Sep 30, 1977 - Determination From Viking Lander Tracking Data ... Radio tracking data from the Viking landers have been analyzed to determine the ...
VOL. 82, NO. 28
JOURNAL OF GEOPHYSICAL
RESEARCH
SEPTEMBER 30, 1977
Lander Locations,Mars PhysicalEphemeris,and Solar SystemParameters: Determination From Viking Lander Tracking Data A. P. MAYO, W. T. BLACKSHEAR, R. H. TOLSON,AND W. H. MICHAEL, JR. NASA Langley ResearchCenter, Hampton, Virginia 23665 G. M.
KELLY
Analytical MechanicsAssociates,Inc., Hampton, Virginia 23665 J. P. BRENKLE AND T. A. K OMAREK
Jet PropulsionLaboratory, Pasadena,California 91103
Radio trackingdata from the Viking landershavebeenanalyzedto determinethe parametersof the Mars physicalephemeris,the radii of Mars at the landingsites,and the landerlocations.The orientation of the Mars rotationaxis,referredto the 1950.0earthmeanequator,equinox,andepoch,wasdetermined to be 317.340+ 0.003ø right ascension and 52.710+ 0.002ø declination.The planet'srotationperiodwas determinedto be 24 h, 37 min, 22.663 + 0.002 s. Analysesindicatethat the determinationof the motions of the Mars rotationaxiswill requireadditionaltrackingdata. The Mars radii at the sitesof landers1 and 2 are 3389.38 + 0.06 km and 3381.91 + 0.08 km, respectively.The areocentriclocation of lander 1 is 22.272 + 0.002øN, 47.94 + 0.2øW. The lander 2 locationis 47.670 + 0.002øN, 225.71 + 0.2øw. The areocentricright ascensionsof the landersare determinedto be 277.314 + 0.002ø for lander 1 and 99.546 + 0.002ø for lander 2 at 0000 hours,January 1, 1977(Julian date 2443144.5). Possibledeterminationsof relativity parameters,solar oblateness,asteroid mass, and variations of the universal gravitational constant,from their effectson the planetary motions, will require the additional tracking data of the Viking extended mission.
INTRODUCTION
The landertrackingdata containinformationon the physical ephemeris of Mars (rotation axis orientation, rotation rate, precession,nutation), orbits of Mars and the earth, and parametersaffecting the orbital motions. Approximately 6 months of data have been analyzedto obtain the resultspresentedin this paper. The resultsobtained to date are the lander locations, the Mars rotation axis orientation, and the rotation rate. The analysesindicatethat the additionaltrackingdata of the Viking extendedmission are required before significant improvementsin the Mars rotation axis motion and parameters affecting the orbital motion can be obtained.
These effects on the tracking data near conjunction (midOctober 1976 to mid-January 1977) will require further analysis. Lander data during this interval were deleted in the data
processing. Approximately 3 weeksof datas•bsequent to conjunction were combinedwith the preconjunctiondata in the analyses. SOFTWARE DESCRIPTION
The lander tracking data software has the capability to estimatethe parametersdefiningthe Mars physicalephemeris, the planetary ephemerisof the earth and Mars, and parameters affectingthe latter. The statisticsare generatedby considering significanterror sources.For the lander data analysisthe estimatedparametersare the latitude, longitude,and radiusof Mars at the lander locations;the right ascensiona0 and decli-
DATA SUMMARY
The lander trackingdata analysespresentedhave usedapproximately the first 6 monthsof S band range and counted Doppler measurementsmade to both landersby the NASA Deep SpaceNetwork tracking stationsat Goldstone, California (Deep SpaceStation(DSS) 14), Canberra,Australia(DSS 43), and Madrid, Spain(DSS 63). Summariesof the data from
nation b0 of the Mars rotation axis; and the Mars sidereal rotation rate V. The axis orientation anglesare referencedto the earth equator and equinox of 1950.0 and the epochof 0000
hours, January 1, 1950.The orientationat any time is expressed[de Vaucouleurs,1964] as
eachstation,asusedin theanalysis, aregivenin Table1.Over the approximately 6-month total data interval the data con-
sistedof 2387 Doppler points for Viking lander 1 (VL 1) and 704 points for lander 2 (VL 2). Eighty-two range measurements were obtained for VL 1 and VL 2. The first 3 months of
data containedan averageof about25 Dopplerpointsper sol for VL 1 and about 15 pointsper sol for VL 2. In this interval, 16 range measurementswere made to VL 1 and 28 to VL 2. On November 25, 1976, about 4 months after VL 1 touchdown,
Mars was in conjunctionwith the sun. Near conjunctionthe lander rangingdata wereaffectedby the chargedpar.ticles of the interplanetarymedium, the solar corona, and relativity. Copyright¸ 1977by the AmericanGeophysicalUnion. Paper number 7S0426.
