Refinement of Parameters and Motion Forecast for High-Orbit Objects with a Big Area to Mass Ratio

Transcription

1 Refeet of Paaete ad Moto Foeat fo Hgh-Obt Objet wth a Bg Aea to Ma Rato Steayat V.A., Agaov V.M., Khutoovky Z.N. Itoduto The uoe of th eot to develo the ethod ad algoth fo deteato of obt of objet wth a bg aea to a ato. Objet wth a bg aea to a ato (AMR> 2 /kg) ae uually loated o hgh Eath obt. The obt geatly evolve ude the fluee of ola adato. The dtubg fluee of ola adato odeably exeed dtubae aued by the eetty of the Eath gavty feld ad the fluee of the Moo ad Su gavtato. Beaue of the lak of kowledge o the oetato ad fo of objet, the aeleato odued by the ola adato eue hage geatly t value ad deto. Th ake t oble to detee the obt fo log teval of te ad eult a odeable eae of the foeatg eo. A eety ae to lude the ube of the efed aaete the veto haatezg the ola adato value ad the deto of t eue, alog wth the objet tate veto. A aoah odeed fo the deteato of the obt ad oto foeatg of objet wth a bg AMR. A lted a of the eaueet teval oe ae uffet fo the deteato of a eeay et of aaete. The deto of the veto of aeleato aued by the ola adato eue a vay odeably, but a lted age, devate fo the deto Su-objet. A algoth ooed fo a jot deteato of the objet obtal aaete ad the veto of addtoal aeleato, aued by the ola adato eue wth a aout of a o foato. A algoth debed fo foato of a weght atx of a o veto takg to aout the age of t vaato value ad deto. The queto ae dued of athg ewly geeated obt of objet wth a bg AMR to the obtal atalog. A athg algoth ooed takg to aout the eo deteato of objet tate veto ad the dety of ouato of a odeed at of obtal aaete ae wth othe objet. A exale gve of deteato ad foeatg of a eal SO obt ad etato of eo foeatg o the ba of eal eaueet obtaed by goud falte.. Jot deteato of a SO obtal aaete ad aaete haatezg the fluee of the ola adato o the SO oto. Geeal foulato of the oble The et of efed aaete Q{ q, P } of oto of a objet wth a bg AMR lude: x SO obtal eleetq ooet of the veto P of addtoal aaete haatezg the fluee of the ola adato eue o the SO oto The deteato of the veto of efed aaeteq by the eaueet data ψ, =, 2,..., N aed out by a axu-lkelhood ethod by ea of the futoal zato whee Φ N T ( Q) ξ ( QWξ ) ( Q) = = ()

2 ξ ( ) Q = ψ ψ ( Q ), =,2,..., N ae deae of the eaued ad the alulated value of all tye of eaueet; ψ eaueet; ψ t alulated aalog; W = eaueet weght atx; K K ovaae atx of the eaueet eo Meaueet ψ odeed a a veto, the ooet of whh ae, geeally eakg, teoeted by a oelato deedee. Deo of veto ψ a dffe. I a atula ae, a be equal to (ala eaueet). It uoed that veto eaueet 2 { ψ, ψ,..., ψ } ψ oog the futoal () ae ot deedet o oe aothe. At the ae j k te, a eee of oelato aued betwee ooet ψ ad ψ of veto ψ. Thee oelato ad the auay of eaueet ψ ae haatezed by the ovaae eo atx K. I th eot the followg odeed a oble eaueet tye: Dtae Radal eed Agula eaueet of ght aeo ad delato Po data o the SO tate veto Po data o the obtal eleet Po data o the ola adato eue veto A algoth fo the alulato of thee eaueet alulated value efed eto 4.. Fo zato of futoal Φ a tadad teato Newto ethod ued. At eah k -th tage a yte of equato olved whh uually alled a yte of oal equato A Q= b (2) whee atx A ad veto b of the yte of equato (2) ae deteed by the foula N T ξ ξ W = Q Q N T ξ Wξ = Q A= b= ad the aaete Q ae efed: Q = Q + Q The oe of foato of a yte of oal equato volve auulato of the atx, ght at ad futoal ooto to the eal oeo of og data. To ake uh auulato oble t eeay to have the followg foato fo eah ueve eaueet Deo of veto of the eaued aaete (t oble that = ). The deo deed o the tye of eaueet ad deteed afte t eteval. Meaued veto ψ (ala a atula ae) (3)

