THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Electrical Engineering Space Physics Research Laboratory Scientific Report MS-2 THE THERMAL TRANSPIRATION EQUATION: TABLES OF F(s) vs. s Y. S. Hsu ORA Project 04304 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GODDARD SPACE FLIGHT CENTER CONTRACT NO. NASr-15 GREENBELT, MARYLAND administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1965

LIST OF SYMBOLS angle of attack i 2kT C: mst probable thermal speed of particles: C = 2 m s2 f(s) = e-S + \ s (1 + erfs) k: Boltzmann's constant (1.3804 x 10 Joules/oK) m: particle mass (Kg) n: number density (Part/M3 number density having a velocity Un for a Maxwellian gas r: number of particles crossing a unit area per unit time Un, Co P = pressure p = density (kg/M3) Vn Co To absolute temperature un: velocity of molecules in the direction of normal to the orifice V: drift velocity (relative velocity between chamber and gas) Vn =V cos a Subscripts i ~ parameters inside gauge o0 parameters outside gauge iii

OBJECTIVE The purpose of this report is to present tables and graphs useful in the interpretation, in terms of ambient conditions, of pressure measurements made from pressure gauges mounted on sounding rockets and satellites. V

INTRODUCTION For a pressure gauge mounted on sounding rockets and satellites, the measured pressure values usually include the effects of the drift velocity of the moving gauge. Such a method was successfully employed by this Laboratory in the TP and Pitot-static series of atmospheric measurements. The basic relationship which holds for a pressure gauge immersed in a gas, in which the mean free path of the molecules is long compared with the dimensions of the gauge, is the well known "thermal transpiration" equation. This equation with drift velocity considerations, has been derived by various authors -' and is presented here for completeness. DERIVATION OF f(s) Consider a chamber with an orifice of area A immersed in a gas which has a drift velocity V (Fig 1.).. Parameters outside the chamber are denoted by subscripts "o" and parameters inside the chamber by subscripts "i". The angle between the drift velocity vector V and the normal to the orifice is denoted by a, and will henceforth be referred to as the angle of attack. For the condition of dynamic equilibrium, the number of gas molecules entering the chamber from the outside must be equal to the number of molecules leaving the chamber. If we assume that the mean free path of the molecules, both inside and outside the chamber, is large compared with the dimensions of the orifice, there will be no interaction between the incoming and exiting flux of particles. It is therefore possible to consider the flows separately and equate the equilibrium relationships. The quantity dri, the incremental number of molecules leaving the chamber per unit time per unit area and having velocities in the range uni and uni+duini, is given by dri = u d i u dNni or di i ni f(u) duni (1) 1

where 3 /2 m u2 f(u) -= k e 2 is the Maxwell-Boltzmann velocity distribution function. In Cartecian coordinates f(u) is given by f(Uxyuz ) = e 2kT (2) For the case of molecules crossing an orifice, only one component need be considered. Consider the x-component. We can disregard molecules having velocities in the y and z directions (uy,uz) because they do not cross the orifice and therefore do not contribute to the overall flow. Adapted to our present notations, the one-dimensional Maxwell-Boltzmann velocity distribution function becomes m un. /m vl/2 m Uni! f(ni) = e 2kT (3) Substituting Eq. (3) into Eq. (1), we have Fi, the number of molecules leaving the chamber per unit time per unit area: rUni=~~! n T/2 muni ri = u: ni (2k e 2kT Uni dini (4) JUn= =i Equation (4) can be further simplified by substituting Ci for /-, which gives 2 Uni /Unii ri uni e 1 dUn (5) o o\ / where Ci is the most probable thermal speed of the molecules inside the chamber. Integrating Eq. (5), we have ni Ci 00 - nU) ni U 2 r. = 21-/o e - 1 2 4) d i) 2

i ni Ci e-Uni/C2 -e-Un/ci ni Ci (6) Next, consider the flux of particles entering the chamber from the outside. In computing this flux, thetotal velocity of the gas molecules will be the sum of the thermal velocity and the drift velocity of the chamber with respect to the gas. Thus do,. the incremental particle flux crossing the orifice from the outside per unit time per unit area and having a velocity range between uno and uno+duno, is given by dro = (Uno+Vn)dNno (7) where Vn is the component of drift velocity in the direction of normal to the orifice. Next substitute for the one-dimensional case of the MaxwellBoltzmann velocity distribution function, or dNn = n0 -(\e 2 o ) TO e COO/ dfo) into i;q. i). upon Integration this gives the particle flux entering the chamber from the outside per unit time per unit area: (uno+Vn) = /uno2 =o u no+Vnno ) ) e d (8) (uno+Vn) = To simplify the notation further, let the ratio of the drift velocity in the direction of the normal to the orifice Vn, to the ambient most probable thermal velocity Co, be denoted by sj and let the ratio of the component of the molecules velocity in the direction of the normal to the orifice, to the most probable thermal velocity, be denoted by X: Vn Uno - = s; = C0 Co Equation (8) can then be written 5

no Co 2 n Vn 2 r f 0 X e-%2 CIX 0 e-%2 dX (9) 0 ^s; -s Rearranging the second integral, we can write Eq. (9) as no 00 o F 0 2 00 2 n0 -o2 n Co 2,.,o o _L2 ro e dX + s e d + e- dJ (10) The above are standard integrals. X e dX e n, CO ro - 2 no Co 0d- s e d e d = - - e -s 2 /i = no Cb s e~S 2 dJ n Co 0 -%2 on CO S s _2 s e d -— = 0 es (12) V -s n co0 2 n C s s / e- d% erfs (12) -s n C f -x00 2 n0o S 2 r -X2 ~ -k22 Therefore no Co (14) r - n~.j e d s=2 (1 ( erfs) (l) no CO j _k2 no CO s 2