a = a0 + # sin/cos/x (secb)T
(1)
b =b0+ # sinI(sin A)T
(2)
where # is the precessionconstantin degreesper century,I is the Mars meanobliqui.ty,A is the anglealongthe Mars mean equator from its ascendingnode on the earth mean equator of 1950.0 to the Mars autumn equinox, and T is Julian centuries sinceepoch. The Mars hour angle V at any time is given by
1/= 1/o+ I'd
(3)
where d is days sincethe epoch of 1/o.The valuesof # and 1/o were -708 are sec/century [Lowell, 1914] and 4.376ø on 0000
hours,November27, 1971(Juliandate (J.D.) 2441282.5)[de 4297
4298
RADIO SCIENCE
TABLE 1. Summary of Lander Tracking Data Used in the Analysis
Station 14 43
63
Lander 1 2
No. Points
No. Points
No. Points
Two-Way Doppler*
Mu 2 Range*
Plop Range*
576 264
0 9
NA NA
1
99I
NA
19
2
23O
NA
21
1 2
820 210
NA NA
24 9
The data intervals are July 20-October 11, 1976, and January 9-30, 1977. The solar conjunction region is deleted. *Mathematical definitions of these measurementsare given by Khatib et al. [1972].
the lander data analysisto be limited to lander locations,Mars rotation axis orientation, and rotation rate.
The ephemeriscorrectionsto the landerdata were,madein the following manner. By using the sixth-order Mariner 9 gravity field of Mars [Danielsand Tolson, 1976] some 18 individual orbits of Viking 1, distributed over the data interval, were determinedfrom the orbiter Doppler data, and the orbiter rangeresidualswere established.After calibration for hardware delays [Komarek, 1976] and charged particle effectsthe residualswere attributed to errors in the pre-Viking Mars and earth ephemerides.Since theseeffectsare similar for both the orbiter and the lander, the orbiter range residualswere usedto calibrate the lander range data for ephemeriserrors. Specifically, the orbiter ranging residuals,relative to ephemerisDE 96, were least squaresfitted with a quartic time polynomial as At> = --956.1 + 34.0738t-
Vaucouleurset al., 1973], respectively.The Mars mean orbit was that of Sturins [1971], and the earth and Mars planetary
0.01335t •
-- 0.0006421 t 3 + 0.0000006357t
4
(4)
ephemerides were obtainedfrom Jet PropulsionLaboratory where Ap is the increment,in meters, to be added to the range ephemeristape DE 96 [Standishet al., 1976]. The anglesI and A are given in terms of the axis orientation angles and the mean orbit by Her Majesty's Nautical Almanac Office [1961]. The range and Doppler observablesare expressedby Moyer [1971] and Khatib et al. [1972]. Batch processing,a Bayes estimation technique,and statisticsinvolving consideredparameters [Moyer, 1971] were used. Any solution parameter could be included
in the considered
as well as the solve-for
category.
