Out-of-plane orbital maneuvers using swing-bys with the Moon

Transcription

1 Jurnal f Physcs: Cnference Seres PAPER OPEN ACCESS Out-f-plane rbtal maneuvers usng swng-bys wth the Mn Related cntent - Pwered Swng-By Maneuvers arund the Mn A F Slva, A F B A Prad and O C Wnter cte ths artcle: J B S Net et al 5 J. Phys.: Cnf. Ser ew the artcle nlne fr updates and enhancements. hs cntent was dwnladed frm IP address n 9/8/8 at :3

2 XII Clóqu Brasler de Dnâmca Orbtal CBDO Jurnal f Physcs: Cnference Seres 64 (5) 4 IOP Publshng d:.88/ /64//4 Out-f-plane rbtal maneuvers usng swng-bys wth the Mn J B S Net, A F B A Prad and J K S Frmga Natnal Insttute fr Space Research INPE, Av ds Astrnautas 758, SJC-SP, Brazl. E-mal: net.jbs9@gmal.cm.br Abstract. hs paper has the gal f shwng sme cases f plane change maneuvers usng a swng-by wth the Mn t decrease the magntude f the mpulses used, when cmpared t a classcal Hhmann maneuver. he analytcal mdel s based n the "patched-cncs" apprach, where a seres f "tw-bdy" prblems s cnsdered t buld the whle maneuver. A study f the effects f the sem-majr axes f the transfer rbts were made, t cmplete sme prevus studes made n the lterature. he results shw that, fr sme fnal nclnatns, the use f the swng-by n the Mn s really advantageus.. Intrductn Durng the desgn phase f artfcal satellte mssns, ne the mst mprtant factrs s the cst. Optmzatn methds are usually appled t search fr lwer cst slutns frm the cnceptn t the end f the useful lfe f the equpment. Mre recently, studes are als made cnsderng the later dspsal f the spacecraft. Lkng fr slutns t reduce the magntude f the mpulses requred n rbtal maneuvers, several methds have been develped and used, such as the classcal transfers f Hhmann and B-ellptcal [], bth f them cnsderng cplanar rbts. Nw, f the bjectve s t make a plane change, t s necessary t apply sme knd f prpulsn wth a cmpnent perpendcular t the rbtal plane. hs type f maneuver are usually very expensve n terms f fuel cnsumptn and dfferent ptns have been cnsdered t reduce the fuel cnsumptn. An alternatve slutn, already cnsdered n the lterature, s t use a gravty asssted maneuver (swng-by) wth the Mn t make the ut-f-plane part f the maneuver, as shwn n []. he swng-by s a type f maneuver that des nt use prpulsn based n engnes, but takes advantage f the gravtatnal nfluence f a celestal bdy when the spacecraft passes thrugh ts zne f nfluence [3-6]. he smaller bdy that s passng by the larger ne s accelerated r braked by the gravty feld f the celestal bdy and s ts trajectry s mdfed. Successful mssns, such as yager and [7], shwed the effcency f the swng-by n savng fuel. An example f the use f ths maneuver t change the rbtal nclnatn f a spacecraft s the Ulysses mssn, presented n [8], where the spacecraft was placed n a near plar rbt arund the Sun, n rder t bserve the ples f the Sun. hs paper ams t shw that, fr sme desred fnal nclnatns, a swng-by wth the Mn (n the Earth-Mn system) can be mre ecnmcal then usng plane change classcal maneuvers. he Cntent frm ths wrk may be used under the terms f the Creatve Cmmns Attrbutn 3. lcence. Any further dstrbutn f ths wrk must mantan attrbutn t the authr(s) and the ttle f the wrk, jurnal ctatn and DOI. Publshed under lcence by IOP Publshng Ltd

