On the PPN 1+2 Body Problem

Transcription

1 On the PPN 1 Body Proble arxiv:gr-qc/971039v1 8 Dec 1997 D Şelaru, I Dobrescu Gravitational esearch Laboratory, 1-5 Mendeleev str, Bucharest, oania e-ail: dselaru@scou1ctro, idobrescu@scou1ctro June 8, 017 Abstract A particular case of the three-body proble, in the PPN foralis, is presented The Hailtonian function is obtained and the proble is reduced to a perturbed two-body one 1 INTODUCTION Several approaches were perfored on the PPN two-body proble, Soffel, 1989 In the PPN three-body proble the ost studied case is the otion of a test particle in the gravitational field of two assive bodies, i e the restricted proble Bruberg, 197 This paper presents a different situation: the otion of a binary syste of two very sall asses around a assive body, when the otion of the assive body not is affected by the presence of the couple The physical situations odeled ay be the otion of a binary asteroid a binary coet fragents or of two artificial objects The special choice of asses the sall bodies have equal asses particularize the last case One uses the PPN foralis in the first approxiation in the Hailtonian forulation In the Jacobi coordinates, the three-body proble splits in two coupled perturbed twobody probles: the otion of the barycenter of the binary syste and the relative otion of the two sall bodies The first is rather well approxiated by the otionofatest particleinthegravitationalfield ofaassive body, e g Daour and Deruelle 1985, Heiberger et al 1990 An iposed circular solution for this proble is substituted in the two-body relative otion Siilar to the case of the restricted three-body proble, one passes to a cooving coordinate syste, where the Ox axis coincides with the direction given by the ass center GL eport 1/04/1997 1

2 of the binary syste and the center of the assive body In this syste the Hailtonian becoes autonoous Conclusions are given to the possibility of a GT experient in this fraework We start with the usual Lagrange function L N of the N-particles oving in a self-consistent field in the PPN foralis see Soffel, 1989 for notations We obtain the Hailton function for the three-body proble by eans of the Legendre transforations: H 3 = i i,j i H 3 = p i p4 i i 8c 3 i i < p i,v i > L 3 with p i = L 3 v i = 1 i,j i G i j r ij G 4γ 3 < p 4c i,p j > 1 r ij rij 3 β 1 c i,j i k i i,j i Gγ 1 j c i r ij p i 1 < p i,r ij >< p j,r ij > G i j r ij k r ik The Hailtonian foralis is used in order to benefit of the Lie-Deprit ethod in the study of the near integrable systes THE PPN 1BODYPOBLEM WITH EQUAL SMALL MASSES This section describes the Hailtonian of a particular case of the three-body proble: a close pair, 1 = =, rotating around the coon center of ass and the third body, 3 = M, far fro these two, with << M We perfor the siplectic transforations Jacobi coordinates: r 1 = r, r = r, r 3 =, p 1 = p r p, p 1 = p r p, p 3 = p p Due to that is not present in the Hailtonian function consequently, p is a constant of otion and considering p = 0, we obtain: H 1 = H Newton H PPN, where H Newton = p r G p M GM r 4M 3

3 and GM 1 1 r r H PPN = 1 p 4 8c 3 r p4 8 p p r < p r,p > p4 8c M 4 3 Gγ 1 p c r p Gγ 1M p r c r p 4 < p 1 r,p > r p r p 4 < p 1 r,p > r Gγ 1p 1 1 c M r r G [ 4γ 3 p c r r p 1r < p ] 4 3,r > < p r,r > G c G c 4γ 3 r 4γ 3 r < p p r, r >< p, r > r 3 < p p r, r >< p, r > r 3 r M 1 1 r r r 1 1 M r r r r < p r,p > p < p r,p > p β 1G c M M We have 1 r n < r, > r = n=0 n n1p n, r 1 r = where P n is the n-th usual Legendre polynoial n=0 r n n n11n P n < r, >, r 3 THE DECOUPLING OF THE TWO POB- LEMS A physical realization for this proble is the otion of two bodies with asses of 10 3 kg each, with 1 relative sei-ajor axis considered unperturbed and situated at the distance of 0 il k fro the Sun The nuerical evaluations 3 5

