THE U NI V ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Aeronautical and Astronautical Engineering High Altitude Engineering Laboratory Technical Report ATMOSPHERIC SOUNDING BY SATELLITE MEASUREMENTS OF STELLAR REFRACTION...F.. F...Ficc.bach M E. X,aves P. -B. J. W Petersn -.,o. ORA.Project 04963 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GODDARD SPACE FLIGHT CENTER CONTRACT NO. NASw-140 GREENBELT, MARYLAND administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR December 1962

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TABLE OF CONTENTS Page LIST OF TABLES: v LIST OF FIGURES vii PARTIAL LIST OF SYMBOLS ix THE UNIVERSITY OF MICHIGAN PROJECT PERSONNEL xi ABSTRACT xiii ACKNOWLEDGAEMNT xv I. INTRODUCTION 1 II. THE THEORETICAL BASIS FOR A SINGLE DENSITY PROFILE 3 A. Astronomical Refraction 3 B. Solution of the Inverse Problem 5 C. Effect of Two Approximations 7 D. Location of Tangent Point 13 E. Magnitude of Refraction 19 F. Summary 21 III. THE THEORETICAL BASIS FOR GLOBAL COVERAGE 22 A. Time Considerations 22 B. Celestial Considerations 23 -IV. PROPOSED EQUIPMENT AND METHODS 34 A. General 34 B. Data Processing 36 C. Telescope, Clock, Tape Recorder, and Telemeter 36 V. AN EMPIRICAL ATTEMPT: TOO EST ITH-E FUNDAMENTAL EQUATIONS 39 VI. ERRORS AND THEIR EFFECTS 51 VII. CONCLUSIONS 54 REFERENCES 55 iii

TABLE OF CONTENTS (Concluded) Page APPENDIX A. COMPUTATION OF LATITUDE AND LONGITUDE OF RAY TANGENCY AND SUBSATELLITE POSITION AT OCCULTATION GIVEN A STAR AND AN ORBIT 56 B. ATMOSPHERIC ATTENUATION OF STARLIGHT 61 I. Introduction 61 II. Attenuation by Refraction 61 III. Scattering - 62 IV. Absorption 71 V. Distribution of Effective Scattering Agents 72 VI. Conclusions 74 REFERENCES 80 C. VARIABILITY OF STARLIGHT DUE TO THE EARTH'S ATMOSPHERE 82 I. Introduction 82 II. Seeing 83 III. Extremes of Seeing 87 IV. Scintillation 88 V. Conclusions 95 REFERENCES 97 iv

LIST OF TABLES Table Page I. Effects of Extreme Atmospheric Gradients 12 II. Approximate Aigles of Refraction 21 III. Distribution of Stars in Each Magnitude Group 27 B-I. /I0 in Percentage Transmission (Molecular Scattering) 65 B-II. I/Io in Percentage Transmission (Water Vapor Scattering) 67 B-III. Average Particle Concentrations in Stratosphere and Troposphere 68 B-IV. I/Io in Percentage Transmission (Haze Particles with Radii < 0.1 L 69 B-V. I/Io in Percentage Transmission (Total Scattering Effect of Air Molecules, Water Vapor and Dust) 69 B-VI. Range of Mean Tropopause Altitude 73 B-VII. Percentage Reduction of a Horizontal Ray Due to Attenuating Agent 75 B-VIII. Reduction in Apparent Magnitude of a Horizontal Ray Due to Attenuating Agents 77 B-IX. Number of Stars Theoretically Detectable at Various Levels Within the Atmosphere, With Scattering Taken into Account 77 C-I. Probability of Exceeding Various Extreme Values of Quivering 88 C-II. Range of Mean Tropopause Altitudes 90 C-III. Altitude of Density Irregularities as Obtained by Various Investigators 91 C-IV. Fourier Spectra Related to Frequency Ranges and Aperture Size 92 C-V. Range/Mean of Total Amplitude of Stars at z=0O and z=90~ 95 C-VI. Maximum Distance Traversed by a Straight Horizontal Ray Within a Layer Az, for Various Layer Thicknesses 94 v

LIST OF FIGURES Figure Page 1. Coordinates employed in non-sphericity analysis. 8 2. Geometry of refraction. 14 3. Total refraction angle as a function of altitude angle. 17 4. Time to occult as a -function of —azimuth and tangent.ray height. 18 5. Motion of geographical position of tangent point during a scan,. -- 19 6. Scan length as a- function of azimuth angle. 20 7. Scan time vs. azimuth, Nimbus Orbital parameters. 24 8. Number of stars as a function of visual magnitude. 26 9. Star data point positions, December 21.. 29 10. Star data point positions, August 21. 30 11. Plot of data point positions, December 21, 32 12. Plot of data point positions, August 21. 33 13. Reflected and unrefracted star images on photographic film. 41 14. Geometry of the orientation of coordinate systems X and X't 44 15. Orientation of coordinate systems X' and X". 45 16. Multiple-exposure frame depicting setting star. 48 17. Illustration of overlapping frames. 50 A. Geometry of unrefracted light rays. 56 A-2. Geometry of occultation. 57 vii

LIST OF FIGURES (Concluded) Figure Page B-1. Geometry of attenuation by refraction. 62 B-2. Transmission ratio I/Io as a function of ray perigee ho. 63 B-3. Percentage transmission I/Io at various- optical wavelengths as a function of ray perigee ho, for water vapor and molecular scattering. 66 B-4. Reduction in apparent magnitude of stars as a function of attenuation 1 - I/Io. 76 B-5. Number of stars theoretically detectable from a satellite with varying gains, as a function of ray perigee ho. 78 C-1. Dispersion of light rays upon passage through a turbulent layer in the atmosphere. 82 C-2. Simplified geometry of horizontal grazing ray viewed from satellite. 83 C-3. Image intensity I as a function of image diameter for typical cases of good and poor seeing. 84 C-4. RMS fluctuations of starlight as a function of zenith angle z, for z < 85~. 86 C-5. Photomultiplier tube as used to obtain Fourier spectra for scintillation. 90 C-6. Simplified geometry of maximum distance traversed by a straight horizontal ray. 94 viii

PARTIAL LIST OF SYMBOLS b departure of refracted ray at tangency from an unrefracted ray g gravitational acceleration GP geographical position (longitude and latitude) of a satellite or light ray tangency h altitude above the earth's surface H - scale height M mean molecular weight p atmospheric pressure r radius from center of the earth R refraction angle R* universal gas constant re earth's radius = 6371 km s distance along a ray path T atmospheric temperature z zenith angle Greek Letters latitude 7 vernal equinox (Aries).. index of refraction *v refractivity = -1 p- atmospheric density ix

PARTIAL LIST OF SYMBOLS (ConclUded) Subscripts L in the horizontal plane, lateral to the rzy o at light ray tangency to the earth, or to a spherically stratified atmospheric shell P in the plane of the refracted ray s as measured on the satellite, or to the satellite position x

THE UNIVERSITY OF MICHIGAN PROJECT PERSONNEL Both Full and Part Time Bodine, Margie S., Secretary Childs, David L., B.S.E., Assistant in Research Fischbach, Frederick F., M.S., Associate Research Mathematician Graves, Maurice E., S.M., Assistant Research Meteorologist Hays, Paul B., M.S.E., Assistant in Research Jones, Leslie M., B.S.E., Laboratory Director Klein, Gretchen S., B.S., Assistant in Research Mizgala, Charles M.:, B.S., Assistant in Research Mosakewicz, Mary C., Secretary Peterson, John W., M.S., Associate Research Engineer xi

ABSTRACT A method of obtaining atmospheric density, temperature, and pressure profiles between 5 and 25 km by measuring the refraction of stellar images with satellite instrumentation is described. The daily location of observations over the earth's surface which may be obtained from a Nimbus-type orbit is shown. Instrumentation and data handling requirements are outlined. Current areas under investigation are described. xiii

ACKNOWLEDGEMENT The authors gratefully acknowledge the financial support of the Aeronomy and Meteorology Branch, Goddard Space Flight Center, NASA, which made the project possible.: The Baker-Nunn ballistic camera measurements, were made with the cooperation of the crew of the Smithsonian Satellite Tracking Station, Maui, especially Messrs. J. T. Williams and William Perry. A great amount of credit also goes to the Smithsonian Astrophysical Observatory personnel in Cambridge, whose generous support of the project -has ranged from permission to use their camera to assistance with the data reduction. xv

I. INTRODUCTION This is the second report on.the refractive technique for atmospheric structure measurements. Several basic ideas have been revised, corrected, and augmented since the first report,1'2 and because it is deemed important that the technique be fully described within a single binding, a certain amount of background material and the fundamentals are reiterated herein. The reader is assured that the present report contains a comprehensive picture of the technique to date. An orbiting vehicle with the means of viewing stars through the earth's atmosphere and of measuring the refraction of the starlight permits deduction of the density, temperature, and pressure of the atmosphere at perigee of the light ray. The limits are approximately 5 km to 25 km, the lower limit due to probable obscuration of the light ray by clouds and the upper limit due to diminution of the effect to be measured. The refraction is a direct measure of density as given in the Dale and Gladstone Law, v = kp, where v is refractivity and p is atmospheric density. Since the orbiting equipment can measure only the total refraction of a ray and there are contributions to refraction by varying densities along the ray path, the means of ascribing the measurement to the perigee location is given. As a particular star is viewed during occulation by the earth, several measurements may be made at different ray heights; thus a density profile is obtained, permitting integration for pressure and temperature structure as well. The time required for a profile is found to be less than 30 seconds, and with a pair of telescopes sufficient measurements may be made to approach an observation density of one profile every 200 miles over the earth's surface. Only a little more than lO5 information bits per orbit are required from telemetry. Data processing in time equal to observation time is required to prevent backlogging of data and might be handled by one IBM 7090 computer. Data reduction requires precise satellite tracking which exceeds present capabilities, but which is anticipated to be available within a year or two. Other requirements appear to be within present capabilities. It should be noted that the refraction technique attacks the meteorological problem by furnishing the data meteorologists are presently equipped to handle: pressure, temperature, and density. To accomplish this requires deyelopment of the refraction measuring equipment. This stands in sharp contrast to the new techniques in forecasting. 1

Neither approach can be called correct or incorrect, Nor does one in any way degrade the value of the other; in fact, cloud photographs in addition to p, t, p, profiles would place the forecaster in a very strong position. However, the information rate.. required for the refractive technique is only about 2% of the others, indicating the great directness of this technique in measuring atmospheric. structure.

II. THE THEORETICAL BASIS FOR A SINGLE DENSITY PROFILE A. ASTRONOMICAL REFRACTION When a ray of light passes through a medium of varying refractive index, t, it undergoes a curvature C where C = grad sin z (1) Here z is the angle between the ray and the gradient; but the gradient lies along an earth's radius, making z the zenith angle. If a stellar ray enters the atmosphere on any, oblique path, its integrated curvature when it reaches an observer is known as the refraction angle, thus: Ro Cds (2) In the case of a ray entering a spherically stratified atmosphere, Eq. (2) becomes Ro = J tan z (2a) 1 Snell's law for a spherically stratified atmosphere provides the equation Or sin z = const. (3) where r is the radius of curvature of the strata, a function of the altitude h. Equation (2a) then becomes R~ = tuOrosin Zo:1.../2 (4) 1. [t2r27_grsinz ]l where ro = re+ho. If the ray strikes an observer located onthe earth's surface, we call its refraction angle the astronomical refraction. However, if the ray misses the earth and passes on through. the atmosphere to a satellite observer, the 13

the total refraction.is. twice the astronomical refraction, Ro, which would be measured by an observer stationed:t the point of ray tangency to the atmosphere. The subscript s will designate measurements made at the satellite. Then, Rs = 2Ro with zo = 90~ to Rs = 2o(re+ho)f. (5)'1 [2(re+h) 2-2(re+ho) 2 ] / In Eq. (5) and subsequent equations the subscript o indicates values at the point of tangency. Density and refractivity are connected by Dale and Gladstone's Law v kp = - 1; (6) thus the connection between satellite refraction and the atmospheric density. The literature contains a great deal of work on the calculation of the astronomical refraction angle, once a model atmosphere is prescribed. Of particular interest is the assumption of an isothermal atmosphere where the scale height, H, is defined by R*T H = - (7) Mg where R* = universal gas -constant T = temperature M = mean.molecular weight g = acceleration of gravity'We may then write _= ~..-(ho-h)/H (

and by substitution in (5) with change of variable R,= 2St~(C~-l) P e(h~-h)/Hdh (9) h... (re.h)2.-rl.] ho', ef r r+h) 2,2.r2,,2 ] which with certain approximations can be manipulated to obtain the classical formula1 Ro = kpJ 2ro (10) 2H Model atmospheres other than isothermal have likewise been utilized, with results considerably more complicated than (10). The inverse problem, that of specifying the atmosphere from a knowledge of refraction, has not been treated extensively. B. SOLUTION OF THE INVERSE PROBLEM Specification of the density profile of the real atmosphere from a knowledge of one refraction angle is patently impossible. However, the satellite will make many observations at various levels during occultation so that a continuous function is provided:.- RsR(ho0) The Rs(ho) data enables one to solve the inverse problem of recovering an atmospheric density profiles This principle may be demonstrated by considering Eq. (11), which is derived from Eq. (5) by dropping.certain higher order terms. O. Rs(ho) r o k d dh (11) h adh J1-ho Equation (11) is quite general in that it incorporates an arbitrary density profile; it is an integral equation similar to Abel's Integral Equation, and has the solution p(h) =- - R()dh (12) 5

This solution can be verified by substitution of (12) into (L1) with ho replaced by x as the variable of integration: Rf1' dh d Rs(x) dx h.:~ dh Deriving the differential limr(x)d Rs(x)dx ft Rs(x)dx dh hB ~ c~O:~' h h 1-0 Xh xx-(h+-) h ] ( The variable of integration of the f irst integral on the right-hand side is now changed, x - x+e, and we have hh:j. - i~(x+)- R((X+ -h h dx( d. / Rs(x)dx dR dx dJXE sXjs dx h h.x-h h (15) When (15) is substituted- into (13) R1 dh 0 dRs dx(16) o4ho h dx rx-h This integral can be solved by inverting the order of. integration as follows:.00 XI =.d - dx (17) dx s"ho dx Jh(h-ho)(x-h) But - dh = -2 tan -zxh, = j (18) ho (h-ho)h(x-h) h'ho 0~~~~6~l

Therefore (17). becomes 00 00 R(ho) d x = -s Rs(ho) (19) ho ho If the star is acquired by the sensor at a high altitude where the refraction is vanishingly small, and is tracked downward to occultation where refraction is a maximum, the scan will provide refraction angle as a function of time. Knowledge of the satellite's position and that of the star, together with a correction for the deflection of the ray, will provide ho as a function of time, Thus Rs(ho) is available for use in Eq. (12) to compute the density, p(h). The two assumptions involved in Eq. (11) and the derivation of ho(t) are discussed at length in Sections II-C and II-D. Once the density profile is obtained, it may be integrated to yield a pressure profile. Since the initial pressure is a boundary condition it must be estimated, but the error due to this estimate is reduced by an order of magnitude after 13 km of integration. The integration may be started above the level for which "accurate" denas-ity data are claimed so that comparable accuracies will be obtained for pressures about 5`km;below densities. Temperature may be deduced from the pressure and density. For example, a constant error in absolute density of +5% at 25 km is equivalent to +18% at 33 km and ~2.3% at 20 km. If integration were begun at 33 km, at 20 km the pressure would be in error less than ~5% due to density inaccuracy. The initial pressure at 33 km would have to be estimated and might be off 15%, adding only 1l5% error in pressure at 20 km. Thus comparable errors of about 5% would obtain in pressure (and temperature) at 20 km and density at 25 km. C. EFFECT OF TWO APPROXIMATIONS 1. Non-Spheric ity* The ability to use the refraction of starlight as an atmosphere's density probe depends critically upon the ability to reproduce mathematically the refractivity of the atmosphere from a discrete set of refraction angle measurements. It has been shown previously that for a spherically stratified exponential atmosphere this inversion is elementary. However, the real atmosphere is not spherically stratified; thus one must consider this effect very carefully. Fortunately, the lateral variations of density (i.e., refractivity) are small compared to the vertical changes, permitting a linear analysis of *This analysis is due to Mr. Paul B. Hays. 7

the phenomenon. Thus one may proceed to study this added problem it-eratively, as was done previously. Rewriting the basic refraction equation (1) dT _ l+ + - = Tx(Grad [ixT) (20) ds In this equation T represents a unit vector tangent to the ray. This differential equation must be integrated from the star to the satellite in order to obtain the total refraction at the satellite. This may be written formally as follows: s + = Ts += TV -i 1 T(s) x[Grad [i(s)xT(s)]ds (21) *t s (s) where subscript s refers to the satellite and subscript -oo represents some position on the far side of the atmosphere where there is no refraction. To simplify this solution.an iterative method is applied which assumes as a zeroth iterative the unrefracted ray. That is, since the refraction is small, the error in following a straight ray will be very small. The only problem remaining is to write the integrand of (21) in such a form that the integrals can.be evaluated. The coordinate system shown in Fig. 1 is introduced, where s(o) is the line y = re+ho, z = O, x = s and s0 = rd. Y S(O) j -(o ho -. / - Fg1 Codaeelyi anays 8

