© Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México eit eiaad srnmı Astrof´ısica Astronom´ıa y de Mexicana Revista hnc n oto,Ntoa nttt o pc Research, Space for Institute Brazil. National INPE, Control, and chanics nttt o pc eerh NE Brazil. INPE, Research, Space for Institute n,Brazil. ing, egr(ooo ta.20) ESNE Team MESSENGER 2007), Mes- al. BepiColombo et missions (Solomon 1977), senger Penzo the & like (Flandro Kohlhase Voyager past, 1978), 1966; missions Burgess the Several & (Dunne in Mariner spacecraft. technique the this body to celestial used a energy of well give gravity as the to thrusts, using low options some and lit- as high the consider in available they astro- are erature; in solutions topic Several important very dynamics. a is consumption fuel 1 2 3 h erhfrobtlmnuesta minimize that maneuvers orbital for search The oreo niern n pc ehooy pc Me- Space Technology, Space and Engineering of Course eea oriao fteGaut col National School, Graduate the of Coordinator General ˜ oPuoSaeUiest,UEP colo Engineer- of School UNESP, University, State S˜ao Paulo OEE EOGAIYASS AEVR RUDVENUS AROUND MANEUVERS AERO-GRAVITY-ASSIST POWERED TDIGDRC N NIETEFCSO MUSSIN IMPULSES OF EFFECTS INDIRECT AND DIRECT STUDYING e Words: Key median individual, fuerzas forma las din´amica direct midiend en de aceleraci´on la hace efectos obtenida en se los presente Esto estudian fuerza impulsiva. Tambi´en se maniobra una espacial. e aplicaci´on de la nave nuevas curvatura, ´angulo tr la opci´on genera de su la Esta cambiar de de puede cambios cual arrastre. los la al cuenta maniobra, s sustentaci´on en y que toma la impulsiva y a maniobra Venus una de con alrededor pa centro planeta el un combina maniobra de Esta gravitacional Venus. de alrededor gravedad por comin accelerations the indivi applic each of the of integration effects the of the by effects measuring dynamics, by indirect the done the the is and of This inclination direct maneuver. ma the the the and studies energy for the also possibilities curvature, new tr of gives the angle option of the This modification drag. the applied and is and lift pass that Venus, maneuver the around impulsive combine approach an that closest with maneuvers planet the are of These field paper. present the .INTRODUCTION 1. lpeet r´cl eefc ne sui elsmaniobra las de estudio el en enfoca art´ıculo se presente El h td far-rvt-sitdmnuesaon Venus around maneuvers aero-gravity-assisted of study The eeta ehnc lnt n aelts atmospheres satellites: and planets vehicles — mechanics celestial .O Murcia O. J. eevdArl1 08 cetdJl 32018 23 July accepted 2018; 10 April Received 1 .F .A Prado A. B. F. A. , , 54 ABSTRACT RESUMEN 8–0 (2018) 485–500 , h rgadlf ocsgnrtdb h atmosphere the by to generated due forces spacecraft lift and the drag of the the & trajectory (ii) 2010); the Qi Prado in 1993; & deviations Gomes Prado 2007; Prado & 2015; Broucke Xu Uphoff grav- 1988; 1965; the Prussing Broucke of & effect 1989; (Hollister the Venus to of due field primary Sun, ity largest the the system, to the respect of with or- vector spacecraft velocity the this the the of from of in rotation involved the stems (i) a are name maneuver: that bital study this aspects to of different is origin three paper The present ma- neuver”. the maneu- aero-gravity-assisted of of “powered this called goal goal maneuver of The the energy are with of options ver. variations literature several the the but travel- increasing in System, spacecraft considered a Solar also of the energy tech- in the powerful ling The change a (2008). is to maneuver al. nique et assisted Jehn gravity (2006), pure Grard (2006), al. et 2 n .M Gomes M. V. and , rmtoeforces. those from g evr hc a vary can which neuver, el nerc´nd la integraci´on de la te pccat h paper The spacecraft. egı ainclinaci´on la nerg´ıa y to fa impulsive an of ation seidrco ela de indirectos e os oiiiae aala para posibilidades ttemmn fthe of moment the at o fco ecada de efectos los o ulfrepeetin present force dual jcoyi aeby made is ajectory odnr e campo del dentro so piae lperi- el en aplica e g ytegravity the by age ycoi debidos ayectoria eoasistidas aero s 3 stefcsof focus the is space — 485 © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México nrygisi h oaino h eoiyvector velocity the of rotation the the is for reason gains physical The energy literature. the in tention for made studies even planets. similar not other mis- with available, a comparisons not of on are based goals results specific general for and in sion, trajectories involved best iden- dynamics the times can simulations tify complex four numerical to the only maneuver, up to this be Due this can of energy maneuvers larger. for of used variations be an- to type, can Mars, that compared eight Regarding planet when to other maneuvers. up energy similar for give of The Earth, can variations planet. Venus the the that with times show collision of will risk results high a as at- energy, as of dense well variations very Besides large a allows which planets. has mosphere, mis- Venus outer for location, and System important inner this Solar the the to is in going it sions position the because strategic selected for a is Venus in planet planet This the stud- approach. considers is close study maneuver This impulsive the by ied. caused of drag trajectories application and the lift of the maneu- modification of the the contribution to in the due particular, them In of indi- each ver. the of reveal contribution to the force, vidual of each integrations by numerical given perform atmo- accelerations to the is obtained. idea of results The the in- contributions changing can further the forces, trajectories spheric decrease the or trajectory. in the crease made change modification also change but only The not vector, that velocity the impulse, the understand the to of is effects objective indirect The involved dynamics. force each the of in effects the of verification tailed made are maneuvers among Venus. the paper, using differences present the the orbits, In is and retrograde them. emphasis existence and 2018), their direct al. showing In comparing et 2018). (Murcia on al. Mars placed et of exam- (Murcia case Mars Some the (Murcia and Earth 2017b) literature. the Prado the & around maneuvers in consider GOMES recent studied & ples well very PRADO, been MURCIA, is not com- has and maneuver effects of single three a more these those in or of bined of two consideration of each is The combinations to There to them. related as 2016). well studies as al. of points, et number Gomes large 2015; 2013a,b; a al. et al. Silva et Venus 1999b; to al. Silva approach et closest Casalino the 1996; of (Prado an point spacecraft of the the application by when passes the time the (iii) at 2006); maneuver impulsive Grard Team 2006; MESSENGER al. 2007; et al. et (Solomon Venus of 486 h uegaiymnue a eevdmr at- more received has maneuver gravity pure The de- a making of goal the has paper present The hspolmtkn noacutteeet fdrag. study of effects 1995b) the Broucke account & into (Gomes taking (Prado problem References and this change 2013b) possible. to al. The body also et celestial is (2006). a maneuver al. of the atmosphere et the Mc- Okutsu of (1999a); also (2003); use al. are al. et approaches et Casalino Conaghy close in Maneuvers shown and like 2008). thrust possible, Prado low & use Gomes that (Gomes 2013b,a; approach dis- close al. new a the et after and particles clouds of the also of the tribution are evolution of the particles members finding of of individual goal Clouds the for with impulse. body geometry studied, the the best the as apply indicated Moon to and the approach Silva used close (2013a,b); They the al. (2015). et al. (1996); Silva et Prado (1999b); like al. et studies, Casalino published approach some close has the also with combined impulse an plying (2002). al. et Heaton in study found to find Trips be to can spacecraft. able Europa a is for that Longuski trajectories method & optimal graphical