Design of Aerogravity-Assist Trajectories

Home , Gravity assist, Launch window, Specific orbital energy

AOO-39774

AIAA-2000-4031 DESIG AEROGRAVITY-ASSISF NO T TRAJECTORIES

Wyatt R. Johnson*and James M. Longuski"^ School of Aeronautics and Astronautics, Purdue University West Lafayette, Indiana 47907-1282

Aero-gravity assist (AGA) trajectories are optimized in the sense of maximizing AV obtaine e flyth by y ,db maximizin g aphelion, minimizing perihelion minimizind an , g e tim th f fligho e t (TOF particulaa r fo ) r destination planet graphicaA . l method based on Tisserand's criterion is introduced to identify potential AGA trajectories. To demonstrate the application of the theory, patched-conic AGA trajectories are compute launca d eaco dan t h , h7 VQ f f planeOe o Solao th D n ri tL/ Systemn a r Fo . 6.0 km/s reachee , b yearPlut 5 y 5. sn odma i usin Venus-Mars-Venuga s AGA.

Nomenclature used Earth and Venus on its way to Saturn. A much- E = heliocentric specific orbital energy, anticipated mission to Pluto, the Pluto-Kuiper Ex- km2/s2 press, will require high launch energy, along with = semimajo R bodyA rm axiAG k , f so a Jupiter gravity assist to reach Pluto in 8 years. Ra = aphelion distance, km Many other trajectories to Pluto have been found, — perihelioRp n distancem k , requiring up to 4 fly by bodies, with flight times be- 1 rp = periapsis at AGA body, km tween 10 and 15 years (compared to the Hohmann = nondimensionaU l heliocentric speed transfe r years)5 4 tim f o e . However a give r fo n, = t/onondimensionao l excess velocity planet and flyby V^, there is a limit to the bending = heliocentri V c velocit spacecraftf yo , (and thus AV) that gravity will supply. Further- km/s more, with each additional flyby, the total flight time Vpi = velocity of planet with respect to often increase requiree th d san d phasin plane th f g-o Sun, km/s becomes et s harde meeto t r . VQ= excesO s velocity, km/s Ove yearse rth , improvement gravite th n so y assist = angl ea between VVOd d pQan ira , idea have been proposed. One of these is to replace 6 = aerodynamic turn angle, rad the slower conic arcs between planets with faster 3 2 = gravitationap, l constant, km /s low-thrust arcs.2 Though this technique has merit, = turtotaA n AG l angled ra , as the VQO of the flyby body increases, the AV gained Superscripts becomes increasingly smaller (as in the case of the = pre-flyby superscript conventional gravity assist). Another idea involves 3 = post-flyb + y superscript flyin liftinga g body, (e.g. waverider,a ), througe hth atmospher flybe th f yeo planet. Aerodynamic forces Subscripts can augmen bendine th t g angl arbitrarilo et y large n = path index subscript values. Lewi McRonaldd san 4 contend tha teche tth - 0 — solar nology exists to build a waverider with L/D ratios greater than 7. The AGA maneuver could dramat- Introduction ically augment the gravity technique.5'6 An AGA RAVITY assist trajectories have become pow- has the added advantage of yielding AV increases Gerful aid enablinn si g mankin exploro dt e eth with higher flyby VOQS. Solar System. The famous Voyager II mission de- Because the turn angle in an AGA is arbitrary, pended on gravity assists from multiple planets. The an AGA can perform nearly as well in one flyby Galileo spacecraft used Earth and Venus in order s severaa l conventional gravity assists. There will to reach Jupiter; recently the Cassini spacecraft has be some energy losses due to drag, but this is more 'Doctoral Candidate, Member AIAA. than made up for by the years shaved from the flight ^Professor, Associate Fellow AIAA, Member AAS. e phasinth time d easiers gi an , . Preliminary work Copyrigh 200© t Wyaty b 0 . JohnsoR t d Jamean n. M s 7 Longuski. Published by the American Institute of Aeronautics by Sims shows that with AGA spacecrafa , t could and Astronautics, Inc. with permission. reac years7 ho t Plut ,5 wit n oi h minimal launch

energy. Bonfiglio8 confirm e worth sf Simo k y b s calculating several AGA trajectories to Pluto with Satellite th e Tour Design Program (STOUR), which 100- Bonfiglio modified for the purpose. 80- In this paper we find the maximum possible AV for an AGA maneuver, compute the performance en- velope of AGA trajectories, apply a graphical technique to gain further insight into available AGA 20- trajectories, and compute theoretical bounds on the 0- minimum tim flighf eo eaco t t h planet. 30 30 AGA Equations 20 10 A model relating pre and post flyby T^s is given 0 0 by:9 Fig. 1 AJ7 as a function of U^, and 2 1/2 AGA. (V~{ V n/r+= + p) exp (~29/(L/D)\ - M/r p} (1) For convenience, we nondimensionalize this equa- 2 tion, using Uoo = V00 /(/i/rp) to obtain:

U+ = (U-+l)eXp[-2B/(L/D)]-l (2) Wthed more an ncalculat th , e convenienAV e eth t non-dimensiona froU A lm

