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DTWG overview
=============
The responsibility of developing analysis techniques by which the
Huygens entry and descent trajectory is reconstructed from the
official NASA/ESA hand-off point at the interface altitude of 1270
km to the surface is given to the Huygens Descent Trajectory
Working Group (DTWG), which was chartered in 1996 as a subgroup of
the Huygens Science Working Team (HSWT) [ATKINSON2007].
Membership includes representatives from each of the Huygens probe
science instrument teams and the orbiter radar, the Huygens and
Cassini project scientists, and engineers from ESTEC, ESOC, and
Alcatel (Cannes).
The primary goals of the Descent Trajectory Working Group are:
- to develop a framework between experiment teams and the Huygens
Mission Team for sharing and exchanging data relevant to the
descent trajectory analysis and modeling;
- to develop methodologies by which the probe descent trajectory
and attitude can be accurately reconstructed from the probe and
orbiter science and engineering data; and
- to provide a single, common descent profile that is consistent
with all the available probe and orbiter engineering and science
data, and that can be utilized by each instrument team for
analysis of experiment measurements, and correlation of results
between experiments.
Reconstruction Strategy and Input Datasets
==========================================
The reconstruction methodology, strategy, and results are described
in detail in [KAZEMINEJAD2007] and summarized in the
following paragraphs.
The reconstruction strategy adapted by the group can be summarized
by the following three phases
(1) The reconstruction of the probe entry phase from the interface
point down to an altitude of ~100 km.
(2) The reconstruction of the probe descent phase from ~145 km down
to surface impact.
(3) The entry and descent trajectory merging process, which ensured
a smooth transit between entry and descent phase in terms of
both altitude and descent speed.
The methodology and results of each step will be discussed
separately in the subsequent sections.
The DTWG product is derived from two sets of input data:
1. The DTWG Input Data, including all the relevant instrument and
housekeeping data made available to the Huygens DTWG to perform
the reconstruction of the probe entry and descent trajectory.
2. The Huygens Probe Initial Conditions and Event File (see
EVENT_FILE_04042005.TXT in the GEOMETRY directory) provided by
the Project Scientist Team (PST) to the DTWG, comprising the
probe initial conditions, physical constants and epochs of
important events during the probe mission.
The entry phase reconstruction is based on the following set
of input data:
- HASI 3-axis accelerometer measurements (only axial measurements
were actually used, due to measurement uncertainties of normal
accelerometers);
- DWE zonal wind measurements in the altitue range ~155 - 100 km;
The descent trajectory reconstruction is based on more than just one
instrument dataset. The altitude and descent speed are derived
from:
- HASI pressure and temperature measurements (dynamically corrected
derived dataset: HASI_L4_ATMO_PROFILE_DESCENT.TAB from the
HASI archive volume HP-SSA-HASI-2-3-4-MISSION-V1.1)
- SSP impact measurements (time of impact)
- GCMS molecular mass measurements
- DWE zonal wind measurements
- DISR meridional image based probe drift measurements
The following instrument data were not directly ingested into the
reconstruction process but only used for comparison of results and
consistency check:
- SSP speed of sound derived altitude profile
- Radar Altimeter measurements of probe altitude
Entry Phase Reconstruction Methodology:
======================================
The probe entry phase is defined as the portion of the trajectory
starting from the official NASA/ESA interface point at nominally
1270 km altitude down to T0, the time at which the Parachute
Deployment Device (PDD) was fired (at an altitude of about 155
km).
The probe entry phase is reconstructed by numerically integrating
the equations of motion in a Titan-centered EME2000 (inertial)
coordinate system which can be written as
a = a_g + a_Ad
where a_g is the gravitational acceleration and a_Ad is the
acceleration due to aerodynamic forces.
The integration of the equations of motion requires the knowledge
of the full probe state vector (i.e., the cartesian position and
velocity vector specified in the integration reference system) at
an initial epoch. The entry state vector is provided by the Cassini
Navigation team to ESA together with the estimated interface epoch
(in Ephemeris Time) and the corresponding state vector
uncertainties in the form of a text kernel. This information is
made available to the DTWG by the PST via the Huygens Probe Initial
Conditions and Event File (also in the form of a NAIF text kernel)
which is also archived.
