Design and Optimization of Gravity-Assisted Trajectories
to Outer Planets
Alberto Lobato Fernández * and Rafael Vazquez†
Escuela Superior de Ingenieros, Universidad de Sevilla, Seville, 41092, Spain
The preliminary design of interplanetary trajectories using gravity assisted maneuvers is studied in this
paper. The method of patched conics and a solver of the Lambert problem—without singularities for 180 degrees transfers—are used to cast the space trajectory design process as an optimization problem, subsequently
solved by using MATLAB. Given the energy requirements to reach outer planets, two additional types of maneuvers are considered: powered flybys and deep space maneuvers. These techniques are applied to design a
mission to Saturn between the years 2016 and 2033, using two deep space maneuvers and a Jupiter flyby. The
results of this analysis are the different favorable launch opportunities, as well as the orbital parameters and
the impulses needed along the mission.
Nomenclature
⊕
X
Y
α∞
C3
δ∞
→
−e
µ
RP
rp
SoI
V∞
= Symbol for Sun
= Symbol for Earth
= Symbol for Jupiter
= Symbol for Saturn
= Right Ascension of the Launch Asymptote
2
)
= Hyperbolic excess velocity squared (V∞
= Declination of the Launch Asymptote
= Eccentricity vector
= Gravitational parameter
= Planet radius
= Periapse radius
= Sphere of Influence
= Hyperbolic excess velocity
I. Introduction
The design of interplanetary trajectories is a complex problem involving orbital mechanics, numerical methods,
and optimization. Since real planetary orbits are neither circular nor coplanar, Hohmann or other simple theoretical
transfers are not appropriate. In addition, for missions to outer planets, maneuvers such as powered flybys or deep
space maneuvers need to be considered to overcome the limits of conventional launchers.
While the theoretical basis of interplanetary trajectories is well-known, there are no simple and freely available
tools that can be used to design them with a modicum of realism. The object of this paper is the development of simple
and accessible algorithms for interplanetary trajectory planning that do not require gross simplifications and are able
to find optimized solutions. The algorithms are implemented in MATLAB, which allows for their straightforward use
in an academic setting. Thus, the developed tools might be used by students to easily explore interplanetary travel
without the simplifying hypothesis usually made in the classroom.
To find an optimal interplanetary trajectory, several steps are necessary. The first is to find the orbit that joins
two points in space with a given flight time, i.e., Lambert’s problem. While many algorithms solve it, many present
singularities for 180 degree transfers. Unfortunately, optimal transfer solutions tend to have transfer angles close to
180. Since it avoids this singularity, this paper uses the algorithm of R.Battin. 1,2
On the other hand, gravity-assist maneuvers are required to reach outer planets such as Saturn. Thus, an initial
proposed trajectory could be modeled as two ballistic segments, joining the Earth, the flyby planet and the destiny.
* Graduate Student, Department of Aerospace Engineering; [email protected].
† Associate Professor, Department of Aerospace Engineering; [email protected].
However, the fuel costs might still be too large even for the optimal solution. Consequently, some extra degrees of
freedom are introduced in the problem by using Deep Space Maneuvers (DSMs) or, as an alternative, a powered flyby
maneuver.
Following the literature, this process is posed as an optimization problem. 3 Two different procedures for an EarthSaturn mission, with a Jupiter flyby, are covered in this paper. The first one consists of dividing the complete heliocentric trajectory into different segments, continuous in position though not in speed. These discontinuities represent
impulses of DSMs; the optimization algorithm then aims to minimize the sum of these impulses (including injection
from Earth and circularization burn in Saturn). In the second approach, the trajectory is composed of two ballistic
orbits (Earth-Jupiter and Jupiter-Saturn) and a powered flyby at Jupiter. Again, the optimization objective is to minimize the total amount of impulses along the mission, including injection from Earth and circularization burn in Saturn.
Results for different launch opportunities between years 2016 and 2033 are presented for both algorithms.
II. Trajectory models
The complete trajectory has been divided in five different segments. Three of them are the planetary segments
around Earth, Jupiter and Saturn, respectively. The classical analysis of scales for interplanetary missions is adopted. 8
This is, since planetary radii are significantly smaller than planetary Spheres of Influence, the limit of the Sphere of
Influence (SoI) is considered (from the point of view of the planetary segments) to be located at infinity. On the other
hand, from the perspective of the heliocentric trajectories Earth-Jupiter and Jupiter-Saturn, the SoIs are reduced to a
point. Finally, using the method of patched conics, the five segments are joined to compose the complete trajectory.
