viking relativity experiment

7/28/2019 Viking Relativity Experiment

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VOL. 82. NO. 28 JOURNAL OF GEOPHYSICAL RESEARCH SEPTEMBER 30. ]977

The Viking Relativity ExperimentI. I. SHAPIRO, R. D. REASENBERG,P. E. MACNEIL, AND R. B. GOLDSTEIN

Massachusells Instit ll le of Technology. Camhridge. Ma.uacll /lse lt .< 02/39

J. P. BRENKLE, D. L. CAIN, T. KOMAREK, AND A. I. ZYGIELBALiMJet Propul.


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4330 RAmo SClEI'CEmost severe where it is most important: near superior con-junction. For larger sun-earth-Mars angles the density ofplasma is lower, the scale of plasma inhomogeneities larger,and the signal path shorter.In the remainder of this paper we discuss the method used tomeasure group delays, the procedures employed in data collec-tion and data analysis. and the results obtained.

2. METHOD OF DELAY MEASUREMENTWe describe here brieRy the equipment and the technique

used to obtain the group delay measurements upon which therelativity experiment is based.Three tracking stations were used for these measurements,one each in Australia. California, and Spain. Each of thesestations has a 64-m-diameter antenna equipped with a trans-mitter-receiver system and various calibration device~. Thelocal oscillator signals used in each transmitter-receiver systemare derived from an atomic frequency standard, or clock.whose instability is about I part in 1012,or less, over timescales corresponding to the round-trip delay of signals propa-gating between station and spacecraft. Time at each station ismaintained to within about 20 JtS of universal time, coordi-nated (UTC).How is this equipment in conjunction with the spacecraft

transponders used to measure group delays? The nearly mono-chromatic, approximately 2.2-G Hz S band 'carrier' signal tobe transmitted toward a spacecraft is first modulated by theintroduction of approximately 650 phase changes, of alternat-ing sign. at evenly spaced intervals. This 'square wave' modu-lation resuh5 in about a 9-dB carrier suppression, the remain-ing 90% of the signal power being in the side bands. At thespacecraft this phase modulation is first separated from thecarrier. the carrier frequency isthen multiplied by 240/221, themodulation is reapplied with 0.9-dB carrier suppression, andthe resultant signal is transmitted. The X band downlink signalis similarly generated except that the frequency multiplier is880/221. The frequency change of the S band carrier signals isnecessary to enable simultaneous transmission and receptionat both the spacecraft and the tracking station.The group delay is determined by cross correlation of themodulation on the signals received on the earth with a delayed'stretched' replica of the transmitted modulation. The delayused for the cross correlation is based on prior knowledge,whereas the stretching, needed to accommodate the effect ofthe Doppler shift, is controlled directly by the frequencychange detected in the received carrier signal. Both the 'in-phase' and the 'quadrature' (900 out of phase) cross-correla-tion coefficients are determined. The expected normalized val-

1"-0"---Fig. I. Estimation of signal delay via cross correlation of themodulation on the received signal with a suitably modilied (see text)replica of the modulation imparted to the transmitted signal. The in-phase I and quadrature Q normalized correlation coel1icients are eachshown as a function of01.the error in the assumed value of the round-

tr ip group delay of the radio signals propagating between the earthand a Viking spacecraft.

ues of these coefficients for a true square wave modulation inthe absence of noise are illustrated in Figure I as a functionofthe difference OTbetween the value of the measured delayandthe value of the delay assumed in forming the replica.Forexample, for OT= 0 the coefficient I for the correlatiQn ofthereceived modulation with the in-phase replica will be unity,whereas the coefficient Q for the corresponding correlationwith the quadrature replica will vanish. The 'triangle' patternstraced by I and Q as functions of OTare periodic with . theperiod T of the modulation. It is easy to show that withintheinterval T the value of OTcan be expressed as

