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\begin{document}                                                              
                                                                              
\title{Production of Reduced Data Records for the                             
Phoenix Atmospheric Structure Experiment}                                     
                                                                              
\author{Paul Withers$^{a}$, D.~C. Catling$^{b}$}                              
                                                                              
\affil{$^{a}$ Center for Space Physics, Boston University,                    
725 Commonwealth Avenue, Boston, MA 02215, USA.}                              
                                                                              
\affil{$^{b}$                                                                 
Department of Earth and Space Sciences, University of Washington,             
Box 351310, Seattle, WA 98195-1310, USA.}                                     
                                                                              
\pagebreak                                                                    
                                                                              
\begin{abstract}                                                              
The purpose of this report is to describe the methodology used to produce     
Reduced Data Records (RDRs) for the Phoenix Atmospheric Structure             
Experiment (ASE) from its Experimental Data Records (EDRs). These RDRs        
include vertical profiles of atmospheric density, pressure, and temperature.  
                                                                              
During its brief flight through the atmosphere of Mars, Phoenix recorded      
acceleration and angular velocity data using accelerometers and gyroscopes    
within an inertial measurement unit. These time series data, which            
constitute the EDRs, are archived at the NASA Planetary Data System (PDS).    
The archived EDRs require correction because of errors associated with the    
use of a preliminary version of a transformation matrix between two           
reference frames. The EDRs, which were acquired at a high rate of 200 Hz,     
are noisy, but this can be improved by smoothing at the cost of worsened      
vertical resolution. Smoothing of the axial acceleration requires care. We    
have described how the running arithmetic mean of the axial acceleration is   
a biased estimate of the true acceleration and explained how this bias can    
be corrected.                                                                 
                                                                              
The accelerations and angular velocities (EDRs) have been used to             
reconstruct the trajectory of Phoenix through the atmosphere and the          
associated atmospheric structure (RDRs). The position and velocity of         
Phoenix were reconstructed by forward integration of the equations of motion  
and its orientation history was determined directly using onboard angular     
velocity measurements. The density profile along the trajectory was           
reconstructed from $\sim$130 km to parachute deployment, $\sim$13.5 km        
above the surface, using the drag equation. The corresponding pressure        
profile was obtained from the equation of hydrostatic equilibrium and the     
corresponding temperature profile was obtained from the ideal gas law.        
                                                                              
The angle of attack of Phoenix during atmospheric flight was determined       
directly using angular velocity measurements and indirectly using the         
ratio of normal to axial accelerations and the spacecraft's aerodynamic       
database. The former was typically 1--2 degrees greater than the latter,      
a discrepancy for which we do not have a good explanation.                    
\end{abstract}                                                                
                                                                              
\pagebreak                                                                    
                                                                              
\section{\label{intro}Introduction}                                           
                                                                              
Data from the entry, descent, and landing (EDL) of the Phoenix spacecraft,    
have been used to obtain a profile of martian atmospheric density,            
pressure, and temperature from $\sim$130 km to $\sim$13.5 km above the        
surface. The thermal structure of the martian atmosphere is sensitive to      
radiative forcing from suspended dust and to diabatic heating associated      
with atmospheric dynamics \citep{zurek1992,leovy2001}. It is also perturbed   
by a wide variety of waves and tides                                          
\citep{leovy1979,banfield2000,withers2003b}.                                  
This is the first such profile of martian atmospheric structure from the      
polar regions. The atmospheric processes that can be observed in such         
profiles were discussed by \citet{magalhaes1999}, who also compared the       
advantages and disadvantages of this measurement technique to those of        
other techniques.                                                             
                                                                              
NASA's Mars Scout program selected the Phoenix mission for flight in 2003.    
Its ``science mission focuses on providing the ground truth for the 2002      
Odyssey discovery of massive ice deposits hidden under surface soils in       
the circumpolar regions'' \citep{smith2008}. The objectives of this mission   
were: (1) to study the history of the ground-ice and its emplacement          
mechanisms, (2) to address the effect that subsurface ice has on the local    
surface geomorphology, (3) to characterize the climate and local weather of   
the landing site, and (4) to address the habitability of the icy soil         
\citep{smith2008, arvidson2009, smith2009}.                                   
Much of the design of the Phoenix spacecraft, including those aspects         
relevant for cruise and EDL, derived from the mothballed ``Mars Surveyor      
2001 Lander'' \citep{guinn2008, desai2008, grover2008}. Phoenix launched on   
4 August 2007 and successfully landed on Mars in the Vastitas Borealis or     
northern plains region on 25 May 2008. The time and position of landing are   
reported in Table~\ref{landedcoords}.                                         
\nl                                                                           
[TABLE~\ref{landedcoords}]                                                    
                                                                              
Phoenix carried two Honeywell inertial measurement units (IMUs, model number  
YG9666BC) \citep{taylor2008}. IMU-A recorded data during EDL; IMU-B, flown    
as a backup, was never turned on after launch and is not discussed further.   
Each IMU consisted of three identical single-axis Allied Signal Q-Flex        
accelerometers and three identical Honeywell GG1320 RLG ring-laser            
gyroscopes. These were mounted orthogonally. Accelerometers measure           
acceleration and gyroscopes measure angular velocity, but the IMU was         
configured to output time-integrated quantities, specifically velocity        
change and angle change, every 5 ms. These measurements, which constitute     
the Experimental Data Records (EDRs) of the Atmospheric Structure Experiment  
(ASE), were archived by the NASA Planetary Data System (PDS) with associated  
documentation \citep{catling2008}. \citet{desai2008} and \citet{grover2008}   
also provide valuable background information on the IMU, spacecraft and EDL   
sequence. Analyses of the IMU data for engineering purposes have been         
undertaken \citep{desai2008, blanchard2009}. The purpose of this report is    
to describe and discuss a reconstruction of the Phoenix EDL trajectory and    
the associated atmospheric structure that has produced a set of Reduced       
Data Records (RDRs) for the Phoenix ASE investigation.                        
                                                                              
Section~\ref{trajreconintro} introduces the principles of trajectory          
reconstruction, Section~\ref{dataprep} describes how the Phoenix EDR data     
were processed, Section~\ref{attitude} describes how the attitude of          
Phoenix was determined, Section~\ref{entrystate} describes the state of       
Phoenix at atmospheric entry, Section~\ref{doingtrajrecon} describes the      
trajectory reconstruction for Phoenix, Section~\ref{atmrecon} describes the   
atmospheric structure reconstruction for Phoenix, Section~\ref{consistency}   
describes some consistency checks on this reconstruction,                     
Section~\ref{future} describes possible future work, and Section~\ref{concs}  
summarizes the main conclusions of this work. Appendix~\ref{frameapp}         
describes how errors in the EDR data were corrected and                       
Appendix~\ref{smoothapp} describes how noise in the axial accelerations was   
reduced by smoothing.                                                         
                                                                              
\section{\label{trajreconintro}Introduction to Trajectory Reconstruction}     
                                                                              
The trajectory reconstruction process used for Phoenix was similar to those   
applied to earlier missions                                                   
\citep[][and references therein]{withers2006c, withers2003}.                  
First, clean time series of angular velocities and accelerations at the       
centre of mass in a suitable frame were produced. Second, spacecraft          
orientation, position and velocity at the time of atmospheric entry were      
obtained. Third, the equations of motion were integrated forward in time      
from entry using the initial conditions, angular velocities, accelerations    
and spatially-varying gravitational field to determine spacecraft             
orientation, position and velocity as functions of time. The existence of     
reliable angular velocity data from the Phoenix gyroscopes enabled direct     
determination of spacecraft orientation, unlike the indirect methods used     
on some earlier spacecraft \citep{spencer1999}.                               
                                                                              
