-..f...."-.. - t_ These above values are used in equation/ t to calculate the error in density at its threshold of measurement. An additional error in the determination of density is caused by the uncertainyv in the sphere's altitude. Assuming that the atmospheric temperature is constant over small changes in altitudes, the percent change in density as a function of error in altitude determination is (~ - l o100 exp (- z/IS) -100 /g where z is altitude error and HS is the scale height. Table II shows the error in density caused by the various factors discussed above. For a typical drag acceleration-altitude profile the vertical drag acceleration will remain small until the sphere reaches the subseqaently denser atmosphere an then increase to a peak of 5 to 10 g decreasing to a (Figure ptlt a steady 1g/((]?BI^ @ Ihe points at which the errors were calculated in

- 13 - Table II correspond to a vertical drag acceleration a of: 1. Density measurement threshold. 2. Wind determination threshold (discussed in next section). 3. A largeacceleration on the rise of the peak while cz is still large. s t a 4. The la celeration after the peak where cz is small. The value of cz used for the first three points was 1500 m/sec and a value of 100 m/sec was used for the last point. These values are smaller than would actually be encountered$ thus the cz)error in Table II is an upper bound. The altitude error in Table II was calculated from equation ^ using a value of(z)- 100 m. So far nothing has been said about the errors in density determination caused by uncertainty in the drag coefficient Cd. If the Cd term in equation _.P is not used, the resulting error would be that of the product Cd. Table II shows that the error in Q Cd then is quite small with the exception of the high and low altitudes where, respectively, the accelerometer sensitivity and lack of knowledge of vertical wind w, cause large errors. durin" t- bljor potion of this teimexit. The drag coefficient is a fuw cnc n aber a. d ReyrlMa- nabr.. Ep erimental da is.a i fo~ Cd i the Teh on blf~iach An1era m O to Ii and Reynolds numnbere fBom to H a(ffon rd rt 1956).^ mable a &I show t trajeqtoy dita nd M tac+ Pi^^ers ibd la 1ifoot difeter, 0 pound sphere drpped ~~frc^oOS j3w. aR^'^ e-e alcuiati were on thd IBM 6y0 coanputer '7

- 13a - Good data on the drag coefficient is lacking for conditions existing For satellite velocities and altitudes a drag coeffice.nt of 2 or 2.3 is usually a sumedA during a large portion of tris experiment.\ TI2e met hod used to calculate the / a drag coefficient depends upon the type of flow across the body. The type of flow is characterized in one of tie following t;iree phases: 1. a free molecular flow region, in which the mean free molecular path of the air molecules is large compared to the sphere dimensions, 2. a slip flow region, in which the molecular mean free path is small compared to the boundary layer about the sphere, but not small enough to be neglected, and 3. the continuous flow region, in which the molecular mean free the medium can be considered as a continuum. path is very small an The free molecular flow region can be defined as the region in wh-ich number M lO 10 ere M is Mach/CaK and Re is the Reynolds number. The ratio 14 is an Re(/ ) /Re indication of the ratio of molecular mean free path to sphere diameter. In this region the drag coefficient is a function of the ratio 9 where e is the (Petersen, 195m) sphered velocity and ci is the mean molecular velocity. Table III shows trajectory data and Mach and Reynolds numbers for a 12 foot diameter, 120 Figure a shws acceleration and velocity curves pound sphere dropped from 300km (no winds)/ The calculations were made on the IBM 650 computer at thle Willow Run Laboratories using the APDC Standard Atmosphere. Calculating M from the computer data indicates that the sphere Re is in the free molecular flow region for altitudes above 145 km. Petersen (1956) d1r