OT= ~ (I- II I:I0 I )sign Q (I)In practice, modifications to this formula are used in theestimation of OTto account for the deviations of the modu-lation from a true square wave.As can be seen from Figure I, OTcan be determined onlyto

within an integral multiple of T. How can this ambiguity inOTbe removed'? Our a priori knowledge of the delay is of in-sulficient accuracy for this purpose when the modulation ofshortest period, ~ 2 JtS, is used. Therefore modulations withperiods longer by successive multiples of two are transmittedsequentially until a period is encountered that is large incom-parison with the a priori uncertainty in the delay and thattherefore insures the proper removal of the ambiguity. Onlythe signs of the values of I for the longer-period modulationsare needed to remove the ambiguity. Thus its removal does notadd appreciably to the time required to make a delay measure-ment.Despite the ~ 2-Jtsextent of the shortest period availablewith the present equipment, the signal-to-noise ratios usuallyattained in a few minutes of integration are so high that thestandard deviation in the estimate of OTdue to random noisealone is often only about 2-3 ns. The uncertainties in the epochand the rate of each station's dock usually introduce a com-parable error in delay measurement. More important, how-ever, is the degradation in measurement accuracy ca4.sedbyuncertainties in the calibrations of the delays through thespacecraft and through the ground equipment: transmitter,antenna, and receiver [Komarek and Otoshi, 1976]. These in-strumental delays are not constant and for the spacecraft, forexample, vary by several tens of nanoseconds over the rangesof signal level and temperature encountered ~uring the mis-sion. The overall uncertainty in these calibrations is estimatedto be about IOns and generally provides the limit on achiev-able measurement accuracy. Many of these calibration prob-lems would be ameliorated if shorter-period modulations andhigher-bandwidth transponders and receivers were available.

3. DATACOLLECTIONIn this preliminary report we discuss only those delay mea-

surements obtained during an approximately I-month periodcentered on November 25, 1976, the date of the superiorconjunction of Mars. Here we describe the procedures that,with only a few exceptions, were used to obtain these data.To estimate with useful accuracy the coronal contribution tothe delay measurements made between tracking station andlander during this period, it was necessary, as stated earlier, tomake simultaneous dual-band delay measurements betweentracking station and orbiter. However, because a given groundstation could track only one spacecraft at a time, such simul-taneous tracking had to involve those two of the three available64-m-diameter antennas that could view Mars at the same time:

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QN W '-L-E 5un 2 3 411/21/76 - Ro----.OhrUT Mars I "_.__5 11/29/76Ohr UTFig. 2. (0) The upper portion shows the daily means of the inferredeffeet of plasma on delay measurements. The lower portion shows thetotal spread (maximum minus minimum) of the plasma correctionsapplied to the individual delay measurements made on a given day. (b)The relative positions on the plane of the sky of the sun and Mars asseen from the earth on the days surrounding the superior conjunctionof Mars on November 25, 1976. Solar equatorial coordinates arcshown.

deep-space stations (DSS) 14and 43 in Goldstone, California.and Canberra, Australia, respectively. The two landers areseparated from one another by nearly 1800 in longitude onMars, so at most one lander was visible during the entire3-hour period that Mars could be observed simultaneouslyfrom both stations. On each of those days on which landerand orbiter tracking was scheduled a series of separate mea-surements was made of the round-trip group delays betweena tracking station and that lander which was in view. Tech-nical constraints on the landers actually limited such mea-surements to a period of about 100 min and 40 min for landerI and lander 2, respectively.Because of certain of its unique features that increase theprobability for successful measurement the Mu-II sequentialranging system [Martin and ZygielbaulIl. 1977], available onlyat DSS 14, was almost always used to make the delay mea-surements between tracking station and lander. Simultane-ously, dual-band delay measurements were made betweenDSS 43 and a conveniently placed orbiter, i.e.. one near itsI\poapsis orbital position, where tracking is least difficult.These latter delay measurements utilized the planetary ranging'assembly (PRA) [Osborn. 1974], which unfortunately has aninherent ambiguity interval of ~2 IlS for X band group delaymeasurements, since the PRA is not configured to accommo-date longer-period modulations at X band. To try to ensurethe accurate removal of this ambiguity, dual-band ranging toan orbiter was first carried out from DSS 14 on each day ofscheduled observations until the time Mars 'rose' at DSS 43.Since the Mu-II ranging systeni has no such ambiguity prob-lem, a comparison of the dual-band data from the two track-ing stations enabled the ambiguity in the DSS 43 delay d~lta

