12
Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.
AIAA Meeting Papers on Disc, 1996, pp. 400-410A9634751, NGT-51129, AIAA Paper 96-3614
Trajectory options to Pluto via gravity assists from Venus, Mars, and Jupiter
Jon A. SimsPurdue Univ., West Lafayette, IN
Andrew J. StauglerPurdue Univ., West Lafayette, IN
James M. LonguskiPurdue Univ., West Lafayette, IN
AIAA/AAS Astrodynamics Conference, San Diego, CA, July 29-31, 1996, Collection of
Technical Papers (A96-34712 09-12), Reston, VA, American Institute of Aeronautics and
Astronautics, 1996, p. 400-410
We use analytic and numeric techniques to assess trajectory options for the Pluto Express science spacecraft to belaunched early in the next decade. These techniques have been shown to be highly efficient and thorough. Theconstraints placed on the Pluto Express trajectory for this study are severe (total flight time to Pluto of 12 years orless using a Delta-class launch vehicle with no upper stage). In addition, no gravity assists involving the Earth arepermitted. Using the aforementioned techniques, we found suitable trajectories with launch windows before, near,and after the date of the baseline launch. We also discovered several asteroid flyby opportunities for the baselinemission and for a backup trajectory. (Author)
Page 1
AIAA-96-3614-CP
TRAJECTORY OPTIONS TO PLUTO VIAGRAVITY ASSISTS FROM VENUS, MARS, AND JUPITER
Jon Andrew Sims* Andrew James Staugler,t and James Michael Longuski*School of Aeronautics & Astronautics
Purdue University, West Lafayette, IN 47907-1282
AbstractIn this paper we use analytic and numeric techniques to assess trajectory options for the Pluto
Express sciencecraft to be launched early in the next decade. These techniques have been developed bythe authors in previous works and have been shown to be highly efficient and thorough. The constraintsplaced on the Pluto Express trajectory for this study are severe — total flight time to Pluto of 12 years orless using a Delta-class launch vehicle with no upper stage. In addition, no gravity assists involving theEarth are permitted. Using the aforementioned techniques, we found suitable trajectories with launchwindows before, near, and after the date of the baseline launch. We also discovered several asteroid flybyopportunities for the baseline mission and for a backup trajectory.
Introduction
Pluto is the only known planet in the solar systemthat has not been visited by an interplanetary space-craft. For many years NASA has been studying mis-sion concepts to explore this distant world. By 1993the concept, known then as the Pluto Fast Flyby mis-sion, was to launch a relatively low mass spacecraft(with a dry mass of approximately 100 kg) on a directtrajectory using a Titan IV or Proton launch vehicleand upper stages. In the prevailing budgetary cli-mate, however, these launch configurations have beendeemed too expensive.
The current concept, known as Pluto Express,evolved from a thorough trade study of various com-binations of launch vehicles, upper stages, trajec-tory types, and spacecraft systems. The study showsthat the most cost effective and lowest risk option(for a launch in 2001 or 2002) uses a Delta or Mol-niya launch vehicle, with no upper stage, to placethe spacecraft on a trajectory with gravity assists atVenus and Jupiter.1 The baseline trajectory (for thistrajectory type in the study) launches in March 2001and flies by Venus three times before using a Jupitergravity assist to reach Pluto in about 12 years. (Thistrajectory is designated by VVVJGA to indicate 3
'Doctoral Candidate. Member AIAA.'Graduate Student. Student Member AIAA.'Associate Professor. Associate Fellow AIAA. Member
AAS.
Copyright © 1996 by Jon A. Sims, Andrew J. Staugler andJames M. Longuski. Published by the American Institute ofAeronautics and Astronautics, Inc. with permission.
Venus gravity assists and 1 Jupiter gravity assist inthat, order.)
From a programmatic point of view it is importantto have a backup trajectory available with a launchdate roughly a year from the baseline to allow forsome schedule slips during spacecraft development.To minimize the impact on the design of the space-craft, the backup trajectory should have substantiallythe same characteristics as the baseline. Similar di-rect trajectories from Earth to another planet occurevery synodic period. However, when more than twoplanets are involved, the relative alignment of theplanets does not repeat for a long time. The backuptrajectory in this case will necessarily use a differ-ent combination of transfers to reach the final des-tination. The mission cannot wait until the originaltrajectory repeats. So a search is required to find asuitable backup to the VVVJGA trajectory for thePluto Express mission.
In this paper, we search for trajectories to Plutowith the following characteristics:2
1. Launch date: October 2001 - December 2002
2. Post-launch deterministic AV < 3500 m/s
3. Cs such that propellant and payload can belaunched on a Delta 7925 with no upper stage
4. Flight time < 12 years
5. No Earth flybys
The 3500 m/s post-launch AV and 12 year flight timeare not hard limits but serve as guidelines for exam-ining and comparing trajectories. Although initially
400
the fly by radius at Jupiter is not constrained, we use5 Jupiter radii as a guideline for the minimum flybyradius in order to mitigate the radiation damage tothe spacecraft.