The gravitational forces a_g acting on the spacecraf's centre of
mass cannot be detected by the onboard accelerometers as those
measurements are made in a frame fixed with respect to the
spacecraft (referred to as the spacecraft reference system). In
other words the spacecraft and the accelerometers are both free
falling at the same rate. The gravitational forces therefore have
to be modelled at each step of the reconstruction process.
For the proper transformation of the measured aerodynamic
accelerations from the spacecraft reference system into the
integration reference system the probe angle-of-attack (AOA) is
required. The AOA is the separation angle of the probe axial body
axis and the flow velocity vector and is time dependent AOA=AOA(t).
In planetary entry probe missions the AOA is typically estimated
from the ratio of measured normal to axial accelerations. Those
equal the ratio of the corresponding aerodynamic coefficients which
in return are available in an pre-flight aerodynamic database as a
function of AOA and Mach number Ma. Due to a sensitivity problem of
the HASI piezoresistive accelerometers however, the measurements
along the normal axis turned out to be too inaccurate for a
derivation of the angle of attack via the aforementioned method and
a zero angle of attack had therefore to be assumed for the
reconstruction of the entire entry trajectory. In this special case
the aerodynamic drag a_D equals the measured axial acceleration a_A.
The lift force value a_L was neglected since the lift force vector
co-rotated with the probe and therefore averaged to zero as the
probe rotated. This was based on the assumption that the lift force
remained essentially constant over an entire roll period.
Once the norm of the drag force is known the corresponding vector
a_Ad can be obtained from the simple fact that the drag force vector
always points in the opposite direction of to the relative velocity
vector v_rel of the spacecraft with respect to the atmosphere
(expressed in the inertial frame of integration).
a_Ad = - a_D * v_rel/norm(v_rel)
v_rel is the relative velocity which was derived from
v_rel = v_in - omega_p cross r_in - v_wind
where r_in and v_in are the probe inertial position and velocity
vectors, omega_p = 4.560678E-06 rad/sec, and v_wind is the velocity
vector of the atmospheric wind in the inertial frame. v_wind was
assumed to be directed into a zonal direction with a norm of 90 m/s
(once DWE measurements were available they were taken into account).
Entry Phase Reconstruction Results:
==================================
The numerical integration was started at the interface point using
the Cassini Navigation estimation of the probe state vector. This
vector was adjusted in the trajectory merging process in order to
ensure a good match of the entry and descent phase.
It is important to keep in mind that the direction of the relative
velocity vector v_rel was influenced by atmospheric winds. The DWE
zonal wind measurements were however constrained to the descent
phase. A parameter study showed that the assumption of a constant 90
m/s prograde zonal wind for altitudes above 143.9 km (for lower
altitudes DWE measurements were available) achieved the best merging
of entry and descent phase.
The entry phase reconstruction based on the HASI accelerometer data
could only be achieved down to an altitude of 100 km. At lower
altitudes the numerical integration of the equations of motion did
not converge anymore due to the built-up error introduced by the
integration of accelerometer measurements, which got strongly perturbed
by the pendulum motions that was introduced by the opening of the various
parachutes. An additional a systematic error was introduced by the
fact that during the parachute sequence the accelerometers were
not located at the center of gravity anymore, as the jettison of
the backcover and the release of the parachutes introduced a non-negligible
change of mass distribution.
The results of the entry phase trajectory reconstruction are
depicted in plots ENTRY_ALT.JPG and ENTRY_VEL.JPG (see BROWSE
directory and description in BROWSE_INFO.TXT). The altitude
profile is shown with respect to the reference surface, which is
considered to be at a radial distance of 2575 km from Titan's
center. The figure shows both the reconstructed profile (solid line)
based on the flight data and the preflight simulation trajectory as
described by [KAZEMINEJAD2004]. (dashed line). It is
important to point out that the preflight simulation is based on the
initial conditions as estimated by the Cassini Navigation team
(i.e., an altitude of exactly 1270 km at interface epoch). The
reconstructed profile (labeled with Flight Data) used the modified
initial state vector based on the correction that was provided by
the entry and descent phase merging algorithm. The altitude
difference between the two state vectors is about 22 km, which
corresponds to about 0.72 sigma of the specified radial error.
