A GLOBAL SOLUTION FOR THE GRAVITY FIELD, ROTATION, LANDMARKS, AND
EPHEMERIS OF EROS
A. S. KONOPLIV, J. K. MILLER, W. M. OWEN, D. K. YEOMANS, J. D. GIORGINI
Jet Propulsion Laboratory
California Institute of Technology
4800 Oak Grove Drive
Pasadena, CA 91109
E-mail: Alex.Konopliv@jpl.nasa.gov
R. GARMIER, J. P. BARRIOT
Observatoire Midi-Pyrenees
Toulouse, France
ABSTRACT - As part of the NEAR Radio Science investigation, a
global solution was generated that includes both spherical and
ellipsoidal harmonic gravity fields of Eros, Eros pole and rotation
rate, Eros ephemeris, and landmark positions from the optical data.
This solution uses the entire one-year in orbit collection of X-band
radiometric tracking (Doppler and range) from the Deep Space Network
and landmark tracking observations generated from the NEAR spacecraft
images of Eros. When compared to a constant density shape model, the
gravity field shows a nearly homogeneous Eros. The Eros landmark
solutions are in good agreement with the Eros shape model and they
reduce the center-of-mass and center-of-figure offset in the z-
direction to 13m. Most of the NEAR spacecraft orbits are determined in
all directions to an accuracy of several meters. The solution for the
ephemeris of Eros constrains the mass of Vesta to 18.3 +- 0.4 km**3/s**2
and reduces the uncertainty in the Earth-Moon mass ratio.
KEYWORDS: NEAR, Eros, gravity, small bodies, orbit determination
INTRODUCTION
The NEAR-Shoemaker spacecraft was in orbit around 433 Eros for one year
from orbit insertion on February 14, 2000 to landing on the asteroid
surface on February 12, 2001. The science objectives included the
measurement of the gravity field of Eros from Doppler tracking and the
determination of the asteroid shape with the NEAR Laser Rangefinder
(NLR). The comparison of the gravity and shape indicates the uniformity
of the asteroid's mass distribution. The spherical harmonic gravity
fields of Eros from the joint effort of the Radio Science investigation
and the JPL Navigation Team have been previously presented [1,2] where
[2] showed results using two independent software sets (Orbit
Determination Program or ODP [3] and PCODP). The PCODP software was
developed by Jim Miller specifically for missions to small bodies and,
in particular, for NEAR navigation. This paper provides an updated
solution to the ODP solution in [2]. Whereas the previous ODP solution
was based only upon a subset of radiometric data, this new solution
uses the entire radiometric and optical landmark data set. This
solution has been archived in the Planetary Data System (PDS) as JPL
gravity solution NEAR15A (or file JGE15A01.SHA). The ellipsoidal
gravity solution for Eros is also based on the same complete data set
and has been previously presented [4]. It provides similar scientific
conclusions on the homogeneity of Eros as the spherical harmonics. This
paper provides the processing details on how this ellipsoidal solution
was generated.
The previous conclusions on the uniformity of Eros were based upon a
shape model developed by the Navigation Team [2] or Jim Miller' s shape
model number 101. This model contains the same long wavelength features
as the model developed by the NLR team [5] and both show nearly the
same results when compared to the gravity field. So the scientific
conclusions for the gravity are unchanged. However, the landmarks
contain the short wavelength information on the shape model and an
accurate high resolution model is needed to evaluate the landmark
solutions. The shape models developed by the Navigation and Radio Science
teams are not accurate enough to fully assess the landmark solutions.
However, when comparing the landmarks to the high resolution 180th
spherical harmonic degree shape model of the NLR team [5], the results
are very good. So we will use this model to present all the results
requiring a shape model.
Table 1. NEAR mission orbit segments from orbit insertion to the
maneuver to initiate landing on February 12, 2001. (Please see
alternate files for table)
The initial orbit for NEAR was nearly circular with a radius of about
350 km and an inclination of 35 degrees to the equator of Eros. The
orbit of NEAR was progressively lowered as the rotation and gravity
field of Eros became better known [6]. Table 1 displays all the
different orbits of the NEAR spacecraft during its mission. The best
orbit for determination of the gravity field of Eros occurred on July
14, 2000 and lasted for 10 days. This orbit was polar and circular with
a radius of 35 km and provides a very strong data set for the gravity
field. The tracking data from the entire remaining mission outside
these ten days give only a slight improvement in the gravity field.
However, all the orbits contribute to determination of the Eros orbit
around the Sun and the landmark solutions. Most orbits provide
information on the rotation of Eros.