In this system: d(1 (ah sin i +s cos iy + \ (22) dx = h — < In order to simplify still further, introduce the small angle assumptions: on (1l), so that 5(1) -= ix + Rpiy + RLiz where Rp = refraction angle in the plane x, y; and RL= refraction angle out of this plane. Thus: Rp = h sin + os 9> dx (23) and 00 R, adx (24) coo Now introduce Dale and Gladstone's Law [Eq. (6)] and neglect the square of kp compared to 1: 00 Rp I k' (h sin K + aP cos G dx (25) and.00 R = k P dx (26) L 6L -P00 At -this point the problem' is completely formulated, and one need only introduce a correct form for p(x,y,z). This is done by introducing a pseudo-exponential atmosphere with variable mean density and scale height: P = Po(sgz)e(hh)H(Z) (27) 9

Then ap Po(s,L) e-(h-4)/H(s@,L) (28) ah H( se, L) ap ao(se, L) -(h-ho) /H( Se,L) ( s L)L) h-ho) /(s L e._ e~h~ho /H s + h-h sH sL) e ~as as LI(sg, L) as-e (29) ___Q -.( h-ho) /H( se, L) h-h0 -(H( L hho) /t( s@, L) L+ po(sL) LH2(sq., L) J) of x, one finds that s x hho x2 sin G 1, cos si) 2( re+ho) re+tho Then: o(s x, i) Rp -k f p2(xi) K + 3E(X,L) X3 7 _ - -H(x, L) x 2E(x,L) (re+ho)j 2 +apo(xL) x e(re+p~)(X'L) ()2) x re+ho R -k apo(x,L) p(xL) H(xL) 2 2(re+ho)H(x,L) LCo H2(xL) 6L 2(r+h) The mean density and scale height arethen expanded along the line x in a Taylor series, i.e., Po =.+'POh(o)x + OP.' (O)x2/2 + p" (O)X3/3 +. (34) + poPL(O)L +'op2(OrX + rP(o)X2/2 + -h 10

where p,(O) = a p (35) Po ax at:x 0 P (O) = 1 ap_ (36) po aL at x= 0 and similarly for H. These derivatives are the density variations noted at the point of closest approach of the unrefracted ray. They are now substituted into the approximate refraction formulas above and the formulas are integrated, resulting in the following expressions for the refraction angles: 0 2 (37) RL RpoHo PL() + (38) where is the refraction for a spherically stratified atmosphere as in E (10 9) is the re fraction for a spherically stratiefed atmosph ere as in E (10 ha). In order to clarify the significance of the additional terms, Table I has been included to indicate the effects of extreme atmospheric gradients. Table I was constructed by scanning meteorological records for extreme gradients, computing the derivatives, and substituting in Eqs. (37) and (38). The importance of the results is threefold. First, it proves the nearly negligible character of the spherical stratification assumption in the development of the general refraction. equations. Second, it proves the negligible error inherent in assigning the refraction-density to the geographical position of ray tangency. Third, it proves the entirely negligible character of refraction in the lateral plane. 11

TABLE I EFFECTS OF EXTREME ATMOSPHERIC GRADIENTS Extreme Case of Non-Spherical Stratification Rp/Rpo RRp Strong extratropical gradient of P and T (300 mb) 1 +.oo6 -.001 Intense cold trough (300 mb) 1 +.008 Strong hurricane (500 mb) 1 +.026.002 Surface frontal zone (1000 mb) 1 +.013 2. Effect of Integrating Along Straight Rather Than Refracted Ray In order to obtain the solution for the inversion, Eq. (11) was written on the basis of the geometry of an unrefracted ray. Likewise, the manipulation required in passing from Eq. (9) to Eq. (10) involves the approximations = A to = 1 and \12r+h-ho ~=2r in the denominator of the integrand, which corresponds again to integration along a straight ray path. To determine the significance of this approximation, the following approach was taken. Equation (9), the exact expression for refraction in an isothermal atmosphere, was evaluated by graphical methods. Equation (9), after application of the approximations, becomes Rs = 2(1o-1l) (r7) e(h40)h)/H ho epsio and this expression was evaluated by both analytic and graphical methods. The evaluations were carried out for a ray tangent at the earth's surface, since this ray's departure from a straight line is greatest. The scale height was 7.5 km and ~o = 1.000277. Upon graphic evaluation, Eq. (9), the exact expression, yielded Rs= 75.48 minutes, while Eq. (40) yielded Rs = 68.76 minutes. Equation (40), evaluated analytically, gave Rs ~= 69.44 minutes. 12

Thus, we conclude from the close agreement (1%) between graphical and analytic results for Eq. (40) that the graphical technique is accurate. KnowiAg this, we further conclude that the comparison between graphical evaluations is valid and that the approximations are seen to be not at all negligible, but equal to 9% at the earth's surface. Clearly, integrating along a straight rather than refracted ray will introduce significant errors. Effort is being directed toward an analytic solution of integration along the refracted ray. The present estimate, however, is that a solution by numerical methods will be required. With the availability of high-speed computers, a numerical method should not be considered undesirable nor necessarily less ideal than an analytic solution for this problem. D. LOCATION OF TANGENT POINT Simple geometry is employed to obtain the geographical position of ray tangency, and is shown in Fig. 2. The satellite's position is assumed known, as well as the refraction angle and direction of the star. Referring to Fig. 2, lines SS' and SB can be drawn with the given information. The normal to SB is drawn through the earth's center, determining P and the normal distance p. Knowing p, OP' can be drawn normal to the star's direction, determining BB' and hence the intersection B. The geographical position of B, GP, yields latitude and longitude. Appendix A contains a complete discussion of the determination of latitude and longitude. Less simple and more. important is the determination of the ray's height, ho, at tangency. The radius OB = re+ho+b is known, Thus, the determination of b, is equivalent to the determination of ho. A direct result of Snell's Law of refraction is that for any ray path in a continuous, spherically-stratified atmosphere rL sin z = constant (41) where, of course, z is measured from the perpendicular to the gradient of ~. Thus, in Fig. 2, rs5sin zs = rolosin Zo (42) 13

Fig. 2. Geometry of refraction. 14

and in triangle OBS sin Zs re+ho+b (43) sin ZB rs But since ts = 1 and sin zo 1, roo = (re+ho+b) sin zB (44) However, sin zB = sin(90~+Ro) = cos Ro and (re+ho) 11o b = - (re+ho) (45) cos Ro or b = ((re+ho+b) (1 c (46) Equation (46) is exact with only the assumption of spherical stratification. Departures from spherical stratification apparently have a negligible effect on the determination of b.; This may be concluded from the analysis in Section II-C-1. The distance b, and consequently ho, is a function of Ro and O only, and independent of the ray path or density distribution along the ray. It can be shown that about 99% of the refraction of a ray tangent at ho occurs within ho+25 km of the surface. Using temperature extremes of the real atmosphere which may exist up to 50 km, and considering limiting isothermal cases where H = 6.4 and 8.4 km, the limit of variation in Lo, and thus in b, may be computed from Eq. (10). The result is a maximum index of refraction change of +.0015%. Equation (46) then shows b to have a maximum, error of 90 m. A 90-meter altitude error is equivalent to about 1.1% error in po or about 1.1% in Rs. In any inversion formula used to retrieve the density function, such as Eg. (12), the integral of Rs(ho) from h to oo will no doubt appear. Since the maximum error in the function Rs(ho) is a constant percentage, 15

the percentage error in the integral can be no more than this. In fact, if the errors in Rs(ho) should be at all scattered, the error in the integral will be considerably smaller. In view of these considerations, one would expect a maximum of 1% density error due to uncertainties in ho, and these would be subject to an- iterative improvement if necessary. The foregoing applies to determination of longitude, latitude, and height of the point of tangency for a particular ray. -Dung one scan, the tangent point moves downward and in the direction of satellite motion. due to satellite motion, toward the satellite due to refraction, and westerly due to the earth's rotation. Using the direction opposite to the satellite velocity vector as a ref.erence, one may speak of the azimuth angle of a ray as the angle in the horizontal plane between the projection of the ray onto the plane and the reference. The altitude is the angle between the ray and the horizontal plane. Referring to Fig. 2, Rs-9 is the altitude, and if ap = satellite's angular velocity component in the OSB plane, 0 = wpt and 9 90+Rs-sin'l[(re+ho+b/rs).cos(Rs/2)]. Rs as a function of 9 is shown in Fig. 3, while in Fig. 4 the time to occult is shown as a function of azimuth, Az, and tangent ray height, ho. From Fig. 4 it may be noted that at 0~ azimuth the time for the scan to move from ho = 35 km to h0 = 5 km is 24 secondswhereas at 300 azimuth the time is 28 seconds. At greater azimuths a proportionally greater time penalty is paid, so that at 45~ the same scan requires almost 40 seconds. The motion of the geographical position of the tangent point in the general direction of the satellite is shown in Fig. 5, where a non-rotating earth is assumed. S is the sub-satellite path and AA' is the locus of geographical positions of light rays tangent to the atmosphere at 25-km height. -B' is the locus of sub-satellite positions where 25-km rays would be seen by the satellite, and CC' is the same for 5-km rays. Angle i is the inclination of the orbital plane to the normal to the direction of the star, which is the angle between planes determined by.S and AAK' If the star is viewed directly aft of satellite, the geographical position moves from T to GP?, which is 49.4 km, or 26!26 subtended at the earth's center, 0. t From the spherical triangles in Fig. 5, we can write the path length GP1 GP2 of the scants geographical position in going from 25- to -5-km height: = cos- (cos 26\26 cosP (47) where TX = ~sin'l(cot i tan 92) 16

60 UJ. 50 z 40 ir z 30 z 20 10 0 I-. 0 -30.5 -31.0 - 31.5 - 32.0 -32.5 -33.0 ALTITUDE ANGLE R -- Fig. 3. Total refraction angle as a function of altitude angle. 17

-30.85 30 sec 40 sec 40 km -30.904 -so.9s' ~~5 sec -30.95 -31.00 w _ _ ___ 30 km w -31.05 -31..10 - 2~ -31.15 20 0eC 00 w 20 kmr M -31.20 -31.25t 15 km 15 sec 03.3 10 sec 10 km -31.35 5 sep 0 sec 5 km -31.40 u-I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 5 10 15 20 25 30 35 40 45 50 55 60 65 Az, AZIMUTH ANGLE (DEGREES RIGHT OR LEFT OF AFT) Fig. 4. Time to occult as a function of azimuth and tangent ray height.

A' 8B C' Fig. 5. Motion of geographical position of tangent point during a scan. and X GP1 = sin'l(cot i tan @1) azimuth = cos-l(sin i cos TX). Scan length as a function of azimuth angle is shown in Fig. 6. These results are for a non-rotating earth. Scan-time may be taken from Fig. 4 and rotation of the earth added vectorially. E. MAGNITUDE OF REFRACTION The angle measured, Rs, must.have sufficient magnitude-to be readily detected by the instrumentation. We may compute the approximate angles by Eq. (10), obtaining. 19

210.0 200.0 190.0 180.0 170.0 160.0 150.0 E -140.0 C 130.0 -J 120.0 110.0 100.0 90.0 80.0 70.0 60.0 50.0 40.0 0 10 20 30 40 50 60 70 AZIMUTH (DEGREES) Fig. 6. Scan length as a function of azimuth angle.

TABLE II APPROXIMATE ANGLES OF REFRACTION ho (km) Rs (minutes) 0 69.4 5 41.7 10 23.4 15 1l.0 20 5.o 25 2.3 F. SUMMARY In summary, a single refraction scan consists of measuring the apparent position of a star relative to a space stabilized reference. While the light ray passes through little or no atmosphere and suffers no refraction, the star's angle with respect to the reference is known, is constant, and may be pre-computed. But if the star is viewed in a direction generally opposite to that of the satellite's velocity, the earth will appear to move upward and will eventually occult the star. Prior to occultation the starlight will pass through ever increasing atmosphere and be measurably refracted. This refraction will be sensed as an angular change relative to the space reference, and will be recorded'in the satellite as a function of time. Probably, the star will be tracked from a tangent height of about 35 km downward to obscuration by clouds at roughly 5 km. The angles measured will vary from a few seconds to almost one degree, and the time involved will be a little less than onehalf minute. Perhaps one angle per second would be recorded in digital form, together with a time reference, so that from 20 to 30 datum points would exist to reconstruct the function Rs(t). The function Rs(t) would be reconstructed by ground based computers, which would compute ho(t), and thus Rs(ho) and p(h). The density function would be integrated for pressure and temperature.: The computation would include determination of the geographical position for each datum point. Tracking data would be an additional input required by the computers. 21

III. THE THEORETICAL BASIS FOR GLOBAL COVERAGE A. TIME CONSIDERATIONS The objective is to provide a density-temperature-pressure profile of the atmosphere from 5 to 25 km at as many locations on the earth's surface as are required by meteorologists for three-to-five day and longer range forecasts. Presently, adequate coverage exists only in North America and Europe, These data are derived from balloon-sondes, making an extension to polar, oceanic, and Communist-bloc areas all but impossible. A consensus of meteorological writers indicates that p-T-p profiles spaced 200 statute miles apart over the entire earth's surface would suffice for forecasting purposes. Among the mneteorological satellites, the Nimbus orbit most nearly approaches the ideal for refractive techniques. Therefore Nimbus orbital parameters have been used to determine the global coverage possible. The Nimbus orbit is assumed to have these characteristics: 1. circular, height 1100 km, period 107 minutes; 2, inclination of orbital plane to equator 99,890; 3. precession of orbital plane in same direction and at same rate as mean sun; 4. injection into orbit over the equator at local noon; and 5. launched southward from Pacific Missile Range. Since this orbit is nearly polar, all areas of the earth may be scrutinized, However, if observations are equally spaced timewise, a tremendous concentration of observations near the poles are a consequence of the polar orbit. The main problem, then, is to obtain an adequate observation density near the equator. The satellite will make between 13 and 14 complete orbits per day and the earth will rotate about 26~ during an orbit, which is about 1800 statute miles at the equator, The satellite's latitude change is roughly 3~ or 200 statute miles per minute, thus it is evident that 9 scans per minute in the equatorial regions would be required for optimal coverage. Since the scan time is a funetion of azimuth angle only and in no case can be reduced beyond 21 seconds, three or four separate star-tracking devices would be required. In the midlatitudes, two star-trackers would suffice and in polar regions, ones 22

For purposes of planning a prototype experiment, a compromise on two, star-trackers is adopted. This will yield less than optimal observation densities in the tropics, but will be adequate or more. so everywhere else. If at some later time additional equatorial observations are required, the number of star-trackers might be increased. As the satellite moves in its orbit, the star-trackers would preferably track successive stars on various azimuths since this will have the effect of spreading the observations uniformly over the earth. On the other hand, as the azimuth.angle increases, the time required for a scan lincreases, and fewer observations are possible. Time versus azimuth is shown in Fig. 7. The maximum azimuth which provides the optimum of observation density.and observation location uniformity is rather arbitrary, but appears to be in the neighborhood of 300. At that angle, the scan time required is only about 15%& more than directly aft, at O~, yet the location of.the tangent point lies 1200 miles from the sub-satellite path. For a-more accurate determination of p and T the scan should be started in the neighborhood of 35 km (as explained in Section II-B). We may also assume an average azimuth of 15~ and, from Fig. 4, determine the average scan time to be 25 seconds. It has been estimated on the basis of similar devices that the time for the tracker to slew and lock on to a star will be under 3 seconds. On the average, then,. one scan per tracker will be made in slightly less than 28 seconds, or about 236 scans per tracker per orbit. B. CELESTIAL CONSIDERATIONS In the preceding discussion,,.two important factors were not taken into account: first, the restrictions on star-tracking caused by sunlight; and second, the availability of a star in the desired position and at the desired time. As for the sunlight, it appears that with present state-of-the-art equipmet, only a few of the very bright stars may be tracked under daylight. illumination. Exactly what angle below the horizon the sun must attain before the luminosity of the atmosphere is decreased sufficiently for tracking is not known. The determination of this angle will be difficult and probably, not well estimated until the star-sensing equipment is selected and tested, Even then the difficulty of testing an. instrument.above the atmosphere may preclude a firm answer prior to an orbital flight. In the absence of other information, one may assume that i is possible to conduct refraction scans in one:half of the orbit-in particular, the ha4lr-rbit in which the points of ray tangency lie in the earth's shadow. Therefore each tracker can make only 118 scans per orbit. 23

80.0 70.0 60.0 50.0 w o 40.0 il.o 20.0 10.0 0.0 10 15 20 25 30 35 40 45 50 SECONDS Fig. 7. Scan time vs. azimuth, Nimbus orbital parameters.

