© Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México eit eiaad srnmı Astrof´ısica Astronom´ıa y de Mexicana Revista ancalne nitrlntr rvli rdcn h eylrevlct cha System. velocity Solar large orbit the very from in the or another producing surface to is Earth’s body travel the one interplanetary from the in observations of challenges from atmos exploration main gained Earth’s that in be work believe not at biologists could mechanisms insig basic and us the geologists given of has astronomers, understanding Ivanov1990) better 1990; a and (McKinnon System Magellan and 1990), (Kerrod hntepro fteobt(ae19) nsc ae,te∆ the cases, such In o time 1994). burn (Hale the orbit when reasonable the is of assumption this period orbit, the new than the som of take velocity will the it Although to instantaneously. occur which systems, propulsive oestain,hwvr h ienee ocmlt h rnfrmyas ean be also may transfer and the ∆ complete initial minimum to the a needed Given time with the transfer however, required. For the situations, orbi be some impulse. perform initial single may to the a orbits generally of with transfer velocity is accomplished the intermediate be and and can orbit transfer impulses final the the intersect, of orbits velocity final the between difference the 3 2 1 h ag muto nomto rmitrlntr isos uhas such missions, interplanetary from information of amount large The otase rmoeobtt nte h eoiycagso h pccataeas are spacecraft the of changes velocity the another to orbit one from transfer To eateto ahmtc,PeaaoyYa,Qsi Unive Qassim Year, Preparatory Mathematics, of Department eateto srnm,Ntoa eerhIsiueo As of Institute Research National Astronomy, of Department eateto srnm,Fclyo cec,Kn Abdulazi King Science, of Faculty Astronomy, of Department NLSSO RNFRMNUESFO NTA CIRCULAR INITIAL FROM MANEUVERS TRANSFER OF ANALYSIS ua sl eo aiba lrset,dsrolmset iod comparacio de gr´aficamente. tipo mostramos este los desarrollamos y respecto, resultados, 10 Al mediante maniobra. mejor mos ´utiles la para cu´al mapeos es obtener importan permite comparaci´on es el´ıpticas. y Esta com- espaciales, y dise˜no misiones se ´optimo circulares de ecuaciones maniobras Consideramo estas las con paran reci´en espaciales. obtenidas; ecuaciones misiones usando las planas en impulsivas maniobras transferencias an´alisis las par este de Desarrollamos problema circular. el´ıptica ´orbita o final una a lar et edvlpdti oprsntruhu e eut,tgte ihsome with together as- results, Words: this ten Key In meaning. throughout other. their comparison show the to than this graphs better o developed is to we maneuver designers ma- mission one pect, elliptic the where for and showing important of circular mappings is newly problem comparison useful of using the This comparisons maneuvers study made. equations, planar to are these neuvers considers developed With was It orbit equations. missions. elliptic derived space or for circular transfers final impulsive a to orbit lar RI OAFNLCRUA RELPI ORBIT ELLIPTIC OR CIRCULAR FINAL A TO ORBIT rsnao na´lssd a aiba etaseecad n ´orbita circu- una de transferencia de maniobras las an´alisis de un Presentamos ntepeetppra nlsso h rnfrmnuesfo nta circu- initial from maneuvers transfer the of analysis an paper present the In eeta ehnc ehd:aayia pc vehicles space — analytical methods: — mechanics celestial eevdMrh1 06 cetdMy1 2016 16 May accepted 2016; 14 March Received .A Sharaf A. M. .INTRODUCTION 1. , 52 ABSTRACT RESUMEN 8–9 (2016) 283–295 , 1 n .S Saad S. A. and rnm n epyis ar,Egypt. Cairo, Geophysics, and tronomy nvriy