Smithsonian Contributions to Astrophysics VOLUME 3, NUMBER 9 GEODETIC USES OF ARTIFICIAL SATELLITES by GEORGE VEIS SMITHSONIAN INSTITUTION Washington, D.C. 1960 Publications of the Astrophysical Observatory This series, Smithsonian Contributions to Astrophysics, was inaugurated in 1956 to provide a proper communication for the results of research con- ducted at the Astrophysical Observatory of the Smithsonian Institution. Its purpose is the "increase and diffusion of knowledge" in the field of astro- physics, with particular emphasis on problems of the sun, the earth, and the solar system. Its pages are open to a limited number of papers by other investigators with whom we have common interests. Another series is Annals of the Astrophysical Observatory. It was started in 1900 by the Observatory's first director, Samuel P. Langley, and has been published about every 10 years since that date. These quarto volumes, some of which are still available, record the history of the Observatory's researches and activities. Many technical papers and volumes emanating from the Astrophysical Observatory have appeared in the Smithsonian Miscellaneous Collections. Among these are Smithsonian Physical Tables, Smithsonian Meteorological Tables, and World Weather Records. Additional information concerning these publications may be secured from the Editorial and Publications Division, Smithsonian Institution, Washing- ton, D.C. FEED L. WHIPPLE, Director, Astrophysical Observatory, Smithsonian Institution. Cambridge, Mass. For sale by tbe Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. ? Price 65 cents Contents Page Introduction 95 The coordinate systems 96 Elliptical coordinate systems 96 Terrestrial rectangular coordinates 97 Sidereal rectangular coordinates .. 97 Geodetic rectangular coordinates 100 Accuracy of the transformations 106 Determination of directions 108 The observations 110 Measurement of distances 110 Optical measurement of directions I l l Electronic measurement of directions 112 Correlations between the observed quantities 112 Keduction of observations to the station 115 Correction for aberration 115 Correction for optical refraction 117 Correction for electronic refraction 122 Observations to objects of known positions 122 Use of angles in space 122 Use of directions and distances 123 Accuracy of determined positions 126 Observations to objects of unknown positions 129 Unconditioned equations of observation 129 Introduction of conditions 132 Connection of geodetic systems 133 Observations to orbiting objects 135 Quasi-simultaneous observations 136 Use of a first-approximation orbit 136 Use of a second-approximation orbit 139 Use of a third-approximation orbit 143 Orbit of unknown mean distance . . . 145 Applications of the various methods 145 The orbit: ephemerides and visibility 145 The observations 146 Method for a known orbit 147 Method of orbital interpolation 148 Methods for simultaneous observations 150 in Page The use of the moon 152 The orbit of the moon: libration 152 The observations 153 Determination of positions and expected accuracy 155 Acknowledgments 157 Symbols used 157 References 159 IV Geodetic Uses of Artificial Satellites By GEORGE V E I S 2 Introduction The geodetic methods heretofore applied to the data provided by the artificial satellites have been based on a dynamic approach. That is, the perturbations in the motion of a satellite have been used to determine the gravitational field of the earth, and thus to obtain informa- tion on the shape of the geoid (Jacchia, 1958b; O'Keefe, 1958; O'Keefe and Eckels, 1958; Kozai, 1959). This paper presents another way of using the satellites in geodesy. The methods depend on a geometric approach, which we may say is related to mathematical geodesy in the same way that the dynamic approach is related to physical geodesy. The geometric method con- sists in performing a triangulation in space and determining the positions of a certain number of observing stations whose positions are un- known. The science of geodesy today covers a wide field, but its main purpose is to determine the size and shape of the earth as a whole, or of large areas on its surface. This problem, at least at first sight, may seem to be a purely geometric one. But as soon as geodesists began using the direction of the vertical as a reference direction, the problem became dy- namic as well. Selecting the vertical as the direction of reference is fully justified for the survey of a small area?say, a few square kilometers?since we can assume that the directions of the vertical are parallel to them- selves over that area. But to relate them over large areas at different points on the earth we need to know the shape of the geoid, since the vertical is defined as being perpendicular to the geoid. Since an approximation for the geoid is an ellipsoid of revolution, the geoid over a large area can be replaced by a reference ellipsoid. We therefore use the vertical as if it were the normal to the ellipsoid, and thus establish the relation between the reference directions. This is the method of "classical" geodesy. The angle between the normal and the ver- tical (deflection of the vertical) can attain several seconds of arc, however, and over very large areas (of the order of continents) the errors accumulate so that the methods of classi- cal geodesy are not adequate. With the methods of physical geodesy (Stokes, 1849;VeningMeinesz, 1928) we try to find the geoid as it actually is. To do this we theoretically need measurements of gravity over the entire earth. Even if these measure- ments do not cover the whole world, we can nevertheless obtain an accuracy to within a few seconds. We can then relate the reference directions over any large area, provided we have measurements of gravity. But even if we have related the reference directions at the different points, another problem remains to be solved: the effect of refraction. The different lines of the geometric figures that we establish on the earth's surface are defined by rays of light. Because of atmos- pheric refraction the light rays are bent, and this bending occurs almost completely on a vertical plane. Although we correct for the effect of refraction, the proper amount of cor- rection is very uncertain, and the errors in vertical angles (for long lines) can attain several seconds. I t is important to note, however, that such errors diminish as the light rays move away from the earth's surface. ? This paper is based on a dissertation, "Geodetic Applications of Observations to the Moon, Artificial Satellites, and Rockets," presented to the Institute of Geodesy, Photogrammetry, and Cartography, of Ohio State University. * Consultant Geodesist, Optical Satellite Tracking Program, Actrophytical Observatory, Smithsonian Institution; and Refearch Associate, Harvard College Observatory. 05 96 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS These difficulties have led to a duality in the methods of geodesy. Determinations of po- sitions are made separately, for the planimetry and for the elevations. The planimetry is based on a more or less geometric procedure, triangulation, while the vertical control is based on a physical method, precise levelling. The methods presented here depend on ob- servations of orbiting objects such as artificial satellites. These methods have the advantage that they use a unique system of reference direction which is not affected by the earth's gravitational field (or the geoid), and that the errors from refraction are reduced to a minimum. The mathematical tools are fairly simple, consisting mainly of analytical geometry in three dimensions. In principle, the method involves our per- forming a three-dimensional space triangula- tion, with the observed objects (satellites) and the observing stations on the earth as the ver- tices. On this space-triangulation net we compute the positions of the stations. Since the observed object will usually be at a high altitude, errors from refraction are reduced to a minimum. An important feature of the methods pre- sented here is that instead of measuring angles to obtain directions, we determine them di- rectly, in relation to a reference system defined by the stars. The relative positions of the stars, corrected for proper motion, aberration, and parallax, will be the same for all observers, everywhere on the earth. Therefore if we con- nect a system of reference to the stars, we can then define the system by the equator and the ecliptic of a certain epoch. For practical reasons, however, we may also use a system of reference that is not fixed with respect to the stars but is moving; i.e., we use a system defined by the instantaneous equator and ecliptic. This motion (precession and nutation) actually will not affect the ac- curacy of the definition of our reference system. The accuracy depends only on the consistency of the positions of the stars, and the accuracy of their proper motions. Finally, this system will be connected to a reference system on the earth. Since the two systems will have a relative motion, we must have continuous astronomic observations from fixed observatories, to connect the two systems. Although the methods presented here were devised specifically for use with the artificial satellites, the same methods can be applied, with some modifications, to such targets as the moon, rockets, and even to terrestrial objects. Because these methods have such general applications, the presentation that follows will commonly use the word "object" rather than "satellite." The coordinate systems The several systems of coordinates and their transformations present various problems. Elliptical coordinate systems.?The system commonly used in geodesy is that of ellipsoidal coordinates. A point Q is defined by projecting it along the normal to a reference ellipsoid. The projection Q' on the ellipsoid is defined by the latitude 4> and longitude X. The distance QQ' defines the height H of the point Q above the ellipsoid. Some question exists whether the curved vertical should be used instead of the normal. For points near the surface of the earth, how- ever, the differences are extremely small. For a height of 10 km, for example, the differences are 4 cm in , 0 in X, 10~4 cm in H. In an ideal system of coordinates, the refer- ence ellipsoid would be centered at the center of the mass of the earth and the 3-axis would coincide with the mean axis of rotation. This is called the terrestrial ellipsoid (Heiskanen and Vening Meinesz, 1958). Since the center of mass of the earth is not known, this ideal system of terrestrial coordinates unfortunately cannot be realized. It is therefore necessary to substitute either the astronomic or the geodetic coordinate system. The astronomic coordinates fa and \A are defined as the angles which the vertical makes with the equator and the meridian of Green- wich. These coordinates are related to the geoid and can be obtained with an accuracy of the order of 0*1 by astronomic observations. They differ by several seconds from the terrestrial coor- dinates on account of the irregularities of the gravitational field. Gravimetrically corrected for the deflection of the vertical, the astronomic GEODETIC USES OF ARTIFICIAL SATELLITES 97 coordinates can