Strange a built in (2002) space. Jupiter three-dimensional & around Felipe maneuvers the Broucke studied Moon. & (1999) the Prado around Prado orbits to approach, mapped mission generic (1991). (1995a) Williams a more & in a Longuski like Using in concept, described can as this planets Pluto, distant using Hol- More visited in as (1965). be well Prussing as & (1989), ma- Braun lister gravity-assisted & pure planet Striepe a in The for neuver spacecraft, (1984). used a Davis been & has by of Venus Dunham in used series in be shown A detailed also as as mission. can spacecraft, Ulysses approaches a the close of for mod- (1986) plane to Carvell possible orbital reference also the is in It ify described 1978). are Burgess Maneuvers & mission (Dunne 1982). Mariner (1981, like Galileo the al. maneuver, The for et this D’Amario 2008). in from shown al. benefited et also Jehn spacecraft al. maneu- 2006; et of Team Grard MESSENGER type 2006; 2007; this al. et using (Solomon literature con- ver also the are in Mercury the sidered to to related Missions Flandro (1977), in mission. Penzo Voyager ap- shown & of Kohlhase like examples and maneuver, (1966) several this are of There plications Broucke 1993). 1988; Prado (Broucke spacecraft & the increase of can trajectory the energy body reduces the the A behind body passage celestial a passage. the while of energy, the front of in passes geometry that depend- the energy, on the decrease This ing or system. increase the of can body change spacecraft primary the the frame. to of respect inertial energy with the the to changes maneuver respect This with spacecraft the of xedn hs ai tde,tecneto ap- of concept the studies, basic those Extending © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México eipi fisobtaon h lnt hscom- This planet. the around orbit the its by of passes to spacecraft periapsis added the is when vector impulse velocity an the approxi- terms, at- those exponential This Besides an mation. Venus. using of modeled atmosphere is mosphere the with of 2012) inclusion (Szebehely the problem three-body stricted Lara 2013; (Prado in 2016). problems before similar done on as literature time, the along accelerations force in- each the from by of coming made effects is cumulative study the by This tegrating made drag trajectories maneuver. and the impulsive lift of the understand modifications of impor- the to contributions relative to is the due topic their in this detail variations of in goal the main show the The to of tance. able dynamics, study the be detailed in involved to a force (ii) each time, atmosphere; of first contribution dense the very be- for a Venus, making, has planet planet this the cause around aero-gravity powered trajectories finding assisted (i) objectives: two has free- Mars (2015) al. a et of Hughes study in ex- described A is as mission Mars, (2015). return in Jesick mission in double-flyby al. plained a et is Returning Henning in as shown (2014), as Mars. possible, planet also are the designed missions in- using (2007) the al. also of et maneuvers atmospheres Armellin the planets. by termediate lift/drag passages ratio the the for values during high using also missions Pluto, designed to and (2002) Longuski Mars for & Earth, Strange solutions planets Venus. same building the problem, using (2005) this al. trajectories et in Lavagna worked and ra- (2000) also lift/drag al. et high type Bongfilio this with in tio. realized large when not the are maneuvers, that drag of showed to due 2000) Venus energy (1995, of Mars, al. losses planets et the Sims with used drag, Earth. They to and lift 10. ratio to (1992), the up values of Randolph values & high used McRonald which like paper, present 2017a). Prado using & possible (Murcia al. are are technique et changes results this Murcia inclination Several (2017b); Even Prado it. (2018). & to Murcia opposite in or points available can planet de- which force, the It this to varia- of maneuver. direction the the the decrease on from pends or obtained increase energy can of as- tions force, “aero-gravity effects lift the the called including of also planet, maneuvers, (AGA) the sisted” of mosphere h qain fmto r eie rmtere- the from derived are motion of equations The paper present the aspects, those all Considering the to linked be can researches other Several at- the including maneuver, complete more The TDIGEFCSO AA RUDVNS487 VENUS AROUND PAGAV OF EFFECTS STUDYING elgbems scalled is mass negligible eerh,tescn oemsiebd scalled is body massive more M second the research), ois h ags n scalled is one largest the bodies: popular future. more maneuver the and of in type guidance this make in may space- as craft, interplanetary to well available It as techniques accuracy control the systems, involved. in propulsion benefits improvements of that and de- expected risks a also after the is mission, can of the to It study for nature tailed necessary in when maneuvers. available used interplanetary potential be in a be- help cited show provide papers to of the is goal to fore, similarly the paper, Therefore, present collision. the of always risks implementations, presenting practical in difficulties has (Murcia Mars and 2018). 2017b), al. et Prado planets & the using (Murcia im- maneuver Earth the complete the of and presence con- pulse, re- the researches without this previous maneuvers of So, sidering continuation a trajectory. are is whole They search the 1991). during ref- 1992, the constant McRonald in & shown (Lewis maximum as Waveriders, erences Those for valid are respectively. values Venus, the in of or Venus direction to opposite lift pointing the components where has situations the representing lift, higher a ihtelf aigacmoetpitn oVenus; to and pointing 9 values component maximum but a the having described, lift just the situation with same the representing igaon eu) h pccati oigaround moving is spacecraft The Venus). around ling hr sn topee pt 5 (when to up zero the atmosphere) from no by is ranges indexed there which is coefficient, The atmosphere ballistic Mars. the from of obtained Earth, results the participation by the given times ones four the to and than up energy larger of times variations eight provide to and out before, turned studied that not were using which Venus, trajectories planet discovered the the with along formed, ge- best the found. if be consumption, can fuel other ometry of to terms compared in efficient options more is maneuver bined ihacmoetpitn poiet Venus; to drag, opposite to pointing equal component is which a lift 1, with where situation lift; the without represents trajectories represent which sas eesr oasm ausfrtelf-odrag lift-to the for ( values ratio assume to necessary also is 2 h yaia ytmi osiue ythree by constituted is system dynamical The maneuver of type this that known is it course Of h td fteeet fec oc ste per- then is force each of effects the of study The Vnsi u iuain)adatidbd with body third a and simulations) our in (Venus L / D .Tevle sdhr for here used values The ). .DNMCLMODEL DYNAMICAL 2. M − 3 ,tebs cnro for scenarios best the 9, tesaerf travel- spacecraft (the . M 0 × 1 10 teSni this in Sun (the − L 7 km / D 2 / r:0, are: g It kg. − 1, © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México h topeeo eu.