edh KSA. Jeddah, University, z v st,Bria,KSA. Buraidah, rsity, BoceadPao 96 atse l,21) In 2012). al., et Santos 1996; Prado, and (Broucke v eurdt efr h aevri simply is maneuver the perform to required 2,3 oa ytmpoie nweg that knowledge provides System Solar iefrtesaerf oaccelerate to spacecraft the for time e ine De ta.17) Voyager 1974), al. et (Dyer Pioneer rudteErh n fthe of One Earth. the around s tit h itr fteSolar the of history the into ht gsncsayt rvlfrom travel to necessary nges oegnrlcss multiple cases, general more motn osdrto.A consideration. important h okti uhsmaller much is rocket the f suirel estudiar a nlobt,teobjective the orbits, final hr n elg.Many geology. and phere ue ob oeusing done be to sumed .We h nta and initial the When t. epr el para te btain trar nes s 283 © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México .Pwrflgtfo h at’ ufc oijc h pccatit iclro circular a into spacecraft the inject to surface Earth’s the from flight Power 1. .A ,a musv eoiyiceetpae h eil nanwelpi ri of orbit elliptic new a in vehicle the places increment velocity impulsive an a, At 3. .A ,a musv eoiyiceetpae h eil na litcobto a of orbit elliptic an in vehicle the places increment velocity impulsive an b, At 2. edtesaerf notescn litclobtwt eipi tterdu of radius the at periapsis with orbit elliptical ∆ second third the a into spacecraft the send lpe ieadascae ulfracntn hutvhcefrcrl-ocrl co circle-to-circle for vehicle too thrust graphical/analytical constant Simple a (1994). for Thorne (2007). and fuel Vallado Alfano associated in and and initial found time the be when elapsed transfers can orbit coplanar for minimum-time and a obtain to study detailed 284 htis, that .Terqie musv eoiyt nettesaerf noteitreit el intermediate the into spacecraft the inject to velocity impulsive required The 3. ftohl-litcobt.Fo h nta ri,a∆ a orbit, initial the From orbits. half-elliptic two of where 05.Teb-litctase sa ria aevrta oe pccatfr spacecraft a moves (Curtis ∆ that circles less maneuver outer require orbital and situations, an inner certain the is in of transfer may radii bi-elliptic the The are ellipse 2015). transfer the of apoapse ri,wt naopi tsm point some at apoapsis an with orbit, ic 2 Since .O h nta iclrobt h eoiyis: velocity the orbit, circular initial the On 1. h ih eunebigcniee silsrtdi iue1 n ti ecie sf as described is it and 1, Figure in illustrated is considered being sequence flight The .O h nemdaeelpi ri h eoiyis velocity the orbit elliptic intermediate the On 2. h omn rnfr(omn 95 sa litclobttnett ohci both to tangent orbit elliptical an is 1925) (Hohman, transfer Hohmann The µ a stegaiainlcntn qa o398598 to equal constant gravitational the is = v r spromdijcigtesaerf notedsrdorbit. desired the into spacecraft the injecting performed is a + r b eget we , C V C V V V a b b b C V V b = = C V b V = a b = √ √ =  2.2 = µ µ  r  r a b  qain ftetase maneuver transfer the of Equations .   r 1 2 2 1   b r r   v .BSCFORMULATIONS BASIC 2. 2 2 a b wyfo h eta oy tti on,ascn ∆ second a point, this At body. central the from away 2 1 − − hnasadr omn rnfr h ielpi rnfrconsists transfer bi-elliptic The transfer. Hohmann standard a than   − − 1 2.1 1 / / 1 2 2 2 a 1 HRF&SAAD & SHARAF a 1   h ih sequence flight The . ∆   r r   a b V r 1 r r 1 r C V   / a / b a b b r r 2 2  a  b = r r b   − − + 1 v . V = 1 1 2km 62 r r µ µ sapidbotn h pccatit h rttransfer first the into spacecraft the boosting applied is / b / V b b 2 2 + 1 r − b r r =  =  a tpitband b point at b r C V + 1 µ r r  √ b √ 3 r a r b , a / b 2 2   b −  sec  . r  r 1 a b / r r r r + 1  − 2 2 a b a b . 