TRACKING AND LANDMARK DATA
The tracking data for NEAR consisted of two-way X-band Doppler tracking
(~7.2 GHz uplink and 8.4 GHz downlink) and two-way range. Although
one-way Doppler (spacecraft to DSN station) and three-way Doppler
(receiving DSN station is different from transmitting) were also
collected, these data were not included in the global solution because
of good two-way Doppler coverage. The two-way Doppler data were
processed with a sample time of 60 seconds. The data arcs had a typical
RMS noise level 0.03 mm/s and the RMS of the arcs varied between 0.2
and 0.5 mm/s. A total number of 317,600 Doppler measurements were
processed. The Doppler data are the most important measurements for the
determination of the gravity field of Eros. The range data were
collected throughout the entire mission with a total of 74,180 range
measurements included in the solution. The RMS accuracy of the range
was less than one meter (about 2 range units or 30 cm), but due to
station calibration biases the actual measurement accuracy was about
3meters for the NEAR spacecraft distance relative to the DSN station.
The range data are critical for the determination of the Eros orbit
around the Sun. The NEAR spacecraft was tracked mostly by the 34-meter
antennas of the Deep Space Network (DSN). The following station
complexes were used: 14, 15, 25 (California), 34, 43, 45 (Spain), and
54 and 65 (Australia).
The other important navigation measurement in addition to the DSN tracking
is the landmark observations of the surface of Eros [7]. While the DSN
tracking measures the velocity and position of the spacecraft in the
line-of-sight (Earth to Eros) direction, each landmark observation
from the camera provides an angle measurement in two directions of the
spacecraft position relative to the Eros surface. The accuracy of the
angle observations is less than one-pixel of the camera. The size of
the pixel for the NEAR camera is rectangular with dimensions of 160x90
microradians. This roughly amounts to a spacecraft position measurement
along the spacecraft direction (along track) and normal to the orbit plane
of about 2-3 meters for the 35x35 km orbit and 20-30 meters for distances
of 200 km. Since the Doppler measurements typically determine the radial
component of the orbit the best (e.g., the MGS, Magellan, and Lunar
Prospector results [8,9,10]) the landmark tracking is an excellent
complement in that it measures the other directions. The end result is an
orbit that is known to several meters in all directions. The 160
microradian pixel direction is called the "line" component and is generally
a measurement along the spacecraft along track direction and is normal to
the Sun direction. The 90 microradian pixel direction is called the
"pixel" component and is generally normal to the spacecraft orbit plane
and is in the direction of the Sun line.
Figure 1. The 1624 landmarks collected during the NEAR mission. These
are plotted as seen from the (a) positive x-axis, (b) positive y-axis,
(c) negative x-axis, (d) negative y-axis, (e) positive z-axis, and (f)
negative z-axis. (Please see alternate files for figure)
The accuracy of the landmark data is limited by the pointing accuracy.
Although the pointing knowledge of the NEAR spacecraft is 50
microradians, the star tracker is located on the opposite end of the
spacecraft relative to the camera. Due to thermal effects and the
flexing of the spacecraft deck, the attitude of the camera continuously
changed by several pixels relative to the attitude specified by the
star tracker. Camera attitude calibrations were performed daily by
rotating the spacecraft to point the camera at a star field. The
observed orientation was then compared to the star tracker orientation.
With the calibrations, the pointing uncertainty was reduced to about
one pixel. The ideal accuracy that one could obtain without the
pointing errors is about the 1/4 pixel level given a well-defined
landmark. The landmarks chosen for the NEAR mission are craters that
varied in size from several kilometers in radius to several meters. The
actual landmark position is defined to be the center of the crater
projected onto the plane tangent to the crater rim. So the landmark
does not reside on the actual surface of Eros but is above the surface
at a distance given by the depth of the crater. For the larger
landmarks this can be significant (several hundred meters) but for the
majority of the landmarks which are smaller it is a several meter
effect. The data weight assigned to each observation depends on the
size of the crater due to errors in finding the center of the crater.
Larger craters have a larger uncertainty. The adopted weight SIGMA in
pixels as a function of the apparent pixel size D of the crater is
SIGMA = [1 + (0.1 x D)**2]**0.5
Figure 1 (or Fig. 3 of ref. [7]) shows all the landmarks selected for
the NEAR mission. Of the 1624 landmarks in the final database, 1554
were used in this study (most of the unused landmarks were only
observed a few times). The landmarks or craters are grouped into
different series labeled "A" to "I". The "A" series are large craters
of radius 1 to 2.5 km that became visible on the north side during the
early approach (Feb. 3, 2000). At this time Eros was about 40 pixels in
size and at a distance of 9000 km. The "B" series began to be collected
on Feb. 12, 2000 and are generally smaller. The "C" series were
selected after orbit insertion in the equatorial region and slightly
north, but mostly equatorial. Craters of all sizes are in this series.