ENTRY_VEL.JPG shows the norm of the inertial velocity vector for
both the reconstructed profile (solid) and the preflight simulation
(dashed). Although the actual velocity at the interface altitude was
very close to the preflight prediction, the two profiles start to
diverge slightly once the probe enters the entry deceleration pulse.
This is due to the previously mentioned altitude difference of 22
km: as the actual probe initial position at interface was lower than
the predicted one (and the one used in the preflight simulation),
the probe encountered denser atmospheric layers during the main
deceleration pulse, which is also depicted in the plot. This implied
a faster velocity decrease and explains the difference between the
two velocity profiles. The two vertical dashed lines mark the T_0
event at about 155 km altitude as well as the start of the probe
relay transmission at about 143 km.
Once the probe dynamics were reconstructed from the numerical
integration of the measured aerodynamic deceleration, the
atmospheric density could be inferred from
rho = -2*m(t)/(C_D*A)*a_Ad/v_rel^2
where m, A, and C_D are respectively the probe mass, cross-section
area and drag coefficient. The probe mass was not constant during
the entry phase due to the ablation process of heat shield material,
which is expressed by the time dependence of m=m(t). As the Huygens heat shield
was unfortunately not equipped with any Thermal Protection System
(TPS) recession sensors, the ablation process could only be modeled
taking into account the integrated mass loss that was estimated from
preflight simulations. The time dependent mass loss was simulated
according to the relation
m(t)=m_0*exp(0.5*sigma*v_rel)^2-v_max^2)
with sigma = 4.18*10E-10 m^{-2}s^2 and the initial mass m_0 = 320
kg. v_max is the flow velocity (relative probe velocity with respect
to the atmosphere) at the time of the start of the ablation process
and was assumed to be the maximum probe velocity during the entry
phase. The sigma value was adjusted to fit the estimated integrated
ablation mass of 9.7 kg. The drag coefficient C_D is also time
dependent and was interpolated from the preflight aerodynamic
database [HUY-EADS-MIS-RE], which provides the Huygens entry module
axial and normal coefficients for free molecular flow (Kn > 10),
transitional flow (0.001 <= Kn <= 10), and continuum flow ( Kn <
0.001), as a function of angle-of-attack AOA and Knudsen number Kn
or Mach number Ma. Kn is given by
Kn=1.2533 sqrt(gamma) Ma/Re
where Re is the dimensionless Reynolds number which was derived
using the relation
Re =v_rel*rho*D_spc/mu_visc
D_spc is the diameter of the probe front shield (i.e., 2.7 m) and
mu_visc is the dynamic viscosity which was calculated (in units of
kg/m/s) according to
mu_visc = 1.458E-06 T^1.5/(T + 110.4)
The Mach number Ma is simply derived from the ratio of the relative
probe velocity and the speed of sound c_s. One can see that the
calculation of aerodynamic parameters like the Reynolds number, the
Mach number and the Knudsen number require the knowledge of the
atmospheric density and temperature. The atmospheric temperature can
be derived from the pressure profile assuming the ideal gas law,
which is a sufficient approximation in the upper parts of the
atmosphere due to the low density. The pressure profile can be
derived from the density profile from the integration of the
equation of hydrostatic equilibrium.
The reconstruction of the atmospheric density however in return
requires the knowledge of the aerodynamic flight regime for the
proper interpolation of the drag coefficient C_D. The reconstruction
of the atmospheric structure must therefore be performed in an
iterative process starting with a constant C_D and AOA=0.