3 Weght atx (weght a atula ae) haatezg the eaueet eo. The atx eleted togethe wth the eaued aaete. ψ Calulated veto ψ ad a atx of t devatve by the efed aaete veto Q Fo eah tye of eaueet a eal odule develoed ovdg alulato of the veto of t alulated value ψ ad a atx of devate by the uet veto of the SO keat ψ aaete. 2. SO oto odel ad equato SO oto odel ode Eath gavtatoal fluee eeeted by a u of etal ad eet at of the gavty feld otetal Moo ad Su gavtatoal fluee Sola adato eue fluee I the etal efeee yte EJ2 oeted wth the equato ad Eath equox of eoh J2 the SO oto equato ae eeeted the fo = v (4) EJ 2 v = µ E + A 3 fgav ( ) + f () + f () whee SV oodate the efeee yte EJ2, = A SV oodate the efeee yte (Geewh oodate yte), EJ 2 oo = AEJ 2oo Moo oodate the efeee yte, v SV eed veto the efeee yte EJ2, v = A v SV eed veto the efeee yte, EJ 2 µ 3 Eath gavtato otat, A atx of tafe fo EJ2 to EJ 2 EJ 2 A = A EJ 2 A EJ 2 T atx of tafe fo to EJ2 taoed elato to atx f gf (gavty feld) futo fo alulato of the SV dtubg aeleato aued by eetty of the gavtato feld f (ot a) futo fo alulato of the SV dtubg aeleato aued by the fluee of the gavtato feld of the Su, Moo, ad laet f (ola eue) futo fo alulato of the SV dtubg aeleato aued by the ola adato eue. The ft thee u the ght at of the equato (4) ae odtoal o the fluee by the Eath, Moo, ad Su gavtato feld o the SV oto. Foula fo alulato of thee dtubae ae well debed lteatue (fo exale []). A detaled deto of a atula

4 alato of algoth beyod the oe of th eot. The ext eto debe a odel of the ola adato fluee o the SO oto. 3. Model of dtubg aeleato aued by the ola adato fluee The odel ued fo the alulato of the dtubg aeleato aued by the ola adato eue fo objet wth a bg AMR baed o the followg auto: The dtubg aeleato deto a oe lt dffe fo the deto Su-SO. O dffeet tajetoy eto the dtubg aeleato value ad deto a hage dotuouly. The dtubg aeleato value ad deto ae debed by a 3-deo veto of aaete whh eeve a otat value wth a eto. O the eleted teval of te the SO tajetoy otuou ad debed by a yte of dffeetal equato wth oe odel ad oe SO tate veto elated to a efed oet of te. I Fgue, a a exale, a eto of the SO tajetoy heatally how oveg the, X, v deteg the SO tajetoy elated to the teval of te [ t4 t ]. The tate veto { } ed of that teval t. The teval [ t4, t ] dvded to fou eto [ t4, t 3], [ t3, t 2], [ 2, ] [ t, t ] ( t 4 < t 3 < t 2 < t < t ). The aaete veto fluee o the SO oto o the eto [, ] t t t t, detee the ola adato eue Fgue. Daga fo akg a odel of the ola adato eue odeato The foe F of the ola adato eue at the dtae AU 5 9 (Eath obtal adu), o a efetly efletg ufae wth the aea of 2 loated othogoally to the deto towad Su, aout aoxately to N. Fo a efetly abobg ufae th foe aoxately equal to N. I a geeal ae F a be exeed by the foula [2] F N whee efleto dex deedg o the ufae oete: = oeod to olete adato aboto, = 2 oeod a olete adato efleto. Let A be the aea of the SO deto elato to the deto towad Su, ad t a. I the ae whe the ufae oletely abob the ola adato flux, the dtubg aeleato 6 A f deted alog the veto Su-SO ad t value aout to 4.5 / 2. A fo a ea Eath SO the deto Su-SO a lttle dffeet fo the deto Su-Eath, the dtubg aeleato veto a be alulated by the foula

5 whee f A = Su geoet oto the efeee yte EJ2. A I the eal tuato the SO ufae doe ot oletely abob lght. The deto of the dtubg aeleato veto f a be dffeet fo the deto of the Su adato flux ad t odulu a be oe tha the odulu of veto f. I th ae the dtubg aeleato veto a be alulated by the foula: whee ( ) f = 4.5 b + b + b = 4.5 B 6 6 e e {,, } e T = a veto of aaete deteg the ola adato eue fluee e a ooet of th veto lyg the elt lae ad othogoal to the deto towad Su a ooet of veto deted othogoally to the elt lae a ooet of veto deted fo the Su towad Eath be Matx B b b b b ooed of veto b e, b, b whh ae alulated by the foula: = a blok veto deted fo Su towad Eath = v v a blok veto deted othogoally to the elt lae be = b b a blok veto lyg the elt lae ad othogoal to the deto towad the Su. Veto b e, b, b eeet thee utually othogoal deto. Pojeto of veto o thee deto wth a auay of a ultle, oly wth the ooet of the dtubg aeleato f. If a objet oletely abob adato, the ooet e ad, deted othogoally to the lght flux, ae equal to zeo, ad the ooet equal to the aea to a ato:,, A = The atual value e,, ae oe o le dffeet fo thee value. (6) (5) 4. Algoth fo the jot deteato of a SO obtal aaete ad aaete haatezg the fluee of the ola adato o the SO oto I eto thee a uay deto of a tadad oedue fo the efeet of a veto of the etated aaete Q by the eaueet data. Foula (2) ad (3) ovde the