\section{\label{dataprep}Preparation of Data}                                 
                                                                              
Characteristics of the IMU and its measurements are summarized in file        
CATALOG/INST.CAT of \cite{catling2008}. The data used in this report come     
from the archived EDRs; specifically, the measured velocity increments and    
angle increments reported in file DATA/IMU\_A\_EDR\_M.TAB of                  
\cite{catling2008}. Zeroes were reported for all IMU data at samples 12031    
to 12040, a duration of 50 ms, due to a downlink transmission problem. This   
data gap was filled by interpolation in this report. These data were          
converted into accelerations and angular velocities by dividing by the        
relevant time interval, 5 ms.                                                 
                                                                              
Many different coordinate frames were associated with Phoenix, as discussed   
in file DOCUMENT/EDRSIS.PDF of \citet{catling2008}, \citet{blanchard2009},    
and the Phoenix frame kernel distributed by NAIF (JPL's Navigation and        
Ancillary Information Facility) in SPICE format                               
(ftp://naif.jpl.nasa.gov/pub/naif/PHOENIX/kernels/fk/phx\_v06.tf).            
The archived data were transformed into the desired frame, the ``C frame'',   
as described in Appendix~\ref{frameapp}. This frame, also called the          
``Phoenix cruise frame'', is defined in Appendix~\ref{frameapp}.              
Appendix~\ref{frameapp} also discusses a small, but significant, correction   
to the EDR accelerations and angular velocities. The symmetry axis of the     
spacecraft, along which the largest accelerations were experienced, is the    
$\pm X$-axis in the ``C frame''.                                              
                                                                              
The 5 ms interval between successive raw data points corresponds at high      
elevations to a change in elevation of 6 m, far smaller than any relevant     
atmospheric lengthscale. Accordingly, noise in the data was reduced by        
averaging. Accelerations along the $Y$ and $Z$-axes of the ``C frame'' were   
smoothed with a 256-point running mean. These axes are normal to the          
spacecraft's symmetry axis. The same uncertainty, 0.0025 ms$^{-2}$, was       
assigned to each of these smoothed data points. This value corresponds to     
the standard deviation of the smoothed normal accelerations at high           
elevation. All uses of uncertainties described in this report were based      
on Monte Carlo techniques assuming normal distributions.                      
                                                                              
Accelerations along the $X$-axis of the ``C frame'', the spacecraft's         
symmetry axis, were smoothed using a more complicated algorithm. The guiding  
principles behind this algorithm were (A) the need for balance between        
reduction in noise and acceptable vertical resolution and (B) consideration   
of the appropriateness of an unweighted arithmetic mean. In theory, the       
axial accelerations at high elevations should increase exponentially with     
time \citep[e.g. Figure 1 of][]{magalhaes1999} due to the descent at          
constant speed into an exponentially-varying atmosphere that occurs prior     
to peak deceleration. In such circumstances, the arithmetic mean of a         
subset of successive accelerations is not the best possible estimate of       
the true acceleration at the centre of this interval. The axial               
accelerations were smoothed as described in Appendix~\ref{smoothapp}.         
                                                                              
The standard deviation of axial accelerations recorded at high elevation,     
prior to the onset of detectable aerodynamic drag, is a measure of the        
uncertainty or error in the acceleration data. This standard deviation        
depends on the size of the smoothing window. Analysis of these data           
(not shown) indicated that the dependence of the standard deviation,          
$\sigma_{aa}$, on the number of points in the smoothing window, $N$,          
satisfies:                                                                    
                                                                              
\begin{eqnarray}                                                              
\label{saaeqn}                                                                
\sigma_{aa} = \frac{a_{0}}{N}                                                 
\end{eqnarray}                                                                
                                                                              
where $a_{0}$ was found to equal $0.3$ ms$^{-2}$. Although this value of      
$a_{0}$ is not derived directly from the instrumental digitization, it is     
similar to the ratio of the digitization limit of 2.7 mm s$^{-1}$ to the      
5 ms interval between successive data points ($0.54$ ms$^{-2}$). Note that    
the $1/N$ dependence in Equation~\ref{saaeqn} differs from the $1/\sqrt{N}$   
trend expected for values sampled from a normal distribution. The             
uncertainties in the smoothed axial accelerations were assumed to equal       
the relevant standard deviation, so the uncertainty in any given              
measurement is determined by the smoothing window used. Variations in the     
width of the window used to smooth the axial accelerations, which are         
outlined in Appendix~\ref{smoothapp}, are shown in                            
Figure~\ref{nusedinsecvstfig}.                                                
\nl                                                                           
[FIGURE~\ref{aaxialfig}]\nl                                                   
[FIGURE~\ref{nusedinsecvstfig}]                                               
                                                                              
At high elevations, the unsmoothed angular velocities had a standard          
deviation of $10^{-3}$ rad s$^{-1}$. This was adopted as the uncertainty      
of the unsmoothed angular velocities. Angular velocities about the $X$,       
$Y$ and $Z$-axes of the ``C frame'' were then smoothed with a 101-point       
running mean.                                                                 
                                                                              
Measured accelerations are valid at the location of the IMU, whereas the      
reconstruction process requires accelerations at the location of the centre   
of mass (CM). Angular corrections were investigated using the positional      
information reported in the file DOCUMENT/EDRSIS.PDF of \citet{catling2008},  
but neglected because their effects were negligible below 100 km. The         
appropriate correction is $\underline{\dot{\omega}} \times \underline{r}      
+ \underline{\omega} \times \left(\underline{\omega} \times \underline{r}     
\right)$, where $\underline{\omega}$ is the angular velocity of the           
spacecraft as measured by the IMU gyroscopes, $\underline{r}$ is the          
separation of the IMU from the centre of mass, and a dot indicates a          
derivative with respect to time \citep{tolson1999}. The typical magnitude     
of $\underline{\omega}$ was $10^{-2}$ rad s$^{-1}$, the typical magnitude     
of $\underline{\dot{\omega}}$ was $10^{-2}$ rad s$^{-2}$, and                 
$\left|\underline{r}\right|$ was $\sim$1 m. Hence the two angular correction  
terms were $10^{-2}$ ms$^{-2}$ and $10^{-4}$ ms$^{-2}$, respectively.         
Comparison with Figure~\ref{aaxialfig} shows that these corrections are       
small. Finally, data points prior to the time of entry were discarded.        
This completed the preparation of the IMU data.                               
                                                                              
\section{\label{attitude}Attitude Determination}                              
                                                                              
The aerodynamic accelerations were measured in a frame defined relative to    
the Phoenix spacecraft, but the trajectory reconstruction process must use    
a frame that is defined relative to Mars. For example, knowledge of the       
aerodynamic accelerations in a spacecraft-fixed frame does not provide        
knowledge as to whether the aerodynamic acceleration is directed ``up''       
or ``down'' relative to the surface of Mars. The attitude of Phoenix          
must be known in a frame that can be related to Mars itself.                  
                                                                              
