AIAA 2009-6470
AIAA SPACE 2009 Conference & Exposition
14 - 17 September 2009, Pasadena, California
Interplanetary Trajectory Analysis for 2020-2040 Mars
Missions including Venus Flyby Opportunities
Takuto Ishimatsu∗, Jeffrey Hoffman†, and Olivier de Weck‡
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139
This paper develops a tool which is capable of calculating ballistic interplanetary trajectories with planetary flyby options based on the knowledge of astrodynamics and analyzes
Mars trajectories in the time frame 2020 to 2040, including transfer trajectories with Venus
flybys. Using the trajectory programs developed in this work, we investigate the relation
between departure and arrival dates and energy required for the transfer trajectories. The
contours of C3 or ∆Vtot for a range of departure dates and times of flight would be useful
for the creation of a long-term Earth-Mars and Mars-Earth transportation schedule for
mission planning purposes. For planetary flybys, we allow simple powered flybys with the
velocity impulse at periapsis to expand the flyby mission windows. Having obtained the
results for Earth-Mars direct trajectories by a full-factorial computation and Earth-VenusMars flyby trajectories by a "pseudo full-factorial" computation, we discuss the nature of
the trajectories and the competitiveness of Earth-Venus-Mars flyby trajectory windows
with Earth-Mars direct trajectory windows.
Nomenclature
C3
hm
r
rm
t
TOF
v
V∞
∆Vd
∆VPFM
∆Vtot
µ
=
=
=
=
=
=
=
=
=
=
=
=
Subscripts
1
=
2
=
3
=
a
=
d
=
e12
=
e23
=
i
=
o
=
2
Characteristic energy (= V∞
), km2 /s2
Minimum passing altitude at planetary flyby, km
Position vector
Minimum passing radius at planetary flyby, km
Date, day
Time of flight, day
Velocity vector
Hyperbolic excess velocity, km/s
∆V required for departure, km/s
∆V required for powered flyby maneuver, km/s
Total ∆V (= |∆Vd | + |∆VPFM |), km/s
Standard gravitational parameter, km3 /s2
Departure point in time
Encounter point in time
Arrival point in time
Arrival point on transfer trajectory
Departure point on transfer trajectory
Encounter point on pre-encounter trajectory
Encounter point on post-encounter trajectory
Inbound trajectory at planetary flyby
Outbound trajectory at planetary flyby
∗ Graduate
Research Assistant, Aeronautics and Astronautics, MIT, 33-409, AIAA Student Member.
of the Practice, Aeronautics and Astronautics, MIT, 37-227, AIAA Member.
‡ Associate Professor, Aeronautics and Astronautics, MIT, 33-410, AIAA Associate Member.
† Professor
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Copyright © 2009 by the American Institute of Aeronautics and
Astronautics,
Inc. All of
rights
reserved. and Astronautics
2
2
1.5
1.5
1
1
0.5
0.5
Y [au]
Y [au]
Mars
Sun
0
Venus
0
Sun
-0.5
-0.5
Earth
Earth
-1
-1
-1.5
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
-2
-2
2
Mars
-1.5
-1
X [au]
-0.5
0
0.5
1
1.5
2
X [au]
Figure 1. Example of Earth-Mars direct trajectory.
I.
Figure 2. Example of Earth-Venus-Mars flyby trajectory.
Introduction
ollowing NASA’s Vision for Space Exploration, the next several decades will see an increasing number
F
of both robotic and human explorations to Mars.
When we consider the Mars missions, in addition to
the system architecture, we should also focus on the transfer trajectories. Planning future missions requires
1, 2
trajectory data years in advance. This paper focuses on Mars trajectories including Venus flyby opportunities
in the time frame 2020 to 2040, during which such missions seem most relevant.
Practically, there are two main types of interplanetary trajectories from Earth to Mars: one is Earth-Mars
direct trajectories (Figure 1) and the other is Earth-Venus-Mars flyby trajectories (Figure 2). Therefore,
we develop a tool which is capable of calculating ballistic interplanetary trajectories with planetary flybys.
Since a flyby trajectory must specify three dates for departure, flyby encounter, and arrival, it is obvious that
a full-factorial computation is not feasible due to its huge amount of computational effort. However, since
Earth-Venus and Venus-Mars trajectories can be calculated independently, the problem can be decomposed
into two separate Lambert’s problems. We can cut corners dramatically by taking advantage of this; we can
perform a "pseudo full-factorial" computation by using memory cache obtained from the two problems that
have been separately computed in advance.