A. Initial and final impulses
Before introducing gravity-assist maneuvers and DSMs, the two impulses needed in the planetary segments around
Earth and Saturn are presented.
To avoid modeling the launch segment, consider the spacecraft starting at an initial circular parking orbit at an
altitude h = 200 km. The initial geocentric trajectory involves an initial impulse ∆V, to transfer the vehicle from the
parking orbit to a hyperbolic trajectory relative to the Earth, in order to scape the gravitational pull of the planet (Figure
1). Considering that point P1 is located in the infinite, to reach the hyperbolic excess velocity VSoI = V∞ (which is a
design parameter to be computed later), the impulse needed is given by
s
s
2µ⊕
2µ⊕
2
+ V∞ −
(1)
∆V =
R⊕ + h
R⊕ + h
The fist heliocentric segment, depicted in Fig. 2, is obtained by propagating the orbit from P1 (Earth’s SoI) with
−
→
−−→
initial speed V1 , which is related to V∞ by
−
→ −→ −−→
V1 = V⊕ + V∞
−−→
where the direction of V∞ depends on the orientation of the exit hyperbola asymptote with respect to the Sun’s position
−−→
−→
(typically oriented so that V∞ is in the same direction as V⊕ , obtaining a tangent exit).
Figure 1. Patched conics. Geocentric phase.
Figure 2. Patched conics. Heliocentric phase.
Finally, during the Saturn orbital segment, another impulse is needed to enter a circular orbit with an altitude of
h = 10RY . The impulse is given by:
v
v
t
t
2µY
2µY
2
+ V∞
(2)
∆V =
−
11RY
11RY
where V∞ is the hyperbolic excess velocity of the arrival orbit in Saturn’s SoI.
B. Gravity assist maneuver
When a vehicle approaches a planet, it follows a hyperbolic flyby trajectory around it. As it enters following the
entry asymptote, it has an excess hyperbolic velocity V∞ . When leaving the planet through the exit asymptote, it
has the same velocity. Although the modulus of speed relative to the planet does not change, to find the heliocentric
(relative to the Sun) velocity one needs to add the planet speed. It is then observed that the maneuver produces a
change in the heliocentric speed of the spacecraft, modifying its kinetic energy. That is the reason why flybys are
commonly used in the design of interplanetary missions, in order to achieve the high level of energies required to
reach outer planets.
First of all, analyzing the maneuver on its orbital plane, the main variables that appear are:
−
→
−
→
• v1H and v2H : approach and exit heliocentric speeds to the planet.
→
−
→
−
• v1P and v2P : approach and scape planetocentric speed.
−→
• VP : heliocentric speed of the planet
→
−
→
−
As v1P and v2P are the hyperbolic excess speed of the planetocentric orbit, their modulus are the same, v1P = v2P . Additionally, if the heliocentric approach speed and the periapse radius are known, the rest of variables can be obtained for
the planar maneuver:
→
−P −
→ −→
v1 = v1H − VP
→
−
If the flyby is plane (does not modify the orbital plane of the orbit), then the scape planetocentric speed v2P is computed
by using the rotation angle δ = 2
arcsin(1/e) (see Figure 3), where e is the eccentricity of the hyperbolic orbit, r p = −µ/(v1P )2 (1 − e). Finally, the
heliocentric scape speed is obtained adding to the planetocentric excess scape speed the planet velocity:
−
→ →
− −→
v2H = v2P + VP
→
−
For a 3-D flyby (with plane change), another parameter has to be introduced in addition to r p and v1P . Define first
→
−
the B plane as the plane passing through the center of the planet and normal to the arrival asymptote, and the vector T
contained in B and parallel to ecliptic plane (see Fig. 4). The new parameter is denoted as ζ, the angle (measured on
→
−
the B plane) between the orbital plane and line T .