1 −   a on a) point (at / 1 2 1 1 / r r / / 2 a b a on b) point (at 2 2 ,    V − + 1 + 1 a 1 / tpita. point at 2 r r r r , a a b . b pogee nlobt r xcl circular exactly are orbits final st eemn h minimum the determine to ls 04 ePtradLissauer, and Pater de 2014; , bta ,wt radius with b, at rbit   moeobtt nte and another to orbit one om h nldsrdobt where orbit, desired final the − − itclobtis: orbit liptical , lnrtase sgvnby given is transfer planar 1 1 apogee / / ce.Tepras and periapse The rcles. 2 2 r a n eie radius perigee and ollows: r c n perigee and v sapidto applied is r b . r (6) (3) (2) (5) (1) (4) r a b . . © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México n iclrfia riswl eobtained. be will orbits final circular and since sn qain()w get we (4) Equation Using sn hseuto n qain() eget we (5), Equation and equation this using .Terqie musv eoiyt nettesaerf notefia litclo elliptical final the into spacecraft the inject to velocity impulsive required The 4. .Terqie musv eoiyt iclrz h ri,a on ,is: c, point at orbit, the circularize to velocity impulsive required The 6. .O h nlelpia ri,a h on ,tevlct is velocity the c, point the at orbit, elliptical final the On 5. rmteaoefruain,teepesosfrtevlct eurmnsf requirements velocity the for expressions the formulations, above the From ∆ V a = 2.3 V fa C V C V eoiyrqieet o h nltase maneuver transfer final the for requirements Velocity . C V ∆ V = c V a a NLSSO RNFRMNUES285 MANEUVERS TRANSFER OF ANALYSIS b b √  = 2 1 = µ    ( + 1 r r r  2 a a c r r  − a b i.1 h ih sequence. flight The 1. Fig. r r 1   a c / r 2  a  ∆ ∆ + 2 − V V 1 2 r 1 2 1 c a / c     2 =  = − 1 | / V r r V + 1 2 C V a b fa cc =   − − b r r C V  a c V + 1 V  c a a | 1 2 , .  −   r r 2 1 1 a b /   2 r r + 1 a a on c) point (at b −   1 / r 2 r a c + 1 −  1 ) − r r a b 1 . / .  rtetocsso elliptical of cases two the or 2 btis: rbit , − 1 / 2 . (12) (11) (10) (8) (9) (7) © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México n qaetersl ozr.W get We zero. to result the equate and nlobtcudb eie.I hsrset oefidnsaet eetbihdadlisted and established be to are findings some respect, method efficient this most In the decided. transfers, be of could modes orbit various final the of (velocity), energy the 1)when (16) scicdn ihtevleof value the with coincident is 286 h rpia ersnaino qain(9 ssoni iue3. Figure in shown is (19) Equation of representation graphical The where hc switnas: written is which h oiiesg nEuto 1)i sdif used is (16) Equation in sign positive The hc switnas: written is which hti,frall for is, that aevr hsmd ftase silsrtdi iue2. Figure in illustrated orbit is parking transfer the of from mode This transfer maneuver. a of case the h on fteitreto ftetocre ∆ curves two the of intersection the of point The aigotie h eoiyrqieet o h nlelpia n iclrorbit circular and elliptical final the for requirements velocity the obtained Having ofidtemxmmvleo ∆ of value maximum the find To eut1 Result eut2 Result For α = β h w uvs∆ curves two The : aho h onso nescino h w uvs∆ curves two the of intersection of points the of Each : qain(4 ol ewitnas written be could (14) Equation , α = β i.e. , α r 1 a 2 2.3.1 2.3.2  = + 1 β r ∆ ∆ ∆ 2 c C V rmtetase oepito iw ehv h result: the have we view, of point mode transfer the From . eoiyrqieet o h nlelpia orbit elliptical final the for requirements Velocity . C V C V T V eoiyrqieet o h nlcrua orbit circular final the for requirements Velocity . ow have: we so , α T V T V T V α .OTMMMD FTRANSFER OF MODE OPTIMUM 3. b c b b  c T V e VC /V e 1 = ( = = / e 2 3.1 VC /V √ ∆  + b C V 1 α β 1 T V h aefrwhich for case The . n ∆ and  ∆ − − ∆ − β < α b b HRF&SAAD & SHARAF α α T V e 1) ti eesr odffrnit qain(9 ihrsetto respect with (19) Equation differentiate to necessary is it , directly T V α 1+ (1 s 1 = α ± s   c T V  e −