For equilibrium to exist, it is necessary that ri = To. Equating Eqs. (6) and (14) yields niCi = noCo f(s) (15) where f(s) = e2 + s l (1 + erfs) (16) Therefore, a relation between inside parameters and outside parameters of a pressure gauge can be obtained as a function of s = Vn/Co. Tables I and II give f(s) = Les + s WK (1 + erfs) versus s. Figure 2 shows f(s) in graphic form. From the graph it can be seen that f(s) becomes linear for large s, and examination of Eq. (16) shows that for large s f(s) = 2 1: s (17) Figure 3 shows f(s) for large negative s. From the graph it can be seen that f(s) becomes a straight line on the semi-log plot, indicating that for large negative s, f(s) approaches an exponential variation. The above is also seen to be true upon examination of Eq. (16). Hence for large negative s S2 f(-s) = e-2 (18) Equation (15) can be interpreted in terms of pressure and temperature. 5

Since P = nkT and C - /2kT where P = pressure k = Boltzmann's constant T = Absolute temperature m = Molecular mass: niCi 2ki T kTo'n C kkTi m Po 2kTo niC-i P K TnoCo Po Ti Therefore Pi Ti (1 pi = f(s) (19 Po - To

Pii P i Po To Po - - - - -- < ^N V = Vcos a Orifice Satellite pressure gage V = Drift velocity of gas with respect to the pressure gage Fig. 1 7

f(s) 0 roN J 4 cn1 - I oo w o 0 I o0 )PO -~ O0 1) Im C

C\j u —(S + ((/) - -I 10 I D - 0_ 10'O'O'O'O'0 00 O'0'0'0 - ~ - (9);

TABLE I -s F(-s) -s F(-s) -3.0000.00000595 -2.5000.00012722 -2.9900.0000634 -2.4900.00013463 -2.9800.00000675 -2.4800.00014245 -2.9700.00000721 -2.4700.00015070 -2.9600.00000771 -2.4600.00015939 -2.9500.00000824 -2.4500.00016856 -2.9400.00000876 -2.4400. C 17821 -2.9300.00000939 -2.4300.00018841 -2.9200.00001000 -2.4200.00019912 -2.9100.00001066 -2.4100.00021042 -2. 9000.00001135 -2.4000.0 0 0022233 -2.8900.00001211 -2.3900.00023485 -2.8800.00001290 -2.3800.00024803 -2.8700.0001377 -2. 3700.00026191 -2.8600.00001466 -2.3600.00027652 -2.8500.00001561 -2. 3500.00029191 -2.8400.00001665 -2. 3400.U00030808 -2.8300.00001771 -2.3300.00032508 -2.8200.00001886 -2.3200.00034298 -2.8100.00002008 -2.3100.00036178 -2.8000.00002137 -2.3000.00038157 -2.7900.00002274 -2.2900.00040234 -2.7800.00002421 -2.2800.00042417 -2.7700.00002573 -2.2700.00044713 -2.7600.00002739 -2.2600.00047121 -2.7500.00002911 -2.2500.00049653 -2.7400.00003094 -2.2400.00052311 -2.7300.00003290 -2.2300.00055101 -2.7200.00003495 -2.2200.00058029 -2.7100.00003716 -2.2100.00061104 -2.7000.00003948 -2.2000.0006432 -2.6900.00004192 -2.1900.00067710 -2.6800.00004449 -2.1800.00071259 -2.6700.00004725 -2.1700.00074980 -2.6600.00005015 -2.1600.00078881 -2.6500.00005323 -2.1500.0u002972 -2.6400.00005651 -2.1400.00087258 -2.6300.00005995 -2.1300.00091749 -2.6200.00006359 -2.1200.00096454 -2.6100.00006743 -2.1100.00101384 -2.6000.00007149 -2.1000.00106546 -2.5900.00007580 -2.0900.00111953 -2.5800.00008037 -2.0800.OC117611 -2.5700.00008515 -2.0700.00123536 -2.5600.00009025 -2.0600.00129736 -2.5500.00009559 -2.0500.00136222 -2.5400.00010125 -2.0400.00143CC8 -2.5300.00010725 -2.030 0.00150105 -2.5200.C00011355 -2. 0200.00157527 -2.5100.00012021 -2.0100.00165287 10