4331to be removed reliably with a few possible exceptions: thecoronal elTecton the (downlink) measurements of group delayat X band never exceeded about 5 J.l.S.and the differencesin the coronal effects on the DSS 14 and DSS 43 measure-ments of delay were usually under I J.l.S.since the two sets ofmeasurements were rarely separated in time by more than~ I hour and never by more than ~2 hours. (For sun-earth-Mars angles under about 0.50. where the effects would havebeen larger. the severe gradients and turbulence in the coronaprevented any useful measurements of delays from being ob-tained. )

4. DATA A=--ALYSISOur preliminary analysis of the delay data proceeded in two

steps. First. we corrected the relevant delay measurements fo~the effects of the solar corona. Second, we compared thecorrected data with theoretical predictions based on the theoryof general relativity.Plasllla Correcliom

In the determination of plasma corrections we have not yetattempted to account for the differences in spatial locationsbetween lander and orbiter and between DSS 14and DSS 43.However. to account for the more important fact that thedual-band delay measurements were available only for thedownlink paths, we did employ the 'thin-screen' model toestimate the uplink plasma delay: we assumed that the entireplasma effect was confined to the planar region perpendicularto the earth-Mars line and passing through the center of thesun. Thus for the plasma correction for the uplink path ofa given measurement we took that value of the plasma delaymeasured on the downlink path at a time earlier by the round-trip travel time of a radio signal propagating between thethin screen and Mars.

The mean plasma corrections are shown in Figure 2a foreach day on which delay measurements were obtained nearsuperior conjunction. The erratic behavior of these dailymeans demonstrates vividly that a simple parameterized modelof the corona could not yield an adequate representation of theplasma delays. The scatter of the daily means about the pre-dicted plasma delays obtained from such a model [Tyler el 01..1977] is approximately a hundredfold greater than the delaymeasurement error. Thus without direct plasma calibrationthe accuracy achievable in the Viking relativity experimentwould be drastically reduced.

On the lower portion of Figure 2a we show for each appro-priate day the difference between the maximum and the min-imum of the plasma corrections applied to the individual delaymeasurements obtained on that day. These differences variedfrom about 20 to 400 ns but by no means varied monotonicallywith the sun-earth-Mars angle. The ratios of these differencesto their corresponding daily means were also far from con-stant. even when they were corrected for the differences in thetime spans of the data obtained on different days. These char-acteristics provide further evidence of the erratic time varia-bility of the corona.

One systematic effect isclearly discernible in Figure 20: theasymmetry of the daily means with respect to reflection aboutthe date of superior conjunction: i.eoo the daily means aresubstantially smaller after superior conjunction. This qual-itative behavior wa5 anticipated because (I) the solar latitudeof the intersection of the earth-Mars line with the thin screenjust before conjunction was nearly equatorial. whereas at thecorresponding time after conjunction the solar latitude of this


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Fig. 3. (lI) Relative delay residuals from lander I for six representative days on which six or more delay measurementswere analYl.ed. (h) Relative values of the plasma corrections applied to the delay measurements for each given day.

point was much higher (see Figure 2b), and (2) at minimumsunspot activity, such as in 1976, the mean coronal electrondensityat a givenradial distancefromthe sun isas much as 10times lower for polar than for equatorial latitudes (see. forexample. Counselman and Rankin [1972] and Berman et al.[1976]). The asymmetry in Figure 2a does not fully exhibit thislatitude variation of the mean electron density partly becausethe coronal effect on delays is proportional to the integral ofthe electron density along the ray path of the radio signalswhich sample only low solar latitudes far from the sun.