3 XII Clóqu Brasler de Dnâmca Orbtal CBDO Jurnal f Physcs: Cnference Seres 64 (5) 4 IOP Publshng d:.88/ /64//4 cntrbutn f the present paper s t analyze the nfluence f the sem-majr axs f the transfer rbt f the spacecraft, whch was a pnt nt cnsdered n prevus studes [].. Methdlgy he cmplete maneuver s dvded n three parts. In the frst phase an mpulse s used t put the satellte n a clse apprach trajectry wth the mn. In the secnd stage the three-dmensnal swngby s made and, n the last phase, tw mpulses are appled t take the satellte back t an rbt wth the same sem-majr axs and eccentrcty f the ntal rbt, but nw wth the fnal desred nclnatn. he analytcal mdel used s presented n [9], and t cnsders a system cnsstng f three bdes: a prmary (M, the Earth), a secndary (M, the Mn) and a thrd bdy f neglgble mass wth ts mtn gverned by the ther tw prevus cted bdes (M 3, the artfcal satellte). he three phases f the prblem s dmnated by the celestal mechancs f the tw bdes, s t s pssble t use the patched-cncs apprach. Intally, t s cnsdered that the satellte s n an rbt arund the Earth that s cplanar wth the rbt f the Mn. Fr the calculatns, t was assumed a canncal system f unts, where: () the gravtatnal cnstant s equal t ne; () the unt f dstances s the Earth-Mn dstance ( km); () the angular velcty f the prmares s equal t ne; (v) the mass f the mn s gven by µ L =m /(m +m ) (where m and m are the masses f the Earth and Mn, respectvely), and the mass f the Earth (µ ) s gven by µ = - µ L, s the ttal mass f the system s equal t ne; (v) the tme unt s set such that the perd f the prmares s π... Intal mpulse he mpulse used n the frst step t generate the transfer rbt (Δ ) and the eccentrcty f ths rbt (e ) are gven by: a e a a e e a e a () a where a s the sem-majr axs f the transfer rbt and ts mnmum value s gven by a = ( + a )/, where a s the sem-majr axs f the ntal rbt f the satellte arund the Earth and e s the eccentrcty f ths rbt... he Swng-by maneuver In the secnd step the calculatn f the swng-by starts when the satellte appraches the Mn and ts mtn s gverned by ths bdy. Fgure shws the gemetry f the swng-by, where: r ap s the dstance f the clsest apprach f the satellte wth the mn (perlune) and p s the velcty at ths pnt; the angles α and β specfy the perapss pstn n the three-dmensnal space and λ s the angle frmed between the vectr and the hrzntal plane passng by the perlune. Fgure shws the sum f the velcty vectrs durng the swng-by, where: and are the velcty vectrs wth respect t the mn befre and after the swng-by, respectvely, and the angle between them s δ, where δ s the s called deflectn angle; and are the satellte velcty vectrs befre and after the swng-by, respectvely, wth respect t the Earth; s the velcty vectr f the Mn wth respect t the Earth; γ s called flght-path-angle, whch s the angle between and ; ϕ s the angle between and. p ()

4 XII Clóqu Brasler de Dnâmca Orbtal CBDO Jurnal f Physcs: Cnference Seres 64 (5) 4 IOP Publshng d:.88/ /64//4 Fgure. Gemetry f the three-dmensnal Swngby. Fgure. he sum f vectrs durng the swng-by. Intally, t s calculated and the true anmaly (θ): (3) a cs a e e It s nw pssble t calculate γ and the magntude f the velcty f the satellte wth respect t the mn ( ): (4) tan e sn e cs cs (5) (6) Here ϕ and δ are calculated by: cs (7) sn r ap L (8) he apprach angle α and the angle λ are gven belw, respectvely, by: 3

5 XII Clóqu Brasler de Dnâmca Orbtal CBDO Jurnal f Physcs: Cnference Seres 64 (5) 4 IOP Publshng d:.88/ /64//4 he vectrs and are gven by [9]: (9) sn tan tan () sn cs cs,cs sn,sn cs ( sn sn cs cs sn, sn sn sn cs cs,cs sn ),, sn cs cs,cs sn,sn cs cs,cs sn cs ( sn sn cs cs sn, sn sn sn,, and allws the calculatn f the mpulse generated by the swng-by ( ), gven by [9]: sn cs cs,cs sn, sn sn he calculatn f the energy change due t the swng-by s gven by: () () (3) (4) E cs sn sn (5) he angular mmentum vectrs befre ( C ) and after the swng-by ( C ) are gven by [9]: C R d L cs sn sn cs sn sn sn ) C R dl,sn sn cs cs sn, (, sn sn cs cs sn, ( cs cs cs (6) cs cs cs (7) cs sn sn cs sn sn sn ) where R d L,, s the pstn vectr f the Mn wth respect t the Earth, and d L s the Earth- Mn dstance. hen, the magntude f the angular mmentum change (ΔC) s gven by [9]: C C d C sn d L cs sn sn sn,sn, cs sn (8) 4