Descent Phase Reconstruction Methodology:
========================================
Vertical Trajectory:
-------------------
The descent phase altitude and descent speed reconstruction was
performed in an upward direction, starting from the time of probe
impact as measured by the SSP impact penetrometer. The conversion of
the HASI atmospheric pressure P and temperature T measurements into
altitude z and descent speed dz/dt was based on the assumption of
hydrostatic equilibrium and the equation of state for a real gas. It
can be shown that the descent velocity is given by
dz/dt = -1/g(z)*R_univ*T*zeta/(mu*P)*dP/dt
where dP is the incremental change in atmospheric pressure, which is
related to the corresponding incremental change in altitude dz. g(z)
is the local acceleration of gravity, which needs to be recalculated
at each step according to g=G*M_0/z^2 (z can be approximated by the
previous reconstruction step z_(i-1) ). R_univ is the universal gas
constant (8314.3 J/kmol/K). mu is the mean molecular mass
(kg/kilomole), which can be inferred from the GCMS measured mole
fractions of nitrogen and methane f_N2and f_CH4 according to
mu=f_N2*mu_N2 + f_CH4*mu_CH4
where mu_N2 and mu_CH4 are the molecular masses of respectively N_2
and CH_4.zeta is the compressibility factor that takes into account
the deviation of the gas behavior from an ideal gas due to particle
interaction (van der Waals forces) and the impact of a finite
particle volume. Especially in the altitude ranges from the surface
up to about 70 km Titan's atmosphere is characterized by a
combination of relatively high densities (in the order of magnitude
of 10E-04 to 10E-02 g/cm^3 and low temperatures (100 to 93 K) which
are both drivers for a deviation of the atmosphere from an ideal gas
behavior. For the computation of the compressiblity we restricted
ourselves to the second virial coefficent B_2 and its relations to
zeta as given by [DYMOND1992]
zeta = 1 + B_2*rho/mu
For a gas mixture of N_2 and CH_4 the second virial coefficent was
composed in the following manner
B_2 (T) = f_N2^2*B_(2,N_2)(T) +
+ f_N2*f_CH4*B_(2,CH_4-N_2)(T) +
+ f_CH4^2*B_(2,CH_4)(T)
where B_(2,N_2), B_(2,CH_4), and B_(2,CH_4-N_2) are the temperature
dependent virial coefficients for the various pure gas and
interaction components, which were evaluated using polynomial fits of
laboratory measurements data as tabulated by [DYMOND1992]. With the
measured mole fractions taken from the GCMS measurements and the
derived virial coefficients from zeta values in the range from very
close to 1 (i.e., almost ideal gas behaviour) at altitude ranges
above 70 km and a continious decrease to values down to 0.965 (i.e.,
a deviation of 3.5 \% from the ideal gas law) close to the surface
were obtained.
Multiplying the expression for dz/dt by dt and integrating on both
sides yields
Delta_z=[z_i-z_(i-1)]=-1/g*R_univ*T_(i-1/2)*zeta/mu*ln[P_i/P_(i-1)]
T was considered to be constant throughout the altitude interval
Delta_z and was approximated by the mean value of two subsequent
measurements, i.e., T_(i-1/2)=1/2(T_i+T_(i-1) ). Starting from the
initial value z_0 at Titan's surface the final altitude was
derived by simple addition of the subsequently derived altitude
intervals Delta_z
z_i=z_0+ sum_i Delta_z_(i-1)
Note that the minus sign in the equation above ensures that for a
reconstruction starting from the surface in an upward direction the
pressure gradient has to be negative (i.e., P_i < P_(i-1) and
Delta_z therefore positive. Assuming a constant descent velocity for
the descent interval Delta_z the descent velocity was approximated
by
v = dz/dt ~ Delta_z/Delta_t
It should be noted that the atmoshperic temperature and pressure
based altitude profile provides the radial probe distance from the
probe impact point neglecting Titan's flattening. The integration of
the equations of motion during the entry phase reconstruction
however provides the distance of the probe to Titan's center, which
requires the assumption of a planetary radius (i.e., 2575 km) to
obtain the so called reference surface altitude.
Horizontal Trajectory
---------------------
Two sources of wind drift measurements are available 1) the DWE
zonal wind measurements, and 2) the image derived zonal and
meridional drifts from the Descent Imager and Radial Spectrometer
(DISR) instrument. Note that both the DWE and the DISR derived drift
measurements required as an input the vertical probe descent profile
(i.e., altitude and descent speed), which was provided to the
instrument teams by the Descent Trajectory Working Group (DTWG). The
probe horizontal motion reconstruction was therefore carried out in
an iterative process between the DTWG, DWE, and DISR teams.