The statisticson the estimated parameters consideredthe uncertaintiesof the tracking station locations,precessioncon-
computedfrom the pre-Viking ephemeristo obtain measured range and t is the number of days from 0000 hours, day 180, 1976, ephemeristime. The root mean square of the fit to the range residualswas 9 m. The polynomial was evaluatedat the lander range and Doppler observationtimes in order to calibrate the lander tracking data for ephemeriserrors. The lander tracking data are corrected for the effect of the earth's troposphere and the small motions of the tracking stationsdue to the wandering of the earth's pole. Differences between measurements of station time, universal time, and
atomic time are also considered.The earth pole wander values and polynomials relating the types of time were obtained by Airy-0 defining the prime meridian, as well as range, Doppler the Tracking System Analytic Calibration (TSAC) group noise,and bias. The data from eachlander could be processed [Fliegeland O'Neil, 1973]from data obtained from the Bureau separately or in combination to obtain estimatesof the solu- International de l'Heure in Paris. The troposphere zenith range correctionswere determinedby the TSAC group from tion parameters. tracking station tropospheredata. Elevation effectswere apThe softwarealso hasthe additional capabilitiesto solvefor the tracking station locations, astronomical unit, corrections plied, and range and Doppler corrected,for the effectsof the earth's troposphere[Madrid et al., 1974]. to the earth and Mars orbits, Mars precessionconstant, Mars The total chargedparticle effectson the range data cannot nutation amplitudes, Mars pole wander, solar oblateness,varibe determined directly from the lander data, sincethe landers ations in the universalgravitational constant, asteroid mass, and relativity parameters affecting the tracking signal and the have only S band radio systems.The orbiter ranging signals, orbital motions.Thesecapabilitiesare applicablemainly to the however, traversedalmost the samechargedparticle environadditional data that will be obtained during the extended ment. The chargedparticle calibrations were determined for mission. the orbiter from its dual frequencyS and X band rangingdata [Madrid, 1974] and were then interpolatedto the lander ranging times and used to calibrate the lander range data. When DATA PROCESSINGTECHNIQUE the data deletednear solar conjunctionare excluded,the accuThe lander tracking data were processedat the end of the racy of the chargedparticle calibrations for the lander range first 3 days [Michael et al., 1976a], 3 months [Michael et al., 1976b], and 6 months. The first 3 days of data obtained from VL I were processedto solvefor the Mars spin axisorientation TABLE 2. A Priori Values and Prelander Standard Deviations of Solution ParametersUsed in the Analysis of 3 Days of Data and the lander location. Processingthe longer arcs of VL 1 data and the VL 2 data in a simultaneoussolution yielded A Priori stant, nutations, earth and Mars orbits, and location of crater
more accurate values of the lander locations, the rotation axis
orientation, and an improved spin rate of Mars. Six months of lander tracking data alone did not contain sufficient orbit information
to allow accurate simultaneous
evaluation
of the
earth and Mars orbits and the Mars rotation axis orientation, the rotation rate, and the lander locations. In order to obtain
the Mars physicalephemerisparameterswith high accuracy the orbiter range data were used to evaluate the effects of ephemeris errors on the tracking data, and the corrections were applied to the lander data as suggestedby Blackshearet al. [1973]. This techniquepermittedthe solutionparametersof
Parameter
Mars spin axis a0, deg b0,deg Lander
A Priori Value
317.32 52.68
Standard Deviation
0.2 0.2
1 location*
u•, km v•, km
X•, deg W
3138.0017 1293.3910
47.5
25.0 60.0
2.5
*Here u and v are componentsperpendicularand parallel, respectively, to the Mars rotation axis and X is the longitude.
MAYOETAL.:VIKINGLANDER LOCATIONS TABLE 3.
A Priori Values and Standard Deviations of Considered
ParametersUsed in the Data Analysis
Parameter
Marsprecession constant, "/century Mars dynamicalellipticity
-708.0 0.0052
measurement is estimated at about 30 m. The effect of the
meanchargedparticleson the Doppler data was small,and Doppler calibrationsfor chargedparticle effectswere not
A Priori Standard Deviation
A Priori Value
4299
made.The Dopplervariationsdue to the highlydynamic portionof the chargedparticleenvironment werealsoprobablysmallasevidenced by smallscatterin theDopplerresidu-
50.0
als.