TABLE I (Continued) -s F(-s) -s F(-s) -2.0000.00173400 -1.5000.01528635 -1.9900.00181877 -1.4900.01589787 -1.9800.00190737 -1.4800.01653108 -1.9700.00199995 -1.4700.01718668 -1.9600.00209665 -1.4600.01786531 -1.9500.00219764 -1.4500.01856768 -1.9400.00230309 -1.4400.01929450 -1.9300.00241317 -1.4300.02004645 -1.9200.00252810 -1.4200.02082428 -1.9100.00264803 -1.4100.02162876 -1.9000.00277317 -1.4000.02246063 -1.8900.00290374 -1.3900.02332066 -1.8800.00303989 -1.3800.02420968 -1.8700.00318192 -1.3700.02512846 -1.8600.00332998 -1.3600.02607787 -1.8500.00348434 -1.3500.02705874 -1.8400.00364523 -1.3400.02807194 -1.8300.00381291 -1.3300.02911836 -1.8200.00398759 -1.3200.03019887 -1.8100.00416957 -1.3100.03131442 -1.8000.00435908 -1.3000.03246591 -1.7900.00455647 -1.2900.03365432 -1.7800.00476195 -1.2800.03488058 -1.7700.00497586 -1.2700.03614573 -1.7600.00519847 -1.2600.03745075 -1.7500.00543012 -1.2500.03879662 -1.7400.00567114 -1.2400.04018444 -1.7300.00592184 -1.2300.04161524 -1.7200.00618258 -1.2200.04309009 -1.7100.00645369 -1.2100.04461010 -1.7000.00673557 -1.2000.04617635 -1.6900.00702854 -1.1900.04779001 -1.6800.00733301 -1.1800.04945219 -1.6700.00764938 -1.1700.05116409 -1.6600.00797806 -1.1600.05292683 -1.6500.00831945 -1.1500.05474169 -1.6400.00867398 -1.1400.05660983 -1.6300.00904209 -1.1300.05653249 -1.6200.00942426 -1.1200.06051095 -1.6100.00982091 -1.1100.06254646 -1.6000.01023253 -1.1000.06464029 -1.5900.01065962 -1.0900.06679379 -1.5800.01110268 -1.0800.06900823 -1.5700.01156221 -1.0700.07128497 -1.5600.01203875 -1.0600.07362538 -1.5500.01253283 -1.0500.07603079 -1.5400.01304504 -1.0400.07850263 -1.5300.01357588 -1.0300.08104227 -1.5200.01412600 -1.0200.08365115 -1.5100.01469595 -1.0100.08633069 11

TABLE I (Concluded) -s F(-s) -s F(-s) 1.0000.08908233 -.5000.35386777 -.9900.09190755 -.4900.36244480 -.9800.09480783 -.4800.37117914 -.9700.09778468 -.4700.38007231 -.9600, 10083956 -.4600.38912583 -.9500.10397404 -.4500.39834120 -.9400.10718961 -.4400.40771990 -.9300.11048785 -.4300.41726337 -.9200.11387031 -.4200.42697309 -.9100.11733855 -.4100.43685044 -.9000.12089417 -.4000.44689683 -.8900.12453876 -.3900.45711365 -.8800.12827395 -.3800.46750225 -.8700. 13210131 -.3700.47806393 -.8600.13602250 -.3600.48880000 -.8500 14003916 -.3500.49971178 -.8400.14415290 -.3400.51080047 -.8300.14836542 -.3300.52206731 -.8200.15267837 -.3200.53351351 -.8100.15709341 -.3100.54514024 -.8000.16161223 -.3000.55694862 -.7900.16623649 -.2900.56893977 -.7800.17096791 -.2800.58111478 -.7700.17580817 -.2700.59347469 -.7600.18075896 -.2600.60602053 -.7500.18582200 -.2500.61875328 -.7400.19099901 -.2400.63167391 -.7300.19629167 -.2300.64478333 -.7200.20170171 -.2200.65808243 -.7100.20723085 -.2100.67157207 -.7000.21288079 -.2000.68525305 -.6900.21865324 -.1900.69912620 -.6800.22454993 -.1800.71319222 -.6700.23057257 -. 1700.72745185 -.6600.23672289 -.1600.74190579 -.6500.24300256 - 1500.75655466 -.6400.24941331 -.1400.77139904 -.6300.25595685 -.1300.78643955 -.6200.26263485 -. 1200.80167668 -.6100.26944902 -.1100.81711093 -.6000.27640104 -.1000.83274276 -.5900.28349260 -.0900.84P57260 -.5800.29072535 -.0800.86460079 -.5700.29810096 -.0700.88082770 -.5600.30562110 -.0600.89725363 -.5500.31328737 -.0500.91387880 -.5400.32110143 -.0400.93070347 -. 5300.32906492 -. 0300.94772782 -. 5200.33717939 -.0200. 96495195 -.5100.34544650 -.0100.98237599 12

