Planetary and Space Science ] (]]]]) ]]]–]]]

Contents lists available at SciVerse ScienceDirect

Planetary and Space Science

0032-06

doi:10.1

n Corr

E-m

PleasPlane

journal homepage: www.elsevier.com/locate/pss

Measurement of the radius of Mercury by radio occultation duringthe MESSENGER flybys

Mark E. Perry a,n, Daniel S. Kahan b, Olivier S. Barnouin a, Carolyn M. Ernst a, Sean C. Solomon c,Maria T. Zuber d, David E. Smith d, Roger J. Phillips e, Dipak K. Srinivasan a, Jurgen Oberst f,Sami W. Asmar b

a Planetary Exploration Group, Johns Hopkins University Applied Physics Laboratory, Laurel, MD 21044, USAb Jet Propulsion Laboratory, Pasadena, CA 91109, USAc Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, DC 20015, USAd Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USAe Planetary Science Directorate, Southwest Research Institute, Boulder, CO 80302, USAf German Aerospace Center, Institute of Planetary Research, D-12489 Berlin, Germany

a r t i c l e i n f o

Article history:

Received 8 November 2010

Received in revised form

28 July 2011

Accepted 29 July 2011

Keywords:

Mercury

MESSENGER

Occultation

RF

Radius

33/$ - see front matter & 2011 Elsevier Ltd. A

016/j.pss.2011.07.022

esponding author.

ail address: mark.perry@jhuapl.edu (M.E. Perr

e cite this article as: Perry, M.E., et at. Space Sci. (2011), doi:10.1016/j.p

a b s t r a c t

The MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) spacecraft

completed three flybys of Mercury in 2008–2009. During the first and third of those flybys, MESSENGER

passed behind the planet from the perspective of Earth, occulting the radio-frequency (RF) transmis-

sions. The occultation start and end times, recovered with 0.1 s accuracy or better by fitting edge-

diffraction patterns to the RF power history, are used to estimate Mercury’s radius at the tangent point

of the RF path. To relate the measured radius to the planet shape, we evaluate local topography using

images to identify the high-elevation feature that defines the RF path or using altimeter data to

quantify surface roughness. Radius measurements are accurate to 150 m, and uncertainty in the

average radius of the surrounding terrain, after adjustments are made from the local high at the tangent

point of the RF path, is 350 m. The results are consistent with Mercury’s equatorial shape as inferred

from observations by the Mercury Laser Altimeter and ground-based radar. The three independent

estimates of radius from occultation events collectively yield a mean radius for Mercury of

2439.270.5 km.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The shape of a differentiated planetary body provides con-straints on its internal structure, its thermal and rotationalevolution, the degree of compensation of large-scale topography,and other physical properties. Planet-shape data are of particularinterest in studies of Mercury’s interior structure because of theplanet’s large core and the indications that at least the outer coreis molten (Margot et al., 2007; Hauck et al., 2007). Before 2008,the best shape information for Mercury came from Earth-basedradar data, which were confined to within 101 of latitude of theequator (Harmon et al., 1986; Anderson et al., 1996). Since 2008,the MErcury Surface, Space ENvironment, GEochemistry, andRanging (MESSENGER) spacecraft has flown by Mercury threetimes (Solomon et al., 2008), providing new data on the planet’sequatorial shape and radius (Zuber et al., 2008; Smith et al., 2010;Oberst et al., 2010).

ll rights reserved.

y).

l., Measurement of the radiuss.2011.07.022

This paper reports on the three radio-frequency (RF) occultationevents that occurred during MESSENGER’s flybys. We derive radiusmeasurements from observations of the occultations, estimateuncertainties, and relate the results to current knowledge ofMercury’s shape. The three occultation events are the ingress andegress during the first flyby of Mercury (M1) and egress during thethird flyby of Mercury (M3). Mercury did not occult the spacecraftduring the second flyby (M2), and a spacecraft anomaly preventedobservation of the ingress during the third flyby.

During the start and the end of an occultation, the RF signalamplitude displays a diffraction pattern that contains informationneeded to extract the time of occultation. Combined with accu-rate position data, the time of occultation defines the path of theRF transmission that just grazes the surface, a line that is tangentto the outer surface of the planet (Fjeldbo et al., 1976; Smith andZuber, 1996; Asmar et al., 1999). The grazing RF path provides theradius of the planet at the point where the grazing path intersectsthe surface.

High-standing topographic relief on the surface can interceptthe RF signal and define a grazing RF path that is 1 km or moreabove the average height of the surrounding terrain and 50 km

s of Mercury by radio occultation during the MESSENGER flybys.

M.E. Perry et al. / Planetary and Space Science ] (]]]]) ]]]–]]]2

or more from a tangent point calculated on the basis of a smoothsphere. To relate an occultation-derived radius measurement tothe general shape of the planet at the point of measurement, wemust understand the topography where the grazing path istangent to the surface. The grazing RF path can be well defined,but the relationship of the grazing path to the long-wavelengthplanet shape may be poorly determined. To find the intersectingfeature, we examine images. If a clear intersecting edge is notfound, we use surface roughness characteristics as revealed byMESSENGER’s Mercury Laser Altimeter (MLA) data to relatestatistically the radius measurement to the broad-scale planetshape and quantify the uncertainty in that relationship.

