Carlos Renato Huaura Solórzano2, Alexander Alexandrovich Sukhanov3,
                 Antonio Fernando Bertachini de Almeida Prado2


        Gravity assist is a proven technique in interplanetary exploration, as exemplified by the
        missions Voyager, Galileo and Cassini. In the present paper, based in this well-known
        technique, an algorithm is developed to optimize missions to the outer planets. Then,
        this algorithm is applied to a mission to Neptune for the mid-term (2008-2020). The
        following schemes are analyzed: Earth–Jupiter–Neptune, Earth–Venus–Earth–
        Jupiter–Neptune, Earth–Venus–Earth–Jupiter–Saturn–Neptune. Transfer trajectories
        that provide a good compromise between the delta-V and the time of flight to Neptune
        are presented. In particular, the effects of the pericenter height for the gravity assist
        with Jupiter are studied in detail, since the final results has a strong dependence on
        this variable.


The technique of gravity-assist maneuvers is studied for several authors. In the
decade of the 60’s, Flandro [1] considered a mission to the exterior Solar System
using the concept of gravity-assist maneuvers with Jupiter, Saturn and Uranus. This
type of trajectory was used by the Voyagers 1 and 2. Hollister and Prussing [2]
considered a Mars transfer through Venus, analyzing the advantages of an
impulsive maneuver during the close approach with Venus. Several procedures
were developed (D’Amario et al. [3-4]) with the goal of minimizing the total impulse
for multiple-flyby trajectories with constraints on flyby parameters and maneuver
times. This procedure successfully optimized the Galileo mission that contains up to
eleven flybys. Later, this procedure was modified to minimize the total impulse for
applications in interplanetary trajectories. Examples of applications of this new
method are given for several types of the Galileo mission. Longuski et al. [5]
considered a new approach to planetary mission design. This new design tool is
applied to the problems of finding multiple encounter trajectories to the outer
planets. Striepe and Braun [6] analyzed missions to Mars using maneuvers assisted
by the gravity of Venus. This maneuver provides a non-propulsive change in the
heliocentric energy of the spacecraft that can reduce the amount of propellant
necessary to complete the interplanetary mission and/or to reduce the duration of
some missions. For certain position of the planets, it incorporates a propulsive
maneuver. Peralta and Flanagan [7] planned the interplanetary trajectories of the
Cassini mission. Besides, the trajectory with multiple gravity assists with Venus-
Venus-Earth-Jupiter supplies the energy necessary to reach Saturn.
Sims et al. [8] analyzed several trajectories to Pluto using gravity-assisted
maneuvers with Jupiter. They also analyze gravity-assist maneuvers with Mars, in
conjunction with three maneuver assisted by Venus. Sukhanov [9] studied a mission
to the Sun, by means of gravity assists with the interior planets. It considers
maneuvers with Earth, Mars, and Venus. There are advantages in the cost with
  Presented at The Malcolm D. Shuster Astronautics Symposium, realized 12-15, June, 2005.
  National Institute for Space Research (INPE).
  Space Research Institute (IKI) of the Russian Academy of Sciences.
respect to gravity-assist maneuvers with Jupiter, being also possible to use multiple
Mercury maneuvers. Other articles (Longuski and Williams [10], Patel et al. [11])
analyzed opportunities for a mission to Pluto using gravity-assist maneuvers.

The next step in the intensive exploration of the outer planets, following the Galileo
and Cassini missions, would be a similar orbiter and atmospheric probe mission to
Neptune. Uranus, another planet that was not much explored in the past, is not in a
good position relative to the orbits of the other planets to be visited, thus a possible
mission to reach this planet has a high consumption of fuel. For this reason, Uranus
is also not in good position to be used for a gravity-assist maneuver to reach
Neptune. Neptune is scientifically important, because of its turbulent atmosphere
and the presence of the large moon Triton. Triton is particularly interesting because
of its size, retrograde orbit, and the insight into the Solar System cosmogony to be
gained from its study and comparison with Pluto and Charon. Hammel et al. [12]
proposed missions to the Neptune system, including an orbiter and a Neptune
atmospheric multi-probe. Venus and Earth gravity assists can also be used to reach
Neptune (Swenson [13]).