Ωi h potential. the by is giving Ω Venus. forces of the the atmosphere (in of the respectively, components vertical plane), and orbital horizontal the are 8 UCA RD,&GOMES & PRADO, MURCIA, where spacecraft, the describe the (1-4) of Equations motion the Venus. from derived of terms atmosphere the problem adding three-body 2012) restricted mathematical the (Szebehely on The based Sun Swing-By. is passage the model called This around maneuver spacecraft a Venus. the in of near orbit passes the it modifies when Sun the 488 rai h ieto ftemto ftespacecraft the of motion the projected of the ( of direction functions the are in orbital lift area the and the Drag in in null. velocity Be- are analyzed and spacecraft. are acceleration the plane, trajectories of the motion the cause to opposite tion h ria ln ftesaerf n perpendicu- motion; and its spacecraft to the lar of plane orbital the pciey,tedniyo h topeeadthe and atmosphere ( Venus the of spacecraft-atmosphere of velocity density the spectively], ieaue ti osbet combine to possible is it literature, h spacecraft, the hsvlei sda nidpnetprmtr re- parameter, independent angle. incident an the as placing used is value this rf,o nl fatc.I h rsn ae tis ratio inci- it the an that paper keeps such present attitude angle the dent spacecraft In the attack. that assumed of angle or craft, L/D n hs offiins h ratio The coefficients. those fine in()sosti dependency. this shows (5) tion C efficient A BL ,teda n itcecet[( coefficient lift and drag the ), oflo h sa oecaueaalbei the in available nomenclature usual the follow To ntoeequations those In safnto fteicdneageo h space- the of angle incidence the of function a is , = C C B D B = Ω C ( ti loueu odfietecoefficient the define to useful also is It . C = BL L /C 2 1 A x 1 2 y ¨ ¨ = ρAC ˙ 2 + − and D x C F ~ .I hswy qain 67 de- (6-7) equations way, this In ). 2 D C ˙ 2 ~ D Atm x y + D B C 2 sda,afrei h direc- the in force a drag, is V Ω = Ω = A y m = L D ~ w 2 2 = ; ntes aldblitcco- ballistic called so the in C slf,wihi oc in force a is which lift, is x y + L ~ D D L L + + + 2 1 = A m F F L/D − D r ~ = yAtm xAtm 2 1 1 C µ F ρAC C 1 L xAtm B /C scntn,and constant, is + C m C C D L h asof mass the , D µ r D L V ,( ), 2 2 Z n hence and , V and w 2 w direction C .Equa- ). L F ,re- ), yAtm (2) (1) (7) (6) (5) (3) (4) aitoso nryaeue:90 of used: geome- terms are in the values energy Regarding extreme of two variations the linear. approach, the of to large try close too not are are variations results the conclusions If the general here. affecting obtained the not of same, but variation the results, are the behaviors the the course changes reduce Of to parameter as this so parameters. km, of 330 number at fixed altitude riapsis nrybfr n fe h asg yVns re- Venus, by passage the 2007). (Prado after spectively and the representing before (8-10), energy equations by frame, inertial in n 270 and tion) ffcso h topee( atmosphere the the including without of Venus effects around peri- orbit the its at of starting apsis spacecraft & the (Neto with steps 1998) time Prado negative inte- using made numerical are conditions, grations initial the defining After lcto fteiplea h eipi fteorbit the of periapsis the at ap- impulse the the including of from maneuvers plication excluded The are study. present trajectories the by those spacecraft and the of planet captures the verify to made Numerical are tests variations. is energy the maneuvers calculate gravity-assisted to pure applied the in used nique fVns h alsi offiin a ausi the in values has 0 coefficient from ballistic range atmosphere The comes the passes spacecraft Venus. now the of but maneuver point, same of the type from this In ver. by pos- provided energy is maneuver. of it gravity-assist variations far the points the very extreme obtain is to those that sible From point another Venus. planet, to from the proceed of atmosphere then near the and pass inside spacecraft not the but Venus, of ma- trajectories gravity-assisted the the of the neuvers, case the in the simulated In is trajectories this paper. the present So, conditions all of initial integrations. point numerical starting the the are velocity for spacecraft and used the position by of initial modeled Venus The vectors from be enough can orbits. far spacecraft-Sun Keplerian be system to the location assumed and a be reaches can spacecraft that the until performed is h ueia iuain r aeuigape- a using made are simulations numerical The h aitoso nrycnb band nthe in obtained, be can energy of variations The fe ht oe h eogaiyass maneu- aero-gravity-assist the comes that, After E + = 2 1 . o5 to 0 ◦ mxmmvrain Boce1988). (Broucke variation) (maximum E X ˙ ∆ − I . 0 ∆ + E = × = 1 2 10 X E ˙ − X I + C ˙ 7 I km 2 2 B + + + ) h integration The 0). = E 2 Y / ˙ − I g h aetech- same The kg. Y 2 . ˙ ◦ I mnmmvaria- (minimum ∆ + Y ˙ I 2 (10) (9) (8) © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México h topeeo eu smdldb nexponen- by an expressed primaries. by con- the modeled tial, spacecraft is of Venus motion of the the atmosphere of with The plane planar, the travel- motion to be The spacecraft strained to the bodies. assumed celestial and is two orbit those around circular ling a in A Sun 1. periapsis. Figure the in at shown applied similar is representation impulse is graphical an It explained, is just next. there maneuver described but aero-gravity-assist is the it and to objective paper main the this is of Venus be around spacecraft can the of figure color The 2016). online. al. viewed et around (Gomes maneuver Venus Powered-aero-gravity-assisted 1. Fig. ??.html pc otesraeo Venus of surface the to spect h in.Tesmltosas sdacntn number constant a colli- used or also captures of simulations generate number The to large sions. a allowed have but that not energy, value do of that a variations is (Murcia significant with It Mars maneuvers and 2018). 2017b) al. Prado et the & using (in made (Murcia value simulations Earth same in the used is units) units), research canonical canonical present (0.5 the km/s in The 17.5 used planet. periapsis the ma- at and of velocity atmosphere the approach the by close through impulse, obtained passage combines trajectories that atmosphere the neuver the on of Venus influence the of show can that tions the here. for performed implemented integrations is numerical 1968) (Fehlberg adapta- size with step tive method 7/8 Runge-Kutta-Fehlberg A ersnsteattd ftesaerf ihre- with spacecraft the of altitude the represents 4 h opeesse a eu riigthe orbiting Venus has system complete The h eut oss falrenme fsimula- of number large a of consist results The http://nssdc.gsfc.nasa.gov/planetary/factsheet/?? ρ .RESULTS 3. 5exp 65 = TDIGEFCSO AA RUDVNS489 VENUS AROUND PAGAV OF EFFECTS STUDYING 4 ( − nk seTbe1). Table (see km in , h/ 15 . 9) kg / m 3 where , oVns(aan ta.20;Lws&McRonald 1992); & Mcronald Lewis & Randolph 2005; 1991; al. 1992, et pointing (Lavagna and Venus literature to the in cited values maximum i sue v ieetvalues: different ( five lift/drag The assumed tio 1988). (Broucke energy of feeg,ad270 and energy, of fapoc f90 of approach of pa- simulations. Venus the the main for presented selected the are rameters affecting 1 Table not In gen- same, the conclusions. the but are for results, behaviors values the eral different of modifies use parameters The parame- these of considered. number be large to a ters already fixed has of problem since study, this use the simplifies The of parameters small those collisions. criterion for a values and same and captures km. energy the of 330 of number on variations is based significant which used having periapsis, is the value of This altitude the for iet eu;9 n ftemxmmvle ie in cited values oppo- maximum for direction the literature of the the one in lift 9, and Venus; pointing drag to lift when site 1, with lift; equal, no are towards is pointing there lift when the 0, with Venus; equal, are lift and drag iueo h mus hti osbet ec with reach to possible mag- is the technology. that for current impulse value the a of with nitude maneuver powered maneuver unpowered a an and 0.0 representing are km/s, used values 0.5 the and Regarding impulse, the 2005). of al. magnitude et the (Lavagna Venus to opposite ai,temgiueo h mus,adteageof angle the the and impulse, pa- plot: the other of each magnitude The for the ratio, constant impulse. kept the are axis of rameters vertical the direction repre- while color the is axis, A shows coefficient horizontal the ballistic units. in the sented canonical where used, in is unit, code mass per energy cl egt1. km 15.9 km 2500 kg/m 65.0 altitude Atmosphere height Scale density Surface ao eixs108 semiaxis Major ria eid247days 224.7 (1 km/s 35 velocity Orbital period Orbital as4 km 6051.8 Mass radius equatorial Mean 3.1 h iuain osdrdvle o h angle the for values considered simulations The h rtsto eut hw h aitosof variations the shows results of set first The aitoso nryOtie rmVenus from Obtained Energy of Variations . EU PARAMETERS VENUS ◦ L ◦ h einwt aiu losses maximum with region the , h einwt aiu gains maximum with region the , / D AL 1 TABLE n onigi h direction the in pointing and . 