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\title{Mars Odyssey Accelerometer: Report on Data Processing, 
Validation of Data Products and Scientific Analysis}

\author{Paul Withers$^{a,*}$}

\affil{$^{a}$ Center for Space Physics, Boston University, 
725 Commonwealth Avenue,
Boston, MA 02215, USA.}

\affil{$^{*}$ Corresponding author email address: withers@bu.edu}

\noindent
Number of pages: \pageref{lastpage} \\
Number of figures: \ref{lastfig} \\
Number of tables: \ref{lasttable} \\

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% Running head of no more than 55 characters
\noindent
{\bf Proposed Running Head:} \\
Odyssey Accelerometer Data

\noindent
{\bf Editorial Correspondence to:} \\
Paul Withers \\
Center for Space Physics, Boston University, \\
725 Commonwealth Avenue, Boston, MA 02215, USA. \\
Email: withers@bu.edu \\
Phone: (617) 353 1531 \\
Fax: (617) 353 6463

\pagebreak

\begin{abstract}
This report serves two purposes. 
First, source material for PDS documentation describing the generation 
of these data products. Second, source material for a scientific paper 
that briefly describes the generation of these data products and spends 
more time describing some analysis and interpretation of these data 
products. 
\end{abstract}

\noindent
Keywords: Mars, Atmosphere; Data Reduction Techniques

\pagebreak

\section{\label{intro}Introduction}

The aim of this paper is to describe the processing of Mars Odyssey 
(ODY) accelerometer data and to present the results of preliminary 
scientific analysis of the derived data products. The structure of 
the paper is as follows: an introduction to the ACC datasets and a 
description of how the ACC dataset was processed 
(Section~\ref{uberprocessing}), 
a comparison to other results (Section~\ref{ubervalidate}), and 
a presentation of scientific results from analysis of the ACC 
data products
(Sections~\ref{scisectionprofs}--\ref{scisectioncaltother}).

ODY reached Mars on 24 October 2001 \citep{smith2005}. ODY was 
inserted into an initial orbit around Mars with a period of 18.6 
hours using chemical propulsion, then transferred into its desired 
science orbit, which has a period of less than two hours, by 
aerobraking.
Aerobraking ended on 11 January 2002.
Aerobraking uses atmospheric drag forces during a brief interval 
around periapsis to make a small adjustment to a spacecraft's orbit. 
Repeated over many periapses, aerobraking can cause significant 
orbital changes \citep{spencer2007}. The ODY accelerometer (ACC) 
instrument was used to measure atmospheric densities during 
aerobraking. These measurements influenced operational decisions 
during aerobraking, such as changes to periapsis altitude. The 
instrument was not part of the mission's scientific payload and
 its data were not archived at the Planetary Data System (PDS) in 
the usual manner. We have acquired ACC measurements and used them 
to obtain measurements of martian atmospheric properties.

Previous work on the ODY ACC dataset includes \citet{tolson2007}, 
\citet{crowley2007},
\citet{bougher2006}, \citet{fritts2006}, \citet{wang2006} and 
\citet[][and references therein]{withers2006b}. Previous publications 
on ODY aerobraking include
\citet{spencer2007}, \citet{chavis2005}, 
\citet[][and references therein]{mase2005}, \citet{smith2005}, 
\citet{tolson2005}, \citet{takashima2002a} and \citet{takashima2002b}.

Figures of orbit number, periapsis latitude, altitude, local 
solar time (LST) and $L_{s}$ are contained in \citet{tolson2005} 
and \citet{tolson2007}. $L_{s}$ is the aerocentric longitude 
of the Sun, a measure of the martian season. $L_{s}$ is 
$0^{\circ}$ at the northern spring equinox, 
$90^{\circ}$ at the northern summer solstice, 
$180^{\circ}$ at the northern autumn equinox and
$270^{\circ}$ at the northern winter solstice.
These figures show that the ODY ACC measurements separate 
naturally into two subsets --- dayside and nightside. In this 
work, ``dayside'' corresponds to 12 hours $<$ LST $<$ 24 hours 
and ``nightside'' corresponds to 0 hours $<$ LST $<$ 12 hours.
The boundary between dayside and nightside is not the same as 
the terminator that separates the sunlit and shadowed hemispheres. 
Sub-solar latitude changed from -24.9$^{\circ}$N to 
-20.1$^{\circ}$N during ODY aerobraking, so all latitudes 
poleward of 65.1$^{\circ}$N (69.9$^{\circ}$N) were in darkness 
in the polar night at the start (end) of aerobraking.

\section{\label{uberprocessing}Data Processing}

\subsection{\label{accdescription}The Accelerometer Instrument 
and its Measurements}

The ODY ACC instrument consisted of two inertial measurement 
units (IMUs). Each IMU contained three single-axis accelerometers, 
each of which measured linear acceleration along the accelerometer's 
axis at the position of the relevant sensor, and three single-axis 
gyroscopes, each of which measured angular velocity about the 
gyroscope's axis at the position of the relevant sensor. These 
measurements were converted into three-axis linear acceleration 
and three-axis angular velocity at a single position within the 
IMU before being transferred out of the IMU. ``The primary IMU was 
used throughout aerobraking. The internal orientation of two of the 
three [single-axis] accelerometers was not aligned with the body axes, 
and so data from these two accelerometers were combined in the ground 
data system to form body axes accelerations'' \citep{tolson2005}.
Data were acquired only near periapsis, so the dataset can easily 
be subdivided by orbit. Aerobraking concluded after several hundred 
orbits. The label ``P076'' refers to orbit number 76, and so on.

In this work we concentrate exclusively on the three-axis linear 
acceleration measurements and do not directly use the three-axis 
gyroscope measurements. ODY produced two distinct data streams of 
three-axis acceleration measurements, low rate and high rate, both 
of which had a nominal interval between measurements of 1 second. 
The generation of the high rate data stream was reported in 
\citet{tolson2005}. ODY's onboard computer sampled the IMU's 
output 200 times per second. For orbits 1--136, the mean of every 
200 samples was sent to the high rate data stream each second. 
For orbits 137--268, the mean of the first 50 of every 200 samples 
was sent to the high rate data stream each second. For orbits 
269--336, the mean of the first 20 of every 200 samples was sent 
to the high rate data stream  each second \citep{tolson2005}. Onboard 
memory limitations and the increase in the duration of aerobraking 
passes as the mission progressed were responsible for this 
deliberate degradation in ACC data quality. The generation of the 
low rate data stream has not been discussed in the scientific 
literature. Its noise characteristics do not change during aerobraking, 
which suggests that the same sampling rate was used throughout 
aerobraking. 
We hypothesize that the low rate data stream contains one of the 
200 samples collected every second, not the mean of a subset of 
the 200 samples. This hypothesis is supported by analysis of the 
noise characteristics of the low rate and high rate data streams.

Low rate data have been delivered to the PDS Atmospheres Node in 
a non-standard format, but have not yet been peer-reviewed nor 
accepted into the PDS (ftp://atmos.nmsu.edu/odyssey/level0/data/). 
Acceleration measurements for orbit XXX were provided in an ASCII 
file (pXXXacc.txt) that contains time in YY/DOY-HH:MM:SS.SSS 
format and x, y, and z-axis accelerations in m s$^{-2}$. There 
is no accompanying documentation, so the time standard and 
reference frame required to use the data have to be identified 
by other means. 
Many different time standards are used by spacecraft missions. 
These times were confirmed to be UTC times, not barycentric 
dynamical times or ephemeris times (ET), by reproduction of 
published plots of acceleration/density versus time since 
periapsis for P076 \citep{tolson2005}.
We assumed that the x, y, and z-axes were those of the 
``M01\_SPACECRAFT'' frame defined by Navigation and Ancillary 
Information Facility (NAIF) SPICE kernel m01\_v28.tf, which is 
identical to the Lockheed Martin (LMA) ``mechanical coordinate 
system'' for ODY \citep{takashima2002a,m01v28tf}. 
The ``M01\_SPACECRAFT'' frame and LMA ``mechanical coordinate 
system'' are identical to the ``spacecraft frame'' described in 
the PDS instrument host file for ODY (versions created after 14 
March 2003).
The LMA ``mechanical coordinate system'' and a POST ``body-frame 
coordinate system'' are also defined in \citet{chavis2005}.

High rate data have been delivered to the PDS Atmospheres Node in 
a non-standard format, but have not yet been peer-reviewed nor 
accepted into the PDS (ftp:atmos.nmsu.edu/odyssey/level0/data/). 
Acceleration measurements for orbit XXX were provided in a MATLAB 
file (LOPXXX.mat). Some documentation was provided in Microsoft 
Excel and Word files. We extracted variables HiACCTime and HiACC 
from each orbit's LOPXXX.mat file. According to the documentation, 
HiACCTime is time expressed as UTC seconds past J2000. These were 
confirmed to be UTC seconds, not ET seconds, by reproduction of 
published plots of acceleration/density versus time since periapsis 
for P076 \citep{tolson2005}. According to the documentation, 
HiACC is x, y, and z-axis accelerations in m s$^{-2}$. These 
accelerations are similar to the low-rate accelerations, which 
suggests that they are expressed in the same coordinate system. 
Therefore we again assumed that the x, y, and z-axes were those 
of the ``M01\_SPACECRAFT'' frame defined by SPICE kernel 
m01\_v28.tf, which is identical to the Lockheed Martin (LMA) 
mechanical coordinate system for ODY 
\citep{takashima2002a,m01v28tf}.

We extracted the low rate and high rate data from these files, 
then discarded some isolated data points. High rate data for 
hundreds of orbits contain gaps of $>$30s towards the end of 
the outbound leg. Data after these gaps were discarded. There 
is an additional gap of $>$5s towards the end of the outbound 
leg in high rate data for both P109 and P173. Data after these 
gaps were discarded. There is a gap of 3s between the last and 
penultimate data points in orbits 26, 27, 28, 38, 69, 74, 123, 
155, 170, 179-189, 249, 273, 276, 285, 292, 297, 299, 301, 303, 
305, and 306. The last data point was discarded in these orbits. 
Gaps of 1s-2s in the high rate data remain in about 20\% of the 
orbits. These are short enough that their effects are minimal, 
so the data were not modified to correct for them. This completes 
step 1 (data cleaning) of the data processing.

\subsection{\label{orbitavail}Orbits with Available Data}

No data were delivered to the PDS for orbits 1, 2, 3, 5, and 
327. The possibility that data were acquired, but omitted from 
the dataset delivered to the PDS, for these orbits can be excluded.
During aerobraking operations, ``Quick Look Reports'' were 
generated by the ACC team within hours of each periapsis pass 
in support of mission operations. These reports summarized ACC 
measurements from a given aerobraking pass.
No ``Quick Look Reports'' were generated for these orbits, which 
suggests that no useful data were received on the ground.

Low rate data, but not high rate data, exist for orbits 4, 57, 
158, 213, 255, 266, 289, and 336. The high rate data stream for 
orbit 137 begins well inside the atmosphere, close to periapsis, 
and contains measurements at 4 s intervals instead of the usual 
1 s intervals. The low rate data for orbit 137 are normal. Data 
for orbits 4 (low rate only, periapsis altitude = 290 km) and 
6 (low rate and high rate, periapsis altitude = 160 km) exist, 
but aerodynamic accelerations cannot be separated from 
non-aerodynamic accelerations when altitudes are so high and 
aerodynamic drag is so weak (see Section~\ref{acccorr}). Results 
for orbits 1-336 used the following data streams: no results for 
orbits 1, 2, 3, 4, 5, 6 and 327; results based on low rate data 
for orbits 57, 137, 158, 213, 255, 266, 289 and 336; and results
based on high rate data for all other orbits.

\subsection{\label{acccorr}Determination of Aerodynamic 
Accelerations from Measured Accelerations}

We wish to derive atmospheric densities, $\rho$, from 
aerodynamic accelerations, $\underline{a_{aero}}$. Measured 
accelerations, $\underline{a_{meas}}$, are not identical to 
$\underline{a_{aero}}$. They are related by:

\begin{equation}
\label{fullacceqn}
\underline{a_{meas}} = \underline{a_{bias}} +
\underline{a_{aero}} +
\underline{a_{gg}} +
\underline{a_{thr}} +
\underline{\omega} \times \left( \underline{\omega} 
\times \underline{r} \right) +
\dot{\underline{\omega}} \times \underline{r} 
\end{equation}

$\underline{a_{bias}}$ represents the effect of sensor 
bias, $\underline{a_{gg}}$, which is called the gravity 
gradient term, represents the effect of differences between 
the acceleration due to gravity at the sensor and at the 
spacecraft's centre of mass, $\underline{a_{thr}}$ represents 
the effect of reaction control system (RCS) thruster firings, 
$\underline{\omega}$ is the angular velocity of the spacecraft, 
$\underline{r}$ is the position of the accelerometer with 
respect to the spacecraft's centre of mass, an overdot 
indicates a derivative with respect to time, and the last 
two terms represent the effects of angular motion of the sensor 
about the spacecraft's centre of mass. The derivation of 
atmospheric densities uses only the y-axis component of 
$\underline{a_{aero}}$, so we focus on the y-axis component of 
Equation~\ref{fullacceqn}. We now address each term in 
Equation~\ref{fullacceqn}.

The two angular acceleration terms can, in principle, be 
calculated. SPICE orientation information is sufficient 
to determine $\underline{\omega}$ and $\dot{\underline{\omega}}$. 
However, we do not know $\underline{r}$ accurately. The positions 
of the accelerometer and the centre of mass were measured before 
launch. The accelerometer has not moved since then, but the 
centre of mass has. Fuel consumption causes a secular change in 
centre of mass position. Accelerations experienced during 
aerobraking cause the fuel to slosh back and forth in the tanks, 
which also alters the centre of mass position. Rapid changes 
in $\underline{r}$ could introduce an additional term, a Coriolis 
term, into Equation~\ref{fullacceqn}. According to 
\citet{tolson2005}, the accelerometer position is (x,y,z) = 
(0.164, -0.544, 1.137) metres relative to the centre of mass. 
However, the reference frame for this position is not defined. 
The frame suggested by Figure 1 of \citet{tolson2005} is not 
consistent with either the ``M01\_SPACECRAFT'' frame or the 
``body coordinate system'' 
\citep{m01v28tf, takashima2002a, chavis2005}.
According to \citet{chavis2005}, the accelerometer position 
at the midpoint of aerobraking was (x,y,z) = (-0.629, -0.0172, 1.1) 
metres relative to the centre of mass in the ``body coordinate 
system.'' This is (x,y,z) = (0.0172, 0.629, -1.1) metres relative 
to the centre of mass in the ``M01\_SPACECRAFT'' frame.
It is likely that the frame used by \citet{tolson2005} differs 
from the ``M01\_SPACECRAFT'' frame by a factor of $-1$, which 
makes their accelerometer position (x,y,z) = (0.164, -0.544, 1.137) 
metres relative to the centre of mass. 
We do not correct $\underline{a_{meas}}$ for the two angular 
acceleration terms, therefore angular accelerations influence 
the uncertainty in $\underline{a_{meas}}$ (Section~\ref{threshold}).

$\underline{a_{thr}}$ is ``two orders of magnitude less 
than the periapsis drag effect'' \citep{tolson2005}.
The typical acceleration at periapsis is $2 \times 10^{-2}$ 
ms$^{-2}$, so the thruster term is on the order of 
$2 \times 10^{-4}$ ms$^{-2}$. 
The nominal impulse, $I$, generated by a single RCS 
thruster firing is approximately $I = 0.00804038 t + 0.02125765$, 
where $t$ is the duration of the thruster firing in milliseconds 
and $I$ is in N-sec \citep{mase2000}.
There are four RCS thrusters, labelled RCS1 to RCS4. The change 
in ODY's velocity due to an RCS thruster firing, 
$\Delta \underline{v}$, equals $I \underline{\hat{e}}/m$ where 
$m$ is ODY's mass and $\underline{\hat{e}}$ is a unit vector. 
$\underline{\hat{e}}_{RCS1}$ = (-X, Y, -Z), 
$\underline{\hat{e}}_{RCS2}$ = (-X, -Y, -Z), 
$\underline{\hat{e}}_{RCS3}$ = (X, -Y, -Z), 
$\underline{\hat{e}}_{RCS4}$ = (X, Y, -Z), where 
$X = 0.8926$, $Y = 0.4162$, and $Z = 0.1736$ \citep{mase2000}.
These unit vectors are in the ``M01\_SPACECRAFT'' frame. 
Thruster firings for orbit XXX are reported in ASCII files 
(pXXXthot.txt) that accompany the low rate accelerometer data. 
The cumulative times that each thruster has fired for are 
listed as functions of time. Typical thruster firings last for 
0.1 sec, much less than the typical interval between cumulative 
times in pXXXthot.txt (1--10 sec). Since the times of thruster 
activity cannot be resolved, their contributions cannot be 
removed from the measured accelerations. Also, the actual 
impulse produced by a thruster will differ from the nominal 
impulse. Past experience has shown that determination of 
thruster impulse to within 50\% is difficult \citep{tolson2005}. 
We do not correct $\underline{a_{meas}}$ for thruster activity.

$\underline{a_{gg}}$ is on the order of 
$1 \times 10^{-6}$ ms$^{-2}$ \citep{cancro1998}.
This is much smaller than $\underline{a_{thr}}$, so 
we neglect it.