In the body-fixed Titan system the Huygens velocity vector can be
decomposed into its radial, zonal, and meridional components
according to where prograde zonal is considered as the motion from
west to east and meridional from south to north. From the kinematic
equations for a spherical coordinate system one can easily derive
the change in longitude and latitude caused by zonal and meridional
drift (in radians/second)
d lambda/dt = v_zon(t)/[ R(t)* cos phi(t) ]
d phi/dt = v_mer(t)/R(t)
where R(t) is the instantaneous distance of the probe from the
center of Titan and is simply given by R=R_P+z_i, lambda is the
probe longitude, and phi is the planetocentric latitude. The
independent reconstruction of the probe vertical trajectory was
previously outlined and yields the altitude profile z_i. It can be
seen that above equation is a system of two coupled first order
differential equations which can be solved by numerical integration
since all time dependent variables are known at each integration
step. It should also be noted that the coupling in vanishes for
small latitude ranges and could therefore in the case of Huygens
also be solved with reasonable accuracy if no meridional drift
measurements were available, and the latitude would have been kept
constant over the integration time span.
Descent Phase Reconstruction Results:
=====================================
The reconstructed descent phase altitude and descent speed profiles
are plotted in DESC_ALT.JPG, DESC_VEL1.JPG, and DESC_VEL2.JPG (see
BROWSE directory). DESC_ALT.JPG compares the reconstructed probe
altitude (solid line) to the preflight simulation (dashed line). It
can be readily seen that the simulated descent trajectory predicted
the probe impact to occur about 14 minutes earlier. In large part,
this is difference can be contributed to the uncertainties in the
main and drogue chute aerodynamic database.
The reconstructed descent speed profile is shown in DESC_VEL1.JPG
and DESC_VEL2.JPG. DESC_VEL1.JPG zooms into the time of main chute
release shown by the vertical dashed line labeled as Drogue. The
solid line is again the reconstructed descent speed profile and the
dashed line the preflight simulation. It can be seen that during the
time of main chute the actual descent speed was about 5 m/s lower
than predicted in the simulations. This is also very likely the
cause for the longer flight time in the actual mission. Also shown
in the same panel is the descent speed profile from the entry phase
reconstruction. One can see that even if the descent phase velocity
shows some oscillations during the main chute phase, the overall
trend agrees well with the smooth velocity profile from the entry
phase reconstruction.
DESC_VEL2.JPG depicts the descent speed profiles during the drogue
cute phase down to probe surface impact. The descent speed
reconstruction predicts a probe impact speed of ~4.6 m/s. This is
consistent with the speed derived from the altitude measurements of
the SSP-APIS sensor, which is also shown as a single triangle.
Comparing the reconstructed altitude profile with the Radar
Altimeter Unit (RAU) meaurements one gets residuals from a few
meters close to the surface up to more than 3.5 km at an altitude of
60 km can be seen. Above 1 km altitude the error bar of the
reconstructed trajectory is smaller than the residuals. The altitude
discrepancy between the RAU measurements and the reconstructed
trajectory can at this point not be fully explained. The following
facts however speak for a higher confidence in the reconstructed
trajectory
- the reconstructed descent trajectory provides a much better
fit in the merging process with the entry phase reconstruction;
- the reconstructed descent trajectory is consistent with the
descent altitude as well as derived impact speed of the SSP acoustic
sounder instrument;
- the reconstructed descent trajectory could be confirmed by
independent reconstruction efforts (B. Blanchard and S. Streepe,
private communication);
The integration of the horizontal probe drift was started at
T_0+118.0 sec (= interface epoch + 6.44 min corresponding to an
altitude of ~144.2 km) and the corresponding initial conditions
are
Epoch = T0+ 118.0 sec
T0 = ET 158965885.184
T0 = UTC 2005-01-14T09:10:20.99
Altitude = 144.2 km w.r.t surface impact point
West Longitude = 195.94 deg
South Latitude = 10.33 deg
Further inputs to the drift integration are the previously
reconstructed vertical descent altitude profile, the zonal wind
speed as provided by DWE, and the meridional drift derived from 236
images of the DISR instrument containing significant surface detail.
Note that even if the DISR image based drift trajectory is only well
constrained below altitudes of 30 km the DISR team was able to
extend the trajectory up to an altitude of 150 km, which could then
be used in the DTWG horizontal probe drift integration.