There are other publications, utilizing propulsive maneuvers, to avoid the
disadvantage of a certain planetary configuration required for the gravity-assist
maneuver. Another possibility considered in the literature (Gershman et al. [14]) for
the exploration of Neptune is to perform an aerocapture maneuver in combination
with radioisotope power source and electric propulsion. Other projects considered
the use of solar electric propulsion, Earth gravity-assist maneuver and aerocapture
to reach Neptune and Triton (Esper [15]). A combination of the electric propulsion
(supplied either by solar arrays or radio isotopic source) with a chemical engine can
improve the transfer characteristics (Fielher and Oleson [16]). Other papers
(Malyshev et al. [17] and Racca [18]) studied the combination of propulsive and
gravity-assist maneuvers for this trip.

The present paper considers the gravity-assist maneuvers, including the possibility
of powered swing-by, as the one used when passing by the Earth. The lack of power
is compensated with several gravity-assist maneuvers. Those trajectories need a
long time for the transfers, necessary for phasing the spacecraft trajectory and the
flyby planets, but the large savings obtained can compensate this larger time of
flight. It is possible, for interplanetary flight trajectories, to approximate the legs of
the flight before and after the gravitational maneuver by arcs of conic sections.
These arcs are unambiguously determined by the dates of launch, flybys, and arrival
to the desired planet.


A previous work performed by the authors of this paper (Solórzano et al. [19]) shows
that the best schemes, without considering the consumption for the braking
maneuver near Neptune, are represented by the Earth-Jupiter-Neptune (EJN),
Earth-Venus-Earth-Jupiter-Neptune (EVEJN) and Earth-Venus-Earth-Jupiter-Saturn-
Neptune (EVEJSN) transfers. The goal of the present paper is to study the
mentioned schemes in more detail. Two cases will be considered in this analysis.
The one that includes the braking maneuver near Neptune and the one that does
not include this maneuver. It is assumed that a small braking delta-V is applied
when inserting the spacecraft into an orbit that has a very high apocenter around
Neptune. After that, the apocenter can be lowered by means of aerobraking in the
Neptune’s atmosphere.

Figure 1 Planetary configuration and transfer trajectory for the EVEJN scheme with
launch in 2015.

          Figure 2 Total delta-V vs. pericenter height for the EJN scheme.

Figure 1 shows the planetary configurations (onto the ecliptic plane) for the 12-year
EVEJSN transfer. However, retrograde trajectories are also possible, but the
transfer time is too long. From the point of view of the energy provided by the
planets, the main contributions come from Jupiter, Saturn, Venus and Earth.
Considering this fact and also that the best trajectories (EJN, EVEJN, EVEJSN)
include a Jupiter flyby, the influence of the variations of the pericenter height near
Jupiter on the fuel consumption will be analyzed in this paper. Figure 2 shows the
behavior of the total delta-V versus the pericenter height.