There are a multitude of possible ways to describe the orientation of a       
rigid body. In this report, we use quaternions to describe the relationship   
of the ``C frame'' to the ``J2000 frame'', which is defined by the Earth's    
mean equator and equinox at the J2000.0 epoch. We adopted the convention      
that the quaternion $\left[q_{0}, q_{1}, q_{2}, q_{3}\right]$ represents      
a rotation about the axis parallel to the vector $\underline{q_{v}} =         
\left[q_{1}, q_{2}, q_{3}\right]$ by angle $2 \cos^{-1} q_{0}$. The time      
dependence of the quaternion that describes the orientation of the            
``C frame'' relative to the ``J2000'' frame satisfies:                        
                                                                              
\begin{eqnarray}                                                              
\label{dq0dteqn}                                                              
\frac{dq_{0}}{dt} = \frac{1}{2} q_{0} \underline{\omega} -                    
\underline{q_{v}} \times \underline{\omega}                                   
\end{eqnarray}                                                                
                                                                              
\begin{eqnarray}                                                              
\label{dqvdteqn}                                                              
\frac{d\underline{q_{v}}}{dt} = \frac{-1}{2} \underline{q_{v}} .              
\underline{\omega}                                                            
\end{eqnarray}                                                                
                                                                              
where $\underline{\omega}$                                                    
is the angular velocity of the spacecraft as measured by the IMU gyroscopes.  
The vector aerodynamic acceleration in the ``J2000'' frame,                   
$\underline{a}_{J2000}$, can be derived from the vector aerodynamic           
acceleration in the ``C frame'', $\underline{a}_{C}$, as follows:             
                                                                              
\begin{eqnarray}                                                              
\label{quattransformeqn}                                                      
\underline{a}_{J2000} =                                                       
\left(                                                                        
\begin{array}{ccc}                                                            
1 - 2 \left(q_{2}^{2} + q_{3}^{2}\right) & % q11                              
2 \left(q_{2} q_{1} - q_{0} q_{3} \right) & % q21                             
2 \left(q_{3} q_{1} + q_{0} q_{3} \right) \\ % q31                            
2 \left(q_{1} q_{2} + q_{0} q_{3} \right) & % q12                             
1 - 2 \left(q_{1}^{2} + q_{3}^{2}\right) & % q22                              
2 \left(q_{3} q_{2} - q_{0} q_{1} \right) \\ % q32                            
2 \left(q_{1} q_{3} - q_{0} q_{2} \right) & % q13                             
2 \left(q_{2} q_{3} + q_{0} q_{1} \right) & % q23                             
1 - 2 \left(q_{1}^{2} + q_{2}^{2}\right) % q33                                
\end{array}                                                                   
\right)                                                                       
\underline{a}_{C}                                                             
\end{eqnarray}                                                                
                                                                              
\section{\label{entrystate}Entry State}                                       
                                                                              
Analysis of radio communications between Earth and Phoenix prior to entry     
were used to determine its trajectory prior to atmospheric entry. Since       
atmospheres do not have sharp edges, ``atmospheric entry'' is a slightly      
arbitrary concept. The Phoenix project adopted a distance of 3522.2 km from   
the centre of mass of Mars as its entry interface. The time, position and     
velocity at the entry interface are listed in Table~\ref{entrystatetable}     
with their uncertainties. For computational convenience, the uncertainty      
on entry time, formally 0.002 s, was set to zero and the uncertainty on       
entry radius, formally zero, was set to the radial distance travelled by      
Phoenix at the entry radius during 0.002 s, or about 2.5 m.                   
\nl                                                                           
[TABLE~\ref{entrystatetable}]                                                 
                                                                              
The orientation of Phoenix during its cruise to Mars was monitored and        
reconstructed by the spacecraft team. This was archived at the NAIF node      
of the PDS in the form of SPICE kernels                                       
(ftp://naif.jpl.nasa.gov/pub/naif/PHOENIX/kernels/ck/). The quaternion at     
the entry interface was obtained from the relevant SPICE kernels and is       
listed in Table~\ref{entryquattable}.                                         
\nl                                                                           
[TABLE~\ref{entryquattable}]\nl                                               
                                                                              
\section{\label{doingtrajrecon}Reconstructed Trajectory}                      
                                                                              
The equations of motion were integrated forward in time using methods         
described in previous papers and associated archives                          
\citep{withers2003, withers2006c} to obtain time series of position and       
velocity. The only significant difference was that the transformation of      
acceleration measurements from a spacecraft-fixed frame into a planet-fixed   
frame used a direct method enabled by the derived quaternions                 
(Section~\ref{attitude}), rather than an indirect method based upon           
acceleration ratios \citep{withers2003}.                                      
                                                                              
The reconstructed trajectory is shown in                                      
Figures~\ref{zvstfig}--\ref{vrelvszfig}. All ``altitudes'' used in this       
report are radial distances above 3376.3 km (Table~\ref{landedcoords}).       
Uncertainties in all the results reported in this work were obtained using    
Monte Carlo techniques (1000 reconstructions) and normally distributed        
uncertainties. The duration of the smoothing window used for the axial        
accelerations, which is shown in Figure~\ref{nusedinsecvstfig}, can be        
expressed as a vertical resolution if multiplied by the rate of change of     
altitude, as shown in Figure~\ref{nusedinkmvszfig}.                           
\nl                                                                           
[FIGURE~\ref{zvstfig}]\nl                                                     
[FIGURE~\ref{latvszfig}]\nl                                                   
[FIGURE~\ref{lonvszfig}]\nl                                                   
[FIGURE~\ref{vrelvszfig}]\nl                                                  
[FIGURE~\ref{nusedinkmvszfig}]                                                
                                                                              
The angle of attack, namely the angle between the spacecraft symmetry axis    
and the vector velocity of the atmosphere relative to the spacecraft, was     
also determined using the assumption that the atmosphere rotates with the     
solid body of the planet. This directly determined angle of attack is         
labelled $\alpha_{D}$. Results are shown in Figure~\ref{alphadzoomvstfig}.    
This angle can be determined directly if gyroscopic data are available, as    
they are for Phoenix, but must be determined indirectly if they are not       
\citep{spencer1999, withers2003, withers2006c}.                               
\nl                                                                           
[FIGURE~\ref{alphadzoomvstfig}]\nl                                            
                                                                              
The moment of parachute deployment is clearly visible in the time series of   
accelerations. The time and reconstructed position of parachute deployment    
are listed in Table~\ref{chutetable}. The moment of impact is also clearly    
visible in the time series of accelerations. The time and reconstructed       
position of the first ground contact are listed in                            
Table~\ref{impactstatetable}. Results are consistent with \citet{desai2008}   
and \citet{blanchard2009}, as well as the coordinates reported in             
Table~\ref{landedcoords}. In particular, the speed at first ground contact    
differs from zero by only 0.1\% of the entry speed, which indicates that the  
inferred orientation history of Phoenix is reliable. We do not address the    
near-surface orientation in this work, but it should be noted that a similar  
reconstruction by \citet{blanchard2009} found that the reconstructed          
orientation appeared unreliable near the surface where it varied rapidly,     
whereas the safe landing of Phoenix on its three legs implies that its        
orientation must have been relatively stable at this time. One possible       
explanation for this discrepancy is that near-surface winds, whose            
magnitude and direction are not known from observations, were not             
included in the reconstruction.                                               
\nl                                                                           
[TABLE~\ref{chutetable}]\nl                                                   
[TABLE~\ref{impactstatetable}]                                                
                                                                              