6 XII Clóqu Brasler de Dnâmca Orbtal CBDO Jurnal f Physcs: Cnference Seres 64 (5) 4 IOP Publshng d:.88/ /64//4 [9]: After the swng-by maneuver s cmpleted t s pssble t calculate the fnal nclnatn ( ) by Cs C C Z sn sn cs cs sn cs cs cs cs sn sn cs sn sn sn (9).3. Fnal mpulses In the thrd step, after the swng-by s cmpleted, tw mpulses are used t return the satellte t ts ntal rbt (wth a and e ), but keepng the mdfed nclnatn. Frst, t s calculated the sem-majr axs f the satellte after the swng-by (a ): a () Next, t s calculated the sem-majr axs f the transfer rbt (a 3 ) and the frst mpulse (Δ ): a a 3 e a e a () () a3 e a a e When the satellte reaches the pergee f the transfer rbt ne last mpulse s used t crcularze the rbt (Δ 3 ): 3 a (3) 3 a e a a e After the cmplete maneuver, all mpulses are added: (4) t 3.4. Classcal Maneuver test the swng-by as an alternatve t the mre cnventnal types f plane change maneuver, ths technque s cmpared t a mn-mpulsve maneuver, gven by []: m a (5) sn e a 5

7 XII Clóqu Brasler de Dnâmca Orbtal CBDO Jurnal f Physcs: Cnference Seres 64 (5) 4 IOP Publshng d:.88/ /64//4 3. Results Accrdng t the ntal bjectves f the present paper and usng the prpsed technque, tw smulatns are made fr each value f the sem-majr axs f the transfer rbt: a (mnmum semmajr axs) and 5a. Fr bth smulatns, the ntal eccentrcty was e =, but n the frst smulatn (smulatn ) the ntal sem-majr axs was a =.7 and the perapss dstance r ap =.46 []. In the secnd smulatn (smulatn ) the ntal value f the sem-majr axs was a =.69 and fr the perapss dstance t was used agan r ap =.46. he ntal sem-majr axs and the perapss dstance are shwn n canncal unts (the Earth-Mn dstance). he smulatns are made fr β rangng frm t π and the results are shwn n fgures 3 t, dsplayng the fnal nclnatn ( ) n the vertcal axs as a functn f β n the hrzntal axs, bth n radans. Sme graphcs shw Δ t - Δ m, the dfference between the ttal mpulse used by the maneuver that ncluded the swng-by and the magntude f the mn-mpulsve maneuver, gven n canncal unts f velcty ( canncal unt f velcty s apprxmately.3 km/s), als as a functn f β n the hrzntal axs. Fgure 3. vs. β, fr smulatn cnsderng the sem-majr axs a. Fgure 4. Δ t - Δ m vs. β, fr smulatn cnsderng the sem-majr axs a. Fgure 5. vs. β, fr smulatn cnsderng the sem-majr axs 5a. Fgure 6. Δ t - Δ m vs. β, fr smulatn cnsderng the sem-majr axs 5a he results fr the slutn, cnsderng the sem-majr axs a, shwn n fgures 3 and 4, shw that the use f the swng-by s justfed nly fr the fnal nclnatns between.9 rad and.95 rad, whch crrespnds t β between.63 rad and.94 rad. Insde ths range the savngs n the magntude f the mpulses reaches.5 canncal unts. Hwever, the results gven by Slutn cnsderng the sem-majr axs 5a, as shwn n fgures 5 and 6, ndcates that the use f the swng-s justfed by the frst famly f slutns fr the fnal nclnatns between.9 rad and.9 rad. Fr values f β between.8 rad and.99 rad, the savngs n the magntude f the mpulse reaches values up t.86. 6