TABLE II s F(s) s F(s). 000 1.00000000.5000 2.12626773. 10() 1.01782398.5100 2.15.329444.0200 1.03584795.5200 2.18047532.300 1. 05407181.5300 2.20780885.0400 1.07249546.5400 2.23529339.0)500. 09111880.5500 2.26292732.0600 1.10994162.5600 2.29070905.0700 1.12896369.5700 2.31863689. 800 1. 14818478.5800 2.34670928. C900 1. 16760460.5900 2.37492457.1000 1.18722275.6000 2.40328097 1100 1.20703891.6100 2.43177697.1200 1.22705266.6200 2.46041080.1300 1.24726354.6300 2.48918077.1400 1.26767-103.6400 2.51808524.1500 1.28827463.6500 2.54712251.1600 1.30907378.6600 2.57629082.1700 1.33006783.6700 2.60558850.1800 1,35125621.6800 2.63501385.1900 1.37263818.6900 2.66456521.2000 1.39421304 7000 2. 69424072.2100 1.41598004.7100 2.72403878.2200 1.43793841.7200 2.75395766.2300 1.46008730.7300 2.78399560.2400 1.48242588.7400 2. 1415096.27 500 1.5049532. 500 84442192.2600 1.52766851,7600 2.87480688.2700 1.55057067.7700 2.90530413.2800 1.57365875.7800 2.93591183.2900 1.59693174.7900 2.96662840.3000 1.62038P58.000 2.99745214.3100 1.64402820.8100 3.02 381 32.3200 1.66784948.8200 3.059414-30.3300 1.69185127.3300 3.09054938.3400 1.71603243.8400 3.12178481.3500 1.74039175.d500 3.15311906.3600 1.76492797.600 3.18455043.3700 1.78963989.8700 3.21607727.3800 1.8 1452622.8800 3.24769786.3900 1.83958562.8900 3.27941069.4000 1.86481678.9000 3.31121406.4100 1.89021841.9100 3.34310648.4200 1.91 578905.9200 3.37508622.4300 1.94152734.9300 3.407151 76.4400 1.96743186.9400 3.43930152.4500 1.99350 116.9500 3.47153395.4600 2.01973379.9600 3.50384751.4 700 2.04 612827.9700 3.53624061.4800 2.07268310.9800 3.56871176.4900 2. 099396 77.9900 3.60125944 13

TABLE II (Continued) s F(s) s F(s) 1.0000 3.63388228 1.5000 5.33248621 1.0100 3.66657859 1.5100 5.36734378 1.0200 3.69934708 1.5200 5.40222186 1.0300 3.73218614 1.5300 5.43711.966 1.0400 3.76509452 1.5400 5.4 7203696 1.0500 3.79807070 1.5500 5.50697273 1.0600 3.83111328 1.5600 5.54192662 1.0700 3.86422086 1.5700 5.57689810 1.0800 3.89739215 1.5800 5.61188656 1.0900 3.93062568 1.5900 5.64689147 1.1000 3.96392021 1.6000 5.68191236 1.1100 3.99727440 1.6100 5.71694881 1.1200 4.03068680 1.6200 5.75200003 1.1300 4.06415635 1. 300 5.78706592 1.1400 4.09768170 1.6400 5.82214582 1.1500 4.13126153 1.6500 5.85723931 1.1600 4.16489470 1.6600 5.89234596 1.1700 4.19857991 1.6700 5.92746526 1.1800 4,23231608 1.6800 5.96259683 1.1900 4.26610190 1.6900 5.99774039 1.2000 4.29993623 1.7000 6.03289539 1.2100 4.33381802 1.7100 6.06806147 1.2200 4.36774594 1.7200 6.10323840 1.2300 4.40171903 1.7300 6. 13 42571 1.2400 4.43573630 1.7400 6.17362303 1.2500 4.46979648 1.7500 6.20883000 1.2600 4.50389862 1.7600 6.24404633 1.2700 4.53804165 1.7700 6.27927172 1.2800 4.5'7222444 1.7800 6.31450570 1.2900 4.60644615 1.7900 6.34974831 1.3000 4.64070576 1.8000 6.38499892 1.3100 4.67500222 1.8100 6.42025745 1.3200 4.70933467 1.8200 6.45552349 1.3300 4.74370223 1.8300 6.49079680 1.3400 4.77810377 1.8400 6.52607703 1.3500 4.81253856 1.8500 6.56136423 1.3600 4.84700572 1.8600 6.59665781 1.3700 4.88 150430 1.8700 6.63195777 1.3800 4.91603357 1.8800 6.66726375 1.3900 4.95059252 1.8900 6.70257550 1.4000 4.98518050 1.9000 6.73789293 1.4100 5.01979661 1.9100 6.77321595 1.4200 5.05444014 1.9200 6.80854392 1.4300 5.08911031 1.9300 6.843877C2 1.4400 5.12380636 1.9400 6.87921494 1.4500 5.15852755 1.9500 6.91455746 1.4600 5. 19 3273 19 1.9600 6.94990450 1. 4700 5.22804254t 1 9700 6.98 525578 1.4800 5.26283491 1.9800 7.02061117 1.4900 5.29764974 1.9900 7.05597061 14