We describe the methods we have developed tosearch for trajectories with the characteristics givenabove, and we present some trajectories resultingfrom our search. We apply these same methods toidentify early launch opportunities (pre-baseline) inlate 20003 and to reexamine the options around thetime of the baseline.
Approach
For this study we use a combination of analytictechniques and numeric software tools to find trajec-tories to Pluto satisfying the given constraints. Thetwo primary mission design software tools that we useare STOUR4 (Satellite Tour Design Program) andMIDAS5 (Mission Design and Analysis Software).Both of these programs were originally developed atJPL.
Numeric Software Tools
STOUR was modified by Williams6 to providethe capacity of automated design of patched-conicgravity-assist trajectories. (The original version wasinteractive — not automated.) The user providessearch parameters including a range (and step size)of launch dates and launch energies and the sequenceof planets to be encountered. The user then executesSTOUR to find all trajectories within the given con-straints. Patel7 incorporated into STOUR the abilityto determine an estimate of local minimum AV forpowered flybys and broken-plane maneuvers.
MIDAS minimizes total AV while using patched-conic trajectory simulation. The program is capableof shifting trajectory event times such as launch date,arrival date, and flyby dates and is able to add ordelete deep space maneuvers and powered flybys inorder to find an optimal solution.
Analytic Techniques
A Jupiter gravity assist has enormous potential toreduce the launch energy, total AV, and flight timefor trajectories to Pluto.8"15 We note that Saturn,Uranus, and Neptune are not in good positions toprovide a gravity assist to Pluto for the launch daterange we are considering and for total flight times of12 years or less. So our search focused on trajectoriesthat use a Jupiter gravity assist. In the time framewe are considering, a Delta 7925 with no upper stagecannot launch the Pluto Express spacecraft directly
5 6 7TOTAL DELTA-V (KM/S)
Fig. 1. Venus Gravity Assist Potential. (Mul-tiple Venus Encounters without Voo Leverag-ing-)
to Jupiter (keeping the flyby radius above 5 Jupiterradii) such that it reaches Pluto in less than 12 years.Since we are searching for trajectories that do not usean Earth gravity assist, we use multiple Venus flybys(as another way) to increase the heliocentric energyof the trajectory to reach Jupiter at the appropriatetime with a sufficient arrival VCQ.
Following an initial Venus flyby, trajectory legswhich reencounter Venus can be either V^ turning orVoa leveraging. With Voo turning there are no ma-neuvers between the Venus encounters, and the Vco(magnitude) at Venus remains the same. The aphe-lion radius that can be achieved with Voo turningusing one or more Venus gravity assists is shown inFigure 1. For this figure we assume that the orbits ofEarth and Venus are circular and coplanar and thatthe launch Voo is directed opposite to Earth's veloc-ity. The minimum flyby altitude at Venus is assumedto be 250 km. The time-of-flight problem betweenVenus gravity assists is not taken into account forthe multiple Venus flybys, so these contours repre-sent potential and may not be realizable.
As the Earth launch energy increases, the Voo atVenus increases. The solid line in Figure 1 representsthe final aphelion radius if the Vco can be turnedparallel to the velocity of Venus (with respect to theSun), Vv- A single gravity assist can turn the Vooa limited amount. This turn angle decreases as theVoo increases. As the launch energy increases andthe corresponding Voo at Venus increases, a point is
401
reached at which a single flyby can no longer turnthe Voo parallel to Vv, and the single flyby curvein Figure 1 leaves the solid curve. The final aphelionradius then reaches a maximum and decreases. Mul-tiple (n) Venus flybys increase the effective turn angleby a factor of n, and the curve shapes are similar tothose for a single Venus flyby.
The dashed lines in Figure 1 represent the per-formance that can be achieved with a maneuver im-mediately after the final Venus gravity assist. Thedashed lines originate from the point on each curvebeyond which it is more efficient to add AV after thefinal Venus flyby.
The solid curve labeled "Direct Launch" indicatesthe aphelion that can be achieved by a launch di-rectly from Earth with no Venus flyby. As can beseen from the figure, a single Venus gravity assist andsubsequent AV requires more total AV than a directlaunch to reach the radius of Jupiter, but multipleVenus gravity assists have the potential to outper-form a direct launch.
The term 14o leveraging refers to the use of a rel-atively small deep-space maneuver to modify the Vooat a body. For the purposes of this study, the maneu-ver occurs near aphelion of a near-resonant transferbetween consecutive Venus flybys to increase the Vooat Venus. These trajectories are analogous to theAV-EGA trajectories introduced by Hollenbeck.16
The potential of these AV-VGA trajectories is shownin Figure 2 where we plot the final aphelion radiusthat can be achieved, without a propulsive maneuverat Venus, as a function of the aphelion AV. The num-bers on the plot correspond to the number of Venusyears between Venus flybys for the nominal resonanttransfer. The "+" ("-") indicates Venus encounterjust after (before) the spacecraft passes through peri-helion. (A more thorough analysis and explanation ofVoo leveraging and Voo turning at Venus is presentedin Reference 17.)