Finally, if an impulse is given to the vehicle during the maneuver (powered flyby), a new free parameter is introduced in the formulation. Thus, the flyby can be divided into two parts, an approximation path and a scape path. The
approximation segment begins in the infinite, where the Jupiter Sphere of Influence is supposed to be, and end in the
periapse of the planetary orbit. The periapse radius is r p = a1 (1 − e1 ), where a1 = −µ/(v1P )2 is the approach orbit
semimajor axis. For the scape trajectory we similarly have r p = a2 (1 − e2 ) and a2 = −µ/(v2P )2 .
If v1P and v2P are known, a1 and a2 can be calculated. In addition, since the approach and scape trajectories coincide
in the periapse, one has
a1 (1 − e1 ) = a2 (1 − e2 )
(3)
This equation has two unknowns, e1 and e2 , hence another equation is needed to define the complete maneuver when
planetary excess velocities are given.
If we take into account that, in hyperbolic orbits, the angle θ∞ (Fig. 5) is defined as
1
θ∞ = arccos(− )
e
Figure 3. Gravity assist speeds definition.
Figure 4. Gravity assist angles definition.
then both flyby segments satisfy
θ∞1
=
arccos(−
1
),
e1
θ∞2 = arccos(−
1
)
e2
(4)
→
−
→
−
In addition, the angle between v1P and v2P is θ∞1 + θ∞2 − π (Fig. 5). Thus it can be stated that:
→
−P→
−
v1 v2P
v1P v2P
= cos [θ∞1 + θ∞2 − π] = − cos [θ∞1 + θ∞2 ]
(5)
Solving numerically the system of equations (3) and (5), the values for e1 and e2 are obtained. Once done this, the
→
−
impulse required in the periapse of the orbit to reach scape speed v2P is
s
s
2µ
2µ
µ
µ
−
−
−
(6)
∆V =
r p a2
r p a1
These gravity assist model allow to formulate one procedure to design the Earth-Saturn trip. The mission will
consist of two ballistic segments, Earth-Jupiter and Jupiter-Saturn. The problem has 3 degrees of freedom, thus
variables T 0 , T 1 and T 2 (Earth departure, Jupiter arrival and Saturn arrival dates) are set as optimization parameters.
Given these dates, the position of the planets can be propagated and the Lambert problem solved to obtain the two
ballistic heliocentric orbits and the vehicle hyperbolic excess speed at each planet. At Jupiter, the values of v1P and v2P
will probably not match, consequently a ∆V is needed in the periapse flyby maneuver. The optimizer selects T 0 , T 1
and T 2 in order to minimize Σ∆V along the mission, using equations (1), (2) and (6).
C. DSM-based trajectory design
Deep Space Maneuvers or DSMs consist in additional impulses given to the vehicle at intermediate points of its
heliocentric orbit. The effectiveness of the impulse depends on the relative position of the spacecraft with respect to
the Sun. Thus, carefully selecting the application point, it is possible to considerably reduce the total impulse amount
along the mission.
As an alternative to powered flybys, Earth-Saturn trajectories with two DSMs—and a (non-powered) flyby at
Jupiter—are computed next. Two DSMs are considered, the first one between the Earth and Jupiter, and the second
one between Jupiter and Saturn. The formulation of the global problem is:
2
• Given a departure date T 0 , a value for the C3 (C3 = V∞
) to scape from Earth’s SoI, the Declination of the Launch
Asymptote DLA or δ∞ and Right Ascension of the Launch Asymptote RLA or α∞ , the initial heliocentric is
propagated for a flight time of η1 (T 1 − T 0 ).
Figure 5. Gravity-assisted maneuver with an impulse in the periapse.
Figure 6. Earth-Jupiter-Saturn mission with DSMs.
• T 1 is the arriving date to Jupiter, where a gravity-assisted maneuver is performed. η1 ∈ [0, 1] is the fraction of
flying time in the first segment Earth-Jupiter when the first DSM impulse is applied.
−
• Once the orbit has been propagated, the position and speed vectors just before the first DSM, →
r1 , −
v→
1i , are calcu→
−
lated. Using planetary ephemeris, the position of Jupiter is also known at T 1 ; it is denoted as r2 . In addition, as
the flight time between this first DSM and the arrival to Jupiter is known, (1 − η1 )(T 1 − T 0 ), the Lambert problem
−
→
is solved to obtain the orbit that joins these to points. The solution provides the values for −
v→
1 f and v2i , which are
the speed after the DSM and the heliocentric arrival speed to Jupiter respectively.