When ODY is far outside the atmosphere, 
$\underline{a_{meas}} = \underline{a_{bias}}$. Pre-entry 
and post-exit measured accelerations can be used to estimate 
instrument bias.
Scatter in $a_{meas,lo,y}$ (the y-component of the low rate 
measured acceleration) was large enough that estimates of 
$a_{bias,lo,y}$ could not be distinguished from zero. 
$a_{meas,lo,y}$ was not corrected for bias. 
$a_{bias,hi,y}$ was calculated from pre-entry and post-exit 
accelerations. The pre-entry value of $a_{bias,hi,y}$ was 
assumed to be the mean of all $a_{meas,hi,y}$ measurements 
from 10s after the start of the data stream to 60s later. 
The post-exit value of $a_{bias,hi,y}$ was assumed to be 
the mean of all $a_{meas,hi,y}$ measurements from 70s 
before the end of the data stream to 60s later. These time 
ranges were chosen to ensure that aerodynamic accelerations 
and any transients associated with the beginning/end of the 
data stream were negligible. The typical pre-entry bias was 
$-2.4 \times 10^{-4}$ ms$^{-2}$ and the typical post-exit 
bias was $-2.2 \times 10^{-4}$ ms$^{-2}$,
as shown in Figures~\ref{preavarray} and \ref{postavarray}. 
$a_{bias,hi,y}$ was assumed to vary linearly with time during 
an aerobraking pass. The post-exit bias was assumed to equal 
the pre-entry bias for orbits 84, 91, 93, and 95 due to 
lack of adequate post-exit data.
We correct $\underline{a_{meas}}$ for bias for high rate data only.

[FIGURE~\ref{preavarray}]
[FIGURE~\ref{postavarray}]

%The estimated bias was similar to the standard error in the mean 
%of the selected 
%measurements (the standard deviation of selected measurements 
%divided by the 
%square root of number of selected measurements). 

Thus $a_{aero,lo,y}$ = $a_{meas,lo,y}$ and 
$a_{aero,hi,y}$ = $a_{meas,hi,y}$ - $a_{bias,hi,y}$. 
Maximum values of $a_{aero,hi,y}$ and $a_{aero,lo,y}$ are shown 
in Figure~\ref{maxaaeroy}. 
This completes step 2 (data correction) of the data processing.

[FIGURE~\ref{maxaaeroy}]

\subsection{\label{averaging}Reduction of Noise in the Data}

We next formed 7-point and 39-point running means of 
$a_{aero,lo,y}$ and $a_{aero,hi,y}$, effectively 7-s and 
39-s averages, and labelled them $a_{aero,lo,y,7}$, 
$a_{aero,lo,y,39}$, $a_{aero,hi,y,7}$, and $a_{aero,hi,y,39}$. 
This reduced the noise in the accelerations at the expense 
of spatial resolution. The duration of the 39-point time 
series is shorter than the duration of the 7-point time series, 
which is shorter than the duration of the unaveraged time 
series, because a 7-point running mean cannot be calculated 
for the first three data points in a series of measurements.

The noise inherent in $a_{aero,lo,y}$, $a_{aero,hi,y}$, 
$a_{aero,lo,y,7}$, $a_{aero,hi,y,7}$, $a_{aero,lo,y,39}$, 
and $a_{aero,hi,y,39}$ was estimated based on the standard 
deviation of selected pre-entry measurements. Measurements 
from 30s after the start of the time series to 60s later 
were used for $a_{aero,lo,y}$, $a_{aero,lo,y,7}$, and 
$a_{aero,lo,y,39}$. Measurements from 10s after the start 
of the time series to 200s later were used for $a_{aero,hi,y}$. 
Measurements from 10s after the start of the time series 
to 100s later were used for $a_{aero,hi,y,7}$. Measurements 
from 30s after the start of the time series to 60s later
were used for $a_{aero,hi,y,39}$. The start times of these 
time series were chosen to ensure that transients associated 
with the beginning of the data stream did not affect any 
of the calculated noise values. The end times of these time 
series were chosen to ensure that aerodynamic accelerations 
did not affect any of the calculated noise values. These 
time series were made as long as possible to increase the 
accuracy of the noise calculation.

Results are shown in Figure~\ref{accnoise} and 
Table~\ref{accnoisevalue}. Noise levels clearly depend on 
the number of samples acquired per second: 200 for orbits 
1--136, 50 for orbits 137--268 and 20 for orbits 269--330.
The ratio of noise levels in $a_{aero,hi,y}$ for orbits 
269--330 to orbits 1--136 is 6.05, significantly greater 
than the ratio of $\sqrt{200/20}=3.16$ that would occur 
for samples from a Gaussian distribution.
The ratio of noise levels in $a_{aero,hi,y}$ for orbits 
137--268 to orbits 1--136 is 3.28, significantly greater 
than the ratio of $\sqrt{200/50}=2$ that would occur for 
samples from a Gaussian distribution.
The ratio of noise levels in $a_{aero,hi,y}$ for orbits 
269--330 to orbits 137--268 is 1.84, close to the ratio of 
$\sqrt{50/20}=1.58$ that would occur for samples from a 
Gaussian distribution.
Ratios of noise levels are also stated in Table~\ref{accnoiseratio}. 
This suggests that scatter in $a_{aero,hi,y}$ for orbits 
137--330, but not for orbits 1--136, can be considered 
to be Gaussian noise. 
%Scatter in measured accelerations is consistent with Gaussian 
%noise for orbits 137--330, but not for orbits 1--136, which 
%suggests that the influence of some other source of scatter 
%is important for data acquired at 200 samples per second, 
%but not at 50 and 20 samples per second.

[FIGURE~\ref{accnoise}]
[TABLE~\ref{accnoisevalue}]
[TABLE~\ref{accnoiseratio}]

\subsection{\label{threshold}Selection of Useful 
Data and Determination of Uncertainties}

Useful data are data that are clearly dominated by 
aerodynamic acceleration, not other factors such as 
instrument noise (Figure~\ref{accnoise}), the effects 
of thrusters, and the effects of angular accelerations 
(Section~\ref{acccorr}). We process the dataset to 
remove those data points that are not useful.

Accelerations due to thrusters are on the order of 
$2 \times 10^{-4}$ ms$^{-2}$ (Section~\ref{acccorr}).
 Let the magnitudes of the y-component of the two angular 
acceleration terms in Equation~\ref{fullacceqn} be 
$Y_{1}$ and $Y_{2}$ respectively. We estimate $Y_{1}$ 
and $Y_{2}$ using angular rate information from SPICE 
and $\underline{r}$. We tested two values for 
$\underline{r}$, both assumed to be in the 
``M01\_SPACECRAFT frame'', 
$\underline{r} = \left(0.164, -0.544, 1.137\right)$ 
metres \citep{tolson2005} and 
$\underline{r} = \left(0.0172, 0.629, -1.1\right)$ 
\citep{chavis2005}. 
Differences in $Y_{1}$ and $Y_{2}$ are relatively 
small, and we adopted the value derived from \citet{tolson2005}.

Data points that were smaller than a threshold were 
deemed to be not useful. The threshold was calculated 
for each time step and each of the six sets of accelerations 
($a_{aero,lo,y}$, $a_{aero,lo,y,7}$, $a_{aero,lo,y,39}$, 
$a_{aero,hi,y}$, $a_{aero,hi,y,7}$, and $a_{aero,hi,y,39}$). 
The threshold was the largest of (A) the relevant noise level, 
(B) $2 \times 10^{-4}$ ms$^{-2}$, (C) $Y_{1}$ and (D) $Y_{2}$. 
Angular accelerations determine threshold values near periapsis 
when accelerations significantly exceed threshold values. Further 
from periapsis, the time at which accelerations cross the threshold 
is controlled by either the effects of thrusters ($a_{aero,lo,y,7}$, 
$a_{aero,lo,y,39}$, orbits 7--136 of $a_{aero,hi,y}$, $a_{aero,hi,y,7}$ 
and $a_{aero,hi,y,39}$) or by noise ($a_{aero,lo,y}$ and 
orbits 137--336 of $a_{aero,hi,y}$). Thus the vertical range of 
ODY density profiles is not affected by angular accelerations or 
choice of $\underline{r}$.
%The threshold was controlled by the effects of thrusters for 
%$a_{aero,lo,y,7}$, $a_{aero,lo,y,39}$, $a_{aero,hi,y}$ (orbits 7--136), 
%$a_{aero,hi,y,7}$ and $a_{aero,hi,y,39}$, and by noise for 
%$a_{aero,lo,y}$ and $a_{aero,hi,y}$ (orbits 137--336). 
%Recall that the noise in the high rate data increased between 
%orbits 136 and 137 (Section~\ref{accdescription}). 
%Angular accelerations are not the limiting factor controlling 
%the quality of ODY ACC data products.

Data points that were smaller than their corresponding 
thresholds were discarded. This sometimes resulted in large 
gaps in the time series of retained data points, which is not 
desirable. To correct this, additional data points were discarded 
as necessary to ensure that the time series of retained data 
points was continuous. For example, 
suppose that data points at $t<-100$s did not exceed 
their thresholds, 
data points at $-100$s $<t<$ $-90$s exceeded their thresholds, 
data points at $-90$s $<t<$ $-80$s did not exceed their 
thresholds, 
data points at $-80$s $<t<$ $+100$s exceeded their 
thresholds and 
data points at $t>+100$s did not exceed their thresholds. 
Data points at $-80$s $<t<$ $+100$s would be retained, 
but all other data points would be discarded, including 
data points at $-100$s $<t<$ $-90$s.
Each retained data point was assigned an uncertainty 
equal to its corresponding threshold. This completes 
step 3 (data selection) of the data processing.

Figures~\ref{p076avst} and \ref{p076avstwithsigmas} show 
$a_{aero,hi,y}$, $a_{aero,hi,y,7}$ and $a_{aero,hi,y,39}$ 
versus time for orbit P076. 
%Figure~\ref{p076avst} can be compared to Figure 10 of \citet{tolson2005}.

[FIGURE~\ref{p076avst}]
[FIGURE~\ref{p076avstwithsigmas}]

\subsection{\label{sctpos}Spacecraft Position, Velocity, and Orientation}

Interpretation of the ACC measurements requires knowledge of 
ODY's position, velocity and orientation.
The position, velocity, and orientation of ODY in an inertial 
reference frame during aerobraking were obtained from SPICE 
kernels m01\_ab.bsp, m01\_sc\_ab0110.bc, m01\_sc\_ab0111.bc, 
m01\_sc\_ab0112.bc, m01\_sc\_ab0201.bc, and m01\_sc\_ab0202.bc 
\citep{odyspicervkernels}. The position and orientation of 
Mars in an inertial reference frame were obtained from SPICE 
kernels de414.bsp and pck00008.tpc. Time-related SPICE kernels 
naif0008.tls and orb1\_sclkscet\_00129.tsc were also used in this 
work. The velocity of the martian atmosphere, assuming that it 
rotates with the solid body of the planet, in an inertial reference 
frame was calculated using these SPICE kernels. The velocity of 
the atmosphere relative to the spacecraft and of the spacecraft 
relative to the atmosphere can be determined from this information. 
Spacecraft altitude was defined as 
$\left|\underline{r_{ODY}} - \underline{r_{A}}\right|$, 
where $\underline{r_{ODY}}$ is the position of the spacecraft 
relative to the centre of mass of Mars, $\underline{r_{A}}$ is 
the position of a point on a pre-defined martian areoid relative 
to the centre of mass of Mars, and $\underline{r_{ODY}}$ is 
parallel to $\underline{r_{A}}$. 
In this work, the MOLA 0.25 x 0.25 degree areoid was used 
\citep{smith2003}.
Periapsis is defined in this work as the point of minimum 
altitude, not minimum radial distance.

SPICE kernels for ODY's orientation are not available on 
28 November 2001, which encompasses aerobraking passes on 
orbits 58 and 59. ODY's attitude on these orbits was assumed 
to be $\phi =0^{\circ}$ and $\theta=-4^{\circ}$, where $\phi$ 
and $\theta$ are defined in Section~\ref{dragsection} 
\citep{tolson2005}.
This attitude, the nominal trim attitude, is consistent 
with ODY's average attitude for neighbouring orbits.

\subsection{\label{dragsection}Derivation of Atmospheric 
Densities}

The atmospheric density, $\rho$, satisfies:

\begin{equation}
\label{drageqn}
m \left|a_{aero,y}\right| = \frac{\rho V^{2} C_{y} A}{2}
\end{equation}

where $m$ is the spacecraft mass, $a_{aero,y}$ is the 
y-component of the aerodynamic acceleration, $V$ is the 
scalar speed of the spacecraft relative to the atmosphere, 
or vice versa, $C_{y}$ is an aerodynamic coefficient, and 
$A$ is a reference area connected with $C_{y}$ 
\citep{withers2003}.
Similar equations exist for all directions. We use the 
y-axis in the M01\_SPACECRAFT frame since acceleration 
measurements along this axis, which is close to the velocity 
of the spacecraft relative to the atmosphere, have the 
greatest signal-to-noise ratio.

$m$ decreased monotonically from 460.8 kg at the start of 
aerobraking to 451.7 kg at the end, as shown in 
Figure~\ref{odymassfig} (pers. comm, Mase, 2006). Uncertainty 
in the absolute mass is 3 kg, although the change in mass from 
one orbit to the next orbit is known much more accurately ($<$1 g). 
$A$ is 11.03 m$^{2}$ 
\citep{chavis2005}. $V$ is known from SPICE information 
(Section~\ref{sctpos}). The product $\rho \times C_{y}$ can 
be found using measured or known quantities ($m$, $a_{aero,y}$, 
$V$, and $A$) at each time step along an aerobraking pass.

[FIGURE~\ref{odymassfig}]

Next, $\rho$ and $C_{y}$ are determined from the product 
$\rho \times C_{y}$. ODY's $C_{y}$, typically $\sim$2, is 
a function of density and the attitude of the spacecraft 
with respect to its velocity relative to the atmosphere. 
Spacecraft attitude can be expressed as two angles, $\phi$ 
and $\theta$. If the velocity of the atmosphere relative to 
the spacecraft is $\underline{U} = U \underline{u}$, where 
$\underline{u}$ is a unit vector expressed in the 
``M01\_SPACECRAFT frame'', then pitch ($\theta$) and yaw 
($\phi$) angles are defined by \citep{takashima2002a}:

\begin{equation}
\label{phithetaeqnx}
u_{x} = \cos \theta \sin \phi 
\end{equation}
\begin{equation}
\label{phithetaeqny}
u_{y} = \cos \theta \cos \phi 
\end{equation}
\begin{equation}
\label{phithetaeqnz}
u_{z} = - \sin \theta
\end{equation}

Values of $C_{y}$ for various $\rho$, $\phi$, and $\theta$ 
have been determined by numerical simulation. These values 
constitute the ``aerodynamic database.'' The aerodynamic 
database was reported in \citet{takashima2002b} and provided to 
us by R. Mase (pers. comm., 2006). Uncertainties in $C_{y}$ 
are estimated to be 3\% \citep{chavis2005}. 
Inputs to the numerical simulations ranged from $10^{-4}$ kg km$^{-3}$ 
to $10^{4}$ kg km$^{-3}$ for $\rho$, from -60$^{\circ}$ to 
60$^{\circ}$ for $\phi$, and -60$^{\circ}$ to 60$^{\circ}$ for 
$\theta$. These ranges encompassed all conditions encountered 
during aerobraking. 
We found the unit vector $\underline{u}$ as a function of time 
for each aerobraking pass (Section~\ref{sctpos}). Angles $\phi$ 
and $\theta$ were then determined using 
Equations~\ref{phithetaeqnx}--\ref{phithetaeqnz}, also as 
functions of time for each aerobraking pass. At each time 
step in each aerobraking pass, the simulated value of $C_{y}$ 
was found as a function of $\rho$ using the aerodynamic 
database. The simulated value of the product $\rho \times C_{y}$ 
was also found as a function of $\rho$. 

$\rho \times C_{y}$ is a \textbf{single-valued function} of 
$\rho$ for any specified $\phi$ and $\theta$, which enables 
both $\rho$ and $C_{y}$ to be found from $\rho \times C_{y}$. 
At each time step in each aerobraking pass, the observed 
value of the product $\rho \times C_{y}$ and the simulated 
function $\rho C_{y}\left(\rho\right)$ were compared to 
determine the value of $\rho$. The value of $C_{y}$ followed 
trivially.
This is illustrated in Figures~\ref{rhocy} and \ref{rhovscy}. 
This technique is also discussed by \citet{takashima2002b} 
and \citet{tolson2005}.

[FIGURE~\ref{rhocy}]
[FIGURE~\ref{rhovscy}]

This process was applied to $a_{aero,lo,y}$, $a_{aero,hi,y}$, 
$a_{aero,lo,y,7}$, $a_{aero,hi,y,7}$, $a_{aero,lo,y,39}$, 
and $a_{aero,hi,y,39}$ to obtain $\rho_{lo}$, $\rho_{hi}$, 
$\rho_{lo,7}$, $\rho_{hi,7}$, $\rho_{lo,39}$, and 
$\rho_{hi,39}$. Uncertainties in each of these densities 
were found using \citep{bevington1969}:

\begin{equation}
\label{srhoeqn}
\left( \frac{\sigma_{\rho}}{\rho}\right)^{2} = 
\left( \frac{\sigma_{m}}{m}\right)^{2} + 
\left( \frac{\sigma_{Cy}}{C_{y}}\right)^{2} + 
%\left( \frac{\sigma_{a,aero,y}}{a_{aero,y}}\right)^{2} 
\left( \frac{\sigma_{a}}{a}\right)^{2} 
\end{equation}

Derived densities that were smaller than their uncertainties 
were discarded. This sometimes resulted in large gaps in the 
time series of retained densities, which is not desirable. 
To correct this, other derived densities were discarded as 
necessary to ensure that the time series of retained 
densities was continuous. 
For example, suppose that densities at $t<-100$s did not 
exceed their uncertainties, 
densities at $-100$s $<t<$ $-90$s exceeded their uncertainties, 
densities at $-90$s $<t<$ $-80$s did not exceed their 
uncertainties, 
densities at $-80$s $<t<$ $+100$s exceeded their 
uncertainties and 
densities at $t>+100$s did not exceed their uncertainties. 
Densities at $-80$s $<t<$ $+100$s would be retained, 
but all other densities would be discarded, including 
densities at $-100$s $<t<$ $-90$s.
This completes step 4 (density derivation) of the data processing.