r r ∆ = α hl h eaiesg sue for used is sign negative the while , ∆ = a b α α 1 √ 1+ (1 e + 1 and T V 1 ) + VC /V 2 β 2 2 otenwobt n iclrzn ta h n fthe of end the at it circularizing and orbit, new the to 2 T V α   V β − c α b α β VC /V   e b ∆ +  ) s = ∆ + + 1 feulvle of values equal of 1 +  2 / α β r r b 2 α s V α c b  α V + n ∆ and a . α + c , α 2  0 = ,  = ( − β α α 1 2 1 β   , / β +  T V 2 1 β − T V / α β ) 2 e VC /V 2  − − c α VC /V ! 1 1 3 1 / β . . , 2 b nesc tvle of values at intersect cusacrigt Equation to according occurs 0 = b fijcigavhceit the into vehicle a injecting of n ∆ and . β > α ,te,b comparing by then, s, T V ssprtdresults. separated as e . VC /V b represents α which , (14) (19) (16) (15) (17) (13) (20) (18) α © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México α hsrtososta h aiu ∆ maximum the that shows ratio This h aiu au f∆ of value maximum The utpyn ohsdso hseuto by equation this of sides both Multiplying nEuto 2)teei nyoecag ntesg ftececet.Te,accor Then, coefficients. the of is: sign root the This in root. change real one one only only exists is there there (21) Equation In = o,ltu osdrtecs where case the consider us let Now, eut3 Result α M 15 = For : . 58176. α = β T V h eoiyrqieeto iclrobti ftemxmmvle0 value maximum the of is orbit circular a of requirement velocity the , e α VC /V = i.3 rpia ersnaino qain(19). equation of representation Graphical 3. Fig. NLSSO RNFRMNUES287 MANEUVERS TRANSFER OF ANALYSIS α b M s therefore is, 4 + 5 = T V F i.2 rnfrmd for mode Transfer 2. Fig. α ( ∆ α e ∞ → √ = ) T V smr hnhl of half than more is ( + 2(1 √ e α VC /V cos 7 o hscs ehv,fo qain(9,ta h velocity the that (19), Equation from have, we case this For . 3 − α b 15 )

| 3 α / α M 1 3 2 2 α tan 0 = − 3 / 2 9 − . α n hnsurn t eobtain we it, squaring then and 1 536258 − √ 37 α C V 3 0 = 1 = ! b . β hc stoexpensive. too is which , 15 = . . . 58176 . igt ecre’rule, Descartes’ to ding . 328at 536258 (21) (23) (22) © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México w ot r eetd ic ecnie h aei which in case the consider we since rejected, are roots two nEuto 3)teeaetrecagsi h ino h offiins hn accordi are: Then, roots coefficients. These the roots. of real sign three the exist in there changes that three implies are there (31) Equation In h ausof values The relpi rnfrmnues iue4sostesproiino h w cur two the of superposition the shows 4 Figure maneuvers. transfer elliptic or nrmn to increment 288 htis that lo h eoiyt eunfo nnt,o qiaetytevlct oma by: to given velocity is the equivalently orbit, or infinity, its from from return to velocity the Also, odtrietepito nescino h w uvs∆ curves two the of intersection of point the determine To h o-iesoa eoiyiceetfrtettlmnue a eotie yadn Eq adding by obtained be can maneuver get: total to the (25), for increment velocity non-dimensional The utpyn by Multiplying utpyn by Multiplying hn qain(6 becomes (16) Equation Then, ec,terqie qainis: equation required the Hence, qain 1)ad(6,t get to (26), and (19) Equations eut4 Result oeaietecs nwhich in case the examine To h w uvs∆ curves two The : saet infinity to escape α M α √ α n qaigbt ie ftersligeuto,w get, we equation, resulting the of sides both squaring and 1 + and n qaigbt ie ftersligeuto,w obtain we