The number of possible scans per orbit. has been based on the assumption that the tracker can slew and lock on to a star at about the -35-km level within 3 seconds. Regardless of the type of star-tracking equipment, it will be advantageous to track brighter stars, and we must.deal with the questions of how bright an available star might be and what effect the launch date might have. A 600 swath centered aft of the satellite would be swept out by the tracker's field of view. This constitutes one-half of the celestial sphere. Since the orbital plane is slowly precessing, tle portion of the celestial sphere in view is constantly changing. Thus the launch date itself is of no consequence and for any given date a particular star field.will be available. To.determine which stars would be available on any given day, the IBM 709 computer was utilized; an outline of its program is given here. A magnetic-tape version of the Albany General (Boss) Catalog was obtained from Goddard Space Flight Center. This catalog has information on about 33,000 stars down to 7:.5 magnitude, the stars ordered by right ascension. Since the primary purpose was to obtain the star-magnitude implications of various procedures, the stars were first sorted and counted by magnitude groups and a new.tape made in which the stars remained ordered by right. ascension within each magnitude group. The star count is shown in Fig. 8. Next, the distribution of stars in each magnitude group by declination north or south,.and for each hour of. right ascension was obtained. The results are shown in TabIe III. It is well known that the star population is most dense around the galactic equator, or Milky Way. Another belt of stars, Gould's Belt of Bright Stars, is also of interest; it contains the nearer and brighter stars of what is known as the local cluster. The plane of Gould's Belt is inclined about 200 to the galactic plane. Both planes run approximately north-south. The galactic plane is inclined approximately 580 and Gould's Belt 65~ to the equator. Since the orbit is inclined 800 to the equator it is evident that with the proper right ascension the bright stars would almost all be within the field of the star tracker, whereas three months later the orbit would intersect the belt sharply and the fewest bright stars would be seen'. The computer was programmed so that the inputs were the orbital elements and the desired star coordinates. The outputs were: geographical position of measurement, position of satellite at occultation, time of occultation, azimuth of star at occultation. The appropriate equations are given in Appendix A. This program was run for each two hours of right ascension or, equivalently, for each month of the year, for all magnitude groups less than 4.0. 25

10 104 — MAGNITUDE NUMBER -2 to 0 2 o0, 1.00 10 X 1.01,, 2.00 27 Ir /2.01,A 3.00 107 3.01 ", 4.00 291 4.01 1 5.00 1005 u. 3 5.150 -- 1203 a, /5.51 it 6.25 3592 0 -2,, 6.25 6237 102. 0 I 2 3 4 5 6 7 8 9 10 11 12 13 MAGNITUDE Fig. 5. Number of stars as a function of visual magnitude. 26

TABLE III DISTRIBUTION OF STARS IN EACH MAGNITUDE GROUP Magnitude Group <2. 00 2.01-3.O0 3.01-4.00 4.01-5.00 5.01-5.50 Total Declination: North 18 47 141 438 384 1248 South 21 60 150 545 619 1395 Right Ascension: (Hours) 0-1 0 7 730 37. 81 1-2 1 4 13 34:41 93 2-3 0 3 6 6 47 92 3-4 1 3 16 44 39 103 4-5 1 1 15 50 62 129 5-6 6 11 15 57 63 152 6-7 5 1 10 44 74 134 7-8 4 2 14 50 79 149 8-9 1 4 10 47 58 120 9-10 1 4 14 44 49 112 10-11 1 3 14 37 41 96 11-12 1 3 10 29 48 91 12-13 4 9 10 35 49 107 13-14 2 4 7 37 33 83 14-15 3 8 9 41 37 98 15,16 0 6 20 48 52 126 16-17 2 7 15 47 49 120 17-18 1 10 20 35 37 103 18-19 2 4 11 46 68 131 19-20 1 2 15 43 68 129 20-21 1 3 13 -43 49- log09 21-22 0 3 9 35 39 86 22-23 1 3 14 49 43 110 23-24 0 2 4 44 41 91 27

The output of geographical positions was counted and sorted on the basis of:.1. azimuth. of star < 30~ when observed; 2. latitude of measurement in 100 groups from pole to pole; and 3. declination of star, north or south. Only results from stars with azimuths < 300 Oere plotted. Results from north-.ern and southern..stars were separated in all cases- because observations are restricted to northern stars when the satellite is injected southward on the equator at midnight, and are restricted to southern stars when. injection is southward at noon. (This is approximate for two reasons: (1) it is a.mere assumption.that.the portion of the orbit unused due to sunlight is exactly one-half; and (2) except at the equinoxes the-dividing circle is not actually -the celestial equator, but. is inclined equally to.the sunts.declination,) The use'of southern stars increases the number of possible observations for any given.magnitude by about 20%; therefore the noon-southward orbit is to be preferred, The number of possible observations were plotted for each orbital plane according to latitude group and declination of star. Orbits were then classed as favorable on.the bases of: 1. largest number of observations; and 2. highest proportion of observations in equatorial regions. By -the use of southern stars, it was determined that December 21 was the -most favorable and, August,21 the least favorable date. However, the star distribution in the various orbits was sufficiently uniform that even in the best orbital plane —that obtaining on December 21_ —only 153 observations per orbit were possible, when 236 were desired* (On August 21, only 88 were possible.) Thus, it was:apparent -that stars fainter than.. 4 0 would have to -be used to approach..the required density, This process was repeated for the stars down to magnitude 5.5, for December 21 and August 21 dates, On the former date 812.southern stars, and on the latter d-ate 647 southern stars were in the field, The data point positions of-these stars" are Bshown in Figs. 9 and 10 respectively. These stars were then subjected to hand.selection of 236 stars from each orbit. spaced timewise so that they could all be scanned by two trackers and spaced geographically so that- the best- data point distribution could be attained. Stars were not divided north or south, but on.the exact basis of the sun.position on that date.

NO. OF OBSERVATIONS 0U I I qI (AI 0I I 0c 0I 0 C (t ---- 9~ *70.................... t P ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,,~ -eee................... eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee ee O/ eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee. l'""eoeeeeeeeeeeeeeeeeeeeeeeeee r ooeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeefeeeeeeee ___j0 0 eeeeee0 e e e e e e e e e e eeeee'eeeee l' —1 OO*000*000 000000.000 eeeeeeeeeeeeeeeeee'eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeel.0.O ~1 -e-eeeeeeee.eeeeeeeeeeeeeeeeeeeeeeee eeee9eeee0 e e e e ~ ~ ~ ~ * ~ O ~~ ~ O O O O O _j__ t eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeOeeeeeeeeeq ~eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 0 0'0 0 0 0 0. 0 0 0 0. 0 0 0 4 O O O O O O O O O O O O Oil, 00 00 000,~ e.eeee -eeeeeee e ee e 0,0,.... leeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee~ eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeJ oeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeI.t. eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeel ff Deeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeq 0000ee000ee'eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee/ g eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeel vw ~e00eeee00ee00eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee0 ee 0. W -, ee eeeeeeeeeee.''O r eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee'eeeeeeeeeeeeel.,'00 -000000#000000*00*00000 0eeeeeeeeeeeeeeeeeeeeeeeeeeOffeeeeeeeeeeeeeeeee0.. -e..v O e e e 0eeeeeeeeeeeeeeeeeeeeeeeeeeeeeee4 ee~) eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeel ~ *~ ~ 0 I-0 "0,eeeeeeeeeeeeeeeee 000000000'0000000'00000000000000'0 0,eeeeeeeeee e e e e e e e e......~.......ill emJ 0 0,eeeeee000eeeeeeeeeeeeeeeeeeeeeeeeeeeeee00000 oeeeee ee 0 0 0 0 0 0 Dm 00 0 0 0 00 0 eeeeee0e0e0e0eee0e0eee0eee0eeeeeeeeeeeeeeeeeeeeeeel ~0 0 00 0,e00e~~ o~~~~oeeee e e e e e e e e e e e e e e e e e e e eeeeee... — C~ 00teeoeeeeeeeeeeeeeeeeeeeeee0ee eeeeeeee00eeeeeeeeeeee —-- O~0 m0 0eeeeeeeeee eeeeeeeee e e eeeee e eeeeeeiI eeeeeeeeeeeeeeeee0eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee~ O e e e eeeee0e eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeel 0000 r01 0eee e e 0e0ee0e0eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee~J 0"00eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee00 eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee/ 000000 0000000000000~0 00000009000 C t70 e eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeel eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee. 000 00 -e-e0e0ee0ee0e e 0eeeeee e 0eeeeeeeeeeeeeoeo eeeeeroo eteeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee0e e e eeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 0eeeeeeeeeeeeeeeeeeeeeeeee0e e e e e eeeee0ee0 e e e eeeeeeeeeeeee~,~.me o'eeeeeeeeeeeeeeee eeeeeeeeeeeeeeeeee eee................... eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee e V1 00 ~00000000000000*000000000000000000000*0000000000 eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeI gJ'l4 eeeeeeeeeeeeeeeeeeee e e e e e e e e Oeeeeeeoeeeeeeeeeeeeeeeeeeeeeee.0000eeeee, eeeeeeeeeeeeeeeeeeeeeeeee0eeee0eeeeeeeeeeeeeeeeeeeee t —0eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeel 0 OD 0000000 ci-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i Fj) e e e e e e e eeeeeeeeee 0....eeeeeeeee0 e e e e e e e e e e e e e e e e e eeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee OD 09.. 0..0 eeee e e e e e e e e e e e e e e e e e e e e e e e e ee e — 0G "- ~....''"......~...,.............................. ~ H...~..~....,.,......1,,,,,,, -r 00~................0#00009000009 0 ~ 0000000 000 000000 ~ ~0 ~ 0 ~ J......~.T. 4~~~~~~~~ ~ 000000~00000 eeeeeee~~~~~~~~~ O eeeeeo " 0000000000009000000 0000000eeeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee-e-e-e-e-e-e/ (D00000000000000*0000000000000900000*00*000000 O 000000000000eeeeeeeeeeeeeeeeeeeeeee e eeeeeeeeee eeeeeeeeeeeeeeeeeeeeeeee'eeeeeeeeeeeeeel.0.eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee0 0 - eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee eeeeeeeee e e e e eeeeeeeeeeeeeeeeeeeeeeeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee. eeeeeeef eeeeeeeeeeeeeeeeeeeeeee e e eeeeeeeeeeeeeeeeee F~eeeeeoeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeee0ee0 e eeeeeeeeeeeee 0000e0000e00eee0eeee0000e0 ee0e00e00 e eeeeeeeeeeeee 0000000000000000000000000 f A eeJe00000000000e0e000e000eeeeeee0eeeeeeeeeJ 4 00-eeeeeeeeeeeeeeeeeeeeeeeeeeeeee,~o,. oeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee e eeeeeeeeeeeeeeeeeeeeee " ~~~~~~~~~~~~~ ~~~, ~ eee0eeeeeeee0eeeeeeeeeeeeeeeeeeeeeeeeeeee4 00000000000000000000009000009000000000000 0 00eeeeeeeeeeoeeeeeeeeeeeeeeeeeeeeeeeeeee 00eeeeeeeeeeeeeeeee0eeeeeee0 e eeeeeeeeeeeee 00e0eeeeeeeeeeeeeeeeee0eeeeeeeeeeeeeee~ 00eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee4 000 0oeeeeeeeeeeeeeeeeeeeeeeeeeeeee0a.e0. ~ 0E0eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee ~ eeeeeeeeeeeeeeeeeeeeeeeeeeeeee'eeeeeee-eeeC ~eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee4 eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee e e 00 ~'eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee......~ eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee! r"- ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,~~~~~U ~eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee e e e e e e L eeeeeeeeeeeeeee0eeeeeeeeeeee0eeeeeeeeeeeee0eee00,0000000 —=,0 00.e-eeeeeeoeeeeeeeeeeeeeeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee40*0000 C,eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee~ O!000 m0 eeeeeeeeeeeeeeeeeeeeeeeeeeeee ooooeeee,eeeeeeeeeeeeeeeeeeeeeoeeeeeeeeeeee0o00000. e-'eeeeeeeeeeeeee...............,eeeeeeeeeeeeeeeeeeeeeeeee e e e e e e eeeeeOOeeeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeee e e Oeeeeeeeeeeeeeeee4 00e0eeeeeeeeeeeeeeeeeee000000eeeeeeeeeeeeeeeeeeeeeeeeeeej00*90

NO. OF OBSERVATIONS r3 *aJ rV <,^ w, i, iJ P P P cn Un cn Q, a) a) a) s V, (D ~ ~ ~ ~ NN( o A 0 M00O: G- ",0.............. - 0.................~ -'1~eeeeem.ewe-,ee*eeeeeeeeeeeeeeeq-,, 00,.0eeeee e e 0e0e e 0eeeeeeeeee —-------------........................................................................... eeeeeeeeeeee.eeeeeeeeeeeeeeeeee'eeeeeeeeeeeeeeeeeeeeee.eeeeeeeeeee (~oseeeeeeeeeeeeeeeeeeeeeeeeeeeeeee~..................................................... rr..........,.....~......................~......................................~....... 0~.~0~~ 0~~ 0~10 0 Os * **,0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0~~ V eee 00000000 e000000e0e00000000e00000000e00000*e0* 0*009000* 0000~ee e0e.0eeee*0eeeeeeeeeeeeeeeeeeeeVeeeeee 0 * F eeeeeeeeeeeeeeeeeeeeeee e e 0eeeeeeeeeeee ee ee..eee.eee.eee.ee.eee..e..eeeee.e.e.eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee.. O meeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee e _q............... m.a.eeeeeeeeeeeeeee.e eeeeeeeee......................... "-I -. @@************v**************v******............. ~. (~ 0 *************************** ^ 0 ID w' @e -01 040 0lD 0106 le -0- boel 0 0le 0 * 916- 0 *le* 0 00 l * G 0 - -01 0 0 * l 00 ~4...,.,eeee e e e e eeee.................-n *' ~............- - (J)mm*****w J e e e e e oeeeeoeeeeeeoeeeeeeee eeeee.. (Aee eeeeeeeeeeeeeeeeeeeee eeee*eee e * 000 00 4~ e e e 0 -ee|000eeee e e e e e e e e 40000000 e e: e e e 0 v R, *0 000 0, 00e~~ ~~ *00 00 00 0* 0 0 0 0 0 00~~00 0 00000 0 00000ee' *L**....~*~~~~~~~*~~~~~~*oo*~0.............. 0~~e~~~~oee00oe.~~~~~eoe~~~0** ~ *||* *** * cr o o...,... c~.o*.. *..~.~ _...........................,~~~ ~~~ n0e0 eeeeeeeeee.~eee000* 00000000000000000 * 0 ieeeeeeeeeeeeeeeee~*eeeeee0000o0000000000000000* 0- 0-eeee eeeeoeee0 ~~eeeeeeeeee 0*e~~*~****s**.. CAeeeee eeeeeeeeeeeeeeeee _ - eeeeeeeeeee*e e e e eee e ws*~~@ @@ @@ @@ @ @@@teeee*****eeeeeo*******eeeeeeeeeeeee o................................*vvvv***vv ~~~~~~~~~~~~~~e~ *eeeeeeeeeeeee**vv***** 00e0e 00o0e 000e0eeooooe~~e e oooo~e eo ee oeeooe eoeoooeeeeee~. ee _. e eeeeeeeee eeeeeeeeee eee*0eeeeeeeeee*0 e e e eeeeee H00*00-0060e*0o0000G -- 0,0* 0 0*******. o m:*000000000010 ~' C H _ 000000000000.......~....... cn........... 0 0*0000ee 60o~~o0oeee00e0eeeeee0ee0 _.....................e............... Le.eeeeeeeeeeeeeeea................................. H ts *n~ eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee e e _eeeee eeeeeeeeeeeeeeeeeeeeeeeeee!0O 4+ 00000*000000.....................v B~~~~~~~~........................ ~.r.U........ C~'00 0000000 _0 0 0000 ~ _e eeeee eeeeeeeeeeeeeeeeeeeeeeeeee e ee,0 00-0eeeeeeeeeee _C~ *0000 oo oo*o o ree eeeeeeeeeeeeeeeeeeee...ee e I~************************ - Pi 0000000. 0004000000000000000*.," )~~~~~~~~~oeooo 0ee 0eee~oeo!ooo _ 0000 0*0*0000000000000.00000..... -_ n W. * -, 0. —******************v** —******z. 0 0 0 0 0 0 0 0 rr t........,,...V C. 000000000* ~. v e OD eeee -..-. - -eee e - e ~ - -........................ ~.. { b.eeeeeeeeeeeeeee........................... eeeeee&eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeI ~eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeseeeeeeee-eeeeei. 00A.eeeeeeeeoeoeeeeeeeeoeeeeeee'e0e'e'e'e'e 0ee 0 oeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee'eeeeeeq 0P0000&0* a000 0000a* 00000 00000 *0400 --- 0000000000000*0*0009.* o U1 ~~~0000000406000 00004, d- P' 0*0900000000000*00*00 000*0*00000000000000*0000*000 ~~~~~~ 000000*000000000000* 0*00 000*00 000000 000 000 000000*0000**000 Cf 0 00000000000*0000000000*0000000 0*0**000040*0000*00000000 0000000000*0000009000000000~ 0*0**0000000060000000000004~~~~~~ H ~ ~~~~~ ~~~~~~~~ ~~~ 000*000*00*0*0000000 ~ ~~i~~~~~~~~~~~~~ ~0 ~ ~i i *~ 0 ~~ ~ ~0 ~~ ~ * ~0 * ~~ ~~ ~ ~3 ~~~~~~~~0 ~i~~0 ~~~~00~~~~~~0~0~~... $00*000000000000000006060 -~ ~ *00*00000000 0490*000000 ~~d ~~~~~0*00-0*000*00*000*0000000000000 ~~~~~~ ~(~~~~~ ~~~~ ~~ ~~~ 00~ 0 O ~~~~~~C00000009000****6000*000000000 Ul ~~~~

The selected 236 stars were then. fed into the computer and data points plotted for a 24-hour period on each of the two days. The results are shown in Figs. 11 and 12. Note that due to solar illumination, polar coverage in summer is impossible. Note also the tropical observation density obtained with two. star-trackers. The most significant conclusion of the star studies is that if stars brighter than magnitude 4,0 are used, the desired number of density scans can not be obtained on any date; whereas some magnitude brighter than 5.5 is sufficient for all orbits. All work to date has been based on a noon-midnight orbit. Some Nimbuses might be flown in a sunrise-sunset or twilight orbit. It so, the star-trackers could be programmed to look away from the sun, in which case tracking on azimuths between, say, 15~ and 60~ would be possible. The consequence of higher azimuth angles is a longer scan and fewer scans per minute. Observations could be made for the entire orbit, however. 31

