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\title {\bf Orbit Determination \\ for  Mars Global Surveyor \\ During Mapping}
\author {F. G. Lemoine\thanks{Code 926, Laboratory for
Terrestrial Physics, NASA Goddard Space Flight Center, Greenbelt, MD 20771; E-mail: flemoine@olympus.gsfc.nasa.gov},\\
D.D. Rowlands   and D. E. Smith \\
 Laboratory for Terrestrial Physics \\ NASA Goddard Space Flight
 Center \\  Greenbelt, Maryland  \and
D. S. Chinn,
\\ D. E. Pavlis  and S. B. Luthcke \\
Raytheon ITSS  \\ Greenbelt, Maryland \and
G. A. Neumann  and M. T. Zuber, \\
Department of Earth, Atmospheric, and Planetary Sciences \\
Massachussetts Institute of Technology  \\
Cambridge, Massachusetts }
\date{}
\begin{document}
\maketitle
\begin{abstract}
The Mars Global Surveyor (MGS) spacecraft reached its designated low-altitude,
and near-circular orbit about Mars after termination of aerobraking
on February 4, 1999.  The Mars Orbiter Laser Altimeter (MOLA) returned
its first data from this low-altitude orbit on March 1, 1999. The
Doppler data acquired from the Deep Space Network were used to
determine the orbit of the MGS spacecraft, so that the altimeter data could
be used to map the topography of Mars.  A detailed nonconservative force
model was implemented for MGS that included articulation of the solar arrays.
The tracking data in February and March 1999 were used to improve the
knowledge of the Mars geopotential by developing MGM0964C20,
a solution to 70x70 in spherical harmonics.
When the improved nonconservative force model and the improved
Mars gravity model, MGM0964C20, are applied in the MGS orbit determination
during the mapping orbit, the overlaps of adjacent arcs in March 1999 show
a consistency of 0.6 m in the radial direction, and 2 meters
in total position.  We demonstrate that altimeter crossovers from MOLA
can be used to further improve the orbit determination for these arcs
resulting in an orbit overlap consistency of 0.3 meters in the radial
direction and 1.7 meters in total position.

\end{abstract}

\section*{{\bf INTRODUCTION}}

The Mars Global Surveyor (MGS) spacecraft reached a low-altitude near-circular
orbit on February 4, 1999, after the termination of the second phase of
aerobraking\cite{refAlbee, esposito98}.  The spacecraft carries the Mars
Orbiter Laser Altimeter (MOLA)
whose primary goal is to derive a global, geodetically referenced 0.2$^\circ$ x 0.2$^\circ$ topographic
grid of Mars with a vertical accuracy of better than 30 meters\cite{zuber92}.
During the interim science orbits in the Hiatus mission phase (October to
November 1997)
and the Science Phasing Orbits (March - April, and June - July 1998), 208 passes
of altimeter data were collected by the MOLA instrument. Starting on March 1,
1999, MOLA obtained altimeter measurements from the low-altitude mapping orbit.
Altimeter data will be collected throughout
the mapping phase of the MGS mission, or for at least
one Mars year (687 days). Whereas the interim orbits of Hiatus and SPO were
highly eccentric, and altimeter data were returned only when the spacecraft
was below an altitude of 785 km, the MGS mapping orbit is near circular
with altitudes ranging from 370 to 430 km. In the mapping orbit, altimeter data
are returned continuously at a rate of 10 Hz.

Since the MGS orbit serves as a reference from which the topography
of the planet is deduced, the proper analysis of the altimeter data
requires that the orbits be known as accurately as possible. In this paper,
we will discuss the force modelling required
to achieve the best possible orbits during the mapping phase of the MGS
mission.


\section*{{\bf DATA AND ORBIT DETERMINATION MODELLING}}
\subsection*{Mission Summary and Data Coverage}
The MGS spacecraft was tracked by the antennae of the Deep Space Network (DSN)
using two-way and three-way ramped Doppler, and two-way ramped range
data\cite{moyer87,moyer95}.  In addition, as described by Lemoine
et al.\cite{lemoine99}, one-way Doppler data were obtained
from the Ultrastable Oscillator (USO) aboard MGS during SPO-1
(March - April 1998) and SPO-2 (June - July 1998).
No one-way Doppler data were used outside of these time periods.
The MGS mission phases and orbit characteristics are summarized
in Table 1.

After the end of aerobraking on February 4, 1999,
most of February 1999
was devoted solely to acquisition of tracking data
to allow for improvement of the gravity models of Mars. During
this mission phase (known as the Gravity Calibration Orbit),
the spacecraft was maintained in a quiescent state to minimize disturbances
in the tracking and obtain as much high quality tracking data as possible.

A contingency period of science data collection took place in March 1999,
prior to the deployment of the High Gain Antenna (HGA) on March 29, 1999.
In this mode, (designated Fixed High-Gain Antenna Mapping) the spacecraft
pointed its instruments at Mars 18 to 20 hours per day,
and then turned toward the Earth four to six hours per day to return
the science data, and obtain tracking.
The first MOLA data after HGA deployment were obtained on April 2, 1999.
After HGA deployment, normal mapping operations involved
pointing the instruments at nadir while the High Gain Antenna
independently tracked the Earth.
The High Gain Antenna is located at the end of a two meter boom
and two axis gimbals maintain the antenna pointed at the
Earth over each orbit.

Mapping operations were interrupted on April 16, 1999
when the azimuth gimbal of the HGA became stuck at 41.5$^\circ$.
No MOLA altimeter data were collected from April 16 to April 28, 1999,
while the causes of the anomaly were investigated.
It was determined that the gimbal would not rotate further than 41.5$^\circ$,
precluding simultaneous collection of science data, and pointing of
the High Gain Antenna at the Earth during certain phases of the mission.