The residuals to the DISR based reconstruction results stay below
0.026 deg in longitude and 7.6E-04 deg in latitude.
The currently estimated surface impact coordinates are 192.32 +-
0.24 deg West longitude and 10.25 +- 0.17 deg South latitude.
Merging of Entry and Descent Phase
==================================
The reconstruction of the entry phase trajectory required as an
essential input the probe state vector at the interface epoch, which
was provided together with the estimated uncertainties by the
Cassini navigation team (see EVENT_FILE_04042005.TXT in GEOMETRY
directory). After separation from the Cassini spacecraft the
Huygens probe was on a more than 20 days coast phase with no direct
earth-based tracking information. Prior to the mission it was
already clear that this would increase the delivery uncertainties at
the interface epoch. A probe imaging activity was therefore
introduced between the period of probe separation and Orbit Delay
Maneuver, which through the use of optical navigation images of the
probe, reduced the state vector uncertainties at the interface
point. Even with the probe post-separation images, the radial
uncertainty still remained at about 30 km (without images the radial
uncertainty was on the order of about 70 km).
The initial state vector uncertainties introduced the following
problems in the reconstruction effort:
1. The initial state vector introduced a systematic error in the
position vector, resulting in an incorrect modeling of the
gravitational forces with respect to the measured accelerometer
values. In other words the equations of motion were filled with
inconsistent data, which introduced an error in the reconstructed
state vector. The errors at each integration step summed up and
finally led to a largely erroneous solution.
2. As a result of 1., the integrated trajectory diverged significantly
from the descent phase trajectory in the regions of overlap.
A strategy to merge the entry and descent phase was therefore
developed. This method is described in detail by [KAZEMINEJAD2007]
and only a brief overview is given here:
The entry and descent phase trajectory portions provided profiles of
altitude and descent speed, which for each phase was based on
independent sets of input data and reconstruction techniques. The
entry phase trajectory was based on the entry state vector and the
measured aerodynamic accelerations whereas the descent phase
trajectory was based on the atmospheric temperature, pressure, and
mole fraction measurements. The important key fact here is that both
trajectories overlap in the time interval from about 6.3 to 22
minutes past interface epoch. This corresponds to an altitude range
of about 145 to 100 km. In this overlap region residuals in
altitude and descent speed were calculated and used as input in a
weighted least-squares estimation algorithm. The task of the
statistical estimation algorithm was then to calculate initial state
vector corrections, which would ensure a smooth transition between
the entry and descent phase trajectory portions in terms of both
altitude and descent speed.
Note that the residuals in the horizontal components (i.e.,
meridional and zonal probe drift) were not considered in the
estimation algorithm. This simply stems from the fact that at the
point of the algorithm design it was not expected to obtain any
reliable measurements in the meridional direction from the probe
scientific payload. An extension of the DTWG tool estimation
algorithm to take into account the horizontal resiudals clearly
remains a possibility of future improvement of the algorithm.
The overlap of the entry and descent phase trajectory can also be
seen by comparing the entry files (i.e.,
HUY_DTWG_ENTRY_EME2000_POS.TAB and HUY_DTWG_ENTRY_EME2000_VEL.TAB
with the corresponding descent files HUY_DTWG_DESCENT_POS.TAB and
HUY_DTWG_DESCENT_VEL.TAB). The altitude difference between entry
and descent phase reconstruction in the overlaping region
remains less than 0.6 km. This remaining inconcistency is due to
1. the different methodologies used for the two reconstruction
portions;
2. the different instrument data sets (and their respective
measurement uncertainties);
3. the different definition of altitude in the two reconstruction
portions: in the entry phase reconstruction the altitude is
considered as height above the reference surface of 2575 km,
whereas in the descent phase reconstruction the altitude is
reconstructed as height w.r.t. the impact point on the surface.
Note that the fact that the altitude residuals do not differ by
more than 0.6 km can be considered as constraint of the
topography in the landing site area of being not higher than 600
m.
For the user it is important to note that both profiles are (even
if slightly different) consistent within the resepective measurement
and reconstruction uncertainties. In case a smooth descent speed
profile is needed in ther overlapping region (i.e., 145 - 100 km)
the user might consider to use the results from the entry phase
reconstruction.