1 (3) 5 4 3 , . 2 1 0

where

1 1 = sin" [(1 + + sin" [(1 + U+-1 + lobe correspond optimathe sto l turn angl maxieto - (4) mize AU (and hence, AV) during the flyby. Further combinee b n ca yiel4 o dEquationt d n da an , 3 , s2 aerodynamic turning decreases U^ to the first val- expression for AU explicitly in terms of U^ and t/j+ : ley, where V+j pointsame th en si directio V^,s na . Turning beyond one revolution will again increase AU the AU, but less than before because of drag. Addi- tional revolutions could be completed until [/+, = 0, at which point the spacecraft would be captured in orbit about the gravity-assist planet. ) J5 ] 1) + Maximizing AV in a Single AGA Figur L/D=15r fo show1 e 5 . grapa s .Eq f ho To maximize the AV obtained by a single AGA, Becaus f draeo g losses, [/+ U^< general,n i e Th . we fin analytin a d c representatio e maximuth f no m special cas U^f U^eo = correspond convena o st - t functioa obviou s no a s i f , U^. o nst I froAU m tional (pure) gravity illustrates assisti d an , d sep- Eq thaincrease5 . U A t s almost linearly t Figbu ,1 . arately in Fig. 2. Note that there is a maximum clearly suggests it. A different approach to this max- l gravital Ar Ufo y assists. e Maximizinth r fo 5 . gEq imizatio dons ni Elices,y eb t wit insigh1e bu 0h th t pure gravity-assis tha e tmaximu e casese th te w , m provided by Fig. 1, we find an alternate approx- A occurd AG an e , th s1 Ar whes Ui Fo . n1 U^= imation. Though Elices's approximation has the case, however, the maximum AU is unlimited and advantag f beineo g simpler s accuracit , y decreases increases with increasing U^. The rippling effect in with increasing L/D. Fig. 1 is caused by the Voo being rotated around We start with the assumption that for the op- planee th t through several revolutions e maiTh n. timal turn angle, £/,+ « k U^, where k is some 2

unknown constant. Next, we make this substitution note W e . tha3 int. U^s a t oEq increases grave th , - itational therturd an n s , ei 5 angle . Eq sf o dro t pou quadratic/logarithmia c dependenc. AU U^f n eo o We e parametetheth n i solvk r refo optimization:

lim - --*™dk (6)

Becaus assumine ar e ew g U^ vers i y large grave th , - itational turn angle tends towards zero. , Fro4 . mEq we see that vectoA AG r e diagramTh Fig 3 . . = (l/2)(L/£>) In !)/([/+ +1)] w -(£,/£> fn c)I (7) Unsurprisingly, the transcendental version is the Equation 6 yields: mosratiosD L/ t ratiosw D accurat, lo L/ t l A al . t ea Elices's approximation is the next most accurate. At k - cos(» - (L/D) sin(<£) = 0 quadratie (8th ), 4 = abouc begin D bettero L/ d t o st . Sinc e Tayloeth r series expansio s donnwa e about This transcendental equation will generaln i , , have fc = 1 (which is the case for L/D —> oo), it is several roots which correspond to the locations in expecte methodo d tw thae th ts developed her- eas Fig wher1 . peaea reacheds ki .lookin e Sincar e ew g ymptotically reach zero error as L/D increases. for the maximum peak solution, we pick k to be the firse Th t error liste maximue Tablth n di s i e1 m value of the largest root. While Eq. 8 can be solved numerically, an explicit form is desired. Intuitively, percen compariso0 3 U^< < 0 t e erroth nn r i range . In all cases the largest errors occur at high values optimae th l turn angle wil somewhae b l t less than of [/oo; at high U^ and low L/D, none of the esti- e th TT .e Furthermoreus e W k —. 1 D *, — L/ oo » s a , mators are accurate. A more representative error is approximations: e meath n error, which show e averagsth e error fo r cos((f>) -1 all examined U^ is fairly small. Finally, the mean squared error (MSE l caseslistes i )al r d.fo sin(^>) TT - T= (L/D)+ T k n I Infc Optimizing for Perihelion or Aphelion Two terms are required for the In k term to solve for . (Thk e linear expressio fairls ni y inaccurate except potentialln Clearlca A yAG y yield dramati- cim for very high L/D ratios.) provements in attainable AV over conventional Substituting, we obtain a quadratic expression for gravity assists t unlesBu . maximua sA AG V mA k: makes it easier to reach a desired destination, the attainable AV is useless. The turn angle that re- 2TT sults in the maximum AV may not necessarily be -4 ) (9 =0 -T the optimal turn angle for reaching the next body. D e majoth Indeedf o r e pointe maximuon th , f o s m fine w d , 9 tha e . rooto on tEq tw Solvinn si e th r gfo AV theory is to provide a benchmark for practical roo always i t sothee lesth s alwaysi d r thaan , ns1 AGAs (e.g., AGAs that actually get the spacecraft d an , greaterI thae < se inspectink ty e B .w , 7 . gEq somewhere). thus the higher root is extraneous. Also as L/D —> To this end examine w , e optimizin ture - gth nan oo, fc —> 1, as expected in both cases. Table 1 has a gles abou planete th t maximizo st spacecraft'e eth s summary errore ofth thesn si e three approximations aphelion minimizo t r o , s perihelion,eit depending (Elices, the transcendental, and the quadratic) for targee th o f ni t bod farthes yi r fro r closee mo th o rt U^s ranging from 0 to 30. Sun. The spacecraft will then be able to reach any In the quadratic and transcendental formulations, body in the Solar System between these two bounds. AU is linearly related to U^ by: derivatione Th s that follow assume circular coplanar planetary orbits. Figure 3 presents a vector diagram AU = - 2fccos(c£ fc+ )2 (10maneuverA ) AG e oth f .