Finally. we remark that the dual-band technique for plasmacalibration is appropriate for this test of gravitation theorybecause at the level of accuracy of concern to us here, generalrelativity and all other well-known theories of gravitation arecolor blind: the same vacuum delays are predicted for all signalfrequencies.CO/ilparison(~fCorrected DataWith Theory

The delays measured between tracking stations and landers,aner correction for plasma contributions, were compared withtheoretical predictions of the corresponding round-trip vac-uum delays. These predictions were based on (I ) pre-Vikingknowledge of the orbits of the earth and Mars, based primariiyon radar observations of the inner planets, radio tracking ofthe Mariner series of spacecraft, laser ranging to the moon,and optical observations of the asteroids and the outer planets:(2) pre-Viking determinations of the locations of the trackingstations o"nthe earth from radio tracking of Mariner space-craft: (3) a standard model of the earth's rotation. includingthe effects of polar motion and variations in the earth's rateof rotation: (4) prelim inary determinations of the locations ofthe landers on Mars and of the rotation vector of Mars fromradio tracking of the landers in the weeks immediately follow-ing their respective arrivals on Mars (the results were in goodagreement with those of Michael et at. [1976] and Mayo et al.

[1977]); and (5) the predicted relativistic contribution to thedelay, given by the generalized equation

.h= 2rO(I+"Y)ln (r.+rm+R ) (2)c r. + rm - R

where ro ==GMs/c2 "" 1.5 km is the length equivalent of thi:sun's mass; G is the universal constant ofgravitation; Ms is themass of the sun; c is the speed of light; r., rm, and R are thedistances between the sun and the earth, the sun and Mars,and the earth and Mars, respectively; and l' isa.parameter thatappears in the generalized Schwarzschild metric (see, for ex-ample. Weinberg [1972] and Misner el 01.[1973]). In Einstein'stheory of general relativity, "yis replaced by unity. In othermetric theories of gravitation, l' can take on values differentfrom unity (see, e.g., Will [1974]). With "y= I, (2) showsthat.IT can reach as high as -250 /lS for signals that just graze thelimb of the sun.

The delay measurements .are also affected by the atmo-spheres and ionospheres of the earth and Mars. Their com-bined contribution to a measured delay was as large as 100ns.But only the variations of these contributions over the set ofdelay measurements affect the relativity experiment. Sincethese variations about the mean were rio greater than onequarter of the maximum contribution, we could safely ignorethe atmospheres and ionospheres in our theoretical model forthis preliminary analysis. (Of course, partial correction for theearth's ionosphere is incorporated automatically with the co-ronal plasma correction and is incomplete mainly insofar asthe ionosphere over DSS 14 differs from that over DSS 43.)We also ignored the precession, nutation, and possible polarmotion of Mars in our theoretical model, again justifiablebecause the changes introduced into the delay measurementsby such motions are not expected to exceed 10 ns for theperiod of interest.As an illustration of the consistency of the results fromseparate delay measurements made during a single day we

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SHAPIRO ET Al. : VIKING RELATIVITY EXPERIMENT

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5 Nov.10 15 20 25 30Dote1976) 5 Dec. 10 15Fig. 4. Daily means of the relative residuals for the delay measure-ments made during the approximately I-month period centered onNovember 25, 1976, the date of the superior conjunction of Mars.Anomalous residuals appear on five different days. On November 20,1976,delay measurements were obtained from both DSS 14and DSS43; the means from the measurements at the two stations are shownseparately. Note that useful data were obtained for signal paths pass-

ing as close as -3 s olar radii from the center of the sun (see Figure 2b).show in Figure 3a the differences ('residuals') between thecorrected measurements and the corresponding theoreticalpredictions for typical days on which six or more individualdelay measurements are available. (For some of these daysthere are additional delay measurements which have not yetbeen analyzed.) In Figure 3b we show the corresponding cor-rections that had beenapplied forcoronal plasma. Each resid-ual and each plasma correction is given relative to its corre-sponding daily mean. These examples are a1\ from lander Imeasurements for the period fo1\owing superior conjunction,since with one exception the earlier samples were a1\ fromlander 2 measurements for which the daily tracking time wasmore severely limited (see section 3).