equation, resulting the of sides both squaring and α I  fEutos(2 n 3) epciey r sdt td h ueirt o superiority the study to used are respectively, (32), and (22) Equations of 1 − α ∆ 1 C V sgvnby: given is   T V 2( T V + 1 α > β + 1 α b α c e + 1 − 3 VC /V 1 − 4 2 α 4(3 α − α α − 1) 1 + ∆ let , (7 α 3.2 ∆ C V 7+4 + (7 C V 2 3 b T V −  α V − 2 = h aefrwhich for case The . n ∆ and 1 ∞ b 2 b α − / β e 6 √ 2 HRF&SAAD & SHARAF √ α ( = = = (3 √ = ∆ − 2) ∆ C V C V 2) 2 + 2) V α α V ( = 1 − √ V √ ∞ I ∞ α + + ∞ a b α 2 2 √ r 11 = 2 1 √ VC /V ε 4 + − = 2 3+ (3 + = where , √ α 4(3 )= 2) α ε 1) √ 2 2 ( √ . (5 √

9387655 b 2 α −  1 2 − 2 nesc t(19,0.534). (11.94, at intersect > α − − √ + 1 − − 0 = T V 1) 2 − √ α 1 √ 2 s 1 ε  2) . √ )+( + 2) (2 e . . 2) ,s h nyro norcs is: case our in root only the so 1, √ sara oiienme hti esta one. than less is that number positive real a is VC /V 146554; + 1  α α 2) 1 . − α 2 = − α  − 2 2 √ b √ β 0 = 1 √ α + 1 . α α n ∆ and 2 2) 3 ε  − α .  0 = etesaerf oecp oinfinity to escape to spacecraft the ke . − 0 = 2)

> 2 .  V , . ∞ e ∆ ves 0 571535; α 1 . VC /V  1 / 2 T V gt ecre’rl,this rule, Descartes’ to ng b . ti osbet equate to possible is it , α e VC /V 11 = b . ain 2)and (24) uations 37.Tefirst The 93876. n ∆ and V circular f ∞ VC /V (31) (32) (28) (24) (26) (25) (29) (33) (27) (30) b . © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México oteaoe fteobt n hnapyn h eesr mus ooti h desir the obtain to impulse necessary the applying then is and requirements orbit, energy the the of apogee the to ri,bttedffrneapoce eoas zero approaches difference the but orbit, hs egttefloigresult: following the get we Thus, .Frcrua ris nalcsstemnm for minima the cases all in orbits, circular For 1. .I h ae fmnm for minima of cases the In 2. o ceti ris h es muto nryi xeddb rnfrigdirectly transferring by expended is energy of amount least the orbits, eccentric For al lutae eut5adiscneune.Tetp ftase srepresented is transfer of type The consequences. its and 5 Result illustrates 1 Table for But, eut5 Result o hsrneof range this for α o all For : and 2 = 360485 30.568656 13 6 0.478357 6 13 β α .728590.539888 0.499627 0.475730 9 7 5 5 3 4 0.474288 0.426663 0.454433 5 3 4 9 7 5 β i.4 h uepsto ftetocre ∆ curves two the of superposition The 4. Fig. α > β ti oeecett rnfrdrcl otefia ri n iclrz it. circularize and orbit final the to directly transfer to efficient more is it , β 0 ehave we 10, = lowest h eoiyrqieetfracrua ri sawy rae hnfra el an for than greater always is orbit circular a for requirement velocity the , α > β NLSSO RNFRMNUES289 MANEUVERS TRANSFER OF ANALYSIS when EUT5ADISCONSEQUENCES ITS AND 5 RESULT M ti oeefficient: more is it , ∆ α < β T V β e VC /V o xml,for example, For . prahsifiiy hc en necp orbit. escape an means which infinity, approaches ∆ ∆ C V C V T V T V AL 1 TABLE b α < β b b e e 0 = 0 = M . . 490 395 β α cu ihrfor either occur . . T V α 0and 10 = e VC /V b n ∆ and α β or 1 = ∆ 2, = V T V ∞ VC /V e VC /V α = b dprge nalcases all In perigee. ed rmteprigorbit parking the from . b β nFgr 5. Figure in hc en that, means which , liptic (35) (34) © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México qain(6 otistoparameters, two contains (36) Equation 290 epsesdo aiaadmnm.T banteeteeprmti member parametric to extreme respect the with obtain differentiated To be minima. (36) and Equation maxima of possessed be where follows. • • • qain(6 ol ewitnas: written be could (16) Equation ieetaigEuto 3)wt epc to respect with (36) Equation Differentiating pnrahn eie,dclrt h eil ocruaieisorbit. its