- 13b - shows theoretical results of Cd vs, c. Using values of velocity c from Ci che computed trajectory, the drag coefficient varies from 2.8 at166 km LO 2~4 a, 145 kmo The error in Cd determination in this region has been taken aslO for use in Table IIo Figure 3 shows a plot of M and Re versus altitude using tue computer data. For flow regions other than free molecular flow, the drag coefficient has been determined mainly by experimental methods, Experimental data to about 2c, accuracy, on Cd'is available for Mach numbers up to 4.0 and Reynolds numbers ranging 1956b from 10 to 106 (Jones and Bartman, XDQG';, May,1957). As shown in TPBE Figure 3 Vffi the Mach numbers of the sphere are above 4 until the sphere has descended to an altitude of about 50 km. hBDDED QSSrDaB8BES BLFor use in Table II the error in Cd has been taken as 2% for altitudes below 50 km and as 10% for altitudes between 50 km and 145 km where data on Cd hal&not yet been found.

2.4 Wind Deteinat 14o 2.4.1 Wind. Error AnalysiF?~ ---~ with respect to e determination. The same iscussion will hold for w 2a tnermitis a oDeteitination 2.4.1 Wind E ror Analyesi c wz (ca s)ct to i on the assumption thate expect errors are indepen ae, a term in equination. The same discussionich wll hold for the n a deterministic bas.is*or on a probabilistic basis; on thle assumption that the errors are indepenrdent. The error terms Vr and z^vill be taken as 0.5 m/sec and 10 m/sec term in equations f^ and contail the ratio a wich will be on the order of 0.2 or smaller (section). Thus, the total contribution of this term will be about 2 m/sec. repcielfrresn ivni hepeedigscio,"ie eod

- 15 -The third term, cz ' )is caused by two sources of error, platform z misal gnment,i, and the accelerometer instrument errors, E. Limiting the discussion to the east-up plane; if the accelerometers are perfect instruments and the platform is in perfect alignment then az where 0 is the angle the drag acceleration vector makes with the vertical. When the platform is misaligned by the angle (E the ratio read by perfect accelerometers will be e / e )m - tan ( ) t 6 ) Solving for the error term we have / - ae/az tan - a m/ a& 1- az tan. aZ a tan" -: aaZ Section 2.3.2 For& BR 0.002 radians (se Se/ ctinX2 32 SX) ~e i a 0.002 Assuming cz - 1500 m/sec, the error due to platform misalignment will be of the order of 3m/sec..

The crucial factor in determining the altitude limit of wind measurement is the instrument error caused by the accelerometers; namely the term cz i(e). In section 2.3it was shown that for small accelerations the accelerometer error is approximately 2x10-5g. Since ae (<, the uncertainty z in the numerator will be the dominant factor and the uncertainty in the denominator could be neglected. The accelerometer instrument error will then be cz )i - CZ - 2.22 It will be assumed arbitrarily that the wind threshold will occur when the error in the horizontal wind component is 10 m/sec. Assuming again the vertical velocity to be 1500 m/sec, the threshold of wind measurement will occur approximately where az = 3x10 g 2 23 Remembering that the density measurement was assumed to occur when a = 10-4g the drag has to increase by a factor of about 30 before winds can be measured. From table he Model A atmosphere,the density measurement thresholdfoccurs at approximately 166 km (assuming c- 1500 m/sec)) where the density is 2.3x10-9 kg/m3; the wind measurement threshold then will occur at about 120 km)where the density is 6.9x10-8 kg/m3. Assuming the ARDC standard atmosphere the density threshold occurs at about 147 km (j- 2.3x10-9 kg/n3) and the wind threshold _i occuirat about 116 km _ e.- = 6.9x10-8 kg/m93. The computed trajectory in table III shows that =3 2 -

- 17 - the thresholdsoccur at 150 km and 117 km respectively, which is in good agreement. Table IV shows the errors in wind measurement. It should be noted that the error cz Ei ewhich determines the threshold becomes insignificant as the altitude decreases. The remaining errors are constant for the points calculated because of the assumption that c and the ratio ce (therefore Li ~ ~ ~ ~z the ratio ae ) remain constant. The value of ce (300 m/sec) used is believed to be an upper limit and will be lower in most cases with a proportionate decrease in the first and third error terms of table IV. It should be pointed out that the error caused by aerodynamic forces related to the spin of the sphere are believed small, provided that the spin rate stays within the limitations of the inertial platform. 33

