Planetary and Space Science ] (]]]]) ]]]–]]]
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Planetary and Space Science
journal homepage: www.elsevier.com/locate/pss
Measurement of the radius of Mercury by radio occultation during
the 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, Jürgen Oberst f,
Sami W. Asmar b
a
Planetary Exploration Group, Johns Hopkins University Applied Physics Laboratory, Laurel, MD 21044, USA
Jet Propulsion Laboratory, Pasadena, CA 91109, USA
c
Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, DC 20015, USA
d
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
e
Planetary Science Directorate, Southwest Research Institute, Boulder, CO 80302, USA
f
German Aerospace Center, Institute of Planetary Research, D-12489 Berlin, Germany
b
a r t i c l e i n f o
abstract
Article history:
Received 8 November 2010
Received in revised form
28 July 2011
Accepted 29 July 2011
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) transmissions. The occultation start and end times, recovered with 0.1 s accuracy or better by fitting edgediffraction 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
2439.2 7 0.5 km.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Mercury
MESSENGER
Occultation
RF
Radius
1. Introduction
The shape of a differentiated planetary body provides constraints on its internal structure, its thermal and rotational
evolution, the degree of compensation of large-scale topography,
and other physical properties. Planet-shape data are of particular
interest in studies of Mercury’s interior structure because of the
planet’s large core and the indications that at least the outer core
is molten (Margot et al., 2007; Hauck et al., 2007). Before 2008,
the best shape information for Mercury came from Earth-based
radar data, which were confined to within 101 of latitude of the
equator (Harmon et al., 1986; Anderson et al., 1996). Since 2008,
the MErcury Surface, Space ENvironment, GEochemistry, and
Ranging (MESSENGER) spacecraft has flown by Mercury three
times (Solomon et al., 2008), providing new data on the planet’s
equatorial shape and radius (Zuber et al., 2008; Smith et al., 2010;
Oberst et al., 2010).
n
Corresponding author.
E-mail address: [email protected] (M.E. Perry).
This paper reports on the three radio-frequency (RF) occultation
events that occurred during MESSENGER’s flybys. We derive radius
measurements from observations of the occultations, estimate
uncertainties, and relate the results to current knowledge of
Mercury’s shape. The three occultation events are the ingress and
egress during the first flyby of Mercury (M1) and egress during the
third flyby of Mercury (M3). Mercury did not occult the spacecraft
during the second flyby (M2), and a spacecraft anomaly prevented
observation of the ingress during the third flyby.
During the start and the end of an occultation, the RF signal
amplitude displays a diffraction pattern that contains information
needed to extract the time of occultation. Combined with accurate position data, the time of occultation defines the path of the
RF transmission that just grazes the surface, a line that is tangent
to the outer surface of the planet (Fjeldbo et al., 1976; Smith and
Zuber, 1996; Asmar et al., 1999). The grazing RF path provides the
radius of the planet at the point where the grazing path intersects
the surface.
High-standing topographic relief on the surface can intercept
the RF signal and define a grazing RF path that is 1 km or more
above the average height of the surrounding terrain and 50 km
0032-0633/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.pss.2011.07.022
Please cite this article as: Perry, M.E., et al., Measurement of the radius of Mercury by radio occultation during the MESSENGER flybys.
Planet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022
2
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or more from a tangent point calculated on the basis of a smooth
sphere. To relate an occultation-derived radius measurement to
the general shape of the planet at the point of measurement, we
must understand the topography where the grazing path is
tangent to the surface. The grazing RF path can be well defined,
but the relationship of the grazing path to the long-wavelength
planet shape may be poorly determined. To find the intersecting
feature, we examine images. If a clear intersecting edge is not
found, we use surface roughness characteristics as revealed by
MESSENGER’s Mercury Laser Altimeter (MLA) data to relate
statistically the radius measurement to the broad-scale planet
shape 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’s
limb (Oberst et al., this issue), and Earth-based radar data. RF
occultation measurements make two primary contributions to
these other data sets: (1) absolutely calibrated estimates of radius
with a set of error sources that are largely independent of the
other planet-shape data; and (2) near-global, albeit sparse, coverage. Of particular importance will be measurements of Mercury’s
radius from occultation events in the southern hemisphere, which
is beyond MLA’s range due to MESSENGER’s elliptical orbit and its
periapsis at 60–701N.