Science data collection resumed on April 28, 1999. From April 28 to May
5, 1999, the spacecraft pointed the instruments at nadir
to collect science data 18 to 20 hours per day,
and then pointed the HGA antenna at Earth for four to six
hours per day to transmit these data to the Earth.
By May 5, 1999 the Earth-Mars-MGS relative geometry had changed sufficiently
to allow resumption of normal mapping operations at least through February 2000.

Following, HGA deployment, a degradation in the quality of the tracking data
was observed\cite{sjogren99}, including an increase in the data noise,
as well as a 19 second periodicity in the tracking data.
Thus, none of the tracking data after HGA deployment were used
to help refine the models of the Mars geopotential.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccc|}
\multicolumn{5}{c}{Table 1: MGS Orbit Characteristics during MOLA Data Acquisition}\\
\hline\hline
\rule{0mm}{3mm} Mission & Dates & Orbital  & Periapsis  & Periapsis\\
           Phase        && Period (hr) &Ht. (km) & Latitude (deg)\\
\hline
Insertion & September 12-16, 1997 & 44.993 & 263 & 31$^\circ$ N\\
Hiatus    & Oct. 13 - Nov. 7, 1997 & 35.43 & 170 &  35$^\circ$ N\\
SPO-1     & March 27 - May 1, 1998 & 11.64 & 171& 61$^\circ$ - 71$^\circ$  N\\
SPO-2     & May 28 - July 30, 1998 & 11.64 & 171& 82$^\circ$ - 86.3$^\circ$ N \\
 & & &  &86.3$^\circ$ - 76$^\circ$ N\\
 GCO \&  & February 4, 1999 - $\cdots$ & 1.97 & 377 & 90$^\circ$ S\\
 Mapping & & & & \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

\subsection*{Data Processing and Analysis}
The GEODYN orbit determination program\cite{geodyn} was used to process
the ramped Doppler and range observations for MGS.
The modelling included use of the DE403 planetary ephemerides\cite{standish95},
in both the measurement and force models.
In addition to the third body perturbations caused by the planets and the Sun,
the third body perturbations due to Phobos and Deimos were also included,
using a $GM$ of 720,000 m$^3$/s$^2$ for Phobos\cite{kolyuka91},
and 120,000 m$^3/s^2$ for Deimos\cite{williams81} and the ephemerides of
Jacobson\cite{jacobson97}.
The Mars radiation pressure was modelled using spherical harmonic
models, as described by Lemoine\cite{lemoine92}, based on analysis of the
Viking Infrared Thermal Mapper data.
The formulation of Knocke et al.\cite{knocke88} was used to define the surface
spots and the location of the source vectors for the Mars radiation
pressure. The Mars radiation pressure is computed separately for 43 surface
spots in three concentric rings and a central cap below the
subspacecraft point that are in view of each spacecraft surface.
The Stewart atmosphere model was used to compute the atmospheric density
at the MGS altitude\cite{stewart87}.

The Mars geopotential is modeled in spherical harmonics using
the expression\cite{kaula66},

\begin{equation}
U = {GM \over r} \sum_{l=0}^{\infty} \biggl({a_e\over r}\biggr)^l
\sum_{m=0}^{l}
\bigl\lbrace
\bar C_{lm}  \cos m\lambda + \bar S_{lm}  \sin m\lambda \bigr\rbrace
\bar P_{lm}(\sin\phi)
\end{equation}
\noindent where the expansion is defined using the spherical coordinates
of radius, $r$, latitude, $\phi$, and longitude, $\lambda$; $\bar C_{lm}$ and
$\bar S_{lm}$ represent the normalized geopotential coefficients; $\bar P_{lm}$
are the normalized associated Legendre functions of degree $l$ and order $m$;
$a_e$ is the reference equatorial radius; $G$ is the  universal constant
of gravitation, and $M$ is the planet mass.
The 1991 IAU system of constants was used to define the orientation
of Mars\cite{davies92}.

The {\it a priori} model was the MGM0890 solution, a model complete
to 70x70, based on the historic Viking Orbiter and Mariner 9 S Band tracking,
and the X Band tracking of MGS in the Hiatus and the Science Phasing Orbits
\cite{lemoine99}.  Updated solutions to 70x70 were
soon developed that incorporated the tracking from MGS in February and
March 1999,
and these improved models were applied in the production of orbits
for analysis of the MOLA altimetry.

MGS periodically fires its thrusters to desaturate the reaction wheels, which
absorb angular momentum from disturbance torques acting on the spacecraft.
Three to four angular momentum desaturations occurred per day after entry into
the low-altitude mapping orbit\footnote{The AMD events are much more frequent
in the low-altitude mapping orbit. In all of Hiatus, SPO-1, and SPO-2 only
29 AMD events were documented.}.  The thruster firings can impart some
velocity impulse effectively altering the orbit.
Since telemetry informs us of the time and duration of these events,
the angular momentum desaturations are modelled using empirical
three-axis accelerations.  Constant radial, along-track, and cross-track
accelerations are applied over the duration of each event
(typically two to there minutes),  and are estimated as part of
each orbit determination solution.


In the first pass through the February and March mapping orbit data, the
tracking data were processed in one-day arcs. Later, as the modelling  of the
Mars geopotential improved,  longer arcs  (three to five days)
were used. Each arc adjusted the spacecraft state, a solar radiation
reflectivity coefficient ($C_r$), a drag coefficient ($C_d$) per day,
as well as the empirical accelerations that modelled each AMD event.

Two distinct sets of orbit analyses were carried out.
First, orbits that did not overlap were computed
using the orbit modelling just described.
These arcs in February and March formed the basis for the improvement
in the modelling of the Mars geopotential. Second,
quasi-reduced dynamic orbits that overlaped in time were computed to support
the analysis of the MOLA altimeter data. These orbits adjusted additional
empirical accelerations daily in the form of along-track once per revolution
parameters. The adjustment
of these extra empirical accelerations
helps to remove residual orbit mismodelling.
The orbit overlaps between adjacent arcs (except when orbit trim maneuvers
occurred), provide a running check on the orbit quality.
Since the frequent AMD events (three or four
per day) would "interrupt"  the period of each once per rev. empirical
acceleration, constraint equations in GEODYN "tied" together the once-per-rev
accelerations each day, effectively resulting in a single once-per-rev
adjustment per day (See Rowlands et al.\cite{rowlands97}
for a description of the application of these constraint equations).

