Rarefied Aerothermodynamics of Mars Odyssey
Naruhisa Takashima1 and Richard G. Wilmoth2
1
2
AMA Inc., Hampton, VA 23666
NASA Langley Research Center, Hampton, VA 23681
Abstract. On January 11 of 2002, Mars Odyssey successfully completed the aerobraking phase of the mission, becoming
the second successful planetary mission designed specifically to utilize aerobraking as a primary means of achieving its
mission objectives. Direct simulation Monte Carlo and free-molecular analyses were used to provide aerothermodynamic
characteristics of the spacecraft for mission planning, flight operations, and atmospheric reconstruction. The results of
these analyses were used to develop an aerodynamic database that was used for numerous trajectory simulations both
prior to and during aerobraking operations, and to reconstruct atmospheric density profiles during each pass. The
aerodynamic database was also used together with data obtained from on-board accelerometers to reconstruct the
spacecraft attitudes throughout each aerobraking pass. The reconstructed spacecraft attitudes are in good agreement with
those determined by independent on-board inertial measurements for all aerobraking passes. The differences in the pitch
attitudes are significantly less than the preflight uncertainties of ±2.9%. The differences in the yaw attitudes are
influenced by zonal winds. When latitudinal gradients of density are small, the differences in the yaw attitudes are
significantly less than the preflight uncertainties. Direct simulation Monte Carlo simulations were also performed to
provide aerodynamic heating inputs for detailed thermal analyses of the Odyssey solar panels. Predictions of solar panel
temperatures compared well with those from thermocouple measurements obtained during aerobraking.
INTRODUCTION
On January 11, 2002, NASA’s Mars Odyssey spacecraft successfully completed its aerobraking phase of the
mission, becoming the second successful planetary mission designed specifically to utilize aerobraking as a primary
means of achieving its mission objectives. Launched on April 7, 2001 aboard Boeing’s Delta II 7925, Mars Odyssey
completed its Mars Orbit Insertion (MOI) burn on October 24, 2001 placing the spacecraft in a highly elliptical
capture orbit. After completing the “walk in” phase of aerobraking, where the periapsis altitude was reduced to
approximately 130 km, the “main phase” of aerobraking was commenced. During the next 75 days, the period of the
orbit was gradually reduced from the initial period of 18 hours to approximately 2 hours, when the “walk out” phase
was initiated and the spacecraft was placed in its final 400 km circular science orbit.
Aerobraking utilizes the atmospheric drag to make gradual changes in the orbit. Prior to the Mars Odyssey
mission, aerobraking was successfully used in two missions. The first application of aerobraking in a planetary
mission was during the Magellan mission at Venus where the eccentricity of the orbit was reduced from 0.39 to 0.03
in about 70 days [1]. The second application of aerobraking was for the Mars Global Surveyor (MGS) mission. For
the MGS mission, aerobraking was an enabling technology, where the reduction in the propulsive capability
afforded by the use of aerobraking was needed to satisfy the payload capabilities of the Delta II launch vehicle,
during a November 1996 Earth-to-Mars launch opportunity. A total of approximately 900 aerobraking orbits, which
decreased the orbit period by approximately 43 hours, were accomplished in two phases during its mission [2].
Like the two predecessors, the primary drag surfaces of Mars Odyssey are its solar arrays and the pace of
aerobraking is dictated by the solar array heating. The rate at which the period of the orbit is reduced by aerobraking
is dictated by achieving desirable local true solar time (LTST) at the end of aerobraking while keeping the
temperature of the solar arrays below of that of the flight allowable solar array temperature of 175° C. This was
accomplished by keeping the spacecraft within a specified freestream heating rate corridor during aerobraking.
Based on MGS aerobraking experience, 80% 2σ orbit-to-orbit natural atmospheric density variation was allocated
for the mission. Applying additional safety margins to those numbers, the top of the corridor was defined as
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
218
providing 100% margin to the flight allowable freestream heating rate. The bottom of the corridor was defined by
subtracting the width of the corridor, which was set at 0.18 W/cm2, from top of the corridor. The spacecraft was kept
within the corridor by monitoring the atmospheric densities and performing periodic aerobraking propulsive
maneuvers (ABMs).