2. Data analysis
There are three steps to the use of RF occultations to derive
knowledge of Mercury’s shape:
1. Obtain the time of occultation ingress and egress by extracting
power levels from the RF data, and then compare the levels to
the calculated diffraction pattern.
2. Use the known position of MESSENGER relative to Mercury to
convert the time of each occultation event to the radius at the
point where the RF path grazes the surface.
3. Use all available information on local topography to relate the
calculated 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 transition
in MESSENGER’s RF transmissions as observed from Earth. The
transmissions diffract around Mercury’s surface, displaying a
classical edge-diffraction pattern as recorded at the ground
antennas of NASA’s Deep Space Network (DSN). To locate the
point of geometric occultation, where the spacecraft, Mercury’s
surface, and the DSN antenna all lie along a single straight line, we
must fit a diffraction pattern to the time history of the RF power.
Mercury’s surface-bounded exosphere is too tenuous to affect
radio-wave propagation or the diffraction pattern.
For each occultation event, we recorded the RF transmissions
using the Radio Science Receiver (RSR), an open-loop receiver
(Kwok, 2010). The RSRs are installed at the DSN antennas and
operated by the Jet Propulsion Laboratory (JPL) Radio Science
Systems Group. RSR data are recorded at a rate between 16 kb/s
and 32 Mb/s. Depending on the signal strength and on the analysis
requirements, the data are integrated to a time resolution that
provides the needed signal-to-noise ratio for the intended analyses.
For the flyby occultations, the data were collected by the RSR at 16
bits per sample at four bandwidths: 1, 16, 50, and 100 kHz. We
used the 1 kHz data, where available. For M1 egress, we used
the 50 kHz data because the 1 kHz bandwidth did not contain the
MESSENGER signal. Communications were coherent entering the
M1 occultation, and they remained coherent through the period
covered by the measurements analyzed in this paper. For M1
egress, communications were non-coherent during the analyzed
period. For M3 egress, the MESSENGER transponder locked onto
the diffracted uplink approximately 0.3 s before the time of
geometric occultation, and the analyzed data are all coherent.
The quality of the extracted RF power history, as measured by
time resolution and noise, degrades for low levels of RF signal
power. Operational constraints require that MESSENGER use its
low-gain antennas (LGAs) during the periods when most occultations occur (Srinivasan et al., 2007). The low power delivered by
the LGAs at the ground tracking stations causes low signal-tonoise ratios and obscures the diffraction patterns. LGA power is
further reduced if the LGA boresight is not pointed toward Earth.
For the occultation ingress and egress times during M1, the angle
between the spacecraft-Earth line and the LGA boresight was
within 601. With a 70 m ground-station antenna, the signal-tonoise spectral density ratio, Pc/No, where Pc is carrier power
(W) and No is the noise spectral density in W/Hz, was 23 dB Hz
during M1. Power was higher for M3 egress. The spacecraft
anomaly, noted above, triggered a reconfiguration of the RF
system onto the medium-gain fan-beam antenna, increasing the
unocculted RF power from 23 dB Hz to 35 dB Hz.
We used both the lower-power M1 occultation data and the
higher-power M3 egress data to evaluate several software techniques for extracting the carrier-frequency power levels from the
RSR data: fast Fourier transform (FFT), total power in the in-phase
(I) and (out-of-phase) quadrature (Q) components, and a software
phase-locked loop (PLL). None of these techniques is ideal. The
FFT routine has good accuracy when power exceeds 30 dB Hz, but
noise spikes of 20 dB Hz prevent monitoring the RF diffraction
pattern at the lower power levels typical of the LGA. Similarly, the
I/Q-power technique provides excellent time resolution at the
higher power levels and minimal information at low power.
Summing the squares of the I and Q components produces the
total power in the bandpass, which puts all noise within the
bandpass into the result, interfering with tracking of the diffraction pattern to low power levels. Results from the FFT technique
and I/Q-power agree for M3 egress, the occultation event with
highest RF power. For the lower-power M1 events, both techniques have noise floors that are within 5 or 6 dB of the unocculted
power level, particularly when using the wider bandpass.