Figures~\ref{p076uxvst}--\ref{p076thetavst} show time 
series of $u_{x}$, $u_{z}$, $\phi$ and $\theta$ for orbit P076.
Figure~\ref{p076cyvst} shows time series of $C_{y}$ for orbit P076.
Figures~\ref{p076rhovstlin} and \ref{p076rhovstlog} show time 
series of $\rho_{hi}$, $\rho_{hi,7}$ and $\rho_{hi,39}$ for
 orbit P076.

[FIGURE~\ref{p076uxvst}]
[FIGURE~\ref{p076uzvst}]
[FIGURE~\ref{p076uxvsuz}]
[FIGURE~\ref{p076phivst}]
[FIGURE~\ref{p076thetavst}]
[FIGURE~\ref{p076cyvst}]
[FIGURE~\ref{p076rhovstlin}]
[FIGURE~\ref{p076rhovstlog}]

Each aerobraking pass has up to six derived density profiles: 
$\rho_{lo}$, $\rho_{hi}$, $\rho_{lo,7}$, $\rho_{hi,7}$, 
$\rho_{lo,39}$, and $\rho_{hi,39}$. Each derived density 
value has an associated altitude. Let $z_{max}$ be the greatest 
altitude for a given density profile. 
Values of $\sigma_{\rho}$ and $\sigma_{aaero,y}$ at 
$z_{max}$, as well as the value of $z_{max}$, are listed 
in Table~\ref{topofatmtable} for the three types of high rate 
density profiles at the start and end of aerobraking.
%Periapsis altitude was 110 km at the start of aerobraking, 
%decreased to 95 km by orbit 100, then increased to 115 km 
%at the end of aerobraking.

[TABLE~\ref{topofatmtable}]

Derived densities rely on knowledge of the speed and 
direction of the spacecraft relative to the atmosphere. 
The velocity of ODY in an inertial frame is determined 
from SPICE information (Section~\ref{sctpos}). The velocity 
of the atmosphere in an inertial frame is determined from 
the assumption that the atmosphere rotates at the same 
angular velocity as the solid planet. This assumption is 
not accurate. Models predict zonal and meridional winds 
of $\sim$100 ms$^{-1}$ at aerobraking altitudes 
\citep{bougher1999b, bougher2000, bougher2002}. These 
speeds are consistent with  
results from the first efforts to determine winds from 
aerobraking measurements \citep{baird2007, crowley2007}.

If, instead of assuming solid body rotation for the 
atmosphere, we assume that there is a zonal wind of 
100 ms$^{-1}$, then derived densities are altered by 
$<$1\%. A meridional wind of 100 ms$^{-1}$ alters derived 
densities by $\sim$5\%. Data users should be aware of 
this potential source of error, which is not included in the 
derived uncertainties (Equation~\ref{srhoeqn}) because the 
accuracy of wind predictions has not been quantified.

\subsection{\label{calt}Derivation of Constant Altitude 
Data Products}

The 1-second sampling rate ($<$1 km vertical resolution) 
in the derived density profiles is not necessary for many 
studies of large-scale atmospheric processes. Another 
dataset of reduced size, called the constant altitude 
dataset, was generated from $\rho_{39}$ data to support 
such studies. Orbits for which $\rho_{hi,39}$ data were 
available used $\rho_{hi,39}$ data. Orbits for which 
$\rho_{hi,39}$ data were not available used 
$\rho_{lo,39}$ data.

$\rho_{39}$ measurements for a given orbit were first 
split in two. All densities before periapsis, including 
the periapsis density, were assigned to the inbound leg. 
All densities after periapsis, including the periapsis 
density, were assigned to the outbound leg. Each leg was 
processed separately. Target altitudes were defined as 
100 km, 110 km, 120 km, 130 km,  140 km, 150 km, and 160 km. 
Periapsis never reached 90 km.
If density measurements existed more than 3 km below the 
target altitude and more than 3 km above the target altitude, 
then we proceeded to determine constant altitude data 
products for that target altitude. 

We assume that densities, $\rho$, within 5 km of the 
target altitude, $z_{0}$, satisfy:
\begin{equation}
\label{fiteqn01}
\rho = \rho_{0} \exp 
\left( \frac{-\left(z-z_{0}\right)}{H_{\rho}}\right)
\end{equation}

where $\rho_{0}$ is the density at $z_{0}$ and 
$H_{\rho}$ is the density scale height between 
$z_{0} - 5$ km and $z_{0} + 5$ km. Equivalently:
\begin{equation}
\label{fiteqn02}
\ln \rho = \ln \rho_{0} + \left( \frac{-1}{H_{\rho}} \right) 
\left(z-z_{0}\right)
\end{equation}

Defining $y = \ln \rho$, $x = z - z_{0}$, $a = \ln \rho_{0}$ 
and $b = -1/H_{\rho}$, we have:
\begin{equation}
\label{fiteqn03}
y = a + b x
\end{equation}

Values of $x$, the independent variable, are known exactly. 
Values of $y$, the dependent variable have uncertainties 
$\sigma_{y} = \sigma_{\rho} / \rho$. We use standard 
least-squares techniques, Equation 6-12 of 
\citet{bevington1969} as implemented by the IDL POLY\_FIT 
procedure, to determine best-fit values of $a$ and $b$, and 
uncertainties $\sigma_{a}$ and $\sigma_{b}$.
Then we have \citep{bevington1969}:
\begin{equation}
\label{fiteqn04}
\rho_{0} = e^{a}
\end{equation}

\begin{equation}
\label{fiteqn05}
\sigma_{\rho_{0}} = \sigma_{a} \rho_{0}
\end{equation}

\begin{equation}
\label{fiteqn06}
H_{\rho} = \frac{-1}{b}
\end{equation}

\begin{equation}
\label{fiteqn07}
\sigma_{H_{\rho}} = H_{\rho}^{2} \sigma_{b}
\end{equation}

Assuming an isothermal atmosphere, temperature, $T_{\rho}$, 
and its uncertainty, $\sigma_{T_{\rho}}$, were estimated from 
the fitted density scale height, $H_{\rho}$, using:

\begin{equation}
\label{calttempeqn}
T_{\rho} = \frac{\mu g H_{\rho}}{k_{B}}
\end{equation}

\begin{equation}
\label{sigmacalttempeqn}
\frac{\sigma_{T_{\rho}}}{T_{\rho}} = \frac{\sigma_{H_{\rho}}}{H_{\rho}}
\end{equation}

where $\mu$ is mean molecular mass, assumed to be 43.49 
daltons, $g$ is the acceleration due to gravity, and 
$k_{B}$ is Boltzmann's constant. The assumed $\mu$, which 
corresponds to Viking Lander measurements at the surface, 
is an overestimate \citep{owen1992, magalhaes1999}. Light 
species such as O are more abundant in the thermosphere 
than near the surface. Data users who wish to make a different 
assumption concerning $\mu$ can simply multiply our 
$T_{\rho}$ and $\sigma_{T_{\rho}}$ by the ratio of their 
desired mean molecular mass to 43.49 daltons. $g$ is given by:

\begin{equation}
\label{graveqn}
g = \frac{GM}{\left( R_{ref} + z \right)^{2}}
\end{equation}

where $R_{ref}$ is the mean equatorial radius of the MOLA 
areoid used in this work, 3396 km, $G$ is the gravitational 
constant, and $M$ is the mass of Mars \citep{smith2003}. 
$GM = 4.2828382332 \times 10^{13}$ m$^{3}$ s$^{-2}$ 
\citep{tyler2000}.
As an example, constant altitude data products for orbit 
P076 are shown in Table~\ref{p076calttable}.

[TABLE~\ref{p076calttable}]

No fitted $\rho$, $H_{\rho}$, or $T_{\rho}$ had a smaller 
absolute value 
than its uncertainty. 
One set of results for $H_{\rho}$ and $T_{\rho}$ was 
negative (orbit 12, outbound 130 km, $H_{\rho}$ = -75 km, 
$\sigma_{H_{\rho}}$ = 29 km, $T_{\rho}$ = -1340K, 
$\sigma_{T_{\rho}}$ = 480K). Recall that aerobraking 
passes are not vertical; they span significant horizontal 
extent. Densities do decrease with altitude at 130 km 
on the outbound leg of this aerobraking pass, presumably 
due to some kind of wave in the atmosphere, so this 
negative scale height is formally accurate. However, 
this illustrates that caution should be used when 
interpreting scale heights derived from non-vertical 
density profiles. Fitted values of $\rho$ cannot be 
negative, due to the exponential fitting approach.

The orbits for which constant altitude results were obtained 
are listed in Table~\ref{calttable}. Results were obtained 
at 110 km and 120 km for almost all orbits, at 100 km for 
$>$50\% of orbits, and at 140 km outbound for $>$20\% of 
orbits. Results at 150 km and 160 km are essentially non-existent. 
Accelerations were measured at 140 km on outbound legs at 
daytime LSTs more frequently than at 140 km on inbound legs, 
and the difference between daytime and nighttime densities 
at 140 km explains why there are so many more outbound 140 km 
measurements than inbound 140 km measurements.
Typical uncertainties in fitted $\rho$, $H_{\rho}$, and 
$T_{\rho}$ are shown in Table~\ref{scalttable}.

[TABLE~\ref{calttable}]
[TABLE~\ref{scalttable}]

\subsection{\label{pandtprofiles}Derivation of 
Pressure and Temperature Profiles}

Vertical pressure, $p$, and temperature, $T$, profiles 
can be derived from a vertical density profile using:

\begin{equation}
\label{hydroeqn}
\frac{dp}{dz} = - \rho g
\end{equation}
\begin{equation}
\label{idealeqn}
p = \frac{\rho k_{B} T}{\mu}
\end{equation}

where $g$ and $\mu$ are as defined in Section~\ref{calt}. 
Equation~\ref{hydroeqn} needs an upper boundary condition. 
If the atmosphere is isothermal, then $p = \rho g H_{\rho}$, 
where $H_{\rho}$ is the density scale height, which can be 
derived from a fit to the vertical density profile. If 
applied to aerobraking density profiles, which are non-vertical, 
this approach will produce biased $p$ and $T$ profiles because 
horizontal gradients in the atmosphere are not properly accounted for. 
Differences between values of the periapsis pressure derived for 
the inbound and outbound legs are a particularly visible 
manifestation of this problem (see below). Nevertheless, we have 
used this approach to produce $p$ and $T$ profiles corresponding 
to the inbound and outbound density profiles on each aerobraking 
pass. $T\left(p\right)$ profiles are so much more common in 
atmospheric modelling than $\rho\left(z\right)$ profiles that 
the production of $p$ and $T$ profiles increases the value of 
the ODY ACC dataset to many potential users despite inevitable 
biases in the $p$ and $T$ profiles.

Pressure and temperature profiles have been derived from the 
unaveraged densities, from the 7-point averaged densities, and 
from the 39-point averaged densities. Orbits for which high rate 
data were available used high rate data. Orbits for which high 
rate data were not available used low rate data.

The density measurements for a given orbit were first split in two. 
All densities before periapsis, including the periapsis density, 
were assigned to the inbound leg. All densities after periapsis, 
including the periapsis density, were assigned to the outbound leg. 
Each leg was processed separately. Any measurement whose altitude is 
greater than the altitude of a density measurement that is less than 
twice the density uncertainty was discarded. This improves the 
application of the upper boundary condition for Equation~\ref{hydroeqn} 
and the forthcoming Monte Carlo analysis. If the altitude range of 
the remaining densities was less than 5 km, then no pressure or 
temperature profile was obtained.

An exponential fit to all densities within 10 km of the altitude of 
the highest measurement was used to determine the density scale height 
at the altitude of the highest measurement. Uncertainties in the 
densities were used to determine the uncertainty in the fitted 
density scale height. Procedures were the same as in Section~\ref{calt}.

A Monte Carlo approach was used to obtain the pressure and 
temperature profiles, with uncertainties. A dummy density profile 
was generated using the true density profile and its uncertainties. 
A dummy density scale height for the top of the profile was generated 
using the fitted density scale height and its uncertainty. All 
uncertainties were assumed to correspond to a Gaussian distribution. 
If any of the dummy density values or the dummy density scale height 
were negative, then the dummy profiles were generated again. A single 
pressure profile and a single temperature profile were then obtained 
using Equations~\ref{hydroeqn}--\ref{idealeqn}. After $N$ 
pressure/temperature profiles were obtained, where $N=1000$, the mean 
and standard deviation pressure/temperature were calculated at each 
altitude. The uncertainty in pressure/temperature was assumed to equal 
the standard deviation in the Monte Carlo ensemble. Any $p$ measurement 
whose altitude is greater than the altitude of a $p$ measurement 
that is less than its uncertainty was discarded. $T$ measurements 
were treated similarly.

Periapsis pressure and temperature, and their uncertainties, need 
special treatment, since they are calculated twice, once for the 
inbound leg and once for the outbound leg. The best estimates of 
periapsis conditions are calculated as follows:

\begin{equation}
\label{sperieqn}
\frac{1}{\sigma_{Xbest}^{2}} = \frac{1}{\sigma_{Xin}^{2}} + 
\frac{1}{\sigma_{Xout}^{2}}
\end{equation}

\begin{equation}
\label{bestperieqn}
\frac{X_{best}}{\sigma_{Xbest}^{2}} = 
\frac{X_{in}}{\sigma_{Xin}^{2}} + 
\frac{X_{out}}{\sigma_{Xout}^{2}}
\end{equation}

where $X$ is periapsis $p$ or $T$, $X_{in}$ and $X_{out}$ are 
the inbound and outbound values of $X$, $\sigma_{Xin}$ and 
$\sigma_{Xout}$ are their uncertainties, $X_{best}$ is the best
estimate of $X$, and $\sigma_{Xbest}$ is its uncertainty.

Differences between $p_{in}$ and $p_{out}$ are shown in 
Figure~\ref{dpperivsorb}. 
The differences are not distributed randomly.
Figure~\ref{dpperivsplat} shows how the differences are affected 
by periapsis latitude and by whether the inbound leg is poleward 
or equatorward of the outbound leg.

[FIGURE~\ref{dpperivsorb}]
[FIGURE~\ref{dpperivsplat}]

Figures~\ref{p076pvsz}--\ref{p076tempvspress} show the vertical 
pressure and temperature profiles for orbit P076.

[FIGURE~\ref{p076pvsz}]
[FIGURE~\ref{p076tempvsz}]
[FIGURE~\ref{p076tempvspress}]

\subsection{\label{infosource}Summary of Data Sources}

This data processing has used many sources of information. For 
convenience, all sources are collected here in one place.
Time series of accelerations were obtained in files pXXXacc.txt 
(low rate) and LOPXXX.mat (high rate) which were provided to 
Jim Murphy and the Atmospheres Node of the PDS by the 
aerobraking team. 
Frame information was obtained from SPICE kernel m01\_v28.tf.
ODY position, velocity and orientation were obtained from 
ODY SPICE kernels 
m01\_ab.bsp, m01\_sc\_ab0110.bc, m01\_sc\_ab0111.bc, 
m01\_sc\_ab0112.bc, m01\_sc\_ab0201.bc, and m01\_sc\_ab0202.bc. 
Additional information on time and planetary motions was obtained 
from generic SPICE kernels de414.bsp, pck00008.tpc, naif0008.tls 
and orb1\_sclkscet\_00129.tsc.
Altitudes were obtained using the 0.25 x 0.25 degree MOLA areoid.
ODY's mass history was obtained from R. Mase in file 
Nav\_reconstr\_thru\_P338.xls. ODY's aerodynamic database was 
obtained from R. Mase in file 01\_aerodatabase.v1.inc.
ODY's area of 11.03 m$^{2}$ was obtained from \citet{chavis2005}

\section{\label{ubervalidate}Validation of Results}

\subsection{\label{validationpos}Validation of Results for 
Time and Position}

These results can be compared against the unpeer-reviewed data 
products at the PDS. These include UTC time, altitude, latitude, 
longitude, LST, solar zenith angle (SZA), $\rho$, $\sigma_{\rho}$, 
$H_{\rho}$, and $\sigma_{H\rho}$ at periapsis, inbound 110 km, 
inbound 120 km, outbound 110 km, and outbound 120 km. We refer 
to these as the ``aerobraking team's results''. In this 
section, absolute differences in results are defined as the 
results of this work minus the aerobraking team's results, 
ratios in results are defined at the results of this work 
divided by the aerobraking team's results, and relative 
differences in results are defined as (the results of this 
work minus the aerobraking team's results) divided by the 
aerobraking team's results.
Three periapsis times in the aerobraking team's results are 
incorrect. The time stated for orbit P017 is stated as zero. 
The time stated for orbit P029 is that of orbit P028. 
The time stated for orbit P068 is that of orbit P027. Therefore 
results involving periapsis time are not calculated for these
orbits in this section.

Differences in periapsis time are shown in 
Figure~\ref{deltatvsorb}.
Differences in periapsis time are +5 seconds at the 
start of aerobraking. 
This time offset changes steadily during aerobraking. 
Differences in periapsis time are -60 seconds at the 
end of aerobraking.
The difference at orbit P209 deviates from the general 
trend, so this orbit is not included in 
Figures~\ref{deltazvsorb}--\ref{redefdeltazvsplat}.