The aphelion on the transfer leg of the 2* AV-VGA is slightly larger than the semi-major axis ofMars. So given the appropriate phasing betweenVenus and Mars, a Mars flyby would occur near aphe-lion and could be used to offset or entirely replace theaphelion AV, thereby making these trajectories veryefficient in terms of propellant usage.
Methods for Discovering Trajectories
We use three methods for discovering completetrajectories from launch to Pluto arrival. In the firstmethod we use STOUR to analyze various segments
510'
1 1.5APHELION DELTA-V (KM/S)
Pig. 2. AV-VGA Performance Versus Aphe-lion AV.
of possible trajectories. We then patch together thesesegments and add Venus-Venus transfers as appropri-ate based on our analytic techniques. In the secondmethod we modify trajectories that we originally de-veloped for the Cassini mission to Saturn. In thethird method we run STOUR by specifying the en-tire sequence of flybys from launch to Pluto arrival.In each case we use MIDAS to optimize the trajecto-ries to minimize the total AV.
Method 1The arrival date at Venus for trajectories launched
from Earth is shown in Figure 3. (This type of plotis known as a pork-chop plot.) The launch date forthe STOUR run ranges from October 1, 2001 (desig-nated 11001) to January 1, 2003 (30101) in steps of 5days. The plotted numbers 0, 2, 3, and 4 correspondto the Earth launch VooS 3.0, 3.5, 4.0, and 4.5 km/s,respectively. Type I and II trajectories (transfer an-gle less than 360°) are clearly distinguished from theType III and IV trajectories (transfer angle between360° and 720°) which have longer flight times.
Figure 4 shows the time of flight (TOF) for trajec-tories from Venus to Pluto which fly by Jupiter. (Thetype of plot in Figure 4 is a generalized pork-chop plotfor trajectories which include a gravity assist.) The"Launch Date" would actually be the date of the fi-nal Venus flyby. The plotted numbers 0, 2, 3, and4 correspond to VooS at Venus of 12, 14, 16, and 18km/s, respectively.
The Venus-Jupiter-Pluto run provides dates onwhich the last Venus flyby should occur in order to
402
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Fig. 3. Earth-Venus Launch Opportunities. Fig. 4. Venus-Jupiter-Pluto Trajectory Op-portunities.
use Jupiter on the way to Pluto. We mark thesedates as arrival dates on the launch date/arrival dateplots of the Earth-Venus trajectories. We can thenpick out trajectories from Earth that arrive at Venusapproximately an integer multiple of Venus years be-fore the required Venus fly by dates for trajectories toPluto via Jupiter. Additional Venus flybys are usedwith phasing such that the final Venus flyby occursat the proper time. Using this process, we discoveredseveral trajectories to Pluto with positive injectionmargins using the Delta 7925. Most of these trajec-tories include Voo leveraging between Venus flybys;that is, there are maneuvers near aphelion which in-crease the Voo at the following Venus fiyby. With Vooleveraging, the flyby dates at Venus (on the Earth-Venus leg) differ somewhat from an integer numberof Venus years from the final Venus flyby dates. Thischaracteristic provides for greater flexibility in possi-ble launch dates (than the Voo turning trajectories)and must be considered in the search for such trajec-tories.
Method 2We found several trajectories to Pluto by mod-
ifying a trajectory with three Venus flybys that weoriginally developed for the Cassini mission to Sat-urn. This trajectory was discovered by starting witha complete analytic solution,17 using STOUR to de-termine the appropriate launch date, and optimizingwith MIDAS. The trajectory had no Earth flybys andused Voo leveraging with Venus. Instead of encoun-tering Saturn after the last Venus flyby, the trajecto-ries are now targeted to fly by Jupiter on their way toPluto. In one case we use a Mars gravity assist to re-
place the Voo leveraging maneuver near aphelion onone of the Venus-Venus legs, resulting in a savings ofmore than 300 m/s of AV. In another case we removethe third Venus flyby and proceed directly to Jupiterafter the second Venus flyby.
Method 3The latest version of the automated STOUR can
include a single maneuver between one pair of flybys.This maneuver, which is locally optimized, is either apowered flyby or a broken-plane maneuver — it is nota Voo leveraging maneuver. In our STOUR runs, weplace this maneuver after the final Venus flyby. Thetrajectories from STOUR are then optimized usingMIDAS. In the MIDAS runs the total AV may bereduced using Voo leveraging by adjusting the flybydates and including a maneuver between Venus fly-bys. .
The following sequences are examined usingSTOUR:
EVVV(AV)JPEVVV(AV)PEVV(AV)JPEVMVV(AV)JPEVMVV(AV)PEVMV(AV)JPEVVMV(AV)JP
A few different trajectories are identified using thisapproach. Only one of the trajectories identified byMethods 1 and 2 was rediscovered. This appears to
403
be due to the inability of STOUR to do VOQ leverag-ing. The one trajectory that was rediscovered has noVoo leveraging maneuver.