\subsection*{Nonconservative force modelling}
The nonconservative force model must take into account the complex geometry and
material properties of the spacecraft.  The spacecraft is treated
as a combination
of flat plates arranged in the shape of a box, with attached solar arrays and
drag flaps (ie. 'Box-Wing' or 'Macromodel').  The solar radiation pressure,
atmospheric drag, and  Mars radiation pressure forces acting on each plate
are computed and vectorially summed to compute the total nonconservative force
acceleration acting on the spacecraft.
The Mars radiation pressure includes the reflected solar radiation, and
the natural thermal emission of the planet.
The same approach has been used to model the nonconservative
forces acting on other
spacecraft, for instance TOPEX/Poseidon\cite{marshall94}, TDRSS\cite{luthcke97},
and GEOSAT-Follow-On\cite{lemoinegfo}. Self-shadowing of spacecraft elements
is not considered.  A ten plate box-wing model for MGS was derived
from engineering drawings of the spacecraft\cite{griffin99, tracadas99}.
The box-wing model
included six plates for the spacecraft bus, and a total of four panels
to represent the front and back sides of the +Y and -Y solar arrays.
Each  +Y or -Y panel is represented as a weighted sum of the area and
material properties of the solar array yoke, the inboard solar
array (which has gallium arsenide solar cells), the outboard solar array
(which has silicon solar cells), and the drag flaps. The areas and
reflectivities are summarized
in Table 2.  The reflectivity was assumed to be 80 percent diffuse and
20 percent specular.
Multi-layer insulation (MLI) was assumed to cover the yoke and the spacecraft
body.
The spacecraft configuration  in Feburary and March 1999 is depicted
in Figure 1.  In the spacecraft body-fixed frame, the +Z direction points
at nadir, when MOLA is collecting data (MOLA, the Mars Orbiter Camera,
and the Thermal Emission Spectrometer are located on the +Z face
of the spacecraft bus). The +X direction is the axis through the
stowed position of the High Gain Antenna.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|cccc|}
\multicolumn{4}{c}{Table 2: MGS Macromodel Characteristics}\\
\hline\hline
\rule{0mm}{3mm} Panel  & Area & Reflectivity  & Comment\\
 & (m$^2$) & & \\
\hline
\multicolumn{4}{|l|}{{\underbar {Solar Arrays}}}\\
\multicolumn{1}{|l}{Yoke} &  0.499 & 0.65 & MLI. \\
\multicolumn{1}{|l}{Inner Panel (front)} & 3.077 & 0.11 & GaAs cells.\\
\multicolumn{1}{|l}{Inner Panel (back)}  & 3.077 & 0.43 & 44 \% white paint;\\
& & & 56 \% black substrate.\\
\multicolumn{1}{|l}{Outer Panel (front)} & 3.152 & 0.25 & Si cells.\\
\multicolumn{1}{|l}{Outer Panel (back)}  & 3.152 & 0.23 & 10 \% white paint;\\
 & & & 90 \% black substrate.\\
\multicolumn{1}{|l}{Drag Flaps} & 1.418 & 0.15 &  black kapton.\\
{\bf Composite (front)} & 8.146 & 0.20  & \\
{\bf Composite (back)}  & 8.146 & 0.32  & \\
\hline
\multicolumn{4}{|l|}{\underbar {Spacecraft Bus}}\\
 +X, -X &  3.04  & 0.65 &  MLI.\\
 +Y, -Y &  3.37  & 0.65 &  MLI.\\
 +Z, -Z &  1.82  & 0.65 &  MLI.\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

\begin{figure}[htbp]
\begin{center}
\strut\psfig{figure=fig1.mgs.ps,height=3.5in}
\vspace*{0.00in}
\end{center}
\caption{Nominal MGS spacecraft configuration during Array Normal Spin.}
\end{figure}

The solar radiation pressure, Mars radiation pressure, and drag accelerations
on Mars Global Surveyor are shown in Figure 2 for a 13 hour period starting
at 11:00 on March 6, 1999. The dominant nonconservative force is the solar
radiation pressure. The average total acceleration
is 3.2 x 10$^{-8}$ m/s$^2$ for
the solar radiation pressure, 5.4 x 10$^{-9}$ m/s$^2$ for the Mars radiation
pressure, and 4.8 x 10$^{-10}$ m/s$^2$ for the atmospheric drag.
The Mars radiation pressure peaks in the vicinity of the subsolar point
due to the concentration of the planet's thermal emission and reflected
solar radiation on the sunlight hemisphere\cite{lemoine92}. It is merely fortuitous
that the atmospheric drag is relatively unimportant compared to the
solar radiation pressure. The Mars areocentric longitude on March 6, 1999
was 107$^\circ$, corresponding to summer in the northern hemisphere of Mars.
Since Mars had past aphelion at this time, a substantial portion of the
atmosphere had condensed onto the southern polar cap of Mars,
reducing the mean density at the MGS altitude (370 to 430 km).
By early 2000, Mars will have past perihelion. This, in combination with
the onset of summer in the southern hemisphere, means that the
the southern polar cap will have largely sublimated, and the expected
atmospheric density at the MGS altitude (and hence drag acceleration)
will increase by a factor of 10 to 100\cite{esposito96}.

\begin{figure}[htbp]
\begin{center}
\strut\psfig{figure=fig2.ps,width=4.0in}
\vspace*{0.50in}
\end{center}
\caption{Nonconservative force model accelerations on  Mars Global Surveyor
computed from the MGS macromodel on March 11, 1999.
The solar radiation pressure acceleration dominates at this time.
The spacecraft operating mode affects the orientation of the
solar arrays and the calculation of the solar radiation
pressure. Through approximately 19:00 on March 11,
the spacecraft was nadir pointing, and the solar arrays
articulated, tracking the Sun.
>From about 19:00 through 22:00, the spacecraft was in Earth point
and the solar arrays were held in a fixed orientation.}
\end{figure}



\subsection*{MGS Attitude Modelling}

An essential component of properly modelling the nonconservative forces
with a macromodel is to define the spacecraft attitude at all times.
Since the solar arrays account for such
a large surface area (16 m$^2$) on MGS, modelling their orientation
as a function of time is also essential.