[FIGURE~\ref{deltatvsorb}]

Differences in periapsis latitude are shown in 
Figure~\ref{deltalatvsorb}.
Differences in periapsis latitude are -0.6$^{\circ}$N 
at the start of aerobraking. 
This meridional offset changes steadily during aerobraking. 
Differences in periapsis latitude are -4.0$^{\circ}$N at 
the end of aerobraking.

[FIGURE~\ref{deltalatvsorb}]

Differences in periapsis altitude are shown in 
Figure~\ref{deltazvsorb}.
Differences in periapsis altitude are +0.1 km at the 
start of aerobraking. 
This vertical offset changes steadily during aerobraking. 
Differences in periapsis altitude are -0.5 km at the 
end of aerobraking.

[FIGURE~\ref{deltazvsorb}]

Differences in periapsis longitude are shown in 
Figure~\ref{deltalonvsorb}. Differences are greatest near the 
pole, when the distance that corresponds to one degree of longitude 
is smallest. Differences in periapsis LST, which are similar, 
are shown in Figure~\ref{deltalstvsorb}.

[FIGURE~\ref{deltalonvsorb}]
[FIGURE~\ref{deltalstvsorb}]

Differences in periapsis SZA are shown in 
Figure~\ref{deltaszavsorb}.

[FIGURE~\ref{deltaszavsorb}]

Figures~\ref{deltalatvsplat} and \ref{deltazvsplat} show that 
differences in periapsis latitude and differences in 
periapsis altitude are correlated with periapsis latitude.

[FIGURE~\ref{deltalatvsplat}]
[FIGURE~\ref{deltazvsplat}]

Differences in periapsis time, periapsis longitude, periapsis LST 
and periapsis SZA are also correlated with periapsis latitude,  as 
shown in Figures~\ref{adjdeltatvsplat}--\ref{adjdeltaszavsplat}, 
if the sign of differences obtained on the dayside is reversed.

[FIGURE~\ref{adjdeltatvsplat}]
[FIGURE~\ref{adjdeltalonvsplat}]
[FIGURE~\ref{adjdeltalstvsplat}]
[FIGURE~\ref{adjdeltaszavsplat}]

If we redefine our periapses to occur at the aerobraking team's 
periapsis times, then differences between our results and the 
aerobraking team's results are reduced by between one and two 
orders of magnitude. Differences in redefined periapsis longitude, 
LST and SZA are negligible. Differences in redefined periapsis 
latitude are clearly correlated with periapsis latitude, as shown 
in Figure~\ref{redefdeltalatvsplat}. 
Differences in redefined periapsis altitude, which are approximately 
100 m, are also influenced by periapsis latitude, but exhibit 
substantial scatter as shown in Figure~\ref{redefdeltazvsplat}. 

[FIGURE~\ref{redefdeltalatvsplat}]
[FIGURE~\ref{redefdeltazvsplat}]

A plausible explanation for these results is that different 
areoids were used in this work and the aerobraking team's work. 
The documentation accompanying the aerobraking team's results 
does not specify which areoid was used. The MGS accelerometer 
archives at the PDS used an ellipsoid with equatorial radius of 
3393.4 km and flattening, f, of 0.0052083 with fourth degree 
and order corrections for gravitational potential. The correction 
coefficients were not stated 
(http://pds-atmospheres.nmsu.edu/PDS/data/mgsa\_0002/catalog/altds.cat). 
The MOLA 0.25 x 0.25 degree areoid was used in this work 
\citep{smith2003}.

The aerobraking team results do not include times for inbound 
110 km or the other fixed altitude levels, so we cannot compare 
times for these results. The latitude differences between our 
results and the aerobraking team's results for these altitude levels 
are less than 0.5 degrees. The use of different areoids will 
clearly lead to some differences in such latitudes. We now demonstrate 
that the scale of the latitude difference is consistent with the 
previous results that motivated the hypothesis of different areoids.
If this work and the aerobraking team's work used two different 
areoids, then the two sets of results will cross the inbound 110 km 
altitude at different points along the trajectory. Consequently, the 
two sets of results will cross the inbound 110 km altitude at 
different radial distances from the centre of mass of Mars, different 
UTC times and different latitudes.
Multiplication of these latitude differences by the rate of change 
in altitude with respect to latitude yields vertical differences of 
about 0.5 km, which is the same scale as the differences in 
periapsis altitude shown in Figure~\ref{deltazvsplat}. Therefore 
the scale of the latitude differences in the constant altitude 
results is consistent with the scale of the differences in 
periapsis altitude.

%Multiplication of these latitude differences by the rate of 
%change in altitude with respect to latitude suggests that the 
%differences in altitude between our fixed altitude levels and 
%those of the aerobraking team are about 0.5 km, consistent 
%with the differences in periapsis altitude. This could also 
%be attributed to the use of different areoids.

Figure~\ref{qplot} shows that periapsis latitude is close 
to the mean of the inbound and outbound 120 km latitudes for 
our results, but not for the aerobraking team's results.
We label periapsis latitude as $L_{p}$, latitude at 120 km on 
the inbound leg as $L_{in120}$, latitude at 120 km on the 
outbound leg as $L_{out120}$, and define $Q$ as $0.5 \times 
\left(L_{in120} + L_{out120}\right) - L_{p}$.
$Q$ is non-zero for our results near the pole, which is expected 
when the aerobraking pass goes across the pole. At all other 
latitudes, $Q$ is zero for our results but not for the 
aerobraking team's results.
Orbit 331 is an example. The aerobraking team's results have 
inbound 120 km at 15.76$^{\circ}$N, periapsis at 
25.67$^{\circ}$N, and outbound 120 km latitude at 
27.96$^{\circ}$N. Inbound 120 km is 10 degrees away from 
periapsis, outbound 120 km is 2 degrees away. By contrast, 
a figure similar to Figure~\ref{qplot} using MGS results at 
130 km, which have been archived and peer-reviewed by the 
PDS, shows that periapsis latitude was within 1 degree of the 
mean of the inbound and outbound latitudes, except when 
MGS was close to a pole. It is difficult to develop a hypothesis 
that can explain the results in Figure~\ref{qplot}; one unlikely 
possibility is that the aerobraking team's periapsis results 
use a different areoid from their fixed altitude results. 

[FIGURE~\ref{qplot}]

If positions such as ``periapsis'' and ``inbound 110 km'' 
refer to different positions along the aerobraking trajectory 
in our work and the aerobraking team's work, then differences 
between our results are inevitable. Given a scale height of 
10 km, a difference in altitude of 0.5 km corresponds to a 
difference in density of 5\%. 

\subsection{\label{validationrho}Validation of Results for 
Density and Density Scale Height}

%rhoin110 = 0.061226440, sdev = 0.0510882
%rhoin120 = 0.099102583, sdev = 0.0672655
%rhoout110 = 0.053632978, sdev = 0.103610
%rhoout120 = 0.090621960, sdev = 0.0861239

Figures~\ref{reldiffrhoin110}--\ref{reldiffrhoout120} and 
Table~\ref{validrhodshtable}
compare our results and the aerobraking team's results for 
densities at 110 km and 120 km (inbound and outbound).
The mean of the relative difference is 6\% at 110 km and 
10\% at 120 km.
The standard deviation of the relative difference is 6\% 
at 110 km and 10\% at 120 km.
Differences $\sim$5\% are expected due to areoid differences. 
Differences due to different data processing approaches (such 
as whether to use a data point in a fit or not, different 
interpolation techniques within the aerodynamic database, or 
differences in uncertainties that subsequently affect a fit) 
might also be expected.
We suggest that these differences could be on the order of 
the uncertainties in the results. Uncertainties in the aerobraking 
team's densities are $\sim$5\% at 110 km and 120 km. Uncertainties 
in our densities are $\sim$1\% at 110 km and $\sim$3\% at 120 km. 
These uncertainties are probably too low because they were 
calculated using the assumption that uncertainties in mass 
($<$1\%) and drag coefficient (3\%) were Gaussian, when they 
are actually systematic uncertainties. The effects of unknown 
winds were also neglected, but this is likely to affect both 
sets of results equally (Section~\ref{dragsection}).
These effects (5\% due to areoid, 5\% due to other causes) are 
sufficient to explain differences of 10\%. However, results at 
120 km differ by more than this on many orbits. 

[FIGURE~\ref{reldiffrhoin110}]
[FIGURE~\ref{reldiffrhoout110}]
[FIGURE~\ref{reldiffrhoin120}]
[FIGURE~\ref{reldiffrhoout120}]
[TABLE~\ref{validrhodshtable}]

%dshin110 = 0.083429092, sdev = 0.186090
%dshin120 = 0.045592559, sdev = 0.127672
%dshout110 = 0.11435090, sdev = 0.358268 - 0.087866293 and 
%0.211089 if P122 junked
%dshout120 = 0.084851017, sdev = 0.183602

Figures~\ref{reldiffdshin110}--\ref{reldiffdshout120} and 
Table~\ref{validrhodshtable}
compare our results and the aerobraking team's results for 
density scale heights at 110 km and 120 km (inbound and 
outbound). 
A large difference is seen in the outbound density scale 
height at 110 km for orbit P122 (Figure~\ref{reldiffdshout110}). 
Figure~\ref{p122dshnotinsane} shows that our density scale 
height of 53 km, which is much larger than the aerobraking 
team's density scale height of 12.5 km, is consistent with 
our density profile.
Note also that the strong waves in Figure~\ref{p122dshnotinsane} 
could cause larger-than-expected differences between our densities 
and the aerobraking team's densities.
Our densities at 120 km are based upon fits to data points 
between 115 km and 125 km. If the aerobraking team used, say, 
data points between 117 km and 123 km and had a difference in 
absolute altitudes of 1 km, then significant differences in 
derived densities at 120 km should be expected when density 
profiles contain strong waves.
This may help explain the relatively large density differences 
at 120 km.
The mean of the relative difference in density scale height 
is 10\% at 110 km and 7\% at 120 km.
The standard deviation of the relative difference is 20\% at 
110 km and 15\% at 120 km.
Negligible differences are expected due to areoid differences 
and differences in altitude of 0.5 km \citep{withers2006b}. 
Uncertainties in the aerobraking team's density scale heights 
are $\sim$10\% at 110 km and 120 km. Uncertainties in our 
density scale heights are $\sim$3\% at 110 km and $\sim$6\% 
at 120 km. 
Uncertainties in our density scale heights are not 
underestimated by the assumption that 
systematic uncertainties in mass ($<$1\%) and drag coefficient 
(3\%) were Gaussian. Such systematic uncertainties lead to 
systematic density uncertainties that do not affect derived 
scale heights.

[FIGURE~\ref{reldiffdshin110}]
[FIGURE~\ref{reldiffdshout110}]
[FIGURE~\ref{reldiffdshin120}]
[FIGURE~\ref{reldiffdshout120}]
[FIGURE~\ref{p122dshnotinsane}]

The relative difference in inbound density at 110 km is 
correlated with inbound density at 110 km ($r = 0.47$, where 
$r$ is the correlation coefficient, using the aerobraking team's 
densities and $r=0.55$ using our densities).
The relative difference in outbound density at 110 km is 
correlated with outbound density at 110 km ($r = 0.49$ using 
the aerobraking team's densities and $r=0.59$ using our 
densities).
Correlations are significantly weaker at 120 km, $r=0.05$ for 
inbound using the aerobraking team's densities, $r=0.27$ for 
inbound using our densities, $r=0.26$ for outbound using the 
aerobraking team's densities, and $r=0.40$ for outbound 
using our densities.

Relative differences in density scale height are strongly 
correlated with our density scale heights, but not with the 
aerobraking team's density scale heights.
At 110 km, inbound, $r=0.38$ using the aerobraking team's 
density scale heights and $r=0.81$ using our density scale 
heights.
At 110 km, outbound, $r=0.29$ using the aerobraking team's 
density scale heights and $r=0.93$ using our density scale 
heights.
At 120 km, inbound, $r=0.34$ using the aerobraking team's 
density scale heights and $r=0.76$ using our density scale 
heights.
At 120 km, outbound, $r=0.37$ using the aerobraking team's 
density scale heights and $r=0.80$ using our density scale 
heights.
These striking correlations involving density scale heights 
could occur if our density scale heights are incorrect, but 
the aerobraking team's density scale heights are not. To 
provide reassurance that this hypothesis is untrue, 
Figure~\ref{demodshworks} shows that our density scale 
heights are consistent with our density profiles. The 
inbound density scale height at 110 km on orbit P081 is 
12.6 km in our results and 9.0 km in the aerobraking 
team's results. The inbound density at 110 km on orbit 
P081 is 11.6 kg km$^{-3}$ in our results and 10.6 kg km$^{-3}$ 
in the aerobraking team's results. Figure~\ref{demodshworks} 
shows that our fitted density and density scale height are 
consistent with our density profile, but the aerobraking 
team's fitted density and density scale height are not 
consistent.

[FIGURE~\ref{demodshworks}]
%demodshworks01.eps

There are several trends in the differences between our 
results and the aerobraking team's results. These trends 
are presumably related to the underlying causes of the 
differences, but we are not able to use the trends to 
identify the causes.

\section{\label{scisectionprofs}Science --- Density Profiles}

\subsection{Small-scale Structure}

Oscillatory small-scale structure is present in the 
7-second density profiles, as shown in Figure~\ref{wavyexamplefig}.
This can be quantified by the following figure-of-merit, $X$:
\begin{eqnarray}
\label{wigglyeqn01}
Y = 2 \frac{\left(\rho_{7} - 
\rho_{39}\right)}{\left(\rho_{7} + \rho_{39}\right)}
\\
\label{wigglyeqn02}
X = \sqrt{ \overline{Y^{2}}}
\end{eqnarray}
where $\rho_{7}$ and $\rho_{39}$ are 7-second and 
39-second densities, respectively, and $\overline{Y^{2}}$ 
is the mean value of $Y^{2}$.
Results are shown in Figure~\ref{wavinessfig}.
Small-scale structure is probably due to gravity waves, 
and is therefore affected by conditions in the lower 
atmosphere that control gravity wave excitation and 
upward propagation \citep{creasey2006a, creasey2006b}. 

[FIGURE~\ref{wavyexamplefig}]
[FIGURE~\ref{wavinessfig}]

The typical value of $X$ for ODY is 0.1, slightly larger 
than typical values for MGS, which are shown in Figure 4a 
of \citet{withers2006b}. 
MGS's dayside values of $X$ were largest at 60$^{\circ}$N 
and 60$^{\circ}$S. ODY's dayside values of $X$ have limited 
meridional range, but are consistent with this trend. 
The dayside MGS Phase 1 data and dayside ODY data are quite 
consistent --- $X$ is 0.15 around 60$^{\circ}$N and 
decreases to 0.10 30$^{\circ}$ away. 
The dayside MGS Phase 2 data, where $X$ is 0.07--0.08 and 
is not peaked at 60$^{\circ}$N, are not consistent with these. 
The consistency between ODY and MGS Phase 1 data makes it 
unlikely that the Noachis dust storm, which affected MGS 
Phase 1 data only, controlled small-scale structure in MGS 
Phase 1. 
The $L_{s}$ ranges of MGS Phase 1, MGS Phase 2 and ODY are 
180$^{\circ}$--300$^{\circ}$, 30$^{\circ}$--90$^{\circ}$ 
and 260$^{\circ}$--310$^{\circ}$, respectively.
The consistency in $X$ between MGS Phase 1 and ODY data, but 
not MGS Phase 2 data, might be due to seasonal effects.
ODY's nightside values of $X$, which have a mean of 0.10, are 
essentially independent of latitude. The meridional dependence 
of MGS's nightside values of $X$, which have a mean of 0.07, 
cannot be determined due to their limited meridional range.

Several MGS profiles contain sudden changes in density that 
appear more step-like than oscillatory. Such changes are less 
common in ODY profiles. We identified all 7-second density 
profiles in which density changed by a factor of two in less 
than five seconds within 100 seconds of periapsis. Six of the 
available 329 ODY profiles (1.8\%) satisfied this criterion, 
compared to 34 of the available 800 MGS profiles (4.3\%). The 
sudden density changes in some of these 34 MGS profiles, such 
as P0041, P0114 and P1123, which are shown in Figures 
4b--4d of \citet{withers2006b}, are visually striking. Only 
one ODY profile, P056, has a comparable appearance, as shown 
in Figure~\ref{orb56r7vst}.

[FIGURE~\ref{orb56r7vst}]

\subsection{Polar Warming}

Scale heights increase as latitude increases towards the 
north pole 
\citep{bougher2006}. This increase is not zonally uniform. 
Figure~\ref{nightdshgrid}
shows that nightside scale heights at 110 km are zonally 
uniform equatorward of 
70$^{\circ}$N, but greatest at longitudes between 
90$^{\circ}$E and 
270$^{\circ}$E for latitudes poleward of 75$^{\circ}$N.
\citet{tolson2005} saw zonal structure in density at polar 
latitudes, also dominated by wavenumber 1.

[FIGURE~\ref{nightdshgrid}]

A lower atmosphere general circulation model, supported by 
MGS Thermal Emission Spectrometer (TES) data, predicted that 
two eastward propagating travelling waves with zonal 
wavenumber 1 would be strong at northern latitudes during 
ODY aerobraking \citep{wilson2002b}. These two modes had 
periods of 6.5 and 20 sols. 
The MGS Mars Horizon Sensor Assembly (MHSA) acquired a global 
map of lower atmospheric temperatures at fixed local times 
once per sol \citep[e.g.][]{martin2001}. MHSA data showed a 
region of high temperatures centred on the pole in the northern
hemisphere during ODY aerobraking. The latitude of the boundary 
of this region varied with longitude. On any given day, the 
zonal variation in boundary latitude was dominated by 
wavenumber 1. The phase and amplitude of this wavenumber 1 
component varied. At times, the phase shifted 
30$^{\circ}$--50$^{\circ}$ eastward each sol 
(pers. comm., T. Martin, 2001).
This might explain why zonal structure with a strong wave 
1 component is common in analyses of short intervals of ODY 
data and why zonal structure is weak in analyses of long 
intervals of ODY data.