Results
Trajectories to Pluto
Characteristics of ten of the trajectories, identi-fied by the methods described above, are presentedin Table 1. Five of the trajectories listed are EVVJP,two are EVMVVJP, and the rest are EVVVJP. Thelaunch dates range from April 3, 2002 to August 23,2002. (Recall that we are considering the time periodfrom October 2001 to December 2002.)
STOUR was used with the sequence EVVV(AV)JP to help find Trajectory X and with the sequenceEVMVV(AV)JP to help find Trajectory VIII. Fol-lowing Method 3, the output from STOUR was usedas input to MIDAS. After several runs of MIDAS,with intervening manipulation of the flyby dates andmaneuver locations, the trajectories in Table 1 werefound. STOUR also independently found a tra-jectory similar to Trajectory V with the sequenceEVV(AV)JP. MIDAS was used to manipulate thetrajectory from STOUR and to eventually convergeon the trajectory in the table, which we had alreadydiscovered by other means. STOUR has found noother comparable trajectories using the complete se-quences of planetary flybys that we have examined.All the other trajectories that are presented were dis-covered with MIDAS by patching together segmentsof the trajectories from STOUR (Method 1) or by ma-nipulating trajectories we had discovered previously(Method 2).
If Mars is in the appropriate place to allow a flybybetween Venus encounters, a Mars gravity assist canbe used to replace the Voo leveraging maneuver. Ex-amples of this can be seen by comparing TrajectoryIX to Trajectory VI and Trajectory VIII to Trajec-tory VII. Trajectory VI uses a maneuver of 254 m/sbetween the first two Venus encounters to increase theVoo from 6.04 km/s to 8.14 km/s and uses a maneu-ver of 465 m/s before the final Venus flyby to increasethe VOQ to 12.53 km/s. Trajectory IX uses a Marsgravity assist between the first two Venus encountersto increase the V^ from 6.73 km/s to 13.42 km/s,resulting in a savings of 350 m/s in total determinis-tic AV. A similar comparison can be made betweenTrajectories VIII and VII, where the savings in totalAV is more than 500 m/s.
Injection Margin
An important parameter in mission design is theinjection margin — the difference in mass betweenwhat the launch vehicle can inject on a given trajec-tory and the total mass of the spacecraft and propel-lants that is to be launched on that trajectory. Thetrajectories listed in Table 1 are summarized in Table2 along with their injection margins.18 In determin-ing the injection margin, we assume a launch vehiclecontingency of 10% and an adapter mass which is 5%of the injected mass. We use the rocket equation todetermine the propellant mass.
AVTPL = /n(mi/mf)
The mass of the propellant tanks is assumed to be15% of the propellant mass, so we have
m; = ms/c + mp + 0.15mp
andmf = 0.15mp
wherem; is the total injected wet mass (initial mass),mf is the total injected dry mass,ms/c is the mass of the spacecraft (excluding the
propellant tanks), andmp is the mass of the propellant.
We assume an Isp of 320 s and the total injected drymass (spacecraft and propellant tanks) to be 235 kg.The total post-launch AV, AVxpL, is the sum of thedeterministic post-launch AV, AVpL, and the navi-gation AV, AVNAV- The navigation AV is a roughestimate based on the number of flybys and rangesfrom 200 m/s to 300 m/s. The injection marginis then the injected mass capability (with a contin-gency) of the Delta 7925 for the given Cs minus themass of the spacecraft, propellant tanks, propellant,and adapter.
We use MIDAS to minimize the total determin-istic AV (launch AV + AVpL,). Although there is acorrelation between total deterministic AV and in-jection margin, the trajectories are not optimized tomaximize injection margin. The propellant systemon the spacecraft has an effect similar to an upperstage on the launch vehicle such that for a given totaldeterministic AV, a larger AVpL results in a largerinjection margin. Spacecraft design considerations,however, dictate an upper limit on AVpL and actu-ally favor a smaller AVpj,. The final mission designmust take into account these trade-offs between tra-jectory design and spacecraft design.