I~'~~~~~...~....'C. I~ E. S S ~~~~~~~~~~~~~~~~~~~~~~~`'~...;c- ~, ~ ri ~ ~~~~~~~~~, e' ~ ~ ~ ~.~ - ~,.is-~,.N' ~ t t~~~ ~ ~ ~~~~~~~~~'/. _/'L, —-''IZ\ ~,. ~~~~~ r,'.... t~.... I:~ 6'.e/ ~. ~. ee ~ e!'- ~ ~ ~',''l~~~ ~. ~ ~'~.'.. ~ r. ('5'~,'~ )'-., ~r C2 ~ ~ ~. ~0.. -. "~~L~''~''~ ~ ~.' S,._ ~~~~~~~~!,:, ue ~e ~ ~ ~ Sb) Southern Fig. 11. Plot of data poi~~~nt psion, DeebeS1 ~S.0 9tt* 0 F S S'~ cc c` ~~~~~~~~~~~~~.~.~~~~~~~~.A. 0 0 S 0 % 00`C?. ~ ~j f' \~ ~~~~~~~hL L~~~~~~t ~~e ~ ~t,~~~~~~~'~ cL-~ ~.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ( A ~.~ ~ ~:~~~~~~~~l S S~~~B tr S.. 0 S 0~ AL. 0 0~:~ S 0 ~0 S 0L U'~ /U - - 0 / ~;~ ~I 1 S SO S S (a) Northern~~~~~~~~~~~< / d 0 S O 0 S ~ ~~~~~~~~~~ S SC ~ ~~~~~~~~~~~~~~~~~, 0 ~/'~ ~ ~ ~~~ ~ 5~~~0 F 40., /.O S~~~ 0 ~ ~. \t-~ S S 0 So 0 0 0 ~ ~ ~

ti~~ ~ ~i ~rl ~ r~ ~\~I I~ i ~ / ~~~ ~ ~* ~ ~ ~ ~ ~ ~ ~ ~ ~I I ~ ~IL~I ~ ~t ~~ ~ ~~ ~ r ~ ~1~~ \~~r'~ ~ ~ ~ ~ ~ ~ ~\, /~ ~ ~ ~~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~.~ ~ ~ ~ ~ Y ~r ~;)<~ ~ ~ ~I~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ X ~ ~ ~ ~ ~ ~-y~ ~~ ~ ~:~~~, ~~ ~ ~ ~; ~i r ~~ ~I~ ~ ~ ~ ~ ~~\ \ir',l')l~ C~rl!;FT`R/ ~9L 2/ -Y~\ ~ ~ ~ ~ ax u a\ ~ ~ ~ ~ ~ ~ ~'\~ ~ r ~ Y ~ ~ I ~ ~ ~r ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~'3 ~ r ~ ~ ~ ~ ~~~ r ~: ~ r ~ ~ ~ ~ ~ I I rYJ,I rcCII ~ SYLPB'I F ~\~~ ~ ~ ~~ ~ ~ ~ ~ ~ rLI r~ ~ -cJ I~*Tf C ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ h, V-~ ~ ~ ~ ~. ~\~ ~ ~~ ~ ~ ~/ r ~~ ~ ~ ~. ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~~ ~~ ~ ~~ ~ ~~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~r ~~, ~ ~1 \ ~ / \/ ~` ~1 3 ~ ~ ~ ~ ~ C ~ ~/ ~/'~ i ~ ~ ~ Z ~~,~ ~ ~ ~ ~ ~ ~ ~ ~/~ ~ ~ ~4cS~ ~ ~ ~ ~~ ~~'~i~ ~ t ~r K~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~.. ~ ~ ~ ~ ~ ~ ~ ~ ~~~ ~,~ ~ ~ ~ ~ ~ ~' ~. ~ ~ \I -I -~ ~ ~ i~ "A ~ (a) Northern ~ ~ ~ ~ ~ ~ ~ ~ ~ t. ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~-s r ~I / zY ~\ J~ ~ ~r ~ ~. ~ / X~r V~ ~ ~ ~ ~ ~ ~~ ~..~ ~.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \ i~- I ~I 7k II /\~'~C~~ ~ ~ s ~ ~ ~ ~ ~ ~ ~ ~ ~. ~ ~, ~ ~ ~ ~ * ~ ~ P ~ ~ ~ ~,- ~ t~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t ~ ~ ~ ~ TzfvVi ~.r. ~ ~ ~ I~ EI~rrPEL ~ r fi\Y`3\nY %/QL\ - \ ~~ ~ ~i~ t ~3~\~ ~~ ~ ~ ~ ~.~ ~~ ~ ~I ~ ~. ~ ~ ~I~,.t ~ ~ ~ ~~ ~ Srf.;I.I ~ ~ ~ ~ ~ ~ ~ ~~.. ~ ~ ~ ~ ~. ~ ~ ~ ~ ~ ~ r -L"- \ ~ ~ /5;VJ 1'r7 ~V L\'/ ;I ~ ~ ~ ~ ~r~ ~ ~ ~ ~ ~ ~ ~, ~ ~, .. r ~/(~ ~ ~ ~~ ~~ ~ ~ ~ ~ ~ t ~ ~ ~~r ~ ~ ~ ~~( ~ ~.'W TJ+' ~ ~r ~~~ ~ r ~~ i! ~ ~;ir ~ ~ ~ Y' ~ ~) ~ ~ ~~ ~ ~i ~ ~ ~ W ~ ~ ~ Vr~ ~ Z .~ ~ r\ ~LY r ~ ~ t , ~ ~ ~.. ~ ~ ~ ~I ~~\~ (b) Southern Fig. 12. Plo of data point positions, August 21. 33

IV. PROPOSED EQUIPMENT AND METHODS A. GENERAL In the first report on this techniquel an attempt was made to describe the instrumentation which might be required. This description was little more than a guess and was included mainly to make the technique more lucid. Since that time significant changes in the'"probable" type of equipment have taken place. Again, the scheme proposed here is based more on conjecture than on equipment specifications, but includes the results of discussion with some of the leaders in the manufacture of star-tracking, equipmento The basis of the system is an automatic star-tracking telescope. For adequate global coverage (see Section III-A) two such telescopes are required, simply doubling the number of observations possible. The telescope (or at least a mirror) is attached to an inertial platform which is gyroscopically space-stabilized. The telescope is gimbal-mounted with digital, angular encoder-decoders and positioning apparatus. Associated with the telescopeare a clock accurate to a millisecond per orbit, two tape recorders, and programmer Prior to launch, the stable platform is aligned in a preferred orientation depending on launch time and orbital elements. After injection into circular orbit, it is expected that the rocket accelerations will have caused some error in the platform alignment. The first act of the programmer is to operate the tape recorder on which are stored a pre-programmed set of star coordinates for each tracker. The first stars observed by both trackers are high in the sky, unrefracted, and by far the brightest in their neighborhood. By use of an acquisition mode with wide field, the trackers lock on and the stable platform is then torqued to the desired alignment. (An alternate scheme is simply to measure the stable platform's orientation, and refer all future ground-based computations to its coordinates..) The star-trackers thus function as stellar monitors for the inertial system. The monitoring function can be repeated as often as necessary to correct for gyro drift while orbiting, After the programmer is satisfied that the stable platform has been correctly oriented, it directs each telescope to the coordinates of the first star in its data-gathering program. The coordinates might be given to the nfearest minute and the telescope could probably lock on in the tracking mode~ During -tracking, the starts angular changes along three gimbal axes would be recorded on magnetic tape, together with the time of one (the first, say) 34

measurement. Each successive measurement of angular change could be recorded at a precise time increment such as 1.000 second. After occultation -or obscuration (persisting for some arbitrary length of time such as.5 seconds) the telescope would be commanded to slew-to the next star in its recorded program, and the process would be repeated through the 118 stars in each telescopegs program. The programmer would begin and end the measurements at the proper time relative to the sun. The initial program of star coordinates could be recorded on a spacecraft tape recorder prior to launch. As the orlital plane precesses at about 10 per dar, all stars remain with the same coordinates relative to the- stable platform. However, the earth will gradually move out of this fixed telescope-star orientation and the field of stars, originally centered at 00 azimuth, will change azimuths at 1~ per day, causing the center of the observations to lie off of the sub-satellite path and the occultations to take longer than desired, After a change of several degrees, it would be desirable to update the star programs; this would be done by the Command and Data Acquisition (CDA) station. We shall now examine the information bits required for a new program. If we wished to identify the star among the 236 (which is not fundamentally re, quired) 8 bits would be necessary. The direction can be resolved into two angles, one a maximum of 120" (azimuth), the other 1800 (altitude). For oneminute accuracy we would need 1:7200 and 1:10,800 accuracies, or 13 and 14 bits respectively, The total required would thus be on the order of 236(8 + 13 + 14) = 8260 bits. The simplicity of transmitting this information to the satellite would allow updating the program daily or oftener. Another possibility which might prove valuable is the ground selection of points of particular interest to be scanned. The ground computer would select, approriate stars which would yield measurements as close as possible to these points. Since the entire star program could be changed with only about 10,000 bits of information, a new program for each orbit might even be advantageous. The data information rate must likewise be examined, As previously described, the measurements would require recording two angles to about onesecond accuracy with neither angle exceeding 75 minutes, The accuracy is thus 1:450:0 or 8 bits for each angle plus a bit to distinguish the axis, for a total of 17 bits per datum point. Additionally, each scan requires a time of one millisecond per orbit or 23 bits, and perhaps star identification with another 8 bits. The average scan is about 25 seconds, so we may assume 25 datum poirnts per scan; consequently (25 x 17) + 23 + 8 = 456 bits per scan. 35

Allowing for sync bits, etc,, we might say the average scan would require reC cording 500 bits~ At 236 scans per orbit the total information stored in an orbit would be only 118,000 bits. The telemetry system is capable of transmitting this information to the ground in a few seconds. B. DATA PROCESSING This information would be sent to the computer center from the CDA station by landline or radio link, The tracking data would likewise be sent to the computer center. Then the computer would reconstruct the geometry of each scan, determine the latitude and longitude of each datum point, and integrate to find p(h), T(h), and p(h). These data would then be rapidly forwarded to the National Meteorological Center for meteorological analysis and dissemination. Feedback at the NMC to the computer center might contain requests for additional coverage of certain areas. This input, together with an ephemeris for the next orbit, would enable the computer to choose a new set of stars. Their coordinates would be sent to the CDA station and relayed to the satellite on its next pass. Obviously, the computers must handle the data from an orbit (and whatever new star commands are necessary) in an orbital period of 107 minutes, to prevent an impossible backlog of data. This means that a scan must be handled in 28 seconds, or that data processing must be done in the same amount of time as data acquisition. Presently, the full program of computerized data reduction cannot be written because the final inversion formula, such as Eq. (12), is not known. Numerical methods are under test and results are expected soon. Preliminary results indicate it may be possible to avoid involved iteration; should this prove true, one IBM 7090 computer would probably be capable of reducing the data in real time. Emphasis must be given to the relative simplicity of the data and command telemetry on the order of 104 bits command and 105 bits readout per orbit, Yet this information has the capability of providing density, temperature, and pressure on a global basis. C. TELESCOPE, CLOCK, TAPE RECORDER, AND TELEMETER The telescope has been grossly described above, It must be mounted on a stable platform and be able to point over a hemispheric solid angles Two telescopes, both mounted on- the same stable platform, are likely to be used.

The importance of the stable platform is threefold: first, it provides a stable mount for the telescope, thus providing greater pointing and tracking accuracy; second, it provides a space reference which enables the telescopes to point quickly without search to any celestial coordinates chosen on the ground, thus saving power and time over a random search; and third, it provides a space reference against which to measure the refraction angle. The telescope should have a tracking accuracy of +6 arc seconds and a capability for tracking stars which appear as dim as the 10.5 magnitude. Some of the error implications of the tracking accuraey are discussed in Section VI. The star magnitude was arrived at on the following basis: from the work reported in Section III-B it was determined that stars of magnitude 5,0-54*5 would be the faintest required. However, since the telescope will be on. a satellite looking through the entire earth's atmosphere, the attenuation in that medium must be accounted for. Appendix B is a complete discussion of this attenuation, the results of which are summarized thus: 1. Refraction causes a prismatic dispersion of the light flux and thus an attenuation. 2. Molecular (Rayleigh) scattering and Mie scattering (water vapor, haze, dust) also attenuate the light ray. 3? The attenuation due to refraction and molecular scattering is predictable and constant, whereas Mie scattering is variable. All are more severe nearer the earth. 4. At ho = 5 km, the lowest measurements expected due to extinction by clouds, the refraction and molecular scattering will attenuate the light ray a total of 4 magnitudes. Mie scattering is estimated to cause about 1 magnitude more attenuation. At ho = 10 km, the attenuation is 2.13 magnitudes, while at ho = 25 km the attenuation is negligible. The telescope should therefore be capable of tracking a star of the 1045 magnitude, Since the magnitude required decreases sharply with altitude (at 10 km-7.8 magnitude), and at low altitudes the refraction angle is much larger, the tracking accuracy requirement can be relaxed when the image is the dimmest, Present indications are that 6 arc-second accuracies are possible on 9th magnitude stars using optics 3 inches in diameter. The weightof a two-telescope system with stable platform might be about 60 lb and its power requirement 40 watts. Two tape recorders will probably be required, one to store the stellar monitor and star coordinate programs as commanded from CDA station, and one to store the angular and time data from the telescopes and clock, There has been no attempt to specify this\ equipment since the modest information rates involved make the requirement well within the state of the art. 37

A crystal clock will be standard equipment aboard Nimbus; a detailed description is given in Ref. 3. The clock has an accuracy of 10-7 or 20 bits, whereas 23-bit accuracy (one millisecond per orbit) is required if clock errors are to be unimportant. Calibration may possibly resolve the difficulty — if not, a better timing device must be employed. Since the output required for the refraction measurements is much less elaborate than for the Nimbus, an additional specialized clock for refraction purposes only might suffice. The Nimbus telemetry system3 could handle the refraction command and data acquisition requirements in a few seconds per pats. No additional telemeters would be required.

V. AN EMPIRICAL ATTEMPT TO TEST THE FUNDAMENTAL EQUATIONS The Baker-Nunn camera on Mt. Haleakala, Maui, Hawaii is being utilized in measuring stellar refraction. This camera has 20" aperture and 20" focal length. The field of view is about 5~ x 30~ with a format 55 mm x 300 mm (approx.), which amounts to 406" - per mm. The camera is a modified Schmidt and the film is stretched over a spherical sector with radius equal to focal length. Film can be transported rapidly and automatically from frame to frame and a time dial is photographed on each frame. Exposures may be taken every 2, 4, 8, 16, or 32 seconds automatically; the length of exposure equals one-fifth of this period. Film is developed on Maui and sent to Cambridge for reduction with Mann comparators. The camera can be pointed so that the optical axis is at + 1/20 altitude with the 5~ field vertical. This means that the field extends from +3~0 down to -2~. The camera is located at an altitude of 3048 m, making the dip of the horizon 1046'. Thus stars can be observed all the way to the horizon. The method of obtaining refraction measurements is as follows. Azimuth is set at 270~ or nearly so, depending on the star field of most interest, and locked. Track angle (camera with respect to gimbal ring) is set at 900 and locked. Altitude is set at 88-1/20 and an exposure is made. The film is advanced to the next frame and the camera altitude lowered quickly to 84-1/2~. Another exposure is taken 8.000 seconds after the first. The procedure is repeated and another.exposure is made at altitude 80-1/20, etc. Upon continuation of this procedure the 22nd frame will be exposed at 4-1/2~ altitude. Again the camera is lowered 40 to about + 1/20. The transport mechanism is turned off and instead of a single exposure, 64 exposures are made on this frame. The shutter opens for 8/5 seconds every 8 seconds. A total exposure time of 64 x 8/5 = 102.4 seconds gives roughly an optimum background density. However, it is better to have the star images further apart than 8 seconds (time) of sidereal motion. Therefore, the operator holds a black flag in front of the aperture on every other shutter opening, making the exposures 16 seconds apart instead of 8. Also, prior to making the multiple-exposure frame the timing photography is set so that the time is recorded only on the first_ of the 64 exposures. These camera adjustments for the multi-exposed frame require time to perform; therefore the time interval between the 22nd and 23rd frames is 16.000 seconds. 39