The "D" craters are smaller northern hemisphere landmarks that were
chosen after the spacecraft was lowered into the 200 km orbit on March
3, 2000. The "E" series are smaller equatorial landmarks from the 200-
km orbit. The "F" series are craters from the first systematic survey
of the south side during the first periapse passage at 100 km on April
4 and 5, 2000. The "G" landmarks are very small northern craters from
the same periapse passage. The "H" and "I" series are craters of all
sizes in the far south that were not visible in the April. These
craters were selected from a systematic observing campaign from 100 km
on June 27. There were four sets of mosaics. "H" came from the first
set and "I" came from the other three.
The total number of pictures used for landmark tracking was 17,352.
From these pictures, 127,593 landmark observations were generated and
used in this analysis. Each landmark observation contains two
measurements (one each in the "pixel" and "line" directions). Each
landmark on average was observed 80 times and six of the best landmarks
were observed over 500 times. The radiometric and landmark data were
divided into separate time intervals or data arcs. Over the data arc,
the parameters that are specific to that arc, such as spacecraft state,
are estimated independently for each arc. For NEAR, the data arcs are
made as long as possible without being corrupted by non-gravitational
forces on the spacecraft such as thrusting. So none of the arcs contain
a maneuver that was performed to change the orbit (as listed in Table
1). Table 1 contains the begin and end times of the arcs used in this
analysis with a few exceptions. The exceptions are the longer orbit
segments for the 50x50 km orbit beginning April 30, 2000, the 100x100
km orbit beginning September 5, 2000, the 200x200 km orbit beginning
November 3, and the 35 km equatorial orbit on December 13, 2000. All
the other arcs are given by the begin and end times of the segment (of
which the longest is 30 days on March 3 and the shortest is 19 hours on
October 25). In the longer segments that are exceptions, the arc size
is reduced to between one week and three weeks in length. Most of the
arc boundaries in these cases are given by the times of momentum wheel
desaturations. These occurred about once a week for the longer orbit
segments mentioned above (they were not needed for any of the other
orbit segments). Several of the longer arcs (2-3 weeks) included the
momentum wheel desaturations within the arc. In this case, delta
velocity increments are estimated to account for the maneuvers. So the
most important data for the gravity (the 10-day polar 35-km circular
orbit segment starting July 14, 2000), are processed in one continuous
arc. The Doppler data weight used in the filter for this arc was
tighter at 0.05 mm/s than all the other arcs (at 0.1 mm/s) to
accentuate the gravity information. The actual Doppler RMS data noise
for the July 14 arc was 0.023 mm/s and was one of the better arcs in
terms of data noise.
SOLUTION TECHNIQUE
As mentioned above, the gravity solution presented in this paper was
determined using the JPL Orbit Determination Program (ODP, Ref. [3])
including the optical navigation software [7] and a technique that was
successfully used for planetary gravity efforts of Venus, Mars, and the
Moon [11,12,13]. The ODP estimates the spacecraft state, gravity, and
other parameters using a square root information weighted least-squares
filter [14,15] in the coordinate system defined by the Earth' s mean
equator at the epoch of J2000. The parameters that are estimated
consist of arc-dependent variables (spacecraft position, etc.) that are
separately determined for each data arc and global variables (e.g.,
gravity coefficients and landmark positions) that are common to all
data arcs. The global parameters are determined by merging only the
global portion of the square root information matrix from all the arcs
of the entire mission, but is equivalent to solving for the global
parameters plus arc-dependent parameters of all the arcs. The technique
is described in [16] using partitioned normal matrices and was first
used to analyze Earth orbiter data, and for the type of filter used in
this work (square root information), the method is outlined in [17].
Initially, we converge the data arcs by estimating only the local
variables using the nominal values for the global variables. For each
data arc the local variables estimated are spacecraft position and
velocity at the data epoch, three solar pressure coefficients, range
biases for each station pass, and a velocity increment in three
directions resulting from a momentum wheel desaturation maneuver. The
latter was required for only a few arcs and not the critical July 14
arc.
The NEAR spacecraft is a simple bus spacecraft 1.7 meters square at the
base and about 2 meter stall. A 1.5 meter high gain antenna is fixed to
the top of the bus with four fixed solar arrays (1.2 x 1.8 meters)
pointing outward from the four sides of the bus. The dry mass of the
spacecraft is 468kg. The solar pressure model has two parts. The
constant model is a simple bus model with a cross-sectional area of
10.3 square meters (this model is just applied and not estimated). For
the most part, the NEAR solar panels are Sun-pointing are so the area
projected in the Sun direction is constant. To account for small
changes in the solar pressure force, a small stochastic variation in
three directions at about 5 percent of the overall force is estimated.