21 . × 87 10 × − 6 ,oeo the of one 9, 10 m(1 km UTU CU/T 24 − L/D 3 kg ,when 1, CU L/D ra- ) ) ) © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México h mus saanttemto ftespacecraft the of motion the 180 and against is impulse the ausfrti nl nraei h lcws sense. 90 clockwise of angle the an way, in this increase In angle this for values iue o9 h eeec o h ieto fthe of 0 direction angle the the for reference defines in The shown impulse are 9. results to The 2 Figures 2018). Prado al. & et Murcia Murcia 1996; 2017b; (Prado studies similar in used i.3 aito feeg e nto asa ucino h direc the of function a as mass of unit per energy of Variation 3. Fig. direc the of function a as mass of unit per energy of Variation 2. Fig. GOMES & PRADO, MURCIA, 490 e of ues topee n on pt 5 to up going and atmosphere) eu.Tefl ag fpsil ietosfrthe for from directions going possible around used, of is maneuvers range impulse the full in The the variables show Venus. to those made of are graphs effects way, this In approach. ψ ψ 90 = 90 = ✁ ✁ ✂ ✂ ✄ ✄ -150 -100 -150 -100 100 150 100 150 -50 -50 50 50 0 0 ◦ ◦ C 0x10 0x10 ◦ B o noee lf)adpwrd(ih)mnues h color The maneuvers. (right) powered and (left) unpowered for color The maneuvers. (right) powered and (left) unpowered for hni si h poiedrcin The direction. opposite the in is it when , -7 -7 r nieteitra triga eo(no zero at starting interval the inside are 1x10 1x10 -7 -7 2x10 2x10 C C B B -7 -7 (km (km 2 ◦ 2 3x10 3x10 /kg) /kg) en htteipleis impulse the that means ◦ − hntedrcinof direction the when , -7 -7 180 . 4x10 4x10 0 ◦ × o180 to -7 -7 10 5x10 5x10 − 7 km ◦ -7 -7 h val- The . 2 ✆ ☎ Venus L/D=1 ✆ ☎ Venus L/D=9 -0.1592 -0.1590 -0.1588 -0.1586 -0.1584 -0.1582 -0.1580 -0.1578 /
V=0.0km/s V=0.0km/s -0.156 -0.152 -0.148 -0.144 -0.140 -0.136 =90° =90°
g as kg,
:U) Variation of Energy (C Energy of Variation Variation of Energy (C.U.) Energy of Variation ✝ ✞ ✟ ✠ ✝ ✞ ✟ ✠ -150 -100 -150 -100 150 100 100 150 -50 -50 aei h ieto fVns while Venus, of direction the in made n with 9 and Venus. to opposite direction the in impulse an eutoso h nryvrain banddeto due smaller obtained the variations and energy drag the by of removed ex- reductions is the that final to due energy the occur tra is that This compensations the forces. of the result atmospheric with the energy coeffi- of less ballistic removes increase re- the maneuver the the with so that decreases cient, show of energy orbits magnitude of unpowered a moval The use maneu- maneuvers km/s. side) 0.5 powered (left pow- The unpowered both and vers. for side) 1988]), (right [Broucke ered energy looses craft 50 50 0 0 0x10 0x10 iue n hwterslswhen results the show 3 and 2 Figures -7 -7 ino h mus o G with AGA for impulse the of tion with AGA for impulse the of tion 1x10 1x10 gr a evee online. viewed be can figure online. viewed be can figure -7 -7 ψ 2x10 2x10 90 = C C B B -7 -7 (km (km ◦ 2 2 3x10 3x10 /kg) /kg) mnuesweetespace- the where (maneuvers , -7 -7 4x10 4x10 -7 -7 5x10 5x10 -7 -7 − ☛ ✡ Venus L/D=9 ☛ ✡ L/D=1 Venus 90 L/D L/D V=0.5km/s V=0.5km/s -0.175 -0.170 -0.165 -0.160 -0.155 -0.150 -0.145 -0.140 -0.135 -0.130 -0.125 -0.172 -0.168 -0.164 -0.160 -0.156 -0.152 -0.148 -0.144 =90° =90° ◦ L/D indicates
and 9 = and 1 =
Variation of Energy (C.U) Energy of Variation Variation of Energy (C.U.) Energy of Variation 1 = © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México ie motn ffcso h aevr.Frval- For maneuvers. the of on ues effects important gives utoso nry ti lose httergosin regions 120 the to that zero seen from also interval re- is the smaller It mean energy. which of colors, ductions blue and green the by h tes h ieto fteiple(angle impulse to the compared of effect direction small The a op- is others. the the this from the also approach but of is value, angle there timal the course, decreases moving Of of which effect curvature. Venus, of since to angle Venus, opposite the of pointing gravity is the from lift effects smaller the feeg bandfo ashv ob divided be to have Mars 2 from by obtained energy of h aitoso nrygvnb eu i nthe in lie Venus by given energy that indicates of research variations present the units The of Venus. system for canonical used equivalent the in 0.0380 to o h oso nry ihtemxmmoccurring maximum the with energy, around of loss the for ryotie ntemnuesaon h at lie interval Earth en- the the of around in maneuvers variations the The in obtained ergy Venus. for results equivalent hc ietyrmvseeg rmtespacecraft. the below intervals from The energy as spacecraft, removes a the well gives directly of as motion which also the planet, direction to the This opposite component of forces. gravity atmospheric the the of increases which effects Venus, to the pointing component a has n as h aitoso nrylei h interval the in lie Regard- energy Venus. of − variations for the used Mars, units ing of system canonical ryi h aevr rudteErhhv obe en- to of have Earth variations 1 the the by around divided that maneuvers km/s the means 35.20 in and This ergy Mars for km/s Venus. km/s 29.78 for 24.07 are velocity. per Earth, the velocities energy of the orbital square for of the the unit for of values unit the the The because is the mass done around of planets is unit the This to of necessary velocities Sun. is it orbital systems), the three use the not all are for systems same canonical the (the units re- same different. the the show in very to sults conversions are necessary variations the mag- perform the energy To but the 2018), of al. et Prado nitudes (Murcia & behavior Mars (Murcia general and Earth 2017b) the same using the maneuvers have the for results maneu- These the of part ver. gravity from the away a reduces spacecraft which the with Venus, sending applied space- another the and is of craft, motion propulsion This the the against pointing where energy. component region of a removal is minimum the indicating fteErhSnsse Mri rd 2017b), Prado & means (Murcia system which Earth-Sun the of ytm(ucae l 08,wihmeans which 2018), al. et (Murcia system 0 . 89to 0809 . 36(35 1386 α α agrta 120 than larger 60 = − . 91(35 3971 0 . − . 20 ◦ 83cnncluiso h Mars-Sun the of units canonical 0813 − hsi h aeweeteimpulse the where case the is This . 0 / 0 . 12t .31i h equivalent the in 0.0301 to 0192 24 . 29to 0269 . − 07) . 20 60 2 / ◦ ◦ 29 h lt r dominated are plots the , ob oprdwt the with compared be to aemil lecolor, blue a mainly have TDIGEFCSO AA RUDVNS491 VENUS AROUND PAGAV OF EFFECTS STUDYING − ◦ . 78) 0 aetelretvalues largest the have . 25cnnclunits canonical 0275 2 n h variations the and − 0 . 