All of the aerobraking took place at altitudes where the densities are sufficiently low that the flow is in the
rarefied transitional regime. To accurately predict the aerothermodynamic environment of the spacecraft in the
rarefied transitional flow regime, Direct Simulation Monte Carlo (DSMC) and free molecular techniques were used.
The results from the calculations were used to create the aero/aeroheating database of the spacecraft that was used
extensively in both pre-flight predictions and in-flight analyses, and played a key role in success of the aerobraking
phase of the mission.
COMPUTATIONAL METHOD
The DSMC calculations were performed using DDAC, which is the parallel implementation of the program
DAC (DSMC Analysis Code) [3]. In DAC, the gas collisions are modeled using the variable-hard-sphere (VHS)
model developed by Bird [4], and the Larsen-Borgnakke model is used for internal energy exchanges. The
geometry surface is represented by unstructured triangular elements that are embedded in a two-level Cartesian grid
for the flow field calculation. The solution from the first level of grid cells, which are uniform in size, is used for
grid refinement to create the second-level cells. The grid is refined based on local conditions, thus allowing the
program to meet the spatial resolution requirements without excessive global refinement. The grid cells are typically
refined such that on average the second-level cells have dimensions less than the local mean free path. The local
simulation parameters are set such that there are 10 simulated molecules in each cell, and the local time step is
typically dictated by the local flow time for the problems considered. Additional details of the code can be found in
Ref. [5].
For all calculations the wall collisions were assumed to be fully diffuse, i.e., an accommodation coefficient of
one was specified, with spacecraft wall temperature of constant 300 K. The composition of Mars atmosphere was
assumed to be 95.37% CO2 and 4.63% N2 by mole with a freestream temperature of 144.7 K and velocity of 4811
m/s. A reference temperature of 300 K and a viscosity-temperature-index of 0.71 were used for the VHS model. The
computational geometry shown in FIGURE 1 was provided by Lockheed Martin Astronautics (LMA) and represents
the best pre-flight estimate of the nominal aerobraking configuration.
Free molecular and continuum results were obtained using DACfree [6]. DACfree is a companion code to DAC,
which utilizes the same unstructured triangular surface mesh. The free molecular forces, moments and heat transfer
are calculated with analytical free-molecular analysis and line-of-sight shadowing technique, and a modified
Newtonian method is used to calculate the continuum forces and moments on the geometry. The continuum results
Z
Y
X
Velocity
of
Spacecraft
Nadir
FIGURE 1. Computational Geometry Model
219
are used mainly to guide the development of curve fitting or bridging-function techniques for the aerodynamic
coefficients since Odyssey aerobraking always took place at Knudsen numbers well above those for continuum
flow.
RESULTS
Aerothermodynamic analyses were performed to support a number of different aspects of the Odyssey mission.
The first objective was to develop a complete aerodynamic model that could be used for various trajectory
simulations both prior to and during aerobraking operations and to extract atmospheric density profiles based on
accelerometer data during each aerobraking pass. The second objective was to develop a complete aeroheating
database that could be used in a detailed thermal analysis to predict and reconstruct temperatures on the Odyssey
solar panels.
Aerodynamic Analyses
To satisfy the first objective, a number of different analysis techniques were required. First, DSMC solutions
were obtained for selected atmospheric densities and spacecraft attitudes (pitch and yaw angles). Aerodynamic
coefficients were extracted from this matrix of DSMC solutions, and the matrix was then enriched using freemolecular methods to provide more detailed variations of the various coefficients with pitch and yaw. In effect, the
free-molecular results were used to develop parametric curve fits that passed through the DSMC results. Additional
curve fits were then determined to account for the variations in aerodynamic coefficients with density provided by
the DSMC results. The complete aerodynamic database was then incorporated into three-degree-of-freedom
(3DOF) and six-degree-of-freedom (6DOF) trajectory simulations and was also used in analysis of flight
accelerometer data to determine atmospheric densities. Finally, the aerodynamic force coefficients were used
together with the three-component accelerometer data to determine the relative wind attitude of the spacecraft. This
attitude could then be compared with that obtained from independent measurements of the inertial attitude of the
spacecraft and the trajectory determined from other navigational data.