The time constant for the stochastic solar pressure part is 1 day. It
is important to minimize the a priori uncertainty of the solar pressure
force being estimated because if it is too loose, it can absorb the
acceleration due to the gravity field of Eros. Anything greater than 5%
seems to degrade the gravity solution when looking at the correlations
with topography. With a 5% a priori uncertainty, the resulting solar
pressure values are at about the 5% level and cannot be absorbed by the
gravity field. This model will also absorb any possible outgassing or
thermal radiation.
The range data from the Earth tracking station to the NEAR spacecraft
provide information on the Eros orbit around the Sun. The Eros
ephemeris is estimated in a separate process described below. With the
new Eros ephemeris included in the estimation process, the range biases
solution values are greatly reduced for every station. These biases are
on the order of several meters and represent the path length
calibration errors at each DSN station. In addition to the estimated
parameters, there are other different models involved in the force on
the spacecraft and in the computation of the tracking and landmark
observables. These include, for example, accurate Earth station
position modeling to the 2-3 cm accuracy, ionospheric and tropospheric
corrections to the Doppler and range data (based upon in-situ GPS and
weather measurements), point mass accelerations due to the Sun and
planets, and relativistic time delay corrections on the radiometric
observables.
The global variables determined in the solution include the pole
direction and rotation rate of Eros, either the spherical harmonic
gravity coefficients or ellipsoidal harmonic gravity coefficients, and
the body-fixed cartesian position of the landmarks. The orbit of Eros
is also estimated with the global data set but since it is not strongly
correlated with the other parameters, it can be estimated
independently. Since Eros is in nearly principal axis rotation, it can
be modeled mostly as a simple right ascension (alpha (0)) and
declination system (delta (0)) (see Ref. [18]). The three parameters
alpha (0), delta (0), and rotation rate (W) are estimated. Each
landmark position involves three parameters, and we estimate from the
global data set the position of 1554 landmark positions for a total of
4662 landmark parameters. Two separate solutions are generated for the
complete list of global parameters. One uses spherical harmonics to
model the gravity field and the other ellipsoidal harmonics. The
spherical harmonic expansion to maximum degree and order N of the
gravity potential is given by (Ref.[16,19])
U = (GM/r) x {Sum n=0 to N [Sum m=0 to n {[a(e)/r]**n x P(nm) x
Sin(phi) x [C(nm) x Cos(m*lamba) + S(nm) x
Sin(m*lambda)]}]}
where r is the radial distance from the coordinate origin, GM is the
gravitational constant times the mass of Eros, n is the degree and m is
the order, P(nm) are the fully normalized associated Legendre
polynomials, a(e) is the reference radius of Eros (16 km for our
gravity models), phi is the latitude, lambda is the longitude, and
C(nm) and S(nm) are the normalized gravity coefficients. The spherical
harmonic model is estimated to degree and order 15 for a total of 253
parameters including the GM. The center of the coordinate system is the
center of mass, so the degree one coefficients are zero. Spherical
harmonics are not an ideal representation for the irregularly shaped
Eros (~17x6x6 km) since the spherical harmonic expansion for Eros
converges outside the smallest sphere that encloses the body [19].
However, all the orbits of NEAR except for the landing are outside the
sphere, and spherical harmonics can be used as a simple straight
forward investigation of the gravity and internal structure of Eros. To
maintain convergence, the Bouguer gravity or differences between the
measured gravity field and a gravity field assuming a constant density
for Eros are displayed on a sphere of 16 km.