0378 α ) 08.Fgr hw htteeuvln eut for results equivalent from the vary that Venus shows 3 Figure 2018). eaaigtonihoigclr.Teylo lines yellow The lines the colors. of neighboring inclinations two the This in separating observed atmosphere. fact be the can of Another point effects stronger case. the of powered is and type the noted this confirms in for even Venus This is maneuver, better values. much how larger quantifies much are which h iia eut o aslei h nevlfrom interval the in lie Mars − for results similar The n gis h oino h pccat h inter- The spacecraft. the below of val motion Venus the towards against pointing and components have impulses at ag rm000 to 0.0000 the from the from range for Earth obtained units variations canonical the of system, terms for Venus-Sun In occur not Mars. does and possibility which Venus energy, the of variations indicate zero particular, Earth of In the using variations. maneuvers energy the for the values different of with for magnitudes but results Mars, the and with Earth similarities the shows space- ob- 2018) Mur- al. 2017b; are the Prado et maneuver & cia of (Murcia the literature from motion The effects tained. the smaller to so craft, opposite and Venus oso nryaround energy of loss esaesoni h ih ieo h gr.The ef- important figure. same when the the observed of has fects side impulse right the maneu- the of Powered direction in shown space- maneuver. are energy the the vers the by takes decreases made lift which removal Venus, higher from away The craft units. canonical h aevr rudVnswhen Venus around maneuvers the in energy the of is maneuver. variations Earth of smaller type The the this shows place. compared that second ones in one three Mars the with of planet here, that best indicate the paper is present Venus the in shown here. results made re- simulations The of the type performing this plan- without predict the to radii sults difficult of it masses, atmospheres makes which the the ets, of of of profiles functions values density are the and variations of variations Those combination the the times Mars. four by obtained and effects giving Earth the the times using eight when other the about to planets, compared two larger much are values Those h e seti h agritra fteenergy the from of now interval is larger increased. which the is variations, is aspect coefficient new ballistic The de- the losses unpow- energy when the the creasing with of similar, behavior is The maneuvers ered Venus. to opposite ing ψ nevlfrom interval 0 90 = . et iue3sosteeuvln eut for results equivalent the shows 3 Figure Next, 24to 0234 ◦ hc sastaino eyhg itpoint- lift high very of situation a is which , − 60 − 0 ◦ − . 48cnncluis(ucae al. et (Murcia units canonical 0468 0 sweeteiple on against point impulses the where is − . 52to 1592 0 . 70to 1750 L/D α 60 = − ,wt h maximum the with 1, = − 0 − ◦ 0 . nti emtythe geometry this In . 58cnnclunits. canonical 1578 0 . 20cnnclunits. canonical 0280 . − 2 aoia units, canonical 125 0 . 50to 1580 L/D and 9 = − 0 . 1360 © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México constant o h itt rgrto( ratio drag to lift the for is grav- Venus. spacecraft the the from reduces away because moved maneuver, force because the lift happens of the effects This ity for sign energy. positive of correspond the losses coefficient smaller ballistic to the for values larger u r eyicie for inclined very are but ls hywr lothrzna when exam- horizontal good almost are were regions They red and ples. green the dividing hyaen ogrhrzna,a hywr when were they as horizontal, since longer examples, L/D good no very be- are are mentioned they lines As yellow the colors. fore, lines atmo- adjacent the the of two inclinations of the separating in effects observed as be larger can locations, The sphere same minimum the explained. at and already occur maximum energy The of variations results. interesting energy more remove spacecraft. gravity) the this in- from of by In effects (indirectly, the lift Venus. creasing and to (directly) points drag ballistic now situation, the lift with increase since positive losses coefficient, with energy maneuvers the and with behav- lift, compared different when a re- have ior, which now The maneuvers influence, spacecraft. the unpowered higher from directly in energy a more altitude. moves longer has periapsis stay drag the reducing to Therefore, This so spacecraft Venus. atmosphere, to the the now causes pointing is situation lift that indicating nraigteblitccecetcngv agrval- larger give can coefficient that ballistic show the colors increasing two between borderlines the stant), i.4 aito feeg e nto asa ucino h direc the of function a as mass of unit per energy of Variation 4. Fig. GOMES & PRADO, MURCIA, 492 ψ 90 = ✝ ✞ iue4cniestestaino eaievalues negative of situation the considers 4 Figure h oee aevr rgtsd)as show also side) (right maneuvers powered The -150 -100 100 150 -50 .Floigaanahrzna ie( line horizontal a again Following 1. = 50 0 ◦ 0x10 o noee lf)adpwrd(ih)mnues h color The maneuvers. (right) powered and (left) unpowered for α -7 hrzna ie,i ses oseta the that see to easy is it line), (horizontal 1x10 -7 2x10 C B -7 (km L/D 2 3x10 /kg) L/D -7 .Floigaln of line a Following 9. = = 4x10 − -7 and 1 5x10 -7 L/D ψ ✠ ✟ Venus L/D=-1 90 = V=0.0km/s -0.170 -0.168 -0.166 -0.164 -0.162 -0.160 =90° α 1, = con- ◦
), Variation of Energy (C.U.) Energy of Variation ✝ ✞ ✟ ✠ -150 -100 100 150 -50 hc ae h nevlo aitoso energy of variations from goes of now force, interval is It lift wider. It the the from makes energy. effects which stronger of with variations situation the a differ- for with magnitudes just behavior, ent same the have maneuvers tywhere etry 07) iue6sosterslso h simula- the of results the shows for 6 tions Figure 2017b). cluis oprdt nitra htge from goes that interval an to − compared units, ical itt rgrto with ratio, drag to lift dependent very be coefficient. ballistic to is the maneuver on consequence the The causes lift spacecraft. that remov- the them from ma- of energy a both ing drag, the is with spacecraft, combined this the are of Since effects energy the maneuver. reduces that the neuver the of of space- effect part the the gravity sends increases lift which Venus, so indicate Venus, toward ratio to craft because drag pointing happens to is lift lift This the that for energy. values of negative losses the the for ues ryo h pccat so spacecraft, the of ergy the for values energy. particu- smaller of in indicates variations color, which blue blue, the dark the by lar that dominated seen now is the is It with plot energy coefficient. ballistic of the losses are of due the There increase are in atmosphere. and increases plots the 4 larger of the now Figures effects in stronger both visible the In forms to triangular lines. the inclined look 5, more Just the effects. at atmospheric stronger also figure) the the identify of side (right maneuvers powered The 50 0 0 0x10 . iue5cniestestaino eaielarge negative of situation the considers 5 Figure enwcnie emtist nraeteen- the increase to geometries consider now We 7 to 172 -7 ino h mus o G with AGA for impulse the of tion 1x10 gr a evee online. viewed be can figure L/D − -7 ψ 0 90 = . 6 aoia nt when units canonical 160 2x10 = C B -7 (km − ◦ ti hw htteunpowered the that shown is It . .Upwrdmnuesshow maneuvers Unpowered 1. 2 3x10 /kg) L/D -7 − ψ 0 4x10 = 270 = . 20to 2200 − -7 ,aani h geom- the in again 9, 5x10 ◦ Mri Prado & (Murcia -7 − L/D 0 ☛ ✡ Venus L/D=-1 . V=0.5km/s 60canon- 1600 -0.190 -0.185 -0.180 -0.175 -0.170 -0.165 -0.160 -0.155 -0.150 -0.145 =90° L/D = −
= and 1 (C.U.) Energy of Variation − 1. © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México i.6 aito feeg e nto asa ucino h direc the of function a as mass of unit per energy of Variation 6. Fig. direc the of function a as mass of unit per energy of Variation 5. Fig. aetemnuesi hsstain h maximum The domi- situation. they this so in strong, maneuvers ballistic the are the nate hor- impulses on nearly The dependency show little coefficient. figure) with the from lines ma- of interval powered izontal side The the (right in units. neuvers canonical now 0.1650 lie to variations 0.1595 increased The the from com- energy larger of drag. losses are extra They the to the maneuver. to direc- pared from the closer This energy of spacecraft of part gains the gravity plots. the moves the increases force which of lift Venus, now the sides are of left colors tion the the red for at that en- values Note located of larger increases using coefficients. The when ballistic be- stronger impulses. expected, are no as ergy are lines, there vertical cause by composed plots ψ ψ 270 = 90 = ✁ ✁ ✂ ✂ ✄ ✄ -150 -100 -150 -100 100 150 100 150 -50 -50 50 50 0 ◦ 0 0.0x10 0x10 ◦ o noee lf)adpwrd(ih)mnues h color The maneuvers. (right) powered and (left) unpowered for o noee lf)adpwrd(ih)mnues h color The maneuvers. (right) powered and (left) unpowered for -7 -7 1x10 0.5x10 -7 2x10 -7 C C B B -7 (km (km 2 2 3x10 /kg) /kg) 1.0x10 TDIGEFCSO AA RUDVNS493 VENUS AROUND PAGAV OF EFFECTS STUDYING -7 -7 4x10 -7 1.5x10 5x10 -7 -7 ✆ ☎ L/D=-1 Venus ✆ ☎ L/D=-9 Venus