For the optimal 12-year transfer, the pericenter height of Jupiter flyby is 4.20x105
km. Several simulations show that, varying the pericenter height of the Jupiter flyby,
the value of the optimal delta-V changes. Thus, with the 12-year flight and the height
of pericenter equals to 6x105 km, the optimal value of the total delta-V is 7.891 km/s.
This is approximately 20% higher than the optimal value for the delta-V. For a 14-
year flight, the height varies between 6.7x105 km and 9x105 km, for the simulations
considered. The optimal value is 6.412 km/s. However, the maximum value
considered in the present simulation suffers an addition of 11%, with respect to the
optimal value. The same scheme applied to the 18-year flight gives the optimal
delta-V of 6.355 km/s. However, the optimal delta-V value suffers an addition of
13%, due to the altitude variation. In general, varying the pericenter height of the
Jupiter flyby, the total delta-V increases with respect to the optimal value. The same
fact happens for the other schemes. This analysis is also important when
considering a mission to flyby Jupiter with several different values of the pericenter
height. Figure 3 shows the launch delta-V vs. time of flight for the best schemes for
missions that include and exclude the braking maneuver near Neptune. In all the
simulations, the time of flight was considered between 12 and 18 years. The goal of
this analysis is to study the behavior of the launch delta-V for each of the considered
schemes. The EVEJSN scheme presents the lowest values for the launch delta-V,
and has the minimal value when comparing with other schemes. The EVEJN
scheme gives intermediate values for the launch delta-V.

Figure 4 shows the total delta-V for the Earth to Neptune transfers, not including the
braking maneuver near Neptune. The curves of the minimum total delta-V are
functions of the time of flight. For transfer times smaller than 14 years, the EJN
scheme is optimal in terms of minimum total delta-V. For situations where the time
of transfer is larger than 14 years, the EVEJSN scheme is optimal. The EVEJSN
scheme has a minimum total delta-V equals to the variation in velocity of the EVEJN
scheme in the case of 13.5-year transfer and to the EJN scheme for the case 13.8-
year transfer. However, the Vinf (incoming velocity near Neptune) for the EVEJSN
scheme is very high. The EJN and EVEJN schemes are more efficient in terms of
low excess velocity near Neptune (Fig. 5). Figure 6 shows the optimal total delta-V
as a function of the optimal launch in the time interval 2008–2020. Of course, the
launch dates to Neptune for each of the considered schemes are discrete (to be
more precise, the launch is possible during rather short launch windows). Curves in
Figs. 6 and 8, shown below, just formally approximate these dates. The optimal
launch dates for the schemes without braking maneuver are shown in Table 1. It
seems that the gravity-assist maneuvers with Venus, Jupiter and Saturn have
enormous potential to reduce the total delta-V for trajectories to Neptune.
Figure 3 Launch delta-V for EJN, EVEJN and EVEJSN schemes (solid lines
represent maneuvers that do not include the braking near Neptune and the dashed
lines represent maneuvers that include the braking near Neptune).

 Figure 4 Total delta-V, without the cost of the braking maneuver near Neptune, for
                        EJN, EVEJN and EVEJSN schemes.
Figure 5 Vinf near Neptune vs. time of flight.

 Figure 6 Optimal launch date for EJN, EVEJN and EVEJSN schemes, without the
                        braking maneuver near Neptune.

Figure 7 shows the total delta-V as a function of the time of flight, which includes the
delta-V for the braking maneuver of the spacecraft near Neptune. As seen in Fig. 7,
for times of flight smaller than 15.5 years the EVEJSN scheme has the highest fuel
consumption. However, for times of flight near 17 and 18 years, this scheme shows
a better performance with respect to the fuel consumption.

                                     Table 1
                Optimal launch date for transfers without braking

             Transfer Scheme       Optimal Launch         Total delta-V
                                        Date                 (km/s)
                   EJN              13/01/2018               6.367
                  EVEJN             28/05/2013               5.642
                  EVEJSN            30/05/2015               5.441

The option EJN shows to be excellent, due to the fact that, for the 12-year transfer, it
has a minimum delta-V of 9.298 km/s and keeps a comparative excellent behavior
with respect to the other schemes, until the time of flight of 17 years.

Figure 7 Total delta-V, including the cost of the braking maneuver near Neptune, for
                         EJN, EVEJN and EVEJSN schemes.

The option EVEJN has also low fuel consumption until the time of flight of 15.6
years, but the EVEJSN option results with lower consumption. The options shown
here were simulated considering dates of launch in 2018 for the Earth-Jupiter-
Neptune transfer, 2016 for the Earth-Venus-Earth-Jupiter-Neptune transfer, and
2015 for the Earth-Venus-Earth-Jupiter-Saturn-Neptune transfer. Figure 10 gives
information about the optimal launch date for trajectories that includes the braking
near Neptune.