6 zato of the futoal () by the teato ethod. Although, fo the oluto of a atula oble t eeay to develo a algoth fo the alulato of the eaueet alulated value ψ, =, 2,..., N by the efed veto of the efed aaete Q, ad a algoth fo ψ the alulato of a atx of atal devatve fo the alulated aaete by the efed Q aaeteq. Thee algoth ae debed below. 4.. Calulato of a eaued aaete of a atx of t atal devatve by the efed veto of the etated aaete t t odeed dvded to ubteval by the oet of te Q q,,...,, A oeo teval [, ] > > >. The total of the efed aaete defed by the veto { } t t... t whee q the SO obtal eleet attbuted to the oet of te t, deteg the lght eue fluee the teval [, ] ae the aaete t t. I the oet of te t, t 2,..., t N the eaueet ψ, ψ2,..., ψ N ae aed out. I the oet of te t k loated wth the teval[ t, t ] the deedee of the alulated value ψ k o the et of the efed aaete Q deteed by the foula ( ( ( ( )))) ( ( t k )) ( ( )) = t k X t t ψ X, q,,,..., ψ X,,,, X... X,, X q... (7) Fo th eeetato t obvou that the alulated value ψ deed o the obtal eleet q, ad the et of aaete,,..., haatezg the ola adato eue all teval afte the oet t k. Exeo of the tye X( t, j, X j ) deote the deedee of the SO tate veto X, attbuted to the oet of te t o the value of the veto j the teval tj, tj ad the tate veto X j attbuted to th teval begg te t j. The alulated ealzato of th deedee aed out by a ueal tegato of the oto equato (4) wth the otat value of the veto. j The eeetato X( q ) exee the deedee of the SO tate veto the oet of te t o the obtal eleet q. Th veto alulated by the foula of Kele udtubed oto. Patal devatve fo the eaued futo ψ by the etated aaete ae alulated by the foula: ψ ψ = Q Q

7 4.2. Calulato of devatve fo the SO uet tate veto by the veto of etated aaete The atx of devatve X fo the SO tate veto X the uet oet of te t by the Q efed veto of the etated aaete Q ot of eveal blok a how Fgue 2. If ψ haged the oet of te t k -th tajetoy ubteval, the k+ =,..., = t t t < k+ k (8) Fgue 2. Matx of devatve X Q Table gve a dea of the devatve loato the atx blok vaou ubteval of the SO tajetoy. The ubteval ae deoted by the ybol τ wth the dex: τ k oeodg to the ubteval[ tk, tk ]. To fo vaou tye of devatve, thee dffeet alulato ethod ae ued. The table ell ae aked dffeet way deedg o the alulato ethod ued fo alulato of devatve: Off-dagoal atx blok ( k > ) ae equal to zeo, beaue the SO tate veto doe ot deed o the lght eue fluee o the evou oto eto. The dagoal table ell ae uouded by the blak fae. The featue that the ube of a eto k, o whh the devatve ae alulated, equal to the ube of the efed veto (k = ). Thee devatve ae alulated the oe of the tegato of the equato vaato wth the eewal of the tal odto atx at eah tafe to a ew ubteval. The double fae ak the devatve fo the veto of the uet SO keat q aaete X by the efed eleet of t obt q. They ae alulated by otuou ueal tegato of the equato vaato, wthout eewal of the tal odto. I the gey bakgoud the foula ae ted the table ell ude the dagoal k <. I th alulato vaat the devatve ae alulated by ultlato of the thee atxe k k k k q = = = q q q k k (9) h ae the devatve by the aaete eto. j ae ued, a obtaed the evou tajetoy