Our understanding of Mercury’s shape comes from MLA(Cavanaugh et al., 2007; Zuber et al., 2008), images of Mercury’slimb (Oberst et al., this issue), and Earth-based radar data. RFoccultation measurements make two primary contributions tothese other data sets: (1) absolutely calibrated estimates of radiuswith a set of error sources that are largely independent of theother planet-shape data; and (2) near-global, albeit sparse, cover-age. Of particular importance will be measurements of Mercury’sradius from occultation events in the southern hemisphere, whichis beyond MLA’s range due to MESSENGER’s elliptical orbit and itsperiapsis at 60–701N.

2. Data analysis

There are three steps to the use of RF occultations to deriveknowledge of Mercury’s shape:

1.

PP

Obtain the time of occultation ingress and egress by extractingpower levels from the RF data, and then compare the levels tothe calculated diffraction pattern.

2.

Use the known position of MESSENGER relative to Mercury toconvert the time of each occultation event to the radius at thepoint where the RF path grazes the surface.

3.

Use all available information on local topography to relate thecalculated radius to the large-scale shape of the planet.

2.1. Diffraction and the time of occultation

Mercury’s surface does not cause an instantaneous transitionin MESSENGER’s RF transmissions as observed from Earth. Thetransmissions diffract around Mercury’s surface, displaying aclassical edge-diffraction pattern as recorded at the groundantennas of NASA’s Deep Space Network (DSN). To locate thepoint of geometric occultation, where the spacecraft, Mercury’ssurface, and the DSN antenna all lie along a single straight line, wemust fit a diffraction pattern to the time history of the RF power.Mercury’s surface-bounded exosphere is too tenuous to affectradio-wave propagation or the diffraction pattern.

For each occultation event, we recorded the RF transmissionsusing the Radio Science Receiver (RSR), an open-loop receiver(Kwok, 2010). The RSRs are installed at the DSN antennas andoperated by the Jet Propulsion Laboratory (JPL) Radio ScienceSystems Group. RSR data are recorded at a rate between 16 kb/sand 32 Mb/s. Depending on the signal strength and on the analysisrequirements, the data are integrated to a time resolution thatprovides the needed signal-to-noise ratio for the intended analyses.For the flyby occultations, the data were collected by the RSR at 16bits per sample at four bandwidths: 1, 16, 50, and 100 kHz. Weused the 1 kHz data, where available. For M1 egress, we usedthe 50 kHz data because the 1 kHz bandwidth did not contain theMESSENGER signal. Communications were coherent entering theM1 occultation, and they remained coherent through the period

lease cite this article as: Perry, M.E., et al., Measurement of the radiulanet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022

covered by the measurements analyzed in this paper. For M1egress, communications were non-coherent during the analyzedperiod. For M3 egress, the MESSENGER transponder locked ontothe diffracted uplink approximately 0.3 s before the time ofgeometric occultation, and the analyzed data are all coherent.

The quality of the extracted RF power history, as measured bytime resolution and noise, degrades for low levels of RF signalpower. Operational constraints require that MESSENGER use itslow-gain antennas (LGAs) during the periods when most occulta-tions occur (Srinivasan et al., 2007). The low power delivered bythe LGAs at the ground tracking stations causes low signal-to-noise ratios and obscures the diffraction patterns. LGA power isfurther reduced if the LGA boresight is not pointed toward Earth.For the occultation ingress and egress times during M1, the anglebetween the spacecraft-Earth line and the LGA boresight waswithin 601. With a 70 m ground-station antenna, the signal-to-noise spectral density ratio, Pc/No, where Pc is carrier power(W) and No is the noise spectral density in W/Hz, was 23 dB Hzduring M1. Power was higher for M3 egress. The spacecraftanomaly, noted above, triggered a reconfiguration of the RFsystem onto the medium-gain fan-beam antenna, increasing theunocculted RF power from 23 dB Hz to 35 dB Hz.

We used both the lower-power M1 occultation data and thehigher-power M3 egress data to evaluate several software tech-niques for extracting the carrier-frequency power levels from theRSR data: fast Fourier transform (FFT), total power in the in-phase(I) and (out-of-phase) quadrature (Q) components, and a softwarephase-locked loop (PLL). None of these techniques is ideal. TheFFT routine has good accuracy when power exceeds 30 dB Hz, butnoise spikes of 20 dB Hz prevent monitoring the RF diffractionpattern at the lower power levels typical of the LGA. Similarly, theI/Q-power technique provides excellent time resolution at thehigher power levels and minimal information at low power.Summing the squares of the I and Q components produces thetotal power in the bandpass, which puts all noise within thebandpass into the result, interfering with tracking of the diffrac-tion pattern to low power levels. Results from the FFT techniqueand I/Q-power agree for M3 egress, the occultation event withhighest RF power. For the lower-power M1 events, both techni-ques have noise floors that are within 5 or 6 dB of the unoccultedpower level, particularly when using the wider bandpass.