Having obtained the results by a "pseudo full-factorial" computation, we discuss the competitiveness of
Earth-Venus-Mars flyby windows with Earth-Mars direct windows. We also look at return trajectories, that
is, Mars-Earth direct trajectories and Mars-Venus-Earth flyby trajectories. In general, Mar-Venus-Earth
flyby trajectories are said to be inefficient since we first head for Venus, an inferior planet, which would
require higher energy for departure than a direct trip to Earth.
One of the important characteristics representing a trajectory is the energy required for departure, which
is called C3. In this paper, for distinction, let C3d and C3a be the energy required for departure and arrival,
respectively. Using the trajectory programs we developed in this work, we investigate the relation between
departure and arrival dates and C3d or ∆Vtot , which is the sum of ∆V required for departure (∆Vd ) and
∆V required for powered flyby maneuver (∆VPFM ).3 As shown later, the contours of ∆Vtot for a range of
departure dates (x-axis) and times of flight (y-axis) would serve as a ”visual calendar” of launch windows
and thus be useful for the creation of a long-term Earth-Mars and Mars-Earth transportation schedule for
mission planning purposes.
Planetary flybys are typically intended to save fuel by taking advantage of a planet’s gravity to alter the
path and speed of a spacecraft. However, an opportunity of such a free flyby is rare. Instead, if we make
some small burn at flyby, we would be able to expand the flyby mission window. Therefore in this study, we
allow simple powered flybys with a velocity impulse of ∆VPFM at periapsis with an altitude of hm .
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Input
- Departure date
- Arrival date
Setting
- planet 1 (departure)
- planet 3 (arrival)
for
for
Planetary
Ephemeris
Time of Flight
Direct Trajectory Data
Lambert's
Problem
C3 Calculation
Figure 4. Flow chart of producing direct trajectory data.
Output
- Departure C3
- Arrival C3
Figure 3. Flow chart of calculating a single direct trajectory.
II.
Full-Factorial Computation for Direct Trajectories
To determine an orbit from a specified transfer time and two position vectors, we need to solve Lambert’s
problem. Over the years a variety of techniques for solving Lambert’s problem have been developed. In
order to solve Lambert’s problem in more general way that is valid for all types of orbits, we introduce a
universal variable in the Lagrange coefficients.
The mission of a direct trajectory is to send a spacecraft directly from planet 1 to planet 3 in a specified
time between the departure and arrival dates. The flow chart in Figure 3 shows the overall structure of this
procedure.
Given planet 1, planet 3, and departure and arrival dates, determine a direct trajectory from planet 1 to
planet 3 and C3d and C3a in the following procedure:
1. Select departure and arrival dates.
2. Calculate time of flight, TOF, using the Julian day numbering system.
3. Calculate planetary ephemeris to determine state vector r1 and v1 of planet 1 at departure and state
vector r3 and v3 of planet 3 at arrival.
4. Use r1 , r3 , and TOF in solving Lambert’s problem to find a spacecraft’s velocity vd at departure from
planet 1 and its velocity va at arrival to planet 3.
2
2
5. Calculate C3d (= V∞d
) and C3a (= V∞a
).
For direct trajectories, the procedure of creating C3 contour plot is straightforward as seen in Figure
4: it is a full-factorial computation just by doubly wrapping Figure 3 in a couple of "for" statements for
departure and arrival dates. It only took less than a hundred hours to produce C3 contour plot for 2020-2040
TM
R
Earth-Mars direct trajectories (IntelCore
2 Duo processor at 2.40 GHz).
III.
Pseudo Full-Factorial Computation for Flyby Trajectories
For powered flyby maneuver, the change in velocity required at periapsis can be calculated as
r
r
2µ
2 + 2µ
2
− V∞i
∆VPFM = V∞o +
rm
rm
(1)
where µ is the standard gravitational parameter for a flyby planet.3 A free flyby corresponds to ∆VPFM = 0
since V∞i = V∞o . Ideally, ∆VPFM should be zero so that we would not need to consume any fuel. But it
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for
for
for
for
for
Direct Trajectory Data
Flyby Trajectory Data
Flyby Calculation
for
for
Direct Trajectory Data
Figure 5. Flow chart of producing flyby trajectory data.
would be worthwhile to look into the possibility of powered flyby, since allowing a small amount of ∆VPFM
might get us a much broader mission window.