In February 1999, after entry into GCO, the spacecraft was maintained
in Array Normal Spin (ANS).
In this attitude, the spacecraft +X axis
(containing the stowed High Gain Antenna) was pointed at the Earth, and
the spacecraft rotated once every 100 minutes about this axis.
The spacecraft solar arrays alternated between two fixed orientations,
as depicted by the solar array quaternions in Figure 3 for the Y+ solar array.

During fixed HGA mapping in March 1999, the  attitude regimes were more
complicated. The spacecraft alternated between
the collection of science data 18 to 20 hours per day
(+Z axis pointed nadir), and collection of tracking data  four to six hours
per day (+X axis pointed at the Earth).
During nadir pointing (mapping mode) the solar arrays
articulated continuously, whereas during the Earth point regimes, the orientation
of the solar arrays was held fixed.
After deployment of the High Gain Antenna, the spacecraft remained nearly
exclusively in nadir pointing (mapping) mode, with the exception of
the period of the HGA anomaly (April 16 to April 28), and the second
period of fixed High Gain Antenna mapping (April 28 to May 5, 1999).

Gaps in the telemetered spacecraft and solar array quaternions, caused
some difficulties in the modelling of the spacecraft attitude.
Our data processing with GEODYN required a complete time history
of the quaternions over each orbital arc.
The quaternions had to be preprocessed and prefiltered to fill in the gaps.
Gaps of a few minutes could be easily filled with linear interpolation,
providing sufficient accuracy for the macromodel and even the
measurement model calculations.
However, gaps of several revolutions  had to be filled in by manufacturing
the appropriate spacecraft or solar array quaternions using some knowledge
of the spacecraft attitude control laws\footnote{While interpolation,
or manufacture of quaternions during data gaps had sufficient accuracy
for the macromodel and measurement calculations, geolocation of and analysis of
altimeter data required the more precise pointing knowledge provided
by actual data.}.
During fixed HGA mapping in March 1999, and again from April 28 to May 5 1999,
the time of transition between the Earth point and nadir pointing regimes
was not always obvious because of the gaps in the spacecraft quaternions.
These transition times had to be inferred from other sources, such as
the start and stop times of actual tracking, the start and stop times
of altimeter data collection, and information in the mission sequence of events
(SOE) files.

\begin{figure}[tbph]
\begin{center}
\strut\psfig{figure=quats.saplus.data.99043.33.ps,width=5.0in}
\vspace*{-0.20in}
\end{center}
\caption{MGS Y+ solar array quaternions in GCO. The quaternions define the
orientation of the Y+ solar array with respect to the spacecraft body-fixed frame.
During GCO, the solar arrays alternated between two fixed orientations.}
\end{figure}

\section*{GRAVITY FIELD IMPROVEMENT}
\subsection*{Data Coverage}
The coverage of the Doppler data from February 5, 1999 to March 27, 1999
is depicted in Figure 4.  The coverage is mostly uniform over the
entire planet. Some coverage is lacking over the southern polar regions
because the spacecraft exited from occultation over this area.
By the time lock up on the two-way Doppler signal occurred,
the spacecraft had already moved further away from the South pole.
Nonetheless, the coverage obtained by MGS in just two months is superior
to the low altitude coverage obtained from the Viking Orbiters,
which was limited to the equatorial regions of the planet ( 40$^\circ$ S to
40$^\circ$ N).

\begin{figure}[t]
\begin{center}
\strut\psfig{figure=./figures/coverage.ps,width=5.0in}
\vspace*{-0.25in}
\end{center}
\caption{Doppler tracking coverage for Mars Global Surveyor from February 5,
1999 through March 27, 1999, after entry into the low-altitude mapping orbit,
and prior to deployment of the High Gain Antenna.}
\end{figure}

\subsection*{Method of Solution}
Least squares normal equations were created for the tracking data in
February 1999 and March 1999, and these normal equations were merged
with the normal equations of the earlier MGS data from Hiatus and
SPO, and the historical data from the Viking Orbiters and Mariner 9.
The SOLVE program, a companion program to GEODYN which combines and manipulates
normal equations, was used to obtain the least squares solutions
\cite{solve99}.
No changes were made to the processing of the historical Mariner 9
and Viking Orbiter data described in Lemoine et al.\cite{lemoine99}.
The Mars gravity solutions were computed using least squares
collocation\cite{schwartz76,schwartz78}.
An {\it a priori} Kaula-type power law constraint was applied
in the derivation of the gravity solutions. The power law 13.0 x 10$^{-5}$/$l^2$
was applied in the MGM0890 and MGM0890I solutions, whereas the power law
$\sqrt{2}$ x  13.0 x 10$^{-5}$/$l^2$ was used in MGM0964C5A, MGM0964C18,
and MGM0964C20.