come much closer to the ter- restrial. Terrestrial rectangular coordinates.?The geo- detic coordinates 4>o and Xo are defined in the same way as the terrestrial coordinates and X, but they refer to a particular computation ellipsoid which may not be centered at the center of gravity of the earth, and may not be oriented appropriately. These coordinates are computed from geodetic surveys on a reference or computation ellipsoid, which is oriented at the origin of the geodetic system (Bomford, 1952). For points referred to the same geodetic system, the relative positions will be correct within the accuracy of the survey. They will not be correct if the points do not refer to the same system. The elliptical coordinates (0, X, H) are not convenient for points far from the earth's sur- face. For such points, it is much more con- venient to use a system of rectangular coor- dinates. The particular rectangular coordinate sys- tem X, presented here, has its origin at the center of gravity of the earth (or of the ter- restrial ellipsoid) and is oriented so that the 3-axis is directed toward the mean north pole, as defined by the International Latitude Serv- ice, and the 1-3 plane is parallel to the mean meridian of Greenwich (the meridian instru- ment at Greenwich does not lie in the 1-3 plane), as defined by the Bureau International de l'Heure (Stoyko, 1955). This coordinate system is fixed with respect to the earth's surface, and the coordinates of any point on the earth are fixed and do not change with time, if we assume no movements of the crust. A point is defined in this system by a vector X* (X1, X3, X3). This system is related to the ideal system of the geographic coordinates by the following formulas of transformation: Xl= (N+H) cos 4> cos X=p cos /3 cos X Xt= (JV-j-H) cos sin X=p cos 0 sin X Z*=[(l? e*)N+H] sin =p sio 0 (1 where AT=radius of curvature in prime vertical, e=eccentricity of the ellipsoid, p=radius vec- tor, and /3=geocentric latitude. The first set of equations is more convenient, since tables (Perrier and Hasse, 1935; Army Map Service, 1944) may be obtained for the values of N. Sidereal rectangular coordinates.?For points that do not rotate with the earth, it is conven- ient to have a coordinate system that does not rotate with the earth. This is the sidereal sys- tem. We can reach this, however, only by means of an intermediate system, the instan- taneous terrestrial coordinate system Y. The axis of rotation of the earth is not fixed with respect to its surface. The motion of the true pole is studied by the International Lati- tude Service, which gives the coordinates of the instantaneous pole with respect to a mean pole (the same pole used to define the ^-axis). If we know the coordinates of the apparent (instantaneous) pole we can define another sys- tem Y of coordinates which uses this instanta- neous axis as the y-axis. The I^J^-plane contains the point where the mean merid- ian of Greenwich intersects the equator, which is also the J^-axis, since the instantaneous zero meridian is thus defined by the Bureau International de l'Heure (B.I.H.) for the time UTl. (According to the decision of the Inter- national Astronomical Union at the 1955 meet- ing in Dublin, UTO is the observed time; UTl is the observed time corrected for the motion of the pole; and UT2 is UTl corrected for the seasonal variations of the rotation of the earth.) Let x and y (fig. 1) be the angular (spherical) coordinates of the instantaneous pole P with respect to the mean pole P. We have taken X1 as the primary axis, but since x and y are less than 1 second of arc, the result is practically the same. The transformation from the X to the Y sys- tem is given by the expression in which M a n , is the matrix: cosX'Y1 cosX'Y2 cosX'F2 SMITHSONIAN CONTBIBUTIONS TO ASTROPHYSICS X3 VOL.* FIGURE 1.?Terrestrial and sidereal coordinate systems. From the spherical triangles in figure 1 we obtain the formulas given in equation (2) and, consequently, the relation in equation (2a). In the sidereal coordinate system Z, the Z*-*xis will coincide with the F'-axis, and the Z*-axis will be directed toward the apparent vernal equinox T. The transformation from the Y to the Z system will be with rcos t ?sin t (T sin t cos t 0 Kfi 0 1J t being the angle P Z 1 or Z2 'cos t ? sin t (T sin t cos t 0 Y2 Y3 (3) cosX2Y1=sinx siny COBX3Y1=?sinx cosy cosJ?2F3=?cosxsiny cosX3F3=cosx cosy. rcos x -f sin x sin y ?sin x cos y" 0 cos y -f-sin y j3in x ?sin y cos x -f-cos x cos y.. (2) (2a) GEODETIC USES OP ARTIFICIAL SATELLITES 99 From the definition, the angle t is clearly the same as the Greenwich apparent sidereal time as defined by the B.I.H. (Stoyko, 1955). We can go directly from the X- to the Z- coordinates, since (4) or as in equation (4a). The sidereal time t must be considered as an angle in space and not as elapsed time. To find t we must use UTl rather than UT2 since UT2 is corrected for seasonal variation in the speed of rotation of the earth and thus is not a true measure of the angle between F1 and Z1. This correction (periodic) has an amplitude of about 0!03 or 0!5. We have now obtained a system of coordi- nates (Z) that is not affected by the rotation of the earth. This system is not fixed in space, however, since the axis of rotation of the earth, which defines our system, is moving under the influence of precession and nutation. There are two methods that could be applied here: 1. Keep the Z system and reduce all systems to the apparent positions. This reduction can be made easily by using the classical methods of reduction to apparent positions (Besselian Star Numbers or Independent Star Numbers). The necessary tables can be found in the various Ephemerides. 2. Use another system of coordinates, W, that will be defined as being the mean sidereal system at a certain fixed epoch To. This system will be defined as follows: The W1 axis will be directed toward the mean equinox ^ p of To and the W* axis will be directed toward the mean pole P of To. The relation between the Z and the W systems will be obtained in two steps through a system which will be defined as being the W system but at epoch To; i.e., the epoch in which the Z sys- tem is defined. We will call this system Z. This step is necessary since the motion of the polar axis is given in two parts, as precession and as nutation. Let K and ? be the angle between the line of intersection of the mean equator at To and the mean equator at T with the W and Z2 axes re- spectively (fig. 2). Let v be the angle between the planes of the two mean equators (same figure). The transformation from the W to Z system will be given by the formula: or by the rigorous expression of equation (5). The values of K, G> and v are given as (Danjon, 1952): K=[23042T53+139!73(r0-1.900) + 0T06(T0-1.900)2] (T-T0)+[Z0:23- 0!27 (To-1.900)1 ( r - T 0 ) 2 + is'.oo(T-Toy ?=[23042T53+139T73 (To-1.900) + OTO6(77o-1.9OO)2] (T-T0)+[l09'.50+ 0T39(7To-1.900)l (T-T o ) 2 + 18T32(7T-r0)3 v= [20046T85-85T33 (To-1.900) ? 0r37(T-1.900)2] ( r - r o ) + [ -42 !67 - 0T37(r-1.900)] ( r - T 0 ) 2 - 4ir8O(2T-7To)8 where To and T are expressed in units of a thousand tropical years. Z2 rcos t cos x cos t sin x sin y ?sin t cos y ?cos t sin x cos y ?sin t sin y"" sin t cos x sin t sin x sin y +cos t cos y ?sin t sin x cos y +cos t sin y sm x ?cos x sin y cos x cos y 'Z1' Z2 r ?sin K sin ?+cos K COS W COS V ?cos K sin ??sin K COS U COS V ?cos w sin v^ sin K cos w+cos K sin u cos v cos K COS W?sin K sin u cos v ?sin ? sin v ?, cos K sm v 512692?60 2 -sm K sm v COSP X2 W2 (4a) (5) 100 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS FIGURE 2.?Instantaneous and mean sidereal coordinate systems. If we denote by An, Av, Ae, respectively, the nutation in right ascension, declination, and obliquity, we get or with sufficient approximation, omitting terms of the order 10~9, zr z? z? = "i AM Av -An l Ae -Av^ ?Ae z* (5a) The values of An, Av, Ae, can be obtained from an almanac for the epoch T. Combining the two rotations we get and where We have now achieved a system of rectan- gular coordinates (W) centered at the center of gravity of the earth. This system is not fixed in space, but?what is more important?it has no rotation. We have also established the formulas for making the transformations from one system to another. All these systems are geocentric, which means that we should know the position of the center of gravity of the earth. Geodetic rectangular coordinates.?Just as we were obliged to use a system of geodetic coordi- nates as substitutes for the geographic coordi- nates, so we are required to use a system of "geodetic rectangular coordinates" as a sub- stitute for the terrestrial geocentric coordinates. The geodetic rectangular coordinate system X is defined by the geodetic coordinates HA be the coordinates of the origin of a geodetic system obtained by astro- nomic methods and leveling, and let ?, % f be the absolute deflections, so that 17= (X^?X) cos

Ay \A, HA. But to those coordinates corresponds another point QA on the terrestrial (geocentric) ellip- soid. Since the point Q is the same physical point on the earth, we shall have a displacement of the system of coordinates. This displace- ment will be a parallel translation since at the origin of the system we make the theoretical normal at QA correspond with the observed vertical at Qo> These two lines are parallel (fig. 3), while the direction of the X* axis is identical with the direction of the pole (which is observed directly). We assume here that there are no errors in the determination of the azimuth. If we express ? and 17 in linear units by using the radii p+H and N+H, the total translation will be (?*+i?2+fs)*. In the X system, the coordinates of the new origin will be X*c and the transformation is given by matrices (6) and (7). It should be noted that the same formulas could be obtained by differentiating equation (1), and that equations (6) and (7) apply only for small values of ?, 17, f. x% XL = ' s in v* cos Xo sin ?po sin Xo . ?COS ^ o 1?1 o o 9> a ?53 o 9- \ i?i o J sin v o o o - co s 1 \ .a ?3 o NO. 9 GEODETIC USES OP ARTIFICIAL SATELLITES 103 parallel displacement), it is clear that MA will be the matrix of rotation so that (X'-Xti-x^MU^ix'-xl), (10) xl being the coordinates of the initial point of the geodetic system. But then M=I (the unit matrix) so that equation (9) gives the matrix 1 -dA3 dA2 dA3 1 -dA1 ?dA2 dA1 1 (11) In our case if dA is the magnitude of the error in A, then dAi=dA(?cos 0o cos Xo, ?cos ^ cos (14) (15) GEODETIC USES OF ARTIFICIAL SATELLITES 105 (since we have eliminated second order terms the order of multiplication does not affect the result), or as in equation (16). If we include all three errors of orientation, dA, dZ, dy, then equation (10) will be written as (17) or (x'-xo>). (17a) Grouping the terms, if g* is the vector (dA, dl-, dti) we get or X 1 = x1 -{- ^ c +[s in cos Xo(x2?x?)] 1^7. (18c) Let us now consider the effect of an error in scale. Since triangulation schemes are scaled by geodetic base lines, nonconnected triangula- tions may be at different scales. Then, again eliminating second-order terms, we should add ? (x*?xj) to equation (18). With Ax' used for (xi?xj1), the final equations of transformation will become X* = xl -\- Xc + G)g} + eAx', (19) or X1 = x1 -f Xh + (sin poAx2 ? cos ^0 sin sin $9o cos \>Ax2)dr) + eAx3; or also, X< = x* + XJ. + ( - /)Ax' + e ? Ax*. (20) Finally, we have another group of systems of rectangular coordinates, which are parallel to the geocentric but have their origin at a partic- ular station Q. Such a system will be called "topocentric" and will be designated by a prime accent. If we refer to terrestrial coordinates, the transformation is given by the relation Similarly, in geodetic coordinates we have -dA sin o sin Xo?d? cosXo?dq sin ^ sinXo dA cos v3o sin Xo?d? sin Xo-f cfij sin v?o cos Xo |? (16) 1 ?