circularize to vehicle the decelerate perigee, reaching per Upon the raise would which velocity, incremental altitude. an orbit add apogee, the reached Having oijc h eil noa litcobtwoeaoe sgetrta h nlci final the than greater is apogee whose orbit elliptic an into vehicle the inject To R ∗ = α and ∆ C V T V R b dR c d = ∗ = β  lo ecnie h case the consider we Also, . ∆ i.5 iettase rmteprigobtt h apogee. the to orbit parking the from transfer Direct 5. Fig. C V +  T V + 1  b 2 c R 1 R  R  ∗ 3.2.1 ∗ 1 = /  2 1 ( h xrm ausof values extreme The . / 2 + ×  { R − 1+ (1 ( R R R and + 1 2 2 2 + HRF&SAAD & SHARAF 1+ (1 R  2 + R ( R ∗ R R  R ∗ R ( ∗ R ∗ ) 2 R RR R ∗ R  ∗ / ∗ 2 o fixed a For . + n qaetersl ozr.Ti ol epromdas performed be could This zero. to result the equate and 2 2 ∗ ∗ 1 > R > eget: we R / )  + ∗ RR 2 2 + ) ∗ 1 − RR } / 2 1 ∗ / 1 ( )  2 ) ∗ / 1 (  ) 2 − . R 2 , R + 1 ∗  ∗ ∆ +

( RR R 1 T V R 1 R / ( tgvsa qainin equation an gives it , ( 2 ∗ R ∗ RR . /R ∗ c 2 ∗ VC /V ) + / + ∗  2 R

+ R 1 ∗ b / 1 ∗ 2 2 / R 2 ) 2 − ) / ∗ 1+2 + (1 2 2 2  ) R 2 + 1

fti aiyo uvs let curves, of family this of s 1 / R 1 2 R ∗ × get h nlcircular final the to igee ) ∗ clrobtaltitude. orbit rcular  + 1 / 2 R ) ∗ hc should which , (37) (36) (38) © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México eut7i hw nFgr ,wt epc to respect with 6, Figure in shown is 7 Result h olwn result: following the hc stesm sEuto 2) hrfr,ti ed st h eutshown result the to us leads this Therefore, (21). Equation as same the is which qain(6 eoe none o h condition: the for unbounded becomes (46) Equation rmteaoe ti la ht h litctase a o eue o all for used be not can transfer elliptic the that, clear is it above, the From root real has which qaigtersl,w obtain we result, the Squaring eetn h hsclyipsil root impossible physically the Rejecting htis: That where and qaigt eow get we zero to Equating utemr,snetenmrtri eaie h eaiesg ntedenominator the in sign negative the negative, is R numerator the since Furthermore, ic o h aeudrinvestigation under case the for Since Then where qaigbt ie fEuto 4) egttesolution: the get we (42), Equation of sides both Squaring ∗ eut7 Result 6 Result > .Teeoe o u ae esol have: should we case, our for Therefore, 0. h obde ag o h litctase is transfer elliptic the for range forbidden The : of value limiting The : ( 1 + 3(1 R Y .Cneunl ehv h eutsonbelow. shown result the have we Consequently 9. = /R = X ) R − 1 ( = / g 2(3 2 1+ (1 = u R NLSSO RNFRMNUES291 MANEUVERS TRANSFER OF ANALYSIS − ( = R + 1 2 R R / R /R R R R 2 (3 − R ∗ ∗ 2 1+ (1 ∗ ∗ ) o litctase is transfer elliptic for ) + R R ) ( 1 + 3(1 ∗ 2 2 = = 1 (( R ( + ((1 2 > R > / R 1)( + (3 2 R R − − ∗ = R 0 = ) 3 ∗ − ( 1 + 3(1 ( 1 + 3(1 3 ∗ − − + / ) /R R R R 2 2 3 1 /R ⇒ / 15 R f , ∗ ,i olw,fo qain(6 n h nqaiy(7,that: (37), inequality the and (46) Equation from follows, it 1, ∗ / 3 2 1 + 3 . ) ) 1 ,w have we 3, ) − / R h , uf / − ) 3 < R < X 2) 1 2) /R /R = 2 R 15 / 2(3 1 − 2 2 1 = = = /R R / 1 + 3 1 + 3 R ) ) / < 2 − 2 9 3 ± − gh, 2 R Y, n − R / n 26 6 − 2 1 R 2(3 2(3 9 ( R / 2 − + R /R /R . R 2 ∗ 9 /R R 3 3 + 1 + 2 R ( 0 = 1 R / − − − R M 2 2 + ) ∗ − 1 + 2 2 0 = 9 1 2 + / 15 = / /R /R 2 1) RR 2 R , 1+2 + (1 R. > R ∗ ) ) R < 1 1 1 ∗ ∗ . / 2 / / ⇒ 58176. ) ) 0 2 2 2 . /R . − 1+2 + (1 . . (3 R R 1 R ∗ / ∗ R 2 ) 3 1)( + , − / R 2 nterne(1 range the in ∗ . R ) below. o ∗ R 3 , / sncsayi re that order in necessary is 2 − o )=0 = 9) . , , ) hsw get we thus 9), (52) (50) (51) (48) (49) (47) (39) (40) (45) (46) (41) (42) (44) (43) © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México 