THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Aeronautical and Astronautical Engineering High Altitude Engineering Laboratory Technical Report ANALYTIC SOLUTION FOR ATMOSPHERIC DENSITY FROM SATELLITE MEASUREMENTS OF STELLAR REFRACTION P. B. Hays F. F. Fischbach ORA Project 04963 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GODDARD SPACE FLIGHT CENTER CONTRACT NO. NASw-140 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1963

I. BACKGROUND The deduction of atmospheric density, from satellite measurements of stellar refraction has been described in two previous University of Michigan Technical Reports.1,2 One gap in the theory existed at the time of printing, namely, an exact expression for retrieval of the density function when the refraction is known. In both reports this inversion was approximated by an integration along a straight, rather than refracted, rar path. For this approximate case, an analytic solution could be obtained. In Reference 2, doubt was expressed that an analytic solution could be obtained in the exact case, and numerical methods were begun. However, an exact analytic solution has now been demonstrated. The appropriate equation for this type of solution was used by Bateman3 in 1909 while investigating the propagation of seismic waves. That the seismic and refraction cases are quite analogous is evident since both types of rays are brachistochrones and traverse spherically-stratified media. Spherical stratification of the atmosphere is assumed throughout; and that this assumption leads to negligible error was proved in Reference 2. II. DEMONSTRATION OF SOLUTION If 4 is the index of refraction at radius r from the earth's center and z is the obtuse angle formed by the ray and the earth's radius at any point, the refraction R measured at the satellite is given by R = ~2po ro sin zf~'2' d Ji R =20 ro sin zo J -[42r2 - i2r2 sin2 z ]1/2 1 00 0 The subscript o refers to the vertex, or point of symmetry, of the ray and subscript s will refer to the satellite position. Evidently zo = </2, so rLo R = 2ro04Of 0 d/ Z (d1) 1 4~.[2r2 - 2 112 Since is a single-valued function of r, we may write r(l) and R =R(lor) Thus:

rmnax d log d(plr) R(1ioro) = or dl )(2) R ( oro> )~2korolf [ 2r2 (i1) - 2r2 ]112 -ro ~00 where r(L) is to be found. The notation rmax refers to any r so large that ut(r) = 1-in particular, rs, which cannot be exceeded in the physical sense. However, rmax may be taken to be co, if desired. For comparison with Bateman, we adopt the substitutions P= 1r i a = ro; = rmax Thus: dlog dn R(a) = 2a (dn (3) (112 - Ce2)i/2 Ct Now substitute: R(cI) = R(s); dlog = -2 1 2 2 1 dn ~2t~(t); -' d~n 5 a t Since dn/dt = -r13/2, if vl = a, t = l/ca2 = s, and if v = A, t = 1/p2 = a, we have a s 2 f" 2tpt)dt 2r d(t)dt F(s) = _2 (t 1V 1/2 J (s - t)I/2 s t s a which is an integral equation solved by Abel. The solution is - 1 d fi F(s)ds i((t) =1 d; 2 Tc dt Ijtss provided: (1) F(s) is continuous in the closed interval (a < t < b); (2) F(a) = 0; and F(s)ds () F(s)ds have a continuous derivative throughout the same a {x - s closed interval. The first condition is obviously met by the nature of F(s) = R(G) =T refraction during a scan being continuous throughout. The refraction angle is O at the upper limit, where t is smallest. To check the third condition note that 2

d, F(s ) = ( dx / ss k'(t) ((t) is proportional to the derivative of atmospheric density and obviously continuous over the interval. Resubstituting in Eq. (5), we have: 1 dC L -1 d rmax ir R(porQ)d(poro) k d(cr) = j d(tr ) 2 2 r - P2rr2 PLr -or er Bateman3 points out the following simplification: r d - rmax 1rR( loro)d(pOro) d f rmax i2r2R(poro)d(Woro) d(2r |u,2r2 -,2r2 dr [) 2r2L22 - L2r2 Mr 0o 0oo t r 000 0 0' rmax r _- | rR((toro)d(poro) d ma R(oro)d(por ) 2r2 lr2 - 12r2 d(Lr) or 2r2 - 2r2 - rmax d(.r) f R( oro)ro )d( r)ro );-d~1 d __ r rmx R( orro)d( pror ) d d( Pr2) et d( r)J [L-r2 - >2r'd(~~Lr ) ~Lr go1ro -!r and 5 and ~ ~r ~gro~r

log L(rmx) - log [(-r) = - i (9 ~r I LI2 _ Lr2 rmax log = - _ f R(Lor)d(por)o) (o) 1,' - 2ro2 _ 2r2 ~ - 2 or exp rlr R(kLoro)d(rro) (1) ltJ'ir 2ro -,r2. which has the form = \r(.r) At any point, r = 4r/i so that r(lJ) is given explicitly, which was required. It should be noted that determination of the density function is based on knowledge of the refraction angle as a function of t-oro, which is the ray path constant. That is, loro = Loro sin zo = r sin z = srs sin Zs and since As = 1 rs sin zs = ro o Measurement of refraction angle R is tantamount to measurement of zs, and satellite tracking yields rs. Thus telemetry of R and knowledge of satellite position is equivalent to the measurement of R(pyro ). Although it seems extremely fortunate that the analytic expression requires knowledge of the precise refraction function, R(poro ), that is available experimentally, this result is not altogether surprising. The ray path constant being fundamental to the general concept of brachistochrones, there is considerable a priori likelihood that it would appear in the inversion equation and would be readily susceptible to measurement, as well. 4

III. IMPLICATIONS OF SOLUTION The demonstration of an analytic solution for index of refraction (i.e., density) as a function of height is of fundamentalJ'importance to the satellite refraction technique. First, it proves the existence of a unique inverse function. Second, it allows a simpler, much quicker, evaluation of the function, virtually assuring that the data reduction can be completed in real time. Third, it allows immediate programming of the data reduction process, and in turn a sophisticated error analysis, including simulation of satellite data and densityfunction retrievals. Fourth, it focusses attention on equipment requirements, which become the only major area in which feasibility has not been established. Variation of parameters in the simulated data will assist in specifying the instruments. REFERENCES 1. Jones, L. M., F. F. Fischbach, and J. W. Peterson, Atmospheric measurements from satellite observations of stellar refraction, Univ. of Mich. Tech. Rpt. 04963-1-T, Jan. 1962. 2. Fischbach, F. F. et al., Atmospheric sounding by satellite measurements of stellar refraction. Univ. of Mich. Tech. Rpt. 04963-2-T, Dec. 1962. 3. Bateman, H., The solution of the integral equation connecting the velocity of propagation of an earthquake-wave in the interior of the earth with the times which the disturbance takes to travel to the different stations on the earth's surface, Phil. Mag. s.6, 19, 576-587, 1910. 5