} 106 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS Accuracy of the transformations.?The trans- formation from one coordinate system to an- other requires that we use a number of para- meters that are derived from observations and theory. We shall try to estimate the expected accuracy of the transformed coordinates, using the accuracy with which the parameters are known. In making transformations from the X sys- tem to the Y, from the expression we get the equation dY<=dM\ ,X>- (21) Taking the differentials of the elements of the matrix Mum), we obtain equation (21a). Since the values of x and y are small, we can eliminate the second-order terms so that 0 0 -dx"*\ 0 0 dy jix ?dy 0 > and therefore dY* ro o 0 0 dy -dx'' dy 0 ?, X* (22) I t is not easy to estimate the accuracy with which the coordinates x and y of the pole are given. We accept ?0?02 as a measure of the accuracy (Melchior, 1954, p. 36). With this value for dx and dy we have, for points on the surface of the earth, errors (expected) of about 0.40 meters for the transformation to the Y system. When we perform the transformation from the Y (or X) to the Z system, we have oJ'. (23) We evaluate: r ?sin tdt ?cos t cos tdt ?sin tdt 0 0 and therefore CdZl~\ f?sin tdt ?cos tdt 0 dZ2 cos tdt ?sin tdt 0 0 0 0. F 2 LF3J (24) This formula is correct if Y* has no errors. But from equation (21) we see that Y* has the error dY* so that equation (23) must be written or, if we replace dY* from equation (21), as Since F'~X<, Or we have the matrix shown in (24a). "?sin xdx sin y cos xdx-\- cosy sin xdx sin x cos ydy?cosy cos xdx 0 ? sin ydy cos ydy ? (21a) .cos xdx ? cos x cos ydy-\- sin x sin ydx ? cos x sin ydy?sin x cos ydxj Therefore cos sin , 0 t t ?sin cos 0 t t or 0 i 0 0 Jx 0 ? dx* 0 dy ?dy 0j = 0 jlx 0 ?cos 0 ?sin ?dy tdx? sin tdx +cos 0 r ?sin nd a'fi'i Z^cos 8' cos a' and Z2=cos 8' sin a' Z?=sin 8' sin 5'=Z8 tan a'=ji* (29) (30) Using equation (3) we can find the direction cosines m* in the Y system by making Zl a unit vector. We then obtain m1=cos 8' cos (a'?t), TO2=COS 8' sin (a'?t), TO8=sin 8'. When we introduce 8=a?t= ? (t?a)=GHA, we obtain TO1=COS 8' cos 8', m*=cos 8' sin 8*, (31) TO8=sin 8', and , _ m 2 TO1 sin 8'=TO8. (32) Similarly, we get the direction cosines n* in the X system by using equation (2). We obtain n 1=+sin x sin S'+cos x cos 8' cos 8', n*? ?cos x sin y sin 5'+sin x sin y cos 5' cos 6'+ cos y cos 5' sin 6', (33) n8= -f-sin tf sin y cos 8'?sin x cos y cos 8' cos 0'+ cos x cos y sin 8'', and n 2 cos y+n3 sin y ^ n1 cos z-f sin x n2 sin y?sin x ns cos y sin 8'=nl sin x?cos x n2 sin y+cos x n8 cos y. (34) To obtain directions in the X system, we get the direction cosines q* in the geodetic system X by using equation (17) in the form (35) or q1=nl + (?n2 sin ?* -f n3 cos , we will have V \ AS' f\ V Aa* X t COS 5 ' FIGURE 7.?Orientation of the coordinate systems x and along the orbit. makes an angle

being the primary axes of the error ellipse, we have (41) and since \Aa' cos i'J \cos

] tp k* cos2 ^ >-f sin*

+p sin2 ip ( 1 ? p j sin

^ cos2 i ^ cos ? sin i r^ - j sin >^ cos i sin i o-| - j cos

wt?w cos i sin wr=w sin t. Then the corrections e to a', 8', r will be v>a 1 In the case of photographic determination of the directions, it will be easy to find the values of wa and Wt bv measuring on the plate the change a and 8' of a' and h' with time. We have then ccos5 (52) c cos 8 . , = * : . (51) (See footnote.1) 'Some explanation most be given for this formula. It may seem ridiculous to apply a correction for the velocity of light to the measure- ment of distances. Actually, no measurement can be made instan- taneously, and e should be the. . . velocity of the used yardstick! Since the distances will be measured with electromagnetic waves we must use the velocity of light. Notice, also, that if for the time of the observation we take the mean time between the emission and the reception of the signal, the correction will be zero. GEODETIC USES OF ARTIFICIAL SATELLITES 117 When the directions are obtained by the use of stars, as described on page 111, we will get the apparent direction of the object if the posi- tions of the stars are also apparent. We then apply the corrections given by equation (51) to find the true position (or true direction) of the object. It will be more convenient, how- ever, and will avoid many computations if we use the true rather than the apparent positions of the stars. In this case (fig. 9) we must add (or subtract the negative of) the effect of the aberration of the stars, i.e., the effect of the motion of the observer with re- spect to the sun (annual and diurnal aberration). This means that we must use V*3 instead of to*. Thus we compute the position of the object by using the true positions of the stars; we then apply the correction for the annual aberration for the position of the object as well as the diurnal aberration, and finally apply the correction for the aberration of the object according to equation (51). Since the diurnal aberration is less than 0?3, it will in most cases be negligible. The annual aberration can be computed by Bessel's method: Cc+Dd in R.A. Cc'-\-Ddr in declination. We could further simplify the computations if instead of using the true position of the stars we use the mean position for a given epoch, but the positions of the stars should be corrected for their proper motions. The position of the object, thus determined, and corrected for the annual aberration as well as for the aberration of its own velocity, will be the mean position for the given epoch, and thus will refer to the W system of coordi- nates. We can then compute the true position of the object by using known methods of reduc- tion, such as Bessel's method, if we want our directions to refer to the Z system rather than the W. Correction for optical refraction.?Although the observed objects will be at rather high altitudes, the optical rays will have to pass through the atmosphere and thus undergo a curvature. The existing formulas for the cor- rections for astronomic and geodetic refraction cannot be applied in our case, because they assume that the observed object is either at infinity or at a rather small distance and low altitude. We will therefore try to find a formula that can be applied for the intermediate case. We will assume that the density of the atmosphere p as a function of altitude H is given by the relation, log P=Kt H + K2. (53) The observed values of p at different alti- tudes H (Whipple, 1954) fit accurately enough to those of equation (53) for heights up to 80 to 100 km. The results from the artificial satellites for the density at heights of 200 to 600 km show an important deviation from the values of equation (53) (Whitney, 1959), but this does not affect our solution (see p. 118). The coefficients Kx and Ka will be determined experimentally. Or we can write P=e 1 "~\ I -K-2 where a=-r4> O=-T-7M M (M=log10?). But if n is the index of refraction, (n? l)=Bp according to Gladstone's law, and so or (n-l)=kettB, (54) where it=.Be6=7io? 1, if iu> is n for H=0. We assume that the index of refraction as a function of the altitude H is given by equation (54), k and a being coefficients whose values will be determined experimentally. Let Q (fig. 10) be the station and S the observed object. The ray of light will follow the curved line SMQ, and the observed zenith distance will be Zo, the angle between the vertical and the tangent to the curve at Q. Also let 6 be the angle between the tangent at any point M with the tangent at Q. Assuming no curvature of the earth (this assumption will hold for Z<45?), we have the relation =?tanZ?? n (55) 118 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TOL.8 Vertical FIGURE 10.?Correction for refraction, when curvature of the earth is not considered. which, since Z will be constant, will give by integration = tan ZoQn nQ?In n). (56) Since n is very near 1, we can expand the equation so that e= t anZ 0 [ (w 0 - l ) - (w- l ) ] . (57) If we introduce a system of coordinates $ and il so that | is the tangent at Q we have ^ = 6 = tan Z0[(n*-1) - (n-1)] (58) or with the help of equation (54) But since the angle e is small, we can replace ? with r (r is the distance QS); thus H=r cos Z. Then equation (58) is written ^ = & tan Zo?k tan Zo^C08 z?, (59) or , = C\k tan Z0- Integrating, we obtain Thus we obtain an equation that expresses the line followed by the ray of light: ?-*??? ^ H- If R is the correction, we have tan i ? = | since R is small, and if we replace $ with r we get an equation which gives the correction for the refraction R as a function of the zenith distance Zo and distance r: This is a general formula and can be applied for any r provided Z<45?. We can find that for r= oo (we have a cos ktan3 Zp p2 cos3 Zo f p cos 3 ^ap cos Zo ( J p cos3 Zo J Ha"dH- 2 2 iap cos2 Zo fg g "^g+ A r t a n 2 Z ? fdf. (68a)J ap cos2 Zo J GEODETIC USES OF ARTIFICIAL SATELLITES 121 Integrating and eliminating second order ( H*\terms containing ?j- j , we obtain , y /k tan Zo , k tan Zo p cos Zo k tan Zo _kJ^nZo\ a a?p cos3 Z o / ' ap cos Zo The constant of integration is found from the condition that at the origin 17=0. We get finally equation (69) and thus, since R=% we obtain equation (70). Introducing the values of the constants, we get equation (71) which gives the correction for the refraction for values of Z up to perhaps 80?. The last term usually can be neglected. (All formulas given in this section are based on normal conditions for the value of k. If the temperature and pressure at the station are different, k should be changed. A nomogram (fig. 23) is given on page 159 for this correc- tion. The height of the station Ho will not change the results except for extremely high altitudes. In such a case the integration must begin from Ho.) It can be seen that for Z0100 km, as we have shown, the path of the ray will, practically, already have taken its shape and will follow the asymptote. The equation of the asymptote will be 17 = (k tan Zo+? tan Zo sec2 Z^) ( +\ ap / - tan Zo sec Zo |~1 +? (2 sec2 Zo+tan2 Zo)T (72) a L ?P J Thus the correction will be: i? =it tan Zo + ? tan Zo sec* Zo+ ap ? tan Zo sec Zofl +? (2 sec2 Zo + tan\Zo)T (73) or [_ ap ? J But as with equation (61), we leave the formula for the correction as given in equation (71) since the additional terms will bring prac- tically no change. Thus we can check the fact that for points at infinity we get or fi=60!25 tanZo?0T0682 tanZ0sec*Zo #=60r i8 tanZ0-0!0682 tansZ0, which is exactly the general formula of astro- nomic refraction. v=(ktfinZo)S+[? tanZosec2Zolf + -tanZoBecZori + ? (2 sec* Zo + tan* Z0)"l ( 1 -\_ap j a [_ ap