V=0.0km/s 0.1600 0.1610 0.1620 0.1630 0.1640 0.1650
=270° V=0.0km/s -0.210 -0.200 -0.190 -0.180 -0.170 -0.160 =90°
Variation of Energy (C.U.) Energy of Variation Variation of Energy (C.U.) Energy of Variation ✝ ✞ ✟ ✠ ✁ ✂ ✄ -150 -100 -150 -100 100 150 -50 150 100 -50 ψ n nydffrne ntemgiue ftevari- the of magnitudes show- the same, in the differences is only behavior general ing The energy. of h oino h pccat oho hmremoving spacecraft. them the of from both energy spacecraft, the against of and motion Venus the to opposite pointing components nteopst ie h iiu aitosoccur variations minimum spacecraft. the around space- the side, the to opposite of the energy On motion gives the directly which of com- craft, direction another the and Venus; in of ponent gravity the increase of which effects Venus, the to pointing impulses the components geometry, have this In before. explained sons aito feeg saround is energy of variation 50 50 0 0.0x10 0 0x10 270 = iue7sosterslswhen results the shows 7 Figure -7 -7 ino h mus o G with AGA for impulse the of tion with AGA for impulse the of tion α 1x10 ◦ gr a evee online. viewed be can figure h iuto feteelf n gains and lift extreme of situation the , gr a evee online. viewed be can figure = 0.5x10 -7 − 2x10 75 -7 C C B B ◦ -7 (km (km nti aeteiple have impulses the case this In . 2 2 /kg) 3x10 1.0x10 /kg) -7 -7 4x10 α -7 1.5x10 120 = 5x10 -7 -7 L/D L/D L/D ◦ o h rea- the for , ☛ ✡ L/D=-9 Venus ✆ ☎ Venus L/D=-1 V=0.5km/s V=0.5km/s =90° =270° -0.28 -0.26 -0.24 -0.22 -0.20 -0.18 -0.16 0.145 0.150 0.155 0.160 0.165 0.170 0.175 0.180 = = =
−
− −
Variation of Energy (C.U.) Energy of Variation and 9 (C.U.) Energy of Variation and 1 and 9 © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México and the atmosphere. with the increase stronger from gets gains effects maneuver the energy since coefficient, the ballistic forces. that atmospheric the also of is Note effects which stronger maneuvers, the powered to the due for made plots the eas h aelnsaenwicie n h tri- the present. and again inclined are now shapes are angular lines same the because nevlfo .40t .80cnncluisfor units canonical 0.1800 an against to units, L/D 0.1450 canonical 0.28 from to interval 0.15 from now is i.7 aito feeg e nto asa ucino h direc the of function a as mass of unit per energy of Variation 7. Fig. GOMES & PRADO, MURCIA, 494 a rm019 ool .60cnncluis when units, canonical inter- 0.1650 L/D the only to to 0.1595 compared canonical from case, 0.2200 val unpowered to the 0.1600 the are in from energy of units interval of increase variations the the the in for now to values The due higher force. lift are They ations. et a ese ntestainwhere situation the in seen ef- be Larger can colors. adjacent fects two between lines border when ob- small very behavior are L/D same atmospheric maneuvers the The the to maneuvers. is contributions powered It of the in effects Venus. served the of decreasing gravity so the Venus, space- directly from the moves away energy which removes craft lift, and which spacecraft, maneuvers. drag, the unpowered to from of due case is ballis- the This the in of coefficient decrease the tic by reduced now are energy L/D ψ 270 = ✁ ✂ ✄ h eut o itpitn poiet Venus to opposite pointing lift for results The -150 -100 ψ 100 150 -50 = = ,wihcnb bevdb h horizontal the by observed be can which 1, = n ,rsetvl.Teicessi the in increases The respectively. 9, and 1 = 50 0 0.0x10 270 = ◦ − − o noee lf)adpwrd(ih)mnues h color The maneuvers. (right) powered and (left) unpowered for .Nt h xsec ftinua om in forms triangular of existence the Note 1. .Frtepwrdmnues h range the maneuvers, powered the For 1. -7 ◦ r hw nFgrs8ad9 when 9, and 8 Figures in shown are 0.5x10 -7 C B (km 2 /kg) 1.0x10 -7 1.5x10 -7 L/D ✆ ☎ L/D=-9 Venus V=0.0km/s =270° 0.160 0.170 0.180 0.190 0.200 0.210 0.220
9, = Variation of Energy (C.U.) Energy of Variation ✝ ✞ ✟ ✠ -150 -100 100 150 h ceeaino ahfre( force each of in acceleration involved the force each present ( the of dynamics in the contribution used The idea the session. to 2016; (Lara similar to orbits 2013), indexes of Prado level integral perturbation pa- of the several forms quantify has deliv- different literature force studying The each pers spacecraft. that the velocity to of ers give increase spacecraft total the of the on trajectory whole integrations term the These over each made spacecraft. of the in- of numerically contribution trajectories are the the Earth show term the to each of tegrated from field et gravity accelerations Sanchez the of the that of where to similar (2014), idea an al. is mathemat- It the in model. included ical force acceler- each by the given of done ations is integrations measurement numerical This performing inside by drag. time con- and the shorter lift decreasing of or or tribution increasing longer so a atmosphere, for the trajec- stay new also may the it tories because indirectly, but orbit instantaneously, the modify- vector changes by velocity directly the orbit the ing changes impulse application The the of maneuvers. powered lift of the contribution in the drag of and variation the atten- to Special given is dynamics. tion the in involved force each of in(1.Ti ie h oa aito fvelocity force. of each variation by equa- spacecraft total by the the shown to gives given as This trajectory, (11). the tion of interval time -50 50 3.2 0 0.0x10 hsscinhstega fmauigteeffects the measuring of goal the has section This dniyn h otiuino ahFrein Force Each of Contribution the Identifying . -7 ino h mus o G with AGA for impulse the of tion gr a evee online. viewed be can figure 0.5x10 C F -7 C B C F (km smaue steitgainof integration the as measured is ) 2 h Maneuvers the /kg) 1.0x10 = Z t -7 i t f a c 1.5x10 dt. a c uigtetotal the during ) -7 L/D ☛ ✡ Venus L/D=-9 V=0.0km/s =270° 0.16 0.18 0.20 0.22 0.24 0.26 0.28 =
− and 9 (C.U.) Energy of Variation (11) © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México i.9 aito feeg e nto asa ucino h direc the of function a as mass of unit per energy of Variation 9. Fig. direc the of function a as mass of unit per energy of Variation 8. Fig. hsi lal hw nFgr 0 aefrteun- the for made system. 10, with the Figure maneuvers on in powered acting shown always clearly Sun is is Sun This the the of Venus and for by short gravity passage Venus is the the of because course motion, gravity the Of dominates and Sun trajectory. the each of gravity drag, kept are the ratio drag impulse, plot. to each the lift for the of constant and magnitude approach of the The angle in impulse the axis. of vertical direction the of the and of function defines axis that a horizontal integral angle the as the in force coefficient of given ballistic value the a from the acceleration indicates the code color a ψ ψ 270 = 270 = ✁ ✁ ✂ ✂ ✄ ✄ h eut uniytecnrbtoso lift, of contributions the quantify results The where 12, to 10 Figures in shown are results The -150 -100 -150 -100 100 150 100 150 -50 -50 50 50 0 0 0x10 0x10 ◦ ◦ o noee lf)adpwrd(ih)mnues h color The maneuvers. (right) powered and (left) unpowered for color The maneuvers. (right) powered and (left) unpowered for -7 -7 1x10 1x10 -7 -7 2x10 2x10 C C B B -7 -7 (km (km 2 2 ψ 3x10 3x10 /kg) /kg) TDIGEFCSO AA RUDVNS495 VENUS AROUND PAGAV OF EFFECTS STUDYING 270 = -7 -7 4x10 4x10 ◦ -7 -7 and 5x10 5x10 L/D -7 -7 ✆ ☎ Venus L/D=9 ✆ ☎ L/D=1 Venus V=0.0km/s V=0.0km/s =270° = =270° 0.132 0.136 0.140 0.144 0.148 0.152 0.156 0.151 0.152 0.153 0.154 0.155 0.156 0.157 0.158 0.159 −
9.