404
Table 1 Trajectory Characteristics Table 1 Trajectory Characteristics (continued)
Trajectory
LaunchPerihelionPerihelionVenus 1ManueverPerihelionVenus 2
Jupiter
Pluto
I EVVJP
4/3/20028/10/20025/23/20036/15/20031/14/20049/11/200410/2/2004
2/7/2006
4/3/2014
[2+AV-VGA]
C3 = 14.77 km2/s2
0.696 AU0.696 AUV<x, = 6.21 km/sAV = 547 m/s, 1.620.664 AU (min)Voo = 10.21 km/s,AV = 2.953 km/sVoo = 13.60 km/s,flyby radius = 9.2Voo = 14.83 km/s
AU
Rj
Trajectory IV EVVJP [3~AV-VGA]_____
Launch 5/13/2002 C3 = 14.53 km2/s2
Maneuver 9/29/2002 AV = 142 m/s, 0.63 AUPerihelion 10/2/2002 0.626 AU (min)Perihelion 7/4/2003 0.626 AU (min)Venus 1 8/7/2003 V^ = 7.94 km/s,
AV = 721 m/sManuever 7/1/2004 AV = 365 m/s, 2.30 AUVenus 2 5/7/2005 Voo = 12.34 km/s,
AV = 2.223 km/sPerhelion 5/15/2005 0.706 AUJupiter 8/1/2006 Voo = 16.11 km/s,
flyby radius = 12.5 RjPluto 5/13/2014 Voo = 15.22 km/s
Total deterministic post-launch AV = 3.452 km/s
Trajectory V EVVJP [2:1 Venus-Venus]
Total deterministic post-launch AV = 3.500 km/s
Trajectory II EVVJP [3+AV-VGA]
C3 = 13.24 km2/s2
AV = 708 m/s, 0.63 AU0.627 AU (min)0.627 AU (min)Voo = 8.90 km/s,AV = 402 m/sAV = 384 m/s, 2.33 AU0.657 AUVoo = 12.73 km/s,AV = 2.102 km/sVoo = 16.31 km/s,flyby radius = 12.2 RjVoo = 15.48 km/s
Total deterministic post-launch AV = 3.595 km/s Total deterministic post-launch AV = 2.534 km/s
LaunchManeuverPerihelionPerihelionVenus 1
ManueverPerihelionVenus 2
Jupiter
Pluto
4/3/20028/18/20028/21/20026/9/20037/12/2003
5/16/20045/11/20055/29/2005
8/4/2006
4/3/2014
Launch 7/17/2002Venus 1 11/1/2002Perihelion 11/20/2002Venus 2 1/25/2004
Perhelion 1/29/2004Jupiter 8/16/2005
Pluto 7/17/2014
C3 = 26.63 km2/s2
Voo = 10.11 km/s0.682 AU (min)Voo = 10.11 km/s,AV = 2.534 km/s0.719 AUVoo = 11.26 km/s,flyby radius = 6.9 RjVoo = 13.79 km/s
Trajectory III EVVJP [2~AV-VGA]
LaunchPerihelionPerihelionVenus 1ManueverVenus 2
PerihelionJupiter
Pluto
5/11/20029/29/20026/30/20038/3/20032/16/20049/17/2004
9/21/20042/6/2006
5/11/2014
C3 = 12.92 km2/s2
0.640 AU (min)0.640 AU (min)Voo = 7.18 km/sAV = 285 m/s, 1.58 AUVoo = 9.19 km/s,AV = 3.328 km/s0.717 AUVoo = 13.46 km/s,flyby radius = 9.4 RjVoo = 14.61 km/s
Total deterministic post-launch AV = 3.613 km/s
Trajectory VI EVVVJP[2+AV-VGA, 3~ A V- VGA]
LaunchVenus 1PerihelionManeuverPerihelionVenus 2PerhelionManueverVenus 3
Jupiter
Pluto
7/20/200212/9/200212/9/20027/27/20033/6/20043/21/20043/22/20042/3/200512/31/2005
2/1/2007
7/20/2014
C3 = 12.74 km2/s2
Voo = 6.04 km/s0.719 AUAV = 254 m/s, 1.600.691 AU (min)Voo = 8.14 km/s0.718 AUAV = 465 m/s, 2.30Voo = 12.53 km/s,AV = 3.062 km/sVoo = 19.14 km/s,flyby radius = 19.Voo = 15.64 km/s
AU
AU
i R j
Total deterministic post-launch AV = 3.781 km/s
405
Table 1 Trajectory Characteristics (continued) Table 1 Trajectory Characteristics (continued)
Trajectory VII EVVVJP[2-AV-VGA, 2:1 Venus-Venus]
LaunchVenus 1PerihelionManeuverVenus 2PerihelionVenus 3
PerhelionJupiter
Pluto
7/29/200212/14/200212/19/20028/10/20032/15/20043/7/20045/10/2005
5/16/20058/6/2006
7/29/2014
C3 = 14.34 km2/s2
Voo = 6.25 km/s0.716 AUAV = 529 m/s, 1.59VOD = 10.04 km/s0.664 AU (min)Voo = 10.04 km/s,AV = 3.511 km/s0.711 AUVoo = 15.91 km/s,flyby radius = 13.Voo = 14.79 km/s
AU
i R j
Trajectory IX EVMVVJP[2+M-VGA, 2.75 Venus-Venus]
Launch 8/14/2002 C3 = 20.24 km2/s2
Venus 1 12/22/2002 Voo = 6.73 km/s,AV = 163 m/s
Perihelion 12/26/2002 0.717 AUMars 5/3/2003 V<x, = 12.22 km/sPerihelion 3/24/2004 0.604 AU (min)Venus 2 4/19/2004 Voo = 13.42 km/sVenus 3 12/28/2005 V^ = 13.42 km/s,
AV = 2.945 km/sJupiter 2/1/2007 V*, = 19.06 km/s,
flyby radius = 19.2 RjPluto 8/11/2014 Voo = 15.50 km/s
Total deterministic post-launch AV = 4.040 km/s Total deterministic post-launch AV = 3.108 km/s
Trajectory VIII EVMVVJP[2~M-VGA, 2:1 Venus-Venus]
LaunchVenus 1PerihelionMarsVenus 2PerihelionVenus 3
PerhelionJupiter
Pluto
8/9/200212/20/200212/25/20025/11/20032/11/20043/5/20045/5/2005
5/14/20058/5/2006
8/9/2014
C3 = 18.21 km2/s2
Voo = 6.62 km/s0.716 AUVoo = 10.78 km/sVoo = 11.47 km/s0.641 AU (min)V^ = 11.47 km/s,AV = 3.336 km/s0.699 AUVoo = 15.85 km/s,flyby radius = 13.1 RjVoo = 14.72 km/s