\section{\label{atmrecon}Reconstructed Atmospheric Structure}                 
                                                                              
Atmospheric density, pressure and temperature were determined from the        
reconstructed trajectory using the techniques established by previous         
missions \citep{withers2003, withers2006c}. The atmospheric reconstruction    
began at approximately 130 km altitude, some distance below the entry         
interface. Data close to the entry interface cannot be usefully smoothed      
because they are too close to the start of the data series, and so are        
not useful for the atmospheric reconstruction. According to the file          
CATALOG/INST.CAT of \citet{catling2008}, the nominal entry mass and           
reference area of Phoenix were 572.743 kg and $\pi D^{2} / 4$,                
respectively, where $D = 2.65$ m. Uncertainties on mass and reference         
area were neglected. An atmospheric mean molar mass of 43.49 g mol$^{-1}$     
was assumed at all altitudes \citep{magalhaes1999}. This is an over-estimate  
at high altitudes due to diffusive separation above the homopause             
($\sim$ 125 km). Readers wishing to adopt a different mass profile can        
straight-forwardly multiply the temperatures reported here by the ratio of    
their desired mean molar mass to 43.49 g mol$^{-1}$. The aerodynamic          
characteristics of Phoenix were modelled using a range of theoretical and     
experimental data sources \citep{edquist2008}. The resultant database was     
used to relate unknown density to known axial aerodynamic acceleration. An    
upper boundary condition is needed to determine pressure from density using   
the equation of hydrostatic equilibrium. First, a reconstruction was          
performed without consideration of uncertainties. An exponential fit to       
data between the onset of atmospheric reconstruction and 10 km below this     
level found a scale height equivalent to a temperature of 150 K for an        
isothermal atmosphere. Since the product of pressure and mean molecular mass  
equals the product of density,  Boltzmann's constant and temperature for an   
ideal gas, this fitted temperature can be used in conjunction with density    
at the upper boundary to determine a pressure at the upper boundary. This     
pressure was used as the upper boundary condition in the integration. The     
uncertainty in this pressure was assumed to be equivalent to an uncertainty   
in the temperature at the upper boundary of 50 K. The effects of the          
assumed upper boundary condition and its uncertainty are negligible more      
than a few scale heights below the onset of atmospheric reconstruction.       
The reconstructed atmospheric profiles are shown in                           
Figures~\ref{rhovszfig}--\ref{smoothtempvspfig}.                              
\nl                                                                           
[FIGURE~\ref{rhovszfig}]\nl                                                   
[FIGURE~\ref{pressvszfig}]\nl                                                 
[FIGURE~\ref{smoothtempvszfig}]\nl                                            
[FIGURE~\ref{smoothtempvspfig}]                                               
                                                                              
\section{\label{consistency}Consistency Checks}                               
                                                                              
The aerodynamic characteristics of entry vehicles such as Phoenix depend      
on their angle of attack. Since Phoenix's angle of attack was determined      
directly during the trajectory reconstruction using gyroscope data, it was    
not necessary to indirectly determine the angle of attack prior to finding    
the appropriate aerodynamic coefficient for the atmospheric structure         
reconstruction. However, the angle of attack can still be indirectly          
determined from the ratio of normal to axial accelerations                    
\citep{spencer1999, withers2003, withers2006c}.                               
This ratio, which can be obtained from observations, equals the ratio of      
the normal force coefficent to the axial force coefficient. The latter ratio  
increases monotonically with increasing angle of attack and can be found      
from the modelled aerodynamic characteristics. The inferred value of the      
angle of attack is therefore the value necessary to make the modelled         
ratio of the normal force coefficent to the axial force coefficient equal     
the observed ratio of normal to axial accelerations. The resultant            
indirectly determined angles of attack, $\alpha_{I}$, are shown in            
Figure~\ref{alphadandivstfig} alongside the directly determined angles of     
attack, $\alpha_{D}$. Values of $\alpha_{I}$ are typically 1--2 degrees       
smaller than $\alpha_{D}$. As a specific example, $\alpha_{D} = 4.0^{\circ}$  
and $\alpha_{I} = 2.7^{\circ}$ at $t=1980$ sec. One possible explanation      
involves winds. Values of $\alpha_{D}$ are calculated under the assumption    
that wind speed is zero everywhere, which is clearly unrealistic, whereas     
inferred values of $\alpha_{I}$ reflect the actual atmospheric winds during   
entry. However, this explanation is not sufficient. If a uniform zonal wind   
is assumed, then a wind speed of 300 ms$^{-1}$ is required to adjust          
$\alpha_{D}$ at $t=1980$ sec from $4.0^{\circ}$ to $2.7^{\circ}$. This is     
unphysically fast. Also, the angle of attack at parachute deployment is       
found to be $46^{\circ}$, much larger than any values consistent with the     
actual safe landing of Phoenix. Other possible explanations (possessing       
varying degrees of plausibility) are that the single axis accelerometers      
have some cross-axis sensitivity, that the modelled aerodynamic               
characteristics are incorrect, and that the transformation between the        
``IMU frame'' and ``C frame'' is incorrect. Resolution of this discrepancy    
is beyond the scope of this report. Note that this issue is different from    
discrepancies between predicted and reconstructed angles of attack            
addressed by \citet{desai2008}.                                               
\nl                                                                           
[FIGURE~\ref{alphadandivstfig}]                                               
                                                                              
Small oscillations in the angle of attack can be seen in                      
Figure~\ref{alphadzoomvstfig}. They have amplitudes on the order of tenths    
of a degree and periods of about two seconds. The period of these             
oscillations depends on the atmospheric density according to                  
\citep{schoenenberger2005b}:                                                  
                                                                              
\begin{eqnarray}                                                              
\label{osceqn}                                                                
\Omega^{2} = \frac{- \rho V^{2} A D C_{m\alpha}}{2 I}                         
\end{eqnarray}                                                                
                                                                              
where $\Omega$ is the angular frequency of the oscillations, $\rho$ is        
atmospheric density, $V$ is atmosphere-relative speed, $A$ is a reference     
area, $D$ is a reference diameter, $C_{m\alpha}$ is the derivative of the     
pitching moment coefficient with respect to angle of attack, and $I$ is a     
moment of inertia. If the observed period of oscillations, reconstructed      
density and speed, previously stated reference area and diameter, and         
$C_{m\alpha} \sim 0.1$ rad s$^{-1}$ \citep{edquist2008} are used with         
Equation~\ref{osceqn} to calculate $I$, then the resultant $I$ is within      
$\sim10$\% of 200 kg m$^{2}$ over much of the trajectory. The actual moment   
of inertia is close to 200 kg m$^{2}$ \citep{prince2008}.                     
                                                                              
\section{\label{future}Future Work}                                           
                                                                              
Scientific analysis of these results is in progress (Withers and Catling,     
Scientific results of the Phoenix Atmospheric Structure Experiment, in        
preparation).                                                                 
                                                                              
A direct radio communications link between Phoenix and Earth was maintained   
during EDL. The time series of frequencies recorded on the ground could be    
used to perform an independent trajectory and atmospheric structure           
reconstruction. These data, which determine the line of sight speed of        
Phoenix relative to Earth, are sufficient for this task if it is assumed      
that aerodynamic acceleration is parallel to the atmosphere-relative          
velocity vector. This technique may also offer the possibility of             
near-real-time trajectory and atmospheric structure reconstruction for        
future missions.                                                              
                                                                              
A radar onboard Phoenix recorded the range to the surface for altitudes of    
0--2 km. Incorporation of this altitude and descent speed data into the       
trajectory reconstruction would correct the current non-zero altitude at      
first ground contact and might also be useful in determining atmospheric      
properties in the dynamic environment encountered after parachute deployment. 
                                                                              
The discrepancy between $\alpha_{D}$ and $\alpha_{I}$ is puzzling. Its        
resolution is more likely to improve understanding of characteristics of      
the spacecraft than understanding of the martian environment.                 
                                                                              
\section{\label{concs}Conclusions}                                            
                                                                              
During its flight through the atmosphere of Mars, Phoenix recorded            
acceleration and angular velocity data using accelerometers and gyroscopes    
within its inertial measurement unit. The archived EDR data require           
correction due to the use of a preliminary version of a transformation        
matrix between two reference frames. We have described a correction           
method using an updated version of this transformation matrix. The EDR        
data, which were acquired at a high rate of 200 Hz, are noisy, but this       
can be improved by smoothing at the cost of worsened vertical resolution.     
Smoothing of the axial acceleration requires care. We have described how      
the running arithmetic mean of the axial acceleration is a biased             
estimate of the true acceleration and explained how this bias can be          
corrected. The processed accelerations and angular velocities have been       
used to reconstruct the trajectory of Phoenix through the atmosphere and      
the associated atmospheric structure.                                         
                                                                              