A sequence of solutions were developed that gradually included more
data, and refined the models. This paper summarizes the performance
of five landmark models developed at NASA GSFC that included
the new tracking data from the MGS, low-altitude, mapping orbit:

\begin{enumerate}
\item MGM0890, the {\it a priori} model\cite{lemoine99}
to 70x70 that included MGS data from Hiatus and SPO, as well as the
historical tracking.
\item  MGM0890I, a solution to 70x70 that included
19 days of GCO data from February 4, 1999 through February 23, 1999, in
the form of one-day arcs.
\item  MGM0964C5A, a 70x70 solution that included
all the available data in February and March 1999. The February
data were processed in three day arcs, whereas the March data
were processed in one-day arcs.
\item MGM0964C18, a refinement of the MGM0964C5A solution, where
the MGS Doppler tracking data from February
were upweighted from an effective weight of 7 mm/s to 0.357 mm/s.
\item MGM0964C20, a 70x70 solution derived from the normals were recreated
based on data reductions using the MGM0964C18 solution  and improved
quaternion files for the spacecraft and the solar arrays.
In this solution the February 1999 data were processed in three day arcs,
whereas the March 1999 data were processed in four to five day arcs.
Both the February and March 1999 data were weighted at 0.357 mm/s (0.01 Hz).

\end{enumerate}

Whereas MGM0890, MGM0890I, and MGM0964C5A continued to incorporate the
historical Mariner 9 and Viking Orbiter tracking.
The MGM0964C18 and MGM0964C20 solutions only included the MGS
data from Hiatus, SPO, and the low-altitude mapping orbit.
The new MGS tracking data from the low-altitude, near-circular orbit
completely supercede the historical data in both quality and coverage.
Indeed, as will be shown it is possible to fit the historical data almost
as well with MGS only gravity solutions (MGM0964C18, and MGM0964C20),
as with  solutions that incorporate the historical data (MGM0890I, MGM0964C5A).

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccccc|}
\multicolumn{7}{c}{Table 3: Summary of Mars Gravity Solutions}\\
\hline\hline
\rule{0mm}{3mm} Model & Include & \multicolumn{2}{c}{No. of observations}
 & \multicolumn{2}{c}{Effective Data Wt.} & Kaula Power  \\
 & historical & Hiatus & & \multicolumn{2}{c}{(mm/s)}
 & Law \\
 &  data   & \& SPO & Mapping &Feb. 99 &March 99 & Applied\\
\hline
MGM0890 & yes. & 257,165 & 0 & 7.14 & 7.14 &
13.0 x 10$^{-5}$/l$^2$\\
MGM0890I& yes. & 257,165 & 66,605 & 7.14 & 7.14 &
13.0 x 10$^{-5}$/l$^2$\\
MGM0964C5A & yes. & 257,165 & 96,486 & 7.14 & 7.14 &
$\sqrt{2}$ 13.0 x 10$^{-5}$/l$^2$\\
MGM0964C18 & no. & 257,165 & 96,483 & 0.36 & 7.14 &
$\sqrt{2}$ 13.0 x 10$^{-5}$/l$^2$\\
MGM0964C20 & no. & 257,165 & 93,030 & 0.36 & 0.36 &
$\sqrt{2}$ 13.0 x 10$^{-5}$/l$^2$\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

\subsection*{Orbit performance measured by orbit overlaps}
One gauge of the fidelity of the resultant geopotential models is
to calculate the RMS difference in position between successive overlapping
arcs.  Three sets of overlaps were used:
\begin{enumerate}
\item Twelve overlaps of
thirteen 30-hour arcs in February 1999. These arcs overlapped by
six hours, and adjusted a drag coefficient ($C_d$) every eight hours,
a solar radiation reflectivity coefficient ($C_r$), the spacecraft
state, and the empirical accelerations for the AMD events (see Table 4).

\item Six overlaps of five to six day arcs in March 1999. These arcs
adjusted a $C_d$ per day, a $C_r$ per arc, the spacecraft state,
the empirical accelerations for the AMD events, and an empirical along-track
acceleration per day (see Table 5).
The overlaps average 2022 minutes in length, and on average 20 to 25 percent
of each arc are in each pair of overlaps.

\item Eight overlaps of five to six day arcs in April through
June 1999 (see Table 6).
These arcs adjusted the same set of parameters as the arcs in the
March 1999 overlap tests.  These data are always
independent (not included in the solution), whereas the data in
February and March 1999 may or may not be independent, depending
on the solution tested (see Table 3).  These arcs are not
as high quality as the arcs in February and March 1999 since (a) only
a mean  measurement offset was applied for the High Gain Antenna after
deployment, and (b) the tracking data show evidence of greater noise
following HGA deployment.
\end{enumerate}

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccc|}
\multicolumn{5}{c}{Table 4: MGS Orbit Overlaps in February 1999}\\
\hline\hline
\rule{0mm}{3mm} Model &  \multicolumn{4}{c|}{Orbit Overlap Differences
(m)\footnote{Average RMS of 12 overlaps.}}\\
 &   Radial & Cross-track & Along-track & Total\\
\hline
MGM0890  & 41.72 & 10034. & 432.98 & 10049.\\
MGM0890I &  1.98 &  200.92 & 10.46 & 201.32 \\
MGM0964C5A & 0.80 & 220.86 & 7.65 & 221.10 \\
MGM0964C18 & 0.35 &  77.88 & 3.40 & 78.04 \\
MGM0964C20 & 0.18 & 42.64 & 2.02 & 42.72 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccc|}
\multicolumn{5}{c}{Table 5: MGS Orbit Overlaps in March 1999}\\
\hline\hline
\rule{0mm}{3mm} Model &  \multicolumn{4}{c|}{Orbit Overlap Differences
(m)\footnote{Average RMS of six overlaps.}}\\
 &   Radial & Cross-track & Along-track & Total\\
\hline
MGM0890  & 38.53 &   196.56 & 156.53 & 266.23\\
MGM0890I &  7.82 &   33.37 & 24.09 &  45.54 \\
MGM0964C5A & 2.28 &  20.30 & 6.74 &  22.68 \\
MGM0964C18 & 1.34 &  19.84 & 5.51 & 21.06 \\
MGM0964C20 & 0.60 &   5.28 & 2.01 &  6.26 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccc|}
\multicolumn{5}{c}{Table 6: MGS Orbit Overlaps after HGA deployment}\\
\hline\hline
\rule{0mm}{3mm} Model &  \multicolumn{4}{c|}{Orbit Overlap Differences
(m)\footnote{Average RMS of eight overlaps in April, May and June 1999}}\\
 &   Radial & Cross-track & Along-track & Total\\
\hline
MGM0890I & 11.89 &  113.48 & 234.80 &  263.01 \\
MGM0964C5A & 6.07 &  24.55 & 52.01 &  58.35 \\
MGM0964C18 & 5.52 &  21.32 & 39.33 & 45.54 \\
MGM0964C20 & 5.23 &   8.59 & 24.76 & 27.12 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}