j ? tan Zo sec Z0(sec* Zo+ tan1 ZJHf*. (69) R=k tan Z o +? tan Zo sec* Zo + ? tan Zo sec Zo [~1 +? (2 sec1 Zo + tan* Zo)l (1 ?ap at [_ ap J apr tan Zo sec Z0(sec2 Zo + tan2Z0)HfB. = 60T25 tanZo?0T0682 tan Zo sec* Zo?43570 sec Zo[l?0.00113(2 sec2 Zo+ tan* Zo)] (1 -e-0-W!Mwr)-0r0682 tan Zo sec Zo(sec2 Zo + tan* Zo) *j-e-?'13%6a. (70) (71) 122 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS When the directions are obtained photo- graphically with the stars as reference (p. I l l ) , the correction will be that given in equation (74). The last term generally is negligible. Also, since the term 0.00113 (tan2Z0+2 sec2Z0) is about 0.04 for Z0=75?, we could say that in most cases equation (65) will be sufficient. Correction for electronic refraction.?The re- fraction of electromagnetic waves in the earth's atmosphere is a very complicated problem. It depends on the wave length and also on the dielectric condition of the air. Different ionized layers exist at altitudes that vary with place and time, where reflections and refractions occur. This greatly complicates the problem. How- ever, the shorter the wave length the less is the influence of those layers. The directions thus obtained with radio methods, as described on page 112, must be corrected. The distances obtained with radio methods need two corrections, the first for the curvature of the path followed by the electromagnetic wave, and the second for the change of the value of the velocity along the path. These corrections have been studied (Jacob- sen, 1951; Williams, 1951) for low altitudes for applications in electronic geodesy (Shoran, etc.), but it is very doubtful whether they can be applied also in the case of high altitudes. With the use of radio methods, however, the distances will probably be more accurately measured than the directions. Observations to objects of known positions Several methods exist for making observations to objects of known positions. Use of angles in space.?Let us consider first the case in which we have observed angles to three known positions. Let Su S2, St (fig. 13) be the known positions of the object, and Q the unknown station where the angles /3<12>, /3UM, pm) have been observed. FIGURE 13.?Determining the position of an object by use of three angles in space. In general there is a solution giving the position of Q, provided Si, Sa, S3 are not colinear. The problem is that of resection in space, and Q will lie on the intersection of three tores. The direct solution gives a system of high order and is not applicable. This problem is of great importance in photogrammetry. Of the many solutions suggested, the best, which has also the advantage of being easily general- ized in a case of more than three known points S, seems to be one in which we compu te corrections dXl to an approximate position X* of the point Q. This principle is very much used ia geodesy. Briefly, the solution is obtained as follows: Let (jj_be the approximate position of Q. Using Q and Si, S2, St, we compute the angles is"2*, ^23), j ? 3 " . __ If Q is near Q and the corrections to Q are dX* we obtain the equation: 5 C ^ 1 2 > dX*=coa /3(12>-cos J3<12) (75) and two more equations with /3(23) and /3(S1). [l-0.00113(tan2 Z 0 + 2 sec2 Z0)l (1 - ? -? ???" ) - Z o + sec2 (74) W0.? GEODETIC USES OF ARTIFICIAL SATELLITES 123 The coefficients of dX* ere of the form d cos /3(>w , r 1 cos cosa *Us,?mrj where a* is the direction angle of the line SQ. The solution of this system of three equations with three unknowns gives the correction dX* to the approximate values ~X*. The method can be generalized for the case in which we have observed more than three angles /3(iW. The solution will be obtained by the method of least squares. We must notice, however, that if equation (75) is applicable in the case of three known points, it is not applic- able if we must make an adjustment, since cos put) is not the measured quantity. In this case we must use equations with /3L<2S, QSk J The accuracy of this method of determining the position of the station depends on the accuracy of the observations and on the net configuration (the positions of the object are assumed to be errorless). An analysis to estimate the accuracy could be made, but since this method will not be of much use we shall not attempt it. The reader may refer, however, to a similar study for photo- grammetric applications (Doyle, 1957). We may note, also, that if the different positions of the object lie near a straight line (as happens with an artificial satellite) the solution is very weak. Use of directions and distances.?Let ZQ (fig. 14) be the coordinates of the station Q, Z*s the coordinates of the object S, r* the vector QS, and I* the direction cosines (unit vector) in the Z system of the vector r1 cor- responding to a! and 8'. z' FIGURE 14.?Relation between the positions of the station and the object. The equation relating the positions of Q and S is or S=ZJQ-\-I r. K*O If the coordinates are referred to the terres- trial system, we will have A.8=j\.Q-x~n r. \io) Again, since we almost always know the ap- proximate positions, we will develop formulas relating the corrections to the approximate co- ordinates Z* or X* and use the method of varia- tion of coordinates. Differentiating equation (77) we get where I* is a function of a' and 6'. Differenti- ating equation (29) we get dll=?(cos a' sin i')dir?(sin a' cos 6')daf, dl*= ? (sin a' sin 6')d8' + (cos a' cos h')da!, (80) dl* = (cos 8')d8'. Then equation (79) can be written in the form shown in equations (81). 512692?60 8 124 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 8 (dZ\?dZ1Q) = ? (cos a' sin V)r d8f? (sin a')r cos V da'-\- (cos a' cos V)dr, (dZ2a-dZl) = - (sin a' sin 5')r d8' + (81) (cos a')7 cos S' 6^'+ (sin a' cos 8~')dr, (dZ%-dZl) = + (coB I')r d8'+ 0+ (sin 8')dr. Inverting the system (81) (the matrix is orthonormal) we get or ( I ) (<*Z| (82) dr=(cos a' cos 3') (dZ^ (sin a' cos 5') (dZI- (Bint')(dZl-dZZ). We will write this system (82a) In the Y system, assuming that t is known, we replace a' with 0'=a'?t, and Z* with Y* and find that (83) (sin ?' cos ? ) (dY%-dY%) + LdrJ (83a) Equations (83) or (83a) can also be used for the terrestrial (X) and geodetic (x) systems, since the omitted terms will be of the order of dXXlO'* (for dX=1000 meters we will omit terms of 1 millimeter). Hence we get da' with (84) cos 0' sin 5' sin 0' sin 8' cos 8' r sin0' 7 cos 5' r COS0' 7 cos 3' cos 0' cos 5' sin 0' cos J' and d8'=8'?5', da'=a'?a', dr=r?r, which are the observed values minus those com- puted from the approximate coordinates. The computed values of 5', a', (or ?'), 7 can be obtained from the rectangular coordinates of the object, with the help of equations (30), (32), (34), or (35), depending on the system in which the coordinates are given. If the position of the object is given in spherical coordinates a, S, B, in the Z system, we can find the ap- parent a', 1', (or 0'), 7 as follows (fig. 15): (tana'?a)=tan (fi'?0)= t 7/ cos (9'-tan o = FA sin 0?F^cosfl = -= ' (85) cos (0?0) cos 5 where A=B cos 8?Tl cos 0??% sin 0. NO. 9 GEODETIC USES OF ARTIFICIAL SATELLITES 125 If from one station we have observed a', b' and r of an object of known position, we can find the position of the station with the help of equation (84). If we have observed from the same station many positions j of the object and we have all a'*, bf), r*, we adjust by the method of least squares for the coordinates of the station by using the appropriate weights and correlations, if any. Y1 FIGURE IS.?Topocentric and geocentric directions. I t may happen that we do not observe both directions and distances. If we observe only directions, i.e., a'* and 8'*, for every observation we will have two equations, the first two of equation (84). If we have measured only the distances r*, we will have for every observation one equation, the last of equation (84). We can also get a solution if we have observed only declinations but we do not get a complete solution if we have observed only R.A., since a is independent of Xs. The same equation (84) can be applied both to problems of resection (i.e., from known positions of the object compute the position of the station) and to problems of intersection (i.e., from known positions of stations compute the position of the object). In the latter case, since the object will be moving, we must make the observations simultaneously from all stations. If we do not want the rectangular coordinates of the station (or the object) but instead want the ecliptic coordinates ?Af"/- or Introducing this expression to equation (92a), we obtain V{XS\-X?P-I (98) where P* is the harmonic mean of (r*)*. Equa- tion (98) is of the same type as equation (97) and thus the same conclusions can be obtained. We can therefore state the following general rules concerning the accuracy of our deter- minations of position: a) When we have observed only distances (having the same weight), we obtain the best result if the positions of the observed object can be grouped by threes so that the lines connecting them with the station are mutually perpendicular. b) When we have observed 8' and a' (and 8' and a' cos 8' have the same weight), we obtain the best result if the positions of the observed object can be grouped by twos, so that the lines connecting them with the station are perpendicular (the distances being assumed to be almost equal). c) When we have observed all three elements (i.e., r, 8', a'), the net configuration will not affect the accuracy, provided we have the same weights. d) If we have to choose between measuring distances and directions, we must compare the weights; that is, compare with P2* where a refers to standard errors. If Pi>Pa it will be more accurate to measure direction, and if PidX=l, (104) where A is a matrix consisting of submatrices G# and Qju, dX a vector consisting of the vec- tors dX^m and dX*Si (corrections to the approxi- mate coordinates XQ% and Xst) and I a vector consisting of the vectors (db', da', dr) obtained from the observations at both known and un- known stations. Needless to say, these vec- - G x 0 . . . ^ 0 - G 2 . . . r-e 0 l o . . q - e ; . . . = Gx? 0 ) i 0 . . . GJGJ . . . or (102a) and -er - e , =?/. 