The procedure of producing a contour plot for flyby trajectories is a bit tricky. If we simply conduct a
full-factorial computation with triple "for" statements for departure, flyby encounter, and arrival dates in
a manner similar to Section II, it will take tens of thousands of hours, which is totally impractical. But,
indeed, this procedure repeats the same calculations many times; after solving a flyby trajectory for specific
t1 , t2 , and t3 , for example, if we later solve for t1 , t2 , and t3 + 1, then we solve the same Lambert’s problem
for the combination of t1 and t2 . This duplicated computation occurs hundreds of times, which wastes a lot
of time.
Instead of such a straightforward approach, we should note the fact that the problem can be decomposed
into two separate problems, that is, Earth-Venus and Venus-Mars direct trajectories, which are independently
determined. A Venus flyby is only an interface between these two trajectories since what we need for a flyby
calculation are ve12 , which is a spacecraft’s velocity vector at Venus arrival from an Earth-Venus trajectory,
and ve23 , which is a spacecraft’s velocity vector at Venus departure on a Venus-Mars trajectory. Therefore,
we can in advance conduct full-factorial computations for these two problems separately and store the data
in memory cache. By doing this beforehand, we can just access the memory cache to obtain the information
of these velocity vectors, ve12 and ve23 , when we later on conduct flyby calculations.
By scanning t2 for given t1 and t3 , we can pick an optimal t2 that will minimize C3d for the combination
of t1 and t3 while satisfying all the constraints in Section IV. To save more time in this process, since we do
not want to calculate infeasible points with large C3d and C3a , we screened them out by the constraints in
Eqs. (2a) and (2b), and skipped a flyby calculation for such infeasible points.
Figure 5 shows this procedure of a "pseudo full-factorial" computation.
IV.
Mission Feasibility Criteria
As the criteria for mission feasibility, we define four constraints as follows:
C3d : C3d determines the launch feasibility. In a report from Jet Propulsion Laboratory (JPL), a feasible
launch assumed that C3d is less than 25 [km2 /s2 ].4 Considering envisioned advances in technology, we
use a C3d of 30 [km2 /s2 ] as a launch feasibility criterion.
C3a : For direct entry or orbit insertion at arrival, propulsive capture generally requires a minimum arrival
velocity, while aerocapture tolerates higher arrival velocities. Given an 8 [km/s] limit on Mars entry
velocity, C3a at Mars up to 40 [km2 /s2 ] is acceptable. On the other hand, from an analysis that was
carried out during the Draper/MIT CE&R project we know that Earth entry velocity up to 13 [km/s]
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is tolerable from a gravity load and heating perspective, which would rule out all trajectories with C3a
at Earth over 45 [km2 /s2 ].5
hm : For flyby missions, considering the Venusian atmosphere, the flyby trajectory must pass well above the
surface. We assumed that a minimum passing altitude hm of a feasible flight must be 100 km above
the surface.
∆VPFM : Allowing a large amount of |∆VPFM | would not make sense because we originally wanted to save
fuel by taking advantage of the Venus gravitational field. In this study, we selected 0.3 [km/s] as a
reasonable upper bound for |∆VPFM |.
2
C3d ≤ 30 [km2 /s ]

40 [km2 /s2 ] (Mars arrival)
C3a ≤
45 [km2 /s2 ] (Earth arrival)
hm ≥ 100 [km]
|∆VPFM | ≤ 0.3 [km/s]
V.
(2a)
(2b)
(2c)
(2d)
Integrated ∆Vtot Contours
Now that we have obtained the trajectory data for both direct and flyby cases, we will put them together
into a single chart to create a complete "launch window calendar." Since we have two contour plots for direct
and flyby on the same range in the t1 -t3 plane, selecting a superior one of the two trajectories at each point
on the grid gives a direct/flyby integrated contour plot.
If we use C3d for integration, however, it might be unfair since a flyby trajectory also requires ∆VPFM
for powered flyby maneuver. Instead, by converting C3d into ∆Vd , which is the ∆V required for departure,
we can uniformly treat the ∆V for departure and powered flyby. Thus we define
∆Vtot = |∆Vd | + |∆VPFM |
(3)
and use ∆Vtot instead of C3d for integration. In conversion from C3d to ∆Vd , we assumed a departure
hyperbola starting from a circular parking orbit with an altitude of 300 km.
VI.
