A&A 397, 353–359 (2003) Astronomy DOI: 10.1051/0004-6361:20021518 & c ESO 2003 Astrophysics An upgraded theory for Helene, Telesto, and Calypso P. Oberti1 and A. Vienne2 1 Observatoire de la Cˆote d’Azur, BP 4229, 06304 Nice Cedex 4, France e-mail: [email protected] 2 Institut de m´ecanique c´eleste et de calcul deseph´ ´ em´erides, UMR 8028 du CNRS and Universit´e de Lille, 1 impasse de l’Observatoire, 59000 Lille, France e-mail: [email protected] Received 1 August 2002 / Accepted 10 October 2002 Abstract. A tridimensional model including the perturbation due to Saturn’s oblateness provides accurate solutions for the motions of Tethys and Dione’s Lagrangian satellites Telesto, Calypso, and Helene. Expanded around the Lagrangian points and compared to numerical simulations to check their reliabilities, the solutions are fitted to almost twenty years of data thanks to recently published observations. User-friendly compact series give Telesto, Calypso, and Helene’s positions and velocities in the mean ecliptic and equinox of J2000. Key words. planets and satellites: individual: Saturn – planets and satellites: general – celestial mechanics 1. Introduction 2. Analytical model Recently published observations of Tethys and Dione’s For the sake of convenience, computations are carried out in di- Lagrangian satellites Telesto, Calypso, and Helene (Veiga & mensionless units (Szebehely 1967). The distance between the Vieira Martins 2000; Veiga et al. 2002), give an opportunity primaries, and the sum of their masses (called µ = m/(M + m) to revisit and improve the theory of their motions to face an for the second primary and 1 µ for Saturn), are assumed to be − almost 2-decade observation time span. unity. The reference frame is a barycentric rotating frame with 6 The mass ratios Tethys/Saturn (m/M = 1.20 10− )and a rotational period of 2π/k (in dimensionless time τ), where Dione/Saturn (m/M = 1.85 10 6) ensure the stability× of the the value of k will be computed in order to insure a consistent × − Lagrangian points L4 and L5 in the Restricted Circular 3-Body model involving fixed primaries. The xy-plane contains the pri- Problem. Due to the 1:1-resonance with the companion, a satel- maries, the x-axis being located along the primaries and posi- lite around these positions shows librations in its motion re- tively oriented from Saturn toward the second primary. sulting in a tadpole-shaped orbit in a rotating reference frame. The theory will be expanded around L4. Dealing with the Expansions around the equilibrium positions provided accurate symmetrical point L5 requires only slight modifications indi- solutions in the bidimensional case (Deprit & Rom 1970). cated in Sect. 7. γ √3 Pointed out as a significant perturbation by numerical sim- With γ = 1 2µ, L4 is located at the point , , 0 . − 2 2 ulations, Saturn’s oblateness (through the first zonal coefficient Let x, y, z be the satellite’s Cartesian coordinates and Px, of Saturn’s potential developed in spherical harmonics) was Py, Pz their conjugate variables in the barycentric rotating then introduced in the model by addition of perturbing terms frame. The Hamiltonian of the motion is: in the second and following orders of the previously expanded 1 non-perturbed Hamiltonian (Oberti 1990). H = P2 + P2 + P2 k xP yP 2 x y z y x This new theory, now developed in a tridimensional con- − − 1 µ J 3z2 µ text, deals with the perturbation at the uppermost level of the − 1 + 1 , − r r2 − r2 − ∆ expansion, the quadratic part of the Hamiltonian (generally " !# called H0), improving the convergence of the solution and of- where: fering an explicit generalization suitable for other studies. Moreover, the model could prove useful for the analysis of r = (x + µ)2 + y2 + z2, the CASSINI space probe’s accurate observations of these faint q satellites that should be obtained during its nearing exploration ∆= (x + µ 1)2 + y2 + z2, of the Saturnian system. − q 1 2 Send offprint requests to: P. Oberti, and J = 2 J2re stands for half the first Saturnian zonal harmonic e-mail: [email protected] coefficient J2 (Campbell & Anderson 1989) multiplied by the Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20021518 354 P. Oberti and A. Vienne: An upgraded theory for Helene, Telesto, and Calypso square of Saturn’s equatorial radius (re = 60 268 km), appro- where: priately scaled. The value of J, depending then on the selected 4 4 2 2 2 second primary, is 3.4088 10− for Tethys and 2.0780 10− r = (ξ + S 1) + (η + C1) + ζ , × × for Dione. q L is no longer an equilibrium position (Sharma & Subba 2 2 2 4 ∆= (ξ + S 2) + (η + C2) + ζ , Rao 1976). The conditions: q γ √3 S √3 + C C √3 S x = ,y= , z = 0, x˙ = 0, y˙ = 0, z˙ = 0, S 1 = , C1 = − , 2 2 2 2 lead to: S √3 C C √3 + S γ 3J 1 + γ S 2 = − , C2 = P˙ = k2 1 , 2 2 · x − 2 − 2 2 ! 