By measurement of the 64 image positions of a star or stars and comparison with 64 calculated positions of an unrefracted image, the refraction as a function of ray height may be obtainedIn order to calculate an unrefracted star's position on the film, some connection between the celestial positions and the film coordinates must be made. If the film were very accurately positioned in the camera and the camera accurately positioned with respect to earth coordinates, this would be no problem. However, neither of these conditions hold: the film "framett is not held fixed with respect to the camera moiunt within a few microns and the camera mount position circles can be read only to about ~6', (In normal satellite tracking any star furnishes an origin for right ascension and declination coordinates and satellite positions are taken with respect to these, Refraction itself is of no consequence except for differential refraction between the star and satellite images. By choice of a star very near the satellite, differential refraction can be neglected.) This is the reason for the overlapping photographs from the zenith down to the final frame. Beginning at the zenith, the measured altitude difference between a star at the top of the frame and one at the bottom of the frame is compared with the actual altitude dif~ference computed from a star catalog, The difference (subject to other corrections discussed later) is the refraction contribution between the two altitudes, The lower star of the first frame is on the top of the second frame; the refraction contribution between it and the star on the bottom of the second frame is found in the same manner, This process is repeated until the top of the final frame is reached. Summing all these individual refraction contributions gives the refraction of a star on the top of the final frame, with the following exception. During the time between exposures sidereal motion has changed the altitudes of the stars, so that the refraction contribution of each of these small angles must also be added, All intervals except the last are 8 seconds; thus the sidereal motion is 120 arc seconds and the altitude change cannot be more. Since the altitude difference between the two measured stars on a frame averages 4V, the refraction contribution of this small angle amounts to less than 120"/4~ = 1/120 of the principal contribution, Furthermore, this increment can be estimated very accurately, so the error due to these intervals is on the order of 1/1200. The final interval is 16 seconds and the error is doubled in this case. Unfortunately, the zenith angle is very great, so that this contribution to refraction completely overshadows the contributions near the zenith; therefore, we must allow the overall error to be considered doubled. The overall error in refraction angle due to non-zero intervals between frames is thus 1/600 or -17%i A discussion of other corrections -which may or may not be necessary will be deferred until' the: method of data:reduction on the final frame has been discussed-. -Assw~ue that the accurate refraction of a star image near the top of the frame is known. (It is assumed below that the star being used has its 40

first image near the top of the frame, It'should be apparent that no difficulty arises in moving from star to star if necessary.) The scheme is to measure the centers of the images of the film with a measuring engine which gives the coordinates in microns relative to an arbitrary reference on the film. Then a computation is made of each position on the film of the unrefracted star. The only reference between film and celestial coordinates is -the image of the refracted star whose refraction is known, Details of the computation are as follows, A photographic film is obtained from the Baker-Munn station and for pictorial reasons has one star image on it, as shown in Fig, 13. The distance yl, measured from the horizontal axis of the film, locates the refracted star image and the distance y" positions the unrefracted image of the same star, The distance yY-y" is then the deviation of a straight light ray from the refracted ray and by conversion to angular measure represents the refraction angle R as observed from the earth station. 0-T; y,' ~ ~ 9 R.FRACTED IMAGE Fig, 13, Reflected and unrefracteRATAED4Aes on photographic film, In order to calculate the distance y", these definitions are made: d declination RA right ascension of a star L latitude of the station As longitude of the station measured east from Greenwich 41

GHA of the vernal equinox (function of time) X= As + Xg LEA of the vernal equinox h altitude of the camera Az aximuth of the camera unit vector pointing in the direction of a star X inertial coordinate system XI station coordinate system X" film coordinate system ei = unit base vectors in the coordinate e i 1,2,3 systems X, Xl, XIt ei xi e l,2,I5 F components of the vector S in the coordinate systems X, XT, X xi transformation equations from X to XT x and Xt to X" respectively t, Xk bkiXi (49) direction cosines between unit base ~ aij vectors of the coordinate systems bki transformned coordinates of the vector xk = ckjxj (50) S from X Xto Xt ckj = bkiaij (51)

The components of 9 in the coordinate system X' are determined by relation (48) and the aij's are obtained from the geometry of the orientation of the coordinate systems X and X', as shown in Fig. 14. The base vectors 3 and el satisfy the following relations: 3 (cos L cos el + (cos L sin k)e2 + (sin L)e3 (52) el = (-sin X) e + (cos) (53) The vector e2 is obtained by taking the cross product of'e and tlto form a right-handed triplet of base vectors: 3. r+ +1 + + +. e2 = e3 X el = el. e2 e3 cos L cos X cos L sin k sin L (54) -sin k cos A 0 e2 (-sin L cos k)el + (-sin L sin k)e2 + (cos L3 (5) The coefficients of these linear combinations are the aij's. In matrix notation /1i ai2 a13 a = a2 a22 a23 (56) a3i a32 a33 all -= -sin x a12 = cos x (57) az3 0 0 443

. NORTH CELESTIAL POLE +~~~~~~~~~~~~~~~~~~~ STAR e, 1000'~~~~~~~\ (x,y,*) COMPRISE INERTIAL COORDINATE SYSTEM X x (xy"') COMPRISE STAT'ION COORDINATE SY-STEM X' VERNAL EQUINOX Fig. 14. Geometry of the orientation of coordinate systems X and XI.

a21 = -sin L cos, X a22 = -sin L sin X (58) a23 = cos L a3l = cos L cos \ a32 = cos L sin k (59) a33 = sin L By the same procedure, the bkits can be determined for the transformation (49) between the coordinate systems XI and X", which are shown in Fig. 15. z OPTICAL AXIS OF THE CAMERA ~,x:::::~:i:i:: e 62 Y' AZ Fig. 15. Orientation of coordinate systems X' and X". 45

e3 (cos Az cos h)el + (sin Az cos h) + (sin h)e (6o) el = (sin Az)e + (-cos Az)2 (61) e3 x el = e2 = (cos Az sin h) e + (sin Az sin h)e + (-cos h)e (62) b/l b12 b13 b = b2l b22 b23 (63) b31 b32 b33 bll = sin Az b12 = -cos Az (64) b13 = O b21 = cos Az sin h b22 = sin Az sin h (65) b23 = -cos h b3z = cos Az cos h b32 = sin Az cos h (66) b33 = sin h The combined transformations (48) and (49) as expressed in Eq. (50) requires the matrix multiplication indicated in Eq. (51). 46

C11 Cl2 C13\ /bii b2 bl3\ 11 a2 al3 C21 C22 c23 = b21 b22 b23 a2l a22 a23 (67) 31 c32 c33/ \b3l b32 b33/ 1 a32 a33 cll -=.- Azsin + osAzsin L-cos X ci2 = sin Az cos X + cos Az sin L sin. (68) c l3 = -cos Az cos L c21 =; cos Az sin..!h sin k - sin Az sin h sin L cos X - cos h cos L cos X C22 = —~cos Az sin h cos A - sin Az sin h sin L sin X - cos h cos L sin X (69) C23 = sin Az sin h cosh L.- cos h sin L c3l =-::-cos Az cos h sin X - sin Az cos....h sin L cos. + sin h cos L cos X oC3 cos Az cos h cos X - sin Az cos h sin L sin / + sin L cos L sin. (70) C33 = sin Az cos h cos L -+ sin h sin L An evaluation of these coefficients and the components of S which are expressed in terms of celestial coordinates x = cos d cos RA y = cos d sin RA (71) =z = sin d will determine x", y" z'" by these transformation equations (50) 47

x" = cllx + c12Y + c13Z y" = c2lx + c22y + c23z (72) z" = c31X + c32y + c33z The'distance y" is, then, seen to be a function of the celestial coordinates of a star, the latitude and longitude of the observer, the orientation of the camera, and the time.. A multiple-exposure frame (Fig. 16) depicts a setting star. As the star sets, the light ray traverses atmospheric layers of increasing density and for equal exposure times the distance d between star images decreases due to increased bending of the ray denoted by the'refraction angles R: d1 < d2 < d3... R1 > R2 > R3 ~~ 3' Y 0 d, |I O $ R.. d2 f,R, 1 _ x" do H ~I R3 0 STAR TRAILS *,.REFRACTED IMAGE O UNREFRACTED IMAGE Fig. 16. Multiple-exposure frame depicting setting star. 48

However, the coordinates of the refracted star are not referenced to the x"y" axis of the film; therefore a subtraction, as indicated previously, is not sufficient to determine the refraction angles R. Thus if the refraction of the first image on the frame in Fig. 16 is known then together.with the distance d between the refracted star images as measured and the position calculations of the unrefracted star y", all the succeeding refraction angles can be calculated, The method of determining:the refraction of the first image, which was described earlier, consists of taking overlappirng photographs of the star field from an altitude angle of 90~ down to the horizon (see Fig, 17). At 900, a star may be considered an unrefracted reference star. The refraction angle of the star labeled 1 can now be calculated: R1 = d - (y Y) (73) Star 1 appears on the top of the 88-1/2~ frame and since its refraction angle is known, the refraction of star 2 is given by R2 = d2 - ( + R1) Y2 (74) All succeeding angles are determined in this way from 900 down to the horizon frame, Summing these individual refraction contributions gives us the refraction of a star on the top of the horizon frame. ~Presently, several refraction films are being reduced, No measurements are available at the time of writing. 49

r UNREFRACTED STAR 900 _____ 0I d ~ 186~ 820 ~3 * 0 HORIZON SHOT STAR TRAILS Fig. 17. Tllustration of overlapping frames.

VI. ERRORS AND THEIR EFFECTS The first results of the satellite refraction technique are the density profiles p(h). These are not vertical profiles, but are nearly so at all except high azimuth angles (see Section II-D), The geographical positions (GP) of the highest and lowest ray tangencies are ordinarily close enough so that the difference is negligible. In any case the motion of the scan is readily calculable. For the purpose of this discussion, it is assumed p(h) applies to a particular GP. Errors in p(h) arise from: la. angular errors in the tape recorded value of Rs(t); lb. time errors in the tape recorded value of Rs(t); 2, errors in satellite position as a function of time; 3. errors in deducing ho(t); and 4. errors in retrieving the inverse function p(h) from Rs(ho). It is assumed that no errors arise from data storage or transmission and that errors in the location of the GP are negligible. Errors la and lb can be resolved into an error of refraction angle only. That is, if an absolutely correct refraction were measured at time t, but the time was presumed to be t2, it can be assumed that the measurement was indeed made at t2 and that the refraction angle is in error. There are several sources for error in measuring the refraction angle: error in measuring real time on the satellite; error in recording the time simultaneously with the angular neasurements; and error in the angular measurement due to the optical sensing equipment, from the optics through the digital encoder. Preliminary investigation into equipment indicates that the state of the art in timing is such that the requirement for the nearest millisecond per orbital period can be met, It may thus be fair to assume the maximum timing error to be.0005 seconds. The time rate of change of refraction angle is greatest at the low end where dR/dt is approximately 2'/sec, so the maximum time'error times the maximum angular rate gives only.06", a negligible error. 51

The angular error due to the sensing device -itself, from optics through digital encoding, is strictly a function of the equipment which constitutes the heart of the system. The goal'at present is ~i6 arc seconds, which isconw sidered quite realistic by several manufacturers of star-tracking equipment~ The question of whether this goal will be,attained or perhaps surpassed remains to be answered. Whatever the particular sources -of error within the instrument, it: may be that most are of an absolute magnitude independent of the size of the refraction' angle. The, expectation is, then, that the angular error should be negligible as a percent of refraction angle at the low end of the scan, that this angular error will become an increasingly important percentage of the refraction angle toward the beginning of the scan, and that it will eventually determine the upper limit of meaningful data. The second error source considered is that involved in satellite trackingr Here again all errors can be resolved into one.-an error in position at any given time. The position is required to draw the geometry of Fig. 2 as described in Section II-Do Reference to Fig. 2 will show that errors in satellite height times the sine of the altitude angleg and in the horizontal plane in the direction of the star times the cosine of the altitude angle, become errors in the distance = re+ho+b and thus become errors in ho. Since the altitude angle is roughly 30~, both sine'and cosine are appreciable and errors in satellite position are translated virtually undiminished into errors of tangent height, ho. It appears that positional accuracy of 50 meters would be desirable. Since systems under development are reported to have even greater accuracy, this goal seems very reasonable, Tracking capability -must also include rapid readout., Final, precise determination of position within one orbital period would be ideal,. longer delays mean simply that the data are passed to the meteorologists that much later, The improvement of tracking systems is beyond the scope of,. although it is vital to, this project. It must be assumed that their development will be carried on by other agencies, and will ultimately be available for the refraction technique. Errors in'deducing the function ho(t) were discussed in Section II-D, where it was shown from' Eq. (46) that one could reasonably assume any particular value of ho- to be correct ~90 meters. That analysis did not take into account the error in ho due to satellite positioning error, since it assumed the geometry of Fig. 2 to be known. The satellite tracking error will be independent of the error in estimating %o, so there is a random combination of errors on the order of 50 and 90 meters in determining h0o. It is only fair to note that these are extreme conditions, the probable error being much less The last source of error cited, that-of retrieving the function p(h), from Rs(ho), is also mentioned in Section II-D, and some general conclusions are drawn from the formn of Eq. (12) 52

In summary, a full scale error analysis has not been conducted because the final data reduction techniques have not been determined. Approximations have now been made to all necessary equations and the errors from each known source have been traced through the process independently. Errors due to the mathematical manipulation of the experimental data appear small, The most important errors will be those generated in the angular measuring device and in the determination of satellite position. 53

VII. CONCLUS IONS The. refraction technique has- been studied with emphasis on areas where problems wexreanticipated. Solutions to egthese were obtained in several cases and indicated in others. No insurmountable problem was uncovered. However, the area of equipment design has been scarcely examined, and the most crucial feasibility questions now lie in that domain. In the immediate future, effort should be concentrated on: 1. Investigation into the feasibility of equipment design to meet the requirements of the refraction technique. 2. Numerical integration of the refraction function and development of an inversion technique for retrieving the density function. 3.. Computer programming of the entire data reduction process to prove real time capability. 4. Completion of the ballistic camera refraction measurements and correlation with atmospheric conditions.

REFERENCES* 1. Jones, L. M., F. F. Fischbach, and J. W. Peterson, "Atmospheric Measurements from Satellite Observations of Stellar Refraction, " University of Michigan ORA Report 04963-1-T, January 1962. 2. Jones, L. M., F. F. Fischbach, and J. W. Peterson, Satellite Measurements of Atmospheric Structure by Refraction," PlanetSpace Sci., 9, 351-2, 1962. 3. Stampfl, R. Ao, The Nimbus Spacecraft and Its Communication System, NASA Goddard Space Flight Center Rpt. X-650-62-201, 1962. See Also: Bandeen, W. R., Earth oblateness and Relative Sun Motion Considerations in the Determination of an Ideal Orbit for the Nimbus Meteorological Satellite, NASA TN D-1045, July 1961. Perkin-Elmer Corp., Long Focal Length Ballistic Camera Study. Eng. Rpt. 5942, October 1961. *A comprehensive list of references on refraction and related topics is found in Reference 1. 55

APPENDIX A COMPUTATION OF LATITUDE AND LONGITUDE OF RAY TANGENCY AND SUBSATELLITE POSITION AT OCCULTATION GIVEN A STAR AND AN ORBIT For any unrefracted star the locus of points on the surface of the earth where the star appears on an observer s horizon, i.e., at altitude = 0~, or where the rays. of light from the star are tangent to the earth s surface, there is a great circle, (SGC). The orientation of this great circle is unique with the star. Now, for a satellite at a height hs above the surface' of the earth, the locus of points where the satellite might see the rays of light from the same star tangent to the earth's surface will be a circle of radius re, where re is the radius of the earth. The plane of this circle is parallel to and a distance W from the plane of the circle which is the locus of points where the light rays are tangent to the surface of the earth. The distance W between the planes is dependent directly on the height of the satellite —and for a satellite at height h. = 0 the planes will coincide, i.e., W = 0. The geometry is shown in Fig. A-1. EART_ W w LIGHT RAYS FROM THE STAR Fig. A-1. Geometry of unrefracted light rays. The same general approach can be made for a star whose light is refracted by the atmosphere. The geometry, however, is slightly different, as indicated in Fig. A-2. Two points on the satellite's orbit where it will "see" the rays of light 56

(NCP) xx~ 1z y m, -n EARTTH+ SATELLITE ORBIT: Xr/' xa+ Ya+ zX rs+ (y') x+by+cz Fig. A-2. Geometry of occultation, from the star tangent to the earth,'s surface can be obtained as the'intersection points of the satellite's orbit and the circle that lies in the parallel plane a distance rs sin G from the plane which passes through the locus of points where the star's rays are tangent to the earth's surface. The equation of the satellite's orbit is a solution of the equations x2 +y +2 - r2 ax + by + cz = 0 The.coordirete system chosen is defined by the x axis positive in the direction of the vernal equinox, -the z axis positive in the direction of the North Celestial Pole, and the y axis orthogonal to these axes. The letters a, b, and c are the direction numbers of the normal to the satellite's orbit. 57

The equation of the plane containing the circle which is the locus of points where the satellite will "see" the star's rays tangent to the surface of the earth is lx + my + nz + rs sin = 0 where 1, m, and n are the direction numbers of the normal to the plane containing the circle and are related to the right ascension and declination of the star under consideration as follows: 1 = cos dec cos RA m = cos dec sin RA n = sin dec The simultaneous solutions of the equations X2 + y2+ z2 = r2 ax + by + cz = 0 lx + my + nz + rs sin = 0 yields the following values of x, y, z, which define the two desired satellite positions: -W[ba + c [ba + cPwja]2 + - + y2]W2(b2 + c2) -2 x - 0a2 + p2 + 72 -W[cy - au] + JW2[c7 - aU]2 - [ao + 52 + y2][W2(a2 + c2) - r 2I] y - e22 + 2 + 72 where a = bl - am = cl - an y = cm - bn W = rs ~ sin G z is calculated according to the following conditions: 58