Systematic trends are apparent in each set of residuals,especially those obtained during the first week following supe-rior conjunction. Systematically changing errors in the plasmacalibration may well be the cause. It is possible, but hardlyassured, that these trends will be lessened after we implementand apply a more sophisticated algorithm for the determina-tion of plasma corrections. -What about the consistency between the daily means of thedelay residuals? Here the story is entirely different. To i1\us-trate, we plot the daily means of the residuals in Figure 4, afteradding a single constant to a1\of the data for a given lander.These constants serve mostly to remove in an ad hoc fashionan apparent average error of about 5 IlS in our theoreticalmodel of the earth-Mars distance during this period. Thedifference between the constants used for the two landers isslight and reflects the difference in the errors in the estimates oftheir locations.

Returning our attention to Figure 4, we note that there arefive days for which the residuals are far removed from the:mean of the others. On one of those days, delay measurementswere obtained sequentially from DSS 14and DSS 43, and theresults from each tracking station were significantly differentfrom each other, as well as from the mean. On a sixth day, thefirst day chronologically, the residual is about I IlS larger thanthe majority of the others obtained from observations of thesame lander; however, its epoch is sufficiently far removed fromthe epochs of the others that orbital errors cannot be ruledout as an explanation. What of the other anomalous results?How can they be explained? Plausible errors in orbits, in track-

4333ing station locations, and in lander locations cannot be thecause; such errors would introduce not erratic behavior butonly a slowly varying drift in the residuals. Errors in the soft-ware used to analyze the delay measurements can also beeliminated as a cause: although the residuals shown in thefigures were based on theoretical values obtained with a com-puter program developed at the Massachusetts Institute ofTechnology, substantially the same results were obtainedfrom the use of a program developed wholly independently atthe Jet Propulsion Laboratory. The consistency of the separatemeasurements on a given day and other evidence make hard-ware errors seem unlikely culprits. It is possible that one ortwo, but no more, of the anomalous results could be attrib-utable to an incorrect removal of the ambiguity in the X banddelay measurements for days in which there was a large gap intime between the DSS 14 and the DSS 43 plasma calibrationmeasurements. Such an error, when it is converted to its equiv-alent in round-trip S band delay, would be a (small) multipleof 4.6 IlS. The only other possibility that we can conceiveas an explanation for the anomalous results involves inter-mittent procedural errors in the setup for ranging, but no sucherror has been positively identified.

Let us assume that the relative residuals in Figure 4 that arefar removed from the overall mean, and only these, are in factaffected by procedural or other errors. If we compare all otherresiduals, as in Figure 5, with the 'excess' delays predicted bygeneral relativity ('Y== ]), we may infer by inspection that if 'Ywere to differ from unity by more than about I%, the rest of (2)remaining valid, then there would be a noticeable systematic'dip' or 'bump' in the relative residuals, centered at the time ofsuperior conjunction. (Recall that the theoretical predictionsused in forming these residuals are completely independent ofthe measurements; i.e., the residuals are prefit, not postfit.)

250 a.200>-.!:!..o~~ 150.; ~~- 100..a::50

b.+ +. . + +.. ++++ +++.>;;0;a:: -02 .-Oil

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Fig. 5. (a) The predicted general relativistic contribution (see (2to delay measurements made near superior conjunction. (b) The rela-t ive residuals (see text) of delay measurements based on the assump-tion that general relativity is the correct theory of gravitation. Notethat the ordinate scale is expanded by a factor of 200 relative to that inFigure 5a. Note also that the theoretical predictions used in theconstruction or the residuals are completely independent or the corre-sponding measurements; i.e. , the residuals are prell t, not postfit.


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4334 RADIO SCIENCESince no such systematic deviation is in evidence, we concludetentatively that 'Ydiffers from uni\}' by no more than I% andequivalently that the Viking data agree with the predictions ofgeneral relativity to within 0.5%.