8 XII Clóqu Brasler de Dnâmca Orbtal CBDO Jurnal f Physcs: Cnference Seres 64 (5) 4 IOP Publshng d:.88/ /64//4 Nw, fr the secnd famly f slutns, the use f the swng-by s nly justfed fr values f β between.44 rad and.99 rad, wth fnal nclnatns between.9 rad and.3 rad, wth the savngs n the magntude f the mpulses up t. canncal unts. Fgure 7. vs. β, fr smulatn cnsderng the sem-majr axs a. Fgure 8. Δ t - Δ m vs. β, fr smulatn cnsderng the sem-majr axs a. Fgure 9. by β, smulatn fr 5a. Fgure. Δ t - Δ m by β, smulatn fr 5a Fgures 7 and 8 shw the results fr the slutn cnsderng the sem-majr axs as a. In ths case the use f the swng-by s nt justfed. he results f the slutn fr a sem-majr axs f 5a are shwn n fgures 9 and. It ndcates that the use f the swng-by s nly justfed fr the frst famly f slutns and fr the fnal nclnatns between.93 rad and.5 rad, whch crrespnds t β between.86 rad and.94 rad. he ecnmy n the magntude f the mpulse can reach up t.98 canncal unts. In general, the results are accrdng t the expected fr ths type f maneuver. here s a cst t send the spacecraft t the Mn and back, even n planar maneuvers. take advantage f ths maneuver, t s necessary t have savngs n the ut-f-plane maneuver that cmpensates ths extra cnsumptn. hs s the physcal reasns f the lmtatns n the regn f advantages f the prpsed technque. It can be appled when n the stuatns where the nclnatn change s larger, s the savngs n ths part f the maneuver cmpensates the extra csts nvlved. Althugh expected, the results shwn n the present paper quantfes the exact lcatns f the regns where the maneuver has gans ver the standard nes and hw much are the gans. In partcular, the present paper extends studes made befre by cnsderng the effects f the sem-majr f the transfer rbt n the savngs btaned. hs s a free mprtant parameter when desgnng a mssn. 7

9 XII Clóqu Brasler de Dnâmca Orbtal CBDO Jurnal f Physcs: Cnference Seres 64 (5) 4 IOP Publshng d:.88/ /64//4 4. Cnclusn After analyzng the results, t s cncluded that, fr sme fnal nclnatns desred fr the rbts, the use f the swng-by maneuver as an alternatve t the classcal maneuvers can be a very gd ptn, generatng savngs even greater than canncal unts f velcty. Regardng the analyss f the nfluence f the sem-majr axs f the transfer rbt, the result shw that larger sem-majr axs allws the exstence f new famles f slutns, wth hgher fnal nclnatns. hs type f analyss can help mssn desgners t desgn mssns that can beneft frm the technque shwn here. 5. References [] Prad A F B A and Rs-Net A 993 Um estud bblgráfc sbre prblema de transferêncas de órbtas Revsta braslera de cêncas mecâncas 5 pp [] rre K S and Prad A F B A 6 Changng nclnatn f earth satelltes usng the gravty f the mn Mat. Prb. Eng. 6 pp -3 [3] Gmes M and Prad A F B A A study f the mpact f the ntal energy n a clse apprach f a clud f partcles WSEAS rans. Mat. 9 pp 8-8 [4] Prad A F B A and Brucke R A 995 Effects f atmspherc drag n swng-by trajectry Acta Astr. 36 pp 85-9 [5] Gmes M and Prad A F B A 8 Swng-by maneuvers fr a clud f partcles wth planets f the slar system WSEAS rans. Ap. her. Mec. 3 pp [6] Prad A F B A and Brucke R A 995 A classfcatn f swng-by trajectres usng the mn App. Mec. Rev. 48 pp 38-4 [7] Khlhase C E and Penz P A 977 yager mssn descrptn Sp. Sc. Rev. pp77- [8] Wenzel K P, Marsden R G, Page D E and Smth E J 99 he ULYSSES mssn Astr. and Astrp. Suppl. 9 pp 7-9 [9] Prad A F B A An analytcal descrptn f the clse apprach maneuver n three dmensns Prc. Int. Astr. Cngress (R de Janer) [] Ry A E 988 Orbtal mtn (Brstl) 8