If c = 0 and n - 0 then no calculation If c O then z = (-ax - by)/c If c = O, a f O, b O, x O, y 1 0 then z = (-lx -my -W)/n If c = O, a = O, b $ 0 then z = (-lx -W)/n If c = O, a $ O, b= 0 then z = (-my -W)/n There are four possible combinations of x and y for which a value of z may be obtained. However, in order to obtain the setting point of the occulation, the following conditions must be satisfied: lx + my + nz < 0 (bn - cm)x + (cI - an)y + (am - bl)z > 0 x2 + y2 + 2 = rs2 In order to obtain the geographical position of the measurement use the values of x, y, z obtained in Eq. (1), then XGP = X +T 1 GP = y + T. m ZGP = Z + T n where T = r (sin @ - cos G ~ tan Rs/2) The satellite positions obtained in Eq. (1) are converted to latitude (B) and longitude (X) by the equations = arc tan x' sec (A) X = arc tan y/x Longitude is measured eastward from the positive x-axis. The latitude and longitude of the geographical position, BGP and. 7GP, are obtained similarly: 5GP = arc tan Zs_ x * se(XGp) 59

"GNp = arc tan y/x The coordinate system based on the satellite is oriented such that the x' axis is positive in the direction of the orbital motion of the satellite, the z' axis points in the direction of the satellite's zenith, and the yf axis is orthogonal to these axes. The transformation from unprimed to primed coordinates is accomplished with the following equations. x' = cos dec cos RA(b cos c - c cos - sin x) + cos dec sin RA(c cos $ cos A a sin B) + sin dec(a cos f sin X -'b cos f cos ) y' = a cos dec cos RA + b cos dec sin RA + c sin dec z' cos dec cos RA cos B cos k + cos dec sin RA cos f sin X.+ sin dec sin f The relations zt = arc tan x' sec + Rs = arc tan y'/x' then give the altitude, 9, and the azimuth,,, of the star in the primed or satellite-based coordinates, 60

APPENDIX B ATMOSPHERIC ATTENUATION OF STARLIGHT by'Maurice E. Graves I. INTRODUCTION When a ray of starlight penetrates the earth's atmosphere, it suffers a significant diminution of intensity. This diminution is caused by refraction within the density stratification of the medium, plus the effective presence of certain constituents which scatter or absorb the radiant energy. These constituents include water in all three phases, ozone, dust and haze particles and the gas molecules of the atmosphere. In this report, we will discuss the relative importance of these attenuating agents with regard to horizontal, grazing beams passing through the atmosphere at minimum heights under 30 km. Some of the major contributions to these fields have come from Europeans, viz., Link,8 who has written upon attenuation by refraction, and Schoenberg,l4 Van de Hulstl7,18 and Waldram,19 who have written extensively on light scattering and absorption within the atmosphere. Other material on the latter two topics are to be found in Johnson,4 Lillestrandaet al.7 and Middleton.ll The dearth of measurements of attenuation for horizontal rays at the earth's surface and aloft necessitates the use of theoretical results. However, the formulae should indicate which attenuators are important, and they give useful estimates of the amount of dimming caused by a particular agent. Tables 139158 in List9 and the Light-scattering Coefficients in Boll, Leacock, Clark and Churchill2 will be employed to gile quantitative results. II. ATTENUATION BY REFRACTION Jones, Fischbach and Peterson5 have pointed out that the refraction of horizontal starbeams reduces the intensity of starlight at an observation point outside the atmosphere,/ e.g., at a satellite, by a factor which exceeds 10 for a ray tangent to the eArth's surface. Thus, the apparent magnitude of Sirius, when very near the horizon, is diminished by 10.6 x 100-1'5 magnitudes, or from -1.6 to +2.2. The reduction factor is: S = 1 + Rsx/H, where Rs = refraction angle in radians for a complete horizontal traverse 61

x = path length from satellite to minimum height of ray H = scale height This formula for S can be traced to Link,8 who used it to study the atmospheres of planets when they occult bright stars. It is important to note that x is the distance from the data point to the satellite, and that it can be taken as constant for a particular satellite altitude (see Fig. B-l). Taking x = 3900 km for a satellite altitude of 1100 km, and taking H = 7.5 km, we find the following variation of transmission ratio I/Iovs. height of ray perigee h (see Fig. B-2), where I = intensity of star ray after penetration Io = intensity of star ray prior to entry Fig. B-1. Geometry of attenuation by refraction. III. SCATTERING Confining our study to the optical range of wave lengths, 0.4i to 0.7[L, we will follow Van de Hulst's definitions of basic terms, which give: Attenuation = extinction = scattering + absorption In considering the effects of scattering, some simplifying assumptions are necessary, namely: 1. Scattered light retains the same wave length as incident light. 2. In the case of Mie scattering by small particles, the particles are independent, e.g., their mutual distance apart is at least 3 times their radii. 62

30 o 28 26 24 22 2018 16 ho (km) 14 12 10;8 6 4 2 0 0 0.2 0.4 0.6 0.8 1.0 I (1S) Fig. B-2. Transmission ratio I/Io as a function of ray perigee ho. 63

3. All scattering is single rather than multiple. Thus, the radiation reaching a particular particle or molecule is essentially that of the original beam. Taking up Rayleigh! (molecular) scattering first, we find that the ratio of I/Io introduced above can be equated to an expression, e-b sec Z. Schoenbergl4 states that b = 3523(j - 1)2/3NX4, where. = refractive index of the, air = f(h) and N = number of molecules/cm3. When z is greater than 80~, however, e-b sec z becomes inapplicable because of the behavior of the secant function, and the ray's curvature (2.96 x 10m- kmnl) and earth's curvature (1.57 x 10-4 km-1) must be considered when nearing z = 90~. Furthermore, p is not easy to evaluate as f(h). 7 These difficulties have been met by Lillestrand et al. in the following way: 1. Compute the optical air mass, which is the mass of air, m(h,z), in a volume of unit cross section, starting at height h, taken in a direction z degrees from the zenith.ol3 The formula applicable for z < 80~ is simply m(ho,z) = m(ho,0) sec z; for greater z values Pressly13 use s 220 km m(h,z) = f p(y) dy 0L R- o sin2 z2 +h + y where p(y) = density along the path and R = radius of the earth = 6368 km. 2. Rewrite the Rayleigh expression for I/Io, e-b sec z, as I/I = o0- kmp/po where k = f(X), m = number of unit vertical air masses, and the pressure p, like m, is f(h). Then, following Moon,l2 one can utilize experimental results such as Fowle's3 for (I/Io)z=0 in a vertical column above a mountain summit. Thus, setting m 1 for z = 0 and setting p = -p to solve for k in the equation I/Io = 10k, Moon plots k vs. X and then obtains a numerical value for kl (=0.00380) such that (I/Io) = kl- -4 The exponent of "-4" on X corresponds to a slope of -4 when k is the ordinate, and this agrees with Rayleigh's original expression for b, where b oc - 4. 3~ Plot I/Io against h, using x as a parameter. Following a procedure used by Lillestrand et al., we have done this for 64

z = 900 for molecular scattering, and have obtained the transmissivities shown in Table B-I. TABLE B-I I/Io IN PERCENTAGE TRANSMISSION (MOLECULAR SCATTERING) ho( km) = O.4 m Oe.5 O.6 0.7i I/IO* 0 0 0 1 8 1 5 0 5 24 46 16 10 16 47 70 82 56 15 68 85 93 96 86 20 92 97 98 99 96 25 100, 100 100 100 100 *Integrated effect found by counting squares. The notable dependence of I/Io on A in Table B-I may also be seen by making a simple calculation for red and violet light components, thus: red light: X = 0.7t violet light: X = 0.4 t I/Io cc k-4; therefore the atmosphere removes (0.7/0.4)4 10 times as much violet light as red light through molecular scattering. Although water vapor consists of gas molecules, Moonl2 finds that true Rayleigh scattering is not operative in this case. Fowle3 had separated the water vapor effect in his Mount Wilson data, and in plotting k vs. X graphically, Moon obtains a kl = 0.0075 and (I/Io)z.= oc k-2 rather than -4. To make a table for water vapor scattering similar to Table B-I, we have assumed (I/Io) = 10-km w/wO where w = amount of precipitable water in mm wo = amount of precipitable water in mm at the surface m = number of air masses; m = 1 for h = 0,k = kl X-2 Assuming a value of wo = 20 mm along with Moon, and a decrease of w from wo at h = 0 to 5 mm at h = 5 km to 0 mm at h = 10 km, we have obtained Figure B-3 and Table B-II for water vapor.

100 X' X~~ 90 / ~~~~x' 80~ l~/ / x/ / 80/ x/ /.7j o.4p 70 - -- X 60 0.4/ / O.5u 1 50 0.6p No 0.7,u 40/ I I 30 / / /- I,.WATER VAPOR SCATTERING 20 / --- MOLECULAR SCATTERING I/ I 0 1/ /'o~~ / 0 5 10 15 20 ho (km) Fig. B-3. Percentage transmission I/Io at various optical wavelengths as a function of ray perigee ho, for water vapor and molecular scattering. 66

TABLE B-II I/Io IN PERCENTAGE TRANSMISSION (WATER VAPOR SCATTERING) ho(km) X = 0.44 0.5C 0o.6I 0.74 Z I/Iot 0 0 1 3 9 6 5 34 50 62 71 51 10 100 100 100 l 100 100 15 100 100 100 100 100 20 100 100 100 100 100 *Integrated effect found by counting squares. Sample computations are as follows: x = o.41 ho = 5 km w/wo = 5/20 = 0.25 m = 2(2.14 x 108/1.07 x 107) = 40 k = 0.0075/(0.4)2 = 0.047 I/Io = 1o-kmw/wo -o.470 = 10 = 0.339.X = 0.71 ho = 10 km w/w~ = 10/20 = 0.50 m = 2(1.10 x 108/1.07 x 107) = 20.6 k = 0.0075/(0.7)2 = 0.015 I/Is = 10-o. 154.702 A 20 mm depth of precipitable water is typical of a maritime tropical air mass, and the sample calculations given above show a significant low-level scattering effect due to water vapor. The x = 0.4k1 and. - 0.71p curves for molecular scattering in a Standard NACA Atmosphere have been recomputed and entered in Fig. 67

B-3 for comparison with water vapor scattering~ In addition to molecular and water vapor scattering, there remains the possibility of scattering by liquid water and by solid particles such as dust, haze or ice crystalso An attempt to evaluate the effects of dust and haze was made by Moon,l2 who compared the combined curve for molecular scattering and 20 mm precipitable water with a mean curve for overall scattering obtained at Washington, D. Co by the Smithsonian Institution.15 Assuming that dust and haze particles account for the difference between the curves, a third scattering coefficient was obtained, with kl = 0.0358. To find some approximate values for dust and haze, one needs to assume a probable distribution of particle count with height. The-re is indirect evidence of nuclei in the stratosphere, as well as lower levels, seen in the occurrence of noctilucent and mother-of-pearl clouds, attenuation associated with meteor showers, twilight phenomena and high level "dust horizons-." Direct measurements have also been completed by the use of rockets, as by Junge, Chagnon and Manson6 in the Northern Plains States. The Aitken nuclei counter and impactor used by these investigators gave the average particle concentrations between 7 and 25 km in the stratosphere and upper troposphere. A separation of the particles into size ranges gives the -results shown in Table B-III. TABLE B-III AVERAGE PARTICLE CONCENTRATIONS IN STRATOSPHERE AND TROPOSPHERE Radius of particle (<) < 0.1 0.1-1.0 > 1.0 Average concentration (cm-3) Stratosphere, 25 km 6 0.1 0.01 Troposphere, 7 km variable 1-5 1-5 Since the mean overall scattering curve to be compared is for Washington, D. C., a mean surface particle concentration must be assumed for the same locality. In 1937, such a mean value was estimated at 800 cm3, so we can proceed as before with (I/Io)d = 10-kmd/do where d = concentration of particles per cm3 do = concentration of particles per cm3 at surface. In this case, the graphical analysis by Moon12 yields a kl = 0.0358 and (I/Io)z=0 cc -0~75; thus there is still some dependence upon wave length despite the larger size of the scattering agent. Table B-IV gives the transmission factors computed for haze particles with r&dii < 0ol p and the concentration of 68

particles decreasing from 800 cm-3 at the surface to 100 cm-3 at 5 km to 10 cm'3 at 10 km. TABLE B-IV I/IO IN PERCENTAGE TRANSMISSION (HAZE PARTICLES WITH RADII < 0.1l t) ho(km) 0 = 0o.4 o0.5 o.6~ 0o.7 z I/I o 0 63 2 1 3 5 81 78 76 73 77 10 99 99 99 98 99 15 100 100 100 100 100 *Integrated effect found by counting squares. Particles with radii exceeding O.l1t are too infrequent to contribute an appreciable amount to Table Bz~IV. The "Bouguer Relation" noted by Middletonll may be used to accumulate the scattering effects of molecules, water vapor and dust. This formula is based upon experimental evidence, and it merely combines terms to give I/Io = [10-0s0380 P/Po - 0075k2 w/wo - 003558~"75 d/do] When we perform this computation for specific X and ho values as before, we obtain Table B.V TABLE B-V I/Io IN PERCENTAGE TRANSMISSION (TOTAL SCATTERING EFFECT OF AIR MOLECULES, WATER VAPOR AND DUST) ho(km) o = 0.4 0-.5~ 0o.6, 0.7i Z I/Io* 0 0 0 0 0 5 0 2 11 23 9 10 16 46 69 81 53 15 68 85 93 96 85 20 92 97 98 99 97 *Integrated effect found by counting squares. 69

From 5 km upward, there is a pronounced improvement of transmissivity with increase in wave length in the visible range. The above computations are based upon the formulae and assumptions for Rayleigh scattering. Dust particles of the order of lI in diameter are larger than optical wave lengths and one should seek to apply Mie theory to arrive at some representative values for I/Io. Boll, Leacock, Clark and Churchill2 have published tables for a scattering coefficient, K, in the relation, I/Io = e-KnLd2/4 where n = number of particles per cm3 L = path length in cm* d = diameter of particles, which are assumed to be spherical and uniform in size K = f(a, m), where a = 7d/k and m = index of refraction of the particles relative to the surrounding media. The results depend a great deal on the choice of m. For the stratosphere near an altitude of 18 km, these tables yield the following values for the observed concentration reported by Junge, Changnon and Manson:6 d = 1.0lp X = 0.4~ o = td/X = 7.8 L = 2.5 x 108 cm n = 0.1 cm-3 m = o0.60 K = 1.89, from table m = 0.90 2 2 =!/Io = 83.5% K = 0.90, from table These transmissivities are notably lower than the high values obtained for the stratosphere in Table B-IV. The K values do not diminish indefinitely with increase inom, excepting when cz < 1. With regard to water droplets with diameters up to l10, Mie scattering area coefficients are given in List.9 In this case, the formula suggested for h (km) 0 5 o10 20 30 L(x108 cm) 3.0 3.0 2.8 2 4 2.0 70

I/Io is i'n knsLd2/4 I/Io = ekL/4 where the summation is taken over all sizes of spheres present. The result for a typical fair weather cumulus cloud top with L = 100 ft (3 x 104 cm) is, according 10 to raindrop size spectra listed in Mason, I/IT = e-(225 cm3) (14 X lo 4cm) (2,o)(3 X 103 cm)/4, e-(60 cm-3)(24 X 10-4 cm)2(2.o)(3 X 103 cm)/4 2,08 1,62 - < 0.025 The concentrations of cloud droplets commonly exceed 300 cm-3 in stratiform clouds, so droplet scattering appears to be 100% effective for most cloud types, even when the path length through the cloud is of the order of 102ft. Ice crystals of decidedly nonspherical form displace liquid droplets as the hydrometeoric scattering agent at altitudes of 15 km or less, depending upon the season and the latitude. The irregularity of such crystals creates an extremely complex problem which cannot be solved by convenient groupings of particle sizes as before. However, it does seem probable that the I/Io ratios with which light is transmitted through ice crystal clouds are not appreciably greater than the I/Io ratios with which it is transmitted through the water droplet types discussed above. IV. ABSORPTION Attenuation was defined in Section III as the combined effects of scattering and absorption. Some of the possible absorbing agents are water vapor, C02, ozone, oxygen, liquid water, ice crystals, dust and haze. The absorption bands of water vapor, CO2 and oxygen do not lie in the optical wave lengths and absorption of light by liquid water and ice crystals is negligible in clouds in comparison to the scattering effect. This: leaves only ozone to be evaluated In the visible range, we find the coefficient, k, in the familiar expression I/Io = e-kAz, increasing from about zero at 0.4[1 to 0.05 at 061 and then decreasing.9 This coefficient applies to a layer of pure ozone Az cm thick at the surface, and its smallness indicates that we can probably omit ozone from the list without introducing much error. 71

It appears, then,' that there are no absorbing agents of any consequence in the visible range with the possible exceptions of dust and haze. No evidence on the absorbing qualities of these aerosols has been located as yet. V. DISTRIBUTION OF EFFECTIVE SCATTERING AGENTS The effective light scatterers were found to include the basic, gaseous atmosphere, water vapor, dust and haze, liquid water droplets and ice crystals. The distribution of the last three items is not very well known. However, some comments will be made on each of the five scattering agents. A. Basic Gaseous Atmosphere The composition of the atmosphere is sufficiently uniform to permit the results in Table B-I to be generally applicable. Since we are dealing with horizontal rays, spherical stratification is also assumed tacitly when Table B-I is used. B. Water Vapor Statements about the distribution of water vapor, e.g., precipitable water, must be largely qualitative. The serious attenuation of starlight in the humid tropics has long been noted by the writer, and it is largely in this region that the larger values of 20-30 mm will be found frequently. However, outside of. the Intertropical Convergence Zone.and. the deep moist layers of tropical disturbances, the high moisture values will often cut off at 3-4 km, making the figures for 5 and 10 km in Table B-II somewhat low. On the other hand, in the deep moist layers which invade the middle latitudes> the actual effect of water vapor will probably exceed these figures. In high latitudes, the precipitable water will be much less, making Table B-II unrepresentative there. C. Dust and Haze The surface concentration of solid particles may exceed 50,000 cm-3 at times, denoting great variability in lower levels, and one cannot assume a linear decrease with height because of the tendency for particles to congregate at the bases of temperature inversions. Thus the tropopause is a susceptible haze level, the range of tropopause altitude being as in Table B-VI, *The Intertropical Convergence Zone (ITC) extends almost continuously around the earth within +- 15~ of the Equator. 72