Simulation Results and Discussions
Figure 6 shows the integrated ∆Vtot contours for 2020-2040 Earth-Mars trajectories. These contours
display ∆Vtot up to 4.8 [km/s]. Note that the contours were filtered by the constraints for C3d , C3a , hm ,
and ∆VPFM . Therefore, the "craters" in the figure mean feasible regions in terms of the above criteria. The
local minimum ∆Vtot in each crater is listed in Table 1 as a representative of the launch opportunity. We
determined the "competitiveness" of each opportunity by the following criteria.
• If two neighbor opportunities have a close departure date (a close arrival date), and one has an earlier
arrival date (a later departure date) than the other, the other is regarded as "dominated" since a short
time of flight would be preferable.
• If an opportunity does not have neighbors, the opportunity is non-dominated and thus regarded as
"competitive" since it would add a new launch window even if it has a relatively high ∆Vtot .
As a result, six out of nine flyby craters are competitive. In terms of launch windows, six out of seven
flyby windows are competitive. Earth-Venus-Mars flyby trajectories tend to have a relatively high ∆Vtot but
give new opportunities. Having more launch windows available gives us a flexibility of mission planning.
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Figure 7 shows the integrated ∆Vtot contours for 2020-2040 Mars-Earth trajectories. These contours
display ∆Vtot up to 4.2 [km/s]. The local minimum ∆Vtot in each crater is listed in Table 2 as a representative
of the launch opportunity.
As a result, four out of seven flyby craters are competitive. In terms of launch windows, all the four
flyby windows are competitive. Mars-Venus-Earth flyby trajectories have a much higher ∆Vtot and a longer
TOF but make more launch windows available. As stated in Section I, Mar-Venus-Earth flyby trajectories,
in general, are inefficient due to higher energy required for departure. However, it is interesting that all the
flyby windows are found to be "competitive" and thus significant since they open up additional opportunities
that cannot be replaced by the direct flight opportunities.
Figures 8 and 9 show 2020-2040 Earth-Mars and Mars-Earth flight opportunities, respectively. By scanning the ∆Vtot contours, we can obtain a bunch of lines between the nodes "LEO" and "LMO," each of
which connects departure node and date with arrival node and date as a feasible combination. Blue lines
represent direct flight opportunities and red lines flyby flight opportunities.
We can use these contours and flight opportunity charts to see the flexibility of mission schedule and
perform a trade-off analysis between departure and arrival dates, time of flight, departure and arrival C3’s,
and ∆Vtot , on a mission-by-mission basis.
VII.
Conclusions and Future Work
Using the trajectory programs we developed in this work, we obtained ∆Vtot contours, trajectory data
tables, and flight opportunity charts for 2020-2040 Mars missions including Venus flyby opportunities. A
"pseudo full-factorial" computation method proposed in this paper enabled us to visualize Venus flyby
windows as well as direct flight windows in the form of ∆Vtot contours. These data will serve as a useful
database to create a long-term Earth-Mars and Mars-Earth transportation schedule for mission planning
purposes. It was found that almost all of the Venus flyby opportunities are of significance since they give
additional launch windows that cannot be covered by the direct flight windows. Venus flyby trajectories
can be even more important if a permanent presence is established on Mars because they can be additional
re-supply windows. One of the future works for Venus flyby trajectories is to estimate the risk of a radiation
exposure due to mission duration and proximity to the sun.
The next step we should take is to apply these results to the actual design of more efficient missions.
The Vision for Space Exploration calls for the sustainable space exploration, which will require appropriate
interplanetary supply-chain management. Therefore, future Mars exploration will have to rely not on a
single mission but on a complex supply-chain network. The trajectory database obtained in this work will
be one of the necessary steps to develop a planning tool for space logistics.
Acknowledgments
The research described in this paper was carried out at the Massachusetts Institute of Technology and
was supported in part by the Japan Ministry of Education and in part by the Jet Propulsion Laboratory.
The author wishes to acknowledge the support of Prof. Richard Battin, Dr. Wilfried Hofstetter, and Shinya
Umeno at MIT.
References
1 President Bush, G. W., ”A Renewed Spirit of Discovery – The President’s Vision for U.S. Space Exploration”, The White
House, Washington, D.C., January 2004.
2 National Aeronautics and Space Administration, ”Exploration Systems Architecture Study (ESAS) – Final Report”,
NASA-TM-2005-214062, NASA, Washington, D.C., November 2005.
3 Ishimatsu, T., ”Interplanetary Trajectory Analysis for 2020-2040 Mars Missions including Venus Flyby Opportunities”,
Master’s thesis, Massachusetts Institute of Technology, Cambridge, MA, May 2008.