• The impulse needed in the first DSM is given by ∆V1 = v1 f − v1i .
• Given a periapse radius for the gravity-assisted maneuver in Jupiter r p and the angle ζ, both of them as parameters of the problem, the heliocentric scape speed from Jupiter is obtained, −
v→
2f .
−
• As the position and speed when leaving Jupiter are now both known, →
r2 and −
v→
2 f , the next orbit segment is
propagated for a flight time η2 (T 2 − T 1 ).
• T 2 is the arriving date to Saturn and η2 ∈ [0, 1] is the fraction of the time between Jupiter and Saturn when the
second DSM is performed.
−
• As before, the position vector →
r3 where the DSM is applied is obtained, as well as the speed vector before the
−
→
−
−
−
maneuver v3i . Propagating, Saturn position →
r4 is calculated in T 2 . Solving Lambert’s problem between →
r3 and →
r4
−
→
for a trip time equal to (1 − η2 )(T 2 − T 1 ) gives us the speed vector after DSM, v3 f and the arrival speed to Saturn
→
−
v4 .
• The impulse in the second DSM therefore is ∆V2 = v3 f − v3i .
• Impulses in the Earth parking orbit and to circularize the orbit in Saturn are respectively given by equations (1)
and (2).
To summarize, the DSM-based formulation of the problem has 10 independent variables, namely:
T 0 , T 1 , T 2 , C3 , δ∞ , α∞ , η1 , r p , ζ, η2
The allowed variation intervals for these variables is:
• C3 : will vary between 0 and 90km2 /s2 . The superior limit has been set according to launchers capabilities. 4
• δ∞ ∈ [−40º, 40º]. This interval has been selected to allow a wide launch window from Kennedy Space Center. 5
• α∞ ∈ [0, 360º].
• η1 and η2 ∈ [0.01, 0.99]. Due to practical aspects related to mission control, 3 DSMs point of application have
been separated from injection and gravity-assisted maneuvers.
• r p > 10RX , in order to avoid atmospheric effects and radiation belts around Jupiter.
• ζ ∈ [0, 360º].
Once the problem has been formulated, it is necessary to determine the values of these 10 variables that minimize the
Σ∆V along the mission. Due to the high non-linearity of the problem, its optimization becomes a complex problem.
The optimization has been carried out with MATLAB “active-set” algorithm. Since there exist many local minima; to
overcome this problem the optimization algorithm is started from different initial guesses. To generate a suitable set
of starting points, all the dimensional variables have been nondimensionalized with reference values rendering their
variation intervals comparable. A large set of points has been generated, to cover the parameter space as much as
possible in the search for the optimal solution.
III. Trajectory studies
A. Favourable planetary positions
Before running the global optimization program for the two trajectory formulations (powered flybys and DSMs),
it is important to analyze which relative planetary positions are associated with most effective transfers. This allows
focusing on the most important launching dates and create a well-conditioned set of initial guesses that may lead to a
fast convergence.
First of all, the gravity-assisted maneuver is considered impulsive, thus a ∆V is given to the spacecraft at periapse.
In Figure 7 different curves are shown , associated to a fixed periapse radius r p = 10RX . These represent the relationship between v1p and v2p for different δ angles. For example, given δ = 100º, r p = 10RX and v1p = 6 km/s, excess
hyperbolic scape speed is v2p = 8.9 km/s. Any v2p < 8.9 km/s would be associated with r p > 10RX , therefore they
complying with the constraint for periapse radius.
50
δ = 50º
δ = 60º
δ = 70º
δ = 80º
δ = 90º
δ = 100º
δ = 110º
45
40
35
VP
2
30
25
20
15
10
5
0
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
P
V1
Figure 7. Hyperbolic excess arrival and departure speeds for a given r p = 10RX and different values of δ angle.