The second solution uses ellipsoidal harmonics. The ellipsoidal
potential to maximum degree and order N is given by (Ref. [20])
U = GM x {Sum n=0 to N [Sum m=0 to m=2n+1 {[alpha(nm) x F(nm) x
lambda(1)] / [F(nm) x (a**2 - c**2)] x E(nm)(lambda(2)) x
E(nm)(lambda(3))}]}
where n and m are again the degree and order, and alpha(nm) are the
ellipsoidal coefficients corresponding to the spherical harmonic C(nm)
and S(nm). For every degree there are the same number of ellipsoidal
harmonics as there are spherical harmonics (2n+1). The ratio involving
F(nm)is a Lame' function of the second kind and plays the same role as
[a(e)/r)**n attenuation factor with distance in the spherical harmonic
expansion. The variable lambda(1) is a kind of radius vector and
lambda(2) and lambda(3) are equivalent to latitude and longitude. The
semi-major axes of the ellipsoid are given by a>b>c. The product of
E(nm) is called a surface harmonic and is equivalent to P(nm) x cos(m x
lambda) or P(nm) x sin(m x lambda). As with the spherical harmonics,
the degree one coefficients are zero, since the coordinate system is
chosen to be the center of mass. These ellipsoidal harmonics are
convergent outside the smallest ellipsoid enclosing the body and can be
used to map the gravity field closer to the surface of Eros. Since Eros
is much closer to the shape of a triaxial ellipsoid, fewer coefficients
are needed to represent the gravity field of Eros and less noise or
"aliasing" is observed in the coefficients. Whereas both the spherical
harmonics and ellipsoidal harmonics give nearly the same results
through roughly degree 6 or 7, the ellipsoidal solution remains much
smoother to higher degrees [4]. However, the ellipsoidal coefficients
are limited in numerical stability to about degree 12 (which is
sufficient for the NEAR data of Eros). As mentioned above, both
expansions result in the same scientific conclusions on the internal
structure of Eros [2,4]. We solve for the ellipsoidal representation to
degree and order 12 (167 parameters including the GM).
EROS EPHEMERIS
Eros (433) is a large near Earth asteroid (NEA) with a semi-major axis
of 1.45 AU, 0.22 eccentricity (1.13 AU perihelion distance), 10.8
degree inclination to the ecliptic, and a 1.76 year orbit period. The
ephemeris or orbit of Eros around the Sun is very accurately determined
from the ranging data to the NEAR spacecraft. The original range data
to the NEAR spacecraft measure very accurately the distance to the
spacecraft from the tracking station to within a few meters. Using the
accurately determined NEAR orbits about Eros from the DSN tracking and
landmark data, the range data are shifted from the NEAR spacecraft to
the center-of-mass of Eros, also to an accuracy of several meters.
These new range tracking data are then processed with the ODP treating
Eros as a spacecraft in orbit about the Sun. In addition, the
telescopic images of Eros since 1964 are also processed as angle data
in the ODP. Images of Eros exist as far back as 1893 but the Earth
orientation data are available for the ODP with a begin date of 1964.
However the range data, with a several meter accuracy for one year,
completely dominate the solution and the optical data are not really
needed. The Eros ephemeris is determined as part of an iterative
procedure with the gravity field and landmarks. As a better gravity
field and landmark solution is obtained, there are more accurate orbits
of Eros. These orbits, in turn, provide more accurate range data to
Eros and a better ephemeris of Eros. This new ephemeris is then used
for the next iterative solution of the gravity field and landmarks.
The most significant perturbation on the Eros orbit during the NEAR
mission other than the Sun is a 0.416 AU flyby of the asteroid Vesta on
July 13, 2000. This allows for an estimate of the mass of Vesta. There
are no other major perturbations on Eros. The next largest effects are
encounters by Sappho (80) at 0.17 AU, Flora (8) at 0.39 AU, Desiderata
(344) at 0.51 AU, and Bruchsalia (455) at 0.40 AU. All these effects
are too small to yield a mass estimate.
Figure 2 shows the residual range data to Eros when the Vesta
perturbation is not included, and the fit with the mass of Vesta
estimated. The only other parameters estimated other than the mass of
Vesta is the initial position and velocity of Eros. However, the
perturbations on Eros from the Sun and planets are included in the
force model. Table 2 shows the different estimates of the mass of Vesta
(from Table 4 in Ref. [21]) including the more recent estimate from the
JPL ephemeris effort [22], and the result of this effort. Our estimate
is consistent with most of the determined values of Vesta. There are
higher values of Vesta (~20.0 km**3/s**2) but our RMS of the range
residuals show an increase of about 22% to 2.2 meters from the best fit
RMS of 1.8 meters in Figure 2(b) when the Vesta mass is fixed to this
higher value. So these higher values are not consistent with the Eros
data. If the mass is fixed to the lower value of 17.8 km**3/s**2 [22],
then the RMS only increases by a modest 2.8%, and so we consider the lower
values in Table 2 to be consistent with our results. The error we give
is about 2.5 times the formal error of 0.16 to give a more realistic
error value of 0.4. The range residuals in Figure 2(b) are the result
of both orbit error and DSN range calibrations at the station. The
signature in the residuals at the 200 to 300 days past the epoch of the
data is mostly due to spacecraft orbit error. At this time the NEAR
orbit at Eros is larger and hence has a larger orbit error because the
landmark tracking is not as accurate at the higher orbits. This is
consistent with the orbit errors we see from different orbit solutions
as discussed in the landmark results. The mass estimate of Vesta does
not change if range biases are solved for each DSN range pass. In this
case the RMS of the residuals reduces to about 20 cm.