Variation of Energy (C.U.) Energy of Variation Variation of Energy (C.U.) Energy of Variation ✝ ✁ ✞ ✂ ✟ ✄ ✠ -150 -100 -150 -100 100 150 150 100 -50 -50 nteitra rm185 o1.8720 to is 1.8650 which Sun, from the interval of the con- contribution in has the 0.1592 Venus of to of 8.5% 0.1583 gravity from about interval the the that simulations, in re- tributions shows the similar It all of for reproduction sults. the typical unnecessary very making is figure This nerto ssopdwe h pccatrahsa reaches spacecraft the when the since stopped space- time, is the integration integration of the at- increases velocity which the the craft, of the reduces increase of forces the mospheric functions because assisted are coefficient, gravity variations ballistic the These of importance maneuvers. applied. the impulse shows the times mag- This eleven of order than in Venus, more km/s 5.6 nitude, of to amounts contribution units, The metric in units). canonical notes 50 50 0 0 0x10 0x10 -7 -7 ino h mus o G with AGA for impulse the of tion with AGA for impulse the of tion 1x10 1x10 gr a evee online. viewed be can figure online. viewed be can figure -7 -7 2x10 2x10 C C B B -7 -7 (km (km 2 2 3x10 3x10 /kg) /kg) -7 -7 4x10 4x10 -7 -7 5x10 5x10 -7 -7 CU L/D L/D ☛ ✡ Int. Sun L/D=9 Venus ✆ ☎ L/D=1 Venus V=0.5km/s V=0.5km/s =270° =270° 0.120 0.130 0.140 0.150 0.160 0.170 0.140 0.145 0.150 0.155 0.160 0.165 0.170 0.175
(
CU and 9 = and 1 =
Variation of Energy (C.U.) Energy of Variation Variation of Energy (C.U.) Energy of Variation CU de- , © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México fteageta enstedrcino h impulse the of direction the ( defines that angle with the km/s, of 0.5 of impulse of ψ magnitude a using vers divided magnitudes the since with 9, just by similar, contribution is The drag dynamics. of the the in indicates role of which important Venus, 37% an of about gravity to the of zero contribution from that means goes in This contribution contribution its coefficient. a ballistic the has on 0.06 force strongly to 0.00 lift from The range the results. the in r oae near located are htteiplei ple ntedrcino Venus, of direction the in applied is impulse the that xddsac rmVns ic hsi h unpow- the angle is the this maneuver, Since ered Venus. from distance fixed α aevr with maneuvers i.1.Cnrbtoso h ceeain rmda,lf,gaiyo gravity lift, drag, from accelerations the of Contributions 10. Fig. GOMES & PRADO, MURCIA, 496 α 270 = scer h aiu ffcsfrlf n drag and lift for effects maximum The clear. is ) = ✁ ✂ ✄ et iue1 sotie o oee maneu- powered for obtained is 11 Figure Next, ✝ ✞ ✟ ✠ -150 -100 -150 -100 100 150 -50 − 100 150 -50 50 50 0 0.0x10 0 0.0 90 ◦ ① ◦ 10 and hsi xetd since expected, is This . -7 -7 L/D L/D ψ 0.5x10 0.5 9. = α 270 = ① 10 90 = = -7 -7 C C − B ❇ ◦ (km (k .Teeeto h direction the of effect The 9. ◦ and ♠ 2 2 /kg) 1.0x10 n h iiu around minimum the and /kg) 1.0 α CU ① L/D osntpa n role any play not does 10 -7 -7 fcus depending course of , = − 1.5x10 1.5 α .Teclrfiuecnb iwdonline. viewed be can figure color The 9. ① 90 = 10 -7 -7 ◦ ✆ ☎ Venus Int. Drag L/D=-9 ☛ ✡ L/D=- Int. Sun Venu V=0.0km/s means =270° V=0.0k =270° 1. 1. 1. 1. 1. 1. 1. 1. 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 ✾ ✽ ✽ ✽ ✡ ✽ ✽ ✡ ✽ 71 72 65 67 6 70 66 6 ♠ ☛ ✽ / ☞ ✎ ✝ ✞ ✟ ✠ -150 -100 100 150 -50 oicesn h topei ih;while flight; atmospheric the increasing so h ffcso h rvt fteSnadVnsshow Venus and Sun the of gravity drag. the and of lift increase effects of can The effects it the used, and flight properly atmospheric is the direction an spacecraft the the is if the of and, This trajectory of the effects changes 9. that indirect impulse the factor of the explanation by interesting divided same just very the has a behavior, Drag Venus, of contribution. gravity important and the varia- large by the given of velocity 75% of the about tion contribute that now means can This force lift The maneuvers. unpowered the maneuvers. by 0.12 unpowered now com- the is effects of maximum maximum those the to to important of pared is magnitudes It the ballistic confirmed. the note course with of contributions is the coefficient of flight. increase atmospheric The the op- decreasing direction so Venus, the to in posite applied is impulse the that means -150 -100 50 100 150 -50 0 0.0 50 0 0.0 ① h u n rvt fVns in Venus, of gravity and Sun the f 10 ① 10 -7 -7 0.5 0.5 ① 10 ① 10 -7 C -7 ❇ C ❇ (k (k ♠ ♠ 2 /kg) 1.0 2 /kg) 1.0 ① CU 10 ① 10 -7 -7 wc h au showed value the twice , 1.5 1.5 ① 10 ① CU 10 -7 -7 o unpowered for , ✍ ✑ Int. Venu Venu L/D=- V=0.0k ✡ ☛ L/D=- Int. Lift Venu =270° V=0.0k =270° 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 α ✾ 0.000 0.010 0.020 0.030 0.040 0.050 0.060 ✾ = ✾ ✾ ✽ ✽ ✽ ✽ ✽✽ ✽✾ ☛ ✡ ♠ 3 4 6 7 0 2 5 1 ♠ / − / 90 ◦ © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México as h eaielf oda ai ed h space- the sends ratio be- drag expected, to is lift negative increase the This in cause used value applied. con- same impulse a the means the about This km/s, 0.42 used. of is tribution ratio drag to lift negative att entdi httecnrbto flf sin- 0.012 is to lift of 0.008 contribution (from the 50% that by is creased noted main be The to explained. fact already behavior the show they i.1.Cnrbtoso h ceeain rmda,lf,gaiyo gravity lift, drag, from accelerations the of Contributions 11. Fig. n eu aeol ml aitos h upper The of values variations. the Sun small consider the only plots of gravity have the Venus of and contributions the follows divided and magnitudes drag 9, the by because with just here, behavior, showed same the force func- only a ( the as drag is force to lift lift the the of of tion contribution the of ations that drag explained. already and as lift spacecraft, of the times effects down integration slow stronger larger the the by to caused due increases, small online. aevr sn niplewt antd f05k/ with km/s 0.5 of magnitude with impulse an using maneuvers ✁ ✂ ✄ h nlse ftepeetsuysostevari- the shows study present the of step final The ✔ ✕ ✎ -150 -100 -150 -100 100 150 -50 100 150 -50 50 50 0 0.0x10 0 0.0 ✌ 10 -7 -7 0.5x10 0.5 ✌ 10 -7 -7 C C B ✎ (km L/D (k ✏ 2 2 /kg) 1.0x10 /kg) 1.0 TDIGEFCSO AA RUDVNS497 VENUS AROUND PAGAV OF EFFECTS STUDYING ai Fgr 2.This 12). (Figure ratio ) ✌ L/D 10 -7 -7 and 1 = 1.5x10 1.5 CU ✌ 10 -7 -7 hnthe when ) − ✆ ☎ Int. Drag L/D=-9 Venus ✖ ✑ Int. Sun Venu L/D=- ,and 1, V=0.5km/s =270° V=0.5k =270° 1. 1. 1. 1. 1. 1. 1. ✑ ✒ 0.000 0.002 0.004 0.006 0.008 0.010 0.012 ✓✓ ✗✗ ✓ ✓ ✓ ✓ ✓ 55 60 65 70 75 ✏ 0 5 / ✑ ✘ ✙ ✚ ✛ -150 -100 100 150 -50 ✝ ✞ ✟ ψ 50 ✠ sipratadi ssoni iue1.I the In 12. Figure from in lift of shown increase is the way, it same increase and at- this important the of is quantification increasing The so Venus, flight. of mospheric direction the in craft nagstecnrbto fti oc.Temax- The 0.008 force. from this goes of imum contribution the enlarges 0.012 when u ti motn odfietelmto h ballistic the of limit the ratio, define drag to to important to lift is expected negative it high are but a They prob- using the Venus. when shows appear with also collisions It of lift of lem maneuvers. effects powered the the figure of in This quantification situation. complete this rel- a in shows high the lift the times of show eight importance numbers ative than These more applied. km/s, impulse 4.22 around tion ofo 0.033 contributions from maximum go the lift, the for values higher 0 0.0 -150 -100 100 150 -50 270 = 50 ✜ 0 0.0 10 h u n rvt fVns in Venus, of gravity and Sun the f -7 ✌ 10 L/D CU ◦ -7 and 0.5 when , = ✜ 0.5 L/D 10 − -7 ✌ C 10 CU .Teevle niaeacontribu- a indicate values These 9. ✢ -7 (k C = ✣ ✎ L/D when , 2 (k − /kg) 1.0 ✏ .Teclrfiuecnb viewed be can figure color The 9. 2 ✜ /kg) 1.0 10 -7 ✌ = CU 10 -7 L/D − when , .We considering When 1. 