130 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS YOU* tore need not have all three components; they will depend on how many and what elements we have observed. The matrix A will be of the general form shown in equation (104a). We notice that in each of the three partitions (the partition with all zeros is excluded), at every row, there is only one submatrix 9, and in the two upper partitions, for every element ?Qi% in the left partition, there is a correspond- ing element QJn on the right, in the same row. An example of a matrix A appears in equation (104b). The solution by the method of least squares for dX will be given by the system of normal equations N-dX=A'Pl, (105) where N=A''P-A, and P = the weight matrix (if the observations are of different weight). From the general form of the matrix A we can see that the matrix N of normal equations will be of the form shown on page 131. Since the subscripts in parentheses are constant numbers, it is understood that there is no summation with respect to those indices. The matrices P are the corresponding weight matrices. The matrix N (equation 106) is symmetric, as would be expected. To every row and every column there corresponds one point Qu and St in sequence. Then every element of the matrix is formed by summing all the terms of the form 0'P9 where 6 is the matrix that relates the two points that correspond to the column and row of that element. Since no observations have been made between the stations or between the objects, the corresponding elements will be zero. It may happen, however, that we also have observations between the unknown stations. In this case the elements of the matrices that correspond to those stations will not be zero. These observations in practice would be mainly distance measurements by Shoran or any other electronic method, since no intervisibility will exist for observations of directions, because of the long distance between the stations. Such distance measurements between the stations by any electronic method will be very desirable because they will strengthen our net. Submatrix with elements ?GJU Submatrix with elements 0 . . . Submatrix with elements Observations from unknown stations Observations from known stations (104a) Observations from unknown stations (104b) Observations from known stations GEODETIC USES OF ARTIFICIAL SATELLITES 131 If we observed all three elements from every station, and they have the same weight, the matrix N will be much simplified. The expression [Qj?PQju] will be (?/?*) I, where v is the total number of observed positions and f* is the harmonic mean of the squares of the corresponding distances, the standard error being unity. In this case the matrix JV will be as shown in equation (107). 0 '(/1)(?1)P 0 (/I) (? N= ' (it) (?ai) (106) ? [ G 0 . O ??? ? [-40 [-40 - [-* (107) 132 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. ? The variance V{dX} of the unknowns will be given by the formula: N~l=(A''P'A)~1. In practice, the weights are given in terms of an appropriate weight unit, in order to have simple numbers. Therefore we have (108) o* being the variance of the weight unit. In all our derivations and formulas we have used the terrestrial coordinate system X. The same formulas hold for any other system pro- vided we change the matrices 6 as indicated on page 124 ff. In case we have geodetic rectangular co- ordinates x*, equations (104) to (108) are exactly the same, since the matrices 0 will be the same. The only restriction is that the coordinates x* of the known stations must be referred to the same system and of course the positions of the unknown stations and of the object will refer to the same system. Also it must be mentioned again that the values of a' and 5' will be obtained from equa- tions (32), (34), or (35), depending on the system of coordinates used; e.g., if we use the X system we will first find the approximate direction cosines n* with the approximate coordinates 3 * and then with the help of equa- tion (34), we find 6 and 5. If we use the X system we have to go through the direction cosines q1 of equations (35) and (36), provided we know the values of dA, d?, dti. Since in most cases those values will be unknown, the directions in the X system will be the same as in the X. We could simplify the computations if all observations were made within a short interval (some days), during which the instantaneous pole will be practically fixed. In this case we could make all the computations in the Y system and afterward make the transformation of the adjusted coordinates. Introduction of conditions.?There may be certain conditions that must be fulfilled between the unknown stations?for example, the distance between the unknown stations must be kept fixed. In making our adjustment we must then impose a number of condition equations be- tween the values dX*. Let the equation CdX=k (109) be the condition equations in matrix notation, while A-dX=l (HO) is, as previously, the observation equation. The solution by least squares will be given by the system dX\ (A'Pl\ where N=Af'P'At the same as in equation (105) and X are the auxiliary unknowns (correlates) equal in number to the conditions. From the solution of equation (111) we get the unknowns dX, while their variance is given by the formula, (112) cr being the standard error of the weight unit. In most cases, the conditions will relate the coordinates of the unknown stations Qu. It is more probable that there are fixed lengths or directions between the unknown stations. There may also, however, be conditions between the coordinates of the object (e.g., we may know the distance between two positions of an object if we know its velocity and the elapsed time). The conditions will be mainly of two kinds: imposed length and imposed direction. Let us first consider the conditions of imposed length. Let Lo be the known and fixed distance between any two points Qt and Q2. Let Z be the computed distance between the approximate positions Q and <&. We have (113) where AX^Xi-Xl Differentiating, we get L dL=&X1(dX\-dX\)+&X2(dXl-dX*l) AX*(dXl-dX\) GEODETIC USES OP ARTIFICIAL SATELLITES 133 or A y*(dXi-dX\)-\-~- (dX\-dX\) + A V 3 or, further, by using n*, the direction cosines of the line QiQ2, we obtain dL=n1(dX12-dX\)+n*(dXl-dX{)+ n*(dXl-dX\). If dL?L0?Z, dX* is the correction to the approximate coordinates Xf, and so the condi- tion equation is written n xdX\-nldX\ +n*dX\-n*dX\+nldX\- n*dX\=Lo-L. (114) For every imposed fixed length, we have one condition equation of the form (114) which could also be written as ntdXl-n'dX^Lo-L. (114a) If we take the approximate positions 7$ so that Lo?Z we have By comparing equation (114) with the last of equation (84), we see that they are the same. We can write the conditions also in the form shown in equation (116). We must be sure that we use the direct distance between the two points (also called chord distance), and not the distance along the terrestrial ellipsoid Qength of the geodesic). Next let us consider conditions imposed on the directions. A line connecting two unknown stations may have a known direction that has to be kept fixed in the adjustment. One direc- tion in space has two freedoms, and so for every imposed direction we will have two condition equations. Instead of using the direction cosines of the imposed direction, we will find it more con- venient to use their equivalent elements, i.e., the angles 0' and 8'. Let d'o and 8'0 be the imposed values and 0' and 5' the computed values from the approximate coordinates. I t will not be necessary to derive the condition equations, since we know beforehand that they will be the same as equations (84a) and (84b). We can therefore write the condition equations by using L for the distance between the two points Qi and Q2, as shown in equations (117) and (118). (117) cos 8 If the imposed directions of 0'o and 8'0 have been obtained with the help of astronomic ob- servations, as they will be in general, they refer to the Y system. If we are using a different system, the computed 6' and j ' must be obtained by using the T* coordinates of the approximate points. Regarding the condition equation, we must make the following remark: if we have measured the distance, e.g., by Shoran, between two of the unknown stations, we must not use this measurement as a condition equation, but rather as an observation equation. We can correctly use it as a condition equation only if we want the distance to be kept fixed after the adjust- ment or, perhaps, if the accuracy of the measure- ment between the stations is much higher than that of the measurement to the object. The connection of geodetic systems.?One important problem that often arises in geodesy is that of connecting two geodetic systems; that is, to find the relative positions of the two com- putation ellipsoids or, equivalently, to find the relative positions of the two X systems of coordi- nates. The connection can be made by com- (cos 0'cos 8')(dX\-dXX) + (sin 0'cos 8')(dX*a-dX\) + (sin 8')(dX\-dXf) ==U~ (116) 134 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TOL.? paring the coordinates of common points of the two systems. If we have observed the positions of an object from stations belonging to both systems, we will be able to make the connection. For that we use the points belonging to the one system as known points, and we find the coordinates of the points belonging to the other system (as if they were unknown), but we introduce the necessary condition equations so that the net will not change as a whole. We then compare the coordinates computed in this way with the coordinates of the same points as given in the second system. Instead of comparing the coordinates of points of the one system, as computed from the other system, with those originally given, we could compare the coordinates of the object as computed from the two systems. This will be much more simple, since we will have to com- pute positions of an object from known stations and will have only groups of normal equations with three unknowns. The relation between the two systems will be given by one translation (3 unknowns), one rotation (3 unknowns), and perhaps one scale factor (1 unknown). If the two geodetic systems have no orienta- tion errors dA, d? and drj, they will be parallel, since both will be parallel to the terrestrial sys- tem (see p. 109). Connecting the two systems will then require only translation, and no rotation. Further, if the two surveys are scaled correctly (or are scaled far better than we can detect), we have only three unknowns. Then only one position of an object, observed and computed from both systems, will give us the solution. More positions will help in eliminating the acci- dental errors since we can make an adjustment. From equation (8) we have the expression x[=Xi?Xci for the coordinates of one point in the first system, and x\==X*?Xic% for the coordinates of the same point in the second sys- tem; therefore i?*J?-Ac,??A-ov?-A-c (119) Thus the relative position X^. , (which is the same as the relative position of the center of the computation ellipsoid of the second system with respect to the first), is simply the difference of coordinates of the same point computed in the two systems. If we have more points we will take the mean. If we want to express this translation in terms of the deflections ?, tj, f, we will use equation (7). The ? and 17 so obtained will be expressed in length units, but we can express them in angular units by using the radii of curvature. We must remember, however, that those deflec- tions are not absolute but relative, i.e., with respect to the computation ellipsoid of the first datum. If in equation (7) are the coordinates of the origin of the second datum, the deflections will correspond to that point. If we want the deflections at any other point (^, X), we have to replace (127) n 2 a 3 =k*en. (128) Here k\ is the geocentric gravitational constant with the value (Herrick, Baker, and Hilton, 1958): k\= 1.4350087 Mm3/min2, (129) where me=mass of earth, m,=mass of orbiting object, and Am=an augmentation of the masses for the effect of the perturbations. For an object of negligible mass orbiting around the earth with an orbit of small eccen- tricity, we have with sufficient approximation GEODETIC USES OF ARTIFICIAL SATELLITES 137 the relation (see also equation (129b), p. 144): - | s i n 8 (129a) where a e = t h e earth's equatorial radius, J= the coefficient of second harmonic of the earth's ellipsoid with the value J=1.624X10-3 (Jac- chia, 1958a). When these six elements are given, the orbit is defined and we can find the position of the object at any time T. The positions can be computed in either rectangular or polar coordi- nates. We shall use the polar coordinates a, 6, and R as more convenient. (Note that the values a and 5 are not the apparent but the mean coordinates, referring to the equinox and the obliquity of the epoch for which the elements of the orbit are given.) They are expressed as follows: a=G+arc tan [cos i tan (?