Table 1 Percent error in approximating maximum AGA At/ by different estimators

L/D ElicesIU Transcendental Quadratic Max Mean MSB" Max Mean MSEa Max Mean MSEa 1 74.4 17.1 3.3 69.0 3.7 0.7 55.3 44.5 21.6 2 61.9 3.7 0.4 58.4 1.4 0.3 68.8 23.3 5.7 3 53.4 3.2 0.3 49.7 0.7 0.2 55.2 6.8 0.6 4 46.9 3.6 0.2 43.0 0.4 0.1 46.3 2.5 0.2 5 41.5 3.6 0.2 37.8 0.3 0.1 40.0 1.0 0.1 10 23.2 1.1 0.1 23.3 0.3 0.0 23.7 0.2 0.0 15 17.8 6.5 0.5 16.7 0.4 0.0 16.8 0.0 0.0 a Mean Squared Error

L/D=oo case can be computed as: The simplest case to consider is L/D = oo where [U/(2Ra,p R2 - U)} = (12) e spacecrafth t drao t woul e g t losV^y du dno ean during the aerodynamic turning. This case puts a Equation 12 is derived assuming the spacecraft de- theoretical upper/lower A bounAG wha n y do an t parts Earth and executes an AGA at the next planet accomplishn ca matteo N waveride.w ho r r technol- in its path sequence. However, because (as shown in ogy progresses a waveride, r will neve e ablb ro t e ) Eqtangentia11 . l Earth departure optimae sar n i l outperform this limit. this sense becausd an , e tangentia planeA - AG lde t planeA e arrivas computeAG i t Th e th l t V^a d partures are optimal, we can patch several of these as: trajectorie soptimae togetheth t ge l o multi-bodt r y trajectory. Thi dons si departiny eb g each body tangentiall maximizo t s ya arrivae eth nexe l th T4 tt a o body. Whe spacecrafe nth t arrive nexe th tt sa body , cos( ) 72 t turn i t leavei o ss nex e s th tangentiall to t n o o g o yt ,2 cos(7i) planet. ,2 cos(7a) Finit casD eeL/ 1 l +2Mo (R^ - R~ ) Of course, infinit ratioD L/ eimpossiblee sar A . parameter optimization problem can be set up for VPI,I +1& v£,2+ - i2(Ri/R2)v Pi,iVpi,2 l 1 the finitD casL/ ee e similainfinitth D o t L/ re +2/J.Q (Rz - Rl } + 2^00,1^,1 cos(ai) case. However, V£ is now a function of V^ and B. (U) Launching tangentially from Eart als s ht i neces ono - sarily optimal. For a single-body AGA, maximizing Now that we have an expression for the arrival V^ aphelion or minimizing perihelion can be formulated at the AGA planet in terms of departure conditions as a two-dimensional parameter optimization prob- finoptimut e n Eartha th dca maximizo e t w , i ma e lem for a given L/D and launch V^. Each additional ^00,2- The first and second derivative rules tell us AGA e patbodth hn y i sequenc e adds anothe- di r that a-i = 0° if R2 > RI, or aa = 180°, if R2 < RI. mension to the problem (the aerodynamic turning In the R2 > RI case, we maximize aphelion, and angle for that planet). An analytic solution appears minimize perihelion in the other. intractable a numerica o s , l solutio s calculatedni . spacecrafa , - oo de f turn o y ca ntan Wit D L/ ha The results are presented in Figs. 4 and 5. sired turn angle without losing V^. The Hohmann Figur illustratee4 e maximuth s m apheliod an n transfer shows tha a tangentiat l A s optimuFi m minimum perihelion possible witt a singlha A eAG for maximizing aphelio r minimizinno g perihelion. Venu r severafo s l differen ratiosD e VenuL/ t Th . s Thus, the optimum turn angle in this case makes Gravity Assist (VGA) and Venus Aerogravity Assist the VOQ parallel to the planet's velocity vector (this (VAGA) contours begin at a launch V^ of 2.5 km/s, is not true for the finite L/D case, as rotating the which corresponds to the Hohmann transfer. Note VQO also decreases its magnitude). Since for a tan- that at this launch V^,, it is impossible to increase gential departure, maximizing heliocentric velocity aphelion, sinc spacecrafe eth t arrive t Venusa s with is equivalent to maximizing aphelion (and similarly, its VOQ aligned with the velocity vector of Venus minimizing heliocentric velocity is equivalent to min- . However0) (i.e.= stil s i a , t li , possibl decreaso et e + imizing perihelion), we know that V2 = Vpi,2±V00f2. perihelion. The more turning that can be accom- Thus, maximum aphelion or minimum perihelion plished (i.e., the higher the L/D ratio), the lower the

Direct L/D= Launch Pluto Neptune Uranus

Saturn

Jupiter

pure GA

Mars Earth Venus

Mercury

Launch V_ [km/s] Fig. 4 Maximizing aphelion or minimizing perihelion using a single Venus aerogravity assist (VAGA).