TABLE II (Continued) s F(s) s F(s) 2.0000 7.09133387 2.5000 8.86212695 2.0100 7.12670076 2.5100 8.89756787 2.0200 7.16207105 2.5200 8.93300915 2.0300 7.19744480 2.5300 8.96845102 2.0400 7.23282188 2.5400 9.00389302 2.0500 7.26820201 2.5500 9.03933537 2.0600 7.30358505 2,5600 9.07477796 2.0700 7.33897120 2.5700 9.11022091 2.0800 7.37435991 2.5800 9.14566410 2.0900 7.40975136 2.5900 9.18110752 2.1000 7.4451453 7 2.6000 9.21655130 2.1100 7.48054165 2.6100 9.25199509 2.1200 7.51594031 2.6200 9.28743935 2.1300 7.55134135 2.6 300 9.32288373 2.1400 7.58674443 2.6400 9.35832822 2.1500 7.62214947 2.6500 9.39377296 2.1600 7.65755659 2.6600 9.42921793 2.1700 7.69296563 2.6700 9.46466291 2.1800 7.72837639 2.6800 9.50010812 2.1900 7.76378900 2.6900 9.53555357 2.2000 7.79920310 2.7000 9.57099915 2.2100 7.83461887 2.7100 9.60644484 2.2200 7.87003613 2,7200 9.64189065 2.2300 7.90545487 2.7300 9.67733669 2.2400 7.94087487 2.7400 9. 71278262 2.2500 7.97629637 2.1500 9.74822891 2.2600 8.01111899 2.7600 9.78367507 2.2700 8.04714286 2.7700 9.81912136 2.2800 8.0825678.1 2.7800 9.85456789 2.2900 8.11799395 2.7900 9.89C01453 2.3000 8.1534 2128 2.8000 9.92546105 2.3100 8.18884933 2.8100 9.96090770 2.3200 8.22427869 2.8200 9.99635458 2.3300 8.25970876 2.8300 10.03180146 2.3400 8.29513979 2.8400 10.06724823 2.3500 8.33057165 2.8500 10.10269535 2.3600 8.36600423 2.8600 10.13814235 2.3700 8.40143764 2.8700 10.17358935 2.3800 8.43687189 2.8800 10.20903659 2.3900 8.47230661 2.8900 10.24448371 2.4000 p.50774205 2.9000 10.27993107 2.4100 8.54317832 2.9100 10.31537843 2.4200 8.57861495 2.9200 10.35082579 2.4300 8.61405206 2.9300 10.38627303 2.4400 8.64949000 2.9400 10.42172062 2.4500 8.68492830 2.9500 10.45716798 2.4600 8.72036707 2.9600 10.49261534 2.4700 8.75580645 2.9700 10.52806306 2.4800 8.79124618 2.9800 10.56351042 2.4900 8.82668626 2.9900 10.59895813 15

TABLE II (Continued) s F(s) s F(s) 3.0000 10.63440573 3.5000 12.40680003 3.0100 10.66985321 3.5100 12.44224787 3.0200 10.70530081 3.5200 12.47769570 3.0300 10.74074841 3.5300 12.51314366 3.0400 10.77619624 3.5400 12.54859185 3.0500 10.81164396 3.5500 12.58403981 3.0600 10.84709156 3.5600 12.61948764 3.0700 10.88253927 3.5700 12.65493584 3.0800 10.91798723 3.5800 12.69038367 3.0900 10.95343494 3.5900 12.72583175 3.1000 10.98888278 3.6000 12.76127970 3.1100 11.02433050 3.6100 12.79672766 3.1200 11.05977833 3.6200 12.83217585 3.1300 11.09522629 3.6300 12.86762357 3.1400 11.13067412 3.6400 12.90307176 3.1500 11.16612184 3.6500 12.93851972 3.1600 11.20156980 3.6600 12.97396779 3.1700 11.23701763 3.6700 13.00941563 3.1800 11.27246535 3.6800 13.04486370 3.1900 11.30791330 3.6900 13.08031178 3.2000 11i34336114 3.7000 13.11575973 3.2100 11.37880898 3.7100 13.15120757 3.2200 11.41425705 3.7200 13.18665576 3.2300 11.44970500 3.7300 13.22210371 3.2400'11.48515284 3.7400 13.25755167 3.2500 11.52060068 3.7500 13.29299974 3.2600 11.55604863 3.7600 13.32844770 3.2700 11.59149659 3.7700 13.36389565 3.2800 11.62694454 3.7800 13.39934361 3.2900 11.66239262 3.7900 13.43479156 3.3000 11.69784045 3.8000 13.47023952 3.3100 11.73328841 3.8100 13.50568771 3.3200 11.76873648 3.8200 13.54113567 3.3300 11.80418432 3.8300 13.57658362 3.3400 11.83963215 3.8400 13.61203170 3.3500 11.87508023 3.8500 13.64747977 3.3600 11.91052806 3.8600 13.68292761 3-.3700 11.94597614 3.8700 13.71837568 3.3800 11.98142409 3.8800 13.75382376 3.3900 12.01687205 3.8900 13.78927171 3.4000 12.05232000 3.9000 13.82471967 3.4100 12.08776796 3.9100 13.86016762 3.4200 12.12321591 3.9200 13.89561570 3.4300 12.15866387 3.9300 13.93106365 3.4400 12.19411194 3.9400 13.96651173 3.4500 12.22955990 3.9500 14.00195968 3.4600 12.26500797 3.9600 14.03740764 3.4700 12.30045605 3.9700 14.07285571 3.4800 12.33590400 3.9800 14.10830367 3.4900 12.37135196 3.9900 14.14375174 16