4 Matousek, S., Sergeyevsky, A. B., ”To Mars and Back: 2002-2020 Ballistic Trajectory Data for the Mission Architect”,
AIAA-98-4396, AIAA/AAS Astrodynamics Specialist Conference, Boston, MA, August 1998.
5 Crawley, E., et al., ”Draper/MIT Concept Exploration and Refinement (CE&R) Study – Final Report”, Massachusetts
Institute of Technology, Cambridge, MA, September 2005.
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700
4.8
600
4.6
TOF [days]
500
4.4
400
4.2
300
4
200
3.8
100
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
3.6
Departure date
700
4.8
600
4.6
TOF [days]
500
4.4
400
4.2
300
4
200
3.8
100
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
3.6
Departure date
Figure 6. 2020-2040 Earth-Mars ∆Vtot contours.
Table 1. 2020-2040 Earth-Mars trajectory data.
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Trajectory
2020 Earth-Mars
2020 Earth-Mars
2021 Earth-Venus-Mars
2021 Earth-Venus-Mars
2022 Earth-Mars
2022 Earth-Mars
2023 Earth-Venus-Mars
2024 Earth-Mars
2024 Earth-Mars
2026 Earth-Mars
2026 Earth-Mars
2028 Earth-Venus-Mars
2028 Earth-Venus-Mars
2028 Earth-Mars
2028 Earth-Mars
2030 Earth-Venus-Mars
2031 Earth-Mars
2031 Earth-Mars
2031 Earth-Mars
2033 Earth-Mars
2033 Earth-Mars
2034 Earth-Venus-Mars
2035 Earth-Mars
2035 Earth-Mars
2035 Earth-Mars
2036 Earth-Venus-Mars
2037 Earth-Mars
2037 Earth-Mars
2039 Earth-Venus-Mars
2039 Earth-Mars
2039 Earth-Mars
Departure
07/17/20
08/24/20
08/21/21
10/28/21
09/07/22
09/13/22
09/12/23
10/04/24
10/12/24
10/30/26
11/13/26
01/23/28
03/23/28
11/30/28
12/08/28
01/24/30
01/27/31
02/22/31
07/04/31
04/04/33
04/28/33
07/22/34
05/08/35
06/23/35
08/14/35
06/11/36
08/21/37
09/06/37
04/19/39
09/26/39
09/30/39
Flyby
03/27/22
03/25/22
02/17/24
08/30/28
08/22/28
07/09/30
12/07/34
11/22/36
08/25/39
-
Arrival
01/27/21
10/10/21
09/07/22
08/24/22
03/30/23
09/30/23
07/08/24
09/13/25
05/20/25
08/20/27
08/10/27
07/24/29
02/22/29
10/12/29
07/18/29
12/22/30
08/05/31
01/08/32
10/10/32
09/29/33
01/27/34
06/29/35
12/20/35
01/05/36
10/03/36
05/13/37
03/07/38
10/07/38
08/15/40
09/20/40
05/01/40
TOF [days]
194
412
382
300
204
382
300
344
220
294
270
548
336
316
222
332
190
320
464
178
274
342
226
196
416
336
198
396
484
360
214
218 + 164
148 + 152
158 + 142
220 + 328
152 + 184
166 + 166
138 + 204
164 + 172
128 + 356
-
departure
dep. (+ flyby)
arrival
C3d [km2/s2]
13.19
16.50
28.15
14.26
18.43
13.79
25.56
11.19
17.72
9.14
10.92
28.38
26.57
8.93
9.05
24.72
9.00
8.24
21.66
8.41
7.78
13.92
18.01
10.20
17.52
24.39
17.07
14.85
19.83
12.18
18.65
∆Vtot [km/s]
3.788
3.931
4.423
3.848
4.013
3.814
4.397
3.701
3.983
3.611
3.689
4.584
4.363
3.601
3.607
4.279
3.604
3.571
4.150
3.579
3.551
3.835
3.996
3.658
3.975
4.410
3.955
3.860
4.088
3.744
4.023
C3a [km2/s2]
8.14
14.55
29.81
35.52
13.39
9.32
39.78
6.41
16.85
7.28
8.47
39.35
39.16
10.28
24.39
39.40
30.70
30.60
39.97
15.96
19.16
33.31
8.02
7.20
16.76
39.91
11.19
11.26
39.87
7.23
15.92
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Competitiveness
competitive
dominated
dominated
competitive
competitive
dominated
competitive
dominated
competitive
dominated
competitive
dominated
competitive
dominated
competitive
competitive
competitive
dominated
dominated
competitive
dominated
competitive
dominated
competitive
dominated
competitive
competitive
dominated
dominated
dominated
competitive
700
4
600
TOF [days]
500
3.5
400
3
300
200
2.5
100
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2
Departure date
700
4
600
TOF [days]
500
3.5
400
3
300
200
2.5
100
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2
Departure date
Figure 7. 2020-2040 Mars-Earth ∆Vtot contours.