Trajectory X EVVVJP[1:1 Venus-Venus, 2:1 Venus-Venus]
LaunchVenus 1PerihelionVenus 2PerihelionVenus 3
PerihelionJupiter
Pluto
8/23/200211/14/20021/2/20036/27/20037/10/20039/18/2004
9/22/20042/13/2006
8/24/2014
C3 = 16.87 km2/s2
Voo = 8.98 km/s0.598 AU (min)Voo = 8.98 km/s0.701 AUVoo = 8.98 km/s,AV = 3.159 km/s0.718 AUVoo = 13.22 km/s,flyby radius = 10.0 RjVoo = 14.07 km/s
Total deterministic post-launch AV = 3.336 km/s Total deterministic post-launch AV — 3.159 km/s
Flight Time Analysis
During our initial search, we looked for trajecto-ries that could have adequate performance. Gener-ally, a shorter flight time requires a larger total AV,so the trajectories presented in Table 1 have flighttimes of approximately 12 years, the longest flighttime that was considered reasonable. After the ini-tial search, we performed a preliminary analysis ofthe effect of shorter flight times for some of the tra-jectories. (See Table 3.) The results indicate thatflight times shorter than 12 years are possible whilemaintaining a positive injection margin for launch ona Delta 7925. For example, the injection margin ofTrajectory V decreases from 80.7 kg to 51.2 kg as the
flight time to Pluto decreases from 12.0 years to 11.0years. In general, as the flight time decreases, theAV immediately following the final Venus flyby in-creases while the Jupiter flyby radius decreases. Theinitial legs of the trajectories remain approximatelythe same.
Launch Window Analysis
We also briefly examined the launch window forTrajectory V, which has been selected as a backup.1
Some characteristics of the trajectory for a 23-dayrange of launch dates and a fixed arrival date aregiven in Table 4. The impact on the injection marginin this case appears to be relatively small, allowing forample launch opportunities during the given window.
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Table 2 Trajectories to Pluto Using the Delta 7925(Flight Time: 12 years)
TrajectoryNumber
IIIIIIIVVVIVIIVIIIIXX
TrajectoryType
VVJVVJVVJVVJVVJVVVJVVVJVMVVJVMVVJVVVJ
LaunchDate
4/3/20024/3/20025/11/20025/13/20027/17/20027/20/20027/29/20028/9/20028/14/20028/23/2002
C3(km2/s2)
14.813.212.914.526.612.714.318.220.216.9
AVpL(km/s)
3.5003.5953.6133.4522.5343.7814.0403.3363.1083.159
AVjjAV Injection(m/s) Margin (kg)
200200200200200250250300300250
52.054.455.568.480.7-1.1
-106.010.233.087.0
Earlier Launch Dates
Having found trajectories satisfying our initialconstraints, we extended our search to include launchdates as early as late 2000. The synodic period be-tween Earth and Venus is 1.6 years, so Earth-Venustrajectories similar to those represented in Figure 3are available with launch dates approximately 1.6years (2.6 Venus years) earlier. Following the pro-cedure in Method 1, we can determine the Venus ar-rival dates that are approximately an integer multi-ple of Venus years before the required Venus flybydates for trajectories to Pluto via Jupiter. (Figure 4is again used for these earlier launch dates.) This pro-cedure indicates that there are Type I and II Earth-Venus transfer legs with low Ca that are 5 or 6 Venusyears from the best Venus-Jupiter-Pluto legs. Ourapproach suggests that the most efficient trajecto-ries to Pluto via Jupiter would use these Earth-Venustransfer legs followed by a 2+ AV-VGA and then ei-ther a 3+ or 4+ AV-VGA. The baseline trajectorydescribed in the introduction uses a 2"1" AV-VGA fol-lowed by a 4+ AV-VGA. It is similar to a trajectorypresented in Reference 19. The characteristics of atrajectory similar to the baseline with a flight time of12.0 years are presented in Table 5.
We discovered several other trajectories to Plutowith launch dates in late 2000 and early 2001. Asexpected, none of these trajectories outperforms thebaseline, but several of them do have substantial in-jection margins. Characteristics of one of these tra-jectories that is well suited for the Pluto Express mis-sion are presented in Table 6. The Earth-Venus trans-fer for this trajectory is Type III, instead of the moreefficient Type I or II as used by the baseline.