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ARDC Atmosphere M|odel A Atmosphere Altitude Velocity ]quiv. free- Density Velocity Iquiv. free- Density fall distance fall distanc z v h q v h ( (k3m) (m /sec.) (I(n) (k/) (/ec.) () (kg3) 130 630 21 1.3xl0~8 446 11 2.6x10'8 140 1090 64 4.3x 0o9 657 24 1.2xlO-8 147 1500 120 2.3x 109 830 74 7.5x10-9 150 1695 150 1.8x10-9 920 48 6.1x1O-9 160 2465 342 8.5xl0'10 1270 88 3.2x10-9 166 1500 120 2.3x10-9 170 1650 177 1.9xl0-9 180 2077 300 1.2 0lO9 required TABLE I -xXc v, and h/to give a412 foot dia., 120 pound sphere, a drag acceleration of IxO0 g 0 [.t wvrious, alt'tudes

Drag Approx. Scale | error in density measurement caus- Altitude. error Acc. Alt. ieight ed by corresponding terms in eq. (11) Error in az | (ma) -s () e) ( z) a z (ms -4 lxlO g 165 37.8 1O, 1.3,1 20ct 41j O.4 23 3x10-3g 120 14.9 10 1.3 0.5 0.1 0.8 10 5g 70 6.6 10 1.3 0.04 0.02 1.5 10 lg 30 6.8 a 20 0.8 0.04 1.5 20 TABLE IT - Error in density measurement caused by various factors ( 1iodel A atmosphere). A7

/do.z.. ",,.3. Altitude Velocity Drag. Acc. MaccL. Reynolds Time z v az No. To. t (km) (m/sec.) (M/sec.2) (sec.) 150 1660 0 xlO'3 3.6 0.27 181 117 1835 3xl02 5.4 9.6 200 100 1920 P2.5xl0'1 6.3 Io4 208 65 1859 50 6.0 8.5xo04 226 55 14oo 105 4.2 1.7xl05 232 45 531 40 1.6 2.3x105 243 30 181 9.7.59 7.3x105 306 0 o 15 9.:8.04 3.7x106 1166 TABLE III - Trajectory of a 12 foot dia., 120 pound sphere dropped frtn 300 'nm (no winds, Ai2DC standard atmosphere).

Approx. Drag Error in wind measurement caused by corresponding Wind. Altitude Acc. terms of equation (17). Error (em) a cz e z) cz z Mae) Eie) (km) (g) _z_ __ __ _z_ 120 3x10 3 2 m/sec. 10 n/sec. 3 m/sec. 0.5 mn/s 10.7 m/sec 100 3xl-2 2 1 3 0.5 3.8 70 5 2 0.03 3 0.5 3.6 LAEL IV - errors in wind determination. Approximate altitudes are for Model A atmosphere.

SYMBOLS A Cross-section area of sphere B Term defined in Eq. 2.12 Cd Coefficient of drag ci lean molecular velocity D Drag force vector g Acceleration of gravity h Free-fall distance H Scale height,m Mass of sphere M Mach number Re Reynolds number z Altitude 6(x) Expected error in x Gm Error due to platform misalignment i; Error due to accelerometer,P Atmospheric density ~ ( Angle of platform misalignment Accelerations a Drag and thrust acceleration vector in the earth-coordinate system ae; an; az East, North, and Up components of a Velocities and Winds v Velocity vector of sphere in the earth-coordinate system ve; vn; vz East, North, and Up components of v c Velocity vector of sphere relative to the ambient air in the earthcoordinate system.