Table 2. 2020-2040 Mars-Earth trajectory data.
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Trajectory
2020 Mars-Earth
2020 Mars-Earth
2021 Mars-Venus-Earth
2022 Mars-Earth
2022 Mars-Earth
2021 Mars-Venus-Earth
2021 Mars-Venus-Earth
2021 Mars-Venus-Earth
2024 Mars-Earth
2024 Mars-Earth
2026 Mars-Earth
2026 Mars-Earth
2027 Mars-Venus-Earth
2028 Mars-Earth
2028 Mars-Earth
2029 Mars-Venus-Earth
2029 Mars-Venus-Earth
2030 Mars-Earth
2030 Mars-Earth
2033 Mars-Earth
2033 Mars-Earth
2035 Mars-Earth
2035 Mars-Earth
2037 Mars-Earth
2037 Mars-Earth
2039 Mars-Earth
2039 Mars-Earth
Departure
06/05/20
06/25/20
01/03/21
07/17/22
07/21/22
10/23/22
11/12/22
11/12/22
07/24/24
08/09/24
08/01/26
08/11/26
06/11/27
09/05/28
09/11/28
03/16/29
04/13/29
10/31/30
11/08/30
01/26/33
02/11/33
02/15/35
05/08/35
07/08/37
07/12/37
07/22/39
08/03/39
Flyby
08/07/21
06/30/23
07/10/23
07/20/23
12/06/27
10/12/29
10/24/29
-
Arrival
12/14/20
03/10/21
01/22/22
04/19/23
02/08/23
02/29/24
12/25/23
09/24/23
05/10/25
03/31/25
06/17/27
05/12/27
05/12/28
08/11/29
06/02/29
03/05/30
06/19/30
07/06/31
09/22/31
09/01/33
11/02/33
11/12/35
11/22/35
01/18/38
04/08/38
05/03/40
03/10/40
TOF [days]
192
258
384
276
202
494
408
316
290
234
320
274
336
340
264
354
432
248
318
218
264
270
198
194
270
286
220
216 + 168
250 + 244
240 + 168
250 + 66
178 + 158
210 + 144
194 + 238
-
departure
dep. (+ flyby)
arrival
C3d [km2/s2]
11.44
14.11
27.56
11.36
14.76
25.00
24.34
25.09
8.61
12.02
7.17
7.27
22.99
6.15
6.46
19.18
20.87
5.79
5.43
5.72
5.81
13.07
8.77
14.31
12.55
9.59
13.60
∆Vtot [km/s]
2.480
2.702
3.720
2.473
2.755
3.908
3.505
3.544
2.234
2.529
2.105
2.114
3.392
2.012
2.040
3.111
3.249
1.978
1.945
1.971
1.980
2.616
2.248
2.719
2.573
2.320
2.660
C3a [km2/s2]
10.85
17.50
17.20
10.14
18.16
44.20
22.74
39.40
7.87
18.45
9.35
19.10
24.98
20.11
31.47
41.76
44.00
31.86
29.57
16.69
24.38
9.57
9.16
15.49
12.62
8.14
20.06
8 of 9
American Institute of Aeronautics and Astronautics
Competitiveness
competitive
dominated
competitive
dominated
competitive
dominated
dominated
competitive
dominated
competitive
dominated
competitive
competitive
dominated
competitive
competitive
dominated
competitive
dominated
competitive
dominated
dominated
competitive
competitive
dominated
dominated
competitive
700
TOF [days]
600
500
400
300
200
100
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
LMO
LEO
2020
Departure date
Figure 8. 2020-2040 Earth-Mars flight opportunities.
700
TOF [days]
600
500
400
300
200
100
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
LMO
LEO
2020
Departure date
Figure 9. 2020-2040 Mars-Earth flight opportunities.
9 of 9
American Institute of Aeronautics and Astronautics