+?)], (130) sin 5=sin i sin (&?+?), (131) R=a(l ? ecosE). (132) Here the auxiliary elements o=true anomaly, E= eccentric anomaly, Af=mean anomaly, are given by the formulae n f (i33) M=E ? e sin E, (134) M=M0+n(T-T0), (135) and a is given by equation (128) .7 The classical methods of Laplace and Gauss may be used for determining the elements of the orbit from three observations (Moulton, 1958), although the orbits of the earth's nearby satellites can be determined also with other more simple and less accurate methods. In all those methods, however, we must assume that we know the positions of the stations with respect to the center of gravity of the earth, since the equations of the orbit are given with respect to that point. ? The element M% can be replaced by Tt, which is both the epoch and the time of a certain perigee crossing. The expressions for this latter can are given in Veis (19S8). By differentiating equations (90) and (92) to (97), and using l=w+v, we obtain the following: ba_ tan I sin i di ~~ 1+tan2 I cos2 i ba cos i = bu cos2 /(1+tan21 cos2 i)ba bR cos i cos2 Z(l+tan8 Zoos2*) = 1? e cos E bR w ?? = ? a cos E de bR oh, a e sin dM bM bs bi~ bs ba bs bv dp bE~ bv be" bM bE be" ba bn bR bn cos i sin I cos 6 sin i cos I cos 5 sin i cos I cos 5 sine "sin 2? sin v 1?ecosE sin 2? 1? ecosE 2a s "3^ 2o(l?ecosE) 3 n 138 But SMITHSONIAN CONTBIBTniONS TO ASTROPHYSICS \bvbEbMbn /dado , da do de da do /babv bE bM\, n /da do bE dM V ^ ^ M d M bRbEb TOL.I da\ . (137) bR,bRbE ( 1 3 8 ) When we introduce the values of the partial derivatives, we obtain equations (139), (140), and (141). We now write equations (139) to (141) in a matrix form: da =u* de dQ di dco (142) Or, if we introduce the vector dbn=(dn, de, dQ, di, do, dM0), we obtain 'di da .dRj =U*db\ (143) Equation (143) relates the differentials of the position of the object with the differentials of the parameters of the orbit bm. If we introduce equation (143) into equation (84b), we obtain dec jir ^ =Qrt-Si-U*'db?-ert-dXiQ. (144) This equation will help us to find the orbit from observations from known stations, with fsin i cos I sin o (T? To) Lcos a sin E(l?e cos E) s t sin L I , . , I sin t cos i*icosl*nv / 1 + 1 cos 8 \1?e2 1?ec sin t cos Z sin o cos a sin E(l?e cosE) (139) da= cost sin V(T?TQ) Lcos2 Z(l+tan21 cos2 i) sin^U?e cos c o s i s i n o / 1 . 1 Lcos2 Z(l+tan2 Z cos21) V j ? ^ l ? cost "J cos2Z(l+tan2Zcos2t)J Hr r ae , , TO , F tan Z sin i "1 , . , d e + +L""l+tan2 Z cos2 i\ + cosisinv cos2 Z(l + tan2 Z cos21) 2a(l?ecoaE) , oesinff o?sin.E Z(l+tan21 cos21) sin E{\-e cos ? )I" ?acosL ag sin2 g 1?ecos? \J (141) GEODETIC USES OF ARTIFICIAL SATELLITES 139 the method of variation of coordinates. In this case dX?=0 and thus /dS>\ Ida' \=e-S'U'db't. (145) \dr / From an approximate orbit ?" we find the elements of the matrices 0, S, U, as well as the computed quantities J / 3 / r. If we have measured 5', a', r for two positions of the object8, we can solve for dbu. If we have observed only the directions, we need three positions. If we have more observations, we will adjust by the method of least squares. We can also introduce as unknowns the cor- rection dX? to the coordinates of the stations and solve simultaneously for both dbu and dX^, using equation (144). If the stations from which the observations have been made belong to the same geodetic system and there is no orientation error, the value dX& will be the same for all the stations; therefore we can introduce only one dX* which will be identical with X?. In this case the observation equations will be The variance of bm (or dbu), Xlc, and g' will be \dr/ Q-S-U-db'-Q-Xh, (146) with a total of 6+3=9 unknowns. If we introduce also the orientation errors gs=(dA, d?, drf), the observation equations will be:9 1 . (147) If we have more than the minimum required number of observations, we will adjust by the method of least squares (as described on p. 122 ff. and p. 129 ff). Whenever this method of adjustment is used, it may be necessary to iterate the solution, since the coefficients of the observation equations depend on the approxi- mate orbit. [(e -s-u, - e , -e-G)'P(es-u, - e , -e-6)]-1, where (0 'S-U, ? 0, ? 0 -0) is a matrix con mating of the submatrices Q-S-U, ?0 and ?Q*Q. The variance V{dbu, X?, g}) can be found only after the observations have been made, since it depends on the configuration of the net. We can, however, notice that the ele- ments of the orbit will be more accurately deter- mined if the positions of the object are not concentrated in only one part of the orbit. For the accuracy of the positions of the stations (or of Xc and g$, which is the same), see page 126 ff. Use of a second-approximation orbit.?To the elliptic orbit (considered on p. 136 ff.) we will introduce the secular perturbations caused by the oblateness of the earth (second har- monic) and by the air drag. The oblateness of the earth introduces a rotation of the line of nodes on the equatorial plane (regression of SI) and a rotation of the line of apsides on the plane of the orbit. The two motions will be uniform. They depend both on the orbit (n, e, i) and on the flattening of the earth (or the constant , the instantaneous values corresponding to time T, and for equation r. One method that could be applied to de- termine the orbit (second approximation), similar to the one on page 138 ff., is the follow- ing: With the new definition for the orbit, equation (143) will be written as in (143a), where db^, db', dSw . . ? are the corrections to the approximate values 6*, 6f, $" (db0, db, db . . . may not be of the same di- mensions; actually M=n so that b does not contain M). Then equation (144) will be written as in (144a). Assuming the stations known (i.e., dX^?0), we solve for db^, db' . . . as on page 138. We have applied this method for orbit deter- minations at the Smithsonian Astrophysics! Observatory (Veis, unpublished) and obtained very satisfactory results. A second method would be to find the mean elements from observations within a short interval of time by using, if necessary, approxi- mate values for the variations of the elements to make the reductions. Provided the time interval is short, errors in the values of b, J . . . will not have much effect. Given the mean elements for different epochs Tit T2, . . ., we can find the value of b as a function of time. We now consider the simultaneous deter- mination of the orbit and the positions of the stations. If we try to solve equation (144a) for both the orbit and the positions of the stations vn will have a large number of un- 'd8 da ==f/7* T7*(T?T~} T7*(T?T~\* db' (143a) da' 'dbJTl db' (144a) GEODETIC USES OF ARTIFICIAL SATELLITES 141 knowns, and thus the solution will be weak. Therefore we solve separately for the orbit at epoch and the positions of the stations, and separately for the variation of the elements of the orbit. The larger the part of the orbit we use and the longer the interval of our observations, the more accurate will be the elements of the orbit, as well as their variations. But on the other hand, the smaller the part of the orbit we use and the shorter the period during which the observations are made, the more likely it is that the computed positions of the orbiting object will be correct. For this reason we will divide the observations into two groups. In the first group, the observations will be made over a long period of time to determine the variation of the elements b, $, . . . If the observations are made from enough stations (which need not belong to those we use for geodetic purposes; on the contrary, they preferably should be spread all over the world), and for a rather long period of time (many revolutions;, they can provide us with accurate enough values for b, f>, . . ., although the observations and the positions of the stations may not be so accurate. The second group of observations will be made as described on page 136 ff. Our purpose will be to find the elements of the mean orbit at epoch and the corrections in the positions of the stations. For this we will use an approxi- mate mean orbit 6" for an arbitrary epoch To (e.g., the middle of the observations). We will then compute the positions of the orbiting object by using T>$+bw(T? To) as orbit, where b' is the vector obtained from the first group of observations. In most cases, b" will be constant in the interval (T? To). If not, we must also include terms of 5", b*, etc. If we assume there are no errors in the value of 6f, the discrepancies will be due only to d&", ie., the correction to the mean orbit of epoch To. Thus the observation equations for stations of the same system will be similar to equation (147): the matrix U being evaluated for the value of Separating the unknowns as described above may not always be the most efficient method. For each situation we must decide which sep- aration will give the strongest solution. For example, the mean anomaly n will often be more accurately determined from the first group of observations than from the second (it should be remembered that n is actually the secular variation of M). With the help of equation (146)?where Xc will be replaced by dX^, which is the correction to the coordinates of the station?we could use stations not belonging to the same sys- tem, but such a method would be less accurate because of the large number of unknowns. In such a case we must also have distance measure- ments to the object. From stations of the same geodetic system we will not usually be able to observe the object over a large part of its orbit, and thus the orbit will not be well determined. This inaccuracy will, in turn, bring large errors in the determination of X*c and g1. There are two possible methods for dealing with the problem. 1) The first method uses two groups of sta- tions, belonging to two different geodetic sys- tems, from which the object can be observed at two different parts of its orbit during the same revolution (fig. 16). The observation equations will be two groups of the form of equation (148), the one referring da' ]= e-S-U>dbl-QXh-QGg1, (148) FIGURE 16.?Connection of two geodetic system* by fir?t method. 142 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS Y O L . 1 to observations from stations of the first system and the other to observations from stations of the second system; or, (d?) = 9SUdb?