The PLL technique can track the RF signal to low power levelsusing a narrow tracking bandwidth of 5–10 Hz. By tracking thefrequency, the PLL routine captures the power in the RF transmis-sions without including other noise in the bandpass. Unfortu-nately, the RF signal is broader than 10 Hz, and a narrowbandwidth distorts the RF power history because it captures onlya portion of the signal power. We evaluated several bandwidthsand found that 25–50 Hz provides a good compromise betweenaccurate capture of the total power and tracking to low powerlevels. The PLL technique enabled the extraction of additionaldetails of the diffraction pattern for the low-power M1 events.

The diffraction pattern of an occultation event is set by thespecific geometry and velocities of that occultation. For a planetarybody occulting RF radiation, the observed power levels follow thepredictions of edge diffraction theory. The diffraction pattern at theobserver – the DSN station – can be parameterized with a scalelength, u, which depends on the wavelength of the radiation and onthe geometry of the source, edge, and observer. For the MESSEN-GER flybys, where the distance to Earth is much greater than thedistance between MESSENGER and Mercury, the appropriate para-meterization of the standard Fresnel diffraction result is

u¼

ffiffiffiffiffiffiffiffid2

El2dS

sfor dEbdS ð1Þ

s of Mercury by radio occultation during the MESSENGER flybys.

M.E. Perry et al. / Planetary and Space Science ] (]]]]) ]]]–]]] 3

where dE is the distance from the Earth to the edge (the surface ofMercury), dS is the distance from the spacecraft to the edge, andl is the RF wavelength. Fresnel’s integrals describe the diffractionpattern as a function of u (Jenkins and White, 1976). The period ofthe first oscillation in the diffraction pattern is approximately 2u.

Incorporating the relative velocities of MESSENGER, Mercury,and Earth, we converted the distance-based diffraction pattern,e(u), to a time-based pattern, e(t), that we compared directly tothe RF power levels observed during the occultations. At the timeof the M3 egress, MESSENGER was 14,380 km from the center ofmass (COM) of Mercury. The scale length, u, for edge-diffractioneffects with this geometry is 4220 km and translates to a timescale of 0.358 s per u for diffraction patterns at the DSN receivingantenna at Goldstone, California.

The RF power for the M3 egress, plotted in Fig. 1, shows a slowrise, a peak above the unocculted power, and oscillations, all ofwhich are signatures of edge diffraction. The desired time, thetime of geometric occultation, is when the power level is one-fourth of the unocculted power level (i.e., down by 6 dB) (Jenkinsand White, 1976). By fitting a diffraction pattern to the RF datasurrounding an occultation event, we can locate the desired powerlevel with an accuracy that is better than the time resolution of theindividual data points. Because (1) the shape of the diffractionpattern is determined by the geometry, and (2) the verticalposition of the pattern is determined by the unocculted powerlevel, the only free parameter is the time, which is the horizontalposition of the diffraction pattern in Fig. 1. A least-squares analysisfor the M3 egress indicates that the time of geometric occultation,the point when the signal was 6 dB below its unocculted value of35 dB Hz, occurred at 23:00:13.5570.03 s UTC. An arrow in Fig. 1shows the time of geometric occultation.

Because of the lower RF power levels for the M1 occultationevents, the diffraction patterns are barely detectable, even withoptimal PLL parameters. The uncertainty in the time of geometricoccultation for these events is estimated to be 0.1 s, from thedispersion among the results using different techniques for

23:00:13.55 ± 0.03 Time of geometric occultation

6 dB-Hz below un-occulted level

Un-occulted level

Seconds

dB-H

z

Fig. 1. RF signal strength (diamond symbols) measured by the RSR at DSS-14, the

70 m DSN antenna at Goldstone, at the time that MESSENGER emerged from its

occultation behind Mercury during M3. The data were acquired at 50 kHz and

averaged over intervals of 0.1 s. The time axis shows the number of seconds past

23:00:00 UTC on 29 September 2009. The power levels were extracted using an

FFT algorithm. The data show a gradual rise, a peak above the final power

(35 dB Hz), and ‘‘ringing,’’ all consequences of diffraction. The solid line shows

the result of a least-squares fit of the diffraction pattern that is determined by the

geometry at the time of egress. Geometric occultation – where the source, edge,

and receiver are all on a straight line – is the point where received power is one-

fourth of the unocculted level. Because the fit involves many points, the

uncertainty in the occultation time is less than the time resolution of the

individual data points.

Please cite this article as: Perry, M.E., et al., Measurement of the radiuPlanet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022

extracting the RF signal levels. With the FFT technique, the timeof geometric occultation occurred when the RF signal was belowthe noise level. Under this condition, the time of geometricoccultation is not observed, and the timing of the diffractionpattern must be determined by the timing and features of thoseportions of the signal that exceed the noise level.