0.0005
Thea prioriestimates andstatistics fortheanalysis of 3 days
(C - A )/C* Tracking station locationst
of data are givenin Table 2. A priori statisticson the solution
u, km v, km
0.0015
0.0150 3.0 X 10 -'• 12800.0 125.0 300.0 0.015 0.01
A, deg Mu 2 rangenoise,rangeunit:J: Ploprangenoise,rangeunitõ
Plop rangebias,rangeunitll Counted Doppler noise, Hz Counted Doppler bias, Hz
parameterswere not used in the analysis of 3 months and 6 months of data. The statistics of the considered error sources
are givenin Table 3. The residuals for the approximately 6monthdata arc are shownin Figures1 and 2. The Doppler residuals aregenerally lessthan0.03Hz, andtherangeresiduals are less than 60 m. ROTATION AXIS ORIENTATION
*The nutationsdependon the dynamicale!lipticity(C - A)/C, whereC is the Mars polar momentof inertiaandA is the equatorial
Numerous determinations of the orientation of the Mars
axisof rotationhavebeenmadesinceSchiaparelli obtainedan there u and t; are components perpendicular andparallel,respec- estimatein 1886from a decadeof observing the position tively, to the earth's 1903.0pole and X is the longitude.SeeJet anglesof the polarcaps.Many determinations havebeenmade PropulsionLaboratory[1976] for their a priori values. from earth-based observations of surface markings and the $One rangeunit approximatelyequals0.0011 m for Mu 2. naturalsatellites, Phobosand Deimos.The Mariner9 photoõOnerangeunit approximatelyequals0.15 m for Plop. II Includesdatabiasandbiasequivalent of errorsin ephemeris and graphsand trackingdata of 1971havealsoyieldedestimates. moment.
chargedparticlecalibrationsusingthe orbiter data.
The rotation axis orientation, obtained from the lander track-
x I0 -t
z z
z
ø
o 8
z z
z -6
•11111111 0
111111111
III
T [tie
IIIIIIIII
Iii111111
FROrl DFIY 200,19'76
i111111
IIII
( DRYS
x I0 -e 6
_
_-
-I'11111111
A 111111111
I,,d',111
TIME
II
IIIIIIIII
FROl'l DRY 200,1976
IIii11111
IIIIIIIll
[ [DRYS
Fig. 1. Dopplerresiduals for processed landerdata.
IIIIIIIII
4300
RADIO SCIENCE
x 10 t 1o
-
oVL
1
-
AVL 2
_
,9
6 _
-
=
A
-
O
z-
'"AA A
z ua
o
A
7C• _
A
0 0
A
A
0 0
Z
0
z z
•.
_q
z z z
-6
z -8 z
z z
lO o
3
6
TIME
:)
12
FROM 0RY 200,1•)76
t5
2l
18
x tO'
(0RYS)
Fig. 2. Range residualsfor processedlander data.
ing data, is comparedin Figure 3 with the axis orientations
.5-
obtainedfromMariner9 radioandlandmarktracking[Born et al., 1972], earth-based observationsof Phobos and Deimos
from1877to 1969[Sinclair, 1972],andpost-Mariner 9 analyses[de Vaucouleurs et al., 1973].With the exceptionof the resultsof de Vaucouleurs et al., for whichno specific uncertaintyis given,thecirclesandellipses shownin thefigureare the uncertainties and demonstrate the highaccuracy of the landerresults. Thisaccuracy of axisorientation is equivalent at thesurface of Marsto a 200-muncertainty in theposition of
.4-
RIGHT ASCENSION
.2 -
the rotation axis. The axis orientation estimates shown in
Table4 for analyzingapproximately 3 days,3 months,and6 monthsof landerdata demonstrate the abilityof thedatato give consistentvaluesof spin axis orientation.The values shownin the tableare for the0000hours,January1, 1950, epoch.The Mars rotationaxisorientation, at anyepoch,was determined
.1 -
--
VIKING LANDERTRACKING(52.710+ 0.002; 317.340 + 0.003)
317.00 -
to be
MARINER 9 LANDMARK TRACKING
-----
TELESCOPICOBS. PHOBOS-DEIMOS
....