TABLE B-VI RANGE OF MEAN TROPOPAUSE ALTITUDE Latitude Altitude of Mean Tropopause O - 25~ 16 km 25 - 60 10 - 16 60 - 90 8. As a first approximation, we may take the data of Junge, Changnon and Manson5 from the Northern Plains States to be representative of the dust and haze distri._ bution in the mean over the continents. The surface value used may be too great for the air over the oceans. D. Liquid Water Droplets and Ice Crystals The high attenuation rate found above for clouds signifies that poor penetration can be expected from cloud-obstructed light. Therefore the presence of small cumulus towers can evidently interrupt a starbeam as surely as a stratified layer, and the great frequency of clouds in the lower and middle troposphere causes much light extinction. The distribution of the higher, cirriform clouds is of interest here, but it is only partly known because ground observers miss seeing over 50% of the true occurrence and aircraft observations are scarce. 16 Stone has analyzed the following data to arrive at some of the characteristics of cirriform cloud distribution: (a) Double-theodolite measurements taken during the International Cloud Year, 1896-1897 (b) 176 British aircraft observations, 1949-1954 (c) 2000 Canadian aircraft observations, 1950-1955 (d) 700 USAF aircraft observations, 1954-1955. 1 Appleman has analyzed 1375 Western European aircraft observations, 19501957, along with the above observations. The two studies yield the following conclusions regarding upper cloud distribution: 73

1. Nontropical Regions (a) The percentage occurrence of cirriform clouds ranges from 15 to 30%, (b) The modal height of the bases in high latitudes is 7 kmo The modal height in low latitudes is 12 km. There is a linear variation between high and low latitudes. (c) The seasonal range of modal height over Canada is < 2 kmo (d) The thickness is < 2 km in a majority of cases, but thickness has occasionally exceeded 5 km. The mean thickness is 6000 ft. (e) Cirrus usually becomes layered when thickness exceeds 3 km. (f) Most cirrus tops are 3000 to 5q00 ft below the tropopauseo 2. Tropical Regions (a) There is an abundance of cirriform clouds flowing out from the Intertropical Convergence Zone. Their:bases are frequently reported by airline pilots to be above 12 kmo In and near the ITC, these clouds must have a percentage occurrence of 50% or more at 12-16 kmo Their origin is due to ascending moist air, particularly in heavy cumuli which have a'nocturnal maximum over the oceanso (b) Stratospheric occurrence of cirrus has been reported occasionally in middle latitudes, but hardly at all in high or low latitudes. Such frequency is possible where there is sufficient moisture and vertical motion, but favorable conditions are seldom found above the tropopause. VI. CONCLUSIONS The attenuation of a ray of starlight within the earth's atmosphere is caused by (1) refraction into a solid angle, and (2) scattering by air molecules, water in any form and haze or dust particles. Absorption by the above scattering agents and other gases is found to be negligible, with the possible exception of haze and dust, Some representative, quantitative results for these attenuators are sunmarized in Table B-VII for the integrated waveband from 0o.4 to 0,o7p. In this table it is more convenient to present the quantity 1 - I/Io, the percentage reduction of the beam intensity, rather than I/Io. A ray path through the entire atmosphere parallel to a tangent grazing ray is considered for heights of 0, 5, 10, 15 and 20 kmo 74

TABLE B-VII PERCENTAGE REDUCTION OF A HORIZONTAL RAY DUE TO ATTENUATING AGENT ho (kilometers) 0 5 10 15 -20 25 Refraction (constant 91 86 78 63 46 27 Scattering Molecular (constant) 99 84 44 14 4 0 Water vapor (variable) 94 49 0 0 0 0 Haze and dust (variable) 97 23 1 0 0 0 Bouguer Relation 100 91 47 15 3 0 Water droplets (100 ft or more of cloud) 100 100 100 100 100 100 Ice crystals (ditto) 100 100 100 100 100 100 Absorption 0 0 0 0 0 0 The values computed by the Bouguer Relation summarize, in effect, the first three items under "Scattering," and can be changed to the form of stellar magnitude reduction. Fig. B-4 gives the curve used for this purpose, whereby a reduction factor of 1001/s (=2.512) in light intensity equals a decrease of 1.0 in apparent magnitude. Table B-VIII gives the converted values from Table B-VII for individual scattering agents and for the Bouguer Relation for combined scattering effects. The latter formula's 100% attenuation at ho = 0 presents a problem, but if the magnitude reduction curves are drawn for each component of the Bouguer Relation, as in the inset of Fig. B-IV an extrapolation to ho = O analogous to the three lower curves gives a magnitude diminution of about 10 at the surface. Thus the total effect of attenuation at a satellite's vantage point ranges from 10 magnitudes for a surface grazing ray, to 0.2 magnitude for a ray with a minimum tangent height of 25 km. With regard to the brighter stars, these totals indicate that the following comments should hold for normal scattering conditions and no optical aid at a satellitic observation point: 1. For a surface grazing ray, the only objects surpassing in brightness the minimum magnitude of +6.0 for visual sighting are the Sun and the Moon. 2. For a ray with minimum tangent height of 5 kkm, the 5 brightest planets and 15 stars are also detectable. 35 For such a ray at 10 km, the number of available stars approaches 300. 73

25 20 WATER VAPOR SCATTERING 15 BOUGUER RELATION | h,(km) 10I 1\/< / MOLECULAR SCATTERING 0 2 4 6 7 10 6,- REDUCTION IN APPARENT MAGNITUDE -4 t2 100 80 60 40 20 0 ()I -) Fig. B-4. Reduction in apparent magnitude -of stars as a function of attenuation 1 - I/Io..76

TABLE B-VIII REDUCTION IN APPARENT MAGNITUDE OF A HORIZONTAL RAY DUE TO ATTENUATING AGENTS, FOR X = 0.4 ~ to X = 0.7 [ ho (kilometers) 0 5 10 15 20 25 (1) Refraction effect 2.6 2.1 1.6 1ol o.6 0.2 Scattering Molecular 5.0 2.0 o,6 0.1 0.0 0.0 Water vapor 3ol 0.7 0.0 0.0 0.0 0.0 Haze and dust 359 0.2 0.0 0,0 0.0 0,0 (2) Bouguer Relation 10.0 2,7 0.7 0.2 0.0 0.0 Total of (1) and (2) 12.6 4.8 2.3 1.3 0.6 0.2 From the totals in Table B-VIII and a table of apparent stellar magnitudes, Fig. B-5 is obtained to show the number of stars detectable, theoretically, with no optical aid at the satellite and with instrumentation giving 2 and 4 magnitudes gain due to magnification. It is also interesting to note the number of stars which can be seen at the horizon, theoretically, by an observer situated at various levels within the atmosphere. In this case, one can omit the refractive attenuation and assume that the unaided eye can detect stars brighter than magnitude +6.0 with-no scattering. Taking scattering into account as was done in Table B-VIII, the results are as shown in Table B-IX: TABLE B-IX NUMBER OF STARS THEORETICALLY DETECTABLE AT VARIOUS LEVELS WITHIN THE ATMOSPHERE, WITH SCATTERING TAKEN INTO ACCOUNT Reduction in Magnitude Number of Stars h01o (kilometers) ~~t~Due to Scattering Detectable o 5 5 8 5 1.5 780 10 0.3 3400 77

25 20 NO OPTICAL AID 15 h(km) 10 100 1000 10,000 N Fig. B-5. Number of stars theoretically detectable from a satellite with varying gains, as a function of ray perigee ho.

The number of detectable stars is much greater in this case, due to the halving of the number of optical air masses traversed by the light ray and the elimination of refractive attenuation. The distributions of some of the important variable scattering agents discussed in this report are not well known. In general, their influence in the upper troposphere is felt in tropical regions mainly, and in midlatitudes secondarily. The high extinction rates of clouds, for example, are of especially great importance near the intertropical Convergence Zones and around tropical disturbances. The frequency of occurrence of cirriform elouds range from 15 to 30% outside of these regions with a modal height of 7-10 km. 79

REFERENCES 1. Appleman, Ho S., Occurrence and. forecasting of cirrostratus clouds, Technical Note No. 40, WMO, Geneva, Switzerland, 1961. 2. Boll, Leacock, Clark and Churchill, Light-scattering functions, Univo of Michigan Press, 1958. 3. Fowle, F. E., The atmospheric scattering of light, Smithsonian Misco Collection, 69, Noo 3, 1918. 4. Johnson, J. C., Physical Meteorology, M.I.T. and John Wiley and Sons, 1954, pp 33 -64. 5 Jones, L. M., Fo F. Fischbach, and J. W. Peterson, Atmospheric measurements from satellite observations of stellar refraction, University of Michigan Technical Report Noo 04963-1-T, Jan. 1962. 6. Junge, Changnon and Manson, Atmospheric aerosols, J. Meteorology, Feb. 1961, pp 81-108. 7. Lillestrand, R. et al., Mechanical Division of General Mills, Inc., Minneapoli Minn., 1960o 8. Link, M= F., Occultations et eclipses des planetes, Bull. Astronomique, 9, 1933, p 227. 9. List, R. Jo, Smithsonian Meteorological Tables, Washington, D. Co, 1958, 10. Mason, B. J., The physics of clouds, Oxford Monographs on Meteorology, London, 1957. 11o Middleton, W.EoKo, Vision Through the Atmosphere, University of Toronto Press, 1952. 12. Moon, P., Proposed standard radiation curve for engineering use, J. Franklin Insto 230, 1940, 588-593. 13. Pressly, E. Co, Air mass between an observer and outer space, Physical Review, 8, NoR 3, 89, No 3, Feb 1, 1953, 654-655 14. Schoenberg, E., Handbuch der Astrophysik, Band II, Julius Springer, Berlin, 1929, 171-22 7, 80

15. Smithsonian Institution, Annals of Astrophysical Observations, 2l, 1908, 113 16. Stone, R. G., A compendium on cirrus and cirrus forecasting, AWS TR-105-150, USAF, 1957. 17. Van de Hulst, H. C., Light Scattering by Small Particles, John Wiley and Son, New York, 1957. 18o Van de Hulst, H. C., The Atmospheres of the Earth and Planets, The University of Chicago Press, 1952, 49-111. 19. Waldram, J. M., Measurement of the photometric properties of the upper atmosphere, Trans. of the Illuminating Engr. Soc., London, Aug. 1945. 81

APPENDIX C VARIABILITY OF STARLIGHT DUE TO THE EARTH'S ATMOSPHERE by Maurice E. Graves I. INTRODUCTION When a beam of starlight reaches the earth's atmosphere, its speed suffers a minute reduction and the light ray bends toward greater medium density. The wavefront, after being planar in space, undergoes corrugation when it passes through a layer of temperature inhomogeneity, and normal lines to the wavefront are no longer parallel (see Fig. C-l). To an observer of these nonparallel rays there may be fluctuations in image position, size and intensity, and distortion of image shape. PLANAR WAVE FRONT ] TURBULENT LAYER DISPERSION OF RAY DIRECTION Fig. C-1. Dispersion of light rays upon passage through a turbulent layer in the atmosphere. This report will summarize some of the literature dealing with fluctuation and distortion phenomena, and will make some estimates for horizontal grazing rays passing through the stratosphere and upper troposphere. The general procedure will be to condense the abundant information on stars observed near the zenith, to discuss the very limited material on stars with large zenith angles, and then to surmise what results may be expected for stars with large zenith angles viewed from high levels (see Fig. C-2). A planetary atmosphere changes the intensity of a light ray as well as the direction of advancement, and it is convenient to speak of a "semi-constant" and a variable component for each, thus: 82

Fig. C-2. Simplified geometry of horizontal grazing ray viewed from satellite. "semi-constant" term: refraction Change of direction variable term: seeing (quivering distortion, pulsation) "semi-constant" term: extinction Change of intensity (scattering, absorption) variable term: scintillation Besides the aperiodic variations of the magnitudes of the changes of directhere are large-scale seasonal, latitudinal and diurnal effects incorporated wli;nln the "semi-constant" term which are related directly to the temperature and density structure of the atmosphere. II. SEEING There is some overlapping of terms currently being used to describe the fluctuation of starlight. In this paper, the definitions used by Stock and Kellerl4 and Tatarski21 will be accepted as recent and authoritative, and parentheses will be used at times to signify related terms not appearing in these references. Thus the three basic image changes and movements will be referred to in the following ways: 1. Variation of magnitude: scintillation, twinkling 2. Variation of image size: distortion, pulsation, spreading, blurring 3. Displacement of image: quivering/ (2) and (3), combined: seeing, poor seeing

Three major factors determine the quality of seeing for a particular stellar object: 1. Telescopic aperture, D 2. Meteorological conditions 3. Zenith distance, z These factors will now be discussed in some detail. At a particular time, a small D of, say, 10 cm employed at the ground will receive nearly parallel rays and form a relatively sharp image. However, at a later time, the parallel rays may have a slightly different direction, producing an image displacement. Rapid changes in the direction of the incoming rays cause an erratically moving image; this motion is called quivering (dancing). On the other hand, a large D of, say, 30 cm may cover the entire range of ray directions, with the average direction being parallel to the wavefront. This gives a steady position, e.g., little quivering, but causes some distortion, e.g., spreading and blurring, amounting to an image diameter of 2-4" arc (see Fig. C-3). 1.0 GOOD SEEING POOR SEEING.5 0 1 2 3 4 5 d (sec of arc) Fig. C-3. Image intensity I as a function of image diameter for typical cases of good and poor seeing. For a given D and z, the seeing is determined by the meteorological conditions, specifically, the presence of temperature (and density) irregularities in horizontal planes. Seeing may originate anywhere in the atmosphere, but since density (and refractive power) decrease with altitude, the source of this type of fluctuation is more likely to be found in the lower levels. Within a turbulent zone near the ground, the eddy cells will act as lenses if their horizontal dimensions are no larger than D. Larger cells act as prisms, so the eddy effects are to encourage blurring in the first case and quivering in the second case. The prisms must change in shape or strength as the ray passes, however, if an image movement is to occur. 84

An expression related to the quality of seeingl4 is: K22 = k2 N 2f where x = wave length of light; N = average number of turbulent elements in path; Lm = average size of elements; 82f = mean square variation of A, the refractive index. When 2 < 1, the quality of seeing is good, that is, any turbulent layer permits the light to pass through in an undiffracted (zero-order) plane wave. When K2 > 1, the quality of seeing is inferior. It should be noted here that 3f in the above expression is simply the difference between the refractive index of the air inside a turbulent element and the refractive index of the environment, thus: -f = P (lO -1) Po where subscript zero refers to p and A at normal T and p. But p = pRT, where p, p and T refer to environment air 0 = 8P' 5p(RT) + (pR)ST P T f = ( -) = _ PT Near sea level, p - 1 = 2.9 x 10-4 and 6f = - 1 x 106 ST. It -is clear that an increase in zenith distance, z, leads to a notable increase in path length of a starbeam passing through the atmosphere. This leads one to expect a positive change in any of the fluctuations when z nears 90~. Tatarski21 has published Russian work on quivering which shows the rms magnitude 85

to be less than 1" arc for z < 85~0 (see Fig. C-4). The frequency of image movement is not discussed therein, but Ellison and Seddon5 state that the frequency of quivering is similar to that of scintillation. 1.00 600 70~ 800 840 z I 2 3 4 5 6 7 8 9 sec z Fig. C-4. RMSfluctuations of starlight as a function of zenith angle z, for z < 85~. For a horizontal path through the entire atmospheric shell tangent to the earth's surface, the rms magnitude must be increased. Since the path length is thereby doubled, the application of the principle of discontinuous, random motion in one dimension gives a factor of 42. Therefore, the quivers would have a total rms magnitude of about 1.0" x {e25 1.5" at zenith angles of 60-85~ and about double this amount at z close to'900. The following calculation gives the percentage rms error at the surface: 3" arc =0. l 69' x 60"/1' I Here, 69' is the representative refraction angle for a complete path. Although the prism-like segments of larger eddies lose their importance to seeing above lower levels, there is some question about horizontal layers in the turbulent wind shear zones near jet streams. When considering a large z and \small D in viewing through such a layer, it appears that some quivering will take place when large eddies are present. On the other hand, the following computation indicates that the percentage error is again very small: Let Rs =. refraction angle for a complete horizontal traverse 86

(Rs )12 km = 18 (Rs)sfc = 69' (RS)12 km 1 (Rs)sfc 4 Let p = air density (P)12 km (P)sfc = 0.3 The ratio Rs/p'z a constant below 12 km and the percentage error due to quivering remains at O.l1% when the prism-like eddy segments maintain the same frequency of occurrence and variability throughout the troposphere.ll The turbulent layers aloft will be taken up again in Section IV. III.o EXTREMES OF SEEING Seeing encompasses both quivering and image distortion, and its extreme form is pulsation. Tombaugh,22 in gusty surface wind conditions, has recorded the swelling of 3rd magnitude 8 Geminorum to three or four times the size of Jupiter. The swelling was folloWed by total disappearance in the telescope for several seconds. Notwithstanding this reported case of pulsation, there is no evidence that large variations in image size, and disappearances in particular, occur during normal viewing, even at surface air density. 21 The individual extreme values in quivering are not given in Tatarski,1 but they may be estimated from the rms when one assumes a certain number of observations, thus: Pr -a < = Pr < i where a = an arbitrary extreme value x = mean = 0 n = number of observations Now suppose a = rms = 3" and n = 100; these assumptions give the following percentage probabilities (see Table C-I) that a quiver will exceed an extreme value, a:

