(c)2000 American Institut Aeronauticf eo Astronautics& r Publisheso d with Permissio Author(sf no ) and/or Author(s)' Sponsoring Organization. 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 <j>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 146 (c)2000 American Institut Aeronauticf eo Astronautics& r Publisheso d with Permissio Author(sf no ) and/or Author(s)' Sponsoring Organization. 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 tech- nique 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 <j) is the total turn angle (gravitational and Fig functio2 .a A[ s 7a gravita f/of nn o i o y assist. aerodynamic) gives i d y nb an , 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 2 147 (c)2000 American Institut Aeronauticf eo Astronautics& r Publisheso d with Permissio Author(sf no ) and/or Author(s)' Sponsoring Organization. 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 tha e 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 —.