160
11
140
10
120
100
∆θ = 70º
∆θ = 90º
∆θ = 110º
∆θ = 130º
∆θ = 150º
∆θ = 170º
80
60
40
20
700
800
900
1000
tv Earth−Jupiter [days]
∆θ = 40º
∆θ = 70º
∆θ = 100º
∆θ = 130º
∆θ = 160º
∆θ = 190º
9
8
7
6
5
4
3
1200
1100
(a)
1400
1600
1800
tv Jupiter−Saturn [days]
2000
2200
(b)
10.5
∆θ = 70º
∆θ = 90º
∆θ = 110º
∆θ = 130º
∆θ = 150º
∆θ = 170º
10
9.5
9
vP
Jupiter [km/s]
1
∆V Saturn [km/s]
12
8.5
8
7.5
18
∆θ = 40º
∆θ = 70º
∆θ = 100º
∆θ = 130º
∆θ = 160º
∆θ = 190º
16
14
vP
Jupiter [km/s]
2
C3 [km2/s2]
13
180
12
10
8
7
6
6.5
4
6
5.5
600
700
800
900
1000
tv Earth−Jupiter [days]
1100
2
1200
1200
(c)
1600
1800
tv Jupiter−Saturn [days]
2000
2200
(d)
210
400
200
350
190
300
180
170
160
∆θ = 70º
∆θ = 90º
∆θ = 110º
∆θ = 130º
∆θ = 150º
∆θ = 170º
150
140
130
120
600
∆θ = 40º
∆θ = 70º
∆θ = 100º
∆θ = 130º
∆θ = 160º
∆θ = 190º
250
Γrelp2 [º]
Γrelp1 [º]
1400
700
800
900
1000
tv Earth−Jupiter [days]
1100
1200
200
150
100
50
0
1200
1400
1600
1800
tv Jupiter−Saturn [days]
2000
2200
(e)
(f)
p
Figure 8. Earth-Jupiter-Saturn mission with coplanar planetary orbits. Plots show C3 of injection from Earth, v1 and arrival Γrelp1 in
p
X
Jupiter for different Earth-Jupiter transfer angles ∆θ as a function of Earth-Jupiter flying time. For the second segment, the values of v
⊕
2
and departure Γrelp2 in Jupiter are shown, as well as the circularization impulse in Saturn ∆V as functions of Jupiter-Saturn flying time for
Y
different ∆θ .
X
After this consideration, the coplanar problem is used for preliminary analysis. The two different segments, EarthJupiter and Jupiter-Saturn have been calculated by solving the Lambert problem with a variety of trip times and relative
positions of the planets. The obtained solutions are presented in Figure 8. The variation of the relative position of
planets, ∆θ, has been constrained to a maximum of 170º due to the fact that, in the real three-dimensional problem,
near 180º transfers result in polar heliocentric orbits with high injection energies needed.
Firstly, Figure 8a is analyzed. It can be seen that, for the first segment, only transfer angles greater than 130º result
in affordable values of the C3 . The minimum value is obtained for the maximum transfer angle. In this case, the
following values are chosen:



∆θ⊕X = 170º


tv⊕−X = 850 days
With these values, from Figures 8c and 8e it is obtained that vX
1 = 6km/s and Γrelp1 = 160º. Since the excess
X−Y
X
hyperbolic arrival speed v is known, now the problem is to find the trip time t
that makes values of δ y vX satisfy
v
1
2
the constraints imposed by Figure 7.
X−Y
In Figure 8b it is shown that, for short values of tv
, the circularization impulse ∆V at Saturn is lower for transfer
X
angles between 70º and 100º. Moreover, during the flyby it is appropriate that vX
2 > v1 , in order to employ the periapse
X
impulse to increase
the kinetic
energy of the vehicle. Additionally, according to Figure 7, for values of v2 > 6km/s
the subtraction Γrelp1 − Γrelp2 = δ should be lower than 110º. Therefore, Γrelp2 should be in the interval [50º, 270º].
Y
Taking this into account and looking at Figure 8f, it can be seen that the curve with ∆θ = 70º satisfies these
X
X−Y
Y
X−Y
constraints for tv
> 1200 days, and curve with ∆θ = 100º does it when tv
> 1900 days. However, for those
X
trip times, the transfer angle of 100º is associated to a value of v2p < v1p .
Y
X−Y
Consequently, in the Jupiter-Saturn segments, transfers of ∆θ = 70º and trip times near tv
= 1800 days are
X
considered:


Y


∆θX = 70º



t⊕−X = 1800 days
v
Having obtained the most favourable planetary configuration for the mission, it is now required to obtain the dates
for these configurations to occur. In Figure 9, the angle between Earth and Jupiter, as well as the one between Jupiter
and Saturn, are presented for different departure times T 0 , considering that T 1 = T 0 + 850 and T 2 = T 1 + 1800.