perihelion can be. The ability to decrease perihelion cal to Hohmann transfers from Venus to that planet, Hohmane foth r n transfe appare cause th th r s i refo - havy ma e d fairlan y lengthy l finit TOFsal r e Fo . t discontinuiten e Hohmanth Fign y— i 4 . n results L/D contours, the departure a will be somewhat in a perihelion at Venus; but we can immediately get larger. Thus, Venus will not be at perihelion (if additional bending to decrease it even further. traveling upwell r aphelio)o travelinf n(i g downwell). Although the TOFs will be shorter, they are not As the launch VTO increases, so does the arrival VQO at Venus. Furthermore, a also increases. This much mortravelinf i o outee s th o rgt planetsd an , makes it possible to increase aphelion by rotating the cose com increasef th o tt ea d launch V^. VOQ back towards a = 0. For launch VooS less than In many ways, Mars behaves oppositel Venuso yt . 3.2 km/s, a pure VGA is able to achieve maximum Figure 5 illustrates the Mars Gravity Assist (MGA) turning (without overturnin e VQO)gth . Thusn a , and the Mars Aerogravity Assist (MAGA). At the t neededAGno As i . Hohmann launch VQO, it is impossible to decrease t forBu launch V^s higher tha km/s2 n3. VAGa , A perihelion; however, aphelion can still be increased. more effectively increases aphelion. A pure VGA can Furthermore, the MGA is capable of full turning for not even get to Jupiter for a launch T^, of 15 km/s. launc s lesx hV s wherc tha km/sar 2 e ne3. gravTh . - For a high launch V^, there is a correspondingly alony it sufficiens ei smalles i t r than tha Venusf o t , high a at the VGA. But with high arrival VQO at because Mars has lower gravity. Because of this Venus, the maximum turn angle is insufficient to ro- lower gravity, the MGA can barely rotate the V^ at tate the Vco enough to increase the spacecraft's he- higher arrival VooS. However, the MAGA is still able liocentric velocity (and therefore, energy) VAGA . A to, since the departure direction is unconstrained. does not suffer this disadvantage, since the V^ can This is the reason behind the large gap in increas- be rotate arbitrarn a o dt y direction aerodye Th . - ing aphelion between the pure MGA case and the namic portio tura f s responsiblno ni r mosefo f o t L/D — 5 case. At the launch V^ for an MGA to the turning at high arrival V^s, unlike low arrival reach Jupiter, a MAGA is easily capable of escaping V^s where gravity dominates. Solae th r System othen I . r words singla , e MAGA Sinc represen5 e d Figsan 4 . t aphelio perid nan - allow a spacecrafs reaco t t h Jupiter instea- re f do quirin traditionae gth nyby3 r o othef so l2 r planets. helion distances pointe th , s wher contouea r inter- sects a planet imply the spacecraft arrives tangen- A MAG Aalss i o more capabl decreasinf eo g peri- tially. Also, the L/D = oo case must always depart helio t highena r launch V^s tha VAGAna . Decreas- Venus tangentially, since this provide extremae sth l ing perihelio equivalens ni reducino t t g heliocentric heliocentric velocity. Therefore, points where the velocity during a flyby. For any gravity-assist body, L/D = co contour intersect a planet are all identi- the goal is to rotate the spacecraft's VOQ to be paral-

r i=10/L/D=5 puree-/ Direct L/B= Launch 7 Pluto Neptune TTT Uranus

Saturn

Jupiter

Earth Venus

Mercury Direct Launch

Launch V_ [km/s] Fig. 5 Maximizing aphelion or minimizing perihelion using a single Mars aerogravity assist (MAGA).

lei, and opposite in direction to the planet's velocity 2fi+ Q/R2E V%+ vector. Since the departure direction is arbitrary 2fj, /R heliocentrie foAGAn th a r d an , c velocit Marf yo s si -2(flp/-R)^ ~~ Q much smaller than that of Venus, a MAGA is able 2E + - 2(RP/R) reduco t e perihelion more tha nVAGa Ae th can r fo , (13) same flyby VOQ. When there are more than one or two AGA plan- In pure gravity assists, the pre and post V^s are considero t s et , this graphical method becomes cum- orbite samee th t th s bu ,(whic - E pointe n ha ar n so bersome doed t providan ,no s e much insight- An . Rp plot differente ar ) . Thi graphicalls si y depicted other drawbac thas ki t this, approach doe t prosno - as following constant VOQ contours on an E-Rp plot, vid e arrivaeth l VQO r doe no t ,provid i s e TOPeth . as show e distanc FigTh n ni contou a . 6 .n eo r that An alternate metho developes di d instead. can be traversed by a single flyby depends (in part) on the radius of the flyby body (i.e., when a flyby ap- E-Rp Analysis proache surfacee sth , further turnin t possible)no s gi . A graphical tour design method based on Tis- The tick marks (dots) on the plot denote contour serand's criterio developes searchinn i n wa d ai o dt g distance at maximum turning. This plot illustrates fo r e Europpathth r fo sa Orbiter mission.11f I '12 tha vera t t Mary no effectiv s si e gravity-assist body, we assume that satellite circularn i e ar s , coplanar sinc tics eit k mark verye sar close togethere th n O . orbits around a central body, then two orbital el- other hand, Jupiter's tick mark spreae sar d outd an , ements completely describ e orbitshape eth th f eo . therefore much more capable of altering a space- For the Europa Orbiter case, period and periapsis craft's orbit. radius were selectedspecifie us e cW . orbital energy An AGA further spreads out the tick marks. Con- instead of period, since many conic arcs to the outer sider the hypothetical L/D = oo case where the VQO planet hyperbolie sar c with respec Sune th .o t Since ca desireturney e nb an o dt d direction. This corre- thes quantitieo tw e s provid e orbieth t shapee th , sponds to moving the tick marks to the endpoints flyby condition knowe sar n when that orbit crosses of the contours (or simply removing them). Finite a given body's path. In particular, the arrival V^ valueD L/ s complicate matters becaus VOe eQth does is know givea r , flybnRp,d fo nE an y planet. From not remain constant. As the aerodynamic turn an- Fig. 3, we have that: gle increases e V+,th , decreases. However, this loss may be worthwhile since the tick mark constraint is no longer valid. Putting this all together, a sample trajector Pluto yt depictes oi Fign di . 7 .