The position and velocity of Phoenix were reconstructed by forward            
integration of the equations of motion and its orientation history was        
determined directly using onboard angular velocity measurements. The          
density profile along the trajectory was reconstructured from $\sim$130 km    
to parachute deployment, $\sim$13.5 km above the surface, using the drag      
equation. The corresponding pressure profile was obtained from the            
equation of hydrostatic equilibrium and the corresponding temperature         
profile was obtained from the ideal gas law.                                  
                                                                              
The angle of attack of Phoenix during atmospheric flight was determined       
directly using angular velocity measurements and indirectly using the         
ratio of normal to axial accelerations and the spacecraft's aerodynamic       
database. The former was typically 1--2 degrees greater than the latter,      
a discrepancy for which we do not have a good explanation.                    
                                                                              
\indent{\bf{Acknowledgments\\}}                                               
PW acknowledges support from NASA (NNX09AG16G). DC acknowledges past          
support from the UK Science and Technology Facilities Council (STFC)          
awarded to the University of Bristol for Phoenix lander data reduction.       
PW and DC also acknowledge help and assistance from many people associated    
with the Phoenix project.                                                     
                                                                              
\newpage                                                                      
                                                                              
\appendix                                                                     
                                                                              
\section{\label{frameapp}Reference frames}                                    
                                                                              
The IMU data were recorded in the ``local IMU coordinate frame'' for IMU-A,   
but were transformed into the ``Phoenix mechanical frame'' before being       
archived as EDRs \citep{catling2008}. These are abbreviated here as the       
``IMU frame'' and ``M frame'', respectively. The ``M frame'' is also called   
the ``PHX\_LANDER'' frame and is assigned the NAIF identifier -84001          
(ftp://naif.jpl.nasa.gov/pub/naif/PHOENIX/kernels/fk/phx\_v06.tf).            
The symmetry axis of the spacecraft, along which the largest accelerations    
occurred, is the $\pm Z$-axis in the ``M frame''. During the course of this   
project, we discovered that a preliminary version of the appropriate          
transformation matrix, $\underline{\underline{M}}_{IMU}^{M,Prelim}$, had      
been used to transform the data from the ``IMU frame'' to the ``M frame''     
for archiving. The preliminary matrix corresponds to the spacecraft as        
designed, not as built. The elements of this matrix, which were reported in   
the file DOCUMENT/EDRSIS.PDF of \citet{catling2008}, are:                     
                                                                              
\begin{eqnarray}                                                              
\label{mimumprelimeqn}                                                        
\underline{\underline{M}}_{IMU}^{M,Prelim} =                                  
\left(                                                                        
\begin{array}{ccc}                                                            
0 & \cos\delta & -\sin\delta \\                                               
-1 & 0 & 0 \\                                                                 
0 & \sin\delta & \cos\delta \end{array}                                       
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
where $\delta = 25^{\circ}$.                                                  
For the analysis described in this report, the ``Phoenix cruise frame'' was   
used because its orientation, but not the orientation of the ``M frame'',     
was readily available at entry. This is abbreviated here as the ``C frame''.  
The ``C frame'' is also called the ``PHX\_LANDER\_CRUISE'' frame and is       
assigned the NAIF identifier -84000. The transformation matrix from the       
``M frame'' to the ``C frame'', $\underline{\underline{M}}_{M}^{C}$, was      
reported in the file DOCUMENT/EDRSIS.PDF of \citet{catling2008}. Its          
elements are:                                                                 
\begin{eqnarray}                                                              
\label{mmceqn}                                                                
\underline{\underline{M}}_{M}^{C} =                                           
\left(                                                                        
\begin{array}{ccc}                                                            
0 & 0 & 1 \\                                                                  
\cos\epsilon & -\sin\epsilon & 0 \\                                           
\sin\epsilon & \cos\epsilon & 0 \end{array}                                   
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
where $\epsilon = 120^{\circ}$.                                               
                                                                              
The transformation matrix from the ``IMU frame'' to the ``C frame'',          
$\underline{\underline{M}}_{IMU}^{C}$, was reported in Lockheed-Martin        
Interoffice Memo ``PHX-SE-07-0196: Phoenix Mars Lander Alignment Results''    
by A. Stoltz. Its elements are:                                               
\begin{eqnarray}                                                              
\label{mimuceqn}                                                              
\underline{\underline{M}}_{IMU}^{C} =                                         
\left(                                                                        
\begin{array}{ccc}                                                            
0.001658000000000 &  0.434532000000000 &  0.900655000000000 \\                
0.865583671733631 & -0.451637796435926 &  0.216304845776395 \\                
0.500761102355330 &  0.779233610045474 & -0.37687298280806 \end{array}        
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
This matrix was determined from pre-launch measurements, not design           
specifications. Its elements differ from those found by combining             
$\underline{\underline{M}}_{IMU}^{M,Prelim}$ and                              
$\underline{\underline{M}}_{M}^{C}$ only at the third decimal place, but      
these small differences have a surprisingly large effect on the               
reconstructed trajectory and atmospheric structure. If the preliminary        
matrix was used, then angles of attack reached ten degrees at parachute       
deployment, first ground contact and parachute deployment occurred several    
kilometres higher than in our final reconstruction, and temperatures          
below 30 km were 5K warmer than in our final reconstruction. We recommend     
that designers of future atmospheric structure investigations carefully       
evaluate how uncertainties in the orientation of accelerometer axes in a      
spacecraft-fixed frame affect their final scientific data products.           
                                                                              
The updated transformation matrix from the ``IMU frame'' to the ``M frame'',  
$\underline{\underline{M}}_{IMU}^{M,Updated}$, corresponds to the             
spacecraft as built. It satisfies:                                            
                                                                              
\begin{eqnarray}                                                              
\label{mimumupdateeqn}                                                        
\underline{\underline{M}}_{IMU}^{M,Updated} =                                 
\left( \underline{\underline{M}}_{M}^{C} \right)^{T}                          
\underline{\underline{M}}_{IMU}^{C}                                           
\end{eqnarray}                                                                
                                                                              
where the superscript $T$ indicates the transpose.                            
                                                                              
We transformed the originally archived accelerations and angular rates        
(EDRs) from the preliminary version of the ``M frame'' into the               
``IMU frame'' using $\underline{\underline{M}}_{IMU}^{M,Prelim}$,             
then from the ``IMU frame'' into the ``C frame'' using                        
$\underline{\underline{M}}_{IMU}^{C}$.                                        
                                                                              
Users of the originally archived EDR data should modify the archived values   
using the matrix $\underline{\underline{M}}_{M,Prelim}^{M,Updated}$, which    
satisfies:                                                                    
                                                                              
\begin{eqnarray}                                                              
\label{mmprelimmupdateeqn}                                                    
\underline{\underline{M}}_{M,Prelim}^{M,Updated} =                            
\underline{\underline{M}}_{IMU}^{M,Updated}                                   
\left( \underline{\underline{M}}_{IMU}^{M,Prelim} \right)^{T} =               
\left( \underline{\underline{M}}_{M}^{C} \right)^{T}                          
\underline{\underline{M}}_{IMU}^{C}                                           
\left( \underline{\underline{M}}_{IMU}^{M,Prelim} \right)^{T}                 
\end{eqnarray}                                                                
                                                                              