2.2. Measured radius and uncertainty calculations

To convert occultation times to radius, and to assess uncer-tainties, we first constructed a line between the spacecraft andthe receiving antenna at Goldstone at the time of the occultationevent. This line is the RF path that just grazes the surface ofMercury. The point on that line that is closest to Mercury’s COM isthe tangent point that we take to be the diffraction edge. Thedistance of the point from Mercury’s COM is the local radius, andthe location of that point in Mercury coordinates is given by itslatitude and longitude. Table 1 lists the calculated radius andlocation for each of the MESSENGER flyby occultation events.Fig. 2 shows the location of each occultation measurement on aglobal image mosaic of Mercury.

There are two sources of uncertainty in these calculations ofradius: uncertainty in the spacecraft position and uncertainty inthe time of occultation. The uncertainty in spacecraft positionwith respect to Mercury, sPosition, contributes directly to uncer-tainty in radius. The MESSENGER navigation team produced thespacecraft trajectory by analyzing data from many weeks beforeand after the encounter to reduce uncertainty in the spacecraftposition. Data included radiometric range and range-rate data,optical images, and results of delta-differential one-way rangingobservations (Lanyl et al., 2007). These long, dense data setsenabled the navigation team to reduce trajectory uncertainties toless than 30 m, at one standard deviation (s), with respect toMercury during the period that the spacecraft was within onehour of closest approach (A.T. Taylor, 2008, personal communica-tion). The calculations were performed in a Mercury-centeredcoordinate system, so the uncertainty in Mercury’s position,which is approximately 1 km relative to Earth, is independent ofthe uncertainty with respect to Mercury and is a negligiblecontribution to dE.

The achievable resolution, sTime, of the time of geometricoccultation depends on the diffraction fit quality, which varieswith RF power. For M1, the gain was sufficient for an uncertainty

Table 1Radius and surface location for each occultation event.

Event Radius (km)a Lat

(1N)

Long

(1E)Measured Adjusted for

topographyb

Deviation

from MLA

fitc

M-10 ingress 2439.5 1.1 67.4

M-10 egress 2439.0 67.6 258.4

M1 ingress 2437.8 2437.3 �1.6 �25.54 225.28

M1 egress 2440.4 2439.9 �0.4 �7.33 41.83

M3 egress 2441.65 2440.5 �0.1 36.06 28.23

Mean radius �0.7 (2439.2)

a The radius is with respect to Mercury’s center of mass.b The adjusted values are estimates of the radius of the surrounding terrain

after subtracting the height of topographic features inferred to have influenced the

measurement.c The difference between the MESSENGER occultation-derived measurements

and the radius at the corresponding longitude as determined by the spherical

harmonic fit to MLA data to degree 2. The mean radius reflects the average

difference of these measurements from the 2439.9 km radius derived from the

MLA fit.

s of Mercury by radio occultation during the MESSENGER flybys.

Fig. 2. Locations of the three measurements of Mercury’s radius from MESSENGER flyby radio occultations, displayed on a cylindrical projection of an image mosaic of

Mercury surface features (Becker et al., 2009). Zero longitude is at the center of the map. Expanded views of each of the boxed areas are shown in Fig. 4. The images for the

area surrounding the radius measurement during M1 ingress are from Mariner 10. The best images for the locations of the other two occultation events are MDIS images

from the flybys, but both locations were near the limbs at the times the images were obtained, so image quality is much less than will be available now that MESSENGER is

in orbit about Mercury.

Table 2Sources and estimates of uncertainty for the M1 and M3 radio occultation

measurements.

Event VPERP

(km/s)

Uncertainties, 1s Comments on

topography

Spacecraft

position (m)

Time (s) Radius,

total (m)

M1 ingress 1.6 30 0.1 160 No obvious

topographic features

M1 egress 1.1 30 0.1 120 No obvious

topographic features

M3 egress 1.4 20 0.03 50 Poor-quality image,

but likely crater rim

Notes: VPERP is the velocity perpendicular to the line of sight and normal to the

surface. Position uncertainty and velocity are orthogonal to the line of sight. This

table does not include uncertainties associated with local topography, which are

7350 m (see text).

M.E. Perry et al. / Planetary and Space Science ] (]]]]) ]]]–]]]4

in the diffraction fit of 0.1 s, 1s, for both ingress and egress.During M3, the higher RF power at egress resulted in a better fit,with an uncertainty of 0.03 s.

Multiplying the time uncertainty by VPERP, the velocity per-pendicular to the line of sight and normal to the surface, convertsthe timing uncertainty to a radius uncertainty. The parameterVPERP is a relative velocity that includes corrections for the relativemotions of Mercury and Earth. The uncertainty in VPERP isnegligible, less than one part in 105. The total uncertainty in thecalculated radius, sTotal, is the root sum of the squares of the twoindependent uncertainties

sTotal ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

PositionþðsTimeVPERPÞ2

qð2Þ

Table 2 summarizes these uncertainties for the three measure-ments of radius by MESSENGER.