Figure 2. Range residuals to Eros for (a) a zero Vesta mass, and (b)
the mass of Vesta estimated. (Please see alternate files for figure)
Additional information is visible in Figure 2(b). Prior to the large
orbit errors that dominate beginning 220 days after the epoch, a
monthly oscillation is visible in the range residuals. The amplitude is
about 1.5 meters. This is due to the motion of the Earth about the
barycenter of the Earth-Moon system. This allows us to put constraints
on the Earth-Moon mass ratio. The solution for the Eros orbit and Vesta
mass used the JPL planetary ephemeris DE403. DE403 uses a mass ratio of
81.300585. The latest constraint on the mass ratio is from the Lunar
Laser Ranging (LLR) and Lunar Prospector results [28] where the value
is 81.300566 +- 0.000020. The result from Figure 2(b) is nearly the
same, except the uncertainty can be reduced. The range data to Eros
results in an Earth-Moon mass ratio of 81.300570 +- 0.000005 and an
improved lunar GM value of 4902.8000 +- 0.0003.
Table 2. Mass estimates of Vesta. (Please see alternate files for
table)
LANDMARK RESULTS
As the result of the landmark observations, the orbits of NEAR are very
well determined. The landmarks are very important for the higher
altitude orbits. For the initial 350-km circular orbit of NEAR, for
example, the landmark tracking lowers the overall orbit error to
hundreds of meters or less, whereas the orbit error with radiometric
tracking alone is about 20 km. The orbit error for the close orbits
(35-km) is several meters in all three directions and for the most part
can be obtained with sufficient radiometric tracking alone. However,
the landmark tracking reduces the time required to redetermine the
spacecraft position after a maneuver. These orbit error results were
determined by differencing orbits where the only change in the solution
procedure is that one contains only radiometric tracking and the other
includes landmark tracking. To quantify the orbit error with landmark
tracking, orbits determined with the ODP (this paper or Radio Science
Team orbits) were differenced with those determined by the Navigation
Team using the PCODP software. Both solutions use the radiometric and
landmark data. However, both procedures are very independent with
different data arc intervals and spacecraft models, and the PCODP used
a subset of the landmarks. These differences together with the
differences of the orbits determined by the NLR Team have been included
in the PDS archive for NEAR [27]. The orbit differences are computed
not as an RMS but as an average of the absolute value of the
difference. The NLR orbits are determined using radiometric data and
NLR altimeter measurements, and are the result of a joint gravity and
shape estimation. The NLR orbit set begins with the 200-km orbit on
March 3, 2000 with the collection of altimetry.
Figure 3. Average orbit differences between (a) Radio Science and
Navigation, and (b) Radio Science and NLR. (Please see alternate files
for table)
Figure 4. Differences of landmark position solution and 180th degree
NLR shape model for (a) original coordinate system of NLR shape model,
(b) NLR shape model coordinate system rotated by +0.155degrees about
the z-axis, and (c) NLR shape model rotated about the z-axis and
shifted down the z-axis by 19 meters (i.e. in the negative direction).
(Please see alternate files for table)
Figure 3 displays the orbit differences in all three components
(radial, transverse, and normal to the orbit) between Radio Science and
the other two sets. The orbit differences for the higher 200-km orbits
are tens of meters for the landmark tracking orbits and hundreds of
meters for the NLR orbits which do not contain landmark tracking. The
initial 350-km orbit (Feb. 14-24) differences for the landmark orbits
are several hundred meters and are off the scale in Figure 3(a). For
the lower orbits, the landmark solutions agree to mostly better than 5
meters. This most likely provides an upper limit on the orbit
uncertainty for the orbit with either the Radio Science orbits or
Navigation orbits possibly being better. The range residuals for the
ephemeris of Eros (Fig. 2) suggest that the Radio Science orbit error
in the Earth line-of-sight direction is less than two meters. The NLR
lower altitude orbits show a hundred meter difference mostly in the
alongtrack direction.
The positions of the smaller craters or landmarks of the global
solution are determined to an accuracy of about 2 meters for all three
body-fixed directions on the surface of Eros. The larger landmarks have
uncertainties of tens of meters and, in several cases, up to two
hundred meters.
Next we compare the landmark positions with the 180th degree and order
Eros spherical harmonic shape model derived by the NLR Team [5]. The x,
y, and z Eros body-fixed position of each landmark is estimated in the
global solution process with the resulting uncertainty being about 2-
3meters in each direction. The corresponding latitude, longitude, and
radius is then computed for each landmark. Using the latitude and
longitude of the landmark, the radius from the NLR shape model is
computed. The radius values from the landmarks and NLR shape model are
then differenced and displayed in Figure 4.