DSMC Solutions
FIGURE 2 shows the non-dimensionalized density contour plots in a plane approximately 1 m above the bottom
of the spacecraft for freestream densities of 10 kg/km3 and 100 kg/km3, where the latter value represents the highest
density expected to be encountered during aerobraking. The plots show the typical diffuse shock layers that occur in
rarefied transitional flow, with the layer getting pressed to the surface as the freestream density increases. FIGURE 3
shows the surface pressure contours for a freestream density of 100 kg/km3 at the nominal attitude. The plot shows
that the spacecraft bus shields the center solar array and the edges of outboard solar arrays from the on-coming flow.
The total number of molecules in the simulations performed varied from 0.5 million for 0.1 kg/km3 runs to 2.5
million for 100 kg/km3 runs. Most cases were run for over 10,000 time steps to ensure adequate sample size.
Aerodynamic Database
The aerodynamic database of Mars Odyssey was constructed by combining results from free
molecular/continuum analysis computed with DACfree and DSMC results computed with DDAC. Free-molecular
analyses were used to provide variations of aerodynamic coefficients vs. spacecraft attitude (pitch and yaw), and
DSMC calculations performed over a limited range of attitudes and atmospheric densities were used to account for
variations in coefficients with freestream density. Once the solutions were obtained, multivariate curve fits were
performed to construct an “enriched” database of aerodynamic coefficients with sufficient resolution for use in both
3DOF and 6DOF trajectory simulations. These curve fits covered a pitch and yaw attitude range of ±60° and density
range of 10-4 to 2500 kg/km3 where the lower value represents the free molecular limit and the higher value
represents the continuum limit.
The aerodynamic computational matrix was defined based on rotation angles in the spacecraft body coordinate
systems, where pitch (θ) is defined as the first rotation about the X-axis and yaw (φ) is defined at the second rotation
about the Z-axis. Free molecular calculations were performed for yaw and pitch angles of -60° to +60° in 5-degree
220
ρ/ρ∞
1.5
2.6
68
92
54.7129
52.044
49.3751
46.7062
44.0372
41.3683
38.6994
36.0305
33.3615
30.6926
28.0237
25.3548
22.6858
20.0169
17.348
14.6791
12.0102
9.34123
6.67231
4.00339
1.33446
55.6101
29.3582
ρ/ρ∞
181.867
172.849
163.83
154.812
145.794
136.775
127.757
118.739
109.72
100.702
91.6835
82.6651
73.6468
64.6284
55.6101
46.5917
37.5734
28.555
19.5367
10.5183
1.5
FIGURE 2. Nondimensionalized density contour plots.
p
2.50
2.32
2.14
1.96
1.79
1.61
1.43
1.25
1.07
0.89
0.71
0.54
0.36
0.18
0.00
FIGURE 3. Surface pressure contours for freestream density of 100 kg/km3 at the nominal attitude.
increments. DSMC calculations were performed for densities of 0.l, 1.0, 3.162, 10.0, 31.62 and 100 kg/km3 at pitch
and yaw angles of -60°, 0° and +60°, resulting in total of 54 DSMC calculations.
The database was enriched by assuming that the shape of each coefficient curve for a given angle sweep at any
density is the same as the free molecular result and that values of each aerodynamic coefficient approach free
molecular values as the density decreases and Newtonian values (which are also calculated by DACfree) as the
density increases. For a given density, the free molecular coefficient curve in pitch was scaled using the DSMC
results, and the curve was offset to match the coefficient value at φ = 0° for each pitch angles as shown in FIGURE
4. By repeating the procedure, but exchanging the direction and performing the scaling and offset for every 5° in
yaw angle, the variations of force and moment coefficients with attitude are determined. FIGURE 5 shows the
contour plots of the force coefficients for a freestream density of 10 kg/km3 and the variation of axial force
coefficient with freestream density for the nominal attitude of yaw and pitch angles of 0 degree. The line in the
density variation plot is formed with the values returned from the interpolation routine that accompanies the
aerodynamic database. The error bars of ±2.9% represent the estimated uncertainties in the aerodynamic database.
The sources of uncertainty include computational errors, physical model errors and boundary condition errors. The
symbols in the figure represent all the DSMC runs that were made to establish the uncertainty due to computational
errors. The uncertainty value of ±2.9% is primarily due to the uncertainty in the accommodation coefficient.