For the optimal launch date, our interest is to determine the time of flight that
provides the minimum fuel consumptions. The EVEJSN scheme has the best values
of delta-V, thus the minimum value is 5.647 km/s for a time of flight of 30.36 years
(Fig. 8 and Table 2). The EJN scheme shows (Fig. 9) smaller fuel consumptions,
when compared to other schemes, for the delta-V required by the braking maneuver
near Neptune. For these schemes, it is considered that the braking near Neptune is
applied in a distance that corresponded to 5% of the radii of Neptune. Thus, the
impulsive maneuver changes the hyperbolic orbit into a parabolic orbit near

Figure 8 Optimal launch date for EJN, EVEJN and EVEJSN schemes, including the
                        braking maneuver near Neptune.
Figure 9 Braking delta-V near Neptune for EJN, EVEJN and EVEJSN schemes.

                                    Table 2
  Optimal launch date for several transfers that includes a braking maneuver
                                near Neptune

             Transfer Scheme       Optimal Launch        Total delta-V
                                        Date                (km/s)
                   EJN              15/01/2018              6.494
                  EVEJN             28/05/2013              5.899
                  EVEJSN            01/06/2015              5.647


In this paper, the minimum total delta-V is obtained as a function of the launch date
and the flight duration for a mission to the Neptune system. This parameter
determines the fuel consumption to launch from LEO (Low Earth Orbit), to perform
midcourse maneuvers, and to break the spacecraft near Neptune. After that, we
analyzed the effects of the variation of the pericenter height near Jupiter in the fuel
consumption. It was considered the Venus, Earth, Jupiter and Saturn gravity-assist
maneuvers, showing their advantages in specific dates. Thus, direct transfers to
Jupiter offers the best alternatives to the impulsive trajectories. Such a transfer is
available for Earth departures in 2005, 2006, and 2007 and, after that, only in about
2017. However, Venus is available and offers good results for launching from Earth
in the mid-term 2012-2016.

The EJN scheme provides minimum total delta-V for the transfers which durations
are smaller than 14 years. This scheme also gives relatively low incoming Vinf near
Neptune. For longer transfers, the EVEJSN scheme is optimal in terms of minimum
total delta-V, but Vinf is high. In very long EVEJSN and EVEJN schemes, the Vinf is
relatively small and the delta-V is close to the minimum among all schemes
considered. For the cases considering the braking maneuver near Neptune, the EJN
scheme provides minimum total delta-V for the transfer duration of less than 17
years. For transfer duration between 17 and 18 years, the EVEJSN scheme is
optimal in terms of minimum total delta-V. The EVEJSN and EVEJN schemes are
most acceptable for longer transfer times. All the previous schemes allow a passage
near Neptune, depending on the objectives of the mission. It is possible to perform a
flyby or to remain in orbit around the planet. In the present case, it is necessary to
change the trajectory of the spacecraft to keep it in orbit around Neptune. Then, the
braking maneuver is applied close to Neptune. However, there is also a possibility to
reoptimize the trajectory with the goal of reaching some asteroid of the main belt.


The authors are grateful to CAPES-Brazil (Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior) for the scholarship given to the first author, to the
National Council for Scientific and Technological development (CNPq), Brazil, for
the research grant receive under Contracts 300828/2003-9 and 308294/2004-1, and
to the Foundation to Support Research in the São Paulo State (FAPESP), for the
research grant received under Contracts 2006/00187-0 and 06-04997-6.


[1] Flandro, G. A. “Fast reconnaissance missions to the outer solar system utilizing
   energy derived from the gravitational field of Jupiter”, Astronautica Acta, Vol. 12,
   n. 4, 1966, pp. 329-337.