The PLL technique can track the RF signal to low power levels
using a narrow tracking bandwidth of 5–10 Hz. By tracking the
frequency, the PLL routine captures the power in the RF transmissions without including other noise in the bandpass. Unfortunately, the RF signal is broader than 10 Hz, and a narrow
bandwidth distorts the RF power history because it captures only
a portion of the signal power. We evaluated several bandwidths
and found that 25–50 Hz provides a good compromise between
accurate capture of the total power and tracking to low power
levels. The PLL technique enabled the extraction of additional
details of the diffraction pattern for the low-power M1 events.
The diffraction pattern of an occultation event is set by the
specific geometry and velocities of that occultation. For a planetary
body occulting RF radiation, the observed power levels follow the
predictions of edge diffraction theory. The diffraction pattern at the
observer – the DSN station – can be parameterized with a scale
length, u, which depends on the wavelength of the radiation and on
the geometry of the source, edge, and observer. For the MESSENGER flybys, where the distance to Earth is much greater than the
distance between MESSENGER and Mercury, the appropriate parameterization of the standard Fresnel diffraction result is
u¼
sffiffiffiffiffiffiffiffi
d2E l
2dS
for dE bdS
ð1Þ
Please cite this article as: Perry, M.E., et al., Measurement of the radius of Mercury by radio occultation during the MESSENGER flybys.
Planet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022
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where dE is the distance from the Earth to the edge (the surface of
Mercury), dS is the distance from the spacecraft to the edge, and
l is the RF wavelength. Fresnel’s integrals describe the diffraction
pattern as a function of u (Jenkins and White, 1976). The period of
the 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 to
the RF power levels observed during the occultations. At the time
of the M3 egress, MESSENGER was 14,380 km from the center of
mass (COM) of Mercury. The scale length, u, for edge-diffraction
effects with this geometry is 4220 km and translates to a time
scale of 0.358 s per u for diffraction patterns at the DSN receiving
antenna at Goldstone, California.
The RF power for the M3 egress, plotted in Fig. 1, shows a slow
rise, a peak above the unocculted power, and oscillations, all of
which are signatures of edge diffraction. The desired time, the
time of geometric occultation, is when the power level is onefourth of the unocculted power level (i.e., down by 6 dB) (Jenkins
and White, 1976). By fitting a diffraction pattern to the RF data
surrounding an occultation event, we can locate the desired power
level with an accuracy that is better than the time resolution of the
individual data points. Because (1) the shape of the diffraction
pattern is determined by the geometry, and (2) the vertical
position of the pattern is determined by the unocculted power
level, the only free parameter is the time, which is the horizontal
position of the diffraction pattern in Fig. 1. A least-squares analysis
for the M3 egress indicates that the time of geometric occultation,
the point when the signal was 6 dB below its unocculted value of
35 dB Hz, occurred at 23:00:13.5570.03 s UTC. An arrow in Fig. 1
shows the time of geometric occultation.
Because of the lower RF power levels for the M1 occultation
events, the diffraction patterns are barely detectable, even with
optimal PLL parameters. The uncertainty in the time of geometric
occultation for these events is estimated to be 0.1 s, from the
dispersion among the results using different techniques for
dB-Hz
Un-occulted level
3
extracting the RF signal levels. With the FFT technique, the time
of geometric occultation occurred when the RF signal was below
the noise level. Under this condition, the time of geometric
occultation is not observed, and the timing of the diffraction
pattern must be determined by the timing and features of those
portions of the signal that exceed the noise level.
2.2. Measured radius and uncertainty calculations
To convert occultation times to radius, and to assess uncertainties, we first constructed a line between the spacecraft and
the receiving antenna at Goldstone at the time of the occultation
event. This line is the RF path that just grazes the surface of
Mercury. The point on that line that is closest to Mercury’s COM is
the tangent point that we take to be the diffraction edge. The
distance of the point from Mercury’s COM is the local radius, and
the location of that point in Mercury coordinates is given by its
latitude and longitude. Table 1 lists the calculated radius and
location for each of the MESSENGER flyby occultation events.
Fig. 2 shows the location of each occultation measurement on a
global image mosaic of Mercury.