-9X<-> /dl'\ [da'] = OS- U- dbl-Q-Xh,-< \dr) (149) This gives a total of 18 unknowns, or 12 unknowns if the geodetic systems have no orientation errors. If we have independent unknown stations between the two systems, we could compute their positions by observing the object while it moves from one geodetic system to the other. For this we will compute the position of the object for the times of observation, using the orbit 6j+t#J+&f (T?To)+. . . , as obtained from the adjustment of equation (149), and continue according to the procedure described on page 122 ff. Or, we could introduce the observations from the unknown stations to the system (149) and adjust as a whole. This would give a more rigorous solution, but would result in a system with more unknowns. Similarly, if we have more than two geodetic systems along the orbit, we will have to introduce for each one an additional group of equations, of the form (149). 2) The second method is to use a small part of the orbit, which is the same for the observa- tions from both geodetic systems. This can be done since different areas will be under the same part of the orbit during each revolution (fig. 17) because of the rotation of the earth. The orbit determined in this manner may not be very accurate but, on the other hand, the poorly determined elements willl not greatly affect the determination of the positions. The observation equations will be the same as those of equation (149). Errors in the orbit and the timing will have certain effects. From equations (143) and (89a) we can see that an error Sdu in the orbit will introduce an error in the position of the object, N revolution N + l revolution FIGURE 17.?Connection of two geodetic systems by second method. given by the expression (150) To find the effect of an error 5Tin the timing of the observations, we must differentiate equations (129) to (134) with respect to T so that we can find the effect of dT in a, 8, R. The result will be /da\ I dd )=u> \dR/ dT where u* is a vector, the last row of the matrix U multiplied by n. So the introduced errors in the position of the object will be dX^Siu^T. (151) If, furthermore, we have an error sb* in the value of bv (we again use the index u to indicate the complete vector b, 5 . . . with zero ele- ments if they do not exist), we will introduce an additional error db*(T? To) to bu, or an error in the position of the object, 6Xla=(T-T0)SU6i>?. (152) From equations (150), (151), and (152) we see that the variance of Xls will be V{Xts}=S>U'V{bl}-U'-S'+S-u-V{T}.u'-S'+ (T-T0)S-U'V{b?}>U'>S'(T-T0). GEODETIC USES OP ARTIFICIAL SATELLITES 143 The variance of XQ, then, can be found with the help of equation (101). The effect of an error bT in bu (T? To) will be of the second order and thus may be neglected. The accuracy with which we can expect to determine the orbit and the positions of the unknown station (or the elements Xc and g*) depends not only on the accuracy of the obser- vations and the net configuration (see p. 139^, but also on the accuracy of the elements 6" (since they are determined separately) and on the correctness of our theory of the orbit. An error 8b* in 6" in the determination of the orbit with the method described on p. 140 ff. will introduce an error to the value of 8bu0=8b?E{(T-T0)}, (153) where E{ (T? To)} is the mean value of (T? To) of the different observations used. If we select To as the mean time of the observations, E{(T? To)} will be zero; thus we will not introduce errors in bo (provided 6*=0). How- ever, an error 8bn will introduce errors 8X3 in the positions given by equation (152). We must keep 8XS as low as possible, and this can be done by keeping 8b" low (by making many observations from many stations well spread along the orbit and over a long period of time), and by keeping short the interval of time used for the geodetic connections (using no more than a few revolutions, if possible). If we desire an accuracy such that the errors in position do not exceed /x', then 8b", and T? To should satisfy the conditions, SU 8om (T? To) +?)+? cos (o+2w) + | ecos(3?+2w) V ^ \ cos t -[ (v?M)+e sin v~ sin 2(t>+?)?^ e sin (v+2a)? i e sin (3v+ 2?) \ ? yK1"!8in2 0 [-^ ?sin'i) (o-^y [(2-?si ' M+e sin - ? sin2 -5Lvw)sin v+ ?J 1? VT?i5)sin 2 o V? f n"~R sin2ijesin (v-\-2u)? \ 2 ~ l 2 S m 2 V S i n 2(t;+)~~6 ( 1 ? s m S ^e s m (3tH-2?)|- GEODETIC USES OF ARTIFICIAL SATELLITES 145 An orbit of unknown mean distance.?We shall now consider the case in which we know the orbit except for the mean parallax (or mean distance, which corresponds to an unknown value for kl). (The moon's orbit is an example; after the perfection of Brown's (1899a, 1899b, 1901, 1908) theory, it is expected that the only doubtful value will be the mean parallax, cr0-) Since the orbit will be based on the unknown mean distance, there will be an unknown scale factor (1+K) by which the given distances Rt should be multiplied to give the correct dis- tance R. Thus or B=0.+K)R, dR=R?Rt=KRt, where K is supposed to be a small quantity. But a difference dR in R will introduce a difference in X% given by equation (89); or dX%=cos 6 cos 8 dR= (Rt cos 6 cos 8)K?S1K, dX%=sm B cos 8 dR= (Rt sin d cos 5)K=S2K, d3T|=sin 8 dR = (Rt sin 8) K =S>K. (156) Introducing equation (156) to equation (84) we get /d8'\ ida'MsU-edXl. These are the observation equations for the computation of both dX*Q and #c. They are the same as equation (84b) with d8=da=0, Applications of the various methods Any of the three methods described earlier can be used to obtain geodetic information from the artificial satellites. The choice of method will depend on the kind of satellite, its orbit, shape, instrumentation, etc. If we know the orbit of the satellite with a good degree of accuracy (of the same order as that with which we want to determine the posi- tions of the stations), we can apply the method described in pages 122 to 126. If we do not know the orbit accurately enough, but we do know that the variation of the elements is more or less smooth, we can use the method of orbital interpolation as described in pages 135 to 144. This more flexible method can be used with any spherical satellite of rather high specific gravity and a perigee height of more than 500 km. If we do not know the orbit at all, we can use the satellite by making simultaneous obser- vations and using the method described in pages 129 to 133. Simultaneity of observations can be ideally achieved if the satellite is specially instrumented to send flashes of very short dura- tion; such an object we shall call a flashing satellite. The orbit: ephemerides and visibility.?Ran- dom variations occur in the orbital accelera- tion,11 because of variations (of geophysical and solar origin) in the densities. Such random variations make it almost impossible to deter- mine an orbit that could be extrapolated for a long period of time and give a sufficiently ac- curate position for the satellite. The accelera- tion varies from 10 per cent (perigee heights of 170 km) to 100 per cent (perigee heights of 700 km) of its value. These variations are sufficient to displace a satellite by several kilo- meters along the orbit (or some seconds in time) in one day, even for a perigee height of 700 km. However, such variations in acceleration do not seem to be of a sudden character. For these reasons, the orbits and ephemerides prepared by the various agencies (e.g., the Smithsonian Astrophysics! Observatory) are revised at least once every week. Although the extrapolation of an orbit may be in error by several kilometers, the a posteriori orbit determination is expected to be much more exact. Preliminary results of orbit determina- tions made with the method described on page 140 without correction for the short period per- turbations and with observations of moderate accuracy, gave residuals in the position of the satellite of the order of 1 km. The final orbits from accurate observations are expected to be accurate to within ? 100 meters, or better. Visibility is an important factor. Unless we measure only distances, or use electronic meth- " By orbital acceleration we mean the variation of the period P or of the mean motion m, per revolution or per day. In the notation of p. 140 the acceleration will be i. 146 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TOL.J ods for determining the directions, the satellites must be visible to be observed or photographed. During the day a satellite is rarely visible. At night it will be visible if it is illuminated, as would be possible if the satellite carried a light source, either continuous or flashing. O'Keefe (1956) has suggested the use of retro- directive reflectors illuminated from the ground by searchlights. The simplest solution would be to let the sun illuminate the satellite. Indeed the satellite can be out of the earth's shadow and thus illuminated by the sun, while an observer underneath it is in the shadow (fig. 18). The Point where the satellite enters the earth's shadow Area of visibility FIGURE 18.?Area of satellite visibility. satellite will enter into the shadow of the earth when at an angular distance D from the line of the terminator given by the formula cos D f where p is the radius of the earth. The effect of refraction has been excluded. The observer, on the other hand, should be in darkness, the degree depending on the apparent magnitude of the satellite. If the satellite is rather bright, the nautical twilight (12? depression of the sun) must have ceased for the observer, but if the satellite is faint, the astronomical twilight (18? depression) should be considered. This means that the observer should be farther than 12? (or 18?) from the terminator. In addition, the satellite must be above the horizon of the observer, preferably more than 15? above, to reduce the errors from refraction. Or, the angular distance d between the satellite and the observer should be sin (75?-d)=psin 105?P+H ' Figure 18 shows the form of the visibility area on the earth, which can be found easily with a graphical solution if we use a polar stereographic projection (fig. 19), and assume the earth to be a sphere. A polar stereographic map of the world may be combined with this graphical solution to give the visibility areas on the same map. This method is now in use for visibility predictions at the Smithsonian Astro- physical Observatory (Schilling, 1958). The observations.?Both optical (photo- graphic) and electronic methods can be used for observing the satellites. For the moment, however, only photographic methods seem to be useful for geodetic purposes, since present electronic methods do not yet give the needed accuracy. Since the apparent angular velocity is fairly important and satellites are not always very bright, the Smithsonian Astrophysical Observa- tory constructed a special camera, the Baker- Nunn Satellite Tracking camera, designed by J. G. Baker and J. Nunn under the direction of F. L. Whipple (Henize, 1957). The instru- ment is a Super-Schmidt F/l camera with FIGURE 19.?Area of satellite visibility on a stereographic projection. GEODETIC USES OP ARTIFICIAL SATELLITES 147 focal length 50 cm and a field of view 5? X 30?, and is expected to photograph satellites fainter than 10th magnitude. The focal field is spherical, and a cinemascope film stretched on a focal spherical surface is used for the emulsion support. At a scale of 406" per mm, an accuracy of ? 