200
150
100
[º]
50
0
−50
−100
−150
Earth−Jupiter
Jupiter−Saturn
−200
5.6
13/3/2012
5.7
8/12/2014
5.8
3/9/2017
5.9
30/5/2020
6
T0 [JD−24·105]
6.1
24/2/2023
20/11/2025
6.2
6.3
6.4
13/5/2031
x 10
6/2/2034
4
16/8/2028
Figure 9. Relative positions between the Earth, Jupiter and Saturn. In dashed lines there are plot the 170º value (blue) and the interval
[60º-80º] (red)
Y
Different possible departure dates T 0 are those that satisfy ∆θ⊕X = 170º at the same time that ∆θ = 70º (it has been
X
established the interval [60º − 80º] to allow some variations around the initial value). The results are T 0 = 2458130 JD
(Julian Date) (1/11/2018), T 0 = 2460550 (8/27/2024) and T 0 = 2460940 (9/21/2025).
B. Optimal trajectories
The different optimal results are presented below. First of all, values of the variables associated with completely
ballistic missions, with a Jupiter flyby, between 2016 and 2033 are shown to demonstrate that this solution is not
practical.
tv⊕−X
T0
2458321
21/7/2018
1133 days
X−Y
tv
tv
C3
Σ∆V
1740 days
7.871 years
1714.800
39.583
(1) Results with departure on 2018
tv⊕−X
T0
2460500
8/7/2024
459 days
X−Y
tv
tv
C3
Σ∆V
2414 days
7.871 years
267.374
17.615
(2) Results with departure on 2024
tv⊕−X
T0
2461013
3/12/2025
494 days
X−Y
tv
tv
C3
Σ∆V
2404 days
7.939 years
595.099
23.620
(3) Results with departure on 2025
Table 1. Completely ballistic missions. C3 in [km2 /s2 ] and speeds in [km/s].
Using ballistic mission with powered flybys, two different sceneries has been covered. In the first one, the flying
X−Y
time has been limited to a maximum of 2000 days for each segment (t⊕−X < 2000 and t
< 2000 ) in order to
v
v
control the length of the mission. The second one has consisted in solving the problem without any time restrictions.
The results for a minimum Σ∆V among 2016-2033 are shown below.
T0
2458132
13/1/2018
tv⊕−X
X−Y
tv
tv
C3
∆V⊕
773 days
2000 days
7.597 years
78.268
6.339
tv⊕−X
X−Y
tv
tv
C3
∆V⊕
829 days
2852 days
10.085 years
76.601
6.279
∆VX
0.000153
∆VY
3.686
Σ∆V
10.025
(1) Bounded trip times.
T0
2458133
14/1/2018
∆VX
−0.000108
∆VY
3.599
Σ∆V
9.878
(2) Non bounded trip times.
Table 2. Ballistic mission with impulse during the flyby. C3 in [km2 /s2 ] and speeds in [km/s].
With regard to DSM-based missions, the same approach has been adopted: both bounded flying time and non
bounded problems have been solved. The results obtained are:
T0
2458132
13/1/2018
tv⊕−X
X−Y
tv
tv
C3
∆V⊕
∆VDS M1
∆VDS M2
773 days
2000 days
7.597 years
78.1277
6.3337
0.0094
0.0061
3.6859
10.0351
∆VY
Σ∆V
∆VY
Σ∆V
(1) Bounded trip times. η1 = 0.0344 and η2 = 0.2055.
T0
2458132
13/1/2018
tv⊕−X
X−Y
tv
tv
C3
∆V⊕
∆VDS M1
∆VDS M2
816 days
2673 days
9.559 years
76.9061
6.2904
0.0051
0.0024
3.5887
(2) Non bounded trip times. η1 = 0.0119 and η2 = 0.2010.
Table 3. Missions with two DSM and ballistic gravity-assist in Jupiter. C3 in [km2 /s2 ] and speeds in [km/s].
9.8866
C. Analysis of results
Starting with the fully ballistic mission, in Table 1 it can be seen that the obtained solutions present high values
of C3 , always greater than the restriction imposed of 90km2 /s2 . It is therefore impossible to launch an spacecraft to
Saturn employing actual launchers if these trajectories are chosen. In addition, the total impulse needed along these
missions is also too large.
