Masses and orbits of minor planets with the GAIA mission
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1 asses ad obits of io laets with the GAIA issio Sege ouet Suevisos : F.igad D.Hestoffe
2 PLAN Itoductio Puose of the PhD Iotace of asses The diffeet ethods to estiate these asses Descitio of close aoach Diffeet ways to see the heoeo ethod fo the ass deteiatio The Least squaes ethod Peculia case : the L.I.P. Hill s oble Posects
3 Puose of Ph.D. Obectives: To develo a ethod which will teat the astoetic easueets fo a asteoid to obtai its ass eseach the ais of close aoach ad idetify the ost efficiet oes fo the ass deteiatio Ioveet of obits
4 Iotace of asses of asteoids (/) Fo the dyaical viewoit : Obtai ecise log-te obit of seveal salle etubed obects I idiectly way, Icease the accuacy of the sola syste eheeides liitig facto: the lagest asteoids aily as eheeis easue oe ecisely: sola quaduole J the PPN aaete β (easuig the o lieaity i sueositio of gavity)
5 Iotace of asteoids (/) Fo the hysical viewoit: ass Diesios Bulk desity Ifoatio about Iteal stuctue Oigi ad Foatio ocess elatioshi Desity taxooic class? Poosity Geetic elatioshi betwee Classes of eteoites ad suosed asteoids souce
6 Iotace of asteoids (/) Fo the hysical viewoit: Bitt et al. ass Diesios ASTEOIDS III Bulk desity Ifoatio about Iteal stuctue Oigi ad Foatio ocess elatioshi Desity taxooic class? Poosity Geetic elatioshi betwee Classes of eteoites ad suosed asteoids souce
7 The diffeet ethods to estiate these asses Satellites of asteoids : Good kowledge of thei obits ca ovide ecise ass of iay (also with GAIA). Aalysis of obital etubatio oduced by utual close aoaches betwee : two asteoids asteoid & sace-obe
8 Descitio of the heoeo ϕ ta G ( bv ) b : iact aaete G : costat of gavitatio V : elative velocity of the ecoute
9 The close aoach (/) Chaacteisatio of efficiecy : Agle of deflectio : ϕ G ( ) ta bv b is the iact aaete ϕ si V G( ) V the elative velocity of the two asteoids Velocity vaiatio of the scatteed asteoid : Δ v G ( ) v v is the elative velocity of the two asteoids at the itat of the closest aoach
10 The close aoach (/) Chaacteisatio of efficiecy : Agle of deflectio : ϕ G ( ) ta bv b is the iact aaete ϕ si V G( ) V the elative velocity of the two asteoids Velocity vaiatio of the scatteed asteoid : Δ v G ( ) v v is the elative velocity of the two asteoids at the itat of the closest aoach
11 The close aoach (/) Chaacteisatio of efficiecy : Agle of deflectio : ϕ G ( ) ta bv b is the iact aaete ϕ si V G( ) V the elative velocity of the two asteoids Velocity vaiatio of the scatteed asteoid : Δ v G ( ) v v is the elative velocity of the two asteoids at the itat of the closest aoach
12 The close aoach (/) Eegy efficiecy ε : ΔV ε ΔV T V. ΔVT V. ΔV ΔV is the total heliocetic velocity chage esultig fo the close aoach ΔVT is the tagetial cooet of ΔV Chage i the ea otio of the scatteed asteoid : a Δ G s av ± G s V. Δ V ε. Δ V s is the ass of the Su a the sei ao axis of the scatteed asteoid
13 The close aoach (/) Eegy efficiecy ε : ΔV ε ΔV T V. ΔVT V. ΔV ΔV is the total heliocetic velocity chage esultig fo the close aoach ΔVT is the tagetial cooet of ΔV Chage i the ea otio of the scatteed asteoid : a Δ G s av ± G s V. Δ V ε. Δ V s is the ass of the Su a the sei ao axis of the scatteed asteoid
14 O-CAΔu O C The Least Squaes ethod O C ( O C) ( O C) AΔu Δu ( A The esidual vecto O-C : whee ( O C) obs c ( ) obs c ( ) is the ube of obsevatios of asteoid is the total ube of studied asteoids obs is obseved gaia logitude (o a scaig geat cicle) of the asteoid c t A) A t ( O C) is calculated gaia logitude of the asteoid
15 gl u u u u Δ Δ Δ Δ O-CAΔu The adustable aaetes vecto Δu Coodiates of ositio Coodiates of velocity asses of k etube asteoids PPN aaete k J β u gl Sola quaduole Fo asteoid Global aaetes z y x z y x u...
16 J J k k D D A β β O-CAΔu The atial deivative atix A asteoid asses of k Petubes Positio-velocity β J
17 J J k k D D A β β O-CAΔu The atial deivative atix A asteoid asses of k Petubes Positio-velocity β J i utual ot ae asteoid ad k asteoid fo k t obs k iteactio
18 Size of atix A Nube of ows : ube of asteoids ~ 4 5 ube of obsevatio a asteoid ~7 Nube of colu : 4 ube of asteoids ~ ube of aaetes ~ 6 fo each asteoid k i coo (k ~ ) 6 k 4 9 cooets but with ~ 6 o zeo
19 Peculia case: the Lage Iact Paaete L.I.P: the iact aaete b stays geat duig whole ecoute Cosequese : slight deflectio with sall ecoute s velocity A «iulse aoxiatio» (two-body aoxiatio) gives icoect esults [Héo 984] : aticles fo laeaty igs with cicula ad colaa obits. Coaaiso of ea toque φ (aticle o the aticle ): φ K G ( 4 b ) is the ea agula velocity of cete of ass b Coefficiet K fo iulse aoxiatio is diffeet fo theoical K.
20 Defiitio: asses,, iteactig to the ewtoia foces Hill s oble Equatios of otio i the syodic fae:... G ( ). G Y ( ) >> η >> >> ~ ξ Cetifuge foce.. ξ... η. η ξ G Coiolis foce ξ G ( ) η ( ) Aalytical solutios ξ, η (Heo,Petit.986) ξ C. Ty to colete the G iulse aoxiatio t with the foulatio of Hill s oble. X
21 Posect Choose a ethod fo the vaiatioal equatios i ode to calculate the atial deivatives atix A Ty to colete the iulsioal teateet i the case L.I.P. with the foulatio of Hill ad the aalytical solutio. Teat by efficiet ethod the big atix A ad esolve it by least squaes
22 The Hill s oble ~ >> >> >> Defiitio: asses,, iteactig to the ewtoia foces dt d dt d Equatios of otio i the ietial fae:, 5 )..( ) ( >> C. X Y ( ). & cete of ass
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