For consistency in these tests, all the results reported
in Tables 4, 5 and 6 used the improved  and refiltered
solar array and spacecraft quaternions, even though the normals
of the February and March 1999 data that contributed to the MGM0890I,
MGM0964C5A, and the MGM0964C18 solutions used the first cut,
unfiltered quaternions.
The overlaps show that the weight selected for the tracking data plays
an important role.  Earlier solutions weighted the data too pessimistically
at 7.14 mm/s (0.1 Hz).
The X Band Doppler data have a precision of 0.1 mm/s, or better.
Upweighting the data to 0.36 mm/s (0.01 Hz), as in the MGM0964C18
and in MGM0964C20 (see Table 3) allows these
data to contribute more fully to the geopotential solution.
The improvement observed with the MGM0964C20 field (in comparison
to the antecedent solution, MGM0964C18) is due both to the reiteration,
and also to the careful refiltering and reconstruction of the
quaternions.  Using the first set of quaternions (unfiltered) resulted
in average overlaps using MGM0964C18 for the February data of 0.46 m
in the radial direction, 108.84 m cross-track, 3.72 m along-track, and
109.05 m in total position. So simply switching quaternion sets improved
the orbits (as reported in Table 4, the average radial overlap
for the February data with the refiltered quaternions and MGM0964C18
is 0.35 m in the radial direction, and 78.04 m in total position).
However, the second order improvement occurred with
the development of the MGM0964C20 model. Nonconservative force
mismodelling that had compromised the earlier solutions was removed
in MGM0964C20.

The radial orbit overlaps with the latest MGM0964C20 solution in
February and March 1999 are on average only a tens of cm.  The dramatic
radial consistency of the orbits is due not just to the fidelity
of the gravity solution, but also the observing geometry. In February
and March 1999 the tracking body (the Earth) was located from -4$^\circ$ to
+14$^\circ$ above the MGS orbit plane. Thus, the MGS orbit was nearly
edge-on as seen from the Earth. This edge-on geometry was propitious
for maximizing the  sensitivity to the Doppler data and
to the determination of the MGS orbit in the radial direction.
The overlaps are overall better in March (see Table 5) rather than
in February 1999 (see Table 4). The adjustment of extra empirical
accelerations in March 1999 compensates for the sparser tracking.

The overlaps are not shown following HGA deployment for the MGM0890
field (see Table 6). It is encouraging, that even with independent
data (none of the post-HGA deployment data were used in the
geopotential solutions), the total consistency of the orbits
in all directions shows the same trends as the overlaps in February
and March 1999.


\subsection*{Orbit performance measured by altimeter crossovers}
Altimeter crossovers from MOLA were used to evaluate the geopotential
solutions MGM0890I, MGM0964C5A, MGM0964C18, and the MGM0964C20.
The altimeter crossovers have the advantage of being completely
independent data, and are globally distributed.
Altimeter crossovers were formed for four five to six day arcs from
starting on March 7, March 13, March 18, and March 23, 1999.
Between 940 to 980 crossovers were computed
internally to  each arc. No crossovers were formed if telemetered
spacecraft quaternions were missing for either the descending or ascending
track of a potential crossover point.
The modelling of the altimeter crossovers is described in Rowlands
et al.\cite{rowlands99}.
Altimeter crossovers from the highly eccentric orbits in SPO
were used to estimate corrections to the preflight measurements of
the orientation of the MOLA instrument with respect to the spacecraft
frame\cite{lemoine99, rowlands99}. These new offset values of
-0.0028$^\circ$ in roll,and -0.0086$^\circ$ in pitch were simply applied in
the current tests  with the mapping orbit altimetry.
The crossovers are not used in these orbit determination solutions, which
were determined solely from the range and the Doppler data for each
gravity field model.  The crossovers are editted for off nadir pointing angles,
as well as the slope and the roughness of the terrain. Large crossover residuals
(greater than 80 m) were eliminated.
Because of the edit criterion of 80 m, the number of crossovers is
not identical in all of the test arcs.
The results are tabulated in Table 7 in aggregate for all four arcs.
The crossovers show the same trend as the orbit overlaps, and the
fits to the Doppler data. The crossover RMS is reduced as the fields are
refined, and show the best performance with the MGM0964C20 field.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccc|}
\multicolumn{5}{c}{Table 7: MGS orbit performance measured by altimeter crossovers}\\
\hline\hline
\rule{0mm}{3mm} Model &  \multicolumn{2}{c}{Doppler} &
\multicolumn{2}{c|}{Crossovers}\\
 & No. of obs. & RMS (mm/s) & No. of obs. & RMS (m) \\
\hline
MGM0890I & 14,838 & 18.93 & 3400 & 21.04 \\
MGM0964C5A & 14,838 & 1.65 & 3455 & 5.61\\
MGM0964C18 & 14,838 & 0.84 & 3449 & 5.57\\
MGM0965C20 & 14,838 & 0.53 & 3455 & 5.01\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

