ANALYSIS OF TRAJECTORIES TO NEPTUNE USING GRAVITY ASSISTS1
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ANALYSIS OF TRAJECTORIES TO NEPTUNE USING GRAVITY ASSISTS1 Carlos Renato Huaura Solórzano2, Alexander Alexandrovich Sukhanov3, Antonio Fernando Bertachini de Almeida Prado2 ABSTRACT 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. INTRODUCTION The technique of gravity-assist maneuvers is studied for several authors. In the decade of the 60’s, Flandro  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  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.  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  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  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.  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  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 1 Presented at The Malcolm D. Shuster Astronautics Symposium, realized 12-15, June, 2005. 2 National Institute for Space Research (INPE). 3 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 , Patel et al. ) 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.  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 ). 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. ) 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 ). 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 ). Other papers (Malyshev et al.  and Racca ) 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. ANALYSIS OF THE SCHEMES A previous work performed by the authors of this paper (Solórzano et al. ) 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 Neptune. 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 CONCLUSIONS 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. ACKNOWLEDGMENTS 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. REFERENCES  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.  Hollister, W.M., Prussing, J.E. “Optimum transfer to Mars via Venus”. Astronautica Acta, Vol. 12, n. 2, 1966, pp. 169-179.  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.  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.  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.  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.
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