292 ueial ofind to numerically where eut9 ehv orinequalities: four have we 9, Result range. codn oRsl ,w aetersl hw below. shown result the have we 7, Result to According a iclrtase sspro oteelpi rnfrfrall for transfer elliptic the to superior is transfer Circular (a) yia rpia lutaino eut8i hw nFgr 7. Figure of in shown trajectories is in 8 example, Result for of suitable, illustration is graphical range Typical This transfer. elliptic the si h bv ae ehv h eutsonnext. shown result the have we case, above the in As b litctase sspro otecrua rnfrfrall for transfer circular the to superior is transfer Elliptic (b) yia rpia lutain fRsl 0aesoni iue n 10. and 9 Figures in shown are 10 Result of illustrations graphical Typical yia rpia lutaino eut9i hw nFgr 8. Figure in shown is 9 Result of illustration graphical Typical transfer. elliptic the to nterange the In ausof values rmteaoeaayi,teeeittociia ais namely, ratios, critical two exist there analysis, above the From eut10 Result h range: The eut8 Result h range: The eut9 Result h range: The R I R = ∗ α R > o all For : o all For : R I o l eogn oterange the to belonging R all For : I 11 = o ie au of value given a For . 1 9 R ≤ I < R < ≤ R R . ≤ 365 nwa olw esaldsussproiyo iclro ellipt or circular of superiority discuss shall we follows what In 938655. R 0 R R . ≤ R ≤ eogn oterne1 range the to belonging eogn oterne9 range the to belonging ≤ R 9 R 3.2.2 M R I h w uvs∆ curves two the , M 1 oprsnbtenelpi n iclrtransfers circular and elliptic between Comparison . < R < i.6 eut7wt epc to respect with 7 Result 6. Fig. ;9 9; R ∗ R > ≤ ∆ HRF&SAAD & SHARAF C V R T V ecnsleteequation the solve can we , ≤ T V R b < R < c ≤ R I − e I ≤ VC /V R ; ∆ R R C V ≤ T V I ,adfrall for and 9, ≤ b R ≤ b n ∆ and e I R n o all for and , R 0 = M R < R R > R ≤ n o all for and , R T V M 0 0 R R c , where , ; VC /V ∗ M R . R > R ∗ and R ihst eu n Mars. and Venus to flights R > b ∗ R nesc at intersect R > ∗ R R M iclrtase sspro to superior is transfer circular , R > 0 I , ntrso hs aisand ratios these of terms In . stero fEuto (54). Equation of root the is iclrtase ssuperior is transfer circular , then: , R ctasesi each in transfers ic = R 0 sy o all for (say) (54) (53) © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México 56 er Grayn 96.Hwvr aaoi ri eursa init an Eart the requires the in orbit from time parabolic the a flight corresponding However, the of i.e. 1996). while duration time, (Gurzadyan, years, the years in 19.33 45.60 example, gain t is for a known trajectory, is gives As, parabolic It it namely, operations. smaller. advantages, rescue significantly space definite and possesses stations orbits space supplying for especially o rfrbe ic ene oktta smr oefladsed oeenergy. more spends and o powerful more point is energetic that the rocket from a therefore, need and, we orbit since preferable, elliptic not corresponding the for that st n h iiu ottaetr,i em ffe osmd otase spacecra a transfer to consumed, fuel of terms in (20 trajectory, cost Prado these minimum by of the developed One find satellites to artificial transfer. is for parabolic transfer containing planar maneuvers bi-parabolic successful propose to authors tti on,i hudb oe httems rtclfco fatransfer a of factor critical most the that noted be should it point, this At h piaiyo aaoi riswt epc otedrto fflight, of duration the to respect with orbits parabolic of optimality The i.8 rpia lutain fRsl for 9 Result of illustrations Graphical 8. Fig. for 8 Result of illustrations Graphical 7. Fig. NLSSO RNFRMNUES293 