Numerous profiles acquired during the early stages of ODY 
aerobraking contained evidence of unusually large horizontal 
density gradients. 
Meridional density gradients are typical near the winter pole 
\citep{tolson2005,bougher2006}.
Such evidence included large changes in density over a small 
change in altitude (orbit P011, Figure~\ref{orb011rhovsz}) and 
periapsis density being significantly smaller than the 
maximum density (orbit P012, Figure~\ref{orb012rhovst}). 

[FIGURE~\ref{orb011latvst}]
[FIGURE~\ref{orb011rhovst}]
[FIGURE~\ref{orb011rhovsz}]
[FIGURE~\ref{orb012rhovst}]

%We consider orbits 8--84 inclusive. The ratio of maximum density 
%(7s) to periapsis density (7s) is greater than 1.4 in 14 of 
%these 77 orbits (18\%). 

We continue to investigate possible wavenumber 1 structure 
in the polar vortex by 
analysing all orbits with periapsis altitude below 130 km, 
periapsis latitude less than 80$^{\circ}$N and periapsis 
LST greater than 12 hours. Orbits 9 to 80 satisfy these 
criteria, and we examine their 7-second density profiles.
The ratio of maximum density to periapsis density is greater 
than 1.4 in 12 of these 72 orbits (17\%). 
If the ratio of maximum density to periapsis density was 
greater than 1.4 for orbit $X$, then this ratio was always 
smaller than 1.4 for orbits $X-1$ and $X+1$. There is only 
a 12\% chance of this occurring randomly. Therefore some 
mechanism is preventing orbits with high ratios of maximum 
density to periapsis density from occurring consecutively. 
Suppose this mechanism is represented as a black box whose 
output depends deterministically on a set of inputs and that 
its output determines whether or not the ratio of maximum 
density to periapsis density exceeds 1.4. The observed series 
of ratios show that the black box's output changes significantly 
from one orbit to the next. Therefore the inputs on which the 
black box's output depends must also change significantly from 
one orbit to the next. Periapsis latitude and periapsis LST, 
for example, do not change significantly from one orbit to 
the next.
Only two of the many characteristics that define an aerobraking 
pass change significantly from one orbit to the 
next --- time (UTC, not LST) and longitude.

If a travelling wave is responsible for this behaviour, then 
we would expect to find orbits with high ratios of maximum 
density to periapsis density occuring preferentially at 
certain ``adjusted longitudes'', where adjusted longitude = 
periapsis longitude - $360^{\circ} \left(t-t_{0}\right) / T$, 
$t$ is time since arbitrary time $t_{0}$, and $T$ is the 
travelling wave's period. A range of periods from -20 to +20 
days was tested, but results were inconclusive \citep{wilson2002b}.

\section{\label{scisectiontides}Science --- Tides in 
Constant Altitude Data}

\subsection{Tides in ODY Data}

Thermal tides were responsible for zonal variations in MGS 
density measurements at fixed latitude, season, altitude and 
LST \citep{withersphd, withers2003b}. Zonal density variations 
attributable to thermal tides are also present in ODY density 
measurements.
``Dayside'' corresponds to 12 hours $<$ LST $<$ 24 hours and 
``nightside'' corresponds to 0 hours $<$ LST $<$ 12 hours.

Figure~\ref{day60n70nzonal} shows outbound dayside densities 
at 130 km between 
60$^{\circ}$N and 70$^{\circ}$N. There is clearly a strong 
influence of longitude on density. Figure~\ref{day60n70nzonal} 
also shows a harmonic fit to the density measurements. Fitted 
parameters are listed in Table~\ref{odywavefit6070}.
The fitting procedure is described in Section~\ref{fitalgebra}.

[FIGURE~\ref{day60n70nzonal}]
[TABLE~\ref{odywavefit6070}]

The zonal structure persists over a broad altitude range.
Figure~\ref{manyzday60n70n} shows outbound dayside densities 
at 120, 130 and 140 km 
between 60$^{\circ}$N and 70$^{\circ}$N. The longitudes of 
peaks and troughs do not change significantly with altitude, 
indicating that the phases of the dominant tidal modes do not 
change significantly with altitude at these altitudes. 
Zonal mean density decreases as altitude increases, as do the 
relative amplitudes of the peaks and troughs in density. Similar 
trends were present in the MGS observations 
\citep{withersphd, withers2003b}.

[FIGURE~\ref{manyzday60n70n}]

Figure~\ref{manyznight60n70n} shows nightside inbound densities 
at 110, 120 and 130 km between 60$^{\circ}$N and 70$^{\circ}$N. 
Fitted parameters from 130 km are listed in 
Table~\ref{odywavefit6070}. In this case, nightside zonal mean 
densities are about half the corresponding dayside values. Relative 
amplitudes of the zonal structure are also smaller on the nightside 
than the dayside.

[FIGURE~\ref{manyznight60n70n}]

Meridional trends in the zonal structure are most apparent in the 
nightside data, despite the zonal structure being stronger on the 
dayside, because the meridional extent of the nightside data is 
much greater than that of the dayside data.
Figure~\ref{nightcontourinrho120} shows the zonal structure as a 
function of latitude for inbound densities at 120 km on the 
nightside.
The zonal structure is coherent over substantial meridional 
distances, as seen in dayside MGS data 
\citep{withersphd, withers2003b}.
The longitudes of some density peaks and troughs change 
significantly more than others, whereas longitudes of all 
density peaks and troughs were relatively uniform in 
dayside MGS data \citep{withersphd, withers2003b}.
The strength of the zonal structure at 120 km appears to 
increase from pole to equator.
However, this trend is not observed at 130 km. There are 
theoretical reasons to expect weaker zonal structure at 
the poles than at the equator, and suggestions of a similar 
trend in dayside MGS data 
\citep[][and references therein]{withers2003b}.

[FIGURE~\ref{nightcontourinrho120}]

Section~\ref{fitalgebra} defines the amplitude and phase 
of each harmonic term. 
The phase of the wave-$n$ harmonic is between zero and 
360$^{\circ}$E/$n$ by definition. 
Figure~\ref{nightalllatphases} shows phases of each term in 
Figure~\ref{nightcontourinrho120}'s harmonic fit. 
Phases are not randomly distributed, as expected for 
zonal structure that is coherent over an extended meridional range.
Phases for wave-2 and wave-3 do not vary greatly with 
latitude. In fact, the wave-2 phase is always between 
0$^{\circ}$E and 35$^{\circ}$E, 20\% of its full range.
The wave-3 phases varies almost monotonically between 
15$^{\circ}$E at 
70$^{\circ}$N--80$^{\circ}$N and 65$^{\circ}$E at 
10$^{\circ}$N--20$^{\circ}$N. 
Although phases for wave-1 and wave-4 in a given latitude 
band are generally similar to their values in adjacent 
latitude bands, their values vary within their full ranges. 
The wave-1 and wave-4 phases are positively correlated, 
which may indicate a physical connection of some kind.

[FIGURE~\ref{nightalllatphases}]

Figure~\ref{nightalllatamps} shows relative amplitudes of 
each term in Figure~\ref{nightcontourinrho120}'s harmonic 
fit. Amplitudes are more variable than the phases. Amplitudes 
of the wave-2 and wave-3 terms are generally largest.

[FIGURE~\ref{nightalllatamps}]

These trends in ODY phases and amplitudes are generally 
consistent with analysis of dayside MGS data. Future analysis 
of the numerical values of the stable phases may be productive.

Simultaneous day/night measurements at the same latitude and 
altitude were made by ODY for a short period.
On orbit 127, ODY made observations at 130 km at 
75.7$^{\circ}$N, 16.5 hr {\it and} 75.5$^{\circ}$N, 2.5 hr. 
We now compare the zonal structure at 130 km between 
70$^{\circ}$N and 80$^{\circ}$N on the dayside 
(outbound, orbits 96--146, LST 16.44--16.68 hrs) and on the 
nightside (inbound, orbits 109--151, LST 2.25--2.67 hrs).
Results are shown in Figures~\ref{simdayzonal} and 
\ref{simnightzonal} and in Table~\ref{odywavefit7080}.
The nightside's zonal structure is weak, if not non-existent. 
The dayside's zonal structure is slightly more significant. 
By contrast, zonal structure at 130 km between 
50$^{\circ}$S and 70$^{\circ}$S during MGS Phase 2 was 
present on the dayside and stronger on the nightside.

[FIGURE~\ref{simdayzonal}]
[FIGURE~\ref{simnightzonal}]
[TABLE~\ref{odywavefit7080}]

\subsection{Tides in MGS Phase 1 Data}

Neither \citet{withers2003b} nor \citet{withersphd} 
discussed MGS Phase 1 data. \citet{keating1998} fitted 
wave-1 and wave-2 components to 21 densities at 125 km 
from orbits 90 to 110.
Here we investigate tides in the MGS Phase 1 data in 
more detail, focusing on outbound data at 130 km which 
give the clearest results.
Data before orbit 70 was neglected, because data at 
130 km from orbits 1--70 are insufficient to sample 
all longitudes. Data from orbits 70--100, 80--110, 
90--120, $\ldots$, 170--200 were fitted. Overlapping 
intervals were used for ease of analysis.
Figure~\ref{mgsp1out130orb130160} shows density as a 
function of longitude for orbits 130--160. The zonal 
structure is readily apparent and obviously dominated 
by wavenumber 2.
Figure~\ref{mgsp1out130relamps} shows that wavenumber 2 
is stronger than wavenumber 1, which is stronger than 
wavenumber 3, which is stronger than wavenumber 4.
Figure~\ref{mgsp1out130phases} shows that all phases 
change slowly during MGS Phase 1.
Similar analysis at higher altitudes shows that the zonal 
structure is less apparent  and its amplitudes and phases 
are more variable at higher altitudes than at 130 km.

[FIGURE~\ref{mgsp1out130orb130160}]
[FIGURE~\ref{mgsp1out130relamps}]
[FIGURE~\ref{mgsp1out130phases}]

Wavenumber 2 is important in the dayside zonal structure 
for both MGS Phase 1 and MGS Phase 2. Wavenumber 1 is 
stronger than wavenumber 3 in the MGS Phase 1 data, but 
weaker in MGS Phase 2 data. Seasonal changes in the 
atmosphere below 130 km are probably responsible for this 
difference.

\subsection{\label{fitalgebra}Harmonic fit to zonal structure}

Harmonic fits to zonal structure are performed using the 
same techniques as \citet{withers2003b}. These techniques 
are also discussed in \citet{withersphd}.
Density as a function of longitude is fitted to this 
function:

\begin{eqnarray}
\label{tides_wavemodel}
\rho = a_{0} + a_{1} \cos (\lambda) +
 b_{1} \sin (\lambda) + \cdots + 
a_{k} \cos (k\lambda) + b_{k} \sin (k\lambda)
\end{eqnarray}

Where $\rho$ is density, $\lambda$ is east longitude, 
and the $a_{i}$ and 
$b_{i}$ are model parameters.
This is a wave-$k$ model as it contains wavenumbers up 
to and including $k$. 
Measurement uncertainties are not used in the fitting 
procedure because the sol-to-sol atmospheric variability 
is typically much greater than the measurement uncertainties. 
With the above assumption about dealing with uncertainties, 
$m$ density 
measurements, and $n$ model parameters, the process 
follows Equation 
7.18 of \citet{neter1974}:

\begin{eqnarray}
\label{tides_yeqn}
\underline{Y} = \underline{\underline{X}} \; \underline{p} + 
\underline{\epsilon}
\end{eqnarray}

Where $\underline{Y}$ is an $m$-element vector of density 
observations, 
$\underline{\underline{X}}$ is an $m$-by-$n$ matrix, 
$\underline{p}$ is an 
$n$-element vector of model parameters, and 
$\underline{\epsilon}$ is an 
n-element vector of uncorrelated random variables of 
mean $0$ and standard 
deviation $\sigma$.

Using the fitting function in Equation~\ref{tides_wavemodel}:

\begin{eqnarray}
\label{tides_fillx}
X_{i0} = 1 \\ \nonumber 
X_{i1} = \cos \left(\lambda_{i}\right) \\ \nonumber 
X_{i2} = \sin \left(\lambda_{i}\right) \\ \nonumber 
\vdots \\\ \nonumber 
X_{in-2} = \cos \left(\frac{(n-1) 
\lambda_{i}}{2}\right) \\ \nonumber 
X_{in-1}   = \sin \left(\frac{(n-1) \lambda_{i}}{2}\right)
\end{eqnarray}

\begin{eqnarray}
\label{tides_fillb}
p_{0} = a_{0} \\ \nonumber 
p_{1} = a_{1} \\ \nonumber 
p_{2} = b_{1} \\ \nonumber 
\vdots \\ \nonumber 
p_{n-2} = a_{\frac{n-1}{2}} \\ \nonumber 
p_{n-1} = b_{\frac{n-1}{2}} 
\end{eqnarray}

The index $i$ labels an individual density measurement.
 The $n$ variables 
($1, \cos\left(\lambda\right), 
\sin\left(\lambda\right), \ldots$) 
being fitted are linearly independent. The least 
squares solution for the model 
parameters, $\underline{p}$, follows Equation 
7.21 of \citet{neter1974}:

\begin{eqnarray}
\label{tides_peqn}
\underline{p} = \left(\underline{\underline{X}}^{T} 
\underline{\underline{X}}\right)^{-1} \; 
\underline{\underline{X}}^{T} \; 
\underline{Y}
\end{eqnarray}

Where a superscript $T$ indicates the transpose of 
a matrix and the superscript 
$-1$ indicates the inverse of a matrix. Note that 
the constant density term, 
$a_{0}$ or $p_{0}$, is not formally identical to 
the mean density. Hence it is referred to as the constant 
density term rather than as the zonal mean term. 
However, the two are usually similar.

The model predictions, $\underline{\hat{Y}}$, for 
atmospheric densities at the 
longitudes at which measurements have been made 
follow Equation 
7.23 of \citet{neter1974}:

\begin{eqnarray}
\label{tides_ypred}
\underline{\hat{Y}} = \underline{\underline{X}} \; 
\underline{p}
\end{eqnarray}

The covariance matrix for $\underline{p}$ follows 
Equation 7.39 of  
\citet{neter1974}:

\begin{eqnarray}
\label{tides_covmatrix1}
\underline{\underline{Cov}}\left(\underline{p}\right) = \frac{
\left(\underline{\hat{Y}} - \underline{Y}\right)^{T} \; 
\left(\underline{\hat{Y}} - \underline{Y}\right)}{m-n} \; 
\left(\underline{\underline{X}}^{T} \underline{\underline{X}}\right)^{-1} 
\end{eqnarray}

1 $\sigma$ uncertainties in the model parameters are given by:

\begin{eqnarray}
\label{tides_covmatrix2}
\sigma_{p_{j}} = Cov_{jj}
\end{eqnarray}

Using sine and cosine terms is useful for forming a linear 
model. However,
it is more useful to interpret paired sine and cosine 
terms as being a single
sinusoid with an amplitude and a phase. The phase of 
a given harmonic 
can be defined as the longitude of its first peak 
east of 0$^\circ$. Thus the phase of the wave-$q$ 
harmonic must lie between 0$^\circ$ and 
360$^\circ$E/$q$ where $q$
labels a certain harmonic.

\begin{eqnarray}
\label{tides_eqn2a}
a_{q}\;\cos\left(q\lambda\right) + 
b_{q}\;\sin\left(q\lambda\right) = 
R_{q}\;\cos\left(q \left[\lambda - 
\alpha_{q} \right] \right) 
\end{eqnarray}

Where $R_{q}$ is the amplitude of the $q$th 
harmonic and $\alpha_{q}$
is the phase of the $q$th harmonic. Note that 
\citet{withersphd} incorrectly call $q\alpha_{q}$ 
the phase of the $q$th harmonic.
Trigonometry yields the following
solutions for $R_{q}$ and $\alpha_{q}$ in terms 
of $a_{q}$ and $b_{q}$.

\begin{eqnarray}
\label{tides_eqn2b}
R_{q} = \left(a_{q}^{2} + b_{q}^{2}\right)^{1/2} 
\end{eqnarray}

\begin{eqnarray}
\label{tides_eqn2c}
\alpha_{q} = \frac{1}{q} \tan^{-1}\left(b_{q}/a_{q}\right)
\end{eqnarray}

Using the usual formula, Equation 4.9 of 
\citet{bevington1969},
for transforming uncertainties of uncorrelated 
variables:

\begin{eqnarray}
\label{tides_eqn3a}
\sigma_{R_{q}}^{2} = \sigma_{a_{q}}^{2} 
\left( \frac{ \partial R_{q}}{\partial 
a_{q}} \right)^{2} + \sigma_{b_{q}}^{2} 
\left( \frac{ \partial R_{q}}{\partial 
b_{q}} \right)^{2} 
\end{eqnarray}

\begin{eqnarray}
\label{tides_eqn3b}
\sigma_{\alpha_{q}}^{2} = \sigma_{a_{q}}^{2} 
\left( \frac{ \partial 
\alpha_{q}}{\partial a_{q}} \right)^{2} + 
\sigma_{b_{q}}^{2} \left( \frac{ 
\partial \alpha_{q}}{\partial b_{q}} 
\right)^{2}
\end{eqnarray}

\begin{eqnarray}
\label{tides_eqn4a}
\sigma_{R_{q}}^{2} = \sigma_{a_{q}}^{2} 
\left( \frac{a_{q}}{R_{q}} \right)^{2} + 
\sigma_{b_{q}}^{2} \left( \frac{b_{q}}{R_{q}} 
\right)^{2} 
\end{eqnarray}

\begin{eqnarray}
\label{tides_eqn4b}
\sigma_{\alpha_{q}}^{2} = \frac{b_{q}^{2} 
\sigma_{a_{q}}^{2} + a_{q}^{2} 
\sigma_{b_{q}}^{2}} {q^{2} \left( a_{q}^{2} + 
b_{q}^{2} \right)^{2}}
\end{eqnarray}

A phase of 3$^\circ$ $\pm$ 15$^\circ$ is a perfectly 
reasonable and 
meaningful result. An amplitude of 3 kg km$^{-3}$ 
$\pm$ 15 kg 
km$^{-3}$ means that the harmonic is not present 
in any statistically 
significant sense and that its corresponding phase 
is meaningless.