This leads to consider mission with powered flybys and trajectories with DSMs. Firstly, focusing on trajectories
with powered flybys, in Table 2 it is shown that associated values of C3 are now affordable for launchers, allowing mission implementation. Furthermore, the sum of impulses suffers an important reduction respect to those of completely
ballistic missions. Even though the impulses during flyby are rather low (∆VX ) and thus the resulting trajectories are
near-ballistic, adding this degree of freedom to the problem allows to find better solutions.
With respect to trip times, when they are bounded, the optimization algorithm converges when the trip time of
Jupiter-Saturn reaches the superior limit (Table 2.1). It means that increases of this time could derive into reductions
of the target variable Σ∆V. That is the reason why the problem has been also solved without any time restrictions, in
order to evaluate if it is worth increasing trip time to reduce the amount of energy required. Comparing Tables 2.1 and
2.2, decreases of C3 and Σ∆V are achieved, although mission times increases in a 30% (7.6 years to 10.1 years).
If DSM missions are analyzed, the same conclusions are drawn. C3 and Σ∆V are very similar to those of the
previous alternative, being also reduced when trip time is not bounded.
Additionally, it can be seen that these solutions converge to departure dates near to those obtained in the previous
subsection for the planar case.
The four affordable solutions are presented in the next table to allow comparisons between them. The maximum
load that could be launched by a Delta IV launcher (including payload and propellant) has also been computed.
T0
tv
Σ∆V
C3
Load
GA1
13/1/2018
7.6 years
10.025 km/s
78.27 km2 /s2
650 kg
GA2
14/1/2018
10.1 years
9.878 km/s
76.60 km2 /s2
750 kg
DS M1
13/1/2018
7.6 years
10.035 km/s
78.13 km2 /s2
650 kg
DS M2
13/1/2018
9.6 years
9.887 km/s
76.91 km2 /s2
750 kg
Table 4. Optimal solutions.
Finally, an important additional factor should be taken into account when comparing the two approaches. Even
though powered flyby missions (GA1 and GA2 ) and DSM-based trajectories (DS M1 and DS M2 ) have similar values
of energy, the possible difficulties of communications between Earth and the vehicle when it is performing a flyby
makes the missions with DSM more convenient .
Finally, having reduced the problem to alternatives DS M1 and DS M2 , it needs to be remarked the important
difference of initial loads that can be launched in every case. In the design of a real mission this would be an important
point to take into account, and designers should come to a compromise between time to complete the whole mission
and payload needs.
IV. Conclusions
The problem of preliminary interplanetary design to outer planets has been studied using different techniques.
Gravity-assisted maneuvers have been introduced as a resource to get the required energy to reach far planets. Deep
Space Maneuvers and impulses at the flyby periapse have also been described as means to increase the degrees of
freedom in the global trajectory design process.
Complete ballistic paths have been proved to be inefficient for real missions, resulting in too high launch energies.
As a result, analysis with impulsed gravity-assists and DSM maneuvers have been developed.
The complete trajectory analysis has been cast into an optimization problem, where the cost function is the total
amount of impulses given along the mission. Due to high non-linearity of the problem, the optimization algorithm
needs to be started from a wide variety of initial guesses, in order to find as many local minimums as possible. A
previous study of the simplified planar problem has been proved to be accurate enough to obtain these initial guesses
for the problem.
Solutions for constrained and non-constrained flight time have been obtained, using either DSM or powered flyby
maneuvers. Bounded flight time missions are attractive when compared to unbounded, as a small increase in fuel
consumption leads to large reductions in their duration; however, they are associated with greater C3 energies, resulting
in lower payload launching capabilities.
Finally, the analysis performed in this paper suggests that the most favourable Earth-Saturn mission with a gravity assist in Jupiter between years 2016 and 2033 should be launched in January 2018 with an injection energy of
77km2 /s2 − 78km2 /s2 . If the heavy payload alternative is chosen, the total mass that could be put in orbit with a Delta
IV launcher would be of 750kg. 4 The flyby around Jupiter would occur during April 2020 and the spacecraft would
finally reach its orbit around Saturn in August 2027.
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