Asteroid Flybys
NASA has a policy that missions to the outer so-lar system will include flybys of main-belt asteroids.20
Galileo flew by two asteroids during its VEEGA(Venus-Earth-Earth Gravity-Assist) trajectory toJupiter. The first spacecraft encounter with an as-teroid occurred on October 29, 1991, when Galileoflew by Gaspra at a relative velocity (V^) of 8 km/snear the aphelion of the Earth-Earth leg of the trajec-tory. After the final Earth flyby, Galileo flew by Ida(with a Voo of 12.4 km/s) and provided images withthe first direct evidence of a natural satellite of anasteroid. These flybys added much to our knowledgeof asteroids, which, in turn, plays an important partin our understanding of the formation and dynamicalevolution of our solar system.
To incorporate an asteroid flyby, we first opti-mize a trajectory to Pluto with planetary flybys andthen search for asteroids that pass "close" to this tra-jectory. These nontargeted asteroid encounters arestrictly a matter of chance. We can, however, givesome rules of thumb based on our experience andprovide some specific examples.
The resonance (or near-resonance for Voo lever-aging) of a Venus-Venus leg determines the aphelionradius of that portion of the trajectory, since the per-ihelion radius is generally close to the orbital radiusof Venus. An orbit with a resonance of 2 Venus yearshas an aphelion radius of around 1.6 AU. Less than5% of all known asteroids have a perihelion radiusbelow 1.6 AU,21 so Venus-Venus legs with 2 year res-onances have very few opportunities for asteroid en-counters. The aphelion radius is near 2.3 AU foran orbit with a resonance of 3 Venus years. About
407
Table 3 Examination of Flight Time to Pluto Table 5 Baseline Trajectory Characteristics
Traj.No.
I
II
III
IV
V
IX
LaunchDate
4/3/024/3/024/5/02
4/3/024/5/024/6/02
5/11/025/12/025/13/02
5/13/025/14/025/15/02
7/17/027/19/027/20/027/19/02
8/14/028/14/028/14/02
TOF(yrs)
12.11,10
121110
12
.0
.0
.0
.0
.0
.0
.011.010
121110
1211
.0
.0
.0
.0
.0
.010.010
121110
.0
.0
.0
.0
C3(W/s2)
14.7714.5914.32
13.2413.2613.30
12.9213.0613.26
14.5314.5914.69
26.6326.1625.9925.81
20.2420.3820.49
AVPL(km/s)
3.3.4.
3.4.4.
3.3.4,
3,3,4
2.2,34
334
.50,82,32
,60,02,68
,61.88,31
.45
.85
.48
.53
.73
.04
.03
.11
.61
.43
Rad.a
(flj)
9.27.15.1
12.29.56.9
9.47.35.3
12.59.77.1
6.95.33.85.0
19.215.011.0
Baseline EVVVJP[2+ A V- VGA, 4+ A V- VGA]
LaunchPerihelionVenus 1ManeuverPerihelionVenus 2ManeuverPerihelionVenus 3
Jupiter
Pluto
3/9/20017/21/20018/29/20013/20/200211/6/200211/25/20021/23/20045/14/20056/1/2005
7/11/2006
3/10/2013
C3 = 15.89 km2/s2
0.605 AU (min)Voo = 8.72 km/sAV = 150 m/s, 1.620.672 AUVoo = 9.70 km/sAV = 430 m/s, 2.970.649 AUVco = 14.43 km/s,AV = 1.674 km/sVoo = 17.75 km/s,flyby radius = 9.3V^ = 18.13 km/s
AU
AU
RJ
Total deterministic post-launch AV = 2.253 km/sNavigationInjection
Table
AVMargin
6 Trajectory
AVNAV = 250 m/s284 kg
XI Characteristics
Flyby radius at Jupiter.
Table 4 Launch Window for Trajectory V
Trajectory XI EVVVJP[2-AV-VGA, 4+AV-VGA]
LaunchDate*
7/7/20027/9/20027/11/20027/13/20027/15/20027/17/20027/19/20027/21/20027/23/20027/25/20027/27/20027/29/2002
C3(km2/s2)
25.1825.4925.4126.0625.7526.7826.7027.1427.1625.5224.6523.89
AVPL(km/s)
2.822.742.682.612.582.532.522.562.572.692.772.86
Total AV(km/s)
7.137.067.006.956.916.916.896.946.967.017.067.11
LaunchPerihelionVenus 1PerihelionManeuverVenus 2PerihelionManeuverPerihelionVenus 3
Jupiter
Pluto
8/10/20001/3/20019/3/20019/14/20014/25/200211/13/200211/19/20022/2/20045/14/20056/2/2005
6/24/2006
8/8/2012
C3 = 12.24 km2/s2
0.644 AU (min)Voo = 7.16 km/s0.705 AUAV = 392 m/s, 1.60 AUVoo = 9.84 km/s0.716 AUAV = 432 m/s, 2.99 AU0.648 AUVoo = 14.52 km/s,AV = 2.044 km/sVoo = 18.82 km/s,flyby radius = 7.7 RjVoo = 19.93 km/s
Total deterministic post-launch AV = 2.868 km/sNavigation AV AV^AV = 25° m/s
Injection Margin_____233 kg__________
a In all cases the arrival date is 7/17/2014 (for aflight time of 12.0 years) and the flyby radius atJupiter is 6.9 Jupiter radii.