TABLE II (Continued) s F(s) sF(s) 4.0000 14.17919970 4.5000 15.95159984 4.0100 14.21464777 4.5100 15.98704779 4.C200 14.25009573 4.5200 16.02249575 4.0300 14.28554368 4.5300 16.05794382 4.0400 14.32099164 4.5400 16.09339166 4.0500 14.35643959 4.5500 16.12883973 4.0600 14.39188755 4.5600 16.16428757 4.0700 14.42133574 4. 5700 16.19913588 4.0800 14.462/8369 4.5800 16.23518372 4.0900 14.49823165 4.5900 16.206 3 1 79 4.1000 14.53367972 4.6000 16.30607963 4.1100 14.56912768 4.6100 16.34152770 4.1200 14.60457563 4.6200 16.37697554 4.1300 14.64002383 4.6300 16.41242385 4.14U0 14.67547178 4.6400 16.44787169 4. 1500 14.71091974 4.6500 16.48331976 4.1600 14.74636781 4.6600 16'.51876760 4. 1700 14.78181577 4.6700 16.55421567 4.1800 14.81726372 4.6800 16.58966351 4.1900 14.85271192 4.6900 16.62511182 4.2000 14.88815987 4.7000 16.66055965 4.2100 14.92360783 4.7100 16.69600773 4.2200 14.95905578 4.7200 16.73145580 4. 2 300 14.99450374 4.7300 16.76690364 4.2400 15.02995169 4.7400 16.80235171 4.2500 15.06539989 4.7500 16.83779979 4.2600 15.10084784 4.7600 16.87324786 4.2700 15.13629580 4. 700 16.90869570 4.2800 15.17174375 4.7800 16.94414377 4.2900 15.20719171 4.7900 16.97959161 4.3000 15.24263966 4.8000 17.01503968 4.3100 15.27808762 4.8100 17.05C48 752 4.3200 15.31353581 4.8200 17.08593583 4.3300 15.34898376 4.8300 17.12138367 4.3400 15.38443172 4.8400 17.15683174 4.3500 15.41987979 4.8500 17.19227958 4.3600 15.45532775 4.8600 17.22772765 4.3700 15.49077570 4.8 00 17.26317549 4.3800 15.52622390 4.8800 17.29862380 4.3900 15.56167185 4.8900 17.334071 64 4.4000 15.59711981 4.9000 17.36951971 4.4100 15.63256776 4.9100 17.40496778 4.4200 15.66801572 4.9200 17.44041562 4.4300 15.70346367 4.9300 17.47586370 4.4400 15.73891187 4.9400 17.51131177 4.4500 15.77435982 4.9500 17.54675984 4.4600 1.5.8098077 4.9600 17.58220768 4.4700 15.84525573 4.9700 1.61765575 4.4800 15.88070369 4.9800 17.65310359 4.4900 15.91615164 4.9900 17.68855166 4 4 1 5 1 4 4 17 5 5 1 17

TABLE II (Continued) s F(s) s F(s) 5.00 0 0 17. 72 39991 4 5.5000 19.49639988 5.U100 17.75944781 5.5100 19.53184772 5.0200 17.79489565 5.5200 19.56729579 5.0300 17.83034372 5.5300 19.60274363 5.0400 17.86579156 5.5400 19.63819170 5.0500 17.90123963 5.5500 19.67363954 5.0600 17.93668771 5.5600 19.70908761 5.0700 17.97213578 5.5700 19.74453568 5.C800 18.00758386 5.5800 19.77998376 5. 0900 18.04303169 5.5900 19.81543159 5.1000 18.07847977 5.6000 19.85087967 5.1100 18.11392760 5.6100 19.88632774 5.1200 18.14937568 5.6200 19.92177558 5.1300 18. 1482375 5.6300 19.95722365 5.1400 18.22027183 5.6400 19.99267173 5.1500 18.25571966 5.6500 20.02811980 5.1600 18.29116774 5.6600 20.06356764 5.1700 18.32661557 5.6700 20.09901571 5.1800 18.36206365 5.6800 20.13446355 5.1900 18.39751172 5.6900 20.16991186 5.2000 18.43295979 5.7000 20.20535970 5.2100 18.46Q40763 5.7100 20.24080777 5. 2200 18.50385571 5.7200 2 0.27625561 5.2300 18.53930354 5.7300 20.31170368 5.2400 18.57475162 5.7400 20.34715152 5.2500 l8.61019969 5.7500 20.38259983 5.2600 18.64 564776 5.7600 20.41804767 5.2700 18.68109584 5.7700 20.45349574 5.2800 18.71654367 5.7800 20.48894358 5.2900 18.75199175 5.7900 20.52439165 5.3000 18.78743958 5.8000 20.55983973 5.3100 18.82288766 5.8100 20.59528756 5.3200 18.85833573 5.8200 20.63073587 5.3300 18.89378381 5.8300 20.66618371 5.3400 18.92923164 5.8400 20.70163178 5.3500 18.96467972 5.8500 20.73707962 5.3600 19.00012755 5.8600 20.77252769 5.3700 19.03557563 5.8700 20.80797553 5.3800 19.07102370 5.8800 20.84342384 5.3900 19.10647178 5.8900 20.87887168 5.4000 19.14191961 5.9000 20.91431975 5.4100 19.17736769 5.9100 20.94976759 5.4200 19.21281552 5.9200 20.98521566 5.4300 19.24826360 5.9300 21.02066350 5.4400C 19.26371167 5.9400 21.05611181 5.4500 19.31915975 5.9500 21.09155965 5.4600 19.35460782 5.9600 21.1270077? 5.4700 19.39005566 5.9700 21.16245556 5.4800 19.425503 73 5.9800 21. 19790363 5.4900 19.46C95157 5. 9900 21.23335171 18