2 √3 3J √3 1 + γ 3. Local solutions P˙y = k 1 , − 2 − 2 2 Using, for example, the software Mathematica, we expand the ! P˙z = 0, functions of the coordinates in the vicinity of ξ = η = ζ = 0: where dots stand for derivation with respect to the dimension- 1 1 = 1 S ξ C η + 3S 2 1 ξ2 less time. Canceling the accelerations would require two differ- r − 1 − 1 2 1 − ent values of k: 3 1 2 2 1 2 + S 1C1ξη + 3 C1 1 η ζ + , 1 + γ 1 + γ 2 2 − − 2 ··· k1 = 1 + 3J , k2 = 1 + 3J 2 2γ · s ! s ! 1 3 = 1 3S ξ 3C η + 5S 2 1 ξ2 The value of k is in fact computed in order to immobilize r3 − 1 − 1 2 1 − the second primary, also perturbed by Saturn’s oblateness. The 15 3 3 + S C ξη + 5 C2 1 η2 ζ2 + , point (1, 0) (or any point on the unit circle) turns into an equi- 2 1 1 2 1 − − 2 ··· librium position for the Hamiltonian: 1 5 1 2 2 1 J 2 2 K = P + P k xP yP 1 + , = 1 5S 1ξ 5C1η + 7S 1 1 ξ 2 x y − y − x − r r2 r5 − − 2 − 35 5 5 where: + S C ξη + 7 C2 1 η2 ζ2 + , 2 1 1 2 1 − − 2 ··· r = x2 + y2, and similar expressions for ∆ with the second index. Then, scal- when:q ing with respect to the total power p + q + r of the monomi- als ξpηqζr, the Hamiltonian can be expanded up to an arbitrary k = √1 + 3J. order n. With k2 > k > k1, L4 becomes an almost-equilibrium position: Neglecting the remainder: 3J 3J √3 1 1 P˙ x = µ, P˙y = µ, P˙z = 0. H ) = H + H + H + + H , − 2 2 (n 0 1 2 2 ··· n! n A first canonical transformation defines a coordinate system in H containing the terms where p + q + r 2. 0 ≤ the vicinity of L4: Defining: γ √3 3 1 + γ 3 1 + γ x = + Cξ S η, Px = k + CPξ SPη, α = 1 + 5J ,β= γ + 5J , 2 − − 2 − 4 2 4 2 √3 γ " !# " !# y = + S ξ + Cη, Py = k + SPξ + CPη, the substitutions into H lead to: 2 2 0 z = ζ, Pz = Pζ, 1 2 2 2 H0 = Pξ + Pη + Pζ k ξPη ηPξ where S and C are the sine and cosine of a rotation devoted 2 − − 1 γ to removing a ξη-term in the expression of the zero-order 3J − (S ξ + C η) − 2 2 2 Hamiltonian below. This kind of coordinate system is gener- ! ally referred to as the normal frame at an equilibrium position. 1 + γ ξ2 α 1 + 2S 2 + 2β √3SC 1 3J Applying the transformation turns the Hamiltonian into: − − − 2 2 " !# 1 2 2 2 2 2 1 + γ η H = Pξ + Pη + Pζ k ξPη ηPξ α 1 + 2C 2β √3SC 1 3J 2 − − − − − − 2 2 " !# 1 2 1 2 k Cγ + S √3 ξ + k S γ C √3 η 2αSC+ β √3 C2 S 2 ξη −2 2 − − − 1 + γ J 3 ζ2 1 γ 1 + γ ζ2 1 + 1 − , 1 9J − 2r r2 − r2 − 2∆ − − − 2 2 · " !# " !# P. Oberti and A. Vienne: An upgraded theory for Helene, Telesto, and Calypso 355 The terms in ξ and η, resulting from the fact that L4 is not ex- The required expression for H0 and the canonicity of the trans- actly an equilibrium position, are small because J (1 γ) /2 = formation provide ten relations leading to: − Jµ. At the precision required by the observations, they can 1 1 safely be ignored, transforming this way the Lagrangian a13 = , a14 = , √v2 2kv a √a + 2ku u2 point L4 in an actual equilibrium position for the theory. − − − Removing the ξη-term leads to: ω1 ω2 a21 = − , a22 = , √u2 + 2ku b √b 2kv v2 − − − δ 1 δ + 1 β 2 a = ua , a = va , a = va , a = ua , S = − , C = ,δ= 1 + 3 , 31 21 32 22 43 13 44 14 − 2 δ 2 δ s α r r where: and H0 finally reads: b a + d b a d u = − ,v= − − 1 4k 4k · H = P2 + P2 + P2 0 2 ξ η ζ Finally, the canonical transformation: a 2 b 2 c 2 2I k ξPη ηPξ ξ η ζ , Q = 1 sin ϕ , P = 2I ω cos ϕ , − − − 2 − 2 − 2 1 ω 1 1 1 1 1 r 1 where: p 2I Q = 2 sin ϕ , P = 2I ω cos ϕ , 1 3 1 + γ 5 2 ω 2 2 2 2 2 a = 1 δ + 3J 3 δ , r 2 2 − 2 2 − 2 p " ! !# 2I3 Q3 = sin ϕ3, P3 = 2I3ω3 cos ϕ3 1 3 1 + γ 5 ω3 b = 1 + δ + 3J 3 + δ , r 2 2 2 2 p " ! !# leads to: 1 + γ H0 = I1ω1 I2ω2 + I3ω3.