5. DISCUSSIONANDCONCLUSIONSThe best prior determination of'Y from analysis of a time

delay experiment was based on the radio tracking of Mariner9. This experiment, as indicated earlier, showed that 'Ywasunity to within the estimated uncertainty of 4% [Anderson etal., 1976; Reasenberg and Shapiro. 1976]. A different experi-ment based on the gravitational deflection of radio waves[Folllalont and Sramek. 1976]yielded a more stringent limit of2% on the uncertainty in any possible deviation of 'Y fromunity, a result that is twofold lower than that of Mariner 9 andtwofold higher than our present preliminary result.What are the prospects for improvement of this Vikingresult? If calibrated delay measurements are accumulatedthroughout the Viking mission, without large gaps, we antici-pate that our final analysis will lower the uncertainty in 'Ytoabout 0.2%. In this analysis wewillalso include alI other usefuldata and will estimate 'Ysimultaneously with alI of the otherrelevant parameters that govern the dynamics of the solarsystem and hence the inertial trajectories of the deep-spacestations and the Viking landers.What alternative theories of gravitation could be proveninvalid by, say, a confirmation that 'Ywas unity to within 0.2%?Unfortunately, none. AII other widely discussed and analyzedtheories of gravitation, such as the theory of Brans and Dicke[1961] and that of Rosen [1974], have a special property: theirpredictions either agree or can be made to agree with those ofgeneral relativity in the post-Newtonian regime that applies tothe Viking relativity experiment. In many of these theories theagreement is secured by the selection of an appropriate valueof an adjustable parameter, such as 'Y(no such parametersexist in general relativity). For example, the scalar-tensor the-ory of Brans and Dicke contains only one additional parame-ter, a dimensionless constant w (>0) related to 'Yby

'Y= (w + I)/(w + 2)A limit of 0.2% on the deviation of'Y from unity would restrictw to being greater than 500, sincew ~ (I - 'Y)-'when0 < I -'Y I. But no matter how accurate the experiment, it couldnot prove such a theory invalid, provided that general relativ-ity is a correct description of gravitation to that level. Only ifgeneral relativity is proven wrong, or if experiments can bemade sensitive to 'post-post-Newtonian' effects, will there be asubstantial winnowing of competing theories.Acknowledgmellls. We thank the Viking project staff, the per-

sonnel at the deep-space stations, and our colleagues on the RadioScience Team for their indispensable aid in this experiment. We wishparticularly to thank D. L. Brunn and W. L. Martin for their contribu-tions to the radio-tracking system and J. S. Martin, G. A. Soffen, andA. T. Young for their generous support. The Massachusetts Insti tuteof Technology experimenters were sponsored in part by the NationalAeronautics and Space Administration (NASA) under contractNASI-9702 and in part by the National Science Foundation undergrant PHY72-05104A05. The portion of the research carried ou(attheJet Propulsion Laboratory, California Institute of Technology, wasunder contract NAS7-100 sponsored by NASA.