221
3
ρ= 31.62 kg/km
Interpolated
Free Molecular
DSMC
2
θ=0
o
1.75
Cy
1.5
1.25
1
θ = -60
o
0.75
θ = +60
0.5
-50
o
0
50
φ
FIGURE 4. Aerodynamic database enrichment.
2.25
1
0
Cx
Cx
0.94
0.80
0.67
0.53
0.40
0.27
0.13
0.00
-0.14
-0.27
-0.40
-0.54
-0.67
-0.81
-0.94
ρ = 10.0 kg/km3
-1
-50
2
50
θ
0
0φ
50
Aero Model
DSMC Data
Mars 2001 Odyssey Aero Database
for Nominal Aerobraking / Safemode Configuratio
For θ = 0° and φ = 0°
Error bars: +/- 2.9 %
Cy
-50
50
0 φ
θ0
50
-50
Cz
1.10
0.94
0.78
0.63
0.47
0.31
0.16
0.00
-0.16
-0.31
-0.47
-0.63
-0.78
-0.94
-1.10
1.75
1
0
Cz
Cy
Cy
1.89
1.62
1.35
1.08
0.81
2
0.54
0.27
1.5
0.00
-0.27
1
-0.54
-0.82
0.5
-1.09
-1.36
-50
-1.63
-1.90
-1
-50
50
θ0
1.5 -4
10
0φ
50
-50
10-3
10-2
10-1
100
ρ (kg/km3)
101
102
103
FIGURE 5. Force coefficient contour plots for freestream density of 10 kg/km3.and variation of axial force
coefficient at the nominal attitude with free stream density
Flight Data Analysis
FIGURE 6 shows the atmospheric density during aerobraking pass 183. The atmospheric density, ρ is
reconstructed using the equation,
ρ V 2C y A
(1)
a =
y
2m
where ay is the axial-acceleration, m is the mass, V is the velocity, Cy is the axial force coefficient and A is the
reference area of the spacecraft. Since the axial-force coefficient is a function of both spacecraft attitude and
freestream density, an iterative process is required to determine the density. The attitude of the spacecraft is
determined from the inertial measurement unit (IMU) data and the spacecraft velocity, which is calculated along an
aerobraking trajectory from the periapsis state with J2 gravitational term and assuming a rigid rotating atmosphere.
The axial acceleration is measured by the accelerometer on the spacecraft. The correct density is determined once
222
OPTG File: optg_ab_od183-183_211_v1;
Orbit: p183
Time (UTC): 14:13: 0 12/27/2001
Density (kg/km3): 30.321
Telemetry Type: HiRate
Dynamic Pressure (N/m2): 0.248
Altitude (km): 104.530
Scale Height (km): 5.672
50
Peak Density (kg/km3): 38.77 at -14.44 s.
3
Density (kg/km )
40
30
20
10
0
-200
-100
0
100
200
Time from Periapsis (s)
FIGURE 6. Density during aerobraking pass 183.
.
20
20
θ (IMU)
θ (Aero)
15
15
10
Yaw Angle (φ)
Pitch Angle (θ)
10
φ (IMU)
φ (Aero)
5
0
-5
5
0
-5
-10
-10
-15
-15
-20
-200
-100
0
100
-20
-200
200
Time from Periapsis (s)
-100
0
100
200
Time from Periapsis (s)
FIGURE 7. Spacecraft attitude comparison for aerobraking pass 183.
20
20
θ (IMU)
θ (Aero)
15
15
10
Yaw Angle (φ)
Pitch Angle (θ)
10
φ (IMU)
φ (Aero)
5
0
-5
5
0
-5
-10
-10
-15
-15
-20
-200
-100
0
100
-20
-200
200
Time from Periapsis (s)
-100
0
100
Time from Periapsis (s)
FIGURE 8. Spacecraft attitude comparison for aerobraking pass 170.
223
200
the product of density, axial force coefficient and known values equal the measured axial-acceleration. Details
concerning density determination and atmospheric modeling for the Mars Odyssey mission can be found in Ref. [7].