It is often useful to normalize the zonally-varying 
terms in each wavefit ($R_{q}$)
by their constant density term ($R_{0}$). This facilitates 
a comparison of the 
strength of the zonal structure between different 
seasons or altitudes. The symbol $R_{q}^{N}$ labels 
normalized harmonic amplitudes.

\begin{eqnarray}
\label{tides_eqn5a}
R_{q}^{N} = \frac{R_{q}}{R_{0}} 
\end{eqnarray}

\begin{eqnarray}
\label{tides_eqn5b}
\sigma_{R_{q}^{N}}^{2} = \sigma_{R_{q}}^{2} 
\left( \frac{ \partial 
R_{q}^{N}}{\partial 
R_{q}} \right)^{2} + \sigma_{R_{0}}^{2} 
\left( \frac{ \partial 
R_{q}^{N}}{\partial R_{0}} \right)^{2} 
\end{eqnarray}

\begin{eqnarray}
\label{tides_eqn5c}
\sigma_{R_{q}^{N}}^{2} = \frac{R_{q}^{2} \; 
\sigma_{R_{0}}^{2}}{R_{0}^{4}} + 
\frac{\sigma_{R_{q}}^{2}}{R_{0}^{2}}
\end{eqnarray}

A 1 $\sigma$ uncertainty about the fitted function 
can be calculated at any 
given longitude following Equation 7.54a of 
\citet{neter1974}:

\begin{eqnarray}
\label{tides_eqn6}
\sigma_{fit}^{2} = 
\left( 1 + \underline{X}_{new}^{T} 
\left( \underline{\underline{X}}^{T} 
\underline{\underline{X}}\right)^{-1}\underline{X}_{new} 
\right) \;
\frac{\left( \underline{\hat{Y}} - \underline{Y} 
\right)^{T} \; 
\left(\underline{\hat{Y}} -
\underline{Y}\right)}{m-n}
\end{eqnarray}

where $\underline{X}_{new}$ is an $n$-element vector 
whose elements are 
calculated identically to those of a single 
column of 
$\underline{\underline{X}}$, as in 
Equation~\ref{tides_fillx}. The longitude 
used to calculate $\underline{X}_{new}$ is that at 
which the 1 $\sigma$
uncertainty is desired. Formally, this is the 1 
$\sigma$ uncertainty on what a 
new observation at that longitude might be. The 
predicted value of that 
observation is as in Equation~\ref{tides_ypred}.

\section{\label{scisectioncaltother}Science --- 
Other Aspects of Constant Altitude Data}

\subsection{\label{boughercomp}Comparison of 
results to predictions}

The Mars Thermospheric General Circulation Model 
(MTGCM) has predicted the state of the atmosphere 
under conditions appropriate for ODY aerobraking 
\citep{bougher1999b, bougher2000, bougher2002}. 
Results from eight MTGCM simulations have been made 
available online at 
http://aoss-research.engin.umich.edu/tgcm\_planets\_archive/predms01.html 
and are summarized in Table~\ref{boughermodels}.
Dust opacities ($\tau$) of 0.3, 1.0 and 3.0 were simulated.
Altitude ranges are 100--150 km or better and vertical 
resolutions are 5 km or better.
Results are available at LSTs of 3, 4, 15, 16, 17 
and 18 hours.
Results at LST=4 hours and LST=15 hours were not used 
because, apart from a small region near the pole, all 
data have either LST$<$3 hours or LST$>$16 hours.
The simulated latitudes, altitudes, local solar times 
and seasons almost completely cover the range of 
conditions in the ODY dataset. Additional simulations 
at equatorial latitudes at $L_{s} \approx 297^{\circ}$ 
and nightside LSTs would have been useful.

[TABLE~\ref{boughermodels}]

%[TABLE~\ref{resdetails}] hasn't been used yet!

ODY and MTGCM dayside densities are compared in 
Figure~\ref{bougherrhoday}.
Multiple MTGCM dayside densities are plotted for a single 
latitude and altitude. These correspond to different 
LSTs (16, 17 and 18 hours), dust opacities 
($\tau$ = 0.3, 1.0 and 3.0) and seasons 
($L_{s} = 256^{\circ}$, 277$^{\circ}$ and 297$^{\circ}$). 
Dust opacity is responsible for most of the range in 
density at fixed latitude and altitude. This is why the 
multiple MTGCM densities at, say, 57.5$^{\circ}$N and 
100 km form three distinct clusters. There are only two 
clusters at 87.5$^{\circ}$N and 100 km because simulated 
densities are available at only two dust opacities.
Observed and simulated densities are very consistent. 
Simulations at $\tau=1.0$ appear most consistent with 
observations.

[FIGURE~\ref{bougherrhoday}]

ODY and MTGCM nightside densities are compared in 
Figure~\ref{bougherrhonight}. Simulated densities are 
available at only two dust opacities ($\tau$=0.3 and 1.0). 
Observed polar densities are consistent with simulations at 
$\tau$=1.0. Observed mid-latitude densities are significantly 
smaller than simulated densities. Densities equatorward of 
70$^{\circ}$N do not vary with latitude, in constrast to the 
strong meridional density gradient in simulations, which 
may indicate difference between simulated and actual 
dynamics.

[FIGURE~\ref{bougherrhonight}]

ODY and MTGCM dayside scale heights are compared in 
Figure~\ref{bougherdshday}. Densities at different altitudes 
are clearly distinct, but scale heights are not. Therefore 
the large filled circles indicate an average of all scale 
height measurements within a ten degree latitude window. 
Altitude is indicated by the colour of the filled circle.
Observed scale heights at 110 km are about a factor of 
two greater than predicted. Equatorward of 80$^{\circ}$N, 
observed scale heights at 120 and 130 km are consistent with 
predictions. Observations poleward of 80$^{\circ}$N show 
that the scale height decreases as altitude increases, 
opposite to the predicted increase in scale height with 
increasing altitude. However, it is important to note 
that the observed scale heights are derived from non-vertical 
density profiles.

[FIGURE~\ref{bougherdshday}]

ODY and MTGCM nightside scale heights are compared in 
Figure~\ref{bougherdshnight}. Observations and simulations 
are reasonably consistent, especially equatorward of 
70$^{\circ}$N. Observed scale heights are somewhat greater 
than predicted poleward of 70$^{\circ}$N \citep{bougher2006}.
Both observations and simulations show that scale heights 
(and, by implication, temperatures) increase poleward of 
about 70$^{\circ}$N. Observations show the scale height 
decreasing as altitude increases, whereas simulations show 
a more uniform scale height.

[FIGURE~\ref{bougherdshnight}]

\subsection{Variations with Latitude and LST}

Figure~\ref{avrho} shows mean dayside and nightside densities 
for a range of latitudes and altitudes.
Differences between nightside and dayside densities increase 
as altitude increases and as latitude moves from pole to 
equator.
ODY data were obtained around northern winter solstice, when 
northern polar regions are not illuminated by the Sun. It is 
therefore reasonable that nightside and dayside densities are 
similar during the polar night.

%The effect of altitude on the difference between nightside and 
%dayside densities can be explained if upper atmospheric 
%temperatures (at 100--140 km) have a greater diurnal range 
%than lower atmospheric temperatures (below 100 km). This is 
%consistent with dynamical heating and cooling playing a larger 
%role in determining temperatures, relative to radiative heating 
%and cooling, at low altitudes and high pressures than at high 
%altitudes and low pressure 
%\citep[e.g.][and references therein]{zurek1992}.

[FIGURE~\ref{avrho}]

Figure~\ref{avdsh} shows a similar plot for scale heights. 
Scale heights are greater in the polar region than at tropical 
and mid-latitudes. Dayside scale heights are greater than 
nightside scale heights. Nightside scale heights at 120 and 
130 km do not vary significantly with latitude equatorward of 
60$^{\circ}$N. Nightside mid-latitude scale heights at 130 km 
are slightly greater than at 120 km. 
A very different picture is presented by analysis of MGS dayside 
scale heights \citep{withers2006b}. Scale heights are greater 
in the tropics in MGS data than in polar regions. Scale heights 
vary significantly with latitude equatorward of 60$^{\circ}$ in 
MGS data.

[FIGURE~\ref{avdsh}]

\subsection{Interannual Variability}

There are no instances where ODY and MGS data were obtained at 
different years, but identical latitudes, LSTs and seasons. 

ODY and MGS data were obtained at similar latitudes and seasons, 
but dissimilar LSTs. MGS measured densities at 130 km, 
55$^{\circ}$N--60$^{\circ}$N, LST=12.0-12.8 hrs, 
Ls=270$^{\circ}$--285$^{\circ}$ in February 1998.
ODY measured densities at 130 km at 130 km, 
62$^{\circ}$N--70$^{\circ}$N, LST=16.4-18.0 hrs, 
Ls=263$^{\circ}$--288$^{\circ}$ in November 2001. There is a 
difference of 6 hours in LST. The mean density was 
2.64 kg km$^{-3}$ for ODY and 3.25 kg km$^{-3}$ for MGS. 
The mean density scale height was 7.1 km for ODY and 7.6 km 
for MGS.

\section{\label{concs}Conclusions}

Density profiles, as well as constant altitude densities and 
density scale heights, have been generated from Odyssey (ODY) 
accelerometer measurements. These form a more complete dataset 
than the unpeer-reviewed data products currently available 
from the Planetary Data System (PDS). They will be archived 
at the PDS in compliant formats with extensive documentation.

Small-scale structure, probably due to gravity waves, is present 
in ODY density profiles. Its amplitudes and meridional trends
are broadly consistent with those seen in MGS density profiles. 
Only one ODY density profile, P056, contains a sudden change in 
density comparable to those present in MGS density profiles.
The previously-observed increase in temperature with latitude 
near the north polar region during winter is not zonally-uniform, 
which suggests that dynamics plays a major role in maintaining 
the high temperatures.

Thermal tides produce zonal variations in atmospheric densities 
during ODY aerobraking, as previously observed during MGS 
aerobraking. Tidal effects persist from 120 km to 140 km, with 
amplitudes, but not phases, changing significantly with altitude. 
Zonal structure is weaker on the nightside than on the dayside.
The zonal structure is coherent over 60 degrees of latitude, 
although it is less stable for ODY than for MGS.

Atmospheric simulations produced in support for ODY aerobraking 
can now be compared to data. Predicted dayside densities were 
accurate. The predicted strong meridional  gradient in nightside 
densities was not observed. Simulations did not predict changes 
in scale height with altitude well. These discrepancies can be 
used to investigate which model boundary conditions were 
inadequate or which representations of physical processes 
were incomplete.

\indent{\bf{Acknowledgments\\}}

We gratefully acknowledge the Odyssey operations team for 
safely guiding Odyssey through its aerobraking phase; Bob Mase, 
Jill Prince and their NASA colleagues for providing mass, 
aerodynamics and orientation information to us; Steve Bougher 
for making MTGCM simulations publicly available; and the 
Odyssey aerobraking team, led by Gerry Keating, for making 
raw data available to the PDS.
Partial support for PW's activities was provided by the Odyssey 
Participating Scientist Program.


\newpage

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  {Woo}, {Armstrong}, {Flasar}, {Simpson}, and {Priest}]{tyler2000}
{Tyler}, G.~L., {Balmino}, G., {Hinson}, D.~P., {Sjogren}, W.~L., {Smith},
  D.~E., {Woo}, R., {Armstrong}, J.~W., {Flasar}, F.~M., {Simpson}, R.~A.,
  {Priest}, P., 2000.
\newblock {PDS volume MORS\_1006}.
\newblock In: MGS-M-RSS-5-SDP-V1.0, NASA Planetary Data System.

\bibitem[{Wang} et~al.(2006){Wang}, {Fritts}, and {Tolson}]{wang2006}
{Wang}, L., {Fritts}, D.~C., {Tolson}, R.~H., 2006.
\newblock {Nonmigrating tides inferred from the Mars Odyssey and Mars Global
  Surveyor aerobraking data}.
\newblock Geophys. Res. Lett.~33, L23201, 10.1029/2006GL027753.

\bibitem[{Wilson} et~al.(2002){Wilson}, {Banfield}, {Conrath}, and
  {Smith}]{wilson2002b}
{Wilson}, R.~J., {Banfield}, D., {Conrath}, B.~J., {Smith}, M.~D., 2002.
\newblock {Traveling waves in the northern hemisphere of Mars}.
\newblock \grl~29, 1684, 10.1029/2002GL014866.

\bibitem[{Withers}(2003){Withers}]{withersphd}
{Withers}, P., 2003.
\newblock {\em {Tides in the Martian atmosphere --- and other topics}}.
\newblock Ph.\ D. thesis, University of Arizona.

\bibitem[{Withers}(2006){Withers}]{withers2006b}
{Withers}, P., 2006.
\newblock {Mars Global Surveyor and Mars Odyssey Accelerometer observations of
  the martian upper atmosphere during aerobraking}.
\newblock Geophys. Res. Lett.~33, L02201, 10.1029/2005GL024447.

\bibitem[{Withers} et~al.(2003){Withers}, {Bougher}, and
  {Keating}]{withers2003b}
{Withers}, P., {Bougher}, S.~W., {Keating}, G.~M., 2003.
\newblock {The effects of topographically-controlled thermal tides in the
  martian upper atmosphere as seen by the MGS accelerometer}.
\newblock Icarus~164, 14--32.

\bibitem[{Withers} et~al.(2003){Withers}, {Towner}, {Hathi}, and
  {Zarnecki}]{withers2003}
{Withers}, P., {Towner}, M.~C., {Hathi}, B., {Zarnecki}, J.~C., 2003.
\newblock {Analysis of entry accelerometer data: A case study of Mars
  Pathfinder}.
\newblock \planss~51, 541--561.

\end{thebibliography}

\newpage

\clearpage
% \clearpage prints all the tables, etc, rather than 
%saving them until the end



\begin{deluxetable}{lrr}
\tablewidth{0pt}
\tablecaption{\label{accnoisevalue}Mean noise in 
high-rate acceleration measurements}
\tablehead{
\colhead{Acceleration} &
\colhead{Orbits} &
\colhead{Noise (m s$^{-2}$)}
}
\startdata
$a_{aero,hi,y}$ & 1--136 & $8.601 \times 10^{-5}$ \nl
 & 137--268 & $2.824 \times 10^{-4}$ \nl
 & 269--330 & $5.200 \times 10^{-4}$ \nl
$a_{aero,hi,y,7}$ & 1--136 & $1.824 \times 10^{-5}$ \nl
 & 137--268 & $5.168 \times 10^{-5}$ \nl
 & 269--330 & $7.960 \times 10^{-5}$ \nl
$a_{aero,hi,y,39}$ & 1--136 & $4.655 \times 10^{-6}$ \nl
 & 137--268 & $8.663 \times 10^{-6}$ \nl
 & 269--330 & $1.262 \times 10^{-5}$
\enddata
\end{deluxetable}

\clearpage

\begin{deluxetable}{lccc}
\tablewidth{0pt}
\tablecaption{\label{accnoiseratio}Ratios of mean 
noise in high-rate acceleration measurements}
\tablehead{
\colhead{Orbits} &
\colhead{$a_{aero,hi,y,1}$:$a_{aero,hi,y,7}$ } &
\colhead{$a_{aero,hi,y,1}$:$a_{aero,hi,y,39}$} &
\colhead{$a_{aero,hi,y,7}$:$a_{aero,hi,y,39}$}
}
\startdata
1--136 & 4.71 & 18.48 & 3.92\nl
137--268 & 2.34 & 5.71 & 2.44\nl
269--330 & 2.56 & 6.42 & 2.51\nl
\hline
Expected ratio\nl
for $\sqrt{N}$ & $\sqrt{7} = 2.65$ & 
$\sqrt{39} = 6.25$ & $\sqrt{39/7} = 2.36$
\enddata
\end{deluxetable}

\clearpage

\begin{deluxetable}{lrrr}
\tablewidth{0pt}
\tablecaption{\label{topofatmtable}Some characteristics 
of derived high-rate density profiles.}
\tablehead{
\colhead{} &
\colhead{$\rho_{hi,39}$} &
\colhead{$\rho_{hi,7}$} &
\colhead{$\rho_{hi}$}
}
\startdata
\underline{Start of Aerobraking} \nl
$z_{max}$ (km) & 146 & 145 & 142 \\
$\sigma_{\rho}$ at $z_{max}$ (kg km$^{-3}$) & 
0.4 & 0.4 & 0.6 \nl
$\sigma_{aaero,y}$ at $z_{max}$ (m s$^{-2}$) & 
$2 \times 10^{-4}$ & 
 $2 \times 10^{-4}$ & $2 \times 10^{-4}$ \nl
\hline
\underline{End of Aerobraking} \nl
$z_{max}$ (km) & 134 & 132 & 122 \\
$\sigma_{\rho}$ at $z_{max}$ (kg km$^{-3}$) & 
0.6 & 0.8 & 4 \nl
$\sigma_{aaero,y}$ at $z_{max}$ (m s$^{-2}$) & 
$2 \times 10^{-4}$ 
 & $2 \times 10^{-4}$ & $1.3 \times 10^{-3}$ 
\enddata
\end{deluxetable}