Reconstruction Uncertainties
============================
In both the entry and descent phase portions an error estimation of
the reconstructed trajectory was done. In the entry phase
reconstruction the reconstruction uncertainties were determined by
numerical propagation of the 14 x 14 covariance matrix, which was
provided to the project by the Cassini Navigation team (see
EVENT_FILE_04042005.TXT in the GEOMETRY directory). The main
contribution of the uncertainties come from the initial dispersion
at the interface epoch.
For the descent phase the altitude error was derived from the sum of
the errors at each altitude interval sigma Delta_zi (see Equation
in the description of the descent phase reconstruction). sigma
Delta_zi was calculated at each time interval t_i based on the law
of error propagation. For the compuation of the altitude error the
1-sigma uncertainties of the temperature measurements (i.e., +- 0.25
K and +- 1 K in the measurement ranges from respectively 60K to 110K
and 110K to 330K), the pressure measurements (i.e., 1\% of the
measurement value), the mean molecular mass (1\% of the inferred
value), and the gravitational acceleration $\sigma(g)$ were taken
into account. For the evaluation of sigma(g) an error of Titan's GM
of 0.2818 km^3/s^2 and Titan's radius of 0.5 km were taken into
account. Note that the error of the compressibility factor zeta, the
universal gas constant R_univ, and the ratio of specific heats gamma
were neglected.
IMPORTANT: The descent velocity uncertainties were simply derived
from the time derivatives of the altitude uncertainties. Even if
mathematically correct, the resulting uncertainties are extremely
large (even getting to the order of magnitude of the reconstructed
value itself). These uncertainties are not clealry not realistic and
were just added for the sake of completeness.
LIST OF DTWG Members
====================
Chair/DWE
David H. Atkinson
Dept. of Electrical & Computer Eng.
University of Idaho, Box 441023
Moscow, Idaho 83844-1023
USA
atkinson@ece.uidaho.edu
+1 (208) 885-6870 (phone)
+1 (208) 885-7579 (fax)
Co-Chair
Bobby Kazeminejad
German Space Operations Center (GSOC)
German Aerospace Center (DLR)
82234 Wessling, Germany
Bobby.Kazeminejad@dlr.de
+49 8153 28 2603 (phone)
+49 8153 28 1450 (fax)
Huygens Project Scientist
Jean-Pierre Lebreton
Huygens Project Division
Keplerlaan 1, PO Box 299
2200 AG Noordwijk, ZH
The Netherlands
+31 71 565 3600
+31 71 565 4697
Jean-Pierre.Lebreton@rssd.esa.int
Huygens Operation Scientist
Olivier Witasse
Huygens Project Division
Keplerlaan 1, PO Box 299
2200 AG Noordwijk, ZH
The Netherlands
olivier.witasse@rssd.esa.int
Cassini Project Scientist
Dennis Matson
Jet Propulsion Laboratory, M/S 230-205
4800 Oak Grove Drive
Pasadena, CA 91109-8099
+1 (818) 354-2253
+1 (818) 393-4495 (fax)
dennis.l.matson@jpl.nasa.gov
ACP
David Coscia
Univ. Paris 12
AV. General de Gaulle
94000 Creteil
France
coscia@lisa.univ-paris12.fr
GCMS
Hasso Niemann
NASA GSFC
Mail Code 915
Greenbelt, Md. 20771
Tel.: 301-614-6381
Fax.: 301-614-6406
Hasso.B.Niemann@nasa.gov
HASI
Francesca Ferri
CISAS “G. Colombo”, Univ. of Padova
Via Venezia 1
35131 Padova, Italy
390498276798 (phone)
390498276788 (fax)
francesca.ferri@unipd.it