Since the accuracy of the atmospheric density data is directly linked to the accuracy of the aerodynamic database,
the performance of the aerodynamic database was monitored during the entire aerobraking phase of the mission by
comparing the “measured” spacecraft attitude with reconstructed data using accelerometer ratios and the
aerodynamic database. The ratios of accelerometer measurements, ax/ay and az/ay, are equivalent to the ratios of
force coefficients Cx/Cy and Cz/Cy, can be used to extract the spacecraft attitude from the database. FIGURE 7
shows spacecraft attitude comparison for aerobraking pass 183. Good agreement between the two sets of data, well
within the uncertainties, which are now a combination of the database uncertainty and the accelerometer uncertainty,
are observed through the entire aerobraking pass. This pass was atypical in that there was very little atmospheric
variability during this pass. Atmospheric analysis showed that there was very little latitudinal variation in density for
this particular pass. The majority of passes showed large atmospheric variability and the existence of zonal winds.
FIGURE 8 shows the spacecraft attitude comparison for aerobraking pass 170, a more typical pass. The influence of
zonal winds can be observed in the difference in the yaw near periapsis. Overall, comparisons from all aerobraking
passes show that the aerodynamic database and the model reconstruct the flight data with the expected accuracy.
Aeroheating Analyses
To satisfy the aeroheating objective, detailed predictions of the local heat flux to the surface of the solar panel
were required. A full thermal analysis requires knowledge of the heat flux at each surface element of its
computational mesh, and this requirement presented a slightly different challenge than that for aerodynamics. Since
the amount of data required for each flight condition was much greater than for the aerodynamic-based analyses, a
more limited number of DSMC computations were performed. These computations used the same computational
geometry as the aerodynamic computations but were run for longer times to increase the local surface sample sizes
in the DSMC simulations. Since all DSMC computations used the same computational mesh for the surface, it was
then possible to develop a multivariate interpolation technique at each surface mesh element to describe the local
heat flux as a function of atmospheric density and spacecraft attitude. Therefore, a complete map of surface heat
flux could be obtained for any flight condition during aerobraking.
Representative surface heating maps from both free molecular and DSMC analyses are shown in FIGURE 9.
The low heat flux on the center panel is caused by shadowing by the spacecraft bus. The details of the thermal
analyses are beyond the scope of this paper since numerous other factors, such as solar flux, self re-radiation,
material thermal properties, etc. had to be incorporated. However, the temperatures predicted by these thermal
analyses generally showed relatively good agreement with thermocouple measurements obtained during
aerobraking, and the availability of detailed temperature maps provided information on the maximum temperatures
reached by the solar panel (which did not typically occur at the thermocouple locations) that increased the
confidence in the heating limits imposed on the mission.
DSMC
FreeMolecular
qi/0.5ρV3: 0.00 0.10 0.20 0.30 0.40 0.51 0.61 0.71 0.81 0.91
FIGURE 9. Typical surface heat flux predictions. Density=100 kg/km3, Pitch=-15 deg, Yaw=+15 deg.
224
CONCLUSIONS
DSMC and free-molecular methods were used to provide aerothermodynamic predictions for the Mars Odyssey
spacecraft. The predictions were used to create an aerodynamic database that was used for numerous trajectory
simulations both prior to and during aerobraking operations and to reconstruct atmospheric density profiles during
each pass. The aerodynamic database was also used together with data obtained from on-board accelerometers to
reconstruct the spacecraft attitudes throughout each aerobraking pass. The reconstructed spacecraft attitudes are in
good agreement with those determined by independent on-board inertial measurements for all aerobraking passes.
The differences in the pitch attitudes are significantly less than the preflight uncertainties of ±2.9%. The differences
in the yaw attitudes are influenced by zonal winds. When the latitudinal gradients of density are small, the
differences in the yaw attitudes are significantly less than the preflight uncertainties. Direct simulation Monte Carlo
simulations were also performed to provide aerodynamic heating inputs for detailed thermal analyses of the Odyssey
solar panels. Predictions of solar panel temperatures compared favorably with those from thermocouple
measurements obtained during aerobraking. These results demonstrate the capabilities of the current DSMC and
free-molecular analysis codes to accurately predict rarefied flow fields about complex spacecraft geometries.
ACKNOWLEDGMENTS
The authors would like to acknowledge Doug Gulick, Lockheed-Martin Astronautics (currently with Ball
Aerospace), for providing the computational geometry
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of Spacecraft and Rockets, 36, No. 3, May-June, 1999.
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