2E + 2fi0/R - 2VplV cos(7) In Fig. 6, the VOQ contours start at the lower-right

200

Rp [AU]

Fig. 6 The E-RP plot. 1000

800 -

600 -

400 -

200

-200 -

-400 -

-600 -

-800

-1000

Rp[AU]

Fig 7 . Illustratio EVMVn a f no P trajectory using AGA.

km/ 1 planets e eacr = th cornes f fo e x h o V Th .t a r Table 2 Fastest potential AGA trajectories to spacing between contours is also 1 km/s. In Fig. 7, Mercury with L/D=7 (ignoring phasing). The contoue th r km/spacing2 t better a se fo ar s r clar- TOF is given in years. ity. This figure depicts a trajectory from Earth to Pluto using AGAs, wit a hlaunc hkm/s6 f Vô . VCQ Mercury This launch conditio Venua alss n i n oo s contour— Path TOF meaning that the spacecraft can coast from Earth 3 EVEVY 0.79 to Venus. If phasing works out, Venus will be there 4 EVY 0.37 when the spacecraft reaches Venus's orbit. Next, 5 EVY 0.30 e spacecrafth t o MarVenua t goe A a n vi o sAG s 6 EVY 0.27 (VAGA). This is graphically depicted as proceeding e firsalonc upth ar e tVenusia gth n contours.a f I VGA were done instead, the spacecraft would be lim- mum TOF path between any desired initial and final turnins crosn iteit ca n di st g(i onltic e kyon mark). condition in the minimal number of comparisons. Wit AGAn ha e spacecraf th t ,som ge en ca "freet " resultA r Mar AG Venud fo s e an se unre Th ar s - gravitational turning; but the rest of the turning will markable, since no gravity assists are needed to reach resul losn i tV^.f sthio n I s case spacecrafe th , - ar t them. (The one exception being a launch V^ higher rives at Venus with a V^ of about 12.3 km/s, but tha Earth-Venue nth s Hohmann t lowebu , r thae nth leaves Venus wit Vha onlf ô y 10.7 km/s. Thi- sal Earth-Mars Hohmann. In this case, a Venus flyby lows the spacecraft to arrive at Mars with a V^, of is required to get to Mars.) The optimal trajectory 18.9 km/s. The spacecraft performs another AGA, from Eart Mercuro ht y involve t leasa sflybe on ty which lowers its Mars V^ to 16.0 km/s, but this al- t Venusa r launcFo . h V^s higheF TO re thath , n4 low s perihelioit s drasticalle b o nt y pumped down saving minimals i s witA hsummarAG A . resultf yo s to 0.086 AU. The spacecraft coasts to Venus, where for trajectorie Mercuro st presentes yi Tabln di . e2 a final VAGA pumps up the heliocentric energy to Fo a launcr km/sh3 f Vô ,a singl e VAGs Ai a hyperbolic orbit. From there, the spacecraft can insufficient to reach Mercury; furthermore, it is im- reach Pluto. possible to reach Mercury with only 2 flybys. The e E-RpTh plot provide a valuabls - eex toor fo l fastest potential 3-flyby trajectory to Mercury uses a amining GA or AGA trajectories. However, the Venus-Earth-Venus combination of AGAs for a TOF maneuvedecayinA AG e gth r V^ n complicatei e sth of 0.79 years. Additional flybys beyon e thirdth d interpretation of the E-RP plot. For this reason, a increase TOFe sth thu d unnecessarye ,an ar s r Fo . computer progra writtes mwa searco nt h througe hth launch y^s of 4 km/s or higher, a single VAGA is E-Rp contours to find the shortest TOF trajectory sufficient to reach Mercury. The TOF cannot be from Eart evero ht y other planet prograe Th . m cal- improved with additional flybys. numbey caseA an culater f sAG fo o r d s an bot A hG e biggesTh missionn i t s advantagi A o st AG f eo flyby bodies t becaus e circularBu . th f eo , coplanar the outer planets. A summary of results is shown in assumption, e lac f th phasink o d an g (timing) con- Table 3. Blank areas mean that no such trajectory siderations hypotheticay an , l trajectory still needs s possibli e (du insufficieno et t launch energy)r Fo . verifiee b o othet y db r means. example, no trajectory to Jupiter using only 1 or 2 Becaus orbitae eth l stat specifie— c energd yan flyby bodies exist launca r sfo h km/s 3 VQ f Oo . perihelion — is a continuous vector, we discretize Since this analysis does not take into account e E-Rpth plot int ocollectioa - or f nodes no e Th . phasing (however t doei , s allo resonanr wfo t flybys), bital state vector afte flyba r thes i y n mappeo dt the existence of a trajectory that returns to a given e nearesth t node. Thus e finee discretizath ,th r - planet is not guaranteed. In general, the greater tion, the smaller the error in the algorithm. There numbee th tunef o r givesa n plane uses i t d (exclud- possibl4 o t p eu coase ar t arcs froplanee mon o t t ing resonant flybys), the less likely such a trajectory another for each point in the E-Rp plot11 (only 2 will exist. possible arc hyperbolir sfo c trajectories) l possiAl . - Ou l planetr al algorith t a sA mAG allown a r fo s ble arcs are considered, keeping the direction of the except Mercury and Pluto (which do not have ap- trajectory consistent for each arc. There are liter- preciable atmospheres). Interestingly enoughe th , ally billions of trajectories that must be considered. time-optimal trajectories rely most heavily on Venus To drastically cut back on search time, while ensur- and Mars, and only occasionally use Earth. For ing each optimal trajector founds yi emploe w , e yth mancasese th optimae f yo ,th l trajector altern a s yi - Viterbi algorithm.13 This algorithm find minie th s - nating serie Venusf so , Earth Mard an , s AGAs until