DISR
Chuck See
Lunar and Planetary Laboratory
Department of Planetary Sciences
University of Arizona
Tucson, AZ 85721-0092
+1 (520) 621-1097
csee@lpl.arizona.edu
GCMS/IDS
Jonathan Lunine
Lunar and Planetary Laboratory
Department of Planetary Sciences
University of Arizona
Tucson, AZ 85721
USA
jlunine@lpl.arizona.edu
SSP
Brijen Hathi
PSSRI
The Open University
Walton Hall
Milton Keynes
MK76AA
UK
B.Hathi@open.ac.uk
+44 (0) 1908-659593
IDS
Daniel Gautier
Obs. de Paris-Meudon
92195 Meudon, Cedex, France
33-1-45077707 (phone)
33-1-45077469 (fax)
daniel.gautier@obspm.fr
IDS
Francois Raulin
Lisa Universite Paris 12
Ave. General de Gaulle
94000 Creteil, France
33-1-45171560 (phone)
33-1-45171564 (fax)
raulin@univ-paris12.fr
Radar
Ralph Lorenz
Lunar and Planetary Laboratory
Department of Planetary Sciences
University of Arizona
Tucson, AZ 85721
rlorenz@lpl.arizona.edu
Additional DTWG Support
Michael Allison
NASA/Goddard Institute for Space Studies
2880 Broadway
New York, NY 10025
Phone: (1) 212/678-5554
FAX: (1) 212/678-5552
pcmda@giss.nasa.gov
Michael K. Bird/DWE
Radioastronomisches Institut der Universität Bonn
Auf dem Hügel 71
D-53121 Bonn
F. R. Germany
Phone: (49) 228 / 73-3651
Fax: (49) 228 / 73-3672
mbird@astro.uni-bonn.de
Robindro Dutta-Roy/DWE
Radioastronomisches Institut der Universität Bonn
Auf dem Hügel 71
D-53121 Bonn
F. R. Germany
Phone: (49) 228 / 73-3783
Fax: (49) 228 / 73-3672
e-mail: duttaroy@astro.uni-bonn.de
Vincent Gaborit/HASI
Observatoire Paris-Meudon
5 Place Jules Janssen
92 Meudon
France
vincent.gaborit@obspm.fr
Nadeem Ghafoor/SSP
University of Kent
Unit for Space Sciences and Astrophysics
Canterbury, Kent UK-CT2 7NR
UK
(01483) 873346
+44 (0) 08 898 8340
nalg1@ukc.ac.uk
n.ghafoor@sstl.co.uk
Guy Israel/ACP
Service d'Aeronomie du CNRS
B.P. No 3
Verrieres-le-Buisson 91371
France
guy.israel@aerov.jussieu.fr
Mark Leese/SSP
PSSRI
The Open University
Walton Hall
Milton Keynes
MK76AA
UK
M.R.Leese@open.ac.uk
+44 (0) 1908-652561
Lisa McFarlane/DISR
Lunar and Planetary Laboratory
Department of Planetary Sciences
University of Arizona
Tucson, AZ 85721-0092
520-626-5363
lmcfar@lpl.arizona.edu
Miguel Perez/ESTEC PST
Huygens Project Division
Keplerlaan 1, PO Box 299
2200 AG Noordwijk, ZH
The Netherlands
mperez@rssd.esa.int
Mike Prout/DISR
Lunar and Planetary Laboratory
Department of Planetary Sciences
University of Arizona
Tucson, AZ 85721-0092
prout@pirl.lpl.arizona.edu
Bashar Rizk/DISR
Lunar and Planetary Laboratory
Department of Planetary Sciences
University of Arizona
Tucson, AZ 85721-0092
+1 (520) 621-1160
bashar@lpl.arizona.edu
Anne-Marie Schipper/Alcatel
100 boulevard du Midi
BP99
06322 Cannes La Bocca
France
anne-marie.schipper@space.alcatel.fr
Claudio Sollazzo/PST
Jet Propulsion Laboratory
4800 Oak Grove Drive - MS 230-250
Pasadena, CA 91109
USA
+1 (818) 393-3811
+1 (818) 393-4217 (fax)
claudio.sollazzo@jpl.nasa.gov
Marty Tomasko/DISR
Lunar and Planetary Laboratory
Department of Planetary Sciences
University of Arizona
Tucson, AZ 85721
+1 (520) 621-6919
mtomasko@lpl.arizona.edu
John Zarnecki/SSP
PSSRI
The Open University
Walton Hall
Milton Keynes
MK76AA
UK
J.C.Zarnecki@open.ac.uk
+44 (0) 1908-659599
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