There are two sources of uncertainty in these calculations of
radius: uncertainty in the spacecraft position and uncertainty in
the time of occultation. The uncertainty in spacecraft position
with respect to Mercury, sPosition, contributes directly to uncertainty in radius. The MESSENGER navigation team produced the
spacecraft trajectory by analyzing data from many weeks before
and after the encounter to reduce uncertainty in the spacecraft
position. Data included radiometric range and range-rate data,
optical images, and results of delta-differential one-way ranging
observations (Lanyl et al., 2007). These long, dense data sets
enabled the navigation team to reduce trajectory uncertainties to
less than 30 m, at one standard deviation (s), with respect to
Mercury during the period that the spacecraft was within one
hour of closest approach (A.T. Taylor, 2008, personal communication). The calculations were performed in a Mercury-centered
coordinate system, so the uncertainty in Mercury’s position,
which is approximately 1 km relative to Earth, is independent of
the uncertainty with respect to Mercury and is a negligible
contribution to dE.
The achievable resolution, sTime, of the time of geometric
occultation depends on the diffraction fit quality, which varies
with RF power. For M1, the gain was sufficient for an uncertainty
6 dB-Hz below un-occulted level
Table 1
Radius and surface location for each occultation event.
23:00:13.55 ± 0.03
Time of geometric occultation
Event
Radius (km)a
Measured
Seconds
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 onefourth 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.
M-10 ingress
M-10 egress
M1 ingress
M1 egress
M3 egress
Mean radius
2439.5
2439.0
2437.8
2440.4
2441.65
Adjusted for
topographyb
2437.3
2439.9
2440.5
Lat
(1N)
Long
(1E)
1.1
67.6
25.54
7.33
36.06
67.4
258.4
225.28
41.83
28.23
Deviation
from MLA
fitc
1.6
0.4
0.1
0.7 (2439.2)
a
The radius is with respect to Mercury’s center of mass.
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.
b
Please cite this article as: Perry, M.E., et al., Measurement of the radius of Mercury by radio occultation during the MESSENGER flybys.
Planet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022
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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.
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 perpendicular to the line of sight and normal to the surface, converts
the timing uncertainty to a radius uncertainty. The parameter
VPERP is a relative velocity that includes corrections for the relative
motions of Mercury and Earth. The uncertainty in VPERP is
negligible, less than one part in 105. The total uncertainty in the
calculated radius, sTotal, is the root sum of the squares of the two
independent uncertainties
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sTotal ¼ s2Position þ ðsTime VPERP Þ2
ð2Þ
Table 2 summarizes these uncertainties for the three measurements of radius by MESSENGER.
2.3. Relating the radius measurements to planet shape
Depending on the number of fitted radii, the spatial resolution
of the global shape model from occultations alone is hundreds to
thousands of kilometers. Relative to these scales, a radius derived
from an occultation event is essentially a point measurement, and
it may not represent the average radius over large areas, or even
the average radius within tens of kilometers of the occultation
point. However, with data available prior to MESSENGER’s entry
into orbit about Mercury, we can estimate the effects of local
topographic variations on the radius measurement, deduce the
average height of the terrain within tens of kilometers of the
radius measurement, and adjust the radius measurement based
on local topography. We take this adjusted height to represent the
planet shape in the vicinity of the occultation-derived radius.
During the orbital phase of the MESSENGER mission, we expect to
recover this shape information in hundreds of occultation locales.
If there are detailed topographic data near the point where the
RF path is tangent to the surface, then those data can be used to
identify the specific feature – the edge – that defines the path of
the RF transmission that just grazes the surface at the time of
occultation. If there are no detailed topography data or if topographic analysis fails to identify the intersection feature, an
alternative approach is to reduce the measured radius by an
amount derived from a statistical analysis of surface roughness.
Table 2
Sources and estimates of uncertainty for the M1 and M3 radio occultation
measurements.
Event
VPERP
(km/s)
Uncertainties, 1s
Comments on
topography
Spacecraft
Time (s) Radius,
position (m)
total (m)
M1 ingress
1.6
30
0.1
160
M1 egress
1.1
30
0.1
120
M3 egress
1.4
20
0.03
50
No obvious
topographic features
No obvious
topographic features
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).