4 fortheperturbativetersfrotheeqs 3and4willbeestiatedinthiscase For the Hailton function given by Eq 4 the Newtonian orders of perturbations for the relative proble 1, are: λ 1 = M r 3 = O and for the otion of the barycenter relative to the assive body: λ = 1 4 r = O In this case, the otion of the barycenter of the couple ight be considered unperturbed in a good approxiation, λ = Oλ 5 1 The proble is reduced to the study of the relative otion in the binary syste The otion of the assive body is not affected by the presence of the couple; this type of proble being considered as 1 body proble By retaining the significant ters, the proble splits in two perturbed two-body probles The otion of the barycenter is described by the following Hailtonian: H c = p M 4M GM p4 β 1G M c O Gγ 1M 64c 3 4c 10 0 p 4 p 8 The perturbing ters aintained of the order of O10 8 p /4 fro Eq 8 are given by the PPN odel of the proble, which decouples fro the relative proble in the binary syste in the considered approxiation The resulting Hailtonian is siilar to the one for the case of the otion of a test particle in the gravitational field of a assive body This proble adits a circular solution Krefetz, 1967 In polar coordinates the reference plane is that of the binary syste barycenter otion = X Y, P, Φ = arctan Y X, P Φ, the canonic change of coordinates is obvious this solution has the for: = 0, P = 0, P Φ = P Φ0, Φ = n tφ 0 9 One denotes with the initial conditions fro the Eq 9, in order to distinguish between the and those of the Newtonian proble, without losing the generality, one ay consider Φ 0 = 0 The relative otion in the binary syste is described by the following Hailton function: H bin = p r G GMr < r, > P r 3 r 4

5 Gγ 1Mr 1 p 8c 3 p r < p r,p > Gp 8c r Gγ 1Mp r c < p c r,p > G < r,p > 8c r 3 Introducing Eq 9 in Eq 10 one obtains: H bin = p r 1 P Φ0 8c 0 GMr xcosn tysinn t P 0 3 r Gγ 1P Φ0 Mr c 3 0 Gγ 1M G c 0 r β 1G M c r O 1 P Φ0 8c p r β 1GM c 0 P Φ0 p 4c 3 0 x sinn tp y cosn t 11 p x sinn tp y cosn t GP Φ0 xsinn tycosn t O 8c r p r The relativistic ters retained are of the order of O10 8 p r / Passing to the cooving coordinate syste which rotates such that the axis Ox joints the assive body with the barycenter of the binary syste, we obtain the autonoous Hailtonian: H bin = p r 1 P Φ0 8c 0 n xp y yp x GM Gγ 1P Φ0 M c 3 0 Gγ 1M G c 0 r 1 P Φ0 8c 0 β 1GM c 0 3x r P Φ0 p 4c 3 0 y 1 rp y GP Φ0 y 8c 0 r O p r The influence of the PPN odel of the 1 proble is: a a direct one due to the presence of the last three ters in Eq 1, b an indirect one, by the odification of the coefficients which appear in the other ters, also present in the Newtonian case 4 CONCLUSIONS The ain purpose of this paper is to present a theoretical proble which ay constitute the preise of a GT experient The preliined developent of Solar Syste space issions could create the technological preises for setting up such a 1 body syste, fored of equal asses, on a precise given orbit, with the purpose to detect the influence of soe PPN paraeters in order 5

6 to evaluate the GT predictions To becoe real, this odel should take into account facts as: the presence of other bodies, the shape of the assive body, a ore realistic orbit of the barycenter than the circular one, the radiation pressure, the influence of the electroagnetic field, etc The study of this type of proble has soe advantages: - the possibility of choosing the appropriate initial conditions: the asses rapport, the sei-ajor axis of the binary syste ass center otion relative to the assive body, the orbital eleents of the relative otion of the two-body syste related to the orbital plane of the ass center otion; - the faster evolution of the tie in the sense that one needs a shorter interval of tie for perforing the experient than in the case of the study of a natural body otion; - the possibility of realizing ore precise easureents for the obtained effects, due to the fact that the bodies in the binary syste are very close and due to the high precision given by the present easuring devices which ay be set up on-board One notices here that, for the chosen initial data, in order to obtain the relativistic correction for the indirect effects, it is necessary a first order theory with the sall paraeter λ 1 In order to obtain the direct effects a second order theory should be perfored by eans of standard procedures eferences [1] Bruberg, V A, elativistic Celestial Mechanics, Nauka, Moscow 197 [] Daour, T, Deruelle, N, Ann Inst Poincare, 43, [3] Heiberger, J, Soffel, M H, uder, H, Celest Mech and Dyna Astron 47, [4] Krefetz, E, Astron J, 7, [5] Soffel, M H, elativity in Astroetry, Celestial Mechanics and Geodesy, Springer-Verlag Berlin Heidelberg New-York London Paris Tokyo