Tabl e3 Fastest potentia trajectorieA AG l e outeth ro t s planets with 'L/D—7 (ignoring phasing). gives i F yearsn i TO .e Th

Voo Jupiter Saturn Uranus Neptune Pluto Path TOF Path TOF Path TOF Path TOF Path TOF EVEMJ 1.99 EVEMS 3.53 EVEMU 8.21 EVEMN 15.33 EVEMP 23.08 3 EVEVMJ 1.64 EVEVMS 2.43 EVEVMU 4.24 EVEVMN 6.35 EVEVMP 8.21 EVEVEMJ 1.56 EVEVMVS 2.20 EVEVMVU 3.46 EVEVMVN 4.91 EVEVMVP 6.18 EVMJ 2.20 EVMS 5.61 4 EVEMJ 1.43 EVEMS 2.44 EVMVU 4.71 EVMVN 7.21 EVMVP 9.44 EVEMVJ 1.36 EVEMVS 1.92 EVEMVU 3.20 EVEMVN 4.65 EVEMVP 5.93 EVEMVEJ 1.32 EVEMVES 1.87 EVEMVEU 3.14 EVEMVEN 4.58 EVEMVEP 5.85 EMJ 2.46 EVMJ 1.46 EVMS 2.61 EVMU 5.52 EVMN 9.14 EVMP 12.45 5 EVEMJ 1.24 EVMVS 2.08 EVMVU 3.53 EVMVN 5.18 EVMVP 6.63 EVEMVJ 1.23 EVEMVS 1.78 EVEMVU 3.04 EVEMVN 4.49 EVEMVP 5.76 EMJ 1.68 EMS 3.64 EMU 14.16 6 EVMJ 1.27 EVMS 2.21 EMVU 4.41 EMVN 6.96 EMVP 9.23 EVEMJ 1.11 EVMVS 1.74 EVMVU 3.00 EVMVN 4.45 EVMVP 5.72 the destination planet is reached. Because the inner tory with the indicated launch date and TOF. The planets have sucsemimajow hlo r axes compareo dt letter itself is an index for the launch V^s that are outee th r planets, using them exclusively (since ew searched. In this case, an "A" represents a launch cat arbitrarnge y bendin t themga s bettei ) r than FO f 4.0Oo 0 km/s" represent"B a ; slaunca h f Vô hopinoutee th l rg al planet s lin rightp eu examr Fo . - 4.50 km/s, and so on. ple, the best trajectory to Pluto with a launch Vô With the given conditions, the fastest trajectory of 6 km/s uses a Venus-Mars-Venus-Mars series of to Pluto has a launch Voo of 6.0 km/s, and takes only AGAs opposes a , usino dt g Jupiter, Saturn, Uranus, launca 5.e 5us hyearse f Vônl w o km/ 5 f I !y5. s or Neptune (even though AGA alse ar so allowet da instead, we can get there only slightly later. Further- these planets). While Jupite powerfua s i r l gravity- more, these trajectorie possible sar e every few years. assis awar fa t effectivelyo planeto t to s i t i , y compete These extremel trajectorieF TO w ylo Pluto st - obe with the inner planets. Another factor evident from gi disappeao nt Pluts ra o moves further away. Even examining Fig. 6 is that if Jupiter is used to pum trajectorpV VM wit e yhlowea th , D rati5 L/ rf oo up a spacecraft's energy, then its semimajor axis is still yields short TOFs, somewhat less tha nyeaa r greatly increased. This lengthens the size and TOF longer than the L/D = 7 case. Earlier work by Bon- of the conic arc to the next planet, and thus, Jupiter figlio yielded AGA trajectories to Pluto that either t usedisno . This mean- t havde no o et so d tha e w t too yeark5 s e samlongeth r e fo rlaunc h energyr o , pen phasinn do g e outewitth ho rt Jupitet ge o t r took 4 years longer for an increased launch energy planets — only on the phasing of Venus, Earth, and okm/s7 f . Clearly E-Rpe th , graphical methoa s di Mars. powerful tool. Considering trajectorie Pluto s t launc a r ofo h V^ Another mission of interest is the Solar Probe thae km/s0 ose 6. ft e addin w , flybd 3r yga lowers which will study the Sun from as close as 4 solar the TOF to 5.7 years. This trajectory is, in fact, the radii. Reducing perihelio thay nb - tdi muca y hb EVMVP discussed previously. However, the actual rect launc vers hi y expensive, requirin km/5 2 ga s muce b y h shorterma F TO , since Plut currentls oi y launch Vôffseo t Earth'e th t s heliocentric velocity. about 30 AU away from the Sun. while our algorithm A purtrajectorA eG achievyo t same eth e goa- re l assumes a constant semimajor axis of approximately quires that the last gravity assist has a flat contour, 40 AU. with tick mark enougr sfa h apart. Examining Fig, .6 This trajectory was examined in more detail us- thae w onle ese th t y practical choics i pur r eA fo eG ing STOUR for a 40-year launch window (2000-2040) Jupiter. In theory, any planet could be used, but and launch V^s ranging from 4.0 km/s to 6.0 km/s. several resonance requirede sar , which drivee th p su The results are shown in Fig. 8. The trajectory is TOF to unreasonable values. Previously, we noted indicate PATe th y Hdb label, wher numbee eth n r trajectorAGA Plutyto VMV graphethao as the t d corresponds to the nth planet from the Sun. The in Fig. 7 has a very small perihelion of 0.086 AU af- LIFT/DRAG fiel ratiodD giveL/ se usesth t eacda h MAGAe th r secone te Th . d VAGA increases energy flyby planet. In this case, an L/D=7 was used dur- and perihelion to get to Pluto. However, suppose ing the VMV. Finally, the plot itself is comprised we use the second VAGA to reduce energy and per- of several letters. Each of these represents a trajec- ihelion. Trajectories to 4 solar radii are computed