The first approach for assessing topography uses data from
images or altimetry to identify the highest section of the terrain
along the RF path. If an identifiable feature intercepts the grazing
RF path, the high terrain then contributes the edge measured in
the occultation analysis, and the location of the radius measurement is well determined. The images may also contain information on the height of the feature relative to the surrounding
terrain (e.g., shadow length), and such data can supply the
correction necessary to estimate the planetary radius corresponding to a representative average of the surrounding terrain.
In the absence of an isolated topographic high along the
grazing RF path, an adjustment of the measured radius for the
effects of local topography can be made on the basis of statistical
assessments of surface roughness for appropriate terrain types.
Fig. 3 shows two assessments of Mercury’s surface roughness
derived from MLA profiles obtained during M1 and M2 (Zuber
et al., 2008; Smith et al., 2010). To obtain a correction for the
measured radius, we modified the standard surface-roughness
analysis, the square of the height differences versus baseline
(this is a form of the Allan variance), which gives the average
peak-to-peak variations in surface topography over different
distance scales (Cheng et al., 2001; Barnouin-Jha et al., 2008).
Instead of peak-to-peak variations, we determine the most-probable distance between the RF grazing path and the average
elevation of the surrounding terrain. This distance is equivalent
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to the mean value of the difference between the highest point
along a path and the average height along that path
mPeak ¼ /maxðh½x : x þ LÞ/h½x : x þ LSS for all x along track
ð3Þ
where h[x:x þL] 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 of
the peak above the average terrain level.
Both roughness assessments in Fig. 3 used all of the MLA flyby
data, irrespective of the type of terrain, and the standard deviation of the mean values of the peak heights is large: 50–80% of the
mean. Assessing the peak heights on the basis of terrain type will
reduce the standard deviation, but the flyby MLA data are not tied
sufficiently well to terrain type for such a characterization.
Features with horizontal dimensions greater than 30–50 km
can be identified in the images, so the distance scale for adjusting
the occultation-derived radii is in this range. For L¼ 40 km, the
adjustment serves to reduce the radius by 5007350 m. Although
this analysis corrects the magnitude of the radius measurement, it
does not determine the along-track location of the edge, which
can be uncertain by 770 km. When available, information on the
topography that has high resolution in both horizontal and
5
vertical coordinates will reduce or remove the uncertainty due
to surface roughness.
To search for local topographic highs, we overlaid the location
of the grazing RF path onto the best available surface images,
which are from Mariner 10 or MESSENGER’s Mercury Dual
Imaging System (MDIS) (Hawkins et al., 2007). Results from stereo
analysis of MDIS images (Oberst et al., 2010; Preusker et al., this
issue) would be helpful, but for the locations of the three
occultation measurements reported here, no stereo topographic
analyses were available. Fig. 4 shows expanded views of the areas
of each occultation event. None of the three occultation events
occurred in an area with high-resolution MDIS images. There are
no distinctive topographic features near the locations of the M1
occultation measurements, so we used the probable-peak relationship 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 the
locus of the grazing path, and the intercepting edge may be at the
rim of the crater. There are no data on the rim height from MLA
data, stereo analyses, or shadow measurements, adding to the
uncertainty in the radius inferred for the terrain surrounding the
crater. From the relationship of Pike (1988) between crater rim
height and crater diameter for Mercury, the rim height of a fresh,
120-km-diameter crater is approximately 1.5 km. The crater may
be degraded, however, in which case the rim height would be less.
With no information on the state of preservation of this crater, we
use 1.5 km as the maximum height and assume the same vertical
uncertainty as for the surface-roughness measurements. Subtracting the vertical uncertainty, 0.35 km, from the maximum
height, 1.5 km, we take the rim to be 1.1570.35 km above the
level of the surrounding terrain. The M3 egress radius has therefore been adjusted by this amount (Table 1) to obtain the average
elevation of the surrounding terrain that we compare to the shape
derived from MLA equatorial topographic profiles expanded to
spherical harmonic degree 2.
3. Discussion
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.
From MLA flyby data (Zuber et al., 2008; Smith et al., 2010) and
ground-based radar observations (Anderson et al., 1996), Mercury’s
center of figure (COF) in the planet’s equatorial plane is offset
approximately 600 m from its COM. These same data also show a
difference between major and minor equatorial radii of 1.3–1.6 km.