8 Table Etated aaete 2 q τ q τ 2 2 q τ q 2 Subteval 5. Softwae leetato of the algoth fo jot deteato of a SO obtal aaete ad aaete haatezg the ola adato fluee o the SO oto Fo the ooed ethod the algoth fo the jot deteato of a SO obtal aaete ad the aaete haatezg the ola adato fluee o the SO oto leeted oftwae. Fo the algoth alablty vefato, a SO eleted, doveed ue 28 (ube 92 the atalog of the Ittute of Aled Matheat of Rua Aadey of See). It takg wa tated, but eveal te t wa lot ad foud aga. To deotate the algoth wok, a teval fo 29/6/7 to 29/7/3 eleted. Fo etato of the ethod effetvee, the alulato wee aed out two vaat:. By the tadtoal ethod: efeet of 7 aaete 6 ooet of the SO tate veto ad oe value of the ola adato eue ato wth the auto that t deted o the le Su-SO; 2. Refeet of 9 aaete ude the develoed ethod:6 ooet of the SO tate veto ad 3 ooet of the veto haatezg the ola adato fluee o the SO oto the followg deto: Alog the le Su-SO, Othogoal to the le Su-SV, the elt lae, Othogoal to the le Su-SV, othogoal to the elt lae. I oeg eaueet thee alway a oble of a eaueet teval eleto. A hot eaueet teval doe ot ovde uffet auay of the efed aaete, haatezed by the a o ovaae eo atx. A log eaueet teval lead to the ouee of deae due to the oofoty of the aeted odel to the eal SO oto. A a teo of the oeg qualty, the followg value take q a a σ C = o x X whee T

9 C x the a o ovaae eo atx a the ajo obtal e-ax X the SO tate veto attbuted to the oet of efeet t σo the oot-ea quae weghted value of the devato of the alulato fo the alulated aalog. The value q gve a dea of the obtaed oluto qualty: t deeae wth a deeae of the a o e-ax eo ad eae wth a eae of the alulato deae, f the aeted odel eae to ofo to the SO eal oto. The oao of the effetvee of the SO oto aaete deteato, ug aaet oble 7 ad 9, wa aed out by the followg atte (Fgue 3). The lat day of the oeo teval wa exluded fo the oeo ad wa ued fo the etato of the oluto qualty a a otol teval. The alle value of the oot-ea quae deae the otol teval eeeted by a bette oluto, fo the ot of alulato of the taget detato fo the followg ght. The efed tate veto wa attbuted to the oet of the lat eaueet luded oeo,.e. oe day befoe the ed of the eaueet teval. The aaete wee efed by the eag eaueet teval by ea of egagg eale eaueet oeo. At eah uh teval 7 ad 9 aaete wee efed. The obtaed data ae efed Table 2 ad 3. Fo oao of the eult of 7 ad 9-aaet oble, the data of thee table ae obed ad eeeted Table 4. Ieag legth of the foeatg teval Cotol teval Fgue 3 Refeet oet Table 2. Refeet of 7 aaete Yea 29 Aea to a ato Mea weghted deae q-qualty teo Aveage eaueet devato the lat teval (eod of a) 7/28-7/ /27-7/ /25-7/ /24-7/

10 7/22-7/ /2-7/ /7-7/ /6-7/ /4-7/ /3-7/ /-7/ /-7/ /9-7/ /8-7/ /7-7/ /5-7/ Table 3. Refeet of 9 aaete Aea to a ato Yea 29 I the deto Su-SO I the elt lae Othogoally to the elt lae MSD q Aveage devato 7/28-7/ /27-7/ /25-7/ /24-7/ /22-7/ /2-7/ /7-7/ /6-7/ /4-7/ /3-7/ /-7/ /-7/ /9-7/ /8-7/ /7-7/ /5-7/ Table 4. Coao of eult of 7 ad 9-aaet oble Poeo teval Aveage devato Aveage devato Yea 29 legth the otol eto the otol eto -28 7/28-7/ /27-7/ /25-7/

11 -24 7/24-7/ /22-7/ /2-7/ /7-7/ /6-7/ /4-7/ /3-7/ /-7/ /-7/ /9-7/ /8-7/ /7-7/ /5-7/ I Fgue 4 thee devato ae eeeted the gahal fo. The o le oeod to the oluto of the 7-aaet oble (oe aaete fo deto of the ola adato fluee). The blue le oeod to the oluto of the 9-aaet oble (3 addtoal aaete). It obvou that the foeatg eo the eod vaat bette a the eaueet teval legth oe tha day Fgue 4. The ax of aba te day fo the begg of the eaueet teval. The ax of odate aveage devato of the eaued value fo the alulated oe (eod of agle). The vetal le ak the otal teval legth. Refeee. Davd A. Vallado, 27, Fudaetal of Atodya ad Alato 2. Wetz, J.R. ad W. J. Lao, 999, Itoduto to Atodya,.45