It is of interest to see whether the crossovers can still contribute
to orbit improvement in the near-circular orbit. Crossovers were processed
simultaneously with the Doppler tracking in four arcs of March 1999.
Empirical once per revolution along-track accelerations
were adjusted daily for these arcs, in addition to a drag coefficient
per day, a reflectivity coefficient per arc, and the empirical accelerations
for the AMD events.
March 1999  is a particularly tough test for the
crossovers since the Doppler data - although sparse, are of high quality.
In addition, the Doppler data in March 1999 were used
in the MGM0964C20 solution. The average RMS overlap
is shown in Table 8 for the three overlaps of these  arcs.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccc|}
\multicolumn{5}{c}{Table 8: MGS Orbit Overlaps with Altimeter Crossovers}\\
\hline\hline
\rule{0mm}{3mm} Model &  \multicolumn{4}{c|}{Orbit Overlap Differences
(m)\footnote{Average RMS of three  overlaps for four arcs from March 6, to March 23, 1999}}\\
 &   Radial & Cross-track & Along-track & Total\\
\hline
No Crossovers & 0.59 & 0.81 & 1.83  & 2.17 \\
With Crossovers & 0.26 & 1.40 & 0.90 & 1.73 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

The addition of the crossover data improves the orbits overall, and
especially in the radial, and the along-track directions. Some degradation
is observed in the cross-track direction.
It is especially encouraging that the consistency in the radial direction
has been further improved from 0.59 to 0.26 meters.  The use of these
additional data in the form of altimeter crossovers allows
extra empirical parameters to be adjusted - for instance cross-track
once per revolution empirical accelerations - that would further
improve these orbit consistency results.

\subsection*{RMS of fit to the Historical Data}
Since we have used nearly all the available MGS data, the only
quasi-independent data are the S Band data from the Viking Orbiters
and Mariner 9. It is a necessary condition that the new models including
MGS data obtain comparable fits to solutions based solely on the
historical data, for instance the
GMM-1\cite{smith93}, and Mars50c\cite{konopliv95} solutions. Both
GMM-1 and Mars50c are 50x50 solutions.
A series of test arcs of the Mariner 9 and Viking Orbiter data
were selected for each distinct orbital inclination and periapse
altitude. The orbit geometry and the number of test arcs selected
are listed in Table 9. All the orbits were highly eccentric, and
the orbit periods ranged from 12 to 24 hours\cite{lemoine92}.
Two observations are readily apparent upon examination of the RMS
of fit for these test arcs. First, the RMS of fit for all the new
models with MGS is broadly similar to the GMM-1 and Mars50c models.
In fact, for the low altitude periapse data (300 km), the fit to
the Viking-1 and Viking-2 data is superior with the MGS derived
models. Even the models that exclude all the historical data
(MGM0964C18 and MGM0964C20) fit these low altitude data better
than GMM-1 and Mars50c. The second observation pertains to
the fit to the high altitude periapse data, in particular those
from Mariner 9 and Viking Orbiter 1. Here the fits are degraded
with the MGS derived models, compared to GMM-1 and Mars50c.
So these results suggest an inconsistency in the long-wavelength
gravity field to which MGS and the Viking Orbiters and Mariner 9
are sensitive. This inconsistency may reflect a discordance
in reference frames, and an inadequacy in the 1991 IAU planet orientation
model\cite{davies92}. New solutions for the Mars orientation are available
based on simultaneous analysis of Viking Lander and Pathfinder tracking
which could be used instead of the 1991 IAU model.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|cccc|}
\multicolumn{4}{c}{Table 9: Mariner 9 and Viking Orbiter Orbit Summary}\\
\hline\hline
\rule{0mm}{3mm} Spacecraft & Periapse & Inclination & No. of Test\\
            & Ht. (km) & (deg) & Arcs\\
\hline
Mariner 9 & 1600 & 64 &  6\\
Viking-1 & 1500  & 39 & 6 \\
Viking-1 &  300 &  39 & 4 \\
Viking-2 & 1500 & 55 \& 75 & 2 \\
Viking-2 & 800 & 80 & 7 \\
Viking-2 & 300 & 80 & 9 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}


\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|cccccccc|}
\multicolumn{8}{c}{Table 10: Mariner 9 and Viking Orbiter: RMS Summary}\\
\hline\hline
\rule{0mm}{3mm} Orbit & \multicolumn{7}{c|}{Gravity Models:
(RMS of fit in mm/s)}\\
 & GMM-1 & Mars50c & 0890 & 0890I & 0964C5A & 0964C18 &
 0964C20\\
\hline
Mariner 9 &  1.52 & 1.47 & 1.96 & 2.72 & 1.93 & 1.73 & 1.89\\
VO-1, 1500 km & 2.36 & 2.42 & 2.35 & 2.56 & 4.20 & 4.20 & 4.03\\
VO-1,  300 km & 1.77 & 1.55 & 1.48 & 1.63 & 1.74 & 1.54 & 1.62 \\
VO-2, 1500 km & 1.93 & 2.19 & 1.54 & 1.56 & 1.67 & 1.97 & 1.82\\
VO-2   800 km & 1.84 & 1.80 & 1.79 & 1.85 & 2.82 & 2.41 &2.00\\
VO-2   300 km & 1.16 & 1.17 & 0.90 & 0.86 & 0.84 & 0.78 & 0.78 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

\subsection*{Error Analysis}
It is useful to project the predicted error from the gravity field
covariances to 70x70, and ascertain the formal commission error
due to uncertainties in the gravity field on the MGS orbit.
These results are summarized in Table 11.
The total radial error has decreased from 44 meters with MGM0890 to
8 to 10 cm with the MGM0964C18 and MGM0964C20 models. Similarly,
the total error is reduced from 205 meters with MGM0890, to 1.6 to 2.0
meters with the latest MGS derived gravity models. These order of
magnitude improvements in the predicted orbit error for MGS beg the
question as to how realistic they really are. They certainly are a lower
bound, since they omit the contribution of terms higher than degree 70.
Unfortunately, other than the orbit overlap tests, we have no means
to really test these orbit error predictions.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccc|}
\multicolumn{5}{c}{Table 11: Mars Global Surveyor Predicted Orbit Error}\\
\hline\hline
\rule{0mm}{3mm} Model & \multicolumn{4}{c|}
{Orbit Error Projections (meters)}\\
 & Radial & Cross-track & Along-track & Total\\
\hline
MGM0890 & 43.95 & 184.33 & 79.68 & 205.57\\
MGM0964C18 & 0.10 & 1.53 & 1.29 & 2.01 \\
MGM0964C20 & 0.08 & 1.17 & 1.09 & 1.60 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