Fig. 5 compares the occultation measurements of radius to the
spherical harmonic solution to the variation in near-equatorial
radius with longitude derived from MLA measurements made
during M1 and M2 (Smith et al., 2010). Results both before and
Fig. 4. Nominal locations (black crosses) of the tangent points of the grazing RF path (red lines) for each of the three occultation measurements of radius. The M1-ingress
image is from Mariner 10, and the other two are MDIS images from MESSENGER flybys. There are no distinctive topographic features in the vicinity of the radius
measurements made during the M1 occultation events, but the location of the radius measurement at the M3 occultation egress appears to lie at or near the rim of a
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 radius of Mercury by radio occultation during the MESSENGER flybys.
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6
M.E. Perry et al. / Planetary and Space Science ] (]]]]) ]]]–]]]
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.
after adjusting for local topography are shown. The M1 egress
results are near the MLA fit for that longitude, a not-unexpected
result given that the occultation measurement is at the nearequatorial latitude of 71S. The M3 egress and M1 ingress are each
farther from the equator, and, depending on the shape of Mercury
at higher latitudes, may deviate from the equatorial radius values
at a given longitude. After adjusting for topography, the M3 radius
also agrees with that from MLA, although this agreement is likely
fortuitous, given the latitude at which it applies (361N). The M1
egress radius is approximately 1 km less than the MLA-derived
equatorial shape, and we have no explanation other than the
likelihood of latitudinal variation in topography.
To extract a measurement of Mercury’s mean radius from the
three occultation-derived radius measurements, we account
approximately for the shape of the planet (Fig. 5) at spherical
harmonic degree 2 by subtracting the fitted equatorial radius
from each occultation radius. This procedure is predicated on
the assumption that the equatorial degree-two shape can be
extended to the latitudes of the occultations. Weighting the three
occultation measurements of radius by their uncertainties gives
an estimated mean radius of 2439.60 70.04 km before adjusting
for local topography and 2439.270.2 km after (Table 1).
However, the dispersion of the results about the mean is much
larger than the uncertainty from known errors, indicating that
there are additional factors affecting the result, including
unmodeled (essentially undersampled) higher-order variations
in planet shape. The standard deviation of the three results
compared to the degree-2 shape is 0.8 km, and we use this value
to estimate the increase in uncertainty due to these additional
factors, yielding a standard deviation of 0.5 km for the derived
mean radius.
There are several other observations of Mercury’s radius for
comparison. There were two occultation events from the first
Mariner 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 10
occultation radii agree with the near-equatorial shape determined
from MLA, within their uncertainties (Fig. 5), despite the fact that
one of the measurements was at high latitude. Oberst et al.
(this issue) described the results of limb analyses of Mercury
flyby images and compared their results to other measurements
of Mercury’s radius. The results of the limb analyses are consistent with MLA data in equatorial regions where they sample the
same longitudes.
Once MESSENGER is in orbit about Mercury, the planet will
occult MESSENGER’s RF transmissions every 12 h for most of the
12-month orbital mission phase (Solomon et al., 2001, 2007;
Srinivasan et al., 2007; Zuber et al., 2007). The mission observation plan includes capturing more than 90% of the 660 expected
occultation events, as shown in Fig. 6.
Please cite this article as: Perry, M.E., et al., Measurement of the radius of Mercury by radio occultation during the MESSENGER flybys.
Planet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022
M.E. Perry et al. / Planetary and Space Science ] (]]]]) ]]]–]]]
Acknowledgments
Details on the mission, flybys, and Mercury orbit insertion are
maintained and updated at the MESSENGER web site: http://
messenger.jhuapl.edu/. The MESSENGER mission is supported by
the NASA Discovery Program under contracts NAS5-97271 to the
Johns Hopkins University Applied Physics Laboratory and NASW00002 to the Carnegie Institution of Washington.
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Please cite this article as: Perry, M.E., et al., Measurement of the radius of Mercury by radio occultation during the MESSENGER flybys.
Planet. Space Sci. (2011), doi:10.1016/j.pss.2011.07.022