MARINER 9 RADIO TRACKING POST MARINER 9 VALUE
1
52.50
.6
.7
.8
.9
53.00
317.340 ø - 0.10106T
(5)
DECLINATION
52.710 ø -
(6)
Fig. 3. Mars rotat{onaxis orientationfor the 1950.0epoch.
0.05706T
MAyo ET AL.: VIKING LANDERLOCATIONS
TABLE 4.
4301
Mars Rotation Axis Orientation, Rotation Rate, and Lander Location Solutions for Various Amounts
of Data
Data Arc Length 81 sols at VL 1,
188 sols at VL 1,
Parameter
Estimated 3 sols at VL 1'
35 sols at VL 2
144 sols at VL 2
a0, deg b0,deg
317.35 + 0.06 52.71 + 0.01
317.34 + 0.006 52.710+ 0.004
317.340 + 0.003 52.710+ 0.002
350.891986 + 12X 10-s
350.891986 + 7 X 10-s
l), deg/d r•, km
v,•,degNI' X•, degW
3389.5 + 0.3
3389.38 + 0.08
22.27 + 0.02 48.0 + 0.2•t
r•., km
3389.38 + 0.06
22.272 + 0.006 47.94 + 0.2
22.272 + 0.002 47.94 + 0.2
3381.88 + 0.22
•., degN• X•.,degW
3381.91 + 0.08
47.669 + 0.006 225.71 + 0.2
47.670 + 0.002 225.71 + 0.2
*A sol is the interval from Martian midnight to the next midnight. l'Areocentric.
:]:Thisuncertaintyis with respectto the locationof crater Airy-0. The longitudeuncertaintyis 0.07ø when referencedto a mathematicallydefinedprime meridian located 148.24ø right ascensionat 0000 hours, January 1, 1950.
where T is measured in Julian centuries from 0000 hours, January 1, 1950, J.D. 2433282.5. The right ascensionand declinationrateswere determinedby usingthe Lowell [1914] precessionconstant. MOTION
OF THE AXIS OF ROTATION
The total motion of the Mars axisof rotation is composedof both precessionand nutation components.The Mars equinox precesses westwardabout 7 arc sec/earthyear, amountingto an equivalentequinox motion at the Mars radius of 115 m/yr. This motion converts to a yearly change of 0.001ø in right ascension of the Mars axis of rotation
and 0.0006 ø in declina-
tion as measured in the earth equator and equinox of the 1950.0 system. These yearly motions are smaller than the presentaccuraciesof the right ascensionand declinationof the axis shownin Table 4. However, the major sourceof the 1950 axis location error is the uncertainty in the precessionconstant, usedto map the axis to the 1950epoch.The uncertainty in the precessionconstant was assumedto be 50"/century, which is conservativelylarger than the 30"/century difference betweenthe precessionconstantsof Lowell [1914]and Lorell et al. [1973].The nutationsof Mars have a miximum equivalent surfacemotion amplitudeof about 30 m in longitudeand 10 m in obliquity, with a period slightlylessthan 1 earth year. The additionaltrackingdata of the extendedmissionwill improve the analysesand may yield measurementsof the axismotion. MARS ROTATIONAL