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REFERENCESAnderson, J. D., P. B. Esposito, W. Martin, C. L. Thornton, andD.O.Muhleman,Experimentalestof generalrelativity using time-delay data fromMariner6 and Mariner7, Astrophys.J.. 200.221-233, 1975.Anderson, J. D., M. S. W. Keesey, E. L. Lau, E. M. Standish, Jr ., andX. X. Newhall, Tests of general relat ivity using astrometric andradiometric observations of the planets, paper presented at theThird International Space Relativity Symposium, 27th Congress,Int. Astronaut. Fed., Anaheim, Calif., 1976.Berman, A. L., J. A. Wackley, S. T. Rockwell, and J. G. Yee, ThePioneer II 1976solarconjunction:A unique opportunity to explorethe heliographic latitudinal variations of the solar corona, DeepSpace Network Progr. Rep. 42-35. pp. 136-147, Jet Propul. Lab.,Pasadena, Calif., 1976.Brans, C., and R. H. Dicke. Mach's principle and a relativistic theoryof gravitation, Phys. Rev.. /24.925-935, 1961.Counselman,C. c., III, and J. M.Rankin, Densityof the solar coronafrom occultations of NP 0532, Astrophys. J.. /75.843-856, 1972.Fomalont, E. B., and R. A. Sramek, Measurements of the solargravitational deftection of radio waves in agreement with generalrelativity, Ph)'s. Rev. Leu.. 36. 1475-1478,1976.Komarek, T., and T. Otoshi, Terminologyof ranging measurementsand DSS calibrations, Deep Space Network Progr.Rep. 42-36. pp.35-40, Jet Propul. Lab., Pasadena, Calif., 1976.Martin, W. L.. and A.I. Zygielbaum, Mu-II ranging, Tech. Memo. 33-768. pp. 1-70, Jet Propul. Lab., Pasadena, Calif., 1977.Mayo, A. P., W. T. Blackshear, R. H. Tolson, W. H. Michael, Jr.,G. M. Kelly, J. P. Brenkle, and T. Komarek, Lander locations,Mars physical ephemeris, and solar system parameters: Determina-tion from Viking lander tracking data, J. Geophys.Res.,82. thisi ssue, 1977.Michael, W. H., Jr., D. L. Cain, G. Fjeldbo, G. S. Levy, J. G. Davies,M. D. Grossi, I. I. Shapiro, and G. L. Tyler, Radio science experi-ments: The Viking Mars orbiter and lander,/carus, /6. 57-74, 1972.Michael, W. H., A. P. Mayo, W. T. Blackshear, R. H. Tolson, G. M.Kelly, J. P. Brenkle, D. L. Cain, G. Fjeldbo, D. N. Sweetnam, R. B.Goldstein, P. E. MacNeil, R. D. Reasenberg, I. I. Shapiro, T. I. S.Boak III, M. D. Grossi, and C. H. Tang, Mars dynamics, atmo-spheric and surface properties: Determination from Viking trackingdata, Scimce, /94. 1337-1339, 1976.Misner, C., K. S. Thorne, and J. A. Wheeler, Gravitation. W. H.Freeman, San Francisco, Calif., 1973.Osborn, G. R., Planetary ranging operational software,DeepSpaceNetwork Progr. Rep. 42-2/. pp. 87-91, Jet Propul. Lab., Pasadena,Calif., 1974.

Reasenberg, R. D., and I. I. Shapiro, Solar system tests of generalrelativity, paper presented at the International Meeting on Experi-mental Gravitation, Acad. Naz. dei Lincei, Coli. Ghislieri, Pavia,Italy, 1976.Rosen, N., A theory of gravitation, Ann. Phys.. 84. 455-473, 1974.Shapiro, I. I., Fourth test of general relativi ty, Phys. Rev.Leu.. /3.789-791, 1964.Shapiro, I. I., G. H. Pettengill,M. E.Ash,M. L. Stone,W.B.Smith,R.P.Ingalls, and R. A. Brockelman, Fourth test of general relativ-ity: Preliminary results, Phys. Rev.Leu.. 20. 1265-1269, 1968.Shapiro, I. I., M. E.Ash, R. P. Ingalls,W.B.Smith, D. B. Campbell,R. B. Dyce, R. F. Jurgens, and G. H. Pettengill,Fourth test ofgeneral relativity:New radar result,Phys.Rev. Leu.. 26. 1132-1135,1971.Tyler, G. L., J. P. Brenkle,T. A. Komarek, A. I. Zygielbaum,TheViking solar corona experiment, J. Geophys.Res.. 82. this issue,1977.Weinberg, S., Gravilation and Cosmology: Principles and Applicationsof the General Theory of Relativity, John Wiley, New York, 1972.Will, C. M., The theoretical tools of experimental gravitation, inExperimelllal Gravitation, edited by B. Bertot ti, pp. 1-110, JohnWiley, New York, 1974.

(Received May 31, 1977;revised June 16, 1977;accepted June 16, 1977.)

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