1.5 ✜ 1.5 10 ,t 0.120 to 9, = -7 ✌ L/D 10 -7 CU L/D ✥ ✦ Int. Venu Venu L/D=- V=0.5k =270° = o powered for , ✡ ☛ Int. Lift Venu L/D=- 0.1600 0.1610 0.1550 0.1560 0.1570 0.15 0.15 V=0.5k ✤ ✧ =270° ± ,to 1, = ★ ✧ to 1 ✣ ✤ 0.00 0.02 0.04 0.06 0.0 0.10 0.12 ✑ ✒ 0 0 / ✤ ✓ ✏ / CU ✑ ± 9 , © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México i.1.Cnrbtoso h ceeain rmda,lf,gaiyo gravity lift, drag, from accelerations the of Contributions 12. Fig. GOMES & PRADO, MURCIA, 498 aevr sn niplewt antd f05k/ with km/s 0.5 of magnitude with impulse an using maneuvers htti ii saon 1 around is limit shows 12 this Figure that problem. this causes that coefficient online. otevr togeeto t topee h ma- The atmosphere. its of using effect neuvers which strong plot, due very Venus the the with in to collisions region of blank occurrence the a indicates is there point this o h aeraos ote r o eetdi the in repeated not are paper. they present so reasons, same the for in feeg fasaerf asn yVnsa a as Venus by passing varia- spacecraft the a measure of to energy is of of tions objective trajectory The the modify spacecraft. and to a Venus maneuver, of field impulsive the gravity an from the forces Venus, a drag of and is lift atmosphere This the uses that Venus. maneuver around maneuvers gravity-assist ✁ ✂ ✄ hspprpeet td fpowered-aero- of study a presents paper This ✭ ✮ ✎ -150 -100 100 150 -50 -150 -100 50 100 150 -50 0 50 0x10 0 0x10 -7 -7 1x10 ψ 1x10 90 = .CONCLUSIONS 4. -7 -7 2x10 2x10 ◦ C C B aeeatytesm results, same the exactly have -7 ✯ (km -7 (k ✰ 2 3x10 /kg) 2 . 3x10 /kg) 5 × -7 -7 10 4x10 4x10 − 7 -7 km -7 5x10 2 5x10 / g After kg. -7 -7 ✆ ☎ Int. Lift L/D=1 Venus ✱ ✑ Venus L/D= Int. Lift V=0.5km/s =270° V=0.5k =270° 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.00 0.032 0.000 0.004 0.00 0.012 0.016 0.020 0.024 0.02 ✲ ✩ ✰ ✴ ✳ /s ✝ ✞ ✟ ✠ ✵ ✶ ✷ ✸ -150 -100 -150 -100 100 150 -50 100 150 -50 ausotie sn h at,adfu ie the Mars. times in the four verified and times results Earth, eight equivalent the about using are variations obtained Venus The values by maneuver. given Earth of energy to type of compared this for results Mars better and gives that planet showing Venus, this around maneuvers in energy. mosphere of maximum losses with maximum one with one the the results, and the gains are obtain geometries function to particular a Two used as variations parameters. energy those of the where show at- figures colors in the expressed the of are effects results the The defines mosphere. that quantity coefficient, a ballistic ap- is the of which and angle spacecraft; of the the and of distance velocity proach periapsis the approach; the close impulse spacecraft; the the the of to magnitude and applied maneu- direction the the describe like that ver, parameters the of function 50 ψ 50 0 0 0x10 0x10 270 = h eut dnie lal h ffcso h at- the of effects the clearly identified results The h u n rvt fVns in Venus, of gravity and Sun the f -7 -7 ◦ 1x10 1x10 and -7 -7 L/D 2x10 2x10 C C ✪ ✹ -7 (k -7 (k .Teclrfiuecnb viewed be can figure color The 9. = ✫ ✺ 2 3x10 2 /kg) 3x10 /kg) -7 -7 4x10 4x10 -7 -7 5x10 5x10 CU -7 -7 o powered for , ☛ ✡ Venus Int. Lift L/D=-1 ✼ ✻ Int. Lift Venus L/D=- V=0.5k =270° V=0.5k =270° 0.000 0.002 0.004 0.006 0.00 0.010 0.012 0.12 0.00 0.02 0.04 0.06 0.0 0.10 ✿ ✫ ❀ ✬ ✺ /s /s © Copyright 2018: Instituto de Astronomía, Universidad Nacional Autónoma de México ihdrcinaround direction with maneuver. the of part lift ob- gravity effects the positive the from increases to tained geometry compared This variations detailed. are energy the in ngtv ausof values (negative ftesaerf,temxmmvle cu o im- near energy for directions the occur with values increase maximum pulses to the the spacecraft, is the In goal of maneuver. the gravity where the another situations of to effects and Venus, the toward spacecraft increase spacecraft the the of sending component motion the against ffc o itadda slctdnear located is drag and lift for effect h mus sapidi h ieto fVns and Venus) of near direction minimum the the in applied is impulse the inmr hneee ie h eoiyincrement velocity impulse. the the times by contribu- given eleven a gives than which more Venus, of tion field is gravity which the Sun, of the of 1.870 of contribution near the gravity of the 8.5% 0.159 that about around shows contributions has dynamics Venus the in volved simi- have Venus of contributions. atmosphere lar the (when by force passage lift and the for values L/D large while impulse, L/D the of the spacecraft. motion increase the to the of help energy components of both direction so spacecraft, the another and in Venus to component pointing component one have opeedet h oicto ftetrajectory the of spacecraft. modification at- the the the of from to obtained due of are effects mosphere effects indirect Those larger the impulse; of analyzed. the illustration trajectories an the are numbers all because 9, for factor constant same the is the by has divided Drag just Venus. behavior, of con- gravity the the of of 75% tribution about to The increases lift maneuvers. of 0.12 unpowered contribution the to by showed increase value the effects maximum The h antdsdvddb ,since with 9, just by similar, divided is important magnitudes drag the an of Venus, contribution The of the gravity value. of the 37% around of of contribution contribution maximum 0.06 a to 0.00 plies from range the CU in contribution a has lift h aiu osso nryocrfrimpulses for occur energy of losses maximum The h motneo sn itpitn oVenus to pointing lift using of importance The h oee aevr hwta h maximum the that show maneuvers powered The nteupwrdmnues when maneuvers, unpowered the In in- forces the of contribution the of analysis The (when force lift the for values low of use The eedn nteblitccecet hsim- This coefficient. ballistic the on depending , )gnrtsmnuesdmntdb the by dominated maneuvers generates 1) = )gnrt aevr hr h impulse the where maneuvers generate 9) = CU hsvlecnrsteimportance the confirms value This . L/D α = α sson n h increases the and shown, is ) − 60 = TDIGEFCSO AA RUDVNS499 VENUS AROUND PAGAV OF EFFECTS STUDYING 90 α ◦ 120 = ◦ opst direction). (opposite ihacomponent a with , L/D ◦ α eas they because , CU 90 = L/D 9. = CU hc is which , ◦ twice , = (when L/D − 9, oe,V,P˜eo,J,Pao .F .A,& Transac- WSEAS A., 2008, A. B. B. F. A. F. Prado, & V. A. Gomes, Prado, J., Pi˜neros, 643 22, V., JGCD, 329 1999, 12, Gomes, A. AcAau, B. 1966, F. G. A. Flandro, Prado, & G. Felipe, and seventh-, sixth-, fifth-, 424 NASSP, Classical 8 1978, 1968, conf., E. E. aiaa Burgess, Fehlberg, 1984, & A. A. S. J. Davis, Dunne, & W. 1982, D. H. R. Dunham, Stanford, & V., D. Byrnes, JGCD, A., L. 1981, D’Amario, R. Stanford, & D., Byrnes, L., D’Amario, ie agrta rg h nraeo h itcon- lift nine the is of 0.032 lift increase from when the goes way, tribution same drag, the than In larger times drag. to equal is e,adterslsso ht hnusing when that, show results the and ied, velocity the impulse. than the larger by contribu- times given numbers increment a eight gives Those than which more lift, used. tion of is importance ratio the drag show to lift negative 0.012 to 0.008 from 50%, negative by when increased is lift o h alsi offiin rud1 around coefficient ballistic the for − aaio . oaud,G,&Psrn,D 1999a, D. Pastrone, & G., Colasurdo, L., 18 82, DOI: 1, AdAnS, Casalino, Space, 1993, 1986, A. 4220, R. B. Carvell, F. A. 88, Prado, & AIAA/AAS, A. R. Broucke, 1988, 2000, X. R. N. Vinh, & Broucke, M., J. Longuski, P., E. Bongfilio, J. M. Lewis, & P., R. Starkey, M., Lavagna, R., Armellin, Education Higher of (CAPES). Improvement Personnel the Coor- the for from dination support financial the Foundation and (FAPESP) Research No. S˜ao Paulo and from No. 2016/24561-0 2016/14665-2 No. Grants Grants (CNPq); Devel- by Technological opment and Scientific Na- provided for the Council from tional support 301338/2016-7 No. and the 406841/2016-0 for ation ,ti rbe perdbyn iiigvalue limiting a beyond appeared problem this 9, ,811 9, 869 3, mechanics, theoretical and applied on tions Applied and 817 Computational 35, Mathematics, 2016, J. Golebiewska, 287 Report, Technical NASA, trol con- stepsize with formulas Runge-Kutta eighth-order 465 5, JGCD, 591 4, GD 2 637 22, JGCD, 1159 10.2514/6.1988-4220 768 37, JSpRo, 1051 44, JSpRo, 2007, h rbe fcliin ihVnsi lostud- also is Venus with collisions of problem The of contribution the that shows also analysis This h uhr iht xrs hi appreci- their express to wish authors The 00 SA rnatoso Mathematics, on Transactions WSEAS 2010, . 156 22, JGCD, 1999b, . 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