PERIOD
4. The dots shown in the figure are the valuesof the rotational period, and the vertical bars are the uncertainties.
The Viking landerresultsgive the siderealrotationalperiod as 24 h, 37 min, 22.663 s, some 8 ms longer than the period given by de Vaucouleurs et al. and 6 ms shorter than the estimateof Ashbrook. The consistencyof the Viking lander to give this rotation period with 3 monthsof data and 6 months of data is evident in the results shown in Table 4. LANDER LOCATIONS
The
lander
locations
distance of the landers from the Mars
vations
22.65
the sidereal
arc
rotation
axis and the
22.68 -
22.66
1659 to 1881 and established
different
right ascensionof the landers are determined by both the Doppler and the range data. The Doppler data are relatively insensitiveto the componentof the lander positionparallel to the Mars spinaxis.This componentis determinedby the range data, and its accuracyis influencedby the accuracywith which the ephemerisand chargedparticle calibrationswere made. The lander tracking data contain no information on the
ASHBROOK•
22.67
rotational period as 24 h, 37 min, 22.655 + 0.013 s. The results of the lander analysis,along with those of Wislicenus,Bakhuyzen, and Ashbrook, as obtained from earth-basedobservations [Michaux, 1967], and those of de Vaucouleurset al. [1973], obtained from Mariner 9 data, are comparedin Figure
from
althoughthe landerperpendiculardistancefrom the Mars spin axis, distancefrom the equator parallel to the rotation axis, and longitude were actually estimated from the data. The
The rotationalperiod of Mars is determinedvery effectively from trackinglandedspacecrafts.The rotation period has also been determined from earth-based observations spanning some 3 centuries.Wislicenus analyzed the earth-basedobsermade from
as determined
lengths of lander tracking data are shown in Table 4. The lander latitudes, longitudes,and radii are shown in the table,
VIKING PRELIMI NARY•.
BAKHUYZEN122.663 + 0.002 secl .1 ' I
de VAUCOULEURS !
NISLICENUS
24hr, 37 min, 22.64 1860
1•90
1•20
1•50
YEAR
Fig. 4.
Mars siderealrotation period.
19180
4302
RADIO SCIENCE CONCLUSION
35X102
Approximately6 monthsof landerdatahavebeenanalyzed. The lander locations, Mars rotation axis orientation, and rota-
30
tion periodwere accuratelydetermined.The analysesindicate that the additional tracking data of the Viking extendedmissionare requiredbeforesignificantimprovementsmay be obtained in the Mars rotation axis motion and parametersaffecting the earth and Mars orbits.
25
=
I
i
j
f-QUARTICPOLYNOMIAL FIT
20
15
REFERENCES
INC REMENTAL
RANGE, meters 10 (MEASUREDPREDICTED)
Blackshear, W. T., R. H. Tolson, and G. M. Day, Mars lander position estimation in the presenceof ephemerisbiases,J. $pacecr. Rockets, 10, 284, 1973. Born, G. H., E. J. Christensen, S. N. Mohan, J. F. Jordan, and T. C.
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0
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I '
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I
:tlillllJi lilll[11[JlllllllllJtllllllll
-150
6
9
rms= 9 meters illlllill
12
IJllllll[
15
JillJill Illlll[11
18
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Duxbury, Determination of the Mars spin axisdirectionfrom Mariner 9, paper presentedat meeting of the AAS Planetary Science Division, Amer. Astron. Soc., Kona, Hawaii, March 20-24, 1972. Daniels, E. F., and R. H. Tolson, Sphericalharmonic representation of the gravity field of Mars using a short arc technique, paper presentedat the AIAA/AAS AstrodynamicsConference, Amer. Inst. of Aeronaut. and Astronaut., Amer. Astron. Soc., San Diego, Calif., Aug. 18-20, 1976. Davies, M. E., and D. W. G. Arthur, Martian surface coordinates, J. Geophys.Res., 78, 4355, 1973. de Vaucouleurs,G., The physicalephemerisof Mars, Icarus, 3, 236, 1964.
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location of the Mars prime meridian.The right ascensionof the Mars prime meridian, passingthrough crater Airy-O, was determined from the Mariner 9 photographsto be at right ascension 4.376 ø at 0000 hours, November 27, 1971 (J.D.