Figure 4(a) shows the original differences. We suspected that the
coordinate systems of this global solution might be different than the
NLR coordinate system. This is due to different pole and rotation
values used and to orbit corrections applied in the shape model
crossover analysis. The radial differences were again calculated after
rotating the body-fixed coordinate system of the NLR shape model by
+0.155 degrees about the z-axis (i.e., features are shifted to the left
in longitude in the map of the shape model in the new coordinate
system). The results are displayed in Figure 4(b). Much of the noise
and structure was removed. Next the NLR shape model was rotated as
above and then shifted along the negative z-axis by 19 meters and
displayed in Figure 4(c). This was very successful in laying the
residuals flat. The discontinuity in Figure 4(b) at landmark numbers
880 and 1150 correspond to "G" craters being located in the northern
hemisphere and then "H" and "I" craters in the south.
The landmark and NLR shape differences are very sensitive to shifts in
the z-component of the shape model but are not as sensitive to
translations in the x and y directions. So the landmarks can be used to
constrain the z-height difference between the center-of-mass and
center-of-figure coordinate systems. The location of the center-of
figure of the shape model before the translation of the z-axis is -13m,
0m, +32m in the x, y, and z direction, respectively. The shape center-
of-figure is defined to be the center-of-mass of the shape' s gravity
assuming a constant density, and it is determined by numerical
integration over the volume. With the translation of the shape z-axis,
the new location of the z-component of the center-of-figure offset is
+13m. So the center-of-mass and center-of-figure offset has reduced
significantly and indicates a more uniform Eros in the z-direction.
With the overall length of Eros in the x-direction of 34-km and z-
length of 11-km, this indicates long wavelength density variations of
less than 1%.
In Figure 4, the landmarks are listed in order of selection. Again,
each landmark position is not on the true surface of Eros. It is the
center of the crater projected upward to the rim of the crater. So it
is above the surface by an amount equal to the depth of the crater. The
initial landmarks were larger and Figure 4 shows the larger depth for
these initial craters. For the smaller craters beyond number 600, the
depths are smaller and mostly below 10 meters and as low as 1 or 2
meters. Negative differences and some positive differences are either
due to errors in the landmark or possible gaps in the NLR data where
results are interpolated. The RMS of the differences is 5.6 meters for
the smaller landmarks (1250 to 1550) with outliers greater than 20
meters deleted. This result is much better than any shape model from
the Navigation Team [1,2,29] or those by the main author (estimated to
degree 120). These models have very accurate long wavelength
information but poor short wavelength information. The NLR model does
very well in both the long and short wavelength features as shown by
the comparison with the landmark solutions.
GRAVITY RESULTS
The gravity field of Eros was modeled with both spherical harmonics and
ellipsoidal harmonics. Although, spherical harmonics are not an ideal
representation for the irregularly shaped Eros, they still can be used
to evaluate the uniformity of Eros. In this paper we mostly discuss the
spherical harmonic results. The ellipsoidal harmonic conclusions are
nearly identical and the ellipsoidal solution generated by this work
has been presented previously [4]. The NLR [5] shape model is used to
display the results in this section, but the Navigation and Radio
Science shape models give nearly identical results.
One way to compare different gravity and topography solutions is to
look at the correlations between the coefficients. The correlations
are dominated by the ellipsoidal shape of the gravity and topography,
and so are nearly equal to one. But small changes in the correlations
indicate which solutions match more closely. The correlations for this
solution are ever so slightly larger than the previous results [2]
which used a radiometric only gravity solution and the Navigation Team
shape model. The correlations through degree 10 are shown in Table 3
for this gravity solution (radiometric plus landmark tracking) and the
NLR shape model, and the previous results. From Table 3, one notes that
the correlation between gravity and shape dramatically reduces at
degree 10. This is because the gravity field is determined to roughly
degree and order 10.
Table 3. Gravity and gravity from shape spherical harmonic
correlations. The shape model used in this paper is from the NLR Team
[5]. (Please see alternate files for table)
Figure 5 shows the RMS magnitude spectrum of the gravity field with
both the RMS of the coefficients and the RMS of coefficient
uncertainty. The a priori constraint in the gravity field (0.005 for
n=11 to 15) is visible in the higher RMS of the uncertainty and gravity
for n>10. The uncertainty in the coefficients or noise matches the
coefficient magnitude or signal at degree 10.
So, the gravity field of Eros is determined to about degree 10 or about
a 5-km half-wavelength resolution. However, the amplitude of the
difference in the coefficients is much smaller, and the differences
between the gravity and shape can be investigated only to degree 7.