The numerical values of the elements of the matrix                            
$\underline{\underline{M}}_{M,Prelim}^{M,Updated}$ are as follows:            
\begin{eqnarray}                                                              
\label{numbersmmprelimmupdateeqn}                                             
\underline{\underline{M}}_{M,Prelim}^{M,Updated} =                            
\left(                                                                        
\begin{array}{ccc}                                                            
      0.99991267 & -0.00087993926 &   -0.013188274 \\                         
   0.00090164837 &     0.99999798 &   0.0016463352 \\                         
     0.013186473 &  -0.0016580001 &     0.99991177 \end{array}                
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
To provide a mechanism by which data users can verify that they have          
modified the EDR data correctly, we now state the values of some modified     
datapoints. The original (from file DATA/IMU\_A\_EDR\_M.TAB of                
\citet{catling2008}) and modified velocity changes in ms$^{-1}$ along the     
$X$, $Y$ and $Z$ axes of the ``M frame'' at the first time step (RELATIVE     
TIME=0.000 seconds) are:                                                      
\begin{eqnarray}                                                              
\label{origandupdateaccatfirstdteqn}                                          
\left(                                                                        
\begin{array}{c}                                                              
0.00013646 \\                                                                 
-0.00271028 \\                                                                
0.00006364 \end{array}                                                        
\right)                                                                       
\rightarrow                                                                   
\left(                                                                        
\begin{array}{c}                                                              
0.00013799366\\                                                               
-0.0027100467\\                                                               
6.9927455 \times 10^{-5}\end{array}                                           
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
The corresponding angle changes (radians) are:                                
\begin{eqnarray}                                                              
\label{origandupdateomegaatfirstdteqn}                                        
\left(                                                                        
\begin{array}{c}                                                              
-0.00000786 \\                                                                
0.00000600 \\                                                                 
-0.00000918 \end{array}                                                       
\right)                                                                       
\rightarrow                                                                   
\left(                                                                        
\begin{array}{c}                                                              
-0.0015487050\\                                                               
0.0011955575\\                                                                
-0.0018585567\end{array}                                                      
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
The peak axial deceleration in the EDRs occurs at RELATIVE TIME=132.340       
seconds. The original and modified velocity changes (ms$^{-1}$) at this       
timestep are:                                                                 
\begin{eqnarray}                                                              
\label{origandupdateaccatpeakdteqn}                                           
\left(                                                                        
\begin{array}{c}                                                              
-0.00430083 \\                                                                
-0.00150571 \\                                                                
-0.42150492 \end{array}                                                       
\right)                                                                       
\rightarrow                                                                   
\left(                                                                        
\begin{array}{c}                                                              
0.0012597930\\                                                                
-0.0022035232\\                                                               
-0.42152195\end{array}                                                        
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
The corresponding angle changes (radians) are:                                
\begin{eqnarray}                                                              
\label{origandupdateomegaatpeakdteqn}                                         
\left(                                                                        
\begin{array}{c}                                                              
-0.00001348  \\                                                               
0.00001100 \\                                                                 
-0.00003497 \end{array}                                                       
\right)                                                                       
\rightarrow                                                                   
\left(                                                                        
\begin{array}{c}                                                              
-1.3027308 \times 10^{-5}\\                                                   
1.0930251 \times 10^{-5}\\                                                    
-3.5162906 \times 10^{-5}\end{array}                                          
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
                                                                              
\section{\label{smoothapp}Reducing noise in axial acceleration measurements}  
                                                                              
The noise in the axial acceleration measurements must be reduced by           
averaging in order for uncertainties in the reconstructed atmospheric         
properties to be small enough that the results have scientific value.         
However, the normal averaging process provides a biased estimate of the       
true axial acceleration at the centre of the time series. We now explain      
the source of this bias and how the bias was corrected.                       
                                                                              
The axial acceleration measured before substantial deceleration occurs        
increases exponentially with time \citep{magalhaes1999}. This can also be     
seen in Figure~\ref{aaxialfig}. This happens because the axial                
acceleration, $a_{a}$, satisfies:                                             
                                                                              
\begin{eqnarray}                                                              
\label{drageqn}                                                               
m a_{a} = \frac{- C_{A} \rho A V^{2}}{2}                                      
\end{eqnarray}                                                                
                                                                              
where $m$ is spacecraft mass, $C_{A}$ is the axial force coefficient,         
$\rho$ is atmospheric density, $V$ is atmosphere-relative speed and $A$ is    
a reference area. Values of $m$ and $A$ are fixed, $V$ is effectively         
constant until substantial deceleration occurs and $C_{A}$ changes only       
slightly during flight through the rarefied upper atmosphere. Hence $a_{a}$   
is proportional to $\rho$, which depends exponentially on altitude, $z$,      
through $\rho \propto \exp\left(-z/H\right)$ where $H$ is the density scale   
height. Since the rate of change of altitude is also approximately constant,  
the magnitude of $a_{a}$ increases exponentially with time. Dropping the      
subscript $a$ for convenience, we have:                                       
                                                                              
\begin{eqnarray}                                                              
\label{sinh01eqn}                                                             
a = a_{0} \exp \frac{t}{\tau}                                                 
\end{eqnarray}                                                                
                                                                              
where $t$ is time, $a_{0}$ is the acceleration at time $t=0$, and $\tau$ is   
the timescale, which equals the ratio of the atmosphere's density scale       
height to the rate of change of altitude.                                     
                                                                              
The mean acceleration between $t=-t_{X}$ and $t=t_{X}$, $a_{mean}$,           
satisfies:                                                                    
                                                                              
\begin{eqnarray}                                                              
\label{sinh02eqn}                                                             
a_{mean} = \frac{1}{2t_{X}} \int^{t=t_{X}}_{t=-t_{X}} a_{0}                   
\exp \left(\frac{t}{\tau}\right) dt                                           
\end{eqnarray}                                                                
\begin{eqnarray}                                                              
\label{sinh03eqn}                                                             
a_{mean} =                                                                    
\frac{a_{0} \tau}{2t_{X}} \left[                                              
\exp \frac{t_{X}}{\tau} - \exp \frac{-t_{X}}{\tau}                            
\right]                                                                       
\end{eqnarray}                                                                
\begin{eqnarray}                                                              
\label{sinh04eqn}                                                             
a_{mean} = a_{0} \frac{\tau}{t_{X}} \sinh\left(\frac{t_{X}}{\tau}\right)      
\end{eqnarray}                                                                
                                                                              
The mean acceleration is not the same as the desired acceleration at the      
centre of the time series. Although the arithmetic mean of the logarithm of   
the acceleration does, in principle, give an unbiased estimate of the         
acceleration at the centre of the time series, this is not useful in          
practice because the noisy measured accelerations can be positive or          
negative. A process that requires finding the logarithm of a negative         
number is not helpful.                                                        
                                                                              
According to Equation~\ref{sinh04eqn}, the mean acceleration is               
significantly greater than the acceleration at the centre of the time         
series unless $t_{X} \ll \tau$. However, if the timescale $\tau$ can be       
determined, then the value of $a_{0}$ can be found from $a_{mean}$ and        
$t_{X}$. If a ``long'' average, $a_{L}$, and a ``short'' average, $a_{S}$,    
are calculated over the ranges $t=-2t_{S}$ to $t=2t_{S}$ and $t=-t_{S}$ to    
$t=t_{S}$, respectively, then these averages satisfy:                         
                                                                              
\begin{eqnarray}                                                              
\label{sinh05eqn}                                                             
a_{L} = a_{0} \frac{\tau}{2t_{S}} \sinh\left(\frac{2t_{S}}{\tau}\right)       
\end{eqnarray}                                                                
\begin{eqnarray}                                                              
\label{sinh06eqn}                                                             
a_{S} = a_{0} \frac{\tau}{t_{S}} \sinh\left(\frac{t_{S}}{\tau}\right)         
\end{eqnarray}                                                                
                                                                              