PATH: 32429 0 0, 0 7. 0 7. LIFT/DRAG 0 7. 0 0. : PATH: 3 2 4 2 10 LIFT/DRAG: 0.0 7.0 7.0 7.0 0.0 V1NF(KM/S): 4.00 4.50 5.00 5.50 6.00 V!NF(KM/S): 5.00 6.50 SEARCH EVENT NO.: 5 ALTM11N = 0 KM SEARCH KIN. ALT.: 0 KM SEARCH EVENT KG. 5 : ALTMIN= 0 KM SEARCH M1N. ALT.: 7.5- pplp-l E c 7.0- r 1 1 1 ^ I

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f jssaasssaRssaaaffiasssssass2£^£2222E^E!:2£2si!: c ooooooooooooooooooooooooooooooooooooooooc c c SoSoooooo2~22S£2ES2S^SSSS™™SSSsSSS!?SSSSS^ NCHDATE LAUNCH DATE (YY/MM/DD) LAUNCH DATES SEARCHED: OO/ I/ 1 TO 40/ I/ 1 BY 15.0 DAYS LAUNCH DATES SEARCHED: OO/ I/ 1 TO 10/ I/ ' BY 10.0 DAYS TFMA X- 2850.0 DAY YSS8 S7. ( ) TFMAX - 540.0 DAYS ( 1.5 YRS)

Fig. 8 VMV AGA trajectories to Pluto trajectorieA . AG V sola4 o st VM r Fig radii 9 . .

using STOUR, and are shown in Fig. 9. Trajectories change in the spacecraft's true anomaly, but little with launch VooS as low as 4.50 km/s are possible, change e shaporbith th f n tei o itself smalleA . r the fastest having a 2-year TOP. For a launch V^ turn angle gives an equivalent turn, but with less of 5.00 km/s to 6.5D km/s, many trajectories have los f T4o.so ) Thus e mosth , t efficien s possiblAV t e TOFs under 1.5 years. The quickest of thes eonee hath sse thaaar A t arrivwit AG depard n ha ean e th t TOF just under a year. flyby body nearly tangentially limitine Th . g cas- eoc proposee Th SolaA dG r ProbF e missioTO a s nha curs when L/D = oo, where the spacecraft arrives of aroun aphelion yearsa d6 d an ,f Jupiter' no s dis- and departs tangentially. The near-tangential ar- e probTh . e woulAU 2 tanc5. d f havo e t leasea t rival/departur e previouslth e n conditioi t yme s ni a 4-year period, wit extremeln ha y fast perihelion discussed EVMVP trajectory of Fig. 7. As seen in flyby. On the other hand, the AGA trajectories in figuree AGA3 th l locatee al , sar d nea endpoine th r t Fig. 9 all have perihelia near Venus. This corre- of a VOQ contour. However, arriving or departing spond abouo st 75-daa t y period, wit sloweha r flyby. a contour near an endpoint is insufficient for max- The AGA trajectories thus allow for more science re- givea r imizinfo n L/D.V gA Becaus optimae eth l turn, since the spacecraft would return to the Sun turn angl functioa s ei L/D,f o n highe ratioD L/ r s more frequentlytrajectorieV VM wits e A .hth o t s permit greater travel along the V^ contours. For Pluto, launch opportunities for a solar mission exist L/D = 7, Eqs. 7 and 9 yield w 174 degrees, so ever yearsw yfe . the AGA maneuver travels through about 97% of a VQO contour. Clearly t e AGAno th , e Fign i ar s 7 . Discussion AF-maximum, sinc AGAe eth s middle begith n ni e Previously, we found the optimal atmospheric ocontoura f . e geometrHoweverth e o t th e f yo du , turn angle to maximize AV; but, we also know that contours, maximum-AV t possiblAGAno e r ar s efo this AV may not be pointing in the right direction the EVMVP case at higher L/D ratios; i.e., an AGA to get to the next planet. We can find the necessary capabls i providinf eo exces n i - V whaf sA o op g a s i t condition maximua r s fo AV accomplisA o t mAG h tima reachinr fo l nexe gth t body. Therefore lowea , r thi inspectiny sb E-Re gth P plot (Fig. 6) . ratiD L/ o exists such thas i maximua tA AG V mA If an AGA begins near one of the endpoints of a possibl optimad ean specifia r fo l c maneuvere Th . x contouV r (far lower-lef upper-right)r fa r o t , then AGA turFigo n smaximui e d n7 .th m amount with- woula maximuA AG d drivV me spacecrafA eth t t overturningou . Moreover, even whe nmaximua m towar othee dth r endpoint (bulowea t ta r contour, AV trajectory exists, it may not be time-optimal. since some V^ is lost). However, a maximum AV Fro thae EVMmn se a te Figw , possiblPs i .7 e using AGA maneuver that begins near the middle of a a maximum AV AGA with a Mars V^ of 10 km/s. x contouV r would follo contoue direce wth on n -i r However, from Tablknoe w w, e3 tha EVMVe tth Ps i tion, then backtrack (overturn), only to end up near faster tha EVMPe nth . Even wit hseriea AGAf so s wher maneuvee eth r began. (This result larga n si e that do not use 100% of the possible AV, the tra-