TABLE C-I PROBABILITY OF EXCEEDING VARIOUS EXTREME VALUES OF QUIVERING a(" arc) Pr [x > a] (%) 4 11.51 6 3.59 8 o0,a IV. SCINMTILLATION Scintillation* was introduced above as a variable quantity associated with change of starlight intensity. More specifically, it is definable as the fre.quency of stellar magnitude variations. Its observable effects and its causes will be discussed as we take up again the three major factors which were important to the quality of the seeing, and which also determine the scintillation, namely: 1o Telescopic aperture 20 Meteorological conditions 35 Zenith distance There has been considerable discussion, even controversy, about the optical nature of scintillation. Rayleigh20 was one of the early investigators in this field, with his refraction theory in 18935 His results indicated that the altitude of scintillation origin was less than 4 kmo Later, Little15 criticised this theory and introduced his diffraction theory. Little's most effective criticism was that in the refraction theory, the density gradient must be of the order of magnitude of 0o5%/cm, a figure which is far greater than what may reasonably be expected in- the upper troposphere. The diffraction theory, on the other hand, needs but 0.0006%/cm gradient, which corresponds to 0.0020C/cm temperature gradient. Diffraction, however, seems to require a rather thick disturbing layer, according to Little, whereas refraction theory conforms well to the notion. of a thin-layered scintillation source region. Although challenged on several points, it appears that Fellgett's discussion7 of the two theories explains the basic differences. Starting with the postulate that scintillation deals with the propagation of light through an atmosphere of varying refractive index, it is clear that one is dealing with the wave properties of light. A boundary-value solution of Maxwell's Equations would be desirable, but this is so complex a problem one seeks an approximation. The *We are concerned here with brightness scintillation, and will ignore the second type, called chromatic or color scintillation. 88

first-order approximation is Huygens' Principle of propagation of light waves, and diffraction enters into the application. The second-order approximation assumes that wave length is negligible in magnitude and no diffraction occurs. It is commonly referred to as the "Ray Theory," and is in the field of geometric optics; Huygens' Principle is in the field of physical optics.10 The former model is adequate for most considerations, but wave diffraction techniques give more realistic results at times and their theoretical base is more rigorous. During the past 15 years, there have been a number of studies on the nature of scintilLation, e.g., its observed effects, frequency origin, etc., by Chandrasekhar, Ellison and Seddon,5 Epstein,6 Gaviola,8 Keller,l2.1 Mikesell, Hoag and Hall,16 Protheroe,l8l9 and Tatarski.21 Before considering the three determining factors explicitly, some attention will be given these facets of the subject. The principal observable effect is the shadow-band of about 8 cm band-width. Shadow-bands form transitory patterns, due to their rapid motion and structural changes. They move in accordance with the wind at their level of origin, leading to some successful efforts19 to find the wind speed at an assumed altitude, usually near the tropopause.* The formula V = aRb gives a good fit between the desired wind speed aloft, V, and the scintillation ratio, R (300 cycle/sec component/10 cycle/sec component). For single frequencies, Gifford9 has shown that the 150 cycle/sec scintillation frequency has its maximum correlation with wind speed at 30-40,000 feet, but the 9 cycle/sec frequency has little correlation below 60,000 feet. Lord Rayleigh's question,20 "Is the ultimate effect (of scintillation) a small residue of neutralizing effects?" has been answered negatively by all investigators who have been active in the field in the past few years. In fact, there is ample evidence that the shadow-bands are related to wind velocity near 10 km;1,6,19 furthermore, they are inclined identically to the Fraunhofer lines, so that a single height or thin layer is the cource of the intensity variations. MeineJi states that the minimum thickness of this layer is 10 cm. Photomultiplier tubes have been used by Barnhart, Keller, and Mitchell,1 Keller,13 Mikesell, Hoag and Hall,16 and Protheroe, 18l19 to study the shadowbands. The schematic diagram in Fig. C-5 illustrates the method employed in conjunction with an analog computer. Whereas the unaided eye detects the lower range of frequencies up to 16 cycle/sec, the instrumental readings give values up to 1000 cycles/sec. The specific cause of scintillation is necessarily a phenomenon which creates multiple refractions. The sole candidate appears to be turbulence, e.g., the eddy motion of parcels of air with differing indices of refraction. *Although the singular form of "tropopause" is used here, duplicity is not uncommon, especially near jet stream cores and in the subtropics. 89

FREQUENCY I 2 3 4 5 6 Fig. C-5. Photomultiplier tube as used to obtain Fourier spectra for scintillation. 1. Objective lens with twin, rotatable apertures 2. Two photomultiplier tubes 5. Analog computer 4. Autocorrelation coefficients obtained 5. Autocorrelation function obtained 6. Fourier spectra obtained This eddy motion is also the cause of aircraft turbulence, and since the advent of jet-turbine engines at least a start has been made toward the discovery of its distribution in space. In general terms, we may say that the altitude of origin is usually near the tropopause, where wind speed, wind shear (in any plane) and the vertical density gradient can be relatively large at times. This altitude ranges as shown in Table C-II. TABLE C-II RANGE OF MEAN TROPOPAUSE ALTITUDES Latitude Altitude of Mean Tropopause O - 25~ 16 km 25 - 60o - 16 60 - 90 8 It is interesting to compare Table C-II with Table C-III, which shows irregularities obtained by checking the transitional motion of the shadow-bands against observed winds. 90

TABLE C-III ALTITUDE OF DENSITY IRREGULARITIES AS OBTAINED BY VARIOUS INVESTIGATORS Year Investigator Altitudde of Density Irregularities; Basis 1893 Rayleigh20 4 km; refraction theory 1952 Ellison and Seddon5 5 km; planetary scintillation 1954 Barnhart, Keller 11 km; observed winds versus transitional and Mitchelll motion of-shadow patterns 1954 Epstein6 10 kmn; ditto 1955 Protheroel8 9 km; ditto 1961 Protheroe!9 9 kin; ditto The altitude values of 9-11 km appearing in Table C-Ill approximate the level of typically strong wind shear just below the tropopause around organized jet streams. They also approximate tropopause altitude in midlatitudes when the tropopause. is at maximum depression. The first of the three major factors determining scintillation is telescopic aperture. Perhaps the simplest way to express the relationship is as follows: Random statistical fluctuations, viz. scintillation, cc (A)1/2, where A = aperture area Aperture diameter D cc A/ Thus scintillation cc (D)-1. Since quivering is also a random fluctuation, the magnitude of image displacements oc (D) 1. A similar result can also be obtained from an analysis of the spectrum function (see Fig. C-5),1 Table C-IV summarizes some of Protheroe's results19 and it shows the contrast between small and large apertures, with zenith angle = 00: The meteorological conditions attendant to pronounced scintillation must be those which produce turbulence at altitudes near 10 km, for turbulence has been designated as the sole cause and the evidence to date favors such altitudes. The eddy motion in the wind shear zones mentioned above is known to be strong at times, and of suitable eddy size to cause significant bumpiness in jet aircraft moving at about 300 mps. This eddy size would be of magnitude 91

TABLE C-IV. FOURIER SPECTRA RELATED TO FREQUENCY RANGES AND APERTURE SIZE Aperture Strength of Fre que nc y Diameter Fourier Spectra 1-3 in. 0-100 cy/sec About constant 1-3 100-500 Transitional 1-3 500-1000 About zero 10-20 10-50 About constant 10-20 50-100 Transitional 10-20 100-500 About zero several tens to a few hundred. meterso However, we know that we must deal with density irregularities of only a -few centimeters magnitude, for planets show little tendency to scintillate when their diameters are as great as 30" arc.12 At 10 km, 30" arc corresponds to just 5 cm,* The aircraft observations of turbulence establish the occasional presence of large eddies and the fact that the turbulent layer is usually evaded by the customary 600-meter change in flight altitude. In addition, the writer has often noticed shear-produced "Cobblestone Turbulence" in jet aircraft, and the eddy sizes would be in the range of several meters to a few tens of meters in these cases. Employing Kolmogoroff's ideas on eddy energy dissipation,2 one can easily visualize a spectrum of eddy sizes ranging on down to zero diameter, thus accounting for the necessary scintillation source, e.go, eddy sizes of 5 cm or less. The importance of zenith angle, z, can be seen by examining the spectrum function (see Fig.. C-5). This may be written as:14 B(w) 16n2Azb(w)sin2( zkw2) ho2 -2 where B(w) = spectrum function for the shadow pattern ho2 = mean square amplitude of intensity variation in pattern *The Ellison and Seddon result of 5-km in Table' C-IIT is based upon a much smaller maximum planetary scintillation diameter, of but 3". It assumes an average eddy diameter-of 8 cm, equal to the average width of shadow-bands. The writer feels that 3" is much too small, and that 30" is about right for the critical arc length. 92

Az = slant distance through layer b(w) = spectrum function for the frequency of elements of different sizes z = slant distance from telescope to turbulent layer k< = wave length in cm w = wave number (1/cm) The turbulence is assumed to be homogeneous and isotropic, and these assumptions seem to be satisfied quite well in natural conditions, excepting ve4ry near the earth's surface. When z is quite large, the sin2(iczXw2) term begins to oscillate between 0 and 1, but Az increases, so that B(w) must tend to increase or ho2 must decrease, or both. The latter quantity does decrease slightly, and this is confirmed by Tatarski, who states that a chromatic effect sets in at z = 600 whereby the scintillation at different wave lengths loses its correlation, producing a lessening of total twinkling effect as z increases further. However, large amplitude, low frequency oscillations increase at low star altitudes, as we: can readily observe by using the unaided eye in the visual range. Some values for range/mean of the total amplitude of stars at z = 0~ and z = 90~ are given by Protheroe19 for a small aperture: TABLE C-V RANGE/MEAN OF TOTAL AMPLITUDE OF STARS AT z = 0~ AND z = 900 D z Range/mean 1-3 in. 00.5-1.5 1-3 in. 90~ About 10 Thus, for small apertures, the total amplitude of the fluctuations increases by a whole order of magnitude when z approaches 90~0'. With larger apertures, the amplitude gain i's not as great. For a horizontal path near the earth's surface, using a light source at a finite distance, Project Michigan investigators3 have shown scintillation-dependence mainly upon the vertical temperature gradient and the turbulence characteristics along the path, but also upon the path length and wind speed and direction. If a surface temperature inversion is present, a fair amount of mixing produces strong scintillation. 93

Above the low-level zone where the above type of scintillation is detected, the occurrence of this phenomenon through a horizontal path is not well known. 14 Meinel mentions manned balloon ascents where diminishing scintillation effects were observed through the first 6 km, with negligible fluctuations of any kind above 15 km. There is little indication of the amount of scintillation one perceives when passing through the tropopause layer or a wind shear zone where the meteorological conditions are likely to encourage its occurrence. If we assume the tropopause to be parallel to the earth's surface in all directions in the neighborhood of a data point, then the maximum distance traversed by an unrefracted light ray within such a layer is related to the layer thickness, Az (see Fig. C-6). An analytical solution gives the values shown in Table C-VI. (AL) max. Fig.. C-6. Simplified geometry of maximum distance traversed by a straight horizontal ray. TABLE C -VI MAXIMUM DISTANCE TRAVERSED BY A STRAIGHT HORIZONTAL RAY WITHIN A LAYER Az, FOR VARIOUS LAYER THICKNESSES Az (AL)max 100 mt 74 km 250 112 km 500 160 km 750 196 km 1000O 226 km 94

Several minor causes have been omitted from this discussion of scintillation and seeing. Water vapor, for example, has a high opacity for infrared radiation, giving it a greater influence than would be estimated from the refractive index of moist air. The mechanism producing temperature and density irregularities is the radiation loss of heat from moist air masses at night, into space. Five km may be a reasonable estimate of the tops of the moist layers in most air masses of this type. Seco.nd, Minnaert17 refers to the relation of scintillation to path water content, cloud edges, isobar curvature and low-level gradients of temperature, density and pressure, but all of these possible causes can be discounted in tangential star-tracking above a few km. Third, the differing physical properties of stars are responsible for scintillation anomalies. One qualitative statement citing such differences is a law of Bijourdan23 which states that the spectral range of a star decreases from white to red, thereby reducing the scintillation in a like manner. V. CONCLUSIONS Variations in the direction of motion and in the intensity of a starbeam fall into the general categories of seeing and scintillation, respectively. In either case, the major factors which determine the amount of observed variation are (1) telescopic aperture, (2) meteorological conditions and (3) zenith distance. Increase of aperture reduces quivering, which is the principal seeing phenomenon, but it distorts the image shape so that a stellar point image then attains a diameter of 2-4" of arc. Increase of aperture also reduces scintillation, beginning with the higher frequencies. Meteorological turbulence, producing prism-like segments which momentarily deflect light rays, is the basic cause of seeing. Near the ground, but probably not aloft, such turbulence sometimes causes changes in image size, called pulsation. Studies of the shadow-bands of scintillation have established high-level turbulence near the tropopause as the chief source of this effect. The additional path length for zenith angles near 900 increases quivering to an rms value of about 3" arc for a complete traverse, but the resulting angular error is only 0.1%. With small apertures, total scintillation amplitude increases by a factor of 10 at these zenith angles, with larger apertures increases somewhat less. At minimum grazing distances of several kilometers, the seeing error should not exceed its- surface magnitude of 0.1% and the scintillation effect may be expected t-o decrease with altitude due to decrease of the air density. However, in turbulent inversion layers such as those commonly encountered near the tropo95

pause at about 10.kin, grazing angles will evidently have horizontal path segments of a few kilometers lying within the perturbed zones. At these particular levels, there is a strong possibility of significant scintillation effects when there is turbulenceo 96

REFERENCES 1.o Barnhart, PO E., G. Eo Keller, and Wo Eo Mitchell, Jr., Investigation of upper air turbulence by the method of analyzing stellar scintillation shadow patternS, Final Technical Report, Air Force Cambridge Research Center and The Ohio'State U. Research Foundation, July 1959. 2. Batchelor, G. K., Quart. J. Ro. Meteorol Soc., 76, 133-146, 1950. 3. Bellaire, F. R., and F. C Elder, Scintillation and visual resolution over the ground, Willow Run Laboratories, The University of Michigan, October 1960. 4. Chandrasekhar, S., Monthly Notices Royal Astron. Sock, 112, 473-483, 1952. 5. Ellison, M. A. and H. Seddon, Monthly Notices Royal Astron. Soc., 112, 73-87, 1952 6. Epstein, E. S., Relation between stellar scintillation and atmospheric phenomena, The Pennsylvania State University, Unpublished Master's Thesis, June 1954. 7. Fellgett, P. B., Quart, J. Royal Meteorol. Soco, April 1956. 8. Gaviola, E., On seeing, fine structure of stellar images, and inversion layer spectra, The Astronomical Journal, 54, 155-161, 1949, 9. Gifford, F., Jr., Bull. American Meteorol. Soc.o, 36, Jan. 1955. 10. Humphreys, W. J., Physics of the Air, McGraw-Hill Book Co,, Inc., New York, 1929. 11l Jones, L. Mo, F. F. Fischbach, and J. W. Peterson, Atmospheric measurements from satellite observations of stellar refraction, Technical Report, The University of Michigan, January 1962. 12. Keller, Go E., Astronomical "seeing" and its relation to atmospheric turbulence, The Astron. J., 58, 113-125, July 1953o 135 Keller, G. E,, Relation between the structure of stellar shadow band patterns and.st~llar scintillation, J. of the Opt. Soc. of America, 45, 845-851, ~ctober, 19355 14. Kuiper, Go P. and B. M. Middlehurst, Telescopes, The University of Chicago Press, Chicago, Ilol, 1960. 97

15o Little, C. G., A diffraction theory of the scintillation of stars oh optical and radio wave-lengths, The Roy. Astron. Soc. Monthly Notices, ill, 289-302, 1951o 16, Mikesell, Ao H., A. A, Hoag, and J. S. Hall, The scintillation of starlight, Jo of the Opt. Soc. of America, 41, 289-295, Oct., 1951. 170 Minnaert, M., The Nature of Light and Colour in the Open Air, Dover Publications, Inc., New York, 1954. 18. Protheroe, Wo M., Determination of shadow band structure from stellar scintillation measurements, JO of the Opt. Soc. of America, 45, 851-855, October, 1955. 19. Protheroe, Wo M,, Stellar scintillation, Science, 134, 1593-15999 17 November 1961. 20. Rayleigh, Lord, On the theory of stellar scintillation, Philosophical Magazine S5, 36, 129-142, July 18935 21. Tatarski, V. I., Wave Propagation in a Turbulent Mediumn, McGraw-Hill Book Company, New York, 1961. 22. Tombaugh, C, W. and B. A. Smith, A seeing scale for visual observers, Sky and Telescope, XVII, 449, July 1958. 23. Vassy, E., Physique de L'Atmosphere, Vol. II, Gauthier-Villars, Paris, 1959. 98

UNIVERSITY OF MICHIGAN'-, I 3 9015 02826 6966 I II 3 9015 02826 6966