2.3. Relating the radius measurements to planet shape

Depending on the number of fitted radii, the spatial resolutionof the global shape model from occultations alone is hundreds tothousands of kilometers. Relative to these scales, a radius derivedfrom an occultation event is essentially a point measurement, andit may not represent the average radius over large areas, or eventhe average radius within tens of kilometers of the occultationpoint. However, with data available prior to MESSENGER’s entryinto orbit about Mercury, we can estimate the effects of localtopographic variations on the radius measurement, deduce theaverage height of the terrain within tens of kilometers of theradius measurement, and adjust the radius measurement basedon local topography. We take this adjusted height to represent theplanet shape in the vicinity of the occultation-derived radius.During the orbital phase of the MESSENGER mission, we expect torecover this shape information in hundreds of occultation locales.

If there are detailed topographic data near the point where theRF path is tangent to the surface, then those data can be used toidentify the specific feature – the edge – that defines the path ofthe RF transmission that just grazes the surface at the time ofoccultation. If there are no detailed topography data or if topo-graphic analysis fails to identify the intersection feature, analternative approach is to reduce the measured radius by anamount derived from a statistical analysis of surface roughness.

Please cite this article as: Perry, M.E., et al., Measurement of the radiuPlanet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022

The first approach for assessing topography uses data fromimages or altimetry to identify the highest section of the terrainalong the RF path. If an identifiable feature intercepts the grazingRF path, the high terrain then contributes the edge measured inthe occultation analysis, and the location of the radius measure-ment is well determined. The images may also contain informa-tion on the height of the feature relative to the surroundingterrain (e.g., shadow length), and such data can supply thecorrection necessary to estimate the planetary radius correspond-ing to a representative average of the surrounding terrain.

In the absence of an isolated topographic high along thegrazing RF path, an adjustment of the measured radius for theeffects of local topography can be made on the basis of statisticalassessments of surface roughness for appropriate terrain types.Fig. 3 shows two assessments of Mercury’s surface roughnessderived from MLA profiles obtained during M1 and M2 (Zuberet al., 2008; Smith et al., 2010). To obtain a correction for themeasured radius, we modified the standard surface-roughnessanalysis, the square of the height differences versus baseline(this is a form of the Allan variance), which gives the averagepeak-to-peak variations in surface topography over differentdistance scales (Cheng et al., 2001; Barnouin-Jha et al., 2008).Instead of peak-to-peak variations, we determine the most-prob-able distance between the RF grazing path and the averageelevation of the surrounding terrain. This distance is equivalent

s of Mercury by radio occultation during the MESSENGER flybys.

M.E. Perry et al. / Planetary and Space Science ] (]]]]) ]]]–]]] 5

to the mean value of the difference between the highest pointalong a path and the average height along that path

mPeak ¼/maxðh½x : xþL�Þ�/h½x : xþL�SS for all x along track ð3Þ

where h[x:xþL] is the set of MLA measurements for a distance L

from point x, ‘‘max’’ denotes the maximum value within the set,/S indicates mean value, and mPeak is the most-probable height ofthe peak above the average terrain level.

Both roughness assessments in Fig. 3 used all of the MLA flybydata, irrespective of the type of terrain, and the standard devia-tion of the mean values of the peak heights is large: 50–80% of themean. Assessing the peak heights on the basis of terrain type willreduce the standard deviation, but the flyby MLA data are not tiedsufficiently well to terrain type for such a characterization.

Features with horizontal dimensions greater than 30–50 kmcan be identified in the images, so the distance scale for adjustingthe occultation-derived radii is in this range. For L¼40 km, theadjustment serves to reduce the radius by 5007350 m. Althoughthis analysis corrects the magnitude of the radius measurement, itdoes not determine the along-track location of the edge, whichcan be uncertain by 770 km. When available, information on thetopography that has high resolution in both horizontal and

Fig. 3. Surface roughness analysis of the M1 and M2 MLA profiles plotted as a

function of the horizontal scale over which roughness is measured. The upper line

is the standard deviation of the peak-to-peak heights (Barnouin-Jha et al., 2008).

The thick blue line is the most-probable height of the highest terrain above the

average level, and we consider this curve as the most appropriate information for

adjustment of the radius measurements from occultations (see text). With 40 km

as a nominal length scale, the RF grazing path is statistically 500 m above the

average terrain. The height of the highest terrain above the average level is quite

variable; the arrow denotes the standard deviation in this height at a 40 km length

scale.

Fig. 4. Nominal locations (black crosses) of the tangent points of the grazing RF path (r

image is from Mariner 10, and the other two are MDIS images from MESSENGER fl

measurements made during the M1 occultation events, but the location of the radius

120-km-diameter impact crater that is barely visible in this image.

Please cite this article as: Perry, M.E., et al., Measurement of the radiuPlanet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022

vertical coordinates will reduce or remove the uncertainty dueto surface roughness.