jectory can be faster than any comparable pure GA References trajectory. 1Sims, J., Staugler, A., and Longuski, J., "Trajectory Op- tion o Plutt s a Gravitovi y Assists from Venus, Marsd an , Conclusions Jupiter," Journal of Spacecraft and Rockets, Vol. 34, No. 3, May-June 1997, pp. 347-353. An AGA can potentially yield much higher AVs 2Petropoulos, A., Longuski, J., and Vinh, N., "Shape- than a pure gravity assist, and the E-RP plot shows Based Analytic Representation f Low-Thruso s t Trajectories whe possibles i t ni rea e appars i Th l. A powe-AG f ro for Gravity-Assist Applications, PapeS . 99-337AA "No r , AAS/AIAA Astrodynamics Specialist Conference, Girdwood, ent when multiple AGA flybys are used, especially Alaska, Aug. 1999. when one body acts as a Voo-leveraging maneuver for 3Nonweiler, T., "Aerodynamic Problems of Manned another MVM)o V r .VM Thes(suc a s ha e trajecto- Space Vehicles," Journal of the Royal Aeronautics Society, ries allow for extremely fast low-energy missions. Vol , Sept63 . . 1959 . 521-528pp , . 4 As seen from the E-Rp analysis, the Earth is not Lewis, M. and McRonald, A., "Design of Hypersonic Wa- veriders for Aeroassisted Interpanetary Trajectories," Journal becausA s onluse ha oftes AG yda mod a r t ei fo n - of Spacecraft d Rockets,an , Sept.-Oct 5 . No Vol , 29 .. 1992, erat eorbie effecth n t othee o shapet th n r O hand. , pp. 653-660. Mars and Venus can be quite effective. Venus is 5Randolph McRonaldd an . J , , "SolaA. , r System 'Fast typically most usefu changinn i l orbitae gth l energy Mission' Trajectories Using Aerogravity Assist," Journal of Spacecraft d Rockets,an , March-Apri2 . No Vol , 29 . l 1992, ospacecrafta f , while Mar typicalls si y most useful pp. 223-232. in changin gspacecraft'a s perihelion. o Eartd n hca 6McRonald, A. and Randolph, J., "Hypersonic Maneu- both, but neither quite as well as Mars or Venus. verin r Planetargfo y Gravity Assist," Journal f Spacecrafto The outer planets are too far away to be useful as and Rockets, Vol. 29, No. 2, March-April 1992, pp. 216-222. 7Sims, J., "Delta-V Gravity-Assist Trajectory Design: AGA bodies. Theory and Practice," Ph.D. Thesis, School of Aeronautics AGA provides three significant advantages. First, and Astronautics, Purdue University, West Lafayette, IN, trajectories do not have to rely on phasing of the Dec. 1996. outer planets (aside fro e targetmth t onlbu )n yo 8Bonfiglio, E., "Automated Design of Gravity-Assist Venus, Earth Marsd an , . Second, TOF smalle sar . and Aerogravity-Assist Trajectories," Master's Thesis, School of Aeronautics and Astronautics, Purdue University, West Sinc e initiatrajectorA th e AG l phasn a y f eo wil l Lafayette , AugIN , . 1999. usually rely onl Venun y o Marsd san time - th , ere 9Anderson, J., Lewis, M., Kothari, A., and Corda, S., quired to build up the spacecraft's orbital energy is "Hypersonic Waveriders for Planetary Atmospheres," Journal kepminimuma o t t . Finally, fast trajectoriel al o st of Spacecraft d Rockets,an , July-Aug4 . No Vol , 28 .. 1991, pp. 401-410. planets exist usin launcw glo h energy. 10Elices, T., "Maximum AV in the Aerogravity Assist Ma- The AGA technique provides exciting new trajec- neuver," Journal f Spacecrafto d Rockets,an , 5 . No Vol , 32 . tories to difficult targets in the Solar System. For Sept.-Oct. 1995 . 921-922pp , . example, Pluto can be reached in only 5.5 years us- "Strange Longuskid an . GraphicaA N " , , J. , l Methor dfo Gravity Assist Trajectory Design," AIAA Paper No. 2000- ing a VMV AGA, for a L/D of 7, with a launch V^ 4030, Denver, Colorado, Aug. 2000. of 6.0 km/s. The trajectories presented here supply 12Heaton, A., Strange, N., Longuski, J., and Bonfiglio, compelling reason develoo st p highypersoniD hL/ c , "AutomateE. d Desig Europe th f no a Orbiter Tour," AIAA vehicles waverider)(suce th s ha developmene Th . f to Pape . 2000-4034No r , Denver, Colorado, Aug. 2000. 13 AGA technology will enable deep space exploration Viterbi, A., "Error bounds for convolutional codes and asymptoticalln a y optimal decoding algorithm," IEEE Trans- launcw lo t ha shorr energfo td flighyan t time. actions n Informationo Theory, Vol. IT-13, April 1967, pp. 260-269.

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