TABLE II (Continued) s F(s) s F(s) 6.0000 21.26879978 6.5000 23.04119968 6.0100 21.30424786 6.5100 23.07664776 6.C200 21.33969569 6.5200 23.11209559 6.0300 21.3514 377 6.5300 23.14754367 6.0400 21.41059160 6.5400 23.18299174 6.0500) 21.44603968 6.5500 23.21 43958 6.0600 21.48148751 6.5600 23.25388765 6.0700 21.5 1693583 6.5700 23.28933573 6.0-00 21.55238366 6.5800 23.32478380 6.0900 21.58783174 6.5900 2336023164 6.1000 21.62327957 6.6000 23.39567971 6.1100 21.65872765 6.6100 23.43112755 6.1200 21.69417548 6.6200 23.46657562 6. 1300 21.72962379 6.6300 23.50202370 6.1400 21.76507163 6.6400 23.53747177 6.1500 21.80051970 6.6500 23.57291961 6.1600 21.S3596778 6.6600 23.60836768 6.1700 21.87141562 6.6700 23.64381552 6.1800 21.90686369 6.6800 23.67926359 6.1900 21.94231176 6.6900 23.71471167 6.2000 21.97775984 6.7000 23.75015974..21 00 22.01320767 6.7100 23.78560758 6.2200 22.04865575 6.7200 23.82105565 6.2300 22.08410358 6.7300 23.85650373 6.2400 22.11955166 6,7400 23.89195156 b.2500 22.15499973 6.7500 23.92739987 0.2600 22.19044781 6.7600 23.96284771 6.2700 22.22589564 6.7700 23.99829578 6.2800 22.26134372 6.7800 24.03374362 6.2900 22.29679155 6.7900 24.06919169 O.3000 22.33223963 6.8000 24.10463953 6.3100 22.36768746 6.8100 24.14008760 6.3200 22.40313578 6.8200 24.17553568 6.3300 22.43858361 6.8300 24.21098375 6.3400 22.47403169 6.8400 24.24643159 6.3500 22.50947976 6.8500 24.28187966 6.3600 22.54492760 6.8600 24.31732750 6.3700 22.58037567 6.8700 24.35277557 6.3800 22.61582375 6.8800 24.38822365 6.3900 22.65127182 6.8900 24.42367172 6.4CC00 22.68671966 6.9000 24.45911980 6.41 G( 22.72216773 6.9100 24.49456763 6.4200 22.75761557 6.9200 24.53001571 6.4 300 22.79306364 6.9300 24.56546354 6.4400 22.82851171 6.9400 24.60091186 6.4500 22.86395979 6.9500 24.63635969 6.4600 22.89940763 6.9600 24.67180777 6. 4700 22.93485570 6.9700 24.70725560 6.4800 22.97300354 6.9800 24.74270368 6.4900 23.00575161 6.9900 24.77815151 19