MANEUVERS TRANSFER OF ANALYSIS R R ∗ ∗ 3.Teojcieo hsmaneuver this of objective The 03). aeo h omn rjcoyis trajectory Hohmann the of case 30. = 10. = smnindaoe epe some tempted above, mentioned as a velocity ial oPuo ntecs fthe of case the in Pluto, to h aevri t uaintime, duration its is maneuver uaino h ih ilbe will flight the of duration a h oini parabolic in motion the hat iw aaoi risare orbits parabolic view, f rpsl steelliptic- the is proposals tfo akn orbit parking a from ft √ ie rae than greater times 2 © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México hnaohraeson nti epc,w eeoe hs oprsn hogottnr ten throughout comparisons meaning. these their developed showi cir show we mappings of respect, to transf Useful Comparisons this graphs In the some made. developed. shown. are of is are designers, orbit analysis another mission elliptic than an for or regard, important circular this are final which In a to equations. orbit derived circular newly with maneuvers 294 oecp rmtemi lntt nepaeaysaewt aiu eoiya higher velocity maximum a with probl to the space or to interplanetary planet, applied to is this planet maneuver of main the satellite the in satellite from natural natural escape a a to using around of orbit idea an the Moreover, to planet a around h uhr r rtflt h eee o i/e aubecmet htimprov that comments valuable his/her for referee the to grateful are authors The h rbe fipliemnuesfrsaemsin sadesd ycnieigteca the considering by addressed, is missions space for maneuvers impulsive of problem The i.1.Gahclilsrtoso eut1 for 10 Result of illustrations Graphical 10. Fig. i.9 rpia lutain fRsl 0for 10 Result of illustrations Graphical 9. Fig. .CONCLUSIONS 4. HRF&SAAD & SHARAF R R ∗ ∗ 0and 20 = 0and 60 = R R gweeoemnue sbetter is maneuver one where ng 0 0 14 = 12 = ua n litcmaneuvers, elliptic and cular infinity. t dteoiia manuscript. original the ed . mo asn spacecraft a causing of em rmnuesfo initial from maneuvers er . 6945. 7972. ri rudteplanet. the around orbit sls oehrwith together esults, eo planar of se © Copyright 2016: Instituto de Astronomía, Universidad Nacional Autónoma de México ae .J 94 nrdcint pc lgt (Englewood Flight, Space to Introduction 1994, J. flights, F. Hale, interplanetary of Theory 1996, A. Jack- G. & Jr., Gurzadyan, J.R. Cowley, R., R. Nunamaker, W., J. (3rd Dyer, Sciences, Planetary 2015, J. J. Lissauer, & I. Pater, de Stu- Engineering 19(2), for Mechanics JGuC, Orbital 1996, 2014, D. A. H. F.B. Curtis, 35 A. 42, Prado, Sci., Astronaut. & J. A. 1994, R. D. J. Broucke, Thorne, & S. Alfano, .A hrf eateto srnm,Fclyo cec,Kn buai Universit Abdulaziz King Science, of Faculty Astronomy, of Department Sharaf: A. M. .S ad eateto srnm,Ntoa eerhIsiueo Astro of Institute Research National Astronomy, of Department Saad: S. A. lff,NJ:Petc Hall) Prentice N.J.: Cliffs, publishers) Breach (Netherlands: 710 11, JSpRo, 1974, W. R. son, CUP) MA: Cambridge ed.; OXF) UK: ed.; (3rd dents 274 [email protected]). ([email protected]). NLSSO RNFRMNUES295 MANEUVERS TRANSFER OF ANALYSIS REFERENCES ald,D .20,Fnaetl fAtoyaisand Astrodynamics of Fundamentals 2007, 2012, A. G. D. Colasurdo, Vallado, & A., Eng., A.F.B. & Prado, S., Sci. P. Mech. D. Soc. Santos, Braz. J. 2003, 676 A. 348, A.F.B. Natur, Prado, 1990, B. W., McKinnon, (London: Voyager, of Journeys The 159 1990, 51, P, Robin. & Kerrod, EM 1990, A. B. Ivanov, Bodies, Heavenly of Attainability The 1960, W. Hohmann, plctos 3de. eln:Springer) Berling: ed.; (3rd Applications, 16 2012, Eng., Probl. 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