[2] Hollister, W.M., Prussing, J.E. “Optimum transfer to Mars via Venus”.
   Astronautica Acta, Vol. 12, n. 2, 1966, pp. 169-179.

[3] D'Amario, L.A., Byrnes, D.V., Stanford, R.H. “A New Method for Optimizing
   Multiple-Flyby Trajectories”, Journal of Guidance, Control and Dynamics, Vol. 4,
   n. 6, 1981, pp. 591-596.

[4] D'Amario, L.A., Byrnes, D.V., Stanford, R.H. “Interplanetary Trajectory
   Optimization with Application to Galileo”, Journal of Guidance Control and
   Dynamics, Vol. 5, n. 5, 1982, pp. 465-471.

[5] Longuski, J.M., Williams, S.N., Steve, N. “Automated design of gravity-assist
   trajectories to Mars and the outer planets”, Celestial Mechanics and Dynamical
   Astronomy, Vol. 52, n. 3, 1991, pp. 207-220.

[6] Striepe, S.A., Braun, R.D. “Effects of a Venus swingby periapsis burn during an
   Earth-Mars trajectory”, Journal of the Astronautical Sciences, Vol. 39, n. 3, 1991,
   pp. 299-312.
[7] Peralta, F., Flanagan, S. “Cassini interplanetary trajectory design”, Control Eng.
   Practice, Vol. 3, n. 11, 1995, pp. 1603-1610.

[8] Sims, J.A., Staugler, A.J., Longuski, J.M. “Trajectory options to Pluto via gravity
   assists from Venus, Mars, and Jupiter”, Journal of Spacecraft and Rockets, Vol.
   34, n. 3, 1997, pp. 347-353.

[9] Sukhanov, A.A. “Close approach to Sun using gravity assists of the inner
   planets”, Acta Astronautica, Vol. 45, n. 4-9, 1999, pp. 177-185.

[10] Longuski, J. M., Williams, S. N. “The last grand tour opportunity to Pluto,” The
   Journal of the Astronautical Sciences, Vol. 39, n. 3, 1991, pp. 359-365.

[11] Patel, M. R., Longuski, J. M., Sims, J. A. “Uranus-Neptune-Pluto opportunity”,
   Acta Astronáutica, Vol. 36, n. 2, 1995, pp. 91-98.

[12] Hammel, H. B., Baines, K. H., Cuzzi, J. D., de Pater, I., Grundy, W. M.,
   Lockwood, G. W., Perry, J., Rages, K. A., Spilker, T., Stansberry, J. A.
   Exploration of the Neptune System 2003-2013. Astronomical Society of Pacific
   Conferences, Vol. 272, 2002, pp. 297-322.

[13] Swenson, B. L. “Neptune atmospheric probe mission,” AIAA paper 92-4371,

[14] Gershman, R., Nilsen, E., Wallace, R. “Neptune orbiter,” paper presented in IAA
   International Conference on Low Cost Planetary Missions, 2000.

[15] Esper, J. “The Neptune/Triton explorer mission: A concept feasibility study,”
   NASA Center for AeroSpace Information and European Space Agency ESA,

[16] Fiehler, D. I., Oleson, S. R. “Neptune orbiters utilizing solar and radioisotope
   electric propulsion,” AIAA paper 2004-3978, 2004.

[17] Malyshev, V. V., Usachov, V. E., Tychinskii, Y. D. “Solar probe mission with
   multiple gravity assist maneuvers realized with conversion launchers”, Cosmic
   Research, Vol. 41, n. 5, 2003, pp. 431-442.

[18] Racca, G. D. “Capability of solar electric propulsion for planetary missions”,
   Planetary and Space Science, n. 49, 2001, pp. 1437-1444.

[19] Solórzano, C. H., Sukhanov, A. A., Prado, A. B. A. “Optimization of transfers to
   Neptune”, Nonlinear Dynamic and System Theory, to be published.
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