408
Table 7 Potential Asteroid Flybys on the FinalVenus-Venus Leg of the Baseline Trajectory
Table 8 Potential Asteroid Flybys Followingthe Final Venus Flyby of Trajectory V
No.
323701838
14071907289729163182521754326324
Name
BruciaOriolaSeraphinaLindelofRudnevaOle RomerVoronveliyaShimanto1966 CL1988 VN1991 DN1
Radius(km)
2023321284414364
Increase inTotal AVa
(km/s)0.280.220.120.270.060.010.020.070.020.170.02
FlybyVo.
(km/s)9.07.17.96.410.811.414.56.515.06.710.7
Not including additional
20% of all asteroids have a semi-major axis less than2.3 AU, and some encounter opportunities usually oc-cur on these 3 Venus-year legs. For an orbit with aresonance of 4 Venus years, the aphelion radius isaround 3.0 AU — larger than the semi-major axis of70% of all asteroids. These orbits generally have sev-eral opportunities for asteroid flybys. The encounterVooS for asteroid flybys on the Venus- Venus legs havea fairly uniform distribution between 5 km/s and 15km/s with relatively few occurring outside this range.
The portion of the trajectory following the lastVenus flyby passes through the main asteroid belt.The cost in AV to add an asteroid encounter on thispart of the trajectory, and the Voo of the flyby, de-pends generally on whether there are a total of twoor three Venus gravity assists. Since the time fromlaunch to final Venus flyby is usually shorter if thereare only two Venus flybys, the heliocentric velocityfollowing the final Venus flyby required to reach Plutowith a total flight time of 12 years tends to be less.Hence the cost in AV and the flyby Voo also tend tobe smaller.
The baseline trajectory has many opportunitiesfor asteroid flybys at a relatively low cost in addi-tional deterministic AV. For example, a flyby of theasteroid Seraphina (#838, radius: 32 km, type: P)can be added near the aphelion of the final Venus-Venus leg, increasing the total deterministic AV by0.12 km/s. The relative velocity of the flyby is 7.9km/s. For an additional cost in total deterministicAV of 0.02 km/s, we can include a flyby of the aster-oid Rudneva (#1907, radius: 8 km) at a V^ of 11.0km/s about 200 days before the flyby of Seraphina.
No.
812176217742718286933515151
Name
AdeleRussellKulikovHandleyNepryadvaSmithWeerstra
Radius(km)
14129151068
Increase inTotal AVa
(km/s)0.310.140.100.270.420.350.12
FlybyVoc
(km/s)17.016.116.318.220.414.813.0
Not including additional
Table 7 lists a few asteroids that could be added tothe original trajectory. The increase in total AV andthe flyby VM listed in the table are for the case inwhich there is only one asteroid flyby.
When flybys are added after the final Venus flyby,the increase in total AV and asteroid encounter V^both tend to be higher (compared to encounters be-fore the final Venus flyby). For example, a flyby ofthe asteroid Thusnelda (#219, radius: 22 km, type:S) can be added 148 days after the final Venus flyby,increasing the total deterministic AV by 0.23 km/s.The relative velocity of the flyby is 25.4 km/s. Or foran additional cost in total deterministic AV of 0.24km/s, we can add a flyby of the asteroid 1983 VM7(#4692, radius: 3 km) at a Voo of 22.3 km/s. A flybyof a larger asteroid, Erida (#718, radius: 38 km) canbe added to the original trajectory with an increasein total deterministic AV of 0.32 km/s. For this flybythe Voo is 23.9 km/s.
Trajectory V has a total of two Venus flybys. TheVenus-Venus leg is a 2 Venus- year resonant orbit withan aphelion radius of 1.6 AU, providing essentiallyno opportunities for asteroid encounters. Followingthe final Venus flyby, however, the trajectory passesthrough the main asteroid belt, and an asteroid flybycan be included for an increase in total determinis-tic AV of about 0.1 km/s or more. A few potentialasteroid flybys are listed in Table 8.
Conclusions
The three methods described in this paper workquite well in discovering trajectories with widely dis-tributed launch windows for the Pluto Express mis-sion. We found many trajectories with positive in-jection margins using the Delta 7925. The backup
409
trajectory has a launch date in July 2002 (sixteenmonths after the baseline), and an excellent launchopportunity exists in August 2000 (seven monthsbefore the baseline). We also found opportunitiesaround the time of the baseline; however, accordingto our study, the baseline trajectory appears to be themost energy efficient opportunity for launch dates in2000-2002.
Acknowledgments
We gratefully acknowledge the contributing workof Steven N. Williams, which was performed at theJet Propulsion Laboratory, California Institute ofTechnology, under contract with the National Aero-nautics and Space Administration. We are alsograteful to Steven E. Matousek for his direction andsupport. Part of this work has been supportedby National Aeronautics and Space AdministrationGrant NGT-51129 (JPL Technical Advisor Steven N.Williams).
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