2 " is expected in the determination of the directions (Henize, 1958). In addition, a time unit, controlled by a crystal clock, gives the time of the middle of the exposure to O'.OOOl (Davis, 1958). How- ever, the accuracy of the timing is not expected to be higher than ?08.001. This error in timing will introduce errors in the direction of the motion, in view of the fact that the satel- lites have rather rapid motion. The variance of d' and a' cos 5' can be computed from equation (42). The Baker-Nunn camera was developed especially to photograph faint satellites. If * they are not particularly faint we can use astro- graphic cameras or even long focus refractors, provided we use a special shutter to interrupt the trail and get the timing. For a flashing satellite no shutter is needed. Markowitz (1959b) has developed a dual-rate camera on the same principle as that of his moon camera (Markowitz, 1954). Ballistic cameras (e.g., of the type made by Wild Heerbrugg Ltd., Switzerland) also can be used for bright satellites. These cameras, developed for missile tracking, can give directions with an accuracy between 3 " and 5" when properly used. For a flashing satellite they can be used with or without a shutter. The direction of a satellite can also be de- termined electronically (see p. 112); there is then no problem of visibility but, on the other hand, the satellite must carry a transmitter. Electronic methods do not as yet give sufficient accuracy and can be used only as described on page 140 ff. No direct distance measurements to the satellites have yet been made, but are expected in the near future. Johns (1958) discusses this possibility and expects an accuracy of ?30 meters in the measured distances, a rather optimistic estimate for long distances. If we also include the effect of errors in timing (p. I l l ) , the accuracy will be still less. The observed directions (or distances) must be corrected for aberration (p. 115 ff.) and refraction (p. 117 ff.). Method for a known orbit.?The method based on the assumption of a known orbit is perhaps the simplest in regard to the geodetic computations, but it is the most difficult to apply because of the difficulty in obtaining an accurate orbit. The theory of this method is discussed on pages 122 to 128. If we know the orbit, to determine the posi- tion of a station we have only to observe the satellite (or satellites) from the unknown sta- tion with an appropriate instrument (e.g., photograph the object with a Baker-Nunn or ballistic camera against the star background). We observe the satellite at a minimum of two positions, and record the times of the observa- tions. The reduction of the observations, after the appropriate corrections have been applied, will furnish the observed elements (say a' and 5') and their variances. We obtain the solution as described on page 122 ff. The corrections to the approximate coordinates will be given by equation (91) and the variance by equation (92). If we have a number of stations all belonging to the same geodetic system, we introduce as unknowns the coordinates X*c of the origin of the system and, perhaps, the rotation g* and the scale factor ?. The observation equations using equations (20) and (84) in this case will be: 'dS") da' ,dr J =-erix t e-e T tGtigt-ert(Xi-x$)e. (157) It is important to know the system of refer- ence in which the orbit is defined and the system in which the observations were made, in order to apply the correct expression and to reduce the observed and computed 5', a', r to the same system, as explained on page 124. Since this method requires that we know the position of the satellite to a high degree of accuracy, we shall use it only for satellites whose orbits can be determined accurately. Since, as we have seen, random variations in 148 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS the accelerations displace the satellite con- siderably along the orbit, we must use an orbit derived from a rather large number of observa- tions made around the same time that the observations from the unknown station Cor stations) were made. For a spherical satellite of a high specific gravity and a perigee height of more than 700 km, and preferably of 1000 km, and using good observations, we can get an orbit accurate to ?100 meters or better, with the method described on page 143. The larger part of this uncertainty will be of an accidental character. This orbit, and thus the unknown stations as well, will refer to a system of reference that may not be geocentric. Indeed, the origin of this system will be a kind of weighted mean of the origins of the different systems to which the stations, used for the determination of the orbit, were referred. Some complications will also arise from the fact that the true orbit does refer to the center of gravity of the earth. If we use many stations belonging to different geodetic systems all over the world to determine the orbit, we can expect that a large part of the errors in the different systems will be compen- sated; thus the origin of the mean system in which the orbit will be given will be close enough to the center of our ideal terrestrial ellipsoid. New positions for the stations could be determined with this orbit and an iteration could be performed to improve the positions of the stations and thus also of the orbit. Another approach is to compute the orbit by using only stations that belong to one and the same system. The positions of the unknown stations thus determined will refer to that same system. The larger part of the uncertainty in the position of the satellite will be accidental in character. Therefore if we use a certain num- ber of observations to determine the position of the station, and those observations are dis- tributed in every direction, we will considerably improve the final result as given by equation (84). The same is also true for the effect of errors in the observations. Combining equa- tion (98), which will hold for a good distribution even if we do not measure distances, with equation (103), then for the standard error of the position of a station aQ we obtain (158) o If we adopt the rather conservative values ?s=?100 meters, in 6> and S'e in e will be, respectively, and a(T?T0)6'e. It follows that for a duration (T-To) of one hour and a height of 700 km, to get an accuracy of 7 meters in the position of the satellite, we must have &i> is the standard error of p. This variance should be combined with the one given by the errors of timing, equation (46). The motion of the moon is about 0T5 per second of time, so even if the error of timing is 0"01, the introduced errors in the positions will be only 0T005. This means that equation (46) can be disregarded and the variance will be given by equation (164). To evaluate (164) GEODETIC USES OF ARTIFICIAL SATELLITES 155 p is o- and the variance of the topography is <*\ov will have the equation, (165) If there is no correction for the topography of the moon, atov will be equal to the RMS of the topography, which is of the order of 0T7. This value will drop to 0T2 with the use of the existing maps of the moon and perhaps to less than Of 1 after the work of Watts is completed. The value of If we make no correction for the topography a ?g we have (see p. 154) a = 0T18, and if we have ?^ *2. 100 observations we will get the following errors ' in the coordinates: ^ = ? 4 8 meters, o*=?48 meters, u*=?36 meters. i With the additional corrections for the topog- raphy from existing maps and from Watt's ? maps (p. 155) the errors will be respectively: ? ax= ? 13 meters, ?b positions of the stars used as reference introduce .9 another source of error. ?. If the standard error of one coordinate of a o star is ?0Tl and we use 10 reference stars, the *j* computed coordinates of the moon will have an / additional error of about ?0!06. This will increase the errors a* by some meters. Only after the final adjustment has been made will we be able to determine the accuracy of our result. However, we may expect that the coordinates of the stations will be computed with an accuracy of tbe order of 30 to 50 meters. GEODETIC USES OF ARTIFICIAL SATELLITES 157 A final source of error that we must consider is the correctness of the ephemeris of the moon. The coordinates of the moon in the Improved Lunar Ephemeris are given to within 0T01 in 5, 0*001 in a, and 0T001 in vs. From this rounding we thus have errors approximately equal to ? 5 meters in 8, ? 7 meters in a, and ?30 meters in a. With a great number of observations, how- ever, as mentioned on page 128, the errors so introduced will be very small. What is important is the correctness of the theory of the moon from which the ephemerides were prepared, since it will introduce systematic errors. As has already been pointed out, the mean parallax of the moon, GJO, is very doubtful. For this reason it will be advisable to use the mean distance M of the moon as an additional unknown. We will then get much better re- sults, considering that the mean distance may be wrong by several kilometers. I t is not easy to estimate the correctness of the theory of the moon and how it will affect our solutions. On the other hand, by analyzing the residuals we should be able to check and probably correct the orbit of the moon. The same holds also for the relative position of the center of figure with respect to the center of gravity. Acknowledgments The author acknowledges his indebtedness for help received during the preparation of this paper, especially to Profs. W. A. Heiskanen, R. A. Hirvonen, and F. J. Doyle of the Institute of Geodesy, Photogrammetry and Cartography, Ohio State University, and to Drs. L. G. Jacchia, C. A. Whitney, Y. Kozai, and Profs. J. A. Hynek and F. L. Whipple of the Smith- sonian Astrophysical Observatory. Symbols used Following is a list of symbols used and their meanings: u This is affected not only by errors of the mean PM?IIM-, bat also by the earth's equatorial radios. a, semimajor axis A, azimuth bu, vector or orbital elements e, eccentricity E, eccentric anomaly H, terrestrial height Ha, geodetic height HA, astronomic height ?/, coefficient of sec- ond harmonic of the earth's potential k\, geocentric gravi- tational con- stant I*, direction cosines in the sidereal system L, geometric dis- tance between the two stations TO', direction cosines in the instan- taneous terres- trial system M, mean anomaly n', direction cosines in the terres- trial system n, mean anomalistic motion N{, matrix of normal equations P{, weight matrix q{, direction cosines in the geodetic system q, perigee distance r, topocentric dis- tance R, geocentric dis- tance (on pp. 118 to 122, R=correction for refraction) /, Greenwich side- real time T, time as independ- ent variable v, true anomaly V, variance W, coordinates in mean sidereal system z', coordinates in geodetic system X', coordinates in ter- restrial system Y*f coordinates in in- stantaneous ter- restrial system Z', coordinates in ap- parent sidereal system Z, zenith distance a, right ascension, geocentric a', rignt ascension, topocentric T , equinox 6, decb'nation, geo- centric 8', declination, topo- centric r, HA-H % deflection of the vertical in prime vertical 6, a?t X, terrestrial longi- tude Xo, geodetic longitude \A, astronomic longi- tude ?, deflection of the vertical in meridian p, radius vector, geocentric