TABLE II (Continued) s F(s) s F(s) 7.0000 24.81359982 7.5000 26.58599973 7.0100 24.84904766 7.5100 26.62144780 7.0200 24.88449574 7.5200 26.65689564 7.0300 24.91994357 7.5300 26.69234371 7.0400 24.95539165 7.5400 26.72779155 7.C500 24.99083948 7.5500 26.76323962 7.0600 25.02628756 7.5600 26.79868746 7.0700 25.06173563 7.5700 26.83413577 7.0800 25.09718370 7.5800 26.86958361 7.0900 25.13263178 7.5900 26.90503168 7.1000 25.16807961 7.6000 26.94047952 7.1100 25.20352769 7.6100 26.97592759 7.1200 25.23897552 7.6200 27.01137567 7.1300 25.27442384 7.6300 27.04682374 7.1400 25.30987167 7.6400 27.08227181 7.1500 25.34531975 7.6500 27.11771965 7.1600 25.38076758 7.6600 27.15316772 7.1700 25.41621566 7.6700 27.18861556 7.1800 25.45166349 7.6800 27.22406363 7.1900 25.48711181 7.6900 27.25951171 7.2000 25.52255964 7.7000 27.29495978 7.2100 25.55800772 7.7100 27.33040762 7.2200 25.59345555 7.7200 27.36585569 7.2300 25.62890363 7.7300 27.40130353 7.2400 25.66435170 7.7400 27.43675160 7.2500 25.69979978 7.7500 27.47219968 7.2600 25.73524761 7.7600 27.50764775 7.2700 25.77069569 7.7700 27.54309559 7.2800 25.80614376 7.7800 27.57854366 7.2900 25.84159160 7.7900 27.61399150 7.300C 25.87703967 7.8000 27.64943957 7.3100 25.91248751 7.8100 27.68488765 7.3200 25.94793582 7.8200 27.72033572 7.3300 25.98338366 7.8300 27.75578380 7.3400 26.01883173 7.8400 27.79123163 7.3500 26.05427957 7.8500 27.82667971 7.3600 26.08972764 7.8600 27.86212754 7.3700 26.12517548 7.8700 27.89757562 7.3800 26.16062379 7.8800 27.93302369 7.3900 26.19607162 7.8900 27.96847177 7.4000 26.23151970 7.9000 28.00391960 7.4100 26.26696754 7.9100 28.03936768 7.4200 26.30241561 7.9200 28.07481551 7.4300 26.33786368 7.9300 28.11026359 7,4400 26.37331176 7.9400 28.14571166 7.4500 26.40875983 7.9500 28.18115973 7.4600 26.44420767 7.9600 28.21660757 7.4700 26.47965574 7.9700 28.25205564 7.4800 26.51510358 7.9800 28.28750372 7.4900 26.55055165 7.9900 28.32295156 20

TABLE II (Concluded) s F(s) s F(s) 8.OOO0 28.35839987 8.5000 30.13079977 8.0100 28.39)384770 8.5100 30.16624761 8.0200 28.42929578 8.5200 30.20169568 8.0300 28.46474361 8.5300 30.23714375 6.0400 28.50019169 8.5400 30.27259159 8.0500 28.53563952 8.5500 30.30803967 8.0600 28.57108760 8.5600 30.34348750 8.70 28.60653543 8. 5700 30.37893558 8.0800 28.64198351 8.5800 30.41438341 8.0900 28.61743134 8.5900 30.44983149 8.1000 28.71287942 8.6000 30.48527932 8.1100 28.74832749 8.6100 30.52D72740 8. 1200 28.78377533 8.6200 30.55617523 8.1300C 28.81922364 8.6300 30. 59 162 378 8.1400 28.85467172 8.6400 30.62707162 8.1500 28.89011955 8.6500 30.66251969 8.1600 28.92556763 8.6600 30.69796753 8.1700 26.96101570 8.6700 30.73341560 8.1800 228.99646354 8.6800 30.76886344 8.1900 29.03191161 8.6900 30.80431151 8.2000 29.06735945 8.7000 30.83975959 8.2100 29.10280752 8.7100 30.87520742 8.2200 29.13825536 8.7200 30.91065550 8,2300 29.17370343 8.7300 30.94610333 8.2400 29.20915127 8.7400 30.98155141 6.2500 29.24459982 8.7500 31.01699972 8.2600 29.28004766 8.7600 31.05244780 8.2700 29.31549573 8.7700 31 08789563 8.2800 29.35094357 8.7800 31.12334371, 2900 29.38639164 8.7900 31.15879154 8.3000 29.42183948 8.8000 31.19423962 8,3100 2.45728755.100 3122968745 8.3200 29.49273562 8.8200 31.26513553 8.3300 29.52818346 8.8300 31. 30058336 8.3400 29.56363153 8.8400 31.33603144 8,3500 29.59907937 8.8500 31.37147951 8.3600 29.63452744 8.8600 31.40692735 8.3700 29.66997528 8.8700 31.44237542 8.3800 29.70542383 8.8800 31.47782373 8.3900 29.74087167 8.8900 31.51327157 8.4000 29.77631974 8.9000 31.54871964 8.4100 29.81176758 8.9100 31.58416772 8.4200 29.84721565 8.9200 31.61961555 8.4300 29.88266349 8.9300 31.65506363 8.4400 29.91811156 8.9400 31.69051147 8.4500 29.95355940 8.9500 31.72595954 8.4600 29.98900747 8.9600 31.76140738 8.4700 30.02445531 8.9700 31.79685545 8.4800 3.0.05990338 8.9800 31.83230329 8.4900 30.09535146 8.9900 31.86775136 21

REFERENCES 1. J. E. Ainsworth, D. F. Fox, and H. E. LaGow, "Measurement of UpperAtmosphere Structure by Means of the Pitot-Static Tube," NASA Technical Note D-670. 2. F. V. Schultz, N. W. Spencer, and A. Reifman, "Upper Air Research Program," Engineering Research Institute, University of Michigan, under contract Air Material Command, Cambridge Field Station, Contract No. W-33038 ac-14050, July 1, 1948. 22

UNIVERSITY OF MICHIGAN 3 9015 03025 4893