The ratio of the averages satisfies:                                          
\begin{eqnarray}                                                              
\label{sinh07eqn}                                                             
\frac{a_{L}}{a_{S}} = \frac{1}{2} \frac{\sinh\left(\frac{2t_{S}}{\tau}        
\right)}{\sinh\left(\frac{t_{S}}{\tau}\right)}                                
\end{eqnarray}                                                                
\begin{eqnarray}                                                              
\label{sinh08eqn}                                                             
\frac{a_{L}}{a_{S}} = \cosh \left(\frac{t_{S}}{\tau}\right)                   
\end{eqnarray}                                                                
                                                                              
Equation~\ref{sinh08eqn} can be rearranged using the trigonometric identity   
$\cosh^{-1}\left(x\right) = \ln \left( x + \left(x^{2} - 1\right)^{1/2}       
\right)$ to yield:                                                            
\begin{eqnarray}                                                              
\label{sinh09eqn}                                                             
\frac{t_{S}}{\tau} = \ln \left(                                               
\frac{a_{L}}{a_{S}} + \sqrt{ \left(\frac{a_{L}}{a_{S}}\right)^{2} - 1}        
\right)                                                                       
\end{eqnarray}                                                                
                                                                              
Thus the timescale $\tau$ can be determined from the two related means,       
$a_{L}$ and $a_{S}$, and then used in either Equation~\ref{sinh05eqn} or      
Equation~\ref{sinh06eqn} to find the true acceleration at the centre of       
the series of data points. Figure~\ref{sinhfig} illustrates differences       
between $a_{L}$ and $a_{S}$ for the Phoenix data. Figure~\ref{sinhcorrfig}    
shows how these differences are drastically reduced after application of      
the correction procedure outlined above.                                      
\nl                                                                           
[FIGURE~\ref{sinhfig}]\nl                                                     
[FIGURE~\ref{sinhcorrfig}]                                                    
                                                                              
For practical application of these theoretical results, two considerations    
are critical. Namely, whether to apply this correction and what duration to   
choose for the averaging window. If $t_{X}$ is sufficiently small in          
comparison to $\tau$, then the uncorrected average is so close to the true    
axial acceleration that correcting it is more likely to worsen it than        
improve it. The duration of the averaging window should be chosen such that   
the combination of uncertainty and vertical resolution is optimal for the     
desired application.                                                          
                                                                              
Creation of the composite series of smoothed axial accelerations for Phoenix  
required several steps. To begin, several series of smoothed, uncorrected     
axial accelerations were calculated using a $2^{i}$ point running mean.       
These were labelled $a_{i}$, where $i$ ranged from 6 to 13. Next, several     
series of smoothed, corrected axial accelerations were calculated by          
taking the $a_{i}$ series, then correcting them using the ratio to the        
$a_{i+1}$ series. These were labelled $a_{i}^{C}$, where $i$ ranged from      
10 to 12. Finally, the composite series was constructed as follows.           
                                                                              
First, unaveraged values were used to fill the first 4096 data points.        
Second, $a_{12}^{C}$ values were used to fill the next 1081 data points.      
Third, $a_{11}^{C}$ values were used to fill the next 2595 data points.       
Fourth, $a_{10}^{C}$ values were used to fill the next 1882 data points.      
The use of corrected values cannot continue indefinitely, since the axial     
acceleration eventually ceases to vary exponentially with time. Also, it      
becomes unnecessary when the averaging window is small by comparison to       
the timescale for changes in the axial acceleration. Accordingly,             
uncorrected averages were used after this point. Fifth, $a_{9}$ values        
were used to fill the next 114 data points. Sixth, $a_{8}$ values were        
used to fill the next 127 data points. Seventh, $a_{7}$ values were           
used to fill the next 337 data points. Eighth, $a_{6}$ values were used       
to fill all remaining data points prior to parachute deployment. The          
duration of the averaging window used to calculate the $a_{6}$ values,        
0.32 sec, corresponds to averaging over a vertical distance of less than      
0.5 km. The use of a shorter averaging window to improve vertical resolution  
further is not justified. Ninth, unaveraged values were used to fill all      
data points after parachute deployment. The atmospheric structure             
reconstruction terminates at parachute deployment, so there is little         
motivation to reduce noise by averaging after this point. In all cases,       
the locations of the transitions between one average and another average      
were fixed based on consideration of the scatter in the average with the      
shorter averaging window. These variations in the width of the window used    
to smooth the axial accelerations are shown in                                
Figures~\ref{nusedinsecvstfig} and \ref{nusedinkmvszfig}. The impact of       
these variations on the uncertainty of the smoothed axial acceleration is     
shown in Figure~\ref{aaxialfig}.                                              
                                                                              
\newpage                                                                      
                                                                              
\bibliographystyle{icarus}                                                    
\bibliography{references}                                                     
                                                                              
\newpage                                                                      
                                                                              
\clearpage                                                                    
                                                                              
\begin{table}                                                                 
\caption{\label{landedcoords}                                                 
Time and position of Phoenix landing.                                         
}                                                                             
\begin{tabular}{lr}                                                           
\hline                                                                        
Time\footnotemark[1] (UTC) & 2008-05-25T23:38:24 \\                           
Aerocentric latitude\footnotemark[2] ($^{\circ}$N) & 68.21878 $\pm$           
0.00006 \\                                                                    
Longitude\footnotemark[2] ($^{\circ}$E) & 234.24845 $\pm$ 0.000096 \\         
Radius\footnotemark[2] (km) & 3376.2915 $\pm$ 0.0014 \\                       
Elevation\footnotemark[3] (m) & -4131 \\                                      
\hline                                                                        
\end{tabular}                                                                 
                                                                              
\footnotemark[1]{\citet{smith2009}.}\\                                        
\footnotemark[2]{Martin-Mur (personal communication, 28 May 2008).            
The landed latitude, longitude and radius, which are shown with 1$\sigma$     
uncertainties, were determined from Doppler tracking.}\\                      
\footnotemark[3]{Elevation is with respect to the areoid defined by the       
Mars Orbiter Laser Altimeter (MOLA) investigation, specifically 16 pixels     
per degree gridded MOLA data acquired from                                    
http://geo.pds.nasa.gov/missions/mgs/megdr.html \citep{smith2003}.}\\         
\end{table}                                                                   
                                                                              
\clearpage                                                                    
                                                                              
                                                                              
\begin{table}                                                                 
\caption{\label{entrystatetable}                                              
Phoenix entry state$^{1}$.                                                    
}                                                                             
\begin{tabular}{lrr}                                                          
\hline                                                                        
Quantity & Value & 1$\sigma$ Uncertainty\\                                    
\hline                                                                        
Time (UTC) & 2008-05-25T23:30:57.733 & 0.002\footnotemark[7] \\               
$t - t_{ref}$\footnotemark[2] (s) & 1857.733 & 0.002\footnotemark[7] \\       
Radius\footnotemark[3] (km) & 3522.2 & 0\footnotemark[8] \\                   
Areocentric latitude ($^{\circ}$N) & 69.3660 & 0.04 \\                        
Longitude ($^{\circ}$E) & 197.7160 & 0.00031 \\                               
Speed\footnotemark[4] (ms$^{-1}$) & 5600.34 & 0.02 \\                         
Flight path angle\footnotemark[5] (deg) & 13.010 & 0.00015 \\                 
Heading angle\footnotemark[6] (deg) & 77.6720 & 0.000055 \\                   
\hline                                                                        
\end{tabular}                                                                 
                                                                              