2441282.5)[de Vaucouleurs et al., 1973].The uncertaintyin the right ascensionof the crater at the 1971 epoch was 10 km [Dat)iesand Arthur, 1973]. Owing to the uncertaintyof the Mars spin rate since 1971, the prime meridian uncertaintyat the 1976 epochis about 11 km. Until improvedknowledgeis obtained on the right ascensionof crater Airy-O, the longitudes
of the'landers arelimitedto an accuracy of about11km or 0.2ø. The right ascensionsof the landers, in contrast, are determined to an accuracyof 0.002ø over the data arc and are 277.314ø (VL 1) and 99.546ø (VL 2) at 0000 hours,January 1, 1977 (J.D. 2443144.5). PLANETARY
EPHEMERIS
AND HELIOCENTRIC
PARAMETERS
The lander tracking data contain information on the earth and Mars planetary ephemeridesas well as on the Mars physicalephemeris.The influenceof planetaryephemeriserrors was removed from the analysesthrough the use of the orbiter range data, as was previously discussed,and the remaining information on the Mars physical ephemeriswas analyzed. The differences between the measured range to Mars, as reducedfrom the orbiter data, and the range to Mars,
as predictedby the pre-Viking developmentephemerisDE 96, are shown in Figure 5. The additional data of the extended Viking missionmay permit a combinedsimultaneoussolution for both the physicaland the planetaryephemeris.In this case the analyseswould investigatethe parameters affecting the planetaryephemerissuchas the solaroblateness,variation of the universalgravitational constant,asteroidmass,and relativistic effecton planetarymotion, as well as the determination of the elements of the earth and Mars orbits.
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Komarek, T. A., Ranging calibration data for June (reported monthly). Rep. IOM 3395-76-139, Jet Propul. Lab., Pasadena, Calif., July 7, 1976. Lorell, J., J. D. Anderson, J. F. Jordan, R. D. Reasenberg,and I. I. Shapiro,Celestialmechanicsexperiment,Mariner Mars 1971project final report, vol. 5, Tech. Rep. 32-1550, p. 22, Jet Propul. Lab., Pasadena,Calif., Aug. 20, 1973. Lowell, P., Precessionof the Martian equinoxes,Astron. J., 28(21), 169, 1914.
Madrid, G. A., The measurementof dispersiveeffectsusingthe Mariner 10 S and X band spacecraftto station link, Deep SpaceNetwork Progr. Rep. 42-22, May-June 1974, p. 22, Jet Propul. Lab., Pasadena, Calif., 1974.
Madrid, G. A., C. C. Chao, H. F. Fliegal, R. K. Leavitt, N. A. Mottinger, F. B. Winn, R. N. Wimberly, K. B. Yip, and J. W. Zielenbach, Tracking systemanalytic calibration activities for the Mariner Mars 1971 mission, internal document, Jet Propul. Lab., March I, 1974. (For update of constants,see Critical planetary constants, l/iking Flight Team Memo. FPAG-14969-WJO KRW FPAG, Jet Propul. Lab., Pasadena,Calif., April 30, 1976.) Michael, W. H., Jr., R. H. Tolson, A. P. Mayo, W. T. Blackshear, G. M. Kelly, D. L. Cain, J.P. Brenkle, I. I. Shapiro, and R. D. Reasenberg,Viking lander locationand spin axisof Mars: Determination from radio tracking data, Science,193, 803, 1976a. Michael, W. H., Jr., A. P. Mayo, W. T. Blackshear, R. H. Tolson, G. M. Kelly, J.P. Brenkle, D. L. Cain, G. Fjelbo, 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, atmosphericand surface properties: Determination from Viking tracking data, Science, 194, 1337, 1976b. Michaux, C. M., Handbook of the physicalpropertiesof the planet Mars, NASA Spec. Publ. 3030, 31, 1967. Moyer, T. D., Mathematicalformulationof the doubleprecisionorbit
MAYO ET AL.: VIKING LANDERLOCATIONS
determinationprogram (DPODP), Tech.Rep. 32-1527, Jet Propul. Lab., Pasadena,Calif., May 15, 1971. Sinclair, A. T., The motionsof the satellitesof Mars, Mon. Not. Roy.
Sturms, F. M., Jr., Polynomialexpressions for the planetaryequators and orbits with respectto the mean 1950.0coordinatesystem,Tech. Rep. 32-1508, p. 8, Jet Propul. ,,ab., Pasadena,Calif., Jan. 15, 1971.
Astron. Soc., 155, 249, 1972.
Standish,E. M., M. S. W. Keesey,and XX Newhall, Jet Propulsion Laboratory developmentephemerisnumber 96, Tech.Rep. 32-1603, Jet Propul. Lab., Pasadena,Calif., 1976.
4303
(Received April 7, 1977; revised May 13, 1977; accepted May 13, 1977.)