\clearpage

\begin{deluxetable}{lrr}
\tablewidth{0pt}
\tablecaption{\label{p076calttable}
Constant altitude data products for orbit P076. 
Uncertainties are $1\sigma$.}
\tablehead{
\colhead{Altitude (km)} &
\colhead{Density (kg km$^{-3}$)} &
\colhead{Scale Height (km)}
}
\startdata
\underline{Inbound} \nl
110 & 17.56$\pm$0.10 & 8.24$\pm$0.12 \nl
120 & 4.47$\pm$0.08 & 6.72$\pm$0.25  \nl
130 & 1.10$\pm$0.08 & 7.37$\pm$1.30 \nl
\hline
\underline{Outbound} \nl
110 & 21.23$\pm$0.12 & 10.30$\pm$0.19 \nl
120 & 7.44$\pm$0.09 & 8.64$\pm$0.28 \nl
130 & 1.98$\pm$0.08 & 7.30$\pm$0.72 
\enddata
\end{deluxetable}

\clearpage

\begin{deluxetable}{lrrr}
\tablewidth{0pt}
\tablecaption{\label{calttable}
Coverage of constant altitude data. }
\tablehead{
\colhead{Altitude (km)} &
\colhead{Number of orbits} &
\colhead{First orbit} &
\colhead{Last orbit}
}
\startdata
\underline{Inbound} \nl
100 & 37 & 95 & 146 \nl
110 & 173 & 46 & 255 \nl
120 & 316 & 13 & 335 \nl
130 & 255 & 11 & 336 \nl
140 & 6 & 7 & 32 \nl
150 & 0 & --- & --- \nl
160 & 0 & --- & --- \nl
\hline
\underline{Outbound} \nl
100 & 37 & 95 & 146 \nl
110 & 173 & 46 & 255 \nl
120 & 316 & 13 & 335 \nl
130 & 280 & 11 & 336 \nl
140 & 77 & 7 & 158 \nl
150 & 3 & 16 & 29 \nl
160 & 0 & --- & --- 
\enddata
\end{deluxetable}

\clearpage

\begin{deluxetable}{lrrr}
\tablewidth{0pt}
\tablecaption{\label{scalttable}
Typical uncertainties in constant altitude data}
\tablehead{
\colhead{Altitude (km)} &
\colhead{$\sigma_{\rho}$ (kg km$^{-3}$)} &
\colhead{$\sigma_{H}$ (km)} &
\colhead{$\sigma_{T}$ (K)}
}
\startdata
\underline{Inbound} \nl
100 & 0.155 & 0.14 & 2.5 \nl
110 & 0.100 & 0.18 & 3.3 \nl
120 & 0.076 & 0.28 & 5.1 \nl
130 & 0.084 & 1.36 & 24.5 \nl
140 & 0.083 & 2.00 & 35.9 \nl
150 & --- & --- & --- \nl
160 & --- & --- & --- \nl
\hline
\underline{Outbound} \nl
100 & 0.176 & 0.51 & 9.3 \nl
110 & 0.104 & 0.33 & 6.0 \nl
120 & 0.078 & 0.34 & 6.1 \nl
130 & 0.083 & 1.32 & 23.7 \nl
140 & 0.089 & 3.01 & 53.9 \nl
150 & 0.095 & 5.07 & 90.4 \nl
160 & --- & --- & --- \nl
\enddata
\end{deluxetable}

\clearpage


%rhoin110 = 0.061226440, sdev = 0.0510882
%rhoin120 = 0.099102583, sdev = 0.0672655
%rhoout110 = 0.053632978, sdev = 0.103610
%rhoout120 = 0.090621960, sdev = 0.0861239


%dshin110 = 0.083429092, sdev = 0.186090
%dshin120 = 0.045592559, sdev = 0.127672
%dshout110 = 0.11435090, sdev = 0.358268 - 0.087866293 
%and 0.211089 if P122 junked
%dshout120 = 0.084851017, sdev = 0.183602

\begin{deluxetable}{lrrr}
\tablewidth{0pt}
\tablecaption{\label{validrhodshtable}
Mean and standard deviation of the relative differences 
between our results and the aerobraking team's results}
\tablehead{
\colhead{Altitude (km)} &
\colhead{Mean} &
\colhead{Standard deviation}
}
\startdata
\underline{Density} \nl
Inbound 110 & 0.06 & 0.05 \nl
Outbound 110 & 0.05 & 0.10 \nl
Inbound 120 & 0.10 & 0.07 \nl
Outbound 120 & 0.09 & 0.09 \nl
\hline
\underline{Density Scale Height} \nl
Inbound 110 & 0.08 & 0.19 \nl
Outbound 110 & 0.11 & 0.36 \nl
Inbound 120 & 0.05 & 0.13 \nl
Outbound 120 & 0.08 & 0.18 \nl
\hline
Outbound 110\tablenotemark{1} & 0.09 & 0.21
\enddata
\tablenotetext{1}{Not including orbit P122 
(Figures~\ref{reldiffdshout110} and \ref{p122dshnotinsane})}
\end{deluxetable}

\clearpage

\begin{deluxetable}{lrr}
\tablewidth{0pt}
\tablecaption{\label{odywavefit6070}Model fit parameters 
for ODY densities measured
at 130 km between 60$^{\circ}$N and 70$^{\circ}$N. 
Uncertainties are $1 \sigma$.}
\tablehead{
\colhead{Parameters} &
\colhead{Dayside (outbound)} &
\colhead{Nightside (inbound)} 
}
\startdata
Constant Amplitude (kg km$^{-3}$)& 2.755 $\pm$ 0.119    
& 1.171 $\pm$ 0.039 \nl
Normalised Wave 1 Amplitude     & 0.099 $\pm$ 0.063     
& 0.076 $\pm$ 0.048 \nl
Normalised Wave 2 Amplitude     & 0.178 $\pm$ 0.062     
& 0.128 $\pm$ 0.048 \nl
Normalised Wave 3 Amplitude     & 0.271 $\pm$ 0.064     
& 0.172 $\pm$ 0.052 \nl
Normalised Wave 4 Amplitude     & 0.122 $\pm$ 0.062     
& 0.122 $\pm$ 0.059 \nl
Wave 1 Phase (degrees)          & 89.9  $\pm$ 35.2      
& 220.9 $\pm$ 36.3  \nl
Wave 2 Phase (degrees)          & 166.4 $\pm$ 9.9       
& 29.1  $\pm$ 10.5  \nl
Wave 3 Phase (degrees)          & 34.8  $\pm$ 4.2       
& 30.5  $\pm$ 5.7   \nl
Wave 4 Phase (degrees)          & 81.4  $\pm$ 7.1       
& 17.8  $\pm$ 5.5
\enddata
\end{deluxetable}

\clearpage

\begin{deluxetable}{lrr}
\tablewidth{0pt}
\tablecaption{\label{odywavefit7080}Model fit parameters for 
ODY densities measured at 130 km between 70$^{\circ}$N and 
80$^{\circ}$N. Uncertainties are $1 \sigma$.}
\tablehead{
\colhead{Parameters} &
\colhead{Dayside (outbound)} &
\colhead{Nightside (inbound)} 
}
\startdata
Constant Amplitude (kg km$^{-3}$)& 1.829 $\pm$ 0.085    
& 1.173 $\pm$ 0.036 \nl
Normalised Wave 1 Amplitude     & 0.135 $\pm$ 0.066     
& 0.069 $\pm$ 0.043 \nl
Normalised Wave 2 Amplitude     & 0.126 $\pm$ 0.062     
& 0.056 $\pm$ 0.045 \nl
Normalised Wave 3 Amplitude     & 0.047 $\pm$ 0.068     
& 0.051 $\pm$ 0.044 \nl
Normalised Wave 4 Amplitude     & 0.177 $\pm$ 0.068     
& 0.042 $\pm$ 0.054 \nl
Wave 1 Phase (degrees)          & 159.7 $\pm$ 27.9      
& 255.7 $\pm$ 36.9  \nl
Wave 2 Phase (degrees)          & 171.8 $\pm$ 16.6      
& 170.5 $\pm$ 23.8  \nl
Wave 3 Phase (degrees)          & 41.2  $\pm$ 27.4      
& 9.7   $\pm$ 17.7  \nl
Wave 4 Phase (degrees)          & 82.2  $\pm$ 5.5       
& 58.0  $\pm$ 15.5
\enddata
\end{deluxetable}

\clearpage



\begin{deluxetable}{lrrrrrr}
\tablewidth{0pt}
\tablecaption{\label{boughermodels}\label{lasttable}
MTGCM Simulations.}
\tablehead{
\colhead{Label} &
\colhead{Run date} &
\colhead{$\tau$} &
\colhead{$L_{s}$ ($^{\circ}$)} &
\colhead{Date} &
\colhead{Latitudes} &
\colhead{LSTs}
}
\startdata
MS01OCC.AB2 & 09/10/01 & 1.0 & 256 & 21 Oct 2001 & 
57.5, 67.5, 77.5 & 17, 18 \nl
\hline
MS01OCD.AB2 & 09/24/01 & 3.0 & 256 & 21 Oct 2001 & 
57.5, 67.5, 77.5 & 17, 18 \nl
\hline
MS01NVC.AB2 & 10/23/01 & 1.0 & 277 & 23 Nov 2001 & 
67.5, 77.5, 87.5 & 15, 16 \nl
\hline
MS01DCC.AB2 & 10/23/01 & 1.0 & 297 & 25 Dec 2001 & 
37.5, 47.5, 57.5 & 3, 4 \nl
&&&&& 67.5, 77.5, 87.5  & 3, 4\nl
\hline
Nov23t03 & 01/25/00 & 0.3 & 277 & 23 Nov 2001 & 
47.5, 57.5, 67.5 & 3, 4 \nl
&&&&& 77.5, 87.5 & 3, 4 \nl
&&&&& 57.5, 67.5, 77.5 & 15, 16, 17 \nl
&&&&&87.5 & 15, 16, 17 \nl
\hline
Nov23t10 & 10/26/00 & 1.0 & 277 & 23 Nov 2001 & 
47.5, 57.5, 67.5 & 3, 4 \nl
&&&&& 77.5, 87.5 & 3, 4 \nl
&&&&& 57.5, 67.5, 77.5 & 15, 16, 17 \nl
&&&&& 87.5 & 15, 16, 17 \nl
\hline
Dec25t03 & 01/25/00 & 0.3 & 297 & 25 Dec 2001 & 
47.5, 57.5, 67.5 & 3, 4 \nl
&&&&& 77.5, 87.5 & 3, 4 \nl
&&&&& 57.5, 67.5, 77.5 & 15, 16, 17 \nl
&&&&& 87.5 & 15, 16, 17 \nl
\hline
Dec25t10 & 10/26/00 & 1.0 & 297 & 25 Dec 2001 & 
47.5, 57.5, 67.5 & 3, 4 \nl
&&&&& 77.5, 87.5 & 3, 4 \nl
&&&&& 57.5, 67.5, 77.5 & 15, 16, 17 \nl
&&&&& 87.5 & 15, 16, 17 
\enddata
\end{deluxetable}

\clearpage













%\begin{deluxetable}{lrrrrr}
%\tablewidth{0pt}
%\tablecaption{\label{resdetails}
%$L_{s}$, periapsis latitude, periapsis LST, and
%beginning and ending periapsis number for the 3:1 -- 8:1 resonances}
%\tablehead{
%\colhead{Resonance} &
%\colhead{$L_{s}$ (degrees)} &
%\colhead{Latitude ($^\circ$N)} &
%\colhead{LST (hrs)} &
%\colhead{Beginning Periapsis} &
%\colhead{Ending Periapsis}
%}
%\startdata
%3:1 & 48 & 45 & 16 & 645 & 656 \nl
%4:1 & 57 & 32 & 16 & 710 & 725 \nl
%5:1 & 64 & 16 & 15 & 784 & 803 \nl
%6:1 & 71 & $-4$ & 15 & 864 & 901 \nl
%7:1 & 78 & $-28$ & 15 & 963 & 989 \nl
%8:1 & 82 & $-45$ & 15 & 1030 & 1057
%\enddata
%\end{deluxetable}

\clearpage



\underline{Figure Captions}

Figure~\ref{preavarray}: Pre-entry values of $a_{bias,hi,y}$ 
for each orbit that has high-rate data.

Figure~\ref{postavarray}: Post-exit values of $a_{bias,hi,y}$ 
for each orbit that has high-rate data.

Figure~\ref{maxaaeroy}: Maximum values of $a_{aero,hi,y}$ 
(crosses) for each orbit that has high-rate data and 
$a_{aero,lo,y}$ (diamonds) for each orbit that has low-rate 
data only.

Figure~\ref{accnoise}: Noise in acceleration measurements. 
Noise in $a_{aero,hi,y}$ (black crosses), $a_{aero,hi,y,7}$ 
(black triangles) and $a_{aero,hi,y,39}$ (black squares) shown 
for each orbit that has high-rate data. Noise in $a_{aero,lo,y}$ 
(grey crosses), $a_{aero,lo,y,7}$ (grey triangles) and 
$a_{aero,lo,y,39}$ (grey squares) shown for each orbit that 
has low-rate data only. The effects of using different sampling 
rates for orbits 1--136, 137--268 and 269--330 can be seen.

Figure~\ref{p076avst}: Time series of $a_{aero,hi,y}$, 
$a_{aero,hi,y,7}$ and $a_{aero,hi,y,39}$ for orbit P076. Plotted 
values of $a_{aero,hi,y,7}$ ($a_{aero,hi,y,39}$) have been 
increased from actual values by 0.01 (0.02) m s$^{-2}$ for 
clarity.

Figure~\ref{p076avstwithsigmas}: Time series of $a_{aero,hi,y}$, 
$a_{aero,hi,y,7}$ and $a_{aero,hi,y,39}$ for orbit P076. The line 
with the largest maximum acceleration is $100\times a_{aero,hi,y,39}$, 
the line with the second-largest maximum acceleration is 
$10\times a_{aero,hi,y,7}$ and line with the third-largest 
maximum acceleration is $1\times a_{aero,hi,y}$. The remaining 
line, whose value is $2 \times 10^{-4}$ m s$^{-2}$ at $\pm$150s, 
is the uncertainty in $a_{aero,hi,y}$. For orbits prior to P137, 
such as this one, the thruster limit of $2 \times 10^{-4}$ m s$^{-2}$ 
is greater than the noise level in $a_{aero,hi,y}$, 
$a_{aero,hi,y,7}$ and $a_{aero,hi,y,39}$, so uncertainties in 
$a_{aero,hi,y}$, $a_{aero,hi,y,7}$ and $a_{aero,hi,y,39}$ are 
identical.
Note how $a_{aero,hi,y,39}$ covers a larger time interval than 
$a_{aero,hi,y,7}$, which in turn covers a larger time interval 
than $a_{aero,hi,y}$. This is due to the reduction in noise by 
averaging.

Figure~\ref{odymassfig}: Mass of ODY during aerobraking.

Figure~\ref{rhocy}: $\rho C_{y}$ as a function of $\rho$ for 
$u_{x}=0$ and $u_{z}=0$. Values from the aerodynamic database 
are marked by crosses and joined by dashed lines. The solid line 
indicates $\rho C_{y} = \rho$, or $C_{y}=1$. 
Since $\rho C_{y}$ is a single-valued function of $\rho$, this 
figure can be used to find $\rho$ from an observed value of 
$\rho C_{y}$. 
As shown in Figure~\ref{rhovscy}, $C_{y} \approx 2$. 

Figure~\ref{rhovscy}: $C_{y}$ as a function of $\rho$ for 
$u_{x}=0$ and $u_{z}=0$. Values from the aerodynamic database 
are marked by crosses and joined by dashed lines.

Figure~\ref{p076uxvst}: $u_{x}$ as a function of time after 
periapsis for orbit P076.

Figure~\ref{p076uzvst}: $u_{z}$ as a function of time after 
periapsis for orbit P076.

Figure~\ref{p076uxvsuz}: $u_{x}$ and $u_{z}$ for orbit P076. 
The dashed (solid) line indicates time before (after) periapsis. 
Values are plotted for times within 200 seconds of periapsis. 
Crosses mark values of $u_{x}$ and $u_{z}$ at -200s, -100s, 0s, 
100s and 200s.

Figure~\ref{p076phivst}: $\phi$ as a function of time after 
periapsis for orbit P076.

Figure~\ref{p076thetavst}: $\theta$ as a function of time 
after periapsis for orbit P076.

Figure~\ref{p076cyvst}: $C_{y}$ as a function of time after 
periapsis for orbit P076.
Values of $C_{y}$ for $\rho_{hi,39}$, $\rho_{hi,7}$ and 
$\rho_{hi}$ at a given time are similar, but not identical. 
The top line shows $C_{y} + 0.2$ for $\rho_{hi,39}$. 
The middle line shows $C_{y} + 0.1$ for $\rho_{hi,7}$. 
The bottom line shows $C_{y}$ for $\rho_{hi}$. 

Figure~\ref{p076rhovstlin}: Density on a linear scale as a 
function of time after periapsis for orbit P076.
The top line shows $\rho_{hi,39} + 10$ kg km$^{-3}$. 
The middle line shows $\rho_{hi,7} + 5$ kg km$^{-3}$. 
The bottom line shows $\rho_{hi}$. 
Note how $\rho_{hi,39}$ covers a larger time interval than 
$\rho_{hi,7}$, which in turn covers a larger time interval 
than $\rho_{hi}$. This is due to the reduction in noise by 
averaging.

