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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