\footnotemark[1]{Information from the file CATALOG/INST.CAT of                
\citet{catling2008}.}\\                                                       
\footnotemark[2]{$t_{ref}$, a UTC reference time, is                          
2008-05-25T23:00:00.000.}\\                                                   
\footnotemark[3]{145.9 km above the landing site at 3376.3 km.}\\             
\footnotemark[4]{Relative to a Mars-centred inertial frame                    
\citep{spencer1999, withers2003}.}\\                                          
\footnotemark[5]{Angle below horizontal of velocity vector in inertial        
frame.}\\                                                                     
\footnotemark[6]{Angle east of north of velocity vector in inertial frame.}\\ 
\footnotemark[7]{Set to zero as discussed in the text.}\\                     
\footnotemark[8]{Set to 2.5 m as discussed in the text.}\\                    
\end{table}                                                                   
                                                                              
\clearpage                                                                    
                                                                              
                                                                              
                                                                              
                                                                              
\begin{table}                                                                 
\caption{\label{entryquattable}                                               
Quaternion representing the rotation between the ``J2000 frame'' and the      
``C frame'' at the entry interface.                                           
}                                                                             
\begin{tabular}{lr}                                                           
\hline                                                                        
%$q_{0}$ & 0.05012570 \\                                                      
%$q_{1}$ & 0.52484401 \\                                                      
%$q_{2}$ & 0.66515558 \\                                                      
%$q_{3}$ & 0.52876671 \\                                                      
$q_{0}$ &      0.050125700 \\                                                 
$q_{1}$ &       0.52484401 \\                                                 
$q_{2}$ &       0.66515558 \\                                                 
$q_{3}$ &       0.52876671 \\                                                 
% Automatically output and formatted by getinitialquat.pro                    
\hline                                                                        
\end{tabular}                                                                 
\end{table}                                                                   
                                                                              
\clearpage                                                                    
                                                                              
                                                                              
                                                                              
\begin{table}                                                                 
\caption{\label{chutetable}                                                   
Reconstructed conditions and their 1$\sigma$ uncertainties at parachute       
deployment.                                                                   
}                                                                             
\begin{tabular}{lrr}                                                          
\hline                                                                        
Altitude (km) &        13.54 &       0.38 \\                                  
Areocentric latitude ($^{\circ}$N) &        68.265 &      0.030 \\            
Longitude ($^{\circ}$E) &        234.034 &      0.058 \\                      
Atmosphere-relative speed (ms$^{-1}$)  &        390.9 &        1.5 \\         
Angle of attack (deg)  &        4.90 &       0.71 \\                          
Mach number &        1.703 &      0.015 \\                                    
\hline                                                                        
\end{tabular}                                                                 
\end{table}                                                                   
                                                                              
\clearpage                                                                    
                                                                              
                                                                              
                                                                              
\begin{table}                                                                 
\caption{\label{impactstatetable}\label{lasttable}                            
Reconstructed conditions and their 1$\sigma$ uncertainties at first           
ground contact.                                                               
}                                                                             
\begin{tabular}{lrr}                                                          
\hline                                                                        
Altitude (km) &        1.10 &        1.49 \\                                  
Areocentric latitude ($^{\circ}$N) &        68.237 &      0.029 \\            
Longitude ($^{\circ}$E) &        234.311 &      0.054 \\                      
Atmosphere-relative speed (ms$^{-1}$)  &        6.1 &        3.6 \\           
\hline                                                                        
\end{tabular}                                                                 
\end{table}                                                                   
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{aaxial.eps}                                      
\end{center}                                                                  
\caption{\label{aaxialfig}                                                    
Time series of smoothed axial acceleration measurements (dots). The           
associated $1\sigma$ uncertainties are shown by the dashed line.}             
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{nusedinsecvst.eps}                               
\end{center}                                                                  
\caption{\label{nusedinsecvstfig}                                             
Time-varying width of window used to smooth axial acceleration                
measurements.}                                                                
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{zvst.eps}                                        
\end{center}                                                                  
\caption{\label{zvstfig}Reconstructed altitude versus time for Phoenix        
with $1\sigma$ uncertainties shown by the grey envelope.}                     
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{latvsz.eps}                                      
\end{center}                                                                  
\caption{\label{latvszfig}Reconstructed altitude versus reconstructed         
latitude for Phoenix with $1\sigma$ uncertainties shown by the grey           
envelope.}                                                                    
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{lonvsz.eps}                                      
\end{center}                                                                  
\caption{\label{lonvszfig}Reconstructed altitude versus reconstructed         
longitude for Phoenix with $1\sigma$ uncertainties shown by the grey          
envelope.}                                                                    
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{vrelvsz.eps}                                     
\end{center}                                                                  
\caption{\label{vrelvszfig}Reconstructed altitude versus reconstructed        
atmosphere-relative speed for Phoenix with $1\sigma$ uncertainties shown      
by the grey envelope.}                                                        
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{nusedinkmvsz.eps}                                
\end{center}                                                                  
\caption{\label{nusedinkmvszfig}Reconstructed altitude versus vertical        
extent of window used to smooth axial acceleration measurements.}             
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{alphadzoomvst.eps}                               
\end{center}                                                                  
\caption{\label{alphadzoomvstfig}Time series of directly determined angle     
of attack, $\alpha_{D}$, with $1\sigma$ uncertainties shown by the grey       
envelope. Results after parachute deployment are not shown.}                  
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{rhovsz.eps}                                      
\end{center}                                                                  
\caption{\label{rhovszfig}Reconstructed altitude versus reconstructed         
density with $1\sigma$ uncertainties shown by the grey envelope.}             
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{pressvsz.eps}                                    
\end{center}                                                                  
\caption{\label{pressvszfig}Reconstructed altitude versus reconstructed       
pressure with $1\sigma$ uncertainties shown by the grey envelope.}            
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{smoothtempvsz.eps}                               
\end{center}                                                                  
\caption{\label{smoothtempvszfig}Reconstructed altitude versus                
reconstructed temperature with $1\sigma$ uncertainties shown by the grey      
envelope. 501-point running mean values of temperatures and temperature       
uncertainties are shown to reduce high frequency noise.}                      
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{smoothtempvsp.eps}                               
\end{center}                                                                  
\caption{\label{smoothtempvspfig}                                             
Reconstructed pressure versus reconstructed temperature with $1\sigma$        
uncertainties shown by the grey envelope. 501-point running mean values of    
temperatures and temperature uncertainties are shown to reduce high           
frequency noise.}                                                             
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{alphadandivst.eps}                               
\end{center}                                                                  
\caption{\label{alphadandivstfig}                                             
Time series of directly determined angle of attack, $\alpha_{D}$, (black      
line) and indirectly determined angle of attack, $\alpha_{I}$ (grey line).    
Results after parachute deployment are not shown.}                            
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{sinh.eps}                                        
\end{center}                                                                  
\caption{\label{sinhfig}                                                      
Two time series of axial acceleration measurements before bias correction.    
Grey dots indicate smoothing with a 1024 point running mean and black dots    
indicate smoothing with a 2048 point running mean.}                           
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\begin{figure}                                                                
\begin{center}                                                                
\includegraphics[width=20pc]{sinhcorr.eps}                                    
\end{center}                                                                  
\caption{\label{sinhcorrfig}\label{lastpage}\label{lastfig}                   
Two time series of axial acceleration measurements after bias correction.     
Grey dots indicate smoothing with a 1024 point running mean followed by       
correction using ratio to a 2048 point running mean. Black dots indicate      
smoothing with a 2048 point running mean followed by correction using         
ratio to a 4096 point running mean.}                                          
\end{figure}                                                                  
                                                                              
\clearpage                                                                    
                                                                              
\end{document}                                                                