Figure~\ref{p076rhovstlog}: Density on a logarithmic scale 
as a function of time after periapsis for orbit P076.
The top line shows $\rho_{hi,39} \times 100$. 
The next line shows $\rho_{hi,7} \times 10$. 
The next line shows $\rho_{hi}$. 
The bottom line shows the uncertainty in $\rho_{hi,39}$. 
Uncertainties in $\rho_{hi,7}$ and $\rho_{hi}$ are very 
similar, but slightly smoother.

Figure~\ref{dpperivsorb}: $2 \times \left(p_{in} - p_{out}\right) /  
\left(p_{in} + p_{out}\right)$ versus orbit number.

Figure~\ref{dpperivsplat}: $2 \times \left(p_{in} - p_{out}\right) /  
\left(p_{in} + p_{out}\right)$ versus periapsis latitude. Dayside 
(nightside) datapoints are indicated by + (X).

Figure~\ref{p076pvsz}: Pressure profiles (solid lines) for orbit 
P076. Pressures at a given altitude are greater on the outbound 
(equatorward) leg than on the inbound (poleward) leg. $1\sigma$ 
uncertainties are marked by the dashed lines. The horizontal 
solid line joining the inbound and outbound legs indicates 
the pressure discontinuity at periapsis.

Figure~\ref{p076tempvsz}: Temperature profiles (solid lines) 
for orbit P076. Temperatures around 110 km are greater on the 
outbound (equatorward) leg than on the inbound (poleward) 
leg. $1\sigma$ uncertainties are marked by the dashed lines. 
The horizontal solid line joining the inbound and outbound 
legs indicates the temperature discontinuity at periapsis.

Figure~\ref{p076tempvspress}: Temperature versus pressures 
for orbit P076.
Temperatures at a given pressure are greater on the outbound 
(equatorward) leg than on the inbound (poleward) leg.

Figure~\ref{deltatvsorb}: Difference in periapsis time between 
the results of this work and the aerobraking team's results 
as a function of orbit number.

Figure~\ref{deltalatvsorb}: As Figure~\ref{deltatvsorb}, but 
for periapsis latitude.

Figure~\ref{deltazvsorb}: As Figure~\ref{deltatvsorb}, but 
for periapsis altitude ($z$).

Figure~\ref{deltalonvsorb}: As Figure~\ref{deltatvsorb}, but 
for periapsis longitude.

Figure~\ref{deltalstvsorb}: As Figure~\ref{deltatvsorb}, but 
for periapsis LST.

Figure~\ref{deltaszavsorb}: As Figure~\ref{deltatvsorb}, but 
for periapsis solar zenith angle (SZA).

Figure~\ref{deltalatvsplat}: Difference in periapsis latitude 
between the results of this work and the aerobraking team's 
results as a function of this work's periapsis latitude. Both 
dayside and nightside differences are shown for periapsis 
latitudes poleward of about 67$^{\circ}$N. Differences in 
periapsis latitude at a given periapsis latitude are only 
slightly dependent on whether datapoints come from the 
dayside or nightside.

Figure~\ref{deltazvsplat}: Difference in periapsis altitude 
between the results of this work and the aerobraking team's 
results as a function of this work's periapsis latitude. 
Both dayside and nightside differences are shown for periapsis 
latitudes poleward of about 67$^{\circ}$N. Differences in 
periapsis altitude at a given periapsis latitude are independent 
of whether datapoints come from the dayside or nightside.

Figure~\ref{adjdeltatvsplat}: Difference in periapsis time 
between the results of this work and the aerobraking team's 
results as a function of this work's periapsis latitude 
(black symbols). Results from dayside periapses are multiplied 
by $-1$ and plotted again (grey symbols). The unadjusted 
nightside differences and the adjusted dayside differences are 
very similar at any given periapsis latitude.

Figure~\ref{adjdeltalonvsplat}: As Figure~\ref{adjdeltatvsplat}, 
but for periapsis longitude.

Figure~\ref{adjdeltalstvsplat}: As Figure~\ref{adjdeltatvsplat}, 
but for periapsis LST.

Figure~\ref{adjdeltaszavsplat}: As Figure~\ref{adjdeltatvsplat}, 
but for periapsis solar zenith angle (SZA).

Figure~\ref{redefdeltalatvsplat}: Difference in redefined 
periapsis latitude as a function of this work's periapsis latitude. 
Dayside and nightside differences poleward of about 
67$^{\circ}$N appear indistinguishable. Differences in this 
Figure are much smaller than differences in 
Figure~\ref{deltalatvsplat}.

Figure~\ref{redefdeltazvsplat}: As Figure~\ref{adjdeltatvsplat}, 
but for periapsis altitude. Differences in this Figure are much 
smaller than differences in Figure~\ref{deltazvsplat}.

Figure~\ref{qplot}: $Q$ as a function of periapsis latitude for 
our results (black crosses) and the aerobraking team's 
results (grey diamonds).

Figure~\ref{reldiffrhoin110}: Relative difference in inbound 
density at 110 km as a function of orbit number.

Figure~\ref{reldiffrhoout110}: As Figure~\ref{reldiffrhoin110}, 
but for outbound density at 110 km.

Figure~\ref{reldiffrhoin120}: As Figure~\ref{reldiffrhoin110}, 
but for inbound density at 120 km.

Figure~\ref{reldiffrhoout120}: As Figure~\ref{reldiffrhoin110}, 
but for outbound density at 120 km.

Figure~\ref{reldiffdshin110}: As Figure~\ref{reldiffrhoin110}, 
but for inbound density scale height at 110 km.

Figure~\ref{reldiffdshout110}: As Figure~\ref{reldiffrhoin110}, 
but for outbound density scale height at 110 km.

Figure~\ref{reldiffdshin120}: As Figure~\ref{reldiffrhoin110}, 
but for inbound density scale height at 120 km.

Figure~\ref{reldiffdshout120}: As Figure~\ref{reldiffrhoin110}, 
but for outbound density scale height at 120 km.

Figure~\ref{p122dshnotinsane}: Density profile ($\rho_{hi,39}$) 
for orbit P122 (dashed line). Black solid line shows extrapolation
based on fitted outbound density and density scale height at 110 km 
using our results.
Grey solid line shows extrapolation based on fitted outbound density 
and density scale height at 110 km using the aerobraking team's 
results.

Figure~\ref{demodshworks}: Density profile ($\rho_{hi,39}$) for 
orbit P081 (dashed line). Black solid line shows extrapolation 
based on fitted inbound density and density scale height at 110 km 
using our results.
Grey solid line shows extrapolation based on fitted inbound 
density and density scale height at 110 km using the aerobraking 
team's results.

Figure~\ref{wavyexamplefig}: Density profiles from orbit P011. 
Both $\rho_{hi,7}$ (solid line) and $\rho_{hi,39}$ (dashed line) 
are shown.

Figure~\ref{wavinessfig}: Measure of small-scale structure, $X$, 
for dayside (squares) and nightside (triangles). Mean values for 
5$^{\circ}$ latitude bins are plotted.

Figure~\ref{orb56r7vst}: Profiles of $\rho_{hi,7}$ from orbit 
P056. Each of the measurements, which are separated by 1 second, 
is marked by a cross.

Figure~\ref{nightdshgrid}: Average nightside density scale 
heights at 110 km. 
All nightside density scale heights (inbound and outbound) at 
110 km within a 5$^{\circ}$ latitude by 30$^{\circ}$ longitude 
region were identified. The mean scale height (km) is indicated 
by the black number. The number of scale heights from which 
this average was calculated is indicated by the grey number. 
Latitude-longitude regions for which no scale heights are 
available were left blank.

Figure~\ref{orb011latvst}: Latitude of ground track near periapsis 
for orbit P011.

Figure~\ref{orb011rhovst}: Time series of density measurements 
from orbit P011. Both $\rho_{hi,7}$ (solid line) and $\rho_{hi,39}$ 
(dashed line) are shown.

Figure~\ref{orb011rhovsz}: Density profiles from orbit P011. 
Both $\rho_{hi,7}$ (solid line) and $\rho_{hi,39}$ (dashed line) 
are shown.

Figure~\ref{orb012rhovst}: As Figure~\ref{orb011rhovst}, but 
for orbit P012.

Figure~\ref{day60n70nzonal}: All outbound dayside density 
measurements at 130 km between 60$^{\circ}$N and 70$^{\circ}$N 
(crosses), wave-4 harmonic model fit to the data (solid line) 
and 1$\sigma$ uncertainties about the fit (dashed lines).

Figure~\ref{manyzday60n70n}: As Figure~\ref{day60n70nzonal}, 
but different colours correspond to different altitudes. Black 
is 110 km, red is 120 km, green is 130 km and blue is 140 km. 
Data only available at 120--140 km.

Figure~\ref{manyznight60n70n}: As Figure~\ref{manyzday60n70n}, 
but for inbound nightside data. Data only available at 
110--130 km.

Figure~\ref{nightcontourinrho120}: Contour plot of normalized 
fitted densities at 120 km for outbound dayside data. Contour 
intervals are 0.2 (dimensionless) and negative regions 
(low densities) are shaded.

Figure~\ref{nightalllatphases}: Phases from the wave-4 harmonic 
fits used in Figure~\ref{nightcontourinrho120}. Results are 
indicated by asterisks for wave-1, crosses for wave-2, 
triangles for wave-3 and squares for wave-4. 
Vertical lines are 1$\sigma$ error bars.
Symbols are slightly offset from the centre of their 
latitude band for clarity.
Recall that the maximum value of the phase of the 
wave-$n$ harmonic is 360$^{\circ}$E/$n$.

Figure~\ref{nightalllatamps}: Normalized amplitudes 
corresponding to the phases in Figure~\ref{nightalllatphases}. 
Symbols as Figure~\ref{nightalllatphases}.

Figure~\ref{simdayzonal}: Dayside (LST = 16.44--16.68 hours) 
density measurements at 130 km between 70$^{\circ}$N and 
80$^{\circ}$N. Symbols as Figure~\ref{day60n70nzonal}.

Figure~\ref{simnightzonal}: Nightside (LST = 2.25--2.67 hours) 
density measurements at 130 km between 70$^{\circ}$N and 
80$^{\circ}$N. Symbols as Figure~\ref{day60n70nzonal}.

Figure~\ref{mgsp1out130orb130160}: Outbound densities at 
130 km from orbits 130--160 of MGS Phase 1. Symbols as 
Figure~\ref{day60n70nzonal}.

Figure~\ref{mgsp1out130relamps}: Relative amplitudes of 
harmonic components from fits to outbound densities from 
MGS Phase 1. 
Fits were performed to densities from orbits 70--100, 
80--110, 90--120, $\ldots$, 170--200. Results are plotted 
near the centre of their corresponding orbit interval, 
different components are offset by slightly different amounts 
to minimize overlap.
Symbols are asterisks for wave-1, crosses for wave-2, 
triangles for wave-3 and squares for wave-4. Vertical 
lines are 1$\sigma$ error bars.

Figure~\ref{mgsp1out130phases}: Phases corresponding to 
the relative amplitudes in Figure~\ref{mgsp1out130relamps}. 
Symbols as Figure~\ref{mgsp1out130relamps}.

Figure~\ref{bougherrhoday}: Dayside MTGCM (crosses) and 
ODY (small filled circles) densities.
MTGCM densities at X$^{\circ}$ latitude and 
100 (110, 120, 130, 140) km  are plotted at 
X$^{\circ}$-2$^{\circ}$ (X$^{\circ}$-1$^{\circ}$, X$^{\circ}$, 
X$^{\circ}$+1$^{\circ}$, X$^{\circ}$+2$^{\circ}$). 
This minimizes the overlap of MTGCM data points. Colours 
indicate altitude. 
For MTGCM densities, dark blue crosses correspond to 100 km 
and 130 km, with large densities at 100 km clearly 
distinguishable from small densities at 130 km. Similarly, 
black crosses correspond to 110 km and 140 km. Pink crosses 
correspond to 120 km. 
For ODY densities, small filled green circles correspond to 
100 km and 130 km. Small filled red circles correspond to 
110 km and 140 km. Small filled light blue circles 
correspond to 120 km.

Figure~\ref{bougherrhonight}: As Figure~\ref{bougherrhoday}, 
but nightside densities. Data and simulations are shown at 
110, 120 and 130 km.

Figure~\ref{bougherdshday}: Dayside MTGCM (crosses) and ODY 
(small and large filled circles) scale heights. Data and 
simulations are shown at 110, 120 and 130 km.
MTGCM data points are offset in latitude and colours indicate 
altitude as in Figure~\ref{bougherrhonight}. Small filled 
circles, which exhibit significant scatter, are individual 
ODY data points as in Figure~\ref{bougherrhonight}. Large 
filled circles indicate the mean value of all ODY data points 
from a given altitude within a ten degree latitude band. 
Altitude is indicated by the colours of large filled circles 
as it is for small filled circles.

Figure~\ref{bougherdshnight}: As Figure~\ref{bougherdshday}, 
but nightside scale heights.

Figure~\ref{avrho}: Mean dayside (grey) and nightside (black) 
densities. Symbols indicate altitude at 100 km (square), 
110 km (X), 120 km (triangle), 130 km (+) and 140 km (diamond). 
All inbound and outbound data points within a five degree 
latitude interval were averaged. Data points from different 
altitudes are offset slightly in latitude to minimize overlap.

Figure~\ref{avdsh}: As Figure~\ref{avrho}, but scale height.

\clearpage

\begin{figure}
\plotone{REPORTFIG01.EPS}
\caption{\label{preavarray}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG02.EPS}
\caption{\label{postavarray}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG03.EPS}
\caption{\label{maxaaeroy}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG04.EPS}
\caption{\label{accnoise}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG05.EPS}
\caption{\label{p076avst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG06.EPS}
\caption{\label{p076avstwithsigmas}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG07.EPS}
\caption{\label{odymassfig}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG08.EPS}
\caption{\label{rhocy}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG09.EPS}
\caption{\label{rhovscy}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG10.EPS}
\caption{\label{p076uxvst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG11.EPS}
\caption{\label{p076uzvst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG12.EPS}
\caption{\label{p076uxvsuz}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG13.EPS}
\caption{\label{p076phivst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG14.EPS}
\caption{\label{p076thetavst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG15.EPS}
\caption{\label{p076cyvst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG16.EPS}
\caption{\label{p076rhovstlin}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG17.EPS}
\caption{\label{p076rhovstlog}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG18.EPS}
\caption{\label{dpperivsorb}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG19.EPS}
\caption{\label{dpperivsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG20.EPS}
\caption{\label{p076pvsz}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG21.EPS}
\caption{\label{p076tempvsz}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG22.EPS}
\caption{\label{p076tempvspress}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG23.EPS}
\caption{\label{deltatvsorb}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG24.EPS}
\caption{\label{deltalatvsorb}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG25.EPS}
\caption{\label{deltazvsorb}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG26.EPS}
\caption{\label{deltalonvsorb}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG27.EPS}
\caption{\label{deltalstvsorb}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG28.EPS}
\caption{\label{deltaszavsorb}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG29.EPS}
\caption{\label{deltalatvsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG30.EPS}
\caption{\label{deltazvsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG31.EPS}
\caption{\label{adjdeltatvsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG32.EPS}
\caption{\label{adjdeltalonvsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG33.EPS}
\caption{\label{adjdeltalstvsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG34.EPS}
\caption{\label{adjdeltaszavsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG35.EPS}
\caption{\label{redefdeltalatvsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG36.EPS}
\caption{\label{redefdeltazvsplat}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG37.EPS}
\caption{\label{qplot}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG38.EPS}
\caption{\label{reldiffrhoin110}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG39.EPS}
\caption{\label{reldiffrhoout110}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG40.EPS}
\caption{\label{reldiffrhoin120}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG41.EPS}
\caption{\label{reldiffrhoout120}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG42.EPS}
\caption{\label{reldiffdshin110}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG43.EPS}
\caption{\label{reldiffdshout110}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG44.EPS}
\caption{\label{reldiffdshin120}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG45.EPS}
\caption{\label{reldiffdshout120}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG46.EPS}
\caption{\label{p122dshnotinsane}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG47.EPS}
\caption{\label{demodshworks}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG48.EPS}
\caption{\label{wavyexamplefig}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG49.EPS}
\caption{\label{wavinessfig}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG50.EPS}
\caption{\label{orb56r7vst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG51.EPS}
\caption{\label{nightdshgrid}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG52.EPS}
\caption{\label{orb011latvst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG53.EPS}
\caption{\label{orb011rhovst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG54.EPS}
\caption{\label{orb011rhovsz}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG55.EPS}
\caption{\label{orb012rhovst}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG56.EPS}
\caption{\label{day60n70nzonal}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG57.EPS}
\caption{\label{manyzday60n70n}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG58.EPS}
\caption{\label{manyznight60n70n}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG59.EPS}
\caption{\label{nightcontourinrho120}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG60.EPS}
\caption{\label{nightalllatphases}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG61.EPS}
\caption{\label{nightalllatamps}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG62.EPS}
\caption{\label{simdayzonal}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG63.EPS}
\caption{\label{simnightzonal}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG64.EPS}
\caption{\label{mgsp1out130orb130160}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG65.EPS}
\caption{\label{mgsp1out130relamps}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG66.EPS}
\caption{\label{mgsp1out130phases}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG67.EPS}
\caption{\label{bougherrhoday}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG68.EPS}
\caption{\label{bougherrhonight}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG69.EPS}
\caption{\label{bougherdshday}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG70.EPS}
\caption{\label{bougherdshnight}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG71.EPS}
\caption{\label{avrho}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\clearpage

\begin{figure}
\plotone{REPORTFIG72.EPS}
\caption{\label{avdsh}\label{lastfig}\label{lastpage}
Author Paul Withers --- Odyssey Accelerometer Data}
\end{figure}

\newpage

\clearpage

\end{document}