To search for local topographic highs, we overlaid the locationof the grazing RF path onto the best available surface images,which are from Mariner 10 or MESSENGER’s Mercury DualImaging System (MDIS) (Hawkins et al., 2007). Results from stereoanalysis of MDIS images (Oberst et al., 2010; Preusker et al., thisissue) would be helpful, but for the locations of the threeoccultation measurements reported here, no stereo topographicanalyses were available. Fig. 4 shows expanded views of the areasof each occultation event. None of the three occultation eventsoccurred in an area with high-resolution MDIS images. There areno distinctive topographic features near the locations of the M1occultation measurements, so we used the probable-peak rela-tionship discussed above to adjust the measured radius (Table 1).

The best images at the tangent location for M3 egress are poor,but there appears to be a 120-km-diameter impact crater near thelocus of the grazing path, and the intercepting edge may be at therim of the crater. There are no data on the rim height from MLAdata, stereo analyses, or shadow measurements, adding to theuncertainty in the radius inferred for the terrain surrounding thecrater. From the relationship of Pike (1988) between crater rimheight and crater diameter for Mercury, the rim height of a fresh,120-km-diameter crater is approximately 1.5 km. The crater maybe degraded, however, in which case the rim height would be less.With no information on the state of preservation of this crater, weuse 1.5 km as the maximum height and assume the same verticaluncertainty as for the surface-roughness measurements. Sub-tracting the vertical uncertainty, 0.35 km, from the maximumheight, 1.5 km, we take the rim to be 1.1570.35 km above thelevel of the surrounding terrain. The M3 egress radius has there-fore been adjusted by this amount (Table 1) to obtain the averageelevation of the surrounding terrain that we compare to the shapederived from MLA equatorial topographic profiles expanded tospherical harmonic degree 2.

3. Discussion

From MLA flyby data (Zuber et al., 2008; Smith et al., 2010) andground-based radar observations (Anderson et al., 1996), Mercury’scenter of figure (COF) in the planet’s equatorial plane is offsetapproximately 600 m from its COM. These same data also show adifference between major and minor equatorial radii of 1.3–1.6 km.Fig. 5 compares the occultation measurements of radius to thespherical harmonic solution to the variation in near-equatorialradius with longitude derived from MLA measurements madeduring M1 and M2 (Smith et al., 2010). Results both before and

ed lines) for each of the three occultation measurements of radius. The M1-ingress

ybys. There are no distinctive topographic features in the vicinity of the radius

measurement at the M3 occultation egress appears to lie at or near the rim of a

s of Mercury by radio occultation during the MESSENGER flybys.

Fig. 5. MESSENGER and Mariner 10 occultation measurements of radius are shown versus longitude together with the spherical harmonic shape (degree 2, black line) fit to

MLA data obtained near Mercury’s equatorial plane during M1 and M2 (Smith et al., 2010). The square symbols show the elevation relative to a sphere of radius 2440 km.

For the MESSENGER measurements, the vertical extent of the squares represents the uncertainties without adjusting for local topography. The circles and their associated

error bars show the results after adjusting the MESSENGER measurements for local topography.

Fig. 6. Calculated locations of the tangent points of the grazing RF paths for the ingress and egress of each occultation during the one-year orbital mission phase of

MESSENGER. Ingress events are red open circles, and egress events are closed blue circles. The orbital operations plan is to capture 90–95% of these occultation events.

M.E. Perry et al. / Planetary and Space Science ] (]]]]) ]]]–]]]6

after adjusting for local topography are shown. The M1 egressresults are near the MLA fit for that longitude, a not-unexpectedresult given that the occultation measurement is at the near-equatorial latitude of 71S. The M3 egress and M1 ingress are eachfarther from the equator, and, depending on the shape of Mercuryat higher latitudes, may deviate from the equatorial radius valuesat a given longitude. After adjusting for topography, the M3 radiusalso agrees with that from MLA, although this agreement is likelyfortuitous, given the latitude at which it applies (361N). The M1egress radius is approximately 1 km less than the MLA-derivedequatorial shape, and we have no explanation other than thelikelihood of latitudinal variation in topography.

To extract a measurement of Mercury’s mean radius from thethree occultation-derived radius measurements, we accountapproximately for the shape of the planet (Fig. 5) at sphericalharmonic degree 2 by subtracting the fitted equatorial radiusfrom each occultation radius. This procedure is predicated onthe assumption that the equatorial degree-two shape can beextended to the latitudes of the occultations. Weighting the threeoccultation measurements of radius by their uncertainties givesan estimated mean radius of 2439.6070.04 km before adjustingfor local topography and 2439.270.2 km after (Table 1).

However, the dispersion of the results about the mean is muchlarger than the uncertainty from known errors, indicating thatthere are additional factors affecting the result, including

Please cite this article as: Perry, M.E., et al., Measurement of the radiuPlanet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022

unmodeled (essentially undersampled) higher-order variationsin planet shape. The standard deviation of the three resultscompared to the degree-2 shape is 0.8 km, and we use this valueto estimate the increase in uncertainty due to these additionalfactors, yielding a standard deviation of 0.5 km for the derivedmean radius.