Some insight into the validity of the covariance may be gauged by
examining the calibrations of the mapping orbit data in the geopotential
solutions. As described by Lerch\cite{lerch91}, the calibration factor
measures the aggregate change in the geopotential coefficients
divided by the aggregate change in the sigmas of these geopotential
coefficients, when a set of data is removed from a geopotential solution.
If the weighting of the data is appropriate, then the calibration
factors should be close to unity. If the calibration factors are greater than
unity, this is an indication that the set of data is contributing
information inconsistent with the information supplied by the other
sets of data in the solution.
All things being equal, higher calibration factors suggest
either overweighting of data or mismodelling problems with the data.

As discussed in Lemoine et al.\cite{egm96ref},
the calibration factors are averaged by degree or by order.
The calibration factors were computed for the MGM0964C18 and the MGM0964C20
solutions, and are shown in Figure 5 and Figure 6.
The average calibration factors for the mapping orbit
data used in MGM0964c18 were: 1.082 (by degree), and 1.036 (by order).
For the data used in the MGM0964C20 field, the calibration factors
were 0.901 (by degree) and 0.818 (by order).
Indeed, with MGM0964C20, the calibration factors suggest that the
MGS data in February and March 1999 could even be slightly upweighted
from 0.01 Hz (0.36 mm/s).
The lower calibrations MGM0964C20 compared to MGM0964C18
are due to the improved nonconservative force modelling.
Mismodelling of the nonconservative forces deleteriously affects the solution
at the higher degrees and orders.  Relatively high calibration factors
at the lower degrees even with the MGM0964C20 model remain to be elucidated.
By this internal consistency test, then the covariance of the MGM0964C20
solution is at least not overly optimistic.

\begin{figure}[htbp]
\begin{center}
\strut\psfig{figure=./figures/fig5.ps,height=2.5in}
\vspace*{-0.5in}
\end{center}
\caption{Calibration factors for the MGS mapping orbit
data used in the MGM0964C18 solution.}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\strut\psfig{figure=./figures/fig6.ps,height=2.5in}
\vspace*{-0.5in}
\end{center}
\caption{Calibration factors for the MGS mapping orbit data used in
the MGM0964C20 solution.}
\end{figure}



\section*{SUMMARY}
We have improved orbit modelling for Mars Global Surveyor in the
low-altitude and near-circular mapping orbit  by developing a detailed
nonconservative force model
for the spacecraft and by developing improved models of
the Mars geopotential.  A ten plate macromodel was developed
for MGS. Telemetered spacecraft and solar array
quaternions were used define the spacecraft attitude and solar array
orientation in the macromodel.
Two months of tracking data in February 1999 and March 1999 were processed
using this macromodel. These data formed the basis for the order of
magnitude improvement in the modelling of the Mars geopotential.
The improved gravity model, MGM0964C20, a model complete to 70x70
in spherical harmonics results in arc to arc overlap consistency
of a few meters in total position, and under a meter in the radial direction.
One of the best gravity models available prior to entry in the mapping
orbit, MGM0890, which included the MGS data from Hiatus and SPO, had arc to arc
overlap consistencies of 38 to 40 meters in the radial direction
and 220 to 260 meters in total position.
Orbits were calculated used the improved nonconservative and
geopotential modelling, and these orbits were distributed to the
MOLA science team\footnote{These orbits refer the gdn-ql13.*  series
of orbits, and are available in NAIF-SPK format via anonymous ftp
to geodesy.gsfc.nasa.gov (IP address 128.183.109.101);
Change directory (cd) to dist/MGS/quick-look-orbits
and fetch the gdn-ql13.* files.}.


The MOLA  altimeter data were used validate the improvement in the
Mars geopotential modelling.  We have also demonstrated that the altimeter
crossovers can be used in the near-circular mapping orbit to improve
the orbit determination of MGS.

The single outstanding issue in orbit modelling for MGS
remains properly correcting the Doppler data after deployment
of the High Gain Antenna for the motion of the antenna over each
orbit. Only a mean offset has been used so far in the data analysis
for the MGS orbits following HGA deployment.

As stated in the MGS  mission plan, tracking will consist of
one 34-meter ground station pass per day, and one 70-meter ground station pass
every third day\cite{missionplan}. Gaps of up to sixteen hours per day in
the tracking will be possible, especially as the Deep Space Network
faces increasing demands for its time from the  Mars 1998 Orbiter and Lander,
as well as other commitments. The incorporation of altimeter  crossovers
into the orbit determination for MGS will prove especially valuable,
as the tracking becomes less dense.
A second order benefit from the incorporation of crossovers,
is that they will allow the adjustment of additional empirical parameters
that will help to further remove residual errors in the orbit modelling.

\section*{ACKNOWLEDGEMENTS}
We acknowledge Susan K. Fricke (Raytheon ITSS, Greenbelt)
for data processing support, Jeffrey Schott (MIT and GSFC) for technical
support, John J. McCarthy (Raytheon ITSS, Greenbelt) for aid with
SOLVE.  Trish Rodan (JPL), and Richard Simpson (Stanford University)
provided useful mission and spacecraft information.
We are grateful to Boris Semenov (JPL) and the Navigation Ancillary
Information Facility for making available the MGS solar array
quaternions.

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\end{thebibliography}
\end{document}