This is demonstrated in Figure 5 by the difference of the gravity
solution with the gravity from shape assuming a constant density. Note
that the uncertainty and differences in Figure 5 are again only
slightly improved over the previous results [2]. While the center-of-
mass and center-of-figure offsets indicate very small large-scale
changes in density (<1%), the differences between the coefficients of
the gravity and shape are larger at 1-5% of the gravity amplitude.
The next task is to investigate the differences in gravity and gravity
from shape in the spatial domain. The Bouguer gravity is defined as the
difference of the radial component of the gravity and gravity from
shape assuming a constant density. The accelerations are determined on
a sphere of 16 km. With NEAR being in a circular orbit about Eros, the
gravity of the ends of the asteroid is much better determined than the
center of the asteroid. The uncertainty in the gravity when mapped on
the 16-km sphere is roughly uniform (less than one milligal), and the
gravity of the ends is more visible than the rest of the asteroid. The
previous Bouguer results [2] had maximum and minimum values of 1.75 and
-3.86 milligals, respectively. The new results as shown in Figure 6 are
nearly the same and show the acceleration differences for the gravity
and shape for spherical harmonics from degree 2 to 6. The range or
maximum and minimum have slightly reduced to 1.26 and -3.28 milligals,
respectively. The locations of the features are unchanged. We still
have negative Bouguer anomalies located at the ends of Eros (-3.28
milligals for the negative x-axis and -2.98 milligals in the positive
x-axis direction) and slightly shifted to the northern hemisphere [2].
The amplitude of the Bouguer gravity at the ends of the asteroid is
about 1% of the gravity amplitude (without the GM and for degrees 2 to
6).
Figure 5. RMS magnitude spectrum of the gravity and gravity
uncertainty. Also included is the RMS difference of the gravity and
gravity from shape assuming constant density. The NLR shape model is
used. (Please see alternate files for figure)
The negative Bouguer values indicate that the density of the asteroid
ends is slightly less than the rest of the asteroid. A regolith can not
account for the entire negative anomaly. For instance, a 100-m regolith
with a density of 2.0 gm/cm3 (versus the mean 2.6 gm/cm3) gives a
Bouguer value of -1.0 milligals at the negative x-axis end and -0.4
milligals at the positive x-axis end. We would need about three times
this effect to account for what is observed. The lower density ends may
also be the result of an increase in density near the center of the
asteroid. The full Bouguer signature can be accounted for by an
increase in density of 5% for 20% of the asteroid volume near the
center of the asteroid or equivalently a 10% increase for 10% of the
volume. At the center of the asteroid is the Psyche crater and Himeros
depression. The small positive anomalies noted in the Bouger gravity in
the ellipsoidal results [4] lead to the suggestion of possible
compression from impact. As with the spectral differences in Figure 5,
the density contrasts that are suggested by the Bouguer analysis are
larger than the center-of-mass and center-of-figure offsets. Whatever
variations we see need to average to nearly zero on the global scale
such as a regolith or radial decrease in density from the asteroid
center. However, the comparison of the gravity and shape models still
indicate (but do not prove) a fairly uniform Eros.
Figure 6. Eros Bouguer radial acceleration map. Differences
betweengravity and topography are shown on a sphere of 16-km. (Please
see alternate files for figure)
Table 4. Eros GM and rotation solution. (Please see alternate files for
table)
Also determined in the global solution is the GM and rotation of the
asteroid Eros. These values are listed in Table 4 along with a
realistic uncertainty. The uncertainty is about 5 times the formal
uncertainty we get in the global solution. The factor of 5 scaling was
determined by looking at subset solutions for the pole and rotation.
The pole and rotation rate are best determined by the 35-km circular
equatorial orbits near the end of the mission. These orbits result in
an uncertainty of 4-5 times lower for the pole and about 2 times less
for the rotation rate. It is expected that the solar gravity gradient
torque will cause a nine month oscillation in the pole of about 0.01
degrees [2]. However, we have less than two months of the data
sensitive to the pole and it is difficult to detect this pole motion.
In the data that zfd not as sensitive to the pole, long term motion of
about five months from minimum to maximum and 0.01 degrees is visible
in the pole right ascension and declination, but this motion is near
the uncertainty in the pole. So detection of the solar gravity gradient
torque is not conclusive.
ACKNOWLEDGMENTS
Other members of the JPL navigation team provided files, tables, and
information needed for this effort. They include T.C. Wang, B.G.
Williams, P.G. Antreasian, and J.J. Bordi. Also, A. Chamberlin of the
small body ephemeris group helped with the conversion of the Eros
ephemeris files and E.M. Standish provided information on the solution
for the Vesta mass and the Earth-Moon mass ratio. The research
described in this paper was carried out at the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.
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