There are several other observations of Mercury’s radius forcomparison. There were two occultation events from the firstMariner 10 encounter with Mercury (Fjeldbo et al., 1976).Spacecraft position knowledge, with an uncertainty of 1.0 km,determined the overall uncertainty in radius. The Mariner 10occultation radii agree with the near-equatorial shape determinedfrom MLA, within their uncertainties (Fig. 5), despite the fact thatone of the measurements was at high latitude. Oberst et al.(this issue) described the results of limb analyses of Mercuryflyby images and compared their results to other measurementsof Mercury’s radius. The results of the limb analyses are consis-tent with MLA data in equatorial regions where they sample thesame longitudes.

Once MESSENGER is in orbit about Mercury, the planet willoccult MESSENGER’s RF transmissions every 12 h for most of the12-month orbital mission phase (Solomon et al., 2001, 2007;Srinivasan et al., 2007; Zuber et al., 2007). The mission observa-tion plan includes capturing more than 90% of the 660 expectedoccultation events, as shown in Fig. 6.

s of Mercury by radio occultation during the MESSENGER flybys.

M.E. Perry et al. / Planetary and Space Science ] (]]]]) ]]]–]]] 7

Acknowledgments

Details on the mission, flybys, and Mercury orbit insertion aremaintained and updated at the MESSENGER web site: http://messenger.jhuapl.edu/. The MESSENGER mission is supported bythe NASA Discovery Program under contracts NAS5-97271 to theJohns Hopkins University Applied Physics Laboratory and NASW-00002 to the Carnegie Institution of Washington.

References

Anderson, J.D., Jurgens, R.F., Lau, E.L., Slade III, M.A., Schubert, G., 1996. Shape andorientation of Mercury from radar ranging data. Icarus 124, 690–697.

Asmar, S.W., Schubert, G., Konopliv, A.S., Moore, W., 1999. Improving on the highlatitude topography of the Moon via precise timing of the Lunar Prospectorradio occultations (abstract 18.03). Bull. Am. Astron. Soc. 31, 1102.

Barnouin-Jha, O.S., Cheng, A.F., Mukai, T., Abe, S., Hirata, N., Nakamura, R., Gaskell,R.W., Saito, J., Clark, B.E., 2008. Small-scale topography of 25143 Itokawa fromthe Hayabusa laser altimeter. Icarus 198, 108–124.

Becker, K.J., Robinson, M.S., Becker, T.L., Weller, L.A., Turner, S., Nguyen, L., Selby, C.,Denevi, B.W., Murchie, S.L., McNutt, R.L., Solomon, S.C., 2009. Near-globalmosaic of Mercury. Eos Trans. Am. Geophys. Un. 90 (52) (Fall Meeting Suppl.,abstract P21A-1189).

Cavanaugh, J.F., Smith, J.C., Sun, X., Bartels, A.E., Ramos-Izquierdo, L., Krebs, D.J.,McGarry, J.F., Trunzo, R., Novo-Gradac, A.M., Britt, J.L., Karsh, J., Katz, R.B.,Lukemire, A.T., Symkiewicz, R., Berry, D.L., Swinski, J.P., Neumann, G.A.,Zuber, M.T., Smith, D.E., 2007. The Mercury Laser Altimeter instrument for theMESSENGER mission. Space Sci. Rev. 131, 451–480.

Cheng, A.F., Barnouin-Jha, O.S., Zuber, M.T., Veverka, J.F., Smith, D.E., Neumann, G.A.,Robinson, M.S., Thomas, P.C., Garvin, J.B., Murchie, S.L., Chapman, C.R.,Prockter, L.M., 2001. Laser altimetry of small-scale features on 433 Eros fromNEAR-Shoemaker. Science 292, 488–491.

Fjeldbo, G., Kliore, A., Sweetnam, D., Esposito, P., Seidel, B., Howard, T., 1976.The occultation of Mariner 10 by Mercury. Icarus 27, 439–444.

Harmon, J.K., Campbell, D.B., Bindschadler, D.L., Head, J.W., Shapiro, I.I., 1986.Radar altimetry of Mercury: a preliminary analysis. J. Geophys. Res. 91,385–401.

Hauck II, S.A., Solomon, S.C., Smith, D.A., 2007. Predicted recovery of Mercury’sinternal structure by MESSENGER. Geophys. Res. Lett. 34, L18201.doi:10.1029/2007GL030793.

Hawkins III, S.E., Boldt, J.D., Darlington, D.H., Espiritu, R., Gold, R.E., Gotwols, B.,Grey, M.P., Hash, C.D., Hayes, J.R., Jaskulek, S.E., Kardian Jr., C.J., Keller, M.R.,Malaret, E.R., Murchie, S.L., Murphy, P.K., Peacock, K., Prockter, L.M., Reiter,R.A., Robinson, M.S., Schaefer, E.D., Shelton, R.G., Sterner II, R.E., Taylor, H.W.,Watters, T.R., Williams, B.D., 2007. The Mercury Dual Imaging System on theMESSENGER spacecraft. Space Sci. Rev. 131, 247–338.