SYMI3OLS (continued) ce; cn; cz East, ITorth, and Up componernts of c w Velocity vector of wind relative to the earth-coordinate system we; wn; wz East, Norti) and Up components of w * sRefers to downleg of trajectory only

RIFi]:;. i ~C., Edwards, Hi. D. 1956, '"Emission from a Sodium Cloud. Artificially Produced by Means of a Rocket," (abstract only) Bull. Am. Pet. Soc. 39, August, p.436. Frieland, S. S., J. Katzenstein, M. R. Zatzick, 1956, "'lT-hIsed Searchlighting the Atmosphere," J. ___.es. bl61, September, p.415. G;ecpihys. rines, C. 0., 1959, "Movions in t ie Ionosphere," Proc. of tn.e IRE, Vol. 47, No. 2, pp. 176-16o. Jones, L. M, 1956, "Transit-Time Accelerometer,"Rev. Sci. Inst., 27, 374-477. Jones, L. M., and F. L. Bartman, 1956, "A Simplified Falling, Sphere Method for _ peF 56- 497, (AST`i Doc. 101326), Eng. Res, Inst., Thne University of Micnigan, June. Jones, L. M. F. F. Fischbach, and T. W. Peterson, 1958, "Seasonal and Latitude Variation in Upper-Air Density," IGY Rocket Report, No. 1, Nat. Academy of Sciences, July, pp. 47-57. Kallman, H_ K., 1_959, "A Preliminary Model Atmosphere Lased on Rocket and Satellite Data,' J_ Geophys. Research, 64, pp. 6-15-623. Ludlam, F. I., 1957, "'loctilucent Clauds," Tellus 9, August, p. 341. May, A., 1957, "4 Supersonic Drag of Spheres at Low Reynolds Niumbers in Free ---—;~= — t1 — PAppl. Phys., 28, pp. 910-912. Minzne R._ and 1956, "The 1R2DC Model Atmosphere, Air Force Surveys in Geophysics," AFCRC-TR-56-204, (JSTIA Doc. 11023o), Bedford, Mass. colet, M., 1959, "The Constitution and Composition of the Upper Atmosphere, Proc. of t.e IR E, Vol. 47, hio. 2, pp. 142-147. Petersen, N. V., 1956, "Lifetimes of Satellites in Near-Circular and Elliptic Orbits," Jet Propulsion, 26, pp. 344-345. Robertson, D. S., D. T. Liddy, W. G. Elford, 1953, "Measurements of Winds in The Upper Atmosphere by Means of Drifting Meteor Trails," J. A~t. Terr. Pny. 4 (4,5) December, p. 255. Smith, L.., 1953, "Measurement of Winds Between 100 and 300,000 feet by Use of Chaff Rockets,:' (Abstract only) Bull. Am. Met. Soc. 39, August, p. 436. Stroud, W. G., W. R. B ndeen, W. Nordberg, F. L. -artman, J. Ottreeman, and P. Titus, 1958, "Temperatures and Winds in ti.e Arctic as Obtained by the Grenade Experiment,' iGY Rocket Report Series, Io. 1, National Academy of Sciences, pp. 58-79. Whipple, F. L., 1952, "lExploration of the Upper Atmosphere by Meteoric Techniques," in iK E. Langsberg, Adv. in Geophy., 1, Academic Press,. Y., p. 119

-2 -REFElRENCES Jones, L. M., "Atmospheric Phenomena at high Altitudes," DA-36-039-sc-15443, Quarterly Progress Reports 3 thru 6, Eng. Res. Inst., iThe University of 1Michigan, 1 July 1952 - 30 June 1953. Kehlet, A. B., and H. G. Patterson, "Free-Flight Test of a Technique for Inflating an FIASA 12-Foot Diameter Sphere at liigh Altitudes', NASA Memo 2-5-59L, January 1959. Arthur D. Little Inc., "Design S-cudy of Falling-Sphere Method for Measuring UpperAtmosphere Density," Contract Io. AF 19(604)-2636, April 15, 1958.

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