Jenkins, F.A., White, H.E., 1976. Fundamentals of Optics, 4th ed. McGraw-Hill, NewYork, pp. 388–396.

Please cite this article as: Perry, M.E., et al., Measurement of the radiuPlanet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022

Kwok, A., 2010. Deep Space Network Telecommunications Link Design Handbook810-005, Module 209, Rev. A, Open-Loop Radio Science. California Institute ofTechnology, Pasadena, Calif. 19 pp.

Lanyl, G., Bagri, D.S., Border, J.S., 2007. Angular position determination of space-craft by radio interferometry. Proc. IEEE 95, 2193–2201. doi:10.1109/JPROC.2007.905183.

Margot, J.L., Peale, S.J., Jurgens, R.F., Slade, M.A., Holin, I.V., 2007. Large longitudelibration of Mercury reveals a molten core. Science 316, 710–714.

Oberst, J., Preusker, F., Phillips, R.J., Watters, T.R., Head, J.W., Zuber, M.T., Solomon, S.C.,2010. The morphology of Mercury’s Caloris basin as seen in MESSENGER stereotopographic models. Icarus 209, 230–238.

Oberst, J., Elgner, S., Turner, S., Perry, M.E., Gaskell, R.W., Zuber, M.T., Solomon, S.C.Radius and limb topography of Mercury obtained from images acquired duringthe MESSENGER flybys. Planet. Space Sci., this issue. doi:10.1016/j.pss.2011.07.003.

Pike, R.J., 1988. Geomorphology of impact craters on Mercury. In: Vilas, F.,Chapman, C.R., Matthews, M.S. (Eds.), Mercury. University of Arizona Press,Tucson, Ariz. pp. 165–273.

Preusker, F., Oberst, J., Head, J.W., Watters, T.R., Robinson, M.S., Zuber, M.T.,Solomon, S.C. Stereo topographic models of Mercury after three MESSENGERflybys. Planet Space Sci., this issue. doi:10.1016/j.pss.2011.07.005.

Smith, D.E., Zuber, M.T., 1996. The shape of Mars and the topographic signature ofthe hemispheric dichotomy. Science 271, 184–188.

Smith, D.E., Zuber, M.T., Phillips, R.J., Solomon, S.C., Neumann, G.A., Lemoine, F.G.,Peale, S.J., Margot, J.-L., Torrence, M.H., Talpe, M.J., Head III, J.W., Hauck II, S.A.,Johnson, C.L., Perry, M.E., Barnouin, O.S., McNutt Jr., R.L., Oberst, J., 2010. Theequatorial shape and gravity field of Mercury from MESSENGER flybys 1 and 2.Icarus 209, 88–100.

Solomon, S.C., McNutt Jr., R.L., Gold, R.E., Acuna, M.H., Baker, D.N., Boynton, W.V.,Chapman, C.R., Cheng, A.F., Gloeckler, G., Head III, J.W., Krimigis, S.M.,McClintock, W.E., Murchie, S.L., Peale, S.J., Phillips, R.J., Robinson, M.S., Slavin,J.A., Smith, D.E., Strom, R.G., Trombka, J.I., Zuber, M.T., 2001. The MESSENGERmission to Mercury: scientific objectives and implementation. Planet SpaceSci. 49, 1445–1465.

Solomon, S.C., McNutt Jr., R.L., Gold, R.E., Domingue, D.L., 2007. MESSENGERmission overview. Space Sci. Rev. 131, 3–39.

Solomon, S.C., McNutt Jr., R.L., Watters, T.R., Lawrence, D.J., Feldman, W.C., Head,J.W., Krimigis, S.M., Murchie, S.L., Phillips, R.J., Slavin, J.A., Zuber, M.T., 2008.Return to Mercury: a global perspective on MESSENGER’s first Mercury flyby.Science 321, 59–62.

Srinivasan, D.K., Perry, M.E., Fielhauer, K.B., Smith, D.E., Zuber, M.T., 2007.The radio frequency subsystem and radio science on the MESSENGER mission.Space Sci. Rev. 131, 557–571.

Zuber, M.T., Aharonson, O., Aurnou, J.M., Cheng, A.F., Hauck II, S.A., Heimpel, M.H.,Neumann, G.A., Peale, S.J., Phillips, R.J., Smith, D.E., Solomon, S.C., Stanley, S.,2007. The geophysics of Mercury: current knowledge and future opportu-nities. Space Sci. Rev. 131, 105–132.

Zuber, M.T., Smith, D.E., Solomon, S.C., Phillips, R.J., Peale, S.J., Head III, J.W.,Hauck II, S.A., McNutt Jr., R.L., Oberst, J., Neumann, G.A., Lemoine, F.G., Sun, X.,Barnouin-Jha, O., Harmon, J.K., 2008. Laser altimeter observations fromMESSENGER’s first Mercury flyby. Science 321, 77–79.

s of Mercury by radio occultation during the MESSENGER flybys.